Commun. Math. Phys. 191, 1 – 13 (1998)
Communications in
Mathematical Physics c Springer-Verlag 1998
Classification and Construction of Quantum Communication Systems ¨ Bernd Muller Institut f¨ur theoretische Physik, Universit¨at T¨ubingen, Auf der Morgenstelle 14, 72076 T¨ubingen, Germany. E-mail:
[email protected] Received: 5 September 1995 / Accepted: 5 November 1996
Abstract: We consider Quantum Communication Systems (QCS’s), introduced by Davies [10] on trace class operators, on general state spaces. These are different from the usual quantum information theory as they deal with input and output being continuous in time. We introduce the concept of a refinement of a QCS, offering the possibility of distinguishing particles, for instance according to their phase or location, and classify refinements of general QCS. We then introduce the notions of a bounded interaction rate and a bounded modulation and give, in special cases, a classification of QCS’s satisfying both conditions. As this classification shows the existence of a class of QCS’s being completely different from the examples studied by Davies, we are able to discuss the cases of an independent detector and a linear modulation. 1. Introduction In classical communication theory the transmission of information is described by means of a Markov kernel P connecting the input and the output system [2], i.e. the input signal w ∈ X (the set of all input signals) leads to a probability distribution E → P (w, E) on the set of possible outputs. Usually one considers the input as generated randomly, so instead of using one signal w ∈ X one takes a probability distribution µ on X. Then µ is transformed into a probability distribution ν on via the formula Z P (w, E)µ(dw). ν(E) = X
In many cases the set of measures on X or is the dual of the C ∗ -algebra C(X) or C() of the continuous complex valued functions on X or , respectively. Then the above map 3∗ : µ → 3∗ µ := ν is the dual of a (completely) positive map 3 : C() → C(X). This definition in terms of C ∗ -algebras and completely positive maps shows an obvious way to generalize the notion of a communication channel to the non-commutative case.
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The use of general W ∗ - or C ∗ -algebras instead of abelian ones is supposed to take into account the quantum mechanical nature of the physical system which is used for signal transmission. This includes a quantum mechanical formulation of information, (relative) entropy and in particular of the transition kernel P [1, 16, 17, 19–22]. The non-commutative analogue to a message, consisting of the letters a1 , . . . , an then is the transmission of a sequence of states ρ1 , . . . , ρn , which is transformed by the channel into a sequence ω1 , . . . , ωn of output states [13]. The transformation ρi → ωi = 3∗ ρi here is the non-commutative generalization of a Markov kernel. In applications the problem is the identification of the states ωi . This can only be done by means of a measurement, the result of which by definition is classical. So by fixing one type of measurement we still have a classical Markov kernel from input to output, which is also the case in the approach presented below, where the measurement always is a counting experiment. The physical situation we look at in this paper is partly different and not covered by the above considerations. In applications one often wants to transmit a signal function w that is continuous or piecewise continuous in time. This is to be achieved by the modulation of the light beam, which is caused by the input system. The light beam itself is described by a state of the photon system. This implies that a possible input is not a discrete chain of letters or (by use of a quantum code ai → ρi ) states, but a real valued function. However, we emphasize that it is possible to restrict the attention to functions that are constant on the time intervals [nc, (n+1)c) and only take values out of a finite set {a1 , . . . , an }. c represents the time necessary for the transmission of one signal ai , and a1 , . . . , an are the possible letters. This restriction shows that a digital (binary) signal transmission is included in the class of models described below. The way the modulating system influences the state is not specified a priori, yet it is proved in 4.2 that, under some additional assumptions, it can always be formulated as a time dependent perturbation of the interaction free dynamics (cf. Eq. (16)). To give a physical example, we think of a light beam entering a modulating system that influences the light. The modulating system for instance is a semiconductor crystal whose transmission coefficient and refraction index at time t are a function of the value w(t) of the signal w that is to be transmitted. If H is the free Hamiltonian of the photon system, the dynamics of the modulated light might be described by ∂ ρt = −i[H, ρt ] + B(w(t))ρt , (1) ∂t where B(w(t)) contains the influence of the modulating system. The above equation assumes that the state ρ is a trace class operator on some Hilbert space H. We remark that, in the sequel, we use a general state space (for instance a predual space of a W ∗ algebra). That way classical features of the light beam can be described as well [15]. At the other end of the channel the receiver performs a measurement on the state “arriving there”. Usually, one considers a number measurement to count the arriving quanta, though, in general this is not necessary. However, most authors do not take into account that, because of the continuity in time, such a measurement cannot be described by a selfadjoint operator on some Hilbert space, in particular not by only taking the spectral decomposition of the Fock number operator in order to get the relevant observable. In this paper we use the theory of counting experiments developed in [6, 9, 10, 18], where the detecting operation itself is assumed to be a quantum stochastic process. So both input and output are continuous in time, and thus it is not at all clear if the resulting channel is memoryless, which is, though not always physically justified, an assumption made implicitly in many articles by only considering the transmission of one letter. However, it is nevertheless possible in models of the type described below to get a memoryless channel, which will be shown in future work. The main advantage of this
Classification and Construction of Quantum Communication Systems
3
approach is, that it can easily be applied in practice. The use of the powerful theory of open systems [7, 9] reduces very efficiently the set of parameters that have to be taken into account. Though it lacks the generality of the investigations on completely positive channels made for instance in [22], the restriction to the special class of counting experiments allows to get a very concrete meaning of any part of the channel and thus easily shows how to construct physical models. And, we remark that because of the possibility of distinguishing particles according to phase, location, energy, . . . (see 3.1), this restriction is not as strong as one might guess. The paper is organized as follows: In Sect. 2 we give the basic definitions – including the one of a quantum communication system (QCS) – which all go back to Davies [10]. For technical reasons the definition given in [10] differs slightly from Definition 2.1. In particular it is shown how to calculate the transition kernel (Eq. (2)). The connections between systems that can distinguish particles (for instance according to energy, phase or location) and systems that can not are studied in Sect. 3. As for counting processes (CP’s, cf. [18]) we call the corresponding constructions coarse–grainings or refinements. In Theorem 3.1 we give a classification of refinements which generally allows to construct a QCS in two steps: First, one has to construct a one point QCS (i.e. a QCS that cannot distinguish particles) and then it can be refined. In Sect. 4 we introduce the ideas of a bounded interaction rate (for CP’s cf. [6, 8, 9, 18]) and a bounded modulation, and call a QCS bounded if both conditions are fulfilled. For a bounded one point QCS having an interaction free semigroup St (2.2) of type R [18] it is shown in Theorem 4.2 that the QCS always can be formulated via the perturbation theoretic equations (10), (11). This implies that Eq. (1), which so far only seems to be a physically reasonable example, can be derived from rather principal considerations, condensed in Definition 2.1. Conversely Theorem 4.3 implies that (10) and (11) always define a QCS. In particular there is an interaction rate operator J(c) (cf. the results for CP’s [10, 18]). Yet J(c) can depend on a real parameter c. In Proposition 4.4 we show that any decomposition of J(c), i.e. a positive operator valued measure (POVM) J(c, .) on a standard Borel space (X, Σ) with J(c, X) = J(c), defines a refinement of the corresponding QCS. For physical applications this implies that one only has to determine the operators J(c, E) (characterizing the instaneous change of state when a particle of type E is detected) in order to get a refinement of a QCS. Finally in Sect. 5 we introduce two important classes: QCS’s with linear modulation and QCS’s with an independent detector. For the first class it is shown that the modulation operators have to be generators of groups of isometries. Both classes are interesting from the physical point of view and we remark that Davies [10] only studied QCS’s satisfying both conditions. 2. Definition of a QCS The first point we are interested in is the set characterizing the input. As we want to have a model respecting the continuity of time, the messages are supposed to be functions of the time. We thus define Yt to be the set of functions w : [0, t) → R, where w is continuous from the right and has finite limits from the left at any point of the interval. For t = ∞ we additionally require that w is bounded. We consider two topologies on Yt : The first one is the uniform topology, the second one is defined by the Skorohod metric d [23]. Important is that the set of step functions is dense in Yt for both topologies. As one can send one message immediately after another one, we introduce the map v(r), r ∈ [0, t) β s,t : Ys × Yt → Ys+t , β s,t (w, v)(r) = w(r − t), r ∈ [t, t + s).
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Usually the indices s, t are neglected and we use β(w, v) or even w × v instead of β s,t (w, v). In the sequel we often will vary t but have a fixed message w. This means that we use one message w ∈ Y∞ and only consider the restrictions of w to the interval [0, t). The output of the communication system shall be produced by a detection system counting the arriving quanta. Thus we use the following notations, introduced in the theory of counting processes [6, 18]: Atn Cnt
:= :=
{((x1 , t1 ), . . . , (xn , tn ))|0 ≤ t1 < . . . < tn < t, xi ∈ X}, {(t1 , . . . , tn )|0 ≤ t1 < . . . < tn < t},
for n ∈ N. Any point of the measurable space (X, Σ) is characterizing a certain type (energy, location, . . .) of particle. The output ((x1 , t1 ), . . . , (xn , tn )) ∈ Atn therefore means that a particle of type xi has been detected at time ti . The event that no particle has been detected during the time interval [0, t) we denote by zt , which is independent from (X, Σ). Therefore C0t = At0 := {zt } and Xt := Un≥0 Atn is the set of possible outputs. Again the possibility of recording two sequences after each other corresponds to a map λ : Xt × Xs → Xt+s defined by (((x1 , t1 ), . . . , (xn , tn )), ((y1 , s1 ), . . . , (ym , sm ))) → ((y1 , s1 ), . . . , (ym , sm ), (s1 , t1 + s), . . . , (sn , tn + s)). The physical channel (for a mathematical realization see Eq. (2)) connecting Yt and Xt for instance may be the quantized electromagnetic field, described by its state ω. More general the set of states of a physical transmission system forms a state space (V, K, τ ), i.e. V is a base normed space with positive cone K, and τ ∈ V ∗ equals the norm on K [7, 9, 12]. Measurements generally are described by means of an instrument. Given a state space, an instrument by definition is a positive operator valued measure (POVM) E: 0 → B+ (V), where (, 0) is the set of possible measurement outputs, satisfying E()∗ τ = τ (for definitions and the necessity of these notations, see [7, 18]). It is important that the probability of a measurement result in A ∈ is given by hE(A)ω, τ i = ||E(A)ω||. if the system is in the normalized state ω ∈ K = V+ (||ω|| = hω, τ i = 1) at the beginning of the measurement, and afterwards the system is in the state E(A)ω. In the above situation any measurement (and henceforth the corresponding instrument) depends on the message w and the time t up to which it is performed. This leads to the following definition: Definition 2.1. A quantum communication system (QCS)is a family of instruments Et (ω, ·) on Xt such that (i) limt↓0 Et (w, Xt )ω = ω in norm for all ω ∈ V and w ∈ Y∞ . (ii) w → Et (w, E) is weakly measurable and even weakly continuous for E = Xt . (iii) Et (w, E)Es (v, F ) = Et+s (w × v, λ(E × F )) for w ∈ Yt , v ∈ Ys and measurable E ⊆ Xt , F ⊆ Xs . We use the uniform topology in (ii) to define the σ-field on Yt . We could also take the Skorohod metric, but as the uniform continuity in most cases is easier to verify we use the weaker condition. Recall that for hω, τ i = ||ω|| = 1 the probability for having an output in the set E ⊆ Xt , provided that the message w is sent, is given by hEt (w, E)ω, τ i. In
Classification and Construction of Quantum Communication Systems
5
this case the state of the channel directly after the measurement has been transformed into Et (w, E)ω. Therefore (iii) expresses that the time evolution has no memory and is homogeneous. Now for any normalized ω ∈ V+ the map (w, E) → hEt (w, E)ω, τ i =: Pt (w, E)
(2)
is a probability transition kernel from Yt to Xt . Therefore for any choice of ω this map defines a channel in the sense of classical communication theory. Note that the above Markov property (iii) of the time evolution does not imply that this channel is memoryless in the sense of communication theory. Remarks 2.2. (i) If g: R → R is continuous and Et is a QCS, then (E, w) → Et (g ◦ w, E) defines another QCS. (ii) For any constant function, i.e. w(t) ≡ c the family Et (c, .) defines a counting process [6]. w w := Et (w, {zt }) and St+s,s := Et (αs (w), {zt }), (iii) We use the following notations: St,0 w where αs (w)(r) := w(s + r). For 0 ≤ s < t St,s is the interaction free dynamics according to w. For any constant c the family Stc := Et (c, {zt }) is a semigroup. 3. Coarse-Grainings and Refinements If the detector counts the particles without distinguishing them, the space (X, Σ) consists of one point only and Atn is isomorphic to Cnt . In this case we call the corresponding QCS a one point QCS. Let us consider a general QCS Et with measurable (X, Σ). It is obvious that via the projection [ Cnt , ((x1 , t1 ), . . . , (xn , tn )) → (t1 , . . . , tn ) π: Xt → n≥0
the QCS Et is transformed into a one point QCS E˜t , where E˜t is defined by E˜t (w, E) = Et (w, π −1 (E)). Physically this corresponds to the neglection of the possibility of distinguishing particles and we therefore call the resulting one point QCS E˜t the coarsegraining of Et . In most cases the inverse way is more important, as one point QCS can often be constructed by use of semigroup theory (cf. next section). Given a one point QCS E˜t the possibility of distinguishing particles corresponds to a refinement of E˜t . In other words: a refinement of E˜t is another QCS Et with measurable space (X, Σ) whose coarse-graining coincides with E˜t . Let Et be such a refinement of E˜t , then obviously the w of both QCS is the same. Moreover it is interaction free dynamics St,s Et (w, X × [0, t)) = E˜t (w, [0, t))
and
w w Et (w, E × [a, b)) = St,b Eb−a (αa (w), E × [0, b − a))Sa,0
(3) (4)
for all E ∈ Σ and 0 < a < b < t. The subsequent theorem shows that these equations essentially are enough to construct a refinement. In the sequel we will often have to deal with sets of the form A = (E1 × [a1 , b1 )) × · · · × (En × [an , bn )),
(5)
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B. M¨uller
P where Ei ∈ and 0 ≤ a1 < b1 < · · · < an < bn < t. Obviously A is a measurable t set in An . We call a finite union of sets of this form a standard set. The standard sets are closed under the formation of finite intersections and generate the σ-field on Atn . Note that it is sufficient to give a QCS on At1 (and At0 ) as the Markov property then determines the QCS on standard sets and henceforth on general sets. Theorem 3.1. Let (X, Σ) be a standard Borel space and E˜t a one point QCS. Then there is a 1:1-correspondence between the following classes: (i) Refinements Et of E˜t with measurable space (x, Σ). (ii) POVM’s Gt (w, .) on X × [0, t) = At1 (Gt (w, E) ∈ B+ (V)) such that (a) w → Gt (w, E) is weakly measurable, (b) Gt (w, X × [0, t)) = E˜t (w, [0, t)) . w w Gb−a (αa (w), E × [0, b − a))Sa,0 for E ∈ Σ and (c) Gt (w, E × [a, b)) = St,b 0 < a < b < t. The correspondence is given by Et (w, A) = Gt (w, A) for measurable A ⊆ X × [0, t) . Proof. The main idea of the proof is the same as in the corresponding proof for CP’s [18], but as the situation here is much more complicated the details need a closer look. The formal bijective map between Gt and Et is given by Gt (w, A) := Et (w, A) for measurable A ⊆ X × [0, t), i.e., Gt is the restriction of Et on the measurable subsets of At1 ⊆ Xt . To show the bijectivity of the map it is obviously enough to show that it is possible to construct Et whenever Gt is given. For fixed t we subdivide the interval [0, t) k n in 2n disjoint intervals Ikn := [ k−1 2n t, 2n t) for k = 1, . . . , 2 and restrict our attention to the set 3ln := {x = ((x1 , t1 ), . . . , (xl , tl )) ∈ Atl | xi ∈ X, for each k there is at most one ti ∈ Ikn }. It is easy to verify that 3ln ⊆ 3ln+1 and Un≥0 3ln = Atl . We say a map φ : {1, . . . , 2n } → {0, 1} is in Qln , iff cardφ−1 ({1}) = l. For each φ ∈ Qln we define Pφ := {ψ ∈ Qln+1 | for each k ∈ φ−1 (1) there is one and only one l ∈ {2k − 1, 2k} with ψ(l) = 1}. Obviously the Pφ are pairwise disjoint. Now let M0n := {zt/2n }, M1n := X × [0, t/2n ), and for n n × · · · × λ(Mφ(2n ) × Mφ(1) ) . . .). If X = {1} we use each φ ∈ Qln let Aφ := λ(Mπ(2n) l l ˜ n instead of 3n . It is easy to verify that Aφ = ∪ψ∈Pφ Aψ and Bφ instead of Aφ and 3
3ln = ∪φ∈Qln Aφ , where the Aφ are pairwise disjoint. The σ-algebra on Aφ is generated n by sets of the form E = λ(E2n × · · · × E1 ) with measurable sets Ei ⊆ Mφ(i) . For these sets we define µ˜ φ (w, E) = Gt,(2n −1)t/2n (w, E2n ) · · · Gt/2n ,0 (w, E1 ), w and Gb,a (w, {zr }) = Sb,a . Then where Gb,a (w, E) = Gb−a (αa (w), E) for E ⊆ Ab−a 1 Q2 n using the fact that i=1 Mφ(i) is Borel isomorphic to Aφ it follows by [7, Theorem 2] that E → µ˜ φ (w, E) can be extended to a POVM on Aφ . Now it is a technical but straightforward calculation, that for measurable E ⊆ Aφ holds: X µ˜ φ (w, E) = µ˜ ψ (w, E ∩ Aψ ).
Thus the definition µln (w, E) := erates a POVM on Atl satisfying
P
ψ∈Pφ φ∈Qln
µ˜ φ (w, E ∩ Aφ ) for measurable E ⊆ Atl gen-
Classification and Construction of Quantum Communication Systems
7
(i) µln+1 (w, E) = µln (w, E) for measurable E ⊆ 3ln , (ii) µln (w, Clt ) = µln (w, 3ln ). The last equation holds as 3ln is the disjoint union of the Aφ and µln by definition is concentrated on this union. In particular we have µln+1 (w, E) ≥ µln (w, E) for all measurable E ⊆ Atl . Moreover for a standard set E = (E1 ×[a1 , b1 ))×· · ·×(El ×[al , bl )) a direct calculation shows that E ⊆ 3ln for sufficient large n and then w Gbl −al (αal (w), El ) · · · Gb1 −a1 (αa1 (w), E1 )Saw1 ,0 . µln (E) = St,b l
(6)
Now by definition one calculates X µln (w, Aφ ) = E˜t (w, ∪φ∈Qln Bφ ) ≤ E˜t (w, Clt ). µln (w, Atl ) = l φ∈qn
Monotony and boundedness of this sequence imply that for all measurable E ⊆ Clt and ω ∈ V+ (and therefore all ω ∈ V) limn→∞ µln (w, E)ω =: Etl (w, E)ω exists in norm. By the Vitali–Hahn–Saks Theorem [11] we conclude that E → Etl (w, E) is weakly, hence (because of the positivity and the state space properties) strongly σ-additive. Thus Etl (w, .) is a POVM on Atl with (all limits are in the strong topology) Etl (w, Atl )
= =
lim µln (w, Atl ) n→∞ E˜tl (w, Clt ).
˜ ln ) = lim E˜t (w, ∪φ∈Qln Bφ ) = lim E˜t (w, 3 n→∞
n→∞
P∞ Now we can define Et (w, E)ω := l=0 Etl (w, E ∩Atl )ω. It is straightforward that Etl (w, .) is a POVM with Etl (w, Xt ) = E˜t (w, ∪n=0 Cnt ). By construction w → Et (w, E) is weakly measurable and it remains to show 2.1(iii) for Et being an QCS. But as it suffices to verify this equation on standard sets this follows from Eq. (6). Thus there is a unique QCS Et the restriction to At1 of which is Gt , and the theorem is proven. 4. Classification and Construction of QCS It the sequel V, K, τ is a fixed state space. Let Et be a one point QCS. We say Et has a bounded interaction rate if for all a > 0 there is a K(a) > 0 such that ||Et (w, Xt \{zt })|| 5 K(a)t
(7)
for all w ∈ Y∞ satisfying ||w|| 5 a. The QCS has a bounded modulation if for all a > 0 there is a K(a) > 0 such that for measurable sets E ⊆ C0t ∪ C1t holds ||Et (w, E) − Et (v, E)|| ≤ K(a)||w − v||∞ t
(8)
if ||w||∞ , ||v||∞ , t ≤ a. Finally the QCS is bounded if it has a bounded interaction rate and a bounded modulation. We explain the ideas one has in mind when given these definitions. First, for a bounded interaction rate the probability of detecting one or more particles during a time interval of length t gets (linearly) small with t, which is definitely physically reasonable. The problem is that Eq. (7), being formulated for the instruments, contains a uniform estimation for all states. Comparing with CP’s the “best” physical condition should have the form
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B. M¨uller
||Et (w, Xt \{zt })ω|| ≤ K(a, ω)t,
(9)
for ω in some dense subset of V. (If (9) is valid for all ω ∈ V the uniform boundedness principle implies that (7) holds.) Besides the fact that it is certainly the appropriate way to investigate first the simpler structure before turning to a more complicated one, there is another argument that justifies to consider bounded interaction rates. For CP’s (always take into account 2.2(ii)) Eq. (7) leads to the theory of unbounded interaction rates for which a main result is [18] that formally the same equations hold as in the bounded case, one only has to replace bounded operators by unbounded ones. The interpretation of the modulation condition (8) is that similar messages should imply similar outputs, at least for small t. Concerning the physical relevance the same considerations can be made as for the IR-condition, and it is reasonable to expect that in general the operators B(c) in Theorem 4.2 just have to be allowed to be unbounded. However, this is not yet proved and this paper deals with the bounded case. As for CP’s [6, 18] we require that the semigroup St := Et (0, {zt }) has additional properties. We say a semigroup St is of type R if its Farvard class, i.e. the set of all ω ∈ V for which the orbit {St ω|t ≥ 0} is Lipschitz continuous [4], equals the domain D(W ) of its generator W . Recall that this is always true for reflexive spaces. Any pure semigroup on the state space of trace class operators of a Hilbert space is of type R [6, 18]. We denote the Farvard class of St by F (W ). Lemma 4.1. Let St be a contraction semigroup of type R on V with generator (W, D(W )). (i) If A ∈ B(V) and Tt := exp[(W + A)t] is contractive, then Tt is of type R. (ii) If Tt is another contraction semigroup satisfying ||Tt − St || ≤ Kt, then Tt is of type R and there is a unique bounded operator A ∈ B(V) such that Tt = exp[(W + A)t]. Rt Proof. (i) If ω ∈ F (W +A) then the perturbation equation Tt ω = St ω + 0 Tt−s ASs ωds implies ||St ω − ω|| ≤ ||(Tt − St )ω|| + ||Tt ω − ω|| ≤ ||A|| ||ω||t + ||Tt ω − ω||, thus ω ∈ F (W ) = D(W ) = D(W + A), hence F (W + A) ⊆ D(W + A). The inverse inclusion is always true so Tt is of type R. (ii) Let (Z, D(Z)) denote the generator of Tt , then ω ∈ D(Z) implies ||St ω − ω|| ≤ ||Tt ω − ω|| + ||(St − Tt )ω|| ≤ ||Zω||t + Kt, thus ω ∈ D(W ). Hence A := Z − W is well defined on D(Z) and ||A|| ≤ K. Therefore Z = W + A, D(Z) = D(W ) and Tt is of type R by (i). We say a QCS is of type R if its interaction free semigroup St := Et (0, {zt }) is of type R. We always use the notation St = exp(W t). Theorem 4.2. Let Et be a bounded one point QCS of type R. Then there is a unique Family (B(c))c∈R in B(V) such that ∞ Z X w St,s = St−tn B(w(tn )) . . . B(w(t1 ))St1 ωdt1 . . . dtn (10) n=0
t−s s+Cn
0 = Et (0, {zt }) = exp(W t)). All the semigroups St1 c = Et (c, {zt }) for all ω ∈ V (St := St,0 are of type R and we have Stc = exp[(W + B(c))t]. Furthermore there is a unique family (J(c))c∈R in B+ (V) such that
Classification and Construction of Quantum Communication Systems
Et (w, E)ω =
∞ Z X n=0
t ∪E Cn
9
w St,t J(w(tn )) . . . J(w(t1 ))Stw1 ,0 ωdt1 . . . dtn . n
(11)
For any a > 0 there is a K(a) > 0 such that for all c, d ∈ R with |c|, |d| ≤ a, ||J(c) − J(d)|| ≤ K(a)|a − d|
and
||B(c) − B(d)|| ≤ K(a)|c − d|.
(12) (13)
Finally we have τ ∈ D(W ∗ ) and for all c ∈ R it holds (W ∗ + B(c)∗ + J(c)∗ )τ = 0.
(14)
Proof. First the modulation equation (8) implies that limt↓0 ||St ω −ω|| = 0 for all ω ∈ V and c ∈ R. Thus Stc is a contractive C0 -semigroup satisfying ||Stc − St || ≤ K(c)t. By Lemma 4.1 Stc is of type R and Stc = exp[(W + B(c))t] with a unique bounded operator B(c). In particular (10) is valid for any constant function w. Because of B(d)ω−B(c)ω = limt→0 1t (Std − Stc )ω the modulation condition implies (13). Thus defining w ω := S˜ t,s
∞ Z X t−s s+Cn
n=0
St−tn B(w(tn )) . . . B(w(t1 ))St1 ωdt1 . . . dtn ,
w one easily verifies that S˜ t,s is a well defined bounded operator satisfying w ˜w w S˜ t,s Ss,r = S˜ t,r
(15)
w w for r < s < t. By perturbation expansion we have S˜ t,s = St,s for constant functions and (15) implies that this is even true for piecewise constant functions. Now by the w is strongly continuous and a direct calculation modulation condition the map w → St,s w . By the density of the piecewise using (13) shows that the same holds for w → S˜ t,s constant functions we have proved (10). The theory of counting processes [18] shows that (11) is true for constant w, if we define J(c) to be the perturbation connecting the two semigroups Stc and Ttc := Et (c, Xt ) (both of type R by Lemma 4.1 and the interaction condition(7)). A similar argument as the above one shows the validity of (12), and (14) follows from [18]. We thus have for any constant c:
Z Et (c, [0, t))ω = 0
t
c c St,s J(c)Ss,0 ωds.
If w is constant on the interval [a, b) ⊆ [0, t), i.e. w(s) = c for s ∈ [a, b), then by (4) Z Et (w, [a, b))ω
=
w w St,b Eb−a (c, [0, b − a))Sa,0 ω=
Z
b
= a
b−a 0
w c c w St,b Sb−a,s J(c)Ss,0 Sa,0 ωds
w w St,b J(w(s))Ss,0 ωds.
Hence, if w is piecewise constant, i.e. w(s) = ci for s ∈ [ai , ai+1 ), where 0 < a = a0 < a1 < . . . < aN +1 = b < t, it follows
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B. M¨uller
Et (w, [a, b))ω
=
N X i=0 Z b
= a
Et (w, [ai , ai+1 ))ω =
N Z X i=0
ai+1 ai
w w St,b J(w(s))Ss,0 ωds
w w St,b J(w(s))Ss,0 ωds.
As both sides of this equation are continuous in w it holds for general w. Finally one verifies by some easy substitutions that for standard sets E = [a1 , b1 ) × . . . × [an , bn ), 0 ≤ a1 < b1 < . . . an < bn < t: Z w Et (w, E)ω = St,t J(w(tn )) . . . J(w(t1 ))Stw1 ,0 ωdt1 . . . dtn . n E
Both sides of this equation are σ-additive and coincide on the generating system of standard sets, hence we get (11). We now give the converse of the above theorem. Theorem 4.3. Let St = exp(W t) be a C0 -semigroup on V and (B(c))c∈R a strongly continuous family of bounded operators on V with B(0) = 0 such that all the semigroups Stc := exp[(W +B(c))t] are positive and contractive. For each c ∈ R, let J(c) ∈ B+ (V) be w and Et (w, E) a positive operator such that (14) holds. Then defining the operations St,s by (10) and (11) we get a one point QCS Et . Et is bounded iff (12) and (13) are valid. Proof. The uniform boundedness principle and the strong continuity of c → B(c) and c → J(c) imply that for any r > 0 there is a constant Ar > 0 such that supu≤r (||B(u)||+ w is a well ||J(u)||) = Ar < ∞. Then for fixed w ∈ Y∞ , it is easy to verify that St,s P∞ n w n defined operator with ||St,s || ≤ n=0 Ar (t − s) /k!. Using the Lebesgue Theorem and w . By hypothesis and this uniform estimation we get the strong continuity of w → St,s w perturbation theory the operators St,s are positive and contractive for constant functions w w w w Ss,r = St,r for r < s < t, therefore St,s is w. A direct calculation shows that St,s positive and contractive for piecewise constant w and by continuity even for arbitrary w. It now is obvious that each Et (w, E) is a well defined positive bounded operator. The strong σ-additivity of E → Et (w, E) the Markov equation 2.1(iii) and 2.1(i) are straightforward calculations. 2.1(ii) follows similar to the corresponding proof for the w . Equation (14) and perturbation theory show that Et (w, Xt )∗ τ = τ for map w → St,s constant functions. As above we extend this equality to piecewise constant functions and then to arbitrary w ∈ Y∞ . Now if the QCS Et is bounded (12) and (13) follow as in the proof of Theorem 4.2. The converse result can be obtained by using the explicit expressions for Et (w, E). We yet have shown how one point QCS’s should be constructed. Now we will concentrate on refinements of such a QCS. As in the case of CP’s any decomposition [18] of the operators J(c) will determine a refinement of the QCS. Proposition 4.4. Let (X, Σ) be a standard Borel space and let Et be constructed as in Theorem 4.3. For each c ∈ R let J(c, .) : Σ → B+ (V), E → J(c, E) be a strongly σadditive map with J(c, X) = J(c). Furthermore let c → J(c, E) be strongly continuous for all E ∈ Σ. Then there is a unique refinement E˜t of Et with measurable space (X, Σ) such that for all E ∈ Σ holds: Z t w ˜ sSt, sw J(w(t), E)Ss,0 ds. Et (w, E × [0, t))ω = 0
Classification and Construction of Quantum Communication Systems
11
Proof. Uniqueness is obvious. To prove existence we use Theorem 3.1. For measurable A ⊆ X × [0, t) and s ∈ [0, t) define As := {x ∈ X|(x, s) ∈ A} and Z
t
Gt (w, A)ω := 0
w w St,s J(w(s), As )Ss,0 ωds.
As in usual measure theory one verifies that the integral is weakly well defined and σadditive in A. Thus Gt (w, .) is a POVM on X × [0, t) which obviously is measurable in w. By Hypothesis 3.1(ii) (b) holds and 3.1(ii)(c) follows with some easy substitutions. We remark that not all refinements of the QCS E˜t necessarily have this form. The classification theorem for arbitrary CP’s with bounded interaction rate [18] shows that for a general refinement the operators J(c, E) have to be formulated as operators on a -dual space [5]. This theorem can even be used to derive structural results similar to those in Theorem 3.2 for arbitrary bounded QCS’s. We made the additional assumptions in Theorem 4.2 because the existence of the operators B(c) ∈ B(V) could not be proved without assuming that St is of type R. And even then it is not yet shown that an arbitrary non-one point QCS can be formulated on V rather than on some -dual space. However, using the classification of counting processes on Hilbert spaces [6, 18] one can show as in the proof of 4.2, that any refinement of a bounded QCS of type R on the state space of trace class operators must have the form which is given in the above proposition.
5. Special Cases Let us consider the QCS Et constructed in 4.3. There are two different differential w ρ) one equations related to Et . First for the interaction free dynamics (i.e. ρt = St,s formally gets ∂ ρt = (W + B(w(t)))ρt . (16) ∂t For the dynamics characterizing the whole system (ρt = Et (w, Xt )ρ) we have ∂ ρt = (W + B(w(t)) + J(w(t)))ρt . ∂t
(17)
Both equations at least are valid on D(W ) for the dense set of piecewise constant functions at any point where w is continuous. The first one shows that the interaction free dynamics contains some information about the transmitted message via the “modulation” B(w(t)). As W essentially contains the energy of a particle, B(w(t)) may be interpreted as an energy modulation caused by the influence of the input system. It then seems to be physically reasonable – at least for small w(t) – that B(c) has the form B(c) = cB0 ,
(18)
i.e. the modulation is linear. We say the QCS Et has a linear modulation if (18) holds. In order to meet the conditions of Theorem 4.3 the operator B0 must have additional properties. Recall that any norm continuous group of isometries on a state space V is positive.
12
B. M¨uller
Proposition 5.1. If Et is a bounded QCS as constructed in Theorem 4.3 with B(c) = cB0 , then B0 is the generator of a group of isometries, in particular B0∗ τ = 0. Conversely if B0 generates a group of isometries, W is the generator of a positive semigroup St of contractions and c → J(c) ∈ B+ (V) is a strongly continuous map with J(c)∗ τ = −W ∗ τ for all c, then Eq. (10) and (11) define a QCS with linear modulation. Proof. Let Et be a QCS with linear modulation, then all the operators W (c) = W + cB0 are generators of contraction semigroups, hence dissipative. Thus for any ω ∈ D(W ) = D(W (c)) with tangent functional x it is 0 ≥ hW (c)ω, xi = hW ω, xi + chB0 ω, xi. As this inequality is valid for all c ∈ R, we conclude hB0 ω, xi = 0. Therefore the operators (±B0 , D(W )) are dissipative and by [3, Prop. 3.1.15] even ±B0 is dissipative, hence B0 is the generator of a group of isometries (which is positive). By use of the Trotter–Kato formula one easily sees that the operators W (c) = W + cB0 generate positive contraction semigroups Stc with W (c)∗ τ = W ∗ τ . Thus we can apply Theorem 4.3 to get the desired result. We see that for a QCS with linear modulation the functional J(c)∗ τ does not depend on c. In particular ||J(c)|| = ||J(c)∗ τ || = ||W ∗ τ || is independent of c. We call a QCS Et as constructed in Theorem 4.3 a QCS with independent detector if J(c) = J for some fixed J ∈ B+ (V). In this case Eq. (11) suggests that J describes the instantaneous change of the state when a particle is detected. For physical applications the most interesting cases will be QCS’s with linear modulation and an independent detector, though one cannot expect that the operators J and B0 are bounded. To give some ideas on applications, we add a few comments on existing models that are not yet published. A direct generalization of Davies’ model [10] to macroscopic coherent light [14, 15] has to use W ∗ -algebraic methods, as the creation and annihilation operators in the relevant representation of the CCR cannot be formulated on Fock space. However, as the center of this representation is non-trivial, classical structures appear and can be detected. If the transmission uses squeezed light, the modulation operators on Hilbert space may have the form a∗2 − a2 or another quadratic expression in the creation and annihilation operators. On Fock space (resp. the trace class operators on Fock space) the existence of such a model has been proved and additional properties could be derived. Using Theorem 4.3 for a dependent model on the set Tsa (H) of trace class operators on a Hilbert space H with J(c)ρ = a(c)uρu∗ , W ρ = −i[H, ρ] − a(0)ρ, B(c)ρ = −i[8(c), ρ] − (a(c) − a(0))ρ, where ρ ∈ Tsa (H), u ∈ B(H) is unitary and a : R → R+ and φ : R → Bsa (H) are continuous functions, we get a very simple expression for the intensity registered by the detector: I(t) = hEt (w, Xt )ω, J ∗ (w(t))τ i = a(w(t)), which easily allows a numeric calculation of the channel capacity when using for instance a binary signal transmission. It is not always clear how to interpret a dependent detector although some considerations imply that this might be connected with a nontrivial movement of the detecting system. However we want to emphasize that there exist QCS without linear modulation or independent detector.
Classification and Construction of Quantum Communication Systems
13
References 1. Accardi, L.: A new class of quantum states: Examples and applications. In: C. Bendjaballah, O. Hirota, and S., Reynaud, ed., Quantum Aspects of Optical Communications. No. 378, Lecture Notes in Physics. Berlin–Heidelberg–New York: Springer-Verlag, 1991, pp. 138–150 2. Ash, R: Information Theory. New York: Interscience, 1965 3. Bratteli, O. and Robinson, D.W. Operator Algebras and Quantum Statistical Mechanics. Volume 1. New York–Heidelberg–Berlin: Springer-Verlag, 1987 4. Butzer, P.L. and Behrens, H.: Semigroups of Operators and Approximation. New York: Springer-Verlag, 1967 5. Clement, P., Diekmann, O., Gyllenberg, M., Heilmann, H.J.A.M., and Thieme, H.R.: Perturbation theory for dual semigroups, I. The sun-reflexive case. Math. Ann. 277, 709–725 (1987) 6. Davies, E.B.: Quantum stochastic processes. Commun. Math. Phys. 15, 277–304 (1969) 7. Davies, E.B.: An operational approach to quantum probability. Commun. Math. Phys. 17, 239–260 (1970) 8. Davies, E.B.: Quantum stochastic processes III. Commun. Math. Phys. 22, 51–70 (1971) 9. Davies, E.B.: Quantum Theory of Open Systems. London–New York: Academic Press, 1976 10. Davies, E.B.: Quantum communication systems. IEEE Trans. on Inf. Th. 23, 530–534 (1977) 11. Dunford, N., and Schwartz, J.T.: Linear Operators, Volume 1. New York–London: Interscience Publishers, 1958 12. Edwards, C.M.: The operational approach to algebraic quantum theory I. Commun. Math Phys. 16, 207–230 (1970) 13. Hall, M.J.W., and O’Rourke, M.J.: Realistic performance of the maximum information channel. Quantum Opt. 5, 161–180 (1993) 14. Honegger, R., and Rapp, A.: General Glauber coherent states on the Weylalgebra and their phase integrals. Physica A 167 945–961 (1990) 15. Honegger, R., and Rieckers, A.: The general form of non-Fock coherent Boson states. Publ. RIMS Kyoto University 26, 397–417 (1990) 16. Ingarden, R.S.: Quantum information theory. Rep. Math. Phys. 10, 43–73 (1976) 17. Lindblad, G.: Quantum entropy and quantum measurements. In: C. Bendjaballah, O. Hirota, and S. Reynaud, ed. Quantum Aspects of Optical Communications. No. 378, Lecture Notes in Physics. Berlin– Heidelberg–New York: Springer-Verlag, 1991, pp. 71–80 18. M¨uller, B.: On generalized quantum stochastic counting processes. To appear in Commun. Math. Phys. 19. Ohya, M.: On compound state and mutual information in quantum information theory. IEEE Trans. on Inf. Th. 29 (5), 770–774 (1983) 20. Ohya, M.: Some aspects of quantum information theory and their applications to irreversible processes. Rep. Math. Phys. 27 (1), 19–47 (1989) 21. Ohya, M.: Information dynamics and its applications to optical communication processes. In: C. Bendjaballah, O. Hirota, and S. Reynaud, ed. Quantum Aspects of Optical Communications. No. 378, Lecture Notes in Physics. Berlin–Heidelberg–New York: Springer-Verlag, 1991, pp. 81–92 22. Ohya, M.: Rigorous derivation of error probability in coherent optical communication. In: In: C. Bendjaballah, O. Hirota, and S. Reynaud, ed. Quantum Aspects of Optical Communications. No. 378, Lecture Notes in Physics. Berlin–Heidelberg–New York: Springer-Verlag, 1991, pp. 203–212 23. Parthasarathy, K.R.: Probability measures on metric spaces. New York–London: Academic Press, 1967 Communicated by H. Araki
Commun. Math. Phys. 191, 15 – 29 (1998)
Communications in
Mathematical Physics c Springer-Verlag 1998
Quantum Dynamical R-Matrices and Quantum Frobenius Group G.E. Arutyunov, S.A. Frolov Steklov Mathematical Institute, Gubkin str. 8, GSP-1, 117966, Moscow, Russia. E-mail:
[email protected];
[email protected] Received: 24 January 1997 / Accepted: 17 March 1997
Abstract: We propose an algebraic scheme for quantizing the rational RuijsenaarsSchneider model in the R-matrix formalism. We introduce a special parametrization of the cotangent bundle over GL(N, C). In new variables the standard symplectic structure is described by a classical (Frobenius) r-matrix and by a new dynamical r-matrix. ¯ Quantizing both of them we find the quantum L-operator algebra and construct its particular representation corresponding to the rational Ruijsenaars-Schneider system. Using the dual parametrization of the cotangent bundle we also derive the algebra for the L-operator of the hyperbolic Calogero-Moser system. 1. Introduction As soon as the classical dynamical r-matrices first appeared [1] on the scene of integrable many body systems, the problem of their quantization became of real interest. The main hope related to this problem is to find a new algebraic structure that ensures the integrability of the corresponding quantum models. We recall [2] that having a finite-dimensional completely integrable system with the Lax representation dL dt = [M, L] one can always write the Poisson algebra of L-operators in the r-matrix form. However, in general, an r-matrix appears to be a nontrivial function of dynamical variables. At present the classical dynamical r-matrices are known for the rational, trigonometric [1, 3] and elliptic [4, 5] Calogero-Moser (CM) systems, as well as for their relativistic generalizations – rational, trigonometric [6, 7] and elliptic [8, 9] Ruijsenaars-Schneider (RS) systems [10]. The problem of quantizing the dynamical r-matrices is rather nontrivial since, in general, such r-matrices do not satisfy a single closed equation of the Yang-Baxter type, from which they can be uniquely determined. Up to now there exists only one example of a quantum dynamical R-matrix related to the quantum spin CM system [11]. This Rmatrix solves the Gervais-Neveu-Felder equation [12, 13] and has a nice interpretation in terms of quasi-Hopf algebras [14].
16
G.E. Arutyunov, S.A. Frolov
A natural way to understand the origin of dynamical r-matrices is to consider the Hamiltonian reduction procedure [3, 15]. Factorizing a free motion on an initial phase space by the action of some symmetry group, we get nontrivial dynamics on the reduced space. An r-matrix appears in the Dirac bracket, which describes the phase structure of the reduced space. In our recent papers [16, 17] we obtained the elliptic RS model, being the most general one among the integrable systems of the CM and RS types, by using two different reduction schemes. In the first scheme, the affine Heisenberg double was used as the initial phase space and in the second one we considered the cotangent bundle over the centrally extended group of double loops. The aim of this paper is to quantize the reduction scheme leading to the dynamical systems of the RS type. Although the most interesting is the spectral-dependent elliptic case [16, 17], to clarify the general approach in this paper we restrict ourselves to considering the simplest rational model. Our construction is based on a special parametrization of the cotangent bundle over the group GL(N, C). This parametrization is similar to the one considered in [18]. However, instead of Euler angles, we use another system of generators to parametrize the “momentum”. These generators obey a quadratic Poisson algebra described by two r-matrices r and r. ¯ The matrix r solves the N -parametric classical Yang-Baxter equation and is related to the special Frobenius subgroup in GL(N, C). We define a special matrix function L on T ∗ G invariant with respect to the action of this Frobenius subgroup. We call this function the “L-operator” since the Poisson algebra of L literally coincides with the one for the rational RS model [7]. Moreover, performing the Hamiltonian reduction leading to the RS model [19], one can recognize in L the standard L-operator of the rational RS model. Then we pass to the quantization. The quadratic Poisson algebra can be quantized by using the R-matrix approach [20, 21]. The compatibility of the corresponding quantum algebra implies the quantum Yang-Baxter equation for R and some new equations ¯ We solve these equations and get an explicit form for R and R. ¯ involving R and R. Coming back to the original generators of T ∗ G we recover the standard commutation relations of the quantum cotangent bundle. We derive a new quadratic algebra which is satisfied by the “quantum” L-operator: −1 −1 −1 L2 R¯ 12 R12 R¯ 21 = R12 L2 R¯ 12 L1 . L1 R¯ 21
We see that the matrices R and R¯ come in this algebra in a nontrivial way. Thus, the dynamical r-matrices (classical and quantum) appear as the composite objects constructed ¯ from the more elementary blocks R and R. It follows from our construction that the quantum L-operator is factorized in the form L = W P . Here W satisfies the defining relations of the quantum Frobenius group, W1 W2 R12 = R12 W2 W1 , where R, being the quantization of r, is an N -parametric solution to the quantum Yang-Baxter equation. The diagonal matrix P plays the role of a generalized momentum. We find the simplest representation of the L-operator algebra and relate it with the rational RS model. The hyperbolic (trigonometric) CM system is known to be dual to the rational RS model [10]. This duality is explained by the existence of the dual parametrization of T ∗ G. In this parametrization the CM model can be easily quantized and we get the commutation relations satisfied by the corresponding quantum L-operator.
Quantum Dynamical R-Matrices and Quantum Frobenius Group
17
2. Frobenius Algebra and Dynamical r-Matrices In this section we introduce a special parametrization of the cotangent bundle T ∗ G over the matrix group G = GL(N, C). As a manifold the space T ∗ G is naturally isomorphic to G ∗ × G, where G ∗ is dual to the Lie algebra G = Mat(N, C). The standard Poisson structure on T ∗ G can be written in terms of variables (A, g), where A ∈ G ∗ ≈ G and g ∈ G, as follows 1 [C, A1 − A2 ], (2.1) 2 {A1 , g2 } = g2 C, (2.2) (2.3) {g1 , g2 } = 0. P Here we use the standard tensor notation and C = i,j Eij ⊗ Eji is the permutation operator. Any matrix A belonging to an orbit of maximal dimension in G ∗ admits a factorization: (2.4) A = T QT −1 , {A1 , A2 } =
where Q is a diagonal matrix with entries qi , qi 6= qj . We also fix the order of qi by using the action of the Weyl group. It is obvious that the matrix T in Eq. (2.4) is not uniquely defined. Indeed, one can multiply T by an arbitrary diagonal matrix from the right. We remove this ambiguity by imposing the following condition: T e = e,
(2.5)
where e is a column with all ei = 1. The choice of (2.5) is motivated by the study of the reduction procedure leading to the Calogero-type integrable systems [22]. Let us note that the condition (2.5) defines a Lie subgroup F ⊂ G. The corresponding Lie algebra F has a natural basis Fij = Eii − Eij , where Eij are the standard matrix unities. The commutation relations of Fij are [Fij , Fkl ] = δik (Fil − Fij ) + δil (Fkj − Fkl ) + δjk (Fij − Fil ). It is worthwhile to mention that F is not only the Lie algebra but also an associative algebra with respect to the usual matrix multiplication: Fij Fkl = δik Fil + δjk (Fik − Fil ). Let us rewrite the Poisson structure (2.1) in terms of the variables T and Q. It is well known that the center of the Poisson algebra (2.1) is generated by the Casimir elements trAn = trQn . Therefore, the coordinates qi Poisson commute with A, T and Q. The only nontrivial bracket one has to calculate is {T, T }. We have {Tij , Tkl } = To find
δTij δAmn
X
δTij δTkl {Amn , Aps }. δAmn δAps mn,ps
we perform the variation of (2.4): T −1 δT Q − QT −1 δT + δQ = T −1 δAT.
This equation can be easily solved, and we obtain the derivatives
18
G.E. Arutyunov, S.A. Frolov
X 1 δTij −1 −1 = (Tia Tnj Tam + Tij Tna Tjm ) δAmn qja
(2.6)
δqi −1 = Tni Tim , δAmn
(2.7)
{T1 , T2 } = T1 T2 r12 (q),
(2.8)
a6=j
and
where qij ≡ qi − qj . By using (2.6) we get where the r-matrix
r12 (q) =
X 1 Fij ⊗ Fji qij
(2.9)
i6=j
appears. It is clear that r12 (q) should be a skew-symmetric solution of the classical Yang-Baxter equation (CYBE). The origin of this r-matrix can be easily understood if we notice that F is a Frobenius Lie algebra, i.e. there is a nondegenerate 2-cocycle (coboundary) on F: ω(X, Y ) = tr(Q[X, Y ]), X, Y ∈ F .
(2.10)
According to [23], to any Frobenius Lie algebra one can associate a skew-symmetric solution of the CYBE by inverting the corresponding 2-cocycle. One can check that the cocycle (2.10) corresponds to r12 (q) ∈ F ∧ F. Coming back to (2.4), we see that any orbit of maximal dimension in G ∗ can be supplied with the structure of the Frobenius group. It is worthwhile to note that ω is the Kirillov symplectic form on the coadjoint orbit of the maximal dimension parametrized by Q. Now, following [18], we introduce a special parametrization for the group element g. To this end we consider an element A0 = gAg −1 , which Poisson commutes with A and possesses the Poisson bracket 1 {A01 , A02 } = − [C, A01 − A02 ]. 2 Diagonalizing A0 = U QU −1 with the help of the matrix U , U e = e, we find that g = U P T −1 ,
(2.11)
where P is some diagonal matrix. It is obvious that the Poisson bracket for U is given by (2.12) {U1 , U2 } = −U1 U2 r12 (q) and that {T, U } = 0. To proceed with the calculation of the brackets {U, P }, {T, P }, δU {P, P } and {P, Q}, we should use the derivative δA0ij that is given by (2.6) with the mn replacement T → U , A → A0 . Performing simple computations, we get {P1 , P2 } = 0, {Q1 , P2 } = P2
X
Eii ⊗ Eii .
(2.13)
i
Introducing pi = log Pi , we conclude that {qi , pj } = δij . We also find the Poisson brackets:
Quantum Dynamical R-Matrices and Quantum Frobenius Group
{U1 , P2 } = U1 P2 r¯12 (q), {T1 , P2 } = T1 P2 r¯12 (q), where we have introduced a new matrix X 1 Fij ⊗ Ejj . r¯12 (q) = qij
19
(2.14) (2.15)
(2.16)
i6=j
The Jacobi identity leads to a set of equations on the matrices r and r. ¯ However, we postpone the discussion of these equations till the next section, where the quantization of T ∗ G will be given in terms of variables Q, T, P, U . Let us define the L-operator as the following function of phase variables, being invariant under the action of the Frobenius group: L = T −1 gT = T −1 U P.
(2.17)
By using (2.8), (2.12) and (2.13-2.15) one can easily find the Poisson brackets containing L: X Eii ⊗ Eii , (2.18) {Q1 , L2 } = L2 i
{T1 , L2 } = T1 L2 r¯12 (q) − T1 r12 (q)L2 , {L1 , L2 } = r12 (q)L1 L2 + L1 L2 (r¯12 (q) − r¯21 (q) − r12 (q)) + L1 r¯21 (q)L2 − L2 r¯12 (q)L1 .
(2.19) (2.20)
We see that the brackets for L and Q are the ones for the L-operator and coordinates of the rational RS model found in [7]. It is obvious that In = trLn = trg n form a set of mutually commuting functions. Let us note that the L-operator (2.17) has the form L = W P , where W = T −1 U . Since both T and U are elements of the Frobenius group F , the element W also belongs to F . Calculating the Poisson bracket for W , we see that it coincides with the Sklyanin bracket defining on F the structure of a Poisson-Lie group: {W1 , W2 } = [r12 (q), W1 W2 ].
(2.21)
The Poisson relations of W and P are {W1 , P2 } = −P2 [r¯12 (q), W1 ].
(2.22)
A well-known property of the Poisson algebra (2.21) is the existence of a family of mutually commuting functions Jn = trW n . Moreover, it turns out that Jn commute not only with themselves but also with P and Q. In Sect. 3 we show that the same property holds in the quantum case. Now we construct the simplest representation of the Poisson algebra (2.21), (2.22) and (2.13) and relate it with the rational RS model. In fact, this representation corresponds to a zero-dimensional symplectic leaf of the bracket (2.21). To this end we employ the Hamiltonian reduction procedure. The 1-form corresponding to the Poisson structure (2.1-2.3) is α = tr(Ag −1 dg). Recall that G acts on T ∗ G by transformations A → hAh−1 , g → hgh−1 in a Hamiltonian way. By using the 1-form α one can easily get the corresponding moment map µ: µ = gAg −1 − A. Performing the Hamiltonian reduction, we fix its value to be
20
G.E. Arutyunov, S.A. Frolov
gAg −1 − A = −γ(ee+ − 1), where γ is an arbitrary constant and e+ is a row with all e+i = 1. In terms of (T, L, Q) variables this equation acquires the form T (LQL−1 − Q)T −1 = −γ(ee+ − 1).
(2.23)
Since T e = e and L = W P the last equation can be written as W Q − QW − γW = −γee+ U. Equation (2.24) can be elementary solved and one gets X γ ei bj Eij , W = γ + qij i,j
(2.24)
(2.25)
where b = e+ U . If we recall that W should be an element of F , i.e. W e = e, then we find the coefficients bj : Q 1 a (qaj + γ) Q , (2.26) bj = γ a6=j qaj and thereby
Q Wij =
a6=i (qaj
Q
+ γ)
a6=j qaj
.
(2.27)
One can check that W given by (2.27) has the desired Poisson brackets (2.21) and (2.22). We do not give an explicit proof of this statement since in Sect. 3 we show that the same function W realizes a representation for the corresponding quantum algebra. The relation of our L-operator with the standard Ruijsenaars L-operator [10] is given by the following canonical transformation: qi → qi , Pi →
Y (qai − γ)1/2 a6=i
(qai + γ)1/2
Pi .
3. Quantum L-Operator Algebra In this section we quantize T ∗ G in terms of variables Q, T, P, U and obtain the permutation relations for the quantum L-operator. The algebra of functions on T ∗ G can be easily quantized and one gets an associative algebra generated by A and g subject to the standard relations: 1 ~[C, A1 − A2 ], 2 [A1 , g2 ] = ~g2 C, [g1 , g2 ] = 0,
[A1 , A2 ] =
(3.1) (3.2) (3.3)
where ~ is a quantization parameter. The commutation relations for the generators T, Q, U, P can be straightforwardly written by using the ideology of the Quantum Inverse Scattering Method [20, 21] (we present only nontrivial relations):
Quantum Dynamical R-Matrices and Quantum Frobenius Group
21
−1 T1 T2 = T2 T1 R12 (q), U1 U2 = U2 U1 R12 (q), T1 P2 = P2 T1 R¯ 12 (q), U1 P2 = P2 U1 R¯ 12 (q), X Eii ⊗ Eii . [Q1 , P2 ] = ~P2
(3.4) (3.5) (3.6)
i
¯ Here R(q) and R(q) are quantum dynamical R-matrices having the following behavior near ~ = 0: ¯ R(q) = 1 + ~r(q) + o(~), R(q) = 1 + ~r(q) ¯ + o(~). It is worthwhile to mention that the matrix elements of the dynamical R-matrices do not commute only with the momenta pi due to the relations [T1 , Q2 ] = [U1 , Q2 ] = 0. It follows from the compatibility conditions that the R-matrices should satisfy the following set of equations R12 (q)R21 (q) R12 (q)R13 (q)R23 (q) R12 (q)R¯ 13 (q)R¯ 23 (q) R¯ 12 (q)P −1 R¯ 13 (q)P2 2
= = =
1, R23 (q)R13 (q)R12 (q), R¯ 23 (q)R¯ 13 (q)P3−1 R12 (q)P3 , = R¯ 13 (q)P −1 R¯ 12 (q)P3 . 3
(3.7) (3.8) (3.9) (3.10)
Let us demonstrate how to get, for example, Eq. (3.9). This equation follows from (3.5) and the following chain of relations: −1 T1 R12 R¯ 13 R¯ T1 T2 P3 = T1 P3 T2 R¯ 23 = P3 T1 R¯ 13 T2 R¯ 23 = P3 T2 T1 R12 R¯ 13 R¯ 23 = T2 P3 R¯ 23 ¯ = T2 T1 P3 R¯ −1 R¯ −1 R12 R¯ 13 R¯ = T1 T2 R−1 P3 R¯ −1 R¯ −1 R12 R¯ 13 R. 13
23
12
13
23
The solution of the Yang-Baxter Eq. (3.8) can be easily found if one notes that 2 = 0. It is known (see e.g. [24] Prop.6.4.13) that if r ∈ Mat(N, C) ⊗ Mat(N, C) r12 satisfies r3 = 0 and solves the classical Yang-Baxter equation then R = er is a solution of QYBE. Therefore, R12 = e~r12 = 1 + ~r12 is a desired solution of the QYBE. The solution of Eq. (3.9) can be found if one supposes that R¯ has the same matrix structure as r¯ does: X r¯ij (~, q)Fij ⊗ Ejj . (3.11) R¯ 12 (q) = 1 + ~ i6=j
¯ Then the following R-matrix is a solution of Eqs. (3.9) and (3.10): R¯ 12 (q) = 1 +
X i6=j
~ Fij ⊗ Ejj . qij − ~
(3.12)
P −1 (q) = 1 − i6=j q~ij Fij ⊗ Ejj . It is not difficult to verify that R¯ 12 Now we should show that the generators A = T QT −1 and g = U P T −1 satisfy the commutation relations (3.1-3.3). From the relations (3.4-3.6) we get [A1 , A2 ] = T2 T1 (R12 Q1 R21 Q2 − Q2 R12 Q1 R21 )T1−1 T2−1 , X −1 Eii ⊗ Eii )R¯ 12 R12 − R12 Q1 )T2−1 T1−1 , [A1 , g2 ] = g2 T2 T1 (R¯ 12 (Q1 + ~ [g1 , g2 ] =
i −1 −1 −1 ¯ ¯ R12 U2 U1 (R12 P1 R21 P2 R12
−1 −1 − P2 R¯ 12 P1 R¯ 21 )T2−1 T1−1 .
22
G.E. Arutyunov, S.A. Frolov
By using the following identities, which can be checked by direct computation: R12 Q1 R21 Q2 − Q2 R12 Q1 R21 = ~(CQ1 R21 − R12 Q1 C), X −1 Eii ⊗ Eii )R¯ 12 R12 − R12 Q1 = ~C, R¯ 12 (Q1 + ~ i −1 −1 −1 ¯ P2 R¯ 12 R12 R12 P1 R21
−1 −1 − P2 R¯ 12 P1 R¯ 21 = 0,
(3.13)
one derives the desired commutation relations for A and g. Let us finally present the permutation relations for Q, T and L = T −1 U P , which can be easily obtained by using (3.4-3.6) and (3.13): X Eii ⊗ Eii , (3.14) [Q1 , L2 ] = ~L2 i
T1 R12 L2 = L2 T1 R¯ 12 , −1 −1 −1 L2 R¯ 12 R12 R¯ 21 = R12 L2 R¯ 12 L1 . L1 R¯ 21
(3.15) (3.16)
It is clear that just as in the classical case the quantities In = trg n form a set of mutually commuting operators. Let us show that In can be expressed in terms of L and Q solely and thereby they can be interpreted as quantum integrals of motion for the rational RS model. By using the definition of L we rewrite In as t
t2 In = trg n = trT Ln T −1 = tr12 C12 T1 Ln1 T2−1 = tr12 C12 T1 Ln1 T2−1 ,
where t2 denotes the matrix transposition in the second factor of the tensor product. It follows from (3.15) that t
t
t2 t2 L1 R¯ 21 . L1 T2−1 =T2−1 R12
Applying this relation, we derive t
t2 t2 t2 t2 t2 T1 T2−1 R12 L1 R¯ 21 · · · R12 L1 R¯ 21 . In = tr12 C12 t
Exchanging T1 and T2−1 with the help of Eq. (3.4), one gets t
t2 t2 t2 t2 In = tr12 C12 T2−1 T1 L1 R¯ 21 · · · R12 L1 R¯ 21 . t
t2 t2 t2 t2 t2 Since C12 T2−1 T1 = C12 and R¯ 21 C12 = C12 we finally arrive at t2 t2 t 2 t2 t 2 t2 t 2 In = tr12 C12 L1 R¯ 21 R12 L1 R¯ 21 R12 L1 · · · L1 R¯ 21 R12 L1 .
It is natural to regard this expression for In as a “quantum trace” of the operator Ln . Just as in the classical case the quantum L-operator has the form L = W P , where W = T −1 U satisfies the defining relations of the quantum Frobenius group: R12 W2 W1 = W1 W2 R12 .
(3.17)
The algebra (3.14), (3.16) rewritten in terms of Q, P and W is given by (3.6), (3.17) and by the relation
Quantum Dynamical R-Matrices and Quantum Frobenius Group
23
−1 −1 W1 P2 R¯ 12 = P2 R¯ 12 W1 .
(3.18)
This shows that the representation theory for L essentially reduces to the one for the quantum Frobenius group. It is known that the algebra (3.17) admits a family of mutually commuting operators given by [25]: Jn = tr1...n Rˆ 12 Rˆ 23 . . . Rˆ n−1,n W1 . . . Wn , where Rˆ ij = Rij Cij . Now we demonstrate that Jn commutes with P . For the sake of clarity we do it for n = 3. It follows from (3.18) that −1 −1 −1 W1 R¯ 14 R¯ 24 W2 R¯ 24 R¯ 34 W3 R¯ 34 . J3 P4 = Rˆ 12 Rˆ 23 P4 R¯ 14
Equation (3.9) written in terms of Rˆ acquires the form −1 ¯ −1 ˆ P3−1 Rˆ 12 (q)P3 = R¯ 23 R13 R12 R¯ 13 R¯ 23 .
By using this equation we can push P4 on the left. Taking into account that R¯ 12 is diagonal in the second space, we get −1 ¯ −1 ¯ −1 ˆ R24 R34 R12 Rˆ 23 W1 W2 W3 R¯ 34 R¯ 24 R¯ 14 . J3 P4 = P4 R¯ 14
Taking the trace in the first, second and third spaces, one gets the desired property. Now we give an example of the representation of the algebra (3.17), (3.18). Namely, −~
∂
we prove that the W -operator given by Eq. (2.27) realizes this algebra with Pj = e ∂qj . It is obvious that [W1 , W2 ] should be equal to zero since W depends only on the coordinates qi . Thus, the following relation has to be valid: [r12 (q), W1 W2 ] = 0. Substituting the explicit form of r12 (q) we have
1 Wmn qkm qkm qln 1 1 1 − Wml Wmn − Wkn − Wkn qkm qkm qln X 1 (Wkl Wmj − Wkj Wml ). + δln qlj
[r12 (q), W1 W2 ]kl mn = Wkl
1
Wmn −
1
Wkn −
(3.19)
j6=l
First we show that when l 6= n the first line in (3.19) cancels the second one. Since γ bj we get Wij = γ+q ij 1 1 γ γ γ γ 1 − − − γ + qkl qkm γ + qmn qkm γ + qkn qln γ + qmn 1 1 γ γ γ γ 1 = 0. − − γ + qml qkm γ + qmn qkm γ + qkn qln γ + qkn In the case l = n the r.h.s. of (3.19) reduces to
24
G.E. Arutyunov, S.A. Frolov
− −
1 qkm
(Wkl − Wml )2 +
X 1 (Wkl Wmj − Wkj Wml ) = qlj j6=l
X 1 γ qkm γ γ γ γ 2 b + ( − )bj bl l (γ + qkm )2 (γ + qml )2 qlj γ + qkl γ + qmj γ + qkj γ + qml 2
X Wmj qmk = Wkl . γ + qml γ + qkj j
j6=l
P W Thus, one has to show that for m 6= k the series S = j γ+qmj vanishes. To this end we kj 1 consider the following integral Q I 1 dz a6=m (qa − z + γ) Q , I= 2πi qk − z + γ a (qa − z) where the integration contour is taken around infinity. Since the integrand is nonsingular at z → ∞, we get I = 0. On the other hand, summing up the residues one finds Q X 1 a6=m (qaj + γ) Q = S. I= γ + qkj a6=j qaj j Now we turn to Eq. (3.18). Explicitly it reads as ~ ~ ~2 ~ ~2 −1 − W + Wjl [W , P ] = − + Pj kl j kl qlj − ~ qkj qkj (qlj − ~) qkj (qlj − ~) qkj X ~ ~2 ~2 − Wki − Wli (.3.20) + δjl qkj (qij − ~) qij − ~ qkj (qij − ~) i6=j
For the sake of shortness in (3.20) we adopt a convention that if in some denominator qij becomes zero, the corresponding fraction is also regarded as zero. Thus Eq. (3.20) is equivalent to the following system of equations: ~
Pj−1 [Wkl , Pj ] =
(qkl Wkl − qjl Wjl ), for k 6= l 6= j; qkj (qlj − ~) ~ Wjl , for j 6= l; Pj−1 [Wjl , Pj ] = (qlj − ~) X ~ Wki ; Pk−1 [Wkk , Pk ] = − qik − ~
(3.21) (3.22) (3.23)
i6=k
Pj−1 [Wkj , Pj ] =
~ ~(~ − qkj ) X 1 Wki (Wjj − Wkj ) + qkj qkj qij − ~
(3.24)
i6=j
−
~2 X 1 Wji , for k 6= j. qkj qij − ~ i6=j
In the sequel we shall give an explicit proof only for the latter case since the other three cases are treated quite analogously. The l.h.s. of (3.24) is 1
We are grateful to N.A. Slavnov for explaining to us the technique of treating such series.
Quantum Dynamical R-Matrices and Quantum Frobenius Group
Q Pj−1 [Wkj , Pj ]
=
Pj−1
a6=k (qaj
Q
a6=j
+ γ)
qaj
25
Q
a6=j,k (qaj
Q
Pj − Wkj = γ
+ γ − ~)
a6=j (qaj
− ~)
− Wkj .
P As to the r.h.s., one needs to calculate the sum i6=j qij1−~ Wki . For this purpose we evaluate the following integral with the integration contour around infinity: 1 I= 2πi
I
Q
dz z − qj − ~
a6=k (qa
Q
− z + γ)
a (qa
− z)
.
The regularity of the integrand at z → ∞ gives I = 0. On the other hand, summing up the residues one finds 1 I=− ~
Q a6=k Q
(qaj + γ − ~)
−
a6=j (qaj − ~)
X i6=j
1 1 Wki + Wkj . qij − ~ ~
From here one deduces the desired series: X i6=j
1 1 Wki = − qij − ~ ~
and X i6=j
1 1 Wji = − qij − ~ ~
Q a6=k Q
(qaj + γ − ~)
a6=j (qaj − ~)
Q a6=j Q
(qaj + γ − ~)
a6=j (qaj
− ~)
+
1 Wkj ~
(3.25)
+
1 Wjj . ~
(3.26)
Now substituting these sums in the r.h.s. of (3.24) one proves (3.24). It follows from our proof that the L-operator L=
X ij
Q
a6=i (qaj
Q
a6=j
+ γ)
qaj
e
∂ −~ ∂q
j
Eij
realizes the representation of the algebra (3.16). Let us briefly discuss the degeneration of the RS system to the rational CM model. To get the CM model, one should rescale ~ → γ~ and consider the limit γ → 0, L → 1 + γL. Then L is the L-operator of the CM model. From Eqs. 3.14), (3.16) one derives the quantum algebra satisfied by the L-operator of the CM model: [Q1 , L2 ] = ~
X
Eii ⊗ Eii ,
(3.27)
i
[L1 , L2 ] = ~[r12 − r¯12 , L1 ] − ~[r21 − r¯21 , L2 ] + ~2 [r12 − r¯12 , r21 − r¯21 ]. (3.28) The last formula can be written in the following elegant form [L1 + ~(r21 − r¯21 ), L2 + ~(r12 − r¯12 )] = 0.
(3.29)
26
G.E. Arutyunov, S.A. Frolov
4. Quantum R-Matrix for the Hyperbolic CM System In this section we describe a dual parametrization of T ∗ G, which is related to the hyperbolic CM system. We start with diagonalizing the group element g = V DV −1 and δVij δDi impose the constraint V e = e. The derivatives δgkl and δg are obtained in the same kl manner as in Sect. 2. Calculating the Poisson brackets of V , D and A we get {V1 , A2 } = V1 V2 s12 V2−1 , X Eii ⊗ Eii V2−1 , {D1 , A2 } = −D1 V2
(4.1) (4.2)
i
where s12 = −
X i6=j
Di Fij ⊗ Eji . Di − Dj
The L-operator corresponding to the hyperbolic CM system is defined as the following function on the phase space: L = V −1 AV.
(4.3)
Calculation of the Poisson algebra of T ∗ G in terms of L, V, D results in {L1 , L2 } = [r˜12 , L1 ] − [r˜21 , L2 ], {V1 , L2 } = V1 s12 , X {D1 , L2 } = −D1 Eii ⊗ Eii ,
(4.4) (4.5) (4.6)
i
where we have introduced the matrix 1 r˜12 = −s12 + C. 2 The matrix r˜12 can be written in the following form: qij
r˜12
1X qij 1X e 2 1X =− cth Eij ⊗ Eji + Eii ⊗ Eii , (4.7) qij Eii ⊗ Eji + 2 2 2 2 i sinh 2 i6=j
i6=j
where qi = log Di . Now we clarify the connection of r˜12 with the dynamical r-matrix found in [1]. By 1 1 1 conjugating L-operator (4.3) with the matrix D 2 : L˜ = D 2 LD− 2 and calculating the Poisson bracket for L˜ with the help of (4.4) and (4.6), we arrive at {L˜ 1 , L˜ 2 } = [R˜ 12 , L˜ 1 ] − [R˜ 21 , L˜ 2 ],
(4.8)
where 1 1 1X −1 −1 Eii ⊗ Eii R˜ 12 = D12 D22 r˜12 D1 2 D2 2 − 2 i 1X qij 1X 1 =− cth Eij ⊗ Eji + q Eii ⊗ Eji . 2 2 2 sinh 2ij
i6=j
i6=j
(4.9)
Quantum Dynamical R-Matrices and Quantum Frobenius Group
27
It is important to note that R˜ differs from the matrix found in [1] by the term 1X 1 q Eii ⊗ (Eji + Eij ). 2 sinh 2ij i6=j
However, this term does not contribute to the bracket (4.8) if we take into account the ˜ representation of the L-operator of the hyperbolic CM system: L˜ =
X
pi Eii +
i
1X 1 q Eij , 2 sinh 2ij
(4.10)
i6=j
where (p, q) form a pair of canonically conjugated variables. Thus, on the reduced space ˜ these matrices define the same Poisson structure for L. ∗ Now we quantize T G in terms of A, V and D variables. We postulate the following commutation relations: [V1 , A2 ] = ~V1 V2 s12 V2−1 , X Eii ⊗ Eii V2−1 . [D1 , A2 ] = −~D1 V2
(4.11) (4.12)
i
One can verify that the compatibility of these relations with (3.2) follows from the following identity satisfied by s12 : X s12 − D1−1 s12 D1 + Eii ⊗ Eii = C. i
By using Eqs. (3.1), (4.11) and (4.12) one derives the commutation relations for the quantum L-operator: [L1 , L2 ] = ~[r˜12 , L1 ] − ~[r˜21 , L2 ] + ~2 [r˜12 , r˜21 ], X Eii ⊗ Eii , [D1 , L2 ] = −~D1
(4.13) (4.14)
i
[V1 , L2 ] = ~V1 s12 .
(4.15)
The relation (4.13) can be also written in the form (3.29): [L1 + ~r˜21 , L2 + ~r˜12 ] = 0.
(4.16)
One can check without problems that the L-operator (4.10) realizes the representation of the algebra (4.16). To complete our discussion let us show the existence of N mutually commuting operators in the algebra composed by the L-operator and the coordinates D. Obviously, In = trAn mutually commute. Applying the technique used in the previous section to derive the quantum integrals of motion, one can show that In can be expressed as the following function of L and D: t2 (L1 + ~st212 )n . In = trAn = trV Ln V −1 = tr12 C12
In the component form these integrals look as
28
G.E. Arutyunov, S.A. Frolov
In =
X
Lj 1 j 2 δj 1 m 1
j1 ,...,jn+1 m1 ,...,mn
L j2 j3 δm1 m2
δj 1 j 2 − 1 +~ δj 2 m 1 Dj1 Dj−1 −1 2
δ m j − δm j + ~ 1 3 −1 1 2 δj3 m2 D j2 Dj 3 − 1
Ljn jn+1 δmn−1 mn + ~
!
! ···
δmn−1 jn+1 − δmn−1 jn Djn Dj−1 −1 n+1
! δjn+1 mn
δjn+1 mn .
Let us note that In can not be expressed as a linear combination of trLn solely. 5. Conclusion The approach to R-matrix quantization of the RS models proposed in the paper seems to be general. The problem of real interest is to apply it to the trigonometric and elliptic cases. As was recently shown, the trigonometric and elliptic RS models are obtained from the cotangent bundles over the centrally extended loop group [19] and double loop group [17] respectively. A natural suggestion is to use for this purpose the above-mentioned phase spaces. It is known that the Heisenberg double [26] can be regarded as a natural deformation of the cotangent bundle T ∗ G. It seems to be interesting to investigate the Poisson structure of the Heisenberg double in the same parametrization. One could expect the appearance of another Poisson structure on the Frobenius group induced by the one on the dual Poisson-Lie group G∗ . The appearance of the quantum Frobenius group F states the problem of developing the corresponding representation theory. Owing to the method of orbits, one can suggest that irreducible representations of F should be in correspondence with the symplectic leaves of the Poisson-Lie structure. On the other hand, it is known [27] that the symplectic leaves of a Poisson-Lie structure are the orbits of the dressing transformation. Studying the orbits of F ∗ and corresponding representations of F , one can hope to obtain the quantum integrable systems being some “spin” generalizations of the RS model [28]. Another open problem related to the representation theory is to find the universal Frobenius R-matrix. As was shown in [29], the Ruijsenaars Hamiltonians can be related to the special L-operator satisfying the fundamental relation RLL = LLR with Belavin’s elliptic R-matrix. It would be interesting to clarify the relationship of this approach with our construction. Acknowledgement. The authors are grateful to L.O.Chekhov, P.B.Medvedev and N.A.Slavnov for valuable discussions. This work is supported in part by the RFFR grants N96-01-00608 and N96-01-00551 and by the ISF grant a96-1516.
References 1. 2. 3. 4.
Avan, J. and Talon, M.: Phys. Lett. B303, 33–37 (1993) Babelon, O. and Viallet, C.M.: Phys. Lett. B237, 411 (1989) Avan, J., Babelon, O. and Talon, M.. Alg. Anal. 6 (2), 67 (1994) Sklyanin, E.K.: Alg. Anal. 6 (2), 227 (1994)
Quantum Dynamical R-Matrices and Quantum Frobenius Group
29
5. Braden, H.W. and Suzuki, T.: Lett. Math. Phys. 30, 147 (1994) 6. Avan, J. and Rollet, G.: The classical r-matrix for the relativistic Ruijsenaars-Schneider system. preprint BROWN-HET-1014 (1995) 7. Suris, Yu.B.: Why are the rational and hyperbolic Ruijsenaars-Schneider hierarchies governed by the same R-matrix as the Calogero-Moser ones ? hep-th/9602160 8. Nijhoff, F.W., Kuznetsov, V.B., Sklyanin, E.K. and Ragnisco, O.: Dynamical r-matrix for the elliptic Ruijsenaars-Schneider model. solv-int/9603006 9. Suris, Yu.B.: Elliptic Ruijsenaars-Schneider and Calogero-Moser hierarchies are governed by the same r-matrix. solv-int/9603011 10. Ruijsenaars, S.N.: Commun. Math. Phys. 110, 191 (1987) 11. Avan, J., Babelon, O. and Billey, E.: The Gervais-Neveu-Felder equation and the quantum CalogeroMoser systems. Preprint PAR LPTHE 95-25, May 1995; hep-th/9505091 (to appear in Commun. Math. Phys.) 12. Gervais, J.L. and Neveu, A.: Nucl. Phys. B238, 125 (1984) 13. Felder, G.: Conformal field theory and integrable systems associated to elliptic curves. hep-th/9407154 14. Babelon, O., Bernard, D. and Billey, E.: A quasi-Hopf algebra interpretation of quantum 3-j and 6-j symbols and difference equations. Preprint PAR LPTHE 95-51, IHES/P/95/91, q-alg/9511019 15. Arutyunov, G.E. and Medvedev, P.B.: Phys. Lett. A223, 66–74 (1996) 16. Arutyunov, G.E., Frolov, S.A. and Medvedev, P.B.: Elliptic Ruijsenaars-Schneider model via the Poisson reduction of the affine Heisenberg double. hep-th/9607170 17. Arutyunov, G.E., Frolov, S.A. and Medvedev, P.B.: Elliptic Ruijsenaars-Schneider model from the cotangent bundle over the two-dimensional current group hep-th/9608013 18. Alekseev, A. and .Faddeev, L.D: Comm. Math. Phys. 141, 413 (1991) 19. Gorsky, A. and Nekrasov, N.: Nucl. Phys. B414, 213 (1994); Nucl.Phys. B436, 582 (1995); Gorsky,A.: Integrable many body systems in the field theories. Prep. UUITP-16/94, (1994) 20. Faddeev, L.D.: Integrable models in (1+1)-dimensional quantum field theory. In: Recent advances in field theory and statistical mechanics. Eds. Zuber, J.B., Stord, R. (Les Houches Summer School Proc. session XXXiX, 1982), Elsevier Sci.Publ., 1984 p. 561 21. Kulish, P.P., Sklyanin, E.K.: Quantum spectral transform method. Recent developments. In: Integrable quantum field theories. Eds. Hietarinta J., Montonen C., Lect .Not. Phys. 51, 1982 p. 61 22. Olshanetsky, M.A., Perelomov, A.M.: Phys. Reps. 71, 313 (1981) 23. Belavin, A.A. and Drinfel’d, V.G.: Funk. Anal. i ego pril. 16 (3), 1–29 (1982) 24. Chari, V. and Pressley, A.: A Guide to Quantum Groups. Cambridge: Cambridge University Press 25. Maillet, J.M.: Phys. Lett. B245, 480 (1990) 26. Semenov-Tian-Shansky, M.A.: Teor. Math. Phys. 93, 302 (1992)(in Russian) 27. Semenov-Tian-Shansky, M.A.: Publ. RIMS Kyoto Univ. 21 (6), 1237–1260 (1985) 28. Krichever, I.: A.Zabrodin, Spin generalizations of the Ruijsenaars-Schneider model, non-abelian 2D Toda chain and representations of Sklyanin algebra. hep-th/9505039 29. Hasegawa, K.: Ruijsenaars’ commuting difference operators as commuting transfer matrices qalg/9512029 Communicated by G. Felder
Commun. Math. Phys. 191, 31 – 60 (1998)
Communications in
Mathematical Physics c Springer-Verlag 1998
Representation Theory of Lattice Current Algebras Anton Yu. Alekseev1,2,? , Ludwig D. Faddeev3 , ¨ Fr¨ohlich2 , Volker Schomerus4,5 Jurg 1 Institute of Theoretical Physics, Uppsala University, Box 803 S-75108, Uppsala, Sweden. E-mail:
[email protected] 2 Institut f¨ ur Theoretische Physik, ETH – H¨onggerberg, CH-8093 Z¨urich, Switzerland 3 Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191011, Russia. E-mail:
[email protected] 4 II. Institut f¨ ur Theoretische Physik, Universit¨at Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany. E-mail:
[email protected] 5 Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, Japan
Received: 25 April 1996 / Accepted: 14 April 1997
Abstract: Lattice current algebras were introduced as a regularization of the left- and right moving degrees of freedom in the WZNW model. They provide examples of lattice theories with a local quantum symmetry Uq (G ). Their representation theory is studied in detail. In particular, we construct all irreducible representations along with a lattice analogue of the fusion product for representations of the lattice current algebra. It is shown that for an arbitrary number of lattice sites, the representation categories of the lattice current algebras agree with their continuum counterparts. 1. Introduction Lattice current algebras were introduced and first studied several years ago (see [2, 13] and references therein). They were designed to provide a lattice regularization of the left- and right-moving degrees of freedom of the WZNW model [30] and gave a new appealing view on the quantum group structure of the model. In spite of many similarities between lattice and continuum theory, fundamental relations between them remain to be understood. In this paper we prove the conjecture of [2] that the representation categories of the lattice and continuum model agree. 1.1. Lattice current algebras. Lattice current algebras are defined over a discretized circle, i.e., their fundamental degrees of freedom are assigned to N vertices and N edges of a 1-dimensional periodic lattice. We enumerate vertices by integers n(modN ). Edges are oriented such that the nth edge points from the (n − 1)st to the nth vertex. Being defined over lattices of size N , the lattice current algebras come in families KN , N a positive integer. A precise definition of these (associative *-)algebras KN is given in the next section. We shall see that elements of KN can be assembled into (s × s)− matrices, Jn and Nn , n ∈ ZmodN , with KN -valued matrix elements such that ?
On leave of absence from Steklov Institute, Fontanka 27, St.Petersburg, Russia
32
A. Y. Alekseev, L. D. Faddeev, J. Fr¨ohlich, V. Schomerus 1
2
2
1
1
1
2
2
2
1
R0 Jn Jn =Jn Jn R ,
Jn R Jn+1 =Jn+1 Jn , 2
1
R0 N n R N n = N n R0 N n R, 2
1
1
2
0 N n Jn =Jn R N n R ,
1
2
(1.1) 2
1
0 Jn N n−1 = R N n−1 R Jn .
elements in End(V ) ⊗ KN , where To explain notations we view the matrices Jn , Nn asP V is an s-dimensional vector space. In this way Jn = mn,ς ⊗ln,ς determines elements mn,ς ∈ End(V ) and ln,ς ∈ KN which are used to define X
1
Jn =
2
mn,ς ⊗ e ⊗ ln,ς , Jn =
X
e ⊗ mn,ς ⊗ ln,ς ,
where e is the unit in End(V ). Similar definitions apply to Nn . Throughout the paper we will use the symbol σ for the permutation map σ : End(V ) ⊗ End(V ) → End(V ) ⊗ End(V ), and application of σ to an object X ∈ End(V ) ⊗ End(V ) is usually abbreviated by putting a prime, i.e. X 0 ≡ σ(X). The matrix R = R(h) ∈ End(V ) ⊗ End(V ) which appears in Eqs. (1.1) is a one-parameter solution of the Yang Baxter Equation (YBE). Such solutions can be obtained from arbitrary simple Lie algebras. The lattice Kac-Moody algebra KN depends on a number of parameters, including the “Planck constant” h in the solution R(h) of the YBE and the “lattice spacing” 1 = 1/N . A first, nontrivial test for the algebraic relations (1.1) comes from the classical continuum limit, i.e., from the limit in which h and 1 = 1/N are sent to zero. Using the rules Nn ∼ 1 − 1η(x) , Jn ∼ 1 − 1j(x) , R ∼ 1 + iγhr with x = n/N , γ being a deformation parameter and the standard prescription {., .} = lim i h→0
[., .] h
to recover the Poisson brackets from the commutators, one finds that 1 2 γ [C, j (x)− j (y)]δ(x − y) + γCδ 0 (x − y) , 2 1 2 1 2 γ {η (x), η (y)} = [C, η (x)− η (y)]δ(x − y) , 2 2 1 2 1 γ {η (x), j (y)} = [C, j (x)− j (y)]δ(x − y) + γCδ 0 (x − y) . 2 1
2
{j (x), j (y)} =
Here C is the Casimir element C = r + r0 = r + σ(r). For clarity, let us rewrite these relations in terms of components. When we express C = ta ⊗ ta and j(x) = ja (x)ta in terms of generators ta of the classical Lie algebra, the relations become c jc (x)δ(x − y) + γδab δ 0 (x − y), {ja (x), jb (y)} = γfab c ηc (x)δ(x − y), {ηa (x), ηb (y)} = γfab c jc (x)δ(x − y) + γδab δ 0 (x − y). {ηa (x), jb (y)} = γfab c 0 c c s are the structure constants of the Lie algebra, i.e., [ta , tb ] = fab t . We easily The fab recognize the first equation as the classical Poisson bracket of the left currents in the WZNW model. Furthermore, the quantity j R (x) ≡ j L (x) − η(x) Poisson commutes
Representation Theory of Lattice Current Algebras
33
with j L (x) ≡ j(x) and satisfies the Poisson commutation relations of the right currents, i.e., 1 2 1 2 γ {j R (x), j R (y)} = − [C, j R (x)− j R (y)]δ(x − y) − γCδ 0 (x − y) , 2 1
2
{j R (x), j L (y)} = 0 .
(1.2)
Hence we conclude that the lattice current algebra as described in Eqs. (1.1) is the quantum lattice counterpart of the classical left and right currents. One would like to establish a close relationship between the lattice current algebra and its counterpart in the continuum model. A first step in this direction is described in this paper. We find that the representation categories of the lattice and the continuum theory coincide. For this to work, it is rather crucial to combine left- and right-moving degrees of freedom. For instance, the center of the lattice current algebra with only one chiral sector changes dramatically depending on whether the number of lattice sites is odd or even. The only *-operation known for such algebras [26] is constructed in the case of Uq (sl(2)) and does not admit straightforward generalizations. However, no such difficulties appear in the full theory. It therefore appears to be rather unnatural to constrain the discrete models to one chiral sector. 1.2. Main results. In the next section we use the methods developed in [4] to provide a precise definition of lattice current algebras. In contrast to the heuristic definition we use in this introduction, our precise formulation is applicable to general modular Hopf algebras G, in particular to Uq (G ), for an arbitrary semisimple Lie algebra G . The main result of Sect. 3 provides a complete list of irreducible representations for the lattice current algebras KN . Theorem A. (Representations of KN ) For every semisimple modular Hopf algebra G and every integer N ≥ 1, there exists a lattice current algebra KN which admits a family IJ on Hilbert spaces WNIJ . Here the labels I, J run of irreducible ∗-representations DN through classes of finite-dimensional, irreducible representations of the algebra G. The two labels I, J that are needed to specify a representation of KN correspond to the two chiralities in the theory of current algebras. In fact, the algebra K1 is isomorphic to the quantum double of the algebra G [21] and the pairs I, J label its representations. These results are in agreement with the investigation of related models in [22]. Next, we introduce an inductive limit K∞ of the family of finite dimensional algebras KN . It can be done using the block-spin transformation [13] KN → KN +1 . Under this embedding, every irreducible representation of KN +1 splits into a direct sum of IJ irreducible representations of KN . It appears that the representation DN +1 always splits IJ into several copies of the representation DN . Thus, representations of the inductive limit K∞ are in one to one correspondence with representations of K1 (or KN for arbitrary finite N ). In order to be able to take tensor products of representations of the lattice current algebras, we introduce a family of homomorphisms 3N,M : KN +M −1 → KN ⊗ KM , which satisfy the co-associativity condition
34
A. Y. Alekseev, L. D. Faddeev, J. Fr¨ohlich, V. Schomerus
(id ⊗ 3M,L ) ◦ 3N,L+M −1 = (3N,M ⊗ id) ◦ 3M +N −1,L . Let us notice that the co-product 3N,M is supposed to provide a lattice counterpart of the co-product defined by the structure of superselection sectors in algebraic field theory [9, 10, 27, 28]. By combining the co-product 3N,M with the block-spin transformation we construct a new co-product 1N : KN → KN ⊗ KN which preserves the number of sites in the lattice (see Subsect. 4.3). This co-product is compatible with the block-spin transformation and, hence, it defines a co-product for the inductive limit K∞ : 1∞ : K ∞ → K ∞ ⊗ K ∞ . Our second result concerns tensor products of representations of the lattice current algebra K∞ . Theorem B (Representation category of the lattice current algebra). The braided tensor categories of representations of the lattice current algebra K∞ with the coproduct 1∞ and of the Hopf algebra K1 with the co-product 11 coincide. In principle, our theory must be modified to apply to Uq (G ), q p = 1. It is well known that Uq (G ) at roots of unity is not semi-simple. This can be cured by a process of truncation which retains only the “physical” part of the representation theory of quantized universal enveloping algebras. The algebraic implementation of this idea has been explained in [4] and can be transferred easily to the present situation. We plan to propose an alternative treatment in a forthcoming publication.
2. Definition of the Lattice Current Algebra Our goal is to assign a family of lattice current algebras (parametrized by the number N of lattice sites) to every modular Hopf-*-algebra G. Before we describe the details, we briefly recall some fundamental ingredients from the theory of Hopf algebras. 2.1. Semisimple, modular Hopf-algebras. By definition, a Hopf algebra is a quadruple (G, , 1, S) of an associative algebra G (the “symmetry algebra ”) with unit e ∈ G, a onedimensional representation : G → C (the “co-unit”), a homomorphism 1 : G → G ⊗G ( the “co-product”) and an anti-automorphism S : G → G (the “antipode”). These objects obey a set of basic axioms which can be found e.g. in [1]. The Hopf algebra (G, , 1, S) is called quasitriangular if there is an invertible element R ∈ G ⊗ G such that R 1(ξ) = 10 (ξ) R for all ξ ∈ G , (id ⊗ 1)(R) = R13 R12 ,
(1 ⊗ id)(R) = R13 R23 .
Here 10 = σ ◦ 1, with σ : G ⊗ G → G ⊗ G the permutation map, and we are using the standard notation for the elements Rij ∈ G ⊗ G ⊗ G. For a ribbon Hopf-algebra one postulates, in addition, the existence of a certain invertible central element v ∈ G (the “ribbon element”) which factorizes R0 R ∈ G ⊗ G (here R0 = σ(R)), in the sense that R0 R = (v ⊗ v)1(v −1 )
Representation Theory of Lattice Current Algebras
35
(see [24] for details). The ribbon element v and the element R allow us to construct a distinguished grouplike element g ∈ G by the formula X S(rς2 )rς1 , g −1 = v −1 P 1 rς ⊗ rς2 of R. The element g is where the elements rςi come from the expansion R = important in the definition of q-traces below. We want this structure to be consistent with a *-operation on G. To be more precise, we require that (2.1) R∗ = (R−1 )0 = σ(R−1 ) , 1(ξ)∗ = 10 (ξ ∗ ), and that v, g are unitary 1 . This structure is of particular interest, since it appears in the theory of the quantized universal enveloping algebras Uq (G ) when the complex parameter q has values on the unit circle [18]. At this point we assume that G is semisimple, so that every representation of G can be decomposed into a direct sum of finite-dimensional, irreducible representations. From every equivalence class [I] of irreducible representations of G, we may pick a representative τ I , i.e., an irreducible representation of G on a δI -dimensional Hilbert space V I . The quantum trace trqI is a linear functional acting on elements X ∈ End(V I ) by trqI (X) = T rI (Xτ I (g)) . Here T rI denotes the standard trace on End(V I ) with T rI (eI ) = δI and g ∈ G has been defined above. Evaluation of the unit element eI ∈ End(V I ) with trqI gives the quantum dimension of the representation τ I , dI ≡ trqI (eI ) . Furthermore, we assign a number SIJ to every pair of representations τ I , τ J , SIJ ≡ N (trqI ⊗ trqJ )(R0 R)IJ
with (R0 R)IJ = (τ I ⊗ τ J )(R0 R) ,
for a suitable, real normalization factor N . The numbers SIJ form the so-called S-matrix S. Modular Hopf algebras are ribbon Hopf algebras with an invertible S-matrix 2 . Let us finally recall that the tensor product, τ × τ 0 , of two representations τ, τ 0 of a Hopf algebra is defined by 0 0 (τ × τ )(ξ) = (τ ⊗ τ )1(ξ) for all ξ ∈ G .
In particular, one may construct the tensor product τ I × τ J of two irreducible representations. According to our assumption that G is semisimple, such tensor products of representations can be decomposed into a direct sum of irreducible representations, τ K . IJ in this Clebsch-Gordan decomposition of τ I × The multiplicities NK τ J are called fusion rules. Among all our assumptions on the structure of the Hopf-algebra (G, , 1, S) (quasitriangularity, existence of a ribbon element v, semisimplicity of G and invertibility of S), semisimplicity of G is the most problematic one. In fact it is violated by the algebras 1 Here we have fixed ∗ on G ⊗ G by (ξ ⊗ η)∗ = ξ ∗ ⊗ η ∗ . Following [18], we could define an alternative involution † on G ⊗ G which incorporates a permutation of components, i.e., (ξ ⊗ η)† = η † ⊗ ξ † and ξ † = ξ ∗ for all ξ, η ∈ G. With respect to †, 1 becomes an ordinary ∗ -homomorphism and R is unitary. 2 If a diagonal matrix T is introduced according to T 2 I IJ = $δI,J dI τ (v) (with an appropriate complex factor $), then S and T furnish a projective representation of the modular group SL(2, Z).
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A. Y. Alekseev, L. D. Faddeev, J. Fr¨ohlich, V. Schomerus
Uq (G ) when q is a root of unity. It is sketched in [4] how “truncation” can cure this problem, once the theory has been extended to weak quasi-Hopf algebras [18]. Example (Hopf-algebra Zq ). We wish to give one fairly trivial example for the algebraic structure discussed so far. Our example comes from the group Zp . To be more precise, we consider the associative algebra G generated by one element g subject to the relation g p = 1. On this algebra, a co-product, co-unit and an antipode can be defined by 1(g) = g ⊗ g , S(g) = g −1 , (g) = 1 . We observe that G is a commutative semisimple algebra. It has p one-dimensional representations τ r (g) = q r , r = 0, . . . , p − 1, where q is a root of unity, q = e2πi/p . We may construct characteristic projectors P r ∈ G for these representations according to Pr =
p−1
1 X −rs s q g for r = 0, . . . , p − 1 . p s=0
r
One can easily check that τ (P s ) = δr,s . The elements P r are employed to obtain a nontrivial R-matrix, X q rs P r ⊗ P s . R= r,s
When evaluated with a pair of representations τ r , τ s we find that (τ r ⊗ τ s )(R) = q rs . The R-matrix satisfies all the axioms stated above and thus turns G into a quasitriangular P −r2 Hopf-algebra. Moreover, a ribbon element is provided by v = q Pr . We can finally introduce a ∗-operation on G such that g ∗ = g −1 . The consistency relations 2.1 follow from the co-commutativity of 1, i.e. 10 = 1, and the property R = R0 . A direct computation shows that the S-matrix is invertible only for odd integers p. Summarizing all this, we have constructed a family of semisimple ribbon Hopf-*-algebras Zq , q = exp(2πi/p). They are modular Hopf-algebras for all odd integers p. 2.2. R-matrix formalism. Before we propose a definition of lattice current algebras, we mention that Hopf algebras G are intimately related to the objects Nn , n ∈ ZmodN, introduced in Eq. (1.1). To understand this relation, let us introduce another (auxiliary) copy, Ga , of G and let us consider the R-matrix as an object in Ga ⊗ G. To distinguish the latter clearly from the usual R, we denote it by N± , N− ≡ R−1 ∈ Ga ⊗ G , N+ ≡ R0 ∈ Ga ⊗ G . Quasi-triangularity of the R-matrix furnishes the relations 2
1
1a (N± ) =N ± N ± , 2
1
2
1
1
2
R N + N − =N − N + R , 1
(2.2)
2
R N ± N ± = N ± N ±R . Here we use the same notations as in the introduction, and 1a (N± ) = (1 ⊗ id)(N± ) ∈ Ga ⊗Ga ⊗G. The subscript a reminds us that 1a acts on the auxiliary (i.e. first) component of N± . To be perfectly consistent, the objects R in the preceding equations should all be equipped with a lower index a to show that R ∈ Ga ⊗ Ga etc. We hope that no confusion will arise from omitting this subscript on R. Equations (2.2) are somewhat redundant:
Representation Theory of Lattice Current Algebras
37
in fact, the exchange relations on the second line follow from the first equation in the first line. This underlines that the formula for 1a (N± ) encodes information about the product in G rather than the co-product 3 . More explanations of this point follow in Subsect. 2.3. Next, we combine N+ and N− into one element N ≡ N+ (N− )−1 = R0 R ∈ Ga ⊗ G . From the properties of N± we obtain an expression for the action of 1a on N , 2
1
1
2
1a (N ) = N + N + (N − )−1 (N − )−1 1
2
1
1
1
2
= R−1 N + N + R(N − )−1 (N − )−1 2
2
= R−1 N + (N − )−1 R N + (N − )−1 1
2
= R−1 N R N . As seen above, the formula for 1a (N ) encodes relations in the algebra G and implies, in particular, the following exchange relations for N : 1
2
R0 N R N = R0 R1a (N ) = R0 10a (N )R 2
1
= N R0 N R . This kind of relations first appeared in [23]. We used them in the introduction when describing the objects Nn assigned to the sites of the lattice. Thus we have shown that, for any modular Hopf-algebra G, one may construct objects N obeying the desired quadratic relations. The other direction, namely the problem of how to construct a modular Hopf-algebra G from an object N satisfying the exchange relations described above, is more subtle. To begin with, one has to choose linear maps π : Ga → C in the dual Ga0 of Ga . When such linear forms π ∈ Ga0 act on the first tensor factor of N ∈ Ga ⊗ G they produce elements in G: π(N ) ≡ (π ⊗ id)(N ) ∈ G for all π ∈ Ga0 . π(N ) ∈ G will be called the π-component of N or just component of N . Under certain technical assumptions it has been shown in [6] that the components of N generate the algebra G. In this sense one can reconstruct the modular Hopf-algebra G from the object N. Lemma 1 ([6]). Let Ga be a finite-dimensional, semisimple modular Hopf algebra and N be the algebra generated by components of N ∈ Ga ⊗ N subject to the relations 1
2
N R N = R1a (N ) , where we use the same notations as above. Then N can be decomposed into a product of elements N± ∈ Ga ⊗ N , 3 The co-product 1 of G acts on N according to 1(N ) = (id ⊗ 1)(N ) = N N ± ± ± ± ˜ ± ∈ Ga ⊗ G ⊗ G. Here N± [N˜ ± ] on the right hand side of the equation have the unit element e ∈ G in the third [second] tensor factor
38
A. Y. Alekseev, L. D. Faddeev, J. Fr¨ohlich, V. Schomerus −1 N = N + N− such that −1 ∈ Ga ⊗ N ⊗ N , 1(N ) ≡ N+ N˜ N− −1 ∈ Ga ⊗ N (N ) ≡ e ∈ Ga , S(N± ) ≡ N±
define a Hopf-algebra structure on N . Here, the action of 1, , S on the second tensor −1 are supposed component of N, N± is understood. In the equation for 1(N ), N+ , N− P to have a trivial entry inPthe third component while N˜ = nς ⊗ e ⊗ Nς with e being the unit in N and N = nς ⊗ Nς ∈ Ga ⊗ N . As a Hopf algebra, N is isomorphic to Ga . Let us remark that the ∗-operation in G induces a ∗-operation in N which looks as follows: (2.3) N+∗ = N− . In our definition of the lattice current algebras below, we shall describe the degrees of freedom at the lattice sites directly in terms of elements ξ ∈ G, instead of working with N (as in the introduction). The δI -dimensional representations τ I of Ga ∼ = G furnish a δI × δI -matrix of linear forms on Ga . When these forms act on the first tensor factor of N , we obtain a matrix N I ∈ End(V I ) ⊗ G of elements in G, N I ≡ τ I (N ) = (τ I ⊗ id)(N ) . These matrices will turn out to be useful. us illustrate all these remarks on the examExample (R-matrix formalism for PZqrs). Let q P r ⊗ P s and that Zp has only one-dimensional ple of G = Zp . Recall that R = representations given by τ r (g) = q r . Evaluation of the objects N± in representations τ r r produces elements N± = (τ r ⊗ id)(N± ) ∈ C ⊗ Zq ∼ = Zq . Explicitly, they are given by X X r q rs Ps = g r and N− = q −rs Ps = g −r . N+r = Together with the property (τ r ⊗ τ s )1 ∼ = τ r+s the relations (2.2) become g ±(r+s) = g ±s g ±r , q rs g s g −r = g −r g s q rs . For N ∈ Zq ⊗ Zq we find N=
X
q 2rs Pr ⊗ Ps and N r = g 2r .
As predicted by the general theory, the elements N r ∈ Zq generate the algebra Zq when p is odd. 2.3. Definition of KN . Next, we turn to the definition of the lattice current algebras KN associated to a fixed modular Hopf algebra. Before entering the abstract formalism, it is useful to analyse the classical geometry of the discrete model. Our classical continuum theory contains two Lie-algebra valued fields, namely η(x) and j(x). To describe a configuration of η, for instance, we have to place a copy of the Lie-algebra at every point x on the circle. On the lattice, there are only N discrete points left and hence configurations of the lattice field η involve only N copies of the Lie algebra. When
Representation Theory of Lattice Current Algebras
39
passing from the continuum to the lattice, we encode the information about the field j(x) in the holonomies along links, Z jn = P exp( j(x)dx) . n
R
th
Here n denotes integration along the n link that connects the (n − 1)st with the nth site. The classical lattice field jn has values in the Lie group. Let us remark that, even at the level of Poisson brackets, the variables jn can not be easily included into the Poisson algebra. The reason is that jn ’s fail to be continuous functions of the currents. Therefore, we should regularize the Poisson brackets (or commutation relations) of the lattice currents. This regularization is done in the most elegant way with the help of Rmatrices. This consideration explains an immediate appearance of the quantum groups in the description of the lattice current algebras. In analogy to the classical description of the lattice field η, the lattice current algebras contain N commuting copies of the algebra G or, more precisely, KN contains an N -fold tensor product G ⊗N of G as a subalgebra. We denote by Gn the subalgebra G n = e ⊗ . . . ⊗ G ⊗ . . . ⊗ e ⊂ G ⊗N , where G appears in the nth position and all other entries in the tensor product are trivial. The canonical isomorphism of G and Gn ⊂ G ⊗N furnishes the homomorphisms ιn : G → G ⊗N for all n = 1, . . . , N. We think of the copies Gn of G as being placed at the N sites of a periodic lattice, with Gn assigned to the nth site. In addition, the definition of KN will involve generators Jn , n = 1, . . . , N . The generator Jn sits on the link connecting the (n − 1)st with the nth site. Definition 1. The lattice current algebra KN is generated by components 4 of Jn ∈ Ga ⊗ KN , n = 1, . . . , N, along with elements in G ⊗N . These generators are subject to three different types of relations. 1. Covariance properties express that the Jn are tensor operators transforming under the action of elements ξm ∈ Gm like holonomies in a gauge theory, i.e., ιn (ξ)Jn = Jn 1n (ξ) for all ξ ∈ G, 1n−1 (ξ)Jn = Jn ιn−1 (ξ), for all ξ ∈ G ιm (ξ)Jn = Jn ιm (ξ) for all ξ ∈ G, m 6= n, n − 1 modN .
(2.4)
The covariance relations (2.4) make sense as relations in Ga ⊗ KN , if ιn (ξ) ∈ Gn ⊂ KN is regarded as an element ιn (ξ) ∈ Ga ⊗ KN with trivial entry in the first tensor factor and 1n (ξ) ≡ (id ⊗ ιn )1(ξ) ∈ Ga ⊗ Gn ⊂ Ga ⊗ KN . 2. Functoriality for elements Jn on a fixed link means that 1
2
Jn Jn = R1a (Jn ).
(2.5)
This is to be understood as a relation in Ga ⊗ Ga ⊗ KN , where 1a : Ga ⊗ KN → Ga ⊗ Ga ⊗ KN acts trivially on the second tensor factor KN and R = R ⊗ e ∈ 4 Recall that a component of J is an element π(J ) ≡ (π ⊗ id)(J ) in the algebra K . Here π runs n n n N through the dual Ga0 of Ga .
40
A. Y. Alekseev, L. D. Faddeev, J. Fr¨ohlich, V. Schomerus
Ga ⊗ Ga ⊗ KN . The other notations were explained in the introduction. We also require that the elements Jn possess an inverse Jn−1 ∈ Ga ⊗ KN such that Jn Jn−1 = e, Jn−1 Jn = e .
(2.6)
3. Braid relations between elements Jn , Jm assigned to different links have to respect the gauge symmetry and locality of the model. These principles require 1
2
2
1
if n 6= m, m ± 1modN,
Jn Jm =Jm Jn 1
2
2
1
Jn R Jn+1 = Jn+1 Jn .
(2.7)
R denotes the element R ⊗ e ∈ Ga ⊗ Ga ⊗ KN as before. The lattice current algebra KN contains a subalgebra JN generated by components of the Jn only. They are subject to functoriality (2) and braid relations (3). The subalgebra JN admits an action of G ⊗N (by generalized derivations) such that the full lattice current algebra KN can be regarded as a semi-direct product of JN and G ⊗N with respect to this action. Our covariance relations (1) give a precise definition of the semi-direct product. Let us briefly explain how Definition 1 is related to the description we used in the introduction. The relation of the Hopf algebras Gn and the objects Nn has already been discussed. Our covariance relations in Eq. (2.4) correspond to the exchange relations between N and J in the third line of Eq. (1.1). They can be related explicitly with the help of the quasi-triangularity of R, using the formula N = R0 R. We have, for instance, 2
1
1
1
0 0 N n Jn = (e ⊗ (R R)n ) Jn =Jn [(id ⊗ 1n )(R R)]213 1
1
2
= Jn R(e ⊗ (R0 R)n )R0 =Jn R N n R0 , where we use the notation (R0 R)n = (id ⊗ ιn )(R0 R) ∈ Ga ⊗ Gn and R = (R ⊗ e) ∈ Ga ⊗ Ga ⊗ KN as usual. [.]213 means that the first and the second tensor factors of the expression inside the brackets are exchanged. For finite-dimensional semisimple modular Hopf algebras G, Lemma 1 implies that one could define KN using the objects Nn ∈ Ga ⊗ Gn ⊂ Ga ⊗ KN instead of elements η ∈ G ⊗N . The generators Nn would have to obey the exchange relations stated in Eq. (1.1) and 1
2
N n R N n = R1a (Nn ) . The functoriality relation for Jn did not appear in the introduction. But we can use it now to derive quadratic relations for the Jn in much the same way as has been done for N , earlier in this section. Indeed we find 1
2
R0 Jn Jn = R0 R1a (Jn ) = R0 10a (Jn )R 2
1
= Jn Jn R . This exchange relation is the one used in the introduction to describe the lattice currents Jn . When the first two tensor factors in this equation are evaluated with representations of Ga ∼ = G one derives quadratic relations for the KN -valued matrices, JnI ≡ (τ I ⊗ id)(Jn ) ∈ End(V I ) ⊗ KN .
Representation Theory of Lattice Current Algebras
41
As we discussed earlier in this section, elements in the algebra KN can be obtained from Jn with the help of linear forms π ∈ Ga0 . To understand functoriality properly one must realize that it describes the “multiplication table” for elements π(Jn ) ∈ KN . If we pick two linear forms π1 , π2 ∈ Ga0 on Ga , the corresponding elements in KN satisfy π1 (Jn )π2 (Jn ) = (π1 ⊗ π2 )(R1(Jn )) . We can rewrite this equation by means of the (twisted) associative product ∗ for elements πi ∈ Ga0 , (π1 ∗ π2 )(ξ) ≡ (π1 ⊗ π2 )(R1(ξ)) . It allows us to express the product π1 (Jn )π2 (Jn ) in terms of the element π1 ∗ π2 ∈ Ga0 , π1 (Jn )π2 (Jn ) = (π1 ∗ π2 )(Jn ) for all π1 , π2 ∈ Ga0 . Remark. It may help here to invoke the analogy with a more familiar situation. In fact, the Hopf algebra structure of G induces the standard (non-twisted) product · on its dual G0, (π1 · π2 )(ξ) ≡ (π1 ⊗ π2 )(1(ξ)) for all ξ ∈ G . Let us identify π ∈ Ga0 with the image π(T ) = (π ⊗ id)(T ) of some universal object T ∈ Ga ⊗ G 0 and insert T into the definition of the product ·, π1 (T )π2 (T ) ≡ (π1 ⊗ π2 )(1a (T )) = (π1 · π2 )(T ) . Here and in the following we omit the · when multiplying elements π(T ). The derived multiplication rules for components of T ∈ Ga ⊗ G 0 are equivalent to the following functoriality: 1
2
T T = 1a (T ) 1
and imply RTT-relations:
2
2
1
R T T =T T R .
Such relations are known to define quantum groups, and hence our variables Jn describe some sort of twisted quantum groups. This fits nicely with the nature of the classical lattice field jn . As we have noted earlier, the latter takes values in a Lie group. We saw above that the definition of a product for components of Jn implies the desired quadratic relations. The converse is not true in general, i.e. functoriality is a stronger requirement than the quadratic relations. In the familiar case of Uq (sl2 ) for example, functoriality furnishes also the standard determinant relations which are usually “added by hand” when algebras are defined in terms of quadratic relations. Due to functoriality we are thus able to develop a universal theory which does not explicitly depend on the specific properties of the Hopf algebra G. We have shown that the mathematical definition of KN represented in this section agrees with the description used in the introduction. The algebra KN now appears as a special example of the “graph algebras” defined and studied in [4] to quantize ChernSimons theories. This observation will enable us to use some of the general properties established there. The most important one among such general properties is the existence of a *operation. On the copies Gn , a *-operation comes from the structure of the modular Hopf-*-algebra G. Its action can be extended to the algebra KN by the formula Jn∗ = Sn−1 Jn−1 Sn−1 ,
(2.8)
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A. Y. Alekseev, L. D. Faddeev, J. Fr¨ohlich, V. Schomerus
where Sn = (id⊗ιn )[1(κ−1 )(κ⊗κ)R−1 ] ∈ Ga ⊗Gn ⊂ Ga ⊗KN . Here ιn : G → Gn ⊂ KN is the canonical embedding, and the central element κ ∈ G is a certain square root of the ribbon element v, i.e. κ2 = v (cp. [4] for details). Let us note that the formula (2.8) can be rewritten using the elements Nn,± ∈ Ga ⊗ Gn constructed from the R-element according to Nn,+ ≡ (id ⊗ ιn )(R0 ) , Nn,− ≡ (id ⊗ ιn )(R−1 ) . The conjugate current is expressed as −1 −1 Jn Nn−1,− (κn−1 κn ) . Jn∗ = (κn−1 κn )−1 Nn,−
(2.9)
Here κn−1 = ιn−1 (κ), κn = ιn (κ). In order to verify the property (Jn∗ )∗ = Jn , one uses the following identities: −1 −1 vn−1 Jn vn−1 = va Nn−1 Jn , vn−1 Jn vn = va−1 Jn Nn ,
where va is the ribbon element in the auxilary Hopf algebra Ga . The ribbon elements v at different lattice sites generate automorphisms of the lattice current algebra which resemble the evolution automorphism in the quantum top [3]. Example (The U (1)-current algebra on the lattice). It is quite instructive to apply the general definition of lattice current algebras to the case G = Zq . Recall that Zq is generated by one unitary element g which satisfies g p = 1. Representations τ s of G were labeled by an integer s = 1, . . . , p − 1, and τ s (g) = q s with q = exp(2πi/p). We can apply the one-dimensional representations τ s to the current Jn ∈ Ga ⊗ KN to obtain elements Jns = τ s (Jn ) = (τ s ⊗ id)(Jn ) ∈ KN . Functoriality becomes Jns Jnt = q ts Jns+t , where we have used that (τ s ⊗ τ t )1a (ξ) = τ s+t (ξ) and (τ s ⊗ τ t )(R−1 ) = q −st (s + t is to be understood modulo p). The relation allows to generate the elements Jns from Jn1 ∈ KN . Observe that Jn0 is the unit element e in the algebra KN . From the previous equation we deduce that the pth power of the generator Jn1 is proportional to e, (Jn1 )p = q p(p−1)/2 Jnp = q p(p−1)/2 e . This motivates us to introduce the renormalized generators wn = q (1−p)/2 Jn1 ∈ Kn which obey wnp = e. In the following it suffices to specify relations for the generators wn and gn = ιn (g) of KN . The covariance equations (2.4) read gn wn = qwn gn , wn gn−1 = qgn−1 wn , since (τ 1 ⊗ id)1(g) = qg. The exchange relations for currents become wn wn+1 = q −1 wn+1 wn . The identity 1(κ−1 )(κ ⊗ κ)R−1 = e ⊗ e with κ =
P
q −r
2
/2
Pr finally furnishes
wn∗ = wn−1 . At this point one can easily recognize the algebra of wn ’s as the lattice U (1)-current algebra [12].
Representation Theory of Lattice Current Algebras
43
2.4. The right currents. Let us stress that there is a major ideological difference between our discussion of lattice current algebras and the work in [4]. In the context of Chern Simons theories, the graph algebras were introduced as auxiliary objects, and physical variables of the theory were to be constructed from objects assigned to the links, i.e. from the variables Jn . Here the Jn ’s represent only the left-currents, and we expect also right- currents to be present in the theory. They are constructed from the variables Jn and the elements ξn ∈ Gn and hence give a physical meaning to the graph algebras. We define a family of new variables JnR ∈ Ga ⊗ KN on the lattice by setting −1 −1 Jn Nn−1,+ . JnR = Nn,−
The JnR turn out to provide the right currents in our theory. For the rest of this section we will use the symbol JnL to denote the original left currents Jn . Proposition 2 (Right-currents on the lattice). With JnR ∈ Ga ⊗ KN defined as above, one finds that L commute for arbitrary n, m, 1. the elements JnR and Jm 1
2
2
1
R L L R Jn Jm =Jm Jn ;
2. the elements JnR satisfy the following exchange and functoriality relations: 2
1
1
2
R R R R Jn+1 R Jn = Jn Jn+1 , 2
(2.10)
1
R R R Jn Jn = R1a (Jn ) .
(2.11)
(Here we are using the same notations as in the definition of the lattice current algebra above.) If we denote the subalgebra in KN generated by the components of left currents JnL by JNL and, similarly, use JNR to denote the subalgebra generated by components of JnR , the result of this proposition can be summarized in the following statement: JNR and JNL form commuting subalgebras in KN , and JNR is isomorphic to (JNL )op . Here the subscript op means opposite multiplication. Both statements have their obvious counterparts in the continuum theory (cp. Eq. (1.2)). Example (The right U (1)-currents). We continue the discussion of the U (1)-current algebra on the lattice by constructing the right currents. Our general theory teaches us to consider wnR = gn wn−1 gn−1 . 1 1 = ιn (g) and gn−1 = Nn−1,− = ιn−1 (g). The reader is invited to check Here gn = Nn,+ L that these elements commute with wn = wn .
2.5. Monodromies. In the continuum R x theory one is particularly interested in the behavior of the chiral fields g C (x) = P exp( 0 j C (x)dx) under rotations by 2π, i.e. the monodromy of g C . Here and in the following, C stands for either L or R. The monodromy of g C is determined by the expression I C m = P exp( j C (x)dx).
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A. Y. Alekseev, L. D. Faddeev, J. Fr¨ohlich, V. Schomerus
Due to the regularizing effect of the lattice, left and right monodromies, M L , M R , are relatively easy to construct and control for our discrete current algebra. They are obtained as an ordered product of the chiral lattice holonomies JnL or JnR along the whole circle, i.e. L M L = va1−N J1L J2L · · · JN
and
R M R = va1−N JN · · · J2R J1R .
(2.12) (2.13)
When we derive relations for the monodromies, it is convenient to include the factors involving va = v ⊗ e ∈ Ga ⊗ KN . The definition in terms of left and right currents produces elements M L and M R in Ga ⊗KN . Their algebraic structure differs drastically from the properties of the currents Jn . We encourage the reader to verify the following list of equations: 1
2
L L L M R M = R1a (M ),
10 (ξ)M L = M L 10 (ξ),
2
1
R R R M R M = R1a (M ) ,
100 (ξ)M R = M R 100 (ξ)
for all ξ ∈ G and with 100 (ξ) = (id ⊗ ι0 )(10 (ξ)) = (id ⊗ ι0 )(σ ◦ 1(ξ)). The functoriality relations are familiar from Subsect. 2.2 and imply that the algebra generated by components of the monodromy M L or M R is isomorphic to G or Gop . There are several places throughout the paper where this observation becomes relevant for a better understanding of our results. As usual, we may evaluate the elements M C in irreducible representations of Ga . This results in a set of KN valued matrices M C,I ≡ (τ I ⊗ id)(M C ). Their quantum traces cIC ≡ tr Iq (M C,I ) are elements in the algebra KN which have a number of remarkable properties. First, they are central elements in the lattice current algebra KN , i.e. the cIC commute with all elements A ∈ KN . Even more important is that cIR , cIL ∈ KN generate two commuting copies of the Verlinde algebra [29]. Explicitly this means that X ¯ K IJ K ∗ NK cC and (cK cIC cJC = C ) = cC ¯ denotes the unique label such that N K K¯ = 1, for C = R, L. Here and in the following K 0 and 0 stands for the trivial representation τ 0 = . If the S-matrix SIJ = N (tr Iq ⊗ tr Jq )(R0 R) is invertible, and N is suitably chosen, the linear combinations χIC =
X
N dI SI J¯ cJC
J
provide a set of orthogonal central projectors in KN , for each chirality C = R, L, i.e., χIC χJC = δIJ χIC , (χIC )∗ = χIC . Proofs of all these statements can be found in [4]. We will see in the next section that products χIL χJR provide a complete set of minimal central projectors in the lattice current algebra or, in other words, they furnish a complete set of characteristic projectors for the irreducible representations of KN .
Representation Theory of Lattice Current Algebras
45
Example (The center of the lattice U (1) current algebra). In terms of the variable wn = q (1−p)/2 Jn1 = −q 1/2 Jn1 (cf. Subsect. 2.3 for notations), the definition of the monodromy M L,1 = (τ 1 ⊗ id)(M L ) becomes M L,1 = (−1)N q 3N/2−1 w1 w2 · · · wN ∈ KN . In this particular example, the quantum trace is trivial so that c1L = M L,1 . It is easily checked that c1L commutes with all the generators wn , gn ∈ KN and that it satisfies (c1L )p = c0L = e . Of course, this relation follows also from the formula crL csL = considerations apply to the right currents.
P
Ntrs ctL = cr+s L . Similar
2.6. The inductive limit K∞ . So far, the lattice current algebras KN depend on the number N of lattice sites, and one may ask what happens when N tends to infinity. A mathematically precise meaning to this question is provided by the notion of inductive limit. The latter requires an explicit choice of embeddings of lattice current algebras for different numbers of lattice sites. They will come from some kind of inverse block-spin transformation [13]. Suppose we are given two lattice current algebras KN and KN +1 with generators Jn , n = 1, . . . , N and Jˆm , m = 1, . . . , N + 1 respectively. The embeddings of G into KN or KN +1 will be denoted by ιn or ιˆm . An embedding γN : KN → KN +1 is furnished by γN (Jn ) = Jˆn for all n < N, γN (JN ) = va−1 JˆN JˆN +1 , and γN (ιn (ξ)) = ιˆn (ξ) for all n < N , γN (ιN (ξ)) = ιˆN +1 (ξ) . The intuitive idea behind γN is to pass from KN to KN +1 by dividing the N th link on the lattice of length N into two new links, so that we end up with a lattice of length ˆ ∈ Ga ⊗ KN +1 , and, N + 1. Observe that γN maps the monodromies M ∈ Ga ⊗ KN to M consequently, the same holds for our projectors χIL , χJR ∈ KN and χˆ IL , χˆ JR ∈ KN +1 , γN (χIL ) = χˆ IL , γN (χJR ) = χˆ JR .
(2.14)
Since the set of numbers N is directed, the collection of KN , together with the maps γN , forms a directed system, and we can define the inductive limit K∞ ≡ lim KN . N →∞
S
By definition, K∞ = N KN / ∼, where two elements AN ∈ KN and AN 0 ∈ KN 0 are equivalent, iff AN is mapped to AN 0 by a string of embeddings γM , i.e., AN 0 = γN 0 −1 ◦. . .◦γN +1 ◦γN (AN ). For the lattice U (1)-current algebra, a detailed investigation of this inductive limit was performed in [5]. We have chosen to define the block-spin transformation by dividing the N th link of the lattice. Now we introduce another block spin operation by dividing the 1st link of the lattice:
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A. Y. Alekseev, L. D. Faddeev, J. Fr¨ohlich, V. Schomerus
γ˜ N (Jn ) = Jˆn+1 for all n > 1, γ˜ N (J1 ) = va−1 Jˆ1 Jˆ2 , and γ˜ N (ιn (ξ)) = ιˆn+1 (ξ) for all n . Notice that the two block-spin operations ‘commute’ with each other: γ˜ N +1 ◦ γN = γN +1 ◦ γ˜ N . While the map γN is used in the definition of the inductive limit, we reserve γ˜ N for the definition of the co-product for lattice current algebras (see Sect. 4). 3. Representations of the Lattice Current Algebra The stage is now set to describe our main result on the representation theory of the lattice current algebras. We will begin with a much simpler problem of representing two important subalgebras of KN . Their representations will serve as building blocks for the representation theory of the lattice current algebra KN . 3.1. The algebra U. The lattice current algebras KN contain N − 1 non-commuting holonomies Uν , ν = 1, . . . , N − 1, Uν ≡ va1−ν J1 · · · Jν ∈ Ga ⊗ KN . The objects Uν commute with all elements in the symmetry algebras Gn , except the ones for n = 0 and n = ν. These properties of Uν remind us of holonomies in a gauge theory, which transform nontrivially only under gauge transformations acting at the endpoints of the paths. Together, the elements in G0 ⊗Gν ⊂ KN and the components of Uν generate a subalgebra, Uν , of the lattice current algebra KN . These subalgebras Uν ⊂ KN are all isomorphic to the algebra U which we investigate in this subsection. We begin with a definition of U . Definition 3. The algebra U is the *-algebra generated by components of elements U, U −1 ∈ Ga ⊗ U together with elements in G0 ⊗ G1 such that 1
2
U U = R1a (U ) , ι1 (ξ)U = U 11 (ξ),
10 (ξ)U = U ι0 (ξ),
and U −1 is the inverse of U . Here we use the same notations as in Subsect. 2.3. In particular, ι0,1 denote the canonical embeddings of G into G0 ⊗ G1 . The ∗-operation on U extends the ∗-operation on G0 ⊗ G1 ⊂ U, so that U ∗ = S1−1 U −1 S0 . Here Si = (id ⊗ ιi )(1(κ−1 )(κ ⊗ κ)R−1 ) ∈ Ga ⊗ U for i = 0, 1, and κ is a certain central square root of the ribbon element, as before. The algebra U admits a very nice irreducible representation, D, which is constructed by acting with elements in U on a “ground state” |0i. The state |0i may be characterized by the following (invariance-) properties: ιi (ξ)|0i = |0i(ξ) for all ξ ∈ G
i = 0, 1,
Representation Theory of Lattice Current Algebras
47
where is the trivial representation (co-unit) of G. Here and in the following we neglect to write the letter D when elements in U act on vectors. Since we are dealing with a unique representation of U, ambiguities are excluded. While the preceding formula means that |0i is invariant under the action of G0 ⊗ G1 , the components of U ∈ Ga ⊗ U create new states in the carrier space, <, of the representation D, |πi ≡ π(U )|0i = (π ⊗ id)(U )|0i ∈ < for all π ∈ Ga0 . In particular, one identifies |i = |0i because (U ) = U 0 = e. We can think of the vectors |πi as coming from a universal object u ∈ Ga ⊗ <, i.e., |πi ≡ π(u) = (π ⊗ id)(u) . In other words, u = U |0i. A complete description of the representation D on < is given in the following proposition. Proposition 4 (Representation of U). There exists an irreducible *-representation D of the algebra U on a carrier space < such that 1
2
U u = R1a (u) , (e ⊗ ι1 (ξ))u = u(ξ ⊗ e),
(e ⊗ ι0 (ξ))u = (S(ξ) ⊗ e)u .
Here u ∈ Ga ⊗ < and 1a (u) = (1 ⊗ id)(u). The representation space < contains a unique invariant vector |0i ∈ <. Proof. The formulas for the action of U on < follow from u = U |0i ∈ Ga ⊗ < by using the invariance of |0i under the action of ι1 (ξ) and ι0 (ξ). In particular, we have that 1
2
1
2
U u = U U |0i = R1a (U )|0i = R1a (u),
and
(e ⊗ ι1 (ξ))u = U 11 (ξ)|0i = U |0i(id ⊗ )(1(ξ)) = u(ξ ⊗ e) . To derive the action of G0 on < we rewrite the equation U ι0 (ξ) = 10 (ξ)U according to ι0 (ξ)U = (S(ξς1 ) ⊗ e)U (e ⊗ ι0 (ξς2 )) . Here we have inserted the expansion 1(ξ) = ξς1 ⊗ ξς2 and used some standard Hopfalgebra properties of the antipode S. Then one proceeds as in the computation of ι1 (ξ)u to obtain the last formula claimed in Proposition 4. Observe that the formulas in the preceding proposition define an action – not a coaction – of the algebra U on the representations space <. Again, it is important to keep in mind that the co-product 1 in the first formula of Proposition 4 acts on the first tensor factor Ga of u ∈ Ga ⊗ <. In terms of the multiplication ∗ in G 0 (cf. Subsect. 2.3) one has that π1 (U )|π2 i = |π1 ∗ π2 i for all π1 , π2 ∈ G 0 . Consequently, the components π(U ) act on < as some kind of (twisted) multiplication operators. Furthermore, the matrix elements of uI = τ I (u) ∈ End(V I ) ⊗ < span a δI2 -dimensional subspace of < which is invariant under the action of G0 ⊗ G1 ⊂ U. With the help of Proposition 4 we deduce
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A. Y. Alekseev, L. D. Faddeev, J. Fr¨ohlich, V. Schomerus
ι1 (ξ)uI = uI τ I (ξ) , ι0 (ξ)uI = τ I (S(ξ))uI for all ξ ∈ G and with ιi (ξ) = (eI ⊗ ιi (ξ)), where eI is the unit element in End(V I ). The two formulas furnish the following decomposition of < into a direct sum of G0 ⊗ G1 modules, M Vˇ I ⊗ V I , (3.1) <∼ = I
where Vˇ I is dual to V I and the sum extends over the classes of irreducible representations of G. Vˇ 0 ⊗ V 0 ⊂ < coincides with the one-dimensional subspace spanned by |0i. All these features of the representation space < resemble those of the algebra of squareintegrable functions on a Lie-group G with its characteristic action of left and right invariant vector fields. This similarity is not too surprising and can be traced back to the analogy between the objects U and T . In Subsect. 2.3, T was found to satisfy “RT T relations” which are the key ingredient in the deformation theory of groups G. Mainly for technical reasons we finally look at a certain subalgebra D of U and its action on <. Lemma 2. Let D be the subalgebra of U which is generated by elements ξ ∈ G1 and components of U . When the representation D of U is restricted to D ⊂ U it furnishes an irreducible representation D of D on <. Proof. To prove this lemma we show that every vector |πi in the representation space < is cyclic under the action of D. Since < contains the cyclic vector |0i, our task simplifies to the following problem: show that for every |πi ∈ < there is a representation operator Aπ ∈ D(D) such that Aπ |πi = |0i. For the proof it is crucial to find the projector on |0i in D(D). It is constructed from the minimal central projector P 0 ∈ G that corresponds to the trivial representation = τ 0 . By definition, P 0 satisfies τ I (P 0 ) = δI,0 , so that ι1 (P 0 )|π 0 i = |0iπ 0 (P 0 ) holds for all π 0 ∈ G 0 . The other ingredient we need below is a distinguished element µ ∈ G 0 – called the right integral of G – with the properties (µ ⊗ id)1(ξ) = µ(ξ) , µ(P 0 ) = 1 . Now we choose ξπ ∈ G such that π(ξπ ) = 1. A short technical computation shows that (e ⊗ ξ0 )1(P 0 ) = (S(u−1 ξ0 ) ⊗ e)R1(P 0 ) , where u = g −1 v ∈ G and g was introduced in Subsect. 2.1. We abbreviate η0 ≡ S(u−1 ξ0 ) and regard η0 as a map from G to G acting by left multiplication so that µ ◦ η0 makes sense as an element in G 0 . Let us define Aπ ≡ ι1 (P 0 )(µ ◦ η0 )(U ) . The following calculation proves that Aπ |πi = |0i and hence completes the proof of the lemma: Aπ |πi = ι1 (P0 )(µ ◦ η0 )(U )|πi = ι1 (P0 )|(µ ◦ η0 ) ∗ πi = |0i((µ ◦ η0 ) ∗ π)(P0 ) = |0i(µ ⊗ π)((η0 ⊗ e)R1(P0 )) = |0iπ(ξ0 )µ(P 0 ) = |0i .
Representation Theory of Lattice Current Algebras
49
The algebra D is a semidirect product of G and its dual G 0 , the latter being supplied with the twisted product ∗ that we discussed in Subsect. 2.3. In this light, D appears as a close relative of the deformed cotangent bundle Tq∗ G over a group G which differs from the structure of D only through the use of the standard product · in G 0 . Example. Let us continue our tradition and illustrate the theory with the example G = Zq . The algebra U is then generated by the unitary elements g0 , g1 and w satisfying wp = 1 , g1 w = qwg1 , wg0 = qg0 w, and g0 commutes with g1 (cf. Subsect. 2.3 for further details). States in < are created from a ground state |0i with invariance properties g1 |0i = |0i and g0 |0i = |0i. Through iterated application of w on |0i we may produce p linearly independent vectors |ri ≡ wr |0i ∈ < for r = 0, . . . , p − 1 . Specializing the proof of Proposition 4 to our example, we obtain g1 |ri = q r wr g1 |0i = q r |ri and similarly for g0 . This shows that the one-dimensional subspace spanned by |ri corresponds to the summand Vˇ r ⊗ V r in the decomposition (3.1) of <. The subalgebra D of U is generated by g = g1 and w, with Weyl commutation relations wg = qgw and g p = e = wp . It acts irreducibly on the p-dimensional space <. 3.2. The algebra K. Before we deal with the general situation, it is helpful to study the simplest example of a lattice current algebra for which the lattice consists of only one (closed) link and one site, i.e., the case N = 1. Strictly speaking, K1 has not been defined above. So we must first give a definition. Definition 5 (The algebra K). The *-algebra K ≡ K1 is generated by components of M, M −1 ∈ Ga ⊗ K and elements ξ ∈ G with the following relations 1
2
M R M = R1a (M ) , 1(ξ)M = M 1(ξ) for all ξ ∈ G M −1 is the inverse of M , so that M −1 M = e = M M −1 . The action of ∗ is extended from G to K by the formula M ∗ = S −1 M −1 S , where S = 1(κ−1 )(κ ⊗ κ)R−1 , as before.
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A. Y. Alekseev, L. D. Faddeev, J. Fr¨ohlich, V. Schomerus
Components π(M ) of the monodromy M can be represented on the carrier spaces V I of the representations τ I of G. This is accomplished by the formula DI (π(M )) = (π ⊗ τ I )(R0 R) ∈ End(V I ) for all linear forms π ∈ Ga0 on Ga . An equivalent universal formulation without reference to linear forms π is DI (M ) = (id ⊗ τ I )(R0 R) ∈ Ga ⊗ End(V I ) . Indeed, one may check that such an action on V I is consistent with the functoriality of M , i.e., with the first relation in Definition 5, 1
2
0 0 R13 R12 R23 R23 ] DI (M R M ) = (id ⊗ id ⊗ τ I )[R13 0 0 = (id ⊗ id ⊗ τ I )[R13 R23 R12 R13 R23 ] 0 0 = (id ⊗ id ⊗ τ I )[R12 R23 R13 R13 R23 ]
= (id ⊗ id ⊗ τ I )[R12 (1 ⊗ id)(R0 R)] = R(1a ⊗ id)(DI (M )) . From our discussion in Subsects. 2.4, 2.5 we know already that Ga ⊗ K contains not only the left monodromy M L = M but also the right monodromy −1 M L N+ ∈ G a ⊗ K M R = N−
N+ = R 0 ,
with
N− = R−1 .
Here N± are regarded as elements in Ga ⊗ G ⊂ Ga ⊗ K, as before. Since the components of M L and M R commute, they can be represented on spaces V I ⊗ V J such that the right/left-monodromies act trivially on the first/second tensor factor, respectively. This action of left and right monodromies on W IJ ≡ V I ⊗ V J can be extended to an action of the entire algebra K. Actually, the algebra K is isomorphic to the Drinfeld double of G [23, 21]. Proposition 6 (Representations of K). The algebra K has a series of irreducible *representations, D IJ , defined on the spaces W IJ . Explicitly, the action is given by DIJ (M ) = (id ⊗ τ I )(R0 R) , J I J DIJ (ξ) = (τ I × τ )(ξ) = (τ ⊗ τ )(1(ξ)) .
Here DIJ (M ) ∈ Ga ⊗ End(V I ) is regarded as an element of Ga ⊗ End(W IJ ) with trivial action on the second tensor factor V J in W IJ . Proof. To check the representation property is left as an exercise to the reader. It may be helpful to consult Theorem 12 of [6]. Irreducibility follows from the fact that the action of the monodromies on the spaces V I is irreducible (cf. Lemma 1 of [6]).
Representation Theory of Lattice Current Algebras
51
Example. For G = Zq , the definition of K furnishes an algebra with generators c = c1L and g such that cp = 1 , g p = 1 and cg = gc . Some explanation can be found at the end of Subsect. 2.5. The abelian algebra K has one-dimensional representations, Dst , on spaces W st labeled by two integers s, t = 0, . . . , p − 1, Dst (c) = q s , Dst (g) = q s+t , where q = exp(2πi/p). For comparison with the general formulas in Proposition 6, one should keep in mind that g is a generator of the algebra G while c coincides with the element τ 1 (M ) ∈ K, up to a scalar factor. 3.3. Representations of KN . In representing the full lattice current algebras KN it is convenient to pass to a new set of generators: Let us define elements Uν ∈ Ga ⊗ KN , ν = 1, . . . , N − 1, as above by Uν ≡ va1−ν J1 · · · Jν . Then the Uν , together with M ≡ M L and the elements ξ ∈ Gn , n = 1, . . . , N, generate KN . They obey the following relations: 1
2
U ν U ν = R1a (Uν ) , 1
2
M R M = R1a (M ) , 1
2
2
1
R0 U ν U µ = U µ U ν for 1 ≤ ν < µ ≤ N − 1 , 1
2
2
1
R0 U ν M = M R0 U ν ,
(3.2)
10 (ξ)M = M 10 (ξ) for ξ ∈ G , ιν (ξ)Uν = Uν 1ν (ξ) for ξ ∈ G , 10 (ξ)Uν = Uν ι0 (ξ) for ξ ∈ G . We infer that KN contains (N − 1) copies of the algebra U generated by Uν , Gν , G0 and one copy of the algebra K generated by M ≡ M L and G0 . These subalgebras do not commute, but the non-commutativity is felt only by Uν , M and G0 . In any case, the exchange relations motivate us to look for representations of the lattice current algebra KN on spaces (3.3) WNIJ ≡ <⊗N −1 ⊗ W IJ . To state the formulas we introduce the notation Dν , ν = 1, . . . , N − 1, which stands for the representation D of the algebra Uν on the ν th factor of the tensor product (3.3), i.e., for every X ∈ Uν , Dν (X) = id1 ⊗ . . . ⊗ idν−1 ⊗ D(X) ⊗ idν+1 ⊗ . . . ⊗ idN −1 ⊗ Id , where idµ acts as the identity on the µth factor < in WNIJ , and Id is the identity on W IJ . Similarly, DIJ denotes the action of the subalgebra K ⊂ KN on the last factor W IJ in WNIJ . The map ι0 embeds elements of G into all the subalgebras Uν of the lattice current algebra KN . Hence G acts on each tensor factor < in WNIJ independently with the help
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A. Y. Alekseev, L. D. Faddeev, J. Fr¨ohlich, V. Schomerus
of representations Dν . We employ the co-product 1 on G to obtain a family ϑν , ν = 1, . . . , N, of G-actions on WNIJ . In the following, ϑ1 is the trivial representation, ϑ1 = , and ϑν (ξ) = (D1 × ... × Dν−1 )(ι0 (ξ)) ⊗ idν ⊗ . . . idN −1 ⊗ Id for all ξ ∈ G. The symbol × denotes the tensor product of representations. With these conventions we are prepared to define representations of KN on WNIJ . The essential idea is borrowed from the well known Jordan-Wigner transformation. In fact, in writing the actions of our generators on WNIJ we have to relate the Uν ’s and M (which obey non-trivial exchange relations among each other) to the operators Dν (Uν ) tensor factors in WNIJ and hence commute. and DIJ (M ). The latter act on different Qn In analogy with the ‘tail’-factors i=0 σi3 of the Jordan-Wigner transformation, we will employ tails of R-matrices to express the Uν in terms of Dν (Uν ) and M in terms of D IJ (M ). Related constructions appear in Majid’s ‘transmutation theory’ (see e.g. [19]). Theorem 7 (Representations of KN ). The algebra KN has a series of irreducible *IJ realized on the spaces WNIJ given in Eq. (3.3). In the representation representations DN IJ DN , the generators of KN act as IJ (Uν ) = (id ⊗ ϑν )(R−1 )Dν (Uν ) , DN IJ DN (M ) = (id ⊗ ϑN )(R−1 )DIJ (M )(id ⊗ ϑN )(R) , IJ DN (ιν (ξ)) = Dν (ιν (ξ)) for ν = 1, . . . , N − 1 , ξ ∈ G , IJ DN (ι0 (ξ)) = (ϑN ⊗ DIJ )(10 (ξ)) for all ξ ∈ G .
Every *-representation of KN can be decomposed into a direct sum of the irreducible IJ . representations DN Proof. The proof is similar to the one of Theorem 15 in [6]. Although we do not present the details, we stress that the factors (id⊗ϑν )(R±1 ) produce “tails of R-elements” which IJ . are responsible for the correct exchange relations of Uν , M in the representations DN From the definition of ϑν and quasitriangularity of R we infer (for ν > 1) −1 −1 −1 R13 . . . R1ν . (id ⊗ ϑν )(R−1 ) = R12
Here it is understood that the second tensor factor of R1µ is represented on the µth factor < in WNIJ with the help of Dν ◦ ι0 . A simple example suffices to illustrate how nontrivial exchange relations between Uν , M arise in our representations 1
2
1
2
−1 IJ 0 0 (R12 DN U 1 U 2 ) = R12 D1 (U 1 )R23 D2 (U 2 ) 1
2
0 0 −1 −1 (R12 ) R23 D1 (U 1 )D2 (U 2 ) = R12 2
1
−1 D2 (U 2 )D1 (U 1 ) = R23 2
1
IJ (U 2 U 1 ) . = DN
These equations are to be understood as equations in Ga ⊗ Ga ⊗ End(WNIJ ). To reach the second line, we insert the exchange relation of U1 with elements ι0 (ξ) and quasitriangularity of R−1 . Then we use that the images of D1 and D2 commute.
Representation Theory of Lattice Current Algebras
53
IJ Irreducibility of the representations DN follows from the case N = 1 which we have treated in Subsect. 3.2, together with Lemma 2 of Subsect. 3.1. L It is quite instructive to evaluate the projectors χK L χR (defined at the end of SubIJ sect. 2.5) in the representations DN . The answer is given by (cf. [6]) IJ L (χK DN L χR ) = δI,K δJ,L . L As we have promised above, the elements χK L χR are characteristic projectors for the irreducible representations of the lattice current algebra KN . The representation theory described here survives the limit in which N tends to infinity. With the help of our embeddings γN : KN → KN +1 (see Sect. 2.6) we can define IJ IJ an action DN +1 ◦γN of the lattice current algebra KN on WN +1 . It is, of course, no longer irreducible, so that WNIJ+1 decomposes into a direct sum of irreducible representations of the algebra KN . The observations made in the preceding paragraph combined with formula (2.14) furnish an isomorphism
WNIJ+1 ∼ = WNIJ ⊗ < of KN modules, with < being a multiplicity space of the reduction. In particular, all the IJ . This irreducible subrepresentations of the KN -action on WNIJ+1 are isomorphic to DN implies that the inductive limit for the directed system (KN , γN ) splits into independent contributions coming from the simple summands of KN . Since inductive limits of simple IJ algebras are simple, we conclude that K∞ possesses irreducible representations D∞ IJ on W∞ . As for the algebras KN , the labels I, J run through the classes of irreducible representations of G. Example (Representations of the U (1)-current algebra). To discuss the representation theory of the U (1)-current algebra, we introduce the generators vν = q (ν−1)/2 w1 · · · wν ∈ KN which agree with Uν1 = (τ 1 ⊗ id)(Uν ), up to a scalar factor. The whole algebra KN is generated from the elements gν ∈ Gν ⊂ KN , the monodromy c ∈ KN and the holonomies vν ∈ KN such that gν vν = qvν gν , v ν vµ = q g0p
=
gνp
p
=c =
−1
vµ vν
vνp
=1
vν g0 = qg0 vν , for ν < µ , for all ν = 1, . . . , N − 1
st are denoted and c commutes with every other element. Vectors in the carrier space of DN by |r1 , r2 , . . . , rN −1 is,t ,
where rν , s, t = 0, . . . , p − 1. We can easily define an action of our generators on such states, vν |r1 , . . . , rν , . . . , rN −1 is,t = q r1 +...+rν−1 |r1 , . . . , rν + 1, . . . , rN −1 is,t , gν |r1 , . . . , rν , . . . , rN −1 is,t = q rν |r1 , . . . , rν , . . . , rN −1 is,t , g0 |r1 , . . . , rN −1 is,t = q r1 +...+rN −1 +s+t |r1 , . . . , rN −1 is,t , c |r1 , . . . , rN −1 is,t = q s |r1 , . . . , rN −1 is,t .
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A. Y. Alekseev, L. D. Faddeev, J. Fr¨ohlich, V. Schomerus
The numerical factor on the right hand side of the first line is an example of the “tail of Relements” discussed above. A similar term usually appears in the action of monodromies M but is absent here. It can be seen that the contributions from (id ⊗ ϑN )(R−1 ) and IJ st (M ), cancel for DN (c), (id ⊗ ϑN )(R), which occur in the general expression of DN r since all irreducible representations τ of Zq are one-dimensional. From the experience with conformal field theories we expect that the diagonal repL L II ¯ ¯ on the Hilbert space H ≡ I WNII is particularly relevant. Here we resentation DN just wish to remark that this representation can be realized by a very natural construction. Indeed, it was observed (cf. [4]) that algebras such as KN admit a distinguished invariant linear functional ω : KN → C, 1
N
ω(JI11 . . . JINN ξ) = (ξ)δI1 ,0 . . . δIN ,0 for all ξ ∈ Gn , n ∈ ZmodN and JnI = (τ I ⊗ id)(Jn ), as usual. When the quantum dimensions dI are positive, this functional is positive and hence furnishes – by the GNS construction – a Hilbert space Hω together with a representation π of KN on Hω . If |0iω denotes the GNS vacuum, states in Hω are obtained from 1
N
I I J11 . . . JNN |0iω .
L ¯ The formula shows that Hω is isomorphic to the diagonal sum H = I WNII ∼ = <⊗N . I An explicit evaluation of cR,L on Hω establishes an isomorphism of the two spaces as KN modules. Proposition 8. The GNS-representation arising from the state ω : KN → C is unitarily L ¯ equivalent to the diagonal representation I WNII of the lattice current algebra. The quantum lattice analog of the group-valued local fields of the WZNW model act in this diagonal representation [2, 8]. 4. Product of Representations All continuous current algebras are equipped with a trivial co-product which can be written for Fourier modes of currents j(n) as 1(j(n)) = j(n) ⊗ 1 + 1 ⊗ j(n).
(4.1)
From the point of view of CFT this co-product is not satisfactory, because it changes the central charge of representations. In the framework of CFT, the central charge is characteristic of the model, and one must define a new co-product which preserves it. Such a co-product is provided by the structure of CFT [15, 20]: X Ckn z n−k j(k). (4.2) 1zCF T (j(n)) = j(n) ⊗ 1 + 1 ⊗ k≤n
Here Ckn ≡ n!/k!(n − k)! are binomial coefficients. Observe that the co-product 1CF T is not symmetric and explicitly depends on the parameter z. The aim of this section is to introduce a lattice counterpart of 1CF T . 4.1. Co-product for lattice current algebras. Because lattice current algebras are labelled by the number of lattice sites, it is not necessary that both current algebras on the right
Representation Theory of Lattice Current Algebras
55
hand side of the co-product correspond to chains of the same length. We shall define a family of embeddings (4.3) 3M,N : KN +M −1 → KM ⊗ KN for any N and M . The homomorphisms 3M,N determine an action of KN +M −1 on the IJ tensor products WM ⊗ WNKL of representation spaces for KM and KN . Pictorially, 3M,N corresponds to gluing two closed chains of length M and N by identifying some site of the first chain with some site of the second chain. In this way, the co-product 3M,N explicitly depends on the positions of the identified points. This property is similar to the z-dependence of 1zCF T . Below, we always assume that the enumeration starts from the gluing points, so that this extra parameter does not show up in our formulas; in the same fashion one can put e.g. z = 1 in the continuum theory. After gluing we cut the resulting eight-like loop at the middle point and get one connected chain of length N +M −1. Similarly to 1zCF T , the co-product 3M,N depends on the order in which M and N appear. Next, we present the constructive description of 3M,N . α ,m = Let us denote the left currents of KN by Jnβ , n = 1, . . . , N, and similarly by Jm β α 1, . . . , M, the left currents of KM . Jn and Jm are regarded as elements in G ⊗KM ⊗KN with the property 1
2
2
1
α β β α Jm Jn =Jn Jm
for all n, m. In addition to the left currents, we need N + M commuting copies of the symmetry algebra G to generate KM ⊗ KN . With these notations we can define the announced embedding 3M,N : KN +M −1 → KM ⊗ KN . It maps the generators Jρ , ρ = 1, . . . , N + M − 1, of KN +M −1 in the following way to generators of KM ⊗ KN : α Jρ J α N αJ β M − 1 3M,N (Jρ ) = β J +1 ρ−M β α −1 JN (N− )
for ρ = 1, . . . , M − 1 for ρ = M for ρ = M + 1, . . . , N + M − 2
,
(4.4)
for ρ = N + M − 1
α α where N± = N0,± . For elements ξ ∈ G, we define
α for ρ = 1, . . . , M − 1 ιρ (ξ) ⊗ e β 3N,M (ιρ (ξ)) = e ⊗ ιρ−M +1 (ξ) for ρ = M, . . . , N + M − 2 , α (ι0 ⊗ ιβ0 )(1(ξ)) for ρ = N + M − 1
(4.5)
where e means the unit element in KM or KN . Proposition 9. The map 3M,N : KN +M −1 → KM ⊗ KN defined through Eqs. (4.4) and (4.5), is an algebra homomorphism. One can prove this proposition by directly verifying the defining relations for KN +M −1 . In order to justify calling 3M,N a co-product, we need some kind of co-associativity. For lattice current algebras this holds in the following form.
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A. Y. Alekseev, L. D. Faddeev, J. Fr¨ohlich, V. Schomerus
Proposition 10. The family of homomorphisms 3M,N satisfies the following property (id ⊗ 3N,L ) ◦ 3M,L+N −1 = (3M,N ⊗ id) ◦ 3M +N −1,L .
(4.6)
Applying our formulas for 3M,N twice to the generators of KL+M +N −2 one easily sees that the homomorphisms on the left and on the right hand sides of (4.6) coincide. ˜ Let us notice that along with the co-product 3 one can introduce a co-product 3 defined by β N+ Jρα for ρ = 1 Jα for ρ = 2, . . . , M − 1 ρ ˜ M,N (Jρ ) = (4.7) 3 β −1 β α JM (N+ ) J1 for ρ = M Jβ for ρ = M + 1, . . . , N + M − 1 ρ−M +1 ˜ M,N acts on elements ξ ∈ G according to Eq. (4.5) with the 1 in the last line being and 3 ˜ are ‘intertwined’ by the ∗-operation. replaced by 10 . These two co-products 3 and 3 To make a precise statement we introduce an element K ∈ KM ⊗ KN by the expression β −1 ˜ M,N (ι0 (κ))(ια K=3 0 (κ) ⊗ ι0 (κ)) . With this notation we have ˜ M,N (x∗ ) K 3M,N (x)∗ = K −1 3
(4.8)
for all x ∈ KN +M −1 . This property is similar to (2.1) where the ∗-operation intertwines 1 and 10 and, in fact, it reduces to the latter on elements ξ ∈ Gn ⊂ KN +M −1 . 4.2. Special cases. There are important special cases of the maps 3M,N which we would like to consider in more detail. First, observe that the map 31,1 : K1 → K1 ⊗ K1
(4.9)
satisfies the co-associativity condition (id ⊗ 31,1 ) ◦ 31,1 = (31,1 ⊗ id) ◦ 31,1 .
(4.10)
This is a specification of Eq. (4.6) for the case of L = M = N = 1. We conclude that the map 11 = 31,1 furnishes a co-product for the algebra K1 . It has been recently shown [21] that as a Hopf algebra K1 is isomorphic to the Drinfeld double of G. In particular, this implies existence of an R-matrix for the Hopf-algebra K1 . As we know (see Sect. 3), the algebra K1 is generated by the elements ξ ∈ G and by the universal element M ∈ G ⊗ K1 . Irreducible representations of K1 are labelled by the pairs (I, J) of irreducible representations of G. This implies that as an algebra K1 is isomorphic to G ⊗ G (see also [23] where an isomorphism of quasi-triangular Hopf algebras is described). There is another interesting special choice of chain lengths N and M : 31,N : KN → K1 ⊗ KN .
(4.11)
The co-associativity condition (4.6) adapted to this case reads (id ⊗ 31,N ) ◦ 31,N = (31,1 ⊗ id) ◦ 31,N .
(4.12)
This shows that 31,N also provides a co-action of the Hopf algebra K1 on KN . Such a structure has been noticed already in [2]. We shall see that it permits us to establish a one-to-one correspondence between representations of KN and K1 .
Representation Theory of Lattice Current Algebras
57
Our last remark on the properties of 3M,N concerns the inductive limit K∞ . Using the block-spin embeddings KN → KN +1 , one can construct a commutative diagram: 3M,N : KN +M −1 → KM ⊗ KN ↓ ↓ 3M,N +1 : KN +M → KM ⊗ KN +1
(4.13)
Commutativity (4.13) ensures that the sequence of homomorphisms 3M,N defines homomorphisms (4.14) 3M,∞ : K∞ → KM ⊗ K∞ . Then Eq. (4.6) implies the co-associativity for 3M,∞ : (id ⊗ 3N,∞ ) ◦ 3M ∞ = (3M,N ⊗ id) ◦ 3N +M −1,∞ .
(4.15)
Thus, 3M,∞ provides a co-module structure for K∞ with respect to the family KM . In particular, K∞ is a co-module over K1 . 4.3. Implications for representation theory. The co-product 3M,N yields a notion of tensor product for representations of the algebras KM and KN . Definition 11. The representation D of the algebra KN +M −1 is called a tensor product of the representations DM of KM and DN of KN if it acts on the tensor product of the corresponding vector spaces WM ⊗ WN according to the following formula: D(x) = (DM ⊗ DN )3M,N (x)
(4.16)
for all elements x of the lattice current algebra KN +M −1 . The resulting representation will be denoted by DM × DN . We would like to analyse the structure of this new tensor product. We denote by 0 the trivial representation of the symmetry Hopf algebra. Proposition 12. For any M and N and for arbitrary labels I and J of the representations of the symmetry algebra the following representations of KM +N −1 are isomorphic: IJ IJ 00 00 IJ DM D N ' DN × DM . +N −1 ' DM ×
(4.17)
00 In this sense, tensoring with the vacuum representation DN is trivial. IJ ⊗ WN00 and To prove this proposition one first checks that on the spaces WM the central elements of KM +N −1 have eigenvalues corresponding to the repIJ resentation DM +N −1 . Then one checks that the dimensions of all these spaces coincide IJ with the dimension of WM +N −1 . This completes the proof. Observe that the tensor product of representations that we have introduced, relates representations of different algebras. For instance, if we take N = M , we obtain a representation of the algebra K2N −1 on the tensor product of representation spaces of KN . The idea now is to embed the algebra KN into K2N −1 with the help of the block spin maps γ˜ M , IJ WN00 ⊗WM
γ˜ 2N −1,N := γ˜ 2N −2 ◦ . . . ◦ γ˜ N +1 ◦ γ˜ N : KN → K2N −1 .
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A. Y. Alekseev, L. D. Faddeev, J. Fr¨ohlich, V. Schomerus
In this way we may represent the algebra KN on tensor products of its own representation spaces. The resulting representation certainly has a huge commutant and is not appropriate to describe the representation theory of KN . We shall define a certain projection α ∈ KN ⊗ KN that projects to a more interesting subrepresentation. operator PN α The construction of PN proceeds as follows. Notice that the lattice current algebra KN contains N − 1 local projectors pi ∈ Gn ⊂ KN where n runs from 1 to N − 1. They are uniquely determined by the property ιi (ξ)pi = (ξ)pi for all ξ ∈ G. An explicit formula for pi in terms of the objects Ni can be obtained along the lines of Subsect. 2.5. When Ni replaces the monodromy M then we obtain pi instead of χ0 . These projectors pi commute with each other, i.e., pi pj = pj pi for all i, j so that their product defines again a projector PN , N −1 Y PN := pi ∈ K N . i=1
From the defining relations of KN it is fairly obvious that PN commutes with N0 and α will denote the projector PN ⊗ e ∈ KN ⊗ KN the monodromy M . In the following, PN β and similarly PN = e ⊗ PN ∈ KN ⊗ KN . With these objects at hand, we are now able to define a new co-product 1N for the algebra KN , a 3N,N (γ˜ 2N −1,N (x)) for all x ∈ KN . 1N (x) := PN
(4.18)
It is easy to check that 1N defines a homomorphism because γ˜ 2N −1,N and 3N,N are α homomorphisms and the projector PN commutes with the image of 3N,N ◦ γ˜ 2N −1,N : β 1N (x) so that the coKN → KN ⊗ KN . Let us also mention that 1N (PN x) = PN associativity of 1N follows from that of 3N,M , (1N ⊗ id)1N (x) = (id ⊗ 1N )1N (x) for all ξ ∈ KN . In deviation from the standard properties of co-products, 1N is not unit preserving, i.e., 1N (e) 6= e ⊗ e ∈ KN ⊗ KN (e ∈ KN denotes the unit element), and there is no one-dimensional trivial representation of KN . The role of the co-unit is actually played 00 of KN . Such algebraic properties are characteristic by the vacuum representation DN for weak Hopf-algebras [7] and the closely related weak quasi-Hopf algebras of [18]. Notice, that the co-product 1N is compatible with the block spin operation: (γN ⊗ γN )1N = 1N +1 γN . This property is ensured by the fact that two block spin operations γN and γ˜ N commute with each other (see Sect. 2). Thus, one can define an operation 1∞ : K∞ → K∞ ⊗ K∞ which provides a co-product for the inductive limit of lattice current algebras. We would finally like to compare the representation category of KN with that of K1 . To this end notice that the formula IJ 00 DN ' D1IJ × DN
(4.19)
provides a one-to-one correspondence between representations of K1 and KN for arbitrary N . In fact, this implies the same kind of correspondence for representations of K1 and K∞ . Because KN is semisimple, for all N , the isomorphism (4.19) induces a map FN (D1 ) = DN
(4.20)
which assigns to each representation D1 of the algebra K1 a representation DN of the algebra KN .
Representation Theory of Lattice Current Algebras
59
To describe the properties of the map F, it is convenient to use the language of the theory of categories (see e.g. [17]). It is clear that the map FN is invertible and that it defines a co-variant tensor functor mapping the category of representations of the Hopf-algebra K1 into the category of representations of the lattice current algebras KN . Actually, on the image of PNα , the tensor product of representations of KN defined through 1N is isomorphic to the representation obtained with the help of the co-action 31,N . Since tensor operators for the latter may be trivially identified with tensor operators of the quasitriangular Hopf-algebra K1 , the functor FN provides an equivalence of braided tensor categories. In the limit N → ∞ we arrive at the following conclusion. Theorem 13. The functor F establishes an isomorphism between representations the Hopf algebra K1 and the lattice current algebra K∞ which is compatible with a coproducts of K1 and K∞ and establishes an equivalence of braided monoidal categories. We can view this fact as the lattice analogue of a theorem in [16, 14] on the equivalence of tensor categories corresponding to quantum groups and current algebras. Here the algebra K∞ replaces the current algebra, and K1 = G ⊗ G is the direct product of two quantum groups corresponding to two chiral sectors. From this point of view, finding an exact relationship between lattice and continuum current algebras emerges as a challenging problem. Acknowledgement. We would like to thank A. Connes and K. Gawedzki, participants and lectures of the 95’ Summer School on Theoretical Physics at Les Houches for an inspiring atmosphere. V.S. would also like to thank T. Miwa and I. Ojima for their hospitality at RIMS. We are grateful to A. Bytsko and F. Nill for their remarks and criticism.
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Commun. Math. Phys. 191, 61 – 70 (1998)
Communications in
Mathematical Physics c Springer-Verlag 1998
Collet-Eckmann Maps are Unstable Alexander Blokh1,? , Michał Misiurewicz2 1 Department of Mathematics, University of Alabama in Birmingham, University Station, Birmingham, AL 35294-2060, USA. E-mail:
[email protected] 2 Department of Mathematical Sciences, IUPUI, 402 N. Blackford Street, Indianapolis, IN 46202-3216, USA. E-mail:
[email protected]
Received: 17 October 1996 / Accepted: 16 April 1997
Abstract: We show that a smooth interval map satisfying the Collet-Eckmann condition at some critical point is not structurally stable in C r topology for any r. 1. Introduction We want to address an old problem of structural stability of smooth interval maps. The reader can find a good account of this problem in [MS]. Basically, one expects that the only way an interval map can be structurally stable is that the trajectories of all critical points are attracted to attracting periodic points. However, this is known only for C 1 -stability ([J]). It is widely believed that this problem resembles the Closing Lemma, that is, a perturbation necessary to change the topological type of a map has to be global (or at least a proof is possible only for global perturbations). Here we show that some maps admit the proof of instability in any C r topology with local perturbations. These maps are the ones satisfying the Collet-Eckmann condition at some critical point. Positive Lyapunov exponent at a critical point causes “sensitivity” at this point, so a local perturbation has global effects. There is a slight difficulty when we speak of structural stability of interval maps. Let us illustrate it with an example. Let f : [0, 1] → [0, 1] be a convex C ∞ unimodal map with f (0) = 0 and a = supx∈[0,1] f (x) < 1. Set gε (x) = f (x) + ε for 0 < ε ≤ 1 − a. The maps gε converge to f in C ∞ topology as ε → 0. None of gε is conjugate to f , since for each of them 0 is not a fixed point while for f it is. Yet this is not the kind of instability of f we want. Basically, the change of behavior of the map was achieved by the change of the interval on which it was defined (and rescaling back). In our example the map may have really changed its topological type, but to establish that is much more difficult than to prove the “instability” as above. ?
He was partially supported by NSF grant DMS 9626303.
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To avoid problems like this, let us define the essential interval for a continuous interval map f as the smallest interval containing trajectories of all interior local extrema of f . Clearly, this interval is invariant for f . Now, continuous interval maps are said to be essentially conjugate if they are conjugate when restricted to their essential intervals. Finally, we say that an interval map f of class C r is C r -structurally stable if there exists a neighborhood of f in C r topology such that every map from this neighborhood is essentially conjugate with f . Note that we can also speak of C r -structural stability of f even if f is not of class r C . Namely, we require that there is a C r -neighborhood of the 0 function such that for every h from this neighborhood f + h is essentially conjugate with f . This may be not too interesting for all maps f , but it makes sense for instance if f is piecewise C r . Sometimes we are interested in changing the topological type of a map by introducing a small local perturbation. To describe this phenomenon, we introduce the notion of structural sensitivity at a point. Namely, we say that a map f is C r -structural sensitive at x if for every C r -neighborhood V of the function 0 and for every neighborhood U of x there exists h ∈ V which is 0 outside U and such that f + h is not essentially conjugate to f . We will use the following terminology. Let f be a continuous interval map. A point x is called a periodic sink (from one side) if there exists n > 0 and a (one-sided) neighborhood U of x such that f n (x) = x, f n (U ) ⊂ U andSthe diameter of f k (U ) tends ∞ to 0 as k → ∞. The basin of attraction of x is then the set k=0 f −k (U ). Note that if f n has a (one-sided) derivative at x, its absolute value is less than or equal to 1. It is well known that any point from the boundary of the basin of attraction of a periodic sink is either periodic or preperiodic. An interval J will be called a wanderval if f n |J is a homeomorphism for every n, the images of J are pairwise disjoint, and the orbit of J does not converge to the orbit of a periodic (even one-sided) sink (we can talk about convergence of the orbit of J since by the second condition all points of J have the same ω-limit set). It is easy to see that this is equivalent to f n |J being homeomorphism for every n and the existence of a point x ∈ J whose ω-limit set is not a periodic orbit. Indeed, if the first set of conditions is satisfied then the absence of flat spots (i.e. intervals on which f is a constant) implies that the orbit of no point x ∈ J converges to a periodic orbit. Suppose now that the second set of conditions holds. In that case if some images of J intersect each other then the union of all images of J is an invariant union of finitely many intervals such that the restriction of f on this union is monotone on every component; this implies that all points of J have periodic orbits as their ω-limit sets. The non-existence of wandervals for smooth interval maps was proven in a series of papers with the most general result obtained in [MMS]. The question of their existence in a piecewise-smooth setting remains unsolved in general, however under the assumption of the exponential growth of the derivative at a point we prove that this point is not contained in a wanderval.
2. Main Theorem We look at the class of all continuous maps f : [0, 1] → [0, 1] for which there exist points 0 = a0 < a1 < . . . < as = 1 such that f is of class C 1 on each of the intervals [ai , ai+1 ], with non-zero derivative on each (ai , ai+1 ). We will refer to these maps as piecewise smooth maps. The points a0 , a1 , . . . , as will be called singular points. Points at which f has local extrema (except 0 and 1) will be called turning points.
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We will say that a piecewise smooth map f satisfies the Collet-Eckmann condition at a turning point c of f if (2.1) |(f n )0 (f (c))| ≥ αλn for all n, where α > 0 and λ > 1 (in other words, if the lower Lyapunov exponent at f (c) is positive). If only one-sided derivatives of f n at f (c) exist, we mean (2.1) for both of them. We introduce a function r : [0, 1]2 → R ∪ {∞} as follows: r(x, y) =
|f (x) − f (y)| |x − y| |f 0 (x)|
if x 6= y, and r(x, x)=1 (if only one-sided derivatives at x exist, we take as f 0 (x) the derivative from the side where y is). We will call this function relative stretching, since it measures how the interval is being stretched relative to the derivative at one of its endpoints. In fact, this makes sense only if x and y belong to the same lap of f (by a lap we mean a maximal interval on which f is monotone; there may be singular points in the interior of a lap). We will call the infimum of r(x, y) over the pairs of points x, y from the same lap the shrinkability of f . Note that if a point x is critical (i.e. the derivative of f at x vanishes) then r(x, y) = ∞ for all y 6= x. If a, b are two consecutive singular points, let us look at r restricted to [a, b]2 . Clearly, r is continuous off the diagonal. If x 6= y and f 0 (x) 6= 0 then r(x, y) = f 0 (z)/f 0 (x) for some z ∈ (x; y) (by (x; y) we mean (x, y) if x < y and (y, x) if y < x). Hence, since f is of class C 1 on [a, b] and f 0 (x) 6= 0 for all x ∈ (a, b) we conclude that r is also continuous on the diagonal, at all points (x, x) such that f 0 (x) 6= 0. Let f be a piecewise smooth map and let c be a turning point of f . For a given ε > 0 and a neighborhood U of c we denote by B(f, ε, U, c) the set of maps g : [0, 1] → [0, 1] such that |g(x)−f (x)| ≤ ε for every x ∈ U , g(x) = f (x) for every x ∈ / U , |g(c)−f (c)| = ε, and c is a local extremum of g. For every point x ∈ [0, 1] its itinerary (for f ) is the sequence (in (x))∞ n=0 , where in (x) is the point f n (x) if this is a turning point and the lap of f to which f n (x) belongs otherwise. Main Theorem. Let f be a piecewise smooth map with non-zero shrinkability, satisfying the Collet-Eckmann condition at a turning point c. Then there is a neighborhood U of c such that for every ε > 0 if g ∈ B(f, ε, U, c) then either g has more local extrema than f , or the g-trajectory of c is attracted to the orbit of a periodic (at least one-sided) sink, or the itineraries of c for f and g are different. Proof. We may assume from the very beginning that we are considering only those g ∈ B(f, ε, U, c) which have c as their only local extremum in U . Indeed, we may choose U such that f has only one local extremum in it. Then, if g has more than one local extremum in U , it has more local extrema than f in the whole interval, and we are done. Since we will often consider behavior of f separately at each side of some point x, we will speak about right and left halves of x. The point 0 has only right half, and the point 1 only left one. Now every half-point has the derivative of f well defined at it. Moreover, we can tell to which half of f (x) a given half of x is mapped by f . Let λ and α be constants from (2.1). We choose η and µ such that η < 1 < µ < ηλ.
(2.2)
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Let p be the number of singular half-points that are not periodic. We choose an integer N > p such that (2.3) αµN ≥ 1 and
µ ηλ
N
<
K η
p ,
(2.4)
where K is the shrinkability of f . For every non-periodic singular half-point we choose its neighborhood (one-sided, of course) such that every piece of trajectory of length N passes through it at most once. Let V be the union of these neighborhoods. Thus, every piece of a trajectory of length N visits V at most p times. For every periodic singular half-point that is a sink we choose its neighborhood (again one-sided) contained in the basin of attraction. Let W be the union of these neighborhoods. Since f satisfies the Collet-Eckmann condition at c, the trajectory of f (c) does not visit W at all. Then we choose δ > 0 such that if x∈ / V ∪ W , |x − y| < δ and x, y do not lie on opposite sides of a periodic singular point then r(x, y) ≥ η. To see that this is possible, notice that for a small δ, if x, y are as above then they belong to some closed interval J on which f 0 is continuous and non-zero. The function r is continuous on J 2 and it is equal to 1 on the diagonal, hence it is larger than η in some neighborhood of the diagonal. We are now ready to choose U required in the theorem. The choice depends on the trajectory of c, so we will consider several cases. In what follows we denote f i (c) by ci . Case 1. The point c is periodic. Then clearly for any g ∈ B(f, ε, U, c) with sufficiently small ε and U the point c will not be periodic of the same period. Therefore the itineraries of c for f and g will be different. Remember that this case is possible, since we allow non-zero one-sided derivatives at the turning points. Case 2. The point c is preperiodic (but not periodic). Then ck is a periodic repelling point for some k > 0. Let us work with the halves of ck and their neighborhoods (i.e. one-sided neighborhoods in the usual sense). Then there are the following two possibilities for the behavior of f . Case 2a. There are no inverse images of turning points in some closed one-sided neighborhood of ck . Then there is an interval (ck ; a) invariant for some iterate of f , and a is a periodic sink from the appropriate side. For sufficiently small ε and U and for any g ∈ B(f, ε, U, c) with g(c) on the appropriate side of c1 the g-trajectory of c will be attracted by the orbit of a which remains a periodic sink from the appropriate side for g. Case 2b. There are inverse images of at least one turning point of f in an arbitrary small one-sided neighborhood of ck . We can choose one of these inverse images sufficiently close to ck , and then we can choose next inverse images by following the periodic orbit of ck backwards. In this way we get inverse images of a turning point arbitrarily close to c with an additional property that their trajectories before hitting a turning point miss some fixed (small) neighborhood U of c. Let us take a sufficiently small ε > 0 and g ∈ B(f, ε, U, c) with g(c) on the appropriate side of c1 . Then there is a / U for i < m. point a ∈ (g k (c); ck ) such that f m (a) is a turning point and f i (a) ∈ Suppose that the itineraries of c for f and g are the same. Then by induction we see that f i (a) ∈ (g k+i (c); ck+i ) for i ≤ m (even if g k+i (c) ∈ U , the induction step works). Since ck+m and g k+m (c) are on the opposite sides of the turning point f k (a), we get a contradiction.
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We have shown that if c is a periodic or preperiodic point then choosing a neighborhood U of c according to Cases 1, 2a or 2b we can guarantee that for every ε > 0 if g ∈ B(f, ε, U, c) and c is the only local extremum of g in U then the itineraries of c under the maps g and f are different, or c is attracted to an orbit of a periodic sink. Case 3. The point c is neither periodic nor preperiodic. Let us show that there are inverse images of turning points arbitrary close to c1 on both sides of c1 . Indeed, suppose that this is not the case. Then there is an interval J containing c1 such that all iterates of f are monotone on J. The discussion following the definition of a wanderval implies now that if ω(c) is not a periodic orbit then J is a wanderval. However, because of (2.1), c cannot be attracted to a periodic (even one-sided) sink. It cannot be also on the boundary of a basin of attraction of a periodic sink (because then it would be periodic or preperiodic). Piecewise-smoothness of f implies that these are the only two ways the set ω(c) can be a periodic orbit. Hence, ω(c) is not a periodic orbit and thus there exists a point a1 in a small neighborhood of c1 such that (c1 ; a1 ) is a wanderval. We denote f i−1 (a1 ) by ai . We have n Y |cn+1 − an+1 | = r(ck , ak ). |c1 − a1 | |(f n )0 (c1 )| k=1
The numbers r(ck , ak ) are usually greater than or equal to η. There are several cases when they may be smaller than η, and then they are larger than or equal to K. The first case is when ck ∈ V ∪W . This can happen at most p times during each of N consecutive steps. The second case is when |ck − ak | ≥ δ. However, the images of a wanderval are pairwise disjoint, so this can happen at most 1/δ times no matter how big n is. The third case would be when ck and ak lie on the opposite sides of a periodic singular point. Then this periodic point would belong to a wanderval, so this case is impossible. Therefore (we use (2.4)) lim inf n→∞
|cn+1 − an+1 | |c1 − a1 | |(f n )0 (c1 )|
1/n
≥ (η N −p K p )1/N >
µ . λ
Together with (2.1) this proves that |cn+1 − an+1 | grows exponentially (at least as a constant times µn ), a contradiction. Therefore there are inverse images of turning points of f arbitrary close to c1 on both sides of c1 . Clearly the same holds for the point c. If the orbit of such an inverse image x comes closer to c (if coming from the other side, we look at the first image to decide whether it is closer), we replace x by this point of the orbit of x. In such a way we see that there are arbitrarily small neighborhoods (a, b) of c such that f (a) = f (b), some image of a is a turning point d of f , and the orbit of a before getting to d does not pass through (a, b). We will refer to such neighborhoods as very nice. By (2.4), we have p/N ηλ K > 1. µ η Therefore we can choose an integer M ≥ N so large that ηλ µ
K η
p/N !M ≥2
η 2+1/δ . K
(2.5)
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Next we choose a very nice neighborhood U of c, such that f k (U ) ∩ U = ∅ for k = 1, 2, . . . , M − 1. Moreover we choose U so that orbits of periodic singular points are disjoint from U . Take some ε > 0 and g ∈ B(f, ε, U, c). Recall that due to the arguments from the beginning of the proof we may assume that c is the only local extremum of g in U . Set bn = gn (c), αn = |(f n )0 (c1 )| and γn =
|cn+1 − bn+1 | . εαn
Suppose that the itineraries of c for f and g are the same. Then cn+1 and bn+1 are in the same lap of f (the laps of g are the same as for f ) and if a is an endpoint of U and x ∈ U then f (x) and g(x) lie on the same side of f (a) (otherwise g would have extra turning points). We claim that for a given n > 0 either both bn and cn are in U or both are outside U . First observe that the g-orbit of a and the f -orbit of a are the same. Let us now show by induction that if an endpoint a of U belongs to [cn ; bn ], then f i (a) = g i (a) ∈ [cn+i ; bn+i ] for i = 0, 1, . . . , m, where m is the smallest integer such that f m (a) is a turning point. Indeed, if f i (a) = g i (a) ∈ [cn+i ; bn+i ] then applying f and g to this we get f i+1 (a) = g i+1 (a) ∈ f [cn+i ; bn+i ] ∩ g[cn+i ; bn+i ]. Since f and g are either both increasing or both decreasing on [cn+i ; bn+i ], we get f i+1 (a) = g i+1 (a) ∈ [cn+i+1 ; bn+i+1 ], which completes the induction step. Thus, f m (a) = g m (a) ∈ [cn+m ; bn+m ]. This is a contradiction, since by our assumptions there is no turning point in [cn+m ; bn+m ]. This proves the claim. Consider again two cases depending on the behavior of the trajectory of c. Case 3a. The point c is non-recurrent. Choose U so small that no cn is in U for n > 0. / U for Let us show that this U satisfies the requirements of the theorem. Indeed, cn ∈ / U for all n. Hence, the f -orbit and the g-orbit of b1 are the same. all n, thus bn ∈ Therefore if the itineraries of c1 and b1 for f and g respectively are the same then there are no inverse images of turning points of f in a one-sided neighborhood [c1 ; b1 ) of c1 , a contradiction. Case 3b. The point c is recurrent. Let U be the set chosen after formula (2.5). We are going to look at how γn behaves when n grows. More precisely, we call n special if cn ∈ U (this includes n = 0) and prove by induction that if n is special then γn ≥ (µ/λ)n . This is clearly true for n = 0. Now we show how to make an induction step. Note that the situation is very similar to what we encountered when we were proving that c1 is not an endpoint of a wanderval. Assume that n is special and n + m is the next special number. By the definition of U we have m ≥ M . Our goal is to estimate γn+m from below; to this end we estimate from below quotients γn+1+k /γn+k for k = 0, 1, 2, . . . , m − 2. We have γn+1+k /γn+k =
|cn+2+k − bn+2+k | εαn+k , · εαn+1+k |cn+1+k − bn+1+k |
on the other hand we have bn+1+k = f k (bn+1 ), and thus bn+2+j = f (bn+1+j ), so γn+1+k /γn+k =
|f (cn+1+k ) − f (bn+1+k )| = r(cn+1+k , bn+1+k ) . |cn+1+k − bn+1+k ||f 0 (cn+1+k )|
Thus we can estimate γn+k+1 /γn+k from below by η or K. As before, we normally use η, but we have to use K in some cases. The first case is when cn+1+k ∈ V ∪ W . This can happen at most p times during each of N consecutive steps. The second case is
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when |cn+1+k − bn+1+k | ≥ δ. This can happen at most 1/δ times. The reason is that the intervals (cj ; bj ) for j = n + 1, . . . , n + m − 1 are pairwise disjoint. Indeed, suppose that they are not. Then one of them intersects the interval (cn+m ; bn+m ) (because the images of non-disjoint intervals are non-disjoint). The latter interval is contained in U , while the one intersecting it is disjoint from U , a contradiction. The third case would be when cn+1+k and bn+1+k lie on the opposite sides of a periodic singular point. We claim that this is impossible. Call this periodic point x. By the choice of U the trajectory of x is disjoint from U . On the other hand, we assumed that the itineraries of c1 and b1 for f and g respectively coincide. This together with the fact that f and g coincide outside U implies that f m−k−1 (x) ∈ [cn+m ; bn+m ] ⊂ U , a contradiction. Hence, the estimate of γn+1+k /γn+k from below by K will be used at most 1+ mp/N times when the first case occurs, and at most 1/δ times when the second case occurs. For γn+m /γn+m−1 we have to modify slightly the estimate, since perhaps bn+m+1 is different from f (bn+m ). Since |bn+m+1 − f (bn+m )| ≤ ε, we get γn+m ≥
|f (cn+m ) − f (bn+m )| − ε 1 ≥ γn η m−j K j − , εαn+m−1 |f 0 (cn+m )| αn+m
where j = mp/N + 1/δ + 2. Using (2.5) and (2.1), and since m ≥ M , we get µ m 1 − . γn+m ≥ γn · 2 λ αλn+m Since µ n+m µ n µ m 1 ·2 − ≥ λ λ αλn+m λ (because of (2.3) and since n+m ≥ M ≥ N ), we get γn+m ≥ (µ/λ)n+m . This completes the induction step. For every special n we get |cn+1 − bn+1 | ≥ γn εαn ≥ εαµn , a contradiction since µ > 1. This completes the proof.
Let us make an observation that leads to a result which seems to be of some interest by itself. Namely, in the beginning of the whole proof and in the beginning of Case 3 we did not use the assumption that the point c is a turning point. We used only the fact that the lower Lyapunov exponent at f (c) was positive. We concluded that this point did not belong to a wanderval. Hence, we get the following result. Proposition 2.1. Let f be a piecewise smooth map with non-zero shrinkability. Then at every point contained in a wanderval the lower Lyapunov exponent is non-positive. Some time ago one of the authors (AB) got interested in the question of existence of wandervals under the assumptions less restrictive than those from [MMS]. In particular it would be nice to find out whether a piecewise smooth map with some singular points at which the order of degeneracy is different to the left and to the right may have wandervals (in the unimodal case this question is quoted in [MS], where it is noted that the proof from [MMS] breaks down under new milder assumptions). We would like to point out that Proposition 2.1 answers the question in a rather specific situation. It also shows that it may make sense to consider a question of whether points with specific properties may be contained in a wanderval (one could call this a “pointwise” approach to the problem of existence of wandervals). Note also that we only need C 1 -smoothness while a higher smoothness is necessary for all previous results (see, e.g., [MMS]).
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3. Discussion of Assumptions We made three assumptions in Main Theorem: piecewise smoothness, non-zero shrinkability and the Collet-Eckmann condition at a turning point c. Now we will discuss them. Our definition of piecewise smoothness is as unrestrictive as possible. In particular, it does not prevent existence of wandervals, even ones that do not come close to the turning points. An example of such a map can be easily derived from Denjoy’s example of a circle diffeomorphism with wandervals (see [CN]). We compensate by assuming that a turning point c satisfies the Collet-Eckmann condition. Then the trajectory of this point has nothing to do with wandervals (see Proposition 2.1). The assumption on non-zero shrinkability is in fact a condition on the behavior of the map close to singular points. Let a be a singular half-point. We will say that f is non-flat at a if either f 0 (a) 6= 0 or there exists a neighborhood W (one-sided) of a and a constant L > 0 such that for every t ∈ W \ {a}, L≤
|f (t) − f (a)| ≤ 1. |t − a| |f 0 (t)|
(3.1)
Lemma 3.1. If a is the only singular point in W = [a; b) and (3.1) holds for all t ∈ (a; b), then r(x, y) ≥ L for every x, y ∈ W . Proof. The sign of f 0 on W is constant; we may assume it is positive. We may also assume that a is the left endpoint of W . The proof in the remaining three cases is similar. We have by (3.1), 0 f 0 (t)(t − a) − (f (t) − f (a)) f (t) − f (a) = ≥ 0, t−a (t − a)2 so the function (f (t) − f (a))/(t − a) is non-decreasing. Hence, if a < x < y < b, then (f (y) − f (a))/(y − a) ≥ (f (x) − f (a))/(x − a). Therefore f (y) − f (a) f (x) − f (a) x − a f (y) − f (x) f (y) − f (a) − = − ≥ 0, y−x y−a y−a x−a y−x so we get
f (y) − f (a) f (x) − f (a) f (y) − f (x) ≥ ≥ . y−x y−a x−a
Hence, by (3.1) we get r(x, y) ≥ L and r(y, x) ≥ L. This completes the proof.
Corollary 3.2. If f is smooth on [a, b], non-flat at a and b, and f 0 is non-zero on (a, b) then f has non-zero shrinkability on [a, b]. To justify our use of the term “non-flat”, we prove the following lemma. Lemma 3.3. Let l > 1 be an integer and let f be a function of class C l−1 defined in a (one-sided) neighborhood of a point a. Assume that f (i) (a) = 0 for i = 1, . . . , l − 1, and that f (l) (a) exists and is non-zero. Then f is non-flat at a. In particular, if a is a non-degenerate critical point of f (that is, f 0 (a) = 0 but f 00 (a) 6= 0) then f is non-flat at a.
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Proof. By the definition of the derivative, we have f (l−1) (t) = f (l) (a). t→a t − a lim
Thus, by l’Hˆopital’s rule, f (l) (a) f 0 (t) = t→a (t − a)l−1 (l − 1)! lim
Therefore lim
t→a
and
f (l) (a) f (t) . = t→a (t − a)l l! lim
1 f (t) = , (t − a)f 0 (t) l
so (3.1) holds in a neighborhood of a with L = 1/(2l). Thus, f is non-flat at a.
On the other hand, assume that (3.1) holds. If t > a and f 0 (t) > 0 (the other three cases are similar) we get from the first inequality of (3.1), [ln(f (t) − f (a))]0 ≤
1 . L(t − a)
Integrating from t (close to a) to some b > a we get ln(f (b) − f (a)) − ln(f (t) − f (a)) ≤ Therefore
1 [ln(b − a) − ln(t − a)]. L
f (t) − f (a) ≥ const.(t − a)1/L .
This means that f really cannot be flat (in the common sense) at a. The assumption on non-zero shrinkability restricts severely possible behaviors near singular points that are not turning points. It is easy to check that if the singularities from both sides of such a point have the same order, shrinkability stays positive. However, if they are of different orders, shrinkability is zero. Thus, our assumptions on the behavior on both sides of a singular point are quite different than the assumptions in [MS]. We don’t care if the orders of the singularities are different if the point is a turning point, but it is important that they are the same if the point is not a turning point. In [MS] this is just the opposite. The third assumption we are making is the Collet-Eckmann condition at one of the turning points. One can ask whether we can replace it by a subexponential expansion. We cannot do it when using our techniques. The trajectory of the turning point may be coming back to U with some fixed frequency, and we have to guarantee sufficient expansion between each two consecutive returns. 4. Corollaries Main Theorem is stated in a rather technical way. However, it has important consequences, that can be stated in more general terms. In order to do it, we need a couple of lemmas. They are very simple, so we state them without proof. Lemma 4.1. If g satisfies one of the conditions from the conclusion of Main Theorem then it is not essentially conjugate to f .
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Lemma 4.2. Let f be a piecewise smooth map and let U be a neighborhood of a turning point c of f . Let h : [0, 1] → [0, 1] be a function of class C ∞ that is 0 outside U and 1 in some neighborhood of c. Then the map εh tends to 0 in the C ∞ topology as ε → 0. Moreover, f + h ∈ B(f, ε, U, c) if (f + h)([0, 1]) ⊂ [0, 1]. Thus, we can always make perturbations of the type described in the Main Theorem. Now, our corollaries are the following. Corollary 4.3. Let f be a piecewise smooth map with non-zero shrinkability, satisfying Collet-Eckmann condition at some turning point. Then f is not C r -structurally stable for any r ≤ ∞. Corollary 4.4. Let f be a piecewise smooth map with non-zero shrinkability, satisfying the Collet-Eckmann condition at a turning point c. Then f is C r -structurally sensitive at c for every r ≤ ∞. Readers for which the term “non-zero shrinkability” is too special, can use the results of the previous section and change the assumptions in the above corollaries from “a piecewise smooth map with non-zero shrinkability” to “a smooth map with non-degenerate critical points”. For unimodal maps the situation is somehow simpler. If a critical point c has different itineraries for f and g, then there is t ∈ (0, 1) such that c is periodic for tf + (1 − t)g. Therefore, in view of Lemma 3.3, we can state another corollary as follows. Corollary 4.5. For any 1 ≤ r ≤ ∞, in the space of C r unimodal interval maps with nondegenerate critical points, the set of Collet-Eckmann maps is nowhere dense and every such map can be approximated by maps with the critical point belonging to a superattracting periodic orbit. References [J]
Jakobson, M.V.: Smooth mappings of the circle into itself. Mat. Sb. (N.S.) 85 (127), 163–188 (1971) (Russian; English translation: Math. USSR - Sbornik 14, 161–185 (1971) [MMS] Martens, M., de Melo, W. and van Strien, S.: Julia-Fatou-Sullivan theory for real one-dimensional dynamics. Acta Math. 168, 273–318 (1992) [MS] de Melo, W. and van Strien, S.: One-Dimensional Dynamics Berlin: Springer Verlag, 1983 [CN] Coven, E. and Nitecki, Z.: Nonwandering sets of the powers of maps of the interval. Ergod. Th. & Dynam. Sys. 1, 9–31 (1981)
Communicated by Ya. G. Sinai
Commun. Math. Phys. 191, 71 – 86 (1998)
Communications in
Mathematical Physics c Springer-Verlag 1998
The Statistics of Burgers Turbulence Initialized with Fractional Brownian Noise Data Reade Ryan Courant Institute of Mathematical Sciences, New York University, New York, N.Y. 10012, USA. E-mail:
[email protected] Received: 27 January 1997 / Accepted: 30 April 1997
Abstract: The statistics of the solution to the inviscid Burgers equation are investigated when the initial velocity potential is fractional Brownian motion. Using the theory of large deviations for Gaussian processes, we characterize the tails of the probability distribution functions (PDFs) of the velocity, the distance between shocks, and the shock strength. These PDFs are shown to decay like “stretched” exponentials of the form log P (θ) ∝ −θ4−2h . Our method of proof can also be used to extend these results to a much larger class of Gaussian potentials. This work generalizes the results of Avellaneda and E [2, 3] on the inviscid Burgers equation with white-noise initial data. 1. Introduction The one-dimensional Burgers turbulence (BT) system, described by the nonlinear wave equation (1.1) ∂t u(x, t) + u(x, t)∂x u(x, t) = µ∂xx u(x, t), µ ≥ 0, initialized at time t = 0 with random data uo (x), has been discussed in various fields in mathematics and physics. The structural similarity between the Burgers equation and the Navier-Stokes equation was an initial motivation behind the study of Eq. 1.1. While it is now clear that the differences between Burgers and Navier-Stokes turbulence are at least as great as the similarities, the study of Burgers systems is still employed as a testing ground for analytical approaches to Navier-Stokes turbulence [5, 8]. The Burgers equation has also been used in the study of shock wave formation in compressible fluids [6], nonlinear waves in long transmission lines, and the formation of large-scale mass clustering in an expanding universe [12, 15]. Researchers in the latter area of study have been interested in the statistics of the solution to the one-dimensional Burgers equation initialized with two general types of Gaussian data. The first type of initial data is a family of Gaussian processes called fractional Brownian motions (FBMs), denoted here as Bh (·), h ∈ (0, 1). The FBMs
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are a one-parameter family of continuous, self-similar processes whose increments, Bh (x) − Bh (y), are mean-zero and have the following second-moment power law: E (Bh (x) − Bh (y))2 = |x − y|2h ,
(1.2)
where E[·] denotes expectation.1 R y The second case of interest specifies that the integral of the initial data φo (y), = uo (x)dx, often referred to as the initial potential, is a fractional Brownian motion. 0 Thus, the initial data uo (x) is a generalized Gaussian process, called fractional Brownian noise. Burgers, himself, worked on the case when h = 1/2 (i.e. white noise initial data) [4]. Burgers [4] showed that the solution to the inviscid Burgers equation with random data resolves itself at time t into a “sawtooth” profile with areas of rarefaction, in which u(·, t) ramps up with a slope of 1/t, followed by downward shocks of random size.2 Thus, in order to obtain a complete understanding of the solution statistics for a BT system, one needs to investigate the PDFs of the velocity u(x, t), the strength of a generic shock, and the rarefaction interval size. Recently, work has been done on the Markovian BT cases, i.e. Brownian motion initial data (h = 1/2 for the first type) and white noise data (h = 1/2 for the second type). Sinai, in his 1992 paper [14], looked at the statistical structure of shocks for inviscid BT with Brownian motion data. By looking at the PDF of the shock strength, he proved that at any fixed time t the points at which shocks occur are dense in Eulerian space and the corresponding shock points in Lagrangian space form a random set of Hausdorff dimension 21 almost surely. Avellaneda and E, on the other hand, did an indepth study of the white-noise case [2, 3]. They investigated the shock structure in this BT system, showing that the shock points form a discrete set almost surely. In addition, they looked at the probability distribution functions of the velocity at a point, the size of a shock, and the size of a rarefaction interval. They proved that the tails of these PDFs all decay like cubic exponentials, i.e. log P (θ) ∝ −tr θ3 , where r = 1 in the former cases and r = −2 in the latter. However, very little rigorous work has been done on the non-Markovian cases3 . In their 1994 paper [15], Vergassola et al., based on numerical and heuristic anaylsis, hypothesized that for Burgers systems, initialized with the second data type, the PDF for the strength of a generic shock at time t, denoted |S(t)|, has the following decay law for s 1: log Prob {|S(t)| > s} ∝ −t2−2h s4−2h .
(1.3)
This law agrees with the work of Avellaneda and E on the white-noise case (h = 1/2). With a few technicalities that shall be discussed later, we shall rigorously prove that the above equation holds ∀h ∈ (0, 1) by finding upper and lower bounds on the probability of a generic shock being very large. We shall also find upper and lower bounds on the tail events for the velocity and an upper bound on the tail of the wavelength size distribution, 1 The parameter h is often called the Hurst exponent, due to Hurst’s work on these processes in the field of hydrology. 2 In general, rarefaction intervals and the intervals between shocks are not equivalent for inviscid Burgers systems. However, for the BT systems we shall be studying, these two entities coincide. Rarefaction intervals will be precisely defined later and shown to be equivalent to the intervals between shocks for the cases under consideration. 3 Handa’s work [7] on the Hausdorff dimension of the shock points in BT systems with the first type of Gaussian data is a notable exception.
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both of which give similar decay laws for the PDFs of these random variables. As of yet, we have not realized a solid lower bound for the wavelength size probabilities. The reason for this has to do with the particular shock structure of these models. This shall be looked at in some depth below. 2. Solution to the Inviscid Burgers Equation Before stating and proving the results of this paper, let us begin by giving a brief description of the solution to the inviscid Burgers equation. In this section, we shall give precise definitions of the velocity u(x, t), the rarefaction intervals, and the velocity shocks in terms of the initial potential φo (y). Given that φo (y) is continuous and that lim|y|→∞ φo (y)/y 2 = 0, it is a standard exercise to show that there exists a unique left-continuous, vanishing-viscosity solution to the inviscid Burgers equation with initial data uo (x) (proof given in [14, Sinai] among others). Note that fractional Brownian motion satisfies the above two conditions. Thus, asking what the solution to the inviscid Burgers equation initialized by fractional Brownian noise, a generalized random process, is a well-posed question. At a fixed time t, this solution is given by u(x, t) =
x − Y (x, t) , t
(2.1)
where
1 x 1 x Y (x, t) = min y : φo (y) + y 2 − y = min φo (z) + z 2 − z . z 2t t 2t t
(2.2)
Given the assumptions on φo (y), the above minimum, taken over z ∈ R, is well-defined and finite ∀x ∈ R, and thus, Y (x, t) is well-defined and finite ∀x ∈ R. It is easy to see that at points x, for which there exists a unique minimum point y ∗ for the function 1 2 y − xt y, Y (x, t) is continuous and non-decreasing in x and is equal to y ∗ . φo (y) + 2t When there exists more than one global minimum point for some x0 (with t fixed), then Y (x, t) is discontinuous at x0 with lim Y (x, t) = Y (x0 , t) = y1 ,
x↑x0
lim Y (x, t) = y2 ,
x↓x0
where y1 is the smallest minimum point and (2.3) where y2 is the largest minimum point.
(2.4)
Thus, u(x, t) has a shock at x0 with the size of the discontinuity = (y2 −y1 )/t. Looking at the definition of Y (x, t), one sees that if a shock exists at x0 in Eulerian space (x-space), then there corresponds in Lagrangian space (y-space) an interval [y1 , y2 ] for which the 0 1 2 function 8(y; t) = φo (y) + 2t y is greater than or equal to the line xt y + C, for some 0 C, with equality holding at the end points and is strictly greater than xt y + C outside this interval. The interval [y1 , y2 ] will be referred to as a Lagrangian shock interval. We shall make much use of this Eulerian/Lagrangian correspondence in what follows. Using a similar Lagrangian-space construction, we can also give a precise definition for rarefaction intervals and, for the systems under consideration, show their equivalence with the intervals between shocks. We define a rarefaction interval as an interval (x1 , x2 ) such that ∀x ∈ (x1 , x2 ), Y (x, t) = yr , for some yr . Such an interval is considered “rarefied” because no Lagrangian points are mapped into (x1 , x2 ) by the function Y −1 (·, t), which is often called the Lagrangian map for obvious reasons.
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As we mentioned before, an interval between two shocks in Eulerian space is not necessarily a rarefaction interval. If, however, such an interval I is not rarefied, there must exist at least one subinterval J ⊆ I for which Y (x, t) is continuous and monotonically increasing in x. However, from geometric considerations, one sees that, lim
y↑Y (x)
x 8(Y (x); t) − 8(y; t) ≤ ≤ Y (x) − y t
lim
y↓Y (x)
8(y; t) − 8(Y (x); t) . y − Y (x)
Therefore, if J exists, 8(y; t) must be continuously differentiable in y for the interval 1 2 y , Y −1 (J, t). However, for the initial data under consideration, 8(y; t) = Bh (y) + 2t and so is nowhere differentiable. In this case, any interval between shocks must be a rarefaction interval.
3. Theorems and Proofs With the above definitions of the velocity at a point x, the strength of a generic shock and the distance between two shocks, we are now ready to state and prove our results. Ry Theorem 1. Given that the initial potential in the inviscid Burgers equation 0 u0 (y 0 )dy 0 = Bh (y), where Bh (·) is a fractional Brownian motion with parameter h, the following holds true for the PDF of the velocity, u(x, t): (3.1) exp −Cv 4−2h t2−2h ≤ Prob {u(x, t) > v} ≤ exp −C 0 v 4−2h t2−2h , where v 1, and C, C 0 > 0 are dependent on the paramenter h. Theorem 2. Let δx(t) be the size of a rarefaction interval at time t, which contains some specified point x. Then for some C > 0, which depends on the Hurst exponent h of the initial potential, and with L 1, (3.2) Prob {δx(t) > L} ≤ exp −CL4−2h t−2 . Our last theorem in this section concerns the distribution of SE , the strength of an Eulerian shock. As with the rarefaction interval size, it will not be hard to find the set which dominates the probability of a shock being large. It is the lower bound that shall be a problem. The crux of the dilemma is that we do not know what the structure of the Eulerian shocks is. Are there a finite number of shocks almost surely in any finite Eulerian interval? Are there accumulations of shocks around certain points? Or are there an infinite number of shocks in any interval? Avellaneda and E in their 1995 paper [3] showed that with white-noise initial data, the solution to the inviscid Burgers equation has a discrete shock structure, meaning that at any time t there is only a finite number of shocks in any bounded Eulerian interval with probability one. Sinai [14] proved, however, that with Brownian-motion initial data, there is an infinite number of shocks in any interval on the x-axis. There may be intervals such that the sum of all the jump lengths is very small, but none with no jumps. In the Sinai case, the probability of the jump of any generic Eulerian shock being greater than zero is zero. But this is a misnomer, since there really is no Eulerian probability distribution to speak of. Without a better understanding of the shock structure, it is impossible to say whether or not there exist any rarefaction intervals, let alone ones larger than some number L.
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For Burgers turbulence initialized with a fractional Brownian motion potential, it is generally believed that the Eulerian shock structure is discrete. Indeed, numerical simulations have given qualitative confirmation of this hypothesis. However, a rigorous proof has yet to be found due to the difficulty in analyzing the almost sure properties of these non-Markovian processes. Getting back to our problem, we see that the best one can do is to look at the distribution of Eulerian shock size, |SE |, conditioned on |SE | > , for some > 0. This distribution exists no matter what the structure of shocks is. Furthermore, with a lower bound on this conditional distribution, proving that the system has discrete shocks would be enough to extend this lower bound to the unconditioned distribution of |SE |. Therefore, our theorem is the following: Theorem 3. Let |SE | be the strength of an Eulerian shock at time t. Then ∃ C, C 0 > 0 such that Prob {|SE | > s} ≤ Prob |SE | > s |SE | > ≤ exp −C 0 s4−2h t2−2h , (3.3) Prob |SE | > s |SE | > ≥ exp −Cs4−2h t2−2h . (3.4)
The basic analytical tool we will use to prove our results is the theory of large deviations for Gaussian processes. In estimating the tails of the PDFs for the velocity, the shock strength, and the distance between shocks, we need to look at the probability of certain rare events for the initial potentials. The decay rates for these PDFs are determined by the probability of the most likely fractional Brownian motion path in each event. Our arguments boil down to estimating for above and below the action of the most likely path in each case. For the upper bounds on the tails (i.e. the lower bounds on the action), we employ the theory of extreme values for Gaussian processes, making use of Adler’s work on this subject [1]. To find the lower bounds, we use the spectral representations of the initial potentials (given below) in conjunction with the Cameron-Martin-Girsanov formula for Brownian motion. The spectral representation of a real Gaussian process G(y) with stationary increments is given by the formula Z∞
Z∞ (1 − cos(yk)) (E [k])
G(y) − G(0) =
1/2
−∞
sin(yk) (E [k])1/2 dZ2 (k),
dZ1 (k) + −∞
(3.5) where the function E [k] is the energy spectrum of G(y) and Z1 (·) and Z2 (·) are independent Brownian motions on R. For stationary processes, the energy spectrum is defined as the Fourier transform of the correlation function. Here, we define the energy spectrum of a real stochastic process with stationary increments in a similar fashion, as the function which satisfies the equation
E (G(y) − G(0))
2
Z∞ = 2
(1 − cos(2yk))E [k] dk.
(3.6)
−∞
For a fractional Brownian motion Bh (y) with Hurst exponent h, it is trivial to show that
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E [k] = Ch |k|
−1−2h
, where
Ch−2
Z∞ = 2
(1 − cos(2k))|k|−1−2h dk.
(3.7)
−∞
Combining Eqs. 3.5 and 3.7, we obtain the spectral representation of FBM. It is evident that the determining factor for the decay rate of large-deviation events for FBMs is the behavior of their energy functions in the infra-red part of the spectrum. Thus, the tail probabilities for BT with FBM initial potentials are determined by the exponent −1 − 2h in the power law in this part of the energy spectrum. Due to this fact, Theorems 1, 2 and 3 hold for a broader range of Burgers systems than those initialized with fractional Brownian noise data. With relative ease, we shall extend the proofs of our theorems to include BT systems whose initial velocity potentials have the same infrared behavior in their energy spectra as the FBMs. More specifically, Theorems 1, 2 and 3 hold for stationary-increment Gaussian processes, having energy spectra E [k] which satisfy the following: • E [k] = Ch |k|−1−2h , • ∃A > 0 : e
−Ak2
for k ∈ (−1, 1)
≤ E [k] ≤ Ch |k|
−1−2h
(3.8) ,
for k ∈ / (−1, 1).
3.1. Proof of Theorem 1: Velocity PDF bounds. By the stationarity of the initial data and the translation invariance of the Burgers equation, {u(x, t), x ∈ R} is a stationary process. Thus, we can focus solely on the tail distribution of u(0, t) without loss of generality. 1 2 y first 3.1.1. Proof of velocity upper bound. Let y ∗ ≡ the value at which Bh (y) + 2t ∗ achieves its minimum on R. Since u(0, t) = −y /t, when Bh (y) is the initial potential, and noting that u(0, t) = −u(0, t) in law, we only need to find an upper bound on the 1 2 y = 0 at y = 0, one has that meausre of {y ∗ ≥ vt}. Since Bh (y) + 2t 1 2 1 2 ∗ (3.9) {y ≥ vt} = min Bh (y) + y ≤ min Bh (y) + y y
Using the following result, due to Adler [1], we shall bound the measure of the right-hand side of the above equation by the desired probability. Lemma 3.1 (Adler). Let G(·) be a continuous mean-zero Gaussian process on R. Define a metric on R, p(x, y) = E[G(x) − G(y)]2 . Given some bounded set I ∈ R, we define NI () = the number of -balls needed to cover the set I under the metric p(·, ·). If ∀ > 0, NI () ≤ K−α for some α, K > 0, then with σI2 = supx∈I E[G(x)2 ], 1 2 2 η (3.11) P sup G(x) ≥ m ≤ C(α, K)m exp − m /σI , 2 x∈I where η is any real number > α. This is not the best possible result for the supremum of a Gaussian process, but it is more than sufficient for our purposes. Before we can apply this lemma, there are two problems that must be overcome. The first is the fact that the boundary in the event we are interested in is not a level set. This is rectified by looking at the process Z(y) = y −2 Bh (y)
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for y ≥ vt. We can now look at the equivalent event, {miny≥vt Z(y) ≤ −1/2t}. The second difficulty is the unboundedness of the interval (vt, ∞) over which the infimum is being taken. To circumvent this technicality we note that Prob
min Z(y) ≤ −
y∈[vt,∞)
1 2t
∞ X
≤
n=0
Prob
min
y∈[vt+n,vt+n+1)
Z(y) ≤ −
1 2t
. (3.12)
In order to apply Lemma 3.1, we need to show that there ∃K and α > 0 such that for every interval In = [vt+n, vt+n+1), NIn () ≤ K−α . Setting g(y, x) = E[Bh (y)−Bh (x)]2 , we have E[Z(y) − Z(x)]2 ( " # ) 2 y 2 x = − 1 g(0, x) + − 1 g(0, y) + g(x, y) /x2 y 2 . x y Assuming that y ≥ x and setting δ = (y/x)2 − 1, 2 δ 2 g(0, y) + g(x, y) /x2 y 2 E[Z(y) − Z(x)] ≤ 1+δ δ2 y 2h−2 + g(x, y)/x2 y 2 x2 (1 + δ) (y − x)2 ≤ 4 6 2−2h + g(x, y)/x2 y 2 . x y ≤
(3.13)
(3.14) (3.15) (3.16)
Thus, ∀x, y > 1 the metric p(x, y) defined by the second moment of Z(x) − Z(y) is dominated by 2 max{4(y − x)2 , g(x, y)}. Since g(x, y) = |x − y|2h , it is clear that ∀n ≥ 0, NIn () ≤ K−1/2h , for some K > 0. With E[Z(y)2 ] = y 2h−4 , Lemma 3.1 yields P
1 min Bh (y) + y 2 ≤ 0 y>vt 2t
−2 ∞ X t C 4−2h (vt + n) ≤ exp − (2t)η 8 n=0 1 4−2h 2−2h C v exp − t . ≤ (2t)η 8
(3.17) (3.18)
This, combined with Eq. 3.10, establishes the upper bound part of Theorem 1. In order to extend this result to any Gaussian process, G(·), with an energy spectrum satisfying Eq. 3.8, one only needs to note that with Z(y) now defined as y −2 G(y), and g(x, y) ≡ E[G(x) − G(y)]2 , the above proof goes through in an identical fashion. 3.1.2. Proof of velocity lower bound. Let us define a set E of FBM paths as follows: E = {Bh (·) : |Bh (y) + φ(y)| < 1, ∀y ∈ (−vt, vt + 1)}, 0
where φ(y) =
for y < 0 + α y 2 /t for y ∈ [0, vt + 1] φ(vt + 1) for y > vt + 1 , 1 2
(3.19)
(3.20)
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1 and α = vt . To establish the velocity PDF lower bound, we need to prove two things: one, that the set E ⊂ {|u(0, t)| > v}, and, two, that (3.21) P (E) ≥ exp −Cv 4−2h t2−2h , for some C > 0. 1 2 y > −α(vt)2 − 1, To prove the first, we note that for every path in E, Bh (y) + 2t 1 2 ∀y ∈ (−vt, vt). On the other hand, Bh (vt + 1) + 2t (vt + 1) < −α(vt + 1)2 + 1, which is 1 equal to −α(vt)2 − 1 − vt . Therefore, y ∗ must be outside (−vt, vt), and |u(0, t)| > v.4 To establish Eq. 3.21, we recall the spectral representation of fractional Brownian motion given by Eqs. 3.5 and 3.7, and write Z∞ (cos(yk) − 1)k −h−1/2 d(Z1 (k) + gc (k)) (3.22) Bh (y) + φ(y) = Ch −∞ Z∞
+ Ch
sin(yk)k −h−1/2 d(Z2 (k) + gs (k)),
−∞
where Z1 (·) and Z2 (·) are standard independent Brownian motions, and gc (k) and gs (k) are given by 1 −1 sin[(vt + 1)k] 2t + α (vt + 1) cos[(vt + 1)k] − k g˙c (k) = , (3.23) πCh |k|3/2−h and 1 −1 (1 − cos[(vt + 1)k]) 2t + α (vt + 1) sin[(vt + 1)k] − k . (3.24) g˙s (k) = πCh |k|3/2−h It is a simple exercise in Fourier analysis to show that gc (k) and gs (k) satisfy Eq. (3.22). Setting W1 (k) = Z1 (k) + gc (k), and W2 (k) = Z2 (k) + gs (k), we employ the CameronMartin formula to obtain a measure where W1 (·) and W2 (·) are standard independent eh (y) ≡ Bh (y) + φ(y) is now a Brownian motions. Under this measure, it is clear that B R mean-zero fractional Brownian motion on (−∞, ∞). Setting Z = g˙c (k)dW1 (k) + R R R g˙s (k)dW2 (k), and noting that [g˙c (k)]2 dk + [g˙s (k)]2 dk = Cφ t−2 (vt)4−2h + O(v 3−2h ), we obtain 1 P (E) = exp − Cφ v 4−2h t2−2h + O(v 3−2h ) (3.25) 2 " # e × E(0,0) exp {Z} ; sup |Bh (y)| < 1
y∈[−vt,vt+1]
1 ≥ exp − Cφ v 4−2h t2−2h − O(v 3−2h ) 2 × P(0,0)
|Z| < v 2−h t1−h ,
sup y∈[−vt,vt+1]
(3.26) ! eh (y)| < 1 |B
.
In order to obtain a sufficiently tight lower bound on the second term in line 3.26, we ˇ ak [13], and the need to make use of two technical lemmas. The first is due to Z. Sid´ second is due to Monrad and Rootzin [9]. They are as follows. 4
Note that symmetry implies that P (|u(0, t)| > v) = 2P (u(0, t) > v).
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ˇ ak). Given any continuous mean-zero Gaussian process X(y), y ∈ R Lemma 3.2 (Sid´ and a correlated mean-zero Gaussian random variable Z, then for any β, η > 0 and some Borel set I ⊂ R ! ! P
sup |X(y)| < β, |Z| < η
≥ P
y∈I
sup |X(y)| < β
· P (|Z| < η).
y∈I
Lemma 3.3 (Monrad and Rootzin). For a fractional Brownian motion Bh (y), with Bh (0) = 0, ! P
sup |Bh (y)| < 1
≥ exp {−C(h)t} .
y∈[0,t]
Applying these two lemmas to the second term in line 3.26, we get ! eh (y)| < 1 sup |B P(0,0) |Z| < v 2−h t1−h , y∈[−vt,vt+1]
≥ P |Z| < v
2−h 1−h
t
,
· P(0,0)
(3.27) !
sup y∈[−vt,vt+1]
eh (y)| < 1 |B
≥ exp {−Cvt} . The last inequality employs the fact Z ∼ N (0, C 0 v 4−2h t2−2h ), which implies that the probability of |Z| < v 2−h t1−h is order one. This completes the proof of Theorem 1. Of some interest is the exponential decay constant, 21 Cφ . The formula for this constant is ∞ 2 Z 1 Ch−2 cos(k) − k −1 sin(k) 1 Cφ = dk 2 8 π2 |k|3/2−h −∞ 2 Z∞ −1 sin(k) − k [1 − cos(k)] dk . + |k|3/2−h −∞
Except in the case where h = 1/2, this is not the best possible constant. With white-noise initial data, 21 Cφ = 1/6, which in a previous paper I have shown to be optimal [11]. We shall discuss further the upper and lower bound constants later in the paper. To extend this result to the more general Gaussian processes referred to earlier, we need only a few minor adjustments. The first problem that arises is the integrability of [g˙c (k)]2 and [g˙s (k)]2 , when the energy spectrum E [k] of our Gaussian process has a very rapid decay as k → ∞. Using the spectral representation of these processes, we see that cos(2vtk) sin(2vtk) Ch 2vt /(E [k])1/2 , − g˙c (k) = 2tπ k2 k3 sin(2vtk) 1 − cos(2vtk) Ch g˙s (k) = 2vt − /(E [k])1/2 . 2tπ k2 k3
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For rapidly decaying E [k], it is clear that these functions are not in L2 (R). Therefore, we must alter the set E slightly. The problem is that the curve φ is not smooth enough. So let us smooth it out by convolving it with a heat kernel, i.e. Z 2 1 e−(y−x) /2A φ(x)dx (3.28) φ∗ (y) = √ 2πA 1 for some A > 0. Setting α = 2t in the definition of φ(y) (Eq. 3.20), we redefine the set E = {Bh (·) : |Bh (y) + φ∗ (y)| < v, ∀y ∈ (−vt, 2vt)}. We see that everything goes through as before. For any stationary increment Gaussian process whose energy spectrum satisfies Eq. 3.8, φˆ∗ (k)/(E [k])1/2 is in L2 with its L2 -norm bounded from above by C 00 v 4−2h t2−2h . Geometry, again, implies that E ⊂ {|u(0, t)| > v} for large enough v. Repeating the steps of the FBM proof, one simply needs to show that P(0,0) [supy∈(−vt,2vt) |G(y)| < v] ≥ C > 0 for all v large enough. This inequality comes easily from a direct application of Lemma 3.1.
3.2. Proof of Theorem 2: Rarefaction interval tails. Without loss of generality, we look at the possibility of a large rarefaction interval surrounding the point x = 0. Following the proof of Avellaneda and E [3] for the white-noise case (h = 1/2), if there is such L , or an interval around x = 0 whose length is greater than L, then either u( L2 , t) ≥ 2t L L u(− 2 , t) ≤ − 2t . Therefore, L L L L + Prob u(− , t) ≤ − (3.29) Prob {δx(t) ≥ L} ≤ Prob u( , t) ≥ 2 2t 2 2t 1 L ≤ 2 exp − ( )4−2h t−2 , (3.30) 8 2 ∀L t. This concludes the proof of Proposition 4.2. We note that the result is also true for the more general Gaussian processes that we are interested in. 3.3. Proof of Theorem 3: Shock Strength tails. As was previously mentioned, we do not know the structure of the Eulerian shocks. Thus, we cannot say whether or not an Eulerian-shock probability distribution exists. We must instead look at the distribution of Eulerian shocks conditioned to be greater in size than some > 0. To get estimates on this distribution, we shall use the connection between the Lagrangian shock intervals and Eulerian shocks. Fixing time t, and setting SL (y) ≡ the Lagrangian shock interval which contains the point y, and {Sn }n∈Z ≡ the sequence of Eulerian shocks such that the size of each shock, denoted |Sn |, is > , one has P |SL (0)| ∈ (s, s + ds) |SL (0)| > t = E 1{|SL (0)|∈(s,s+ds)} |SL (0)| > t (3.31) 1 dy {|SL (y)|∈(s,s+ds)} −l = E lim l→∞ Rl 1{|SL (y)|> t} dy
Rl
−l
(3.32)
Statistics of Burgers Turbulence
81 N P
n=−N = E lim N →∞
s t
1{|Sn |∈[s/t,(s+ds)/t]} 2N + 1
2N + 1 N P |Sn |
(3.33)
n=−N N P
E[1{|Sn |∈[s/t,(s+ds)/t]} ] s n=−N lim = t E[|Sn |] N →∞ 2N + 1 s = P |SE | ∈ [s/t, (s + ds)/t] |SE | > . t E[|Sn |]
(3.34) (3.35)
The equality in line 3.32 uses the stationarity of the initial data and the fact that we are only looking at shocks whose size is greater than . In line 3.33, we ignore the end effects of the interval (−l, l), which are negligible as l and, thus, N → ∞. In line 3.34, we employ the stationarity and ergodicity of the sequence {Sn }n∈Z , which follow from the stationarity and ergodicity of the initial data (details given in my doctoral thesis [10]). Thus, P |SL (0)| > s
Z∞ 1 s0 0 |SL | > t = |SE | > ds0 . s P |SE | = t E[|Sn |] t s
(3.36) This implies that, for any K > 1, s t E[|Sn |] P |SL (0)| > s |SL | > t ≥ P |SE | ≥ |SE | > (3.37) s t t E[|Sn |] ≥ P |SL (0)| > s |SL | > t − P |SL (0)| > Ks |SL | > t . Ks This inequality, combined with the following proposition, which gives bounds on the Lagrangian shock interval PDF, establishes Theorem 3. Proposition 1. With initial potential being a fractional Brownian motion of parameter h, the PDF of the Lagrangian shock interval SL (y), around any point y, at a fixed time t, satisfies the following for s 1: exp −C(h)s4−2h t2−2h ≤ P (|SL (y)| > s) ≤ exp −C 0 (h)s4−2h t2−2h , (3.38) for some C(h), C 0 (h) > 0. 3.3.3. Proof of Propostion 1: Upper bound. Let SL be the Lagrangian shock interval which contains the point y = 0. Our method of proof will be to find a large set F , which is a subset of {SL > s}c . We will then obtain a lower bound on the probability of this set, thus, yielding an upper bound on {SL > s}. To this end, we make the following claim: Claim 3.1. Let F be the following set of fractional Brownian motion paths, with Bh (0) = 0 : l2 1 − 2 1 2 l2 1 + 2 + y ≤ Bh (y) + y ≤ + y , F = Bh (·) : − (3.39) 2t 2t 2t 2t 2t
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s(1−) where l = √ and is some constant in (0, 1). 4
2 (1+)
Then, F ⊂ {SL > s}c , given that Bh (·) is the initial potential for the inviscid Burgers equation. Furthermore, for some C() > 0, (3.40) P (F ) ≥ 1 − exp −C()s4−2h t−2 . With this claim established, the upper bound on Prob {SL > s} is immediate. Proof of Claim 3.1. Let us first show that F ⊂ {SL > s}c . To do this, we must show that for every path, Bh (·), in F , there can be no interval [y1 , y2 ] 3 y = 0, whose length 1 2 y ≥ βy + C, for any constants β, C, with is greater than s, and for which Bh (y) + 2t equality holding at the endpoints. By the convexity of the quadratic bounds on Bh (y), one sees that if such an interval exists then the line segment z = βy+C, y ∈ [y1 , y2 ] must also be contained in these bounds (see Fig. 1). However, it is an easy task to calculate that the largest interval surrounding the point y = 0, for which √ any such line segment 4
(1+)
2 l, which we have lies entirely between the above quadratic bounds, has length (1−) conveniently set equal to s. Therefore, every path Bh (·) ∈ F also belongs to the set {SL > s}c .
z-axis
z = (1+ε)y2/2t + εl2/2t
z = Bh(y) + y2/2t
z = (1-ε)y2/2t - εl2/2t
y-axis
Shock Interval SL 0
Fig. 1. The Lagrangian picture when Bh (·) ∈ F . The length of SL is less than
1/2 4 2 (1+)
(1−)
l = s
The second part of this proof is a straightforward application of Lemma 3.1. l2 + y 2 , ∀y ∈ R (3.41) P (F ) = P(0,0) |Bh (y)| ≤ 2t 2t 2 l2 = 1 − P(0,0) |Bh (y)| ≥ + y , for some y ∈ R (3.42) 2t 2t
Statistics of Burgers Turbulence
83
≥ 1 − 2P(0,0)
l2 + y 2 , for some y ∈ (0, l) |Bh (y)| ≥ 2t 2t
l2 + y 2 , for some y ∈ (l, ∞) − 2P(0,0) |Bh (y)| ≥ 2t 2t l2 ≥ 1 − 2P(0,0) Bh (y) ≥ , for some y ∈ (0, l) 2t −2 −2P(0,0) y Bh (y) ≥ for some y ∈ (l, ∞). 2t 2 η s exp −C()s4−2h t−2 , ≥ 1−C t h
(3.43)
(3.44)
(3.45)
4−2h
(1−) . The best choice of for each h is ≈ h2 , with “≈” becoming where C() = 29−3h (1+)2−h “=” as h & 0. We note that the only conditions necessary in the above proof were the continuity of fractional Brownian motion, and the fact that E[Bh (x)−Bh (y)]2 ≤ C|x−y|2h , ∀x, y ∈ R. Therefore, this proof holds for Burgers systems initialized with one of the related Gaussian potentials.
3.3.4. Proof of Proposition 1: Lower bound. This proof follows a line of reasoning similar to that of the velocity PDF lower bound proof. Let a set of fractional Brownian motion paths, denoted E, be defined as follows: o s n s (3.46) E = Bh (·) : |Bh (y) + φ(y)| < 1, ∀ ∈ − − 1, + 1 , 2 2
where φ(y) =
1 2t 1 2t 1 2t
2 + ε s2 + 1 for y < − s2 − 1, + ε y2 for y ∈ − s2 − 1, s2 + 1 , s 2 + ε 2 + 1 for y > s + 1,
(3.47)
1 2 and ε = s2 . For any path Bh (·) ∈ E, Bh (y) + 2t y ≥ −ε(s/2)2 − 1, ∀y ∈ (−s/2, s/2). 1 2 2 For y = ±(s + 1), Bh (y) + 2 y ≤ −ε(s/2) − 2 − ε + 1. Thus, there exists a Lagrangian shock interval [y1 , y2 ] 3 y = 0 such that y1 < −s/2 and y2 > s/2 (see Fig. 2), and E ⊂ {SL > s}. To establish the appropriate lower bound on the measure of E, we shall again use the spectral representation for FBMs and the Cameron-Martin-Girsanov formula. From this representation and a little Fourier analysis, we have
Z∞ Bh (y) + φ(y) = Ch
(cos(yk) − 1)k −h−1/2 d(Z1 (k) + g(k))
−∞ Z∞
+Ch
(3.48)
sin(yk)k −h−1/2 dZ2 (k),
−∞
˙ = where Z1 (·) and Z2 (·) are standard independent Brownian motions, and g(k) ˆ φ(k)/E [k]1/2 , i.e.
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z = Bh(y)+ 1 y2 2t
z-axis
z = - εy2 + 1
z = - εy2 - 1 y-axis
0
-s/2-1 -s/2
s/2
s/2+1
Fig. 2. The Lagrangian picture when Bh (·) ∈ E, displayed with the bounds of set E. The shock interval around y = 0 is greater than s
g(k) =
2
1 2t
+ε Ch π
"
# [ s2 + 1] cos [ s2 + 1]k − k −1 sin [ s2 + 1]k . |k|3/2−h
(3.49)
Setting W1 (k) = Z1 (k) + g(k), we employ the Cameron-Martin formula to obtain a measure where W1 (·) is a standard Brownian motion. Under this measure, it is clear eh (y) ≡ Bh (y) + φ(y) is now a centered fractional Brownian motion on (−∞, ∞). that B R R 2 ˙ dk = C 0 (h)t−2 s4−2h + O(s3−2h ), Setting Z = g(k)dW ˙ 1 (k) and noting that [g(k)] we get P (E)
"
(3.50) #
1 eh (y)| < 1 = E(0,0) exp − C 0 (h)t−2 s4−2h − O(s3−2h ) + Z ; sup |B 2 y∈(−s/2−1,s/2+1) 1 0 −2 4−2h 3−2h 2−h −1 ≥ exp − C (h)t s (3.51) − O(s )−s t 2 ! eh (y)| < 1, . × P(0,0) |Z| < s2−h t−1 , sup |B y∈(−s/2−1,s/2+1)
Applying Lemmas 3.2 and 3.3 to the second term in the above inequality, we get the desired lower bound on the measure of E and, thus, on the probability of {SL > s}. The formula for the constant in the leading exponential term is 1 1 0 C (h) = 4−2h 2 2 2 2 Ch π
Z∞ −∞
2 cos(k) − k −1 sin(k) dk. |k|3−2h
(3.52)
Statistics of Burgers Turbulence
85
In order to extend this result to general Gaussian initial potentials with the abovestated spectra, one follows the same line of reasoning used in extending Theorem 1 to cover such cases. Without going through the details a second time, it is clear, by comparing the proofs of Theorem 1 and Proposition 1, that this reasoning works here in an identical manner. 4. Discussion Using large-deviation methods, we were able to find bounds on the tail probabilities for BT systems with FBM velocity potentials. The PDFs we studied were found to decay like stretched exponentials, with the rates of decay determined by the infrared power laws in the energy spectra of the velocity potentials. The question of whether the bounds we found are optimal, however, must be answered in the negative. The exponential constants in the upper and lower bounds for the PDFs of the velocity and the shock strength do not match. Looking closely at the the velocity PDF decay rate, one can see, heuristically, where the problem lies. In a heuristic sense, large-deviation theory implies that the probability that a given realization of the velocity potential φo = φ occurs is proportional to Z 1 1 2 ˆ exp − S(φ) , S(φ) = |φ(k)| /E [k] dk. (4.1) 2 2 Here, S(φ) is the classical action associated with the Gaussian process, in our case the velocity potential, whose energy spectrum is E [k]. The probability of u(0, t) > v, for v 1, is, thus, estimated by finding the path φ∗ with the minimum action that satisfies this condition. In other words, log Prob {u(0, t) > v} ≈ −
inf
S(φ).
φ∈{u(0,t)>v}
(4.2)
law
Using the self-similarity of fractional Brownian motion, Bh (·) = ah Bh (·/a), and the initial velocity-potential description of the set {u(0, t) > v}, given by Eq. 3.9, it is not hard to show that 4−2h 2−2h S(φ) = v t S(φ) . (4.3) inf inf φ∈{u(0,t)>v}
φ∈{u(0,1)>1}
Thus, we see that the true constant in the stretched exponential decay of the velocity PDF should be Cu = inf φ∈{u(0,1)>1} S(φ). In essence, we found the velocity-PDF upper bound by minimizing S(φ) over the 1 2 y ≤ 0} ⊃ {u(0, t) < −v}. This minimization problem is much set {miny>vt φ(y) + 2t more tractable and is solved by the path 1 φ(y) = E Bh (y) Bh (0) = 0, Bh (vt) = − (vt)2 , (4.4) 2t whose action is = 18 v 4−2h t2−2h . For the lower bound on the probability of {u(0, t) > v}, we found at the action 21 Cφ of a particular path φ ∈ {u(0, t) < −v} (defined by Eq. 3.20), giving us an upper bound Cu . Thus, we proved that 1 1 ≤ Cu ≤ Cφ . 8 2
(4.5)
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We conjecture that for h ≥ 21 , the optimal path for {u(0, 1) > 1} is given by 1 φ∗ (y) = E Bh (y) Bh (z) = − z 2 , ∀z ∈ [0, 1] , (4.6) 2 with S(φ∗ ) = Cu =
( 23 − h)0( 25 − h) 1 1 , 1 2 (4 − 2h)0(h + 2 )0(3 − 2h) 2h(2 − 2h)
(4.7)
where 0(·) is the Gamma function. This conjecture was proved for h = 21 in [11]. However, the non-Markovian nature of fractional Brownian motion (h 6= 21 ) makes analysis of the action operators S(·) of these processes and, thus, the above minimization problems difficult at best. References 1. Adler, R.J.: An Introduction to Continuity, Extrema and Related Topics for General Gaussian Processes. Inst. Math. Statist. Lecture Notes – Monograph Series, Vol. 12, 1990 2. Avellaneda, M.: PDFs for velocity and velocity gradients in Burgers Turbulence. Commun. Math. Phys. 169, 134 (1995) 3. Avellaneda, M. and E, W.: Statistical properties of shocks in Burgers Turbulence. Commun. Math. Phys. 172, 13 (1995) 4. Burgers, J.M.: The nonlinear diffusion equation. Dordrecht: D. Reidel Pulishing Co., 1974 5. Gotoh, T. and Kraichnan, R.: Statistics of decaying Burgers turbulence. Phys. Fluids A 5, 445 (1993) 6. Gurbatov, S., Malakhov, A. and Saichev, A.: Nonlinear random waves and turbulence in nondispersive media: Waves, rays and particles. New York: Manchester Univ. Press, 1991 7. Handa, K.: A remark on shocks in inviscid Burgers turbulence. In: Fitzmaurice, N., Gurarie, D., McCaughan, F., Woyczy´nski, W. (eds.), Nonlinear waves and weak turbulence with applications in oceanography and condensed matter physics. Progress in Nonlinear Differnetial Equations and their applications, Vol. 11, Boston, Berlin: Birkh¨auser, 1993, pp. 339–345 8. Kraichnan, R.: Models of intermittency in hydrodynamic turbulence. Phys. Rev. Lett. 65, 575 (1990) 9. Monrad, D. and Rootzin, H.: Small value of Gaussian processes and functional laws of the iterated logarithm. Probab. Theory Related Fields 101, no. 2, 173–192 (1995) 10. Ryan, R.: Large Deviation Analysis of Gaussian Fields and Statistics of Burgers Turbulence. Doctoral thesis at Courant Institute of Mathematical Sciences at New York University, accepted September 1996 11. Ryan, R.: Large deviation analysis of Burgers turbulence with white-noise initial data. To appear in Commun. Pure Applied Math. 12. Shandarin, S.N. and Zel’dovich, Ya.B.: The large-scale structure of the Universe: Turbulence, intermittency and structures in a self-gravitating medium. Rev. Mod. Phy. 61, 185 (1989) ˇ ak, Z.: Rectangular confidence regions for the means of multivariate normal distributions. J. Am. 13. Sid´ Statist. Assoc. 62, 626–633 (1967) 14. Sinai, Y.: Statistics of shocks in solutions of inviscid Burgers equation. Commun. Math. Phys. 148, 601–622 (1992) 15. Vergassola, M., Dubrulle, B., Frisch, U. and Noullez, A.: Burgers’ Equation, Devil’s Staircases and the Mass Distribution for Large-Scale Structures. Astron. Astrophys. 289, 325–356 (1994) Communicated by Ya. G. Sinai
Commun. Math. Phys. 191, 87 – 136 (1998)
Communications in
Mathematical Physics c Springer-Verlag 1998
Vertex Operators – From a Toy Model to Lattice Algebras Andrei G. Bytsko1,2 , Volker Schomerus3 1 Institut f¨ ur Theoretische Physik, Freie Universit¨at Berlin, Arnimallee 14, 14195 Berlin, Germany. E-mail:
[email protected] 2 Steklov Mathematical Institute, Fontanka 27, St.Petersburg 191011, Russia. E-mail:
[email protected] 3 II. Institut f¨ ur Theoretische Physik, Universit¨at Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany. E-mail:
[email protected]
Received: 16 December 1996 / Accepted: 5 May 1997
Abstract: Within the framework of the discrete Wess–Zumino–Novikov–Witten theory we analyze the structure of vertex operators on a lattice. In particular, the lattice analogues of operator product expansions and braid relations are discussed. As the main physical application, a rigorous construction for the discrete counterpart gn of the group valued field g(x) is provided. We study several automorphisms of the lattice algebras including discretizations of the evolution in the WZNW model. Our analysis is based on the theory of modular Hopf algebras and its formulation in terms of universal elements. Algebras of vertex operators and their structure constants are obtained for the deformed universal enveloping algebras Uq (G). Throughout the whole paper, the abelian WZNW model is used as a simple example to illustrate the steps of our construction. Contents 1 2 2.1 2.2 2.3 2.4 2.5 2.6 3 3.1 3.2 3.3 3.4 3.5 4 4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Hopf Algebras and Vertex Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Semi-simple modular Hopf algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Universal elements and R-matrix formalism . . . . . . . . . . . . . . . . . . . . . . 94 Vertex operators and their structure data . . . . . . . . . . . . . . . . . . . . . . . . . 96 Gauge transformations of vertex operators . . . . . . . . . . . . . . . . . . . . . . . 99 On the construction of vertex operators . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Vertex operators for Zq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 A Toy Model for the Discrete WZNW Theory . . . . . . . . . . . . . . . . . . . . 102 Properties of chiral vertex operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Second chirality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 ∗-operation for chiral vertex operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Quantum group valued field g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Toy model for Zq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Review on Lattice Current Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Definition of lattice current algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
88
A. G. Bytsko, V. Schomerus
4.2 4.3 4.4 5 5.1 5.2 5.3 5.4 5.5 6 6.1 6.2 6.3 6.4 A.1 A.2 A.3 A.4 A.5 A.6
Left currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Holonomies and monodromies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Current algebra for Zq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Vertex Operators on a Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Definition of WN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Vertex operators at different sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Extension on a covering of the circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Construction of the local field gn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Lattice vertex operators for Zq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Automorphisms and Discrete Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 121 Remarks on the classical continuum limit . . . . . . . . . . . . . . . . . . . . . . . . 121 Automorphisms induced by the ribbon element . . . . . . . . . . . . . . . . . . . 122 Discrete dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 U (1)-WZNW model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Proof of Proposition 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Proof of Proposition 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Structure data for left chiral vertex operators . . . . . . . . . . . . . . . . . . . . . 132 Properties of the field g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Properties of lattice vertex operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Properties of lattice field gn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
1. Introduction Quantization of the WZNW model. The Wess–Zumino–Novikov–Witten (WZNW) model [50, 46, 51, 41] is one of the most famous examples of a rational conformal field theory (CFT) [14, 42, 44]. On the classical level it describes some time evolution for a field g(x) mapping points x of the circle S1 into a compact Lie group G. Among the dynamical variables of the theory, the currents j r (x) = g −1 ∂− g, j l = (∂+ g)g −1 are of particular interest. In contrast to the field g, the currents j r and j l are chiral, so that ∂+ j r = 0 and ∂− j l = 0. Moreover, their Poisson structure is well known to give rise to two commuting copies of Kac–Moody (KM) algebras. Even though numerous papers have been devoted to the quantization of the WZNW-model (e.g. [16, 26, 10, 36, 13, 20, 21, 35]), a rigorous construction of the continuum theory (which requires field strength renormalization) is not fully understood. This motivates the search for lattice regularizations of the theory (i.e., the circle S1 is replaced by a periodic lattice with lattice spacing a) which preserve much of the symmetry structure of the continuum WZNW-model. One may construct appropriate discretizations of the classical model (i.e., ~ = 0) first and then quantize the classical lattice theory to obtain a well defined discrete quantum theory (i.e., ~ 6= 0, a 6= 0). Investigation of the latter is expected to provide insights into the structure of the continuum model. A final step would involve performing the limit a → 0 while keeping ~ 6= 0. The realization of this program was started in [4, 5, 27] where a lattice regularization of the Kac–Moody algebra has been proposed. Classical and quantum lattice current algebras were further investigated in [33, 6]. Our aim here is to extend the analysis of [6] by introducing chiral vertex operators. In comparison with the current algebra, the algebras of vertex operators contain (a finite number of) additional generators. Within these larger algebras we will be able to prepare a discrete analogue of the group valued field g(x) by combining left- and right chiral vertex operators.
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Quantum symmetry structure of the WZNW model. Let us recall that solutions of the classical Yang–Baxter equation appear already in the Poisson structure of the classical lattice current algebras (see [33] and references therein). After quantization, quantum groups and quantum universal enveloping algebras G 1 are expected to emerge. Throughout this paper we will meet (global and local) objects (the monodromies M α and a discrete field Nn , see below) whose nature reflects a quantum algebraic structure as well as an object (the discrete field gn ) which displays features of a quantum group. The corresponding deformation parameter is of the form q = exp{iγ~} with γ = π/(k + ν) (where k is the level of the KM algebra and ν is the dual Coxeter number of G) and does not depend on the lattice spacing a. Therefore, the quantum group structures of the continuum and lattice WZNW model coincide. It is also worth mentioning that some aspects of the quantum symmetry structure survive reductions to other theories so that part of what we describe below may be compared with studies of the quantum Liouville and Toda models [37, 29, 11, 23]. Remarks on lattice current algebras. Before we summarize our results, let us briefly review the discretization used in [6] for the chiral currents j l (x), j r (x). Recall that the latter are Lie-algebra valued fields which depend periodically on the variable x. Instead of working with these standard variables, we prefer to pass to the fields j r (x) and η(x) = j r (x) − j l (x) and describe their lattice counterparts. Our lattice divides the circle into N links of length a = 2π/N . So there are N vertices at the points x = an which are numbered by n = 0, . . . , N − 1, N ≡ 0 and the nth link runs from the (n − 1)st vertex to the nth . We may discretize the field η(x) by the simple prescription R (n+ 1 )a ηn := (n− 21 )a η(x)dx = aη(na) + O(a2 ) so that the lattice field ηn has values in a tensor 2 product of N copies of the Lie algebra which are assigned to the N vertices on the lattice. For the right chiral current j r (x) the discretization scheme is different. In this case we encode the information about the field in the holonomies along links, i.e., we define the lattice field jnr by Z jnr
j r (x)dx) .
:= P exp( n
R Here n denotes integration along the nth link. By construction, this classical lattice field jnr has values in the Lie group. The rather different treatment of the fields η(x) and j r (x) may be understood from the Poisson structure of the classical theory, which is ultralocal for η(x) but contains terms proportional to δ 0 (x − y) if the field j r (x) is involved (see [4, 33, 6]). When we pass to the quantum theory, the functions on the space of field configurations become operators and generate some non-commutative algebra KN . More concretely, the algebra KN is generated from the quantum lattice fields Jnr , Nn which correspond to the classical fields jnr , ηn described above. We review the explicit definition of lattice current algebras in Sect. 4. Let us only mention here that a very elegant formulation for commutation relations of the quantum operators can be given in the R-matrix language. In mathematical terms, one has to regard the quantum fields Nn and Jn = Jnr as objects in the tensor product Ga ⊗ KN of the deformed universal enveloping algebra Ga = Uq (G) with the lattice current algebra KN . We can understand this by looking at 1
For shortness, we will often refer to G as quantum algebra.
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the classical lattice field jnr , for instance. It was constructed as the holonomy of the Liealgebra valued field j r (x) and may be evaluated with irreducible representations of the Lie algebra. Let us denote such representations by τ I and introduce the symbols V I , δI for their carrier spaces and dimensions, respectively. Then we see that jn = jnr gives rise to δI × δI -matrices jnI of dynamical variables. Accordingly, the corresponding quantum operators JnI are matrices of generators for KN which is to say that JnI ∈ End(V I )⊗KN . All these objects JnI may be assembled back into one universal element Jn ∈ Ga ⊗ KN . More details will be presented later; we anticipated this heuristic discussion of universal elements only to prepare for some formulae below. One of the main aims in [6] was to develop a complete representation theory for lattice current algebra KN . It turned out that KN possesses a family of irreducible ∗-representations on vector spaces WNIJ with labels I, J running through classes of finite-dimensional, irreducible representations of Uq (G). Two such labels are needed because of the two chiralities in the current algebra. Furthermore, the algebra KN was found to admit two families of local co-actions 3rn , 3ln : KN 7→ Ga ⊗ KN of the Hopf algebra Ga . They may be considered as a special case of the more general lattice fusion products in [6] and give rise to a notion of tensor products for representations of KN (see also [45] for related results). Vertex operators on a lattice. Product structures in the representation theory are precisely what is needed to initiate a theory of vertex operators. More technically, we employ the homomorphisms 3rn , 3ln in extending the lattice current algebra KN by chiral vertex operators 8rn , 8ln so that the following intertwining relations hold for both chiralities α = r, l, α α for all A ∈ KN . (1.1) A 8α n = 8n 3n (A) The elements 8α n generate an extension WN of the lattice current algebra KN ⊂ WN . Since 3α n (A) is an element of Ga ⊗ KN and hence also of the extension Ga ⊗ WN , the product on the r.h.s. of (1.1) is well defined for 8α n ∈ Ga ⊗ WN . On the l.h.s., A = e ⊗ A ∈ Ga ⊗ KN with e ∈ Ga being the unit element. Our vertex operators 8α n on the lattice possess a number of properties which are all closely related to properties of vertex operators in the continuum theory. Let us highlight some of them without going into a detailed discussion.2 1. Lattice vertex operators 8α n at a fixed lattice site obey operator product expansions of the form 2
1
8rn 8rn = Fr 1a (8rn )
1
and
2
8ln 8ln = Fl 1a (8ln ) .
(1.2)
1
α As usual, the notation 8α n means that we regard the vertex operator 8n as an element of Ga ⊗ Ga ⊗ WN with trivial entry in the second tensor factor etc. We have also α used the shorthand 1a (8α n ) = (1 ⊗ id)(8n ) ∈ Ga ⊗ Ga ⊗ WN for the action of the co-product on the first tensor factor of 8α n . The objects Fα are analogues of the fusion matrix in the continuum theory. We describe their general properties and, in particular, their relation with 6j-symbols in Sect. 2. 2 A construction of (non-chiral) vertex operators for infinite open lattices has been suggested in [45]. Some properties of these vertex operators are similar to what we shall consider here. However, these are different structures, in particular, because for a finite lattice the current algebra KN has a non-trivial center C. An action of our vertex operators on elements from C will play a crucial role in the theory.
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2. Lattice vertex operators 8α n assigned to different lattice sites obey braid relations 1
2
2
1
8rn 8rm = Rr− 8rm 8rn
2
and
1
1
2
8ln 8lm = Rl+ 8lm 8ln
(1.3)
for all 0 ≤ n < m < N . Here the objects R± play the role of the braiding matrix in the continuum theory. Let us add that lattice vertex operators of different chirality commute for all n, m. Furthermore, 8α n commute with Nm for m 6= n and with Jm for m 6= n, n + 1, that is, the vertex operators have local exchange relations with elements of the current algebra. 3. Lattice vertex operators 8α n , α = r, l, satisfy the following difference equation: α α 8α n+1 = 8n Jn+1 .
(1.4)
In the naive continuum limit, we have Jnα = e ⊗ e − aJ α (x) + O(a2 ) with x = an and the difference equation becomes a differential equation which expresses ∂x 8α (x) as a (normal ordered) product of 8α (x) and J α (x). Such an equation is well known for the quantized continuum theory. As one may infer from the third property in this short list, lattice vertex operators (much like their continuum counterparts) cannot be periodic. Indeed, starting from 8α 0 an iterated application of Eq. (1.4) gives α α 8α N ≡ 80 M
with
α M α = J1α . . . JN .
The objects M α , α = r, l, are called chiral monodromies. Actually, the lattice rotation n 7→ n + N gives rise to an inner automorphism of the algebra of vertex operators which acts trivially on the lattice fields Jnα and Nn . We show in Sect. 6 that this automorphism can be generated by conjugation with a unitary element v. The latter is constant on the irreducible representation spaces WNIJ of the lattice current algebra KN and its value vIJ = e2πi(hJ −hI ) can be expressed in terms of the conformal dimensions hI of the WZNW model. This leads us to identify v with the operator exp{2πi(L0 − L¯ 0 )} which generates rotations by 2π in the continuum theory. In the lattice theory v is obtained from quantum traces of chiral monodromies M α and is related to the ribbon element of Uq (G). It will be shown in Sects. 3 and 5 that the field Sa (8ln )8rn 3 can be restricted to the L ¯ KK . Let us denote this restriction by gn which suggests that diagonal subspace K W N it is a quantum lattice analogue of the group valued field g(x) in the WZNW model. In fact, our analysis will reveal that gn is a local and quantum group valued field, i.e. 1
2
2
1
gn gm = gm gn
(n 6= m)
and
2
1
1
2
R gn gn = gn gn R .
(1.5) α
Moreover, gn turns out to be periodic. In contrast to the chiral currents j (x), the time evolution of the group valued field g(x) is described by a nontrivial second order differential equation. Its discrete analogue is discussed in Sect. 6. Before we address the full lattice theory we explain some basic constructions in a simple toy model (cf. Sect. 3). Here one studies the algebra generated by the monodromies M r , M l instead of the whole (lattice) current algebra and universal (deformed) tensor operators for the quantum algebra G as simple examples of vertex operators 3
l
l
We use the notation Sa (8n ) = (S ⊗ id)(8n ) with S being the antipode of G.
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[2, 22, 3, 19]. This finite dimensional toy model may be regarded as a special case of the discrete WZNW theory where N = 1 and it describes the non-local degrees of freedom for an arbitrary number N of lattice sites. Objects and relations of the toy model admit for a nice pictorial presentation which, in particular, brings new light into the shadow world [39]. 2. Hopf Algebras and Vertex Operators 2.1. Semi-simple modular Hopf algebras. By definition, a Hopf algebra is a quadruple (G, , 1, S) of an associative algebra G (the “symmetry algebra ”) with unit e ∈ G, a onedimensional representation : G 7→ C (the “co-unit ”), a homomorphism 1 : G 7→ G ⊗G (the “co-product ”) and an anti-automorphism S : G 7→ G (the “antipode ”). These objects obey a set of basic axioms which can be found, e.g., in [1, 49]. The Hopf algebra (G, , 1, S) is called quasi-triangular if there is an invertible element R ∈ G ⊗ G such that R 1(ξ) = 10 (ξ) R for all ξ ∈ G, (id ⊗ 1)(R) = R13 R12 ,
(1 ⊗ id)(R) = R13 R23 .
Here 10 (ξ) = P 1(ξ)P , with P being the permutation, i.e., P (ξ ⊗ η)P = η ⊗ ξ for all ξ, η ∈ G, and we are using the standard notation for the elements Rij ∈ G ⊗ G ⊗ G. For a ribbon Hopf-algebra one postulates, in addition, the existence of a certain invertible central element v ∈ G (the “ribbon element ”) which factorizes R0 R ∈ G ⊗ G ( here R0 = P RP ), in the sense that R0 R = (v ⊗ v) 1(v −1 ) ,
S(v) = v ,
(v) = 1
(2.1)
(see [48, 40] for details). We want this structure to be consistent with a ∗-operation on G. To be more precise, we require that 4 R∗ = (R−1 )0 = P R−1 P , 1(ξ)∗ = 10 (ξ ∗ ) , v ∗ = v −1 .
(2.2)
This structure is of particular interest, since it appears in the theory of the quantized universal enveloping algebras Uq (G) when the complex parameter q has values on the unit circle [43]. At this point we assume that G is semi-simple, so that every finite dimensional representation of G can be decomposed into a direct sum of finite dimensional, irreducible representations. From each equivalence class [I] of irreducible representations of G, we may pick a representative τ I , i.e., an irreducible representation of G on a δI -dimensional Hilbert space V I . The quantum trace tr Iq is a linear functional acting on elements X ∈ End(V I ) by tr Iq (X) = tr I (Xτ I (w)) . Here tr I denotes the standard trace on End(V I ) with tr I (eI ) = δI and w ∈ G is a distinguished group-like element P constructed from the ribbon element v and the element R by the formula w−1 = v −1 S(rς2 )rς1 , where the elements rςi come from the expansion P 1 R= rς ⊗ rς2 . 4 We fix ∗ on G ⊗ G by (ξ ⊗ η)∗ = ξ ∗ ⊗ η ∗ . Following [43], we could define an alternative involution † on G ⊗ G which involves a permutation of components, i.e., (ξ ⊗ η)† = η † ⊗ ξ † and ξ † = ξ ∗ for all ξ, η ∈ G. With respect to †, 1 becomes an ordinary ∗ -homomorphism and R is unitary.
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Evaluation of the unit element eI ∈ End(V I ) with tr Iq gives the quantum dimension, dI := tr Iq (eI ), of the representation τ I . Furthermore, we assign a number SIJ to every pair of representations τ I , τ J : SIJ := ϑ(tr Iq ⊗ tr Jq )(R0 R)IJ
with (R0 R)IJ = (τ I ⊗ τ J )(R0 R) ,
with a suitable, real normalization factor ϑ. The numbers SIJ form the so-called S-matrix S. Modular Hopf algebras are ribbon Hopf algebras with an invertible S-matrix.5 ×τ 0 , of two representations τ, τ 0 of a Let us finally recall that the tensor product, τ Hopf algebra is defined by 0 0 ×τ )(ξ) = (τ ⊗ τ )1(ξ) for all ξ ∈ G . (τ
In particular, one may construct the tensor product τ I ×τ J of two irreducible representations. According to our assumption that G be semi-simple, such tensor products of representations can be decomposed into a direct sum of irreducible representations. Among all our assumptions on the structure of the Hopf-algebra (G, , 1, S) (quasitriangularity, existence of a ribbon element v, semi-simplicity of G and invertibility of S-matrix S), semi-simplicity of G is the most problematic one. In fact it is violated by the algebras Uq (G) when q is a root of unity. It is sketched in [7] how “truncation” can cure this problem, once the theory has been extended to weak quasi-Hopf algebras [43].
Example (Modular Hopf-algebra Zq [6]). We wish to give one fairly trivial example for the algebraic structure discussed so far which comes from the group Zp . To be more precise, we consider the associative algebra Zq generated by one element h subject to the relation hp = e. Co-product, co-unit and antipode for this algebra can be defined by 1(h) = h ⊗ h , S(h) = h−1 , (h) = 1 . We observe that Zq is a commutative semi-simple algebra. It has p one-dimensional representations τ t (h) = q t , t = 0, . . . , p − 1, where q is a root of unity, q = e2πi/p . We may construct characteristic projectors P t ∈ Zq for these representations according to 1 X −tm m q h for t = 0, . . . , p − 1 . P = p p−1
t
(2.3)
m=0
One can easily check that τ t (P s ) = δt,s . The elements P t are employed to obtain a nontrivial R-matrix: p−1 X R= q ts P t ⊗ P s . (2.4) t,s=0
It is easy to see that (τ ⊗ τ )R = q . The R-matrix satisfies all the axioms stated above and thus turns Zq into a quasi-triangular Hopf algebra. Moreover, a ribbon element is P 2 given by v = q −t P t . It is natural to introduce a ∗-operation on Zq such that h∗ = h−1 . The relations (2.2) hold due to the co-commutativity of 1, i.e., 10 = 1, and the property R = R0 . A direct t
s
ts
If a diagonal matrix T is introduced according to TIJ = $δI,J d2I τ I (v) (with an appropriate complex factor $), then S and T furnish a projective representation of the modular group SL(2, Z). 5
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computation shows that the S-matrix S is invertible only for odd integer p. Summarizing all this, the algebra Zq , q = exp(2πi/p) is a semi-simple ribbon Hopf-∗-algebra. It is a modular Hopf algebra for all odd integer p. The reader is invited to check that for Zq the quantum trace trqI coincides with the standard one. 2.2. Universal elements and R-matrix formalism. Modular Hopf algebras admit a very elegant R-matrix description. For its presentation, let us introduce another (auxiliary) copy, Ga , of G and let us consider the R-matrix as an object in Ga ⊗ G. To distinguish the latter clearly from the usual R, we denote it by N− ≡ (R0 )−1 ∈ Ga ⊗ G .
N + ≡ R ∈ Ga ⊗ G ,
At the same time let us introduce the standard symbols R+ = R and R− = (R0 )−1 ∈ Ga ⊗ Ga . Quasi-triangularity of the R-matrix furnishes the relations 1
2
1
1a (N± ) =N ± N ± ; 1
2
2
1
R+ N + N − =N − N + R+ ,
2
2
(2.5)
1
R + N ± N ± = N ± N ± R+ . Here we use the same notations as in the introduction, and 1a (N± ) = (1 ⊗ id)(N± ) ∈ Ga ⊗ Ga ⊗ G. The subscript a reminds us that 1a acts on the auxiliary (i.e., first) component of N± . To be perfectly consistent, the objects R± in the preceding equations should all be equipped with a lower index a to show that R± ∈ Ga ⊗ Ga , etc. We hope that no confusion will arise from omitting this subscript on R± . The Eqs. (2.5) are somewhat redundant: in fact, the exchange relations in the second line follow from the first equation in the first line. This underlines that the formula for 1a (N± ) encodes information about the product in G rather than the co-product.6 Next, we combine N+ and N− into one element N := N+ (N− )−1 ∈ Ga ⊗ G . From the properties of N± we obtain an expression for the action of 1a on N , 1
2
2
1
2
1
2
1
2
1
−1 −1 −1 R+ 1a (N ) = R+ N + N + N −1 − N − =N + N + R+ N − N − = 2
2
1
1
−1 =N + N −1 − R+ N + N − =N R+ N .
(2.6)
As seen above, the formula for 1a (N ) encodes relations in the algebra G and implies, in particular, the following exchange relations for N : 2
1
1
2
−1 −1 −1 0 −1 R− N R+ N = R− R+ 1a (N ) = R− 1a (N )R+ =N R− N R+ .
(2.7)
This kind of relations appeared first in [47] to describe relations in Uq (G). One may in fact also go in the other direction, which means to reconstruct a modular Hopf algebra G from an object N satisfying the above exchange relations. To begin with, one has to choose linear maps π : Ga 7→ C in the dual Ga0 of Ga . When such linear forms π ∈ Ga0 act on the first tensor factor of N ∈ Ga ⊗ G they produce elements in G: π(N ) ≡ (π ⊗ id)(N ) ∈ G for all π ∈ Ga0 . 0 N 00 ∈ G ⊗ G ⊗ G. The co-product 1 of G acts on N± according to 1(N± ) = (id ⊗ 1)(N± ) = N± a ± 0 00 Here N± and N± have the unit element e ∈ G in the third and second tensor factor, respectively. 6
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π(N ) ∈ G will be called the π-component of N or just component of N . It has been shown in [9] that the components of N generate the algebra G, that is, one can reconstruct the modular Hopf algebra G from the object N . A more precise formulation is given by the following lemma. Lemma 1 ([9]). Let Ga be a finite-dimensional, semi-simple modular Hopf algebra and N be the algebra generated by components of N ∈ Ga ⊗ N subject to the relations 1
2
N R+ N = R+ 1a (N ) ,
(2.8)
where we use the same notations as above. Then N can be decomposed into a product −1 , such that of elements N± ∈ Ga ⊗ N , N = N+ N− 0 −1 ) 1(N ) ≡ N+0 N 00 (N−
∈ Ga ⊗ N ⊗ N ,
−1 S(N± ) ≡ N± ∈ Ga ⊗ N ,
(N± ) ≡ e ∈ Ga ,
∗ N± ≡ N∓
define a Hopf-algebra structure on N . Here, the action of 1, , S on the second tensor 0 and N 00 are regarded as elements of Ga ⊗ component of N, N± is understood. N+0 , N− G ⊗ G with trivial entry in the third and in the second tensor factors, respectively. As a Hopf algebra, N is isomorphic to Ga . There is another object, similar to N , that is equally natural to consider and that will appear later in the text, e := N+−1 N− ∈ Ga ⊗ G . N Its properties are derived in complete analogy with our treatment of N , 1
2
e ) =N e R− N e, R− 1a (N
1
2
2
1
e R− N e =N e R+−1 N e R− . R+−1 N
(2.9)
e An appropriate version of Lemma 1 establishes an isomorphism between the algebra N e and the algebra Gop . The latter stands for the quantum generated by components of N algebra G with opposite multiplication, i.e., elements ξ, η ∈ Gop are multiplied according to ξ · η := ηξ. e . We wish to rewrite this simple Observe that property (2.2) implies that N ∗ = N formula for the action of ∗ on N in a more sophisticated way which proves to be useful in the sequel. For this purpose, let us introduce an element S ∈ Ga ⊗ G as follows S := N+ 1(κ) (κ ⊗ κ)−1 = N− 1(κ−1 ) (κ ⊗ κ) ,
(2.10)
where κ is some central square root of the ribbon element v ∈ G, i.e., κ2 = v and κ commutes with all ξ ∈ G. The two expressions for S given in (2.10) are equivalent due to (2.1). It is easy to check that S∗ = S ,
S 0 ≡ P SP = S −1 .
(2.11)
e with the help of S: Now we are able to rewrite the ∗-operation on N and N N ∗ = S −1 N −1 S ,
e∗ = S N e −1 S −1 . N
(2.12)
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Example (The universal elements for Zq ). The notion of universal elements can be illustrated with the example P of Zq . The elements P N± ∈ Ga ⊗ G are constructed P from the R-matrix (2.4): N± = t,s q ±ts P t ⊗P s = s P s ⊗h±s and hence N = s P s ⊗h2s . The functorial (2.5), (2.8) can be verified by using the obvious identity P properties k s−k P ⊗ P . In order to make these properties more transparent, we 1(P s ) = k p introduce an Hermitian operator pb such that h = qb . It follows from the definition of τ s s that τ (b p) = s and that the co-product, antipode and co-unit act on pb according to 1(b p) = pb ⊗ e + e ⊗ pb , S(b p) = −b p , (b p) = 0 . In these notations, the characteristic projector (2.3) acquires the form P p−s) e are given by and the universal elements N± , N, N P s = p1 m q m (b p⊗b p N± = q ± b ,
p⊗b p N = q2 b ,
p⊗b p e = q −2 b N .
(2.13)
These expressions simplify the task of checking the functoriality relations in (2.5), p⊗b p p ⊗ e+e ⊗ b p) ⊗ b p = q ± (b =N ± N ± . 1a (N± ) = q ±(1⊗id) b 1
2
P 2 p2 and hence Observe that the ribbon element v = s q −s P s can be written as v = q −b − 21 b p2 . A simple calculation gives S = e ⊗ e for the element S we may choose κ = q defined in (2.10). Thus, formulae (2.12) simplify for Zq and become N ∗ = N −1 and e∗ = N e −1 . N 2.3. Vertex operators and their structure data. Our next aim is to recall the theory of tensor operators for a semi-simple modular Hopf algebra G. To this end, we combine the carrier spaces V I of its finite dimensional irreducible ∗-representations τ I into the model space M = ⊕I V I . Each subspace V I ⊂ M appears with multiplicity one. The model space M comes equipped with a canonical action of our modular Hopf algebra so that we can think of G as being contained in the associative algebra V = End(M) of endomorphisms on M. Let us also introduce C ⊂ V to denote the center of G ⊂ V and e for the unit element of V. Definition 1 ((Vertex operator)). An invertible element 8 ∈ Ga ⊗ V is called a vertex operator for G, if 1. 8 intertwines the action of G on the model space M in the sense that ξ 8 = 8 10 (ξ) for all ξ ∈ G .
(2.14)
Here ξ = e ⊗ ξ on the l.h.s. and 10 (ξ) = P 1(ξ)P on the r.h.s. are both regarded as elements in Ga ⊗ V. 2. 8 obeys the following generalized unitarity relation 8∗ = S −1 8−1 = κa κN+−1 8−1 κ−1 ,
(2.15)
where S ∈ Ga ⊗ G was defined in (2.10)–(2.11). On the r.h.s., κ±1 = (e ⊗ κ±1 ) and κa = (κ ⊗ e), so that all these factors are elements of Ga ⊗ V. Invertibility of 8 means that there exists an element 8−1 ∈ Ga ⊗ V such that 8 8−1 = e ⊗ e = 8−1 8.
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Since Definition 1 is fundamental to what follows below, let us discuss it in more detail. In Eqs. (2.14)–(2.15) it would be possible to replace 10 by 1 and at the same time S by S −1 . We shall meet elements 8 with such properties later and call them vertex operators as well. The relation (2.14) describes the covariance property of 8. It means that 8 is a universal tensor operator for G (see, e.g., [43]). More precisely, we may evaluate the element 8 ∈ Ga ⊗ V with representations τ I of G to obtain matrices 8I = (τ I ⊗ id)(8) ∈ End(V I ) ⊗ V. The rows of these matrices form tensor operators which transform covariantly according to the representation τ I of G. The relation (2.14) may be rewritten in the R-matrix formalism of Subsect. 2.2 (see [6], where a similar calculation was discussed) as follows: 1
2
2
1
N ± 8 =8 R ± N ±
1
or
2
2
1
N 8 R− = 8 R+ N .
(2.16)
These relations are equivalent [17] to the definition of deformed tensor operators in terms of generalized adjoint actions of G which is often used in the theory of (q-deformed) tensor operators (see, e.g., [15]). Our formula (2.15) for the ∗-operation on 8 certainly deserves a more detailed explanation.7 Both expressions we have provided describe 8∗ in terms of 8−1 . Using the intertwining relation (2.14) one concludes that the conjugated vertex operator obeys a transformation law which differs from the covariance properties of the inverse 8−1 : 8−1 ξ = 10 (ξ) 8 while
8∗ ξ = 1(ξ) 8∗ .
The second relation follows from our assumption (2.2) on the behaviour of the co-product under conjugation. Comparison of the two transformation laws motivates to multiply 8−1 with a factor N+−1 so that we obtain two objects with identical covariance properties, namely 8∗ and N+−1 8−1 . In addition, the operation ∗ is supposed to be an involution, i.e., (8∗ )∗ = 8. This requires to dress the operator N+−1 8−1 with factors of κ as we did in the second expression for 8∗ in (2.15). All these factors can be moved to the left of 8−1 with the help of Eq. (2.14), so that 8∗ = S −1 8−1 . The identity (8∗ )∗ = 8 holds then as a consequence of (2.11). Suppose for the moment that we are given a vertex operator 8 in the sense of our Definition 1. Then we can use it to construct the following structure data of 8, 2
1
F := 8 8 1a (8−1 ) ∈ Ga ⊗ Ga ⊗ V, −1
σ(f) := 8 (e ⊗ f) 8
for all f ∈ C ⊂ V ,
D := 8 N 8−1 ∈ Ga ⊗ V.
(2.17) (2.18) (2.19)
As they are defined, the last tensor components of F, D and σ(f) belong to the algebra V. However, with the help of relation (2.14) and standard axioms of Hopf algebra it is easy to see that F, D and σ(f) commute with all elements ξ ∈ G ⊂ V and hence that F ∈ Ga ⊗ Ga ⊗ C while σ(f), D ∈ Ga ⊗ C. Before we give a comprehensive list of properties of the structure data, we introduce some more notations, R± ≡ F 0 R± F −1 ∈ Ga ⊗ Ga ⊗ C and 1F (ξ) ≡ F (1(ξ) ⊗ e) F 7
−1
∈ Ga ⊗ Ga ⊗ C.
∗-operations of a similar form have appeared in [43, 3, 7].
(2.20) (2.21)
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Here F 0 = (P ⊗ e)F (P ⊗ e). As a consequence of Eqs. (2.17)–(2.18) and our definition (2.20) we obtain the following exchange relations for vertex operators: 2
1
1
2
2
1
1
2
R± 8 8 = 8 8 R± , R± σ σ (f) =σ σ (f) R± .
(2.22)
It is also worth noticing that one may think of R and 1F as being obtained from R and 1 through a twist with F in the sense of Drinfeld [25]. Proposition 1 (Properties of the structure data). Let the structure data be defined as in Eqs. (2.17)–(2.19). Then it follows from Definition 1 that 1. the element D ∈ Ga ⊗ C may be expressed in terms of σ and the ribbon element v so that (2.23) D = va v−1 σ(v) . Here va = (v ⊗ e) ∈ Ga ⊗ C and v = (e ⊗ v) ∈ Ga ⊗ C, that is, we denote the ribbon element by va and v when it is regarded as an element of Ga or C, respectively. 2. The elements F, R± ∈ Ga ⊗Ga ⊗C and D ∈ Ga ⊗C together with the homomorphism σ : C → Ga ⊗ C obey the following set of relations: 3 (2.24) (e ⊗ F ) (id ⊗ 1a )(F ) = σ (F ) (1a ⊗ id)(F ) , 1
2
2
D R− = R+ σ (D), 2
1
σ σ (f) = 1F (σ(f)) 2
1
R− D = σ (D) R+ ,
(2.25)
for all f ∈ C,
(2.26)
1
3
R±,12 σ (R±,13 ) R±,23 = σ (R±,23 ) R±,13 σ (R±,12 ). 2
1
(2.27) 2
The symbol σ (D) denotes (id ⊗ σ)(D) ∈ Ga ⊗ Ga ⊗ C and σ (D) = (P ⊗ e) σ (D) (P ⊗ e) with P being the permutation. Similar conventions apply to Eqs. (2.24), (2.27). 3. The behaviour of the structure data with respect to the ∗-operation is given by F ∗ = Sa F −1 R∗± = R−1 ± , ∗
∗
σ(f) = σ(f ),
with
Sa = (R+ 1(κ) (κ ⊗ κ)−1 ) ⊗ e ∈ Ga ⊗ Ga ⊗ C , D∗ = D−1 , ∗
(2.28) ∗
(1F (ξ)) = 1F (ξ ),
for all ξ ∈ G and f ∈ C. It means, in particular, that D, R± are unitary while σ, 1F act as ∗-homomorphisms. A proof of the main statements can be found in Appendix A.1. It should be mentioned that some of the relations given in Proposition 1 have appeared in the literature before. Equation (2.27) is probably the most characteristic in our list as it generalizes the usual Yang–Baxter equation. It appeared first in connection with the quantum Liouville model [37]; later some universal solution for Eq. (2.27) in the case of G = Uq (sl(2)) has been found [11]. More recently in [12], the elements F and R and their relations were reinterpreted in the language of quasi-Hopf algebras [25]. As we remarked already, F may be regarded as a twist and it follows from Eq. (2.24) that the twisted co-product 3 −1 . The latter can be used to 1F is quasi-coassociative with co-associator φ = σ (F12 ) F12 rewrite Eq. (2.27) as a quasi Yang–Baxter equation (more details are discussed, e.g., in [12, 18]).
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Relations (2.19), (2.25) and the first equation in (2.22) have been introduced in [2] in a description of deformed cotangent bundles Tq∗ G. There, an object N was defined in terms of 8 and D through Eq. (2.19). The relation (2.25) allowed to derive exchange relations for N which guaranteed that coordinate functions for the fibers of Tq∗ G could be obtained from N . We shall see later that the equations in Proposition 1 have a number of important implications for the lattice theories. Reversing this logic, many of the relations in Proposition 1 were conjectured as natural properties of a coordinate dependent braiding matrix in the continuum WZNW-model [26, 20, 21]. 2.4. Gauge transformations of vertex operators. There exists a large gauge freedom in the choice of vertex operators 8. In fact, one may replace 8 7→ 38 with 3 ∈ Ga ⊗ C being invertible and unitary. This transformation does not change the general properties (2.14)–(2.15) of vertex operators but certainly effects their structure data. Namely, after the action of 3 on 8 the initial structure data transform into the following ones: 2
F 7→ 3 σ (3) F 1a (3−1 ), 2
D 7→ 3 D 3−1 ,
σ(f) 7→ 3 σ(f) 3−1 for all f ∈ C, 2
where σ (3) = (id ⊗ σ)(3) ∈ Ga ⊗ Ga ⊗ C, as before. One may reduce such a gauge freedom by additional requirements on the structure data or on the vertex operators. For instance, the gauge freedom allows to normalize the vertex operators in the following sense. Consider the element w := a (8) ≡ ( ⊗ id)8 ∈ V, where : G 7→ C stands for the co-unit of G. An application of the Hopf algebra axiom ( ⊗ id)1 = id to (2.14) furnishes the identity ξw = wξ and hence w ∈ C. From this and Eqs. (2.18)–(2.19) we conclude that ( ⊗ id)D = e , ( ⊗ id)σ(f) = f for all f ∈ C . Moreover, (2.15) implies unitarity of w (observe that ( ⊗ id)(S) = e). Therefore, we can perform the gauge transformation 8 7→ (e ⊗ w−1 )8, which does not change σ and D but normalizes F and 8 so that, without loss of generality, we may assume a (8) = e ∈ C ,
(id ⊗ ⊗ id)F = ( ⊗ id ⊗ id)F = e ⊗ e ∈ Ga ⊗ C .
The normalization of F follows from the normalization and operator product expansion of 8 with the help of ( ⊗ id)1 = id = (id ⊗ )1. It also leads to the identities ( ⊗ id)R± = (id ⊗ )R± = e ⊗ e. Finally, let us notice that multiplication of vertex operators 8 by element f ∈ Ga ⊗ G from the right, i.e., 8 7→ 8F, corresponds to twisting the co-product of G. 8 Transformations of this kind relate vertex operators 8q = 81 Fq for the deformed universal enveloping algebras Uq (G) with unitary vertex operators 81 of the undeformed algebras U (G) [25]. 2.5. On the construction of vertex operators. So far, we have considered the vertex operators as given objects. In the spirit of Lemma 1, however, we can reverse our approach and think of them as being defined through Eqs. (2.17)-(2.19) with an appropriate set of structure data. This is made more precise in the following proposition. 8 The object F should not be confused with our F ∈ G ⊗ G ⊗ C. We use similar letters mainly for a a historical reasons.
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Proposition 2 (Reconstruction of 8 from structure data). For a modular Hopf algebra G ∼ = Ga with center C, let F ∈ Ga ⊗ Ga ⊗ C and a homomorphism σ : C → Ga ⊗ C be given. Define the elements D ∈ Ga ⊗ C, R± ∈ Ga ⊗ Ga ⊗ C through Eqs. (2.23), (2.20), respectively, and suppose that F, σ (together with D, R± ) satisfy the relations (2.24)–(2.28). Then there exists a vertex operator 8 ∈ Ga ⊗ V for G such that 2
1
8 8 = F 1a (8)
,
8 f = σ(f) 8 .
(2.29)
In particular, the invertible element 8 ∈ Ga ⊗ V has the properties (2.14)–(2.15) and the algebra V generated by its components is associative. V may be identified with the L algebra of operators on the model space M = I V I , as before. Proof. Let us only sketch the proof since it is based on the same computations that are involved in the proof of Proposition 1. The construction of 8 starts from Eqs. (2.29). In e which is generated by components fact, one can use them to build an abstract algebra V e of an object 8 ∈ Ga ⊗ V and elements in C such that the two relations (2.29) hold. The properties (2.24), (2.26) ensure this algebra to be well defined and associative. Due e admits a consistent ∗-operation which makes 8 unitary in the sense to Eqs. (2.28), V e is defined by Eq. (2.19). With of Eq. (2.15). In the next step, an element N ∈ Ga ⊗ V the help of Eqs. (2.25) one proves that N obeys the relations (2.8), (2.16) and hence e contains G as a subalgebra. This subalgebra is finally used to analyze a concrete that V e and to show that V e ∼ representation of V = V = End(M); hence, components of 8 become operators on the model space M. Let us apply Proposition 2 to the example of G ∼ = Uq (G). To this end we need to define appropriate candidates for F and σ which is achieved with the help of the Clebsch–Gordan maps C[T L|S] : V T ⊗ V L → V S and the 6j-symbols { .. .. .. } of Uq (G). Within the space V L of highest weight L, we fix a basis of eigenvectors eL λ for the Cartan subalgebra with eigenvalues λ and denote the associated Clebsch–Gordan S ]. Now define F, σ such that coefficients by [ Tϑ L λ ς F T L = (τ T ⊗ τ L )(F )
and
σ L (b p) = (τ L ⊗ id)(σ(b p))
TL Fϑλ,ϑ 0 λ0 =
X
∗ T S { Tpˆ L pˆ +ϑ+λ pˆ +λ } [ ϑ0
have matrix elements
L S ],
λ0 ς
(2.30)
S,ς
σ L (b p)λ,λ0 = (b p + λ) δλ,λ0 .
(2.31)
Here b p is a rank(G)-dimensional vector of elements in C with τ K (b p) = K. Other notations and conventions are explained in Appendix A.2. Proposition 3. (Vertex operators for Uq (G)) There exist vertex operators 8q for the deformed universal enveloping algebras Uq (G) such that 2
1
8q 8q = F 1a (8q )
,
8q f = σ(f) 8q .
Here F is built up from the 6j-symbols and the Clebsch–Gordan maps of Uq (G) as in Eq. (2.30) and σ is given by (2.31).
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The statement follows directly from Proposition 2 once the relations (2.24)–(2.28) have been checked to hold for F, σ. The latter is done in Appendix A.2. Let us mention that formulae similar to (2.30) were considered in [12, 18]. 2.6. Vertex operators for Zq . To conclude our discussion of vertex operators, let us provide an explicit formula for 8 in our standard example G = Zq . Let us fix a set of normalized basis vectors |si, s = 0, . . . , p − 1, for the one-dimensional carrierL spaces V s of the representations τ s . They span the p-dimensional model space M = s V s . b ∈ End(M) by On this space one can introduce a unitary operator Q b |p − 1i = |0i Q
and
b |si = |s + 1i Q
for all s = 0, . . . , p − 2. This operator obeys Weyl commutation relations with the b h = h Q. b With the help of Q b and the characteristic projectors generator h ∈ Zq , i.e., q Q s P introduced in Subsect. 2.1 we are able to define 8: X X bs = 1 b s ∈ Ga ⊗ End(M) . 8 := Ps ⊗ Q q −st ht ⊗ Q p s s,t b and the Weyl relations of Q b and h that 8 obeys all the It follows from the unitarity of Q defining properties of a vertex operator (as we explained in Subsect. 2.2, the element S in Eq. (2.15) becomes trivial for G = Zq ). One may then compute the structure data. To this end it is convenient to employ the operator pb introduced in Subsect. 2.2 such that p . Since the commutative algebra Zq is isomorphic to its center C, all elements in h = qb Zq can be regarded as elements of C and we use our standard notational conventions p for h, b whenever we do so, in particular we shall use h = qb p ∈ C. We also introduce b ς b b and h imply an anti-Hermitian operator ςb by Q = e , so that the Weyl relations for Q [b p , ςb] = e. Within these notations our basic objects look as follows: p ∈G, h = qb
p ⊗b ς p 2 ∈ C , 8 = eb v = q −b ∈ Ga ⊗ End(M) .
Now expressions for the structure data may be obtained by short computations, F =e⊗e⊗e,
p ) = h−1 ⊗ qb p = q −b p ⊗ e+e ⊗ b p, σ(qb
p D = q −2 (b
2
⊗ e−b p⊗b p)
p⊗b p⊗e and R± = qb . Let us remark that, although the example of vertex operators for Zq is fairly trivial, it nevertheless shares some features with the case of G = Uq (G). Indeed, the ribbon element p (b p+ρ) [25], where b p ∈ C ⊗r is a r = rank(G)-dimensional of Uq (G) is given by v = q −b K vector such that τ (b p) = K and ρ is the sum of the positive roots. Our above formula (2.31) means that
b b p, b ⊗e+e⊗b σ(b p) = H p , D = (χ ⊗ e) · q −2H⊗ b is a vector of elements in the Cartan subalgebra such that He b L = λeL and where H λ λ the element χ ∈ G = Uq (G) can be worked out easily with the help of Eq. (2.23). Such expressions, or special cases thereof, may be found in [23, 20, 2, 21, 18]). The element F and the vertex operators 8 are certainly quite non-trivial for Uq (G) (for some explicit examples see [33, 22, 19, 18]).
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3. A Toy Model for the Discrete WZNW Theory In the rest of this paper we shall apply the theory of modular Hopf algebras and their vertex operators to construct and investigate the lattice WZNW-model. We start with a simple toy model for which the lattice consists of only one site and one edge (see Fig. 1). When we discuss the general notion of lattice current algebras in Sect. 4, we shall understand that they contain chiral observables M (the chiral monodromies) being assigned to the edge. r8
M
Fig. 1. Single-vertex lattice. Chiral observables M are assigned to the edge while chiral vertex operators 8 sit on the vertex
3.1. Properties of chiral vertex operators. Later in the text we shall find that the global chiral observable M in the lattice current algebra obeys the following relation 2
1
M R+ M = R− 1a (M ) ,
(3.1)
where R± , 1a are attributes of the modular Hopf algebra G as before. Components of M generate an algebra J with center denoted by C. Equation (3.1) reminds us of the defining relation (2.6) for the universal element N , which contains all the information about the structure of G. Indeed, the only difference is that the R+ on the l.h.s. of Eq. (2.6) has been replaced by R− . A short computation reveals that we can pass from Eq. (3.1) to (2.6) by rescaling M with the ribbon element va = (v ⊗ e) ∈ Ga ⊗ G. This implies that N 7→ va M provides an isomorphism of the algebras G and J . In particular, the commutation relations for M , 2
1
1
2
−1 −1 R± M R+ M = M R− M R∓
(3.2)
coincide with Eqs. (2.7) for the element N . The isomorphism of J and G certainly implies that there is a ∗-operation on J given by the formula (2.12) with N replaced by M (notice that the factor va is unitary). The lattice theories, however, choose a different conjugation which we discuss in Subsect. 3.3 below. Now let us introduce a vertex operator 8 for J ∼ = G. It will be called chiral vertex operator of the toy model and its properties can be copied from the relations (2.14) -(2.19) when we keep in mind to replace N by va M , η 8 = 8 10 (η), 2
1
8 8 = F 1a (8), D 8 = va 8 M,
1
2
2
1
M 8 R− = 8 R+ M , 2
1
1
(3.3)
2
R± 8 8 =8 8 R± ,
(3.4)
8 f = σ(f) 8 for all f ∈ C.
(3.5)
Here η ∈ J , C stands for the center of J , and we used the same notations as in the previous section. The components of 8 ∈ Ga ⊗ V give rise to the algebra V of chiralL vertex operators. Together with components of M , they act on the model space M = I V I.
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103
We refer to the first equation in (3.4) as operator product expansions (OPE) for 8 and call F the universal fusion matrix. The second formula in (3.4) follows from the operator product expansions; it describes braid relations for the chiral vertex operators and hence leads to interpret R± as the braiding matrix of our model. 9 There exists a nice pictorial presentation for the described algebraic structure. Definitions for the basic objects – except from D, M – are given in Fig 2. Pictures for M and D may, in principle, be constructed with the help of Eq. (2.23), Eqs. (3.5) and an appropriate presentation of the ribbon element. From the basic blocks we can built up the equations (3.3)–(3.5) as in Fig 3. All these pictures are separated by a thick solid line into left and right halves with dotted lines appearing on the left side while thin solid lines exist only on the right side. Our graphical rules are the same as in [39], and, in their terminology, the dotted lines may be said to live in the shadow world. ppppp
pppp @
:= 8
ppppp ppppp ppp ppp p p p p p p p p : = R+ ppppp ppppp
ppppp
ppppp
: = 1a (8)
ppppppp p p p p p p p p p ppppfp
ppppp ppppp p p pppppppp : = R− ppppppppppppppppp : = F ppppp pppp pppp pppppppp ppp
η
ppppp
p @ @
:= f
:= η
ppppp
p pppppp p p p p p p p p p p p p p p ppp p p fp p p p p ppp
: = σ(f)
η@
: = 1 (η)
@
0
Fig. 2. Graphical presentation of our basic objects. Pictures for D and M exist as well, but they are more complicated (cf. remarks in the text)
3.2. Second chirality. What we have discussed so far will be relevant for right chiral objects in the discrete WZNW model. Now we have to describe an analogous construction for the left chiral sector of the theory. To distinguish the two chiralities, we mark the objects of the previous subsection by an extra index r so that M r = M, 8r = 8, Fr = F, σr = σ . . . etc. Their left chiral counterparts will have an index l. To introduce left chiral vertex operators 8l we follow the same strategy as in the previous subsection. Namely, we postulate algebraic relations for an object M l (which will be justified in Sect. 4) and use them as the basic input for our left chiral theory. So let us assume that we are given some object M l such that 1
2
l l l M R− M = R+ 1a (M ) .
(3.6)
The algebra generated by components of M l will be denoted by J l and we use the symbol C l for its center. It is easy to see that the properties of va−1 M l coincide with those of the element e introduced in Subsect. 2.2, Eq. (2.9). This holds, in particular, for the commutation N relations, 1 2 2 1 −1 l l l −1 l (3.7) R± M R − M = M R+ M R∓ . 9 This will become clearer in the full lattice theory where braid relations of vertex operators assigned to different sites contain only R± and the factor R± is absent. Observe also that in the quantum non-deformed limit, i.e., γ → 0, ~ 6= 0, q = ei~γ → 1, the R-matrix R± approaches e ⊗ e whereas the limit of R± is non-trivial (cf. also [2, 19]).
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A. G. Bytsko, V. Schomerus
ppppp ppppp
ppppp
ppppp
p ppp @ @
(3.4)
=
pppp pp p pp pp p p p p pppp p pfp @ @ pppp
pppp
ppppp ppppp ppppppp ppppp p @ @
=
pppp
(3.3)
=
pppp
pp pp pp pp pp pp p
=
8−1 8 = e ⊗ e
pppp p p p p p p p p p p p p p p pfp p p ppp @ @
(3.5)
pp η @@
@ppppp ppppppppp @ ppppp
pp η@ @
=
pppp @ @ppppp pppp @
e⊗e=88
η
p pp pp p p p p p p p p pfp p
ppppp
=
−1
p p ppp p p p p p p p p p p pfp η
Fig. 3. Pictorial presentation of some basic relations. Only the left equations in (33), (34) and the right equation in (35) are depicted. The figure in the lower right corner means that f ∈ C is central in G. More rules are explained in the text.
e , i.e., Thus, the algebra J l is isomorphic to the algebra generated by components of N to Gop ( op means the opposite multiplication, cf. Subsect. 2.2). Since Eqs. (3.6)–(3.7) differ from the properties of M r , the relations for the left chiral vertex operators will differ from those we had in the right chiral sector. The consistent definition of the left vertex operators is provided by the following list of fundamental relations: 1
η 8l = 8l 1(η), 1
2
1
2
8l R− M l =M l 8l R+ ,
2
1
2
2
(3.8)
1
8l 8l = Fl 1a (8l ),
Rl± 8l 8l =8l 8l R± ,
Dl 8l = va−1 8l M l ,
8l f = σl (f) 8l for all f ∈ C l .
(3.9) (3.10)
Components of 8l ∈ Ga ⊗ V l generate algebra V l of left chiral vertex operators and L the l ∼ I act on the left model space M = I V . Starting from the defining equation (3.6) for M l one may check that the exchange relations (3.8) describe a consistent transformation law of the vertex operators 8l . It is then clear that the left vertex operators obey Eqs. (3.9)–(3.10) with some appropriate structure data Fl , σl , Dl , Rl± . The consistency relations for the left structure data can be worked out in analogy to our discussion of Proposition 1. For more detailed explanations see Appendix A.3. Let us now combine the two chiral theories by constructing their tensor product so that all operators act on the space Ml ⊗ Mr with trivial action of the right chiral objects on the first tensor factor and vice versa. In terms of exchange relation this corresponds to 1
2
2
1
8r 8l =8l 8r , 1
2
2
1
8r M l = M l 8r ,
1
2
2
1
r l l r M M =M M , 1
2
2
(3.11)
1
8l M r = M r 8l .
(3.12)
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105
Components of the chiral vertex operators 8l , 8r generate an algebra W = V l ⊗ V r . Although this combination of chiral theories appears to be quite trivial, it sets the stage for the construction of the quantum group valued field g that we are about to discuss in Subsect. 3.4. Before we get there, let us explain how to incorporate our new left chiral objects into the graphical presentation discussed at the end of the previous subsection. The pictures for the left chiral theory are simply mirror images of those in Figs 2, 3, that is, left chiral objects have their dotted lines on the right side and thin solid lines on the left side of the thick solid line. To present the tensor product of the left- and right theory, we draw all objects into the same pictures. Now there are dotted and solid lines on both sides. If we add the rule that these lines of different style do not interfere, we obtain commutativity of the two chiralities as expressed in Eqs. (3.11)–(3.12). 3.3. ∗-operation for chiral vertex operators. In principle, a ∗-operation for M l , M r and the associated vertex operators could be introduced along the lines of Sect. 2. But as we indicated the lattice models choose a slightly different conjugation. Its description requires to introduce a new object. By definition, the models spaces Ml , Mr carry an action of the modular Hopfalgebra G. With the help of the co-product 1 this gives rise to a canonical action of G on the tensor product Ml ⊗ Mr and hence to an embedding ι of the quantum algebra G into the algebra W = V l ⊗ V r of chiral vertex operators. For the exchange relations of ι(ξ) and chiral vertex operators, our construction implies: ι(ξ) 8r = 8r 10ι (ξ),
ι(ξ) 8l = 8l 1ι (ξ)
for all ξ ∈ G; we used 1ι (ξ) = (id ⊗ ι)1(ξ) and similarly for 10ι . These relations imply that 8l , 8r transform covariantly with respect to our new action ι of G on W. They can be rewritten in the R-matrix formulation, 1
2
2
1
r r N ± 8 =8 R± N ± ,
1
2
2
1
l l N ± 8 =8 N ± R± ,
where N± = (id⊗ι)(R± ) ∈ Ga ⊗W. In our pictorial presentation the objects N± would appear as over-/under- crossings of thin and thick solid lines. Hence, they have thin solid lines on both sides of the boundary between the left and the right world. This corresponds to the fact that components of N± act nontrivially on both factors in Ml ⊗ Mr , that is, they are not chiral. The same holds true for the product N = N+ (N− )−1 . Now we are prepared to describe the ∗-operation which is relevant for the toy model. To this end, we build an object Sι with the help of ι by Sι = (id ⊗ ι)(S) ∈ Ga ⊗ W and S ∈ Ga ⊗ G is defined as in Subsect. 2.2. It is used to extend the ∗-operation on G∼ = ι(G) ⊂ W to the algebra of chiral vertex operators: (8r )∗ = Sι−1 (8r )−1 , (8l )∗ = Sι (8l )−1 . The first formula looks familiar already and since the exchange relations of ι(ξ) with 8r coincide with Eq. (2.14), consistency need not to be checked again. The second formula is a variant of Eq. (2.15) which is adapted to the algebraic properties of the left chiral theory. To prove that it is consistent one has to modify our discussion in Subsect. 2.3 slightly. We leave this to the reader. It remains to show that the adjoints of 8l and 8r commute; this is not obvious at all, since Sι is not a chiral object. Commutativity of the adjoints may be seen most easily if we rewrite the adjoints in the form (2.15) which involves conjugation with κ (which is ι(κ) in our case). Then the desired consistency
106
A. G. Bytsko, V. Schomerus
follows from the transformation law of vertex operators under the action of ι(ξ) and the Yang–Baxter equation (see also [6]). It follows from Eqs. (3.5), (3.10) that the conjugation acts on the chiral monodromies M r , M l according to (M r )∗ = Sι−1 (M r )−1 Sι ,
(M l )∗ = Sι (M l )−1 Sι−1 .
We shall rediscover such a behaviour for the chiral monodromies of the lattice theory in Subsect. 4.3. 3.4. Quantum group valued field g. So far we have reached a good level of understanding for our right- and left chiral theories which act naturally on the tensor product Ml ⊗ Mr of chiral model spaces. In this subsection we would like to have a closer look at the diagonal subspace M ¯ V K ⊗ V K ⊂ Ml ⊗ M r . H= K
While components of M , M leave H invariant, this is certainly not the case for the vertex operators 8l , 8r . Nevertheless, the vertex operators can be combined into a new object g which admits restriction to the diagonal subspace H. The construction of g requires careful preparation. Let us begin this with some remarks on the center C of G (recall that C r ∼ = Cl ∼ = C). First, observe that C is spanned by J the characteristic projectors P of irreducible representations τ J of G, i.e., by projectors P J ∈ C which obey τ K (P J ) = δK,J . Notice also that the antipode S maps the element ¯ ¯ P K ∈ C to the characteristic projector P K ∈ C of the conjugate representation τ K , i.e., ¯ 10 K K S(P ) = P . Returning to our toy model, we combine the canonical isomorphism ν : C r → C l and action of the antipode S into a map Slr : C r → C l , Slr (f) = S(ν(f)). With the help of this map we can characterize the diagonal subspace H as a subspace generated by all vectors φ ∈ Ml ⊗ Mr such that fφ = Slr (f)φ holds for all f ∈ C r . In this language, the restriction to H means to impose the constraint f = Slr (f) for all f ∈ C r . This constraint couples the two chiralities and it seems natural to restrict the choice of the left- and right structure data Fα , σα , Dα , Rα ± at the same time. Notice that they were completely independent until now, as long as they solved the appropriate consistency relations. So let us agree to adjust the choice of the structure data for the left chirality to whatever we use in the right chiral part such that l
(2) Fl = Slr (Fr0
Rl±
=
r
−1
),
(2) Slr (Rr± 0 ),
(1) Dl = Slr (Dr−1 ) ,
σl (f) =
(n) (n−1) := (S −1 ⊗ Slr ) and with Slr
(1) Slr (σr
◦
(3.13) −1 Slr (f))
(0) Slr := Slr ,
,
(3.14) (3.15)
and the prime on Fr and Rr± denotes permutation of the first two tensor factors in Ga ⊗ Ga ⊗ C. It is not difficult to show that these formulae give consistent structure data for the left chiral theory (cf. also Appendix A.3). The motivation for Eqs. (3.13), (3.14) comes from the construction of the quantum group valued field g. So let us define g := Sa (8l ) 8r ∈ Ga ⊗ W , 10 Strictly speaking, the conjugate of τ K is obtained with the help of a transpose ¯ ¯ is isomorphic to τ K (this property defines the label K).
(3.16) t
as t τ K ◦ S. The latter
Vertex Operators – From a Toy Model to Lattice Algebras
107
where Sa (8l ) ≡ (S ⊗ id)(8l ). The element g indeed preserves the constraint which we discussed above, that is then g f = g Slr (f) for all f ∈ C r .
if f g = Slr (f) g,
(3.17)
Therefore, components of g map the diagonal space H into itself. This remarkable property is established by a straightforward computation (see Appendix A.4). To study properties of g it is helpful to have some knowledge about the object Sa (8l ). Simple applications of the standard Hopf algebra axioms allow to deduce 2
Sa (8l ) ξ = 1ι (ξ) Sa (8l ),
1
2
1
R+ Sa (8l ) M l = M l R− Sa (8l ),
Sa (8l ) = (8l )−1 θl
with
θl ∈ G a ⊗ C l .
(3.18) (3.19)
Here ξ ∈ J l ∼ = G in the first equation, Sa (8l ) is a shorthand for (id ⊗ Sa )(8l ), and the relation (3.19) may be regarded as a definition of θl . The transformation laws of vertex operators show that θl commutes with e ⊗ ξ ∈ Ga ⊗ J l , and hence θl ∈ Ga ⊗ C l . We in terms of Fl . If we assume for simplicity can actually give an explicit formula for θl P that a (8l ) = e (cf. Subsect. 2.4), then θl = ς fς1 S(fς2 ) ⊗ fς3 , where fςi come from the P expansion Fl = ς fς1 ⊗ fς2 ⊗ fς3 . 2
2
Proposition 4 (Properties of g). Let g denote the object defined in Eq. (3.16) and restricted to the subspace H. This element g ∈ Ga ⊗End(H) obeys the following relations: 2
1
g g = 1a (g) , 1
2
2
1
r r M g R− = g R+ M , l
r
M g=gM ,
2
1
1
2
R± g g = g g R± , 1
2
2
1
l l M R− g = R+ g M , −1
vgv
= g,
(3.20) (3.21) (3.22)
where in the last line v = vr v−1 is a combination of the ribbon elements vr ∈ C r and l l vl ∈ C . Moreover, g is normalized, a (g) = e, and invertible with inverse g −1 = Sa (g). Proofs of all these relations are given in Appendix A.4. Equations (3.20) mean that g obeys the defining relation of a quantum group F = Funq (G). More precisely, components of g generate the dual of the quantum algebra G. The elements M r , M l furnish algebras of left- and right-invariant vector fields for F and they are related to each other by means of Eq. (3.22). All these equations are well known in the theory of quantum groups. In more geometric terms, they describe the deformed co-tangent bundle Tq∗ G [2]. 11 Let us now explain the pictorial presentation of the object g (see Fig 4). First, recall that so far left and right chiral objects lived on the same plane but on different sides of the thick solid line and there was no interaction between them. But if we want to consider the restriction from Ml ⊗ Mr to the diagonal subspace H, we have to modify the rules. Namely, the restriction f = Slr (f) enforces us to change the topology by gluing the plane into a cylinder. Then we can join ends of dotted lines from both sides of the thick solid line and thus combine objects of different chirality. This is demonstrated by the graphical presentation of g in Fig. 4 (the dashed line continues the dotted line around 11
Similarly, the algebra generated by components of bundle T ∗ B of the Borel subgroup of G [19].
8r , M r
only, is a deformation of the co-tangent
108
A. G. Bytsko, V. Schomerus
the back side of the cylinder and, hence, cannot interfere with any line on the front side). Fig. 4 also sketches the proof of the operator product expansions of g in Eq. (3.20). Before concluding this subsection we would like to compare our construction of g with the one discussed in [2]. There, two decompositions of g into triple products of elements, g = u Q−1 v = u0 Qv0 , have been provided. All operators which appear in these relations act on the diagonal subspace H. The variables v0 , u0 are chiral observables, i.e., 1 2 2 1 u0 ∈ Ga ⊗ J l , v0 ∈ Ga ⊗ J r , and hence they commute with each other: u0 v 0 = v 0 u0 . ¯ Notice that components of u0 , v0 leave the subspaces V I ⊗ V I ⊂ H invariant and hence their actions are, in principle, expressible through the chiral objects M α (in particular, u0 , v0 are not to be confused with our vertex operators). The exchange relations of u0 (respectively v0 ) can be controlled only after multiplication with Q ∈ Ga ⊗ End(H). In fact, the elements u = u0 Q and v = Q v0 possess the same exchange relations as our chiral vertex operators. On the other hand, they are certainly not chiral any more (because chiral vertex operators cannot act on H). In particular, u does not commute with v. One may think of u (and similarly of v) as a left chiral vertex operator dressed with a right chiral factor which leaves the quadratic relations unchanged and, at the same time, produces an operator acting on H. Our construction in terms of chiral vertex operators and the restriction from M to the diagonal subspace H is similar to [26, 35].
@ @ ppp
pppp
pppp
l
pp
: = Sa ( 8 )
@ @ ppp pppp
p @ @
pppp
pp
1a (g)
pppp
=
@ @ ppp pppp p @ @
pppp
@ @p @ @ ppp ppppppp pppp p pppp pp pppp ppppp pppp p @ @ @ @
:= g
r
pp
pppp
=
=
@ @pp @ @pppppppp p p p
p ppp p p p p p p @ @ @ @
@ @p ppp pp @ @p @ ppp pp @ @ @
=
pp
=
2 1
gg
l
Fig. 4. The definition of g in terms of 8 and Sa (8 ) is shown on the left side. The right side of the figure is a pictorial proof of the operator product expansion for g (first equation in (3.20))
3.5. Toy model for Zq . It is instructive to realize the constructions of the toy model in the case of G = Zq . Now we have two commuting copies, hα , α = r, l, of the element h (see Subsect. 2.1) generating the chiral algebras J α . We can also introduce Hermitian pα . To introduce the chiral monodromies M r and M l we operators pbα such that hα = qb e . Since M r and M l differ from use the expressions (2.13) for the elements N and N them only by factors va−1 and va , we get p M r = qb
2
⊗ e+2 b p⊗b pr
,
p M l = q −b
2
⊗ e−2 b p⊗b pl
.
(3.23)
Vertex Operators – From a Toy Model to Lattice Algebras
109
The reader is invited to verify the functorial properties (3.1), (3.6) for these objects (in fact, the check repeats the computations performed in Subsect. 2.2). As was explained in Subsect. 2.2, the element Sι is trivial in the case of Zq , therefore the chiral monodromies are unitary. The components of M α act on the model spaces Mα . ¯ Next we need to construct the diagonal subspace H = ⊕V K ⊗V K . It can be seen from the explicit formula for the characteristic projectors (2.3) that Sa (P s ) = P −s , i.e., the representation conjugate to τ s is τ −s (where s is taken modulo p, q p = 1). Therefore, H = ⊕|− si ⊗ |si is a p-dimensional subspace in p2 -dimensional space Ml ⊗ Mr . pl φ for Using the operators pbα , we can characterize the subspace H as follows: pbr φ = −b all φ ∈ H. Now we employ the construction for vertex operators which we provided in Subb α , α = r, l be unitary operators acting on Ml ⊗ Mr such that sect. 2.5. Let Q 0 00 b l |s0 i ⊗ |s00 i = |s0 + 1i ⊗ |s00 i. It is convenient to b r |s i ⊗ |s i = |s0 i ⊗ |s00 + 1i and Q Q ςα b α = eb introduce also two operators ςbα by Q . With these notations it is easy to verify that p−1 X α p ⊗b ςα b αs = eb Ps ⊗ Q ∈ Ga ⊗ End(Mα ) , α = r, l 8 = s=0
are right- and left chiral operators obeying all the properties spelled out in Subsects 3.1 and 3.2, respectively. In particular, the R-matrix commutation relations in (3.3), (3.8) boil p ⊗b ςα ±2 b pr p⊗b p ⊗ e ±2 b pr e ⊗ b ςα q p⊗e⊗b = q ±2b q p⊗e⊗b e p ⊗b . down to Weyl relations: ee ⊗ b Recall that the universal R-matrices in (3.3), (3.8) are regarded as elements in Ga ⊗ p⊗b p⊗e Ga ⊗ End(H) with trivial entry in the third tensor factor; hence, the factor q ±2b converts R∓ into R± (cf. Subsect. 2.2). Now, applying (3.16), we get an explicit expression for g: g=
p−1 X
p ⊗ (b ςr −b ςl ) b rs Q b l−s = eb Ps ⊗ Q .
(3.24)
s=0
This object manifestly maps the diagonal subspace into itself and hence we may regard g as an element in Ga ⊗ End(H). The operator product expansion (3.20) is obvious (see the analogous computation for N± in Subsect. 2.2). Moreover, the first equation in (3.22) is again of Weyl-type (notice that here the factors va∓1 of M α are essential): 2 p 2 ⊗ e −2 b pl b ςr −b ςl ) p 2⊗ e b ςr −b ςl ) −2 b pl 2 b M l g = q −b q p⊗b ep ⊗ (b = q −b ep ⊗ (b q p⊗b q p ⊗e = 2 2 p ⊗ (b ςr −b ςl ) b p⊗b pl p ⊗ (b ςr −b ςl ) b p⊗b pr q p ⊗ e q −2 b = eb qp ⊗ e q2 b = g Mr . = eb
In the last line we used the constraint pbr = −b pl valid on the diagonal subspace. To conclude, we notice that in the Zq case the vertex operators and the field g are unitary. 4. Review on Lattice Current Algebras In the previous section we have considered the toy model for the WZNW theory which certainly did not go much beyond the theory of vertex operators for quasi-triangular modular Hopf algebras (except that we had two commuting copies of this theory). Vertex operators for the infinite dimensional current algebras of the WZNW-model depend in addition on a spatial coordinate x. This brings new locality features into the theory. Our
110
A. G. Bytsko, V. Schomerus
aim is to describe them for a lattice regularization of the WZNW-model developed in [4, 5, 26, 33, 6], where the spatial coordinate assumes the discrete values, x = 2πn/N , n = 0, .., N − 1. We begin this discussion with a brief review on lattice current algebras KN . Our notations are close to those adopted in [6]. 4.1. Definition of lattice current algebras. Let us consider a one-dimensional periodic lattice which consists of N vertices. It is convenient to enumerate the vertices from 0 to N − 1 and the corresponding edges from 1 to N as shown below. G0
G1
r
G2
r
J1
GN −1
r
G N ≡ G0
r
J2
r
JN
Fig. 5. N -vertex periodic lattice. Each site is equipped with a copy of the symmetry algebra G. The discrete currents Jn are assigned to edges
According to the ideology of [6], the definition of the algebra KN involves two kinds of objects – those associated with the sites and those associated with the edges. The nth site of the lattice is equipped with a copy Gn of the algebra G and copies for different sites commute. In other words, KN contains Gn and the whole tensor product G ⊗N as subalgebras. The canonical isomorphism of G and Gn ⊂ G ⊗N furnishes the embeddings ιn : G 7→ G ⊗N for n = 0, .., N − 1: ιn (ξ) = e ⊗ . . . ⊗ ξ ⊗ . . . ⊗ e ∈ G ⊗N for all ξ ∈ G , where the only nontrivial entry of the tensor product on r.h.s. appears in the nth position. The definition of KN also involves generators Jnr , n = 1, .., N, (the right currents) which are discrete analogues of the continuum holonomies along the edges (cf. Introduction). Definition 2 ([6]). The lattice current algebra KN is generated by components of invertible elements Jnr ∈ Ga ⊗ KN , n = 1, .., N along with elements in G ⊗N . These generators are subject to the following relations : 2
1
r r r r ∗ −1 r −1 J n J n = R− 1a (J ) , (Jn ) = Sn (Jn ) Sn−1 , 1
2
2
1
1
2
2
r r r r r r r r J n+1 J n =J n R+ J n+1 , J n J m =J m J n f or n 6= m, m ± 1 (modN ),
ιn (ξ) Jnr = Jnr 10n (ξ) , 10n−1 (ξ) Jnr = Jnr ιn−1 (ξ) for all ξ ∈ G, ιm (ξ) Jnr
=
Jnr ιm (ξ)
(4.1)
1
(4.2) (4.3)
for all ξ ∈ G, m 6= n, n − 1 (modN ).
Here R± denote the elements R± ⊗ e ∈ Ga ⊗ Ga ⊗ KN , Sn = (id ⊗ ιn )(S) ∈ Ga ⊗ Gn ⊂ Ga ⊗ KN with S defined as in (2.10), and 10n (ξ) = (id ⊗ ιn )(10 (ξ)), where 10 (ξ) = P 1(ξ)P as usual. Invertibility of Jnr means that there exists an element (Jnr )−1 ∈ Ga ⊗ KN such that Jnr (Jnr )−1 = e ⊗ e = (Jnr )−1 Jnr . The lattice current algebra KN contains a subalgebra JNr generated by components of the currents Jnr only. They are subject to relations (4.1)–(4.2). The full lattice current algebra KN can be regarded as a semi-direct product G ⊗N n JNr , where the action of G ⊗N on JNr is given by the covariance relations (4.3). Taking into account the quasi-triangularity of R± , we obtain the following consequence of (4.1) 1
2
2
1
R± J rn J rn R∓ =J rn J rn .
(4.4)
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111
These R-matrix relations for the description of the lattice Kac–Moody algebras have been introduced first in [4]. Following our discussion in Subsect. 2.2, one can introduce the objects Nn,± = (id ⊗ ιn )(R± ) ∈ Ga ⊗ Gn ⊂ Ga ⊗ KN , which obey the standard relations (2.5). They are used to rewrite the relations (4.3) in the following R-matrix form: 1 2 2 1 2 1 1 2 r r r r (4.5) N n,± J n =J n R± N n,± , J n N n−1,± = R± N n−1,± J n . 4.2. Left currents. The continuum WZNW model possesses two chiral subalgebras, that is, along with the (right) current j r (x) it involves the left current j l (x) such that left and right currents commute. A nice feature of the lattice current algebra KN is that it already contains the second chirality in an encoded form. Indeed, one may introduce the following new variables Jnl ∈ Ga ⊗ KN : −1 Jnl := va2 Nn−1,+ Jnr Nn,− .
(4.6)
In the notations of Definition 2 they obey (see [6] for details) 1
2
l l l l ∗ l −1 −1 J n J n = R+ 1a (Jn ), (Jn ) = Sn (Jn ) Sn−1 , 1
2
2
1
1
2
2
(4.7)
1
l l l l l l l l J n R− J n+1 =J n+1 J n , J n J m =J m J n for n 6= m, m ± 1 (modN ) , (4.8) 1
2
2
1
l r r l J n J m =J m J n for all m, n,
ιn (ξ) Jnl = Jnl 1n (ξ),
1n−1 (ξ) Jnl = Jnl ιn−1 (ξ) for all ξ ∈ G,
ιm (ξ) Jnl = Jnl ιm (ξ)
for all ξ ∈ G, m 6= n, n − 1 (modN ).
(4.9) (4.10)
Due to these properties, the objects Jnl may be interpreted as left counterparts of the right currents Jnr . Notice that there is a manifest symmetry between the defining relations for the right currents and the properties of left currents. It underlines the fact that left and right chiralities in the WZNW model appear on an equal footing. In fact, (4.7)–(4.10) could be regarded as an alternative definition of the lattice current algebra KN . It also follows that JNl and JNr , i.e., the algebras generated by components of Jnl and r Jn , respectively, are commuting subalgebras in KN and JNr is isomorphic to (JNl )op . Here the subscript op means opposite multiplication as before. 4.3. Holonomies and monodromies. The currents Jnα , α = r, l were defined as discrete analogues of holonomies along the nth edge. Similarly, one may introduce the holonomies along the link connecting the 0th and the nth sites : Unα := J1α . . . Jnα ,
n = 1, . . . , N − 1 .
(4.11)
As one might expect, the properties of such holonomies are similar to those of chiral currents.12 Namely, it is easy to verify that 12 Let us mention here some subtle point in the definition of the lattice current algebra. Notice that relations (4.4) would not change if we replaced R− by R+ in the definition (4.1). However, this ambiguity disappears if we demand that Unr and Jnr have the same functoriality relation (compare (4.1) and the first equation in (4.12)). A similar subtlety appears once more in the construction of the left currents. Indeed, we could replace factor va2 by va in the definition (4.6); then we would obtain the relation (4.7) with R− instead of R+ . But in this case functorial properties of Jnl and Unl would be different.
112
A. G. Bytsko, V. Schomerus 2
1
1
r r r U n U n = R− 1a (Un ),
(Unr )∗ = Sn−1 (Unr )−1 S0 , 1
2
2
1
2
l l l U n U n = R+ 1a (Un ),
(4.12)
(Unl )∗ = Sn (Unl )−1 S0−1 ,
(4.13)
2
R± U rn U rn R∓ = U rn U rn ,
1
1
2
R± U ln U ln R∓ = U ln U ln ,
(4.14)
100 (ξ) Unr = Unr ι0 (ξ),
ιn (ξ) Unr = Unr 10n (ξ) for all ξ ∈ G, (4.15)
10 (ξ) Unl = Unl ι0 (ξ),
ιn (ξ) Unl = Unl 1n (ξ) for all ξ ∈ G, (4.16)
and Unα commute with ιm (ξ) for all m 6= 0, n. However, there is an important difference between currents and holonomies: since the latter are localized on the chain of edges that runs from the 0th vertex to the nth , the localization domains of all holonomies overlap. This is reflected in their mutual exchange relations for 1 ≤ n < m ≤ N −1: 2
1
1
2
1
r r r r U n U m = R− U m U n ,
2
2
1
l l l l U n U m = R+ U m U n .
(4.17)
As we have argued in the introduction, holonomies of chiral fields along the whole circle (i.e., the chiral monodromies) are of particular interest. In the continuum case they H are given by mα = P exp{ j α (x)dx}. Monodromies for the quantum lattice theory are defined by a natural discrete analogue of this formula, α . M α = J1α J2α . . . JN
(4.18)
Simple calculations allow to derive the following properties of the monodromies M α : 2
1
r r r M R+ M = R− 1a (M ), r ∗
(M ) = 1
2
S0−1 (M r )−1 S0 , 2
1
R+ U rn M r =M r R+ U rn ,
1
2
l l l M R− M = R+ 1a (M ) , l ∗
l −1
(M ) = S0 (M ) 2
1
S0−1
1
,
(4.20)
2
R− U ln M l = M l R− U ln , 1
2
2
R± N 0,± M r =M r R± N 0,± ,
10 (ξ) M l = M l 10 (ξ),
l l N 0,± R± M =M N 0,± R±
2
2
(4.21)
1
100 (ξ) M r = M r 100 (ξ),
1
(4.19)
(4.22)
1
(4.23)
for all ξ ∈ G and M α commute with ιm (ξ) for all m 6= 0. Now we see that the relations (3.1) and (3.6) which we postulated in the toy model construction indeed describe properties of the chiral monodromies. Our next aim is to extend the toy model to the full lattice theory. Recall that the structure data of the toy model were built from elements in the center C α of the algebra J α spanned by components of M α . Elements in these algebras C α are still central in the full lattice theory. In fact, it follows from (3.2), (3.7) that the algebras C α are spanned by the elements cIα = tr Iq τ I (M α ), where τ I runs through irreducible representations of G, and τ I (M α ) = (τ I ⊗ id)(M α ) [7]. Equipped with this explicit description of C α one concludes from Eqs. (4.21)–(4.23) that the elements cIα commute with Unα , Nn,± for all n and hence they are central elements in KN . Actually, the following stronger statement holds [6]: the elements cIα ∈ KN with I running through the classes of irreducible representations of G generate the full center of the lattice current algebra KN . This explains why the structure data for vertex operators on the lattice will be built from the commuting subalgebras C α exactly as in the toy model.
Vertex Operators – From a Toy Model to Lattice Algebras
113
4.4. Current algebra for Zq . Let us consider the current algebra in the case of Zq . The algebras Gn assigned to the sites are generated by the elements hn = ιn (h). As pn usual, we can introduce pbn such that hn = qb . For Nn,± ∈ Ga ⊗ Gn we have Nn,± = P s ±s ±b p⊗b pn P ⊗ h = q (cf. Subsect. 2.2). Next we build the chiral currents n s Jnr =
p−1 X
1 2 c r )s , q 2 s P s ⊗ (W n
s=0
Jnl =
p−1 X
1 2 c l )s q − 2 s P s ⊗ (W n
(4.24)
s=0
c r and W c l := h−1 W c r −1 from a family of unitary elements W n n n−1 n hn which obey the following Weyl-type relations c α hn , hn−1 W cnα = q W cnα = q −1 W cnα hn−1 , hn W n
(4.25)
c α hm for m 6= n, n − 1 , cnα = W hm W n r l cr W cl W cr W cnr = q W cn+1 cnl = q −1 W cnl W cn+1 , W , W n+1 n n+1 α cα W cα W cm cnα for m 6= n ± 1 . = W W n m
(4.26)
c α )p is obviously central for this algebra, we additionally impose Since the element (W n α c )p = e for all n (which is, in fact, a choice of a normalization). the condition: (W n c α is known as a lattice U (1)-current algebra [31, 28]. The algebra generated by W n c r is a special c l from the hm and W The relation we have used to obtain the elements W n n cl W cr cr cl case of formula (4.6) and it implies that W n m = Wm Wn for all pairs n, m. The functorial properties (4.1) and (4.7) of currents (4.24) can be checked in the same way as we did this for the elements N± in Subsect. 2.2. The exchange relations (4.2), (4.5) and (4.8) are again reduced to Weyl relations. Since Sn = e ⊗ e, the chiral cα. currents are unitary, (Jnα )∗ = (Jnα )−1 , which is in agreement with the unitarity of W n α b n .13 In c α = e$ To proceed, we introduce anti-Hermitian operators $ b nα such that W n these notations the commutation relations given above acquire the form: r [$ bm ,$ b nr ] = ln q (δm,n+1 − δm,n−1 ),
l [$ bm ,$ b nl ] = ln q (δm,n−1 − δm,n+1 ) ,
b nα ] = δm,n − δm,n−1 , [ pbm , $
l [$ bm ,$ b nr ] = 0 .
(4.27)
The chiral currents now can be rewritten in the following form: b p⊗$ b nr = e( 21 ln q) b p 2 ⊗ e+b p⊗$ b nr , Jnr = κ−1 a e
p⊗$ b nl = e−( 21 ln q) b p 2 ⊗ e+b p⊗$ b nl . Jnl = κa eb (4.28) Next we can construct the chiral holonomies. For this purpose the variables $ b nα are more convenient. Indeed, applying the special case of the Campbell-Hausdorff formula, 1 ea eb = ea+b e 2 [a,b] valid if [a, [a, b]] = [b, [a, b]] = 0, we easily obtain: Pn r Pn l 1 2 b k , U l = q − 21 b p⊗ $ bk . p ⊗e b p2 ⊗e b k=1 Unr = q 2 b ep ⊗ k=1 $ e n
It is obvious now why relations (4.12)–(4.16) for the holonomies copy those for the currents. The exchange relations (4.17) are again reduced to Weyl-type relations. 13 Strictly speaking, the algebra generated by $ b nα and hn (see, b nα and pbn is larger than one generated by W e.g., [28]). The latter is called the compactified form of the former.
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A. G. Bytsko, V. Schomerus
Proceeding in the same way, we can construct the chiral monodromies as M α = α α UN −1 JN , which needs again an application of the Campbell-Hausdorff formula. The result reads PN r PN l b k , M l = q −b bk . p2 ⊗e b p2 ⊗e b ep ⊗ k=1 $ ep ⊗ k=1 $ (4.29) M r = qb Bearing in mind that for Zq the quantum trace coincides with the standard one (see Subsect. 2.1), we conclude from (4.29) that the algebras C α are generated by exponentials PN of the elements b pα = k=1 $ b kα . Indeed, using the commutation relations given above, it is easy to verify that these operators commute with all elements of the currents algebra. PN b kα → ±(2 ln q) pbα in(4.29) (the sign depends Performing a formal replacement k=1 $ on the chirality), we recover the formulae (3.23) of the toy model.
5. Vertex Operators on a Lattice 5.1. Definition of WN . In Sect. 3 we have considered algebras J α and V α , α = r, l generated by components of the chiral monodromies M α and the chiral vertex operators 8α , respectively. Both chiralities together were used to generate the algebra W = V l ⊗V r of our toy model. Below we shall define an algebra WN of vertex operators on a lattice. For this purpose, we shall replace the algebras J α in the definition of W by their lattice counterparts JNα . So we assume that we are given the lattice current algebra KN with center C l ⊗ C r (recall that C α ∼ = C ∼ = center of G) and two sets of structure data , D , α = l, r, which obey the standard relations. The last tensor components Fα , σ α , R α α ± of the structure data are regarded as elements in the center of the lattice current algebra KN , i.e., we have Fα ∈ Ga ⊗ Ga ⊗ C α ⊂ Ga ⊗ Ga ⊗ KN , etc. Definition 3. [Algebra of vertex operators on a lattice] The algebra WN is generated by elements in KN and components of the vertex operators 8α 0 ∈ Ga ⊗ WN . Generators Nn , Jnα ∈ Ga ⊗ KN obey the defining relations (4.1), (4.2) and (4.5) for lattice current α algebras. The elements 8α 0 ∈ Ga ⊗ V ⊂ Ga ⊗ WN , α = r, l are subject to the following conditions: 1. They satisfy operator product expansions and exchange relations with elements in the center C l ⊗ C r ⊂ KN given by 2
1
8r0 8r0 = Fr 1a (8r0 ), 1
2
8l0 8l0 = Fl 1a (8l0 ),
8r0 fr = σr (fr ) 8r0 ,
(5.1)
8l0 fl = σl (fl ) 8l0 .
(5.2)
Here Fα ∈ Ga ⊗ Ga ⊗ C α ; σα are homomorphisms from C α to Ga ⊗ C α and fα ∈ C α . Moreover, 8α 0 are invertible and vertex operators of different chirality commute: −1 α α α −1 (8α 0 ) 80 = e ⊗ e = 80 (80 ) , α = l, r,
1
2
2
1
8r0 8l0 =8l0 8r0 .
(5.3)
2. 8r0 and 8l0 are chiral vertex operators for the algebra KN in the sense that the following exchange relations with Jnr , Nn,± ∈ Ga ⊗ KN hold:
Vertex Operators – From a Toy Model to Lattice Algebras 2
1
2
115 2
1
r r r r J 1 80 =80 R+ J 1 , 1
2
1
2
2
2
2
1
1
2
8r0 J rn =J rn 8r0 for n 6= 1, N, 1
1
8r0 J rN =J rN 8r0 R− , 1
2
r l l r J n 80 =80 J n for all n ,
1
2
1
r r N 0,± 80 =80 R± N 0,± ,
(5.4)
2
(5.5)
1
l l N 0,± 80 =80 N 0,± R± ,
(5.6)
and components of Nm,± commute with components of the vertex operators for m 6= 0. 3. The ∗-operation on KN can be extended to WN by the following prescription (8r0 )∗ = (S0 )−1 (8r0 )−1 ,
(8l0 )∗ = S0 (8l0 )−1 ,
(5.7)
where S0 = (id ⊗ ι0 )(S) ∈ Ga ⊗ G0 ⊂ Ga ⊗ KN with S ∈ Ga ⊗ G being constructed by formula (2.10). Further relations involving left currents Jnl and the monodromies M α follow and will be spelled out below. Let us underline once more that the structure data for 8α 0 are constructed from elements in the center C l ⊗ C r of KN . This is possible because both algebras C α are isomorphic to the center of G [6] (see our short discussion at the end of Subsect. 4.3). Next we would like to supplement our definition of WN by a list of consequences which follow from the stated relations. They concern exchange relations of chiral vertex operators with left currents Jnl and elements ξ ∈ Gn ⊂ KN , 1
2
1
2
1
2
1
2
l l l l l l l l J 1 80 =80 R− J 1 , 80 J N =J N 80 R+ , 1
2
2
1
8l0 J ln =J ln 8l0 for n 6= 1, N,
1
2
1
(5.8)
2
l r r l J n 80 = 80 J n for all n,
(5.9)
ι0 (ξ) 8r0 = 8r0 100 (ξ),
ι0 (ξ) 8l0 = 8l0 10 (ξ) ,
(5.10)
ιm (ξ) 8r0 = 8r0 ιm (ξ),
ιm (ξ) 8l0 = 8l0 ιm (ξ) for m 6= 0 ,
(5.11)
for all ξ ∈ G and we used the same notations as in Subsect. 4.1. The first set of relations, i.e. Eqs. (5.8), (5.9), are obtained with the help of Eq. (4.6). From our earlier discussion it is clear that the relations (5.10) are equivalent to Eqs. (5.6). All exchange relations of the chiral vertex operators with elements in KN are local in the sense that objects assigned to sites n 6= 0 or links m 6= 1, N commute with 8α 0 . This means that we can think of 8α 0 as being assigned to the vertex n = 0 and hence explains the subscripts 0 . The precise form of the nontrivial exchange relations involving 8α 0 may be understood in terms of co-actions of G on KN (see remarks in the introduction and [4, 6]). Now let us compare the definition of algebras WN of vertex operators on the lattice with our toy model. To this end we derive exchange relations between the chiral vertex α operators 8α 0 and the chiral monodromies M (see Eqs. (4.18)), 1
M
r
2
2
1
8r0 R− = 8r0 R+ M r ,
2
1
1
2
l l l l M 80 R+ = 80 R− M .
(5.12)
The answer is to be compared with the relations (3.3), (3.8) in the toy model and shows α that the objects (8α 0 , M , N0 ) of the algebra WN obey the same exchange algebra as (8α , M α , N ) in the toy model. Thus, the toy model may not only be considered as a special case of a lattice theory with N = 1 but also it is embedded as a subalgebra in all lattice algebras WN for arbitrary N . We can use this insight to rewrite some of the
116
A. G. Bytsko, V. Schomerus
relations we discussed for the toy model in terms of the corresponding lattice objects. In particular, one has 2
1
1
2
1
2
2
1
Rr± 8r0 8r0 =8r0 8r0 R± ,
Rl± 8l0 8l0 =8l0 8l0 R± ,
(5.13)
va−1
va (8l0 )−1
l
(5.14)
Dl = (va−1 vl ) σl (v−1 l ) .
(5.15)
(8r0 )−1
Dr 8r0
r
=M ,
Dr = (va v−1 r ) σr (vr ),
Dl 8l0
= M with
Here, vα ∈ C α are images of the ribbon element v ∈ G under the canonical isomorphisms from the center of G into the subalgebras C α ⊂ KN (cf. Eq. (2.23)). The latter are generated by quantum traces of monodromies, i.e., by elements of the form tr Iq τ I (M α ) (for notations see Subsect. 4.3). The elements Rα ± in Eqs. (5.13) are given through the 0 −1 α = F R (F ) ∈ G ⊗ G standard formula Rα a a⊗C . ± α ± α 5.2. Vertex operators at different sites. Definition 3 involves only vertex operators assigned to the 0th site of the lattice. We may now try to construct vertex operators 8α n ∈ Ga ⊗ WN from elements in the algebra WN which are assigned to other sites n 6= 0. In particular, they are required to satisfy the characteristic fusion and braid relations of vertex operators and, moreover, we want them to commute with all elements in KN which are assigned to sites m 6= n or edges m 6= n, n + 1. The solution to this problem is certainly not unique. In the following, we shall describe just one possible construction. The idea is to introduce the vertex operators 8α n with the help of the holonomies Unα ∈ JNα by the simple formulae: 8rn := 8r0 Unr ,
8ln := 8l0 Unl for
n = 1, . . . , N − 1 .
(5.16)
Using the relations (4.12)–(4.16) for chiral holonomies, it is easy to verify the following properties of 8α n: 1
2
2
1
r r N n,± 8n =8n R± N n,± ,
(8rn )∗ = (Sn )−1 (8rn )−1 , ιn (ξ) 8rn = 8rn 10n (ξ),
1
2
2
1
l l N n,± 8n =8n N n,± R± ,
(5.17)
(8ln )∗ = Sn (8ln )−1 ,
(5.18)
ιn (ξ) 8ln = 8ln 1n (ξ) for all ξ ∈ G ,
(5.19)
and ιm (ξ) commute with 8α n for any m 6= n. Here Sn = (id ⊗ ιn )(S) with S ∈ Ga ⊗ G as before. Next, one has to investigate fusion and braiding properties of 8α n . The obey the same relations computation (see Appendix A.5) reveals that the elements 8α n th site, i.e. as our vertex operators 8α 0 at the 0 2
1
8rn 8rn = Fr 1a (8rn ), 2
1
1
2
Rr± 8rn 8rn =8rn 8rn R± , 8α n fα
=
σα (fα ) 8α n
1
2
8ln 8ln = Fl 1a (8ln ) , 1
2
2
(5.20)
1
Rl± 8ln 8ln =8ln 8ln R±
(5.21)
for all fα ∈ C , α = l, r
(5.22)
α
α hold with structure data Fα , Rα ± , σα being identical to the structure data of 80 in Eqs. (5.1), (5.2) and (5.13). In order to get an analogue of Eqs. (5.14), we introduce the monodromies α α α . . . JN J1 . . . Jnα = (Unα )−1 M α Unα for α = r, l . Mnα := Jn+1
(5.23)
Vertex Operators – From a Toy Model to Lattice Algebras
117
They are holonomies along the whole circle which begin and end at the nth site. It is now obvious that va−1 (8rn )−1 Dr 8rn = Mnr , va (8ln )−1 Dl 8ln = Mnl
(5.24)
hold for all 0 ≤ n < N and the elements Dα ∈ Ga ⊗ C α are the same as in Eqs. (5.14), (5.15). Let us remark that the quantum traces tr Iq τ I (Mnα ) are elements of the algebras C α ⊂ KN from which we constructed our structure data. Moreover, they do not depend α ) for all pairs n, m [7]. on the index n, i.e., one can prove that tr Iq τ I (Mnα ) = tr Iq τ I (Mm It still remains to investigate the exchange relations of the vertex operators 8α n with currents Jnα ∈ Ga ⊗ KN . Details are explained in Appendix A.5; here we only state the results: 1
2
2
1
r r r r J n+1 8n =8n R+ J n+1 , 2
1
1
2
l l l l J n+1 8n =8n R− J n+1 , 1
2
2
1
2
1
2
1
1
2
1
2
2
1
8ln J ln =J ln 8ln R+ , 1
8rn J rm =J rm 8rn , 1
2
8rn J rn =J rn 8rn R− ,
2
2
1
2
1
8ln J lm =J lm 8ln for m 6= n , n + 1 , 1
8rn J lm =J lm 8rn ,
2
8ln J rm =J rm 8ln for all n, m .
(5.25) (5.26) (5.27) (5.28)
Finally, as a consequence of these relations and (5.3) we derive that 1
2
2
1
8rn 8lm = 8lm 8rn
for all n, m .
(5.29)
To summarize, we established that the construction (5.16) provides us with chiral vertex operators 8rn and 8ln which are naturally assigned to the nth site of the lattice. These vertex operators share the same structure data Fr , Rr± , . . . and Fl , Rl± , . . .. Their exchange relations with elements of the current algebra KN are local in the sense discussed above. Although the vertex operators have local relations with the observables, one should expect that they themselves are non-local. Indeed, it is easy to derive the following exchange relations (see Appendix A.5): 1
2
2
1
2
1
8rn 8rm = Rr− 8rm 8rn , 1
2
8rn 8rm = Rr+ 8rm 8rn ,
2
1
2
1
1
2
8ln 8lm = Rl+ 8lm 8ln for 0 ≤ n < m < N ,(5.30) 1
2
8ln 8lm = Rl− 8lm 8ln for 0 ≤ m < n < N .(5.31)
α th and mth site at which they So, elements 8α n and 8m do not commute even if the n are localized are far apart. The relations (5.30), (5.31) demonstrate clearly that Rα ± play the role of braiding matrices in local quantum field theory.
5.3. Extension on a covering of the circle. In Subsect. 5.2 we have listed properties of the vertex operators 8α n which are valid for 0 ≤ n, m < N . However, unlike the generators of KN , the vertex operators live on a covering of the circle, i.e., if we want α to make sense of objects 8α n with n ∈ Z, the operator 8n+N necessarily differs from α α 8n . Indeed, 8n may be defined for n ∈ Z by the following difference equation which is encoded in Eqs. (5.16): α α (5.32) 8α n+1 = 8n Jn+1 . Here we assume that Jnα has been extended periodically to n ∈ Z. Periodicity properties α of the objects 8α n can be expressed through the monodromies Mn introduced in (5.23),
118
A. G. Bytsko, V. Schomerus α α k 8α n+kN = 8n (Mn ) ,
0≤n
(5.33)
To proceed, we observe that properties of Mnα are similar to those of M α ≡ M0α . Using relations spelled out in Sect. 4, we easily find that Mnα obey the functorial relations 2
1
1
r r r M n R+ M n = R− 1a (Mn ),
2
l l l M n R− M n = R+ 1a (Mn )
(5.34)
which coincide with (4.19). Therefore, Mnr and Mnl obey the exchange relations (3.2) and (3.7). Bearing this in mind, we employ (4.21) to derive 2
1
2
1
1
8rm R+ M rn =M rn 8rm R− ,
2
2
1
8lm R− M ln =M ln 8lm R+
(5.35)
for 0 ≤ n < N and m = n (mod N ), i.e., m = n + kN , k ∈ Z. Using the properties of the monodromies Mnα , we can establish (see Appendix A.5) that relations (5.18)–(5.22), (5.24)–(5.28) are valid for 8α n with the coordinate n being replaced by n0 = n + kN . Thus, the local properties of vertex operators 8α n+kN living outside of the interval 0 ≤ n < N coincide with those of 8α n living inside this interval. The extension of the exchange relations between vertex operators to the covering of our discrete circle is slightly more subtle. For instance, the braid relation of the vertex α operator 8α n and its counterpart 8n+N does not coincide with (5.21). Instead, we find (see Appendix A.5): 2
1
1
2
Rr+ 8rn 8rn+N =8rn+N 8rn R− ,
1
2
2
1
Rl− 8ln 8ln+N =8ln+N 8ln R+ .
(5.36)
A similar situation is found for the braid relations (5.30)-(5.31). It turns out that here we need to apply Eqs. (2.25) for the structure data of the vertex operators. Let us demonstrate their role by investigating the first equation in (5.30) (i.e., the case n < m) with n replaced by n + N , 1
2
1
2
1
1
1
2
2
1
1
r r −1 r r r 8rn+N 8rm =8rn M rn 8rm =v−1 a D r 8n 8m =va D r R− 8m 8n = 2
1
1
2
1
1
1
2
1
r r r r r −1 r r r r = v−1 a R+ σ (Dr ) 8m 8n = R+ 8m va D r 8n = R+ 8m 8n+N . 1
2
1
We see that the result coincides with the first equation in (5.31), which is natural since n+N > m. Proceeding in the same way, one can show that the braid relations (5.30) and (5.31) hold, for all n, m ∈ Z such that |n−m| < N , n 6= m. Thus, the Eqs. (2.25) became an important ingredient for a self-consistent extension of the lattice theory beyond the interval 0 ≤ n < N . 5.4. Construction of the local field gn . As we have shown above, the local properties of lattice vertex operators are the same as those we studied in the toy model case. Therefore, we can repeat the construction of Subsect. 3.4 and introduce the objects gn := Sa (8ln ) 8rn ∈ Ga ⊗ WN .
(5.37)
To proceed, we need some more information about the representation theory of lattice current algebras. As we mentioned before, the algebras KN admit a series of irreducible representations on spaces WNIJ , where I, J run through classes of irreducible representations of the quantum algebra G. These spaces WNIJ are of the form M ¯ VK ⊗VK . WNIJ = V I ⊗ V J ⊗ <⊗N −1 where < = K
Vertex Operators – From a Toy Model to Lattice Algebras
119
Suppose that we describe KN in terms of the holonomies Unα , n = 1, . . . , N − 1, the monodromies M0α and the local elements Nn , n = 0, . . . , N − 1 (notice that the currents can be reconstructed from holonomies and monodromies). We divide these generators into two sets, the first containing all Unα and Nm for m 6= 0 while we put M0α and N0 into the second set. This choice is made so that objects which were not part of the toy model are separated from objects we met in Sect. 3 already. In [6], an action of KN on WNIJ was constructed for which objects in the first set, i.e., holonomies Unα and elements Nm , m 6= 0, act trivially on the factor V I ⊗ V J in WNIJ and irreducibly on <⊗N −1 . It is then straightforward to see that our algebra WN of vertex operators on the lattice possesses only one irreducible representation on the total space M WNIJ ∼ MN = = M ⊗ <⊗N −1 , I,J
where each summand WNIJ appears with multiplicity one. By now, the picture resembles very much the situation in the toy model: we have the model space MN on which WN acts irreducibly. Therefore, we may look for operators that can be restricted to the diagonal subspace M ¯ WNJJ ∼ HN = = H ⊗ <⊗N −1 ⊂ MN . J
This is certainly possible for all elements in KN . But in addition, we may restrict the field gn to HN . As in Subsect. 3.4, the diagonal subspace is characterized by the constraint f = Slr (f) for all f ∈ C r ⊂ KN . If we adjust left and right structure data according to Eqs. (3.14) 14 , the constraint to HN is compatible with the construction of gn , i.e., (3.17) holds with g replaced by gn , n = 0, . . . , N − 1. The properties of the restricted field are spelled out in the following proposition. Proposition 5 (Properties of gn ). When restricted to the diagonal subspace HN , the element gn ∈ Gn ⊗ End(HN ) obeys the following relations: 2
1
g n g n = 1a (gn ), Mnl gn = gn Mnr , gn+N = gn , 1
2
2
1
r r M n g n R − = g n R+ M n ,
1n (ξ) gn = gn 10n (ξ),
2
1
1
2
R± g n g n =g n g n R± ,
(5.38)
Sa (gn ) = gn−1 ,
(5.39)
1
2
2
1
g n g m =g m g n 1
for n 6= m ,
2
2
l l M n R− g n = R+ g n M n , 1
2
2
(5.40)
1
(5.41)
1
N n,± R± g n =g n R± N n,±
(5.42)
for all ξ ∈ G and gn commutes with all ιm (ξ) ∈ Gn ⊂ KN for m 6= n. The properties listed above, and in particular the locality and periodicity relations (5.40), allow to regard gn as an observable in the lattice WZNW-model. It is a discrete analogue of the group valued field g(x). Some remarks on the proof of Proposition 5 can be found in Appendix A.6. To complete the description of gn , let us give its exchange relations with the chiral currents. Using (5.25), we obtain 14 This can be done simultaneously for all sites, since the structure data do not depend on the lattice site n (see Subsect. 5.2).
120
A. G. Bytsko, V. Schomerus 2
1
1
2
g n J rn =J rn g n R− , 1
2
2
1
r r J n+1 g n =g n R+ J n+1 , 1
2
2
1
α α J m g n =g n J m , α = l, r
2
1
1
2
g n J ln =J ln R− g n , 1
2
2
1
l l J n+1 g n = R+ g n J n+1 ,
(5.43)
for m 6= n, n + 1 (mod N ).
5.5. Lattice vertex operators for Zq . Let us construct the algebra WN in the case of G = Zq . To this end we have to add the chiral vertex operators introduced in Subsect. 3.5 to the lattice U (1)-current algebra discussed in Subsect. 4.4. As a result we get the algebra generated by components of the following elements belonging to Ga ⊗ WN : 8α n =
p−1 X
p ⊗b ςnα b α )s = eb P s ⊗ (Q , Nn,± = n
s=0
Jnr =
p−1 X
p−1 X
±b p⊗b pn P s ⊗ h±s , n =q
s=0
1 2 p⊗$ b nr , J l = c r )s = κ− 21 eb q 2 s P s ⊗ (W n n
s=0
p−1 X
1 2 p⊗ $ b nl c l )s = κ 21 eb q − 2 s P s ⊗ (W n
s=0
where α = r, l and n = 0, .., N − 1. According to Eqs. (4.6) and (5.16), not all the generators are independent. Namely, the following relations are to be fulfilled: cnr h−1 cnl = h−1 W W n , n−1
bα = Q bα W cα . c1α . . . W Q n 0 n
Due to the Campbell-Hausdorff formula these equalities may be re-expressed in terms of the generators $ b nα , ςbnα as follows: $ b nl = $ b nr − ln q (b pn + pbn−1 ),
ςbnα = ςb0α +
n X
$ b kα .
(5.44)
k=1 α It is easy to see that all the formulae between 8α n , Jn and Nn,± spelled out in Subsects. 5.1–5.3 are satisfied if we add to Eqs. (4.25)–(4.26) or, alternatively, to Eqs. (4.27) the following relations:
bα bα bα bα hn Q n = q Qn hn , hm Qn = Qn hm for m 6= n , br cr cr br br cr cr Q br W n+1 n = q Qn Wn+1 , Wn Qn = q Qn Wn , −1 b l c l −1 b l c l cl Q cl Q bl bl W Qn Wn+1 , W Qn W n , n+1 n = q n n =q
cα Q bα bα cα W m n = Qn Wm for m 6= n, n + 1 , which can be rewritten as follows: r l [b pm , ςbnα ] = δm,n , [$ bm , ςbnr ] = −[$ bm , ςbnl ] = ln q (δm,n+1 + δm,n ) .
(5.45)
Since we already discussed properties of the vertex operators at a fixed site for the Zq theory in Subsections 2.6 and 3.5, we shall concentrate on the aspects of locality and periodicity here. Actually, the latter simplify in the case of Zq due to the circumstance that all our monodromies Mnα of the same chirality coincide (since all they are given by (4.29)). This allows to rewrite Eqs. (5.30)–(5.31) and (5.36) in the form (recall that in the case of Zq we have R± = R± = R±1 with R given in Subsects. 2.1 and 2.2):
Vertex Operators – From a Toy Model to Lattice Algebras 1
2
2
1
8rn 8rm = Rβ(n−m) 8rm 8rn ,
2
121 1
1
2
8ln 8lm = R−β(n−m) 8lm 8ln
(5.46)
for all n, m ∈ Z. Here β(n−m) = 1+2[ n−m N ] ([x] stands for the entire part of x) for n 6= ] for n = m (mod N ). In the derivation we have also m (mod N ) and β(n−m) = 1+[ n−m N PN PN b kr , ςbnr ] = −[ k=1 $ b kl , ςbnl ] = used the following consequence of Eqs. (5.45): [ k=1 $ p 2 ln q. Notice that, since R = e ⊗ e, the above relations are actually periodic with a period N 0 = pN for odd p. That is, the theory lives on a p-fold covering of the circle so that vertex operators for Zq are periodic on a lattice of size N 0 = pN . Now let us introduce the field gn . We repeat the construction of Subsect. 3.5 and define gn as follows: gn =
p−1 X
p ⊗ (b ς nr −b ς nl ) b r )s (Q b l )−s = eb P s ⊗ (Q . n n
s=0
It obviously admits restriction to the diagonal subspace HN of the model space MN (cf. Subsections 3.5 and 5.4). The locality of gn is evident from Eqs. (5.46) and its periodicity gn+N = (M l )−1 gn M r = gn is, in fact, reduced to the Weyl-type relation which we explained in detail at the end of Subsect. 3.5. 6. Automorphisms and Discrete Dynamics In this section we shall demonstrate that the lattice theory which we constructed above indeed may be regarded as a discretization of the WZNW model. For this purpose we investigate the exchange relations of currents and some automorphisms of our lattice algebra in the classical continuum limit and recover the Poisson structure and the dynamics of the classical WZNW model, respectively. 6.1. Remarks on the classical continuum limit. Let us briefly discuss the classical continuum limit of the algebra of vertex operators. Following ideology of [4], we rewrite the exchange relations (4.2), (4.4) and (4.7), (4.8) for the chiral currents in a more compact form: 2
1
1
2
−1 −1 r r r r Rn−m,+ J n Rn−m+1,+ J m = J m Rn−m−1,− J n Rn−m,− , 1
2
2
(6.1)
1
−1 −1 l l l l Rn−m,+ J n Rn−m+1,− J m = J m Rn−m−1,+ J n Rn−m,− ,
(6.2)
where Rn,± := δn,0 R± + (1 − δn,0 )e ⊗ e is, as usual, an element of Ga ⊗ Ga . Now we consider these relations in the limit where a = 2π/N → 0 and ~ → 0. Since for our theory q = exp{iγ~} (cf. Introduction), we can expand the universal R-matrix according to R± = e ⊗ e + iγ~ r± + O(~2 ) . On the other hand, the lattice fields approach their continuum counterparts as a becomes small: Jnα → e ⊗ e − a j α (x), where x = an. Bearing in mind that Poisson brackets from (6.1)–(6.2):
1 a δn,0
α 8α n → 8 (x) ,
gn → g(x) ,
→ δ(x) when a → 0, we obtain the following
1 2 γ [C, j r (x)− j r (y)] δ(x − y) + γ Cδ 0 (x − y) , 2 1 2 1 2 γ l l {j (x), j (y)} = − [C, j l (x)− j l (y)] δ(x − y) − γ Cδ 0 (x − y) , 2 1
2
{j r (x), j r (y)} =
(6.3)
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A. G. Bytsko, V. Schomerus
where C = (r+ − r− ) ⊗ e. These are the standard brackets for the chiral WZNW currents [41, 51], and the deformation parameter γ is identified with the coupling constant. 15 The exchange relations (5.25)–(5.26) can be treated similarly. Namely, we rewrite them as follows: 2
1
2
1
r r r r J n 8m Rn−m,− = 8m Rn−m−1,+ J n ,
1
2
1
2
l l l l J n 8m Rn−m,+ = 8m Rn−m−1,− J n ,
and get the following Poisson brackets for vertex operators in the classical continuum limit: 1
2
2
{j r (x), 8r (y)} = γ 8r (x) C δ(x − y),
1
2
2
{j l (x), 8l (y)} = −γ 8l (x) C δ(x − y) .
These relations are classical counterparts of the commutation relations known for the chiral primary fields in the continuum WZNW model [41]. α 2 Substitution of the expansions Rα ± = e ⊗ e ⊗ e + iγ~ r± + O(~ ) into Eqs. (5.21) 16 and passing to the classical continuum limit gives 1
2
1
2
1
2
1
2
{8r (x), 8r (y)} = −χr (x−y) 8r (x) 8r (y), {8l (x) 8l (y)} = χl (x−y) 8l (x) 8l (y) , α , and ε(x) = 1 if x > 0 and ε(x) = 0 if where χα (x − y) = ε(x − y) γ r+α + ε(y − x) γ r− x < 0. Such brackets were obtained for the classical WZNW model in [26, 13, 20, 27]. The same technique may finally be applied to the relations (5.43) involving the lattice field g and the resulting formulae for the classical counterpart of Eqs. (5.43) coincide with formulae in [41], namely, 1
2
2
1
2
2
{j r (x), g (y)} = γ g (x) C δ(x − y) , {j l (x), g (y)} = γ C g (x) δ(x − y) . Thus, in the limit ~ → 0, a → 0, our main exchange relations for the chiral currents and chiral vertex operators reproduce the Poisson structure known for the classical WZNW model. 6.2. Automorphisms induced by the ribbon element. The ribbon element, due to its specific properties, allows to obtain certain inner automorphisms of the algebra WN . These are the subject of the present subsection. Non-local automorphism induced by global ribbon elements. Consider an automorphism of the form: −1 for all A ∈ WN . (6.4) A 7→ v−1 r vl A vr vl , Here vr ∈ C r and vl ∈ C l . We call the ribbon elements vα global because they are constructed from the monodromies M α , which are non-local. Since the subalgebras C α constitute the center of KN , all the elements of the current algebra KN ⊂ WN are invariant under the transformation (6.4). For the vertex operators this transformation is nontrivial and may be rewritten with the help of (5.15), (5.22) and (5.24) so that it becomes 15 One may prefer to renormalize the currents by 1/γ so that the δ 0 -term acquires a coefficient 1/γ which, in the classical theory, coincides with the level k of the KM algebra. The quantum correction 1/γ → k + ν is explained, e.g., in [4]. 16 In general, the classical r-matrices rα keep a non-trivial dependence on variables belonging to C α . ±
Vertex Operators – From a Toy Model to Lattice Algebras
123
α α α 8α n+kN 7→ 8n+kN Mn = 8n+(k+1)N ,
0 ≤ n < N, k ∈ Z .
Thus, the automorphism (6.4) is non-local, i.e., it corresponds to a shift n 7→ n + N or, in other words, it rotates the lattice by angle 2π. Being restricted on the diagonal subspace, the field gn is periodic (see Proposition 5), and hence it is invariant under the transformation (6.4). In this sense, the automorphism (6.4) separates “physical” variables living on the circle from “non-physical” ones (like the vertex operators) living on a covering of the circle. Local automorphism induced by local ribbon elements. Recall that the nth site of the lattice is supplied with a copy Gn of the symmetry algebra G. Therefore we can use the local ribbon elements vn ∈ Cn ⊂ Gn to construct the following transformation: A 7→ v0 v1 . . . vN −1 A (v0 v1 . . . vN −1 )−1 ,
for any A ∈ WN .
(6.5)
Here the product is taken over all sites of the lattice. To obtain more explicit formulae for the automorphism (6.5), we have to use relations (4.3), (4.10), (5.19), employ Eq. (2.1) and remember that vn belongs to the center of Gn . As a result we get e −1 J l N e Jnr 7→ Nn−1 Jnr Nn−1 , Jnl 7→ N n−1 n n , en , 8r 7→ va 8r N −1 , 8l 7→ va 8l N n
n
n
n
(6.6)
n
−1 e , N = N+−1 N− . The elements where we used notations of Subsect. 2.2, i.e., N = N+ N− of Gn and C α remain invariant under (6.5), in particular, Nn,± 7→ Nn,± . With the help of these explicit expressions we also obtain
Mnr 7→ Nn Mnr Nn−1 ,
e −1 M l N e Mnl 7→ N n n n,
en gn N −1 . gn 7→ N n
(6.7)
We know already that these formulae describe an automorphism of the algebra WN because they were obtained by conjugation with a unitary element, namely the product of local ribbon elements, in formula (6.5). Without this knowledge, it would be a quite non-trivial task to check the automorphism property directly for the expressions in (6.6)– (6.7). To do this, one would need to apply the relations (4.5) and (5.17) many times.
Local automorphism induced by κn . To construct one more inner automorphism of WN we employ the square roots of the local ribbon elements, A 7→ κ0 κ1 . . . κN −1 A (κ0 κ1 . . . κN −1 )−1 ,
for all A ∈ WN .
Gn , κ2n
(6.8)
= vn and the product is taken over all sites. Here κn ∈ Cn ⊂ Computations similar to those performed above (and making use of (2.10)) allow to rewrite the transformation (6.8) in the following explicit form: −1 −1 = Nn−1,+ ((Jnr )∗ )−1 Nn,+ , Jnr 7→ Nn−1,+ (Sn−1 )−1 Jnr Sn Nn,+ −1 −1 Jnl 7→ Nn−1,− Sn−1 Jnl (Sn )−1 Nn,− = Nn−1,− ((Jnl )∗ )−1 Nn,− , −1 8rn 7→ κa 8rn Sn Nn,+
−1 = κa ((8rn )∗ )−1 Nn,+ ,
8ln 7→ κa 8ln (Sn )−1 Nn,−
= κa ((8ln )∗ )−1 Nn,− ,
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A. G. Bytsko, V. Schomerus −1 −1 Mnr 7→ Nn,+ ((Mnr )∗ )−1 Nn,+ , Mnl 7→ Nn,− ((Mnl )∗ )−1 Nn,− ,
and Nn,± 7→ Nn,± , as before. Here the r.h.s. of all formulae have been rewritten with the help of the ∗-operation introduced for elements of WN in Sects. 4 and 5. Having done so, we see that the image of all basic objects X ∈ Ga ⊗ WN under the automorphism (6.8) coincides with (X ∗ )−1 up to a multiplication with factors Nn,± . We can now accept an inverse logic – we may say that the automorphism (6.8) together with the rules (N± )∗ = N∓ defines the ∗-operation on WN . This picture reveals the naturalness of our ∗-operation, which might have appeared somewhat artificial in the previous sections. It also makes the role of the ribbon element in our theory even more remarkable. To conclude this discussion, we would like to mention that for a lattice of even length, i.e., for N = 0 (mod 2), one may also consider automorphisms of WN generated by the alternating products of vn±1 or κ±1 n . 6.3. Discrete dynamics. As we saw above, the exchange relations of the algebra WN allowed to recover the Poisson structure of the classical WZNW-model in the classical continuum limit. However, this is certainly not sufficient for a construction of the lattice WZNW model. Indeed, the complete description of a classical theory involves an evolution equation for the dynamical variables in addition to the specification of the Poisson structure. Similarly, the formulation of a discrete quantum model requires not only a set of exchange relations between quantum operators but also some one parameter family of automorphisms of the algebra generated by operators in the quantum theory. The parameter is interpreted as time variable. For a theory on a discrete space it is natural to discretize the time as well so that the parameter essentially runs through the set of integers only. In this case the whole family of automorphisms can be reconstructed from the automorphism which provides the evolution for an elementary step in time. Such an automorphism of a lattice model must be local, i.e., the result of its action on the variables assigned to a given site (or link) can only involve variables assigned to some neighboring sites (or links). In the previous subsection we considered three automorphisms of the algebra WN . The first of them was non-local and hence did not correspond to any dynamics.17 The second and the third automorphism were local and, in principle, one may use them in constructing the corresponding classical continuum models. However, the dynamics of such models do not reproduce the dynamics of the WZNW theory. In this subsection we are going to consider local automorphisms which can be interpreted as dynamics of the discrete WZNW model. Let us recall that in the continuum WZNW model the equation of motion for the G-valued field g(x) takes the form: ∂+ ∂− g = (∂+ g) g −1 (∂− g) ,
(6.9)
where ∂± = 21 (∂0 ± ∂x ). From the field g(x) one may construct the following Lie algebra valued currents j r = g −1 ∂− g, j l = (∂+ g)g −1 . (6.10) They turn out to be chiral objects in the sense that their equations of motion are trivialized: ∂ + j r = ∂− j l = 0 .
(6.11)
In the Hamiltonian approach, the initial data are provided by the values of g(0), j r (x) and j l (x) at time t = 0. To recover the dynamics of g(x) one solves the equations ∂0 g = j l g + g j r , ∂x g = j l g − g j r . 17
However, one can use it to describe dynamics of the toy model (see [3]).
(6.12)
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125
Equations (6.11) and (6.12) can be derived with the help of the Poisson brackets given in Subsect. 6.1 if the Hamiltonian and the total momentum are chosen as follows: Z Z r 1 1 2 l 2 tr (j (x)) + (j (x)) dx , P = tr (j r (x))2 − (j l (x))2 dx , H= 2γ 2γ (6.13) where the integration is taken over the whole circle and tr is the usual trace in the corresponding Lie algebra. Let us develop an analogue of the given picture in the quantum lattice theory. More precisely, we shall consider the “physical” subalgebra PN of WN generated by components of the chiral currents Jnα and the field gn , n = 1, .., N which are subject to the relations spelled out in Sects. 4 and 5. As we have indicated in our general discussion above, it is natural to work with a discrete time with a minimal time interval τ (see also [28, 31]), so that the evolution of the quantum theory is described by a single automorphism of PN . In addition to this, we shall also introduce an automorphism which is responsible for the shifts by one lattice unit a = 2π/N in space. Lemma 2 (Shift and evolution automorphisms). Let PN denote the algebra generated by components of Jnα and gn (restriction to the diagonal subspace HN is understood). Then the following two transformations TV , TU , α , α = l, r , TV (Jnα ) = Jn+1
(6.14)
l r TV (gn ) = (Jn+1 )−1 gn Jn+1 ,
(6.15)
r l , TU (Jnl ) = Jn+1 , TU (Jnr ) = Jn−1
(6.16)
l )−1 gn (Jnr )−1 TU (gn ) = (Jn+1
(6.17)
and
extend to automorphisms of the algebra PN . We call TV the shift automorphism and TU the evolution automorphism of the lattice WZNW-model. It is straightforward to verify that the transformations TV and TU preserve all the relations for Jnα and gn given above. Notice also that one may extend TU , TV to the whole algebra WN by Eqs. (6.14), (6.16) and, in addition, the formulae α α TV (8α n ) = 8n Jn+1 ,
TU (8rn ) = 8rn (Jnr )−1 ,
l TU (8ln ) = 8ln Jn+1 .
These automorphisms are actually combined of two chiral automorphisms (cf. Subsect. 6.4). After restriction to the diagonal subspace and, hence, to the algebra PN , we recover Eqs. (6.15) and (6.17). 18 Assume now that Lemma 2 describes inner automorphisms of PN . That is, suppose that there exist operators V, U ∈ PN such that TV (A) = V A V−1 and TU (A) = U A U−1
for any A ∈ PN .
V and U are usually called shift and evolution operators, respectively. In the classical continuum limit a → 0, τ → 0 they reproduce the momentum and the Hamiltonian (6.13): V → e + ~i a P , U → e + ~i τ H. l
l
l )−1 S (8 ); it is evident if we take This needs the following variant of Eq. (5.32): Sa (8n+1 ) = (Jn+1 a n relation (3.19) into account. 18
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A. G. Bytsko, V. Schomerus
Our interpretation of the transformations in Lemma 2 as discrete shifts in space and time, motivate to introduce the the objects Jnα (t), gn (t) ∈ Ga ⊗ PN such that r (t), Jnr (t + τ ) = Jn−1
l Jnl (t + τ ) = Jn+1 (t) ,
(6.18)
l l r gn (t + τ ) = (Jn+1 (t))−1 gn (t) (Jnr (t))−1 , gn+1 (t) = (Jn+1 (t))−1 gn (t) Jn+1 (t) (6.19) α α and Jn (0), gn (0) coincide with our usual generators Jn , gn , respectively. These expressions define Jnα (t) and gn (t) for t = kτ with k being integer. Below we shall also need the equations inverse to (6.18)–(6.19): r (t), Jnr (t − τ ) = Jn+1 r (t), gn (t − τ ) = Jnl (t) gn (t) Jn+1
l Jnl (t − τ ) = Jn−1 (t) ,
(6.20)
gn−1 (t) = Jnl (t) gn (t) (Jnr (t))−1 .
(6.21)
→ e⊗e−aj (x, t) and gn (t) → g(x, t) from Subsect. 6.1 (here x = na = 2πn/N as before) to establish that in the classical continuum limit Eqs. (6.18)-(6.19) become precisely the Eqs. (6.11)–(6.12)19 . Further, combining (6.19) and (6.21), we obtain the following relations:
We can now use the rules Jnα (t)
r , (gn−1 (t))−1 gn (t − τ ) = Jnr Jn+1
α
l gn−1 (t) (gn (t + τ ))−1 = Jn+1 Jnl .
(6.22)
These are lattice analogues of the definitions (6.10) of the chiral currents. Notice that we can express only products of currents on neighboring links through the field gn (t) (nevertheless, in the classical continuum limit, Eq. (6.10) is certainly recovered). 20 Equation (6.22) allow to obtain the dynamics of the lattice model in terms of the field gn only. Indeed, since their r.h.s. are manifestly chiral objects, the combinations of gn -variables on the l.h.s. are to be invariant under the substitutions t → t + τ , n → n + 1 and t → t+τ , n → n−1, respectively. Thus, we derive a lattice analogue of the equation of motion (6.9): gn+1 (t) (gn (t − τ ))−1 = gn (t + τ ) (gn−1 (t))−1 .
(6.23)
Being a discrete analogue of an equation of second order in both variables, this relation involves four different points on the space-time lattice (see Fig. 6). A natural choice of the initial data for Eq. (6.23) is provided by the set gn (t−τ ) and gn (t), n = 0, .., N − 1 (here t is fixed). It is interesting to notice that this set is divided into two subsets (black and white circles on Fig. 6) which have an independent evolution;21 that is, the solution constructed according to Eq. (6.23) from one of the sets never interacts with that constructed from the other set. According to Eqs. (6.22), the initial data gn (t − τ ), gn (t) (at fixed time t) can be restored if we are given the set of currents Jnα , n = 1, .., N and two values of the g-field taken at two arbitrary points of the independent subsets, e.g. g0 (0) and g0 (−τ ). This is a lattice analogue of the initial data usually used in the continuum Hamiltonian approach (see above). 19 In the continuum limit the quotient c := a/τ (speed of light) is supposed to be fixed. In fact, Eq. (6.9) implies that we put c = 1. 1 20 Formally, we can split Eqs. (6.22) into the following relations: J r = (g −1 g n−1 (t)) n n− 1 (t − 2 τ ) and 2
r = (gn− 1 (t − 21 τ ))−1 gn (t − τ ). However, the variables assigned to half integer sites or times are Jn+1 2
not defined in the lattice formalism. To avoid this we could consider these relations as relations in WN and re-express the involved g-fields through vertex operators, while using that vertex operators are chiral to replace formal variables on half integer space-time points by true objects of the lattice theory. As a result we would α α get the obvious relations Jnα (t) = (8n−1 (t))−1 8n (t). 21 Let us stress that it is not necessary to impose a continuity condition on the initial data, i.e., to demand that they possess smooth continuum limit. Moreover, it seems interesting to study the case when the two independent subsets of initial data have different continuum limits (cf. also [28]).
Vertex Operators – From a Toy Model to Lattice Algebras
x−a q
qt+τ x+a q qt−τ
` q
x−a q
`
127
` x+a q ` qt−τ ` q
x+2a q
`
Fig. 6. Graphical presentation of the discrete equation (6.23) and a possible choice of the initial data. The two subsets of the initial data have independent evolutions
To summarize, in this section we have demonstrated that the elements Jnα and gn which constitute the “physical” variables in the algebra WN are indeed quantum lattice analogues of the chiral currents and the group valued field in the WZNW model. 6.4. U (1)-WZNW model. We conclude this section with some comments on the Zq -case. Recall that the g-field constructed from lattice vertex operators in the case of G = Zq is p ⊗ φn given by (cf. Subsect. 5.5) gn = eb , where φn = ςbnr − ςbnl is an operator acting on the physical space HN (see Subsects. 3.5 and 5.5). In the classical limit φn (t) becomes a lattice variable which, according to (6.23), obeys to the following equation of motion: φn (t + τ ) + φn (t − τ ) = φn+1 (t) + φn−1 (t) .
(6.24)
This relation discretizes the equation of motion ∂+ ∂− φ(x, t) = 0 of a free field. The latter is known to arise, in particular, for the continuum U (1)-WZNW model. Moreover, in the classical continuum limit the standard Poisson structure of the abelian WZNW theory is easily recovered from the exchange relations of our Zq lattice model. These two observations allow to identify the Zq lattice theory as a quantized lattice U (1)-WZNW model. In spite of its simplicity, the U (1)-theory has a lot of structure in common with the more complicated nonabelian models. In fact, the abelian model was used here to illustrate many elements of our general theory. It is also worth mentioning that the abelian lattice theory itself has non-trivial mathematical aspects. In particular, explicit formulae for shift operators in chiral theories have been worked out in [31, 28, 8]. We may use these results to present expressions for the the shift and evolution operators V and U . The latter can be decomposed into the chiral components: V = Vl Vr and U = Vl Vr−1 . When acting on elements of the algebra WN , α the operators Vα ∈ WN generate shifts for the chiral sectors, i.e., α , (e ⊗ Vα ) Jnα (e ⊗ Vα )−1 = Jn+1
−1 (e ⊗ Vα ) 8α = 8α n (e ⊗ Vα ) n+1 ,
(6.25)
r where α = r, l, n ∈ Z, and e ⊗ Vl , e ⊗ Vr commute with any element from Ga ⊗ WN l and Ga ⊗ WN , respectively.
Proposition 6 (Shifts operators for Zq ). Let WN be the algebra of lattice vertex opeb nα (chiral c α = e$ rators as defined in Subsect. 5.5, i.e., it is generated by the elements W n ςnα b α = eb currents) and Q (vertex operators) obeying the relations spelled out in Subn sects. 4.4 and 5.5. Let the lattice length N be odd. Then the chiral shift operators obeying (6.25) are given by Vα = Zα
N −1 Y
ρα (N −k),
(6.26)
k=1
b kr )2 } and ρl (k) = exp{ 2 ln1 q ($ b kl )2 }. The function Zα where ρr (k) = exp{− 2 ln1 q ($ Q N2+1 c α Q N2−1 c α −1 α depends only on the element Cα = ( k=1 . W2k−1 ) ( k=1 W2k ) ∈ KN
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A. G. Bytsko, V. Schomerus
To verify that Vr and Vl obey Eqs. (6.25) one proceeds in two steps. The first of them concerns the relations between Vα and the chiral currents and it has been performed in [8]. The computation is based on the following relations: 22 b n ρ (n + 1) ρ (n) = ρ (n + 1) ρ (n) e$ b n+1 , n = 1, .., N − 2 , e$ α α α α α
α
which hold due to Eqs. (4.27). The remaining relations for n = N − 1, N are consequences of the others if the function Zα is chosen in a specific way (see [8] for details). In the second step, one checks the desired properties of Vα with respect to chiral vertex operators. To this end we derive the following relation: α α α 1 b n+1 ςnα ςnα ςnα +$ b n+1 ςn+1 ρα (n + 1) = ρα (n + 1) e− 2 ln q e$ eb = ρα (n + 1) eb = ρα (n + 1) eb . eb (6.27) For the first equality we used the commutation relations (5.45). After this, the CampbellHausdorff formula was employed before we could insert Eqs. (5.44). Notice that it suffices to prove Eq. (6.27) for n = 1. Due to (5.44), the relations between Vα and the vertex operators assigned to other sites are consequences of this case (as soon as the relations for Vα with chiral currents are established). This completes the proof. Let us comment on the construction of the chiral shift operators for a lattice of even length N suggested in [31, 28]. In this case the shift operators are also given by (6.26) but without the factor Zα . When checking the relations between these operators Vα and QN/2 c α QN/2 c α chiral currents, Faddeev and Volkov had to assume that k=1 W 2k−1 = k=1 W2k . Unfortunately, such a constraint is incompatible with the exchange relations in the full theory which includes the objects Nn in addition to chiral currents. One way to bypass this problem would involve shifts by two lattice units. Let us finally mention that the function ρ (which can be identified as a θ-function, if cnα ) appearing in (6.26) admits factorization into a product of two written in terms of W functions of a q-dilogarithm type (see [28]). Actually, these objects (the θ-function and the q-dilogarithm) turn out to be quite universal building blocks for shift operators. They were employed in the recent work [32] to construct shift operators for the SU (2)-lattice KM algebra. Since the expressions involving θ-functions and q-dilogarithms resemble those used in the abelian theory, one expects that the new operators of [32] serve as shifts not only for the current algebras but for the whole algebras of vertex operators as well.
Conclusion In the present paper we have described the construction of lattice vertex operators for a given modular Hopf algebra. The investigation of the classical continuum limit reveals a clear relation between the lattice algebras and the WZNW-model. Since the latter can be reduced to the affine Toda model, our technique may be applied to this theory as well (with certain modifications). Furthermore, there exist many connections with Chern– Simons theory in 2 + 1 dimensions (see [7] for lattice constructions of Chern-Simons observables) which motivate to extend our framework to two spatial dimensions. Let us briefly list some aspects of the presented theory which have not been developed. As we mentioned before, formulae for vertex operators are known only for some particular cases. It would be interesting to work out explicit presentations for universal 22 Such relations were used first in [31, 28] in the construction of shift operators for U (1)-current algebra for even N (cf. remarks in the text).
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129
vertex operators 8 of the deformed universal enveloping algebras Uq (G). Alternatively, one may try to find universal structure data F , σ which solve the discussed set of equations. A further natural extension is to incorporate infinite dimensional structures such as the deformed affine algebras. This might allow for a comparison with the approaches in [34, 38] (see also references therein). Another problem which is to be solved to complete the description of the quantum lattice WZNW model is an explicit construction of the shift and evolution operators. By now, exact formulae have been found for the cases of U (1) and SU (2) [31, 28, 8, 32]. These examples, however, hint at some uniform structures (such as the appearance of q-dilogarithms) that might lead to new formulae for shift operators in more general theories. Appendix: Some Proofs and Further Relations A.1. Proof of Proposition 1. It is not difficult to obtain the relations stated in Proposition 1 from the defining properties of 8. For instance, the formula (2.23) for D follows from the definition (2.19) when N = RR0 is re-expressed in terms of the ribbon element according to Eq. (2.1). To derive Eq. (2.24) one needs no more than associativity of the multiplication in V along with co-associativity of the co-product 1a on Ga , 3
3
3
3
2
1
σ (F12 ) (1a ⊗ id)(F ) (1a ⊗ id)1a (8) =σ (F12 ) 8 1a (8)12 =8 (8 8) = 3
2
1
1
= (8 8) 8= F23 1a (8)23 8= F23 (id ⊗ 1a )(F ) (id ⊗ 1a )1a (8) . The first relation of Eqs. (2.25) is a consequence of the covariance property (2.16) of 8, 2
1
1
1
1
1
2
1
1
2
1
1
2
1
D R− 8 8 = 8 N 8−1 R− 8 8 = 8 N 8 R− = 8 8 R+ N = 2
1
2
1
1
1
2
2
1
= R+ 8 8 N = R+ 8 D 8 = R+ σ (D) 8 8 . We have inserted the definition (2.19) twice and used commutation relations (2.22). Next, using definitions (2.17)–(2.18) of the structure data, we easily check (2.26): 2
1
1
2
σ σ (f) =8 8 (e ⊗ f) 8−1 8−1 = 2
1
= F 1a (8) (e ⊗ f) 1a (8−1 ) F −1 = F 1a (8 (e ⊗ f) 8−1 ) F −1 = 1F (σ(f)) . Let us finally discuss the computation of F ∗ . It is based on the second identity in (2.11) and on the relation (e⊗S)(id⊗1)(S) = (S ⊗e)(1⊗id)(S) which can be checked in a straightforward way. Applying them and the property (2.14) to the definition (2.17), we derive 2
1
1
2
F ∗ = (8 8 1a (8−1 ))∗ = 10a (8 S) (S −1 )13 8−1 (S −1 )23 8−1 1 2 = 10a (8) (10 ⊗ id)(S) (e ⊗ S −1 ) (id ⊗ 10 )(S −1 ) 213 8−1 8−1 1 2 = 10a (8) (10 ⊗ id)(S) (id ⊗ 1)(S −1 ) (e ⊗ S −1 ) 213 8−1 8−1 1
2
1
2
= 10a (8) (S ⊗ e) 8−1 8−1 = Sa 1a (8) 8−1 8−1 = Sa F −1 .
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A. G. Bytsko, V. Schomerus
The index on [.]213 refers to a permutation of tensor factors in the expression enclosed by the brackets. All other relations in Proposition 1 are either obvious or they follow directly from the derived equations. This applies in particular to Eq. (2.27). A.2. Proof of Proposition 3. In this subsection we want to construct consistent structure data for vertex operators of the deformed universal enveloping algebras G = Uq (G) from their Clebsch–Gordan maps and 6j-symbols. To fix our notations, let us recall that the Clebsch-Gordan maps C[T L|S] : V T ⊗ V L 7→ V S have the following properties: C[T L|S] (τ T ⊗ τ L )(1(ξ)) = τ S (ξ) C[T L|S]
for all ξ ∈ G ,
κS ±1 TL ) C[T L|S] R± C[T L|R]∗ = δR,S with κL = τ L (κ) , κT κL X ± L T JT BQS [ R J ] C[JQ|R] C23 [T L|Q] = C[JS|R] C23 [T L|S] (R± ⊗ eL ) ,
(
Q
X
FQS [ L R
T J
] C[QL|R] C12 [JT |Q] = C[JS|R] C23 [T L|S] .
(A.1)
Q
In the order of appearance here these equations describe the intertwining properties of the Clebsch–Gordan maps, their normalization and the definition of the braiding and fusion matrices (or 6j-symbols). It is well known that certain (polynomial) relations for . . . . the numbers B ± . . [ . . ] and F . . [ . . ] follow from their definitions and properties of the quasi-triangular Hopf algebra G (see, e.g., [39]). In particular, one has X T L Q T J S J L T FQS [ L R J ] FN R [ P I ] FM Q [ N I ] = FM R [ P I ] FN S [ P M ] , Q
X
FN S [ L P
T M
] FN R [ L P
T ]∗ M
= δS,R ,
(A.2)
N − NT L + −1 − ( P ) BN N 0 [ P
T M
+ L ] = BN N0 [ P
T M
MT ] + −1 − ( N ).
Here we used the notation ± L ± ( LT P ) = BT L [ P
T 0
−1 LT LT LT ] and + −1 − ( P ) = + ( P ) − ( P ) =
vL v T . vP
(A.3)
To proceed, we parameterize the labels N, M, I and N 0 in terms of new variables λ, ϑ, ι and ϑ0 so that N = P + λ , M = P + λ + ϑ , I = P + λ + ϑ + ι , N 0 = P + ϑ0 . Let us introduce the following matrices C{T L|S}(P ), RT±L (P ) and DT (P ) , C{T L|S}(P )ς,ϑλ = FP +λS [ L P
T P +ϑ+λ ] δς,ϑ+λ
RT±L (P )ϑ0 λ0 ,ϑλ = BP±+λP +ϑ0 [ L P
,
T P +ϑ+λ ] δϑ+λ,ϑ0 +λ0
P +ϑT ) δ 0 DT (P )ϑ0 ,ϑ = + −1 ϑ ,ϑ . − ( P
, (A.4)
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131
We may think of the P -dependent matrices C{T L|S}(P ), RT±L (P ) and DT (P ) as matrices with entries in the algebra C. Whenever we do so, we will neglect to write the P -dependence explicitly and use the symbols C{T L|S}, RT±L and DT . If we introduce in addition the matrix valued map σ L on C by σ L (P )λ0 ,λ = (P + λ) δλ0 ,λ , then Eqs. (A.2) for the fusion and braiding matrices become X L T FQS [ L R J ] C{QL|R} σ (C12 {JT |Q}) = C{JS|R} C23 {T L|S} , Q
C{T L|S} C{T L|R}∗ = δR,S , (DT ⊗ eL ) RT−L = RT+ L σ L (DT ) .
(A.5)
The last of these equations appears already as a close relative of Eq. (2.25). In fact, for semi-simple G one can construct universal objects R± ∈ G ⊗ G ⊗ C and D ∈ G ⊗ C so that RT±L = (τ T ⊗ τ L )(R± ) and DT = τ T (D). Then Eq. (A.5) turns into 1
2
D R− = R+ σ (D) , where σ : C 7→ G ⊗ C is defined so that σ L = (τ L ⊗ id) ◦ σ. When + −1 − is expressed in terms of the ribbon element v as in Eq. (A.3), the definition (A.4) of D becomes D = σ(v)v−1 va . To build the universal element F ∈ G ⊗ G ⊗ C, we combine the matrices C{..|.} with the Clebsch-Gordan maps so that F T L = (τ T ⊗ τ L )(F ) is given by X FTL ≡ C{T L|S}∗ C[T L|S] . S
Multiplying the adjoint of the first equation in (A.2) with Eq. (A.1), taking the sum over R, S and using the intertwining properties of the Clebsch–Gordan maps, we obtain Eq. (2.24) for F . In the same way, one may combine the normalizations for the Clebsch– Gordan maps C[T L|S] and the matrices C{T L|S} to derive that F (1(κ−1 )(κ ⊗ κ) R+−1 ) ⊗ e F ∗ = e ⊗ e ⊗ e , and hence F has the required property under the ∗-operation. Finally, we use that the matrix C{T L|S}(P ) is proportional to δς,ϑ+λ so that (f(P + ς)) C{T L|S}(P )ς,ϑλ = C{T L|S}(P )ς,ϑλ (f(P + ϑ + λ)) . Here f(P ) is an arbitrary function of P , i.e., f may be regarded as an element in C. With our standard conventions, this can be stated as a matrix equation σ S (f) C{T L|S} = C{T L|S} σ L σ T (f) . Keeping in mind that 1F (ξ) = F (1(ξ) ⊗ e)F −1 , we discover Eq. (2.26). All other properties of the structure data follow easily from the relations we have discussed here.
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A.3. Structure data for left chiral vertex operators. We can obtain the relations for the left structure data Fl , σl , Dl , Rl± from Eqs. (2.23)–(2.28) if we substitute 1a → 10a 0 R± → R±
F → Fl0 σ → σl
M → (M l )−1 R± → (Rl± )0
D → Dl−1
va → va
8 → 8l .
The prime on Fl , Rl± ∈ Ga ⊗ Ga ⊗ C l denotes permutation of the first two tensor factors. Once the validity of these rules has been checked for the defining relations (2.17)–(2.19) of structure data, we can apply them to Eqs. (2.23)–(2.28). Within the notations of Proposition 1, the result looks as follows: σl σl (fl ) = 1Fl (σl (fl )) , Dl = (va−1 v) σl (v−1 ), 1 [Fl ⊗ e]1243 (1a ⊗ id)(Fl ) = σ (Fl ) (id ⊗ 1a )(Fl ) , 1
2
1
1
Dl Rl+ = Rl− σl (Dl ), 2
2
2
Rl+ Dl = σl (Dl ) Rl− , 3
1
Rl±,23 σl (Rl±,13 ) Rl±,12 = σl (Rl±,12 ) Rl±,13 σl (Rl±,23 ), Fl∗ = Sa−1 Fl−1 ,
(Rl± )∗ = (Rl± )−1 ,
σl (fl )∗ =
σl (f∗l ),
Dl∗ = Dl−1 ,
1Fl (ξ)∗ = 1Fl (ξ ∗ ).
(A.6) (A.7) (A.8) (A.9) (A.10) (A.11)
Here 1Fl (ξ) ≡ Fl (1(ξ) ⊗ e)Fl−1 ∈ Ga ⊗ Ga ⊗ Vl analogously to definition (2.21). Using the fact that (Sa ⊗ Sa )(1a (ξ)) = 10a (Sa (ξ)), it is simple to see that the objects (3.13), (3.14) satisfy the relations (A.6)–(A.11) if F, σ, 1, R solve Eqs. (2.23)–(2.28). (n) The former equations are actually obtained from the latter by acting with the maps Slr defined in Eq. (3.15). A.4. Properties of the field g. The consistency of the object g with the constraint to the diagonal subspace H is certainly its most important property. It was formulated more precisely in (3.17) using notations from Subsect. 3.4. A formal proof may be given as follows. Suppose that fg = Slr (f)g holds for all f ∈ C r . Then one finds for arbitrary f ∈ Cr, g f = Sa (8l )8r f = Sa (8l ) σr (f) 8r = Sa (S −1 ⊗ id)(σr (f))8l 8r = Sa (S −1 ⊗ Slr )(σr (f))8l 8r = Sa σl (Slr (f)) 8l 8r = Sa (8l ) Slr (f) 8r = Sa (8l ) 8r Slr (f) = g Slr (f) . The computation makes use of the choice (3.14) to replace σr by σl . From now on, we think of g as being restricted to H. We begin our proof of Proposition 4 with the operator product expansion (3.20) of g ∈ Ga ⊗ End(H), 2
1
2
1
2
1
g g = Sa (8l ) Sa (8l ) 8r 8r = (S ⊗ S ⊗ id)(Fl0 10a (8l )) Fr 1a (8r ) = (S ⊗ S ⊗ id) (S −1 ⊗ S −1 ⊗ id)(Fr−1 )10a (8l ) Fr 1a (8r ) = 1a (Sa (8l )) Fr−1 Fr 1a (8r ) = 1a (g) .
Vertex Operators – From a Toy Model to Lattice Algebras 1
133
2
1
2
Here Sa (8) and Sa (8) are shorthands for (Sa ⊗ id)(8) and (id ⊗ Sa )(8), respectively. The exchange relations for g, the formula g −1 = Sa (g) and the normalization a (g) = e follow immediately from the functoriality relation in (3.20). The exchange relations (3.21) are derived from (3.3), (3.8) and the explicit construction of g as a product of Sa (8l ) and 8r . Let us check the first of them: 1
2
1
2
2
2
2
1
2
2
1
2
1
r r l r l r r l r r r M g R− =M Sa (8 ) 8 R− = Sa (8 ) M 8 R− = Sa (8 ) 8 R+ M =g R+ M .
The second relation in (3.21) can be obtained similarly if we take the covariance properties (3.18) of Sa (8l ) into account. Verification of the relations (3.22) makes use of the equality vl = vr which is valid on H and follows with the help of S(v) = v, if the constraint Slr (fr ) = fl is evaluated on the ribbon element. The second relation in (3.22) then is obvious, and for the first we check explicitly: gn Mnr = gn (8rn )−1 va−1 Dr 8rn = Sa (8ln ) va−1 Dr 8rn −1 r l −1 = (8ln )−1 θl v−1 θl 8rn vl r σr (vr ) 8n = vl (8n )
=
(8ln )−1
σl (v−1 l ) vl θl
(A.12)
8rn
= (8ln )−1 va Dl 8ln (8ln )−1 θl 8rn = Mnl Sa (8ln ) 8rn = Mnl gn . In this computation it was convenient to insert the formula (3.19) which expresses Sa (8l ) in terms of (8l )−1 . A.5. Properties of lattice vertex operators. Let us prove that the structure constants Fα , Rα ± and σα appearing in Eqs. (5.20)–(5.22) coincide with those of the vertex operators α α 8α 0 . To this end, we exploit the construction of 8n as a product of 80 and the holonomies Unα (see eq. (5.16)). Equation (5.22) is actually obvious, since fα commute with all the elements Unα . Furthermore, Eq. (5.21) is a simple consequence of (5.20). Hence, we need to prove only Eq. (5.20) which we do for the right chirality ( the left one works analogously), 2
1
2
1
−1 1a (8rn ) = 1a (8r0 ) 1a (Unr ) = Fr−1 8r0 8r0 R− U rn U rn = 2
2
1
2
1
1
= Fr−1 8r0 U rn 8r0 U rn = Fr−1 8rn 8rn . The exchange relations (5.25)–(5.28) are established by induction. Indeed, for n = 0 they are part of Definition 3. Assume now that Eqs. (5.25)–(5.28) hold for a certain α α n < N so that, in particular, 8α n has non-trivial exchange relations with Jn and Jn+1 α α α α only. Then 8n+1 = 8n Jn+1 necessarily commutes with all currents Jm except from Jnα , α α Jn+1 and Jn+2 . It is easy to verify that the exchange relations with Jnα become trivial as well. We demonstrate this for α = r: 1
2
1
1
2
1
2
1
2
1
1
2
1
8rn+1 J rn =8rn J rn+1 J rn =8rn J rn R+ J rn+1 =J rn 8rn J rn+1 =J rn 8rn+1 . α α It can be checked similarly that the relations between 8α n+1 and Jn+1 , Jn+2 coincide with α α those between 8α and J , J . This completes the induction. n n n+1 Now we have to prove Eqs. (5.30)–(5.31). For instance, using (4.17) and (5.25), we derive the first relation in (5.30) for 0 ≤ n < m < N :
134
A. G. Bytsko, V. Schomerus 1
2
1
2
1
1
2
2
1
2
8rn 8rm =8r0 U rn 8r0 U rm =8r0 8r0 R+ U rn U rm = 2
1
2
2
1
1
2
2
1
1
−1 = Rr− 8r0 8r0 R− U rm U rn = Rr− 8r0 U rm 8r0 U rn = Rr− 8rm 8rn .
The relations (5.18)–(5.22), (5.24)–(5.28) which involve vertex operators 8α n outside of the initial interval n = 0, .., N − 1, are derived with the help of Eqs. (5.34)–(5.35) for the monodromies Mnα . Since the derivation uses the same technique as above, we prove only the functoriality relation for 8rn . As a first step, we check the following: 2
1
2
1
−1 1a (8rn+N ) = 1a (8rn ) 1a (Mnr ) = Fr−1 8rn 8rn R− M rn R+ M rn = 2
1
2
2
1
1
= Fr−1 8rn M rn 8rn M rn = Fr−1 8rn+N 8rn+N . Then we use an induction and Eq. (5.35) to get the same property for 8rn+kN . Finally, we establish relations (5.36) directly with the help of Eq. (5.35): 2
1
2
1
1
Rr+ 8rn 8rn+N = R+ 8rn 8rn M rn 1
2
1
1
1
2
1
2
= 8rn 8rn R+ M rn =8rn M rn 8rn R− = 8rn+N 8rn R− . Detailed computations for the other relations in (5.18)–(5.22), (5.24)–(5.28) can be worked out easily. A.6. Properties of lattice field gn . Let us notice that the equality (3.19) holds in the lattice case for all 8ln with the same θl ∈ Ga ⊗ C l (as we explained in Subsect. 5.2, vertex operators of the same chirality assigned to different sites possess the same structure data). Therefore, we can proceed as in the toy model case and rewrite the expression for gn as follows: gn = (8ln )−1 θl 8rn . This relation allows to express gn in terms of g0 and the holonomies Unα ∈ Ga ⊗ KN : gn = (Unl )−1 g0 Unr .
(A.13)
Bearing in mind that elements from KN , and hence, in particular, components of the ¯ holonomies Unα , leave the subspaces WNKK of the full representation space MN = L IJ equality (A.13) explains why all gn can be IJ WN invariant (see Subsect. 5.4), theL ¯ restricted on the diagonal subspace HN = K WNKK simultaneously. Among the properties of the lattice field gn in Proposition 5, only the relations (5.42) and (5.40) have not been considered in the toy model case. Equations (5.42) follow immediately from the covariance properties (5.19) of the vertex operators and the remark that, due to Eq. (3.19), the second relation in (5.19) can be rewritten as follows: Sa (8ln ) ιn (ξ) = 1n (ξ) Sa (8ln ) , Sa (8ln ) ιm (ξ) = ιm (ξ) Sa (8ln ) for m 6= n (modN ) for all ξ ∈ G. The periodicity of gn is derived with the help of relations (5.15), (5.24) and the second equation in (3.22):
Vertex Operators – From a Toy Model to Lattice Algebras
135
gn+N = Sa (8ln+N ) 8rn+N = (8ln+N )−1 θl 8rn+N = (8ln Mnl )−1 θl 8rn Mnr = (8ln )−1 (va Dl )−1 θl va−1 Dr 8rn −1 r = (8ln )−1 σl (vl ) v−1 l θl σr (vr ) vr 8n −1 r = vl (8ln )−1 v−1 l θ l v r 8n v r −1 r l r = vr (8ln )−1 v−1 r θl vl 8n vl = Sa (8n ) 8n = gn .
Due to periodicity, it is sufficient to check the locality of gn only for 0 ≤ n, m < N . Taking, for definiteness, n < m, we derive: 1
1
2
1
1
2
2
2
1
2
1
2
1
2
−1 g g r r g n g m = (U ln )−1 g0 U rn (U lm )−1 g0 U rm = (U ln )−1 (U lm )−1 R− 0 0 R+ U n U m 2
1
2
2
1
1
−1 r r = (U lm )−1 (U ln )−1 R+ g0 g0 R− Um Un 2
2
1
2
1
1
2
1
= (U lm )−1 g0 (U ln )−1 U rm g0 U rn = g m g n . Here we used Eq. (A.13) and the commutation relations between g0 and the holonomies Unα which are obvious consequences of Eqs. (5.43). Acknowledgement. We would like to thank A.Yu. Alekseev, L.D. Faddeev, A. Fring, J. Fr¨ohlich, P.P. Kulish, F. Nill, A.Yu. Volkov for useful discussions and E. Jagunova for preparing the pictures. A.B. is grateful to Prof. R. Schrader for the hospitality at the Institut f¨ur Theoretische Physik, Freie Universit¨at Berlin, and to the T¨opfer Stiftung for financial support.
References 1. Abe, E.: Hopf algebras. Cambridge: Cambridge University Press 1980 2. Alekseev, A.Yu., Faddeev, L.D.: Commun. Math. Phys. 141, 413 (1991) 3. Alekseev, A.Yu., Faddeev, L.D.: An involution and dynamics for the q-deformed quantum top, preprint hep-th/9406196 (1994) 4. Alekseev, A.Yu., Faddeev, L.D.: Semenov–Tian–Shansky, M.A.: Commun. Math. Phys. 149, 335 (1992) 5. Alekseev, A.Yu., Faddeev, L.D.:Semenov–Tian–Shansky, M.A., Volkov, A.Yu.: The unraveling of the quantum group structure in the WZNW theory. Preprint CERN-TH-5981/91 (1991) 6. Alekseev, A.Yu.: Faddeev, L.D., Fr¨ohlich, J., Schomerus, V.: Representation theory of lattice current algebras, preprint q-alg/9604017 (1996), to appear in Commun. Math. Phys 7. Alekseev, A.Yu., Grosse, H., Schomerus, V.: Commun. Math. Phys. 172, 317 (1995) 8. Alekseev, A.Yu., Recknagel, A.: Lett. Math. Phys. 37, 15 (1996) 9. Alekseev, A.Yu., Schomerus, V.: Duke Math. Journal 85, 457 (1996). 10. Alekseev, A.Yu., Shatashvili, S.L.: Commun. Math. Phys. 133, 353 (1990) 11. Babelon, O., Commun. Math. Phys. 139, 619 (1991) 12. Babelon, O., Bernard D., Billey E.: Phys. Lett. B 375, 89 (1996) 13. Balog, J., Dabrowski, L., Feher, L.: Phys. Lett. B 244, 227 (1990) 14. Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Nucl. Phys. B 241, 333 (1984) 15. Biedenharn, L.C., Lohe, M.A.: Quantum group symmetry and q-tensor algebras. Singapore: World Scientific 1995 16. Blok, B.: Phys. Lett. 233, 359 (1989) 17. Bytsko, A.G.: Tensor operators in R-matrix approach. Preprint q-alg/9512025 (1995) 18. Bytsko, A.G.: Fusion of q-tensor operators: Quasi-Hopf-algebraic point of view. Preprint q-alg/9609007 (1996) 19. Bytsko, A.G., Faddeev, L.D.: J. Math. Phys. 37, 6324 (1996). 20. Chu, M., Goddard, P., Halliday, I., Olive, D., Schwimmer, A.: Phys. Lett. B 266, 71 (1991) 21. Chu M., Goddard P.: Nucl. Phys. B 445, 145 (1995)
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22. Chu M., Goddard P.: Phys. Lett. B 337, 285 (1994) 23. Cremmer, E., Gervais, J.-L.: Commun. Math. Phys. 134, 619 (1990) 24. Drinfeld, V.G. Quantum groups. In: Proc. of ICM, Berkeley 1986. AMS 1987; Sov. Math. Dokl. 32, 254 (1985) 25. Drinfeld, V.G.: Algebra and Analysis 1, 1419 (1990) 26. Faddeev, L.D.: Commun. Math. Phys. 132, 131 (1990) 27. Faddeev, L.D. Quantum symmetry in conformal field theory by Hamiltonian methods. In: New symmetry principles in quantum field theories. New York: Plenum Press 1992 28. Faddeev, L.D.: Current-like variables in massive and massless integrable models. Preprint hepth/9408041 (1994) 29. Faddeev, L.D., Takhtajan L.A.: Lect. Notes in Phys. 246, 166 (1986) 30. Faddeev ,L.D., Reshetikhin, N.Yu., Takhtajan L.A.: Algebra and analysis 1, 193 (1990) 31. Faddeev, L.D., Volkov, A.Yu.: Phys. Lett. B 315, 311 (1993) 32. Faddeev, L.D., Volkov, A.Yu.: Shift operators for non-abelian lattice current algebra. Preprint hepth/9606088 (1996) 33. Falceto ,F., Gawedzki, K.: J. Geom. Phys. 11, 251 (1993) 34. Frenkel, I.B., Reshetikhin N.Yu.: Commun. Math. Phys. 146, 1 (1992) 35. Furlan, P., Hadjiivanov, L.K., Todorov, I.T.: Nucl. Phys. 474, 497 (1996) 36. Gawedzki, K.: Commun. Math. Phys. 139, 201 (1990) 37. Gervais, J.-L., Neveu, A.: Nucl. Phys. B 238, 125 (1984). 38. Jimbo, M., Miwa, T.: Algebraic analysis of solvable lattice models. CBMS Regional Conference Series in Mathematics, v. 85. AMS 1994 39. Kirillov, A.N., Reshetikhin, N.Yu.: Adv. Series in Math. Phys. 11, 202. Singapore: World Scientific 1990 40. Kirillov, A.N., Reshetikhin, N.Yu.: Commun. Math. Phys. 134, 421 (1991) 41. Knizhnik, V.G., Zamolodchikov, A.B.: Nucl. Phys. B 247, 83 (1984) 42. Kanie, Y., Tsuchiya, A.: Adv. Studies in Pure Math. 16, 297 (1988) 43. Mack, G., Schomerus, V.: Nucl. Phys. B 370, 185 (1992) 44. Moore, G., Seiberg, N.: Commun. Math. Phys. 123, 177 (1989) 45. Nill, F., Szlachanyi, K.: Quantum chains of Hopf algebras with order-disorder fields and quantum double symmetry, preprint hep-th/9507174 (1995); Quantum chains of Hopf algebras with quantum double cosymmetry, preprint hep-th/9509100 (1995), to appear in Commun. Math. Phys 46. Novikov, S.P.: Russ. Math. Surveys 37:5, 1 (1982) 47. Reshetikhin, N.Yu., Semenov-Tian-Shansky, M.A.: Lett. Math. Phys. 19, 133 (1990) 48. Reshetikhin, N.Yu., Turaev, V.G.: Commun. Math. Phys. 127, 1 (1990) 49. Sweedler, M.E.: Hopf algebras. New York: Benjamin Press 1969 50. Wess, J., Zumino, B.: Phys. Lett. B 37, 95 (1971) 51. Witten, E.: Commun. Math. Phys. 92, 455 (1984) Communicated by T. Miwa
Commun. Math. Phys. 191, 137 – 181 (1998)
Communications in
Mathematical Physics c Springer-Verlag 1998
Quantum Double Actions on Operator Algebras and Orbifold Quantum Field Theories ¨ ? Michael Muger II. Institut f¨ur Theoretische Physik, Universit¨at Hamburg, Luruper Chaussee 149, D-22761 Hamburg, Germany. E-mail:
[email protected] Received: 4 September 1996 / Accepted: 6 May 1997
Abstract: Starting from a local quantum field theory with an unbroken compact symmetry group G in 1+1-dimensional spacetime we construct disorder fields implementing gauge transformations on the fields (order variables) localized in a wedge region. Enlarging the local algebras by these disorder fields we obtain a nonlocal field theory, the fixpoint algebras of which under the appropriately extended action of the group G are shown to satisfy Haag duality in every simple sector. The specifically 1+1 dimensional phenomenon of violation of Haag duality of fixpoint nets is thereby clarified. In the case of a finite group G the extended theory is acted upon in a completely canonical way by the quantum double D(G) and satisfies R-matrix commutation relations as well as a Verlinde algebra. Furthermore, our methods are suitable for a concise and transparent approach to bosonization. The main technical ingredient is a strengthened version of the split property which is expected to hold in all reasonable massive theories. In the appendices (part of) the results are extended to arbitrary locally compact groups and our methods are adapted to chiral theories on the circle. 1. Introduction Since the notion of the “quantum double” was coined by Drinfel’d in his famous ICM lecture [30] there have been several attempts aimed at a clarification of its relevance to two dimensional quantum field theory. The quantum double appears implicitly in the work [19] on orbifold constructions in conformal field theory, where conformal quantum field theories (CQFTs) are considered whose operators are fixpoints under the of a symmetry group on another CQFT. Whereas the authors emphasize that “the fusion algebra of the holomorphic G-orbifold theory naturally combines both the representation and class algebra of the group G” the relevance of the double is fully recognized only in [20]. There the construction is also generalized by allowing for an arbitrary 3-cocycle in ?
Supported by the Studienstiftung des deutschen Volkes
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Michael M¨uger
H 3 (G, U (1)) leading only, however, to a quasi quantum group in the sense of [31]. The quantum double also appears in the context of integrable quantum field theories, e.g. [7], as well as in certain lattice models (e.g. [67]). Common to these works is the role of disorder operators or “twist fields” which are “local with respect to A up to the action of an element g ∈ G” [19]. Finally, it should be mentioned that the quantum double and its twisted generalization also play a role in spontaneously broken gauge theories in 2 + 1 dimensions (for a review and further references see [6]). Regrettably most of these works (with the exception of [67]) are not very precise in stating the premises and the results in mathematically unambiguous terms. For example it is usually unclear whether the “twist fields” have to be constructed or are already present in some sense in the theory one starts with. As a means to improve on this state of affairs we propose to take seriously the generally accepted conviction that the physical content of a quantum field theory can be recovered by studying the inequivalent representations (superselection sectors) of the algebra A of observables (which in the framework of conformal field theory is known as the chiral algebra). This point of view, put forward as early as 1964 [45] but unfortunately widely ignored, has proved fruitful for the model independent study of (not necessarily conformally covariant) quantum field theories, for reviews see [46, 49]. Using the methods of algebraic quantum field theory we will exhibit the mechanisms which cause the quantum double to appear in every quantum field theory with group symmetry in 1 + 1 dimensions fulfilling (besides the usual assumptions like locality) only two technical assumptions (Haag duality and split property, see below) but independent of conformal covariance or exact integrability. As in [21] we will consider a quantum field theory to be specified by a net of von Neumann algebras, i.e. a map O 7→ F(O), (1.1) which assigns to any bounded region in 1 + 1 dimensional Minkowski space a von Neumann algebra (i.e. an algebra of bounded operators closed under hermitian conjugation and weak limits) on the common Hilbert space H such that isotony holds: O1 ⊂ O2 ⇒ F(O1 ) ⊂ F(O2 ).
(1.2)
The quasilocal algebra F defined by the union F=
[
k·k
F(O)
(1.3)
O∈K
over the set K of all double cones (diamonds) is assumed to be irreducible, i.e. F 0 = C1.1 The net is supposed to fulfill Bose-Fermi commutation relations, i.e. any local operator decomposes into a bosonic and a fermionic part F = F+ + F− such that for spacelike separated F and G we have [F+ , G+ ] = [F+ , G− ] = [F− , G+ ] = {F− , G− } = 0.
(1.4)
The above decomposition is achieved by F± =
1 (F ± α− (F )), 2
(1.5)
1 In general M0 = {X ∈ B(H)|XY = Y X ∀Y ∈ M} denotes the algebra of all bounded operators commuting with all operators in M.
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139
where α− (F ) = V F V and V = V ∗ = V −1 is the unitary operator which acts trivially on the space of bosonic vectors and like −1 on the fermionic ones. To formulate this locality requirement in a way more convenient for later purposes we introduce the twist operation F t = ZF Z ∗ , where Z=
1 + iV , (⇒ Z 2 = V ), 1+i
(1.6)
which leads to ZF+ Z ∗ = F+ , ZF− Z ∗ = iV F− implying [F, Gt ] = 0. The (twisted) locality postulate (1.4) can now be stated simply as F(O)t ⊂ F (O0 )0 .
(1.7)
Poincar´e covariance is implemented by assuming the existence of a (strongly continuous) unitary representation on H of the Poincar´e group P such that α(3,a) (F(O)) = Ad U (3, a)(F(O)) = F(3O + a).
(1.8)
The spectrum of the generators of the translations (momenta) is required to be contained in the closed forward lightcone and the existence of a unique vacuum vector invariant under P is assumed. Covariance under the conformal group, however, is not required. Our last postulate (for the moment) concerns the inner symmetries of the theory. There shall be a compact group G, represented in a strongly continuous fashion by unitary operators on H leaving invariant the vacuum such that the automorphisms αg (F ) = Ad U (g)(F ) of B(H) respect the local structure: αg (F(O)) = F(O).
(1.9)
The action shall be faithful, i.e. αg 6= id ∀g ∈ G. This is no real restriction, for the kernel of the homomorphism G → Aut(F ) can be divided out. (Compactness of G need not be postulated, as it follows [27, Thm. 3.1] from the split property which will be introduced later.) In particular, there is an element k ∈ Z(G) of order 2 in the center of the group G such that V = U (k). This implies that the observables which are now defined as the fixpoints under the action of G A(O) = F(O)G = F (O) ∩ U (G)0
(1.10)
fulfill locality in the conventional untwisted sense. In 1+1 dimensions the representations of the Poincar´e group and of the inner symmetries do not necessarily commute. In the appendix of [57] it is, however, proved that in theories satisfying the distal split property the translations commute with the inner symmetries whereas the boosts act by automorphisms on the group Gmax of all inner symmetries. As we will postulate a stronger version of the split property in the next section the cited result applies to the situation at hand. What we still have to assume is that the one parameter group of Lorentz boosts maps the group G of inner symmetries, which in general will be a subgroup of Gmax , into itself and commutes with V = U (k). This assumption is indispensable for the covariance of the fixpoint net A as well as of another net to be constructed later. This framework was the starting point for the investigations in [21] where in particular properties of the observable net (1.10) and its representations on the sectors in H, i.e. the G-invariant subspaces, were studied, implicitly assuming the spacetime to be of dimension ≥ 2 + 1. While it is impossible to do any justice to the deep analysis which derives from this early work (e.g. [22–25, 61, 28] and the books [46, 49]) we have to sketch some of the main ideas in order to prepare the ground for our own work in
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the subsequent sections. One important notion examined in [21] was that of duality designating a certain maximality property in the sense that the local algebras cannot be enlarged (on the same Hilbert space) without violating spacelike commutativity. The postulate of twisted duality for the fields consists in strengthening the twisted locality (1.7) to (1.11) F(O)t = F(O0 )0 , which means that F(O0 ), the von Neumann algebra generated by all F (O1 ), O0 ⊃ O1 ∈ K contains all operators commuting with F(O) after twisting. From this it has been derived [21, Thm. 4.1] that duality holds for the observables when restricted to a simple sector: (1.12) (A(O) H1 )0 = A(O0 ) H1 . A sector H1 is called simple if the group G acts on it via multiplication with a character: U (g) H1 = χ(g) · 1 H1 .
(1.13)
Clearly the vacuum sector is simple. Furthermore it has been shown [21, Thm. 6.1] that the irreducible representations of the observables on the charge sectors in H are strongly locally equivalent in the sense that for any representation π(A) = A Hπ and any O ∈ K there is a unitary operator XO : H0 → Hπ such that XO π0 (A) = π(A) XO ∀A ∈ A(O0 ).
(1.14)
The fundamental facts (1.12) and (1.14), which have come to be called Haag duality and the DHR criterion, respectively, were taken as starting points in [23, 24] where a more ambitious approach to the theory of superselection sectors was advocated and developed to a large extent. The basic idea was that the physical content of any quantum field theory should reside in the observables and their vacuum representation. All other physically relevant representations as well as unobservable charged fields interpolating between those and the vacuum sector should be constructed from the observable data. The vacuum representation and the other representations of interest were postulated to satisfy π0 (A(O)) = π0 (A(O0 ))0 , π A(O0 ) ∼ = π0 A(O0 ) ∀O ∈ K,
(1.15) (1.16)
respectively. It may be considered as one of the triumphs of the algebraic approach that it has finally been possible to prove [28, and references given there] the existence of an essentially unique net of field algebras with a unique compact group G of inner symmetries such that there is an isomorphism between the monoidal (strict, symmetric) category of the superselection sectors satisfying (1.16) with the product structure established in [23] and the category of finite dimensional representations of G. Before turning now to the two dimensional situation we should remark that the duality property (1.11) upon which the whole theory hinges has been proved to hold for free massive and massless fields (scalar [3] and Dirac [21]) in ≥ 1 + 1 dimensions (apart from the massless scalar field in two dimensions) as well as for several interacting theories (P (φ)2 , Y2 ). Furthermore, there is a remarkable link [61] between Haag duality and spontaneous symmetry breakdown. For the rest of this paragraph we assume that only a subgroup G0 of G is unbroken, i.e. unitarily implemented on H. Then the net B(O) = F(O)G0 satisfies Haag duality in restriction to H0 = HG0 whereas A(O) = F (O)G , being a true subnet of B, does not. Yet, defining the dual net (on H0 ) by
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Ad (O) = A(O0 )0 ,
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(1.17)
the fixpoint net A still satisfies essential duality: Ad (O) = Add (O).
(1.18)
(Haag duality, by contrast, is simply A(O) = Ad (O).) Furthermore, one finds Ad (O) = B(O). These matters have been developed further in [28, 15] in the context of reconstruction of the fields from the observables. In 1 + 1 dimensions a large part of the analysis sketched above breaks down due to the following topological peculiarities of 1 + 1 dimensional Minkowski space. Firstly, there is a Poincar´e invariant distinction between left and right, i.e. for a spacelike vector x the sign of x1 is invariant under the unit component of P. This fact accounts for the existence of soliton sectors which have been studied rigorously in the frameworks of constructive and general quantum field theory, see [39] and [37, 38, 63], respectively. We intend to make use of the latter in a sequel to this work. In the present paper we focus on the other well known feature of the topology of 1+1 dimensional Minkowski space, viz. the fact that the spacelike complement of a bounded connected region (in particular, a double cone) consists of two connected components. The implications of this fact are twofold. On one hand, in the adaption of the DHR analysis [23, 24] based on (1.15, 1.16) to 1 + 1 dimensions [35, 58, 36] the permutation group S∞ governing the statistics is replaced by the braid group B∞ , as anticipated, e.g., in [40]. It is still not known by which structure the compact group appearing in the higher dimensional situation has to be replaced if a completely general solution to this question exists at all. Besides the appearance of braid group statistics the disconnectedness of O0 manifests itself also if one starts from a field net F with unbroken symmetry group G. It was mentioned above that in ≥ 2 + 1 dimensions the restriction of the fixpoint net A to the simple sectors in H satisfies Haag duality provided the field net F fulfills (twisted) Haag duality. Since questions of Haag duality have been studied only in the framework of the algebraic approach the third peculiarity of quantum field theories in 1 + 1 dimensions (besides solitons and braid group statistics/quantum symmetry) is less widely known. We refer to the fact that the step from (1.11) to (1.15) may fail in 1+1 dimensions. This means that one cannot conclude from twisted duality of the fields that duality holds for the observables in simple sectors, which in fact is possible only in conformal theories. The origin of this phenomenon is easily understood. Let O ∈ K be a double cone. One can then construct gauge invariant operators in F(O0 ) which are obviously contained in A(O)0 but not in A(O0 ). This is seen remarking that the latter algebra, belonging to a disconnected region, is defined to be generated by the observable algebras associated with the left and right spacelike complements of O, respectively. This algebra does not contain gauge invariant operators constructed using fields localized in both components. We now come to the plan of this paper. Our aim will be to explore the relation between a quantum field theory with symmetry group G in 1+1 dimensions and the fixpoint theory. In addition to the general properties of such a theory stated above, twisted duality (1.11) is assumed to hold for the large theory. As explained above, in this situation duality of the fixpoint theory fails even in the case of unbroken group symmetry. Yet there is a local extension which satisfies Haag duality and one would like to obtain a complete understanding of this dual net. To this end we will need one additional postulate concerning the causal independence of one-sided infinite regions (wedges) which are separated from each other by a finite spacelike distance. This property rules out conformal theories and singles out a (presumably large) class of well-behaved
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massive theories. In Sect. 2 we prove the existence of unitary disorder operators which implement a global symmetry transformation on one wedge and act trivially on the spacelike complement of a slightly larger wedge. Using these operators we will in Sect. ˆ ˆ 3 consider a non-local extension F(O) of the field net F(O). The fixpoint net A(O) ˆ of the enlarged net F(O) under the action of G is shown to coincide with the dual net Ad (O) (1.17) in restriction to the simple sectors. In conjunction with several technical results on actions of G this leads to an explicit characterization of the dual net. In Sect. 4 we will show that there is an action of the quantum double D(G) on the extended net Fˆ and that the spacelike commutation relations are governed by Drinfel’d’s R-matrix. Since massive free scalar fields satisfy all assumptions we made on F this construction provides the first mathematically rigorous construction of quantum field theories with D(G)-symmetry for any finite group G. The quantum double may be considered a “hidden symmetry” of the original theory since it is uncovered only upon extending the latter. The D(G)-symmetry is spontaneously broken in that only the action of the subalgebra CG ⊂ D(G) is implemented in the Hilbert space H. In analogy to Roberts’ analysis this might be interpreted as the actual reason for the failure of Haag duality for the fixpoint net A. The aim of the final Sect. 5 is to show that the methods introduced in the preceding sections are well suited for a discussion of Jordan-Wigner transformations and bosonization in the framework of algebraic quantum field theory. Three appendices are devoted in turn to a summary of the needed facts on quantum groups and quantum doubles, a partial generalization of our results to infinite compact groups and an indication how an analysis similar to Sects. 2 to 4 can be done for chiral conformal theories on the circle.
2. Disorder Variables and the Split Property 2.1. Preliminaries. For any double cone O ∈ K we designate the left and right spacelike O O and WRR , respectively. Furthermore we write WLO and WRO for complement by WLL O 0 O 0 WRR and WLL . These regions are wedge shaped, i.e. translates of the standard wedges WL = {x ∈ R2 | x1 < −|x0 |} and WR = {x ∈ R2 | x1 > |x0 |}. We will not distinguish between open and closed regions, for definiteness one may consider O and all W-regions O O ∪ WRR which as open. With these definitions we have O = WLO ∩ WRO and O0 = WLL graphically looks as follows:
@
@
O WLL
@
O @ O @WL @W R @ O O @ WRR @ @ @ @ @ @ @ @
(2.1)
Whereas, as we have shown in the introduction, Haag duality for double cones is violated in the fixpoint theory A, one obtains the following weaker form of duality.
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Proposition 2.1. The representation of the fixpoint net A fulfills duality for wedges A(W )0 = A(W 0 )
(2.2)
and essential duality (1.18) in all simple sectors. Proof. The spacelike complement of a wedge region is itself a wedge, thus connected, whereby the proof of [21, Thm. 4.1] applies, yielding the first statement. The second follows from wedge duality via O O ) ∨ A(WRR ))0 = A(WRO ) ∧ A(WLO ), Ad (O) = A(O0 )0 = (A(WLL
as locality of the dual net is equivalent to essential duality of A.
(2.3)
We will now introduce the central notion for this paper. Definition 2.2. A family of disorder operators consists, for any O ∈ K and any g ∈ G, of two unitary operators ULO (g) and URO (g) verifying O O Ad ULO (g) F(WLL ) = Ad URO (g) F(WRR ) = αg , O O O O Ad UL (g) F(WRR ) = Ad UR (g) F(WLL ) = id.
(2.4)
O (g) on fields located in the left and right spacelike In words: the adjoint action of UL/R complements of O, respectively, equals the global group action on one side and is trivial on the other. As a consequence of (twisted) wedge duality we have at once
ULO (g) ∈ F(WLO )t , URO (g) ∈ F (WRO )t .
(2.5)
On the other hand it is clear that disorder operators cannot be contained in the local algebras F(O), F (O)t nor in the quasilocal algebra F , for in this case locality would not allow their adjoint action to be as stated on operators located arbitrarily far to the left or right. R Heuristically, assuming U (g) arises from a conserved local current via i j 0 (t=0,x)dx , one may think of ULO (g) as given by U (g) = e R x0 0 i j (x)dx ULO (g) = e −∞ , (2.6) where integration takes place over a spacelike curve from left spacelike infinity to a point x0 in O. The need for a finitely extended interpolation region O arises from the distributional character of the current which necessitates a smooth cutoff. We refrain from discussing these matters further as they play no role in the sequel. In massive free field theories disorder operators can be constructed rigorously (e.g. [43, 1]) using the CCR/CAR structure and the criteria due to Shale. O O (g), UL,2 (g) be disorder operators associated with the same douLemma 2.3. Let UL,1 O O (g) = F UL,2 (g) with F ∈ F(O)t ble cone and the same group element. Then UL,1 unitary. An analogous statement holds for the right-handed disorder operators. O O∗ (g) UL,2 (g). By construction F ∈ F(WLO )t . On the other hand Proof. Consider F = UL,1 O O O (g) implement the same automorphism Ad F F(WLL ) = id holds as UL,1 (g) and UL,2 O of F (WLL ). By (twisted) wedge duality we have F ∈ F (WRO )t and (twisted) duality for double cones implies F ∈ F (O)t .
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Remarks. 1. This result shows that disorder operators are unique up to unitary elements of F (O)t , the twisted algebra of the interpolation region. The obvious fact that ULO (g) ULO (h) and U (g) ULO (h) U (g)∗ are disorder operators for the group elements gh and ghg −1 , respectively, implies that a family of disorder operators constitutes a projective representation of G with the cocycle taking values in F (O)t . 2. Later on we will consider only bosonic disorder operators, which leads to the stronger result F ∈ F(O)+ . For the purposes of the present investigation the mere existence of disorder operators is not enough, for we need them to obey certain further restrictions. Our first aim will be to obtain such operators by a construction which is model independent to the largest possible extent, making use only of properties valid in any reasonable model. To this effect we reconsider an idea due to Doplicher [26] and developed further in, e.g., [27, 14]. It consists of using the split property [12] to obtain, for any g ∈ G and any pair of double cones 3 = (O1 , O2 ) such that O1 ⊂ O2 , an operator U3 (g) ∈ F (O2 ) such that U3 (g)F U3 (g)∗ = U (g)F U (g)∗
∀F ∈ F(O1 ).
(2.7)
In order to be able to do the same thing with wedges we introduce our last postulate. Definition 2.4. An inclusion A ⊂ B of von Neumann algebras is split [27], if there exists a type-I factor N such that A ⊂ N ⊂ B. A net of field algebras satisfies the “split O O ) ⊂ F (WLO ) and F (WRR ) ⊂ F (WRO ) property for wedges” if the inclusions F(WLL are split for every double cone O. (In our case, where wedge duality holds, the split property for one of the above inclusions entails the same for the other as is seen by passing to commutants and twisting.) This property is discussed at some length in [57] and shown to be fulfilled for the free massive scalar and Dirac fields. In quantum field theories where there are lots of cyclic and separating vectors for the local algebras by the Reeh-Schlieder theorem, the split property is equivalent [27] to the existence, for any double cone O, of a unitary operator O O t ) ∨ F(WRR ) and Y O : H → H ⊗ H implementing an isomorphism between F (WLL O O t the tensor product F(WLL ) ⊗ F (WRR ) (in the sense of von Neumann algebras) Y O F1 F2t Y O∗ = F1 ⊗ F2t
O O ∀F1 ∈ F(WLL ), F2 ∈ F(WRR ).
(2.8)
O O That one of the algebras F(WLL ) and F(WRR ), which are associated with spacelike separated regions, has to be twisted in order for an isomorphism as above to exist is clear as in general these algebras do not commute while the factors of a tensor product do O t O ) ∨F (WRR ) and commute. Analogously, there is a spatial isomorphism between F(WLL O t O O O F (WLL ) ⊗ F(WRR ) implemented by Y˜ . We will stick to the use of Y throughout. In order not to obscure the basic simplicity of the argument we assume for a moment that the theory F is purely bosonic, i.e. fulfills locality and duality without twisting. Using the isomorphism implemented by Y O we then have the following correspondences: O O )∼ )⊗ 1, F(WLL = F(WLL O O F(WRR )∼ 1 ⊗ F(W = RR ), F(WLO ) ∼ = B(H) ⊗ F (WLO ), F(WRO ) ∼ = F(WRO ) ⊗ B(H),
(2.9)
whereas Haag duality for double cones yields F(O) = F(WLO ) ∧ F(WRO ) ∼ = F(WRO ) ⊗ F(WLO ).
(2.10)
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(Taking the intersection separately for both factors of the tensor product is valid in this situation as can easily be proved using the lattice property of von Neumann algebras M ∧ N = (M0 ∨ N 0 )0 and the commutation theorem for tensor products (M ⊗ N )0 = M0 ⊗ N 0 .) We thus see that in conjunction with the well known fact [29] that the algebras associated with wedge regions are factors of type III1 the split property for wedges implies that the algebras of double cones are type III1 factors, too. The following property of the maps Y O will be pivotal for the considerations below. Given any unitary U implementing a local symmetry (i.e. U F (O)U ∗ = F(O) ∀O) and leaving invariant the vacuum (U = ) the following identity holds: Y O U = (U ⊗ U ) Y O .
(2.11)
For the construction of Y O as well as for the proof of (2.11) we refer to [27, 14], the difference that those authors work with double cones being unimportant. 2.2. Construction of disorder operators. The operators Y O will now be used to obtain disorder operators. To this purpose we give the following Definition 2.5. For any double cone O ∈ K and any g ∈ G we set ULO (g) = Y O∗ (U (g) ⊗ 1) Y O , URO (g) = Y O∗ (1 ⊗ U (g)) Y O .
(2.12)
As an immediate consequence of this definition we have the following Proposition 2.6. The disorder operators defined above satisfy O UL (g), URO (h) = 0, ULO (g)
URO (g)
= U (g),
O O U (g) UL/R (h) U (g)∗ = UL/R (ghg −1 ).
(2.13) (2.14) (2.15)
Proof. The first statement is trivial and the second follows from (2.11). The covariance property (2.15) is another consequence of (2.11). Remark. We have thus obtained some kind of factorization of the global action of the group G into two commuting true (i.e. no cocycles) representations of G such that the original action is recovered as the diagonal. Furthermore, these operators transform covariantly under global gauge transformations. In particular they are bosonic since k ∈ Z(G). O indeed fulfill the requirements of Definition It remains to be shown that the UL/R 2.2. The second requirement follows from Definition 2.5, which with (2.9) obviously yields (2.16) ULO (g) ∈ F(WLO ), URO (g) ∈ F (WRO ). O ): The first one is seen by the following computation valid for F ∈ F(WLL
ULO (g)F ULO∗ (g) ∼ = (U (g) ⊗ 1)(F ⊗ 1)(U (g) ⊗ 1)∗ = (U (g)F U (g)∗ ⊗ 1) ∼ = U (g)F U (g)∗ , appealing to the isomorphism ∼ = implemented by Y O .
(2.17)
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Returning now to the more general case including fermions we have to consider the apparent problem that there are now two ways to define the operators ULO (g) and URO (g), depending upon whether we choose Y O or Y˜ O . (By contrast, the tensor product factorization (2.10) of the local algebras is of a purely technical nature, rendering it irrelevant whether we use Y O or Y˜ O .) This ambiguity is resolved by remarking that the element k ∈ G giving rise to V by V = U (k) is central, implying that the operators O O ), F2 ∈ F(WRR ) we U (g), g ∈ G, are bosonic (even). For even operators F1 ∈ F(WLL have F1 = F1t , F2 = F2t and thus Y O F1 F2 Y O∗ = Y˜ O F1 F2 Y˜ O∗ = F1 ⊗ F2 ,
(2.18)
so that the disorder variables are uniquely defined even operators. The first two equations of (2.9) are replaced by O ) F(WLL O t F(WRR )
O ∼ )⊗ 1, = F(WLL O t ∼ ). 1 ⊗ F(WRR =
(2.19)
∼ = B(H) ⊗ F(WLO ), ∼ = F(WRO )t ⊗ B(H),
(2.20)
By taking commutants we obtain F(WLO ) F(WRO )t
and an application of the twist operation to the second equations of (2.19) and (2.20) yields O ) F(WRR F (WRO )
O O ∼ 1 ⊗ F(WRR )+ + V ⊗ F (WRR )− , = O O ∼ = F(WR ) ⊗ B(H)+ + F (WR ) V ⊗ B(H)− .
(2.21)
The identity F (O) = F(WLO )∧F(WRO ), which is valid in the fermionic case, too, finally leads to (2.22) F(O) ∼ = F (WRO ) ⊗ F (WLO )+ + F (WRO ) V ⊗ F (WLO )− . While this is not as nice as (2.10) it is still sufficient for the considerations in the sequel. That F(O), O ∈ K is a factor is, however, less obvious than in the pure Bose case and will be proved only in Subsect. 3.3. The following easy result will be of considerable importance later on. Lemma 2.7. The disorder operators ULO (g) and URO (g) associated with the double cone O implement automorphisms of the local algebra F(O). Proof. In the pure Bose case this is obvious from Definition 2.5, (2.10) and the fact that Ad U (g) acts as an automorphism on all wedge algebras. In the Bose-Fermi case (2.22) the same is true since U (g) commutes with V = U (k). Definition 2.8. αgO = Ad ULO (g), g ∈ G, O ∈ K. We close this section with one remark. We have seen that the split property for wedges implies the existence of disorder operators which constitute true representations of the symmetry group and which transform covariantly under the global symmetry. Conversely, one can show that the existence of disorder operators, possibly with group cocycle, in conjunction with the split property for wedges for the fixpoint net A implies the split property for the field net F. This in turn allows to remove the cocyle using the above construction. We refrain from giving the argument which is similar to those in [26, pp. 79, 85].
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3. Field Extensions and Haag Duality 3.1. The extended field net. Having defined the disorder variables we now take the next step, which at first sight may seem unmotivated. Its relevance will become clear in the ˆ sequel. We define a new net of algebras O 7→ F(O) by adding the disorder variables associated with the double cone O to the fields localized in this region. Definition 3.1. ˆ F(O) = F(O) ∨ ULO (G)00 .
(3.1)
Remarks. 1. In accordance with the common terminology in statistical mechanics and conformal field theory, the operators which are composed of fields (order variables) and disorder variables might be called parafermion operators. 2. We could as well have chosen the disorder operators acting on the right-hand side. As there is a complete symmetry between left and right there would be no fundamental difference. We will therefore stick to the above choice throughout this paper. Including both the left and right-handed disorder operators would, however, have the unpleasant consequence that there would be translation invariant operators (namely the U (g)’s) in the local algebras. ˆ 3. The local algebra F(O) of the above definition resembles the crossed product of F(O) by the automorphism group αgO , the interesting aspect being that the automorphism group depends on the region O. These two constructions differ, however, with respect to the Hilbert space on which they are defined. Whereas the crossed product F(O)oαO G lives on the Hilbert space L2 (G, H), our algebras Fˆ (O) are defined on the original space H. For later purposes it will be necessary to know whether these algebras are isomorphic, but we prefer first to discuss those aspects which are independent of this question. The first thing to check is, of course, that Definition 3.1 specifies a net of von Neumann algebras. ˆ Proposition 3.2. The assignment O 7→ F(O) satisfies isotony. ˆ Proof. Let O ⊂ Oˆ be an inclusion of double cones. Obviously we have F (O) ⊂ Fˆ (O). ˆ O) ˆ we observe that U O (g) is a disorder operator for the In order to prove ULO (g) ∈ F( L ˆ too. Thus, by Lemma 2.3 we have U O (g) = F U Oˆ (g) with F ∈ F (O). ˆ larger region O, L L O ˆ ˆ Now it is clear that UL (g) ∈ F(O). Remark. From this we can conclude that the net Fˆ (O) is uniquely defined in the sense that any family of bosonic disorder operators gives rise to the same net Fˆ (O) provided such operators exist at all. For most of the arguments in this paper we will, however, need the detailed properties proved above which follow from the construction via the split property. It is obvious that the net Fˆ is nonlocal. While the spacelike commutation relations of fields and disorder operators are known by construction we will have more to say on this subject later. On the other hand it should be clear that the nets Fˆ and A are local relative to each other. This is simply the fact that the disorder operators commute with the fixpoints of αg in both spacelike complements.
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Proposition 3.3. The net Fˆ is Poincar´e covariant with the original representation of P. In particular αa (ULO (g)) = ULO+a (g) whereas for the boosts we have α3 (ULO (g)) = UL3O (h),
(3.2)
if U (3) U (g) U (3)∗ = U (h) . Proof. The family Y O : H → H ⊗ H of unitaries provided by the split property fulfills the identity (3.3) Y 3O+a = (U (3, a) ⊗ U (3, a)) Y O U (3, a)∗ , as is easily seen to follow from the construction in [27, 14]. This implies ∗
α3,a (ULO (g)) = U (3, a) Y O (U (g) ⊗ 1) Y O U (3, a)∗ ∗
= Y 3O+a (U (3, a) U (g) U (3, a)∗ ⊗ 1) Y 3O+a = UL3O+a (h), where U (3) U (g) U (3)∗ = U (h).
(3.4)
ˆ Proposition 3.4. The vacuum vector is cyclic and separating for F(O). Proof. Follows from
ˆ F(O) ⊂ F(O) ⊂ F(WLO )
since is cyclic and separating for F(O) and
F(WLO ).
(3.5)
Proposition 3.5. The wedge algebras for the net Fˆ take the form O 0 ˆ LO ) = F (WLO ), Fˆ (WRO ) = F(WRO ) ∨ U (G)00 = A(WLL F(W ).
(3.6)
O ˆ As a consequence is not separating for F(W R )! ˆ Proof. The first identity is obvious, while the second follows from F (WRO ) 3 URO (g) ∀Oˆ ∈ WRO and the factorization property (2.14). The last statement is equivalent to not O ). being cyclic for A(WLL
Proposition 3.6. Let Fˆ ∈ F (O)ULO (g). Then the following cluster properties hold. w − lim αx (Fˆ ) = h, Fˆ i · 1,
(3.7)
w − lim αx (Fˆ ) = h, Fˆ i · U (g).
(3.8)
x→−∞ x→+∞
O ˆ Proof. The first identity follows from Fˆ ∈ F(W L ) and the usual cluster property. The second is seen by writing Fˆ = F URO (g −1 ) U (g) and applying the weak convergence of URO as above, the translation invariance of U (g) and the invariance of the vacuum under U (g).
3.2. Haag duality. Observing by (2.15) that the adjoint action of the global symmetry group leaves the ‘localization’ (in the sense of Definition 2.2) of the disorder operators invariant it is clear that the automorphisms αg = Ad U (g) extend to local symmetries of ˆ We are thus in a position to define yet another net, the fixpoint net the enlarged net F. ˆ of F .
Quantum Double Actions on Operator Algebras and Orbifold QFTs
Definition 3.7.
ˆ ˆ A(O) = F(O) ∧ U (G)0 .
149
(3.9)
Remark. We then have the following square of local inclusions: ˆ A(O) ⊂ Fˆ (O) (3.10) ∪ ∪ A(O) ⊂ F (O). R ˆ clearly restricts The conditional expectation m(·) = dg αg (·) from Fˆ (O) to A(O) to a conditional expectation from F(O) to A(O). In Sect. 4 we will see that there is also a conditional expectation γe from Fˆ (O) to F (O) which restricts to a conditional ˆ expectation from A(O) to A(O), provided the group G is finite. Since γe commutes with m the square (3.10) then constitutes a commuting square in the sense of Popa. ˆ Proposition 3.8. The net O 7→ A(O) is local. ˜ Proof. Let O < O be two regions spacelike to each other, O˜ being located to the right of ˆ O. From A(O) ⊂ A(WLO ) and the relative locality of observables and fields we conclude ˆ ˜ On the other hand the operators U O˜ (g) commute with that A(O) commutes with F(O). L ˜ ˆ ˆ A(O) ⊂ Fˆ (WLO ) = F(WLO ) since Ad ULO (g) F (WLO ) = αg and A(O) is pointwise gauge invariant. We have just proved that the net Aˆ constitutes a local extension of the observable net A, thereby confirming our initial observation that A does not satisfy Haag duality. The elements of Aˆ being gauge invariant they commute a fortiori with the central projections in the group algebra, thereby leaving invariant the sectors in H. We will now prove a nice result which serves as our first justification for Definitions 3.1 and 3.7. Lemma 3.9.
A(O 0 )0 = F(O) ∨ ULO (G)00 ∨ URO (G)00 .
(3.11)
Proof. We already know that A(O0 )0 ⊃ F(O) ∨ ULO (G)00 ∨ URO (G)00 .
(3.12)
In order to prove equality we consider the following string of identities, making use of the spatial isomorphisms due to the split property and omitting the superscript O on the wedge regions. A(O0 )0 = (A(WLL ) ∨ A(WRR ))0 = ((F(WLL ) ∧ U (G)0 ) ∨ (F (WRR ) ∧ U (G)0 ))0 0 ∼ = (F(WLL ) ∧ U (G)0 ) ⊗ (F (WRR ) ∧ U (G)0 ) = (F (WR )t ∨ U (G)00 ) ⊗ (F (WL )t ∨ U (G)00 ), = (F (WR ) ⊗ F (WL )) ∨ (U (G)00 ⊗ 1) ∨ (1 ⊗ U (G)00 ) ∼ (3.13) = F (O) ∨ ULO (G)00 ∨ ULO (G)00 . In the third step we have used the identities F (WLL )∧U (G)0 ∼ = F(WLL )∧U (G)0 ⊗1 and F (WRR ) ∧ U (G)0 ∼ = 1 ⊗ F (WRR ) ∧ U (G)0 which are easily seen to follow from (2.20) and (2.21), respectively. The fourth step is justified by F(·)t ∨ U (G)00 = F (·) ∨ U (G)00 .
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Theorem 3.10. In restriction to a simple sector H1 the net Aˆ satisfies Haag duality, i.e. it coincides with the dual net Ad in this representation. Proof. Let P1 be the projection on a simple sector, i.e. fulfilling U (g)P1 = P1 U (g) = χ(g) · P1 ,
(3.14)
where χ is a character of G. Making use of ULO (G)00 ∨ URO (G)00 = ULO (G)00 ∨ U (G)00 and (3.14) we have P1 (F(O) ∨ ULO (G)00 ∨ URO (G)00 ) P1 = With m(F ) =
R
P1 (F(O) ∨ ULO (G)00 ∨ U (G)00 ) P1 = P1 Fˆ (O) P1 .
(3.15)
dg U (g)F U (g)∗ and once again using (3.14) we obtain
On the other hand
ˆ ˆ P1 F(O) P1 = P1 m(Fˆ (O)) P1 = P1 A(O) P1 .
(3.16)
0 P1 A(O0 )0 P1 P1 H = P1 A(O0 ) P1 P1 H .
(3.17)
The proof is now completed by applying the preceding lemma.
Remark. The above arguments make it clear that Haag duality cannot hold for the net A(O) even in simple sectors. This is not necessarily so if the split property for wedges does not hold. In conformally invariant theories gauge invariant combinations of field operators in the left and the right spacelike complements of a double cone O may well be contained in A(O0 ) due to spacetime compactification. One would think, however, that this is impossible in massive theories, even those without the split property. ˆ 3.3. Outerness properties and computation of A(O). While the above theorem allows ˆ us in principle to construct the dual net A one would like to know more explicitly how the elements of Aˆ look in terms of the fields in F and the disorder operators. In the case of an abelian group G this is easy to see. As a consequence of the covariance property (2.15) we then have O O O (h) U (g)∗ = UL/R (ghg −1 ) = UL/R (h), U (g) UL/R
(3.18)
ˆ that is the disorder operators are gauge invariant and thus contained in A(O). It is then obvious that ˆ (3.19) A(O) = A(O) ∨ ULO (G)00 , (G abelian!) ˆ as A(O) is spanned by operators of the form F ULO (g), F ∈ F(O) which are invariant iff F ∈ A(O). The case of the group G being non-abelian is more complicated and we limit ourselves to finite groups leading already to structures which are quite interesting. In order ˆ to proceed we would like to know that every operator Fˆ ∈ F(O) has a unique representation of the form X F (g) ULO (g), F (g) ∈ F(O). (3.20) Fˆ = g∈G
While this true for the crossed product M o G on L2 (G, H) (only for finite groups!) it is not obvious for the algebra M ∨ U (G)00 on H. The latter may be considered as the
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image of the former under a homomorphism which might have a nontrivial kernel. In this case there would be equations of the type X F (g) ULO (g) = 0, (3.21) g∈G
where not all F (g) vanish. Fortunately at least for finite groups (infinite, thus noncompact, discrete groups are ruled out by the split property) this undesirable phenomenon can be excluded without imposing further assumptions using the following result due to Buchholz [16]. Proposition 3.11. The automorphisms αg = Ad U (g) act outerly on the wedge algebras. Proof. Let W be the standard wedge W = {x ∈ R2 | x1 > |x0 |} and assume there is a unitary Vg ∈ F(W ) such that Ad Vg F (W ) = αg . Define Vg,x = αx (Vg ) for all x ∈ W . Obviously Vg,x ∈ F (Wx ). By the commutativity αx ◦ αg = αg ◦ αx of translations and gauge transformations we have Ad Vg,x F (Wx ) = αg . By the computation (for x ∈ W ) Vg Vg,x Vg∗ = αg (Vg,x ) = αg ◦ αx (Vg ) = αx ◦ αg (Vg ) Vg,x = αx (Vg Vg Vg∗ ) = αx (Vg ) =
(3.22)
Vg Vg,x = Vg,x Vg ∀x ∈ W.
(3.23)
V = {Vg,x , x ∈ W }00
(3.24)
we obtain The von Neumann algebra
is mapped into itself by translations αx where x ∈ W and the vacuum vector it is separating for V as we have V ⊂ F (W ). This allows us to apply the arguments in [29] to conclude that V is either trivial (i.e. V = C1) or a factor of type III1 . The assumed existence of Vg , which cannot be proportional to the identity due to the postulate αg 6= id, excludes the first alternative whereas the second is incompatible with (3.23) according to which Vg is central. Contradiction! Remark. This result may be interpreted as a manifestation of an ultraviolet problem. The automorphism αg being inner on a wedge W , wedge duality would imply it to be inner on the complementary wedge W 0 , too, giving rise to a factorization U (g) = VL (g) VR (g), VL (g) ∈ F(W ), VR (g) ∈ F(W 0 ). This would be incompatible with the distributional character of the local current from which U (g) derives. We cite the following well known result on automorphism groups of factors. Proposition 3.12. Let M be a factor and α an outer action of the finite group G. Then the inclusions MG ⊂ M, π(M) ⊂ M o G are irreducible, i.e. M o G ∩ π(M)0 = 0 M ∩ MG = C1. In particular M o G and MG are factors. If the action α is unitarily implemented αg = Ad U (g) then M o G and M ∨ U (G)00 are isomorphic. 0
Proof. The irreducibility statements M o G ∩ π(M)0 = M ∩ MG = C1 are standard consequences of the relative commutant theorem [65, §22] for crossed products. Remarking that finite groups are discrete and compact the proof is completed by an application of [48, Corr. 2.3] which states that M o G and M ∨ U (G)00 are isomorphic if the former algebra is factorial and G is compact.
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We are now in a position to prove several important corollaries to Prop. 3.11. Corollary 3.13. The algebras F(O), O ∈ K are factors also in the Bose-Fermi case. Proof. Since Ad V acts outerly on the factor F (WRO ) by Prop. 3.11, M1 = F (WRO )∨{V } is a factor and there is an automorphism β of M1 leaving F (WRO ) pointwise invariant such that β(V ) = −V . The automorphism β ⊗αk of M1 ⊗F (WLO ) clearly has Y O F (O) Y O∗ as fixpoint algebra, cf. (2.22). Since αk is outer the same holds [65, Prop. 17.6] for β⊗αk . Thus the fixpoint algebra is factorial by another application of Prop. 3.12. Corollary 3.14. Let O ∈ K. The automorphisms αg = Ad U (g) and αgO = Ad ULO (g) act outerly on the algebra F (O). Proof. The pure Bose case is easy. F (O), αgO and αg are unitarily equivalent to F (WRO ) ⊗ F(WLO ), αg ⊗ id, and αg ⊗ αg , respectively. Since αg = Ad U (g) is outer on F (WRO ) the same holds by [65, Prop. 17.6] for the automorphisms αg ⊗ id and αg ⊗ αg of the above tensor product. Turning to the Bose-Fermi case let Xg ∈ F (O) be an implementer of αg or αgO and define Xˆ g = Y O Xg Y O∗ . Then (1 ⊗ V )Xˆ g (1 ⊗ V ) also implements αg ⊗ id or αg ⊗ αg , respectively, since k is central. F (O) being a factor this implies (1 ⊗ V )Xˆ g (1 ⊗ V ) = cg Xˆ g with c2g = ±1 due to k 2 = e. Xˆ g is thus contained either in F (WRO ) ⊗ F (WLO )+ or in F(WRO ) V ⊗ F (WLO )− . In the first case the restriction of αg ⊗ id or αg ⊗ αg to F (WRO ) ⊗ F(WLO )+ is inner which can not be true by the same argument as for the Bose case. (Observe that F(WLO )+ is factorial.) On the other hand, no Xˆ g ∈ F (WRO ) V ⊗ F (WLO )− can implement αg ⊗ id or αg ⊗ αg since both automorphisms are trivial on 0 the subalgebra 1 ⊗ F(WLO ) ∩ U (G)0 which requires Xˆ g ∈ B(H) ⊗ F (WLO )G . This, 0 0 however, is impossible: F (WLO )− ∩F (WLO )G = [F (WLO )∩F(WLO )G ]− = [C1]− = ∅, where we have used the irreducibility of F (WLO )G ⊂ F(WLO ). ˆ Corollary 3.15. Let the symmetry group G be finite. Then the enlarged algebra F(O) = O 00 F (O) ∨ UL (G) is isomorphic to the crossed product F(O) oαO G and the inclusions ˆ A(O) ⊂ F(O), F (O) ⊂ F(O) are irreducible. Proof. Obvious from Prop. 3.12 and Cors. 3.13, 3.14.
Remark. If G is a compact continuous group, outerness of the action does not allow us to draw these conclusions. In this case an additional postulate is needed. It would be sufficient to assume irreducibility of the inclusion A(W ) ⊂ F(W ), for, as shown by Longo, this property in conjunction with proper infiniteness of A(W ) implies dominance of the action and factoriality of the crossed product. ˆ We are now able to give an explicit description of the dual net A. ˆ Theorem 3.16. Every operator Aˆ ∈ A(O) can be uniquely written in the form X A(g) ULO (g), (3.25) Aˆ = g∈G
where the A(g) ∈ F (O) satisfy A(kgk −1 ) = αk (A(g)) ∀g, k ∈ G.
(3.26)
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Conversely, every choice of A(g) complying with this constraint gives rise to an element ˆ ˆ O ) is obtained by replacing of A(O). An analogous representation for the algebras A(W R O UL (g) by U (g). Remark. Condition (3.26) implies A(g) ∈ F (O) ∩ U (Ng )0 , where Ng = {h ∈ G | gh = hg} is the normalizer of g in G. ˆ Proof. By Prop.P 3.11 any Aˆ ∈ A(O) can be P represented uniquely according to (3.25). O ˆ Using αk (A) = g αk (A(g)) UL (kgk −1 ) = g αk (A(k −1 gk)) ULO (g), Eq. (3.26) follows by comparing coefficients. It is obvious that the arguments can be reversed. The ˆ O ) follows from the fact that Fˆ (W O ) is the crossed statement on the wedge algebras A(W R R O product of F (WR ) by the global automorphism group, cf. Prop. 3.5. 3.4. The split property. The prominent role played by the split property in our investigations so far gives rise to the question whether it extends to the enlarged nets Aˆ and Fˆ . As to the net Fˆ it is clear that a twist operation is needed in order to achieve commutativity of the algebras of two spacelike separated regions. Let O1 < O2 be double cones. Then ˆ 2 )T ⊂ F(O ˆ 1 )0 , where one has F(O !T X X X O F (g) UL (g) := F (g)t ULO (g) U (g −1 ) = F (g)t URO (g)∗ , (3.27) g
g
g
and the t on F (g) denotes the Bose-Fermi twist of the introduction. (By the crossed ˆ product nature of the algebras F(O) it is clear that this map is well defined and invertible.) That commutativity holds as claimed follows easily from Fˆ (O1 ) ⊂ F(WLO1 ) and Fˆ (O2 )T ⊂ F (WRO2 )t . It is interesting to observe that the twist has to be applied to the algebra located to the right for this construction to work. This twist operation lacks, however, several indispensable features. Firstly, there is no unitary operator S implementing the twist as in the Bose-Fermi case. The second, more important objection refers to the fact that the map (3.27) becomes non-invertible when extended to right-handed wedge regions, for the operators URO (g) are contained in F(WRO ). Concerning the net Aˆ which, in contrast, is local there is no conceptual obstruction to ˆ O ) = A(W O ). Furthermore, proving the split property. We start by observing that A(W LL LL ˆ O ) H1 = in restriction to a simple sector H1 wedge duality (Prop. 2.1) implies A(W RR O A(WRR ) H1 . As the split property for the fields carries over [26] to the observables in the vacuum sector there is nothing to do if we restrict ourselves to the latter. We intend to prove now that the net Aˆ fulfills the split property on the big Hilbert space H. To this purpose we draw upon the pioneering work [26] where it was shown that the split property (for double cones) of a field net with group symmetry and twisted locality follows from the corresponding property of the fixpoint net provided the group G is finite abelian. (The case of general groups constitutes an open problem, but given nuclearity for the observables and some restriction on the masses in the charged sectors nuclearity and thus the split property for the fields can be proved.) ˆ Proposition 3.17. The net O 7→ A(O) satisfies the split property for wedge regions, provided the group G is finite. Proof. The split property for wedges is equivalent [11] to the existence, for every double ˆ O ), B ∈ cone O, of a product state φO satisfying φO (AB) = φO (A)·φO (B) ∀A ∈ A(W LL
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ˆ O ). For the rest of the proof we fix one double cone O and omit it in the formulae. A(W RR We have already remarked that for the net A product states φ0 are known to exist. In order to construct a product state for Aˆ we suppose γe is a conditional expectation from ˆ RR ) to A(WLL ) ∨ A(WRR ) such that γe (A(W ˆ RR )) = A(WRR ). Then A(WLL ) ∨ A(W γe (AB) = γe (A) γe (B), where A, B are as above, implying that φ = φ0 ◦ γe is a product state. It remains to find the conditional expectation γe . To make plain the basic idea we consider abelian groups G first. In this case γe is given by X ˆ χ, ˆ = 1 γe (A) ψχ∗ Aψ |G|
(3.28)
ˆ χ∈G
where ψχ ∈ F(O) is a unitary field operator transforming according to αg (ψχ ) = χ(g) · ψχ under the group G. This map has all the desired properties. The pointwise ˆ LL ) follows from the fact that this algebra commutes with the unitaries invariance of A(W ψχ . On the other hand ˜ ˜ O ψχ∗ ULO (g) ψχ = χ(g) · ULO (g), O˜ ⊂ WRR
in conjunction with the identity
(3.29)
P
ˆ χ(g) = |G| δg,e (valid also for non-abelian χ∈G O˜ ˆ RR ), g 6= e are annihilated by γe . UL (g) ∈ A(W
groups) implies that the operators Finally, the existence of ψχ ∈ F(O) for all χ (i.e. the dominance of the group action α on F(O)) is well known to follow from the outerness of the group action α. The generalization to non-abelian groups is straightforward. The unitaries ψχ are replaced by multiplets ψr,i of isometries for all irreducible representations r of G. They fulfill the following relations of orthogonality and completeness:
dr X
∗ ψr,j = δi,j 1, ψr,i
(3.30)
∗ ψr,i ψr,i =1
(3.31)
i=1
and transform according to αg (ψr,i ) =
X
Dir0 ,i (g) ψr,i0
(3.32)
i0
under the group. That the conditional expectation γe given by ˆ = γe (A)
dr 1 XX ∗ ˆ Aψr,i , ψr,i |G|
(3.33)
ˆ i=1 r∈G
does the job follows from dr X
∗ ψr,i ULO (g) ψr,i = tr Dr (g) · ULO (g) = χr (g) · ULO (g). ˜
˜
˜
(3.34)
i=1
Again the existence of such multiplets is guaranteed by our assumptions.
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Remark. Tensor multiplets satisfying (3.30, 3.31) were first considered in [25] where the relation between the charged fields in a net of field algebras and the inequivalent representations of the observables was studied in the framework of [21]. Multiplets of this type will play a role in our subsequent investigations, too. 3.5. Irreducibility of A(O) ⊂ Fˆ (O). The inclusions A(O) ⊂ F(O) ⊂ Fˆ (O) are of the form (3.35) N = P K ⊂ P ⊂ P o L = M, where K and L are finite subgroups of Aut P, as studied in [8] (albeit for type II1 factors). There P K ⊂ P o L was shown to be irreducible iff K ∩ L = {e} in Out P and to be of finite depth if and only if the subgroup Q of Out P generated by K and L is finite. Furthermore, the inclusion has depth two (i.e. N 0 ∧ M2 is a factor where N ⊂ M ⊂ M1 ⊂ M2 ⊂ · · · is the Jones tower corresponding to the subfactor N ⊂ M) in the special case when Q = K · L (i.e. every q ∈ Q can be written as q = kl, k ∈ K, l ∈ L). In our situation, where K = Diag(G × G) and L = G × 1, all these conditions are fulfilled, as we have Q = G × G and g × h = (h × h) · (h−1 g × e). The interest of this observation for our purposes derives from the following result, discovered by Ocneanu and proved, e.g., in [69, 55]. It states that an irreducible inclusion N ⊂ M arises via N = MH = {x ∈ M | γa (x) = ε(x)1 ∀a ∈ H} from the action of a Hopf algebra H on M iff the inclusion has depth two. In the next section this Hopf algebra will be identified and related to our quantum field theoretic setup. ˆ For the irreducibility of A(O) in F(O) we now give a proof independent of any sophisticated inclusion theoretic machinery. Proposition 3.18. For any O ∈ K we have ˆ F(O) ∧ A(O)0 = C1.
(3.36)
Proof. All unitary equivalences in this proof are implemented by Y O . With the abbreviations M1 = F(WRO )t and M2 = F(WLO ) we have M01 ∨ M02 ∼ = M01 ⊗ M02 . By (2.10) if F is bosonic or (2.22) in the Bose-Fermi case we have ∼ ˆ F(O) = F (WRO ) ∨ U (G)00 ⊗ F(WLO ) = M1 ∨ U (G)00 ⊗ M2 ,
(3.37)
where we have used Mt ∨ U (G)00 = M ∨ U (G)00 (which is true for every von Neumann algebra M). Furthermore, O O ) ∨ F (WRR ))t ∨ U (G)00 A(O)0 = F(O)0 ∨ U (G)00 = (F(WLL
= =
O O ) ∨ F(WRR )∨ F (WLL M01 ∨ M02 ∨ U (G)00 ∼ =
(3.38)
00
O O t U (G) = F (WLL ) ∨ F (WRR ) ∨ U (G)00 (M01 ⊗ M02 ) ∨ {U (g) ⊗ U (g), g ∈ G}00 .
The relative commutant Fˆ (O) ∧ A(O)0 is thus equivalent to (M1 ∨ U (G)00 ⊗ M2 ) ∧ [(M01 ⊗ M02 ) ∨ {U (g) ⊗ U (g), g ∈ G}00 ].
(3.39)
The obvious inclusion (M01 ⊗ M02 ) ∨ {U (g) ⊗ U (g), g ∈ G}00 ⊂ B(H) ⊗ M02 ∨ U (G)00 in conjunction with the irreducibility property M2 ∧ (M02 ∨ U (G)00 ) = C1 (Cor. 3.15) yields [(M01 ⊗ M02 ) ∨ {U (g) ⊗ U (g), g ∈ G}00 ] ∧ (B(H) ⊗ M2 ) ⊂ B(H) ⊗ 1.
(3.40)
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Now let X be an element of the algebra given by Eq. (3.39). By the same arguments as used earlier, every operator X ∈ P (M01 ⊗ M02 ) ∨ {U (g) ⊗ U (g), g ∈ G}00 has a unique representation of the form X = g Fg (U (g) ⊗ U (g)), where Fg ∈ M01 ⊗ M02 . The condition X ∈ B(H) ⊗ 1 implies Fg = 0 for all g 6= e and thereby X ∈ M01 ⊗ 1. We thus have X ∈ (M01 ∧ (M1 ∨ U (G)00 )) ⊗ 1 and, once again using the irreducibility of the group inclusions, X ∝ 1 ⊗ 1. 4. Quantum Double Symmetry ˆ 4.1. Abelian groups. As we have shown above the algebras F(O) may be considered as crossed products of F(O) with the actions of the respective automorphism groups αO . In the case of abelian (locally compact) groups there is a canonical action [70] of the dual (character-) group Gˆ on M o G given by αˆ χ (π(x)) = π(x) ˆ , χ ∈ G. αˆ χ (Ug ) = χ(g) · Ug
(4.1)
Making use of U O1 (g) U O2 (g)∗ ∈ F , ∀Oi one can consistently define an action of Gˆ on ˆ the net O 7→ F(O), respecting the local structure and thus extending to the quasilocal ˆ algebra F. The action of Gˆ commutes with the original action of G as extended to Fˆ , implying that the locally compact group G × Gˆ is a group of local symmetries of the extended theory O 7→ Fˆ (O). The square structure (3.10) can now easily be interpreted in terms of the larger symmetry: ˆ ˆ Aˆ = Fˆ G , F = Fˆ G , A = Fˆ G×G .
(4.2)
The symmetry between the subgroups G and Gˆ of G× Gˆ is, however, not perfect, as only the automorphisms αg , g ∈ G are unitarily implemented on the Hilbert space H. That there can be no unitary implementer U (χ) for αˆ χ , χ ∈ Gˆ leaving invariant the vacuum is shown by the following computation which would be valid for all A ∈ A(O): h, AULO (g)i = h, U (χ) AULO (g) U (χ)∗ i =
h, Aαˆ χ (ULO (g))i
= χ(g) ·
(4.3)
h, AULO (g)i.
This can only be true if χ(g) = 1 or h, AULO (g)i = 0 ∀A ∈ A(O). The latter, however, can be ruled out, since the density of A(O) in H0 would imply ULO (g) ⊥ H0 which is impossible, being unitary and gauge invariant. This argument shows that the vacuum ˆ in other state ω = h, · i is not invariant under the automorphisms α(χ), ˆ χ ∈ G, words, the symmetry under Gˆ is spontaneously broken. The preceding argument is just a special case of the much more general analysis in [61], where non-abelian groups were considered, too. There, to be sure, the field net acted upon by the group was supposed to fulfill Bose-Fermi commutation relations, whereas in our case the field net is nonlocal. Furthermore, whereas the net F (O), the point ˆ of departure for our analysis, fulfills (twisted) duality, the extended net F(O) enjoys no obvious duality properties. Nevertheless the analogy to [61] goes beyond the above argument. Indeed, as shown by Roberts, spontaneous breakdown of group symmetries is accompanied by a violation of Haag duality for the observables, restricted to the vacuum sector H0 . Defining the net B(O) = F(O)G0 , the fixpoint net under the action of the
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unbroken part G0 = {g ∈ G | ω0 ◦ αg = ω0 } of the symmetry group, a combination of the arguments in [21] and [61] leads to the conclusion that (in the vacuum sector H0 ) B(O) is just the dual net Ad (O) which verifies Haag duality. Our analysis in Sect. 2, leading to the identification of the dual net as Ad = Aˆ = Fˆ G , is obviously in accord with the general theory as we have shown above that G is the unbroken part, corresponding ˆ to G0 , of the full symmetry group G × G. In the case of spontaneously broken group symmetries it is known that, irrespective of the nonexistence of global unitary implementers leaving invariant the vacuum, one can find local implementers for the whole symmetry group. This means that for each double cone O there exists a unitary representation G 3 g 7→ VO (g) satisfying Ad VO (g) F (O) = αg , the important point being the dependence on the region O. (Due to the large commutant of F (O) such operators are far from unique.) A particularly nice construction, which applied to an unbroken symmetry g automatically yields the global implementer (VO (g) = U (g) ∀O), was given in [15]. The construction given there applies ˆ without change to the situation at hand where the action of the dual group Gˆ on F(O) is spontaneously broken. An interesting example is provided by the free massive Dirac field which as already mentioned fulfills our postulates, including twisted duality and the split property. Its symmetry group U (1) being compact and abelian, the extended net Fˆ and the action of the dual group Z can be constructed as described above. By restriction of the net Aˆ to the vacuum sector H0 one obtains a local net fulfilling Haag duality with symmetry group Z. Wondering to which quantum field theory this net might correspond, it appears quite natural to think of the sine-Gordon theory at the free fermion point β 2 = 4π as discussed, e.g., in [53].
4.2. Non-abelian groups. We refrain from further discussion of the abelian case and turn to the more interesting case of G being non-abelian and finite. (Infinite compact groups will be treated in Appendix B.) For non-abelian groups the dual object is not a group but either some Hopf algebraic structure or a category of representations. Correspondingly, the action of the dual group in [70] has to be replaced by a coaction of the group or the action of a group dual in the sense of [62]. For our present purposes these highbrow approaches will not be necessary. Instead we choose to generalize (4.1) in the following straightforward way. We observe that the characters of a compact abelian group constitute an orthogonal basis of the function space L2 (G), whereas in the nonabelian case they span only the subspace of class functions. This motivates us to define an action of C(G), the |G|-dimensional space of all complex valued functions on G, on Fˆ (O) in the following way: X X x(g) ULO (g) = F (g) x(g) ULO (g), x(g) ∈ F(O), F ∈ C(G). (4.4) γF g∈G
g∈G
Again this action of C(G) is consistent with the local structure of the net O 7→ Fˆ (O) and ˆ In general, of course, γF is no homomorphism extends to the quasilocal C ∗ -algebra F. but only a linear map. (That the maps γF are well defined for every F ∈ C(G) should be obvious, see also the next section.) Introducing the “deltafunctions” δg (h) = δg,h any P function can be written as F = g F (g) δg , and γδg will be abbreviated by γg . The latter are projections, i.e. they satisfy γg2 = γg . The images of Fˆ (O) and Fˆ under these will be
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designated Fˆ g (O) and Fˆ g , respectively. Obviously we have Fˆ g (O) = F(O) ULO (g) and Fˆ g = F ULO (g) with O ∈ K arbitrary. It should be clear that the decomposition M Fˆ g (4.5) Fˆ = g∈G
represents a grading of Fˆ by the group, i.e. Fˆ g Fˆ h ⊂ Fˆ gh ∀g, h ∈ G.
(4.6)
(In fact we have equality, but this will play no role in the sequel.) This group grading which is, of course, not surprising as it holds for every crossed product by a finite group allows us to state the behavior of γg under products: X γg (AB) = γh (A) γh−1 g (B). (4.7) h
The novel aspect, however, is that Fˆ is at the same time acted upon by the group G, these two structures being coupled by αg (Fˆ h ) = Fˆ ghg−1
(4.8)
as a consequence of (2.15). This is equivalent to the relation αg ◦ γh = γghg−1 ◦ αg .
(4.9)
In this context it is of interest to remark that several years ago algebraists studied (see [18] and references given there) analogies between group graded algebras and algebras acted upon by a finite group. Similar studies have been undertaken in the context of inclusions of von Neumann algebras. As it turns out the situation at hand, which is rather more interesting, can be neatly described in terms of the action, as defined, e.g., ˆ The relations fulfilled in [68], of a Hopf algebra (in our case finite dimensional) on F. by the αg and γh , in particular (4.9), motivate us to cite the following well known Definition 4.1. Let C(G) be the algebra of (complex valued) functions on the finite group G and consider the adjoint action of G on C(G) according to αg : f 7→ f ◦ Ad(g −1 ). The quantum double D(G) is defined as the crossed product D(G) = C(G) oα G of C(G) by this action. In terms of generators D(G) is the algebra generated by elements Ug and Vh , g, h ∈ G with the relations
and the identification Ue =
Ug Uh = Ugh , Vg Vh = δg,h Vg , Ug Vh = Vghg−1 Ug ,
P g
(4.10) (4.11) (4.12)
Vg = 1.
It is easy to see that D(G) is of the finite dimension |G|2 , where as a convenient basis one may choose V (g)U (h), g, h ∈ G, multiplying according to V (g1 )U (h1 ) V (g2 )U (h2 ) = δg1 ,h1 g2 h−1 ·V (g1 )U (h1 h2 ). This is just a special case of a construction given by Drinfel’d 1 [30] in greater generality which we do not bother to retain. For the purposes of this work it suffices to state the following well known properties of D(G), referring to [30, 59, 20] for further discussion, see also Appendix A. In order to define an action of a Hopf algebra on von Neumann algebras we further need a star structure on the former which in our case is provided by the following
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Proposition 4.2. With the definition Ug∗ = Ug−1 , Vh∗ = Vh and the appropriate extension, D(G) is a *-algebra. D(G) is semisimple. Proof. Trivial calculation. Finite dimensional *-algebras are automatically semisimple. Before stating how the quantum double D(G) acts on Fˆ we define precisely the properties of a Hopf algebra action. Definition 4.3. A bilinear map γ : H × M → M is an action of the Hopf *-algebra H on the *-algebra M iff the following hold for any a, b ∈ H, x, y ∈ M: γ1 (x) = x, γa (1) = ε(a)1, γab (x) = γa ◦ γb (x), γa (xy) = γa(1) (x)γa(2) (y), (γa (x))∗ = γS(a∗ ) (x∗ ).
(4.13) (4.14) (4.15) (4.16) (4.17)
We have used the standard notation 1(a) = a(1) ⊗ a(2) for the coproduct where on the right side there is an implicit summation. The map γ is assumed to be weakly continuous with respect to M and continuous with respect to some C ∗ -norm on H (which is unique in the case of finite dimensionality). After these lengthy preparations it is clear how to define the action of D(G) on Fˆ . Theorem 4.4. Defining γa (Fˆ ), Fˆ ∈ Fˆ for a ∈ {U (g), V (h)|g, h ∈ G} by γUg (Fˆ ) = αg (Fˆ ), γVh (Fˆ ) = γh (Fˆ ),
(4.18) (4.19)
using (4.15) to define γ on the basis V (g)U (h) and extending linearly to D(G) one obtains an action in the sense of Definition 4.3. P Proof. Equation (4.13) follows from 1D(G) = g Vg , (4.14) from 1Fˆ ∈ Fˆ e and (A.4), whereas (4.15) is an obvious consequence of the definition. Furthermore, (4.16) is a consequence of αg being a homomorphism, the coproduct property (4.7) and the definition (A.5). The statement (4.17) on the *-operation finally follows from (αg (x))∗ = αg (x∗ ) and S(Ug∗ ) = Ug on the one hand and (Fˆ g )∗ = Fˆ g−1 and S(Vg∗ ) = Vg−1 on the other. Remarks. 1. It should be obvious that the action of D(G) on Fˆ commutes with the translations and that it commutes with the boosts iff the group G does. Otherwise, U (3) U (g) U (3)∗ = Uh implies α3 ◦ γg = γh ◦ α3 . P 2. In the case of G being abelian Uχ = g∈G χ(g) · Vg , χ ∈ Gˆ constitutes an alternative basis for the subalgebra C(G) ⊂ D(G). The resulting formulae Uχ Uρ = Uχρ , 1(Uχ ) = Uχ ⊗ Uχ and γUχ (·) = αˆ χ (·) establish the equivalence of the quantum double with the ˆ The abelian case is special insofar as D(G) is spanned by its grouplike group G × G. elements, which is not true for G non-abelian. 4.3. Spontaneously broken quantum symmetry. Having shown in the abelian case that the symmetry under the dual group Gˆ is spontaneously broken it should not come as a surprise that the same holds for non-abelian groups G where, of course, the notion of unitary implementation has to be generalized.
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Definition 4.5. An action γ of the Hopf algebra H on the *-algebra M is said to be implemented by the (homomorphic) representation U : H → B(H) if for all a ∈ H, x ∈ M (4.20) U (a) x = γa(1) (x) U (a(2) ) or equivalently γa (x) = U (a(1) ) x U (S(a(2) )).
(4.21)
The representation is said to be unitary if the map U is a *-homomorphism. In complete analogy to the abelian case we see that only a subalgebra of D(G), namely the group algebra CG is implemented in the above sense. A similar phenomenon has already been observed to occur in the Coulomb gas representation of the minimal models [44] and in [7] where two dimensional theories without conformal covariance were considered. It would be interesting to know whether there exists, in some sense, a “quantum version” of Goldstone’s theorem for spontaneously broken Hopf algebra symmetries. In an earlier section we defined a twist operation (3.27) which bijectively maps T ˆ which commutes with all field operators localized in the Fˆ (O) into an algebra F(O) O ofPO. With the notation introduced in this chapter this left spacelike complement WLL operation can be written as F T = g γg (F )t U (g −1 ). One might wonder whether there O is a map T¯ which achieves the same thing for the right spacelike complement WRR . If the quantum symmetry were not spontaneously broken, such a map would be given by X ¯ αg (F )t V (g), (4.22) FT = g
where the V (g) are the projectors implementing the dual C(G) of the group G. Using the spacelike commutation relations and the property U O (g) V (h) = V (gh) U O (g) this claim is easily verified. In the discussion of the abelian case we have mentioned that one can construct, e.g. ˆ For the quantum by the method given in [15], local implementers of the dual group G. double D(G) of a non-abelian group G, however, which is not spanned by its grouplike elements, another approach is needed. Proposition 4.6. For every double cone O ∈ K there is a family of orthogonal projections VO (g) fulfilling X VO (g) VO (h) = δg,h VO (g) , VO (g) = 1, (4.23) g
ˆ = γg F(O)
X
VO (gh) · VO (h)
(4.24)
h
and transforming correctly under the (unbroken) group G, U (g) VO (h) U (g)∗ = VO (ghg −1 ).
(4.25)
Proof. In order to obtain operators with these properties we make use of the isomorphism, for every wedge W , between F(W ) ∨ U (G)00 and F (W ) oα G. We briefly recall the construction of the crossed product M o G. It is represented on the Hilbert space H¯ = L2 (G, H) of square integrable functions from G to H. The algebra M acts according to (π(x)f )(g) = αg−1 (x) f (g) whereas the group G is unitarily represented by (U¯ (k)f )(g) = f (k −1 g). With these definitions one can easily verify the equation
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U¯ (k) π(x) U¯ (k)∗ = π ◦ αk (x). If the group G is finite one can furthermore define the pro¯ ¯ ¯ = E(gk) U¯ (g). jections (E(k)f )(g) = δg,k f (g) for which one obviously has U¯ (g) E(k) As already discussed above there is, as a consequence of the outerness of the action of the group, an isomorphism between the algebras M ∨ U (G)00 and M oα G sending P P ¯ g xg U (g) to g π(xg ) U (g). As both algebras are of type III and live on separable Hilbert spaces this isomorphism is unitarily implemented and can be used to pull back ¯ the projections E(k) to the Hilbert space H, where we denote them by E(k). (E(e) is nothing but the Jones projection in the extension M2 of the inclusion M ⊂ M∨U (G)00 .) Applying these considerations to the algebras of the wedges WLO and WRO we obtain O (k), satisfying the families of projections EL/R O O U (g) EL/R (k) U (g)∗ = EL/R (gk),
which we use to define VO (g) = Y O∗ (
X
O ER (gh) ⊗ ELO (h)) Y O .
(4.26)
(4.27)
h
The properties (4.23) of orthogonality and completeness are obvious whereas covariance (4.25) follows from (4.26) and U (k) = Y O∗ U (k) ⊗ U (k) Y O as follows: X O ER (kgh) ⊗ ELO (kh)) Y O Ad U (k)(VO (g)) = Y O∗ ( h
=Y
O∗
(
X
O ER (kgk −1 h) ⊗ ELO (h)) Y O
h −1
= VO (kgk
(4.28)
).
It remains to show the implementation property (4.24). Using the fact that ELO (g) F(WLO ) ELO (h) = {0} if g 6= h and Fˆ (O) ∼ = F(WRO ) ∨ U (G)00 ⊗ F (WLO ) we obtain X VO (gh) Fˆ VO (h) Y O∗ YO h
=
X
O O ER (ghk) ⊗ ELO (k) F1 ⊗ F2 ER (hl) ⊗ ELO (l)
h,k,l
=
X
O O ER (ghk) ⊗ ELO (k) F1 ⊗ F2 ER (hk) ⊗ ELO (k)
h,k
=(
X
O O ER (gh) F1 ER (h)) ⊗ (
X
h
=
X
ELO (k) F2 ELO (k))
k O ER (gh) F1
O ER (h)
⊗ F2 ,
(4.29)
h
P O where we have written (abusively) F1 ⊗ F2 for Y O Fˆ Y O∗ . Since we have h ER (gh) O U (k) ER (h) = δg,k U (k) it is clear that the above map projects Fˆ (O) onto F(O)ULO (g), ˆ thus implementing the restriction of γg to F(O). O Remark. It should be remarked that the simpler definition V˜O (g) = Y O∗ (ER (g)⊗1)Y O , which also satisfies (4.24), does not lead to a representation of D(G) as these VO ’s do not transform according to the adjoint representation (4.25).
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4.4. Spectral properties. The above discussion was to a large extent independent of the quantum field theoretic application insofar as the action of the quantum double on a certain class of *-algebras was concerned. As we have seen, any *-algebra which is at the same time acted upon by a finite group G and graded by G supports an action of the double provided the relation (4.8) holds. The converse is also true. Let M be a *-algebra on which the double acts. Then Mg = γg (M) induces a G-grading satisfying (4.8). It may however happen that Mg = {0} for g in a normal subgroup. This possibility can be eliminated by demanding the existence of a unitary representation of G in M : G 3 g 7→ U (g) ∈ Mg . In the situation at hand this condition is fulfilled by construction. We now turn to the spectral properties of the action of the double. To this purpose we introduce the following notion [62], already encountered implicitly in the proof of Prop. 3.17. Definition 4.7. A normclosed linear subspace T of a von Neumann algebra M is called a Hilbert space in M if x∗ x ∈ C1 for all x ∈ T and x ∈ M and xa = 0 ∀a ∈ T implies x = 0. The name is justified as hx, yi1 = x∗ y defines a scalar product in T . One can thus choose a basis ψi , i = 1, . . . , dT satisfying the requirements (3.30, 3.31). The interest of this definition stems from the following well known lemma, the easy proof of which we omit. Lemma 4.8. Let T be a finite dimensional Hilbert space in M globally invariant under the action γH of H on M. A basis of the above type gives rise to a unitary representation of H according to γa (ψi ) =
d X
Di0 i (a) ψi0 .
(4.30)
i0 =1
ˆ Our aim will now be to show that the extended algebras F(O), O ∈ K in fact contain such tensor multiplets for every irreducible representation of D(G). In order to do this we make use of the representation theory of the double developed in [20]. (D(G) being semisimple, every finite dimensional representation decomposes into a direct sum of irreducible ones.) The (equivalence classes of) irreducible representations are labeled by pairs (c, π), where c ∈ C(G) is a conjugacy class and π is an irreducible representation of the normalizer group Nc . Here Nc is the abstract group corresponding to the mutually isomorphic normalizers Ng for g ∈ c, already encountered in the remark following Thm. 3.16. The representation πˆ labeled by (c, π) is obtained by choosing an arbitrary g0 ∈ c and inducing up from the representation π(V ˆ g Uh ) = δg,g0 π(h)
(4.31)
of the subalgebra Bg0 of D(G) generated by V (g) , g ∈ G and U (h) , h ∈ Ng0 . The representation space of πˆ (c,π) is thus V(c,π) = D(G) ⊗Bg0 Vπ . For a more complete discussion we refer to [20] remarking only that πˆ (c,π) (Vg Uh ) = 0 if g 6∈ c. Definition 4.9. The action γ of a group or Hopf algebra on a von Neumann algebra M is dominant iff the algebra of fixed points is properly infinite and the monoidal spectrum of γ is complete, i.e. for every finite dimensional unitary representation π of the group or Hopf algebra, respectively, there is a γ-invariant Hilbert space T in M such that γ T is equivalent to π.
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ˆ be a von Neumann algebra supporting an action of the quanProposition 4.10. Let M ˆ where tum double D(G). Assume further that there is a unitary representation of G in M, ˆ is dominant if ˆ g and αh (U¯ (g)) = U¯ (hgh−1 ). Then the action of D(G) on M U¯ (g) ∈ M ˆ is dominant. and only if the action of G on M = γe (M) ˆ D(G) the conditions of proper infiniteness of the Proof. As a consequence of MG = M fixpoint algebras coincide. The “only if” statement is easily seen by considering the representations of the double corresponding to the conjugacy class c = {e}. For these Nc ∼ = G holds, implying that the representations of D(G) with c = {e} are in one-to-one ˆ transforming according correspondence to the representations of G. A multiplet in M to ({e}, π) is nothing but a π-multiplet in M. The “if” statement requires more work. We have to show that for every pair (c, π), where π is an irreducible representation of the normalizer Nc , there exists a multiplet of isometries transforming according to πˆ (c,π) . To begin with, choose g ∈ c arbitrarily and find in M a multiplet of isometries ψi , i = 1, . . . , d = dim(π) transforming according to the representation π under the action of Ng ⊂ G. The existence of such a multiplet follows from the dominance of the group action on M. Now, let x1 , . . . , xn be representatives of the cosets G/Ng , where n = [G : Ng ] = |c|. Furthermore, the proper infiniteness of the fixpoint algebra allows usPto choose a family V1 , . . . , Vn of isometries ∗ ˆ D(G) satisfying Vi∗ Vj = δi,j , in MG = M i Vi Vi = 1. Defining 9ij = Vi αxi (U¯ (g) ψj ), i = 1, . . . , n, j = 1, . . . , d
(4.32)
one verifies that the 9ij constitute a complete family of mutually orthogonal isometries spanning a vectorspace of dimension nd = dim(πˆ (c,π) ). That this space is mapped into itself by the action of the double follows from the fact that, for every k ∈ G, k xi can uniquely be written as xj h, h ∈ Ng . Finally, the multiplet transforms according to the representation (c, π) of D(G), which is evident from the definition of the latter in [20, (2.2.2)]. Remark. Since in our field theoretic application the conditions of the proposition are satisfied thanks to Lemma 3.14 and the discussion in Subsect. 4.2 we can conclude that ˆ F(O), O ∈ K has full D(G)-spectrum. 4.5. Commutation relations and statistics. Up to this point our investigations in this section have focused on the local inclusion A(O) ⊂ Fˆ (O) for any fixed double cone O. Having clarified the relation between these algebras in terms of the action of the quantum double we can now complete our discussion of the latter. To this purpose we recall that the double construction has been introduced in [30] as a means of obtaining quasitriangular Hopf algebras (quantum groups) in the sense defined there, i.e. Hopf algebras possessing a “universal R-matrix,” cf. Appendix A. As it turns out the latter appears very naturally in our approach when considering the spacelike commutation relations of irreducible D(G)-multiplets as defined in the preceding subsection. Proposition 4.11. Assume the net O 7→ F (O) is bosonic, i.e. fulfills untwisted locality. Let O2 < O1 and ψi1 , i = 1, . . . , d1 and ψj2 , j = 1, . . . , d2 be D(G)-tensors in ˆ 1 ), F(O ˆ 2 ), respectively. They then fulfill C-number commutation relations F(O X ψi1 ψj2 = ψj20 ψi10 (Di10 i ⊗ Dj20 j )(R), (4.33) i0 j 0
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where D1 , D2 are the matrices of the respective representations of D(G) and X R= Vg ⊗ Ug ∈ D(G) ⊗ D(G). Proof. The equation
(4.34)
g∈G
P
P Vg = 1 in D(G) implies g γg = id. We can thus compute X X ψi1 ψj2 = γg (ψi1 ) ψj2 = αg (ψj2 ) γg (ψi1 ) (4.35) g
g∈G
=
XX
g∈G
ψj20
ψi10
Dj20 j (Ug ) Di10 i (Vg ),
g∈G i0 j 0
where the second identity follows from γg (ψi1 ) ∈ F (O1 ) ULO1 (g) and Ad ULO1 (g) Fˆ (O2 ) = αg . The rest is clear. Remarks. 1. Commutation relations of the above general type have apparently first been considered in [40]. For the special case of Z(N ) order disorder duality they date back at least to [66]. 2. By this result the field extension of Definition 3.1 in conjunction with Thm. 4.4 may be considered a local version of the construction of the double. (If we had used the URO (g) we would have ended up with R−1 which would do just as well.) 3. If the net O 7→ F (O) is fermionic an additional sign ± appears on the right-hand side of (4.33) depending on the Bose/Fermi nature of the fields. Using the bosonization prescription of the next section this sign can be eliminated. We now turn to a discussion of the localized endomorphisms of the observable algebra A which are implemented by the charged fields in Fˆ as in [25]. Let ψi , i = ˆ 1, . . . , dψ be a multiplet of isometries in F(O) transforming according to the irreducible representation r of D(G). Then the map ρ(·) =
dψ X
ψi · ψi∗
(4.36)
i=1
ˆ The relative locality of A and Fˆ implies the defines a unital *-endomorphism of F. restriction of ρ to A to be localized in O in the sense that ρ(A) = A ∀A ∈ A(O0 ). Furthermore, ρ maps A(O1 ) into itself if O1 ⊃ O as follows from the D(G)-invariance of ρ(x) for x ∈ A. (The conventional argument using duality would allow us only to ˆ 1 ).) conclude ρ(A(O1 )) ⊂ A(O Proposition 4.12. In restriction to A(O1 ), O1 ⊃ O the endomorphism ρ is irreducible. Proof. The proof is omitted as it is identical to the proof of [54, Prop. 6.9], where compact groups are considered. Remarks. 1. In application to the net Aˆ the endomorphisms ρ are localized only in wedge regions, i.e. they are of solitonic character. 2. Due to the spontaneous breakdown of the quantum symmetry the endomorphisms ρ which arise from non-group representations of D(G) should not be considered as true superselection sectors of the net A H0 . This would be justified if the symmetry were unbroken. Nevertheless, one can analyze their statistics, as will be done in the rest of this subsection.
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Whereas the endomorphisms ρ defined above need not be invertible one can always find left inverses [21] φ such that φ ◦ ρ = id. For ρ as defined by (4.36) the left inverse is easily verified to be given by dρ 1 X ∗ ψ i · ψi . φρ = dψ
(4.37)
i=1
In order to study the statistics of endomorphisms one introduces [23, 35] the statistics operators (4.38) ε(ρ1 , ρ2 ) = U2∗ ρ1 (U2 ) ∈ (ρ1 ρ2 , ρ2 ρ1 ), where U2 is a charge transporter intertwining ρ2 and ρ˜2 , the latter being localized in the left spacelike of the localization region of ρ1 . Such an intertwiner is given P complement 2 2∗ G ˜ ˆ O), ˜ O˜ < O1 transforming by U2 = i ψi ψi ∈ F = A, where ψ˜ i is a multiplet in F( according the same representation of D(G) as ψi , such that U2 is D(G)-invariant and thus in A. Lemma 4.13. Let ψi1 ∈ Fˆ (O1 ), i = 1, . . . , d1 and ψj2 , j = 1, . . . , d2 be D(G)-multiplets corresponding to the representations D1 , D2 and let ρ1 , ρ2 be the associated endomorphisms. Then the statistics operator is given by X 1 2 ψi2 ψl1 ψj2∗ ψk1∗ (Dlk ⊗ Dij )(R). (4.39) ε(ρ1 , ρ2 ) = ijkl
The statistics parameter [21] for the morphism ρ which is implemented by the irreducible D(G)-tensor ψi , i = 1, . . . , dψ is ωρ λρ = , (4.40) dρ P with dρ = dψ and Dlj (X) = δlj ωρ , where X = g Vg Ug is a unitary element in the center of D(G). P Proof. With U2 = i ψ˜ i2 ψi2∗ we have X ψi2 ψ˜ i2∗ ψj1 ψ˜ k2 ψk2∗ ψj1∗ . (4.41) ε(ρ1 , ρ2 ) = ijk
Then (4.39) follows by an application of (4.33) to ψj1 and ψ˜ k2 and appealing to the orthogonality relation ψ˜ i2∗ ψ˜ j2 = δij 1. With the identification ψ 1 = ψ 2 = ψ in (4.39), (4.37) and using once more the orthogonality relation we compute the statistics parameter as follows: 1 X 1 X λρ 1 = φρ (ερ,ρ ) = ψl ψj∗ (Dli ⊗ Dij )(R) = ψl ψj∗ Mlj , (4.42) dψ dψ ijl
where Mlj =
X i
jl
(Dli ⊗ Dij )(R) = Dlj (
X g
Vg Ug ).
(4.43)
P An easy calculation shows that X = g Vg Ug is a unitary element in the center of D(G) such that it is represented by a phase times the unit matrix in the irreducible representation D: Mlj = δlj ω, ω ∈ S 1 .
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Remarks. 1. The statistical dimension of the sector ρ, defined as dρ = |λρ |−1 coincides with the dimension of the corresponding representation of the quantum double. This was to be expected and is in accord with the fact [55] that the action of finite dimensional Hopf algebras cannot give rise to non-integer dimensions. 2. Recalling Lemma 4.8 we see that in restriction to a field operator in a multiplet transforming according to the irreducible representation r the action of γX amounts to multiplication by ωr . The unitary X ∈ D(G) may thus be interpreted as the quantum double analogue of the group element k which distinguishes between bosons and fermions. This is reminiscent of the notion of ribbon elements in the framework of quantum groups, seeP Appendix A. In fact, the operator X defined above is just the inverse of Drinfel’d’s u = g Vg Ug−1 which itself is a ribbon element due to S(u) = u. 3. Appealing to the representation theory of D(G) as expounded in [20] it is easy to compute the phase ωr for the representation r = (c, π). It is given by the scalar to which g ∈ c, obviously being contained in the center of the normalizer Ng , is mapped by the irreducible representation π of Ng . As an immediate consequence [19] ωr is an nth root of unity where n is the order of g. We now turn to the calculation of the monodromy operator εM (ρ1 , ρ2 ) = ε(ρ1 , ρ2 ) ε(ρ2 , ρ1 ),
(4.44)
which measures the deviation from permutation group statistics and of the statistics characters [58] (4.45) Yij 1 = di dj φi (εM (ρi , ρj )∗ ). In the latter expression ρi , ρj are irreducible morphisms such that the right-hand side is a C-number since φi (εM (ρi , ρj )∗ ) is a selfintertwiner of ρj and due to the irreducibility of the latter, cf. Prop. 4.12.) We thus obtain a square matrix of complex numbers indexed by the superselection sectors, i.e. in our case the irreducible representations of the quantum double D(G). Proposition 4.14. In terms of the fields the monodromy operator is given by X 1 2 εM (ρ1 , ρ2 ) = ψi2 ψj1 ψk1∗ ψl2∗ (Djk ⊗ Dil )(I),
(4.46)
ijkl
where I = R σ(R).
(4.47)
The statistics characters are given by Yij = (tri ⊗ trj ) ◦ (Di ⊗ Dj )(I ∗ ).
(4.48)
Proof. Inserting the statistics operators according to (4.39) and using twice the orthogonality relation we obtain X 1 2 2 1 ψi2 ψj1 ψk1∗0 ψl2∗ (4.49) εM (ρ1 , ρ2 ) = 0 (Djl ⊗ Dik )(R) (Dkl0 ⊗ Dlk 0 )(R). ijkl k 0 l0 The numerical factor to the right (including the summations over k, l) can be simplified to
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167
1 2 1 2 Djk 0 (Vg Uh ) Dil0 (Ug Vh ) = (Djk 0 ⊗ Dil0 )(I).
(4.50)
g,h∈G
Omitting the primes on k 0 , l0 we obtain (4.46). The formula (4.48) follows in analogy to the computation of λρ from (4.46), (4.37) and Yij ∝ 1. Remark. I = R σ(R) can be considered as the quantum group version of the monodromy operator. 1 In [2] it was shown that Y is invertible, in fact |G| Y is unitary. In conjunction with the known facts concerning the representation theory one concludes [2] that the quantum double D(G) is a modular Hopf algebra in the sense of [60]. We are now in a position to complete our demonstration of the complete parallelism between quantum group theory and quantum field theory (which we claim only for the quantum double situation at hand!). What remains to be discussed is the Verlinde algebra structure [71] behind the fusion of representations of the double and the associated endomorphisms of Fˆ , respectively. The fusion rules are said to be diagonalized by a unitary matrix S if
k = Nij
X Sim Sjm S ∗
km
m
S0m
.
(4.51)
(For a comprehensive survey of fusion structures see [41].) One speaks of a Verlinde algebra if, in addition, S is symmetric, there is a diagonal matrix T of phases satisfying T C = CT = T (Cij = δi¯ is the charge conjugation matrix) and S and T constitute a representation of SL(2, Z) (in general not of P SL(2, Z) = SL(2, Z)/Z2 ), i.e. S 2 = (ST )3 = C.
(4.52)
On the one hand the representation categories of modular Hopf algebras are known [60] to be modular, i.e. to satisfy (4.51) and (4.52), where the phases in T are given by the values of the ribbon element X in the irreducible representations. On the other hand this structure has been shown [58] to arise from the superselection structure of every rational quantum field theory in 1 + 1 dimensions. In this framework the phases in T are given by the phases of the statistics parameters (4.40), whereas the matrix S arises from the statistics characters T =
σ |σ|
1/3
Diag(ωi ), S = |σ|−1 Y.
(4.53)
P P For nondegenerate theories the number σ = i ωi−1 d2i satisfies |σ|2 = i d2i . Using the result P [2] σ = |G| this condition is seen to be fulfilled, for the semisimplicity of D(G) gives i d2i = dim(D(G)) = |G|2 . We thus observe, for the orbifold theories under study, a perfect parallelism between the general superselection theory [58] for quantum field theories in low dimensions and the representation theory of the quantum double [20]. This parallelism extends beyond the Verlinde structure. One observes, e.g., that Eqs. (2.4.2) of [20] and (2.30) of [58], both stating that the monodromy operator is diagonalized by certain intertwining operators, are identical although derived in apparently unrelated frameworks.
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5. Bosonization In this section we will show how the methods expounded in the preceding sections can be used to obtain an understanding of the Bose/Fermi correspondence in 1+1 dimensions in the framework of local quantum theory. This is so say, we will show how one can pass from a fermionic net of algebras with twisted duality to a bosonic net satisfying Haag duality on the same Hilbert space, and vice versa. Our method amounts to a continuum version of the Jordan-Wigner transformation and is reminiscent of Araki’s approach to the XY-model [4]. Our starting point is as defined in the introduction, i.e. a net of field algebras with fermionic commutation relations (1.4) and twisted duality (1.11) augmented by the split property for wedge regions introduced in Sect. 2. As before there exists a selfadjoint unitary operator V distinguishing between even and odd operators. For the present investigations, however, the existence of further inner symmetries is ignored as they are irrelevant for the spacelike commutation relations. Therefore we now repeat the field extension of Sect. 3 replacing the group G by the subgroup Z2 = {e, k}. This amounts to simply extending the local algebras by the disorder operator associated with the only nontrivial group element k, ˆ F(O) = F(O) ∨ {V O },
(5.1)
ˆ is isotonous, i.e. a net. This where V O = ULO (k). Again, the assignment O 7→ F(O) is of course the simplest instance of the situation discussed at the beginning of Sect. 4 where it was explained that there is an action of the dual group Gˆ on the extended net. We thus have an action of Z2 × Z2 on the quasilocal algebra Fˆ generated by α = Ad V and β, α(F + GV O ) = F+ − F− + (G+ − G− )V O , O
O
β(F + GV ) = F − GV ,
(5.2) (5.3)
˜ where F, G ∈ F. We now define F(O) as the fixpoint algebra under the diagonal action α ◦ β = β ◦ α: ˜ ˆ F(O) = {x ∈ F(O) | x = α ◦ β(x)}. (5.4) ˜ Obviously F(O) can be represented as the following sum: ˜ F(O) = F(O)+ + F (O)− V O .
(5.5)
˜ It is instructive to compare F(O) with the twisted algebra F(O)t = F(O)+ + F(O)− V,
(5.6)
the only difference being that in the former expression V O appears instead of V . This reflects just the difference between Jordan-Wigner and Klein transformations. It is well known that the net F t is local relative to F . That the former cannot be local itself, however, follows clearly from the fact that it is unitarily equivalent to the latter by F(O)t = ZF (O)Z ∗ . Lemma 5.1. Let WL and WR be left and right wedges, respectively. Then the wedge algebras of F˜ are given by ˜ L ) = F(WL ), F(W ˜ R ) = F(WR )t . F(W ˜ Wedge duality holds for the net F.
(5.7) (5.8)
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Proof. V O is contained in F (WL )+ for any O ⊂ WL . Thus, F (WL )− V O = F (WL )− , whence the first identity. Similarly we have VRO ∈ F (WR )+ for O ∈ WR , from which we obtain F (WL )− V O = F(WL )− V . Wedge duality for F˜ now follows immediately from twisted duality for F. ˜ Proposition 5.2. The net O 7→ F(O) is local. Proof. Let O1 , O2 be mutually spacelike double cones. We may assume O1 < O2 such that WLO1 and WRO2 are mutually spacelike. The commutativity of F˜ (O1 ) and F˜ (O1 ) follows from the preceding lemma and twisted locality for F since O1 ⊂ WLO1 and O2 ⊂ WRO2 . Remark. A more intuitive proof goes as follows. Let Fi ∈ F (Oi )− , i = 1, 2. Then commuting F1 V O1 through F2 V O2 gives exactly two factors of −1. The first arises from F1 F2 = −F2 F1 and the other from V O2 F1 = −F1 V O2 , whereas V O1 F2 = F2 V O1 . Proposition 5.3. The net F˜ fulfills Haag duality for double cones. ˜ Proof. We have to prove F(O) = F˜ (WLO ) ∧ F˜ (WLO ). Using the lemma the right-hand O side is seen to equal F(WL ) ∧ F(WRO )t which by (2.20) is unitarily equivalent to F(WRO )t ⊗ F(WLO ). On the other hand (2.22) leads to ˜ F(O) = F(O)+ + F(O)− V O ∼ = F(WRO )+ ⊗ F (WLO )+ + F (WRO )− V ⊗ F(WLO )− + [F(WRO )− ⊗ F(WLO )+ + F (WRO )+ V ⊗ F(WLO )− ] V ⊗ 1 = F(WRO )t ⊗ F (WLO ), which completes the proof.
(5.9)
It is obvious that the net F˜ is Poincar´e covariant with respect to the original representation of P. Finally, the group G acts on F˜ via the adjoint representation g 7→ Ad U (g). In particular Ad U (k) = Ad V acts trivially on the first summand of the decomposition (5.5) and by multiplication with −1 on the second, i.e. the bosonized theory carries an action of Z2 in a natural way. It should be clear that the same construction can be used to obtain a twisted dual fermionic net from a Haag dual bosonic net with a Z2 symmetry. It is not entirely trivial that these operations performed twice lead back to the net one started with, as the operators V O constructed with the original and the bosonized net might differ. That this is not the case, however, can be derived from Lemma 5.1, the easy argument is left to the reader. 6. Conclusions and Outlook In this final section we summarize our results and relate them to some of those in the literature. Starting from a local quantum field theory in 1 + 1 dimensions with an unbroken group symmetry we have discussed disorder operators which implement a global symmetry on some wedge region and commute with the operators localized in the spacelike complement of a somewhat larger wedge. Whereas disorder operators are only localized in wedge regions, they can in a natural way be associated to the bounded region where the interpolation between the global group action and the trivial
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action takes place. Extending the local algebras F (O) of the original theory by the disorder operators corresponding to the double cone O gives rise to a nonlocal net Fˆ which is uniquely defined. We have shown that for every quantum field theory fulfilling a sufficiently strong version of the split property disorder operators exist and can be chosen such as to transform nicely under the global group action. As a consequence, the extended theory supports an action of the quantum double D(G) which, however, is spontaneously broken in the sense that only the subalgebra CG is implemented by operators on the Hilbert space. Nevertheless, all other aspects of the quantum symmetry, like R-matrix commutation relations and the Verlinde algebra, show up and correspond nicely to the structures expected due to the general analysis [35, 58]. The spontaneous breakdown of the quantum symmetry is in accord with the findings of [50] where it was argued (in the case of a cyclic group Z(N )) that the vacuum expectation values of order and disorder variables can vanish jointly, as they must in the case of unbroken quantum symmetry, only if there is no mass gap. Massless theories are, however, ruled out by the postulate of the split property for wedges upon which our analysis hinges. The fact that in the situation studied in this paper “one half” of the quantum double symmetry is spontaneously broken hints at an alternative construction which we describe briefly. Given a local net of C ∗ -algebras with group symmetry there may of course be vacuum states which are not gauge invariant. Let us assume that ωe is such that ωg = ωe ◦αg 6= ωe ∀g 6= e, i.e. the symmetry is completely broken. One may now consider the reducible representation ⊕g πg of F on the Hilbert space Hˆ = L2 (G, H), where πg is the GNS-representation corresponding to a soliton state which connects the vacua ωe and ωg . The existence of such states follows from the same set of assumptions as was used in the present investigation [63]. Again, one can construct operators U O (g) enjoying similar algebraic properties as the disorder operators appearing in this paper. Their interpretation is different, however, in that they are true soliton operators intertwining the vacuum representation and the soliton sectors. Extending the local algebras according to (3.1) ˆ The details of the construction gives rise to a field net Fˆ which acts irreducibly on H. outlined above, which is complementary in many ways to the one studied in the present work, will be given in a forthcoming publication. In the solitonic variant there is also an action of the quantum double D(G), where the action of C(G) is implemented in the obvious way, whereas the group symmetry is spontaneously broken. Although the split property for wedges should be satisfied by reasonable massive quantum field theories it definitely excludes conformally invariant models, which via [19, 20] provided part of the motivation for the present investigation. Concerning this somewhat disturbing point we confine ourselves to the following remarks. It is well known that quantum field theories in 1 + 1 dimensions, like the P(φ)2 models, possess a unique symmetric vacuum for some range of the parameters whereas spontaneous symmetry breakdown and vacuum degeneracy occur for other choices. The construction sketched above shows that the algebraic structure of order/disorder duality is the same in both massive regimes. It is furthermore known that the P(φ)2 theory with interaction λφ4 − δφ2 possesses a critical point at the interface between the symmetric and broken phases. Unfortunately, little is rigorously known about the possible conformal invariance of the theory at this point. The case of conformal invariance is quite different anyway, for Haag duality of the net F and of the fixpoint net A are compatible in contrast to the massive case. In the framework of lattice models things are easier as the local degrees of freedom are amenable to more direct manipulation. The authors of [67] considered a class of models where the disorder as well as the order operators were explicitly defined by
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171
specifying their action on the Hilbert space associated to a finite region. They then had to assume the existence of a vacuum state which is invariant under the action of the quantum double. In his approach [4] to the XY-model Araki similarly defines an automorphism of the algebra of order variables which is localized in a halfspace and then constructs the crossed product. In the continuum solitonic automorphisms can be defined for some models [39], but for a model independent analysis there seems to be no alternative to our abstract approach. As to the interpretation of the structures found in the present work and outlined above, we have already remarked that they may be considered as a local version of the construction of the quantum double. The quantum double was invented by Drinfel’d as a means to obtain quasitriangular Hopf algebras, and in [59] it was shown to be “factorizable,” see Appendix A. Furthermore, every finite dimensional factorizable Hopf algebra can be obtained as a quotient of a quantum double by a two-sided ideal. One may therefore expect that quantum doubles will play an important role in an extension of the constructions in [28] to low dimensional theories.
A. Quantum Groups and Quantum Doubles A Hopf algebra is an algebra H which at the same time is a coalgebra, i.e. there are homomorphisms 1 : H → H ⊗ H and ε : H → C satisfying (1 ⊗ id) ◦ 1 = (id ⊗ 1) ◦ 1,
(A.1)
(ε ⊗ id) ◦ 1 = (id ⊗ ε) ◦ 1 = id,
(A.2)
with the usual identification H ⊗ C = C ⊗ H = H. Furthermore, there is an antipode, i.e. an antihomomorphism S : H → H for which m ◦ (S ⊗ id) ◦ 1 = m ◦ (id ⊗ S) ◦ 1 = ε(·)1,
(A.3)
where m : H ⊗ H → H is the multiplication map of the algebra. Remark. By (A.2) the counit, which is simply a one dimensional representation, is the “neutral element” with respect to the comultiplication. For the quantum double D(G) defined in Definition 4.1 these maps are given by ε(V (g)U (h)) = δg,e , X V (hk)U (h) ⊗ V (k −1 )U (h), 1(V (g)U (h)) =
(A.4) (A.5)
k
S(V (g)U (h)) = V (h−1 g −1 h)U (h−1 )
(A.6)
on the basis {V (g)U (h) g, h ∈ G} and extended to D(G) by linearity. A Hopf algebra H is quasitriangular, or simply a quantum group, if there is an element R ∈ H ⊗ H satisfying 10 (·) = R 1(·) R−1 , where 10 = σ ◦ 1 with σ(a ⊗ b) = b ⊗ a and
(A.7)
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(1 ⊗ id)(R) = R13 R23 , (id ⊗ 1)(R) = R13 R12 .
(A.8) (A.9)
Here R12 = R ⊗ 1, R23 = 1 ⊗ R and R13 = (id ⊗ σ)(R ⊗ 1). As a consequence, R satisfies the Yang-Baxter equation R12 R13 R23 = R23 R13 R12 .
(A.10)
It is easy to verify that the R-matrix (4.34) satisfies these requirements. Remark. As shown by Drinfel’d, for quantum groups the square of the antipode is inner, i.e. S 2 (a) = uau−1 , where u is given by u = m ◦ (S ⊗ id) ◦ σ(R). The operator u satisfies ε(u) = 1, 1(u) = (σ(R)R)−1 (u ⊗ u) = (u ⊗ u)(σ(R)R)−1 . For quantum doubles of finite groups the antipode is even involutive (S 2 = id, equivalently u is central). This holds for all finite dimensional Hopf-*-algebras, whether quantum groups or not. A quantum group is called factorizable [59] if the map H ∗ → H given by H ∗ 3 x 7→ hx ⊗ id, Ii is nondegenerate, where I is as in (4.47). Quantum doubles are automatically factorizable. A quasitriangular Hopf algebra possessing a (non-unique) central element v satisfying the conditions (A.11) v 2 = u S(u), ε(v) = 1, S(v) = v, 1(v) = (σ(R)R)−1 (v ⊗ v),
(A.12)
where u is the operator defined in the above remark, is called a ribbon Hopf algebra [60]. Finally, modular Hopf algebras are defined by some restrictions on their representation structure, the most important of which is the nondegeneracy of the matrix Y defined in (4.48). Obviously, the conditions of factorizability and modularity are strongly related. B. Generalization to Continuous Groups In this appendix we will generalize our considerations on quantum double actions to arbitrary locally compact groups (the quantum field theoretic framework gives rise only to compact groups.) In Sect. 4 we identified von Neumann algebras acted upon by the double D(G) of a finite group with von Neumann algebras which are simultaneously graded by the group and automorphically acted upon by the latter, satisfying in addition the relation (4.8). The concept of group grading, however, loses its meaning for continuous groups. This problem is solved by appealing to the well known fact (see e.g. the introduction to [52]) that an algebra A (von Neumann or unital C ∗ ) graded by a finite group G is the same as an algebra with a coaction of the group. A coaction is a homomorphism δ from A into A ⊗ CG satisfying (δ ⊗ id) ◦ δ = (id ⊗ δG ) ◦ δ,
(B.1)
where δG : CG → CG ⊗ CG is the coproduct given by g 7→ g ⊗ g. The correspondence between these notions is as follows. Given a G-graded algebra A = ⊕g Ag , Ag Ah ⊂ Agh and defining δ(x) = x ⊗ g for x ∈ Ag , one obtains a coaction. The converse is also true. The relation αg (Ah ) = Aghg−1 between the group action and the grading obviously translates to
Quantum Double Actions on Operator Algebras and Orbifold QFTs
δ ◦ αg = (αg ⊗ Ad g) ◦ δ.
173
(B.2)
The concept of coaction extends to continuous groups, where the group algebra CG is replaced by the von Neumann algebra L(G) (here we will treat only quantum double actions on von Neumann algebras) of the left regular representation which is generated by the operators (λ(g)ξ)(h) = ξ(g −1 h) on the Hilbert space L2 (G). In the next step we give a precise definition of the double of a continuous group. To this purpose we have to put a topology on the crossed product of some algebra of functions on the group by the adjoint action of the latter. There are many ways of doing this, as is generally the case with infinite dimensional vector spaces. For compact Lie groups two different constructions, one of which appears to generalize to arbitrary compact groups, have been given in [10]. The most important virtue of this work is that the topological Hopf algebras obtained there are reflexive as topological vector spaces, making the duality between D(G) and D(G)∗ very explicit. From the technical point of view, however, the Fr´echet topologies on which this approach relies are not very convenient. In the following we will define the quantum double in the framework of Kac algebras [32, 33]. The latter has been invented as a generalization of locally compact groups which is closed under duality. As the C ∗ and von Neumann versions of Kac algebras have been proved [33] equivalent (generalizing the equivalence between locally compact groups and measurable groups) it is just a matter of convenience which formulation we use. We therefore consider first the von Neumann version which is technically easier. We start with the von Neumann algebra M = L∞ (G) of essentially bounded measurable functions acting on the Hilbert space H = L2 (G) by pointwise multiplication. With the coproduct 0(f )(g, h) = f (gh) and the involution κ(f )(g) = f (g −1 ) it is a coinvolutive Hopf von Neumann algebra. This means 0 is a coassociative isomorphism of M into M ⊗ M , κ is an anti-automorphism (complex linear, antimultiplicative and κ(x∗ ) = κ(x)∗ ) and 0 ◦ κR= σ ◦ (κ ⊗ κ) ◦ 0 holds where σ is the flip. The weight ϕ, defined on M+ by ϕ(f ) = G dg f (g), is normal, faithful, semifinite (n.f.s.) and fulfills 1. For all x ∈ M+ one has (ı ⊗ ϕ)0(x) = ϕ(x)1. 2. For all x, y ∈ nϕ one has (ı ⊗ ϕ)((1 ⊗ y ∗ )0(x)) = κ((ı ⊗ ϕ)(0(y ∗ )(1 ⊗ x))). ϕ 3. κ ◦ σtϕ = σ−t ◦ κ ∀t ∈ R. This makes (M, 0, κ, ϕ) a Kac algebra in the sense of [32], well known as KA(G). The ˆ κ, dual Kac algebra [32] of KA(G) is KS(G) = (L(G), 0, ˆ ϕ), ˆ the von Neumann algebra ˆ of the left regular representation equipped with the coproduct 0(λ(g)) = λ(g) ⊗ λ(g), −1 the coinvolution κ(λ(g)) ˆ = λ(g ) and the weight ϕˆ which we do not bother to state (see e.g. [47]). Defining now an action of G on M by the automorphisms αg (f )(h) = f (g −1 hg) it is trivial to check weak continuity with respect to g. Furthermore, αg is unitarily implemented by ug = λ(g)ρ(g), where (ρ(g)ξ)(h) = 1(g)1/2 ξ(hg) is the right regular representation. We can thus consider the crossed product (in the usual von Neumann ˜ = M oα G on H ⊗ L2 (G) (= L2 (G) ⊗ L2 (G)), generated by algebraic sense [70]) M π(M ) and λ1 (g) = 1M ⊗ λ(g), g ∈ G. ˜ such that the quadruple (M ˜ , 0, ˜ κ, ˜ κ, Proposition B.1. There are mappings 0, ˜ ϕ˜ on M ˜ ϕ) ˜ is a Kac algebra, which we call the quantum double D(G). On the subalgebras π(M ) and λ1 (G)00 = 1M ⊗ L(G) the coproduct and the coinvolution act according to ˜ 0(π(x)) = (π ⊗ π)(0(x)), κ(π(x)) ˜ = π(κ(x)), x ∈ M, ˜0(λ1 (g)) = λ1 (g) ⊗ λ1 (g), κ(λ ˜ 1 (g)) = λ1 (g −1 ), g ∈ G.
(B.3) (B.4)
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The Haar weight ϕ˜ is given by the dual weight [47] ˜ ϕ˜ = ϕ ◦ π −1 ◦ (ıM˜ ⊗ ϕ)( ˆ δ(x)),
(B.5)
˜ to M ˜ ⊗ L(G) which acts according to where δ˜ is the dual coaction from M ˜ δ(π(x)) = π(x) ⊗ 1L(G) , x ∈ M, ˜δ(λ1 (g)) = λ1 (g) ⊗ λ(g), g ∈ G.
(B.6) (B.7)
Proof. The automorphisms αg of M are easily shown to satisfy 0 ◦ αg = (αg ⊗ αg ) ◦ 0 and κ ◦ αg = αg ◦ κ. (The first identity is just g −1 (hk)g = (g −1 hg)(g −1 kg), the second (g −1 hg)−1 = g −1 h−1 g.) Thus α : G → Aut M constitutes an action of G on the Kac algebra (M, 0, κ, ϕ) in the sense of [17]. We can now apply [17, Thm. 1] to conclude ˜ such that the axioms that there exist a coproduct, a coinvolution and a Haar weight on M of a Kac algebra are satisfied. Equations (B.3, B.4) are restatements of [17, Props. 3.1, 3.3] whereas the Haar weight is as in [17, D´ef. 1.9]. [ = (L(G) ⊗ Proposition B.2. The dual Kac algebra of the quantum double is D(G) ˆ ∞ ˜ ˆ ˜ ϕˆ ⊗ ϕ). The coproduct and the counit are L (G), 0, κ, ˜ˆ 0(x) = R (1 ⊗ σ ⊗ 1) (0ˆ ⊗ 0)(x) (1 ⊗ σ ⊗ 1) R∗ , κ(x) ˜ˆ = V ∗ (κˆ ⊗ κ)(x) V,
(B.8) (B.9)
where R and V are given by (Rξ)(g, h) = (uh ⊗ 1) ξ(g, h), (V ξ)(g) = ug ξ(g).
(B.10) (B.11)
Proof. This is just the specialization of [17, Thm. 2] to the situation at hand. According to this theorem the von Neumann algebra underlying the dual of the crossed product ˆ ⊗ L∞ (G), where M ˆ is the von Neumann algebra of K. ˆ In Kac algebra K oα G is M ˆ ∞ ˆ = L(G). The formulae for 0 ˜ and κ˜ˆ are stated in [17, our case M = L (G) such that M Prop. 4.10]. Remark. If the group G is not finite the quantum double is neither compact nor discrete, ˆ˜ = ϕˆ ⊗ ϕ are both infinite. for the weights ϕ, ˜ ϕ [ on an algebra We are now in a position to define a coaction of the dual double D(G) A, provided A supports an action α and a coaction δ satisfying (B.2) (with g replaced by λ(g)). In order to remove the apparent asymmetry between α : A × G → A and δ : A → A ⊗ L(G) we write the former as the homomorphism α : A → A ⊗ L∞ (G) which maps x ∈ A into g 7→ αg (x) ∈ L∞ (G, A). We now show that the maps α and δ can be put together to yield a coaction. [ is defined by Definition B.3. The map 1 : A → A ⊗ L(G) ⊗ L∞ (G) = A ⊗ D(G) 1 = (ıA ⊗ σ) ◦ (α ⊗ ıL(G) ) ◦ δ,
(B.12)
where σ : x ⊗ y 7→ y ⊗ x is the flip map from L∞ (G) ⊗ L(G) to L(G) ⊗ L∞ (G).
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[ on A, i.e. it satisfies Theorem B.4. The map 1 is a coaction of D(G) ˜ˆ ◦ 1. (1 ⊗ ıDˆ ) ◦ 1 = (ıA ⊗ 0)
(B.13)
Proof. Appealing to the isomorphism A⊗L∞ (G) ∼ = L∞ (G, A) we identify A⊗L(G)⊗ ∞ ∞ ∞ L (G)⊗L(G)⊗L (G) with L (G×G, A⊗L(G)⊗L(G)). We compute (1⊗ı)◦1(x) as follows (abbreviating ıL(G) by ıL ) ((1 ⊗ ı) ◦ 1(x))(g, h) = (αg ⊗ ıL ⊗ ıL ) ◦ (δ ⊗ ıL ) ◦ (αh ⊗ ıL ) ◦ δ(x) (B.14) = (αg ⊗ ıL ⊗ ıL ) ◦ (αh ⊗ Ad λh ⊗ ıL ) ◦ (δ ⊗ ıL ) ◦ δ(x) ˆ ◦ δ(x). = (ıA ⊗ Ad λh ⊗ ıL ) ◦ (αgh ⊗ 0) The second equality follows from the connection (B.2) between the action α and the ˆ coaction δ whereas the third derives from the defining property (B.1) of the coaction 0. ˆ ˆ Now (αgh ⊗ 0) ◦ δ(x) is seen to be nothing but [(1 ⊗ σ ⊗ 1) (0 ⊗ 0)(x) (1 ⊗ σ ⊗ 1)](g, h), and the adjoint action of R in (B.8) is seen to have the same effect as Ad (ıA ⊗Ad λh ⊗ıL ) due to ρ(g) ∈ L(G)0 . Proposition B.5. The fixpoint algebra under the coaction 1, defined as A1 = {x ∈ A | 1(x) = x ⊗ 1Dˆ }, is given by (B.15) A1 = Aα ∩ Aδ , where Aα , Aδ are defined analogously. Proof. Obvious consequence of Definition B.3.
[ on A constructed above is exactly the kind The coaction of the dual double D(G) of output the theory of depth-2 inclusions [55, 34] would give when applied to the inclusion AD(G) ⊂ A, which in the quantum field theoretical application corresponds to A(O) ⊂ Fˆ (O). Nevertheless it is perhaps not exactly what one might have desired from a generalization of the results of Sect. 4 to compact groups. At least to a physicist, some kind of bilinear map γ : A × D(G) → A, as it was defined above for finite G, would seem more intuitive. This map should be well defined on the whole algebra A. Such a map can be constructed, provided the von Neumann double D(G) is replaced by its C ∗ -variant, which is uniquely defined by the above mentioned results [5, 33]. The details will be given in a subsequent publication. The representation theory of the quantum double in the (locally) compact case was studied in a recent preprint [51] of which I became aware after completion of the present work. An application of the results expounded there in analogy to Sect. 4 should be possible but is deferred for reasons of space. C. Chiral Theories on the Circle For the foregoing analysis in this chapter the split property for wedges was absolutely crucial. While this property has been proved only for free massive fields it is expected to be true for all reasonable theories with a mass gap. For conformally invariant theories in 1 + 1 dimensions, however, it has no chance to hold. This is a consequence of the fact that two wedges W1 ⊂ W2 “touch at infinity.” More precisely, there is an element of the conformal group transforming W1 , W2 into double cones having a corner in common. For such regions there can be no interpolating type I factor, see e.g. [11]. On the other
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hand, for chiral theories on a circle, into which a 1+1 dimensional conformal theory should factorize, an appropriate kind of split property makes sense. For a general review of the framework, including a proof of the split property from the finiteness of the trace of e−τ L0 , we refer to [42]. We restrict ourselves to a concise statement of the axioms. For every interval I on the circle such that I 6= S 1 , there is a von Neumann algebra A(I) on the common Hilbert space H. The assignment I 7→ A(I) fulfills isotony and locality: I1 ⊂ I2 ⇒ A(I1 ) ⊂ A(I2 ), I1 ∩ I2 = ∅ ⇒ A(I1 ) ⊂ A(I2 )0 .
(C.1) (C.2)
Furthermore, there is a strongly continuous unitary representation of the M¨obius group SU (1, 1) such that αg (A(I)) = Ad U (g)(A(I)) = A(gI). Finally, the generator L0 of the rotations is supposed to be positive and the existence of a unique invariant vector is assumed. Starting from these assumptions one can prove, among other important results, that the local algebras A(I) are factors of type III1 for which the vacuum is cyclic and separating. Furthermore, Haag duality [42] is fulfilled automatically: A(I)0 = A(I 0 ).
(C.3)
Given a chiral theory in its defining (vacuum) representation π0 one may consider inequivalent representations. An important first result [13] states that all positive energy representations are locally equivalent to the vacuum representation, i.e. π A(I) ∼ = π0 A(I) ∀I. This implies that all superselection sectors are of the DHR type and can be analyzed accordingly [37, 36]. As a means of studying the superselection theory of a model it has been proposed [64] to examine the inclusion A(I1 ) ∨ A(I3 ) ⊂ (A(I2 ) ∨ A(I4 ))0 = A(I341 ) ∧ A(I123 ),
(C.4)
where I1,...,4 are quadrants of the circle and Iijk = Ii ∪ Ij ∪ Ik :
I2
I1 '$ @ I4
(C.5)
@ &% I3 At least for strongly additive theories, where A(I1 ) ∨ A(I2 ) = A(I) if I1 ∪ I2 = I, the inclusion (C.4) is easily seen to be irreducible. In the presence of nontrivial superselection sectors this inclusion is strict as the intertwiners between endomorphisms localized in I1 , I3 , respectively, are contained in the larger algebra of (C.4) by Haag duality but not in the smaller one. Furthermore, for rational theories the inclusion (C.4) is expected to have finite index. While we have nothing to add in the way of model independent analysis the techniques developed in the preceding sections can be applied to a large class of interesting models. These are chiral nets obtained as fixpoints of a larger one under the action of a group. I. e. we start with a net I 7→ F (I) on the Hilbert space H fulfilling isotony and locality, the latter possibly twisted. The M¨obius group SU (1, 1) and the group G of inner symmetries are unitarily represented with common invariant vector . Again,
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the net F is supposed to fulfill the split property (with the obvious modifications due to the different geometry). The net I 7→ A(I) is now defined by A(I) = F (I) ∧ U (G)0 and A(I) = A(I) H0 , where H0 is the space of G-invariant vectors. The proof of Haag duality for chiral theories referred to above applies also to the net A, implying that there is no analogue of the violation of duality for the fixpoint net as occurs in 1+1 dimensions. This is easily understood as a consequence of the fact that the spacelike complement of an interval is again an interval, thus connected. However, our methods can be used to study the inclusion (C.4). It is clear that due to the split property A(I1 ) ∨ A(I3 ) ∼ = F (I1 ) ⊗ F (I3 )G×G H0 ⊗ H0 .
(C.6)
Our aim will now be to compute (A(I2 )∨A(I4 ))0 . In analogy to the 1+1 dimensional case we use the split property to construct unitaries Y1 , . . . , Y4 : H → H ⊗ H implementing the following isomorphisms: t t Yi∗ = Fi ⊗ Fi+2 ∀Fi ∈ F(Ii ). Yi Fi Fi+2
(C.7)
(One easily checks that Yi+2 = T Yi , where T x ⊗ y = y ⊗ x.) These unitaries can in turn be used to define local implementers of the gauge transformations Ui (g) = Yi∗ (U (g) ⊗ 1) Yi
(C.8)
with the localization Ui (g) ∈ F (Ii+2 )0 . (The index arithmetic takes place modulo 4.) These operators satisfy Ad Ui (g) F(Ii ) = αg , [Ui (g), Ui+2 (h)] = 0, Ui (g) Ui+2 (g) = U (g).
(C.9) (C.10) (C.11)
In a manner analogous to the proof of Lemma 3.9 one shows (Fi ≡ F(Ii ) etc.) (A2 ∨ A4 )0 = (F2 ∨ F4 )0 ∨ U2 (G)00 ∨ U4 (G)00 .
(C.12)
At this point we strengthen the property of Haag duality for the net F by requiring (F1 ∨ F3 )0 = (F2 ∨ F4 )t ,
(C.13)
which by the above considerations amounts to F having no nontrivial superselection sectors. This condition is fulfilled, e.g., by the CAR algebra on the circle which also possesses the split property. The chiral Ising model as discussed in [9] is covered by our general framework (with the group Z2 ). While (C.13) is a strong restriction it is the same as in [19] where the larger theory was supposed to be “holomorphic.” At this place it might be appropriate to emphasize that the requirement of (twisted) Haag duality (1.11) made above when considering 1+1 dimensional theories by no means excludes nontrivial superselection sectors. Making use of (C.13) we can now state quite explicitly how (A2 ∨ A4 )0 looks. In analogy to Thm. 3.10 we obtain (A2 ∨ A4 )0 = m F1 ∨ F3 ∨ U2 (G)00 H0 . (C.14) Again, using (C.13) one can check that α2 (g) = Ad U2 (g) restrict to automorphisms of F1 ∨ F3 rendering the algebra F1 ∨ F3 ∨ U2 (G)00 a crossed product. Recalling
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A1 ∨ A3 = m(F1 ) ∨ m(F3 ) H0 ,
(C.15)
we have the following natural sequence of inclusions: A1 ∨ A3 ⊂ m(F1 ∨ F3 ) H0 ⊂ (A2 ∨ A4 )0 ,
(C.16)
both of which have index |G|. It is interesting to remark that the intermediate algebra m(F1 ∨F3 ) H0 equals (m(F2 ∨F4 ) H0 )0 . For general chiral theories the existence of such an intermediate subfactor between A1 ∨ A3 and (A2 ∨ A4 )0 is not known. In the case of G being abelian where the Ui (g) are invariant under global gauge transformations we obtain a square structure similar to the one encountered in Sect. 3.: (A2 ∨ A4 )0 A1 ∨ A3 ∨ U2 (G)00 ⊂ ∪ ∪ ⊂ m(F1 ∨ F3 ) H0 . A 1 ∨ A3
(C.17)
It may be instructive to compare the above result with the situation prevailing in 2 + 1 or more dimensions. There, as already mentioned in the introduction, the superselection theory for localized charges is isomorphic to the representation theory of a (unique) compact group. Furthermore, there is a net of field algebras acted upon by this group, such that the observables arise as the fixpoints. The analogue of the inclusion (C.4) then is (C.18) A(O1 ) ∨ A(O2 ) ⊂ A(O10 ∩ O20 )0 , where O1 , O2 are spacelike separated double cones. Under natural assumptions it can be shown that the larger algebra equals m(F(O1 ) ∨ F (O2 )) H0 , implying that the inclusion (C.18) is of the type (F1 ⊗ F2 )G×G ⊂ (F1 ⊗ F2 )Diag(G) just as the first one in (C.16). That the index of the inclusion (C.4) is |G|2 instead of |G| as for (C.18) is a consequence of the low dimensional topology comparable to the phenomena occurring in 1 + 1 dimensions. Acknowledgement. I am greatly indebted to K.-H. Rehren for his stimulating interest, many helpful discussions and countless critical readings of the evolving manuscript. Special thanks are due to D. Buchholz for the proof of Prop. 3.11.
Note added in proof. In Appendix C we claimed that the combination of Haag duality and the split property for wedges is weaker than the requirement of absence of charged sectors which was made in [19] where conformal orbifold theories were considered. After submission of this paper we discovered that this claim is wrong! While this does not affect any result of the present work it shows that the analysis of massive models based on the former assumptions is even stronger related to the one in [19] than expected. Furthermore, if the vacuum sector satisfies HD+SPW then Haag duality holds in all irreducible locally normal representations. In particular, on can replace “simple sector” by “irreducible A-stable subspace of H” in Thm. 3.10. The proofs as well as applications to the theory of quantum solitons will be found in [56]. I thank Prof. B. Schroer for drawing my attention to [72], where massive quantum field theories in 1 + 1 have been considered. In particular it has been shown that the statistics of charged fields is arbitrary in the sense that the same particle states can be created by Bose and by Fermi fields. Even though scattering theory aspects have not been discussed in the present work, the cited result fits well with that of our Sect. 5.
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65. Strˇatilˇa, S.: Modular theory in operator algebras. Abacus Press, 1981 66. Swieca, J. A.: Fields with generalized statistics: An exercise in order and disorder in two dimensional systems. In: Turko, L. et al. (eds.): Fundamental Interactions, Karpacz Winter School 1980 67. Szlach´anyi, K., Vecserny´es, P.: Quantum symmetry and braid group statistics in G-spin models. Commun. Math. Phys. 156, 127–168 (1993) 68. Szyma´nski, W., Peligrad, C.: Saturated actions of finite dimensional Hopf algebras on C ∗ -algebras. Math. Scand. 75, 217–239 (1994) 69. Szyma´nski, W.: Finite index subfactors and Hopf algebra crossed products. Proc. Am. Math. Soc. 120, 519–528 (1994) 70. Takesaki, M.: Duality for crossed products and the structure of von Neumann algebras of type III. Acta Math. 131, 249–310 (1973) 71. Verlinde, E.: Fusion rules and modular transformations in 2D conformal field theory. Nucl. Phys. B300, 360–376 (1988) 72. Schroer, B., Swieca, J. A.: Spin and statistics of quantum kinks. Nucl. Phys. B121, 505–513 (1978) Communicated by H. Araki
This article was processed by the author using the LaTEX style file pljour1 from Springer-Verlag.
Commun. Math. Phys. 191, 183 – 218 (1998)
Communications in
Mathematical Physics c Springer-Verlag 1998
The Scattering Amplitude for the Schr¨odinger Equation with a Long-Range Potential D. Yafaev Department of Mathematics, Universit´e de Rennes I, Campus Beaulieu, 35042 Rennes, France Received: 29 January 1997 / Accepted: 6 May 1997
Abstract: We consider the Schr¨odinger operator with a long-range potential V (x) in the space L2 (Rd ). Our goal is to study spectral properties of the corresponding scattering matrix and a diagonal singularity of its kernel s(ω, ω 0 ) (the scattering amplitude). It turns out that in contrast to the short-range case the Dirac-function singularity of s(ω, ω 0 ) at the diagonal ω = ω 0 disappears and the spectrum of the scattering matrix covers the whole unit circle. For an asymptotically homogeneous function V (x) of order −ρ, ρ < 1, we show that typically s(ω, ω 0 ) = w(ω, ω 0 ) exp iψ(ω, ω 0 ) , where the module w and the phase ψ are asymptotically homogeneous functions, as ω − ω 0 → 0, of orders −(d − 1)(1 + ρ−1 )/2 and 1 − ρ−1 , respectively. Leading terms of asymptotics of w and ψ at ω = ω 0 are calculated. In the case ρ = 1 our results generalize (in the limit ω −ω 0 → 0) the well-known formula of Gordon and Mott. As a by-product of our considerations we show that the long-range scattering fits into the theory of smooth perturbations. This gives an elementary proof of existence and completeness of wave operators in the theory of long-range scattering. In this paper we concentrate on the case ρ > 1/2 when the theory of pseudo-differential operators can be extensively used. Contents 1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
2
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
3
The Eikonal and Transport Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
4
Wave Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
5
The Scattering Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
6
The Diagonal Singularity of the Scattering Amplitude . . . . . . . . . . . . . . 204
7
Examples. Classical mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
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1. Introduction We consider the Schr¨odinger operator H = −1 + V (x) in the space L2 (Rd ) with a long-range potential V (x) satisfying a usual assumption |∂ α V (x)| ≤ Cα (1 + |x|)−ρ−|α| ,
ρ > 0.
(1.1)
Our goal is to develop the stationary scattering theory for the operator H and, in particular, to study the associated scattering matrix. We compare H with the operator H0 = −1 but consider the wave operators with a non-trivial identification J± (depending on the sign of t) (1.2) W± = W± (H, H0 ; J± ) = s − lim eiHt J± e−iH0 t . t→±∞
Such operators were introduced in [6], where the Enss method was used for the proof of their existence and completeness. Following [6], we construct J± as a Fourier integral operator with the phase function satisfying the eikonal equation and the amplitude satisfying the transport equation. We prefer to work with approximate but explicit solutions of these equations. Actually, J± enjoys all properties of pseudo-differential operators (pdo) since its phase function is sufficiently close to hx, ξi. The eikonal equation and the transport equations do not have global solutions so that we satisfy these equations outside of some cone where x is proportional to −ξ for the sign “+00 and to ξ for the sign “−00 . Neighbourhoods of these “bad” directions are cut off by some homogeneous function ζ± of degree 0. In contrast to [6] we use the stationary techniques for construction of wave operators (1.2) and show that long-range scattering fits into the theory of smooth (in the sense of Kato) perturbations. Indeed, the “effective” perturbation T± = HJ± − J± H0 is again a pdo with symbol vanishing as |x|−1 (because of the cut-off ζ± ) at infinity. Our proof of the existence and completeness of wave operators relies on the standard limiting absorption principle and the radiation conditions estimates introduced in [19] for the proof of asymptotic completeness for multi-particle systems. The radiation conditions estimates replace a rather tiresome construction of eigenfunctions of the operator H used (see e.g. [4]) in the first proof of the completeness in the long-range case. Once the existence and completeness of wave operators (1.2) are verified, it is easy to check (see the middle of Sect. 4) that they coincide with more traditional (see e.g. [4]) wave operators obtained by time-dependent modification of the free dynamics. Our basic goal is to study spectral properties of the corresponding scattering matrix S = S(λ), λ > 0, and the diagonal singularity of its kernel s(ω, ω 0 ) (the scattering amplitude). We stress that our paper is to a large extent motivated by [7]. Recall that in the short-range case (when V (x) = O(|x|−ρ ), ρ > 1, as |x| → ∞ and J± = I) the operator S − I is compact. Hence the spectrum of S consists of eigenvalues of finite multiplicity accumulating possibly at the point 1 only and the leading singularity of its kernel is the Dirac-function. As we shall see, the structure of S is completely different in the long-range case. Our study of the scattering matrix relies on its stationary representation. First, using the resolvent estimates, called usually microlocal or propagation estimates (see [14, 10, 9] or [5, 8]) we show, similarly to [7], that the part of S containing the resolvent of the operator H is an integral operator with smooth kernel. The remaining, singular, part S1 of S is quite explicit and can be obtained as the restriction of the operator J+∗ (HJ− −J− H0 ) on the sphere of radius λ1/2 in the momentum representation. It is convenient to treat S1 as a pdo (this view-point was adopted in [2] in the short-range case). We check that the
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principal symbol of S1 is an oscillating function determined by the asymptotics of V at infinity. In particular, this implies that the spectrum of the scattering matrix covers the whole unit circle. Note that in the short-range case the principal symbol of S equals 1 which corresponds to the Dirac-function in its kernel. Thus, in the long-range case this singularity disappears. The diagonal singularity of the scattering amplitude is given by the Fourier transform of the symbol of the operator S1 . It turns out that for an asymptotically homogeneous function V (x) of order −ρ, ρ < 1, X wj (ω, ω 0 ) exp iψj (ω, ω 0 ) , (1.3) s(ω, ω 0 ) = where the modules wj and the phases ψj are asymptotically homogeneous functions, as ω −ω 0 → 0, of orders −(d−1)(1+ρ−1 )/2 and 1−ρ−1 , respectively. The precise result is formulated in Theorem 6.7. Its proof relies on the stationary phase method applied to our integral representation of the scattering amplitude. In the case ρ = 1 (see Theorem 6.10) the sum (1.3) consists of one term only, the module w is asymptotically homogeneous of order −d + 1 and the phase ψ has a logarithmic singularity at the diagonal. We always assume that a potential V (x) is a C ∞ -function which satisfies the estimate (1.1) for all multi-indices α. Actually, only some finite number of derivatives is required but we imposed condition (1.1) for all α in order to use freely the calculus of pseudodifferential operators. We emphasize that the operator J± belongs to the H¨ormander 0 with ρ determined by (1.1) and δ = 1 − ρ. The basic calculus of pseudoclass Sρ,δ differential operators is well developed for this class if ρ > 1/2 > δ. Thus we can lean directly on this theory only if (1.1) is fulfilled for ρ > 1/2. All our results remain valid for an arbitrary ρ > 0 but this requires a separate study of a special class of pdo with oscillating symbols. In this paper we concentrate on the case ρ > 1/2 in order to avoid additional technicalities. Let us mention that regularity of the scattering amplitude away from the diagonal was announced in [1] and its proof can be found in [7]. In the latter paper there is also an upper bound on |s(ω, ω 0 )| at the diagonal. However, as our asymptotic formula shows, this bound is not optimal. Let us say some words about terminology. In the short-range case the scattering amplitude f (ω, ω 0 ; λ) is defined (see e.g. [12]) as the coefficient at the outgoing spherical wave |x|−(d−1)/2 eik|x| in the asymptotics as |x| → ∞ of the scattering solution of the Schr¨odinger equation; the parameters ω 0 and ω = x/|x| are incident and outgoing directions, respectively. In terms of the scattering amplitude the kernel of the operator S(λ) − I equals (2π)−(d−1)/2 ieiπ(d−3)/4 λ(d−1)/4 f (ω, ω 0 ; λ). This relation between the scattering matrix and the scattering amplitude is preserved for ω 6= ω 0 in the long-range case although the plane and spherical waves are distorted. Thus, the scattering amplitude should be defined as f (ω, ω 0 ; λ) = −i(2π)(d−1)/2 e−iπ(d−3)/4 λ−(d−1)/4 s(ω, ω 0 ; λ),
(1.4)
0
but we often use this term with respect to the kernel s(ω, ω , λ) of the scattering matrix. According to (1.4) the scattering cross-section in an angle dω around ω is 6(ω, ω 0 , λ)dω, where X (1.5) (ω, ω 0 ; λ) = (2π)d−1 λ−(d−1)/2 |s(ω, ω 0 ; λ)|2 . Abusing somewhat terminology we call the function 6 itself the scattering cross-section. A choice of operators J± in (1.2) is, of course, not unique. Actually, let U± be multiplication by a function exp(i8± (ξ)) in the momentum representation and set J˜± =
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J± U± . Clearly, the wave operators (1.2) and W± (H, H0 ; J˜± ) exist at the same time and the scattering amplitude s(ω, ˜ ω 0 ; λ) corresponding to J˜± is related to the original one by the equality s(ω, ˜ ω 0 ; λ) = exp(−i8+ (λ1/2 ω))s(ω, ω 0 ; λ) exp(i8− (λ1/2 ω 0 )).
(1.6)
Apparently, there is no preferable choice of identifications J± . Nevertheless, if the diagonal singularity of the function s(ω, ω 0 ; λ) is found for some operators J± , then, by (1.6), it is automatically known for all operators J˜± . In particular, the scattering cross-section 6(ω, ω 0 ; λ) does not depend at all on the choice of identifications. As shown by Gordon and Mott (see e.g. [12]), for the purely Coulomb potential (and d = 3) the quantum cross-section equals the classical one (its definition can be found in Sect. 7) for all scattering angles. According to our analysis, the quantum and classical cross-sections coincide in the limit of small scattering angles for all central potentials V (x) = v|x|−ρ , ρ < 1, and all dimensions d. On the other hand, for the Coulomb potential this is true in the case d = 3 only.
2. Preliminaries 1. We need some elementary facts about pseudo-differential operators (pdo) defined by the equality Z eihξ,xi a(x, ξ)fˆ(ξ)dξ, (2.1) (Af )(x) = (2π)−d/2 Rd
where fˆ(ξ) = (2π)−d/2
Z
e−ihξ,xi f (x)dx Rd
is the Fourier transform of f ∈ S(Rd ). We suppose that the symbol a ∈ C ∞ (Rd × Rd ) and (2.2) |∂xα ∂ξβ a(x, ξ)| ≤ C(1 + |x|)m−ρ|α|+δ|β| , C = Cα,β , for some numbers m ≤ 0, ρ > 1/2 > δ and all multi-indices α, β. Here and below C and c are different positive constants, whose precise values are of no importance. Moreover, we assume that a(x, ξ) = 0 for sufficiently large |ξ|. The class of symbols m . satisfying these conditions is usually denoted by Sρ,δ The adjoint operator A∗ to A is defined by Z ∗ d (A f )(ξ) = (2π)−d/2 e−ihξ,xi a(x, ξ)f (x)dx, Rd
and thus fits into a usual framework of the theory (see e.g. [3, 16]) of pseudo-differential operators (note that the roles of the variables x and ξ are interchanged). This allows to recover all results of the theory for the operator A itself. Necessary information on the calculus of pdo is collected in the four following assertions. Let us denote by X the operator of multiplication by (1 + |x|2 )1/2 . Proposition 2.1. The operator AX −m is bounded and AX −m1 , m1 > m, is compact in the space L2 (Rd ).
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Proposition 2.2. Suppose that symbols aj of pdo Aj , j = 1, 2, belong to classes Sρ,δj . Then A∗2 A1 is the pdo with symbol a(x, ξ) which admits for any N the representation X a(x, ξ) = (−i)α ∂xα ((∂ξα a2 )(x, ξ)a1 (x, ξ))/α! + a˜ (N ) (x, ξ), |α|≤N m ˜ with m ˜ = m1 + m2 − (N + 1)(ρ − δ). where a˜ (N ) (x, ξ) ∈ Sρ,δ 0 and let A be the pdo Proposition 2.3. Let Aj , j = 1, 2, be pdo with symbols aj ∈ Sρ,δ ∗ with symbol a1 (x, ξ)a2 (x, ξ). Then the operator A − A1 A2 is compact. m Proposition 2.4. Suppose that A is a pdo with symbol a ∈ Sρ,δ and G is a pdo (actually, P a differential operator) with symbol g(x, ξ) = |s|≤s0 gs (x)ξ s , where gs ∈ C ∞ (Rd ) and ∂ α gs (x) = O(|x|n−|α| ) as |x| → ∞ for some n and all α. Let B be the pdo with symbol b(x, ξ) = |g(x, ξ)|2 a(x, ξ). Then the operator X p (G∗ AG − B)X p is bounded if 2p = ρ − m − 2n. m is invariant with respect to a change of variables (a In the case ρ > 1/2 the class Sρ,δ diffeomorphism in ξ). This allows to define pseudo-differential operators on manifolds. Moreover, the principal symbol of a pdo (its symbol modulo a term from the class m+1−2ρ ) is invariantly defined on the cotangent bundle of a manifold. We will need to Sρ,δ consider pdo on the unit sphere Sd−1 = {|ξ| = 1}.
2. Here we collect necessary estimates on the resolvent R(z) = (H − z)−1 (and its powers) of the Schr¨odinger operator H = −1 + V in the space H = L2 (Rd ). Various definitions of H-smoothness are discussed e.g. in [15] or [17]. We start with the limiting absorption principle. Proposition 2.5. Let assumption (1.1) be fulfilled, n = 1, 2, . . . and p > n − 1/2. Then the operator-function X −p Rn (z)X −p is continuous in norm with respect to the parameter z in the closed complex plane cut along [0, ∞), possibly with the exception of the point z = 0. In particular, the operator X −p is H-smooth for p > 1/2 on any interval 3 = (λ0 , λ1 ) for 0 < λ0 < λ1 < ∞. Proposition 2.5 implies, of course, that the positive spectrum of H is absolutely continuous. Its proof is naturally divided into two parts. The continuity of X −p Rn (z)X −p outside of the point spectrum of H is proven (for an arbitrary n), for example, in [9] and the absence of positive eigenvalues - in [15], v.4. The following resolvent estimates borrowed from [19, 20] are called radiation conditions-estimates there. Proposition 2.6. Let assumption (1.1) be fulfilled and p > 1/2. Set Gj u = X −1/2 (∂j u − |x|−2 h∇u, xixj ),
j = 1, . . . , d.
(2.3)
Then the operator-functions Gj R(z)G∗k , Gj R(z)X −p for all j, k = 1, . . . , d are weakly continuous with respect to the parameter z in the closed complex plane cut along [0, ∞), possibly with the exception of the point z = 0. strong operator In particular, the operator Gj is H-smooth on any interval 3 = (λ0 , λ1 ) for 0 < λ0 < λ1 < ∞. Propositions 2.5 for n = 1 and 2.6 are sufficient for the proof of existence and completeness of wave operators. However, our study of the scattering matrix requires additional resolvent estimates called usually microlocal or propagation estimates.
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Proposition 2.7. Let assumption (1.1) be fulfilled and n = 1, 2, . . .. Suppose that the m , a± (x, ξ) = 0 in some ball symbol a± (x, ξ) of a pdo A± belongs to some class Sρ,δ |ξ| ≤ c and that the support of a± (x, ξ) is contained in the cone ∓hξ, xi ≥ |x| |ξ|,
ξˆ = ξ/|ξ|, xˆ = x/|x|.
> 0,
Then the operator-functions X p−σ A∗+ Rn (z)X −p ,
X −p Rn (z)A− X p−σ ,
p > n − 1/2, σ > n + m,
are bounded and continuous in norm with respect to the parameter z in the region Re z ≥ λ0 > 0, Im z ≥ 0. Proposition 2.8. Under the assumptions of Proposition 2.7 for all p the operatorfunctions X p A∗+ Rn (z)A− X p are bounded and continuous in norm with respect to the parameter z in the region Re z ≥ λ0 > 0, Im z ≥ 0. Proofs of these assertions can be found either in [14, 10, 9] or in [5, 8]. The proof of [14, 10, 9] relies on the Mourre estimate and is easily accessible. 3. For construction of the scattering matrix we need to discuss restrictions of integral or pseudo-differential operators to the spheres ξ 2 = λ. Let (00 (λ)f )(ω) = 2−1/2 k (d−2)/2 fˆ(kω),
λ = k 2 > 0, ω ∈ Sd−1 ,
(2.4)
be (up to the numerical factor) the restriction of fˆ onto the sphere of radius k. Then the formally adjoint operator is defined by the equality Z eikhω,xi g(ω)dω. (2.5) (0∗0 (λ)g)(x) = 2−1/2 k (d−2)/2 (2π)−d/2 Sd−1
Denote N = L2 (Sd−1 ). The first of the following assertions is a direct consequence of the Sobolev trace theorem (or of Proposition 2.5). The second can be deduced from Proposition 2.6, see [20] for details. Proposition 2.9. For p > 1/2 the operator-function X −p 0∗0 (λ) : N → H is compact and continuous in norm with respect to the parameter λ > 0. Proposition 2.10. The operator-functions Gj 0∗0 (λ) : N → H are bounded and strongly continuous with respect to the parameter λ > 0. We call the operator A[ (λ), defined formally by the equality A[ (λ) = 00 (λ)A0∗0 (λ), the restriction of A to the sphere |ξ| = k. If A is a pdo and A[ (λ) is considered as an integral operator, then, by (2.4), (2.5), the kernel Z 0 [ 0 −1 −d d−2 eikhω −ω,xi a(x, kω 0 )dx (2.6) p (ω, ω ; λ) = 2 (2π) k Rd
(such integrals are understood, of course, in the sense of distributions) of the operator A[ (λ) is an infinitely differentiable function of ω and ω 0 (and λ > 0) for ω 6= ω 0 . However due to a possible strong singularity of function (2.6) at the diagonal ω = ω 0 a
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precise definition of the restriction A[ (λ) requires some assumptions on the symbol of A. The operator A[ = A[ (λ) can also be considered as a pdo on Sd−1 . Actually, let κ be a local diffeomorphism of Sd−1 on an open set κ ⊂ Rd−1 and ωκ = κ(ω). Then Z Z 0 (A[ g)(ω) = (2π)−d+1 eihωκ −ωκ ,bi a[κ (ωκ , b)gκ (ωκ0 )dωκ0 db, gκ (ωκ ) = g(ω). κ
Rd−1
m , where 2ρ > 1, the symbols a[κ are connected for different κ by In the case a[κ ∈ Sρ,δ a usual formula of a change of variables for pdo. In particular, the principal symbol a[ m+1−2ρ (its symbol modulo a term from the class Sρ,δ ) of the pdo A[ is invariantly defined on the cotangent bundle of the unit sphere. The spectral family E0 (λ) of the operator H0 = −1 is described in terms of operators (2.4), (2.5). Actually, set δε (H0 − λ) = (2πi)−1 R0 (λ + iε) − R0 (λ − iε) ,
where R0 (z) = (H0 − z)−1 . Then for any λ > 0 and p > 1/2, lim X −p δε (H0 − λ)X −p = dX −p E0 (λ)X −p /dλ = X −p 0∗0 (λ) · 00 (λ)X −p .
ε→0
Let A be a pdo with symbol a(x, ξ) satisfying (2.2). If m < −1, then the operator A[ (λ) = 00 (λ)X −p X p AX p X −p 0∗0 (λ) , p = −m/2, is correctly defined as a bounded operator in N. In the case m = −1 a definition of the operator A[ (λ) requires vanishing of its symbol on the conormal bundle of the sphere |ξ| = k. −1 and a(tω, kω) = 0, ω ∈ Sd−1 , at least for Proposition 2.11. Suppose that a ∈ Sρ,δ sufficiently large |t|. Then for any functions f, g from the Schwartz class S(Rd ) the double limit
lim (Aδε (H0 − λ)f, δη (H0 − λ)g) =: (A[ (λ)00 (λ)f, 00 (λ)g)
ε,η→0
0 exists. The operator A[ (λ) is a pdo on Sd−1 with symbol from the class Sρ,δ so that [ [ A (λ) is a bounded operator in the space N. The principal symbol of A (λ) is given by the absolutely convergent integral Z ∞ a(zω + k −1 b, kω)dz, |ω| = 1, hω, bi = 0. a[ (ω, b; λ) = (4π)−1 k −1 −∞
This assertion was proved in [13], where it was supposed that ρ = 1, δ = 0. However, the proof of [13] extends automatically to the case ρ > 1/2 > δ (but not to the case ρ ≤ 1/2). Note that the existence of A[ can also be deduced from Proposition 2.6 (see Lemma 4.1 below). 4. Here we study the essential spectrum of some special class of pdo with oscillating symbols. In view of our applications we consider pdo on the unit sphere Sd−1 but the
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problem reduces by a diffeomorphism to operators acting in a domain ⊂ Rd−1 . We need the well-known formula for the action of a pdo A, Z Z 0 −d+1 0 eihω −ω,bi a(ω, b)f (ω 0 )dω 0 db, a ∈ Sρ,δ , (Af )(ω) = (2π)
Rd−1
on the exponential function. Actually, if f ∈ C0∞ (), b0 ∈ Rd−1 , b0 6= 0, and u,t (ω) = −(d−1)/2 f (−1 (ω − ω0 ))eithb0 ,ωi ,
(2.7)
then (Au,t )(ω) = a(ω, tb0 )u,t (ω) + r,t (ω), where kr,t kL2 ≤ C−1 t−ρ
(2.8)
as long as tρ → ∞. 0 Proposition 2.12. Let A be a pdo with symbol from the class Sρ,δ , where ρ > 1/2 > δ. Suppose that for some point (ω0 , b0 ), |ω0 | = 1, hω0 , b0 i = 0, b0 6= 0, the principal symbol a(ω, b) of A admits the representation
a(ω0 , tb0 ) = eiθ(t) (1 + o(1)),
t → ∞,
(2.9)
where θ(t) is a continuous function and lim sup θ(t) = ∞
t→∞
or
lim inf θ(t) = −∞.
t→∞
(2.10)
Then the spectrum of the operator A in the space N covers the unit circle. Proof. It suffices to construct for every point µ = eiϑ a sequence un such that ||un || = 1, un converges weakly to zero and lim ||Aun − µun || = 0.
n→∞
(2.11)
Consider a neighbourhood G ⊂ Sd−1 of the point ω0 , a diffeomorphism of G onto ⊂ Rd−1 and a unitary operator U : L2 (G) → L2 () corresponding to this diffeomorphism. Then the principal symbols of the operators U AU ∗ and A are the same. Since by the proof of (2.11) compact terms can be neglected, we may suppose that the symbol of U AU ∗ coincides with its principal symbol. Abusing somewhat notation we denote the operator U AU ∗ again by A and its symbol by a(ω, b), ω ∈ , b ∈ Rd−1 . This function 0 and satisfies condition (2.9). belongs to the class Sρ,δ 0 , we have that By the condition a ∈ Sρ,δ |a(ω, tb0 ) − a(ω0 , tb0 )| ≤ C|ω − ω0 | sup |(∇ω a)(ω, tb0 )| ≤ C|ω − ω0 | tδ . ω∈
Combining this estimate with (2.8) we obtain for functions (2.7) that ||Auε,t − a(ω0 , tb0 )uε,t || ≤ C(ε−1 t−ρ + εtδ ).
(2.12)
Set now ε = t−(ρ+δ)/2 . Then the right-hand side of (2.12) equals t−(ρ−δ)/2 and hence tends to zero as t → ∞. Let n → ∞ (or n → −∞) under the first (second) assumption (2.10). Choose a sequence tn → ∞ such that θ(tn ) = ϑ + 2πn and set un = uεn ,tn , −(ρ+δ)/2 . Then relation (2.11) follows from (2.9) and (2.12). where εn = tn
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3. The Eikonal and Transport Equations In this section we collect standard material on approximate but explicit solutions of the eikonal and transport equations. These equations arise in an attempt to construct eigenfunctions of the Schr¨odinger operator in the form ψ = eiϕ a. 1. For proving existence and completeness of wave operators and a study of the scattering matrix up to compact terms it suffices to set a(x) = 1. Since (−1 + V )(eiϕ ) = eiϕ (|∇ϕ|2 + V − i1ϕ),
(3.1)
we require that ϕ be a special (approximate) solution of the eikonal equation |∇ϕ|2 + V (x) = λ,
λ > 0.
For an arbitrary unit vector ω ∈ Sd−1 we set ξ = λ1/2 ω and seek ϕ(x, ξ) in the form ϕ(x, ξ) = hξ, xi + 8(x, ξ),
(3.2)
where (∇8)(x, ξ) tends to zero as |x| → ∞ in some cone around ω. Let us first consider an auxiliary linear equation 2hξ, ∇8(x, ξ)i + V (x) = 0,
∇ = ∇x ,
(3.3)
ˆ xi and construct its solution 8(x, ξ) such that (∇8)(x, ξ) tends to zero in any cone hξ, ˆ ≥ κ. Here and below κ is an arbitrary number from the interval (−1, 1). For a given ξ = λ1/2 ω we decompose x = zω + b, where hω, bi = 0. In other words, we choose a coordinate system with the axis z directed along ω. Then (3.3) reduces to 2|ξ| ∂8(z, b)/∂z + V (z, b) = 0 and has a solution −1
Z
z
8(z, b) = −(2|ξ|)
V (z 0 , b)dz 0 + c(b),
(3.4)
(3.5)
0
where c(b) is an arbitrary function of b. According to (3.4) the function ∂8(z, b)/∂z vanishes as |x| → ∞ but the behaviour of Z z (∇b V )(z 0 , b)dz 0 + ∇c(b) (3.6) (∇b 8)(z, b) = −(2|ξ|)−1 0
is determined by a choice of the function c(b). Set Z ∞ c(b) = (2|ξ|)−1 (V (z 0 , b) − V (z 0 , 0))dz 0 .
(3.7)
0
Then, by (3.6), (∇b 8)(z, b) = (2|ξ|)−1
Z
∞
(∇b V )(z 0 , b)dz 0
z
tends to zero as |x| → ∞ in any cone z ≥ κ(z 2 + b2 )1/2 (or, equivalently, z ≥ κ(1 − κ2 )−1/2 |b|). Equalities (3.5), (3.7) for the function 8(x, ξ) =: 8+ (x, ξ) can be, of course, rewritten in an invariant (not depending on a coordinate system) form: Z ∞ V (x ± tξ) − V (±tξ) dt. (3.8) 8± (x, ξ) = ±2−1 0
The function 8− (x, ξ) = −8+ (x, −ξ) also satisfies Eq. (3.3) and (∇8− )(x, ξ) tends to ˆ xi zero in any cone hξ, ˆ ≤ κ. We always suppose that |ξ| ≥ c > 0.
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Proposition 3.1. Let assumption (1.1) hold. Define the function ϕ± (x, ξ) by equalities (3.2), (3.8). Then (0) , (3.9) (−1 + V − ξ 2 )(eiϕ± ) = eiϕ± q± where (0) = |∇8± |2 − i18± . (3.10) q± ˆ For all multi-indices α, β, any κ ∈ (−1, 1) and ±hξ, xi ˆ ≥ ±κ (or |x| ≤ c), |∂xα ∂ξβ 8± (x, ξ)| ≤ C|ξ|−1−|β| (1 + |x|)1−ρ−|α| , (0) |∂xα ∂ξβ q± (x, ξ)| ≤ C|ξ|−1−|β| (1 + |x|)−2ρ−|α| ,
|α| ≥ 1,
(3.11)
ρ ≤ 1.
(3.12)
Proof. Equalities (3.9), (3.10) follow directly from (3.1) and (3.2), (3.3). Differentiating (3.8) we find that Z ∞ t|β| (∂ α+β V )(x ± tξ)dt, |α| ≥ 1. ∂xα ∂ξβ 8± (x, ξ) = 2−1 (±1)1+|β| 0
Note that 2|x ± tξ|2 ≥ (1 − κ2 )(x2 + t2 ξ 2 ) Therefore (1.1) implies that
Z
|∂xα ∂ξβ 8± (x, ξ)| ≤ C
∞
if
± hξ, xi ≥ ±κ|x| |ξ|.
(3.13)
t|β| (1 + |x| + |ξ|t)−ρ−|α|−|β| dt.
0
Making the change of variables t = |ξ|−1 (1 + |x|)s in the last integral we see that it equals the right-hand side of (3.11). Estimate (3.12) for function (3.10) is a consequence of (3.11). Note that, in the case ρ < 1, inequality (3.11) for α = 0 follows from its validity for |α| = 1. 2. A description of the diagonal singularity of the scattering matrix requires a better approximation to eigenfunctions of H. It follows from (3.9) that (−1 + V − ξ 2 )(eiϕ± a± ) = eiϕ± q± ,
(3.14)
where
(0) a± − 2ih∇ϕ± , ∇a± i − 1a± . (3.15) q ± = q± The equality q± = 0 gives us the transport equation for a± . Its approximate solution can be constructed by a procedure similar to that of introduction. Define functions b(±) n (x, ξ) = 1, inductively by relations b(±) 0 (0) (±) (±) fn(±) = 2h∇8± , ∇b(±) n i + iq± bn − i1bn , Z ∞ −1 (x, ξ) = ±2 fn(±) (x ± tξ, ξ)dt, b(±) n+1
(3.16) (3.17)
0
and set ) a± (x, ξ) = a(N ± (x, ξ) =
N X
b(±) n (x, ξ).
(3.18)
n=0
Note that
(±) 2hξ, ∇b(±) n+1 i + fn = 0.
(3.19)
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Proposition 3.2. Let assumption (1.1) with ρ ∈ (1/2, 1] hold. Then functions b(±) n , n≥ (N ) 1, and q± = q± defined by Eqs. (3.16), (3.17) and (3.15), (3.18) satisfy in the region ˆ xi ±hξ, ˆ ≥ ±κ (or |x| ≤ c) the estimates −n−|β| (1 + |x|)−ε0 n−|α| , |∂xα ∂ξβ b(±) n (x, ξ)| ≤ C|ξ|
and
ε0 = 2ρ − 1 > 0,
(N ) |∂xα ∂ξβ q± (x, ξ)| ≤ C|ξ|−N −|β| (1 + |x|)−1−ε0 (N +1)−|α| .
(3.20) (3.21)
Proof. Suppose that (3.20) holds for some n. Then it follows from (3.16) and Proposition 3.1 that |∂xα ∂ξβ fn(±) (x, ξ)| ≤ C|ξ|−n−|β| (1 + |x|)−2ρ−ε0 n−|α| .
(3.22)
Differentiating (3.17) we see that ∂xα ∂ξβ b(±) n+1 (x, ξ) consists of terms Z ∞ t|β1 | (∂xα+β1 ∂ξβ2 fn(±) )(x ± tξ, ξ)dt, β1 + β2 = β. 0
According to (3.22) this integral is bounded by Z ∞ t|β1 | (1 + |x| + |ξ|t)−2ρ−ε0 n−|α|−|β1 | dt. C|ξ|−n−|β2 | 0
Using inequality (3.13) and making the change of variables t = |ξ|−1 (1 + |x|)s we obtain bound (3.20) for n + 1. This proves inductively (3.20) for all n. It follows from (3.16), (3.19) that (0) (±) (±) (±) 2ihξ, ∇b(±) n+1 i + 2ih∇8± , ∇bn i − q± bn + 1bn = 0. (N ) Summing these equalities over n = 0, 1, . . . , N we find that q± = 2ihÊξ, ∇b(±) N +1 i. Thus, (3.21) is a consequence of (3.20) for n = N + 1.
Corollary 3.3. For functions ϕ± (x, ξ) and a± (x, ξ) constructed in Propositions 3.1 and 3.2, respectively, Eq. (3.14) holds with the function q± (x, ξ) satisfying estimates (3.21). Remark. If ρ > 1, then ρ should be replaced by 1 in estimates (3.12), (3.20) and (3.21). On the other hand, in this case one can set ϕ± (x, ξ) = hx, ξi and construct a± (x, ξ) as an approximate solution of the transport equation 2ihξ, ∇a± i + 1a± = 0. Then Eq. (3.14) remains true with a function q± (x, ξ) satisfying (3.21). 4. Wave Operators 1. Here we consider wave operators for the pair of Hamiltonians H0 = −1, H = −1+V and a suitable identification J± . To avoid unnecessary remarks we may suppose that (1.1) holds with ρ ∈ (1/2, 1). Let σ± ∈ C ∞ (−1, 1) be such that σ± (τ ) = 1 in a neighbourhood of the point ±1 and σ± (τ ) = 0 in a neighbourhood of the point ∓1 and let η ∈ C ∞ (Rd ) be such that η(x) = 0 in a neighbourhood of zero and η(x) = 1 for large |x|. Set ˆ xi) ζ± (x, ξ) = η(x)σ± (hξ, ˆ
(4.1)
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and choose some function ψ ∈ C0∞ (R+ \{0}). Due to the function ψ(ξ 2 ) all our considerations will be localized on a bounded energy interval disjoint from zero. The function η is introduced only to get rid of the singularity of the function |x|−1 x at the point x = 0. We construct J± as a pseudo-differential operator, Z (J± f )(x) = (2π)−d/2 eiϕ± (x,ξ) a± (x, ξ)ζ± (x, ξ)ψ(ξ 2 )fˆ(ξ)dξ, (4.2) Rd
where ϕ± and a± are the functions defined in Sect. 3. According to (3.11), the function exp(8± (x, ξ)) satisfies estimates (2.2) with m = 0, the same ρ as in (1.1) and δ = 1 − ρ. It follows that the symbol ei8± (x,ξ) a± (x, ξ)ζ± (x, ξ)ψ(ξ 2 ) 0 of pdo (4.2) belongs to the class Sρ,δ . Thus, by Proposition 2.1, J± is a bounded oper) ator in the space H. Moreover, by virtue of estimates (3.20), the operator J± (a(N ± )− (0) (0) J± (a± ), a± = 1, is compact. Therefore the wave operators (1.2) (and the corresponding scattering matrices) are the same for all N so that in this section we may set a± (x, ξ) = 1. We consider also “inverse” wave operators
W± (H0 , H; J ∗ ) = s − lim eiH0 t J ∗ e−iHt E(R+ ), t→±∞
where J = J+ or J = J− . Recall (see Proposition 2.5) that E(R+ ) coincides with the projection on the absolutely continuous subspace. Our goal in this section is to show that the triple H0 , H, J± fits into the framework of the theory of smooth perturbations so that wave operators (1.2) exist and are complete. To that end we need the following Lemma 4.1. Let T be a pdo with symbol 2 ˆ ), t(x, ξ) = g(x, ξ)w(hx, ˆ ξi)η(x)ψ(ξ −1 where g ∈ Sρ,δ , w ∈ C ∞ [−1, 1] and w(±1) = 0. Then T admits the representation
T =
d X
G∗j B (s) Gj + X −p B (r) X −p ,
(4.3)
j=1
where Gj are defined by (2.3), p = (1 + ρ)/2 and the operators B (s) , B (r) are bounded. Proof. Let us define B (s) as a pdo with symbol ˆ xi ˆ 2 )−1 t(x, ξ). b(s) (x, ξ) = (1 + |x|2 )1/2 |ξ|−2 (1 − hξ, 0 The function (1 − τ 2 )−1 w(τ ) is C ∞ on [−1, 1] so that b(s) ∈ Sρ,δ and the operator B (s) Pd is bounded. Set T0 = j=1 G∗j B (s) Gj . Since
(1 + |x|2 )−1/2
d X
(ξj − |x|−2 hξ, xixj )2 b(s) (x, ξ) = t(x, ξ),
j=1
Proposition 2.4 implies that the operator X p (T − T0 )X p is bounded. Let us calculate the perturbation
Scattering Amplitude for Schr¨odinger Equation with Long-Range Potential
T± = HJ± − J± H0 . According to (3.9) and (4.2), we have that Z eiϕ± (x,ξ) τ± (x, ξ)ψ(ξ 2 )fˆ(ξ)dξ, (T± f )(x) = (2π)−d/2
195
(4.4)
(4.5)
Rd
where τ± (x, ξ) = q± (x, ξ)ζ± (x, ξ) − 2ih∇ϕ± (x, ξ), ∇ζ± (x, ξ)i − 1ζ± (x, ξ)
(4.6)
(0) with ∇ = ∇x , 1 = 1x and q± = q± . This expression is O(|x|−1 ) as |x| → ∞ because of the cut-off function (4.1). Let us single out the term (s) ˆ xi)i, (x, ξ) = −2iη(x)hξ, ∇σ± (hξ, ˆ τ±
(4.7)
which decays as |x|−1 only. Other terms in (4.6) decay more rapidly. Actually, the (r) , defined by the equality function τ± (s) (r) (x, ξ) + τ± (x, ξ), τ± (x, ξ) = τ±
(4.8)
satisfies, according to Proposition 3.1, the estimates (r) |∂xα ∂ξβ τ± (x, ξ)| ≤ C(1 + |x|)−2ρ−|α| ,
x ∈ Rd , 0 < c0 ≤ |ξ| ≤ c1 < ∞.
(4.9)
Let us introduce pdo T±(s) , T±(r) with symbols i8± (x,ξ) (s) τ± (x, ξ)ψ(ξ 2 ), t(s) ± (x, ξ) = e
respectively. Then T± = T±(s) + T±(r) .
i8± (x,ξ) (r) t(r) τ± (x, ξ)ψ(ξ 2 ), ± (x, ξ) = e
(4.10)
Proposition 4.2. Operator (4.4) admits representation (4.3) with p = ρ. −1 Proof. Let us consider the operators T±(s) and T±(r) separately. Since t(s) ± ∈ Sρ,1−ρ and 0 σ± (τ ) = 0 in neighbourhoods of points −1 and 1, Lemma 4.1 can be directly applied to −2ρ T±(s) . It follows from (4.9) that t(r) ± ∈ Sρ,1−ρ and consequently, by Proposition 2.1, the (r) operator X ρ T± X ρ is bounded.
Taking into account that, by Propositions 2.5 and 2.6, the operators X −ρ , ρ > 1/2, and Gj are H0 - and H- smooth on any bounded disjoint from zero interval we arrive (see e.g. [15] or [17]) at the following result. Theorem 4.3. Suppose that condition (1.1) is fulfilled for ρ > 1/2. Let J± = J± (ζ± , ψ) be defined by (4.2). Then all wave operators W± (H, H0 ; J± ),
∗ W± (H0 , H; J± )
(4.11)
W± (H, H0 ; J∓ ),
∗ W± (H0 , H; J∓ )
(4.12)
and
exist. Operators (4.11) as well as (4.12) are adjoint to each other.
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2. Let us show that the wave operators W± (H, H0 ; J± ) are isometric (on some subspace of H) and complete. Lemma 4.4. The following relations hold: ∗ J± − ψ 2 (H0 ))e−iH0 t = 0, s − lim (J±
(4.13)
∗ s − lim J± J± e−iH0 t = 0.
(4.14)
t→±∞
t→∓∞
∗ Proof. By Propositions 2.1 and 2.2 for N = 0, up to a compact term, J± J± is the pdo 2 2 2 (we denote it by A) with symbol ζ± (x, ξ)ψ (ξ ). Thus, Z 2 2 (Ae−iH0 t f )(x) = (2π)−d/2 eihξ,xi−iξ t ζ± (x, ξ)ψ 2 (ξ 2 )fˆ(ξ)dξ. (4.15) Rd
The stationary point ξ = x/(2t) of this integral does not belong to the support of the function ζ± (x, ξ) if t → ∓∞. Therefore supposing that f ∈ S(Rd ) and integrating by parts we estimate integral (4.15) by CN (1 + |x| + |t|)−N for an arbitrary N . This proves (4.14). To prove (4.13) we apply the same arguments to the pdo with symbol 2 (x, ξ) − 1)ψ 2 (ξ 2 ). (ζ± Below we consider an interval 3 = (λ0 , λ1 ), where 0 < λ0 < λ1 < ∞, and choose a function ψ ∈ C0∞ (R+ ) such that ψ(λ) = 1 on 3. Proposition 4.5. The operators W± (H, H0 ; J± ) are isometric on the subspace E0 (3)H and ∗ ) = 0. (4.16) W± (H, H0 ; J∓ ) = 0, W± (H0 , H; J∓ Proof. The results on the operators W± (H, H0 ; J± ) and W± (H, H0 ; J∓ ) are immediate consequences of (4.13) and (4.14), respectively. The second equality (4.16) is a ∗ consequence of the first because W± (H0 , H; J∓ ) = W± (H, H0 ; J∓ )∗ . Now it is easy to prove completeness of the operators W± (H, H0 ; J± ). Theorem 4.6. Suppose that condition (1.1) is fulfilled for ρ > 1/2. Then the asymptotic completeness holds: Ran (W± (H, H0 ; J± )E0 (3)) = E(3)H.
(4.17)
Proof. We have to check that for any f ∈ E(3)H there exists f0 ∈ E0 (3)H such that lim ||e−iHt f − J± e−iH0 t f0 || = 0.
t→±∞
(4.18)
∗ ∗ ) shows that for f0 = W± (H0 , H; J± )f , The existence of the operator W± (H0 , H; J± ∗ −iHt lim ||J± e f − e−iH0 t f0 || = 0,
t→±∞
and hence
∗ −iHt e f − J± e−iH0 t f0 || = 0. lim ||J± J±
t→±∞
(4.19)
∗ −iHt e f || = 0 and, consequently, The second equality (4.16) implies that limt→±∞ ||J∓ ∗ −iHt e f || = 0. lim ||J∓ J∓
t→±∞
(4.20)
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∗ ∗ By Proposition 2.3, up to a compact term, J± J± + J∓ J∓ is the pdo with symbol 2 2 2 2 (ζ± (x, ξ) + ζ∓ (x, ξ))ψ (ξ ). Choosing the functions σ± (see (4.1)) in such a way that 2 (τ ) = 1, we deduce from (4.19), (4.20) that σ+2 (τ ) + σ−
lim ||ψ 2 (H0 )e−iHt f − J± e−iH0 t f0 || = 0.
t→±∞
Since the operator ψ 2 (H) − ψ 2 (H0 ) is compact, this yields (4.18).
∗ ) Remark. Equality (4.17) is equivalent to isometricity of the operator W± (H0 , H; J± on E(3)H.
3. Let us, finally, show that W± (H, H0 ; J± ) coincide with wave operators defined in terms of a time-dependent modification of e−iH0 t . Following [18], we choose a modification of this group in x-representation. First, we check Lemma 4.7. Let a family of unitary operators be defined by the equality (U0 (t)f )(x) = e∓πdi/4 eiΞ(x,t) (2|t|)−d/2 fˆ((2t)−1 x), where −1 2
Ξ(x, t) = (4t)
Z
±t > 0,
(4.21)
1
x −t
V (sx)ds.
(4.22)
0
Then for any function ζ± , lim ||J± (ψ, ζ± )e−iH0 t f − U0 (t)ψ(H0 )f || = 0,
t→±∞
∀f ∈ L2 (Rd ).
Proof. Consider the representation Z 2 −iH0 t −d/2 f )(x) = (2π) eiϕ± (x,ξ)−iξ t ζ± (x, ξ)ψ(ξ 2 )fˆ(ξ)dξ, (J± e Rd
f ∈ S(Rd ). (4.23)
Stationary points ξ± (x, t) of the phase function are determined by the equation x + (∇ξ 8± )(x, ξ± (x, t)) − 2ξ± (x, t)t = 0.
(4.24)
Due to the function ζ± (x, ξ)ψ(ξ 2 ) we are interested only in points ξ± (x, t) such that 2 ≤ C < ∞ and ±hξˆ± , xi ˆ ≥ ±κ for some κ ∈ (−1, 1). Using estimate 0 < c ≤ ξ± (3.11) on ∇ξ 8± we see that for large |t| Eq. (4.24) has a unique solution ξ± (x, t) and ξ± (x, t) = (2t)−1 x + (2t)−1 (∇ξ 8± )(x, (2t)−1 x) + O(|t|−2ρ ).
(4.25)
Applying the stationary phase method to integral (4.23) and taking into account that ζ± (x, ξ± ) = η(x) we find that 2 (x, t))fˆ(ξ± (x, t)), (J± e−iH0 t f )(x) ∼ e∓πid/4 eiΞ± (x,t) (2|t|)−d/2 ψ(ξ±
t → ±∞. (4.26) Here “ ∼00 means that the difference of the left- and right-hand sides tends to zero in L2 (Rd ) and 2 Ξ± (x, t) = hξ± (x, t), xi + 8± (x, ξ± (x, t)) − ξ± (x, t)t
= (4t)−1 x2 + 8± (x, ξ± (x, t)) − (4t)−1 |(∇ξ 8± )(x, ξ± (x, t))|2 ,
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D. Yafaev
where we have used Eq. (4.24). Formula (4.25) allows us to simplify this expression neglecting terms which in the case 2ρ > 1 tend to zero as t → ±∞: Ξ± (x, t) = (4t)−1 x2 + 8± (x, (2t)−1 x)). Using Eq. (3.8) we see that Ξ± (x, t) = Ξ(x, t) for ±t > 0. Finally, again by (4.25), we 2 ˆ )f (ξ± ) by ψ((2t)−2 x2 )fˆ((2t)−1 x) in (4.26). can replace ψ(ξ± Lemma 4.7 allows us to reformulate the results of this section as Proposition 4.8. Suppose that condition (1.1) is fulfilled for ρ > 1/2. Define U0 (t) by Eqs. (4.21), (4.22). Then the wave operators W± = s − lim eiHt U0 (t) t→±∞
exist, are isometric and Ran W± = E(R+ )H. Furthermore, for any ψ ∈ C0∞ (R+ \ {0}) and any function (4.1), W± ψ(H0 ) = W± (H, H0 ; J± (ψ, ζ± )). 4. Local singularities of a potential can be easily accommodated. Actually, suppose that V (x) satisfies condition (1.1) outside of some ball |x| ≤ r0 only. Let η0 ∈ C ∞ (Rd ) be such that η0 (x) = 0 for |x| ≤ r0 and η0 (x) = 1 for |x| ≥ R0 + 1. Set V0 (x) = η0 (x)V (x) and construct the phases 8± (x, ξ) by formula (3.8), where V is replaced by V0 . This gives an additional term (1 − η0 (x))V (x)ζ± (x, ξ) in the right-hand side of (4.6). However, it disappears if the function η(x) in (4.1) is chosen in such a way that η(x) = 0 for |x| ≤ r0 + 1. Then all proofs of this section work without any modification as long as singularities of V (x) inside the ball |x| ≤ r0 are not strong enough to violate estimates of Propositions 2.5 and 2.6. Quite similarly, all constructions are preserved if a potential is a sum V + Vs of a function V satisfying (1.1) and of a short-range term Vs (x) = O(|x|−ρs ), ρs > 1, as |x| → ∞. In this case an additional term Vs J± arising in (4.4) admits, according to Proposition 2.1, the factorization Vs J± = X −p B± X −p , p = ρs /2, with a bounded operator B± . 5. The Scattering Matrix 1. It follows from Theorems 4.3, 4.6 and Proposition 4.5 that the scattering operator S = W+∗ (H, H0 ; J+ )W− (H, H0 ; J− ) commutes with H0 and is unitary on the subspace E0 (3)H. Thus, in a diagonal representation of H0 it reduces to the operator of multiplication by the operator-function S(λ) : N → N called the scattering matrix. We consider the standard diagonal representation of H0 . Let the unitary operator U : H → L2 (R+ ; N) be defined by equalities (U f )(λ) = 00 (λ)f and (2.4). Since U H0 U ∗ acts in the space L2 (R+ ; N) as multiplication by the independent variable λ, the operator U SU ∗ acts as multiplication by the operator-function S(λ). It is unitary for λ ∈ 3.
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199
We need a stationary formula for the scattering matrix in the case when identifications J+ and J− for t → +∞ and t → −∞ are different. Let us introduce auxiliary wave operators (5.1) Ω± = s − lim eiH0 t J+∗ J− e−iH0 t . t→±∞
The operator Ω± commutes with H0 and hence U Ω± U ∗ acts in the space L2 (R+ ; N) as multiplication by the operator-function ± (λ) : N → N. The scattering matrix admits two representations which can be formally written as (5.2) S(λ) = + (λ) − 2πi00 (λ) J+∗ T− − T+∗ R(λ + i0)T− 0∗0 (λ) and
S(λ) = − (λ) − 2πi00 (λ) T+∗ J− − T+∗ R(λ + i0)T− 0∗0 (λ)
with T± defined by (4.4). Both representations can be given a precise meaning in the framework of the smooth scattering theory. For definiteness, we use below representation (5.2). Let us formulate a precise result assuming that H0 = −1, H = −1 + V with a relatively compact perturbation V and some bounded operators J± . Note, however, that Proposition 5.1 has, actually, a sense in the abstract framework. Proposition 5.1. Suppose that T ± = K ∗ B± K
for both signs and
˜ J+∗ T− = K ∗ BK,
(5.3)
where B± , B˜ are bounded operators in some auxiliary Hilbert space G and K : H → G 1/2 is H0 -bounded. Assume that U0 (λ; K) = K0∗0 (λ) : N → G are bounded operators strongly continuous in λ > 0. Let, finally, the operator-function R(z; K) = KR(z)K ∗ : G → G be weakly continuous with respect to the parameter z in the closed complex plane cut along [0, ∞), possibly with the exception of the point z = 0. Then the wave operators W± (H, H0 ; J), W± (H0 , H; J ∗ ) (where J = J+ or J = J− ) and (5.1) exist and the scattering matrix S(λ) admits the representation ˜ 0 (λ; K) + 2πiU0∗ (λ; K)B+∗ R(λ + i0; K)B− U0 (λ; K). S(λ) = + (λ) − 2πiU0∗ (λ; K)BU (5.4) In particular, the operator-function S(λ) − + (λ) is weakly continuous in λ > 0. Equality (5.4) is formally the same as (5.2) but in the right-hand side of (5.4) we have a combination of bounded operators. This gives a correct meaning to (5.2). Below we usually write representation (5.2) keeping in mind that its precise form is given by (5.4). A proof of Proposition 5.1 is practically the same as the proof of the corresponding assertion in [17] in the case J+ = J− . Therefore we shall give only a sketch of the proof concentrating on formula representations and omitting details of their justification. Under assumptions of Proposition 5.1 wave operators W± (J) = W± (H, H0 ; J) admit stationary representations Z ∞ lim (T R0 (λ ± iε)f0 , δε (H − λ)f )dλ (5.5) (W± (J)f0 , f ) = (Jf0 , f ) + −∞ ε→0
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D. Yafaev
and Z (E(3)W± (J)f0 , f ) =
lim (δε (H0 − λ)f0 , J ∗ g − T ∗ R(λ ± iε)f )dλ,
3 ε→0
(5.6)
where J = J+ or J = J− and T = HJ − JH0 . Representation (5.6) ensures that lim (δε (H − λ)W+ (J)f0 , f ) = lim ((J − R(λ − iε)T )δε (H0 − λ)f0 , f ).
ε→0
ε→0
(5.7)
It follows from (5.5) for J = J− and f = W+ (J+ )g0 that (W+ (J− )f0 − W− (J− )f0 , W+ (J+ )g0 ) Z ∞ lim (T− δε (H0 − λ)f0 , δε (H − λ)W+ (J+ )g0 )dλ. = 2πi −∞ ε→0
Equality (5.7) implies (at least formally) that δε (H − λ)W+ (J+ )g0 in the last integral may be replaced by (J+ − R(λ − iε)T+ )δε (H0 − λ)g0 so that for any f0 , g0 ∈ H Z = 2πi
∞
(W+ (J− )f0 − W− (J− )f0 , W+ (J+ )g0 ) lim ((J+∗ − T+∗ R(λ + iε))T− δε (H0 − λ)f0 , δε (H0 − λ)g0 )dλ
−∞ ε→0 Z ∞
= 2πi −∞
((J+∗ − T+∗ R(λ + i0))T− dE0 (λ)f0 /dλ, dE0 (λ)g0 /dλ)dλ.
Since dE0 (λ)/dλ = 0∗0 (λ)00 (λ), this is equivalent to representation (5.2). 2. Let us apply Proposition 5.1 to our triple H0 , H and J± . Thus, we assume that V is multiplication by a function satisfying (1.1) for ρ ∈ (1/2, 1) and J± is given by (4.2), ) where a± = a(N ± for some N . Note, first that according to (4.14) s − lim J− exp(−iH0 t) = 0 t→+∞
so that Ω+ = 0 and hence + (λ) = 0 for all λ > 0. The operator T± is determined by (N ) . Properties of the operator J+∗ T− are summarized in Eqs. (4.5), (4.6), where q± = q± the following Lemma 5.2. The operator J+∗ T− is a pdo with symbol a(x, ξ) = eiφ(x,ξ) wN (x, ξ) + uN (x, ξ), where φ(x, ξ) = 2−1
Z
∞ −∞
V (tξ) − V (x + tξ) dt,
(5.8)
ˆ ˆ + wN (x, ξ) ˆ ξi)hξ, ∇σ− (hx, ˆ ξi)i wN (x, ξ) = −2iη 2 (x)ψ 2 (ξ 2 )σ+ (hx,
(5.9) (5.10)
and −2ρ wN ∈ S1,0 ,
−1−(N +1)ε0 uN ∈ Sρ,1−ρ ,
ε0 = 2ρ − 1.
(5.11)
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201
Proof. Let us apply Proposition 2.2 to pdo A1 = T− and A2 = J+ with symbols t− (x, ξ) and j+ (x, ξ), respectively. It follows from (4.2), (4.5) that ∂xα ((∂ξα j+ )(x, ξ)t− (x, ξ)) = eiφ(x,ξ) vα (x, ξ). −1−ε0 |α|
Here φ(x, ξ) = 8− (x, ξ)−8+ (x, ξ) satisfies (5.9) according to (3.8) and vα ∈ S1,0 according to (3.11), (3.20), (3.21). In particular, ˆ ˆ v0 (x, ξ) = −2iη 2 (x)ψ 2 (ξ 2 )σ+ (hx, ˆ ξi)hξ, ∇σ− (hx, ˆ ξi)ia + (x, ξ)a− (x, ξ).
Equality (3.18) and estimate (3.20) allow us to replace a± by 1 here. This gives the representation (5.8)–(5.10) for a(x, ξ). ˆ 0 (x, ξ), where ˆ ξi)a Let A0 be pdo with symbol σ+ (hx, ˆ a0 (x, ξ) = −2ieiφ(x,ξ) η 2 (x)ψ 2 (ξ 2 )hξ, ∇σ− (hx, ˆ ξi)i.
(5.12)
It follows from Lemma 5.2 and Proposition 2.1 that J+∗ T− − A0 = X −ρ BX −ρ ,
(5.13)
where B is some bounded operator. Furthermore, by Lemma 4.1, the operator A0 admits representation (4.3) with p = ρ. Representations (4.3) for T± and J+∗ T− give factorizations (5.3) for these operators. Now the space G consists of several copies of H, K is a “vector” operator with components X −ρ , Gj , j = 1, . . . , d, and B± or B˜ are “matrix” operators constructed in terms of different operators B (r) and B (s) . Therefore weak continuity of R(z; K) and strong continuity of U0 (λ; K) are consequences of Propositions 2.5, 2.6 and 2.9, 2.10, respectively. Thus, all assumptions of Proposition 5.1 are fulfilled and we can reformulate it for our case. We also take into account that for unitary operator-functions the weak continuity implies the strong one. Theorem 5.3. Let condition (1.1) with ρ > 1/2 hold and let J± be defined by (4.2), ) where a± = a(N ± for some N . Then the scattering matrix S(λ) for the triple H0 , H and J± admits representation (5.2), where + (λ) = 0. In particular, S(λ) is strongly continuous in λ > 0. Remark. A representation of S(λ) to a large extent similar to (5.2) appeared first in [7]. However it seems to us that 00 (λ)J+∗ T− 0∗0 (λ) was not well defined there as a bounded operator in N. Indeed, its definition requires either Proposition 2.6 or 2.11 but assertions of such type were not used in [7]. 3. Now we can start our analysis of singularities of the scattering matrix. We may suppose that a± = 1 in (4.2) till the end of this section. Remark that the terms S1 (λ) = −2πi00 (λ)J+∗ T− 0∗0 (λ),
(5.14)
S2 (λ) = 2πi00 (λ)T+∗ R(λ + i0)T− 0∗0 (λ)
(5.15)
depend on the choice of the cut-off functions ζ± in definition (4.2) of the identifications J± , but their sum S(λ) = S1 (λ) + S2 (λ) does not depend on it. Below we always suppose that σ+ (τ ) = 1 for τ ∈ [−, 1] and σ− (τ ) = 1 for τ ∈ [−1, ], (5.16) where ∈ (0, 1). Then the term S2 (λ) is in some sense negligible.
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Lemma 5.4. With the above choice of J± the operator S2 (λ) is compact and normcontinuous in λ for λ > 0. Proof. Recall that T± = T±(r) + T±(s) , where T±(r) , T±(s) are pseudo-differential operators (s) with symbols t(r) ± , t± defined by (4.10). Let us write (5.15) as S2 (λ) = 2πi 00 (λ)X −p X p B(λ)X p X −p 0∗0 (λ) ,
(5.17)
where p ∈ (1/2, ρ], B(λ) = B1 (λ) + B2 (λ) + B3 (λ) : H → H
(5.18)
B1 (λ) = (T+(r) )∗ R(λ + i0)T−(r) ,
(5.19)
B2 (λ) = (T+(r) )∗ R(λ + i0)T−(s) + (T+(s) )∗ R(λ + i0)T−(r) ,
(5.20)
B3 (λ) = (T+(s) )∗ R(λ + i0)T−(s) .
(5.21)
and
In view of estimates (4.9) Proposition 2.1 implies that the operators X p T±(r) X p are bounded. Therefore, applying Proposition 2.5, we find that the operator X p B1 (λ)X p is bounded and norm-continuous in λ. According to (4.7), (4.10) the support of the ˆ ≤ − and, similarly, the support of the ˆ ξi symbol t(s) + (x, ξ) belongs to the cone hx, ˆ (x, ξ) belongs to the cone h x, ˆ ξi ≥ . Both functions t(s) symbol t(s) − ± (x, ξ) satisfy (2.2) with m = −1 and ρ > 1/2, δ = 1 − ρ. Therefore Proposition 2.7 can be applied to the operators X −p R(λ + i0)T−(s) X p , X p (T+(s) )∗ R(λ + i0)X −p , and Proposition 2.8 can be applied to the operator X p B3 (λ)X p . It follows that the operators X p Bj (λ)X p , j = 2, 3, and hence X p B(λ)X p are bounded and norm-continuous in λ. To conclude the proof we return to representation (5.17) and use that, by Proposition 2.9, the operator 00 (λ)X −p : H → N is compact and norm-continuous in λ. Let us now consider operator (5.14). Its genuinely non-compact part is determined by the operator A0 . Proposition 5.5. Let A0 be the pdo with symbol (5.12) and set S0 (λ) = −2πi00 (λ)A0 0∗0 (λ).
(5.22)
Then the operator S(λ) − S0 (λ) is compact and norm-continuous in λ for λ > 0. Proof. By virtue of (5.13), Proposition 2.9 implies that the operator 00 (λ)(J+∗ T− − A0 )0∗0 (λ) is compact and norm-continuous in λ > 0. Under assumption (5.16) the function ˆ xi) ˆ xi) σ+ (hξ, ˆ equals 1 on the support of ∇σ− (hξ, ˆ and hence it may be omitted in the symbol of A0 . So it remains to take Lemma 5.4 into account.
Scattering Amplitude for Schr¨odinger Equation with Long-Range Potential
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An obvious drawback of this assertion is that the operator A0 contains the cut-off function σ− although the scattering matrix does not depend on it. In the next part we shall show that the principal of the pdo (5.22) does not, actually, depend on this cut-off. ˆ ωi) = 0 in the cones hx, ˆ ωi ≥ κ for some κ ∈ (0, 1) and hx, ˆ ωi ≤ 0, 4. Since ∇σ− (hx, Proposition 2.11 can be applied to the operator A0 . Therefore the operator 00 (λ)A0 0∗0 (λ) (note that Proposition 2.11 gives also another proof of its existence) is a pdo on Sd−1 with the principal symbol Z
s(ω, b; λ) = (4πk)−1
∞ −∞
a0 (zω + k −1 b, kω)dz,
|ω| = 1,
hω, bi = 0.
(5.23)
To calculate this integral we remark that, by (5.9), φ(zω + k −1 b, kω) = 2−1 k −1 V(ω, k −1 b), where the function Z V(ω, b) =
∞ −∞
V (tω) − V (b + tω) dt,
|ω| = 1,
k = λ1/2 ,
hω, bi = 0,
(5.24)
(5.25)
ˆ xi ˆ = does not depend on z. Clearly, η(zω + k −1 b) = 1 for sufficiently large b and hξ, z(z 2 + k −2 b2 )−1/2 for ξ = kω, x = zω + k −1 b. Since σ− (1) = 0 and σ− (−1) = 1, we have that Z ∞ Z ∞ ˆ xi)idz hξ, ∇σ− (hξ, ˆ =k ∂σ− (z(z 2 + k −2 b2 )−1/2 )/∂z dz = −k. −∞
−∞
Taking into account that under assumption (5.16) the symbol of the pdo A0 equals a0 (x, ξ), ψ(λ) = 1 for λ considered and comparing (5.12), (5.23) and (5.24) we find that s(ω, b; λ) = exp i2−1 k −1 V(ω, k −1 b) ,
|ω| = 1,
hω, bi = 0.
(5.26)
This gives us Proposition 5.6. The operator S0 (λ) is a pdo on Sd−1 with the principal symbol 0 . s(ω, b; λ) defined by (5.25), (5.26). In particular, s ∈ Sρ,1−ρ Note that the operator S0 (λ) is determined by its principal symbol up to a compact term. Combining Propositions 5.5 and 5.6 we obtain Theorem 5.7. Let condition (1.1) with ρ > 1/2 hold. Then the scattering matrix S(λ) for the operators H0 , H, identifications J± and λ ∈ 3 admits the representation ˜ S(λ) = S0 (λ) + S(λ), where S0 (λ) is a pdo on Sd−1 with the principal symbol (5.25), (5.26) and the operator ˜ S(λ) is compact.
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According to (5.26), Proposition 2.12 can be applied to the operator S0 (λ) if we require that lim sup V(ω0 , tb0 ) = ∞
t→∞
or
lim inf V(ω0 , tb0 ) = −∞
t→∞
(5.27)
for some point ω0 , b0 6= 0, |ω0 | = 1, hω0 , b0 i = 0. Of course, this condition is satisfied for all points ω0 , b0 if V (x) is an asymptotically homogeneous function of order −ρ. Taking into account that the spectrum of a unitary operator belongs to the unit circle and that, by Weyl’s theorem, a compact operator can not change the essential spectrum we arrive at the following result. Theorem 5.8. Let condition (1.1) with ρ > 1/2 hold. Suppose that function (5.25) satisfies condition (5.27) for some point ω0 , b0 , |ω0 | = 1, hω0 , b0 i = 0. Then for all λ > 0 the spectrum of the scattering matrix S(λ) coincides with the unit circle. 5. Here we collect some additional remarks. 1. We do not have any information on the structure of the spectrum of the scattering matrix. Note, however, that in the radial case V (x) = V (|x|) it consists of eigenvalues. By Theorem 5.8, they are dense on the unit circle. 2. Let U be multiplication by a function exp(i8(ω)), where 8 ∈ C ∞ (Sd−1 ). Proposition 2.12 can be applied also to the operator U S0 (λ). It follows that, under assumptions of Theorem 5.8, the spectrum of S(λ) covers the unit circle for any choice of identifications J± . 3. Theorems 5.7 and 5.8 can be easily extended to more general classes of potentials discussed near the end of Sect. 4 provided the resolvent estimates of Propositions 2.5, 2.7 and 2.8 remain true. In particular, the spectrum of S(λ) coincides with the unit circle if a potential is a sum of a function obeying assumptions of Theorem 5.8 and of a short-range term. 4. If condition (1.1) is fulfilled for ρ > 1, then function (5.25) satisfies Z ∞ V (tω)dt. (5.28) V(ω, b) = c(ω) + O(|b|−ρ+1 ), where c(ω) = −∞
˜ Representation (5.2) remains, of course, true in this case and the operator S(λ) is compact. Conditions (5.27) and (5.28) are incompatible and S0 (λ) differs by a compact term from the operator of multiplication by exp(i2−1 k −1 c(ω)). On the other hand, in the case ρ > 1 one can set 8± = 0 (cf. the remark at the end of Sect. 3). Then representation (5.2) holds with + (λ) = I. In particular, one recovers the standard representation of the scattering matrix if a± = 1. 6. The Diagonal Singularity of the Scattering Amplitude 1. In this section we suppose that identifications J± = J± (a± ) are given by formula ) (4.2), where the function a± = a(N ± is constructed in the middle of Sect. 3 and N is sufficiently large. We proceed from the stationary representation of the scattering matrix S(λ) given by Theorem 5.3. Let us start with analysis of the regular part S2 (λ) = S2(N ) (λ) of the scattering matrix defined by formula (5.15). We shall show that S2 (λ) is an integral operator with smooth kernel s2 (ω, ω 0 ; λ) in variables ω, ω 0 ∈ Sd−1 and λ > 0. Actually,
Scattering Amplitude for Schr¨odinger Equation with Long-Range Potential
205
) the function s2 = s(N is getting more and more regular as N increases. The proof of 2 the following assertion is similar to that of Lemma 5.4. ) Lemma 6.1. Suppose that condition (1.1) with ρ ∈ (1/2, 1) holds. Let J± = J± (a(N ± ) and let condition (5.16) be satisfied. Then the operator S2(N ) (λ) is an integral operator ) ∈ C M (Sd−1 × Sd−1 × R+ ), where an integer M satisfies with kernel s(N 2
M < (2ρ − 1)(N + 1) − d + 1.
(6.1)
Proof. According to (5.15) and (2.4), (2.5), the kernel s2 (ω, ω 0 ; λ) is formally given by the formula s2 (ω, ω 0 ; λ) = 2−1 ik d−2 (2π)−d+1 (T+∗ R(λ+i0)T− ψ0 (kω 0 ), ψ0 (kω)),
k = λ1/2 , (6.2)
) where ψ0 (x, ξ) = exp(ihξ, xi) and the operators T± = T± (a(N ± ) are determined by Eqs. (4.5), (4.6). Formula (6.2) is automatically justified if its right-hand side is, say, a continuous function of ω, ω 0 ∈ Sd−1 and λ > 0. Actually, we shall check that this function belongs to the class C M . To that end we differentiate (6.2) and remark that 0 (∂ωα ∂ωα0 ∂λm s2 )(ω, ω 0 ; λ) consists of terms 0
(T+∗ Rn (λ + i0)T− ψ0(β ) (kω 0 ), ψ0(β) (kω)),
(6.3)
where ψ0(β) (x, ξ) = xβ ψ0 (x, ξ), |β| + |β 0 | + n = |α| + |α0 | + m ≤ M
(6.4)
and |β| ≥ |α|, |β 0 | ≥ |α0 |, n ≤ m. Clearly, X −|β|−p ψ0(β) (kω) ∈ L2 (Rd ) if p > d/2. Therefore we need to check boundedness of the operator X |β|+p T+∗ Rn (λ + i0)T− X |β
0
|+p
,
p > d/2.
(6.5)
We decompose again the function (4.6) into a sum (4.8) but define now the regular (r) (r) (x, ξ) by the equality τ± = q± ζ± . Then T± = T±(r) + T±(s) , where T±(r) , T±(s) are part τ± pdo with symbols (4.10). By estimates (3.11), (3.21), m t(r) ± ∈ Sρ,1−ρ , where m = −1 − ε0 (N + 1),
and
−1 t(s) ± ∈ Sρ,1−ρ ,
ε0 = 2ρ − 1,
ˆ ≤ . t(s) ˆ ξi ± (x, ξ) = 0 if ∓ hx,
Therefore the operator B(λ) = T+∗ Rn (λ + i0)T− satisfies equalities (5.18) - (5.21) with 0 R(λ + i0) replaced by Rn (λ + i0). Set Gj (λ) = X |β|+p Bj (λ)X |β |+p , j = 1, 2, 3. By Proposition 2.5, the operator G1 is bounded if n + p + max{|β|, |β 0 |} < ε0 (N + 1) + 3/2.
(6.6)
By virtue of (6.4) this condition is, of course, satisfied under assumption (6.1). By Proposition 2.7, the operator G2 is bounded if, additionally to (6.6), n + 2p + |β| + |β 0 | < ε0 (N + 1) + 2. This condition is again satisfied under assumption (6.1). Finally, by Proposition 2.8, the operator G3 is bounded for all n, p and β, β 0 .
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D. Yafaev
2. Let us proceed to the analysis of the singular part S1 (λ) of the scattering matrix. Recall that S1 (λ) is defined by equality (5.14) and J+∗ T− is a pdo with symbol a(x, ξ), given −1 so that the kernel s1 (ω, ω 0 ; λ) of the operator by (5.8) - (5.10). In particular, a ∈ Sρ,1−ρ S1 (λ) is determined by formula (2.6): s1 (ω, ω 0 ; λ) = −i2−1 (2π)−d+1 k d−2
Z
0
eikhω −ω,xi a(x, kω 0 )dx,
k = λ1/2 ,
Rd
ω 6= ω 0 .
(6.7) Thus, s1 (ω, ω 0 ; λ) is an infinitely differentiable function of ω, ω 0 for ω 6= ω 0 and of λ > 0. Using now Lemma 6.1 and taking into account that M is arbitrary there, we arrive at the following assertion. Proposition 6.2. Under assumption (1.1), where ρ > 1/2, the kernel s(ω, ω 0 ; λ) of the scattering matrix S(λ) is an infinitely differentiable function of ω, ω 0 for ω 6= ω 0 and of λ > 0. By virtue of Lemma 6.1, formula (6.7) determines the singular part of s(ω, ω 0 ; λ). Moreover, the second inclusion (5.11) implies that the term uN in the right-hand side of (5.8) is negligible. Let us formulate the precise result. Proposition 6.3. Let assumption (1.1) with ρ ∈ (1/2, 1) hold. Set s0 (ω, ω 0 ; λ) = −i2−1 (2π)−d+1 k d−2
Z
0
0
eikhω −ω,xi eiφ(x,kω ) w(x, kω 0 )dx,
(6.8)
Rd
where φ and w = wN are defined by (5.9) and (5.10), respectively. Then s − s0 ∈ C M (Sd−1 × Sd−1 × R+ ), where M is the same as in Lemma 6.1. Remark. The function w in (6.8) is constructed in terms of approximate solutions a± = ) (N ) contains a(N ± of the transport equation and hence it is quite explicit. Thus s0 = s0 all diagonal singularities of the scattering matrix. This result remains meaningful in the case ρ > 1 when one can set 8± = 0. Our goal is to find explicitly the leading singularity of the kernel s(ω, ω 0 ; λ) at the diagonal ω = ω 0 . Below we fix the vector ω 0 =: ω0 and study s0 (ω, ω0 ; λ) as ω → ω0 . Set z = hω0 , xi and b = x − zω0 (so that b belongs to the hyperplane 3ω0 orthogonal to ω0 ). It is convenient to introduce the vector ϑ = ω − hω, ω0 iω0 ∈ 3ω0 .
(6.9)
Clearly, hω0 − ω, bi = −hϑ, bi,
hω0 − ω, ω0 i = 1 − (1 − |ϑ|2 )1/2 =: f (ϑ) = O(|ϑ|2 ) (6.10)
as ϑ → 0. Dependence of different expressions on ω0 and k will be often omitted. Let us formulate an intermediary result.
Scattering Amplitude for Schr¨odinger Equation with Long-Range Potential
Lemma 6.4. Function (6.8) admits the representation Z −1 −1 −d+1 d−1 k e−ikhϑ,bi ei2 k V(ω0 ,b) p(b, ϑ)db, s0 (ω, ω0 ; λ) = (2π)
207
(6.11)
Rd−1
where V is given by (5.25) and p(b, ϑ) = −i2−1 k −1
Z
∞ −∞
w(zω0 + b, kω0 )eikf (ϑ)z dz.
(6.12)
Moreover, p(b, ϑ) = P0 (f (ϑ)b) + p1 (b, ϑ),
(6.13)
where P0 ∈ S(R), P (0) = 1, the function p1 (b, ϑ) is infinitely differentiable with respect to b and (6.14) |∂bα p1 (b, ϑ)| ≤ Cα (1 + |b|)−ε0 −|α| , ε0 = 2ρ − 1 ∈ (0, 1), with constants Cα not depending on ϑ. Proof. Equalities (6.11), (6.12) follow directly from (5.24) and the first equality (6.10). Consider function (5.10) for x = zω0 + b, ξ = kω0 . Remark that ψ(ξ 2 ) = 1 for λ ˆ = 1 on the support of ˆ ξi) considered, under assumption (5.16) the function σ+ (hx, ˆ and ∇σ− (hx, ˆ ξi) ˆ = −2ik∂σ− (z(z 2 + b2 )−1/2 )/∂z. ˆ ξi)i − 2ihξ, ∇σ− (hx,
(6.15)
Let p0 (b, ϑ) be defined by integral (6.12), where w(zω0 + b, kω0 ) is replaced by function (6.15). Changing the variable of integration z by z|b|, we see that p0 (b, ϑ) = P0 (f (ϑ)b), where Z ∞ P0 (t) = −
eiktz ∂σ− (z(z 2 + 1)−1/2 )/∂z dz.
−∞
−2ρ According to (5.10), estimates (6.14) on p1 = p−p0 follow from the inclusion w ∈ S1,0 .
3. Assume now for simplicity that V (x) is a homogeneous function for sufficiently large |x|: −ρ ˆ , v ∈ C ∞ (Sd−1 ), |x| ≥ r0 . (6.16) V (x) = V0 (x) := v(x)|x| Lemma 6.5. If (6.16) is fulfilled for some ρ < 1, then, up to some constant ν(ω), function (5.25) is homogeneous of degree 1 − ρ for sufficiently large |b|: ˆ 1−ρ + ν(ω), V(ω, b) = v(ω, b)|b| Z
where
∞
ˆ = v(ω, b)
−∞
Z
and
r
ν(ω) = −r
does not depend on b (and r).
V0 (tω) − V0 (bˆ + tω) dt
(6.17)
V (tω) − V0 (tω) dt,
|b| ≥ r0 ,
r ≥ r0 ,
(6.18)
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D. Yafaev
Proof. It suffices to make a change of variables s = |b|t in the representation Z ∞ V0 (sω) − V0 (b + sω) ds + ν(ω), |b| ≥ r0 . V(ω, b) = −∞
Since integral (6.11) over compact domain is bounded uniformly in ϑ, we may suppose that (6.17) holds for all b. Let us set γ = ρ−1 and make the change of variables b = λ−γ |ϑ|−γ y in (6.11). Using also (6.13), we see that Z ˆ −(d−1)γ 2γ−1 −d+1 (2πk ) µ(ω0 , k) eiLψ(ω0 ,ϑ,y) q(y, ϑ)dy, (6.19) s0 (ω, ω0 ; λ) = |ϑ| Rd−1
where µ(ω0 , k) = exp i(2k)−1 ν(ω0 ) , L = k 1−2γ |ϑ|1−γ , q(y, ϑ) = P0 (λ−γ |ϑ|−γ f (ϑ)y) + p1 (λ−γ |ϑ|−γ y, ϑ),
(6.20)
ˆ y) = −hϑ, ˆ yi + 2−1 v(ω0 , y)|y| ψ(ω0 , ϑ, ˆ 1−ρ .
(6.21)
It is important that L → ∞ as ϑ → 0 if ρ < 1. Therefore the asymptotics of integral ˆ y) = 0, or, according to (6.19) is determined by stationary points y, where ∇y ψ(ω0 , ϑ, (6.18), Z ∞ ˆ (∇V0 )(y + tω0 )dt = 0, y ∈ 3ω0 . (6.22) 2ϑ + −∞
Let us denote by H(ω0 , y) the Hessian of the function V(ω0 , y), i.e. H(ω0 , y) is the (d − 1) × (d − 1) - matrix with elements Z ∞ ∂ 2 V0 (y + tω0 )/∂yi ∂yj dt, y ∈ 3ω0 . Hij (ω0 , y) = − −∞
h(ω0 , y) = | det H(ω0 , y)|−1/2 exp iπ sgn H(ω0 , y)/4 .
Set also
(6.23)
ˆ y) equals 2−1 H(ω0 , y). By (6.21), the Hessian of ψ(ω0 , ϑ, ˆ there is a unique point Lemma 6.6. Let ρ ∈ (1/2, 1). Suppose that, for a given ϑ, ˆ 6= 0 satisfying (6.22) and that det H(y(ϑ)) ˆ 6= 0. Then y(ϑ) Z ˆ ˆ ϑ)) ˆ ˆ iLψ(ϑ,y( eiLψ(ϑ,y) q(y, ϑ)dy = (4π)(d−1)/2 L−(d−1)/2 h(y(ϑ))e (1 + O(|ϑ|ε )) Rd−1
(6.24)
for some ε > 0 as ϑ → 0.
Proof. According to Lemma 6.4, function (6.20) is bounded in a neighbourhood of ˆ uniformly in ϑ and its derivatives in y are even bounded by C|ϑ|2−γ . a point y(ϑ) Therefore, applying the stationary phase method we find that integral (6.24) over a ˆ equals neighbourhood of the point y(ϑ) ˆ iLψ(ϑ,y(ϑ)) q(y(ϑ), ˆ ϑ)(1 + O(L−1 )). (4π)(d−1)/2 L−(d−1)/2 h(y(ϑ))e ˆ
ˆ
(6.25)
ˆ ϑ) = 1 + O(|ϑ|2−γ ) and hence (6.25) equals the According again to Lemma 6.4, q(y(ϑ), right-hand side of (6.24).
Scattering Amplitude for Schr¨odinger Equation with Long-Range Potential
209
ˆ can be estimated by direct inteIntegral (6.24) outside of a neighbourhood of y(ϑ) gration by parts (m times). Hereby one should take into account that the point y = 0 is singular for function (6.21) and use the bounds |∇ψ(y)| ≥ c max{1, |y|−ρ }
and
|∂ α ψ(y)| ≤ C|y|1−ρ−|α| ,
|α| ≥ 2.
Together with the estimate on q(y, ϑ) used above this shows that the integral considered is bounded by L−m , which is negligible if m > (d − 1)/2. Of course, Lemma 6.6 can be extended to the case of a finite number of stationary points. Combine now the results obtained. Theorem 6.7. Let assumptions (1.1) and (6.16) with ρ ∈ (1/2, 1) hold. Fix k > 0, ω0 ∈ Sd−1 and let ω and ϑ be related by (6.9). Suppose that for a given ϑˆ there is a finite ˆ . . . , yn (ϑ) ˆ satisfying (6.22) and that det H(ω0 , yj (ϑ)) ˆ 6= 0 for all number of points y1 (ϑ), j = 1, . . . , n. Define the functions ψ and h by equalities (6.21) and (6.23), respectively. Then the kernel of the scattering matrix admits as ω → ω0 or, equivalently, ϑ → 0 the representation s(ω, ω0 ; λ) = (πk 2γ−1 )−(d−1)/2 |ϑ|−(d−1)(1+γ)/2 µ(ω0 , k) n X ˆ exp ik 1−2γ |ϑ|1−γ ψ(ω0 , ϑ, ˆ yj (ϑ)) ˆ (1 + O(|ϑ|ε )), h(ω0 , yj (ϑ)) ×
(6.26)
j=1
where γ = ρ−1 , ε = ε(ρ) > 0. Remark. It is possible, of course, that for some ϑˆ there are no points y satisfying (6.22). In this case s0 (ω, ω0 ; λ) → 0 as ϑ → 0 quicker than any power of |ϑ| so that the kernel s(ω, ω0 ; λ) of the scattering matrix remains bounded. 4. The case ρ = 1 is essentially different. Lemma 6.8. If (6.16) is fulfilled for ρ = 1, then function (5.25) admits the representation ˆ + ν1 (ω), V(ω, b) = v(ω) ln |b| + ν(ω, b) where v(ω) = v(ω) + v(−ω), Z Z ˆ ˆ V0 (sω) − V0 (b + sω) ds − ν(ω, b) = |s|≥1
Z ν1 (ω) =
r −r
|s|≤1
V (tω)dt − v(ω) ln r,
|b| ≥ r0 ,
(6.27)
V0 (bˆ + sω)ds,
(6.28)
r ≥ r0 .
Proof. Replacing in (5.25) the potential V by its asymptotics V0 for |b| ≥ r or |t| ≥ r we see that Z r Z Z r V0 (tω) − V0 (b + tω) dt. (6.29) V (tω)dt − V0 (b + tω)dt + V(ω, b) = −r
−r
|t|≥r
Changing in the integral over |t| ≥ r the variable t = |b|s, we rewrite it as
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D. Yafaev
Z
Z V0 (sω)ds +
|s|≤r/|b|
r/|b|≤|s|≤1
ˆ V0 (bˆ + sω)ds + ν(ω, b),
(6.30)
ˆ is defined by (6.28). The first integral in (6.30) equals v(ω) ln(|b|/r). The where ν(ω, b) second integrals in (6.29) and (6.30) cancel each other. Therefore substituting (6.30) into (6.29) we arrive at (6.27). We emphasize that, in contrast to (6.17), the coefficient v(ω) in (6.27) does not ˆ depends on it. Note also that the expression depend on bˆ and, on the contrary, ν(ω, b) for ν1 (ω) does not depend on the parameter r ≥ r0 . Using (6.27) and changing the variable b = |θ|−1 y we rewrite (6.11) as Z ˆ −d+1 d−1 −d+1−iv(ω0 )/2k k µ1 |ϑ| eiΞ(y,ϑ,ω0 ,k) q(y, ϑ)dy, (6.31) s0 (ω, ω0 ; λ) = (2π)
Rd−1
where µ1 (ω0 , k) = exp i(2k)−1 ν1 (ω0 ) , ˆ yi + (2k)−1 v(ω0 ) ln |y| + (2k)−1 ν(ω0 , y) ˆ ω0 , k) = −khϑ, ˆ Ξ(y, ϑ,
(6.32)
and q(y, ϑ) = p(|ϑ|−1 y, ϑ). Let us show that the function q(y, ϑ) in integral (6.31) can be replaced by 1 in the limit ϑ → 0. Lemma 6.9. In the case ρ = 1 for any < 1 Z ˆ eiΞ(y,ϑ,ω0 ,k) (q(y, ϑ) − 1)dy = O(|ϑ| )
as
Rd−1
|ϑ| → 0.
Proof. Let η ∈ C ∞ be such that η(y) = 0 for |y| ≤ 1 and η(y) = 1 for |y| ≥ 2. We consider separately the integrals Z ˆ eiΞ(y,ϑ,ω0 ,k) (q(y, ϑ) − 1)(1 − η(y))dy I1 (ϑ) = Rd−1
Z
and I2 (ϑ) =
e−ikhϑ,yi r(y, ϑ)η(y)dy, ˆ
(6.33)
Rd−1
where r(y, ϑ) = G(y)(q(y, ϑ) − 1),
G(y) = |y|i(2k)
−1
v i(2k)−1 ν(y) ˆ
e
.
Below we use the bounds on p formulated in Lemma 6.4; note that (6.14) holds with any ε0 < 1. Let us estimate q(y, ϑ) − 1 = (P0 (f0 (ϑ)y) − 1) + p1 (|ϑ|−1 y, ϑ),
(6.34)
where f0 (ϑ) = |ϑ|−1 f (ϑ) = O(|ϑ|) as ϑ → 0. The first term in the right-hand side is bounded by C|ϑ||y| and the second - by (1 + |ϑ|−1 |y|)− . Therefore Z |q(y, ϑ) − 1|dy ≤ C1 |ϑ| . |I1 (ϑ)| ≤ C |y|≤2
To estimate integral (6.33) we integrate m times by parts. This gives
Scattering Amplitude for Schr¨odinger Equation with Long-Range Potential
|I2 (ϑ)| ≤ C
X |α|+|β|=m
211
Z Rd−1
|∂ α r(y, ϑ)| |∂ β η(y)|dy.
(6.35)
Now we need estimates on derivatives of the function (6.34) for |y| ≥ 1. Clearly, |∂ α (P0 (f0 (ϑ)y) − 1)| = |f0 (ϑ)|α| P0(α) (f0 (ϑ)y)| ≤ C|ϑ| |y|−|α|+1
(6.36)
because |s||α|−1 P0(α) (s), |α| ≥ 1, is a bounded function. Inequality (6.36) holds also for α = 0. According to (6.14), −1 −|α|− , |∂yα p1 (|ϑ|−1 y, ϑ)| = |ϑ|−|α| |p(α) 1 (|ϑ| y, ϑ)| ≤ C|ϑ| |y|
|y| ≥ 1.
It follows that ∂ α (q(y, ϑ) − 1) is bounded by C|ϑ| |y|−|α|+1 for |y| ≥ 1. Since ∂ α G(y) = O(|y|−|α| ), the same bound holds for the function r(y, ϑ): |∂ α r(y, ϑ)| ≤ C|ϑ| |y|−|α|+1 ,
|y| ≥ 1,
∀α.
Choosing m = d + 1, we estimate the right-hand side of (6.35) by C|ϑ| .
Therefore representation (6.31) combined with Proposition 6.3 implies Theorem 6.10. Let assumptions (1.1) and (6.16) with ρ = 1 hold. Fix k > 0, ω0 ∈ Sd−1 and let ω and ϑ be related by (6.9). Set v(ω0 ) = v(ω0 ) + v(−ω0 ) and Z ˆ −d+1 d−1 ˆ k eiΞ(y,ϑ,ω0 ,k) dy, (6.37) c(ϑ, ω0 , k) = µ1 (ω0 , k)(2π) Rd−1
where the function Ξ is defined by (6.28) and (6.32). Then the kernel of the scattering matrix admits as ω → ω0 or, equivalently, ϑ → 0 the representation ˆ ω0 , k)|ϑ|−d+1−iv(ω0 )/2k (1 + O(|ϑ| )), s(ω, ω0 ; λ) = c(ϑ,
∀ < 1.
(6.38)
5. Let us make some comments on Theorems 6.7 and 6.10. 1. In formulas (6.26), (6.38) factors µ and µ1 which do not depend on ϑ and have modulus 1 are inessential because they can be changed by a choice of the operator J. 2. Condition (6.16) on V can be replaced by conditions (6.17) or (6.27) on V. Actually, for validity of (6.26) or (6.38) at some point ω0 it suffices to require (6.17) or (6.27) at this point ω0 only. 3. Theorems 6.7 and 6.10 can be extended to the case −ρ + V1 (x), V (x) = v(x)|x| ˆ
where V1 (x) satisfies (1.1) for some ρ1 > ρ (and |x| ≥ 1). 4. Local singularities of V are treated automatically as long as resolvent estimates of Propositions 2.5, 2.7 and 2.8 are preserved.
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5. If a potential is a sum V + Vs of a function V satisfying (1.1) and of a short-range function Vs (x) = O(|x|−ρs ) with ρs > d as |x| → ∞, then the additional term Vs (x)ζ± (x, ξ) appears in (4.6). Therefore we can not make (see Lemma 6.1) the ) 0 regular part s(N 2 (ω, ω ; λ) of the scattering amplitude as smooth as we want. To prove that this is a continuous function of ω, ω 0 ∈ Sd−1 (and λ > 0) we need to check boundedness of operator (6.5) for β = β 0 = 0 and n = 1. This requires the results of Proposition 2.7 for n = 1, m = 0 and some p > d/2 and the results of Proposition 2.8 for n = 1 and some p > d/2−1. We expect that this is true although we have not found the proper reference in the literature. Under the assumption ρs > d the contribution of Vs to the singular part (5.14) is also a continuous function of ω, ω 0 ∈ Sd−1 (and λ > 0). Thus, the results of Theorems 6.7 and 6.10 remain true. 7. Examples. Classical mechanics 1. Let us consider several examples. We start with asymptotically central potentials when a leading term of s(ω, ω0 ) as ω → ω0 is a function of |ϑ| only. Note that Z ∞ (1+t2 )−(ρ+2)/2 dt = 2ρ−1 I(ρ), where I(ρ) = π 1/2 0((1+ρ)/2)0(ρ/2)−1 (7.1) −∞
and 0 is the Gamma-function. Example 7.1. Suppose that v(x) ˆ = v = const in (6.16) and ρ ∈ (1/2, 1). Then as ϑ → 0, s(ω, ω0 ; λ) = µ0 w0 |ϑ|−(d−1)(1+γ)/2 eiψ0 |ϑ|
1−γ
(1 + O(|ϑ|ε ),
γ = ρ−1 , ε > 0,
(7.2)
where µ does not depend on ϑ, |µ| = 1, w0 = (2π)−(d−1)/2 ρ−1/2 k (1−2γ)(d−1)/2 (I(ρ)|v|)γ(d−1)/2 , ψ0 = ρ(1 − ρ)−1 k 1−2γ (I(ρ)|v|)γ sgn v. In particular, the scattering cross-section (1.5) :
6(ω, ω0 ; λ) = ρ−1 (I(ρ)|v|λ−1 )γ(d−1) |ϑ|−(d−1)(1+γ) 1 + O(|ϑ|ε ) .
(7.3)
Indeed, using (7.1), we obtain for function (5.25) and large |b| Z ∞ Z ∞ (∇V)(b) = − (∇V0 )(b + tω)dt = ρvb (b2 + t2 )−(ρ+2)/2 dt = 2vI(ρ)b|b|−ρ−1 . −∞
−∞
Therefore the coefficient v in (6.17) does not depend on ω and bˆ and it equals v = 2(1 − ρ)−1 I(ρ)v.
(7.4)
ˆ for any ϑ: ˆ Equation (6.22) reads now as I(ρ)v|y|−ρ yˆ = ϑˆ and has a unique solution y(ϑ) ˆ = Y ϑˆ sgn v, y(ϑ)
Y = (I(ρ)|v|)γ .
(7.5)
Calculating the Hessian of the function |y|1−ρ , y ∈ Rd−1 , we find that function (6.23) equals
Scattering Amplitude for Schr¨odinger Equation with Long-Range Potential
where µ(0)
213
h(y) = µ0 (1 − ρ)−(d−1)/2 ρ−1/2 (|v|−1 |y|ρ+1 )(d−1)/2 , = exp iπ(d − 3)sgn v/4 . It follows from (7.4), (7.5) that ˆ = µ0 2−(d−1)/2 ρ−1/2 (I(ρ)|v|)γ(d−1)/2 . h(y(ϑ))
Similarly, we obtain for function (6.21) ˆ y(ϑ)) ˆ = −2hϑ, ˆ y(ϑ)i ˆ + vY 1−ρ = 2ρ(1 − ρ)−1 (I(ρ)|v|)γ sgn v. ψ(ϑ, Thus, formula (7.2) with µ0 = µ(0) µ is a consequence of (6.26). Consider now asymptotically Coulomb potentials. Example 7.2. Suppose that V (x) = v|x|−1 for sufficiently large |x|. Then formula (6.38) holds with v(ω0 ) = 2v and c = µ0 π −(d−1)/2 0((d − 1)/2 + iv/2k)0(−iv/2k)−1 ,
(7.6)
where µ0 does not depend on ϑ and |µ| = 1. In particular, the scattering cross-section 6(ω, ω0 ; λ) = 2d−1 λ−(d−1)/2 |0((d − 1)/2 + iv/2k)0(−iv/2k)−1 |2 |ϑ|−2d+2 (1 + O(|ϑ| )) (7.7) for any < 1 as ϑ → 0. Indeed, under the assumptions above equality (6.27) is fulfilled with v(ω) = 2v and ˆ By (6.32), integral (6.37) reduces to the coefficient ν(ω), which does not depend on b. calculated explicitly the Fourier transform of the function |y|iv/k and can be in terms of −1 the Gamma-function. This gives (7.6) with µ0 = µ1 exp −ivk ln(k/2) . In the case d = 3 (7.7) implies that 6(ω, ω0 ; λ) = v 2 λ−2 |ϑ|−4 (1 + O(|ϑ| )),
ϑ → 0.
(7.8)
Thus we recover (in the limit ϑ → 0) the celebrated formula of Gordon and Mott. Let us compare formulas (7.3) for ρ < 1 and (7.7) for ρ = 1. The right-hand side of (7.3) is well-defined also for ρ = 1 and contains the same power |ϑ|−2d+2 as the right-hand side of (7.7). On the other hand, the asymptotic coefficient (|v|λ−1 )d−1 in (7.3) for ρ = 1 coincides with that in (7.7) in the case d = 3 only. 2. Let us also consider an example of an essentially non-central potential. We choose ˆ = −hx, ˆ ni in a potential of a dypole type. Let n ∈ Rd be some given vector and v(x) (6.16). We compute all quantities in the right-hand side of (6.26) for this case. Calculating ˆ in (6.17) integral (6.18) and using (7.1) we find that the asymptotic coefficient v(ω0 , b) equals ˆ = πI(ρ)−1 hn, bi, ˆ b ∈ 3 ω0 . (7.9) v(ω0 , b) It is convenient to replace n here by its orthogonal projection m = m(ω0 ) = n−hn, ω0 iω0 on the hyperplane 3ω0 . We assume that m 6= 0. It follows from (7.9) that ˆ b)|b| ˆ −ρ . (∇b V)(ω0 , b) = πI(ρ)−1 (m − ρhm, bi ˆ y) = 0 for function (6.21) reads as The equation ∇y ψ(ϑ, ˆ m ˆ − ρhm, ˆ yi ˆ yˆ = M −1 |y|ρ ϑ,
where
M = π|m|(2I(ρ))−1 .
(7.10)
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Considering the scalar product of (7.10) with m ˆ we see that this equation may have ˆ ˆ solutions y = y(ϑ) only if p := hm, ˆ ϑi > 0. We seek the angular part yˆ of y in the form ˆ yˆ = αm ˆ + β ϑ.
(7.11)
Comparing coefficients at m ˆ in (7.10) we obtain that α and β are connected by the relation αρ(α + βp) = 1. (7.12) Solving this equation together with the normalization condition α2 + β 2 + 2αβp = 1 we get α4 − (q + 2γ)α2 + (q + 1)γ 2 = 0, q = p2 (1 − p2 )−1 , γ = ρ−1 , so that
1/2 α2 = 2−1 q + γ ± q 1/2 4−1 q + γ − γ 2 .
(7.13)
Thus, Eq. (7.10) has solutions only if q ≥ 4(γ 2 − γ), which gives the condition ˆ ≥ 2(1 − ρ)1/2 (2 − ρ)−1 . hm, ˆ ϑi
(7.14)
Under this assumption the right-hand side of (7.13) is non-negative for both signs so that there are four numbers ±αj , j = 1, 2, (we suppose that αj ≥ 0 and α1 corresponds to the sign “+00 ) satisfying (7.13). Now we obtain ±βj from (7.12) and then determine ±yˆj , j = 1, 2, by (7.11). To find |yj | we remark that, according to (7.12), ˆ − ρhm, ˆ = p − ρ(αj + βj p)(αj p + βj ) = −βj α−1 = p−1 (1 − γα−2 ). hm, ˆ ϑi ˆ yˆj ihyˆj , ϑi j j Hence it follows from Eq. (7.10) that |yj | = M γ p−γ (1 − γαj−2 )γ . This determines the ˆ := ψ(ϑ, ˆ yj ) = −ψ(ϑ, ˆ −yj ) : phase ψj (ϑ) ˆ = −hyj , ϑi ˆ + 2−1 πI(ρ)−1 hm, yj i|yj |−ρ . ψj (ϑ)
(7.15)
It remains only to find the Hessian H(ω0 , y) of the function V(ω0 , y) = πI(ρ)−1 hm, yi|y|−ρ ,
y ∈ 3 ω0 .
Suppose for simplicity that d = 3 and choose in 3ω0 ' R2 a system of orthogonal coordinates with the first axis directed along m. Denote by y (1) , y (2) the corresponding coordinates of y ∈ R2 . Then H(ω0 , y) = πρI(ρ)−1 |m||y|−1−ρ h(y), where h(y) is a 2 × 2 - matrix with elements h11 (y) = |y|−3 ((ρ − 1)(y (1) )3 − 3y (1) (y (2) )2 ), h22 (y) = |y|−3 ((ρ + 1)y (1) (y (2) )2 − (y (1) )3 ), h12 (y) = h21 (y) = |y|−3 ((ρ + 1)(y (1) )2 y (2) − (y (2) )3 ), Let us now calculate h(yj ). By (7.12), |y|−1 yj(1) = hyˆj , mi ˆ = αj + βj p = ρ−1 αj−1 =: tj , and hence
|y|−1 yj(2) = (1 − t2j )1/2 ,
Scattering Amplitude for Schr¨odinger Equation with Long-Range Potential
215
h11 (yj ) = (ρ + 2)t3j − 3tj ,
h22 (yj ) = −(ρ + 2)t3j + (ρ + 1)tj , h12 (yj ) = h21 (yj ) = (1 − t2j )1/2 (ρ + 2)t2j − 1 .
Therefore tr h(yj ) = (1 − 2γ)αj−1 < 0 and det h(yj ) = (2 − ρ)ρ−2 αj−2 − 1. It follows from (7.13) that det h(y1 ) < 0 and det h(y2 ) > 0. Taking into account the equality H(ω0 , y) = −H(ω0 , −y), we see that sgn H(ω0 , ±y1 ) = 0, sgn H(ω0 , ±y2 ) = ∓2. Let us formulate the result obtained. Example 7.3. Let V be given by (6.16), where v(x) ˆ = −hx, ˆ ni, ρ ∈ (1/2, 1), and d = 3. Then s(ω, ω0 ; λ) remains bounded if ϑ → 0 in the cone ˆ mi hϑ, ˆ < 2(1 − ρ)1/2 (2 − ρ)−1 . If, on the contrary, the strict inequality (7.14) holds, then as ϑ → 0, ˆ ϑi−1−γ |m|γ k 1−2γ π γ−1 ρ−1 (2I(ρ))−γ s(ω, ω0 ; λ) = µhm, 1−γ 1−γ ˆ cos(k 1−2γ ψ1 (ϑ)|ϑ| ˆ ˆ sin(k 1−2γ ψ2 (ϑ)|ϑ| ˆ × w1 (ϑ) ) + w2 (ϑ) ) (1 + O(|ϑ|ε )), ˆ are defined by (7.15) and where ε > 0, µ does not depend on ϑ, |µ| = 1, ψj (ϑ) wj = [(2 − ρ)ρ−2 αj−2 − 1|−1/2 (1 − ρ−1 αj−2 )1+γ ,
ˆ αj = αj (ϑ).
Let us also write down a final answer in the case d = 2 when 3ω0 is a line. Example 7.4. Let V be given by (6.16), where v(x) ˆ = −hx, ˆ ni, ρ ∈ (1/2, 1), and d = 2. Then s(ω, ω0 ; λ) remains bounded if ϑ → 0 from the direction of −m and s(ω, ω0 ; λ) = µw0 |ϑ|−(1+γ)/2 cos(k 1−2γ ψ0 |ϑ|1−γ −π/4)(1+O(|ϑ|ε )),
ε > 0, (7.16)
as ϑ → 0 from the direction of m. Here µ does not depend on ϑ, |µ| = 1, w0 = π −1/2+γ/2 2−γ/2+1/2 ρ−1/2 (1 − ρ)γ/2 k −γ+1/2 (I(ρ)−1 |m|)γ/2 , ψ0 = ρ(1 − ρ)γ−1 (2I(ρ))−γ (π|m|)γ .
(7.17) (7.18)
Indeed, Eq. (7.10) reduces now to (1 − ρ)m ˆ = M −1 |y|ρ ϑˆ and has solutions if and only if ϑˆ = m. ˆ In this case there are two solutions ±y0 determined by y0 = (1 − ρ)γ (2I(ρ))−γ (π|m|)γ . ˆ ±y0 ) = ±ψ0 with ˆ ±y0 ) = ±ρ(1 − ρ)−1 y0 , so that ψ(ϑ, By virtue of (7.15), ψ(ϑ, ψ0 defined by (7.18). In the case 3ω0 ' R the Hessian of the function V(ω0 , y) = ±πI(ρ)−1 |m||y|1−ρ for ±y > 0 reduces to the second derivative, which allows us to calculate easily function (6.23). Therefore application of (6.26) leads to (7.16). Let us consider the exceptional case ρ = 1.
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Example 7.5. Let V (x) = −hx, ˆ ni|x|−1 for sufficiently large |x| (and d is arbitrary). Then the asymptotics of s(ω, ω0 ; λ) is given by formulas (6.37), (6.38), where v(ω0 ) = 0 and ˆ = −khϑ, ˆ yi − π(2k)−1 hn, yi ˆ Ξ(y, ϑ) (note that this expression does not depend on ω0 ). ˆ = Indeed, v(ω0 ) = 0 since V0 (x) = −V0 (−x). Calculating (6.28) we find that ν(b) ˆ −πhn, bi. So the result formulated above is an immediate consequence of Theorem 6.10. 3. Let us compare asymptotic formula (7.3) for the quantum scattering cross-section with the corresponding classical result. Recall (see e.g. [11]) that for the classical motion in a central potential V (|x|) the scattering cross-section Σ(θ; λ) at the energy λ may be determined by the dependence of the scattering angle θ ∈ (0, π] on the impact parameter l (the minimal distance of a free particle from the coordinate center): Σ(θ; λ) = (sin θ)−d+2 l(θ; λ)d−2 |dl/dθ|.
(7.19)
We emphasize that, as in the quantum case, the physical differential cross-section in some angle equals the product of Σ and of the measure of this angle. The following result is known in the physics literature (see e.g. [11]) but we have not found its correct justification. Proposition 7.6. Suppose that V (x) = V (|x|) and V (r) = vr−ρ , where ρ is an arbitrary positive number, for sufficiently large r. Then in the limit θ → 0, (7.20) Σ(θ; λ) = ρ−1 (I(ρ)|v|λ−1 )(d−1)γ θ−(d−1)(1+γ) 1 + O(θ) , where the coefficient I(ρ) is defined by (7.1). Proof. Recall (see e.g. [11]) that the scattering angle θ is determined by the formula θ = |π − 2ϕ|, where Z ∞ −1/2 r−2 1 − λ−1 V (r) − l2 r−2 dr (7.21) ϕ=l rmin
and
rmin = sup{r : 1 − λ−1 V (r) − l2 r−2 < 0}.
Clearly, V (r) = vr−ρ for large l and r ≥ rmin . Changing in (7.21) the variable r = ly −1 we find that Z y0 −1/2 ϕ= 1 − y 2 − 2εy ρ dy, (7.22) 0
where ε = v(2λlρ )−1 and y0 = y0 (ε) is the positive root of the equation 1−y02 −2εy0ρ = 0. Our goal is to find the asymptotics of integral (7.22) as ε → 0. Fix some c ∈ (0, 1) and consider separately the intervals (0, c) and (c, y0 ). By the Taylor formula, Z c Z c Z c −1/2 −1/2 −3/2 2 ρ 2 1 − y − 2εy 1−y 1 − y2 dy = dy + ε y ρ dy + O(ε2 ). 0
0
In the integral over (c, y0 ) we put z = y 2 + 2εy ρ . Then
0
(7.23)
Scattering Amplitude for Schr¨odinger Equation with Long-Range Potential
Z
y0 (ε)
1 − y 2 − 2εy ρ
2
−1/2
Z
1
dy = c2 +2εcρ
c
217
9ε (z)(1 − z)−1/2 dz,
(7.24)
where 9ε (z) = (y +ερy ρ−1 )−1 and y is considered as a function of z. An easy calculation shows that, away from the point z = 0, 9ε (z) = z −1/2 + ε(1 − ρ)z (ρ−3)/2 + O(ε2 ), and hence integral (7.24) equals Z 1 Z z −1/2 (1 − z)−1/2 dz + ε(1 − ρ) c2 +2εcρ
1
z (ρ−3)/2 (1 − z)−1/2 dz + O(ε2 ).
c2 +2εcρ
Comparing this result with (7.23) we find that ϕ(ε) = π/2 − I(ρ)ε + O(ε2 ),
(7.25)
where 2I(ρ) = 2c
ρ−1
2 −1/2
(1−c )
Z
c2
−
(1−z)−3/2 z (ρ−1)/2 dz−(1−ρ)
Z
1
(1−z)−1/2 z (ρ−3)/2 dz.
c2
0
Since I(ρ) does not depend on c, we can calculate it taking the limit c → 1. This gives Z 1 −1 (1 − z)−3/2 (z (ρ−1)/2 − 1)dz = π 1/2 0((1 + ρ)/2)0(ρ/2)−1 , I(ρ) = 1 − 2 0
so that the numbers I(ρ) and I(ρ) are the same. Formula (7.25) implies the following dependence of the scattering angle θ on the impact parameter l: θ(l) = I(ρ)|v|λ−1 l−ρ + O(l−2ρ ).
(7.26)
The same arguments as above show that this equality may be differentiated: dθ(l)/dl = −ρI(ρ)|v|λ−1 l−ρ−1 + O(l−2ρ−1 ). Solving Eq. (7.26) with respect to l we find that l(θ) = (I(ρ)|v|λ−1 )γ θ−γ 1 + O(θ) ,
γ = ρ−1 .
Combining the results obtained we get asymptotics (7.20) for function (7.19).
Comparing (7.3) and (7.20), we see that in the case ρ < 1 the quantum and classical cross-sections coincide in the limit of small scattering angles (when |ϑ| = sin θ → 0). According to (7.8) the same is true for the Coulomb potential if d = 3. Actually, in this case the formula of Gordon and Mott for the quantum scattering cross-section for an arbitrary angle is the same as the classical Rutherford formula. On the other hand, asymptotics (7.7) and (7.20) show that for the Coulomb potential both quantum and classical cross-sections increase as the same power θ−2d+2 of θ for any d, but the coefficients at this power are different if d 6= 3. It is curious that the classical coefficient is a limit of quantum coefficients for ρ < 1 as ρ → 1. We emphasize that the small angles asymptotics of the scattering cross-section is given in the classical mechanics by the same formula (7.20) for all ρ > 0. In the quantum case 6(ϑ) behaves as |θ|−2d+2ρ if ρ ∈ (1, d) and hence it grows less rapidly than the classical cross-section as ϑ → 0. If ρ > d, then 6(ϑ) has even a finite limit as ϑ → 0.
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References 1. Agmon, S.: Some new results in spectral and scattering theory of differential operators in Rn . Seminaire Goulaouic Schwartz, Ecole Polytechnique, 1978 2. M. Birman, Sh. and Yafaev, D.R.: Asymptotics of the spectrum of the scattering matrix. J. Soviet Math. 25, no. 1, (1984) 3. H¨ormander, L.: The Analysis of Linear Partial Differential Operators III, Berlin–Heidelberg–New York: Springer-Verlag, 1985 4. H¨ormander, L.: The Analysis of Linear Partial Differential Operators IV, Berlin–Heidelberg–New York: Springer-Verlag, 1985 5. Isozaki, H., Kitada, H.: Micro-local resolvent estimates for 2-body Schr¨odinger operators. J. Funct. Anal. 57, 270–300 (1984) 6. Isozaki, H., Kitada, H.: Modified wave operators with time-independent modifies. J. Fac. Sci, Univ. Tokyo, 32, 77–104 (1985) 7. Isozaki, H., Kitada, H.: Scattering matrices for two-body Schr¨odinger operators. Sci. Papers College Arts and Sci., Univ. Tokyo, 35, 81–107 (1985) 8. Isozaki, H., Kitada, H.: A remark on the micro-local resolvent estimates for two-body Schr¨odinger operators. Publ. RIMS, Kyoto Univ., 21, 889–910 (1986) 9. Jensen, A.: Propagation estimates for Schr¨odinger-type operators. Trans. Am. Math. Soc. 291, 129–144 (1985) 10. Jensen, A., Mourre, E., Perry, P.: Multiple commutator estimates and resolvent smoothness in quantum scattering theory. Ann. Inst. Henri Poincar´e, Phys. th´eor. 41, 207–225 (1984) 11. Landau, L.D. and Lifshitz, E.M.: Classical mechanics. London: Pergamon Press, 1960 12. Landau, L.D. and Lifshitz, E.M.: Quantum mechanics. London: Pergamon Press, 1965 13. Lerner, N., Yafaev, D.: Trace theorems for pseudo-differential operators. Universit´e de Rennes, Pr´epublication 96–18, 1996 14. Mourre, E.: Op´erateurs conjugu´es et propri´et´es de propagation. Comm. Math. Phys. 91, 279–300 (1983) 15. Reed, M. and Simon, B.: Methods of Modern Mathematical Physics III, IV, New York: Academic Press, 1979, 1978 16. Shubin, M.A.: Pseudodifferential Operators and Spectral Theory. Berlin–Heidelber–New York: Springer-Verlag, 1987 17. D. Yafaev, R.: Mathematical Scattering Theory, Am. Math. Soc., 1992 18. D. Yafaev, R.: Wave operators for the Schr¨odinger equation. Theor. Math. Phys. 45, 992–998 (1980) 19. Yafaev, D.R.: Radiation conditions and scattering theory for N -particle Hamiltonians. Comm. Math. Phys. 154, 523–554 (1993) 20. Yafaev, D.R.: Resolvent estimates and scattering matrix for N -particle Hamiltonians. Int. Eq. Op. Theory 21, 93–126 (1995) Communicated by B. Simon
Commun. Math. Phys. 191, 219 – 248 (1998)
Communications in
Mathematical Physics c Springer-Verlag 1998
On Summability of Distributions and Spectral Geometry R. Estrada1 , J. M. Gracia-Bond´ıa2,? , J. C. V´arilly3,? 1
P.O. Box 276, Tres R´ıos, Costa Rica. E-mail:
[email protected] Departamento de F´ısica Te´orica, Universidad de Zaragoza, 50009 Zaragoza, Spain 3 Centre de Physique Th´ eorique, CNRS–Luminy, Case 907, 13288 Marseille, France. E-mail:
[email protected]
2
Received: 10 February 1997 / Accepted: 8 May 1997
Abstract: Modulo the moment asymptotic expansion, the Ces`aro and parametric behaviours of distributions at infinity are equivalent. On the strength of this result, we construct the asymptotic analysis for spectral densities arising from elliptic pseudodifferential operators. We show how Ces`aro developments lead to efficient calculations of the expansion coefficients of counting number functionals and Green functions. The bosonic action functional proposed by Chamseddine and Connes can more generally be validated as a Ces`aro asymptotic development.
1. Introduction Most approaches to spectral geometry rely on the asymptotic expansion of the heat kernel and Tauberian theorems. In this work, motivated by a string of recent papers by Connes, we develop spectral geometry from a more fundamental object. According to a deep statement by Connes [10], there is a one-to-one correspondence between Riemannian spin geometries and commutative real K-cycles, the dynamics of the latter being governed by the spectral properties of its defining Dirac operator. On ordinary manifolds, gravity (of the Einstein and the Weyl variety) is the only interaction naturally described by the K-cycle [1, 27, 28]. That is to say, in noncommutative geometry, existence of gauge fields requires the presence of a noncommutative manifold structure, whose “diffeomorphisms” incorporate the gauge transformations. Connes’ new gauge principle points thus to an intrinsic coupling between gravity and the other fundamental interactions. In a recent formulation [7], the Yang–Mills action functional is replaced by a “universal” bosonic functional of the form: Bφ [D] = Tr φ(D2 ), ?
On leave from Department of Mathematics, Universidad de Costa Rica, 2060 San Pedro, Costa Rica.
220
R. Estrada, J. M. Gracia-Bond´ıa, J. C. V´arilly
with φ being an “arbitrary” positive function of the Dirac operator D. Chamseddine and Connes’ work on the universal bosonic functional has two main parts. In the first one, they argue that Bφ has the following asymptotic development: Bφ [D/3] ∼
∞ X
fn 34−2n an (D2 )
as 3 → ∞,
(1.1)
n=0 2 the an are the R ∞coefficients of the heat kernel0 expansion [19] for D and f0 = Rwhere ∞ xφ(x) dx, f1 = 0 φ(x) dx, f2 = φ(0), f3 = −φ (0), and so on. Then they proceed 0 to compute the development for the K-cycle currently [9, 32] associated to the Standard Model, indeed obtaining all terms in the bosonic part of the action for the Standard Model, plus gravity, plus some new ones. Their approach gives prima facie relations between the parameters of the Standard Model, in terms of the cutoff parameter 3, falling rather wide of the empirical mark. In the second part of their paper, they enterprise to improve the situation by use of the renormalization group flow equations [2]. This need not concern us here. Formula (1.1) can be given a quick derivation, by assuming that φ is a Laplace transform. This condition, however, will almost never be met in practice. In order to see that the asymptotic development of Bφ cannot be taken for granted, let us consider, as Kastler and coworkers have done [6, 26] the characteristic functions φ3 := χ[0,3] . This looks harmless enough, giving nothing but ND2 (32 ), the counting number of eigenvalues of D2 below the level 32 . However, it has been known for a long time —see for instance [24]— that there is no asymptotic development for the counting functional beyond the first term. Therefore Eq. (1.1), as it stands, is not applicable to that situation. One of our aims in this paper is to decrypt the meaning of “arbitrary functional”; a related one is to put on a firm footing the development (1.1). Our contribution turns around the Ces`aro behaviour of distributions, and its relation with asymptotic analysis. Most results are new, or seem to be ignored in the literature; the paper is written with a pedagogical bent. The article is organized as follows. Section 2 is the backbone of the paper; there the Ces`aro behaviour of distributions and Ces`aro summability of evaluations are examined. The distributional theory of asymptotic expansions [15] is summarized. The latter is brought to bear by finding the essential equivalence between the Ces`aro behaviour and the parametric behaviour of distributions at infinity. Also we prove that a distribution satisfies the moment asymptotic expansion iff it belongs to K0 , the dual of the space of Grossmann–Loupias–Stein operator symbols [20]. These results are new, having been obtained very recently by one of us [RE, 12]. We try to enliven this somewhat technical section with pertinent examples. Next we consider elliptic, positive pseudodifferential operators; let H be one of those; the functional calculus for H can be based on the spectral density, formally written as δ(λ − H). This is arguably a more basic object than the heat kernel, and its study is very rewarding. In Sect. 3, we show that δ(λ − H) is an operator-valued distribution in K0 . With that in hand, one can proceed to give a meaning to the universal bosonic action for a very wide class of functionals. Following some old ideas by Fulling [17], insufficiently exploited up to now, we emphasize that the Ces`aro behaviour of the spectral density for differential operators is local, i.e., independent of the boundary conditions. This is practical for computational purposes, as it sometimes allows to replace an operator in question by a more convenient local model. In Sect. 4, we reach the heart of the matter: let dH (x, y; λ) denote the distributional kernel of δ(λ − H); a formula for dH is given and immediately applied to compute the
On Summability of Distributions and Spectral Geometry
221
coefficients of its asymptotic expansion on the diagonal, in terms of the noncommutative residues [38] of certain powers of H. We hope to have clarified in the paper that the identification of the higher Wodzicki terms is essentially a “finite-part” calculation. The spectral density is actually a less singular object for operators with continuous spectra than for operators with discrete spectra, and all of the above applies to operators associated to noncompact manifolds: for that purpose, taking account of locality, we work with densities of noncommutative residues throughout. We go on to extend Connes’ trace theorem [8] to noncompact K-cycles. The case of generalized Laplacians is then treated within our procedure. In the light of the preceding, the last two sections of the paper are concerned, respectively, with the counting number and the heat kernel expansions. The counting functional NH (λ) is treated mainly by way of example. Then we reexamine the status of arbitrary smoothing asymptotic expansions, in particular the Laplace-type expansions like the Chamseddine–Connes Ansatz. We point out conditions for the expansions to be valid without qualification, and to be valid only in the Ces`aro sense. Also we exemplify circumstances under which the formal Laplace-type expansion does not say anything about the true asymptotic development. The Chamseddine–Connes expansion is derived and reinterpreted. 2. Ces`aro Computability of Distributions Besides the standard spaces of test functions and distributions, the space K first introduced in [20] and its dual K0 play a central role in our considerations. Familiarity with the properties of K and K0 and with some of their elements will be convenient. For all general matters in distribution theory, we refer to [18]. As our interest is mainly in spectral theory, we consider Grossmann–Loupias– Stein symbols in one variable, almost exclusively. A smooth function φ of a real variable belongs to Kγ for a real constant γ if φ(k) (x) = O(|x|γ−k ) as |x| → ∞, for each k ∈ N. A topology for Kγ is generated by seminorms kφkk,γ = supx∈R { max(1, |x|k−γ ) |φ(k) (x)| }, and so Kγ ,→ Kγ 0 if γ ≤ γ 0 . Notice that φ(k) ∈ Kγ−k if φ ∈ Kγ . The space K is the inductive limit of the spaces Kγ as γ → ∞. Since every polynomial is in K, a distribution f ∈ K0 has moments µn := hf (x), xn i,
n∈N
of all orders; this is an indication that f decays rapidly at infinity in some sense. Denote by D00 (T) the space of periodic distributions with zero mean. They constitute a first class of examples: if f ∈ D00 (T), then, for n suitably large, the periodic primitive with zero mean fn of f of order n is continuous and defines the evaluation of f at φ ∈ K by a convergent integral: hf (x), φ(x)i = (−1)n hfn (x), φ(n) (x)i. Note that in this case all the moments are zero. The algebra K is normal (i.e., S is dense in K) and is a subalgebra of the multiplier algebras OM , M of S, respectively for the ordinary product and the Moyal star product [16]. Other properties of K and K0 will be invoked opportunely. The usefulness of K in phase-space Quantum Mechanics lies in the similitude of behaviour of the ordinary and the Moyal product, when applied to elements of K. The link between both appearances of K is still mysterious to us.
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The natural method of studying generalized functions at infinity is by considering the parametric behaviour. The moment asymptotic expansion of a distribution [15] is given by ∞ X (−1)k µk δ (k) (x) as λ → ∞. (2.1) f (λx) ∼ k! λk+1 k=0
The interpretation of this formula is in the distributional sense, to wit hf (λx), φ(x)i =
N X µk φ(k) (0) k=0
k! λk+1
+O
1 λN +2
as λ → ∞,
for each φ in an appropriate space of test functions. Such an expansion holds only for distributions that decay rapidly at infinity, in a sense soon to be made completely precise; it certainly does not hold for all tempered distributions, as their moments do not generally exist. Distributions endowed with moment asymptotic expansions are said to be “distributionally small at infinity”. We are not happy with this terminology and invite suggestions to improve it. On the other hand, the classical analysis [23] notion of Ces`aro or Riesz means of series and integrals admits a generalization to the theory of distributions, that we intend to exploit in this paper. It turns out that Ces`aro limits and “distributional” ones are essentially equivalent; this will enable us to apply the simpler ideas of parametric analysis to complicated averaging schemes. We begin now in earnest by introducing the basic concept of Ces`aro behaviour of the distributions; justification will follow shortly. Assume f ∈ D0 (R), β ∈ R \ {−1, −2, . . .}. Definition 2.1. We say that f is of order xβ at infinity, in the Ces`aro sense, and write f (x) = O(xβ )
(C)
as x → ∞,
if there exists N ∈ N, a primitive fN of f of order N and a polynomial p of degree at most N − 1, such that fN is locally integrable for x large and the relation fN (x) = p(x) + O(xN +β )
as x → ∞
(2.2)
holds in the ordinary sense. The relation f (x) = o(xβ ) (C) is defined similarly. The notation (C, N ) can be used if one needs to be more specific; if an order relation holds (C, N ) for some N , it also holds (C, M ) for all M > N . The assumption β 6= −1, −2, . . . is provisionally made in order to avoid dealing with the primitives of x−1 , x−2 and such (see Sect. 6 for the general case). If β > −1, the polynomial p is arbitrary and thus irrelevant. We shall suppose when needed that our distributions have bounded support, say, on the left. In that case, we denote by I[f ] the first order primitive of f with support bounded on the left. When f is locally integrable, then, Z x f (t) dt. I[f ](x) = −∞
The notation
f (x) = o(x−∞ ) β
(C)
as x → ∞
will mean f (x) = O(x ) (C) for every β. For the proof of the following workhorse proposition we refer to [12].
On Summability of Distributions and Spectral Geometry
223
Lemma 2.1. (a) Let f ∈ D0 such that f (x) = O(xβ )
as x → ∞.
(C, N )
Then for k = 1, 2, 3, . . . we have: f (k) (x) = O(xβ−k )
as x → ∞.
(C, N + k)
(b) Let f ∈ D0 such that f (x) = O(xβ )
(C)
as x → ∞,
and let α ∈ R. Provided that α + β is not a negative integer, we have: xα f (x) = O(xα+β )
(C)
as x → ∞.
Definition 2.2. We write limx→∞ f (x) = L (C) when f (x) = L + o(1) (C) as x → ∞. That is, limx→∞ f (x) = L (C, k) when fk (x) k!/xk = L + o(1), for fk a primitive of order k of f . For example, if f is periodic with zero mean value, there exists n ∈ N and a continuous (thus bounded) periodic function fn with zero mean value such that fn(n) = f ; then clearly as x → ∞, f (x) = o(x−∞ ) (C) a fact that yields, for f periodic with mean value a0 : lim f (x) = a0
x→∞
(C).
Let f ∈ D0 be a distribution with support bounded on the left and let φ be a smooth function. The following is a key concept of the theory. Definition 2.3. We say that the hf (x), φ(x)i has the value L in the Ces`aro sense, and write hf (x), φ(x)i = L (C) if there is a primitive I[g] for the distribution g(x) = f (x)φ(x), satisfying lim I[g](x) = L
x→∞
(C)
as x → ∞.
A similar definition applies when f has support bounded on the right. If f is an arbitrary distribution, let f = f1 + f2 be a decomposition of f , where f1 has support bounded on the left and f2 has support bounded on the right. Then we say that hf (x), φ(x)i = L (C) if both hfi (x), φ(x)i = Li (C) exist for i = 1, 2 and L = L1 + L2 : this definition is seen to be independent of the decomposition. For instance, let f be a periodic distribution of zero mean and let f1 , f2 , . . . , fn+1 denote the periodic primitives with zero mean of f , up to the order n + 1. Then xn f1 (x) − nxn−1 f2 (x) + n(n − 1)xn−2 f3 (x) − · · · + (−1)n n! fn+1 (x) is a first order primitive of xn f (x), and since fi (x) = o(x−∞ ) (C) for i = 1, . . . , n as x → ∞, it follows that hf (x), xk i = 0 (C)
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for all k ∈ N. To perceive the point of our hitherto abstract definitions, it is worthwhile to recall here briefly the classical theory [23]. Let {an }∞ n=1 be a sequence of real or complex numbers. Often it has no limit, but the sequence of averages Hn(1) := (a1 + · · · + an )/n does. Then people write lim an = L (C, 1). n→∞
Hn(1)
If still does not have a limit, then one may apply the averaging procedure again and again, hoping that eventually a limit will be obtained. There are two main procedures to perform such higher order averages: the H¨older means and the Ces`aro means. The H¨older means are single-mindedly constructed as Hn(k) :=
H1(k−1) + · · · + Hn(k−1) , n
and limn→∞ Hn(k) = L is written lim an = L
n→∞
(H, k).
The properly named Ces`aro means are defined as follows: let A(0) n := an and define (k−1) (k−1) (k) k recursively A(k) = A + · · · + A . If lim k! A /n = L, we write n→∞ n n n 1 lim an = L
n→∞
(C, k),
so that the (C, 1) and the (H, 1) notions are identical. The Ces`aro limits have nicer analytical properties. The good news, at any rate, is that both procedures are equivalent: lim an = L
n→∞
(C, k) ⇐⇒ lim an = L n→∞
(H, k).
One uses the simpler notation limn→∞ an = L (C) if limn→∞ an = L (C, k) for some k ∈ N. A third averaging procedure is equivalent to Ces`aro’s, the so-called Riesz typical means. For real µ, one writes lim an = L
n→∞
if
(R, k, n)
n k−1 1 X 1− an = L. µ→∞ µ µ lim
n≤µ
Riesz originally studied this formula for integral µ, but the means have more desirable P∞ properties with µ real. Now, one may study the summability P of a series n=1 an by ∞ studying the generalized function of a real variable f (x) = n=1 an δ(x − n). The definition of Ces` a ro limits of distributions is tailored in such a way that hf, 1i (C) P∞ P∞ a (C) coincide: a primitive of order k of a δ(x − n) is given by and n=1 n n n=1 P fk (x) = n≤x (x − n)k−1 an /(k − 1)! Note that one could consider distributions of the P∞ form h(x) = n=1 an δ(x − pn ), with pn ↑ ∞; this gives rise to the (R, k, pn ) means. In summary, we have demonstrated the following equivalence.
On Summability of Distributions and Spectral Geometry
Theorem 2.2. The evaluation X ∞
225
an δ(x − n), φ(x)
=L
(C)
n=1
holds iff
P∞ n=1
an φ(n) = L in the Ces`aro sense of the theory of summability of series.
In the same vein: Theorem 2.3. If f is locally integrable and supported in (a, ∞), then hf (x), φ(x)i = L if and only if
Z
(C)
∞
f (x)φ(x) dx = L a
in the Ces`aro sense of the theory of summability of integrals. As shown below, if f ∈ K0 and φ ∈ K, then the evaluation hf (x), φ(x)i is always (C)-summable. We pause an instant to show by example just how useful is the concept of Ces`P aro computability of evaluations. An interesting periodic distribution is the Dirac ∞ comb n=−∞ δ(x − n). Its mean value is 1; therefore ∞ X
δ(x − n) = 1 + f (x),
(2.3)
n=−∞
with f ∈ D00 (T). The distributions ∞ X
δ(x − n) − H(x − 1),
n=1
∞ X
δ(x − n) − H(x),
n=1
where H is the Heaviside function, belong to K0 . In effect, take aPfunction φ1 ∈ K such ∞ that φ1 (x) = 1 for x > 1/2, φ1 (x) = 0 for x < 1/4. Then φ1 (x) n=−∞ δ(x − n) − 1 P∞ P∞ only differs from n=1 δ(x − n) − H(x − 1) or n=1 δ(x − n) − H(x) by a distribution of compact support. It follows that the evaluation X X Z ∞ ∞ ∞ δ(x − n) − H(x − 1), φ(x) = φ(n) − φ(x) dx n=1
1
n=1
is Ces`aro summable whenever φ ∈ K. Now, xα does not belong to K unless α ∈ N, but the previous argument, using φα (x) = φ1 (x) xα , allows us to conclude that the evaluation Z(α) :=
X ∞
δ(x − n) − H(x − 1), x
α
n=1
is (C)-summable for any α ∈ C. Also, Z(α) is an entire function of α, since φα is. We find a formula for Z(α) by observing that if < α < −1, then the evaluation is given by the difference of a series and an integral, so that
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Z(α) =
∞ X
Z
∞
nα −
xα = ζ(−α) +
1
n=1
1 , α+1
<α < −1.
We have learned a simple proof that Riemann’s zeta function is analytic in C \ {1}, with residue at s = 1 equal to 1, and one realizes that the evaluation of the ζ function can be done by Ces`aro means (it is only because the zeta function is the outcome of a regularization that it is useful for renormalization in quantum field theory). The Pprocess ∞ evaluation h n=1 δ(x − n) − H(x), xα i is slightly more involved. However, we may write [14]: X ∞
δ(x − n) − H(x), xα
Z
1
:= Z(α) − F.p.
xα dx, 0
n=1
where F.p. stands for the Hadamard finite part of the integral. Now, Z
1
xα dx =
F.p. 0
1 , α+1
α 6= −1,
therefore, if α 6= −1, ζ(−α) =
∞ X
Z
∞
nα − F.p.
xα dx
(C),
0
n=1
in the sense that ζ(−α) = lim
X bxc
x→∞
Z
x
nα − F.p.
tα dt
(C).
0
n=1
This formula gives a nice representation for ζ(α) when < α < 1. For instance, ζ(0) = −1/2 simply because the fractional part {x} = x − bxc of x is periodic of mean 1/2. For α = −1: ζ(−1) = lim
x→∞
X bxc
Z
x
n−
t dt 0
n=1
1 bxc(bxc x→∞ 2
= lim
+ 1) − 21 x2
(C);
we find that 1 (x − {x})(x − {x} + 1) x2 {x}2 − {x} x(1 − 2{x}) − = + = − + o(x−∞ ) 2 2 2 2 12
(C),
since (1−2{x}) and ({x}2 −{x}+1/6) are periodic of mean zero; we get ζ(−1) = −1/12. Also, the logarithm of the “functional determinant” can be obtained by this method: ζ 0 (0) = − lim
x→∞
X bxc
Z
n=2
on using Lemma 2.1. Stirling’s formula gives
x
log n −
log t dt 0
(C),
On Summability of Distributions and Spectral Geometry
227
√ x log x − x − log(bxc!) = x log x − x − (bxc + 21 ) logbxc + bxc − log 2π + O(x−1 ) {x} − {x} + ({x} − 21 ) logbxc − 21 log(2π) + O(x−1 ) = −x log 1 − x = − 21 log(2π) + O(x−1 ) (C), since x log(1 − x−1 {x}) + {x} = O(x−2 ) and ({x} − 21 ) is periodic of mean zero. From this it follows that ζ 0 (0) = − 21 log(2π). This business of Riemann’s zeta function is not merely amusing; it will be useful later. We make ready for the main equivalence result. Theorem 2.4. Let f ∈ D0 . If α > −1 then f (x) = O(|x|α )
as x → ±∞
(C)
if and only if
as λ → ∞
f (λx) = O(λα )
(2.4) (2.5)
in the topology of D0 . If −j − 1 > α > −j − 2 for some j ∈ N, then (2.4) holds if and only if there are constants µ0 , . . . , µj such that f (λx) =
j X (−1)k µk δ (k) (x) k=0
k! λk+1
+ O(λα )
in the topology of D0 as λ → ∞. Proof. We prove the theorem in the case f has support bounded on the left. The general case follows by using a decomposition f = f1 + f2 , where f1 has support bounded on the left and f2 has support bounded on the right. First we have to clarify the meaning of (2.5). It is a weak or distributional relation: we write f (x, λ) = O(λα ) as λ → ∞ whenever as λ → ∞, hf (x, λ), φ(x)i = O(λα ) for all φ ∈ D. Note that this yields ∂f (x, λ) , φ(x) = −hf (x, λ), φ0 (x)i = O(λα ). ∂x Now, if (2.5) holds, there exists N such that the primitive of order N of f (λx), with respect to x, exists and is bounded by M λα , say for |x| ≤ 1 and λ ≥ λ0 . We have then a primitive fN of order N of f (x), such that |fN (λx)| ≤ M λα+N ,
|x| ≤ 1, λ ≥ λ0 .
Taking x = 1 and replacing λ by x we obtain |fN (x)| ≤ M xα+N , and thus f (x) = O(xα )
(C, N ),
x ≥ λ0 , as x → ∞.
Reciprocally, assume α > −1 and f (x) = O(x ) (C, N ), as x → ∞. Then, if fN is the (locally integrable for x large) primitive of order N of f with support bounded on the α
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left, an obvious estimate gives fN (λx) = O(λα+N ), as λ → ∞, and on differentiating N times with respect to x one obtains λN f (λx) = O(λα+N ), so that (2.5) follows. The case when α is nonintegral and less than −1 is more involved, as one has to deal with the polynomial p in (2.2). Then one shows that the moments hf (x), xk i = µk
(C)
up to a certain order exist, those being essentially the coefficients of p. For the gory details, we refer once again to [12]. A characterization of the distributions that have a moment asymptotic expansion follows. Theorem 2.5. Let f ∈ D0 . Then the following are equivalent: (a) f ∈ K0 . (b) f satisfies
f (x) = o(|x|−∞ )
as x → ±∞.
(C)
(c) There exist constants µ0 , µ1 , µ2 , . . . such that f (λx) ∼
µ0 δ(x) µ1 δ 0 (x) µ2 δ 00 (x) − + − ··· λ λ2 2! λ3
as λ → ∞
in the weak sense. Proof. It is proven in [15] that the elements of K0 satisfy the moment asymptotic expansion. For the converse, it is enough, as customary, to consider distributions with support bounded on one side. We show that if (b) holds, then f ∈ Kγ0 for all γ. From the hypothesis it follows that f (x) = O(x−γ−2 ) (C) as x → ∞. Thus, for a certain n, the nth order primitive fn of f with support bounded on one side is locally integrable and satisfies fn (x) = p(x) + O(x−γ−2+n ) as x → ∞, where the polynomial p has degree at most n − 1. We conjure up a compactly supported continuous function g whose moments of order up to n − 1 coincide with those of f . If gn is the primitive of order n of g −γ−2+n ). If φ ∈ Kγ−n , the with support R ∞ bounded on the left, then fn (x) − gn (x) = O(x integral −∞ (fn (x) − gn (x))φ(x) dx converges. Hence f = (fn − gn )(n) + g ∈ Kγ0 . The rest is clear. We get at once a powerful computational method for duality evaluations. Corollary 2.6. If f ∈ K0 and φ ∈ K, the evaluation hf (x), φ(x)i is Ces`aro summable. Proof. It is enough to check for φ = 1. But, according to the previous theorem, if f ∈ K0 , then f (x) = o(x−∞ ) (C) as x → ∞. By the proof of Theorem 2.4, hf (x), 1i is (C)-summable. Fourier transforms are defined by duality and, in general, if f ∈ S 0 , we cannot make sense of fˆ(u) because the evaluation heixu , f (x)i is not defined. However, if φ ∈ K and u 6= 0, Corollary 2.6 guarantees that the Ces`aro-sense evaluation heixu , φ(x)i (C) is well defined. Thus ˆ φ(u) = heixu , φ(x)i
(C)
when
φ ∈ K, u 6= 0.
b ⊂ K0 ; this follows also from Proposition 4 of [20]. It is clear that K Note as well that the moments of f ∈ K0 are (C)-summable. The converse is true:
On Summability of Distributions and Spectral Geometry
229
Theorem 2.7. Let f ∈ D0 . If all the moments hf (x), xn i = µn (C) exist for n ∈ N, then f ∈ K0 . For the easy proof, we refer to [12]. It is clearly worthwhile to characterize spaces of distributions in terms of their Ces`aro behaviour. Particularly important is the characterization of tempered distributions: Theorem 2.8. Let f ∈ D0 . Then the following statements are equivalent: (a) f is a tempered distribution. (b) There exists α ∈ R such that as λ → ∞
f (λx) = O(λα ), in the weak sense. (c) There exists α ∈ R and k ∈ N such that f (k) (x) = O(|x|α−k )
(C)
as x → ∞.
Proof. Again, it is enough to consider the case when f has support bounded on one side. It is well known that if f ∈ S 0 then there is a primitive F of some order N of slow growth at infinity; it follows that f (x) = O(|x|α ) (C). The rest is clear, in view of the equivalence theorem 2.4 and the fact that distributional order relations can be differentiated at will. We finish by giving several estimates that we will need later. The first one is just a rewording of the properties of the distribution (2.3). R∞ Lemma 2.9. If g ∈ K and if −∞ g(x) dx is defined, then ∞ X
1 g(nε) = ε n=−∞
Z
Lemma 2.10. If g ∈ K(Rn ) and if g(kε) = ε−n
R Rn
Z
Lemma 2.11. If g ∈ K and if
n=1
g(nε) =
1 ε
as ε ↓ 0.
g(x) dx is defined, then g(x) dx + o(ε∞ )
as ε ↓ 0.
Rn
k∈Zn
∞ X
g(x) dx + o(ε∞ )
−∞
By the same token:
X
∞
Z
R∞ 0
g(x) dx is defined, then
∞
g(x) dx + 0
∞ X ζ(−n)g (n) (0) n=0
n!
Proof. This follows from the zeta function example.
εn + o(ε∞ )
as ε ↓ 0.
(Results of this type were used to prove some formulas by Ramanujan in [15].)
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3. Spectral Densities Let H be a concrete Hilbert space, the space of square integrable sections of an Euclidean vector bundle over a Riemannian manifold M , and let H be an elliptic positive selfadjoint pseudodifferential operator on H, with domain X . We consider the derivative, in the distributional sense, of the spectral family of projectors EH (λ) associated to H: dH (λ) :=
dEH (λ) . dλ
For instance, if H is defined on a compact manifold, and 0 < λ1 ≤ λ2 ≤ · · · is the complete set of its eigenvalues, with orthonormal basis of eigenfunctions uj , the kernel of the spectral family is given by [25]: X |uj )(uj |, EH (λ) := λj ≤λ
and the derivative is dH (λ) :=
X
|uj )(uj | δ(λ − λj ).
j
This spectral density is a distribution with values in L(X , H). The defining properties of E(λ): Z Z ∞
∞
dE(λ),
I=
H=
−∞
λ dE(λ) −∞
(in the weak sense) become, in the language of the previous section: I = hdH (λ), 1i,
H = hdH (λ), λi.
The spectral density is used to construct the functional calculus for H. Indeed, we can define φ(H) whenever f is a distribution such that the evaluation hdH (λ), f (λ)i makes sense, by φ(H) := hdH (λ), φ(λ)i, with domain the subspace of the x ∈ H for which the evaluation h(y | dH (λ)x), φ(λ)iλ is defined for all y ∈ H. Especially, one is able to deal with the “zeta operator": H −s := hdH (λ), λ−s i
(3.1)
(for 0 ∈ / sp H), the heat operator: e−tH := hdH (λ), e−tλ i,
t>0
(3.2)
and the unitary group of H, which is just the Fourier transform of the spectral density: UH (t) := hdH (λ), e−itλ i. The useful symbolic formula dH (λ) = δ(λ − H) recommends itself, and we shall employ it from now on.
(3.3)
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We want to studyTthe asymptotic behaviour of δ(λ − H). Let Xn be the domain of ∞ H n and let X∞ := n=1 Xn . The fact that X∞ is dense has, in view of the theory of Sect. 2, momentous consequences. We have H n = hδ(λ − H), λn i in the space L(X∞ , H). Hence, δ(λ − H) belongs to the space K0 (R, L(X∞ , H)). Therefore the moment asymptotic expansion holds: δ(λσ − H) ∼
∞ X (−1)n H n δ (n) (λ) n=0
as σ → ∞,
n! σ n+1
and δ(λ − H) vanishes to infinite order at infinity in the Ces`aro sense: δ(λ − H) = o(|λ|−∞ )
(C)
as |λ| → ∞.
Of course, the last formula is trivial when H is bounded. The space D(M ) of test functions is a subspace of X∞ . We can then realize the spectral density by an associated kernel dH (x, y; λ), an element of D0 (R, D0 (M × M )). Ellipticity actually implies that dH (x, y; λ) is smooth in (x, y). The expansion dH (x, y; λσ) ∼
∞ X (−1)n (H n δ)(x − y) δ (n) (λσ) n=0
as σ → ∞
n! σ n+1
holds in principle in the space D0 (R, D0 (M × M )). We also get dH (x, y; λ) = o(|λ|−∞ )
as |λ| → ∞
(C)
(3.4)
in the space D0 (M × M ). Eq. (3.4) is the mother of all incoherence principles. For instance, passing to the primitive with respect to λ, for an elliptic operator on a compact manifold with eigenfunctions ψn , n ∈ N, one concludes: X ψ¯ n (x)ψn (y) = o(|λ|−∞ ) (C) as |λ| → ∞, λn ≤λ
for x 6= y, which is Carleman’s incoherence relation [5]. It should be clear that the expansions cannot hold pointwise in both variables x and y, since we cannot set x = y in the distribution δ(x − y). In fact, our interest in this paper lies in the coincidence limit dH (x, x; λ), which is not distributionally small. However, it is proven in [13] that, away from the diagonal of M × M , the expansions are valid in the sense of uniform convergence of all derivatives on compacta. On the other hand, if H1 and H2 are two pseudodifferential operators whose difference over an open subset U of M is a smoothing operator, and if d1 (x, y; λ) and d2 (x, y; λ) are the corresponding spectral densities, then [13]: d1 (x, y; σλ) = d2 (x, y; σλ) + o(σ −∞ )
as σ → ∞
in D0 (U × U ). Also, it can be shown that d1 (x, y; λ) = d2 (x, y; λ) + o(λ−∞ )
(C)
uniformly on compacts of U × U , even at the diagonal.
as λ → ∞
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We exemplify the reported behaviour with the simplest possible examples. Let H denote first the Laplacian on the real line. Its spectral density is dH (x, y; λ) =
√ 1 √ cos λ(x − y) , 2π λ
and therefore it is clear that dH (x, x; λ) is not distributionally small, but rather dH (x, x; λ) =
1 √ + o(λ−∞ ) 2π λ
(C)
as λ → ∞.
Let H denote now the Laplacian on the circle; the eigenvalues are λn = n2 , n = 0, 1, 2, . . ., with multiplicity 2 from n = 1 on, with normalized eigenfunctions ψn± (x) = (2π)−1/2 e±inx . Therefore X 1 δ(λ) + 2 cos n(x − y) δ(λ − n2 ) . 2π ∞
dH (x, y; λ) =
n=1
Then ∞ ∞ X X 1 δ (2j) (x − y) δ (j) (λ) δ(λσ) + 2 cos n(x − y) δ(λσ − n2 ) ∼ 2π j! σ j+1 n=1
as σ → ∞
j=0
in D0 (R, D0 (S1 × S1 )), while X 1 δ(λ) + 2 cos n(x − y) δ(λ − n2 ) = o(λ−∞ ) 2π ∞
(C)
as λ → ∞
n=1
if x and y are fixed, x 6= y. On the other hand, X 1 δ(λ) + 2 δ(λ − n2 ) 2π ∞
dH (x, x; λ) =
n=1
does not belong to K0 (R, C ∞ (S1 )). For the first time in this paper, but not the last, we have to find out what the Ces`aro behaviour of a given spectral kernel is. We shall have recourse to a variety of tricks. For now, applying Lemma 2.11 to g(x) := φ(x2 ), for φ a Schwartz function, say, we get: Z ∞ ∞ X 1 φ(εn2 ) = √ x−1/2 φ(x) dx − 21 φ(0) + o(ε∞ ) as ε ↓ 0. 2 ε 0 n=1
It is then clear that dH (x, x; λ) =
1 √ + o(λ−∞ ) 2π λ
(C)
as λ → ∞,
and it is also immediately clear that the distributional and Ces`aro behaviour of the spectral density and its kernel are exactly the same as in the previous example. That the manifold be compact or not and the spectrum be discrete or continuous is immaterial for that purpose. If we seek a boundary problem for the Laplacian, say on a bounded
On Summability of Distributions and Spectral Geometry
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interval of the line, we obtain still the same kind of behaviour (off the boundary, where a sharp change takes place). Note also the estimate: √ X λ (C) as λ → ∞. |ψn± (x)|2 ∼ π ±; λn ≤λ
As an aside, we turn before closing this section to the functional calculus formulas and compare (3.2) with (3.3). Obviously e−t(·) has an extension belonging to K, so there is no difficulty in giving a meaning to the heat operator. Also, as we shall see in Sect. 6, it is comparatively easy to study the asymptotic development of the corresponding Green function as t ↓ 0. One of the motivations of the present approach to spectral asymptotics is to define a sense for expansions of Schr¨odinger propagators and the like, that do not possess a “true” asymptotic expansion. Such an approach can be based in the following idea: Theorem 2.8 points to a rough duality between K0 and S 0 . Let g ∈ S 0 (R) and find α so that g(λx) = O(λα ) weakly as λ → ∞. For any φ ∈ S(R), the function 8 defined by 8(x) := hg(λx), φ(λ)iλ is smooth for x 6= 0 since 8(x) = |x|−1 hg(λ), φ(λx−1 )iλ , and satisfies 8(n) (x) = O(|x|α−n )
as |x| → ∞.
/ supp f , we can define hf (x), g(λx)ix as a tempered Therefore, if f ∈ K0 with 0 ∈ distribution. When 0 ∈ supp f , we need to ascertain independently smoothness of 8 at the origin. It turns out that, for this purpose, it is enough to demand distributional smoothness of g, i.e., the existence of the distributional values g (n) (0), in the sense of [31], for n = 0, 1, 2, . . .. Then g(tH) admits a distributional expansion in L(X∞ , H) as t ↓ 0. This can eventually lead to a proper treatment of some questions in quantum field theory. We say no more here and refer instead to the forthcoming [13]. In Sect. 6 of this paper, results will be stated for g belonging to S(R); for the rest of the paper we will venture outside safe territory only in examples.
4. The Ces`aro Asymptotic Development of dH (x, x; λ) In this section we obtain the asymptotic expansion for the coincidence limits of spectral density kernels. We are fortified with the results of the previous section, implying that the Ces`aro behaviour of the spectral density of pseudodifferential operators is a local matter. Let A be any pseudodifferential operator of order a positive integer d, with complete symbol σ(A), on the Riemannian manifold M . To simplify the discussion, we consider only operators acting on scalars; the treatment of matrix-valued symbols presents no further difficulty. The noncommutative or Wodzicki residue of A is defined by integrating (the trace of) the partial symbol σ−n (A)(x, ξ) of order −n over the cosphere bundle { (x, ξ) : |ξ| = 1 }: Z Z σ−n (A)(x, ξ) dξ dx. Wres A := M
Sn−1
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Here dx denotes the canonical volume element on M . If M is notR compact, Wres A may not exist, but there always exists the local density of the residue Sn−1 σ−n (A)(x, ω) dω, that we denote by wres A(x). We recall that σ(AB) − σ(A)σ(B) ∼
X (−i)|α| ∂ξα σ(A)∂xα σ(B). α!
|α|>0
The kernel kA of A is by definition: kA (x, y) := (2π)−n hei(x−y)·ξ , σ(A)(x, ξ)iξ . In particular, on the diagonal: kA (x, x) := (2π)−n h1, σ(A)(x, ξ)iξ .
(4.1)
In order to figure out the symbol for a spectral density, we start by considering (the selfadjoint extension of) an elliptic operator H with constant coefficients. In this case σ(H n ) = σ(H)n and we assert: σ δ(λ − H) = δ(λ − σ(H)), justified by the identities: Z λn δ(λ − σ(H)) dλ = σ(H n ),
λ = 0, 1, 2, . . . .
In the general case of nonconstant coefficients, we make the Ansatz that: σ δ(λ−H) ∼ δ(λ−σ(H))−q1 δ 0 (λ−σ(H))+q2 δ 00 (λ−σ(H))−q3 δ 000 (λ−σ(H))+· · · (4.2) R in the Ces`aro sense. Computation of λn σ(δ(λ − H)) dλ for λ = 0, 1, 2, . . . then gives q1 = 0; q2 = 21 σ(H 2 ) − σ(H)2 ; q3 = 16 σ(H 3 ) − 3σ(H 2 )σ(H) + 2σ(H)3 , (4.3) and so on. This development, it turns out, gives ever lower powers of λ in the asymptotic expansion of σ(δ(λ − H)). We are interested in explicit formulas for the Ces´aro asymptotic development of the coincidence limit for the kernel of a positive operator H as λ → ∞. From (4.1) and (4.2) with p := σ(H), we get dH (x, x; λ) ∼ (2π)−n h1, δ(λ − p(x, ξ)) + q2 (x, ξ) δ 00 (λ − p(x, ξ)) − · · ·iξ
(C).
In polar coordinates on the cotangent fibres, ξ = |ξ|ω with |ω| = 1, this becomes Z dω h|ξ|n−1 , δ(λ − p(x, |ξ|ω)) + q2 (x, |ξ|ω) δ 00 (λ − p(x, |ξ|ω)) − · · ·i|ξ| . (2π)−n |ω|=1
Hence, if we denote by |ξ|(x, ω; λ) the positive solution of the equation p(x, |ξ|ω) = λ, we need to compute: Z ∂2 n−1 |ξ|n−1 (x, ω; λ) + ∂λ (x, ω; λ) − · · · 2 q2 (x, |ξ|(x, ω; λ)ω)|ξ| −n . dω (2π) p0 (x, |ξ|(x, ω; λ)ω) Sn−1 (4.4)
On Summability of Distributions and Spectral Geometry
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Write: p(x, |ξ|ω) ∼ pd (x, ω)|ξ|d + pd−1 (x, ω)|ξ|d−1 + pd−2 (x, ω)|ξ|d−2 · · · . To solve p(x, |ξ|ω) = λ amounts to a series reversion. In order to see how that is done, let us assume for a short while that H is a firstorder operator with constant coefficients —for instance, the absolute value of the Dirac operator on Rn . We then expect |ξ|(x, ω; λ) ∼
p0 (ω) 1 λ− − p−1 (ω) λ−1 + · · · . p1 (ω) p1 (ω)
Integration over |ω| = 1 gives dH (x, x; λ) ∼ (2π)−n a0 λn−1 + a1 λn−2 + a2 λn−3 + · · ·
(C),
where, clearly, a0 = wres H −n . To compute a1 , a2 , . . . we can as well assume that the development of p is analytic as |ξ| → ∞. Let ψ(z) := z n−1 /p0 (z), so that Z ψ(|ξ|(x, ω; λ)) dω. a0 λn−1 + a1 (x)λn−2 + a2 (x)λn−3 + · · · ∼ Sn−1
If 0 is a circle containing |ξ|(x, ω; λ), wound once around ∞, we have the Cauchy integral: I 1 ψ(z)p0 (z) dz ψ(|ξ|(x, ω; λ)) = ψ(p−1 (λ)) = − 2πi 0 p(z) − λ I I 1 1 ψ(ζ −1 )p0 (ζ −1 ) dζ dζ = . = 2 −1 n+1 2πi 0−1 ζ (p(ζ ) − λ) 2πi 0−1 ζ (p(ζ −1 ) − λ) R Thus aj (x) = Sn−1 cj (x, ω) dω, where I 1 cj (ω) = sn−j−2 ψ(p−1 (1/s)) ds 2πi |s|=ε I I dζ 1 n−j−2 s ds = n+1 (p(1/ζ) − 1/s) (2πi)2 |s|=ε ζ −1 0 I I dζ 1 sn−j−1 ds = 2 n+1 (2πi) 0−1 ζ |s|=ε s p(1/ζ) − 1 I 1 dζ , = 2πi 0−1 ζ n+1 p(1/ζ)n−j which is the coefficient of ζ n in the expansion of p(1/ζ)j−n . Integrating over |ω| = 1 yields thus aj = wres H j−n , so, finally: dH (x, x; λ) ∼
1 (wres H −n λn−1 + wres H −n+1 λn−2 + · · ·) (2π)n
(C),
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where the densities of Wodzicki residues are constant for a constant-coefficient operator. It is amusing that we have arrived at a version of the classical Lagrange–B¨urmann expansion [29], with Wodzicki residues in the place of ordinary residues. Notice that an = 0. This is a very simple “vanishing theorem” (see for instance [3]). Returning to the general case, if H is a positive pseudodifferential operator of order d, then A := H 1/d is a positive pseudodifferential operator of first order. Setting µ = λ1/d , we have δ(µ − A) δ(λ1/d − H 1/d ) = , δ(λ − H) = δ(µd − Ad ) = dµd−1 dλ(d−1)/d and so dH (x, x; λ) ∼
1 a0 (x)λ(n−d)/d + a1 (x)λ(n−d−1)/d d (2π)n + a2 (x)λ(n−d−2)/d + · · · (C).
(4.5)
Clearly, a0 = wres H −n/d . Now, the order of q2 is at most 2d − 1, therefore its higher order contribution to this development is in principle to a1 ; the order of q3 is at most 3d − 2, so it contributes to a2 at the earliest, and so on. Formula (4.5), obtained through fairly elementary manipulations, is the main result of this section. To illustrate its power, we show how to reap from it a rich harvest of classical results (with a little extra effort). Corollary 4.1 (Connes’ trace theorem). For positive elliptic pseudodifferential operators of order −n on a compact n-dimensional manifold, the Dixmier trace and the Wodzicki residue are proportional: Dtr H =
1 Wres H. n (2π)n
Proof. Let H be of order d = −n in (4.5). We get dH (x, x; λ) ∼ −
1 wres H(x) λ−2 + · · · n (2π)n
(C).
Assume the manifold is compact. We then know that H is a compact operator. Now, 0 (λ) ∼ −λ−2 , ergo NH (λ) ∼ λ−1 , ergo heuristically the argument goes as follows: NH λl (H) ∼ l−1 . A Tauberian argument can be used at this point [37] to ensure that the second asymptotic estimate is valid without the Ces`aro condition; and then the result follows. But this is by no means necessary. One can steal a look at Sect. 6 and, by approaching step functions by elements of S, prove in an elementary way that for any given ε > 0 there is l(ε) such that C(1 + ε) C(1 − ε) < λl (H) < , l(ε) l(ε) where C = n−1 (2π)−n Wres H.
On a noncompact spin manifold, consider now the Dirac operator on the space of spinors L2 (S). The noncommutative integral of |D|−n does not exist. However, if R a(x) dx is defined, it is computable by a noncommutative integral:
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Theorem 4.2. Let a be an integrable function with respect to the volume form on M . Then Z 1 a(x) dx = Wres(a|D|−n ), Cn n (2π)n M where on the right hand side A is seen as a multiplication operator on L2 (S). The constants are C2k = (2π)−k /k! and C2k+1 = π −k−1 /(2k + 1)!! Proof. That follows from Theorem 5.3 of [37] if a is a smooth function with compact support. For a positive and integrable, use monotone convergence on both sides; the general case follows at once. The former is a small step in the direction of a theory of K-cycles (or “spectral triples”, as they are nowadays called) over noncompact manifolds. Corollary 4.3 (Weyl’s estimate). Let NH (λ) denote the counting function of H, a Laplacian on a compact manifold or bounded region M acting on scalar functions. Then n vol M n/2 λ , NH (λ) ∼ n(2π)n where n is the surface area of the unit ball in Rn . Proof. The same type of arguments as in Corollary 4.1 work. Indeed, this estimate is a corollary of it [37]. Next consider the Schr¨odinger operators −1 + V (x), with symbol p(x, ξ) = |ξ|2 + V √(x). We can take a slightly different tack and solve the equation p(x, ξ) = λ by |ξ| = (λ − V (x))+ . Corollary 4.4 (The correspondence principle). For Schr¨odinger operators: Z n n/2 NH (λ) ∼ (λ − V (x))+ dx. n(2π)n See [22], for instance, for the reasons for the terminology. A word of caution is in order here. The development (4.5) cannot be integrated term by term in general. Consider, for instance, the harmonic oscillator hamiltonian H = 21 (−d2 /dx2 + x2 ) on R: according to the theory developed here, its spectral density √ behaves as 1/ λ. If ψn , n ∈ N denote the normalized wavefunctions, then indeed, like in Fourier series theory, √ X λ 2 ψn (x) ∼ π 1 n+ 2 ≤λ
is true and can be independently checked. But wres H√−1/2 is not integrable over the real line, λ. Actually, as we saw in Sect. 2, H (λ) behaves as P∞ so one cannot1 conclude that N−∞ δ(λ − (n + )) = H(λ) + o(λ ) (C), so N (λ) = λH(λ) + o(λ−∞ ) (C). Now, H n=0 2 Corollary 4.2 applies, so we have 2 NH (λ) ∼ 2π
Z
√
2λ
√ − 2λ
p 2λ − x2 dx = λH(λ)
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precisely as it should. (See the discussion in [30].) Consider n-dimensional Schr¨odinger operators with (continuous) homogeneous potentials V (x) ≥ 0, V (ax) = ta V (x). The previous formula gives Z NH (λ) ∝ λn/2+n/a V (x)−n/a dx, Sn−1
and this means that if the cone { x ∈ R : V (x) = 0 } is too big, in the counting number estimate we are heading for trouble [36]. But the “nonstandard asymptotics” that might then intervene do not detract from the validity of the nonintegrated formula (4.5). In the remainder of the section, we focus on the computation of spectral densities for Laplacians. Nothing essential is won or lost by considering general vector bundles, so we work on scalars. The more general Laplacian operator on a Riemannian manifold is (minus) the Laplace–Beltrami operator 1 plus potential vector and scalar potential terms, with symbol n
p(x, ξ) = −g ij (x) ξi ξj + (i0kij (x)ξk + 2Ai (x)ξj )
+ (Ai (x)Aj (x) + i(0kij (x)Ak (x) − ∂i Aj (x))) + V (x)
=: −g ij (x)ξi ξj + B i (x)ξi + C(x). Formula (4.5) would seem to give for this case: dH (x, x; λ) ∼
1 a0 (x)λ(n−2)/2 + a1 (x)λ(n−3)/2 + a2 (x)λ(n−4)/2 + · · · (C). n 2 (2π)
In fact, it will be seen in a moment that a1 = a3 = · · · = 0. Also we know already that a0 (x) = Wres 1−n/2 = n . Our task is to compute the next coefficients; it is a rather exhausting one, whose results can be inferred from the extensive work already carried out [19] on heat kernel expansions (see Sect. 6), so we will limit ourselves to the computation of a2 (x) to illustrate the relative simplicity of our approach. Let n ≥ 3. Write a for g ij (x)ωi ωj , then b for B i (x)ωi and c for C(x). Our method calls for solving for the positive root of a|ξ|2 + b|ξ| + (c − λ) = 0 and substituting this in |ξ|n−1 /(2a|ξ| + b). In diminishing powers of λ, we obtain for the latter the development: n(n − 2)b2 (n − 2)c 1 (n−2)/2 (n − 1)b (n−3)/2 (n−4)/2 − λ − λ + λ +· · · . (4.6) 8a 2 2an/2 2a1/2 One sees that odd-numbered terms in this expansion contain odd powers of ω and thus give vanishing contributions, after the integration on the cosphere. Also, the contribution of the q2 term in (4.2) will start at order 21 n − 2 in λ, the contribution of q3 will start at order 21 n − 3 and so on: the terms in the asymptotic expansion of the density kernels of √ Laplacian operators differ by powers of λ, not of λ, as one would expect on general grounds. It is convenient now to use geodesic coordinates at each point; this is justified by the nature of the result. In these coordinates 0kij (x0 ) = 0 and we have the Taylor expansion X 1 (x − x0 )α ∂ α g(x0 ) gij (x) ∼ δij + Riklj (x0 ) (x−x0 )k (x−x0 )l + 3 α!
as x → x0 ,
|α|≥3
where RikljPdenotes the Riemann curvature tensor. Recall P that the Ricci tensor is given l by Rkj := l Rklj and the scalar curvature by R := kj g kj Rkj .
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From (4.3) one obtains for q2 (x0 , ξ), 1 X i−|α| α ∂ξ −g ij (x0 )ξi ξj + B i (x0 )ξi ∂xα |x=x0 −g ij (x)ξi ξj + B i (x)ξi + C(x) . 2 α! |α|>0
(4.7) Let us take for a moment Ai = 0. Then in geodesic coordinates B i (x0 ) = 0 and it is not hard to see that the only surviving term in (4.7) is equal to 13 Rkj (x0 )ξ k ξ j . Also b = 0 in (4.6). So, in view of (4.4) we are left with two terms at order λ(n−4)/2 , to wit: Z (n − 2)C(x0 ) (n−4)/2 λ dω − 2 n−1 S that comes from the third term in (4.6), and the first order contribution of Z ∂2 n−1 (x, ω; λ) ∂λ2 q2 (x, |ξ|(x, ω; λ)ω)|ξ| . dω p0 (x, |ξ|(x, ω; λ)ω) Sn−1 In effect, q2 contributes here a factor of order λ, so the second derivative in the previous formula gives rise to a term of order λ(n−4)/2 also. To finish the computation, we use Z n ij g Aij , dω Aij ω i ω j = n Sn−1 to get
(n − 2)n 1 (4.8) 6 R(x0 ) − C(x0 ) . 2 Notice that for a pure Laplace–Beltrami operator, the contribution to a2 , when computed in geodesic coordinates, comes exclusively through the q2 term. It remains to convince ourselves that vector potentials give no contribution at this stage. On one hand, the c term in (4.6) would contribute now the extra terms a2 (x0 ) =
−
(n − 2)n j (A Aj + i∂j Aj ). 2
On the other, the term in b2 in the same formula would contribute a term of the form 1 j j 2 (n − 2)n A Aj , and in the computation of q2 there appears now a term (2i/n) ∂j B i j that contributes 2 (n − 2)n ∂j A and thereby cancels the rest. Therefore (4.8) stands also in that case. Actually the coefficients of the Ces`aro asymptotic expansion of d(x, x; λ) are all (local densities of) Wodzicki residues for n odd: a2k (x) = wres 1−n/2+k (x), for k ∈ N. For n even we have a2k = wres 1−n/2+k only as long as −n/2 + k < 0 (the Wodzicki residues of nonnegative powers of a differential operator being of course zero); the following coefficients for the parametric expansion are, in our terminology of Sect. 2 (further explained in the next two sections), not “residues” but “moments”. Note that for n = 2, the coefficient a2 is already a “moment” and cannot be computed by a Ces`aro development. This strikingly different behaviour of the odd-dimensional and the evendimensional cases is concealed in the uniformity of the usual heat kernel method, but it reflects itself in the fact that the corresponding zeta functions have an infinite number of poles, corresponding to the residues, in the odd-dimensional case; and a finite number in the even-dimensional case. One has [38]:
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Res ζH (s) =
s=n/2−k
Z
where ζH (s) =
M
1 2
Wres H k−n/2 ,
hdH (x, x; λ), λ−s iλ dx
(< s 0)
is the kernel of the zeta operator (3.1). A direct, “elementary” proof of the essential identity between Wodzicki residues and residues of the poles of the zeta functions is obviously in the cards, but we will not go further afield here. For a nontrivial use of the noncommutative residue in zeta function theory, have a look at [11].
5. Ces`aro Developments of Counting Functions We consider here operators on compact manifolds without boundary and look at the behaviour of the counting function X 1. N (λ) := λl ≤λ
In order to refresh our intuition, we shall follow a deliberately na¨ıve approach and temporarily forget some of what we learned at the end of last section. Envisage first the scalar Laplacian on T2 with the flat metric; then the counting function is given by the following table: λ 0 1 2
4
5
8
9 10 13 16 17 18 20 25 26 · · ·
N (λ ) 1 5 9 13 21 25 29 37 45 49 57 61 69 81 89 · · · +
No doubt, N (λ) ∼ πλ is a reasonable first approximation; but it is also plain that the remainder undergoes wild oscillations. The precise determination of this remainder is a difficult problem, not unlike the problem of determining the next-to-main term in the asymptotic development of prime numbers. An even simpler and more telling example is provided by the eigenvalues λl of the Laplacian on the n-dimensional sphere. They are given by l+n l+n−2 − , (5.1) λl = l(l +n−1) with respective multiplicities ml = n n for l ∈ N. For example, if n = 2, the eigenvalues are l(l + 1) and the multiplicities are (2l + 1). The leading term is N (λ) ∼
2 n/2 λ n!
as λ → ∞.
On the other hand, asymptotically: N (λ+ ) − N (λ− ) ∼ and so
2 ln−1 , (n − 1)!
λ(1−n)/2 N (λ+ ) − N (λ− ) ∼
2 . (n − 1)!
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Plainly, we cannot find an asymptotic formula for N (λ) with error term o(λ(n−1)/2 ) and continuous main term. The example is taken from H¨ormander’s work [24, 25]. The foregoing is a “Gibbs phenomenon” related to the lack of smoothness of the characteristic function. The problem is “solved” if one is prepared to look at the expansions in the Ces`aro sense. The fact that higher order terms in the asymptotic expansion of the eigenvalues of the Laplacian were to be understood in an averaged sense was pointed out by Brownell [4] many years ago. Going back to tori, consider the distribution of nonvanishing eigenvalues {λl }∞ l=1 of the scalar Laplacian on an n-dimensional torus Tn , with the flat metric. The eigenfuncn tions {φl }∞ l=1 can be seen as nonzero smooth functions in R that satisfy 1φl + λl φl = 0 and the periodicity conditions φl (x1 + 2k1 π, . . . , xn + 2kn π) = φl (x1 , . . . , xn ), where the girths of the torus are taken to be 2π in all directions. Those eigenvalues are given by λk = k12 + · · · + kn2 for k = (k1 , . . . , kn ) ∈ Zn , with corresponding eigenfunctions φk (x1 , . . . , xn ) = eik·x . Thus the λl are the nonnegative integers ql that can be written as a sum of n squares. The multiplicity of each such value is the number of integral solutions of the Diophantine equation ql = k12 + · · · + kn2 . We wish to compute the terms in the parametric and Ces`aro developments of N (λ) next to leading Weyl term (which in fact for this problem goes back to Gauss): N (λ) ∼
n n/2 λ n
as λ → ∞.
To do so, we start with the derivative N 0 (λ); this is nothing but (2π)n d(x, x; λ), but, as advertised, it is more instructive to forget for a while the discussion in Sect. 4. We have: ∞ X X δ(λ − λl ) = δ(λ − k12 − · · · − kn2 ). N 0 (λ) = k∈Zn
l=1
Let φ ∈ D(R), let σ be a large real parameter and set ε = 1/σ, so that ε ↓ 0. Then X φ(ε|k|2 ) hN 0 (σλ), φ(λ)iλ = εhN 0 (x), φ(ελ)iλ = ε k∈Zn
Z
φ(|x|2 ) dx + o(ε∞ ) Z ∞ r(n−2)/2 φ(r) dr + o(ε∞ ). = 21 n ε1−n/2
= ε1−n/2
Rn
0
The third equality is just Lemma 2.10. Hence, weakly: −1+n/2
N 0 (σλ) = 21 n σ −1+n/2 λ+
+ o(σ −∞ )
as σ → ∞,
and upon integration N (σλ) =
n n/2 n/2 λ+ σ + o(σ −∞ ) n
as σ → ∞.
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R. Estrada, J. M. Gracia-Bond´ıa, J. C. V´arilly
Observe that the constant of integration µ0 vanishes, as do all the other moments. Then Theorem 2.4 yields: n n/2 λ + o(λ−∞ ) n
N (λ) =
as λ → ∞.
(C)
Hence the error term, although definitely not small in the ordinary sense, is of rapid decay in the (C) sense. We turn to examine P∞ some cases of spheres. The derivative of the counting function for S2 is N 0 (λ) = l=0 (2l + 1) δ(λ − l(l + 1)). To deal with this case, we need a heavier gun than Lemmata 2.9–2.11. This is provided by: Lemma 5.1. Let f ∈ K0 (Rn ), so that it satisfies the moment asymptotic expansion. If p is an elliptic polynomial and φ ∈ S, then hf (x), φ(tp(x))i ∼
∞ X hf (x), p(x)m i φ(m) (0)
m!
m=0
as t → 0.
tm
Proof. The proof consists in showing that the Taylor expansion φ(tp(x)) =
N X φ(m) (0)p(x)m
m!
m=0
tm + O(tN +1 )
holds not only pointwise, but also in the topology of K(Rn ). Consider now the distribution X ∞
f (λ) := (2λ + 1)
δ(λ − l) − H(λ) ,
l=1
that lies in K0 . Notice that hf (λ), φ(t(λ2 + λ))i =
∞ X
Z
l=1
=
∞ X
∞
(2l + 1) φ(t(l2 + l)) − Z
(2λ + 1)φ(t(λ2 + λ)) dλ
0 ∞
(2l + 1) φ(t(l2 + l)) −
φ(tµ) dµ. 0
l=1
From Lemma 5.1 we conclude that, for φ ∈ S, hN 0 (λ), φ(tλ)i =
∞ X
(2l + 1) φ(t(l2 + l))
l=0 ∞
Z ∼
φ(tµ) dµ + φ(0) + 0
∞ X hf (λ), (λ2 + λ)j i φ(j) (0)
j!
j=0
tj
The parametric expansion of N 0 (λ) is thus N 0 (λ/t) ∼ H(λ) + δ(λ)t +
∞ X (−1)j µj δ (j) (λ) j=0
j!
tj+1
as t ↓ 0,
as t ↓ 0.
On Summability of Distributions and Spectral Geometry
243
where the “generalized moments” µj are given by µj = hf (λ), (λ2 + λ)j i =
∞ X
Z
∞
(2l + 1)(l2 + l)j −
(2λ + 1)(λ2 + λ)j dλ
(C).
0
l=1
It follows that N 0 (λ) ∼ H(λ) + o(λ−∞ ) (C) as λ → ∞. In view of our gymnastics with Riemann’s zeta function in Sect. 2, the computation of the µj presents no difficulties. We obtain 2 µ0 = 2ζ(−1) + ζ(0) = − , 3
µ2 = 2ζ(−5) + 4ζ(−3) =
1 , 15 and so on. On integrating, we get µ1 = 2ζ(−3) + ζ(−1) = −
N (λ/t) ∼
8 , 315
µ3 = 2ζ(−7) + 9ζ(−5) + ζ(−3) = −
1 1 4 0 λ H(λ) + H(λ) + δ(λ) t + δ (λ) t2 + · · · t 3 15 315
2 , 105
as t ↓ 0,
(5.2)
and N (λ) ∼ λ H(λ)+ 13 H(λ)+o(λ−∞ ) (C). Note that the λ0 th order term in the Ces´aro development for N (λ) comes from the first moment. The curvature of a sphere Sn is given by R = n(n − 1), so the second term in the development is precisely what we had expected. We look now at the derivative of the counting function for the Laplace–Beltrami simpler to consider the operator 1 − 1, for which we have, operator on S3 . It is slightly P ∞ according to (5.1): N 0 (λ) = l=0 (l + 1)2 δ(λ − (l + 1)2 ). Consider the distribution X ∞ 2 f (λ) := (λ + 1) δ(λ − l) − H(λ + 1) , l=0
lying in K0 . We have: hf (λ), φ(t(λ + 1) )i = 2
∞ X
Z (l + 1) φ(t(l + 1) ) − 2
2
l=0
∞
(λ + 1)2 φ(t(λ + 1)2 ) dλ.
−1
One sees that the moments all cancel: hf (λ), (λ + 1)2j i = ζ(−2j − 2) = 0, for j ∈ N. Therefore we get simply Z ∞ √ 1 φ(u) u du as t ↓ 0, hN 0 (λ), φ(tλ)i ∼ 3/2 2t 0 and thus in this case we collect just the Weyl term N (λ) ∼
λ3/2 H(λ) 3
(C)
as λ → ∞.
(5.3)
We may reflect now that the counting number for these Laplacians on S2 , S3 behave in the expected way for even and odd dimensional cases, respectively. For a generalized Laplacian which is the square of a Dirac operator the qualitative picture is the same. In particular, the Chamseddine–Connes expansion corresponds to n = 4, whereupon the counting functional behaves in much the same way as the one for S2 . Therefore, formal application of the Chamseddine–Connes Ansatz to the characteristic function of the spectrum, as done in [6, 26] misses the terms involving δ and its derivatives —whose physical meaning, if any, is unclear to us.
244
R. Estrada, J. M. Gracia-Bond´ıa, J. C. V´arilly
6. Spectral Density and the Heat Kernel Now we tackle the issue of the small-t behaviour of the Green functions associated to an elliptic pseudodifferential operator H. These are the integral kernels of operator-valued functions of H, of the form G(t, x, y) = hdH (x, y; λ), g(tλ)iλ , where g, as already advertised, will in this section belong (or can be extended) to the Schwartz space S (i.e., we deal with the standard theory as opposed to the framework sketched at the end of Sect. 3). The basic question is whether G(t, x, y) has an asymptotic expansion as t ↓ 0. In effect, we shall see immediately how to obtain from the (C) asymptotic expansion for the spectral density an ordinary asymptotic expansion for Green functions. The emphasis in recent years has been on Abelian type expansions, the so-called heat kernel techniques [19]. It is common folklore that Ces`aro summability implies Abel summability, but not conversely. As we just claimed, one can go from the Ces`aro expansion to the heat kernel expansion. The reverse implication does not work quite the same. If we know the coefficients of the heat kernel expansion and we independently know that a Ces`aro type expansion for the spectral density exists, we can infer the coefficients of the latter from the former. But it may happen that the formal Abel–Laplace type expansion does not say anything about the “true” asymptotic development. √ For instance, if f (λ) := sin λ e λ for λ > 0, then limλ→∞ f (λ) (C) does not exist, since no primitive of f can have polynomial order in λ. Even so, one can show that k(t) = hf (λ), e−tλ i has a Laplace expansion k(t) ∼ a−1 t−1 + a0 + a1 t + · · · as t ↓ 0, that is, limλ→∞ f (λ) = a−1 (A). To get an example of a bounded function with this behaviour, one uses the fact that fm (λ) = sin λ1/m obeys limλ→∞ fm (λ) = 0 (C, N ) only for N > m, together with Baire’s theorem, to construct a bounded function f (λ) = P −k fmk (λ) that does not have a Ces`aro limit as λ → ∞, but for which f (λ) → 0 k≥1 2 in the Abel sense. In order to relate our Ces`aro asymptotic expansions with heat kernel developments, we need to examine expansions of distributions f (λ) that may contain nonintegral powers of λ. Suppose that {αk }k≥1 is a decreasing sequence of real numbers, not including negative integers, and suppose further that f ∈ S 0 , supported in [0, ∞), has the Ces`aro asymptotic expansion X X c k λα k + bj λ−j (C) as λ → ∞. f (λ) ∼ j≥1
k≥1
It follows from Theorem 32 of [15] and from Theorem 2.5 that f has the following parametric development: f (σλ) ∼
X k≥1
ck (σλ+ )αk +
X
bj Pf((σλ)−j H(λ)) +
j≥1
X (−1)m µm δ (m) (λ) m! σ m+1
as σ → ∞, where the “generalized moments” µm are given by X X k µm = hf (x) − c k xα bj Pf(x−j H(x)), xm i, + − k≥1
(6.1)
m≥0
(6.2)
j≥1
and where Pf denotes a “pseudofunction” R ∞ [14] obtained by taking the Hadamard finite part, that is: hPf(h(x)), g(x)i := F.p. 0 h(x)g(x) dx if supp h ⊆ [0, ∞). In particular,
On Summability of Distributions and Spectral Geometry
hPf(x
−j
Z
∞
H(x)), g(x)i = F.p. 0
245
g(x) dx xj
j−1 (k) j−2 X X 1 g (0) k g (k) (0) x . g(x) − dx − j k! k!(j − k − 1) 1 0 x k=0 k=0 (6.3) Notice that taking the finite part involves dropping a logarithmic term proportional to g (j−1) (0). This has the consequence that Pf(x−j H(x)) fails to be homogeneous of degree −j by a logarithmic term; indeed, Z
∞
=
g(x) dx + xj
Z
1
Pf((σλ)−j H(σλ)) = σ −j Pf(λ−j H(λ)) +
(−1)j δ (j−1) (λ) log σ . (j − 1)! σ j
Consequently, hf (λ), g(tλ)iλ ∼
X k≥1
+
ck t
−αk −1
X
∞
F.p.
Z bj tj F.p.
λαk g(λ) dλ
0 ∞ 0
j≥1
+
Z
g (j−1) (0) g(λ) log t dλ − λj (j − 1)!
X µm g (m) (0) tm . m!
(6.4)
m≥0
−λ The heat kernel development R ∞may be recovered by taking g(λ) = e for λ ≥ 0. In that case, dαk is integral, F.p. 0 λαk g(λ) dλ = 0(αk + 1) and g (j−1) (0) = (−1)(j−1) . From this it is clear that the heat kernel of a pseudodifferential operator may generally contain logarithmic terms. Indeed, by harking back to (4.5), on using (6.4) we prove:
Corollary 6.1. The general form of the (coincidence limit of) the heat kernel for an elliptic pseudodifferential operator of order d on a compact manifold M of dimension n is given by X
K(t, x, x) ∼
γj−n (x)t(j−n)/d +
j−n∈dN / +
X
βj−n (x)t(j−n)/d log t +
γj−n (x) =
rm (x)tm
r=1
j−n∈dN+
as t ↓ 0, where
∞ X
0((n − j)/d) aj (x), d(2π)n
and similarly for the other coefficients. (See [21, Cor. 4.2.7].) Now suppose we know a priori that f (λ) has a Ces`aro asymptotic expansion in falling powers of λ, and that we also know that 8(t) := hf (λ), e−tλ iλ has an asymptotic expansion as t ↓ 0 without log t terms. Then it follows that all bj = 0 in (6.1), i.e., there are no negative integral exponents in the Ces`aro development of f , and consequently the constants µm are the moments of f . Thus (6.4) simplifies to 8(t) ∼
X k≥1
ck 0(αk + 1) t−αk −1 +
X (−1)m µm tm . m!
m≥0
246
R. Estrada, J. M. Gracia-Bond´ıa, J. C. V´arilly
This is precisely the case for a (generalized) Laplacian: if n is odd, only half-integer powers of λ appear in the spectral density and logarithmic terms in the heat kernel are thereby ruled out. Notice that the Ces`aro development for an odd dimensional Laplacian need not terminate. For even dimensions, the term k = n/2 is proportional to wres H 0 λ−1 and later terms are proportional to wres H r λ−r−1 . However, since H r is a differential operator, its local Wodzicki residue vanishes for r ∈ N, and the Ces`aro development terminates at the λ0 term. However, as we have seen, at this point the moments (6.2) enter the picture. It has become a habit to write the diagonal of the heat kernel for a Laplacian in the form ∞ X bk (x, x) tk/2 , K(t, x, x) ∼ (4πt)−n/2 k=0
where n is the dimension of the manifold and b0 (x, x) = 1. We see now that bk (x, x) = 0 for k odd, whereas 2k a2k (x) n (n − 2)(n − 4) . . . (n − 2k)
b2k (x, x) =
for k > 0.
A similar formula holds off-diagonal. As we have noted, these expansions are local in the sense that they do not distinguish between a finite and an infinite region of Rn , say. However, the smallness of the terms after the first is not uniform near the boundary, and hence the “partition function” Z
K(t, x, x) dx ∼ (4πt)−n/2
K(t) := M
∞ X
bk tk/2 ,
(6.5)
k=0
with b0 = vol(M ) for scalars, has an expansion with nontrivial boundary terms in general, starting to contribute in b2 [33]. As for the examples, the expansion (6.5) for S2 was first obtained as the partition function of a diatomic molecule [34] and is well known to physicists. On using vol(S2 ) = 4π, we read Mulholland’s expansion directly by looking at (5.2): KS2 (t) ∼
4 2 1 1 1 + + t+ t + ··· t 3 15 315
as t ↓ 0.
As for the SU (2) group manifold, from (5.3), on using vol(S3 ) = 2π 2 and et1 = et e−t(1−1) , the partition function is seen immediately to be √ π KS3 (t) ∼ 3/2 et . 4t We turn at last to the Chamseddine–Connes expansion. The theory of Ces`aro and parametric expansions justifies (1.1), in the following way. We work in dimension n = 4 and take H = D2 , a generalized Laplacian, acting on a space of sections of a vector bundle E, over a manifold without boundary. The kernel of its spectral density satisfies dD2 (x, x; λ) ∼
rk E 1 λ+ wres D−2 (x) 16π 2 32π 4
(C)
as λ → ∞.
Integrating over M and using the formulas of this section with t = 3−2 , we then get
On Summability of Distributions and Spectral Geometry
247
Z ∞ Z ∞ 1 4 2 2 Tr φ(D /3 ) ∼ λφ(λ) dλ + b2 (D )3 φ(λ) dλ rk E 3 16π 2 0 0 X + (−1)m φ(m) (0) b2m+4 (D2 ) 3−2m as t ↓ 0, 2
2
m≥0
where (−1)m b2m+4 (D2 ) = 16π 2 µm (D2 )/m! are suitably normalized, integrated moment terms of the spectral density of D2 . Thus, we arrive at (1.1). We finally take stock of the status of the Chamseddine–Connes development. If φ ∈ S, then the development becomes a bona fide asymptotic expansion. However, if one wishes to use (for instance) the counting function ND2 (λ ≤ 32 ), which does not lie in S, then the present formulae are not directly applicable and one must proceed like in Sect. 5; moreover the expansion beyond the first piece is only valid in the Ces`aro sense. We close by noting that third piece of the Chamseddine–Connes Lagrangian has interesting conformal properties; this is better studied through the corresponding zeta function at the origin [35]. That term is definitely not a Wodzicki residue but a moment; as to whether this fact has any physical significance, the present authors are not of one mind. Acknowledgement. Heartfelt thanks to S. A. Fulling for sharing his ideas with us prior to the publication of [13]. We wish to thank M. Asorey, E. Elizalde, H. Figueroa, D. Kastler, F. Lizzi, C. P. Mart´ın, A. Rivero, T. Sch¨ucker and J. Sesma for fruitful discussions and G. Landi for a question that motivated the paragraph on harmonic oscillators in Sect. 4. JMGB and JCV acknowledge support from the Universidad de Costa Rica; JMGB also thanks the Departamento de F´ısica Te´orica de la Universidad de Zaragoza and JCV the Centre de Physique Th´eorique (CNRS–Luminy) for their hospitality.
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14. Estrada, R., Kanwal, R. P.: Regularization, pseudofunction and Hadamard finite part. J. Math. Anal. Appl. 141, 195–207 (1989) 15. Estrada, R., Kanwal, R. P.: Asymptotic Analysis: a Distributional Approach. Boston: Birkh¨auser 1994 16. Figueroa, H.: Function algebras under the twisted product. Bol. Soc. Paran. Mat. 11, 115–129 (1990) 17. Fulling, S. A.: The local geometric asymptotics of continuum eigenfunction expansions. I. Overview. SIAM J. Math. Anal. 13, 891–912 (1982) 18. Gelfand, I. M., Shilov, G. E.: Generalized Functions I. New York: Academic Press, 1964 19. Gilkey, P. B.: Invariance Theory, the Heat Equation and the Atiyah–Singer Theorem, 2nd edition. Boca Raton: CRC Press, 1995 20. Grossmann, A., Loupias, G., Stein, E. M.: An algebra of pseudodifferential operators and quantum mechanics in phase space. Ann. Inst. Fourier (Grenoble) 18, 343–368 (1968) 21. Grubb, G.: Functional Calculus of Pseudodifferential Boundary Problems. Boston: Birkh¨auser, 1986 22. Gurarie, D.: The inverse spectral problem. In: Forty More Years of Ramifications: Spectral Asymptotics and its Applications, Fulling, S. A., Narcowich, F. J., eds., Texas A&M University, College Station (1991), pp. 77–99 23. Hardy, G. H.: Divergent Series. Oxford: Clarendon Press, 1949 24. H¨ormander, L.: The spectral function of an elliptic operator. Acta Math. 121, 193–218 (1968) 25. H¨ormander, L.: The Analysis of Linear Partial Differential Operators III. Berlin: Springer, 1985 26. Iochum, B., Kastler, D., Sch¨ucker, T.: On the universal Chamseddine–Connes action. I. Details of the action computation. Preprint hep-th/9607158 27. Kalau, W., Walze, M.: Gravity, noncommutative geometry and the Wodzicki residue. J. Geom. Phys. 16, 327–344 (1995) 28. Kastler, D.: The Dirac operator and gravitation. Commun. Math. Phys. 166, 633–643 (1995) 29. Lagrange, J.-L.: Nouvelle m´ethode pour r´esoudre les e´ quations litt´erales par la moyen des s´eries. M´em. Acad. Royale des Sciences et Belles-lettres de Berlin 24, 251–326 (1770) 30. Landi, G.: An introduction to noncommutative spaces and their geometry. Preprint hep-th/9701078 31. Łojasiewicz, S.: Sur le valeur et la limite d’une distribution en un point. Studia Math. 16, 1–36 (1957) 32. Mart´ın, C. P., Gracia-Bond´ıa, J. M., V´arilly, J. C.: The Standard Model as a noncommutative geometry: The low energy regime. Preprint hep-th/9605001, Phys. Rep., in press 33. McKean, H. P., Singer, I. M.: Curvature and the eigenvalues of the Laplacian. J. Diff. Geom. 1, 43–69 (1967) P∞ 1 2 (2n + 1)e−σ(n+ 2 ) . Proc. Camb. Philos. Soc. 24, 34. Mulholland, H. P.: An asymptotic expansion for 0 280–289 (1928) 35. Rosenberg, S.: The Laplacian on a Riemannian manifold. Cambridge: Cambridge University Press 1997 36. Solomyak, M. Z.: Asymptotics of the spectrum of the Schr¨odinger operator with nonregular homogeneous potential. Math. USSR Sbornik 55, 19–37 (1986) 37. V´arilly, J. C., Gracia-Bond´ıa, J. M.: Connes’ noncommutative differential geometry and the Standard Model. J. Geom. Phys. 12, 223–301 (1993) 38. Wodzicki, M.: Noncommutative residue I: Fundamentals. In Manin, Yu. I. (ed.) K-theory, Arithmetic and Geometry. Lecture Notes in Mathematics Vol. 1289, Berlin: Springer, 1987, pp. 320–399 Communicated by A. Connes
Commun. Math. Phys. 191, 249 – 264 (1998)
Communications in
Mathematical Physics c Springer-Verlag 1998
Enriques Surfaces, Analytic Discriminants, and Borcherds’s 8 Function Jay Jorgenson1,? , Andrey Todorov2 1
School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA Department of Mathematics, University of California, Santa Cruz, CA 95064, USA, and Institute of Mathematics, Bulgarian Academy of Sciences
2
Received: 24 July 1995 / Accepted: 21 March 1997
Abstract: In [Bor 96], Borcherds constructed a non-vanishing weight 4 modular form 8 on the moduli space of marked, polarized Enriques surface of degree 2 by considering the twisted denominator function of the fake monster Lie algebra associated to an automorphism of order 2 of the Leech lattice fixing an 8-dimensional subspace. In [JT 94] and [JT 96], we defined and studied a meromorphic (multi-valued) modular form of weight 2, which we call the K3 analytic discriminant, on the moduli spaceQof marked, polarized, K3 surfaces of degree 2d; in certain cases, including when d = pk , where pk are distinct primes, our meromorphic form is actually a holomorphic form. Our construction involves a determinant of the Laplacian on a polarized K3 surface with respect to the Calabi-Yau metric together with the L2 norm of the image of the period map with respect to a properly scaled holomorphic two form. Since the universal cover of any Enriques surface is a K3 surface, we can restrict the K3 analytic discriminant to the moduli space of degree 2 Enriques surfaces. The main result of this paper is the observation that the square of our degree 2 analytic discriminant, viewed as a function on the moduli space of degree 2 Enriques surfaces, is equal to the Borcherd’s 8 function, up to a universal multiplicative constant. This result generalizes known results in the study of generalized Kac-Moody algebras and elliptic curves, and suggests further connections with higher dimensional Calabi-Yau varieties, specifically those which can be realized as complete intersections in some, possibly weighted, projective space. 1. Introduction Let E be an elliptic curve over C, viewed as the complex plane C modulo the Z lattice generated by 1 and τ where τ = a + ib with b > 0. Let det ∆E be the zeta regularized product of the non-zero eigenvalues of the Laplacian relative to the unit area flat metric on E which acts on the space of smooth functions on E, and let dz be the canonical ?
Partially supported through a Sloan Fellowship.
250
J. Jorgenson, A. Todorov
holomorphic one form on E. By a direct calculation, which is essentially Kronecker’s first limit formula, it can be shown that det ∆E = |η(τ )|4 , kdzk2L2 where η(τ ) is the Dedekind eta function, defined by the infinite product η(τ )24 = qτ
∞ Y
1 − qτn
24
with qτ = e2πiτ .
n=1
One of the many fascinating features of this formula is the fact that the spectral invariant det ∆E and the L2 norm of the holomorphic one form dz can be used to obtain the Dedekind eta function η(τ ), which has numerous algebraic, among other, properties. In [JT 94] and [JT 96], we used the above realization of the Dedekind eta function as a guide in order to define and study a meromorphic form on the moduli space of marked, polarized K3 surfaces. Briefly, our construction is as follows. Let (X, e) be a polarized, algebraic K3 surface of degree 2d equipped with its canonical Calabi-Yau metric which is compatible with the polarization e and which gives X total volume one. Let {ω} be a meromorphic family of holomorphic two forms on MdK3,mpa , the moduli space of marked, polarized, algebraic K3 surfaces. Let det ∆(X,e) denote the zeta regularized product of the non-zero eigenvalues of the Laplacian which acts on the space of smooth functions on X. In [JT 94] it was shown that there exists a meromorphic (multi-valued) function fK3,ω,2d on MdK3,mpa such that (det ∆(X,e) )2 = |fK3,ω,2d |2 . kωk2L2 The moduli space MdK3,mpa is a Zariski open set in h2,19 , the symmetric space associated Nd to the group SO0 (2, 19), and there is an integer Nd such that the function fK3,ω,2d extends to a meromorphic form on h2,19 with respect to a certain arithmetic subgroup which depends on d (and the index of the abelianization of this arithmetic subgroup). Nd vanishes on h2,19 \ MdK3,mpa and possibly has additional zeros and The form fK3,ω,2d poles coming from the family of forms {ω}. In certain cases, incloding when d = 2, it is shown in [JT 94] that one can construct a normalized family of forms {ωnor } such that Nd has divisor whose support is contained in h2,19 \ MdK3,mpa . fK3,ω nor ,2d In this article, we shall prove that in the case d = 2, there is a particular choice of 2 when restricted to a certain subforms {ωnor } such that our analytic discriminant fK3,ω,4 2 variety of h2,19 satisfies a product formula. More specifically, we shall consider fK3,ω nor ,4 when restricted to the subvariety of marked, polarized, algebraic K3 surfaces of degree 2 which admit a fixed point free involution which is compatible with the polarization 2 as a function on the moduli space of and marking. Equivalently, we will view fK3,ω nor ,4 marked, polarized Enriques surfaces of degree 2. Using results from [JT 94] and [JT 95], 2 satisfies a few basic characterizing properties. In the recent we will show that fK3,ω nor ,4 article [Bor 96], Borcherds constructed a non-vanishing weight 4 modular form 8 on the moduli space of marked, polarized Enriques surfaces of degree 2 by considering the twisted denominator function of the fake monster Lie algebra associated to an automorphism of order 2 of the Leech lattice fixing an 8-dimensional subspace. Borcherds’s 8
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function is defined through a product formula which immediately extends to the symmetric space h2,10 associated to the group SO0 (2, 10), in which the moduli space of marked, polarized, Enriques surfaces of degree 2 is a Zariski dense subvariety. Our main 2 = c4 8. In other words, the result is that there exists a constant c4 such that fK3,ω nor ,4 holomorphic form which we construct by considering the spectral theory on marked, polarized Enriques surfaces of degree 2 coincides, up to a universal multiplicative constant, with the function constructed by Borcherds in his study of the fake monster Lie algebra. The contents of this article are as follows. In Sect. 2, we state the basic properties from the study of K3 surfaces and Enriques surfaces which we will need in our work. In Sect. 3, we will recall the construction of the analytic discriminant associated to a marked, polarized K3 surface of degree 2d, and then we show how the analytic discriminant can be used to construct a meromorphic modular form on the moduli space of marked, polarized Enriques surfaces of degree d. Finally, in Sect. 4, we shall recall results from [Bor 96] and prove that the Borcherds’s 8 function is related to our analytic discriminant in the case d = 2. In Sect. 5 we study our analytic discriminant for general more degree d polarized Enriques surfaces, and we relate different degree Enriques discriminants. 2. Basic Properties of K3 and Enriques Surfaces Let us review some basic properties of K3 surfaces and Enriques surfaces. For a more general and complete discussion, the reader is referred to [Ast 85]. A K3 surface X is a compact, complex two dimensional manifold with the following properties. a) There exists a non-zero holomorphic two form ω. b) H 1 (X, OX ) = 0. For the purposes of this article, we will assume that all surfaces are projective varieties. From the defining properties, one can prove that the canonical bundle on X is trivial. In [Sh 67], the following topological properties of K3 surfaces are proved. The surface X is simply connected, and the homology group H2 (X, Z) is a torsion free abelian group of rank 22. The intersection form h , i on H2 (X, Z) has the properties: a) hu, ui = 0 mod (2); b) det(hei , ej i) = −1, where {ei } is a basis of H2 (X, Z); c) the symmetric form h , i has signature (3,19). Theorem 5 from p. 54 of [Ser 73] implies that as an Euclidean lattice H2 (X, Z) is isomorphic to the K3 lattice 3K3 , where H2 (X, Z) ∼ = 3K3 = H3 ⊕ (−E8 )2 with
h H=
0 1
1 0
i3
being the hyperbolic lattice. Let α = {αi } be a basis of H2 (X, Z) with intersection matrix 3K3 . The pair (X, α) is called a marked K3 surface. Let e ∈ H 1,1 (X, R) be the class of a hyperplane section. The triple (X, α, e) is called a marked, polarized K3 surface. The degree of the polarization is the integer 2d such that he, ei = 2d. From [PSS 71] and [Ku 77] we have that the moduli space of isomorphism classes of marked,
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polarized, algebraic K3 surfaces of a fixed degree is equal to an open dense set in the symmetric space h2,19 = SO0 (2, 19)/[SO(2) × SO(19)]. Let
0K3,d = {φ ∈ Aut(H 2 (X, Z)) | hφ(u), φ(v)i = hu, vi and φ(e) = e}.
The moduli space of isomorphism classes of polarized, algebraic K3 surfaces of a fixed degree 2d, which we denote by MdK3,pa , is isomorphic to a Zariski open set in the quasiprojective variety 0K3,d \h2,19 . If we allow our surfaces to have singularities which are at most double rational points, then the corresponding moduli space of isomorphism classes of marked, polarized, algebraic surfaces is equal to the entire symmetric space h2,19 . In other words, marked and polarized surfaces corresponding to points in the space h2,19 \ MdK3,mpa are those surfaces for which the projective embedding corresponding to any power of the polarization is singular with singularities which are double rational points. Specifically, the relation between MdK3,mpa and h2,19 is through the period map, which we now describe. The period map π for a marked K3 surface (X, α) is defined by integrating the holomorphic two form ω along the basis α of H2 (X, Z), meaning Z π(X, α) = (. . . , ω, . . .) ∈ P21 . αi
The Riemann bilinear relations hold for π(X, α), meaning hπ(X, α), π(X, α)i = 0 and hπ(X, α), π(X, α)i > 0. Choose a primitive vector e ∈ H2 (X, Z) such that he, ei = 2d, for some integer d. As in [PSS 71], one has the description of h2,19 as ¯ > 0 and hu, ei = 0}. h2,19 = {u ∈ P(H2 (X, Z) ⊗ C) : hu, ui = 0, hu, ui Results from [PSS 71] and [Ku 77] combine to prove that the period map is a surjection d d → MK3,pa be the natural map onto a Zariski open set in h2,19 . Let πmar,d : MK3,mpa which forgets the marking. From the surjectivity of the period map, it follows that πmar,d d coincides with the action of 0K3,d on MK3,mpa . d The set h2,19 \ MK3,mpa can be described as follows: Given a polarization class e ∈ 3K3 , set Te to be the orthogonal complement to e in 3K3 (i.e., Te is the transcendental lattice), so then we have the realization of h2,19 as one of the components of ¯ > 0}. {u ∈ P(Te ⊗ C) : hu, ui = 0 and hu, ui Define the set 1(e) = {δ ∈ 3K3 : he, δi = 0 and hδ, δi = −2}, and, for each δ ∈ 1(e), define the hyperplane H(δ) = {u ∈ P(Te ⊗ C) : hu, δi = 0}. Let
He =
[ δ∈1(e)
H(δ) ∩ h2,19 ,
Enriques Surfaces, Analytic Discriminants, and Borcherds’s 8 Function
and set
253
De = 0K3,d \He .
Results from [Ma 72], [Si 83], [Si 84] and [To 80], to name a few references, combine to give the relation DK3,e = 0K3,d \h2,19 \ MdK3,pa . Geometrically, points in the space DK3,e correspond to algebraic K3 surfaces with a class e which is pseudo-ample, meaning a divisor class for which a sufficently large multiple of the class yields a projective embedding of X with double rational singularities (see [Ma 72]). In Sect. 3, we shall show that for certain d, the divisor DK3,e is an ample divisor on the quasi-projective variety 0K3,d \h2,19 . An Enriques surface Y is a compact, complex surface for which the canonical class KY is not trivial but KY⊗2 is trivial, and H 1 (Y, OY ) = 0. Using the exponential cohomology sequence together with topological considerations, it can be shown that all Enriques surfaces are projective (in contrast to the case of K3 surfaces). In particular, as stated on p. 270 of [BPV 84], one can show that the map Pic(Y ) → H2 (Y, Z) is an isomorphism. As in the case of K3 surfaces, the structure of the homology group H2 (Y, Z) can be canonically determined. The intersection form h , i on H2 (Y, Z) is such that as an Euclidean lattice H2 (Y, Z) satisfies the (non-canonical) isomorphism H2 (Y, Z) ∼ = 3Enr ⊕ Z/2Z, where 3Enr is the Enriques lattice 3Enr = H ⊕ (−E8 ). Define H2 (Y, Z)f = H2 (Y, Z)/Tor(H2 (Y, Z)), where Tor(H2 (Y, Z)) denotes the torsion subgroup of H2 (Y, Z), so then we have the (non-canonical) isomorphism H2 (Y, Z)f ∼ = 3Enr . In particular, H2 (Y, Z)f is a torsion free abelian group of rank 10 and signature (1, 9). Let α = {αi } be a basis of H2 (Y, Z)f with intersection matrix 3Enr . The pair (Y, α) is called a marked Enriques surface. An H-marked Enriques surface is a pair (Y, η), where Y is an Enriques surface and η is an embedding of the lattice H into 3Enr . As shown in [Dol 84], every Enriques surface admits an H-marking. The two-torsion subgroup of H2 (Y, Z) implies the existence of an unramified degree two cover of Y which, by Noether’s formula, can be shown to be a K3 surface. Therefore, many aspects in the study of Enriques surfaces can be derived from the study of K3 surfaces. In particular, following the discussion in [BPV 84] and [Na 85], we have the following Torelli theorem for Enriques surfaces. Let % be the involution on the K3 lattice 3K3 defined by %(z1 ⊕ z2 ⊕ z3 ⊕ x ⊕ y) = (−z1 ⊕ z3 ⊕ z2 ⊕ y ⊕ x). Let 3+K3 and 3− K3 be the %-invariant and %-anti-invariant subspaces, respectively. Observe that the unimodular lattice 21 3+K3 is isometric to the Enriques lattice 3Enr . The class represented by the holomorphic two-form ω is anti-invariant; this follows from the fact that the canonical class of an Enriques surface is not trivial. Therefore,
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if γ ∈ 3+K3 then h[ω], γi = 0. Let X be a K3 surface which admits a fixed point free involution σ, and let Y = X/hσi be the quotient Enriques surface. Lemma 19.1 of [BPV 84] proves the existence of an isometry φ : H2 (X, Z) → 3K3 such that φ ◦ σ ∗ = % ◦ φ. In other words, one can choose a marking of the K3 surface X which is consistent with a marking of the underlying Enriques surface Y . Therefore, there is a one to one correspondence between marked Enriques surfaces and pairs (Y, φ), where Y is an Enriques surface and φ is as above. Since every Enriques surface is projective, we define a marked, polarized Enriques surface of degree d to be the triple (Y, φ, e), where (Y, φ) is a marked Enriques surface and e is the class of an ample divisor with he, eiY = d. Equivalently, one can choose the class of an ample divisor e˜ on X which is invariant under the involution σ and with he, ˜ ei ˜ X = 2d, since σ induces a map σ ∗ : H2 (Y, Z)f → H2 (X, Z) which satisfies the property hσ ∗ (x), σ ∗ (y)iX = 2hx, yiY (see p. 203 of [Na 85]). There is a one to one correspondence between marked, polarized Enriques surfaces of degree d and marked, polarized K3 surfaces of degree 2d with fixed point free involution and whose polarization class lies in 3+K3 and is invariant under the involution. For simplicity, we shall say that a marked, polarized K3 surface with the above restrictions on marking and polarization covers the marked, polarized Enriques surface. Define the space Enr = P(3− K3 ⊗ C) ∩ K3 . From p. 282 of [BPV 84], it can be shown that Enr is isomorphic to two copies of h2,10 , the symmetric space associated to the group SO0 (2, 10). Let 0Enr be the discrete group defined by 0Enr = restr3− {g ∈ Aut(3K3 ) : g ◦ % = % ◦ g}. K3
We define the period map of marked, polarized Enriques surfaces to be the restriction of the period map for marked, polarized K3 surfaces which cover marked, polarized Enriques surface with image into 0Enr \Enr . As with the study of K3 surfaces, one can consider a Torelli theorem for marked, polarized Enriques surfaces. For the current considerations, it suffices to study Enriques surfaces with an almost polarization of degree 2, by which we mean a class of a line bundle of self-intersection 2 which has non-negative intersection with the class of any hyperplane section. It is stated on p. 5 of [St 91] that the results from [Ho 78a] and [Ho 78b] imply that any Enriques surface can be endowed with an almost polarization of degree 2. Following [Na 85], one has the following result. Let l ∈ 3− K3 be such that hl, li = −2. It is shown on p. 283 of [BPV 84] such that no point of the hyperplane Hl = {p ∈ Enr : hp, liY = 0} can be the period point of a marked Enriques surface. Further, Theorem 21.4 on p. 286 of [BPV 84] asserts that all points in the variety
Enriques Surfaces, Analytic Discriminants, and Borcherds’s 8 Function
0Enr \Enr \
0Enr \
[
255
! Hl
l
occur as period points of Enriques surfaces (with an almost polarization of degree 2). However, as stated on p. 168 of [Dol 84], the period map is a finite to one map, hence additional data associated to any Enriques surface is necessary. Let us denote the above variety by M4K3,% , since the above variety is indeed the moduli space of degree 4 K3 covers of Enriques surfaces, which are necessarily of degree 2. In order to obtain a moduli space of Enriques surfaces, with possible additional data thus yielding a variety which covers M2K3,% , we utilize the following result from [Dol 84]: Let M2Enr,H denote the moduli space of H-marked Enriques surfaces with an almost polarization of degree 2. Then there is a finite index subgroup 0Enr,H of 0Enr such that we have the isomorphism ! [ 2 ∼ M = 0Enr,H \Enr \ 0Enr,H \ Hl . Enr,H
l
Finally, let us note that the set DEnr = 0Enr \
[
! Hl
l
is an irreducible divisor on M2K3,% (see also p. 702 of [Bor 96]). When descending from the moduli space of polarized, algebraic K3 surfaces of degree 4, one can describe the moduli space M2Enr,H as follows. ˜ Enr denote the subgroup of 0Enr which is defined as follows. Let e1 and e2 be Let 0 two vectors which generate a copy of H in 3K3 , i.e. e21 = e22 = 0 and e1 e2 = 1. Set e = e1 + e2 and take the polarization on the K3 surface to be given by two distinct, orthogonal embeddings of e into 3K3 , meaning in the isomorphism 3K3 ∼ = H3 ⊕ (−E8 )2 the polarization is given by the vector E = (0, e, e, 0, 0). Note that the polarization is invariant, meaning ρ(E) = E. Define ˜ Enr = restr − {g ∈ Aut(3K3 ) : g ◦ % = % ◦ g and g(E) = E}. 0 3K3
Then we claim that one has the isomorphism M2Enr,H
∼ ˜ Enr \ Enr \ =0
[
! Hl
,
l
˜ Enr ∼ i.e. 0 = 0Enr,H (see p. 40 of [St 88]). We prefer this description of M2Enr,H since the discussion readily generalizes to the setting of general degree polarization. Equivalently, − the subgroup 0Enr,H of 0Enr can be described as follows. Let 3−,∗ K3 be the dual to 3K3 , so then we have an isomorphism 1 − ∼ 1 H(2)/H(2) ⊕ E8 (−2)/E8 (−2). 3−,∗ K3 /3K3 = 2 2 Then 0K3,H is the subgroup of 0K3 corresponding to those elements which respect the above decomposition into direct summands (see p. 41 of [St 88]). For more discussion of this point, see [Dol 84] or [St 88]. Finally, we note that M2Enr,H is a finite cover of − M2K3,% with covering group 3−,∗ K3 /3K3 .
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3. Analytic Discriminants for Enriques Surfaces In this section we shall recall the definition of the analytic discriminant associated to any marked, polarized, algebraic K3 surface. The results in this section are proved in complete detail in [JT 94] and [JT 95]. When the given marked, polarized, algebraic K3 surface covers a marked, polarized Enriques surface, we shall relate our K3 discriminant to information constructed from the Enriques surface. From this, we shall define an analytic discriminant for Enriques surfaces. To begin, let us recall the definition of the Weil-Petersson metric on the moduli space MdK3,mp . Let (X, e) be a polarized K3 surface of degree 2d, and let T(X,e) be the sheaf of holomorphic vector fields on (X, e). From Kodaira-Spencer deformation theory, we can of the moduli space of MdK3,mp at the point (X, e) identify the tangent space TMd K3,mp
with H 1 (X, T(X,e) ). The existence of the non-vanishing holomorphic two-form ω on X implies that we can identify H 1 (X, T(X,e) ) with H 1 (X, ), where is the sheaf of to holomorphic one forms on X. One can then deduce that the tangent space TMd K3,mpa
the moduli space MdK3,mpa at the point (X, α, e) can be identified with the space H 1 (X, )0 = {u ∈ H 1 (X, ) | hu, ei = 0}.
We view any φ ∈ H 1 (X, T(X,e) ) as a linear map from 1,0 to 0,1 pointwise on X. Given φ1 and φ2 in H 1 (X, T(X,e) ), the trace of the map φ1 φ2 : 0,1 → 0,1 at a point x ∈ X with respect to the unit volume Calabi-Yau metric g (meaning a K¨ahler-Einstein metric compatible with the given polarization class e) is simply X nl¯ (φ1 )kl¯ ((φ2 )m Tr(φ1 φ2 )(x) = ¯ . n ¯ )g gkm k,l,m,n
The existence of a Calabi-Yau metric on X compatible with the polarization e is guaranteed by Yau’s theorem [Y 78]. We define the Weil-Petersson metric on MdK3,mpa via the inner product Z hφ1 , φ2 i =
Tr(φ1 φ2 )volg X
d on the tangent space of MK3,mpa at (X, α, e). It is shown in [To 89] that the Weil-Petersson metric on MX,α,e is equal to the restriction of the Bergman metric on h2,19 . Therefore, the Weil-Petersson metric is a K¨ahler metric with K¨ahler form µWP . Since h2,19 is simply connected, there exists a non-vanishing holomorphically varying family of holomorphic two-forms over MdK3,mpa . For any such family {ω}, consider the function on MdK3,mpa defined by Z 2 ¯ kωkL2 = hω, ωi = ω ∧ ω. X
In [To 89] and [Ti 88] it was proved that log kωk2L2 is a potential for the Weil-Petersson metric. The following result from [JT 94] proves the existence of a second potential for the Weil-Petersson metric.
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Proposition 1. Let (X, e) be a polarized, algebraic K3 surface of degree 2d, and let µ denote the unit volume K¨ahler-Einstein form on X which is compatible with the polarization e. Let {ω} be a non-vanishing, holomorphically varying family of holomorphic two forms on h2,19 . a) Let det ∆(X,e) denote the zeta regularized product of the non-zero eigenvalues of the Laplacian which acts on the space of smooth functions on X. Then (det 1(X,e) )2 c = 0, dd log kωk2L2 or, equivalently −ddc log(det 1(X,e) )2 = −ddc log kωk2L2 = µWP . In other words, − log(det 1(X,e) )2 is a potential for the Weil-Petersson metric on MdK3,mpa . b) There is a holomorphic function (possibly multi-valued) fK3,ω,d on h2,19 which vanishes on the codimension one set h2,19 \ MdK3,mpa such that |fK3,ω,d ([X, α, e])| =
(det 1(X,e) )2 kωk2L2
;
whence fK3,d does not vanish on MdK3,mpa . The reader is referred to [JT 94] and [JT 95] for complete details of the proof of Proposition 1. Important Note: The function fK3,ω,d constructed in Proposition 1(b) is a possibly multi-valued function with divisor contained in h2,19 \ MdK3,mpa . At this point, we do not assert any behavior of fK3,ω,d with respect to the discrete group 0K3,d . For this, one would need to impose further restrictions on the family of forms {ω}, which may require the family of forms to be a meromorphic family, so then fK3,ω,d would have additional zeros and poles corresponding to the divisor of {ω} (see Remark 3). Remark 2. Let ∆(0,1) (X,e) denote the Laplacian which acts on the space of smooth (0, 1) forms relative to the unit volume Calabi-Yau metric which is compatible with the polar2 ization e. Using the Hodge decomposition, it is argued in [JT 95] that ∆(0,1) (X,e) = (∆(X,e) ) . With this in mind, our variational formula [JT 95] proves the following result, which is an analogue of Mumford’s theorem for curves. Let L0 = π∗ KX /Mdmpa and L1 = det R1 π∗ . NK3,d If NK3,d = 0K3,d / 0K3,d , 0K3,d , then the sheaf L20 ⊗ L−1 is trivial. 1 Remark 3. Proposition 1 is a generalization of a known result which exists in the setting of elliptic curves. An analogous result in the setting of Calabi-Yau manifolds was established in [JT 95]. For general degree K3 varieties, one can choose a family of NK3,d is a forms {ωd }, which possibly has zeros or poles, for which the function fK3,ω d ,d meromorphic form of weight 2NK3,d with respect to the action by 0K3,d . The ability NK3,d to choose such an integer NK3,d so that the function fK3,ω is single-valued on h2,19 d ,d follows from the Kodaira embedding theorem as applied to the bundle corresponding to automorphic forms of a particular weight, which is an ample bundle (see [Ba 70] and
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p. 191 of [GH 78]). Note that the family of forms {ωd } may have additional zeros or NK3,d poles, which will contribute to the divisor of fK3,ω . By Proposition 1(b), we can view d ,d N
K3,d as a section of the line sheaf (π∗ KX /Md )NK3,d . The holomorphic function fK3,ω d ,d K3,pa fK3,ωd ,d will be called the analytic discriminant for the marked, polarized, algebraic K3 surfaces of degree 2d, relative to the family of forms {ωd }. Finally, if one picks N a different family of forms {ω˜ d }, the modular form fK3,K3,d ω ˜ d ,d will differ from the form
N
K3,d by a multiplicative factor which is a rational function on 0K3,d \h2,19 . fK3,ω d ,d
Remark 4. For certain degree d, one can construct a family of forms {ωd } such that NK3,d d the associated analytic discriminant fK3,ω has divisor which is supported on DK3 = d ,d d h2,19 \ MK3,mpa . For example, in the case 0K3,d has one zero dimensional cusp in BailyQ Borel compactification of 0K3,d \h2,19 , which occurs when d = pj , where {pj } is a set of distinct primes (see [Sc 87]), then one can construct a normalized family of forms {ωd } as follows: 1. Any marked, polarized, algebraic K3 surface is an element of a family of K3 surfaces E → D, where D is the unit disc, such that the monodromy has a Jordan cell of dimension 3; i.e., if T is the monodromy operator, on H2 (X, Z), then (T − id)3 = 0 and (T − id)2 6= 0 (see [To 76] and [JT 94] for details). 2. On the generic fibre Xt of this family, we have, up to sign, a unique cycle γ such that T γ = γ and any other T invariant cycle is an integer multiple of γ. Further, there exists a cycle µ such that T µ = γ + µ. 3. Since h2,19 is contractible, there exists a globally defined, non-vanishing, holomorphically varying family of holomorphic two forms, say ωt ∈ H 0 (h2,19 , π∗ KE d
K3,mpa
where KE d
K3,mpa
/h2,19
/h2,19 ),
is the relative canonical sheaf.
4. In [JT 94], it is shown that the function
Z ωt
φ(t) = γ
is non-vanishing on h2,19 . The normalized family of holomorphic two-forms is defined by {ωnor } = {ωt /φ(t)}. We remark that when following the identical steps in the case of elliptic curves, one constructs the family of holomorphic one forms {dz}. When the family {ωnor } exists, it is immediate that the function |fK3,d ([X, α, e])| · kωnor k2L2 = (det ∆(X,e) )2 is 0d invariant since det ∆(X,e) is independent of the marking of (X, e). Also, we follow NK3,d the standard definition of automorphic form, meaning that fK3,d satisfies a multiplicative transformation law with respect to 0d which involves a power of the functional determinant j(Z, g) appearing as a multiplicative factor (see, for example, [Ba 70] or NK3,d is 0d invariant. [Ba 72]). Equivalently, the function fK3,d
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Let fK3,2 denote the K3 analytic discrminant associated to the normalized family of forms {ωnor } constructed in Remark 4. We now use the function fK3,2 to define a function on M2Enr,mp , hence an analytic discriminant associated to any marked Enriques surface with almost polarization of degree 2. Let (Y, φ, eY ) be any marked, polarized Enriques surface with almost polarization of degree 2, and let (X, α, eX ) be the marked, polarized K3 surface which covers (Y, φ, eY ). We define the degree 2 Enriques analytic discriminant fEnr,2 by fEnr,2 ([Y, φ, eY ]) = fK3,2 ([X, α, eX ]). By Proposition 1(b) and Remark 4, the function fEnr,2 extends to a (multi-valued) holomorphic function on h2,10 (which is one component of Enr which vanishes on DEnr . Remark 5. Using the methods of [Ko 88], one can prove 0Enr,2 / 0Enr,2 , 0Enr,2 has order 4. Indeed, since the Enriques lattice has roots of 2 different norms, its abelianization has at least 2 different elements of order 2, when we conclude 0Enr,2 / 0Enr,2 , 0Enr,2 ∼ = (Z/2Z)2 . 2 Therefore, the function fEnr,2 is a single-valued holomorphic modular form of weight 4 on the quasi-projective variety 0Enr,2 \Enr whose divisor is a positive multiple of the irreducible divisor DEnr .
4. Tube Domains and Borcherds’s 8 Function In this section we shall recall the construction presented in [Bor 96] of a holomorphic modular form on h2,10 relative to 0Enr,E , which is defined to be the image in O(3− K3 ) of the subgroup of O(3K3 ) which consists of those isometries of 3K3 that commute with the involution ρ and leave the degree 4 polarization class E invariant. To begin, we follow the tube domain construction of h2,10 as given in Sect. 1 of [Bor 96]. Let L be the lattice L = −2E8 ⊕ H, so the vector (v, m, n) with v ∈ −2E8 and n, m ∈ Z has norm v 2 + 2mn. In particular, the vectors ρ = (0, 0, 1) where ρ0 = (0, 1, 0) have norm zero. Let M be the lattice M = L ⊕ 2H, and let OM (Z) be the automorphism group of M . Let ¯ M > 0}. M = {w ∈ P(M ⊗ C) : hw, wiM = 0 and hw, wi + The space M has two connected components, and we let OM (Z) be the subgroup of OM (Z) which does not interchange the components. The positive norm vectors of L⊗R form two open cones; choose one cone, √ to be called CL . One of the two components of M can be identified with L ⊗ R + −1CL by the map √ L ⊗ R + −1CL → M ⊗ C
defined by
v 7→ (v, 1, −v 2 /2).
The identification depends on a choice of the cusp, which is equivalent to a choice of a point in the boundary of the moduli space of K3 surfaces corresponding to the
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deepest degeneration of polarized K3 surfaces. Any automorphism of M ⊗ R induces the components an automorphism of M , and if the automorphism does not interchange √ of M , then there is an induced automorphism of L ⊗ R + −1CL . Therefore, as in Theorem 5.3 of [JT 96], we have an isomorphism √ + (Z)\(L ⊗ R + −1CL ) → 0Enr,E \h2,10 . π : OM If d is a vector of M of norm −2, let Hd be the set of points of M which are orthogonal to d. Then results from [BPV 84] and [Na 85] show that ! [ −1 ∼ Hd . π (DEnr ) = OM (Z)\ d
With this, we now have an explicit construction of the moduli space of Enriques surfaces as a tube domain, and we have identified the divisor DEnr in this tube domain. We shall view our Enriques discriminant as a function on this tube domain. We now recall Borcherds’s construction of an automorphic form 8 on the moduli space of Enriques surfaces. Definition 6. Let W be the reflection group √ of the lattice L generated by the reflections of vectors of norm −2. For y ∈ L ⊗ R + −1CL , define (−1)n 8 Y X 1 − e2πinhρ,w(y)iL det(w)e2πihρ,w(y)iL . 8(y) = n>0
w∈W
As stated in [Bor 96], results from [Bor 92] show that the function 8(y) satisfies the following product formula. Let 1+ be the set of vectors of L which have positive inner product with ρ or are positive multiples of ρ. Let {c(n)} be the coefficients of the function ∞ X c(n)qτn = η(τ )−8 η(2τ )8 η(4τ )−8 . f (τ ) = n=−1
Then for all y with Im(y) sufficiently large, we have 8(y) = e2πihρ,yiL
Y
1 − e2πihr,yiL
(−1)hr,ρ+ρ0 iL ·c(hr,riL /2)
.
r∈1+
In other language, Borcherds studies the above functions by considering the twisted denominator formula for an automorphism of the monster Lie algebra coming from an involution of the Leech lattice with an 8-dimensional fixed subspace. The group W is the Weyl group of the fake monster Lie superalgebra, ρ is the Weyl vector, and 1+ is the set of positive roots. We refer the reader to [Bor 92] and [Bor 96] for further discussion of these notions. For now, let us quote the following result from [Bor 96]. Proposition 7. The holomorphic function 8(y) is an automorphic form of weight 4 on √ + (Z). the tube domain L ⊗ R + −1CL with respect to a finite index subgroup 0 of OM The function 8 vanishes to first order along the divisor ! [ Hd D8 = 0\ d
and is non-zero elsewhere on the quasi-projective variety 0\(L ⊗ R +
√
−1CL ).
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The main result of this paper is the following theorem which relates the Borcherds’s 8 function to the analytic discriminant for Enriques surfaces. Theorem 8. Let fEnr,2 be the analytic discriminant for marked, polarized Enriques surfaces with an almost polarization of degree 2, and let 8 be the Borcherds’s 8 function. 2 . Then there is a constant c2 such that 8 = c2 fEnr,2 Proof. Let N be the order of the finite group 0Enr,E / 0Enr,E , 0Enr,E N 2 and consider the quotient fEnr,2 /8 . By Proposition 1 and Proposition 7, the quotient √ is a modular function on the quasi-projective variety 0\(L ⊗ R + −1CL ). Indeed, both forms are sections of the bundle π∗ KM2Enr /h2,10 ⊗ Lχ , where Lχ is the flat bundle corresponding χ ∈ H 1 (M2Enr , Z) which is a finite group isomorphic to the character to 0Enr,E / 0Enr,E , 0Enr,E . Furthermore, both sections vanish on the same irreducible divisor (see [Na 85] and [Bor 96]), so the only possible pole of the quotient lies in the cusp of the Baily-Borel compactification. By Koecher’s principle, any such modular function must be constant, thus concluding the proof. Remark 9. Borcherds used the 8 function to show that the moduli space M2Enr is a quasi-affine variety (see [Bor 96]). We remark here that the realization of M2Enr as a quasi-affine variety also follows from our study of analytic K3 discriminant fK3,2 (see [JT 94] and Sect. 4 of [JT 95]). Specifically, the existence of the normalized family of forms {ωnor } on M2K3,pa which varies holomorpically and does not vanish then implies 2 . that the associated K3 analytic discriminant has divisor which is supported on DK3 2 Therefore, by arguing as in [Bor 96], one concludes that the moduli space MEnr is quasi-affine. 5. Degree d Analytic Discriminants for K3 Covers of Enriques Surfaces In this section we shall compare degree d analytic discriminants when restricted to covers of Enriques surfaces, i.e. marked K3 covers of Enriques surfaces. To compare with the notation of previous sections, we are considering the restriction of the analytic K3 discriminants to the analogue of the space M2K3,% , which is the moduli of degree 2 polarized K3 covers of Enriques surfaces, rather than the space M2Enr,H , which is the moduli of H-marked degree 2 Enriques surfaces. Consider the moduli space of marked, polarized K3 surfaces of degree 2d which admit a fixed point free involution % which preserves the polarization class, i.e., the moduli space of marked, degree d covers of Enriques surfaces. Denote this space by MdK3,%,m . The argument on p. 283 of [BPV 84] applies to show that for any l ∈ 3− K3 such that hl, liY = −2, then no point of the hyperplane Hl = {p ∈ Enr : hp, liY = 0} can be the period point of an Enriques surface of any degree. Therefore, the Torelli theorem for K3 surfaces of degree 2d asserts that one has the isomorphism [ MdK3,%,m = h2,10 \ Hl . l
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Let GdK3,%,e be the image in O(3− K3 ) of the subgroup of O(3K3 ) which consists of isometries of 3K3 which commute with the involution ρ and which leave the polarization class e invariant. Then MdK3,% = GdEnr,%,e \MdK3,%,m . Let fK3,ω,d be a K3 analytic discriminant associated to the family of forms {ω} such that fK3,ω,d is a meromorphicmodular form with respect to same finite index subgroup of GK3,%,e (see Remark 3). Let fEnr,ω,d be the restriction of the K3 analytic discriminant to the subvariety of K3 surfaces which cover Enriques surfaces, and view fEnr,ω,d as a function on MdK3,% . With this, we have the following result. Theorem 10. Let fEnr,ωd1 ,d1 and fEnr,ωd2 ,d2 be analytic discriminants for marked, polarized Enriques surfaces of degrees d1 and d2 , respectively, viewed as meromorphic functions on h2,10 . Then there is a quasi-projective variety Z which is a finite degree 1 2 and MdK3,% , and integer N , and a rational function h on Z such cover of both MdK3,% that N
fEnr,ωd1 ,d1 /fEnr,ωd2 ,d2
= h.
Proof. Both analytic discriminants fEnr,ωd1 ,d1 and fEnr,ωd2 ,d2 can be viewed as meromor1 2 phic functions on h2,10 . Since the groups GdK3,% and GdK3,% are arithmetic, the subgroup 1 2 ∩ GdK3,% 0K3,%,d1 ,d2 = GdK3,% 1 2 is of finite index in both GdK3,% and GdK3,% . Therefore, we can view fEnr,ωd1 ,d1 and fEnr,ωd2 ,d2 as meromorphic forms of the same weight on the quotient space
Z = 0K3,%,d1 ,d2 \h2,10 . Indeed, both discriminants are sections of the bundle π∗ KZ/h2,10 ⊗ Lχ , where Lχ is the flat bundle corresponding to the character χ ∈ H 1 (Z, Z) which is a finite group isomorphic to 0K3,%,d1 ,d2 /[0K3,%,d1 ,d2 , 0K3,%,d1 ,d2 ]. By choosing N so that χN is the trivial character, the proof of the theorem is complete. 6. Concluding remarks One of the interesting aspects of Theorem 8 is that we have now connected two seemingly different fields of investigation: One field being the study of analytic discriminants and the construction of holomorphic functions via spectral invariants; the other field being Borcherds’s theory of fake monster Lie superalgebras and “monstrous moonshine”. Further study of this connection, as well as related questions, is certainly warranted. Along this line, we offer the following speculation which extends the above observations into the setting of K3 surfaces. Let (X, e) be a polarized, algebraic K3 surface. In [JT 94], we show the existence of three cycles γ0 , γ1 and γ2 such that the following holds. If (X, e) is a family of degenerating K3 surfaces with monodromy operator T for which (T − id)2 6= 0 yet (T − id)3 = 0, then
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1 T (γ0 ) = γ0 T (γ1 + γ0 ) = γ1 + γ0 T (γ2 + γ1 + γ0 ) = γ2 + γ1 + γ0 . 2 Define He = {u ∈ H2 (X, Z) : hu, ei = 0} and set 3K3,e = {u ∈ He : hu, γ0 i = hu, γ2 i = 0}. One can show that the cup product restricts to 3K3,e to an inner product of signature (1, 18). Let 1K3,e = {` ∈ 3K3,e : h`, `i = −2}, and write 1K3,e = 1+K3,e ∪ (−1+K3,e ) (see [JT 96]). Define B(3K3,e ) as the Lie algebra associated to the Cartan matrix formed from 1+K3 . With all this, we speculate the existence of a constant cd and integer Nd such Nd is equal to the denominator of the Weyl character formula associated to the that cd fK3,d algebra B(3K3 ). Given the results from [JT 97] and Theorem 8 above, we can now speculate the existence of generalized Kac-Moody algebras associated to any polarized Calabi-Yau variety such that an associated projective embedding realizes the variety as a complete intersection. With such a construction, one can then study the generalization of Borcherds’s 8 function and the possibility of extending Theorem 8 to the setting of general complete intersections. Acknowledgement. Both authors benefited greatly through support from NSF grants and from the Institute for Advanced Study. Both authors thank the referee for comments which improved the exposition of the paper and for pointing out a gap in a previous version. We are grateful to V. Nikulin for writing the article [Ni 96], which pointed out that we omitted an important point in our definition of the K3 analytic discriminant, thus leading to a correction in the case when the degree of the K3 surface is such that the corresponding moduli space has more than a single zero dimensional boundary component in its Baily-Borel compactification (see [Sc 87]). Further ongoing work due to Gritsenko and Nikulin are related to results in this and a subsequent article by the authors.
References [Ast 85] G´eom´etrie des surfaces K3: Modules et p´eriodes. Ast´erisque 126, Paris: Soci´et´e math´ematique de France (1985) [Ba 70] Baily, W. L. Jr.: Eisenstein series on tube domains. In: Problems in Analysis: A Symposium in Honor of Salomon Bochner, Gunning, R. C. ed., Princeton: Princeton University Press (1970), pp. 139–156 [Ba 72] Baily, W.L.Jr.: Introductory Lectures on Automorphic Forms. Publications of the Math. Soc. of Japan 12, Princeton: Princeton University Press, 1972 [BPV 84] Barth, W., Peters, C. and van de Ven, A.: Compact Complex Surfaces. Ergebnisse der Math. 4, New York: Springer-Verlag, 1984 [Bor 92] Borcherds, R.: Monstrous moonshine and monstrous Lie superalgebras. Invent. Math. 109, 405–444 (1992) [Bor 95] Borcherds, R.: Automorphic forms on Os+2,2 (R) and infinite products. Invent. Math. 120, 161–213 (1995) [Bor 96] Borcherds, R.: The moduli of Enriques surfaces and the fake monster Lie superalgebra. Topology 35, 699–710 (1996) [BR 75] Burns, D. Jr., and Rapoport, M.: On the Torelli problem for K¨ahlerian K3 surfaces. Ann. scient. ¨ Norm. Sup. 4e s´erie, t. 8, 235–274 (1975) Ec. [CD 89] Cossec, F. and Dolgachev, I.: Enriques Surfaces I. Progress in Mathematics 79, Boston: Birkh¨auser, 1989 [Dol 84] Dolgachev, I.: On automorphisms of Enriques surfaces. Invent. Math. 76, 163–177 (1984) [GH 78] Griffiths, P. and Harris, J.: Principles of Algebraic Geometry. New York: John Wiley and Sons, 1978 [Ho 78a] Horikawa, E.: On the periods of Enriques surfaces. I. Math. Ann. 234, 73–88 (1978) [Ho 78b] Horikawa, E.: On the periods of Enriques surfaces. II. Math. Ann. 235, 217–246 (1978)
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Jorgenson, J., and Todorov, A.: An analytic discriminant for polarized algebraic K3 surfaces. To appear in Proceedings of the Montreal Conference on Complex Geometry and Mirror Symmetry ’95 [JT 95] Jorgenson, J., and Todorov, A.: A conjectured analogue of Dedekind’s eta function for K3 surfaces. Mathematical Research Letters. 2, 359–376 (1995) [JT 96] Jorgenson, J., and Todorov, A.: Analytic discriminants for manifolds with zero canonical class. In: Manifolds and Geometry, ed. P. de Bartolomeis, F. Tircerri, and E. Vesentini, Symposia Mathematica 36, 223–260 (1996) [JT 97] Jorgenson, J., and Todorov, A.: Algebraic properties of the K3 analytic discriminant Part I: Elliptic K3 surfaces of degree 2. In preparation [Ko 88] Kond¯o, S.: On the Albanese variety of the moduli space of polarized K3 surfaces. Invent. Math. 91, 587–593 (1988) [Ku 77] Kulikov, V.: Degenerations of K3 surfaces and Enriques surfaces. Math. USSR Izv. 11, 957–989 (1977) [Ma 72] Mayer, A.: Families of K3 surfaces. Nagoya Math. J. 48, 1–17 (1972) [Na 85] Namikawa, Y.: Periods of Enriques surfaces. Math. Ann. 270, 201–222 (1985) [Ni 96] Nikulin, V.: The remark on discriminants of K3 surfaces moduli as sets of zeros of automorphic forms. J. Math. Sci. 81, 2738–2743 (1996) [PSS 71] Piatetski-Shapiro, I. I. and Shafarevich, I.: A Torelli theorem for algebraic surfaces of type K3. Math USSR Izv. 5 547–588 (1971) (Collected Mathematical Papers. New York: Springer-Verlag, 1989, pp. 516–557) [Sc 87] Scattone, F.: On the compactification of moduli spaces for algebraic K3 surfaces. Memorirs of the AMS 374 (1987) [Ser 73] Serre, J.-P: A Course in Arithmetic. Graduate Texts in Mathematics. 7, New York: Springer-Verlag, 1973 [Si 83] Siu, Y.-T.: Every K3 surface is K¨ahler. Invent. Math. 73, 131–150 (1983) [Si 84] Siu, Y.-T.: Some recent developments in complex differential geometry. Proceedings of the International Congress of Mathematicians. 1 Warsaw 1983. Warsaw: PWN, 1984, pp. 287–297 [Sh 67] Shafarevich, I., et. al.: Algebraic Surfaces. Proc. of the Steklov Institut 75 (1965) (translated by the AMS (1967)) [St 88] Sterk, H.: Compactifications of the period space of Enriques surfaces: Arithemtic and geometric aspects. Proefschrift Nijmegen 1988 [St 91] Sterk, H.: Compactifications of the period space of Enriques surfaces, Part I. Math. Z. 207, 1–36 (1991) [Ti 88] Tian, G.: Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric. In: Math. Aspects of String Theory. Yau, S.-T. ed., Singapore: World Scientific, 1988, pp. 629–646 [To 76] Todorov, A.: Finiteness conditions for monodromy of families of curves and surfaces. Izv. Akad Nauk USSR, 10, 749–762 (1976) [To 80] Todorov, A. N.: Applications of K¨ahler–Einstein–Calabi–Yau metric to moduli of K3 surfaces. Invent. Math. 61, 251–265 (1980) [To 89] Todorov, A.: The Weil-Petersson geometry of the moduli space of SU (n ≥ 3) (Calabi-Yau) manifolds I. Commun. Math. Phys. 126, 325–346 (1989) [To 94] Todorov, A.: Applications of some ideas of mirror symmetry to moduli spaces of K3 surfaces. Preprint (1994) [Y 78] Yau, S.-T.: On the Ricci curvature of a compact K¨ahler manifold and the complex Monge-Ampere equation I. Commun. Pure Appl. Math. 31, 339–411 (1978)
[JT 94]
Communicated by S.-T. Yau
Commun. Math. Phys. 191, 265 – 282 (1998)
Communications in
Mathematical Physics c Springer-Verlag 1998
Associativity of Quantum Multiplication Gang Liu Department of Mathematics, State University of New York at Stony Brook, Stony Brook, NY 11794, USA Received: 17 November 1994 / Accepted: 8 May 1997
Abstract: We proved the associativity of the multiplication of quantum cohomology for a monotone compact symplectic manifold V for which c1 (A) > 1 for any effective class A ∈ H2 (V ). The same proof also works for any positive compact symplectic manifold with c1 (A) > 1.
1. Introduction This paper is a short version of my dissertation on Quantum cohomology. Quantum cohomology was introduced by the physicist C.Vafa for compact Kahler manifolds (cf. [V]). Recently several efforts have been made to build up a mathematical foundation for it in the setting of symplectic geometry by using the Gromov-Witten invariant (cf. [MS1, RT1, RT2]). The purpose of this paper is to give a proof of the associativity of this new multiplication for a monotonic compact symplectic manifold V for which, c1 (A) > 1 for every effective class A ∈ H2 (V ). A symplectic manifold (V, ω) is called monotonic, if there exists a constant λ > 0 such that for any f ∈ π2 (V ), ω(f ) = λc1 (f ∗ T V ). By rescaling the symplectic form ω, we may assume that λ = 1. Quantum cohomology QH ∗ (V ) of a monotonic symplectic manifold is additively just the usual cohomology of V with coefficients in R[q], the polynomial ring in q. However, its multiplicative structure is a certain deformation of the ordinary cup-product which can be described as follows. If a01 , a02 , a03 are three cocycles in V with Poincar´e duals a1 , a2 , a3 of cycles respectively, then the quantum multiplication a01 ∗a02 can be defined by specifying with all a03 . This triple index can be defined P its pairing 0 0 0 ω(f ) sign(f )q , where f is running through all discrete as follows: < a1 ∗ a2 , a3 >= f
J-holomorphic sphere f : (S 2 , 0, 1, ∞) → (V, a1 , a2 , a3 ). Intuitively this means that we are counting discrete J-holomorphic spheres with three marked points 0, 1, ∞ mapping to the three cycles a1 , a2 , a3 respectively. In order to justify this definition and to see
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what is involved in order to prove the associativity of this multiplication, we need to introduce some basic notations first. Let J (ω) be the set of all ω-compactible almost complex structures for a given symplectic manifold (V, ω). It is well-known that J (ω) is a non-empty contractible set. For a given almost complex structure J ∈ J (ω) and A ∈ H2 (V ), we define the moduli space M(A, J) = {f |f : S 2 −→ V is J − holomorphic and simple, [f ] = A}. It is proved, for example in [G] and [M], that there exists a dense set Jreg of the second category in J (ω) such that for any J ∈ Jreg and A ∈ H2 (V ) the moduli space M(A, J) is a smooth manifold of dimension 2c1 (A) + dim(V ). The p-fold evaluation map eA,J : M(A, J) × (S 2 )p−3 −→ V p is given by: (f, z1 , · · · , zp ) 7→ (f (0), f (1), f (∞), f (z1 ), · · · , f (zp−3 )). Now we can give a formal definition of the quantum multiplication for monotonic symplectic manifolds. Assume that dim(V ) = 2n and that a1 , a2 and a3 are cycles of codimension 2α1 , 2α2 and 2α3 respectively, which are represented by submanifolds of V and which are Poincar´e dual to a01 , a02 and a03 of cocycles in H ∗ (V, R) respectively. Fix a generic J. Then the triple index is defined as follows: Definition 1.1. < a01 ∗ a02 , a03 >=
X
ω(A) #(e−1 , A,J (a1 × a2 × a3 ))q
A∈H2 (V )
where the sum is running over all such A that c1 (A) + n = α1 + α2 + α3 . Here eA,J is the 3-fold evaluation map and #(e−1 A,J (a1 × a2 × a3 )) is the oriented intersection number of eA,J and a1 × a2 × a3 . (This makes sense since MA,J , V 3 and a1 × a2 × a3 are oriented). Remark 1.2. (1) Using the Gromov compactness theorem and the monotonicity asω(A) is sumption, one can show that the coefficient #e−1 A,J (a1 × a2 × a3 ) before q finite for generic J. (2) Since α1 + α2 + α3 ≤ 3n, we have ω(A) = c1 (A) ≤ 2n. Therefore the triple index < a01 ∗ a02 , a03 > is in R[q]. (3) The zero order term of a01 ∗ a02 is just the ordinary cup product a01 ∪ a02 . This can be seen as follows: The condition that ω(A) P = c1 (A) = 0 implies that f is a constant map for any f ∈ MA,J . Therefore A,ω(A)=c1 =0 #e−1 A,J (a1 × a2 × a3 ) is nothing but the triple intersection number of a1 , a2 and a3 . According to our definition, in order to know (a01 ∗ a02 ) ∗ a03 one needs first to know what the Poincar´e dual (a01 ∗ a02 )0 is. For generic J, consider the 2-fold evaluation map eA,J : M(A, J) → V 2 given by f 7→ (f (0), f (1)). Let M(A, J; a1 , a2 ) = e−1 A,J (a1 × a2 ), which is a smooth manifold of dimension 2(c1 (A) + n − α1 − α2 ). If we define ‘singular cycle’ 8A,J (a1 , a2 ) : M(A, J; a1 , a2 ) → V given by f 7→ f (∞), then it is easy to see that
Associativity of Quantum Multiplication
Lemma 1.3.
267
(a01 ∗ a02 )0 =
X
8A,J (a1 , a2 )q ω(A) .
A∈H2 (V )
Let
eA,B,J : M(A, J) × M(B, J) → V 6
be the evaluation map, given by (f, g) 7→ (f (0), f (1), g(0), g(1), f (∞), g(∞)). For two spherical classes A, B ∈ H2 (V ) and cohomology class a0i ∈ H ∗ (V ) with its Poincar´e dual ai of codimension 2αi , i = 1, · · · , 4, we define the moduli space of (A, B)-cuspcurves to be M(A, B, J; a1 , a2 ; a3 , a4 ) = e−1 A,B,J (a1 × a2 × a3 × a4 × 1), where 1 is the diagonal. This moduli space is a smooth manifold of dimension 2(c1 (A + B) + n − α1 − α2 − α3 − α4 ) for generic J. It is easy to see that when c1 (A + B) + n = α1 + α2 + α3 + α4 , #M(A, B, J; a1 , a2 ; a3 , a4 ) = #(8A,J (a1 , a2 ) ∩ 8B,J (a3 , a4 )). Lemma 1.4. X
=
< (a01 ∗ a02 ) ∗ a03 , a04 > #(8A,J (a1 , a2 ) ∩ 8B,J (a3 , a4 ))q ω(A+B) ,
A,B∈H2 (V )
the sum being taken over c1 (A + B) + n =
P4
i=1
αi .
Proof. By definition, X
< (a01 ∗ a02 ) ∗ a03 , a04 >=
sign(f )q ω(B) · q ω(A) ,
A,B∈H2 (V ), f ∈M(B,J)
with 8A,J (a1 , a2 ) of (a01 ∗ a02 )0 which is equal to P f (0) being in the component ω(A) , f (1) being in a3 , f (∞) being in a4 and A∈H2 (V ) 8A,J (a1 , a2 )q 2(c1 (B) + n) = codim8A,J (a1 , a2 ) + 2α3 + 2α4 . Geometrically, 8A,J (a1 , a2 ) is just the collection of A-curves intersecting with a1 and a2 at 0 and 1 respectively, and its dimension is 2(c1 (A) + n − α1 − α2 ). Therefore, each such B-curve f which is counted above, determines an A-curve g, so gives rise to an (A, B)-cusp-curve (f, g), which intersects the given 4-cycles and satisfies the condition that c1 (A + B) + n = α1 + α2 + α3 + α4 . In other words, each such f determines an element of M(A, B, J; a1 , a2 ; a3 , a4 ). The conclusion follows immediately. Now for any P ∈ H2 (V ) with c1 (P ) + n = α1 + α2 + α3 + α4 we define the extended Gromov-Witten invariant X 0 #(8A,J (a1 , a2 ) ∩ 8B,J (a3 , a4 )), 9P,J (a1 , a2 ; a3 , a4 ) = A,B∈H2 (V )
the sum being taken over A + B = P. Then Lemma 1.4 can be restated as
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X
< (a01 ∗ a02 ) ∗ a03 , a04 >=
90P,J (a1 , a2 ; a3 , a4 )q ω(P ) ,
P ∈H2 (V )
the sum being taken over c1 (P ) + n =
P4
i=1
αi . Similarly,
< a01 ∗ (a02 ∗ a03 ), a04 >= (−1)2α1 (2α2 +2α3 ) < (a02 ∗ a03 ) ∗ a01 , a04 > X = (−1)2α1 (2α2 +2α3 ) 90 (P, J)(a2 , a3 ; a1 , a4 )q ω(P ) , P ∈H2 (V )
the sum running over such P that c1 (P ) + n = α1 + α2 + α3 + α4 . Therefore, associativity will follow if we can prove that 90P,J (a1 , a2 ; a3 , a4 ) = (−1)2α1 (2α2 +2α3 ) 80P,J (a2 , a3 ; a1 , a4 ). This is equivalent to the fact that 90P,J is graded-commutative. To this end, we will relate 90P,J to the Gromov-Witten invariant 9P,J , which is graded-commutative by its definition. It can be defined as follows: Fix P, ai , i = 1, · · · , 4 as above with c1 (P ) + n = α1 + α2 + α3 + α4 . Consider the 4-fold evaluation map eP,J : M(P, J) × (S 2 − {0, 1, ∞}) → V 4 . Let M(P, J; a1 , a2 , a3 , a4 ) be e−1 P,J (a1 ×a2 ×a3 ×a4 ) which is a smooth submanifold of M(P, J)×(S 2 −{0, 1, ∞}) of dimension 2. Consider the restriction of the projection π2 : M(P, J) × (S 2 − {0, 1, ∞}) → S 2 − {0, 1, ∞} to M(P, J; a1 , a2 , a3 , a4 ) and denote it by π. Picking up a generic point z ∈ S 2 − {0, 1, ∞}, we define 9P,J (a1 , a2 , a3 , a4 ) = #(π −1 (z)). We will use Mz (P, J; a1 , a2 , a3 , a4 ) to denote π −1 (z). It is proved in [MS1] that when c1 (A) > 1, π −1 (z) is finite for generic J and that the Gromov-Witten invariant 9P is well defined, independent of the particular choices of z, J and representatives ai of cycles. It is also easy to see that 9P is graded-commutative. Therefore the associativity will follow if one can prove the following special decomposition rule: Theorem 1.5. If V is monotonic with c1 (A) > 1 for any effective class A ∈ H2 (V ), then we have 9P,J (a1 , a2 , a3 , a4 ) = 90P,J (a1 , a2 ; a3 , a4 ). To prove this, we will construct a family of gluing maps with a gluing parameter z ∈ C∗ , a M(A, B, J; a1 , a2 ; a3 , a4 ) → Mz (P, J; a1 , a2 , a3 , a4 ) #z : A,B∈H2 (V )
when |z| is large enough, where the disjoint union is taken over A+B = P and c1 (P )+n = α1 + α2 + α3 + α4 . The existence of these maps is established in Sect. 3 by using the gluing technique there. We can prove the special decomposition rule by showing that #z is an orientation-preserving bijection when |z| large enough. The injectivity of #z is
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more or less obvious since the domain of #z is a finite set. The surjectivity will follow from the uniqueness part of Lemma 3.9 of the Picard method if one can prove that when |z| is large enough, for any f ∈ Mz (P, J; a1 , a2 , a3 , a4 ) there exists an approximate J-holomorphic curve g = g1 χz g2 ,which is made from a cusp-curve, (g1 , g2 ) ∈ M(A, B, J; a1 , a2 ; a3 , a4 ), such that the C 0 − distance of f and g is small. This can be shown by analyzing what could happen for a sequence fn ∈ Mzn (P, J; a1 , a2 , a3 , a4 ) when |zn | tends to infinity, which is done in Sect. 4. The proof given in this paper was carried out during the fall of 1993 and completed in early 1994 as my dissertation. During the preparation of this paper, we learned that Y. Ruan and G. Tian proved the general decomposition rule, therefore the associativity in [RT1, RT2]. A different proof was given by D. McDuff and D.Salamon in [MS]. Our proof is independent of the above two proofs. 2. Transversality We start with a discussion about the smoothing of S 2 ∨ S 2 and pre-gluing for cusp curves. Let a 1 Si2 = ((C, wi ) (C, wi0 ))/(wi = 0 ), wi i = 1, 2, where w and w0 are the complex coordinates. Then S12 ∨ S22 is given by identifying w10 = 0 in S12 with w20 = 0 in S22 . Let y be the cuspidal point of S12 ∨ S22 , 0L , 1L and 0R , 1R be the points of w1 = 0, 1 in S12 and w2 = 0, 1 in S22 respectively. When |z| ˜ is small enough, the complex sphere S12 #z S22 with gluing parameter z = 2 ∗ z˜ 2 ∈ C and four ‘marked’ points 0L ,1L ,0R ,1R can be constructed as follows: ˜ |z| ˜ 0 2 2 One first cuts off |w10 | ≤ |z| 2 and |w2 | ≤ 2 from S1 and S2 respectively, then glues 2 |z| ˜ | z| ˜ the two remains along 2 < |w10 | < |z| ˜ and 2 < |w20 | < |z| ˜ by the formula w10 ·w20 = z˜2 . It has a ‘left’ and a ‘right’ complex coordinate w1 , w2 respectively with the relation w1 ·w2 = z˜22 . In w1 -coordinate, the points 0L ,1L ,0R ,1R have coordinates w1 = 0, 1, ∞, z˜22 respectively. Therefore the cross-ratio of these four points is z˜22 . Since the cross-ratio is the only invariant for 4-tuples in S 2 under the P SL(2, C) action, we may consider the moduli space Mz (P, J; a1 , a2 , a3 , a4 ) equally as Mz˜ (P, J; a1 , a2 , a3 , a4 ) f is J − holomorphic and simple, = f : S12 #z S22 → V | , f (0L ) ∈ a1 , f (1L ) ∈ a2 , f (0R ) ∈ a3 , f (1R ) ∈ a4 where z = z˜22 . There is another description of S12 #z S22 given by using a cylindrical coordinate, which is what we need in order to do gluing in Sec. 3. We may think of S 2 − {y} as a union of a half sphere Hi and a half infinite cylinder R+ × S 1 with cylindrical
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coordinate (τi , ti ), i = 1, 2, with ∂Hi identified with {0} × S 1 . In this coordinate the previous construction S12 #z S22 will become the following: ˜ + log 2 is cut The part of Si2 − {y}, i = 1, 2, with cylindrical coordinate τ > − log |z| off, and the rest of them are glued along the collars of the length log 2 of the cylinders twisted with an angle arg z˜ 2 . In a cylindrical coordinate, the pre-gluing f1 χz f2 of cusp curve (f1 , f2 ) is defined as follows: Let β be the ‘bump’ function supported in [0, log2 2 ] with β(τ ) being equal to 1 when τ < 0 and β(τ ) being equal to 0 when τ > log2 2 , and βz be the shifting of β by the amount −log|z|, ˜ where z = z˜22 , i.e. βz (τ ) = β(τ − log |z|), ˜ ˜ i = 1, 2. fi (w) if τi (w) < − log |z|, f1 χz f2 (w) = expy (βz (τ1 (w))f˜1 (w) + βz (τ2 (w))f˜2 (w)) otherwise, where (τi , ti ) is the cylindrical coordinate of Si2 − {y} and f˜i is the lifting of fi under the exponential map expy for i = 1, 2. Now we are ready to give the basic analytic setup. Fix p > 2 and A ∈ H2 (V ). Definition 2.1. We define the mapping space p = {f |f : S 2 → V, [f ] = A, kf k1,p < ∞}, B1,A
where kf k1,p is measured by the standard metric on S 2 and some fixed metric on V . By making a suitable choice of a metric on V such that ai is a geodesic submanifold of V for i = 1, · · · , 4, we can similarly define p (z; a1 , a2 , a3 , a4 ) B1,A f : S12 #z S22 → V, kf k1,p < ∞, = f| . f (0L ) ∈ a1 , f (1L ) ∈ a2 , f (0R ) ∈ a3 , f (1R ) ∈ a4 p ’s. To simplify our notation, We will omit the subscript A in B1,A
Definition 2.2. For any f in B1p , the fiber of the tangent bundles T B1p = W1p of B1p at f is W1p (f ) = {ξ|ξ ∈ Lp1 (f ∗ T V )}. Similarly for f in B1p (z; a1 , a2 , a3 , a4 ), the fiber of the tangent bundles of T B1p (z; a1 , a2 , a3 , a4 ) = W1p (z; a1 , a2 , a3 , a4 ) of B1p (z; a1 , a2 , a3 , a4 ) at f is p W 1 (f, z; a1 , a2 , a3 , a4 ) ξ(0L ∈ Tf (0L ) a1 , ξ(1L ) ∈ Tf (1L ) a2 , p ∗ = ξ|ξ ∈ L1 (f T V ), . ξ(0R ) ∈ Tf (0R ) a3 , ξ(1R ∈ Tf (1R ) a4 p p p The bundle L over B1 or B1 (z; a1 , a2 , a3 , a4 ) is defined to be the bundle whose fiber over f is Lp (0,1 (f ∗ T V )). Note that when p > 2, Lp1 (E) ,→ C 0 (E) for any vector bundle E over S 2 . Since in the marked point case the formulae of local charts and trivializations for these Banach manifolds and bundles are similar to the non-marked point case, we will only deal with the later case. Let ι be the injective radius of a fixed metric on V . Consider Uf = {ξ|ξ ∈ W1p (f ), kξk∞ < ι}. Then the maps Expf : Uf → Expf (Uf ) ,→ B1p given by ξ(τ, t) 7→ expf (τ,t) (ξ(τ, t)) form smooth local charts for B1p . Their derivatives DExpf : Uf × W1p → W1p given by
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(Exp(ξ, η))(τ, t) = Dexpf (τ,t) (ξ(τ, t))(η(τ, t)) will give local trivializations for W1p . The local trivializations for Lp1 can be obtained by using a J-invariant parallel transformation coming from a corresponding J-invariant connection which is described in detail, for instance, in [MS], Sect. 3.3. There they also showed that the connection ∇ can be chosen in such a way that T or = 41 N , where N is the O’Neil tensor of J. Now the ∂ J -operator can be thought of as a section of the bundle Lp over Bbp1 given by f 7→ df + J(f ) ◦ df ◦ i. Let ∂ J,f : Uf → Lp be the corresponding non-linear map under the above local charts Expf of W1p and the local trivializations of Lp over Uf , then we have Lemma 2.3. ∂ J,f has the following Taylor expression: ∂ J,f (ξ) = ∂ J (f ) + D∂ J,f (0)ξ + N (ξ), where (1) the first order term E = D∂ J,f (0) : Lp1 (f ∗ T V )) → Lp (0,1 (f ∗ T V ) is given by: 1 E(ξ) = ∇ξ + J(f ) ◦ ∇ξ ◦ i + NJ (∂J (f ), ξ), 4 where the connection ∇ is a J-invariant connection with its torsion proportional to the Nijenhuis tensor NJ . (2) the non-linear part is of the form: N (ξ) = L1 (ξ) ◦ ∇ξ + L2 (ξ) ◦ ∇ξ ◦ i + Q1 (ξ) ◦ du + Q2 (ξ) ◦ du ◦ i, where Li and Qi are linear and quadratic respectively in the sense that there exists a constant C(f ) depending only on f and the ‘geometry’ of V such that kLi (ξ)k∞ ≤ C(f )kξk∞ and kQi (ξ)k∞ ≤ C(f )kξk2∞ respectively for kξk∞ < ι when i = 1, 2. Proof. See [MS], Sect. 4 and [F1], Sect. 2 for the proof of (1) and (2) respectively.
Our next goal in this section is to state results on transversality. In this aspect, the result of [M] about deformation of J-holomorphic curves plays a fundamental role. Following [MS1], we will state it in its linearized form. Lemma 2.4. Given J ∈ J (V, ω), and a J-holomorphic sphere f : S 2 → V, there exists a constant δ such that for every v ∈ Tf (z0 ) V and every pair 0 < ρ < r < δ there exists a smooth vector field ξ(z) ∈ Tf (z) V along f and an infinitesimal almost complex structure Y ∈ C ∞ (End(T V, J, ω)) such that the following hold: (1) Df (ξ) + Y (f ) ◦ du ◦ i = 0, (2) ξ(z0 ) = v, (3) ξ is supported in Br (z0 ) and Y is supported in an arbitrary small neighborhood of f (Br (z0 ) − Bρ (z0 )). Proof. See [MS], Chapter 6.
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In order to state the transversality we need to give some formal definitions about cusp-curves and their evaluation maps. Let Ai ∈ H2 (V ), i = 1, · · · , n, jν ∈ {1, · · · , n − 1}, ν = 2, · · · , n, with jν < ν and D = {A1 , · · · , AN , j1 , · · · , jn }. Then the moduli space of simple cusp-curves of type D is defined as the following: M(D, J) = ( (f, w, z)|f ∈
n Y
) M (Ai , J), w, z ∈ CP n−1 , fjν (wν ) = fν (zν ), ν = 2, · · · , n ,
i=1
where w = (w2 , · · · , wn ) and z = (z2 , · · · , zn ). Note that if Ai = Aj , for some i 6= j in the above definition, we require that fi 6= fj ◦φ for any φ ∈ P SL(2, C). The p-fold evaluation map eD,T,J : M(D, J) × CP p → M p is defined as follows: eD,T,J (f, w, z, m) = (fT (1) (m1 ), · · · , fT (p) (mp )), where T : {1, · · · , p} → {1, · · · , n}. Now we are ready to state the main result on the transversality of eD,T,J . Lemma 2.5. Given P ∈ H2 (V ) and submanifolds a1 , · · · , ap of V in a general position, there exists a dense subset Jreg (ω) of the second category of J (ω) such that for any J ∈ Jreg (ω), eD,T,J is transversal to a1 × · · · × ap for all (D, T ) with c1 (D) = c1 (A1 + A2 + · · · + AN ) ≤ c1 (P ) when restricted to the set of all (f, w, z, m) ∈ M(D, J) × CP p which satisfy the conditions that for any i ∈ {1, · · · , p}, (1) fT (i) (mi ) 6= fT (i) (zT (i) ) and (2) If T (i) = jν for some ν ∈ {2, · · · , n}, fT (i) (mi ) 6= fT (i) (wν ). Proof. Let M(Ai , J ) =
a
M(Ai , J)
J∈J
denote the universal moduli space of class Ai , which is a Banach manifold (see, for example, [MS], Chapter 3). Consider the evaluation map eD,T :
n Y
M(Ai , J ) × CP 2(n−2) × CP p → V 2n−2 × V p
i=1
given by: (f, w, z, m) 7→ (fj2 (w2 ), f2 (z2 ), · · · fjn (wn ), fN (zn ), fT (1) (m1 ), · · · fT (p) (mp )) and the two relevant evaluation maps π1 ◦eD,T and π2 ◦eD,T , where π1 : V 2n−2 ×V p → V 2n−2 and π2 : V 2n−2 × V p → V p are the two projections. From Lemma 3.4, arguing similarly to [MS], we conclude that eD,T is transversal to 1n−1 × (a1 × · · · × ap ) when restricted to the corresponding domain with the same conditions (1) and (2) in the lemma for fixed J, and π1 ◦eD,T and π2 ◦eD,T are transversal to 1n−1 and a1 × · · · × ap respectively. By taking the inverse images of the above three
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submanifolds under the corresponding evaluation maps, which are transversal to them, and then projecting to J , we find a dense subset Jreg (ω) of second category of J (ω) , which has the that for any J ∈ Jreg (ω), the above three evaluation maps, when Qproperty n restricted to 1 M(Ai , J)×CP 2(n−2) ×CP p , are also transversal to the corresponding submanifolds as before. Now M(D, J) = (π1 ◦ eD,T,J )−1 (1n−1 ), and the claim of this lemma is just that π2 ◦ eD,T,J is transversal to a1 × · · · × ap when restricted to the open subset of M(D, J) in the lemma. This follows from the following elementary fact: Let M1 , M2 and M be three manifolds and fi : M → Mi ,i = 1, 2, be smooth maps. Assume that Ni is a submanifold of Mi such that fi is transversal to Ni , i = 1, 2 and (f1 , f2 ) is transversal to N1 × N2 . Let Wi = fi−1 (Ni ) i = 1, 2, and gj = fj |Wi , j 6= i, then gj is transversal to Nj , j = 1, 2. The proof of this fact follows from the corresponding linear algebra lemma by replacing everything above with a linear one. Remark 2.6. For those ‘bad’ points (f, w, z, m) in M(D, J) × CP p , at which at least one of the conditions in the lemma is not satisfied, the above argument is not applicable. Since the two bad cases we need to deal with are similar, we only consider the case where the condition (1) is violated, i.e., we need to consider those points (f, w, z, m) at which fT (i) (zT (i) ) = fT (i) (mi ) for some i ∈ {1, · · · , p}. In this case if we assume, without loss of generality, that i = 1, T (i) = 1, then we can form the (p − 1)-fold evaluation map eD,T 0 ,J from eD,T,J by deleting the T (1) factor, where T 0 : {2, · · · , p} → {2, · · · , n} given by T 0 (i) = T (i), i = 2, · · · , p. Now we can require that for generic J, eD,T 0 ,J is transversal to 1a1 × 1n−1 × (a2 × · · · × ap ) in the proper domain similar to the one in the lemma above. Here 1a1 = 1 ∩ (a1 × a1 ). Proceeding in this way inductively, we can form (p − i)-fold evaluation maps,i = 1, · · · , p, and achieve transversality for generic J. It is easy to see that all the ‘bad ’ points could be covered by an inverse image at some stage. This is what we need in Sect. 4. Now we have special cases of the above lemma. Lemma 2.7. Given P ∈ H2 (V ) and submanifolds ai of V , i = 1, 2, 3, 4, consider all evaluation maps eA,B,J : M(A, J) × M(B, J) → V 6 with A + B = P given by: eA,B,J (φ, ψ) = (φ(0), φ(1), ψ(0), ψ(1), φ(∞), ψ(∞)). Then for generic J, every eA,B,J is transversal to a1 × a2 × a3 × a4 × 1. As its corollary we have Corollary 2.8. If f = (f1 , f2 ) ∈ M(A, B, J; a1 , a2 ; a3 , a4 ), then for generic J Df = (Df1 , Df2 ) : W1p (f, a1 , a2 ; a3 , a4 ) → Lp (0,1 (f ∗ (T V ))) is surjective with kernel of dimension 2(n + c1 (A + B) − α1 − α2 − α3 − α4 ). Here as in Definition 2.2, we write ξ ∈ Lp1 (f ∗ T V ), ξ(0 ) ∈ Tf1 (0) a1 , ξ(1L ) ∈ Tf1 (1) a2 , W1p (f, a1 , a2 ; a3 , a4 ) = ξ| L . ξ(0R ) ∈ Tf2 (0) a3 , ξ(1R ) ∈ Tf2 (1) a4 , ξ(∞L ) = ξ(∞R ) ∈ Tf1 (∞) V = Tf2 (∞) V Proof. The proof of the surjectivity of Df is basically the same as the one for Dg when g ∈ M(P, J), which can be found, for example, in [MS], chapter 3. The formula for the dimension of the kernel comes from Lemma 3.7 and the fact that KerDf = Tf M(A, B, J; a1 , a2 , a3 , a4 ). This is what we need in order to do gluing in the next section.
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3. Gluing In this section we will construct a gluing map a M(A, B, J; a1 , a2 ; a3 , a4 ) → Mz (P, J; a1 , a2 , a3 , a4 ) #z : A+B=P
when |z| is large enough, J generic, and c1 (P ) + n = α1 + α2 + α3 + α4 . Given a cusp-curve f = (f1 , f2 ) ∈ M(A, B, J; a1 , a2 ; a3 , a4 ) and zα ∈ C ∗ we will use the following short notations: Sα2 = S12 #zα S22 , fα = f1 χzα f2 , Dfα = Dα = D∂ J,fα (0), Df = D∞ = (Df1 , Df2 ). We will use wα (s, t) or (s, t) with (s, t) ∈ R×S 1 to denote the cylindrical coordinate starting from the ‘middle’ of Sα2 and (τi , ti ) ∈ R+ × S 1 , i = 1, 2 to denote the one with τi = 0 at the boundary of the hemisphere ∂Hi . Now we define the following norms on Lp (fα ) and Lp1 (fα , a1 , a2 , a3 , a4 ), which is essentially due to Floer. R If ξα is in Lp (fα ) or in Lp1 (fα , a1 , a2 ; a3 , a4 ), then we define ξ˜α0 = S 1 ξα ◦ wα (0, t)dt ∈ Tx V , where x = f1 (∞) = f2 (∞). Now we switch to the (τ, t)-coordinate. Fix a bump function β on Sα2 such that β(τi ) = 1 when τi > 1 and β(τi = 0 when τi < 1/2, i = 1, 2. Define ξα0 (τ, t) = Dexpx (f˜α (τ, t))(β(τ )ξ˜α0 ) and ξα1 = ξα − ξα0 , where f˜α is the lifting of fα under expx and ξ˜α0 is considered as a vector field along f˜α . Definition 3.1. Let 0 < ε < 1. For any ηα in Lp (0,1 (fα∗ (T V )) and ξα in W1p (fα , a1 , a2 , a3 , a4 ), we define kηα kχ,0 = kηα k0,p;ε = keετ ηα k0,p , kξα kχ,1 = kξα1 k1,p;ε + |ξα0 | = keετ ξα1 k1,p + |ξα0 |, where |ξα0 | = |ξ˜α0 |. Note that the metric on Sα2 we used in the above definition of k · k’s is the metric induced from the cylindrical coordinate wα (s, t). The main estimate in this section is Lemma 3.2. Given a generic J and a cusp-curve f ∈ M(A, B, J, a1 , a2 ; a3 , a4 ) with c1 (A + B) + n = α1 + α2 + α3 + α4 such that for this J, Lemma 2.7 and its corollary hold so that Df : W1p (f, a1 , a2 ; a3 , a4 ) → Lp (0,1 (f ∗ (T V )) is isomorphic in spherical coordinate. Then there exists a constant C independent of zα such that for |zα | large enough, Dα : W1p (fα , a1 , a2 , a3 , a4 ) → Lp (0,1 (fα∗ (T V )) with the norms above has a uniform inverse Gα such that kGα (η)kχ,1 ≤ Ckηkχ,0 for any η ∈ Lp (0,1 (fα∗ (T V ). Proof. We only need to prove that when |zα | is large enough, there exists a constant C such that kξα kχ,1 ≤ CkDα ξα kχ,0 for any ξα ∈ W1p (fα , a1 , a2 , a3 , a4 ). If this is not true, then there exists a sequence {ξα } ∈ W1p (fα , a1 , a2 , a3 , a4 ) with |zα | → ∞ such that
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(i) kξα kχ,1 = kξα1 k1,p;ε + |ξα0 | = 1, (ii) kDα ξα kχ,0 = kDα ξα k0,p;ε → 0 when α → ∞. We will prove that (i) and (ii) contradict each other. In the proof we will repeatedly use the following fact: Lemma 3.3. Let B be a Banach space with a norm k.kB and f : B → R+ be a convex continuous function. If {xi } is a sequence in B such that xi → x weakly for some x in B, then f (x) ≤ lim inf f (xi ). i
Lemma 3.4. (i) and (ii) above imply that there exists a subsequence {ξα0 } such that |ξα0 | → 0, when α → ∞. Proof. By definition, |ξα0 | = |ξ˜α0 |. From (i) we know that |ξ˜α0 | ≤ 1. This implies that 0 0 there exists a subsequence {ξ˜α0 } → ξ˜∞ , for some ξ˜∞ ∈ Tx V . Therefore we only need 0 ˜ to prove that ξ∞ = 0. The idea of the proof is to construct a ξ = (ξ1 , ξ2 ) ∈ W1p (f, a1 , a2 ; a3 , a4 ) such that 0 in the spherical coordinate. The assumption of Df ξ = 0, and that ξ1 (∞) = ξ2 (∞) = ξ˜∞ 0 = 0. Lemma 4.2 implies that ξ = 0. Therefore ξ˜∞ 0 0 = Dexpx (f˜(τ, t))(β(τ )ξ˜∞ ) in the same way as defining To this end, we define ξ∞ 0 0 ∈ 0(f ∗ (T V )). It is easy to see that ξα0 is locally C ∞ -convergent to ξ∞ . ξα0 . Then ξ∞ 2 2 2 Let DR be the domain in Sα or in S1 ∨ S2 of the union of the two half spheres at two ends plus the cylindrical part up to τ = R. From (1) we know that kξα1 k1,p;ε ≤ 1. This implies that for any R > 0, there exists a constant C(R) depending on R such that kξα1 k1,p ≤ C(R) for all α. Note that when R is 1 weakly fixed, all these ξα1 |DR ’s live in same space for large α. Therefore ξα1 |DR → ξ∞;R p p 1 in L1 - space for some ξ∞;R ∈ L1 (f |DR ). By letting R → ∞ and taking a subsequence 1 ’s of the diagonal, we conclude, by the Sobolev embedding argument, that all these ξ∞;R p 1 can be pasted together to give rise to a single section ξ∞ ∈ L1,loc (f, a1 , a2 , a3 , a4 ) such 1 1 |DR = ξ∞;R weakly in Lp1 -space. that ξα1 |DR → ξ∞ 0 1 + ξ∞ . Then ξα |DR → ξ∞ |DR weakly in Lp1 - space. Therefore Let ξ∞ = ξ∞ Dα ξα |DR → D∞ ξ∞ |DR weakly in Lp -space. Here we have used the fact that Dα = D∞ on DR when α is large enough. Our assumption (ii), which says that kDα ξα k0,p;ε → 0 as α → ∞, implies that k(Dα ξα )|DR k0,p → 0 as α → ∞ for any fixed R. From Lemma 3.3 we conclude that k(D∞ ξ∞ )|DR k0,p ≤ lim inf k(Dα ξα )|DR |0,p = 0. α
This implies that (D∞ ξ∞ )|DR = 0 for any R, therefore D∞ ξ∞ = 0. In spherical coordinate, this gives us a solution of D∞ ξ = 0 with a singularity at the cuspidal point y. Since the L2 -norm of ξ∞ is bounded, this singularity is removable. 0 is already smooth. We conclude that as τ → ∞, all these three sections, ξ∞ , Note that ξ∞ 0 1 ξ∞ and ξ∞ , are convergent uniformly with respect to t under the trivialization Dexpx of T V near x. Combining this and the fact that Z 1 p 1 epετ |ξ∞ | dτ dt ≤ kξ∞ k0,p;ε ≤ lim inf kξα1 k0,p;ε ≤ 1, R+ ×S 1
α
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1 we conclude that lim ξ∞ = 0. Therefore τ →∞
0 0 lim ξ∞ (τ, t) = lim ξ∞ (τ, t) = ξ˜∞ .
τ →∞
τ →∞
This proves that in the spherical coordinate ξ∞ has the required properties, and therefore finishes the proof of the lemma. Using Lemma 3.4, by a simple calculation one can show that Lemma 3.5. (i) and (ii) also imply that lim kDα ξα0 k0,p;ε = 0. α→∞
From Lemmas 3.4 and 3.5, we conclude that (I) kξα1 k1,p;ε = 1, (II) kDα ξα1 k0,p;ε → 0 when α → ∞. We will prove that (I) and (II) contradict each other. To do this, we need to have an estimate on the middle part of ξα1 . Let β2 be a ‘bump’ function on Sα2 which is supported in −2 < s < +2 and equal to 1 on − log 2 < s < + log 2, where (s, t) or wα (s, t) is the cylindrical coordinate of Sα2 starting from the middle. Lemma 3.6. lim kβ2 ξα1 k1,p;ε = 0. α→∞
Proof. Let ρα = − log z˜α + log 2, which is the length of the cylindrical coordinate wα (s, t) along the t-direction. Define ηα : [−ρα , +ρα ] × S 1 → Tx V by Dexpx (f˜α ◦ wα (s, t))(ηα (s, t)) = e|ρα |ε · ξα1 (wα (s, t)). Extend ηα trivially over the whole cylinder. Then from (I) there exists a constant C such that (1) ke−ε|s| · ηα kp ≤ C for all α. Let ηα;R be the restriction of ηα to the domain ZR = [−R, +R] × S 1 , then from (I) again there exists a constant C(R) depending on R such that kηα;R k1,p ≤ C(R) for all α. Therefore as α → ∞, (2) ηα;R → η∞,R weakly in Lp1 (ZR , Tx V ) for some η∞;R ∈ Lp1 (ZR , Tx V ). The same argument as in the proof of Lemma 3.4 will prove that when R → ∞, all these ηα;R ’s agree with each other on their overlaps to form a single element η∞ ∈ Lp1,loc (R × S 1 , Tx V ) such that η∞ |ZR = η∞;R . Now (2) implies that when α → ∞, ∂ J0 ηα;R → ∂ J0 η∞;R weakly in Lp (ZR , Tx V ), where ∂ J0 is the standard Cauchy-Riemann operator. Let E˜α be the lifting of Eα under expx as before. Then it is of the form ∂ηα;R ∂ηα;R ∂ηα;R E˜α ηα;R = + J0 + (J˜ − J0 )(f˜α ) + Aα;R · ηα;R , ∂s ∂t ∂t
(3)
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where Aα;R is the restriction to ZR of some zero order operator Aα . It is easy to see that when R is fixed, lim |Aα;R | = 0 and lim |(J˜ − J0 )|f˜α;R | = 0. α→∞ α→∞ From (II) we have lim kE˜α ηα;R kp = 0. Therefore α→∞
lim k∂ J0 ηα;R kp n o ≤ lim kE˜α ηα;R kp + |Aα;R |kηα;R kp + |(J˜ − J0 )|f˜α;R |kηα;R k1,p = 0.
α→∞
(4)
α→∞
From this, (3) and Lemma 3.3, we have k∂ J0 η∞;R kp ≤ lim inf k∂ J0 ηα;R kp = 0. α→∞
This implies that ∂ J0 η∞;R = 0 for any R > 0. Therefore ∂ J0 η ∞ = 0
(5)
From (1) and Lemma 3.3, we have that kη∞;R k0,p;(−ε) ≤ lim inf ke−ε|s| ηα kp is α→∞
bounded not depending on R. This implies that kη∞ k0,p;(−ε) < ∞. This together with (5) and the fact that the constant Fourier component of η∞ |{0}×S 1 is zero imply that η∞ = 0. By a Sobolev embedding argument we conclude that for any fixed R > 0, ηα;R is uniformly C 0 -convergent to zero. Therefore, when α → ∞, kβ2 ηα k1,p ≤ Ck∂ J0 (β2 ηα )kp ≤ C(kβ20 ηα kp + kβ2 ∂ J0 ηα kp ) → 0. This implies that lim eερα kβ2 ξα1 k1,p = 0. Hence α→∞
lim kβ2 ξα1 k1,p;ε = 0.
α→∞
p (f, a1 , a2 , a3 , a4 ) be the weighted Sobelev Finishing the proof of Lemma 3.2. Let W1;ε 2 2 space of sections over S ∪ S − {y} with cylindrical coordinate, which consists of ξ with kξk1,p;ε < ∞ and obvious constraints at the four marked points. p It is proved in [F1] and [LM] that when ε < 1, Df : W1;ε (f, a1 , a2 , a3 , a4 ) → p 0,1 ∗ L0,ε; ( (f (T V ))) is Fredholm. A removing singularity argument will show that any element ξ in the kernel of the above operator will be in W1p (f, a1 , a2 , a3 , a4 ). But by the assumption of this lemma this is impossible unless ξ = 0. p (f, a1 , a2 , a3 , a4 ). Therefore there exists a constant C Now (1 − β2 )ξα1 is in W1,ε independent of α such that
k(1 − β2 )ξα1 k1,p;ε ≤ CkDf {(1 − β2 )ξα1 }k0,p;ε = CkDα {(1 − β2 )ξα1 }k0,p;ε ≤ C kDα ξα1 k0,p;ε + kDα (β2 ξα1 )k0,p;ε ≤ C 2kDα ξα1 k0,p;ε + kβ20 ξα1 k0,p;ε → 0
when α → ∞.
Therefore kξα1 k1,p;ε ≤ k(1−β2 )ξα1 k1,p;ε +kβ2 ξα1 k1,p;ε → 0 when α → 0. This contradicts with (I).
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Lemma 3.7. lim k∂ J fα kχ,0 = 0. α→∞
Proof. If we use ∼ to denote the corresponding lifting maps and operators under expx , then f˜α = β1 f˜1 + β2 f˜2 , where β1 and β2 are supported in the middle part of Sα2 of length log 2. Since ∂˜ J f˜i = 0, |f˜i (τ, t)| ∼ e−τ when τ large, we have k∂ J fα kχ,0 ≤ Ck∂˜ J f˜α kχ,0 ≤ C(kβ10 f˜1 kχ,0 + kβ20 f˜2 kχ,0 ) ∼ e(ε−1)ρα → 0 when α → ∞.
By using Lemma 2.3 together with a simple calculation, one can show that Lemma 3.8. There exists a constant C1 only depending on f such that for any ξα , ηα ∈ W1p (fα , a1 , a2 , a3 , a4 ), kN (ξα )kχ,0 ≤ C1 kξα k∞ kξα kχ,1 , kN (ξα ) − N (ηα )kχ,0 ≤ C1 (kξα kχ,1 + kηα kχ,1 )kξα − ηα kχ,1 . Lemma 3.9 (Picard method). Assume that a smooth map f : E → F from Banach spaces (E, k·k) to F has a Taylor expansion f (ξ) = f (0)+Df (0)ξ+N (ξ) such that Df (0) has a finite dimensional kernel and a right inverse G satisfying kGN (ξ) − GN (η)k ≤ 1 . If kG ◦ f (0)k ≤ δ2 , then the zero C(kξk +kηk)kξ − ηk for some constant C. Let δ = 8C set of f in Bδ = {ξ|kξk < δ} is a smooth manifold of dimension equal to the dimension of ker Df (0). In fact, if Kδ = {ξ|ξ ∈ ker Df (0), kξk < δ} and K ⊥ = G(F ), then there exists a smooth function φ : Kδ → K ⊥ such that f (ξ + φ(ξ)) = 0 and all zeros of f in Bδ are of the form ξ + φ(ξ). The proof of this lemma is an elementary application of Banach ’s fixed point theorem. Applying this to our case we have Lemma 3.10. If A, B ∈ H2 (V ) with A + B = P , c1 (P ) + n = α1 + α2 + α3 + α4 , and f = (f1 , f2 ) ∈ M(A, B, J; a1 , a2 ; a3 , a4 ), then for generic J and a parameter z = z˜22 ∈ C ∗ with |z| large enough, there exists a gluing map #z : M(A, B, J, a1 , a2 ; a3 , a4 ) → Mz˜ (P, J, a1 , a2 , a3 , a4 ) with f = (f1 , f2 ) 7→ f1 #z f2 . Moreover, if gz is another element in Mz˜ (P, a1 , a2 , a3 , a4 ) ‘close’ to the pre-gluing 1 f1 χz f2 in the sense that kg˜z kχ,0 ≤ δ = 8CC , then gz = f1 #z f2 . Here g˜z is a vector field 1 along fz = f1 χz f2 defined by Dexp(fz (τ, t))(g˜ z (τ, t)) = gz (τ, t), and C and C1 are the constants which appeared in Lemma 3.2 and 3.8. Proof. By Lemma 3.2 and Lemma 3.8 we have kGN (ξ) − GN (η)kχ,1 ≤ CkN (ξ) − N (η)kχ,0 ≤ CC1 kξ − ηkχ,1 (kξkχ,1 + kηkχ,1 ), for any ξ and η over fz . Let δ =
1 8CC1 .
Then by Lemma 3.7 we have
kG(∂ J fz )kχ,1 ≤ Ck∂ J fz kχ,0 <
δ 2
when |z| is large enough. The conclusion of the lemma follows by applying the Picard method to the above situation.
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Corollary 3.11. If the gz in Lemma 3.10 is C 0 -close to fz in the sense that kg˜z k∞ ≤ 1 2CC1 , then gz = f1 #z f2 . Proof. Since 0 = ∂ J,fz g˜z = ∂ J (fz ) + Dz (g˜z ) + N (g˜z ), we have g˜z = −G(∂ J (fz )) − GN (g˜z ). Therefore kg˜z kχ,1 ≤ C(k∂ J (fz )kχ,0 + kN (g˜z )kχ,0 ) 1 ≤ k∂ J (fz )kχ,0 + CC1 kg˜z k∞ kg˜z kχ,1 ≤ Ck∂ J (fz )kχ,0 + kg˜z kχ,1 . 2 This implies that kg˜z kχ,1 ≤ 2Ck∂ J (fz )kχ,0 < δ when |z| is large enough.
Note that if we give the orientation to the moduli spaces involved as in [M] and [F2], then the gluing map #z becomes an orientation preserving map.
4. Compactness In this section, we use Wolfson’s version of the Gromov compactness theorem to analyze the convergence of sequences of parametrized J-curves. As a consequence of this analysis, we will see how the condition of Corollary 3.11 could be satisfied. We assume throughout this section that (V, ω) is monotonic with c1 (A) > 1 for every effective class A ∈ H2 (V ) for generic J. Here ‘generic’ means that all transversality about cusp-curves stated in Sect. 3 hold. Fix a generic J and a class P ∈ H2 (V ) with c1 (P ) + n = α1 + α2 + α3 + α4 . Since the only J-holomorphic sphere of class zero are the constant maps, we may assume that P is not zero. Assume that the given four cycles a1 , a2 , a3 , a4 , which have been assumed to be submanifolds of V as we remarked in Sect. 1, are put in a general position in V so that all possible intersections among them are still submanifolds of V . 2 → Lemma 4.1. Consider a sequence {fn } ∈ Mz˜n (P, J; a1 , a2 , a3 , a4 ) with |z| = |z| ˜ 2 ∞. Each such fn gives rise to the two J-holomorphic spheres fL,n and fR,n under the ‘left’ and ‘right’ coordinates of S 2 #zn S 2 respectively, both mapping the ‘standard’ sphere S 2 to V . Then there are two possibilities:
(1) a1 ∩ a2 is not empty and {fR,n } is C ∞ -convergent to some fR ∈ M(P, J, a1 ∩ a2 , a3 , a4 ); or a3 ∩ a4 is not empty and {fL,n } is C ∞ -convergent to some fL ∈ M(P, J, a1 , a2 , a3 ∩ a4 ). (2) There exists a parametrized cusp-curve (fL , fR ) ∈ M(A, B, J; a1 , a2 ; a3 , a4 ) for some A + B = P with fL not equal to fR as unparametrized curves such that {(fL,n , fR,n )} is locally C ∞ - convergent to (fL , fR ) as parametrized curves. Proof. We will use ˆ· to denote the corresponding unparametrized curve and moduli ˆ space. By the Gromov compactness theorem, we have that {fˆn } → fˆ∞ = ∪m i=1 fi,∞ Pm 0 ˆ ˆ ˆ weakly with P = [fn ] = i=1 [fi,∞ ], and that for any C -neighborhood U of f∞ , fˆn is contained in U when n is large enough. The last statement implies that fˆ∞ has a non-empty intersection with any of the a0i s, i = 1, 2, 3, 4. By a detailed combinatorial
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analysis of the intersection pattern of fˆi,∞ ’s and a dimension counting argument, we conclude that m = 1 or 2 . The cases that m = 2, fˆ1,∞ = fˆ2,∞ and that m = 1 but the curve is multiplycovered do not occur. To rule out the first possibility, let [fˆi,∞ ] = A, i = 1, 2. Then ˆ fˆi,∞ ∈ M(A, J; a1 , a2 , a3 , a4 ), whose dimension is 2(c1 (A) + n − α1 − α2 − α3 − α4 ) = 2(c1 (P ) + n − α1 − α2 − α3 − α4 ) − 2(c1 (A)) which is less than zero by our assumption. It is similar for the second case. Now we can use Wolfson’s version of the Gromov compactness theorem to analyze the limit behavior of the sequence {fn }, as parametrized curves. Let lim fn (0L ) = l1 ∈ a1 , lim fn (1L ) = l2 ∈ a2 , n
n
lim fn (0R ) = l3 ∈ a3 , lim fn (1R ) = l4 ∈ a4 . n
n
(1) If l1 = l2 ∈ a1 ∩ a2 or l3 = l4 ∈ a3 ∩ a4 , then we have m = 1. Otherwise, for example, in the case l1 = l2 , fˆ∞ = (fˆ1,∞ , fˆ2,∞ ) is in the moduli space of (A, B)-cusp-curves with A + B = P , which intersects with a1 ∩ a2 ,a3 and a4 . A dimension counting argument shows that the dimension of this moduli space is less than zero. In the case l1 = l2 , we claim that {fR,n } is C ∞ - convergent to fR . If this is not true, then we have only one bubble at some point x1 in S 2 , and {fR,n }|S 2 −{x1 } is locally convergent to a constant map. x1 must be 0R , or 1R . Otherwise l3 = lim fn (0R ) = n ˆ lim fn (1R ) = l4 . This implies that fˆ∞ is in M(P, J; a1 ∩ a2 , a3 ∩ a4 ), which is empty n
for dimension reason. If x1 , for example, is 0R , then a similar argument show that ˆ l1 = l2 = l4 . Therefore fˆ∞ is in M(P, J; a1 ∩ a2 ∩ a4 , a3 ), which is empty again. Similarly, in the case l3 = l4 , we have that {fL,n } is C ∞ -convergent to fL . This gives the possibility (1) of the lemma. (2) Now we can assume that l1 6= l2 , and l3 6= l4 . Consider the sequence {fL,n }. When n tends to ∞, zn tends to the point ∞ in S 2 . But lim fL,n (zn ) = l3 6= l4 = lim fL,n (∞). n→∞ n→∞ This implies that the derivative of fL,n at ∞ blows up when n tends to ∞. Therefore we have one bubble at ∞. We claim that this is the only bubble {fL,n } could have, therefore that {fL,n } is locally C ∞ -convergent. Suppose that this is not true. Let x1 be another bubble point. Then x1 must be 0L or 1L . Otherwise since fL,n |S 2 −{x2 ,∞} is locally convergent to a constant map, we have l1 = lim fL,n (0L ) = lim fL,n (1L ) = l2 , which contradicts the assumption. n→∞
n→∞
If x1 , for example, is 0L , then lim fL,n |S 2 −{0,∞} = lim fL,n (1L ) = l2 locally. n→∞ n→∞ This implies that the limit curve fL is a union of the constant curve l1 with two bubbles f1,∞ and f2,∞ . It follows from the proof of Gromov’s compactness theorem that f1,∞ and f2,∞ are “lying” on l1 in the sense that f1,∞ (∞) = f2,∞ (∞) = l1 (see, for example, [P] or [PW]). This implies that for A = [f1,∞ ], B = [f2,∞ ], the limit cusp-curve is in the moduli space of such (A, B)-cusp-curves that intersect with the given four cycles and that their cuspidal points lie on a2 . But for generic J, this moduli space is empty for the dimension reason. We get a contradiction again. The proof of the other case of (2) is similar. Lemma 4.2. Assume that {fn } is weakly convergent to (fL , fR ) as in the case (2) of Lemma 4.1. Then for any given δ > 0, we have d(fn , fL χzn fR ) < 2δ when |zn | is large enough. Here d(fn , fL χzn fR ) = maxx∈S12 #zn S22 d(fn (x), fL χzn fR (x)) measured by a metric on V .
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Proof. We first prove that fL (∞) = fR (∞). Note that if we view fL as the “base” curve of the cusp-curve (fL , fR ), then fR becomes the bubble. But its parametrization may differ from that coming from the bubbling procedure described in [PW]. Therefore the statement does not immediately follow from that part of the compactness theorem concerning bubble intersections. Assume that d(fL (∞), fR (∞)) = 5δ > 0. Let B1 (R) and B2 (R) be the open balls of radius R in the ‘left’ and ‘right’ coordinates of S12 #zn S22 respectively. Note that for a fixed R, B1 (R) does not intersect with B2 (R) when |zn | big enough. Denote B1 (r) ∪ B2 (R) by B(R) and ∂Bi (R) by Ci (R), i = 1, 2. By using the fact that fn is locally convergent to (fL , fR ) and that the area A(fn ) = A(fL ) + A(fR ), it is easy to see that when R and |zn | large enough, (a) d(fn (C1 (R)), fL (∞)) < δ, d(fn (C2 (R)), fR (∞)) < δ, (b) A(fn |S 2 −B(R) ) < Cδ 2 for some fixed constant C which we will specify soon. Now fn : (S 2 − B(R)) → V is a minimal surface with respect to the metric gJ . Its two boundaries lie on the two disjoint balls BfL (∞) (δ) and BfR (∞) (δ) respectively. Since BfL (∞) (2δ) ∩ fn (S 2 − B(R)) and BfR (∞) (2δ) ∩ fn (S 2 − B(R)) are two disjoint open subsets of the connected surface fn (S 2 − B(R), there exists a point x1 in fn (S 2 − B(R)) such that the distance between x1 and fL (∞) or fR (∞) is larger than 2δ. Therefore Bx1 (δ) does not intersect with the two boundary components of fn (S 2 − B(R)). Now we can apply the monotonicity for minimal surface to conclude that A(fn |S 2 −B(R) ) > A(Bx1 (δ) ∩ fn (S 2 − B(R))) > Cδ 2 for some constant C, which only depends on the geometry of (V, J, gJ ). If we choose the constant C which appeared in (b) above to be the same as the one here, then we get a contradiction. This proves that fL (∞) = fR (∞) = x, the cuspidal point of f . A similar argument, by using monotonicity for the minimal surface again, will show that when R and |zn | are large enough, fn (S 2 − B(R)) is contained in Bx ( δ4 ). From this the conclusion of the lemma follows immediately. Proof of Theorem 1.5. Let P ∈ H2 (V ) with c1 (P ) + n = α1 + α2 + α3 + α4 . By Lemma 3.10, when |zn | is big enough, there exists a gluing map a #zn : M(A, B, J, a1 , a2 ; a3 , a4 ) → Mz˜n (P, J; a1 , a2 , a3 , a4 ). A+B=P
Note that for any cusp-curve f = (fL , fR ), fL #zn fR is locally convergent to f . This plus that the domain of #zn is a finite set implies that #zn is injective. The surjectivity of #zn is a consequence of Lemma (4.2) and Corollary 3.11. As we remarked before, #zn is also preserving the orientation. This proves the special decomposition rule, therefore the associativity of quantum multiplication. Acknowledgement. I would like to express my hearty thanks to my advisor Professor Dusa McDuff for introducing me to the subject and for her encouragement and various kinds of help. This work is indebted to her works and her general ideas about the subject very much. I also wish to acknowledge my debt to Floer, via his unpublished notes. In [F] he tried to established the key technique of this paper, the gluing of J-holomorphic curves, by imitating the corresponding construction in Floer homology. Because the gluing used in this paper is quite different from the one in Floer homology, his effort did not succeed . The key ingredients, Lemmas 3.4 and 3.5, are missing there. However, his general idea on gluing has much influenced my work in Sect. 3.
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References [F1] [F2] [F] [G] [LM] [M] [MS] [PW] [RT1] [RT2] [V] [W]
Floer, A.: The unregularized gradient flow of the symplectic action. Comm. Pure Appl. Math. 41, (775–813) (1988) Floer, A.: Symplectic fixed points and holomorphic spheres. Commun. Math. Phys. 120, (575–611) (1989) Floer, A.: Lecture notes, unpublished Gromov, M.: Pseudo holomorphic curves in symplectic manifolds. Invent. Math. 82, (307–347) (1985) Lockhard, R.B. and McOwen, R.C.: Elliptic operators on noncompact manifolds. Ann. Sci. Norm. Sup. Pisa IV-12, (409–446) (1985) Mcduff, D.: Examples of symplectic structures. Invent. Math. 89, (13–36) (1987) Mcduff, D. and Salamon, D.A.: J-holomorphic curves and Quantum Cohomology. In print (1994) Parker, T.H. and Wolfson, J.G.: Pseudo-holomorphic maps and bubble trees. J. Geom.Anal. 3, 63–98 (1993) Ruan, Y. and Tian, G.: A mathematical theory of quantum cohomology(announcement). Math. Res. Lett. Vol 1 no 1, 269–278 (1994) Ruan, Y. and Tian, G.: A mathematical theory of quantum cohomology. Preprint (1994) Vafa, C.: Topological mirrors and quantum rings. In: Essays on Mirrors Manifolds, edited by S.-T. Yau, Hong Kong: International Press, 1992 Wolfson, J.G.: Gromov’s compactness of pseudo-holomorphic curves and symplectic geometry. J. Diff. Geom. 28, 383–405 (1988)
Communicated by S.-T. Yau
Commun. Math. Phys. 191, 283 – 298 (1998)
Communications in
Mathematical Physics c Springer-Verlag 1998
Complex Matrix Models and Statistics of Branched Coverings of 2D Surfaces Ivan K. Kostov1,? , Matthias Staudacher2 , Thomas Wynter1 1 Service de Physique Th´ eorique, C.E.A. - Saclay, F-91191 Gif-Sur-Yvette, France. E-mail:
[email protected],
[email protected] 2 CERN, Theory Division, CH-1211 Geneva 23, Switzerland. E-mail:
[email protected]
Received: 11 April 1997 / Accepted: 9 May 1997
Abstract: We present a complex matrix gauge model defined on an arbitrary twodimensional orientable lattice. We rewrite the model’s partition function in terms of a sum over representations of the group U (N ). The model solves the general combinatorial problem of counting branched covers of orientable Riemann surfaces with any given, fixed branch point structure. We then define an appropriate continuum limit allowing the branch points to freely float over the surface. The simplest such limit reproduces two-dimensional chiral U (N ) Yang-Mills theory and its string description due to Gross and Taylor. 1. Introduction Recently, two of the authors have considered a complex matrix model which describes the ensemble of branched coverings of a two-dimensional manifold [1]. The model has been interpreted as a string theory invariant with respect to area-preserving diffeomorphisms of the target space. In this paper, we continue the investigation of this model. The geometrical problem we solve consists in the enumeration of the branched coverings of a two-dimensional manifold with a given number of punctures. The target manifold is characterized by its topology, the number of punctures, and its total area. All structures we are considering are invariant under area-preserving diffeomorphisms of the targ et manifold. We assume that the covering surfaces can have branch points located at the punctures. Two covering surfaces related by an area-preserving diffeomorphisn are considered identical. The problem will be reformulated in terms of a lattice gauge theory of N ×N complex matrices defined on a lattice representing a cell decomposition of the target manifold. The vertices of the lattice are the punctures of the target surface. The N1 perturbative expansion of this model generates the covering surfaces with the corresponding combinatorial ?
Member of CNRS
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factors. These discrete surfaces can be interpreted as cell decompositions of continuum surfaces covering the target space. It sh ould be intuitively clear that the combinatorics of these surfaces should not depend on the cellular decomposition of the target manifold, but only on global features like the topology of the manifold, the topology of the covering Riemann surface, and the number and types of branch points allowed. Our solution below will confirm this intuition. The solution of the model is given in terms of a sum over polynomial representations of the group U (N ). This sum is, by construction, a generating function for the combinatorics of covering maps. The order of the representation gives the degree of the (connected and disconnected) coverings. The sum depends on the genus G of the target manifold, and a number of variables tied to geometrical data of the covering maps: N −2 is the genus expansion parameter of the covering surfaces, and we associate, for each cell corner p, a weight t(p) k corresponding to a branch point at p of order k. For some applications, one would consider a slightly more general problem of counting coverings with branch points that can occur anywhere on the smooth target manifold. In order to allow for this possibility, we are led to take a continuum limit: We simply cover the target manifold by a microscopically small cell decomposition, with the weights of these branch points tuned correspondingly. We will find that the combinatorics of the resulting statistics of movable branch points actually simplifies significantly over the general case of fixed branch points: Many special configurations of enhanced symmetry, corresponding to coalescing branch points, are scaled away. The weights associated with the punctures survive scaling; this makes the difference between the punctures and the rest of the points of the cell decomposition. Apart from the obvious mathematical interest of our approach, we are able to connect our results to recent work on the QCD string in two dimensions. In the case of a target space with nonnegative global curvature, the chiral (i.e., orientation preserving) sector of the Gross and Taylor [2] string theory describing two dimensional Yang-Mills theory, is a particular case of the string theory defined by our matrix model. We discuss the problem of the “ factors” in the string interpretation of the Yang-Mills theory and interpret the “-points” as punctures in the target space. 2. Definition of the Model Consider a smooth, two dimensional, compact, closed and orientable manifold MG of genus G with N0 marked points (punctures), which we denote by p = 1, ..., N0 . The manifold is compact in the followingR sense: We assume that there is a volume form dA on MG and the total area AT = dA is finite. We will consider the ensemble of nonfolding surfaces covering MG and allowed to have branch points at the punctures. These surfaces are smooth everywhere on MG and are given a volume form inherited from the embedding. In this way, the area of a surface covering n times the target manifold is equal to nAT . We will resolve the problem by discretizing the target manifold. Introduce a cell decomposition of the original target manifold MG such that each cell is a polyhedron homeomorphic to a disc. The vertices of the cell decomposition are by construction the N0 punctures of MG . The resulting polyhedral surface (cellular complex) MG is thus characterized by its genus G, and by its set of points p, links ` and cells c. The numbers of points, links and cells which we denote correspondingly by N0 , N1 and N2 , are related by the Euler formula (2.1) N0 − N1 + N2 = 2 − 2G.
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Each cell P contributes a fraction Ac to the total area AT of the target manifold, so that AT = c Ac . A given manifold can be discretized in many different ways, but the choice of discretization is irrelevant for our problem. A branched covering of MG of degree n is a surface Σ obtained by taking n copies of each of the polygons of MG and identifying pairwise the edges of the n polygons on either side of each link. The Riemann surface obtained in this way can have branch points of order k (k = 1, 2, ..., n) representing cyclic contractions of edges. They are located at the Ppoints p ∈ MG . The discretized surface Σ has nN1 links, nN2 polygons and nN0 − p bp points (with bp the winding number minus one at point p). Its total area is nAT . Its genus g is given by the Riemann-Hurwitz formula: X bp . (2.2) 2g − 2 = n(2G − 2) + p
The partition function is defined as the sum over all possible coverings Σ → MG conserving the orientation. A factor N 2−2g is assigned to the genus g of the covering surface. Furthermore, we introduce Boltzmann weights associated with its branch points. The weight of a branch point of order k is t(p) k , where k ≥ 2. A regular (analytic) point . gets a weight t(p) 1 The partition sum is now defined as a sum over all (not necessarily connected) coverings Σ → MG : Z=
X
e
−nAT
N
2−2g
N0 Y Y
nk (t(p) k ) ,
(2.3)
p=1 k≥1
Σ→MG
where, associated with the point p ∈ MG , nk (p) (with k ≥ 2) is the number of the branch points of order k of Σ and n1 (p) the number of regular points. The symmetry factor of the map is understood in the sum. Now we introduce a matrix model whose perturbative expansion coincides with (2.3). To each link ` = hpp0 i we associate a field variable 8` representing an N × N matrix with complex elements. By definition 8
= 8† . In order to be able to associate arbitrary weights to the branch points we will associate an external matrix field with the corners of the cells. Let us denote by (c, p > the corner of the cell c associated with the point p. The corresponding matrix will be denoted by B(c,p> . The partition function of the matrix model is defined as Z Y Y [D8` ] exp(e−Ac N Tr8c ), (2.4) Z= c
`
where 8c denotes the ordered product of link and corner variables along the oriented boundary ∂c of the cell c Y 8` B(c,p> , (2.5) 8c = `,p∈∂c
and the integration over the link variables is performed with the Gaussian measure [D8` ] = (N/π)N
2
N Y i,j=1
d(8` )ij d(8` )∗ij e−N
Tr 8` 8†`
.
(2.6)
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I. K. Kostov, M. Staudacher, T. Wynter
3 2
Φ31 Φ13
Φ12 Φ21
B312 B413
1
B214
Φ41 Φ14
4
Fig. 1. The matrices associated with the links < 12 >, < 13 >, < 14 > and the corners (214 >, (312 >, (413 > associated with the point 1
The perturbative expansion of (2.4) gives exactly the partition function (2.3) of branched surfaces covering MG . The weight t(p) k of a branch point of order k (k ≥ 2) or regular point (k = 1) associated with the vertex p ∈ MG equals t(p) k =
1 Tr[Bpk ], N
where we have defined the matrices Bp as the ordered product Y Bp = B(c,p>
(2.7)
(2.8)
c
of the B-matrices around the vertex p.
3. Exact Solution by the Character Expansion Method The method consists in replacing the integration over complex matrices by a sum over polynomial representations of U (N ). Applying to (2.4) the same strategy as in ref.[3], we expand the exponential of the action for each cell c as a sum over the Weyl characters χh of these representations: exp(e−Ac N Tr8c ) =
X 1h h
h
χh (8c ) e−Ac |h| .
(3.1)
The representations are parametrized by the shifted weights h = {h1 , h2 , . . . , hN }, where hi are related to the lengths m1 , ..., mN of the rows of the Young tableau by hi = N − i + mi and are therefore subjected to the constraint h1 > h2 > . . . > hN ≥ 0. Here i = 1, 2, . . . denotes the first, second, . . . row of the tableau as counted down from
Complex Matrix Models and Statistics of Branched Coverings
287
the top. We will denote by |h| = Σi mi the total number of boxes of the Young tableau. The dimension 1h of the representation h is given by 1h =
Y hi − hj
,
(3.2)
hi ! . (N − i)!
(3.3)
j−i
i<j
and the “Omega factor” h by h = N −|h|
N Y i=1
An explicit representation of the Weyl characters χh is only needed for the derivation of two essential integration formulas (fission and fusion rule, respectively): Z h χ (81 ) χh (82 ), [D8] χh (81 8 82 8† ) = 1h h Z (3.4) h χh (81 82 ). × [D8] χh (81 8) χh0 (8† 82 ) = δh,h0 1h For a simple proof of these identities, see [1]. It is now possible to exactly perform the integration with respect to the gaussian measure (2.6) at each link. Using the fusion rule one progressively eliminates links between adjoining cells until one is left with a single cell. The remaining links along that plaquette are integrated out by applying the fission rule. A similar procedure was first used in the exact solution of two-dimensional Yang-Mills theory [4, 5]. Employing the very useful relation χh (A1 ) =
1h h
with
1 TrAk1 = δk,1 , N
(3.5)
one finds the exact solution of the matrix model (2.4): Z=
N0 X 1h 2−2G Y χh (Bp ) −|h|AT e . h χh (A1 ) h
(3.6)
p=1
The characters χh (Bp ) are related to the branch point weights t(p) k at the site p through the Frobenius formula (we omit the index p for clarity): X chh (1n1 , 2n2 , 3n3 , . . .) χh (B) = n1 +2n2 +3n3 ...=|h|
|h|! N n1 +n2 +n3 +... tn1 tn2 tn3 . . . . 1n1 n1 ! 2n2 n2 ! 3n3 n3 ! . . . 1 2 3
(3.7)
The sum is over all partitions of the covering number |h|, and the chh (1n1 , 2n2 , 3n3 , . . .) are the characters of the symmetric group S|h| corresponding to the representation h and the partition class with cycle structure (1n1 , 2n2 , 3n3 , . . .) . Let us emphasize that (3.6), together with (3.7), is the complete solution of the combinatorial problem of counting branched covers (2.3). In Appendix A, we have given the first few terms of the expansion (3.6), in the special case where the branch point weights are identical on all N0 sites. We have also given the first few terms of the free energy F = log Z, corresponding to connected surfaces.
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A special case of (3.6) is obtained when no branch points are allowed on the target manifold. This is done by choosing all matrices Bp = A1 . Then the solution (3.6) becomes completely independent of the cell decomposition of the manifold: The cell corners are indistinguishable from any other point on the surface. We obtain X h 2G−2 e−|h|AT . Z= 1h
(3.8)
h
This is completely trivial for the sphere and very simple (see [1],[2]) for the torus, but highly non-trivial for higher target space genus G ≥ 2: 2 −A for G = 0; N eP T , ∞ 1 −kmAT 0 for G = 1; (3.9) log Z = N k=1,m=1 k e 2−2G −AT N e + (N 2−2G )2 21 (4G − 1)e−2AT + . . . for G ≥ 2. For G ≥ 2, log Z counts the number of smooth and locally invertible maps of a surface of genus g onto a surface of genus G. It would be interesting to find an analytic expression for log Z for all G ≥ 2.
4. Continuum Limit: Statistics of Movable Branch Points In the preceding section we have presented the full solution to the problem of counting the branched covers of a smooth, closed and orientable target manifold of any genus G and N0 punctures. By construction, the solution (3.6) depended, aside from G and the genus g of the branched cover, on the number and types of branch points at the punctures of MG : At each puncture p we are to specify a set of weights t(p) k associated with the winding numbers of the covering surface. It is natural to consider a related, but different combinatorial problem: Allow the covering surfaces to have branch points anywhere on the target manifold. In this case we have to sum over the positions of the branch points with an appropriate integral measure. The solution of this problem is actually contained in the solution of the previous one. Since branching in our model is constrained to occur at the sites of the cell decomposition, we are led to consider a continuum limit: The cell decomposition has to densely cover the manifold. The simplest choice is to assign identical weights at each branching sites, that T is, choose Bp = B everywhere, and assume that all cells have the same area Ac = A N0 . Set 1 t1 = TrB = 1, N (4.1) 1 τk for k ≥ 2, tk = TrB k = N N0 and take N0 → ∞, while holding AT and the continuum couplings τk fixed. Thus the probability for branching at a specific site p goes to zero in a prescribed way. In this continuum limit the configuration space of the branch points becomes infinite and their statistics drastically simplifies due to the fact that the probability to have more than one branch point associated with the same point of MG tends to zero. Mathematically, this simplification is immediately seen from the partition function (3.6), and the Frobenius formula (3.7). The product of quotients of characters exponentiates, and (3.6) becomes:
Complex Matrix Models and Statistics of Branched Coverings
Z=
X 1h 2−2G h
h
exp
X ∞
289
τk N 1−k ξkh e−|h|AT .
(4.2)
k=2
Here, in view of (3.7), the tableau dependent numbers ξkh are given by |h|! chh (1|h|−k , k 1 ) . k(|h| − k)! chh (1|h| )
(4.3)
N N Y k 1X 1− , hi (hi − 1) . . . (hi − k + 1) k hi − hj j=1
(4.4)
ξkh = An explicit formula is ξkh =
i=1
j6=i
which is quickly derived from (3.5),(4.1), and the identity (coming from the Schur definition of the characters, see [4]) N N X χh˜ k (B) ∂ log χh (B) = ∂tk k χh (B)
with
k h˜ i = hi − δki k.
(4.5)
i=1
The ξkh are actually symmetric polynomials of degree k in the weights hi (see Appendix B, where we have also listed a few examples). They are N independent as long as |h| < N . In Appendix B, we have given the first few terms of the expansion (4.2), as well as the first few terms of the free energy F = log Z, corresponding to connected surfaces. Finally, let us mention that we can keep the weights associated with a given number of points unscaled. Then we obtain the partition function of the ensemble of branched coverings of a target manifold of area AT , genus G and N0 punctures. The evident generalization of Eqs. (3.6) and (4.2) is X N0 ∞ X 1h 2−2G Y χh (Bp ) 1−k h exp τk N ξk e−|h|AT . Z= h χh (A1 ) h
p=1
(4.6)
k=2
5. Relation to Chiral 2D Yang-Mills Theory The solution of the map counting problem with movable branch points considered in the last section has an interesting physical interpretation. Let us restrict ourselves to the case of only simple branch points: τk = 0 for k ≥ 3. Then, using the explicit result for ξ2h (see Appendix B), and choosing 1 τ2 = − AT , 2 we can rewrite the partition function (4.2) as X 1h 2−2G AT Z= e− N h h
(5.1)
C2h
,
(5.2)
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where C2h is the second Casimir of the group U (N ). This is very nearly the partition function of two-dimensional U (N ) Yang-Mills theory. This theory was shown to be solvable on any target manifold in [4, 5]. Recently, it was demonstrated by Gross and Taylor [2], that Y M2 can be interpreted as a string theory. This was achieved by the tour de force approach of expanding the exact solution in N1 and interpreting the terms as string maps. Here we are inverting the Gross-Taylor program: we start with a theory whose interpretation as a string theory generating covering maps from a worldsheet to the target manifold is manifest, and aim to derive Y M2 . There remain, however, two subtle differences between (5.2) and Y M2 : (1) In U (N ) Y M2 the sum over polynomial representations h is extended to a sum over coupled (non-polynomial) representations. This difference is easy to understand: The missing representations clearly correspond to orientation reversing maps, which have been eliminated from the start by our chiral matrix model. Indeed it became evident from the work of [2], that Y M2 factorizes into two rather weakly interacting chiral sectors. Our class of models furnishes a precise realization of one such sector. , absent (2) Our partition function contains, for genus G 6= 1, the extra factor 2G−2 h in Y M2 . This factor eliminates the so-called “ points” and “−1 points” introduced originally by Gross and Taylor in order to sustain the string picture for G 6= 1. For G = 0, we can add two punctures and associate weight 1 with the windings around them (i.e. take a matrix source B = 1). The answer is given by (4.6) with N0 = 2 and B1 = B2 = 1 and coincide with the partition function of the Yang-Mills theory on the sphere. The two punctures are the two “ points”. In summary, in the case of a target space representing a sphere with two punctures our theory is exactly equivalent1 to chiral Y M2 . For G ≥ 2 however, it is not possible to eliminate the extra factor ( the “−1 points”) by adding punctures. On the other hand, the “−1 factors” have been given an interesting re-interpretation in the work of Cordes, Moore and Ramgoolam [5]. These authors showed that the Y M2 partition function could be rewritten as a sum over covering maps without non-movable, special singularities, but weighted instead with special topological invariants (the “Euler character of Hurwitz moduli space”). It would be interesting if this result could be reproduced by a simple matrix model, similar in spirit to (2.4) and possessing an equally clear surface interpretation.
6. Sphere-to-Sphere Maps: Saddle Point Analysis In the case of a target space of spherical topology G = 0 we can apply a saddle point technique to the sum over representations to extract the contribution from coverings with spherical topology (i.e. g = 0). This was explained in detail in [3, 6], and leads in the case of the model with fixed branch points (3.6) to the general Riemann-Hilbert problem discussed in [4]. Fortunately, the case of movable branch points (4.2) is much simpler. Introducing a continuous coordinate h = N1 hi and a density ρ(h) = −∂i/∂hi , (remember our convention hi = N − i + mi ) one finds, with the help of (4.4), Z a 1 1 ρ(h0 ) = log(h − b) + AT + V˜ 0 (h), − dh0 0 h − h 2 2 b
(6.1)
1 Adding a puncture to the target space of the Yang-Mills theory does not change anything. For example, the gauge theory defined on a cylinder with Dirichlet boundaries is identical to the gauge theory on the sphere.
Complex Matrix Models and Statistics of Branched Coverings
291
where the effective potential V˜ 0 (h) is, for a finite number of non-zero τn ’s, a polynomial in h whose coefficients are selfconsistently dependent on the first few moments Hn = Ra P ρ(h0 ) N −1−n hni of the resolvent H(h) = 0 dh0 h−h 0 . It is found as follows. Define the potential ∞ X 1 τ k hk . (6.2) V (h) = k k=2
Then
1 V˜ 0 (h) = 2πi
I
ds V (s e−H(s) ). (h − s)2
(6.3)
Here the contour surrounds the cut [b, a] of H(h). The simplest non-trivial example consists of taking τk = 0 for k ≥ 3. The analysis then becomes very similar to the one for chiral Y M2 , obtained in [6]. Equation (6.1) then reads: Z a A T τ2 dh0 ρ(h0 ) + (1 − h). (6.4) = ln(h − b) + − 0 h − h 2 2 b The term log(h − b) is a consequence2 of the fact that the hi are a set of ordered integers. The density is therefore constrained to have a maximum value of 1 (see [4]). This equation is solved in the standard way by a contour integral Z a dh0 ρ(h0 ) H(h) = h − h0 0 (6.5) I h p ds ln(s − b) + A2T + τ22 (1 − s) √ . + (h − a)(h − b) = ln h−b 2πi (h − s) (s − a)(s − b) Expanding the contour out to infinity we catch a pole at s = h, the discontinuity across the cut of the logarithm, and a contribution from the contour at infinity. The final result is p τ2 τ2 AT − + (h − (h − b)(h − b)) H(h) = ln h + 2 2 √2 √ (6.6) h−a+ h−b √ . − 2 ln a−b The density ρ(h) is as given by the discontinuity of H(h) across its cut: 3 r τ2 p 2 h−b −1 − ρ(h) = cos (a − h)(h − b), π a−b 2
(6.7)
and the cut points a and b are determined from the behaviour of H(h) for large h: H(h) =
1 1 +O 2 . h h
(6.8)
Imposing this asymptotic behaviour on (6.6) leads to the two equations 2 A brief argument shows that for the phase corresponding to the sum over surfaces, a part of the density, starting at the origin and finishing at a point b, is saturated at its maximum value. The sum over P coverings is −A
an expansion in powers of e−AT and τ2 . For small τ2 and small e−AT , i.e. large AT , the e T in (4.2) attracts all the hi towards the origin saturating the constraint hi+1 < hi , ρ(h) ≤ 1. 3 Specifically ρ(h) = i (H(h + i) − H(h − i)), where is a small positive real number. 2π
i
hi
term
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I. K. Kostov, M. Staudacher, T. Wynter
χ = τ22 e−AT eχ and η − τ2 e−AT +χ = 1,
(6.9)
where we have defined χ = AT + 2 log((a − b)/4) and η = (a + b)/2. Differentiating ∂ F = − < h > + 21 , whereRthe free energy F is defined (4.2) w.r.t. to AT leads to ∂A a 2 here by F = 1/N ln Z. The expectation value < h >= 0 dhρ(h)h can be calculated from the expansion for H(h) (6.6). Using (6.9) it is possible to integrate up to obtain an explicit expression for the spherical contribution to F , 1 3 F = e−AT +χ 1 − χ + χ2 . 4 6
(6.10)
From (6.9) it is clear that χ is a power series in τ22 e−AT . We can thus perform a standard Lagrange inversion on [9, 8], and obtain after a short calculation: F=
∞ X nn−3 n=1
n!
τ22n−2 e−nAT .
(6.11)
This result was first obtained in [7],[6]. To discuss the convergence properties of (6.11), it is natural to take the continuum coupling proportional to AT , since the branch points can be located anywhere on the manifold (see also (5.1)): τ2 = tAT . Note that the series is only convergent for t2 A2T e−AT < e−1 . Beyond this point the boundary conditions (6.9) lead to a non-physical complex value for χ. We see that the sum over branched coverings is convergent for both large and small areas. For large enough t and intermediate values of the area AT , however, the entropy of the branch points is sufficient to cause the sum to diverge. It is interesting to understand this divergence in terms of the Young tableau density ρ(h). Along the critical line (τ2 , AT ) and τ2 > 0 the density becomes flat at its upper end point a, i.e. the singularity at the end point changes from ρ(h) ∼ (a − h)1/2 to ρ(h) ∼ (a − h)3/2 . Along the critical line (τ2 , AT ) and τ2 < 0 it is the singularity of the density at the point b that changes from 1/2 to 3/2. This is just as occurs in matrix models of pure 2D gravity, and indeed for t2 A2 e−AT ∼ e−1 the free energy (6.11) behaves as F ∼ (e−1 − t2 A2T e−AT )5/2 . (6.12) So far in our analysis we have ignored the constraint that we are summing over positive Young tableaux, i.e. that b > 0. Since b = 0 is not a singular point in the boundary conditions (6.9) it does not correspond to a singularity in the sum over surfaces. If, however, we take the sum over representations (4.2) as fundamental (as would be the case for YM2 ) then one should take this constraint into account. In full YM2 , the inclusion of this constraint triggers the Douglas-Kazakov phase transition [8]. Unlike our case, the phase transition in YM2 can be related to the proliferation of branch points in the sum over surfaces. Setting b = 0 in (6.9) leads to the pair of equations √ (6.13) t2 A2 e−AT = χe−χ and AT = χ + 2 ln(2 − χ). which determine t and AT parametrically in terms of χ. This curve is plotted in Fig. 2. It connects the dot on the left to the shaded area on the right. Immediately below this line the support of the density is entirely positive, i.e. it starts at a positive value of h. In addition it is less than one for the full range of its support, see Fig. 2. The phase transition across this line is the analogue of the large N phase transition that occurs in the one-plaquette Wilson lattice gauge theory (see [11, 12]). There there is a phase
Complex Matrix Models and Statistics of Branched Coverings
ρ ’(b)=0
1 0 0 1 0 1
d=0
0110 11001100 b
d c b a
d c
ρmax =1
b a
0110 10 -2
AT 6
a
5
4
3
b=0 2
1
b
a
-1
293
11111 00000 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 ρ’(a)=0
1
2
t
Fig. 2. Phase diagram in the t, AT plane. A typical density, ρ(h), is sketched in each phase
transition separating a strong coupling regime from a weak coupling phase, and as is the case here, there is no singularity in either phase indicating the transition point. Indeed it is possible to formulate the Gross-Witten [9] and Br´ezin-Gross [10] models in terms of a sum over representations [11] and the large N phase transition is precisely the point at which the density just begins to touch (or pull away from) the origin. Note, however, that in our case the phase transition is second order, while it is third order in Gross-Witten. Further analysis shows that there are two further phases for the model, as first observed in [6]. There is one phase where the density has an entirely positive support but attains its maximal value ρ(h) = 1 over a single finite interval, and another phase where the density starts at the origin and has two separated intervals where ρ(h) = 1. Both of these phases involve two separated nontrivial cuts and can be calculated in terms of elliptic functions. We do not present here explicit results for any of these extra phases. The complete phase diagram for the partition function is shown in Fig. 2. In each phase is sketched a typical density. Along the transition lines are indicated the corresponding critical behaviours of the density. The exact position of the line along which the point d equals zero has not been calculated. We indicate this by using a dotted line. The sum over representations is thus seen to have a much richer phase structure than the perturbative sum over surfaces it generates. 7. Concluding remarks We have introduced a new class of matrix gauge models which solve the general problem of counting branched covers of orientable two-dimensional manifolds with specified branch point structure. The result is given as a weighted sum over polynomial representations of U (N ). We conjecture that a similar methodology could be applied to non-orientable surfaces by replacing the complex matrices by real matrices and the group U (N ) by O(N ).
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Our approach actually treats also the case of manifolds with boundaries. Indeed, due to the invariance with respect to area-preserving diffeomorphisms, the problem does not depend on the length of the boundaries and the latter can be considered as punctures. As an interesting by-product of our investigation, we have derived in a constructive and transparent fashion important features of the original Gross-Taylor interpretation of Y M2 as a string theory. An interesting problem which we did not consider in this paper is the calculation of Wilson loops. The Wilson loop functional defined as the sum of all nonfolding branched surfaces spanning a closed contour C ∈ MG should satisfy a set of loop equations with a contact term similar to those considered in [12]. Appendix A. Free energies for N0 fixed, identical branch points In this appendix we give some examples of how to extract the combinatorics of connected coverings of a smooth manifold with N0 fixed, identical branch points. The partition sum (3.6) for this case becomes X 1h 2−2G χ (B) N0 h e−|h|AT . Z= h χh (A1 )
(A.1)
h
Define Z =1+
∞ X
Zn e−nAT .
(A.2)
n=1
Using the Frobenius formula for characters, the first few Zn ’s, up to order four, are found to be: 0 Z1 = N 2−2G tN 1 1 Z2 = ( N 2 )2−2G 2 1 Z3 = ( N 3 )2−2G 6 1 +( N 3 )2−2G 3 1 4 2−2G Z4 = ( N ) 24
,
1 N0 1 t2 ) + (t21 − t2 )N0 , N N 3 3 2 3 2 N0 (t1 + t1 t2 + 2 t3 ) + (t31 − t1 t2 + 2 t3 )N0 + N N N N 3 1 N0 (t1 − 2 t3 ) , N 4 6 2 8 3 6 (t1 + t1 t2 + 2 t1 t3 + 2 t22 + 3 t4 )N0 + N N N N 6 8 3 6 + (t41 − t21 t2 + 2 t1 t3 + 2 t22 − 3 t4 )N0 + N N N N h 2 1 2 1 +( N 4 )2−2G (t41 + t21 t2 − 2 t22 − 3 t4 )N0 8 N N N i 2 1 2 + (t41 − t21 t2 − 2 t22 + 3 t4 )N0 + N N N 4 3 2 N0 1 4 2−2G 4 (t1 − 2 t1 t3 + 2 t2 ) . +( N ) 12 N N (t21 +
(A.3)
These expressions allow us to obtain explicit results for the map counting problem. The free energy counting connected surfaces is
Complex Matrix Models and Statistics of Branched Coverings
F = log Z =
∞ X
Fn e−nAT .
295
(A.4)
n=1
Let us present some explicit low order results: G = 0: 0 F1 =N 2 tN 1 ,
[ 21 N0 ]−1
F2 =
X
N 2−2g
g=0
1 N0 2 N0 −2g−2 2g+2 (t ) t2 2 2g + 2 1
1 N0 3N0 −6 2 N0 3N0 −8 4 N0 3N0 −7 2 t t t F3 =N 4 t2 + t2 t 3 + t3 + 4 1 2, 1 1 3 2 1 N0 3N0 −12 6 3 N0 3N0 −11 4 + N 0 40 t t t2 + t2 t3 + 6 1 2 2, 2, 1 1 N 1 N0 3N0 −9 3 0 −2 0 −10 2 2 +2 t3N t t t + t ), 2 3 3 + O(N 3 3 1 2, 2 1 N 5 N 5 N N 1 2N0 4N0 −12 6 0 0 0 0 2 F4 =N + 13 + + − t1 124 t2 + 6 1, 4 4 2, 2 16 3 16 6 N N0 0 0 −11 4 + 27 +3 t4N t2 t 3 + 1 1, 4 1, 1, 2 N0 N0 4N0 −10 2 2 + 6 + t1 t2 t3 + 2, 2 2, 1 N N 1 N0 4N0 −8 2 0 0 0 −9 0 −9 3 t4N t4N t2 t 3 t 4 + t t3 + t4 + + 1 1 1, 1, 1 3 4 2 1 N0 1 N0 4N0 −10 3 + 4 + t1 t2 t4 + O(N 0 ). 3, 1 2 1, 1, 1 (A.5) G = 1: 2
0 F1 =tN 1 ,
[ N0 ]+1 N 3 2N0 2X 0 F2 = t1 + (t2 )N0 +2−2g t2g−2 N 2−2g 2 , 2 2 2g − 2 1 g=2 (A.6) 4 0 N0 3N0 −4 2 3N0 −3 −2 −4 F3 = t3N t + N t + 3N t t ), 16 + O(N 0 1 3 2 3 1 2 1 N 7 4N0 0 4N0 −4 2 4N0 −3 −2 F 4 = t1 + N t1 7N0 + 60 t2 + 9N0 t1 t3 + O(N −4 ). 4 2 1
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G = 2: 0 F1 =N −2 tN 1 ,
[ 2 N0 ]+3 N X 0 −4 15 2N0 F2 =N t1 + (t2 )N0 +6−2g t2g−6 N 2−2g 8 , 2 2 2g − 6 1 g=4 N 0 3N0 −4 2 3N0 −3 −6 220 3N0 −8 F3 =N t t 640 +N t2 + 135N0 t1 t3 + O(N −10 ), 3 1 2 1 N 5275 4N0 0 4N0 −3 0 −4 2 F4 =N −8 t1 +N −10 3760N0 +41280 t4N t + 8505N t t 0 1 3 + 2 1 4 2 1
+ O(N −12 ). (A.7) Here have employed the standard notation for binomial and multinomial coefficients. It is instructive to draw the Riemann surfaces corresponding to the various terms in (A.5),(A.6),(A.7). As will be seen, the combinatorics involved increases rapidly in complexity. The above examples are easily checked to be in agreement with the Riemann-Hurwitz formula. Of course, this formula only gives a necessary condition for the existence of a Riemann surface. The above results can be used to decide whether a Riemann surface 2 N3 of given G,g and branch point structure tN 2 t3 . . . actually exists. For example, we see from F4 in (A.5) that there exists a fourfold cover of the sphere by a sphere with exactly six simple branch points, where two branchpoints are located at each of three locations 5 ). As a second example, we of the target manifold (the associated symmetry factor is 16 see from F3 in (A.6) that there exists a triple cover of the torus by a double torus with exactly one branch point of order 2 (the associated symmetry factor is 3).
Appendix B. Free Energies for an Arbitrary Number of Movable Branch Points In this appendix we give some concrete examples of how the maps are counted in the continuum limit we defined above. Define Z =1+
∞ X
Zn e−nAT .
(B.1)
n=1
The first few auxiliary ξkh (see (4.4)) are: X
1 hi − N (N − 1), 2 i X 1 2N − 1 X N (N − 1)(2N − 1) ξ2h = , h2i − hi + 2 i 2 6 i 1X 3 1X X 2N − 1 X 2 ξ3h = hi − hi hj − hi + 3 i 2 i 2 j i ξ1h =
X 1 9 9 N (N − 1)(3N 2 − 3N + 2) + ( N 2 − N + 2) . hi − 3 2 2 8 i
(B.2)
Complex Matrix Models and Statistics of Branched Coverings
297
They are easily computed from the following representation of Eq. (4.4): ξnh =
1 n2
I
∞
X (−n)p ∂ p−1 dh h(h − 1) . . . (h − n + 1) exp H(h) , 2πi p! ∂h
(B.3)
p=1
where we have introduced H(h) =
PN
1 i=1 h−hi .
The first few Zn ’s, up to order four, are:
Z1 =N 2−2G , 1 1 1 Z2 =( N 2 )2−2G e N τ2 + e− N τ2 , 2 3 2 3 2 1 Z3 =( N 3 )2−2G e N τ2 + N 2 τ3 + e− N τ2 + N 2 τ3 + 6 1 1 + ( N 3 )2−2G e− N 2 τ3 3 6 8 6 1 4 2−2G N6 τ2 + 82 τ3 + 63 τ4 N N Z4 =( N ) + e − N τ2 + N 2 τ3 − N 3 τ4 + e 24 2 2 2 2 1 + ( N 4 )2−2G e N τ2 − N 3 τ4 + e− N τ2 + N 3 τ4 + 8 4 1 + ( N 4 )2−2G e− N 2 τ3 . 12
(B.4)
This gives the following continuum free energies (as can be checked easily from the results of Appendix A): G = 0: F1 =N 2 , ∞ X 1 1 τ 2g+2 , N 2−2g F2 = 2 (2g + 2)! 2 g=0 h1 1 1 i (B.5) F3 =N 2 τ24 + τ22 τ3 + τ32 + 6 2 6 h1 3 1 1 i τ26 + τ24 τ3 + τ22 τ32 + τ33 + O(N 0 ), + 18 8 2 18 h1 27 3 1 2 1 i F4 =N 2 τ26 + τ24 τ3 + τ22 τ32 + τ2 τ3 τ4 + τ33 + + τ23 τ4 + τ42 + O(N 0 ). 6 24 2 6 3 8 G = 1:
F1 =1, ∞
3 X 2−2g 2 τ 2g−2 , N F2 = + 2 (2g − 2)! 2 g=2 h i 4 F3 = + N −2 8τ22 + 3τ3 + O(N −4 ), 3 h i 7 F4 = + N −2 30τ22 + 9τ3 + O(N −4 ). 4 G = 2:
(B.6)
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F1 =N −2 , ∞
15 X 2−2g 8 + τ 2g−6 , N 2 (2g − 6)! 2 g=4 h i 220 F3 =N −6 + N −8 320τ22 + 135τ3 + O(N −10 ), 3 h i −8 5275 + N −10 20640τ22 + 8505τ3 + O(N −12 ). F4 =N 4
F2 =N −4
(B.7)
References 1. Kostov, I. and Staudacher, M.: Phys. Lett. B 394, 75 (1997) 2. Gross, D.: Nucl. Phys. B 400, 161 (1993); Gross, D. and Taylor, W.: Nucl. Phys. B 400, 181 (1993); Nucl. Phys. B 403, 395 (1993) 3. Kazakov, V.A., Staudacher, M. and Wynter, T.: Advances in Large N Group Theory and the Solution of Two-Dimensional R2 Gravity. hep-th/9601153, 1995 Carg`ese Proceedings 4. Migdal, A.A.: Zh. Eksp. Teor. Fiz. 69, 810 (1975) (Sov. Phys. JETP 42, (A 75), 413) 5. Rusakov, B.: Mod. Phys. Lett. A 5, 693 (1990) 6. Kazakov, V.A., Staudacher, M. and Wynter, T.: Commun. Math. Phys. 177, 451 (1996); Commun. Math. Phys. 179, 235 (1996); Nucl. Phys. B 471, 309 (1996) 7. Cordes, S., Moore, G. and Ramgoolam, S.: Large N 2-D Yang-Mills Theory and Topological String Theory. hep-th/9402107, and Lectures on 2D Yang-Mills Theory, Equivariant Cohomology and Topological Field Theories. hep-th/9411210, 1993 Les Houches and Trieste Proceedings; Moore, G.: 2-D Yang-Mills Theory and Topological Field Theory. hep-th/9409044 8. Taylor, W. and Crescimanno, M.: Nucl. Phys. B 437, 3 (1995) 9. Taylor, W.. MIT-CTP-2297 hep-th/9404175 (1994) 10. Douglas, M.R. and Kazakov, V.A.: Phys. Lett. B 312, 219 (1993) 11. Gross, D. and Witten, E.: Phys.Rev. D 21, 446 (1980) 12. Gross, D. and Br´ezin, E.: Phys. Lett. B 97, 120 (1980) 13. Staudacher, M. and Wynter, T.: Unpublished (1996) 14. Kazakov, V.A. and Kostov, I.: Nucl. Phys. B 176, 199 (1980) Communicated by R. H. Dijkgraaf
Commun. Math. Phys. 191, 299 – 323 (1998)
Communications in
Mathematical Physics c Springer-Verlag 1998
On GNS Representations on Inner Product Spaces I. The Structure of the Representation Space Gerald Hofmann Universit¨at Leipzig, NTZ, Augustusplatz, D-04109 Leipzig, Germany. E-mail:[email protected] Received: 21 May 1996 / Accepted: 12 May 1997
Abstract: A generalization of the GNS construction to hermitian linear functionals W defined on a unital *-algebra A is considered. Along these lines, a continuity condition (H) upon W is introduced such that (H) proves to be necessary and sufficient for the existence of a J-representation x → πW (x), x ∈ A, on a Krein space H. The property whether or not the Gram operator J leaves the (common and invariant) domain D of the representation invariant is characterized as well by properties of the functional W as by those of D. Furthermore, the interesting class of positively dominated functionals is introduced and investigated. Some applications to tensor algebras are finally discussed.
1. Introduction One of the most powerful theorems frequently used in mathematical physics is that about the GNS representation (Gelfand, Neumark, Segal) stating that for every i) positive functional W defined on ii) a unital C ∗ -algebra there is a cyclic *-representation by bounded operators on a Hilbert space ([26, Chap. 17.4, Theorem 2]). In order to apply that theorem to General (axiomatic) QFT, the following generalizations were considered about 30 years ago. While a generalization of ii) to topological *-algebras leading to unbounded *-representations was given by Borchers ([7]), Uhlmann ([34]) and Powers ([33]), a generalization of i) to hermitian linear functionals yielding representations on (possibly indefinite) inner product spaces ([4, 5]) was studied by Scheibe ([29], cf. Proposition 1, below). The last is of increasing interest because the investigations of some models (e.g., gauge fields ([6, Chap. 10], [21, 25, 14, 15]), massless fields ([31, 24])) considered within the Borchers-Uhlmann approach to General QFT lead to GNS representations on indefinite inner product spaces. Starting with a unital *-algebra A and an hermitian linear functional W defined on A, the following new features enter the theory in contrast to the well-known reconstruction theorem for positive functionals.
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– A priori, the (common and invariant) domain D of the operators of the representation πW (.) does not carry any scalar product [., .] such that the indefinite metric (., .) inherited by W on D satisfies (., .) = [., J.], where J is a bounded Gram-operator (cf. (1), (2), below). However, for as well mathematically (see ad 1) and ad 2) in Chap. 2) as physically (see [19, ch. 3]) motivated reasons , it is desirable to introduce a Hilbert space structure (H, J) on D. – If there are Hilbert-space structures on D, then even the maximal one (cf. Proposition 2) is not uniquely defined in general. Let us mention that a whole family of nonequivalent maximal Hilbert-space structures was explicitly constructed by Araki ([2]) in the case of A = (C2 )⊗ (tensor algebra over C2 ). – Furthermore, if there is a Hilbert-space structure on D, then in general, the representation πW is not involution preserving with respect to the Hilbert-space adjoint operators πW (.)[∗] , i.e., πW is not any *-representation ([30]). However, x → πW (x), x ∈ A, is a J-representation in the sense of Neumark (cf. [26, §41.6], [32], Definition 1). – In contrast to the case of *-representations it was observed by Yngvason that D 6⊂ D[∗] = JD (domain of the adjoint operators π(.)[∗] ) is possible. For applications, both cases i) D ⊂ D[∗] and ii) D 6⊂ D[∗] are of interest since there are models in General QFT such that respectively, i) (see [17, 22, 18, 12]) and ii) (see [13]) apply. Along these lines and guided by the idea that the whole theory is encoded in the functional W , the following questions arise. (Q1) Which conditions must the functional W satisfy such that a Hilbert space structure exists on D? (Q2) Under which conditions does the Gram operator J satisfy J : D → D ? (Q3) Under which conditions does exactly one maximal Hilbert space structure exist on D ? In the case of the tensor algebra S⊗ (field algebra of axiomatic QFT for one hermitian scalar field), (Q1) was answered by Yngvason in [35]. For the general case of the above *algebra A, question (Q1) will be answered in Theorem 3. More precisely, it is shown that a Hilbert space structure exists on D, if and only if W satisfies the Hilbert-space structure condition (H) introduced in Chap. 2. In Theorem 1 and Proposition 3 the structure of D and of its completion is considered as both an inner product space and a locally convex vector space. Using these results, a generalization of the GNS representation to Jrepresentations on Krein-spaces is given in Theorem 2. Since the proofs are constructive, there are given two possibilities for a construction of the corresponding Krein space (see Remark 4). (Q2) was also posed and discussed by Yngvason in [35, chapter 4]. In Theorem 4 an answer to (Q2) will be given in the settings of respectively, the inner product space D and the functional W . Concerning (Q3) it was mentioned by Araki (cf. [2]) that while the indefinite metric (., .) on D is supposed to be relevant to physical interpretation the positive definite metric [., .] is not intrinsic. However, if there is an affirmative answer to (Q2), then sufficient conditions for an affirmative answer to (Q3) are given in Proposition 4 and Corollary 2. Guided by the significance of positive functionals for the investigations on *algebras, the interesting class of positively dominated hermitian functionals is introduced. Among others, it is proven in Theorem 5 that every positively dominated hermitian functional satisfies (H) and consequently leads to a J-representation on a Krein space. It is further shown by two counter-examples (Examples 1, 2) that there are as
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well hermitian linear functionals W not satisfying (H) as such functionals W satisfying (H) but not being positively dominated. On the other hand, for the interesting case of tensor algebras having a countable (algebraic) basis it is shown that the class of positively dominated hermitian functionals coincides with that of hermitian linear functionals (Proposition 5). The pattern of the present paper is as follows. The prerequisites for the following considerations are briefly recalled in Chap. 2. After introducing the Hilbert-space condition (H), quadratic majorants are constructed in Chap. 3. In Chap. 4 there is investigated the structure of the state space obtained by GNS representation. A discussion of (H) including answers to (Q1), (Q2), and (Q3) is given in Chap. 5. While the class of positively dominated functionals is introduced and studied in Chap. 6, some applications to tensor algebras are finally given in Chap. 7. Let us mention that an application of condition (H) to QFT with indefinite metric and transformations of linear functionals including truncation is given in [16]. Further, continuity properties of the representation x → πW (x) will be studied in a subsequent paper.
2. Preliminaries Throughout the present paper let A denote an (associative) *-algebra with unity 1 and W a linear and hermitian functional on A satisfying W (1) = 1. Recall the following GNS-like reconstruction theorem due to Scheibe. Proposition 1. Under the above assumptions there are (i) a vector space D with an inner product (., .), (ii) a vector ψ0 ∈ D satisfying (ψ0 , ψ0 ) = 1, (iii) a representation f 7→ πW (f ) of A by linear operators on D such that W (f ) = (ψ0 , πW (f )ψ0 ), D = span {πW (f )ψ0 ; f ∈ A}, cyclicity of ψ0 , (φ, πW (f )ψ) = (πW (f ∗ )φ, ψ), f ∈ A, φ, ψ ∈ D. Furthermore, D, ψ0 , and πW (.) are uniquely defined by (i),. . .,(iii) up to linear isomorphisms. Proof. See [29, 35].
In order to make the theory mathematically manageable (see ad 2), below) one has to define a Hilbert space structure on D, i.e., there is a positive definite inner product e k.k , k.k = √[., .], such that [., .] and a linear operator J = J ∗ (Gram operator) on H = D (f, g) = [f, Jg], kJk := sup kJxk < ∞,
(1) (2)
kxk≤1
where the continuously extended inner products are also denoted by (., .), [., .] on H. (Here and in the following, lete· k.k (resp.e· τ ) denote the completed hull of a set · relative
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to the (locally convex) topology defined by k.k (resp. τ ).) Let the above Hilbert space structure be denoted by (H, J). Recall further that two Hilbert space structures (Hj , Jj ), j = 1, 2, are called equivalent, if the two positive definite inner products [., .]j satisfy (f, g) = [f, J1 g]1 = [f, J2 g]2 , p and both norms k.kj = [., .]j , (j = 1, 2) are equivalent on D, thus H1 = H2 . Remember also that if the inverse operator J −1 exists as a bounded linear operator on H, then there is an equivalent Hilbert space structure (H0 , J 0 ) given by [x, y]0 = [x, |J|y], J0 =
J , |J|
x, y ∈ H ([9, §2.7(3)]). Noticing that J 0 is a symmetry on H0 (i.e., J 0 = J 0∗ = J 0−1 ), (H0 , J 0 ) is called a Krein space structure. The existence of a Hilbert space structure has the following consequences: 1) there is a maximal Hilbert space structure, 2) the theory of unbounded representations applies, and consequently, the theory is well-understood and mathematically manageable (see [30]). ad 1): A Hilbert space structure (H, J) on D is called maximal if there is no other ⊂
one (H1 , J1 ) satisfying H 6= H1 . Recall the following. Proposition 2. The following are equivalent. (i) (H, J) is a maximal Hilbert space structure, (ii) J −1 is a bounded operator on H, (iii) H is a Krein space. Proof. (i) ⇔ (ii) : [23, Theorem 5], (ii) ⇔ (iii) : [5, V.1.3].
Remark 1. In General QFT there are considered both a) a representation π of a unital *-algebra A and b) representations ρ of certain groups G (gauge groups, Poincar´e group) on some inner product space H, (., .). While the present paper is concerned with a), a detailed analysis of indecomposable representations ρ of G with invariant inner product such as it is found, e.g., in the situation of Gupta-Bleuler QED was given by Araki in [3]. There were considered subspaces H1 ⊂ H such that ρ|H1 (restriction of ρ to H1 ) is irreducible and H1 = H1⊥⊥ (bi-orthogonal complement in H, (., .)), see [3, Theorem 1]. Since H, (., .) is given by π considered in a), there is the following interplay between a) and b). Recalling that if H, (., .) is a Krein space then H1 = H1⊥⊥ holds for every subspace H1 closed relative to the Krein-space topology, Araki’s theory immedeately applies to every closed subspace of Krein space H and topological complications are avoided. ad 2): If a Hilbert space structure is introduced on D, the theory of unbounded representations on Hilbert spaces applies. Let us recall some notions and results from the theory of unbounded representations of *-algebras on Hilbert spaces. Let A be an (associative) *-algebra with unity 1. A (closable) representation π of A on a Hilbert space H, [., .] with domain D(π) ⊂ H is a linear mapping π of A into the set of closable linear operators defined on D(π) such that
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a) π(1) = I (identity operator on H), b) π(x)π(y)ξ = π(xy)ξ for all ξ ∈ D(π), x, y ∈ A, c) D(π) is dense in H, and π(x)ξ ∈ D(π) for all ξ ∈ D(π), x ∈ A. In order to obtain involution-preserving representations on Hilbert spaces, the concept of *-representations was introduced (see [30]). If one considers representations on inner product spaces, there is the concept of J-representation introduced in the case of algebras of bounded operators in [26, §41.6] and generalized to the case of unbounded ˆ ([32]). Following the idea that while the inner product (., .) is intrinsic operators by Ota the (positive definite) scalar product [., .] is only auxiliary, let us give the following reformulation of [32, Def. 2.3]. Definition 1. A representation π of a unital *-algebra A with domain D(π) is called a J-representation, if there is a Krein space (H, J) such that (i) π is a representation on (H, J), (ii) (1) holds, (iii) π ⊂ π (∗) (for an explanation see the following remarks). Remark 2. a) Assuming that (i), (ii) of Definition 1 apply, for every densely defined operator T there are the both adjoint operators T (∗) and T [∗] defined by (T ξ, ζ) = (ξ, T (∗) ζ), [T ξ, ζ] = [ξ, T [∗] ζ], ξ ∈ dom(T ), ζ ∈ dom(T (∗) ) resp. ∈ dom(T [∗] ), and T [∗] = JT (∗) J is satisfied. Setting D(π (∗) ) =
\
dom(π(x)(∗) ),
x∈A
π (∗) (x) = π(x∗ )(∗) |D(π(∗) ) , the restriction of π(x∗ )(∗) to D(π (∗) ), (iii) reads as π(x)ξ = π (∗) (x)ξ for all x ∈ A, ξ ∈ D. Hence, if π is a J-representation, then (ξ, π(x)ζ) = (π(x∗ )ξ, ζ), ξ, ζ ∈ D, and π(x∗ ) = π(x)(∗) , x ∈ A on D. b) Assume that Definition 1 applies. Considering the algebra of operators {π(a); a ∈ A} defined on D and endowed with the *-operation π(a)+ = π(a)(∗) |D , π becomes a *homomorphism between *-algebras A and {π(a); a ∈ A}. c) Continuity properties of π will be considered in a subsequent paper. In order to investigate J-representations the concept of P-functionals was introduced ˆ ([1]), and further investigated in [12, 13]. Setting by Antoine and Ota NW = {f ∈ A; W (g ∗ f ) = 0 for all g ∈ A}, let us consider the following generalization of that concept for answering (Q2) in Chap. 5.
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Definition 2. An hermitian linear functional W defined on a unital *-algebra A is called a generalized P-functional, if W (1) = 1 and there is a linear mapping α : A → A such that α2 αn α1 W (xy) W (α(x∗ )) W ((αx)∗ x)
= id (identity mapping on A), = n for n ∈ NW , = 1, = W (α(x)α(y)), = W ((αx)∗ ), ≥0
for all x, y ∈ A. Remark 3. Considering the projection P =
1 (α + id), 2
it was shown in [1] that P : A → B is an abstract conditional expectation ([27]) of A onto some *-subalgebra B. (Let us mention that the notation "P-functional" refers to the above projection P .) At the end of this section let us recall some notions from the geometry of inner product spaces frequently used in the following ([5]). An inner product space E, (., .) is called decomposable if it admits a fundamental decomposition, i.e., there are positive and negative definite (linear) subspaces E + and E − such that .
.
E = E + (+)E − (+)E (0) , where (x+ , x− ) = 0 for every x± ∈ E ± , E (0) := E ∩ E ⊥ (set of isotropic vectors of E). Further, E, (., .) is called non-degenerate, if E (0) = {0}. Letting E, (., .) be nondegenerate and decomposable, every fundamental decomposition of E is of the form .
E = E + (+)E − , where E ± are as above. Fixing such a fundamental decomposition of E and letting P ± : E → E ± denote the fundamental projectors belonging to it, define the fundamental symmetry J := P + − P − relative to the fundamental decomposition chosen above. Noticing now that [., .]J := (., J.) is a positive definite inner√product (scalar product) on E, consider the topology τJ defined by the norm k.kJ := [., .]J . Since |(x, y)|2 ≤ kxkJ kykJ , x, y ∈ E, the inner product is (jointly) continuous relative to τJ , and thus τJ is called a decomposition majorant assigned to J.
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3. Condition (H) and Quadratic Majorants Let us be given a *-algebra A with unity 1 and an hermitian linear functional W on A satisfying W (1) = 1. Let us consider the following Hilbert-space structure condition. (H) There is a quadratic seminorm p on A such that for each g ∈ A there is a constant Cg ≥ 0 and |W (g ∗ f )| ≤ Cg p(f )
(3)
(f, g) = W (g ∗ f )
(4)
is satisfied for all f ∈ A. Noticing that defines an inner product on A, (H) means that p defines a quadratic and partial majorant on A. Let us also consider the isotropic part A(0) = NW . Let us further introduce the topologies τn (n = 1, 2, . . .) defined by the seminorms P(τn ) = {p(n) }, where p(1) = p, p(n+1) (f ) = sup{|(f, g)|; p(n) (g) ≤ 1}
(5)
(the existence will follow from Lemma 1). Notice that the topologies τn are not separated in general, and furthermore, τn+1 is the polar topology of τn with respect to (4). For any seminorm q, let us put ker(q) = {f ∈ A; q(f ) = 0}. Some of the properties of the topologies τn and the seminorms p(n) used extensively afterwards are collected in Lemma 1. Lemma 1. Assuming that (H) applies, the following are implied. i)
It holds ∞ > p(1) (f ) ≥ p(3) (f ) = p(5) (f ) = p(7) (f ) = . . . , p (f ) = p(4) (f ) = p(6) (f ) = . . . < ∞ (2)
for each f ∈ A. ii) The seminorms p(n) are quadratic on A, n = 1, 2, . . . iii) τn are partial majorants on A, and |(f, g)| ≤ p(n) (f )p(n+1) (g) hold for each f, g ∈ A, n = 1, 2, . . . iv) ker(p(1) ) ⊂ A(0) , ker(p(m) ) = A(0) for m = 2, 3, . . .
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Proof. The proof uses some ideas taken from [5, Lemmata III.4.1-3] i), iii): Using (H), there is some constant Cg(1) < ∞ such that |(f, g)| ≤ Cg(1) p(1) (f ) for each f ∈ A. Hence, p(2) (g) = sup{|(f, g)|; p(1) (f ) ≤ 1} ≤ Cg(1) < ∞ for each g ∈ A. Further,
|(f, g)| ≤ p(1) (f )p(2) (g)
(6)
for each f, g ∈ A.Using |(f, g)| = |(g, f )|, interchanging f and g, and setting Cg(2) = p(1) (g) < ∞, (6) implies
|(f, g)| ≤ Cg(2) p(2) (f ), p(3) (g) = sup{|(f, g)|; p(2) (f ) ≤ 1} ≤ Cg(2) = p(1) (g), |(f, g)| ≤ p(2) (f )p(3) (g)
for each f, g ∈ A. Arguing as above, p(n+2) (g) ≤ p(n) (g), |(f, g)| ≤ p(n) (f )p(n+1) (g)
(7) (8)
follow for n = 1, 2, . . .. (8) proves iii). Noticing that (5) and (7) also yield p(n+3) (g) ≥ p(n+1) (g), g ∈ A, n = 1, 2, . . . , i) follows. iv): Using iii), for each g ∈ A there are Cg(n) < ∞ such that |(f, g)| ≤ Cg(n) p(n) (f ) for all f ∈ A, n = 1, 2, . . .. Hence, f ∈ ker(p(n) ) implies (f, g) = 0 for all g ∈ A, i.e., f ∈ A(0) . Thus, (9) ker(p(n) ) ⊂ A(0) . Conversely, letting y ∈ A(0) , (x, y) = 0 holds for all x ∈ A, and (3) yields p(n+1) (y) = sup{|(x, y)|; p(n) (x) ≤ 1} = 0
(10)
for n = 1, 2, . . . . Hence A(0) ⊂ ker(p(m) ) for m = 2, 3, . . . . Now, iv) follows from (9), (10). ii): Assuming that there is an s ∈ N such that p(s) is quadratic, there is a positive inner product (., .)s on A such that p p(s) (f ) = (f, f )s , f ∈ A. Consider the quotient space E = A/ker(p(s) ) and note that a positive definite inner product is defined by (fˆ, g) ˆ (s) = (f, g)s
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on E, where f, g ∈ A and fˆ, gˆ denote the residue classes containing f, g, respectively. Noticing that g ∈ ker(p(s) ) ⊂ A(0) implies W (g ∗ f ) = 0, W (f ∗ g) = W (g ∗ f ) = 0 for each f ∈ A, an inner product (x, ˆ y) ˆ E = W (y ∗ x), x, y ∈ A, is defined on E. Consider the linear functional ˆ = (x, ˆ y) ˆE Lyˆ (x) on E. Using (8), the k.k(s) -continuity of Lyˆ (.) follows from ˆ = |W (y ∗ x)| = |(x, y)| ≤ Cy(s) p(s) (x) = Cy(s) kxk ˆ (s) , |Lyˆ (x)| p ˆ x) ˆ (s) , xˆ ∈ E, Cy(s) := p(s+1) (y). Hence there is a unique extension where kxk ˆ (s) = (x, ˜ where E˜ denotes the completed hull of E with respect to the norm k.k(s) . L˜ yˆ to E, Applying the Riesz representation theorem, there is zˆ ∈ E˜ such that L˜ yˆ (x) ˆ = (x, ˆ z) ˆ (s) . Moreover, kzk ˆ (s) = kL˜ yˆ k = sup{|W (y ∗ x)|; p(s) (x) ≤ 1} = p(s+1) (y).
(11)
Since the above mapping y 7→ zˆ is linear and k.k(s) is quadratic, (11) implies that the parallelogram-identity applies to p(s+1) . Hence, p(s+1) is quadratic, too. Recalling that p(1) is quadratic due to (H), ii) follows. Using the seminorms p(n) introduced above there are two further interesting topologies ρj on A defined by the seminorms r 1 (j) 2 (j) q = ((p ) + (p(j+1) )2 ), 2 j = 1, 2, respectively. Lemma 1 readily implies the following. Corollary 1. ρj are quadratic majorants on A, and ker(q (1) ) ⊂ ker(q (2) ) = A(0) hold. Furthermore, ρ2 ≤ ρ1 . In order to show that there are hermitian linear functionals not satisfying (H) let us consider the following example. Example 1. Letting B denote the set of all sequences x = (xn )∞ n=0 , xn ∈ C, and introducing algebraic operations by setting (x + y)n = xn + yn , (xy)n = xn yn , (x∗ )n = x¯ n , n = 0, 1, 2, . . . , B becomes a *-algebra. Let
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A = {(λ, x); λ ∈ C, x ∈ B} be the *-algebra with unity (1, 0) obtained from B by adjunction of unity ([26, §7.2]). Consider now the subspace L = {x ∈ B; there exists nx ∈ N with xν = 0 for ν = nx , nx + 1, . . .} . ˇ where (L) ˇ ∗ = L. ˇ Define then an and an (algebraic) direct decomposition A = L + L, hermitian linear functional W on A by setting P∞ λ + j=0 xj for x ∈ L W ((λ, x)) = . λ for x ∈ Lˇ
Statement. W does not satisfy condition (H). Proof. Assuming that W satisfies (H), there is a norm k.k on A such that for each a ∈ A there is a constant Ca < ∞ with |W (a∗ b)| ≤ Ca kbk
(12)
(n) for all b ∈ A. Choosing e(n) = (δn,j )∞ = (0, e(n) ) ∈ A, n = 0, 1, 2, . . . , x = j=0 ∈ B, y ∞ (xm )m=0 ∈ B with xm = mCy(m) , m = 0, 1, 2, . . . , z = (0, x) ∈ A, where Cy(m) is taken from (12). Noticing e(n)∗ x ∈ L,
W ((0, e(n)∗ x)) = xn = n Cy(n) , and setting a = y (n) , b = z in (12), it follows n Cy(n) ≤ Cy(n) kzk,
n = 0, 1, 2, . . . , which is impossible.
4. GNS Representation on Krein Spaces Throughout the present chapter let us assume that (H) applies. In order to apply the well-developed theory of non-degenerate inner product spaces (see [5, ch. IV.4]), let us consider the vector space D = A/A(0) endowed with the non-degenerate inner product (x, ˆ y) ˆ D = W (y ∗ x), where x ∈ x, ˆ y ∈ y, ˆ and ˆ. denotes the residue class of . in D. Furthermore, q (2) inherits a normed and quadratic majorant also denoted by q (2) on D. Setting kf k(1) = q (2) (f ), f ∈ D, let us recursively define kf k(n+1) =
r
1 ((kf k(n) )2 + (kf k0(n) )2 ), 2
where k.k0(.) = sup{|(., g)|; kgk(.) ≤ 1} denotes the polar norm.
(13)
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Lemma 2. k.k(n) , n = 1, 2, . . . , are quadratic majorants on D satisfying (i) (ii)
|(f, g)| ≤ kf k(n) kgk(n) , khk(n) ≥ khk(n+1) ≥ √12 khk(n) ≥
√1 khk0 , (1) 2
f, g ∈ D, 0 6= h ∈ D, n = 1, 2, . . . . Proof. Recall first that every k.k(n) , n = 1, 2, . . . , defines a majorant satisfying (i) on D, see [5, Lemma IV.4.1]. Applying now [5, Lemma III.4.3], Corollary 1 and (13) imply that k.k(n) , n = 1, 2, . . . , are quadratic. Noticing that (i) readily implies khk0(n) ≤ khk(n) , Eq. (13) yields the first two inequalities of (ii). Noting now that khk(n) ≥ khk(n+1) implies khk0(n+1) ≥ khk0(n) , it follows khk(n) ≥ khk0(n) ≥ khk0(1) proving the remainder of (ii).
Applying Lemma 2, let us consider kf k(∞) = lim kf k(n) , n→∞
f ∈ D. Noting that k.k(∞) is a quadratic norm on D, the following hold. Lemma 3. It holds kf k(∞) = limn→∞ kf k0(n) , f ∈ D, and k.k(∞) is a quadratic, minimal and self-polar (i.e., k.k(∞) = k.k0(∞) ) majorant on D. Proof. See [5, Lemma IV.4.1, Theorem IV.4.2].
Noticing that Lemmata 2, 3 imply that the 0-neighborhood U = {f ∈ D; kf k(1) ≤ 1} is k.k(∞) -closed, D ⊂ H(1) ⊂ H(∞)
(14)
follow, where e k.k(1) , H(1) = D e k.k(∞) , H(∞) = D (see [20, §18.4(4)]). Using Lemmata 2, 3, let us k.k(1) − (resp. k.k∞ −) continuously extend the inner product (.,.) of D onto H(1) (resp. H(∞) ), and let us also denote these extensions by (.,.). Let further τ∞ denote the topology defined by k.k(∞) on H(∞) . For the inner product spaces so obtained, the following hold. Theorem 1. The inner product spaces H(∞) and H(1) are i) decomposable and ii) nondegenerate. Furthermore, iii) H(∞) is a Krein space, and iv) there is a fundamental . decomposition H(∞) = H+ (+)H− such that 1ˆ ∈ H+ .
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Proof. i): Noticing that Lemmata 2, 3 imply that k.k∞ (resp. k.k(1) ) are Hilbert majorants on H(∞) (resp. H(1) ), the first assertion to be shown follows from [5, Theorem IV.5.2]. ii): For the following let H and k.k stand for one of the Hilbert spaces under consideration and its norm, respectively. In order to show ii), let us consider f ∈ H and a sequence {fm }∞ m=1 , fm ∈ D, such that lim kf − fm k = 0
m→∞
(15)
and (f, g) = 0
(16)
for all g ∈ H. Using (15), (16) and Lemmata 2, 3, it follows that |(fm , g)| = |(f − fm , g)| ≤ kf − fm kkgk → 0
(17)
for each g ∈ H as m → ∞. Since k.k(1) ≥ k.k(∞) , lim kf − fm k(∞) = 0
m→∞
(18)
applies to both cases under consideration. Considering the seminorms Pm (h) = |(fm , h)|, h ∈ H, m = 1, 2, . . . , and noticing that for each fixed h ∈ H, {Pm (h)}∞ m=1 is bounded (since limm→∞ Pm (h) = 0, see (17)), the uniform boundedness principle applies. Hence for each > 0 there is some k.k(∞) −neighborhood V of 0 such that |Pm (v)| < , for all v ∈ V, m = 1, 2, . . . , i.e., kfm k0(∞) = sup |(fm , g)| → 0 g∈V
(19)
as m → ∞. Using k.k0(∞) = k.k(∞) (see Lemma 3), f = 0 now follows from (18) and (19). iii): Since k.k(∞) defines a minimal Hilbert majorant on H(∞) , iii) follows from ˆ is an ortho-complemented Proposition 2 (i) ⇒ (iii). iv): Noticing that L = span{1} (∞) and positive definite subspace of H , there is a fundamental decomposition H(∞) = . H+ (+)H− such that L ⊂ H+ (see [5, Theorems V.3.4, V.3.5]), and 1ˆ ∈ H+ . Recalling that in Krein spaces all the interesting l.c. topologies such as all the decomposition majorants and the Mackey topology τM (H(∞) , H(∞) ) of the duality (H(∞) , H(∞) ) defined by the inner product (., .) coincide there is as well a natural notion of continuity in H(∞) as further equivalent descriptions of the Krein space topology τ∞ . Along these lines, the following considerations are aimed at further descriptions of the Krein-space norm k.k(∞) . Let us consider the positive definite inner products [., .](n)
(resp. [., .]0(n) , [., .](∞) )
which are defined by k.k(n) (resp. k.k0(n) ,k.k(∞) ) on H(1) , n = 1, 2, . . . . Lemma 2 implies that there is an hermitian and bounded operator G (Gram operator) such that (x, y) = [Gx, y](1) ,
(20)
On GNS Representations
311
kGk(1) = sup |[Gx, x](1) | = sup |(x, x)| ≤ sup kxk2(1) = 1, kxk(1) =1
kxk(1) =1
kxk(1) =1
x, y ∈ H . A relation between [., .](∞) and [., .](1) is given in the following. (1)
Lemma 4. It holds [x, y](∞) = [|G|x, y](1) , x, y ∈ H(1) , where G is taken from (20). Proof. Let us introduce hermitian and bounded operators Hn,n−1 , Hn by [x, y](n) = [Hn,n−1 x, y](n−1) , [x, y](n) = [Hn x, y](1) , Hn = H1,2 H3,2 . . . Hn,n−1 , x, y ∈ H(1) , n = 2, 3, . . .. Notice that the inverse operators (Hn )−1 , (Hn,n−1 )−1 exist due to Lemma 2. Since kxk0(1) =
sup |(x, y)| =
kyk(1) ≤1
it follows that
sup |[Gx, y](1) | = kGxk(1) ,
kyk(1) ≤1
[x, y]0(1) = [G2 x, y](1) , 1 [x, y](2) = [ (I + G2 )x, y](1) , 2
x, y ∈ H(1) . Hence, H2,1 =
1 (I + G2 ). 2
Then, kxk0(2) = =
sup |(x, y)| =
kyk(2) ≤1
sup |[Gx, y](1) |
kyk(2) ≤1
sup |[(H2,1 )−1 Gx, y](2) | = k(H2,1 )−1 Gxk(2) ,
kyk(2) ≤1
[x, y]0(2) = [(H2,1 )−2 G2 x, y](2) = [(H2,1 )−1 G2 x, y](1) , 1 [x, y](3) = [ (I + (H2,1 )−2 G2 )x, y](2) . 2 Hence, 1 (I + (H2,1 )−2 G2 ), 2 1 H3 = H2,1 H3,2 = (H2,1 + (H2,1 )−1 G2 ) 2
H3,2 =
follow. Arguing as above it follows that [x, y]0(n) = [(Hn )−2 G2 x, y](n) , 1 Hn+1,n = (I + (Hn )−2 G2 ), 2 1 Hn+1 = (Hn + (Hn )−1 G2 ), 2
(21)
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where the operators Hn and G are commuting, and Hn , Hn−1 are strictly positive due to mn+1 := m1 =
inf [Hn+1 x, x](1) ≥
kxk(1) =1
1 1 inf [Hn x, x](1) =: mn , 2 kxk(1) =1 2
inf [x, x](1) = 1,
kxk(1) =1
R1 n = 1, 2, . . . . Using the spectral resolution G = −1 λdEλ , it follows straightforwardly that G2 ≤ (H2,1 )2 and (22) 0 ≤ G2 ≤ Hn+1 ≤ Hn . √ 2 2 Hence, H = limn→∞ Hn exists, and (21) yields H = G , and also H = |G| (=: G2 ) due to (22). The proof of the lemma under consideration is completed. Starting with k.k(1) and using Theorem 1i), ii), there is a special fundamental symmetry G (23) J˜ = |G| satisfying J˜ = J˜∗ = J˜−1 on H(1) , where G is taken from (20). Considering the positive definite inner product ˜ (24) [x, y](J)˜ = (x, Jy), the decomposition majorant τJ˜ is defined by the norm q kxk(J)˜ = [x, x](J)˜ , x, y ∈ H(1) . Furthermore, let τJ denote any decomposition majorant on H(1) . τ
τ
J J˜ g g (1) (1) , and ii) τ = τ = τ (H(∞) , H(∞) ) = Proposition 3. It holds i) H(∞) = H =H J M J˜ τ∞ on H(∞) , (where τJ , τJ˜ also denote the l.c. topologies on H(∞) obtained by continuous extension of the corresponding norms).
Proof. i): Noticing that (20), (23) and (24) imply ˜ = [x, GJy] ˜ (1) = [x, |G|y](1) , [x, y](J)˜ = (x, Jy) x, y ∈ H(1) , Lemma 4 yields τJ˜ = τ∞ on H(1) . Recalling also that every fundamental decomposition of H(1) leads to a decomposition majorant τJ which is equivalent to τJ˜ (see [5, IV.6.4]), i) follows from [5, V.2.1]. ii): Since H(∞) is a Hilbert space with scalar product [., .](J)˜ , (24) implies that τM (H(∞) , H(∞) ) is defined by the norm ˜ (J)˜ , p(x) = kJxk x ∈ H(∞) (cf. [5, ch. IV.8]). Since J˜ is a symmetry q q ˜ J˜2 x) = (Jx, ˜ x) = kxk(J)˜ ˜ (J)˜ = (Jx, kJxk follows. Hence, τJ˜ = τM (H(∞) , H(∞) ) on H(∞) completing the proof of ii).
Remark 4 (to Proposition 3). Assuming that (H) applies and noticing that the foregoing considerations are constructive, there are the following two constructions leading to equivalent Krein space topologies.
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313
1. Using Lemmata 1, 2, a Krein space topology is given by kf k(∞) = lim kf k(n) = lim kf k0(n) . n→∞
n→∞
2. Using the Gram operator (see (20)), Lemma 4 yields p kf k(∞) = k |G|f k(1) , f ∈ D. Combining Proposition 1 and Theorem 1, the following generalization of the GNS construction to J-representations on indefinite inner product spaces is implied. Theorem 2. Let us be given A and W as above. a) If W satisfies condition (H), then there is a J-representation π with an algebraically cyclic vector ψ0 ∈ D(π) such that W (f ) = (ψ0 , π(f )ψ0 ), f ∈ A, and Jψ0 = ψ0 . b) Conversely, if π is a J-representation on D, then the hermitian linear functional T (f ) = (ψ, π(f )ψ), f ∈ A, satisfies (H) for each ψ ∈ D. ˆ Theorem 1 implies now that Proof. a) Using Proposition 1, set π = πW , ψ0 = 1. x, a ∈ A, xˆ ∈ D, Definition 1 Definition 1 (i), (ii) apply. Recalling that πW (a)xˆ = ac (iii) follows from ∗ x, y) ˆ = (x, ˆ a cy) = W ((ay)∗ x) = W (y ∗ (a∗ x)) = (ad ˆ = (πW (a∗ )x, ˆ y), ˆ (x, ˆ πW (a)y)
a, x, y ∈ A. Further, Theorem 1 iv) implies Jψ0 = ψ0 . b) The linearity of T is obvious. Applying Definition 1 (iii), the hermiticity of T follows from T (x∗ ) = (ψ, π(x∗ )ψ) = (π(x)ψ, ψ) = (ψ, π(x)ψ) = T (x). Note further that |T (x∗ y)| = |(ψ, π(x∗ )π(y)ψ)| = |(π(x)ψ, π(y)ψ)| = |[Jπ(x)ψ, π(y)ψ]| ≤ kJπ(x)ψk kπ(y)ψk = Cx kπ(y)ψk, where Cauchy-Schwarz inequality was applied to the scalar product [., .], and k.k = √ [., .], Cx := kJπ(x)ψk < ∞. Noticing finally that kπ(.)ψk is a quadratic seminorm on A, it follows that (H) applies to T . Remark 5 (to Theorem 2). a) Noting that every P-functional on A (for definition see [1]) obviously satisfies all the assumptions of Theorem 2a), the main result of [1, Theorem 3] is readily implied by our Theorem 2. b) In contrast to the well-known case of a positive functional W , a crucial distinction enters the theory concerning the uniqueness of the "reconstructed GNS-data" (π(.), ψ0 , D, H). While for positive W the GNS-data are uniquely defined up to a unitary intertwiner U , the situation is quite different in the case at hand in the following way.
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1) The Krein-space structure is not uniquely determined in general, (see also Proposition 5 and Corollary 2, below). 2) If (π1 (.), ψ1 , D1 , H1 ) is a second set of GNS-data also satisfying Theorem 2a), then due to Proposition 1 there is a linear and invertible operator T : D → D1 such that T ψ0 = ψ1 , π1 (.) = T π(.)T −1 , T −1 = T (∗) (pseudo-unitary), and however, T is not extendable to H as a bounded operator in general. Notice also that T is unitary with respect to [., .], if and only if T J = JT . c) Using the constructions explained in Remark 4, there is a further distinction in contrast to the well-known case of a positive functional W . While if the functional W under consideration is positive, the scalar product < ., . > of the state space obtained by GNS construction satisfies ∗ h >, < fcg, hˆ >=< g, ˆ fd
f, g, h ∈ A, in the case at hand [., .] does not satisfy such a condition in general. 5. Some Further Properties of the Representation Space There is the following answer to (Q1). Theorem 3. Considering (i) (ii) (iii) (iv) (v)
Condition (H) is satisfied, there is a quadratic majorant on D, there exists a Hilbert-space structure (H, J) on D, there exists a Krein-space structure (H0 , J 0 ) on D, e τJ , there is a non-degenerate and decomposable subspace E ⊂ D such that D ⊂ E where τJ refers to the corresponding decomposition majorant on E, (vi) statement (iv) applies and D ∩ J 0 D is τJ 0 -dense in D,
the following implications are implied: (i) ⇔ (ii) ⇔ (iii) ⇔ (iv) ⇐ (v) ⇔ (vi). Proof. (i) ⇒ (ii) : follows from Lemma 2. (ii) ⇒ (iii) : Corollary 1 implies that q (2) inherits a positive definite inner product [., .] on D. Since ρ2 is a majorant on A, there is a Gram operator J such that (1), (2) apply. (iii) ⇒ (iv) : Assuming (iii), Eq. (1) and (2) yield the spectral resolution Z m λ dEλ , J= −m
0 < m < ∞, in Hilbert space H, [., .]. Considering a second scalar product [x, y]0 := [x, |J|y], x, y ∈ H, it follows
Setting P + :=
Rm 0
kxk0 :=
p p p √ [x, |J|x] ≤ kJxk kxk ≤ m kxk.
dEλ and J 0 := 2P + − I, it follows (x, y) = [x, Jy] = [x, |J|(2P + − I)y] = [x, J 0 y]0 .
On GNS Representations
315
e k.k0 , J 0 ) is a Krein-space structure Noticing J 0 = J 0∗ = J 0−1 , it follows that (H0 = H on D. Hence (iv) applies. (iv) ⇒ (i) : Since a Krein space structure is a special Hilbert space structure (1), (2) apply. Then, ˆ = |[fˆ, J g]| ˆ ≤ kfˆkkJ gk ˆ = Cg p(f ) |W (f ∗ g)| = |(fˆ, g)| q ˆ < ∞, p(f ) = [fˆ, fˆ]. (v) ⇒ (i) : If E, (., .) is a nonyield (H), where Cg = kJ gk degenerate. and decomposable inner product space with fundamental decomposition E = E + (+)E − and fundamental projections P ± : E → E ± , then the corresponding decomposition majorant τJ is given by p kxkJ = (x, Jx), J = P + − P − , x ∈ E. Let k.kJ be continuously extended onto E˜ τJ , and this extended quadratic norm also be denoted by k.kJ . Using [5, Lemma II.11.4], the proposition under consideration now follows from ˆ y)| ˆ ≤ kxk ˆ J kyk ˆ J = Cy p(x), |W (y ∗ x)| = |(x, where Cy = kyk ˆ J , p(x) = kxk ˆ J , x ∈ x, ˆ y ∈ y, ˆ x, ˆ yˆ ∈ D. (v) ⇒ (vi) : Assuming (v), . consider a fundamental decomposition E = E + (+)E − , the corresponding fundamental symmetry J = P + − P − and decomposition majorant τJ . Since J : E → E, E = E ∩ JE ⊂ D ∩ JD e τJ , it follows that D∩JD is τJ -dense in D. Considering H = E e τJ , follows. Due to D ⊂ E let the τJ -continuous extensions of respectively, J and the inner product (., .) from E and E × E onto H and H × H be denoted by J 0 and (., .)0 . Noticing J 0 = J 0∗ = J −1 , it follows that (H, [., .]J 0 = (., J 0 .)0 ) is a Krein space structure on D. (vi) ⇒ (v) : Assuming (vi) and setting E = D ∩ JD, (v) follows. Remark 6. a) If Theorem 3 (iv) applies, then Theorem 1 (iv) implies that there is an ˆ ˜ satisfying J˜1ˆ = 1. equivalent Krein-space structure (D, J) b) Let us mention that Theorem 3 (v) ⇒ (i) yields a sufficient criterion useful for constructing non-decomposable inner product spaces. More precisely, if an inner product space does not satisfy (H), then necessarily, it is non-decomposable. Hence, the inner product space considered in Example 1 is non-decomposable. c) The proof of Theorem 3 (iii) ⇒ (iv) is based on the idea of introducing the new scalar product [., .]0 on H which is due to Ginsburg and Iokhvidov (see [9, §2 7(3)] and [23, Remark to Theorem 5]). More precisely, if the inverse J −1 of the Gram operator J considered in (iii) ⇒ (iv) of the above proof is unbounded in H, [., .], then both scalar products [., .] and [., .]0 are inequivalent, and thus the Hilbert-space structure (H, J) is inequivalent to the Krein-space structure (H0 , J 0 ). An answer to (Q2) is given in the following. Theorem 4. The following are equivalent: (i) there is a Krein-space structure (H, J) on D such that J : D → D, (ii) D, (., .) is a decomposable inner product space, (iii) W is a generalized P-functional (see Def. 2).
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Proof. (i) ⇒ (ii) : Assuming (i), consider D± := {x ± Jx; x ∈ D} ⊂ D. .
Then, D = D+ (+)D− yields a fundamental decomposition of the inner product space D, (., .), and hence (ii) is satisfied. (ii) ⇒ (iii) : Assuming (ii), there is a fundamental decomposition .
D = D+ (+)D− with 1ˆ ∈ D+ . The corresponding fundamental symmetry J = P + − P − then satisfies ˆ Considering now a direct decomposition J 1ˆ = 1. .
A = NW + B with 1 ∈ B, there is a linear bijection η : B → D. Define now a linear mapping α : A → A by setting x for x ∈ NW . α(x) = η −1 Jη(x) for x ∈ B It straightforwardly follows that Definition 2 applies to W and α. (iii) ⇒ (i) : Applying [1, Lemma 2, Theorem 3], the implication under consideration follows. The following is concerned with (Q3). Recalling the definition of intrinsic topology τint on definite subspaces of inner product spaces (see, e.g., [5, ch. III.3]), there is the following supplement to Theorem 3 (v) ⇒ (i). Proposition 4. Assume that there is a decomposable subspace E ⊂ D with fundamental . decomposition E = E + (+)E − such that D ⊂ E˜ τJ . If E + or E − are intrinsically complete, then up to equivalence, there exists exactly one Krein-space structure on D. ˜ on D. Hence, Proof. Theorem 3 implies that there is some Krein-space structure (H, J) τJ˜ is a minimal majorant on E. Assume now that there is a further Krein-space structure (H0 , J 0 ) on D being non-equivalent to the above. Consequently, τJ˜ |D 6= τJ 0 |D. However, recalling that there is only one minimal majorant on E ([5, Theorem IV.6.2]), τJ˜ |E = τJ 0 |E follows. Now D ⊂ E˜ τJ implies τJ˜ |D = τJ 0 |D which is a contradiction to the above. For the important class of quasi-positive (resp. quasi-negative) inner product spaces, i.e., D does not contain any negative definite (resp. positive definite) subspace of infinite dimension, the following corollary readily follows from Proposition 4 and the fact that every quasi-positive (resp. quasi-negative) inner product space is decomposable. Corollary 2. If D is quasi-positive or quasi-negative, then there exists exactly one Kreinspace structure on D. There is the following interesting application of Corollary 2 to axiomatic quantum field theory. Remark 7. Recalling that the one-particle space D1 of the free massless quantum field of space-time dimension 2 is quasi-positive ([8]) there is exactly one Krein-space structure on D1 . Hence, all the n-particle spaces of the corresponding Fock-space are uniquely determined.
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6. On Positively Dominated Functionals Introducing the cone of positive elements ( A+ =
M X
) f (i)∗ f (i) ; f (i) ∈ A, M ∈ N ,
i=1
a linear functional T on A is called positive if T (f ) ≥ 0, f ∈ A+ . Inspired by the significance of positive functionals within the investigations on *-algebras (see [26, 30] and references cited there), let us consider an interesting subclass of hermitian functionals leading to Krein-space structures. Definition 3. A hermitian linear functional W on A is called positively dominated, if there is some positive functional T on A such that |W (g ∗ f )|2 ≤ T (g ∗ g)T (f ∗ f ), g, f ∈ A. A characterization of positively dominated functionals is given in the following. Theorem 5. Considering (i) W is positively dominated, (ii) there exist a Krein-space structure on D and a positive functional T on A such that kfˆk(∞) ≤
p T (f ∗ f ), f ∈ fˆ,
q where kfˆk(∞) = (J fˆ, fˆ), fˆ ∈ D, (iii) there are a quadratic majorant k.k on D and a positive functional T on A such that p kfˆk ≤ T (f ∗ f ), f ∈ fˆ, (iv) there are positive functionals T (j) on A, j = 1, 2, such that W = T (1) − T (2) , (v) there is a locally convex topology τ on A such that W is continuous and A+ is normal, the following implications are implied: (v) ⇒ (i) ⇔ (ii) ⇔ (iii) ⇔ (iv). Remark 8. Concerning (v) let us mention that in [11] there are explicitly constructed some families of normal topologies in the case of tensor algebras A = E⊗ .
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√ Proof. (i) ⇒ (ii) : Noticing that p(f ) := T (f ∗ f ), f ∈ A, is a quadratic seminorm satisfying |W (g ∗ f )| ≤ Cg p(f ), where Cg = p(g), (H) follows. Hence, there exists a Krein-space structure on D due to Theorem 3. In order to show the remainder of (ii), notice that the polar seminorm p0 satisfies p0 (f ) ≤ p(f ) because of |W (g ∗ f )| ≤ p(g)p(f ), g, f ∈ A. Hence, r 1 00 2 ˆ ˆ kf k(∞) ≤ kf k(1) = p (f ) + p0 (f )2 2 r r 1 1 2 0 2 ≤ p(f ) + p (f ) ≤ p(f )2 + p(f )2 = p(f ) 2 2 p = T (f ∗ f ) completes the proof of (ii). (ii) ⇒ (iii): Lemma 3 yields (iii) with k.k = k.k(∞) . (iii) ⇒ (iv): Assuming that (iii) apply, |W (f ∗ f )| ≤ kfˆk2 ≤ T (f ∗ f ), (1) (2) f ∈ A, follows. p Setting T = Tp+ W, T = T , (iv) is implied. (iv) ⇒ (i): Assuming (j) ∗ (j) ∗ (iv), set aj = T (f f ), bj = T (g g), j = 1, 2, and consider the positive linear functional T = T (1) + T (2) .
Now, (i) follows from |W (g ∗ f )|2 = |T (1) (g ∗ f ) − T (2) (g ∗ f )|2 ≤ ≤ a21 b21 + a22 b22 + 2|T (1) (g ∗ f )T (2) (g ∗ f )| ≤ ≤ a21 b21 + a22 b22 + 2a1 b1 a2 b2 ≤ (a21 + a22 )(b21 + b22 ) = = T (f ∗ f )T (g ∗ g), f, g ∈ A. (v) ⇒ (iv) : follows from [28, V.3.3 Cor. 3].
In Proposition 5 it will be shown that in the case of tensor algebras having a countable algebraic basis every hermitian linear functional is positively dominated. On the other hand, the existence of non-positively dominated functionals follows from the following. Example 2. Consider
B = {1, e, f, ef, f e, ee}
and the vector space A = span (B) of dimension 6. Defining b = b∗ , 1 · b = b, b ∈ B, and e · f = ef, f · e = f e, e · e = −f · f = ee, ef · b0 = b0 · ef = f e · b0 = b0 · f e = ee · b0 = b0 · ee = 0, b0 ∈ B \ {1}, A becomes a unital *-algebra. Consider further the hermitian linear functional W defined by W (b) = 1, b ∈ B. Noticing A+ = {λ1 + µee; λ ≥ 0, µ ∈ R}, it follows that every positive functional T on A satisfies T (ee) = 0,
On GNS Representations
319
and thus Theorem 5 (iv) does not apply to W . Furthermore, the inner product (f, g) = W (g ∗ f ), g, f ∈ A, satisfies (f, g) = [f, Ag], here [., .] denotes the Euclidean metric and the transformation A is given by the matrix A = (aij )6i,j=1 with respect to the basis B, where a1j = aj1 = a22 = a23 = a32 = −a33 = 1, j = 1, . . . , 6, and aik = 0 otherwise. Introducing an eigenbasis of A, it follows (f, g) = [V f, DV g], where D = diag {λ1 , . . . , λ6 }, λ1 ≈ 3.07, λ2 ≈ −1.84, λ3 ≈ −1, 15, λ4 ≈ 0.92, λ5 = λ6 = 0, and V describes the corresponding transformation of coordinates. Hence, A(0) = {x ∈ A; (V x)j = 0, j = 1, . . . , 4} and D = A/A(0) is a Krein space, where [x, y]0 =
4 X
|λi |(V x)i (V y)i
i=1
and J = diag {1, −1, −1, 1} is the corresponding fundamental symmetry.
7. Applications to Tensor Algebras Let us apply the preceding to the special case of tensor algebras A = E⊗ := C ⊕ E1 ⊕ E2 ⊕ . . . , where En = E ⊗ E ⊗ . . . ⊗ E ( n copies). (For definitions and notions the reader is referred to [10, 11].) There is the following immediate consequence of Theorem 3. Corollary 3. If A = E⊗ , then the following are equivalent. (i) Condition (H) applies, (ii) there is a sequence of quadratic semi norms pn on En such that for each gm ∈ Em there are constants Cgm < ∞ with ∗ |Wn+m (gm ⊗ fn )| ≤ Cgm pn (fn ),
fn ∈ En (m, n = 0, 1, 2, . . .).
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Proof. (i) ⇒ (ii) : Using the assertion of Theorem 3 (iii), ∗ |Wn+m (gm ⊗ fn )| = |[fˆn , J gˆm ]| ≤ kJ gˆm kkfˆn k = Cgm pn (fn ),
where P Cgm = kJ gˆm k, pn (fn ) = kfˆn k. (ii) ⇒ (i) : Considering any g ∈ E⊗ , put ∞ Cg := n=0 Cgn and notice Cgn < ∞. Defining the quadratic semi norm v u∞ uX p(f ) = t 2n pn (fn )2 n=0
on E⊗ , |W (g ∗ f )| ≤
X
|Wm+n (gn∗ ⊗ fm )| ≤
m,n
X
Cgn pm (fm ) ≤
m,n
v uX ∞ X √ u∞ ≤ Cg t 2−j 2m pm (fm )2 = 2Cg p(f ) j=0
yields (H).
m=0
Since there are no non-trivial relations between the elements of E⊗ , the following being of some interest with respect to Theorem 5 is implied. Proposition 5. Let us be given a tensor algebra E⊗ having a countable (algebraic) basis. The following are equivalent. (i) W = (1, W1 , W2 , . . .) is an hermitian linear functional on E⊗ , (ii) W is positively dominated. Proof. (ii) ⇒ (i) is obvious. (i) ⇒ (ii) : Let {b(n) }∞ n=1 denote a basis of the basic space E. Setting [n1 , n2 , . . . , nm ] := b(n1 ) ⊗ b(n2 ) ⊗ . . . ⊗ b(nm ) , a basis of E⊗ is given by B = {1} ∪ {∪∞ m=1 [n1 , . . . , nm ]}, nj ∈ N, (j = 1, . . . , m). Let us also use the abbreviations ˜ := [nm , . . . , n1 ], [n] := [n1 , . . . , nm ], [n] [n] ∪ [r] := [n1 , . . . , nm , r1 , . . . , rs ], where [r] := [r1 , . . . , rs ], ri ∈ N, (i = 1, . . . , s). Setting ˜ wn1 ,...,nm := w[n] = |W ([n])| = |W ([n])|, the following is aimed at a recursive definition of constants cn1 ,...,nm := c[n] > 0 such that the following two systems of inequalities are satisfied: |w[s]∪[t] |2 ≤ c[s] c[t] ,
(25)
|c[s]∪[t] |2 ≤ c[s] c[t]∪[˜t]∪[˜s] ,
(26)
for all [s], [t] ∈ B. In order to introduce an ordering in B \ {1} let us consider the decomposition B = ∪∞ m=0 Dm
On GNS Representations
321
(m) into nonintersecting classes, where D0 = {1}, Dm = {∪m l=1 1l },
1(m) m = {[n1 , . . . , nm ]; nj ≤ m (j = 1, . . . , m)}, 1(m) = {[n1 , . . . , nl ]; nj ≤ m (j = 1, . . . , l), there is a l j0 ∈ {1, . . . , l} with nj0 = m}, (mj )
(j) l = 1, 2, . . . , m − 1. Considering any [n(j) 1 , . . . , nlj ] ∈ 1lj ordering " ≺ " by setting
, j = 1, 2, define an
(1) (2) (2) [n(1) 1 , . . . , nl1 ] ≺ [n1 , . . . , nl2 ] (1) if i) m1 < m2 , or ii) m1 = m2 and l1 < l2 , or iii) m1 = m2 , l1 = l2 and [n(1) 1 , . . . , n l1 ] (2) stands before [n(2) 1 , . . . , nl2 ] with respect to lexicographic ordering. Define now c[1] = max{1, |w[1] |2 }. Let us then consider some [n] = [n1 , . . . , ns ] ∈ B \ {1} and assume that c[m] are defined for all [m] ∈ B \ {1} with [m] ≺ [n]. Let us put
˜ Z[n] := {([r], [b]) ∈ B × B; [r] ≺ [n], [b] ≺ [n], [r] ∪ [n] = [b] ∪ [b]}, 2 |c[b] | ; ([r], [b]) ∈ Z[n] , d[n] := max c[r] |w[m]∪[n] |2 e[n] := max ; [m] ≺ [n] , c[m]
(27) (28)
and notice that the max in (27), (28) is only taken over finitely many items. Define then c[n] = max{1, d[n] , e[n] }, and note that (25), (26) are satisfied. Setting now T0 = 1, T2s−1 = 0, T2s ([m1 , . . . , ms ]∗ ⊗ [n1 , . . . , ns ]) := δn1 m1 δn2 m2 . . . δns ms c[n1 ,...,ns ] , s = 1, 2, . . . , (δ.. denotes Kronecker’s δ) (28), (29) imply after linear extension of T2s , |Wν+µ (fν∗ ⊗ gµ )|2 ≤ T2ν (fν∗ ⊗ fν )T2µ (gµ∗ ⊗ gµ ) , |Tν+µ (fν∗
⊗ gµ )| ≤ 2
T2ν (fν∗
⊗
fν )T2µ (gµ∗
⊗ gµ ) ,
(29) (30)
for all fν ∈ Eν , gµ ∈ Eµ (ν, µ ∈ N). Recall that there are sequences {αn }∞ n=0 , αn ≥ 0, α0 > 0, α2s+1 = 0 (s = 0, 1, . . .), such that the inequality of matices G ≥ E (unity matrix) is satisfied, where α0 0 −α2 0 −α4 0 . . . 0 α2 0 −α4 0 −α6 . . . G= −α2 0 α4 0 −α6 0 . . . ...........................
(see [11, Construction 3.6]). Thus,
322
G.Hofmann ∞ X
T2n (fn∗ ⊗ fn ) ≤
∞ X
n=0
α2n T2n (fn∗ ⊗ fn )
n=0 ∞ X
−
n=0
≤
∞ X
X p T2r (fr∗ ⊗ fr )T2s (fs∗ ⊗ fs )
α2n
r+s=2n r6=s
α2n T2n (fn∗ ⊗ fn ) +
n=0
=
∞ X
∞ X
α2n
n=0
α2n T2n (
n=0
X
X
Tr+s (fr∗ ⊗ fs )
r+s=2n r6=s
fr∗ ⊗ fs ),
(31)
r+s=2n
f = (f0 , f1 , . . . , fN , 0, 0, . . .) ∈ E⊗ , are implied. Considering T˜ = (T˜0 , T˜1 , . . .) with T˜n = αn Tn , n = 0, 1, . . ., (29), (31) imply |W (f ∗ g)|2 ≤
∞ X X
|Wµ+ν (fµ∗ ⊗ gν )|
n=0 µ+ν=n
≤
∞ X X
T2µ (fµ∗ ⊗ fµ )T2ν (gν∗ × gν )
n=0 µ+ν=n
≤
∞ X
T2µ (fµ∗ ⊗ fµ )
µ=0
∞ X
T2ν (gν∗ × gν )
ν=0
! ∞ ∞ X X X X ∗ ∗ α2µ T2µ ( fr ⊗ fs ) α2ν T2ν ( fr ⊗ fs ) ≤ µ=0 ∗
r+s=2µ
ν=0
r+s=2ν
∗
= T˜ (f f )T˜ (g g) for all f, g ∈ E⊗ . Hence W is positively dominated by T˜ . The proof is completed.
Acknowledgement. The author is indebted to Professor H. Araki for helpful and stimulating discussions. The suggestions of the referee are gratefully acknowledged.
References ˆ S.: Unbounded GNS representations of a *-algebra in a Krein space. Lett. Math. 1. Antoine, J.-P., Ota, Phys. 18, 267–274 (1989) 2. Araki, H.: On a pathology in indefinite inner product spaces. Commun. Math. Phys. 85, 121–128 (1982) 3. Araki, H.: Indecomposable representations with invariant inner product. A theory of the Gupta-Bleuler triplet. Commun. Math. Phys. 97, 149–159 (1985) 4. Azizov, T.Y., Iokhvidov, I.S.: Linear operators in spaces with indefinite metric. New York: John Wiley & Sons Inc., 1989 5. Bogn´ar, J.: Indefinite inner product spaces. Berlin: Springer-Verlag, 1974 6. Bogoljubov, N.N., Logunov, A.A., Oksak, A.I., Todorov, I.T.: General principles of quantum field theory. Dordrecht: Kluwer, 1990
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7. Borchers, H.J.: On the structure of the algebra of field operators. Nuovo Cimento 24, 214–236 (1962) 8. Dubin, D.A., Tarski, J.: Indefinite metric resulting from regularization in the infrared region. J. Math. Phys. 7, 574–577 (1966) 9. Ginsburg, Ju.P., Iokhvidov, I.S.: Investigations on the geometry of infinite-dimensional vector spaces with bilinear metric. Uspechi Matem. Nauk 17, no.4, 3–56 (1962) (in Russian) 10. Hofmann, G.: On topological tensor algebras. Wiss. Z. Univ. Leipzig, Math.-Naturw. Reihe 39, 598–622 (1990) 11. Hofmann, G.: On algebraic #-cones in topological tensor algebras, I. Basic properties and normality. Publ. RIMS, Kyoto University 28, 455–494 (1992) 12. Hofmann, G.: An explicite realization of a GNS representation in a Krein-space. Publ. RIMS, Kyoto University 29, 267–287 (1993) 13. Hofmann, G.: On the cones of α-positivity and generalized α-positivity for quantum field theories with indefinite metric. Publ. RIMS, Kyoto Univ. 30, 641–670 (1994) 14. Hofmann, G.: On the GNS representation of generalized free fields with indefinite metric. Rep. Math. Phys. 38, 67–83 (1996) 15. Hofmann, G.: Generalized free field like U (1)-gauge theories within the Wightman framework. Rep. Math. Phys. 38, 85–103 (1996) 16. Hofmann, G.: The Hilbert space structure condition for Quantum Field Theories with indefinite metric and transformations with linear functionals. To appear in Lett. Math. Phys. 17. Ito, K.R.: Canonical linear transformation on Fock space with an indefinte metric. Publ. RIMS, Kyoto Univ. 14, 503–556 (1978) 18. Jak´obczyk, L.: Borchers algebra formulation of an indefinite inner product quantum field theory. J. Math. Phys. 29, 617–622 (1984) 19. Jak´obczyk, L., Strocchi, F.: Euclidean formulation of quantum field theory without positivity. Commun. Math. Phys. 119, 529–541 (1988) 20. K¨othe,G.: Topological vector spaces, I. Berlin, New York, Heidelberg: Springer-Verlag, 1984 21. Kugo, T., Ojima, I.: Local covariant operator formalism of non-abelian gauge theories and quark confinement problem. Suppl. of the Progr. of Theor. Phys. 66, 1–130 (1979) 22. Mintchev, M.: Quantization in indefinite metric. J. Phys. A: Math. Gen. 13, 1841–1859 (1980) 23. Morchio, G., Strocchi, F.: Infrared singularities, vacuum structure and pure phases in local quantum field theory. Ann. Inst. Henri Poincar´e 33, 251–282 (1980) 24. Morchio, G., Pierotti, D., Strocchi, F.: Infrared and vacuum structure in two-dimensional local quantum field theory models. The massless scalar field. J. Math. Phys. 31, 1467–1477 (1990) 25. Nakanishi, N., Ojima, I.: Covariant operator formalism of gauge theories and quantum gravity. Singapore: World Scientific, 1990 26. Neumark, M.A.: Normierte Algebren. Thun, Frankfurt am Main: Verlag Harri Deutsch, 1990 27. Sakai, S.: C ∗ -Algebras and W ∗ -Algebras. Berlin: Springer-Verlag, 1971 28. Schaefer, H.H.: Topological vector spaces. London: Collier-Macmillan Limited, 1966 ¨ 29. Scheibe, E.: Uber Feldtheorien in Zustandsr¨aumen mit indefiniter Metrik. Mimeographed notes of the Max-Planck Institut f¨ur Physik und Astrophysik in M¨unchen, M¨unchen, 1960 30. Schm¨udgen, K.: Unbounded Operator Algebras and Representation Theory. Berlin: Akademie-Verlag, 1990 31. Strocchi, F.: Selected Topics on the General Properties of QFT. Lecture Notes in Physics Vol. 51, Singapore, New Jersey, Hong Kong: World Scientific, 1993 ˆ S.: Unbounded representation of a *-algebra on indefinite metric space. Ann. Inst. Henri Poincar´e 32. Ota, 48, 333–353 (1988) 33. Powers, R.T.: Self-adjoint algebras of unbounded operators, I. Commun. Math. Phys. 21, 261–293 (1971), II. Trans. Am. Math. Soc. 167, 85 (1974) ¨ 34. Uhlmann, A.: Uber die Definition der Quantenfelder nach Wightman und Haag. Wiss. Zeitschr. d. Univ. Leipzig 11, 213–217 (1962) 35. Yngvason, J.: Remarks on the reconstruction theorem for field theories with indefinite metric. Rep. Math. Physics 12, 57–64 (1977) Communicated by H. Araki
This article was processed by the author using the LaTEX style file pljour1 from Springer-Verlag.
Commun. Math. Phys. 191, 325 – 395 (1998)
Communications in
Mathematical Physics c Springer-Verlag 1998
Rogers–Schur–Ramanujan Type Identities for the M (p, p0 ) Minimal Models of Conformal Field Theory Alexander Berkovich1 , Barry M. McCoy2 , Anne Schilling2 1
Physikalisches Institut der Rheinischen Friedrich-Wilhelms Universit¨at Bonn, Nussallee 12, D-53115 Bonn, Germany. E-mail: berkov [email protected] 2 Institute for Theoretical Physics, State University of New York, Stony Brook, NY 11794-3840, USA. E-mail: [email protected]; [email protected] Received: 23 July 1996 / Accepted: 15 May 1997
Dedicated to the memory of Poline Gorkova
Abstract: We present and prove Rogers–Schur–Ramanujan (Bose/Fermi) type identities for the Virasoro characters of the minimal model M (p, p0 ). The proof uses the continued fraction decomposition of p0 /p introduced by Takahashi and Suzuki for the study of the Bethe’s Ansatz equations of the XXZ model and gives a general method to construct polynomial generalizations of the fermionic form of the characters which satisfy the same recursion relations as the bosonic polynomials of Forrester and Baxter. (p,p0 ) for many We use this method to get fermionic representations of the characters χr,s classes of r and s.
1. Introduction Rogers–Schur–Ramanujan type identities is the generic mathematical name given to the identities which have been developed in the last 100 years from the work of Rogers [0], Schur [1] and Ramanujan [2] who proved, among other things, that for a = 0, 1, 2 ∞ X q j +aj
j=0
(q)j
=
∞ Y j=1
1 (1 − q 5j−1−a )(1 − q 5j−4+a )
∞ 1 X j(10j+1+2a) = (q − q (5j+2−a)(2j+1) ), (q)∞ j=−∞
(1.1)
where (q)k =
k Y j=1
(1 − q j ), k > 0; (q)0 = 1.
(1.2)
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A. Berkovich, B. M. McCoy, A. Schilling
These identities are of the greatest importance in the theory of partitions [4–5] and number theory and by the early 50’s at least 130 of them were known [5]. The emergence of these identities in physics is much more recent, starting with the work of Baxter [6], Andrews, Baxter and Forrester [7] and the Kyoto group [8] in the 80’s on the order parameters of solvable statistical mechanical models. An even more recent relation to physics is the application to conformal field theories invented by Belavin, Polyakov and Zamolodchikov [9] in 1984. Here the left hand side is obtained by using a fermionic basis and the right hand side is obtained by using a bosonic basis for the Fock space. The bosonic constructions are done in a universal fashion using the methods of Feigin and Fuchs [11, 12]. The construction of the fermionic basis is more involved. The earliest example of such a fermionic representation is for modules of the affine Lie algebra A(1) 1 [12], but the general theory of this application of the identities has only been explicitly developed in the last several years [13]-[17]. However some of the mathematics of these constructions is already present in the original identities (1.1) which are now recognized as being the fermi/bose identities for the conformal field theory M (2, 5). Thus in some sense one might say that the 1894 paper of Rogers [0] is one of the first mathematical contributions to conformal field theory even though conformal field theory as a physical theory was invented only in 1984 [9]. The theory of bosonic representations of conformal field theory characters is well developed. In particular the characters of all M (p, p0 ) minimal models are given by the formula [18] (p,p0 ) (p,p0 ) (q) = q 1r,s −c/24 Br,s (q), (1.3) χˆ r,s where Br,s (q) =
∞ 0 1 X j(jpp0 +rp0 −sp) (q − q (jp +s)(jp+r) ), (q)∞ j=−∞
(1.4)
with conformal dimensions 0
(p,p ) 1r,s =
(rp0 − sp)2 − (p − p0 )2 (1 ≤ r ≤ p − 1, 1 ≤ s ≤ p0 − 1), 4pp0
and central charge c=1−
6(p − p0 )2 . pp0
(1.5)
(1.6)
p and p0 are relatively prime and we note the symmetry property Br,s (q) = Bp−r,p0 −s (q). It is obvious that (1.4) generalizes the sum on the right-hand side of (1.1). The generalization of the q-series on the left-hand side of (1.1) has a longer history. The first major advance was made in the 70’s when Andrews realized [19] that there were generalizations of (1.1) in terms of multiple sums of the form X m1 ,···,mk−1
1
q2m
T
Bm+AT m
k−1 Y i=1
1 (q)mi
(k ≥ 2)
(1.7)
where B is a (k − 1) by (k − 1) matrix, A is a (k − 1)–dimensional vector and the summation variables mi run over positive integers. These results are now recognized as the characters of the M (2, 2k + 1) models. In the interpretation of [14–18] we say that each mi represents the number of fermionic quasi-particles of type i. A second generalization of the form (1.7) is that the summation variables may obey restrictions such as mi being even or odd or having linear combinations being congruent to some
Rogers–Schur–Ramanujan Identities for Minimal Models of CFT
327
value Q (mod N ) (where N is some integer). The odd/even restriction is present in the original work of Rogers [0] and the (mod N ) restrictions were first found by Lepowsky and Primc [12]. Further generalizations of (1.7) are needed to represent the most general character. In particular we need the general form of what we call the “fundamental fermionic form” which was first found in [14] (and generalizes the special case (5.5) of [20] which in retrospect is M (5, 5k + 2)) fr,s (u, q) =
X
q
T 1 T 2 m Bm+A m
m,restr.
k−1 Y i=1
((Ik−1 − B)m + u)i mi
,
(1.8)
q
where Ik−1 is the (k−1) by (k−1) dimensional unit matrix and u is a (k−1)–dimensional vector with components (u)i . (In general we impose the notation that the components of a vector u are either denoted by (u)i or ui . ui would denote a vector labeled by i and not its ith component). We define the q-binomial coefficients for nonnegative m and n as (q)m+n m+n m+n (1.9) = = (q)m (q)n if m, n ≥ 0, n q m q 0 otherwise. There exist generalizations of (1.9) to negative n, and their use in the context of fermionic characters was first found in [21]. We also note that using the property 1 m+n = (1.10) lim n→∞ m q (q)m the general form (1.8) reduces to (1.7) when ui → ∞ for all 1 ≤ i ≤ k − 1. Then in terms of these fundamental fermionic forms the generic form of the generalization of (1.1) is now given as the linear combination X q ci fr,s (ui ; q) = q Nr,s Br,s (q), (1.11) Fr,s (q) = i
where Nr,s is a normalization constant. Character identities of this form which generalize the results of [19] were conjectured for some special cases of M (p, p0 ) in [16] including p0 = p + 1. Proofs of the identities for M (p, p+1) are given in [23–25]. Several other special cases of p and p0 and particular values of r and s are proven in [25]. In a previous letter [26] two of the present authors gave results for the case of arbitrary p and p0 for certain selected values of r and s. Here we generalize and prove the results of that letter. Our method of proof is to generalize the infinite series for the bosonic and fermionic forms of the characters in (1.11) to polynomials Br,s (L, q) and Fr,s (L, q) whose order depends on an integer L and then to prove that both Br,s (L, q) and Fr,s (L, q) satisfy the same difference equations in L with the same boundary conditions. The generalization from infinite series to a set of polynomials is referred to as “finitization” [27]. For the proof of the L–difference equations we utilize the technique of telescopic expansion first introduced in [22] to prove the conjecture of [17, 18] for the M (p, p + 1) model and subsequently used to prove identities for the N = 1 supersymmetric model SM (2, 4ν) [21] and for general series of the A(1) 1 coset models with integer levels [28]. This method is the extension to many quasi particles of the recursive proof of (1.1) given by [2, 4 and 30]. (Somewhat different methods have been used to prove polynomial analogues of the Andrews–Gordon identities in [31–33].)
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There are many ways to finitize the fermionic and bosonic forms of the characters. For example the bosonic character (1.4) has a polynomial generalization in terms of q-binomial coefficients Br(b),s (L, b; q) ∞ X 0 0 L L (jp+r(b))(jp0 +s) q j(jpp +r(b)p −sp) L+s−b , − q = L−s−b − jp0 q − jp0 q 2 2 j=−∞ (1.12) where L + s − b is even and r(b) is a prescribed function of b with 1 ≤ b ≤ p0 − 1 (see (3.4) below). This generalization first appeared in the work of Andrews, Baxter and Forrester [7] for p0 = p + 1 and for general p p0 in the work of Forrester and Baxter [33]. The bosonic polynomials in (1.12) have the symmetry Br(b),s (L, b; q) = Bp−r(b),p0 −s (L, p0 − b; q).
(1.13)
Thanks to (1.10), Eq. (1.12) tends to (1.4) as L → ∞. Notice that in this limit the dependence on b drops out and hence for each character identity there are several different polynomial identities with the same limit. The polynomials (1.12) generalize the polynomials used by Schur [1] in connection with difference two partitions and is the finitization we use in this paper. They satisfy a simple recursion relation in L and can be interpreted as the generating functions for partitions with prescribed hook differences [34]. However, there are several other known polynomial finitizations [22, 25, 33, 38] which satisfy other L difference equations and prove to be useful in other contexts. In this paper we will present a method which allows the construction of fermionic polynomials which satisfy the identities Fr(b),s (L, b; q) = q Nr(b),s Br(b),s (L, b; q)
(1.14)
for the general minimal model M (p, p0 ) in principle for all b and s. It is however difficult to find a notation which allows for a compact treatment of all values of b and s at the same time. Consequently even though our methods in this paper are general, we present results only for certain classes of b and s. However, we emphasize that all cases can be treated by the same methods. Additional results will be presented elsewhere. The polynomials appearing on the left-hand side of (1.14) generalize the polynomials originally used by MacMahon [3] in his analysis of (1.1). In contrast to the bosonic polynomials (1.12) the form of the fermionic polynomials depends on the values of b and s. This is because the fermionic polynomials depend on the continued fraction decomposition of p0 /p. We will present the formalism of this decomposition in Sect. 2 and defer the presentation of our results to Sect. 3. The proof of our results for p0 > 2p is given in Sects. 4–11. The case p < p0 < 2p is obtained from the case p0 > 2p in Sect. 12 by the method of the dual transformation discussed in [26]. We close in Sect. 13 with a discussion of several ways our results can be extended and with an interpretation of our polynomial identities in terms of new Bailey pairs.
Rogers–Schur–Ramanujan Identities for Minimal Models of CFT
329
2. Summary of the Formalism of Takahashi and Suzuki We begin with the observation that the models M (p, p0 ) are obtained as a reduction of the XXZ spin chain X y x z (σkx σk+1 + σky σk+1 + 1σkz σk+1 ), (2.1) HXXZ = − k
where σki (i = x, y, z) are the Pauli spin matrices and 1 = − cos π
p . p0
(2.2)
Consequently we may use the results of the classic study of the thermodynamics of the XXZ chain made by Takahashi and Suzuki [36] in 1972. This treatment begins by introducing, for p0 > 2p, the n + 1 integers ν0 , ν1 , · · · , νn from the following continued fraction decomposition of p0 /p, p0 = ν0 + 1 + p ν1 +
1 1 ν2 +···+ νn1 +2
,
(2.3)
where νn ≥ 0 and all other νj ≥ 1. For the case p0 < 2p we replace p by p0 − p. We say that this is a n + 1 zone decomposition and that there are νj types of quasi particles in zone j. From these integers we define (where µ is an integer, 0 ≤ µ ≤ n + 1) Pµ−1 for 1 ≤ µ ≤ n + 1 . j=0 νj (2.4) tµ = −1 for µ = 0 We refer to tn+1 as the number of types of quasi particles in the system. When an index j satisfies (2.5) tµ + 1 ≤ j ≤ tµ+1 + δn,µ , we say that the index j is in the µth zone and that 1 + tµ and t1+µ are the boundaries of this zone. Note that by definition zone 0 (n) has ν0 + 1 (νn + 1) allowed values of j while all other zones have νµ allowed values of j. We will sometimes refer to j = tn+1 + 1 and j = 0 as “virtual” positions. We will explicitly consider below the case p0 > 2p. The case p0 < 2p will be treated separately in Sect. 12. According to Takahashi and Suzuki [36], there are also νi types of quasi particles in zones i = 0, . . . , n − 1 for the XXZ chain. In zone n there are, however, νn + 2 types of quasiparticles, and in addition there is an extra zone n + 1 with one quasi particle in the XXZ chain. It is the omission of the three quasi particles of zone n and n + 1 which truncates the XXZ chain to the model M (p, p0 ). From the νj we define the set of integers yµ recursively as y−1 = 0, y0 = 1, y1 = ν0 + 1, yµ+1 = yµ−1 + (νµ + 2δµ,n )yµ , (1 ≤ µ ≤ n), (2.6) and further set lj = yµ−1 + (j − 1 − tµ )yµ for 1 + tµ ≤ j ≤ tµ+1 + δn,µ .
(2.7)
(µ) |0 ≤ µ ≤ n, j is in µth zone } We then define what we call the Takahashi length {l1+j
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A. Berkovich, B. M. McCoy, A. Schilling
(µ) l1+j
=
j+1 yµ−1 + (j − tµ )yµ
for µ = 0 and 0 ≤ j ≤ t1 for 1 ≤ µ ≤ n and 1 + tµ ≤ j ≤ t1+µ + δn,µ ,
(2.8)
(µ) which differs from l1+j only at the boundaries j = tµ . Note that l1+j is monotonic in j th while l1+j is not. To indicate that j lies in the µ zone, i.e. 1 + tµ ≤ j ≤ t1+µ , we will in the following always write jµ instead of j. We also define a second set of integers zµ for as
z−1 = 0, z0 = 1, z1 = ν1 + 2δ1,n , zµ+1 = zµ−1 + (νµ+1 + 2δµ+1,n )zµ , (1 ≤ µ ≤ n − 1) (2.9) and zµ−2 + (jµ − tµ )zµ−1 for 1 ≤ µ ≤ n and 1 + tµ ≤ jµ ≤ t1+µ + δn,µ (µ) = l˜1+j µ 0 for µ = 0. (2.10) (µ) We refer to l˜1+j as a truncated Takahashi length. It is clear that z is obtained from µ µ the same set of recursion relations as the yµ except that zone zero is removed in the partial fraction decomposition of p0 /p. The removal of this zone zero is equivalent to 0 considering a new XXZ chain with an anisotropy 10 = − cos π{ pp }, where {x} denotes the fractional part of x. We note that the l(µ) (l˜(µ) ) are the dimensions of the unitary 1+jµ
1+jµ
p
p0
representations of the quantum group su(2)q± with q+ = e p0 (q− = eiπ{ p } ). The final result we need from [36] is the specialization of their equation (1.10) to the case of the 0th Fourier component. Then, using the notation where the integers nk (mk ) with 1 ≤ k ≤ tn+1 are the number of particle (hole) excitations of type k we find what we call (m, n) system (Eq. (2.18) of [26]) iπ
1 1 (mk−1 + mk+1 ) + u¯ k for 1 ≤ k ≤ tn+1 −1 and k 6= ti , i = 1, · · · , n, 2 2 1 1 nti + mti = (mti −1 + mti − mti +1 ) + u¯ ti , for i = 1, · · · , n, 2 2 1 1 ntn+1 + mtn+1 = (mtn+1 −1 + mtn+1 δνn ,0 ) + u¯ tn+1 , 2 2 (2.11) where by definition m0 = L, all u¯ k are integers and all mk are nonnegative integers. Let us emphasize here that whereas mk is always nonnegative, nk may at times take on negative values. We denote (2.11) symbolically as n k + mk =
n = Mm +
1 L ¯ e¯ 1 + u, 2 2
where we define the tn+1 -dimensional vectors e¯ k by δj,k 1 ≤ k ≤ tn+1 (¯ek )j = 0 k = 0, 1 + tn+1 . Additionally, we shall require the 1 + tn+1 -dimensional vectors ek defined as n (ek )j = δj,k 1 ≤ k ≤ 1 + tn+1 0 k = 0. Solving (2.12) for m yields
(2.12)
(2.13)
(2.14)
Rogers–Schur–Ramanujan Identities for Minimal Models of CFT
m = M−1 (n −
331
L 1 ¯ e¯ 1 − u). 2 2
(2.15)
The asymmetric matrix −2M is almost of block diagonal form with each block except the last being the tadpole Cartan matrix Aνi , and the last block being a regular Aνn Cartan matrix (unless νn = 0). Note that in the simplest case p = 1 the matrix −2M becomes the Cartan matrix Ap0 −3 . For these reasons it is natural to think of −2M as a new mathematical construct: a Cartan matrix of fractional size. Algebraic structures associated with −2M will be investigated elsewhere. From the system of equations (2.11) one may deduce the following partition problem for L by multiplying equation i in (2.11) by li and summing all equations up, tn+1 X
(ni −
i=1
u¯ i m(tn+1 ) L )li + l(1+tn+1 ) = , 2 2 2
(2.16)
where lj is given by (2.7) and l(1+tn+1 ) = p0 − 2yn . This partition problem can be employed to carry out an analysis along the lines of [24]. We will also need it later in the proof of the initial conditions and to determine allowed variable changes in the proof of the recurrences. Finally we note that it is useful to consider the cases p0 > 2p and p < p0 < 2p separately because the case p0 > 2p may be obtained from the case p0 < 2p by a “duality map” [26]. We concentrate first on the case p0 > 2p where ν0 ≥ 1. In this case the ν0 types of particles in the 0th zone are treated differently than the remaining particles. At times we will find it necessary to use nk , 1 ≤ k ≤ ν0 as the independent variables in the 0th zone and mk as the independent variables in all other zones. Thus we define the vector of independent variables as ˜ = {n1 , n2 , · · · , nν0 , mν0 +1 , mν0 +2 , · · · , mtn+1 } m
(2.17)
and the vector of dependent variables as n˜ = {m1 , m2 , · · · , mν0 , nν0 +1 , · · · , ntn+1 }.
(2.18)
From (2.11) we find that ¯ 1,ν0 + ˜ + LE n˜ = −Bm
L B 1 1 e¯ ν +1 + u¯ + + e¯ ν0 +1 (u¯ +T · V) + u¯ − 2 0 2 2 2
(2.19)
where the nonzero elements of the matrix B are given in terms of the ν0 × ν0 matrix, 1
1 2 2 .. .
1 CT−1 = 1 .. . 1 2
1 2 3 .. .
... ... ... .. .
1 2 3 .. .
3
...
ν0
(2.20)
(where − 21 CT is the matrix of the first ν0 rows and columns of the matrix M of (2.12)) as
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A. Berkovich, B. M. McCoy, A. Schilling
Bi,j = 2(CT−1 )i,j for 1 ≤ i, j ≤ ν0 , Bν0 +1,j = Bj,ν0 +1 = j for 1 ≤ j ≤ ν0 , n ν0 1X δj,ν0 +1 + (1 − δj,ti ) for ν0 + 1 ≤ j ≤ tn+1 − 1, 2 2
t
Bj,j =
i=2
(2.21)
Btn+1 ,tn+1 = 1, 1 X δj,ti for j > ν0 , Bj,j+1 = − + 2 n
i=2
Bj+1,j
1 = − for j > ν0 . 2
The vector u¯ is decomposed as u¯ = u¯ + + u¯ −
(2.22)
with ν0 X
u¯ + =
u¯ i e¯ i ,
(2.23)
i=1 tn+1 X
u¯ − =
u¯ i e¯ i ,
(2.24)
i=ν0 +1
V=
ν0 X
i¯ei ,
(2.25)
i=1
and ¯ a,b = E
b X
e¯ i .
(2.26)
i=a
The splitting of (2.22) can be done for any vector. We will, for example, also use this ˜ in the following. splitting for the vector m
3. Summary of Results Now that we have summarized the needed results of [36] we may complete the specification of the bosonic and fermionic sums which appear in our identities. We first complete the bosonic specification in Subsect. 3.1 and give the definitions needed for the fermionic sums in Subsect. 3.2. The final results will be outlined in subsection C, but the detailed identities will be presented in Sect. 10. We conclude in Subsect. 3.3 with a discussion of the special case q = 1. 3.1. Bosonic polynomials. In order to complete the specification of the bosonic polynomials Br,s (L, b) of (1.12) we represent both b and s as series in Takahashi lengths. Thus we write
Rogers–Schur–Ramanujan Identities for Minimal Models of CFT
b=
β X i=1
(µi ) l1+j µ
333
(3.1)
i
with 0 ≤ µ1 < µ2 < · · · < µβ ≤ n and 1 + tµi ≤ jµi ≤ t1+µi + δn,µi with the further restriction that if jµi = t1+µi then µi+1 ≥ µi + 2.
(3.2)
This decomposition is unique. We often say that a b of this form lies in zone µβ . Hence if b is in zone µ then (µ) (µ+1) ≤ b ≤ l2+t − 1. l2+t µ µ+1
(3.3)
A similar decomposition is made for s. From the decomposition of b we specify r(b) as follows: ( Pβ r(b) =
˜(µi ) i=1 l1+jµi Pβ (µi ) ˜ i=1 l1+jµi
+ δµ1 ,0
if 1 ≤ b ≤ p0 − ν0 − 1, if p0 − ν0 ≤ b ≤ p0 − 1,
(3.4)
(µ) (µ) where l˜1+j as defined by (2.10). This generalizes the case considered in [26] of b = l1+j µ µ being a single Takahashi length with µ ≥ 1 where we had (µ) (µ) r(l1+j ) = l˜1+j . µ µ
(3.5)
One may prove that X (µ ) p l˜1+ji µ − aµ1 , c = i p0 β
bb
(3.6)
i=1
where bxc is the greatest integer contained in x and n aµ 1 =
1 0
if µ1 ≥ 2 and µ1 even, otherwise.
(3.7)
Using (3.6) the map (3.4) may be alternatively expressed as r(b) = where
bb pp0 c + θ(µ1 even) bb pp0 c n θ(A) =
1 0
for 1 ≤ b ≤ p0 − ν0 − 1 , for p0 − ν0 ≤ b ≤ p0 − 1 if A is true, if A is false.
(3.8)
(3.9)
When r is expressed using relation (3.8) we refer to the expression as the Forrester-Baxter form for r after the closely related formula of [33]. The proof of (3.6) is straightforward and will be omitted. The following elementary properties of the b → r map are clear from (3.4) and (2.10): 1. 2.
r(b + 1) = r(b) or r(b) + 1, if r(b + 1) = r(b) + 1 then r(b + 2) = r(b + 1)
(3.10) (3.11)
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A. Berkovich, B. M. McCoy, A. Schilling
3. if b is as in (3.1), (3.2) with µ1 ≥ 1 then and r(b + x)−r(x) =
4.
r(b − 1) = r(b) = r(b + 1) − 1 = r(b + 2) − 1 ( Pβ
˜(µi ) i=1 l1+jµi (n) l˜2+t −θ(yn 1+n
(3.12)
(n) for b < p0 − yn = l2+t , 1 ≤ x ≤ yµ1 1+n
− y1 < x)
(n) for b = l2+t , 1 ≤ x ≤ yn −1 1+n 0
r(1) = r(2) = 1, r(p − 1) = p − 1.
,
(3.13) (3.14)
Whereas the bosonic polynomials Br,s (L, b; q) depend on the Takahashi decomposition of b only through the map r(b), the corresponding fermionic polynomials depend sensitively on the details of the Takahashi decomposition of b and s. In [26] we treated the simplest case where both b and s consist of one single Takahashi length. In this paper we will still consider s to be of the form (µs ) (µs ) (or p0 − l1+j ) 1 + tµs ≤ js ≤ tµs +1 s = l1+j s s
(3.15)
but b is often left arbitrary (µs and js are from now on reserved to specify s as in (3.15) and are not to be confused with µi or jµi of the Takahashi decomposition (3.1)). However, the complexity of the final fermionic polynomials will depend on the details of the decomposition of b. 3.2. Fermionic polynomials. Our fermionic sums Fr,s (L, b; q) will be constructed out of two elementary fermionic objects fs (L, u; q) and f˜s (L, u˜ ; q) where the vectors u and u˜ depend on b. The objects fs (L, u; q) and f˜s (L, u˜ ; q) are polynomials (in q 1/4 ) and differ from the polynomials of [26] in that the value at q = 0 is not normalized to 1. This choice of normalization is made for later convenience and we trust that no confusion will result from referring to fs (L, u; q) and f˜s (L, u˜ ; q) as polynomials in the sequel. In order to define fs (L, u; q) and f˜s (L, u˜ ; q) we need to discuss two generalizations of the q-binomials in (1.9) which allow n to be negative for n + m < 0, but will automatically vanish if m < 0.
Definition of q-binomials. We use here the definitions of [37] (1) n+1 (0) (q )m m+n m+n for m ≥ 0, n integer, (q)m = = m q n q 0 otherwise, where (a; q)n = (a)n =
n−1 Y
(1 − aq j ), 1 ≤ n; (a)0 = 1.
(3.16)
(3.17)
j=0
The reason for distinguishing between the two q-binomials with superscript zero or one is that the first one is more convenient if the variables nj are taken as independent in the (m, n)-system whereas the latter one is more convenient if the mj are taken as independent. We remark that when m and n are nonnegative we have the symmetry
(0,1)
m+n n
= q
(0,1)
m+n m
. q
(3.18)
Rogers–Schur–Ramanujan Identities for Minimal Models of CFT
335
We also note that we have the special values for L positive (0) (0) −L −L = 1, = 0, 0 q −L q (1) (1) −L −L = 0, = 1. 0 q −L q
(3.19)
Both functions satisfy the two recursion relations without restrictions on L or m (0,1) (0,1) (0,1) L−1 L L−m L − 1 = +q , (3.20) m q m−1 q m q (0,1) (0,1) (0,1) L L−1 m L−1 =q + . m q m q m−1 q
(3.21)
This extension of the q-binomial coefficients is needed for the fermionic function Fr(b),s (L, b; q) for general values of b and s. In [26] we used the definition (1.9) and thus obtained more limited results. Definition of fs (L, u; q). Let us now define fs (L, u; q) with u ∈ Z 1+t1+n , where L is a nonnegative integer. There are actually several equivalent forms, differing only by which set of mk and nk in the (m, n)–system (2.11) is taken to be independent. In the following ˜ n and m as independent variables, three equivalent forms for fs (L, u; q) are given with m, ˜ m and n can be determined by Eqs. (2.19), respectively. The corresponding variables n, (2.15) and (2.12), respectively. X
fs (L, u; q) =
×
tY n+1
˜
¯ 1,ν0 + ˜ + LE ((Itn+1 − B)m
+ 21 Bu¯ + + 21 e¯ ν0 +1 (u¯ +T · V) + 21 u¯ − )j m ˜j
L ¯ 2 eν0 +1
X
T
θ(mtn+1 ≡ u1+tn+1 (mod 2))q Q(n,m)+A
n∈Ztn+1
=
˜ m
˜ − ∈2Ztn+1 −ν0 +w− (u1+t ¯) m ,u n+1 m ˜ + ∈Zν0
j=1
=
T
˜ q Q(m)+A
X ¯ m∈2Ztn+1 +w(u1+tn+1 ,u)
T
q Q(n,m)+A
˜ m
tY n+1 j=1
˜ m
tY n+1 j=1
mj + nj nj
((Itn+1 + M)m + mj
u¯ 2
+
(θ(j>ν0 )) q
(0) q
(1) L ¯ 2 e1 )j . q
(3.22) ¯ a,b and V have been defined in Sects. 1, 2 and Z is the set of integers Here B, M, Itn+1 , E ¯ {. . . , −2, −1, 0, 1, 2, . . .}. The vectors m, n satisfy the system (2.11) with u, ¯ + u¯ = u(s)
tn+1 X i=1
where for s as in (3.15) and 1 ≤ k ≤ t1+n ,
e¯ i ui ,
(3.23)
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A. Berkovich, B. M. McCoy, A. Schilling
¯ (u(s)) k =
δk,js − δk,js
Pn
for 1 + tµs ≤ js ≤ tµs +1 and µs ≤ n − 1, (3.24) for tn < js , µs = n.
i=µs +1 δk,ti
¯ Note that the vectors u¯ and u(s) in (3.23) have dimension tn+1 while u has dimension 1 + tn+1 . ˜ m) ˜ and ˜ and Q(n, m), the linear term AT m Let us now define the quadratic forms Q( ¯ the parity restriction vector w(u1+tn+1 , u): ˜ We define the quadratic form Q( ˜ m) ˜ as Definition of the quadratic forms Q and Q. 1 T ˜ m) ˜ = m ˜ Bm, ˜ Q( 2
(3.25)
where B is defined by (2.21). An equivalent form makes use of (2.17)–(2.19) to write 1 T ¯ 1,ν0 − L e¯ ν0 +1 − B u¯ + − 1 e¯ ν +1 (u¯ +T · V) − u¯ − ). ˜ (˜n − LE Q(n, m) = − m 2 2 2 2 0 2
(3.26)
˜ We write The linear term AT m. A = A(b) + A(s) ,
(3.27)
where the tn+1 -dimensional vector A(b) is obtained from u as 1 (b) Ak = − 2 uk for k in an even, nonzero zone, 0 otherwise,
(3.28)
¯ as and the tn+1 -dimensional vector A(s) is obtained from u(s) 1 ¯ − 2 u(s) ¯ k − 21 VT · u(s)δ k,ν0 +1 for k in an odd zone, = A(s) 1 T k ¯ + − 2 e¯ k Bu(s) for k in an even zone.
(3.29)
¯ The summation variables m ˜ j (j = The parity restrictions and the vector w(u1+tn+1 , u) ν0 + 1, . . . , t1+n ) in the first line of (3.22) and mj (j = 1, . . . , t1+n ) in the third sum of (3.22) are subject to even/odd restrictions which are determined from u and s by the requirement that the entries of all q-binomials in (3.22) are integers as long as u ∈ Z1+tn+1 . To formulate this analytically we define wk(j) for 1 ≤ k ≤ tn+1 , tµ0 + δµ0 ,0 + 1 ≤ j ≤ tµ0 +1 + δµ0 ,n , for some 0 ≤ µ0 ≤ n as wk(j)
0 = j−k w(j) + w(j) 1+tµ+1 k+1
and then define ¯ = w(u1+tn+1 , u)
tn+1 X k=1
k≥j tµ0 ≤ k ≤ j − 1 tµ ≤ k < tµ+1 , for some µ < µ0
e¯ k u1+t1+n wk(1+tn+1 ) +
tn+1 X j=1
(3.30)
(3.31)
wk(j) u¯ j .
(3.32)
Rogers–Schur–Ramanujan Identities for Minimal Models of CFT
337
Then the variables mj in (3.22) satisfy ¯ (mod 2Zt1+n ). m ≡ w(u1+t1+n , u)
(3.33)
Note that mtn+1 ≡ u1+tn+1 ≡ P (mod 2), where parity P ∈ {0, 1}. Clearly, the 1 + t1+n component of the vector u determines the parity of the mt1+n variable in the fundamental fermionic polynomials (3.22). The limit L → ∞. In order to obtain character identities from the polynomial identities we need to consider the limit L → ∞. Only the first expression for fs (L, b; q) in (3.22) is suitable for this limit since Q(n, m) which appears in the other two expressions depends on L. Hence in the limit L → ∞ the expression for the polynomial (3.22) reduces to X T ˜ ˜ ˜ m q Q(m)+A fs (u; q) = lim fs (L, u; q) = L→∞
˜− m
ν ˜ + ∈Z 0 m ≥0 t −ν 0 +w(u ∈2Z n+1
¯) 1+tn+1 ,u
(1) νY tn+1 0 +1 ˜ + 21 u¯ − )j 1 Y ((Itn+1 − B)m × , (q)m˜ j m ˜j q j=1
(3.34)
j=2+ν0
which depends on the vector u¯ + only through the linear term A. Definition of f˜s (L, u˜ ; q). The fermionic polynomial f˜s (L, u˜ ; q) is defined for vectors of the form u˜ = eν0 −j0 −1 − eν0 + u10 , where 0 ≤ j0 ≤ ν0 is in the zone 0 and u10 ∈ Z 1+t1+n is any vector with no components in the zero zone, i.e. (u10 )i = 0 for 1 ≤ i ≤ ν0 . We define f˜s (L, eν0 −j0 −1 − eν0 + u10 ; q) L+j0 q − 2 [fs (L + 1, eν0 −j0 − eν0 + u10 ; q) −fs (L, e1+ν0 −j0 − eν0 + u10 ; q)] for 1 ≤ j0 ≤ ν0 , = L L (q 2 − q − 2 )fs (L − 1, u10 ; q) L +q − 2 fs (L, eν0 −1 − eν0 + u10 , q) for j0 = 0.
(3.35)
When j0 = ν0 , the left-hand side of this definition loses its meaning because we have no e−1 . However, for conformity, we introduce the notation f˜s (L, e−1 − eν0 + u10 ; q) to mean the first line of the right-hand side of (3.35) with j0 = ν0 . We will show in Sect. 6 that for j0 = ν0 this definition reduces to f˜s (L, e−1 − eν0 + u10 ; q) = 0,
(3.36)
while for j0 = ν0 − 1, f˜s (L, e0 − eν0 + u10 ; q) = q =q
L−(ν0 −1) 2
fs (L − 1, e1 − eν0 + u10 ; q)
L−(ν0 −1) 2
fs (L, −eν0 + u10 ; q).
(3.37)
In the limit L → ∞ we have lim f˜s (L, u˜ ; q) = 0.
L→∞
(3.38)
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A. Berkovich, B. M. McCoy, A. Schilling
The identities. The polynomial identities proven in this paper are all of the form Fr(b),s (L, b; q) = q Nr(b),s Br(b),s (L, b; q),
(3.39)
where r(b) as given in (3.4) and Fr(b),s (L, b; q) is of the form Fr(b),s (L, b; q) =
X
L
q cu + 2 gu fs (L, u, q) +
X
q c˜u˜ f˜s (L, u˜ , q).
(3.40)
˜ U˜ (b) u∈
u∈U (b)
Here U (b) and U˜ (b) are sets of vectors determined by b. The sets U (b), U˜ (b) and the exponents cu , c˜u˜ , gu ≥ 0 and Nr(b),s will be explicitly computed. We note however that there are very special values for b for which the representation of Fr(b),s (L, b; q) in (3.40) is not correct. This phenomenon will be discussed in Sect. 8. The results depend sensitively on the details of the expansion of b (3.1). For the cases of 1,2 and 3 zones the identities for all values of b are given in Sect. 7. However for the general case of n + 1 zones with n ≥ 3 there are many special cases to consider. The process of obtaining a complete set of results is tedious and in this paper we present explicit results only for the following special cases of b (and r): (µ) (µ) , r = l˜1+j + δµ,0 1 : b = l1+j µ µ (µ) (µ) (µ) 2a : l1+j − ν0 + 1 ≤ b ≤ l1+j − 1, r = l˜1+j µ µ µ
2b :
1 ≤ µ, tµ + 1 ≤ jµ ≤ t1+µ − 1 + δµ,n + 1 ≤ b ≤ l(µ) + ν0 + 1, r = 1 + l˜(µ)
(µ) l1+j µ
1+jµ
1+jµ
1 ≤ µ, tµ + 1 ≤ jµ ≤ t1+µ − 1 + δµ,n 3: b=
β X
(µ) l1+j , r =1+ µ
µ=0
β X
(µ) l˜1+j , 1≤β≤n µ
µ=1
0 ≤ j0 ≤ ν 0 − 1 1 + tµ ≤ jµ ≤ tµ+1 − 3 1 ≤ µ ≤ β − 2 1 + tβ−1 ≤ jβ−1 ≤ tβ − 2 1 + tβ ≤ jβ ≤ tβ+1 − 1 + δβ,n 4: b=
β X µ=α
(µ) l1+j , r= µ
β X
(µ) l˜1+j , 1 ≤ α, α + 1 ≤ β ≤ n µ
µ=α
1 + tµ ≤ jµ ≤ tµ+1 − 3 α ≤ µ ≤ β − 2 1 + tβ−1 ≤ jβ−1 ≤ tβ − 2 1 + tβ ≤ jβ ≤ tβ+1 − 1 + δβ,n , where α = 1 and α ≥ 2 are treated separately. The results are given in Sect. 10 as follows:
(3.41)
Rogers–Schur–Ramanujan Identities for Minimal Models of CFT
339
Polynomial Identities. 1: 2a : 2b : 3: 4:
10.3 10.4 10.5 10.10 10.15.
(3.42)
Character identities. 1 and 2a : 2b : 3: 4, α = 1 : 4, α ≥ 2 :
10.20 10.21 10.22 10.23 10.24.
(3.43)
These results hold for νi 6= 1, i = 1, · · · , n − 1 and in addition (10.3) and (10.20) are valid even if some or all νi = 1 or νn = 0 (or both) with minor modifications of overall factors. Further results are to be found in Sect. 11 and 12 and (C.19) of Appendix C. (µs ) (µs ) (p0 − l1+j ). The The identities of Sect. 10 (11) are valid for p0 > 2p and for s = l1+j s s identities of Sect. 12 are valid for p0 < 2p. 3.3. The special case q = 1. We close this section with a brief discussion of the special case q = 1. The details of this case will be presented elsewhere, but it is useful to sketch the results here in order to give a characterization of the vectors u and u˜ that appear in (3.40) which is complementary to the constructive procedure developed in Sects. 7 and 8. When q = 1 the bosonic function (1.12) simplifies because the dependence on r vanishes and the fermionic form (3.40) simplifies because f˜s (L, u˜ ; q = 1) = fs (L, u˜ ; q = 1).
(3.44)
For these reasons many of the distinctions between the special cases noted in the previous section disappear and we find the single identity valid for all b, X fs (L, u; q = 1), (3.45) Br(b),s (L, b; q = 1) = u∈W (b)
where the set of vectors W (b) is obtained from the Takahashi decomposition of b (3.1) as follows: If in (3.1) β = 1 so that (µ) , (3.46) b = l1+j µ then we define as in [26] W (b) having only the single element u = e jµ −
n X
eti , 1 + tµ ≤ jµ ≤ tµ+1 + δµ,n , µ = 0, 1, · · · , n.
(3.47)
i=µ+1
This vector was called an r string in [26]. For the general case we write the Takahashi decomposition of an arbitrary b as
340
A. Berkovich, B. M. McCoy, A. Schilling
b=
(µ ) l1+jβµ β
+ x where x =
β−1 X i=1
(µi ) l1+j , µ
(3.48)
i
where β is defined from (3.1). If x = 0 we define a vector v(0) as: v(0) = ejµβ .
(3.49)
If x 6= 0 and it is not true that µβ = n and jn = 1 + t1+n , then we define two vectors as for tµβ + 1 ≤ jµβ ≤ −1 + t1+µβ , e1+jµβ v(0) = e + (1 − δ )e for jµβ = t1+µβ , 1+t1+µβ µβ ,n t1+µβ (3.50) v(1) = ejµβ . Furthermore if x 6= 0 and we do not have µβ = n and jn = 1 + t1+n we define the two numbers (3.51) b(0) = x and b(1) = yµβ − x. In the exceptional case where x 6= 0, µβ = n and jn = 1 + t1+n we define only one vector and one number v(1) = e1+t1+n , (3.52) b(1) = yn − x. We then expand both b(0) and b(1) (and in the exceptional case only b(1) ) again in a Takahashi series and repeat the process as many times as needed until x = 0 in which case the process terminates. This recursion leads to a branched chain of vectors v(i1 ) , v(i1 ,i2 ) , · · · , v(i1 ,i2 ,···,if ) , where 1 ≤ f ≤ µβ + 1; i1 , · · · , if = 0, 1. Notice however that f might vary from branch to branch. Let us define µf to be the lowest µ such that (i ,...,if ) there exist an i such that tµ < i ≤ t1+µ + δµ,n and vi 1 6= 0. Then the set W (b) consists of all vectors v(i1 ) + v(i1 ,i2 ) + · · · + v(i1 ,i2 ,···,if ) −
n X
et k ,
(3.53)
k=µf +1
where all vectors v are determined by the above described recursive procedure. The explicit solution of this recursion relation involves the recognition that there are many separate cases of the Takahashi decomposition of b which lead to sets W (b) which may differ even in the number of vectors in the set. Certain of these special cases correspond to the cases distinguished in the previous section. The complete solution of this recursive definition will be given elsewhere where we will use it to give explicit Rogers-Schur-Ramanujan identities for general values of b. We note in the cases consid(0) is present in the Takahashi decomposition ered in the previous subsection that when l1+j 0 then there are vectors which have components in zone zero of the form u˜ = eν0 −j0 −a − eν0 + u10 , a = 1, 2,
(3.54)
where the vector u10 is defined above (3.35). These are the vectors u˜ ∈ U˜ (b) of (3.40). Note that the set U˜ (b) is empty if in the Takahashi decomposition of b (3.1) we have µ1 ≥ 1. The remaining vectors are the vectors u ∈ U (b) of (3.40).
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4. The Bosonic Recursion Relations In this section we derive recursion relations for the bosonic polynomials Br,s (L, b; q) defined by (1.12) and the b → r map (3.4). Here and in the remainder of the paper we will often suppress the argument q of all functions unless explicitly needed. Moreover we find it convenient to remove an L independent power of q by defining 1 B˜ r(b),s (L, b) = q 2 (φr(b),s −φr(s),s ) Br(b),s (L, b),
(4.1)
where φx,y is defined as φx+1,y − φx,y = 1 − y +
ξ X
(ηi ) l1+j η
i=1
i
(4.2)
when x has the decomposition in terms of truncated Takahashi lengths x=
ξ X i=1
(ηi ) l˜1+j , η i
(4.3)
with 1 ≤ η1 < η2 < · · · < ηξ ≤ n and 1 + tηi ≤ jηi ≤ t1+ηi + δn,ηi . We note that because φx,y appears only as a difference in (4.1) that boundary conditions on φx,y are not needed. This change in normalization allows us to prove the following recursion relations which contain no explicit dependence on s. For 2 ≤ b ≤ p0 − ν0 − 1, ˜ Br(b),s (L − 1, b + 1) + B˜ r(b),s (L − 1, b − 1) +(q L−1 − 1)B˜ if µ1 = 0, j0 ≥ 1 r(b),s (L − 2, b) , B˜ r(b),s (L, b) = L−1 ˜ ˜ 2 B q (L − 1, b + 1) + B (L−1, b −1) if µ1 ≥ 1 r(b)+1,s r(b),s L B˜ r(b),s (L−1, b + 1) + q 2 B˜ r(b)−1,s (L − 1, b −1) if µ1 = 0, j0 = 0 (4.4) where µ1 and j0 are obtained from the Takahashi decomposition of b (3.1). For the remaining cases we have B˜ 1,s (L, 1) = B˜ 1,s (L − 1, 2)
(4.5)
and
B˜ p−1,s (L−1, b + 1) + B˜ p−1,s (L−1, b −1) B˜ p−1,s (L, b) = +(q L−1 − 1)B˜ p−1,s (L − 2, b) ˜ Bp−1,s (L − 1, p0 − 2)
if p0 − ν0 ≤ b ≤ p0 − 2 if b = p0 − 1. (4.6) The recursion relations (4.4)–(4.6) have a unique solution once appropriate boundary conditions are given. One set of boundary conditions which will specify B˜ r(b),s (L, b) as the unique solution are the values obtained from Br(b),s (0, b) = δs,b , 1 ≤ b ≤ p0 − 1.
(4.7)
However, it will prove useful to recognize that this is not the only way in which boundary conditions may be imposed on (4.4)–(4.6) to give (1.12) as the unique solution. One alternative is the condition which is readily obtained from the term j = 0 in (1.12),
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Br(b),s (L, b) =
0 1
for L = 0, 1, 2, · · · , |s − b| − 1 . for L = |s − b|
(4.8)
To prove (4.4)–(4.6) for the b → r map of Sect. 3 we first consider the case where b and r are unrelated and recall the elementary recursion relations for q–binomial coefficients (3.20) and (3.21). If we use (3.20) in the definition (1.12) for Br,s (L, b) we find Br,s (L, b) = Br,s (L − 1, b − 1) + q
L+b−s 2
Br+1,s (L − 1, b + 1),
(4.9)
Br−1,s (L − 1, b − 1) + Br,s (L − 1, b + 1).
(4.10)
and if we use (3.21) we find Br,s (L, b) = q
L−b+s 2
Furthermore if we write (4.9) as q−
L+b−s 2
[Br,s (L, b) − Br,s (L − 1, b − 1)] = Br+1,s (L − 1, b + 1)
(4.11)
[Br,s (L − 1, b + 1) − Br,s (L − 2, b)] = Br+1,s (L − 2, b + 2)
(4.12)
and q−
L+b−s 2
and subtract (4.12) from (4.11) we may use (4.10) on the right-hand side to find q−
L+b−s 2
[Br,s (L, b) − Br,s (L − 1, b − 1) − Br,s (L − 1, b + 1) + Br,s (L − 2, b)] =q
L+s−b −1 2
Br,s (L − 2, b)
,
(4.13) and thus we have Br,s (L, b) = Br,s (L − 1, b − 1) + Br,s (L − 1, b + 1) + (q L−1 − 1)Br,s (L − 2, b). (4.14) If we now relate r to b using the map of Sect. 3 and use the Definition (4.1) then Eqs. (4.9), (4.10) and (4.14) are the three equations of (4.4) and the first equation of (4.6). We also note from (1.12) that ! ∞ X L L j 2 pp0 −jsp j 2 pp0 +jsp −q q B0,s (L, 0) = L+s L−s 0 0 2 − jp q 2 − jp q j=−∞ ! ∞ X L L j 2 pp0 −jsp j 2 pp0 +jsp = q = 0, −q L+s L+s 0 0 2 − jp q 2 + jp q j=−∞ (4.15) where to get the second line we first use (3.18) and then let j → −j. Combining this with (4.10) at r = 1, b = 1 we obtain (4.5). Finally we note that in an analogous fashion we may prove (4.16) Bp,s (L, p0 ) = 0, and thus the last equation of (4.6) follows.
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5. Recursive Properties for Fundamental Fermionic Polynomials fs (L, u) Our goal is to construct fermionic objects from the fundamental fermionic polynomials (3.22) which will satisfy the bosonic recursion relations (4.4)-(4.6) obeyed by B˜ r(b),s (L, b). To do this we will use the following recursive properties for the fundamental fermionic polynomials fs (L, u), where here and in the remainder of the paper we restrict our attention to νi ≥ 2 for i = 1, · · · , n − 1. The analysis for νi = 1 is analogous to that for νi ≥ 2, but since the recursion relations given below should be slightly modified, this case will not be treated here for reasons of space. Let us first introduce the following notation for tµ + 1 ≤ jµ ≤ tµ+1 + δµ,n : u0 (jµ ) = ejµ + u0 − et1+µ (1 − δµ,n ), u±1 (jµ ) = ejµ ±1 + u0 − et1+µ (1 − δµ,n )
,
(5.1)
Pν 0 0 ui = 0 for µ = 0 and for µ ≥ 1 where u0 is an arbitrary vector only restricted by i=1 0 the components must satisfy uj = 0 for j ≤ jµ + 1 − δjµ ,t1+n . These conditions are used in the proof of the recursive properties. Further define (t) = Ea,b
b X
eti .
(5.2)
i=a
Then if j0 is in the zone µ = 0, where 0 ≤ j0 ≤ ν0 = t1 we find (for L ≥ 1) fs (L − 1, u1 (0)), j0 = 0 fs (L − 1, u1 (j0 )) + fs (L − 1, u−1 (j0 )) fs (L, u0 (j0 )) = +(q L−1 − 1)fs (L − 2, u0 (j0 )), 1 ≤ j0 ≤ ν0 − 1 − 1, u1 (ν0 ) + eν0 ) + fs (L − 1, u−1 (ν0 )) fs (LL−1 +(q − 1)fs (L − 2, u0 (ν0 )), j0 = ν0 = t1 .
(5.3)
For jµ in the zone 1 ≤ µ ≤ n, where 1 + tµ ≤ jµ ≤ tµ+1 + δµ,n we have four separate recursive properties (for L ≥ 1): 1. for jµ = 1 + tµ , fs (L, u0 (1 + tµ )) = q + q θ(µ≥2)(
ν0 +1 3 L−1 2 − 4 + 4 θ(µ
ν0 −θ(µ even) L−1 ) 2 − 4
+ θ(µ ≥ 2)q + 2θ(µ ≥ 3)
q
(t) fs (L − 1, u1 (1 + tµ ) − E1,µ )
(t) fs (L − 1, u−1 (1 + tµ ) − E1,µ )
ν0 −θ(µ odd) L−1 2 − 4
µ−1 X
odd)
(t) fs (L − 1, e−1+tµ + u0 (1 + tµ ) − E1,µ )
ν0 −θ(i odd) L−1 2 − 4
(t) fs (L − 1, −E1,i + e−1+ti + u0 (1 + tµ ))
i=2
+ [1 + θ(µ ≥ 2)]fs (L − 1, e−1+ν0 − eν0 + u0 (1 + tµ )) + (q L−1 − 1)fs (L − 2, u0 (1 + tµ )); (5.4) 2. for 2 + tµ ≤ jµ ≤ −1 + tµ+1 + δµ,n ,
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A. Berkovich, B. M. McCoy, A. Schilling ν0 +1 3 L−1 2 − 4 + 4 θ(µ
fs (L, u0 (jµ )) = q +q
ν0 −θ(µ even) L−1 2 − 4
µ X
+ 2θ(µ ≥ 2)
odd)
(t) fs (L − 1, u1 (jµ ) − E1,µ )
(t) fs (L − 1, u−1 (jµ ) − E1,µ ) ν0 −θ(i odd) L−1 2 − 4
q
(t) fs (L − 1, u0 (jµ ) − E1,i + e−1+ti )
(5.5)
i=2
+ 2fs (L − 1, eν0 −1 − eν0 + u0 (jµ )) + (q L−1 − 1)fs (L − 2, u0 (jµ )); 3. for 1 ≤ µ ≤ n − 1, and jµ = tµ+1 , fs (L, u0 (t1+µ )) = q +q
ν0 −θ(µ odd) L−1 2 − 4
ν0 −θ(µ even) L−1 2 − 4
+ 2θ(µ ≥ 2)
µ X
(t) fs (L − 1, −E1,µ + et1+µ + u1 (tµ+1 ))
(t) fs (L − 1, −E1,µ + u−1 (tµ+1 ))
q
ν0 −θ(i odd) L−1 2 − 4
(t) fs (L − 1, −E1,i + e−1+ti + u0 (t1+µ ))
i=2
+ 2fs (L − 1, e−1+ν0 − eν0 + u0 (t1+µ )) + (q L−1 − 1)fs (L − 2, u0 (t1+µ )); (5.6) 4. for µ = n and jn = 1 + tn+1 , fs (L, e1+t1+n ) = q
ν0 −θ(n even) L−1 2 − 4
+ 2θ(n ≥ 2)
n X
q
(t) fs (L − 1, et1+n − E1,n )
ν0 −θ(i odd) L−1 2 − 4
(t) fs (L − 1, −E1,i + e−1+ti + e1+t1+n ) ,
(5.7)
i=2
+ 2fs (L − 1, eν0 −1 − eν0 + e1+t1+n ) + (q L−1 − 1)fs (L − 2, e1+t1+n ) where we remind the reader that the 1 + t1+n component of the vector u determines the parity of the variable mt1+n in the fundamental fermionic polynomial (3.22). We prove these recursive properties in Appendix A. 6. Properties of f˜s (L, u˜ ) In this section we will prove the Recursive properties of f˜s (L, u˜ ). The function f˜s (L, u0 (ν0 − j0 − 1)), defined by (3.35) and (5.1) with u0 → u10 satisfies the recursive properties for 1 ≤ j0 ≤ ν0 − 1 f˜s (L, u0 (ν0 − j0 − 1)) = f˜s (L − 1, u−1 (ν0 − j0 − 1)) + f˜s (L − 1, u1 (ν0 − j0 − 1)) + (q L−1 − 1)f˜s (L − 2, u0 (ν0 − j0 − 1)). (6.1) To prove (6.1) we adopt the simplified notation fj0 (L) = fs (L, u0 (j0 )). Then for 1 ≤ j0 ≤ ν0 − 1 the relation (5.3) reads fj0 (L) = f1+j0 (L − 1) + f−1+j0 (L − 1) + (q L−1 − 1)fj0 (L − 2) which we rewrite as
(6.2)
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fj0 (L) − f1+j0 (L − 1) = f−1+j0 (L − 1) − fj0 (L − 2) + q L−1 fj0 (L − 2).
(6.3)
Then if we replace j0 by ν0 − j0 , L by L + 1 and set I(L, j0 ) = fν0 −j0 (L + 1) − f1+ν0 −j0 (L),
(6.4)
I(L, j0 ) = I(L − 1, j0 + 1) + q L fν0 −j0 (L − 1).
(6.5)
we have We now eliminate fν0 −j0 (L − 1) by first writing (6.5) as [I(L, j0 ) − I(L − 1, 1 + j0 )]q −L = fν0 −j0 (L − 1),
(6.6)
and then by setting L → L − 1 and j0 → j0 − 1 to get the companion equation [I(L − 1, j0 − 1) − I(L − 2, j0 )]q −L+1 = f1+ν0 −j0 (L − 2). Subtracting (6.7) from (6.6) and multiplying by q [I(L, j0 ) − I(L − 1, 1 + j0 )]q −
L−j0 2
(6.7)
we obtain
L+j0 2
− [I(L − 1, −1 + j0 ) − I(L − 2, j0 )]q −
L−2+j0 2
=q
L−j0 2
I(L − 2, j0 ).
(6.8)
Recalling the definition (3.35) we see that f˜s (L, u0 (ν0 − j0 − 1)) = q −
L+j0 2
I(L, j0 ),
(6.9)
and using this along with (6.8) we obtain (6.1). Note from (5.3) that f0 (L) = f1 (L − 1) which implies that (6.10) f˜s (L, e−1 − eν0 + u10 ) = 0 = I(L, ν0 ), and using (6.6) we have f˜s (L, e0 − eν0 + u10 ) = q
L−(ν0 −1) 2
fs (L − 1, e1 − eν0 + u10 ) = q
L−(ν0 −1) 2
fs (L, −eν0 + u10 ). (6.11)
We also need the
Limiting Relation. lim f˜s (L, u˜ ) = 0.
L→∞
(6.12)
To prove (6.12) we use (6.6) and (6.10) to obtain the system of ν0 − j0 equations I(L, j0 ) − I(L − 1, j0 + 1) = q L fν0 −j0 (L − 1), I(L − 1, j0 + 1) − I(L − 2, j0 + 2) = q L−1 fν0 −j0 −1 (L − 2), I(L − 2, j0 + 2) − I(L − 3, j0 + 3) = q L−2 fν0 −j0 −2 (L − 3), ··· I(L − (ν0 − j0 − 1), ν0 − 1) − 0 = q L−(ν0 −j0 −1) f1 (L − (ν0 − j0 )), which, if added together give
(6.13)
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I(L, j0 ) =
ν0 −j 0 −1 X
q L−i fν0 −j0 −1 (L − 1 − i).
(6.14)
i=0
Thus multiplying by q −
L+j0 2
we find
L f˜s (L, u0 (ν0 − j0 − 1)) = q 2
ν0 −j 0 −1 X
j0
q − 2 −i fν0 −j0 −i (L − 1 − i),
(6.15)
i=0
and hence since limL→∞ fj0 (L) is finite we see from (6.15) that the limiting relation (6.12) holds for u = u0 (ν0 − j0 − 1), 1 ≤ j0 ≤ ν0 − 1. The case j0 = 0 requires a separate treatment. First we note that L f˜s (L, u0 (ν0 − 1)) = f˜s (L, u0 (ν0 − 2)) + q 2 fs (L − 1, u0 (ν0 )),
(6.16)
which follows from (3.35). In this equation we let L → ∞ and using (6.15) with j0 = 1 we obtain (6.12) for u˜ = u0 (ν0 − 1). 7. Construction of Fr(b),s (L, b) We now turn to the details of the construction of the fermionic representations Fr(b),s (L, b), L ≥ 0 of the bosonic polynomials B˜ r(b),s (L, b). Our method is to construct ˜ which by construction fermionic functions Fr(b),s (L, b) in terms of fs (L, u) and f˜s (L, u) satisfy the bosonic recursion relations (4.4) and (4.6). We begin by choosing as a starting value for b = 1, (t) ). F1,s (L, 1) = fs (L, −E1,n
(7.1)
The boundary bosonic recursion relation (4.5) requires that (t) ) F1,s (L − 1, 2) = fs (L, −E1,n
(7.2)
from which if we let L → L+1 and use the first fermionic recursive properties in (5.3) we (t) find with u0 = −E2,n , (t) F1,s (L, 2) = fs (L, e1 − E1,n ).
(7.3)
We now continue this procedure in a recursive fashion. We construct Fr(b),s (L, b) for all b by defining Fr(b),s (L, b) in terms of Fr(b−1),s (L, b − 1) and Fr(b−2),s (L, b − 2) through the bosonic recursion relations (4.4) for the b → r map defined in (3.4) and then simplifying the expressions by using the fermionic recursive properties of Sects. 5 and 6. The b → r map is important since it prescribes which bosonic recursion relation is being used to construct Fr(b),s (L, b) (i.e. whether one uses the depth-2 recurrence or one of the depth-1 recurrences). This process is continued until we reach b = p0 − 1 for which the last identity of (4.6) must hold. This recursive construction can be carried out for any starting function F1,s (L, 1) but the last equation of (4.6) will only hold if F1,s (L, 1) has been properly chosen. The recursive process used to generate fermionic polynomials will be referred to as an evolution. The map from two initial polynomials F1,s (L, 1), F1,s (L, 2) to polynomials Fr(b),s (L, b), Fr(b+1),s (L, b + 1) will be called a flow of length b.
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After Fr(b),s (L, b) has been constructed for all p0 − 1 values of b to satisfy the bosonic recursion relations (4.4) –(4.6) we will complete the analysis by studying the behavior for small values of L to show that with a suitable normalization q Nr,s the initial conditions (4.7) or (4.8) are satisfied. For 1 ≤ b ≤ ν0 + 1 (i.e. b in zone 0) this general recursive construction can be explicitly carried out for an arbitrary n + 1 zone problem. In this case the Takahashi decomposition (3.1) of b consists of the single term (0) = 1 + j 0 , 0 ≤ j0 ≤ ν0 . b = l1+j 0
(7.4)
As long as 1 ≤ j0 ≤ ν0 , F1,s (L, b) satisfies the first equation of (4.4) because r = 1 does not change. Comparing this recursion relation with the fermionic recursive property (5.3) and using the values for F1,s (L, 1) and F1,s (L, 2) of (7.1) and (7.3) we conclude that for an n + 1 zone problem with b in zone 0. (t) (0) ), b = l1+j = 1 + j0 and 0 ≤ j0 ≤ ν0 . F1,s (L, b) = fs (L, ej0 − E1,n 0
(7.5)
(1) we have entered into zone one. However we still have r = 1 When b = 2 + ν0 = l2+ν 0 and an identical computation gives (t) ). F1,s (L, 2 + ν0 ) = fs (L, e1+ν0 − E2,n
(7.6)
To proceed further into zone 1 we must cross a boundary where r changes from 1 to 2. Here we use the second bosonic recursion relation in (4.4) for b = 2 + ν0 which has the Takahashi expansion (3.1) with β = 1 and µ1 = 1, j1 = 1 + ν0 , (1) b = 2 + ν0 = l2+ν 0
(7.7)
and find q
L−1 2
F2,s (L − 1, ν0 + 3) = F1,s (L, ν0 + 2) − F1,s (L − 1, ν0 + 1)
(7.8)
which, after using (7.6) and (7.5) becomes q
L−1 2
(t) (t) F2,s (L − 1, ν0 + 3) = fs (L, e1+ν0 − E2,n ) − fs (L − 1, −E2,n ).
(7.9)
(t) ) by an expression in terms of fermionic To reduce this we replace fs (L, e1+ν0 − E1,n polynomials with arguments L − 1 and L − 2 using relation (5.4) with µ = 1. In this case several of the terms in the general expression (5.4) vanish and we have (t) )=q fs (L, e1+ν0 − E2,n
ν0 −2 L−1 2 − 4
(t) (t) fs (L − 1, e2+ν0 − E1,n ) + fs (L − 1, −E2,n )
(t) (t) + fs (L − 1, e−1+ν0 + e1+ν0 − E1,n ) + (q L−1 − 1)fs (L − 2, e1+ν0 − E2,n ). (7.10) Using this in (7.9) and setting L → L + 1 we find
F2,s (L, ν0 + 3) = q −
ν0 −2 4
(t) fs (L, e2+ν0 − E1,n )
(t) (t) ) + (q 2 − q − 2 )fs (L − 1, e1+ν0 − E2,n ) + q − 2 fs (L, e−1+ν0 + e1+ν0 − E1,n L
=q
L
ν −2 − 04
L
(t) (t) fs (L, e2+ν0 − E1,n ) + f˜s (L, e−1+ν0 + e1+ν0 − E1,n ),
(7.11)
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where in the last line we have used the case j0 = 0 in Definition (3.35). In a similar fashion we consider (1) b = 3 + ν0 = l2+ν + l1(0) . 0
(7.12)
We use the third recursion relation in (4.4) written as L
F2,s (L − 1, ν0 + 4) = F2,s (L, ν0 + 3) − q 2 F1,s (L − 1, ν0 + 2)
(7.13)
which, after using (7.6) and (7.11) becomes F2,s (L − 1, ν0 + 4) = q −
ν0 −2 4
(t) fs (L, e2+ν0 − E1,n )
(t) (t) ) − fs (L − 1, e1+ν0 − E2,n )]. + q − 2 [fs (L, e−1+ν0 + e1+ν0 − E1,n L
(7.14)
We now reduce this by using the first equation in (5.3), the first line in the definition (3.35) and setting L → L + 1 to obtain the result F2,s (L, ν0 + 4) = q −
ν0 −2 4
(t) (t) fs (L, e1 + e2+ν0 − E1,n ) + f˜s (L, eν0 −2 + e1+ν0 − E1,n ). (7.15)
Let us review what has been done. We started with F1,s (L, 1) and constructed all the values of F1,s (L, b) with 1 ≤ b ≤ 2 + ν0 to satisfy the first bosonic recursion relation in (4.4) where r does not change. We refer to these recursion relations where r does not change as “moving b on the plateau”. In this process we did not create any new terms in the linear combination. We then constructed F2,s (L, 3 + ν0 ) by using the second recursion relation in (4.4) in which the term with b + 1 has r + 1. In this step we created the new term f˜. We refer to this new term as a reflected term (much as there is a reflected wave at a boundary in an optics problem). We then created F2,s (L, 4 + ν0 ) by using the third recursion relation in (4.4) where b − 1 has r − 1. This process did not create any additional terms. We refer to this process of using the two equations in (4.4) where r changes by one as “transiting an r boundary”. In most cases when b moves on the plateau we apply (5.3) to the fs – terms and (6.1) to the f˜s –terms in (3.40). Note that the fermionic recurrences we employ may still vary from term to term in (3.40). Again in most cases while transiting the r– boundary we use (5.4)–(5.7). However, there are important exceptional cases (related to the so–called dissynchronization effect discussed in Sect. 8 and Appendix B) where this rule breaks down and, as a result, we are forced to apply (5.3) to some terms in (3.40) and (5.4)– (5.7) to others (for examples of this phenomena see (B.2), (B.8) and (B.13)). With this overview in mind we can continue the construction process as far as we like. To be explicit we present the results of the construction for b in zones 1 and 2 in a problem with four or more zones.
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b in zone 1. (1) = (j1 − t1 )(ν0 + 1) + 1, t1 + 1 ≤ j1 ≤ t2 , 1 : b = l1+j 1 (1) r = l˜1+j = j1 − ν 0 , 1 (t) Fr,s (L, b) = q c(j1 ) fs (L, ej1 − E2,n ).
(1) (0) (1) 2 : b = l1+j + l1+j = l1+j + 1 + j0 , 0 ≤ j0 ≤ t1 − 1, t1 + 1 ≤ j1 ≤ t2 − 1, 1 0 1 (1) r = l˜1+j + 1 = j1 − ν0 + 1, 1 ν0−2 4
Fr,s (L, b) = q c(j1 )−
(t) (t) fs (L, ej0 +ej1+1 −E1,n ) + q c(j1 ) f˜s (L, eν0−j0−1 +ej1 −E1,n ).
(1) (0) 3 : b = l1+t + l1+j = ν1 (ν0 + 1) + 2 + j0 , 0 ≤ j0 ≤ t1 − 1, 2 0 (1) r = l˜1+t + 1 = ν1 + 2, 2
Fr,s (L, b) = q c(t2 )−
ν0 −1 4
(t) fs (L, ej0 − et1 + e1+t2 − E3,n )
(t) + q c(t2 ) f˜s (L, eν0 −j0 −1 + et2 − E1,n ,)
(7.16) where 1 c(j1 ) = − (ν0 − 2)(j1 − ν0 − 1) for t1 + 1 ≤ j1 ≤ t2 . 4
(7.17)
b in zone 2. Here we distinguish separate cases depending on whether 1 ≤ j1 ≤ t2 − 2 or j1 = t2 − 1. The restriction (3.2) says that j1 = t2 does not occur. We also do not consider the cases b > y3 . (2) 1 : b = l1+j = y1 + (j2 − t2 )y2 , 1 + t2 ≤ j2 ≤ t3 , 2 (2) r = l˜1+j = 1 + (j2 − t2 )ν1 , 2 (t) Fr,s (L, b) = q c(j2 ) fs (L, ej2 − E3,n ).
(2) (1) (0) (2) 2 : b = l1+j + l1+j + l1+j = l1+j + (j1 − t1 )y1 + 2 + j0 , 2 1 0 2
0 ≤ j0 ≤ t1 − 1, t1 + 1 ≤ j1 ≤ t2 − 2, t2 + 1 ≤ j2 ≤ t3 − 1, r = l˜(2) + l˜(1) + 1 = l˜(2) + (j1 − t1 ) + 1, 1+j2
1+j1
Fr,s (L, b) = q
1+j2
c(j2 )+c(j1 )−
+ q c(j2 )+c(j1 )− + q c(j2 )+c(j1 )− + q c(j2 )+c(j1 )−
ν0 −2 4 ν0 2
+ 41
ν0 +1 4
ν0 2
+ 21
(t) fs (L, e1+j0 + et2 −(j1 −t1 )−1 + ej2 − E1,n )
(t) f˜s (L, eν0 −j0 −2 + et2 −(j1 −t1 ) + ej2 − E1,n ) (t) fs (L, ej0 + ej1 +1 + ej2 +1 − E1,n )
(t) f˜s (L, eν0 −j0 −1 + ej1 + ej2 +1 − E1,n ).
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(2) (0) (2) 3 : b = l1+j + l1+j = l1+j + 1 + j0 , 0 ≤ j0 ≤ t1 , t2 + 1 ≤ j2 ≤ t3 − 1, 2 0 2 (2) r = l˜1+j + 1, 2 ν0
(t) ) Fr,s (L, b) = q c(j2 )− 4 fs (L, ej0 + et2 −1 + ej2 − E1,n (t) + q c(j2 ) f˜s (L, eν0 −j0 −1 − et1 + ej2 − E3,n )
+ q c(j2 )−
ν0 +1 4
(t) fs (L, ej0 + ej2 +1 − E1,n ).
(2) (1) (2) +l1+j = l1+j +(j1 − t1 )y1 + 1, t1 +1 ≤ j1 ≤ t2 − 2, t2 + 1 ≤ j2 ≤ t3 − 1, 4 : b = l1+j 2 1 2 (2) (1) (2) r = l˜1+j + l˜1+j = l˜1+j + (j1 − t1 ), 2 1 2 ν0
(t) ) Fr,s (L, b) = q c(j2 )+c(j1 )− 4 fs (L, et1 −1 + et2 −(j1 −t1 ) + ej2 − E1,n
+ q c(j2 )+c(j1 )− + q c(j2 )+c(j1 )−
ν0 2
+L 2
ν0 +1 4
(t) fs (L, et2 −(j1 −t1 )−1 + ej2 − E1,n )
(t) fs (L, ej1 + ej2 +1 − E2,n ).
(7.18) (2) (2) + lt(1) = l1+j + (ν1 − 1)y1 + 1, t2 + 1 ≤ j2 ≤ t3 − 1, 5 : b = l1+j 2 2 2 (2) (2) r = l˜1+j + l˜t(1) = l˜1+j + ν1 − 1, 2 2 2 ν0
(t) ) Fr,s (L, b) = q c(j2 )+c(t2 −1)− 4 fs (L, ej2 − E2,n
+ q c(j2 )+c(t2 −1)−
ν0 +1 4
(t) fs (L, e−1+t2 + e1+j2 − E2,n ).
(2) (0) (2) 6 : l1+j + lt(1) + l1+j = l1+j + (ν1 − 1)y1 + 2 + j0 , 2 2 0 2
0 ≤ j0 ≤ t1 − 1, t2 + 1 ≤ j2 ≤ t3 − 1, r = l˜(2) + l˜t(1) + 1 = l˜(2) + ν1 , 1+j2
1+j2
2
Fr,s (L, b) = q
c(j2 )+c(t2 −1)−
+ q c(j2 )+c(t2 −1)−
ν0 2
+ q c(j2 )+c(t2 −1)−
ν0 +1 4
+ 41
ν0 4
(t) f˜s (L, eν0 −j0 −1 + ej2 − E1,n )
(t) fs (L, ej0 + et2 + ej2 +1 − E1,n )
(t) f˜s (L, eν0 −j0 −1 + et2 −1 + ej2 +1 − E1,n ),
(7.19) where
1 3 + ν1 (ν0 − 2) − (j2 − t2 ) for t2 + 1 ≤ j2 ≤ t3 . (7.20) 2 4 Thus far we have used our constructive procedure to generate all polynomials Fr(b),s (L, b) for b in zones 0, 1 and 2 where the total number of zones is 4 or greater. However, to complete the process we must carry out the construction for b in the final zone and show that the closing relation in (4.6) is satisfied. The construction for (n) has two features not present in any other zone. The first is that the map b → r b > l1+t 1+n of (3.4) has changed and the second is that the parity restriction on mt1+n , which was even before, is now sometimes allowed to be odd. We recall that this parity is specified in our notation by the parity of the 1 + t1+n component of the vector u. Thus we compute the following results for the final zone. c(j2 ) =
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351
b in zone 1 in a two zone problem. 1: The first equation of (7.16) now may be extended to all 1 + t1 ≤ j1 ≤ 1 + t2 with (t) replaced by zero and c(t2 + 1) given by (7.17). −E2,n (t) replaced 2: The second equation of (7.16) now holds for all 1+t1 ≤ j1 ≤ t2 with −E2,n by zero. 3: Equation three of (7.16) is omitted and replaced by (1) (0) + l1+j = (ν1 + 1)(ν0 + 1) + 2 + j0 , 0 ≤ j0 ≤ t1 − 1, b = l2+t 2 0 (1) r = l˜2+t = ν1 + 1 = p − 1, 2
Fr,s (L, b) = q
c(t2 +1)
(7.21)
fs (L, et1 −j0 −1 − et1 + e1+t2 ),
where c(t2 + 1) is given by (7.17) instead of (7.20).
b in zone 2 in a three zone problem. 1: Equation 1 in (7.18) is now valid for 1 + t2 ≤ j2 ≤ t3 + 1 and Eqs. 2–6 in (7.18) and (7.19) are now valid for t2 + 1 ≤ j2 ≤ t3 + 1 with the convention that wherever e2+t3 ˜ the term is omitted and c(t3 + 1) appears in the argument of some fs (L, u) or f˜s (L, u) is given by (7.20). 2: We have the following additional closing equation: (2) (0) b = l2+t + lt(1) + l1+j = p0 − ν0 + j0 − 1, 2 3 0
0 ≤ j0 ≤ t1 , (2) r = l˜2+t + l˜t(1) = ν1 (ν2 + 2) = p − 1, 2 3
(7.22)
1
Fr,s (L, b) = q c(t3 +1)+c(t2 )− 2 fs (L, eν0 −j0 − et1 − et2 + e1+t3 ), where c(t3 + 1) is given by (7.20). For the problem with 2 and 3 zones we have now constructed a complete set of fermionic polynomials for all 1 ≤ b ≤ p0 − 1 which satisfy the bosonic recursion relations (4.4), (4.5) and the first equation in (4.6) by construction and the second equation in (4.6) by use of the first equation of (5.3). In order to complete the proof of the bose/fermi identities it remains to show that the fermionic polynomials satisfy the boundary conditions for Br(b),s (L, b) at L = 0 (4.7) and to compute the normalization constant in (1.11). This is easily done and thus we obtain the final result that for the (µs ) , cases of 2 and 3 zones with s = l1+j s Fr(b),s (L, b) = q 2 (φr(b),s −φr(s),s )+c(js ) Br(b),s (L, b) 1
where Fr(b),s (L, b) is given by the formulae of this section.
(7.23)
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8. The Inductive Analysis of Evolution In the previous section we constructed all fermionic polynomials Fr(b),s (L, b), 1 ≤ b ≤ p0 − 1 in the problem with 1,2 and 3 zones. The procedure we used is completely general, but becomes somewhat tedious to execute when the number of zones increases beyond three. This is because the results are sensitive to the details of the Takahashi decomposition (3.1). On the other hand, there are certain cases of (3.1) such as b = (µ) , 1+tµ ≤ jµ ≤ t1+µ +δµ,n , 0 ≤ µ ≤ n, where the form of Fr(b),s (L, b) remains very l1+j µ simple for any µ. The question arises if one can treat certain classes of decompositions (3.1) without having explicit formulas for all fermionic polynomials Fr(b),s (L, b), 1 ≤ b ≤ p0 − 1. In this section we shall provide a positive answer to this question by proving inductively a set of explicit formulas for certain flows of length yµ − 1. This inductive analysis is possible because as can be seen from (3.13) the construction of (µ) , 1 + tµ ≤ jµ ≤ t1+µ − 1 + δµ,n , µ ≥ 2 from the pair Fr(b),s (L, b), b = l2+j µ (µ) involves exactly the same steps {Fr(b0 )+1,s (L, b0 + 1), Fr(b0 )+1,s (L, b0 + 2)}, b0 = l1+j µ as that of Fzµ−1 ,s (L, yµ ) from the pair {F1,s (L, 1), F1,s (L, 2)}. Furthermore, recalling that y1+µ = yµ−1 + νµ yµ , it is natural to decompose the flow of length yµ+1 − 1 into flows of smaller length and to take this decomposition as a starting point of our inductive analysis of evolution. In this direction, we first discuss the notation for the flows in terms of the b → r map, whose properties are summarized in (3.10)–(3.14).
Notation. The flow −→x of length x + 1 denotes the sequence of steps corresponding to the b → r map as defined in (3.4) (or equivalently (3.8)) between b = 1 and b = 2 + x. r
(8.1) 1 1
2+x
b
Piece of b → r map between 1 and 2+x (schematic)
According to (3.10) and (3.11) this sequence is made up of three steps: a)
b)
c)
(8.2) Different kinds of steps in the b → r map
(where we note that steps like
(8.3) Two kinds of steps which are not allowed
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353
are not allowed). The b → r map specifies exactly which bosonic recursion relation is being used to define the (in this case fermionic) polynomials. Case a) in (8.2) denotes that r is the same for all three objects Fr,s (L, b − 1), Fr,s (L, b) and Fr,s (L, b + 1) involved and hence one uses the first recursion relation (4.4). Case b) indicates the use of the second and case c) the use of the last recursion relation in (4.4). Let us further denote by =⇒2 the flow according to the steps as defined by the following graph: (8.4) 2
piece of the b → r map of length 3 defining
The b → r map as defined by (3.4) (or (3.8)) has an unambiguous initial point (r1 , b1 ) = (1, 1). By −→x we mean only the sequence of x steps found on the b → r map as specified in (8.1). The notation −→x does not fix the initial point (r1 , b1 ). The initial point (r1 , b1 ) can be placed anywhere . Having agreed on that, we can now piece together flow 1 and flow 2 such that the last segment of the first flow is identical to the first segment of the second flow by identifying the two final points of the first flow as the two initial points of the second flow. For example adjoining flow c of (8.2) to flow b of (8.2) gives flow (8.4) and adjoining a of (8.2) to c of (8.2) gives (8.5) We then denote by −→x(1) −→x(2) the flow given by piecing together the flow −→x(1) and −→x(2) . Note that in general it is not true that −→x(1) +x(2) = −→x(1) −→x(2) . With these conventions we now show that there is a decomposition of −→yµ+1 −2 in terms of −→yµ−1 −2 , −→yµ −2 and =⇒2 given by the following graph: yµ+1-2
= yµ-2
2 yµ-1-2
2
yµ-2
2
yµ-2
2
yµ-2
(8.6)
A decomposition of the flow of length yµ+1-1, µ>1 with νµ>1
In other words we need to show that piecing together −→yµ −2 , −→yµ−1 −2 and =⇒2 as shown in (8.6) amounts to −→yµ+1 −2 . This can be easily done by recalling (3.12) with b = yµ and b = yµ−1 + kyµ for 1 ≤ k ≤ νµ − 1, 2 ≤ µ and proving two additional results: r(b + yµ ) − r(b) = r(yµ ) = zµ−1 , for 1 ≤ b ≤ yµ−1 (8.7) and (µ) r(b + yµ−1 + kyµ ) − r(b) = r(yµ−1 + kyµ ) = l˜1+t for 1 ≤ b ≤ yµ . µ +k
(8.8)
To check (8.7) we use the Takahashi decomposition of b (3.1) with µβ ≤ µ − 2. Pβ (µi ) (µ−1) + l1+t is a valid decomposition in Takahashi If b 6= yµ−1 then b + yµ = i=1 l1+j µi µ lengths and according to (3.4) we have r(yµ + b) − r(b) = l˜(µ−1) = zµ−1 . Similarly for 1+tµ
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A. Berkovich, B. M. McCoy, A. Schilling
(µ) (µ−2) b = yµ−1 we have r(yµ−1 + yµ ) − r(yµ−1 ) = l˜2+t − l˜1+t = zµ−1 . Equation (8.8) may µ µ−1 be checked in a similar fashion. Notice however that the order of the arrows in the decomposition as shown in the previous figure is crucial. If we moved for example −→yµ−1 −2 =⇒2 to the one before last position as shown in the next figure the decomposition does not agree with all the steps as defined by −→yµ+1 −2 . yµ+1-2
= yµ-2
2
yµ-2
yµ-2
2
yµ-2
2
2 yµ-1-2
2 yµ-2
(8.9)
A wrong decomposition of the flow of length yµ+1-1, µ>1
This can be easily seen from (3.12) with b = yµ−1 + (νµ − 1)yµ and b = kyµ , 1 ≤ k ≤ νµ − 1, 2 ≤ µ and the following lemma: Lemma 1.1. a. For 1 ≤ k ≤ νµ − 2, µ ≥ 2 we have r(b + kyµ ) − r(b) = kzµ−1 for all 1 ≤ b ≤ yµ except for b = bµ,−1 =
Pµ−1 i=0
(8.10)
yi for which
r(bµ,−1 + kyµ ) − r(bµ,−1 ) = kzµ−1 + (−1)µ .
(8.11)
b. for 1 ≤ b ≤ yµ−1 we have r(b + (νµ − 1)yµ ) − r(b) = (νµ − 1)zµ−1 .
(8.12)
r(b + lt(µ) ) − r(b) = l˜t(µ) . 1+µ 1+µ
(8.13)
c. for 1 ≤ b ≤ yµ
We shall also require the companion of (8.10) and (8.11) Lemma 1.2. For µ ≥ 3 we have r(b + yµ−1 ) − r(b) = zµ−2 for all 1 ≤ b ≤ yµ except for b = bµ,1 =
Pµ−2 i=0
(8.14)
yi for which
r(bµ,1 + yµ−1 ) − r(bµ,1 ) = zµ−2 + (−1)µ−1 .
(8.15)
These results can be checked in the same fashion as Eqs. (8.7) and (8.8). The decomposition (8.9) fails precisely because there is a special value b = bµ,−1 for which (8.10) is not valid. However, one can find a slightly modified version of (8.9) which does indeed hold. For this we need to define two further types of arrows.
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355
y −2
µ Definition. We denote by −→−1 the flow according to the steps defined by the b → r map (3.4) between 1 + yµ and 2yµ .
r
(8.16) 1 1
2yµ
1+yµ
b
b → r map between 1+yµ and 2yµ (schematic) y −2
We denote by −→1 µ the flow according to the steps defined by the b → r map (3.4) between 1 + yµ−1 and yµ−1 + yµ . r
(8.17) 1 1
yµ-1+yµ
1+yµ-1
b
b → r map between 1+yµ-1 and yµ-1+yµ (schematic) We further identify yµ −2
yµ −2
−→ = −→ 0
(8.18)
Slightly generalizing the above discussion we recapitulate. Let Oi (L, bi , q) for i = 1, 2 be polynomials (in q 1/4 ) depending on L where L is a nonnegative integer. Let Oi (L, bi , q) for i = 3, 4 be polynomials depending on L obtained recursively from O1 (L, b1 , q), O2 (L, b2 , q) by the flow −→xt of length 1 + x. We denote this as x
{O1 (L, b1 , q) , O2 (L, b2 , q)} −→ {O3 (L, b3 , q), O4 (L, b4 , q)} , t
(8.19)
where the symbol x above the arrow denotes that O4 (L, b4 , q) follows from O1 (L, b1 , q) and O2 (L, b2 , q) after x steps. Parameters bi , i = 1, 2, 3, 4 with b2 = b1 + 1, b3 = b1 + x, b4 = b1 + x + 1 and q associated with Oi will often be suppressed. The symbol t below the arrow denotes which sequences of the recursion relations in (4.4) are
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A. Berkovich, B. M. McCoy, A. Schilling
being used. The sequence t can be thought of pictorially as a continuous graph made up of horizontal and diagonal segments where by horizontal segment we mean only the segment between (r, b) and (r, b + 1) and by diagonal segment we mean the segment between (r, b) and (r + 1, b + 1). The only restriction on the sequence t is that a diagonal segment must be preceded and followed by horizontal segments unless the diagonal segment is the first or last segment. As in the case of the notation −→x , the notation −→xt does not fix the initial point. Each pair of adjacent segments, called a step, in the flow −→xt represents one of the three recursion relations in (4.4) in a manner exactly analogous to the discussion following (8.2). More precisely, 1
{O1 (L, b1 , q), O2 (L, b1 + 1, q)} −→{O3 (L, b1 + 1, q), O4 (L, b1 + 2, q)}, t
(8.20)
where O3 (L, b, q) = O2 (L, b, q)
(8.21)
and
O2 (L + 1, b1 + 1, q) − O1 (L, b1 , q) + (1 − q L )O2 (L − 1, b1 + 1, q) if t is a of (8.2) −L/2 [O2 (L + 1, b1 + 1, q) − O1 (L, b1 , q)] O4 (L, b1 +2, q) = q if t is b of (8.2) (L+1)/2 O1 (L, b1 , q) O2 (L + 1, b1 + 1, q) − q if t is c of (8.2). (8.22) Note that for L = 0 the last term in the top line of (8.22) vanishes. Therefore one does not need to know the polynomials O1 (L, b1 , q), O2 (L, b1 + 1, q) for L < 0 to determine O4 (L, b1 + 2, q) for L ≥ 0. To compare three different flows (8.16)–(8.18), let us set the initial points of the yµ −2 y −2 , −→0 µ to be (r1 , b1 ) = (1, 1). Then according to (8.10), (8.11) and flows −→±1 y −2 (8.14) ,(8.15) the pieces of the b → r map used to define −→aµ , a = ±1 differ from y −2 the one used to define −→0 µ at exactly one point. We refer to this phenomena as the Pµ−1−δ dissynchronization effect. Recalling the definition for a = ±1 that bµ,a = i=0 a,1 yi , y −2 y −2 µ ≥ 2 + δa,1 we find that the difference between −→aµ , a = ±1 and −→0 µ can be illustrated by the following two figures. Here the solid line denotes the piece of the y −2 y −2 b → r map for −→aµ , a = ±1 and the dashed line that for −→0 µ .
bµ,a-1
bµ,a
bµ,a+1
Dissynchronization effect for µ-δa,1 even with a=-1,+1
bµ,a-1
bµ,a
bµ,a+1
(8.23)
Dissynchronization effect for µ-δa,1 odd with a=-1,+1
More precisely, all elementary segments in these two flows are identical except in the y −2 interval (bµ,a − 1, bµ,a + 1). For µ − δa,1 even (respect. odd) the flow −→aµ , a = ±1 restricted to the interval [bµ,a − 1, bµ,a + 1] is given by c of (8.2) (b of (8.2)). The flow, y −2 −→0 µ , restricted to the same interval is given by b of (8.2) (c of (8.2)).
Rogers–Schur–Ramanujan Identities for Minimal Models of CFT
357 y
We now may prove the following two decompositions of −→aµ+1 −1, 0, 1
−2
, where a =
yµ+1-2 a
= yµ-2
2 yµ-1-2
-δa,1
2
2
yµ-2 δa,-1
0
2
yµ-2 0
yµ-2 0
(8.24)
The decomposition of the flow of length yµ+1-1 for νµ>1 in the cases µ >1, a=0,1 and µ >2, a=-1 used to prove Proposition 1 and
yµ+1-2 a
= yµ-2
2
-δa,1
yµ-2 -δa,0-δa,1
2
yµ-2 -1
2
yµ-2 -1
2 yµ-1-2 0
2 yµ-2 0
(8.25)
The decomposition of the flow of length yµ+1-1, µ>1 with νµ>2 used to prove Proposition 2 y −2
In (8.24) the arrows =⇒2 −→0 µ appear νµ − 2 times and in (8.25) the arrows yµ −2 =⇒2 −→−1 appear νµ − 3 times. The decomposition (8.24) for a = 0 is (8.6). To prove (8.24) for a = ±1 we use (8.24) with a = 0 and (8.23) with a = ±1, µ replaced by µ + 1 to convert the top graph of (8.24) with a = 0 into the top graph of (8.24) with a = ±1, by altering just the single step centered at bµ+1,a , a = ±1 as prescribed by (8.23). It is easy to see from (8.23) that this procedure converts the first (fifth) arrow of the bottom yµ−2 y (−→1 µ−2 ) for a = 1(−1) and has no effect on other graph (8.24) with a = 0 into −→−1 arrows of the bottom graph (8.24). This completes the proof of (8.24). The decomposition (8.25) is the correct modification of (8.9). When a = 0 it follows from (8.10)-(8.13) and (8.23) with a = −1. To prove (8.25) for a = 1(−1) we again replace one single step of the top graph of (8.25) with a = 0. This converts the first yµ −2 y −2 (−→0 µ ) and thus completes the proof. (third) arrow of the bottom graph into −→−1 Let us finally define some useful sets of terms Definition. For 1 ≤ µ ≤ n, P−1 (L, µ, uµ0 ), P0 (L, µ, uµ0 ) and P1 (L, µ, uµ0 ) are defined as follows: P−1 (L, µ, uµ0 ) =
µ X
q2− L
ν0 −θ(i odd) 4
(t) fs (L, −E1,i + e−1+ti + uµ0 ) + fs (L, e−1+t1 − et1 + uµ0 ),
i=2
=
µ X i=2
q
ν0 −1−θ(i even) 4
(t) f˜s (L, −E1,i + e−1+ti + uµ0 ) + fs (L, e−1+t1 − et1 + uµ0 )
(8.26)
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A. Berkovich, B. M. McCoy, A. Schilling
P0 (L, µ, uµ0 ) =
µ X
q−
ν0 −θ(i odd) 4
(t) fs (L, −E1,i + e−1+ti + uµ0 ) + f˜s (L, eν0 −1 − eν0 + uµ0 )
,
(8.27)
i=2
and P1 (L, µ, uµ0 ) =
µ X
q−
ν0 −θ(i odd) 4
(t) fs (L, e1 − E1,i + e−1+ti + uµ0 ) + f˜s (L, eν0 −2 − eν0 + uµ0 )
, (8.28)
i=2
where uµ0 is any 1 + tn+1 -dimensional vector with non-zero entries only in zone µ or higher, i.e. (uµ0 )i = 0 for i ≤ tµ . (We consider here the case that νi ≥ 2, 1 ≤ i ≤ n − 1.) Equipped with the above notations and explanations we are now in the position to formulate two propositions which will be important in the sequel. Proposition 1. Let a = −1, 0, 1 and P−1 (L, µ, uµ0 ) be defined as in (8.26). Then we have for a = 0, µ ≥ 1 and a = −1, µ ≥ 2 and a = 1, µ ≥ 3, o n (t) (t) + uµ0 ), fs (L, e1 − E1,µ + uµ0 ) fs (L, −E1,µ (8.29) . µ 1 yµ −2 × −→ q c(tµ )+ 2 a(−1) P−1 (L, µ, uµ0 ), fs (L, uµ0 ) a
Proposition 2. Let a = −1, 0, 1 and P0 (L, µ, uµ0 ) and P1 (L, µ, uµ0 ) be defined as in (8.27) and (8.28). Then we have for a = 0, µ ≥ 1 and a = −1, µ ≥ 2 and a = 1, µ ≥ 3, P0 (L, µ, uµ0 ), P1 (L, µ, uµ0 ) n L−ν0 +θ(µ odd) o yµ −2 (t) +c(tµ )+ 21 a(−1)µ 2 fs (L, −E1,µ + uµ0 ), 0 . × −→ q (8.30) a o n µ 1 1 (t) = q c(tµ )− 2 θ(µ even)+ 2 a(−1) f˜s (L, −E1,µ + uµ0 ), 0 Here c(tµ ) is defined recursively by c(tµ+1 ) = c(tµ−1 ) + νµ c(tµ ) −
ν0 − θ(µ even) ν0 + 1 3 + (νµ − 1)(− + θ(µ odd)), (8.31) 4 4 4
where c(t0 ) =
ν0 , 4
c(t1 ) = 0.
(8.32)
We will also need c(j0 ) = 0 for 1 ≤ j0 ≤ t1 ,
1 ν0 + 1 3 µ + θ(µ odd) + c(tµ ) + c(tµ−1 ) c(jµ ) = (−) + (jµ − tµ ) − 2 4 4 for tµ + 1 ≤ jµ ≤ tµ+1 + 2δµ,n and 1 ≤ µ ≤ n.
(8.33)
Propositions 1 and 2 are very important because they enable us to prove many identities without explicitly constructing all fermionic polynomials in a one by one fashion. From them we will prove Theorem 1 (8.50) and as a result obtain the Rogers–Schur– (µ) . The reason for the introduction Ramanujan identities for M (p, p0 ) models at b = l1+j µ
Rogers–Schur–Ramanujan Identities for Minimal Models of CFT
359
of the several sets of terms P0 and P±1 is that (as can be seen quite explicitly in (10.5) (µ) with j0 = 0, 1) the polynomials Fr(b+a),s (L, b + a) for b = l1+j , a = 1, 2, 1 ≤ µ ≤ µ n, 1 + tµ ≤ jµ ≤ tµ+1 − 1 + δµ,n can be written as {Fr(b+1),s (L, b + 1), Fr(b+2),s (L, b + 2)} = q c(jµ )− +q
c(jµ )
ν0 +1 3 4 + 4 θ(µ
odd)
(t) (t) {fs (L, e1+jµ − E1,n ), fs (L, e1 + e1+jµ − E1,n )} ,
{P0 (L, µ, ejµ −
(t) Eµ+1,n ), P1 (L, µ, ejµ
−
(8.34)
(t) Eµ+1,n )}
and then we see from (8.29) and (8.30) that the first pair on the right-hand side of (8.34) may be studied independently from the second pair under the flow −→b−2 , b ≤ yµ (but we note this independence is not true when b > yµ ). We note too that the proofs given below of the two propositions are also quite independent. In particular, for (8.29) we use decomposition (8.24) while for (8.30) we use (8.25). The final tool we need for our inductive proof are the following lemmas which we use to treat the evolution along =⇒2 in the decompositions (8.24) and (8.25): µ
Lemma 2.1. Define g(µ, n, jµ ) = − ν04+1 + 43 θ(µ odd) + (−1) 4 θ(n > µ)δjµ ,tµ+1 , 1 ≤ (t) + pe ˜ 1+t1+n ) = 0 µ ≤ n, 1 + δµ,1 + tµ ≤ jµ ≤ tµ+1 + δµ,n , and fs (L, ea + e2+tn+1 − E1,n for a = 0, 1 and p˜ ∈ Z. Then we have P−1 (L, µ − δ1+tµ ,jµ , ejµ − θ(n > µ)et1+µ + u00 (jµ )) +q2− L
ν0 −θ(µ even) 4
(t) fs (L, e−1+jµ − θ(n > µ)et1+µ − E1,µ + u00 (jµ )), − θ(n > µ)et1+µ + u00 (jµ ))
fs (L, ejµ n (t) fs (L, e1+jµ + θ(n > µ)δt1+µ ,jµ etµ+1 − E1,µ+θ(n>µ) =⇒q + u00 (jµ )), , (8.35) o (t) fs (L, e1 + e1+jµ + δt1+µ ,jµ θ(n > µ)etµ+1 − E1,µ+θ(n>µ) + u00 (jµ )) + P0 (L, µ, ejµ − θ(n > µ)et1+µ + u00 (jµ )), P1 (L, µ, ejµ − θ(n > µ)et1+µ + u00 (jµ )) 2
g(µ,n,jµ )
where u00 (jµ ) =
0 0 uµ+1 − δjµ ,t1+µ (uµ+1 )1+t1+n e1+t1+µ pe ˜ 1+t1+n , p˜ ∈ Z
if µ < n if µ = n.
(8.36)
Lemma 2.2. For 1 + tµ ≤ jµ ≤ t1+µ − 1 − δµ,1 , 1 ≤ µ ≤ n − 1 we have 0 {P−1 (L, µ, et1+µ −(jµ −tµ ) − et1+µ + uµ+1 )+
q2− L
ν0 +1 3 4 + 4 θ(µ
odd)
(t) 0 fs (L, et1+µ −(jµ −tµ )+1 − E1,µ+1 + uµ+1 ),
0 fs (L, et1+µ −(jµ −tµ ) − et1+µ + uµ+1 )} 2
=⇒ q −
ν0 −θ(µ even) 4
(t) 0 {fs (L, et1+µ −(jµ −tµ )−1 − E1,µ+1 + uµ+1 ),
(t) 0 fs (L, e1 + et1+µ −(jµ −tµ )−1 − E1,µ+1 + uµ+1 )} 0 + {P0 (L, µ − δjµ ,t1+µ −1 , et1+µ −(jµ −tµ ) − et1+µ + uµ+1 ), 0 )}. P1 (L, µ − δjµ ,t1+µ −1 , et1+µ −(jµ −tµ ) − et1+µ + uµ+1
(8.37)
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A. Berkovich, B. M. McCoy, A. Schilling
These lemmas follow immediately from the recursive properties (5.3)-(5.7) of fs and the definition (3.35) of f˜s . Proof of Propositions 1 and 2. We prove Propositions 1 and 2 by induction on µ. For y −2 the proof of Proposition 1 we use the decomposition of −→aµ+1 as given in (8.24) and for the proof of Proposition 2 the decomposition (8.25). Since in the decompositions yµ −2 y −2 yµ−1 −2 and −→0 µ−1 but never −→±1 Propo(8.24) and (8.25) one uses flows −→±1,0 sitions 1 and 2 for µ + 1 and all a will follow if Propositions 1 and 2 are true for a = 0 and µ, µ − 1 and a = −1, 1 and µ. One may check that it is sufficient to prove the following initial conditions for Propositions 1 and 2: a) Propositions 1 and 2 for a = 0 and µ = 1, 2, b) Propositions 1 and 2 for a = −1 and µ = 2, c) Proposition 1 for a = −1 and µ = 3. With these initial conditions Propositions 1 and 2 with a = 0 follow for all µ ≥ 1, Propositions 1 and 2 with a = −1 follow for all µ ≥ 2 and Propositions 1 and 2 with a = 1 follow for all µ ≥ 3. Proof of the initial conditions. Here we prove point a) only. Points b) and c) are treated in Appendix B. We first consider Proposition 1 for µ = 1, 2 and a = 0. Taking into account (7.5) with j0 = 0, 1, ν0 − 1, ν0 and case 2 of (7.16) with j0 = −1 + t1 , j1 = −1 + t2 along with case 1 of (7.16) with j1 = t2 and recalling definition of P−1 (8.26) we see that (t) (t) (t) (t) − E1+i,n ), fs (L, e1 − E1,i − E1+i,n ) {fs (L, −E1,i yi −2 c(ti )
−→ q 0
(t) (t) {P−1 (L, i, −E1+i,n ), fs (L, −E1+i,n )}
(8.38)
(t) by ui0 in (8.38) we obtain Proposition 1 for with i = 1, 2. Now if we replace −E1+i,n µ = 1, 2. Note that the above replacement is legitimate because the recursive properties of Sects. 5 and 6 allow us to repeat the constructions of Sect. 7 for any vector ui0 . The proof of Proposition 2 for µ = 1, 2 and a = 0 is only slightly more involved. Making use of case 2 of (7.16) with j0 = 0, 1, ν0 −1, case 1 of (7.16) with j1 → j1 +1 on one hand and case 3 of (7.18) with j0 = 0, 1, 1 + t2 ≤ j2 ≤ t3 − 2, case 6 of (7.19) with j0 = −1 + t1 and case 1 of (7.18) with j2 → j2 + 1 on the other hand we obtain upon (i) < yn , recalling the definitions (8.26)-(8.28) and property (3.13) with b = l1+j i
n
o (t) (t) (t) (t) fs (L, −E1,i + e1+ji − E1+i,n ), fs (L, e1 − E1,i + e1+ji − E1+i,n ) n o (t) (t) + P0 (L, i, eji − E1+i,n ), P1 (L, i, eji − E1+i,n ) n o yi −2 c(ti )− ν0 +1 + 3 δi,1 (t) (t) 4 4 P−1 (L, i, e1+ji − E1+i,n −→ q ), fs (L, e1+ji − E1+i,n ) 0 o n 1 (t) ), 0 + q c(ti )− 2 δi,2 f˜s (L, eji − E1,n (8.39) (t) to derive from with i = 1, 2. Next we use Proposition 1 with µ = i, ui0 = e1+ji − E1+i,n (8.39), q−
ν0 +1 3 4 + 4 δ1,i
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361
o
n
(t) (t) ), P1 (L, i, eji − E1+i,n ) P0 (L, i, eji − E1+i,n o n 1 yi −2 (t) × −→ q c(ti )− 2 δi,2 f˜s (L, eji − E1,n ), 0 .
(8.40)
0
Thus, replacing eji −
(t) E1+i,n
by ui0 in (8.40) we obtain Proposition 2 with µ = 1, 2.
We conclude this subsection with the following comment. In deriving (8.40) from (8.39) we succeeded in separating the evolution of a {P0 , P1 } pair from that of a {fs , fs } (2) +1 ≤ pair. This separation can be made in the formulas (7.18) and (7.19) as long as l1+j 2 (2) b ≤ l2+j2 , 1 + t2 ≤ j2 ≤ t3 − 1. All descendents of the {fs , fs } pair will have e1+j2 in their arguments and all descendants of the pair {P0 , P1 } will have ej2 in their arguments instead. Using this identification principle we easily obtain for ν1 ≥ 3, ν0 ≥ 2, {P0 (L, 2, u20 ), P1 (L, 2, u20 )} y1 +1
−→ q − + q−
ν0 −1 2
(t) (t) {fs (L, e1 + e−2+t2 − E1,2 + u20 ), fs (L, e2 + e−2+t2 − E1,2 + u20 )}
ν0 −2 4
(t) (t) {f˜s (L, e−2+t1 + e−1+t2 − E1,2 + u20 ), f˜s (L, e−3+t1 + e−1+t2 − E1,2 + u20 )} (8.41) by comparing case 2 of (7.18) with j0 = 0, 1; j1 = 1 + t1 and case 3 of (7.18)with (t) by u20 . This result will be used in Appendix B to prove j0 = 0, 1 and replacing ej2 − E3,n Proposition 2 with a = −1, µ = 2.
Proof of Proposition 1 for µ − 1, µ → µ + 1. Let us now show that Proposition 1 with y −2 a = −1, 0, 1 follows inductively for µ + 1. We will decompose −→aµ+1 according to (8.24) which allows us to use Proposition 1 and 2 for µ and µ − 1 for which they are true by assumption. Let us start by applying proposition 1 with µ, a0 = −δa,1 and uµ0 = n o (t) (t) 0 0 0 uµ+1 − etµ+1 to fs (L, −E1,µ+1 + uµ+1 ), fs (L, e1 − E1,µ+1 + uµ+1 ) and subsequently using Lemma 2.1 with µ → µ − 1, jµ−1 = tµ to obtain n o (t) (t) 0 0 fs (L, −E1,µ+1 + uµ+1 ), fs (L, e1 − E1,µ+1 + uµ+1 ) yµ −2 c(t )+ 1 δ (−1)µ+1 0 0 −→ q µ 2 a,1 ), fs (L, −etµ+1 + uµ+1 ) P−1 (L, µ, −etµ+1 + uµ+1 −δa,1 ν0 −θ(µ even) n µ+1 1 2 (8.42) (t) 0 4 q− fs (L, −E1,µ−1 =⇒ q c(tµ )+ 2 δa,1 (−1) + e1+tµ − etµ+1 + uµ+1 ), o (t) 0 fs (L, e1 − E1,µ−1 + e1+tµ − etµ+1 + uµ+1 ) 0 0 + P0 (L, µ − 1, −etµ+1 + uµ+1 ), P1 (L, µ − 1, −etµ+1 + uµ+1 ) . where we have noticed from (8.26) that P−1 (L, µ, uµ0 ) = P−1 (L, µ − 1, etµ − etµ + uµ0 ) +q2− L
ν0 −θ(µ−1 even) 4
(t) fs (L, e−1+tµ − etµ − E1,µ−1 + uµ0 ).
(8.43)
We point out that the appearance of µ−1 instead of µ in the right-hand side of (8.42) after yµ steps explains the use of decomposition (8.24) instead of (8.25) for the proof of Proposition 1. We now apply Proposition 1 with µ → µ−1, a0 = 0 to the first pair and Proposition 2 with µ → µ − 1 a0 = 0 to the second pair in the right-hand side of (8.42) which yields
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o (t) (t) 0 0 + uµ+1 ), fs (L, e1 − E1,µ+1 + uµ+1 ) fs (L, −E1,µ+1 yµ−1 +yµ −2 c(1+t )+ 1 δ (−1)µ+1 0 µ 2 a,1 −→ q ) P−1 (L, µ − 1, e1+tµ − etµ+1 + uµ+1 n
(8.24)
+q 2 − L
ν0 −θ(µ even) 4
(8.44) o
(t) 0 0 fs (L, −E1,µ−1 − etµ+1 + uµ+1 ), fs (L, e1+tµ − etµ+1 + uµ+1 ) .
even) + c(tµ ) + c(tµ−1 ) which follows from (8.33). Here we have used c(1 + tµ ) = − ν0 −θ(µ 4 The symbol (8.24) under the arrow in (8.44) means that we have evolved the initial state according to the first yµ−1 + yµ − 2 steps of the decomposition given by (8.24). Next applying Lemma 2.1 with µ and jµ = 1 + tµ we obtain o n (t) (t) 0 0 + uµ+1 ), fs (L, e1 − E1,µ+1 + uµ+1 ) fs (L, −E1,µ+1 ν0 +1 3 n yµ−1 +yµ c(1+t )+ 1 δ (−1)µ+1 (t) 0 µ 2 a,1 q − 4 + 4 θ(µ odd) fs (L, −E1,µ −→ q + e2+tµ − etµ+1 + uµ+1 ), (8.24) o (t) 0 fs (L, e1 − E1,µ + e2+tµ − etµ+1 + uµ+1 ) 0 0 + P0 (L, µ, e1+tµ − etµ+1 + uµ+1 ), P1 (L, µ, e1+tµ − etµ+1 + uµ+1 ) . (8.45) Now we may use Propositions 1 and 2 again with µ and a0 = δa,−1 which yields o n (t) (t) 0 0 + uµ+1 ), fs (L, e1 − E1,µ+1 + uµ+1 ) fs (L, −E1,µ+1 yµ−1 +2yµ −2 c(2+t )+ 1 a(−1)µ+1 0 µ 2 −→ q ) P−1 (L, µ, e2+tµ − etµ+1 + uµ+1 (8.24) o ν0 −θ(µ even) L (t) 0 0 4 +q 2 − fs (L, −E1,µ + e1+tµ − etµ+1 + uµ+1 ), fs (L, e2+tµ − etµ+1 + uµ+1 ) , (8.46) where we used c(2 + tµ ) = c(1 + tµ ) − ν04+1 + 43 θ(µ odd) + c(tµ ) which again follows from µ+1 1 0 ) (8.33). Notice that the final entry of (8.44) q c(1+tµ )+ 2 (−1) δa,1 fs (L, e1+tµ − etµ+1 + uµ+1 0 and the final entry of (8.46) q c(2+tµ )+ 2 a(−1) fs (L, e2+tµ − etµ+1 + uµ+1 ) only differ in that e1+tµ has become e2+tµ and the phase factor has changed. Applying now repeatedly y −2 =⇒2 and −→0 µ according to the decomposition (8.24) and using Lemma 2.1 and Propositions 1 and 2 for µ we obtain o n (t) (t) 0 0 + uµ+1 ), fs (L, e1 − E1,µ+1 + uµ+1 ) fs (L, −E1,µ+1 yµ−1 +(jµ −tµ )yµ −2 c(j )+ 1 a(−1)µ+1 0 −→ q µ 2 ) P−1 (L, µ, ejµ − etµ+1 + uµ+1 (8.24) o ν0 −θ(µ even) L (t) 0 0 2 +q 2 − fs (L, −E1,µ + e−1+jµ − etµ+1 + uµ+1 ), fs (L, ejµ − etµ+1 + uµ+1 ) . (8.47) The arrow in (8.47) denotes the flow after the first yµ−1 + (jµ − tµ )yµ − 2 steps (tµ + 1 < jµ ≤ tµ+1 ) according to the decomposition (8.24). Finally setting jµ = tµ+1 in (8.47) and using the easily verifiable identity 1
µ+1
L − ν0 + θ(µ odd) L ν0 − θ(µ even) + c(tµ ) − c(t1+µ ) = − (8.48) 2 2 4 along with (8.43) with µ → µ + 1 we obtain Proposition 1 for µ + 1. This concludes the proof of Proposition 1. c(−1 + t1+µ ) +
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363
Proof of Proposition 2 for µ − 1, µ → µ + 1. Let us show that Proposition 2 holds for µ + 1 and a = −1, 0, 1 inductively. First we assume νµ > 2. Recalling the definition of uµ0 we see from (8.27) and (8.28) that 0 0 ), P1 (L, µ + 1, uµ+1 ) P0 (L, µ + 1, uµ+1 n ν0 −θ(µ even) (t) 0 4 fs (L, −E1,µ+1 + e−1+tµ+1 + uµ+1 ), = q− o (t) 0 0 0 fs (L, e1 − E1,µ+1 + e−1+tµ+1 + uµ+1 ) + P0 (L, µ, uµ+1 ), P1 (L, µ, uµ+1 ) .
(8.49)
We evolve these polynomials according to the decomposition (8.25). Thus we first evolve the first pair on the rhs of (8.49) using Proposition 1 and the second pair using Proposi0 followed by Lemma 2.2 with tion 2 for µ, a0 = −δa,1 and uµ0 = e−1+tµ+1 − etµ+1 + uµ+1 jµ = 1 + tµ to obtain 0 0 ), P1 (L, µ + 1, uµ+1 ) P0 (L, µ + 1, uµ+1 yµ −2 c(t )− ν0 −θ(µ even) + 1 δ (−1)µ+1 0 4 2 a,1 −→ q µ ) P−1 (L, µ, e−1+tµ+1 − etµ+1 + uµ+1
−δa,1
+q 2 − L
ν0 +1 3 4 + 4 θ(µ
ν0 −θ(µ even)
2
odd)
o
(t) 0 0 fs (L, −E1,µ + uµ+1 ), fs (L, e−1+tµ+1 − etµ+1 + uµ+1 ) µ+1
1
+ 2 δa,1 (−1) 4 =⇒q c(tµ )− ν0 −θ(µ even) n (t) 0 4 × q− fs (L, −E1,µ + e−2+tµ+1 − etµ+1 + uµ+1 ),
o
(t) 0 fs (L, e1 − E1,µ + e−2+tµ+1 − etµ+1 + uµ+1 )
0 0 + P0 (L, µ, e−1+tµ+1 − etµ+1 + uµ+1 ), P1 (L, µ, e−1+tµ+1 − etµ+1 + uµ+1 ) . (8.50) In the next step we apply Proposition 1 and Proposition 2 with µ and a0 = −δa,0 − δa,1 which yields 0 0 ), P1 (L, µ + 1, uµ+1 ) P0 (L, µ + 1, uµ+1 µ+1 1 2yµ −2 c(2+t )−c(t 0 µ µ−1 )+ 2 a(−1) −→ q ) P−1 (L, µ, e−2+tµ+1 − etµ+1 + uµ+1
(8.25)
+q
ν0 +1 3 L 2 − 4 + 4 θ(µ
odd)
(t) fs (L, −E1,µ
fs (L, e−2+tµ+1
+ e−1+tµ+1 − etµ+1 + 0 − etµ+1 + uµ+1 ) ,
(8.51)
0 uµ+1 ),
2y −2
µ where −→(8.25) denotes the flow according to the first 2yµ −2 steps of the decomposition
even) a,1 (8.25) and we used the identity 2c(tµ ) − ν0 −θ(µ + (−1)µ+1 = c(2 + tµ ) − 2 2 1 µ+1 c(tµ−1 ) + 2 a(−1) which follows from (8.33). Applying now further jµ − tµ − 2 (tµ + 3 ≤ jµ ≤ tµ+1 − 1) times the combination yµ −2 =⇒2 −→−1 using Lemma 2.2 and Proposition 1 and 2 with µ and a0 = −1 we obtain δa,0+2δ
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0 0 ), P1 (L, µ + 1, uµ+1 ) P0 (L, µ + 1, uµ+1 µ+1 1 (jµ −tµ )yµ −2 c(j )−c(t 0 µ−1 )+ 2 a(−1) −→ q µ ) P−1 (L, µ, etµ+1 −(jµ −tµ ) − etµ+1 + uµ+1
(8.25)
+q2−
ν0 +1 3 4 + 4 θ(µ
(t) 0 fs (L, −E1,µ + etµ+1 −(jµ −tµ )+1 − etµ+1 + uµ+1 ), 0 fs (L, etµ+1 −(jµ −tµ ) − etµ+1 + uµ+1 ) . (8.52) Setting jµ = tµ+1 − 1 in the last formula gives 0 0 P0 (L, µ + 1, uµ+1 ), P1 (L, µ + 1, uµ+1 ) (νµ −1)yµ −2 v 0 −→ q P−1 (L, µ, e1+tµ − etµ+1 + uµ+1 ) (8.25) (8.53) ν0 +1 3 L (t) 0 + q 2 − 4 + 4 θ(µ odd) fs (L, −E1,µ + e2+tµ − etµ+1 + uµ+1 ), 0 ) , fs (L, e1+tµ − etµ+1 + uµ+1 L
odd)
where v = c(−1 + tµ+1 ) − c(tµ−1 ) + 21 a(−1)µ+1 . Doing the next step we enter zone µ − 1. Using Lemma 2.2 with jµ = tµ+1 − 1 we get 0 0 ), P1 (L, µ + 1, uµ+1 ) P0 (L, µ + 1, uµ+1 n (νµ −1)yµ v− ν0 −θ(µ even) (t) 0 4 −→ q − etµ+1 + uµ+1 ), fs (L, −E1,µ−1 (8.25) o (t) 0 fs (L, e1 − E1,µ−1 − etµ+1 + uµ+1 ) 0 0 + q v P0 (L, µ − 1, e1+tµ − etµ+1 + uµ+1 ), P1 (L, µ − 1, e1+tµ − etµ+1 + uµ+1 ) . (8.54) The appearance of µ − 1 instead of µ on the right-hand side of (8.54) which occurred after (νµ − 1)yµ steps explains the choice of decomposition (8.25) instead of (8.24) in the proof of Proposition 2. Now using Proposition 1 and Proposition 2 with µ → µ − 1 and a0 = 0 we find 0 0 P0 (L, µ + 1, uµ+1 ), P1 (L, µ + 1, uµ+1 ) ν0 −θ(µ even) (νµ −1)yµ +yµ−1 −2 v+c(t 0 µ−1 )− 4 −→ q ) P−1 (L, µ − 1, −etµ+1 + uµ+1 (8.25) o ν0 −θ(µ even) L (t) 0 0 4 +q 2 − fs (L, −E1,µ−1 + e1+tµ − etµ+1 + uµ+1 ), fs (L, −etµ+1 + uµ+1 ) . (8.55) To complete the proof, we evolve (8.55) according to =⇒2 and use the fermionic recursive property (5.6), the first equation of (5.3) and definition (3.35) to obtain 0 0 ), P1 (L, µ + 1, uµ+1 ) P0 (L, µ + 1, uµ+1 (νµ −1)yµ +yµ−1 v− ν0 −θ(µ even) +c(t 0 0 µ−1 ) 4 −→ q ), P1 (L, µ, −etµ+1 + uµ+1 ) , P0 (L, µ, −etµ+1 + uµ+1 (8.25)
and then apply Proposition 2 with µ and a0 = 0 to arrive at 0 0 P0 (L, µ + 1, uµ+1 ), P1 (L, µ + 1, uµ+1 ) n o ν0 −θ(µ even) L−ν0 +θ(µ odd) yµ+1 −2 (t) +c(tµ−1 )+ +c(tµ ) 0 4 2 −→ q v− fs (L, −E1,µ+1 + uµ+1 ), 0 , (8.25)
(8.56)
(8.57)
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365
where we used (νµ − 1)yµ + yµ−1 + yµ − 2 = yµ+1 − 2. Finally from (8.33) we derive ν0 − θ(µ even) L − ν0 + θ(µ odd) + c(tµ−1 ) + + c(tµ ) 4 2 L − ν0 + θ(µ + 1 odd) 1 = + c(tµ+1 ) + a(−1)µ+1 . 2 2 v−
(8.58)
Combining (8.57) and (8.58) we derive Proposition 2 for µ + 1. This concludes the proof of proposition 2 for νµ > 2. When νµ = 2 a separate treatment is needed because the decomposition (8.25) does not hold. However, in this case the decomposition (8.24) with νµ = 2 can be used and with this the proof of Proposition 2 for νµ = 2 follows. Recalling lemma 2.1, the equations (8.44), (8.47) and the definition of Takahashi length (2.8) we organize the results proven in this section as Theorem 1. For 1 ≤ µ ≤ n, 1 + δµ,1 + tµ ≤ jµ ≤ tµ+1 + δµ,n , we have n o (t) (t) fs (L, −E1,n + P e1+t1+n ), fs (L, e1 − E1,n + P e1+t1+n ) n µ (t) −→ q c(jµ ) P−1 (L, µ − δ1+tµ ,jµ , ejµ − Eµ+1,n + P e1+t1+n )
(µ) l1+j −2
o (t) (t) fs (L, e−1+jµ − E1,n + P e1+t1+n ), fs (L, ejµ − Eµ+1,n + P e1+t1+n ) n 2 (t) =⇒ q c(jµ ) q g(µ,n,jµ ) fs (L, e1+jµ + δt1+µ ,jµ θ(n > µ)etµ+1 − E1,n + P e1+t1+n ), o (t) fs (L, e1 + e1+jµ + δt1+µ ,jµ θ(n > µ)etµ+1 − E1,n + P e1+t1+n ) n o (t) (t) P0 (L, µ, ejµ − Eµ+1,n + P e1+t1+n ), P1 (L, µ, ejµ − Eµ+1,n + P e1+t1+n ) , (8.59) (t) = 0, P = 0, 1 and g(µ, n, jµ ) is defined just above (8.35). where En+1,n +q 2 − L
ν0 −θ(µ even) 4
Before we move on, we observe that the r(b) graph of Sect. 3 for yµ −y1 −1 ≤ b ≤ yµ is (up to a shift) for µ ≥ 2 (8.60) yµ-y1-1 yµ-y1
yµ
Using the fermionic recursive properties of Sect. 5 for fs and (6.1) for f˜s as well as definition (3.35) one can easily show that o y n o n 1 (t) (t) (t) + uµ0 ), fs (L, −E2,µ + uµ0 ) −→ f˜s (L, −E1,µ + uµ0 ), 0 (8.61) fs (L, e1+t1 − E2,µ (8.60)
Notice that if we use (8.62) yµ-y1-1
instead, we obtain
yµ-1
yµ
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A. Berkovich, B. M. McCoy, A. Schilling
o
n
(t) (t) + uµ0 ), fs (L, −E2,µ + uµ0 ) fs (L, e1+t1 − E2,µ o n y1 −1 (t) (t) −→ fs (L, e1 − E1,µ + uµ0 ), fs (L, −E1,µ + uµ0 ) (8.62) n o 1 (t) −→ fs (L, −E1,µ + uµ0 ), 0 .
(8.63)
(8.2) b
Comparing (8.61) with Proposition 2 with a = 0 we infer that for µ ≥ 2, P0 (L, µ, uµ0 ), P1 (L, µ, uµ0 ) n o yµ −y1 −2 c(t )− 1 θ(µ even) (t) (t) fs (L, e1+t1 − E2,µ −→ q µ 2 + uµ0 ), fs (L, −E2,µ + uµ0 ) .
(8.64)
We are now ready to discuss the evolution along the final stretch of the r(b) map
r
p-1
(8.65)
1 1
p’-2y1+1 p’-1-y1
p’-1
b
Last stretch of the b→ r map (n) ≤ b ≤ p0 − y1 − 1 is the same Notice that the piece of the b → r map with 1 + l2+t n+1 (up to a shift) as the map b → r of Sect. 3 with 1 ≤ b ≤ yn − y1 − 1. The piece of the b → r map restricted to the interval [p0 − y1 − 1, p0 − 1] is graph (8.62) with µ = n and the last segment removed. Recalling Theorem 1 (with µ = n and jn = 1 + tn+1 ) we have o n (t) (t) + P e1+t1+n ), fs (L, e1 − E1,n + P e1+t1+n ) fs (L, −E1,n (8.66) (n) 0 l2+t
=p −yn
−→
n+1
q c(1+tn+1 ) {P0 (L, n, (1 − P )e1+tn+1 ), P1 (L, n, (1 − P )e1+tn+1 )} .
Applying (8.64) and (8.63) with µ = n, un0 = (1 − P )e1+tn+1 to the right-hand side of (8.66), we arrive at Theorem 2. o n (t) (t) + P e1+t1+n ), fs (L, e1 − E1,n + P e1+t1+n ) fs (L, −E1,n p0 −3 c(tn+1 +1)+c(tn )− 1 θ(n even) 2
−→q
n
o
(8.67)
(t) (t) × fs (L, e1 − E1,n + (1 − P )e1+t1+n ), fs (L, −E1,n + (1 − P )e1+tn+1 ) .
If we now identify the fermionic polynomials generated in the evolution (8.67) with (o) (L, b) (for P = 1), then we have Fr(b),s (L, b) (for P = 0) and with Fr(b),s
Rogers–Schur–Ramanujan Identities for Minimal Models of CFT
367
(t) F1,s (L, 1) = fs (L, −E1,n ), (t) F1,s (L, 2) = fs (L, e1 − E1,n ),
···
(8.68) 0
and
Fp−1,s (L, p − 2) = q
c(1+t1+n )+c(tn )− 21 θ(n even)
fs (L, e1 −
Fp−1,s (L, p0 − 1) = q
c(1+t1+n )+c(tn )− 21 θ(n
(t) fs (L, −E1,n + e1+t1+n )
even)
(t) E1,n
+ e1+t1+n ),
(o) (t) (L, 1) = fs (L, −E1,n + e1+t1+n ), F1,s (o) (t) F1,s (L, 2) = fs (L, e1 − E1,n + e1+t1+n ),
···
(8.69)
(o) Fp−1,s (L, p0
− 2) = q
(o) Fp−1,s (L, p0 − 1) = q
c(1+t1+n )+c(tn )− 21 θ(n even)
fs (L, e1 −
c(1+t1+n )+c(tn )− 21 θ(n
(t) fs (L, −E1,n ).
even)
(t) E1,n ),
Finally we use the first equation in (5.3) to verify that Fp−1,s (L, p0 −2), Fp−1,s (L, p0 − (o) (o) (L, p0 − 2), Fp−1,s (L, p0 − 1)) satisfy the closing equation in (4.6). Thus we 1) (Fp−1,s have shown that the constructive procedure defined in Sects. 7 and 8 with the initial values for b = 1 and b = 2 specified by the first two equations in (8.68) or in (8.69) gives rise to fermionic polynomials which satisfy all bosonic recursion relations (4.4)–(4.6) for 1 ≤ b ≤ p0 − 1. 9. Normalization Constants and Boundary Conditions (µs ) From the conclusion of the previous section it follows that when s = l1+j , 1 + tµs ≤ s js ≤ t1+µs , L + b + s ≡ 0(mod2) the identity 0 pX −1
Fr(b),s (L, b) =
ks,s0 B˜ r(b),s0 (L, b)
(9.1)
s0 =1 s0 ≡s(mod2)
will hold for L > 0, provided constants ks,s0 can be chosen so that (9.1) holds for L = 0. Using (4.1), (4.7) it is trivial to verify that for ks,s0 = Fr(s0 ),s (0, s0 )
(9.2)
with s0 ≡ s(mod2), 1 ≤ s0 ≤ p0 − 1 Eq. (9.1) indeed holds for L = 0. However, (9.2) is of very little use because Fr(b),s (L, b) have not been explicitly constructed for all b ∈ [1, p0 − 1]. Fortunately, it turns out that the constants ks,s0 can be determined from (9.1) with L = 0, 1, · · · , p0 − 1 and b = 1, s, p0 − 1. In this direction we first calculate the threshold values of L, i.e. the lowest values of L such that Fr(b),s (L, b)6=0 for b = 1, s, p0 − 1. We know that Fr(b),s (L, b) with b = 1, p0 − 1, s is given by X
Fr(b),s (L, b) = q N
n∈Zt1+n mt
1+n
≡P ( mod 2)
T
q Q(n,m)+A
˜ m
tY n+1 j=1
nj + mj nj
(0) .
(9.3)
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A. Berkovich, B. M. McCoy, A. Schilling
˜ is defined in (2.17), Here m ( 0 N = c(js ) c(tn+1 + 1) + c(tn ) − 21 θ(n even)
and P =
0 1
if b = 1 if b = s if b = p0 − 1
(9.4)
if b = 1, s if b = p0 − 1.
(9.5)
n and m are related by (2.11) with 0 ¯ (t) ¯ (t) + e¯ js − E u¯ = −E 1,n 1+µs ,n for b = 1, p − 1,
¯ (t) u¯ = 2¯ejs − 2E 1+µs ,n for b = s where ¯ (t) = E a,b
b X
,
(9.6)
e¯ ti .
(9.7)
i=a
Even though n, m ∈ Zt1+n , a bit of analysis shows that effectively For b = 1, s : ni ≥ δi,js − δi,tj , tj ≥ js mi ≥ 0
(9.8) 0
For b = p − 1 : ni ≥ 0, mi > 0. Moreover, if ni takes on negative values then mi = 0. Therefore, the threshold configurations for the three cases are ¯ (t) n = e¯ js − θ(µs < n)E 1+µs ,n , mtn+1 ≡ 0 (mod 2) for b = 1,
n = 0, mtn+1 ≡ 1 (mod 2) for b = p0 − 1, ¯ (t) n = e¯ js − θ(µs < n)E 1+µs ,n , mtn+1 ≡ 0 (mod 2) for b = s,
(9.9)
with the corresponding thresholds from (2.16) Ltr = Ltr =
µs X i=1 n X i=1 0
(yi − yi−1 ) + ljs = s − 1 for b = 1, (yi − yi−1 ) +
n X
(yi − yi−1 ) − ljs + l1+tn+1
(9.10)
i=µs +1 0
=p − 1 − s for b = p − 1, Ltr =0 for b = s, with lj defined by (2.7) and µs by (3.15). Finally the threshold for Br(b),s (L, b) is Ltr = |s − b|.
(9.11)
Hence for b = 1, L = 0, 1, . . . , s − 2 and L + 1 + s ≡ 0( mod 2) we have from (9.1)
Rogers–Schur–Ramanujan Identities for Minimal Models of CFT 0 pX −1
0=
ks,s0 B˜ 1,s0 (L, 1).
369
(9.12)
s0 =1 s0 ≡s( mod 2)
Clearly, (9.12) and (9.11) imply that ks,s0 = 0 for 1 ≤ s0 ≤ s − 1, s0 ≡ s( mod 2).
(9.13)
Analogously, if we evaluate (9.1) at b = p0 − 1, L = 0, 1, 2, . . . , p0 − 2 − s and L + p0 − 1 + s ≡ 0( mod 2) we obtain
0=
0 pX −1
ks,s0 B˜ p−1,s0 (L, p0 − 1).
(9.14)
s0 =1 s0 ≡s( mod 2)
Equations (9.11) and (9.14) imply that ks,s0 = 0 for 1 + s ≤ s0 , s0 ≡ s( mod 2).
(9.15)
Combining (9.13) and (9.15) yields ks,s0 = δs0 ,s q a(js ) .
(9.16)
To determine a(js ) we evaluate (9.1) at b = s, L = 0 to get Fr(s),s (L = 0, b = s) = q a(js ) B˜ r(s),s (L = 0, b = s),
(9.17)
or since Fr(s),s (L = 0, b = s) = q c(js ) and B˜ r(s),s (L = 0, b = s) = 1 we have a(js ) = c(js ).
(9.18)
Fr(b),s (L, b) = q c(js ) B˜ r(b),s (L, b),
(9.19)
This finally leads to
where L + b + s ≡ 0( mod 2). Analogously, we can show that Fr(b),s (L, b) = 0 for L + b + s ≡ 1( mod 2).
(9.20)
This concludes the proof of the boundary conditions which leads to the equality of the fermionic and the bosonic form of M (p, p0 ) polynomial character identities.
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A. Berkovich, B. M. McCoy, A. Schilling
10. Rogers–Schur–Ramanujan Type Identities for M (p, p0 ) Minimal Models We now may give the Rogers–Schur–Ramanujan type identities for the the four cases of (3.41). From (9.19) we see that all identities are of the form Fr(b),s (L, b) = q 2 (φr(b),s −φr(s),s )+c(js ) Br(b),s (L, b). 1
(10.1)
We thus state the results by giving the appropriate functions Fr(b),s (L, b). (µ) 10.1. Rogers–Schur–Ramanujan identities for b = l1+j a pure Takahashi lengths. µ (µ) a pure Takahashi length follow The Rogers–Schur–Ramanujan identities for b = l1+j µ immediately from Theorem 1 of Sect. 8 and the proof of the boundary conditions of Sect. 9. Since (t) ), F1,s (L, 1) = fs (L, −E1,n (10.2) (t) F1,s (L, 2) = fs (L, e1 − E1,n ),
we infer from (8.59) that (µ) (t) ) = q c(jµ ) fs (L, ejµ − Eµ+1,n ), Fr(b),s (L, l1+j µ
(10.3)
(µ) where c(jµ ) was defined in (8.33) and r(b) = δµ,0 + l˜1+j . Thus we have explicitly proven µ the result announced in [26]. We would like to stress that (10.3) holds even in the cases where some or all νi = 1 or νn = 0 (or both) with appropriate modification of the exponent c(j µ ).
10.2. The vicinity of the Takahashi length. Using the fermionic recursive properties of Sect. 5 and 6 and the Theorem 1 (of Sect. 8), it is easy to extend the Rogers–Schur– Ramanujan identities of the previous subsection to case 2 of (3.41). (µ) (µ) − ν0 + 1 ≤ b ≤ l1+j − 1, µ ≥ 1. Case 2a: l1+j µ µ ν0 −1−θ(µ odd) (µ) (t) c(jµ ) 4 f˜s (L, ej0 −1 − E1,n q − j ) = q + e−1+jµ ) Fr(b),s (L, l1+j 0 µ + θ(µ ≥ 2)
µ X
q
ν0 −1−θ(i even) 4
(t) (t) f˜s (L, ej0 −1 − E1,i + e−1+ti + ejµ − Eµ+1,n )
i=2
(t) ) +fs (L, eν0 −j0 − et1 + ejµ − Eµ+1,n
(10.4)
1 ≤ j0 ≤ ν0 − 1.
(µ) (µ) + 1 ≤ b ≤ l1+j + ν0 + 1, µ ≥ 1. Case 2b: l1+j µ µ
ν0 +1 3 (µ) (t) c(jµ ) + 1 + j ) = q + e1+jµ ) q − 4 + 4 θ(µ odd) fs (L, ej0 − E1,n Fr(b),s (L, l1+j 0 µ + θ(µ ≥ 2)
µ X
q−
ν0 −θ(i odd) 4
(t) (t) fs (L, ej0 − E1,i + e−1+ti + ejµ − Eµ+1,n )
(10.5)
i=2
(t) ) , 0 ≤ j0 ≤ ν 0 , +f˜s (L, eν0 −j0 −1 − et1 + ejµ − Eµ+1,n where recalling (3.36) we note that case 2a agrees with (10.3) if j0 = 0. r in this section is a function of b as given in (3.41).
Rogers–Schur–Ramanujan Identities for Minimal Models of CFT
371
10.3. Further cases. The Propositions 1 and 2 of Sect. 8 are very powerful and can be used to to study all cases 1 ≤ b ≤ p0 − 1. As an illustration we present the results for the fermionic forms of cases 3 and 4 of (3.41). The details of the derivation are in Appendix C. To state the results we need several definitions (with α and β defined by the Takahashi decomposition of case 4 in (3.41)). Definitions.
j = |jα , jα+1 , . . . , jβ >, i = |iα , iα+1 , . . . , iβ−1 , 0 >,
(10.6)
with ik = 0, 1 for α ≤ k ≤ β − 1. When iα = 0 we define ai and bj from i = | 0, · · · , 0, 1, · · · , 1, 0, · · · , 0, · · · , 1, · · · , 1, 0, · · · , 0 >, | {z } | {z } | {z } | {z } | {z } a1
a2
b2
(10.7)
al
bl
where 1 ≤ ai , 1 ≤ i ≤ l and 1 ≤ bj , 2 ≤ j ≤ l. If l = 1, then i = | 0, · · · , 0 >. Note | {z } a1 Pl that α − 1 + a1 + j=2 (aj + bj ) = β. Similarly when iα = 1 we write i = | 1, · · · , 1, 0, · · · , 0, 1, · · · , 1, 0, · · · , 0, · · · , 1, · · · , 1, 0, · · · , 0 >, | {z } | {z } | {z } | {z } | {z } | {z } a1
b1
b2
a2
bl
(10.8)
al
where 1 ≤ aj , bj with 1 ≤ j ≤ l. From i we further define if iµ = 0 jµ + 1 Riµ (jµ + 1) = tµ+1 − (jµ − tµ ) − 1 for iµ = 1.
(10.9)
Results. With these definitions we may have the following results for the fermionic polynomials of cases 3 and 4 of (3.41). Case 3 of (3.41). Here α = 0 and we have X q rf(3,1) (i) fs (L, ej0 +i1 − et1 + u(3) (i, j)) Fr(b),s (L, b) = q c(3) (j) i1 ,...,iβ−1 =0,1 i0 =0
X
+
(10.10)
q rf(3,2) (i) f˜s (L, eν0 −j0 −i1 −1 − et1 + u(3) (i, j)) ,
i1 ,...,iβ−1 =0,1 i0 =1
where c(3) (j) =
β X µ=1
u(3) (i, j) =
β−1 X µ=1
for i0 = 0 we set
(c(jµ ) −
ν0 + 1 3 + θ(µ odd)), 4 4
(t) eRiµ (jµ +1)+|iµ+1 −iµ |−|iµ −iµ−1 | + e1+jβ −i−1+β − E2,n ,
(10.11)
(10.12)
372
A. Berkovich, B. M. McCoy, A. Schilling l Pj−1 1 1X rf(3,1) (i) = δa1 ,1 + (−1)a1 + k=2 (ak +bk ) θ(bj even), 4 2
(10.13)
j=2
and for i0 = 1 Pj−1 δb ,1 1 X ν0 + (−1)b1 − 1 + (−1) k=2 (ak +bk ) θ(bj even), 4 4 2 l
rf(3,2) (i) =
(10.14)
j=2
where we have used the convention that
Pb i=a
= 0 if b < a.
Case 4 of (3.41). Here α ≥ 1 and we have X q rf(4,1) (i) fs (L, u(4,1) (i, j)) Fr(b),s (L, b) = q c(4) (j) iα+1 ,...,iβ−1 =0,1 iα =0
X
+
q rf(4,2) (i)+δ1,α ( 2 − L
ν0 4
)
fs (L, u(4,2) (i, j))
iα+1 ,...,iβ−1 =0,1 iα =1
+q
rf(4,2) (i)−
where c(4) (j) = c(jα ) +
β X
(−1)α +δ1,α 4
fs (L, u(4,3) (i, j))
(c(jµ ) −
µ=α+1
,
ν0 + 1 3 + θ(µ odd)), 4 4
(10.16)
u(4,1) (i, j) = ejα +iα+1 + u(4) (i, j), u(4,2) (i, j) = −etα + etα+1 −(jα −tα )−1−iα+1 + u(4) (i, j), u(4,3) (i, j) = etα −1 − etα + etα+1 −(jα −tα )−iα+1 + u(4) (i, j), u(4) (i, j) =
β−1 X
(10.15)
(10.17)
(t) eRiµ (jµ +1)+|iµ+1 −iµ |−|iµ −iµ−1 | + ejβ +1−iβ−1 − Eα+1,n
µ=α+1
for iα = 0 rf(4,1) (i) =
l Pj−1 X 1 (−1)α (−1)a1 + k=2 (ak +bk ) θ(bj even) 2
(10.18)
j=2
and for iα = 1
rf(4,2) (i) = (−1)α
b1
(−1) 4
+
l 1X
2
Pj−1 (−1)
k=1
(ak +bk )
θ(bj even) .
(10.19)
j=2
10.4. Character identities. It remains to take the limit L → ∞ to produce character identities from the polynomial identities. These character identities are somewhat simpler because f˜s (L, u˜ ) vanishes as L → ∞. We remark that in this limit the explicit dependence on b vanishes and only the dependence on r remains. All polynomial identities which have different values of b but the same value of r give the same character identity. Thus we obtain the following fermionic forms for the bosonic forms of the
Rogers–Schur–Ramanujan Identities for Minimal Models of CFT
373
characters q 2 (φr(j),s −φr(s),s )+c(js ) Br(j),s (q). In all formulas of this section r is a function of j as given in (3.41). 1
Cases 1 and 2a. (t) ). Fr(j),s (q) = q c(jµ ) fs (ejµ − Eµ+1,n
(10.20)
Case 2b. ν0 +1 3 (t) + e1+jµ ) Fr(j),s (q) = q c(jµ ) q − 4 + 4 θ(µ odd) fs (−E2,n +θ(µ ≥ 2)
µ X
q
ν −θ(i odd) − 0 4
!
(t) (t) fs (−E2,i + e−1+ti + ejµ − Eµ+1,n ) .
(10.21)
i=2
Case 3. X
Fr(j),s (q) = q c(3) (j)
q rf(3,1) (i) fs (u(3) (i, j)).
(10.22)
i1 ,...,iβ−1 =0,1 i0 =0
Case 4 with α = 1. X
Fr(j),s (q) = q c(4) (j)
q rf(4,1) (i) fs (u(4,1) (i, j))
i2 ,...,iβ−1 =0,1 i1 =0
+
X
(10.23)
q rf(4,2) (i) fs (u(4,3) (i, j)) .
i2 ,...,iβ−1 =0,1 i1 =1
Case 4 with α ≥ 2. X
Fr(j),s (q) = q c(4) (j)
q rf(4,1) (i) fs (u(4,1) (i, j))
iα+1 ,...,iβ−1 =0,1 iα =0
+
X
(10.24)
q rf(4,2) (i) fs (u(4,2) (i, j))
iα+1 ,...,iβ−1 =0,1 iα =1
+q rf(4,2) (i)−
(−1)α 4
fs (u(4,3) (i, j))
.
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A. Berkovich, B. M. McCoy, A. Schilling
11. Reversed Parity Identities For all the identities presented thus far we have started with the term with b = 1, where the fermionic polynomial (3.22) has mt1+n restricted to even values. Indeed mt1+n has (n) = p0 − 2yn . even parity for all the results presented in this paper as long as b ≤ l1+t 1+n (e) These fermionic sums are what were called even sums Fr(b),s (L, b) in [26]. According to Theorems 1 and 2 of Sect. 8 it is equally possible to start with the fermionic polynomial which appears in the first equation in (8.69) with mt1+n odd and the entire construction of this paper may be carried out exactly as before. The only difference is that the parity of mt1+n will be reversed in all the formulas presented earlier, which amounts to the replacement u → u + e1+tn+1 . The step which must be different is the analysis of the boundary conditions and the computation of the normalization constant. This is done (o) (L, b) (8.69) we below and calling the reversed parity fermionic polynomials Fr(b),s find from each of the identities proven above the corresponding reversed parity identity (o) ¯ (L, b) = q a(s,r) Br(b),s¯ (L, b), Fr(b),s
(11.1)
(µs ) where s¯ = p0 − s with s = l1+j and a(s, ¯ r) is given by the two equivalent expressions s
1 a(s, ¯ r) = c(js ) + c(1 + tn+1 ) + c(tn ) − θ(n even) 2 1 + (φr,s¯ − φr(s),s + φ1,s − φp−1,s¯ ) 2 1 1 = c(js ) + (φr,s¯ − φr(s),s ) + (φp−1,s − φp−1,s¯ + φ1,s − φ1,s¯ ). 2 4
(11.2)
(µ) (11.1) gives the result announced in [26]. We note that when b = l1+j µ
Computation of the normalization constant. From Sect. 9 it suffices to determine the constant a(s) ¯ in the identity (o) ¯ ˜ Br(b),s¯ (L, b), (L, b) = q a(s) Fr(b),s
(11.3)
since from the definition of B˜ r(b),s (L, b) (4.1) it follows that 1 a(s, ¯ r) = a(s) ¯ + (φr,s¯ − φr(s), ¯ s¯ ). 2
(11.4)
In (11.3) we first set b = p0 − 1, r = p − 1 to obtain (o) ¯ 2 (φp−1,s¯ −φr(s), ¯ s¯ ) Fp−1,s (L, p0 − 1) = q a(s)+ Bp−1,s¯ (L, p0 − 1). 1
(11.5)
On the other hand from (7.1) and the last equation in (8.69), (o) (L, p0 − 1) = q c(1+t1+n )+c(tn )− 2 θ(n even) F1,s (L, 1) Fp−1,s 1
= q c(1+t1+n )+c(tn )− 2 θ(n even)+c(js )+ 2 (φ1,s −φr(s),s ) B1,s (L, 1), 1
1
(11.6)
Rogers–Schur–Ramanujan Identities for Minimal Models of CFT
375
where in the last line we used (10.1). Comparing (11.4) and (11.6) and using the symmetry of Br(b),s (L, b) (1.13) with r(b) = b = 1, B1,s (L, 1) = Bp−1,s¯ (L, p0 − 1),
(11.7)
we find the first expression for a(s) ¯ 1 a(s) ¯ = c(1 + t1+tn ) + c(tn ) − θ(n even) + c(js ) 2 1 + (φ1,s − φr(s),s + φr(s), ¯ s¯ − φp−1,s¯ ). 2
(11.8)
A second expression for a(s) ¯ is obtained by setting b = 1, r = 1 in (11.3) (o) ¯ 2 (φ1,s¯ −φr(s), ¯ s¯ ) (L, 1) = q a(s)+ B1,s¯ (L, 1). F1,s 1
(11.9)
However we also find from the second line of (8.68), the first line of (8.69) and (10.1) (o) (L, 1) = q −c(1+t1+n )−c(tn )+ 2 θ(n even) Fp−1,s (L, p0 − 1) F1,s 1
= q −c(1+t1+n )−c(tn )+ 2 θ(n even)+c(js )+ 2 (φp−1,s −φr(s),s ) Bp−1,s (L, p0 − 1). (11.10) Comparing (11.9) and (11.10) and using the symmetry (1.13) with s = b = 1 and s → p0 − s B1,s¯ (L, 1) = Bp−1,s (L, p0 − 1), (11.11) 1
1
we obtain
1 a(s) ¯ = c(js ) − c(1 + t1+n ) − c(tn ) + θ(n even) 2 1 + (φp−1,s − φr(s),s + φr(s), ¯ s¯ − φ1,s¯ ). 2 Equations (11.8) and (11.12) imply the following identity 1 1 c(1 + t1+n ) + c(tn ) − θ(n even) = (φp−1,s + φp−1,s¯ − φ1,s − φ1,s¯ ), 2 4
(11.12)
(11.13)
and using this identity we obtain (11.2) from (11.8) and (11.4). 12. The Dual Case p < p0 < 2p In the XXZ chain (2.1) we see from (2.2) that the regime p < p0 < 2p corresponds to 1 > 0 and since the XXZ chain has the symmetry HXXZ (1) = −HXXZ (−1) the (m, n) systems for ±1 are the same. Thus we consider the relation between M (p, p0 ) and M (p0 − p, p0 ) which is obtained by the transformation q → q −1 in the finite L polynomials Fr(b),s (L, b; q) and Br(b),s (L, b; q). To implement this transformation it is mandatory that the L → ∞ limit be taken only in the final step. (p,p0 ) (L, b; q) given by (1.12) where we have made the dependence Consider first Br(b),s on p and p0 explicit. Then if we note that from the definitions (3.16)
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we find
(0,1)
n+m m
= q −mn
q −1
0
(p,p ) Br(b),s (L, b, q −1 ) = q −
L2 2
q
(0,1) n+m , m q
(b−s)2 4
0
(12.1)
0
(p −p,p ) Bb−r(b),s (L, b; q).
(12.2)
Similarly we let q → q −1 in fs (L, u; q) and f˜s (L, u˜ ; q) given by (3.22) and (3.35) to obtain P 2 fs (L, u; q −1 ) = q
− L4 − L 2
ν0
j=1
uj
fs(d) (L, u; q),
(12.3)
2
L f˜s (L, u˜ ; q −1 ) = q − 4 f˜s(d) (L, u˜ ; q),
where
X
fs(d) (L, u; q) =
q− 2 m 1
T
Mm+A0 T m+C 0
¯ m∈2Zt1+n +w(u1+tn+1 ,u)
×
tY n+1 j=1
((Itn+1 + M)m + mj
u¯ 2
+
(1) L ¯ 2 e1 )j ,
(12.4)
q
M was defined in (2.11), (2.12), and the tn+1 -dimensional vector A0 is 0
0
A0 = A (b) + A (s)
with 0
Ak(b) 0
Ak(s)
= =
− 21 uk 0
for k in an odd zone , for k in an even zone
− 21 u¯ k (s) 0
(12.5)
(12.6)
for k in an even zone . for k in an odd zone
and
1 T ¯ + + y¯ T B¯y . u¯ (s)+ Bu(s) 8 u¯ is given by (3.23) and (3.24), and C0 = −
y¯ =
ν0 X
e¯ i ui ,
(12.7)
(12.8)
i=1
f˜s(d) (L, eν0 −j0 −1 − eν0 + u10 ; q) 0 j0 = ν0 L+j0 2L+1 − (d) 0 q 2 [q 4 fs (L + 1, eν0 −j0 − eν0 + u1 ; q) −fs(d) (L, e1+ν0 −j0 − eν0 + u10 ; q)] 1 ≤ j0 ≤ ν0 − 1, = L 0 2 f (d) (L, e q − e + u , q) ν0 −1 ν0 s 1 1 −(q L − 1)q − 4 fs(d) (L − 1, u10 ; q) for j0 = 0
(12.9)
and the vector u10 ∈ Z 1+t1+n was defined as any 1 + t1+n -dimensional vector Pν0 with no uj = 0 components in the zero zone. We note that for the vectors u in (3.40) that j=1 unless (12.10) u = −eν0 + u10
Rogers–Schur–Ramanujan Identities for Minimal Models of CFT
377
in which case the sum is −1. Thus since the L2 dependent factors in the transforms (12.2) and (12.3) are the same we obtain from all the polynomial identities of M (p, p0 ) of the form (10.1) polynomial identities for M (p0 − p, p0 ) of the form (d) (L, b; q) = q − 2 (φr(b),s −φr(s),s )− 2 c(js )+ Fr(b),s 1
1
(b−s)2 4
0
0
(p −p,p ) Bb−r(b),s (L, b; q),
(12.11)
(d) where Fr(b),s (L, b; q) for M (p0 −p, p0 ) is obtained from the corresponding Fr(b),s (L, b; q) of Sect. 10 as (d) (L, b; q) = Fr(b),s
X
X
q −cu fs(d) (L, u; q) +
q −c˜u˜ f˜s(d) (L, u˜ ; q)
(12.12)
˜ U˜ (b) u∈
u∈U (b)
where the exponents cu , c˜u˜ and the sets U (b), U˜ (b) as defined in (3.40) are explicitly given in Sect. 10. Pν0 uj in fs (L, u; q −1 ) in (12.3) is needed We note in particular that the term − L2 j=1 in case 4 of (3.41) with α = 1 in order to cancel the explicit factor of q − 2 in (10.15). In the limit L → ∞ we find from (12.4), L
lim fs(d) (L, u; q) = fs(d) (u; q)
(12.13)
L→∞
with fs(d) (u; q) =
X
q− 2 m 1
T
Mm+A0 T m+C 0
¯ m∈2Zt1+n +w(u1+t1+n ,u)
(1) tn+1 1 Y ((Itn+1 + M)m + u2¯ )j × , (q)m1 mj q
(12.14)
j=2
and from (12.9) lim f˜s(d) (L, eν0 −j0 −1 − eν0 + u10 ; q) =
L→∞
j0
q 2 − 4 fs(d) (eν0 −j0 − eν0 + u10 ; q) 0 1
j0 = 6 ν0 j0 = ν 0 . (12.15)
Thus we find from (12.11), (12.12) the corresponding character identities (d) Fr(b),s (b; q) = q − 2 (φr(b),s −φr(s),s )− 2 c(js )+ 1
1
(b−s)2 4
0
0
(p −p,p ) Bb−r(b),s (b; q)
(12.16)
and (d) (b; q) = Fr(b),s
X u∈U (b)
q −cu fs(d) (u; q) +
X
j0
q −c˜u˜ + 2 − 4 fs(d) (u˜ 0 ; q), 1
(12.17)
˜ U˜ (b) u∈
where u˜ 0 is obtained from (3.54) by replacing j0 by j0 − 1. We note that in contrast to the case p0 > 2p the terms involving u˜ contribute in the L → ∞ limit.
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13. Discussion The techniques developed in this paper are of great generality and may be extended to derive many further results which extend those presented in the previous sections. We will thus conclude this paper by discussing several extensions and applications which will be elaborated elsewhere. 13.1. Extension of the fundamental fermionic polynomials (3.22). There are two features in the definition of the polynomials fs (L, u; q) which may strike one as perhaps arbitrary and unmotivated; namely the fact that the vector A of (3.27)- (3.29) is of a different ¯ form depending on whether the components of u(b) and u(s) are in odd or even zones and the choice of the branch (0) or (1) of the q-binomial coefficients and it may be asked whether other choices are possible. For the choice of branch of the q-binomials it is clear from appendix A that the fermionic recursive properties of Sect. 5 will be equally valid if the interchange of branches (0) ↔ (1) is made everywhere. The only place the branch is explicitly used is in Sect. 9 where the polynomials Fr(b),s (L, b) are identified with a linear combination of the bosonic polynomials B˜ r(b),s (L, b). A change of branch will lead to different linear combinations. In a similar fashion different linear terms are possible which will also lead to different linear combinations. An analogous phenomena was studied in some detail for the model SM (2, 4ν) in [38]. 13.2. Extension to arbitrary s. Throughout this paper we have for simplicity confined our attention to values of s given by a pure Takahashi length. However, an examination of the proof of Appendix A shows that in fact the fundamental fermionic recursion ¯ relations are valid for all vectors u(s). With this generalization all values of s can be treated in a manner parallel to the way general values of b were treated above and the ¯ final result is a double sum over both sets of vectors u(b) and u(s). This generalization will be presented in full elsewhere. 13.3. Extension of m,n system. The (m, n) system (2.11) of this paper has the feature that the term L/2 appears only in the first component of the equation. However it is also useful to consider L/2 in other (possibly several) positions. Each choice will lead to further identities. Certain features of this extension have been seen in [22, 29, 33, 40]. As long as L/2 remains in the zero zone we are in the regime of weak anisotropy in the sense (1) (1) of [40] and we will obtain identities for coset models (A(1) 1 )N × (A1 )M /(A1 )N +M with N integer and M fractional. However if L/2 is in a higher zone we are in the region of strong anisotropy and identities for cosets with both fractional levels can be obtained. 13.4. Bailey pairs. We conclude by demonstrating that all polynomial identities of this paper may be restated in terms of new Bailey pairs and thus, by means of Bailey’s lemma [20], [41], [42], we may use the results of this paper to produce bose/fermi character identities for models other than M (p, p0 ). Definition of Bailey pair. A pair of sequences (αj , βj ) is said to form a Bailey pair relative to a if βn =
n X j=0
αj . (q)n−j (aq)n+j
(13.1)
Rogers–Schur–Ramanujan Identities for Minimal Models of CFT
379
All of our polynomial identities may be cast into this form. This is easily seen, following [25], by writing the bosonic polynomial (1.12) (where we suppress the dependence of r on b) with L = 2n + b − s (when L + b − s even) as Br,s (2n + b − s, b; q) = (q
b−s+1
)2n
∞ X j=−∞
0
0
q j(jpp +rp −sp) (q)n−jp0 (q b−s+1 )n+jp0 0
q (jp+r)(jp +s) − (q)n−(jp0 +s) (q b−s+1 )n+(jp0 +s)
!
(13.2) .
Thus comparing the identity (1.14) with (13.1) we obtain the following Bailey pair relative to a = q b−s : βn = q − 2 (φr,s −φr(s),s )−c(js ) Fr,s (2n + b − s, b, q)/(aq)2n , 0 0 q j(jpp +rp −sp) for n = jp0 , (j ≥ 0) j(jpp0 −rp0 +sp) for n = jp0 + s − b, (j ≥ 1) . αn = q (jp+r)(jp0 +s) for n = jp0 + s, (j ≥ 0) −q (jp−r)(jp0 −s) for n = jp0 − b, (j ≥ 1) −q 1
(13.3)
If two of the restrictions in (13.3) are the same the formula should be read as the sum of both. The utility of the Bailey pair follows from the lemma of Bailey [41] Lemma. Given a Bailey pair and a pair (γn , δn ) satisfying γn =
∞ X j=n
then
∞ X n=0
δj , (q)j−n (aq)j+n
αn γn =
∞ X
β n δn .
(13.4)
(13.5)
n=0
We note in particular two sets of (γn , δn ) pairs. One is the original pair of Bailey [41] (ρ1 )n (ρ2 )n (aq/ρ1 ρ2 )n , (aq/ρ1 )n (aq/ρ2 )n (q)N −n (aq)N +n (ρ1 )n (ρ2 )n (aqρ1 ρ2 )N −n (aq/ρ1 ρ2 )n , δn = (q)N −n (aq/ρ1 )N (aq/ρ2 )N
γn =
(13.6)
which has been used to produce characters of the N = 1 and N = 2 supersymmetric models [43] and a new pair of [44] which gives characters of the coset models (A(1) 1 )N × (1) ) /(A ) , where M may be fractional. (A(1) 1 M 1 N +M The presentation of the detailed consequences of this observation are too lengthy for inclusion in this paper and will be published separately [45].
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Appendix A. Proof of Fundamental Fermionic Recursive Properties In this appendix we prove the fundamental fermionic recursive properties (5.3) for 0 ≤ j0 < ν0 and (5.5) in detail. The proof of the remaining cases can be carried out in a similar fashion and will be left to the reader. We begin by introducing some shorthand notations. We use the vectors u0 (jµ ) and u±1 (jµ ) as defined in (5.1). Furthermore we need the operator which projects onto the components of a vector in an odd zone n πi =
for t1+2l + 1 ≤ i ≤ t2+2l , otherwise
1 0
(A.1)
and π˜ = 1 − π,
(A.2)
which projects onto components of even zones. We also need the vector Oa,b =
0P b i=a
if b < a πi e¯ i
.
(A.3)
We also make the following definition which generalizes the fermionic polynomial (3.22): q
δ(n,m)
¯ A ¯ = B
X
q
δ(n,m)+8s (n,m,u,L)
tY n+1
n,mt1+n ≡P ( mod 2)
j=1
(0) nj + mj + A¯ j , nj + B¯ j q
(A.4)
where P = 0, 1 with P ≡ u1+t1+n (mod 2), and nj , mj are related by (2.11) for given L ¯ and u, tn+1 X ¯ (u0 (jµ ))i e¯ i + u(s), 1 + tµ ≤ jµ ≤ t1+µ (A.5) u¯ = i=1
and 8s (n, m, u, L) = Q(n, m) + Lf (n, m),
(A.6)
where the quadratic term Q is defined in (3.26) and the linear term Lf is defined through A in (3.27)-(3.29). We note in particular 0 = fs (L, u0 (jµ )). 0
(A.7)
Finally we define ¯ L = set of solutions to (m, n) − system (2.11) with L, u. ¯ {n, m, u}
(A.8)
It will be necessary to make variable changes in n, m in (A.4). Hence we will need some identities relating different objects. For example it is easy to check that the set of ¯ L is equal to the set of solutions {n, m, u¯ + e¯ 1 }L−1 . One may also solutions {n, m, u} show that 8s (n, m, u, L) = 8s (n, m, u + e1 , L − 1). Thus fs (L, u) = fs (L − 1, u + e1 ),
(A.9)
Rogers–Schur–Ramanujan Identities for Minimal Models of CFT
381
which proves (5.3) for j0 = 0. One may also show that ¯ L−2 , ¯ L − {¯e1 , 0, 0} = {n, m, u} {n, m, u} m1 + 8s (n, m, u, L) = (L − 1) + 8s (n − e¯ 1 , m, u, L − 2), P ν0
where u is any vector such that
i=1
q m1
−¯e1 −¯e1
(A.10)
ui = 0. From this follows
= q L−1 fs (L − 2, u0 (jµ )).
(A.11)
Similarly one can prove the identities:
¯ 1,ν0 − e¯ ν0 +1 −2E −¯eν0 +1
1:
¯ 1,ν0 +1 −¯eν0 − E −¯eν0 +1
2:
¯ 1,j0 −E 0
3:
= fs (L − 2, u0 (jµ )),
= fs (L − 1, eν0 −1 − eν0 + u0 (jµ )),
= fs (L − 1, u1 (j0 )),
4 : q mj0 −mj0 −1 +1
5 : q mj0 −mj0 −1 +1
¯ 1,j0 −1 + e¯ j0 −1 − e¯ j0 −E e¯ j0 −1 − e¯ j0
¯ 1,j0 −1 + e¯ j0 −1 − e¯ j0 −2E e¯ j0 −1 − e¯ j0
¯ 1,jµ −E 0
ν0 +1 3 L−1 2 − 4 + 4 θ(µ
6: q
OT 1,jµ n
7: q
OT 1,jµ −1 n+(mjµ −mjµ −1 +1)θ(µ even)
=q
=q
ν0 −θ(µ even) L−1 2 − 4
T
ν0 −θ(k odd) L−1 2 − 4
= fs (L − 1, u−1 (j0 )), j0 > 1,
odd)
= fs (L − 2, u0 (j0 )), j0 > 1,
(t) fs (L − 1, u1 (jµ ) − E1,µ ), 0 < µ ≤ n,
¯ 1,jµ −1 + e¯ jµ −1 − e¯ jµ −E e¯ jµ −1 − e¯ jµ
(t) fs (L − 1, u−1 (jµ ) − E1,µ ), 0 < µ ≤ n,
8 : q O1,tk n+(ntk +m1+tk )θ(k even) =q
¯ 1,1+tk −¯etk − E −¯e1+tk
(t) fs (L − 1, u0 (jµ ) + e−1+tk − E1,k ), 2 ≤ k ≤ µ ≤ n.
(A.12) The derivation of (A.12) requires the verification the following intermediate equations:
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A. Berkovich, B. M. McCoy, A. Schilling
¯ 1,ν0 , 0} = {n, m, u} ¯ L − {¯eν0 +1 , 2E ¯ L−2 , 10 : {n, m, u} ¯ 1,ν0 , u, L − 2), 8s (n, m, u, L) = 8s (n − e¯ ν0 +1 , m − 2E ¯ 1,ν0 , 0} = {n, m, e¯ ν0 −1 − e¯ ν0 + u} ¯ L − {¯eν0 +1 , e¯ ν0 + E ¯ L−1 , 20 : {n, m, u} ¯ 8s (n, m, u, L) = 8s (n − e¯ 1+ν0 , m − e¯ ν0 − E1,ν0 , u + eν0 −1 − eν0 , L − 1), ¯ 1,j0 , 0} = {n, m, u¯ + e¯ j0 +1 − e¯ j0 }L−1 , ¯ L − {0, E 30 : {n, m, u} ¯ 1,j0 , u + ej0 +1 − ej0 , L − 1), 8s (n, m, u, L) = 8s (n, m − E ¯ 1,j0 −1 , 0} = {n, m, u¯ + e¯ j0 −1 − e¯ j0 }L−1 ¯ L + {¯ej0 −1 − e¯ j0 , −E 40 : {n, m, u} (mj0 − mj0 −1 + 1) + 8s (n, m, u, L) ¯ 1,j0 −1 , u + ej0 −1 − ej0 , L − 1), = 8s (n + e¯ j0 −1 − e¯ j0 , m − E ¯ 1,j0 −1 , 0} = {n, m, u} ¯ L + {¯ej0 −1 − e¯ j0 , −2E ¯ L−2 , j0 > 1 50 : {n, m, u} 8s (n, m, u0 (j0 ), L) + 1 + mj0 − mj0 −1 ¯ 1,j0 −1 , u0 (j0 ), L − 2), = 8s (n + e¯ j0 −1 − e¯ j0 , m − 2E ¯ 1,jµ , 0} = {n, m, u¯ + e¯ jµ +1 − e¯ jµ − E ¯ (t) }L−1 ¯ L − {0, E 60 : {n, m, u} 1,µ 8s (n, m, u0 (jµ ), L) + OT1,jµ n =
L − 1 ν0 + 1 3 ¯ 1,jµ , u1 (jµ ) − E(t) , L − 1), µ 6= 0, − + θ(µ odd) + 8s (n, m − E 1,µ 2 4 4
¯ 1,jµ −1 , 0} = {n, m, u¯ + e¯ jµ −1 − e¯ jµ − E ¯ (t) }L−1 ¯ L + {¯ejµ −1 − e¯ jµ , −E 70 : {n, m, u} 1,µ 8s (n, m, u0 (jµ ), L) + OT1,jµ −1 n + θ(µ even)(1 + mjµ − mjµ −1 ) =
L−1 2
ν0 − θ(µ even) ¯ 1,jµ −1 , u−1 (jµ ) − E(t) , L − 1), + 8s (n + e¯ jµ −1 − e¯ jµ , m − E 1,µ 4 (t) 0 ¯ 1,tk , 0} = {n, m, u¯ + e¯ −1+tk − E ¯ }L−1 ¯ L − {¯e1+tk , e¯ tk + E 8 : {n, m, u} 1,k −
8s (n, m, u0 (jµ ), L) + OT1,tk n + (ntk + m1+tk )θ(k even) =
L−1 2
ν0 − θ(k odd) ¯ 1,tk , u0 (jµ )+e−1+tk −E(t) , L − 1), + 8s (n − e¯ 1+tk , m − e¯ tk − E 1,k 4 (A.13) ¯ (t) are given by (2.26), (5.2) and (9.7). ¯ a,b , E(t) , E where E a,b a,b After these preliminaries we are in the position to prove the fermionic recurrences (5.3) for 0 ≤ j0 < ν0 and (5.5). Our method is the technique of telescopic expansion introduced in [22] and further developed in [21] and [28]. With this technique we may expand fs (L, u0 (jµ )), −
fs (L, u0 (jµ )) =
X jµ ¯ ¯ 1,l T −E 0 OT n −E1,jµ + , = q 1,jµ q O1,l n+π˜ l ml 0 −¯el 0
(A.14)
l=1
where (3.20) has been used in odd zones and (3.21) has been used in even zones. We shall further need the telescopic expansions
Rogers–Schur–Ramanujan Identities for Minimal Models of CFT
q
θ(k odd)nt1+k −1+t X1+k
¯ 1,1+t1+k −¯et1+k − E −¯e1+t1+k
383
=
¯ l,1+t1+k ¯ 1,t1+k − E −E q −¯el − e¯ 1+t1+k l=dk ¯ dk ,1+t1+k ¯ 1,t1+k − E T −E , + q Odk ,t1+k n −¯e1+t1+k
and
OT l+1,t
1+k
n+(ml −1)θ(k even)
(A.15)
¯ 1,jµ −1 + e¯ jµ −1 − e¯ jµ −E = e¯ jµ −1 − e¯ jµ jµ −1 X OT ¯ ¯ ejµ −1 )+π˜ l ml −E1,jµ − El,jµ −2 l+1,jµ −1 (n+¯ q −¯el + e¯ jµ −1 − e¯ jµ l=dk ¯ ¯ θ(µ even)+OT ejµ −1 ) −E1,jµ −2 − E1+tµ ,jµ 1+tµ ,jµ −1 (n+¯ , +q e¯ jµ −1 − e¯ jµ
q θ(µ even)
(A.16)
with dk = 1 + tk + δk,0 , where again (3.20) has been used in odd zones and (3.21) has been used in even zones. Using case 6 in (A.12) we may rewrite (A.14) as fs (L, u0 (jµ )) = q
ν0 +1 3 L−1 2 − 4 + 4 θ(µ
odd)
(t) fs (L − 1, u1 (jµ ) − E1,µ )+
µ X
Ik ,
(A.17)
k=0
where Ik =
gk X
T
q O1,l n+π˜ l ml
l=dk
¯ 1,l −E , 0≤k≤µ −¯el
(A.18)
t1+k for k < µ and may now process the terms Ik (0 ≤ k ≤ µ − 1) as for k = µ jµ follows. In the lth term (l = 6 dk ) in (A.18) we perform the change of variables and gk =
n → n + e¯ l − e¯ l−1 − e¯ 1+t1+k , ¯ l,t1+k m → m − 2E
(A.19)
to obtain
T
Ik = q O1,tk n+mdk θ(k even) +
−1+t X1+k l=dk
¯ 1,dk −E −¯edk
q (O1,t1+k +O1+l,t1+k )n+m1+t1+k θ(k odd)+(ml −1)θ(k even) T
T
¯ 1,1+t1+k ¯ l,t1+k − E −E . −¯el − e¯ 1+t1+k (A.20)
For k = 0, µ > 0 we use (A.11) and (A.12) 1,2 and obtain
384
A. Berkovich, B. M. McCoy, A. Schilling
I0 = q
m1
−¯e1 −¯e1
¯ 1,1+ν0 ¯ 1,ν0 − e¯ 1+ν0 −¯eν0 − E −2E + − −¯eν0 +1 −¯e1+ν0
(A.21)
= fs (L − 1, eν0 −1 − eν0 + u0 (jµ )) + (q L−1 − 1)fs (L − 2, u0 (jµ )). Applying the telescopic expansion (A.15) to the last sum in (A.20), we obtain for k 6= 0,
¯ 1,dk −E −¯edk ¯ 1,1+t1+k T −¯et1+k − E + q O1,t1+k n+m1+t1+k θ(k odd) q nt1+k θ(k odd) −¯e1+t1+k ¯ ¯ T −E1,t1+k − Edk ,1+t1+k . −q Odk ,t1+k n −¯e1+t1+k T
Ik = q O1,tk n+mdk θ(k even)
(A.22)
For k 6= 0, we perform one more change of variables n → n + e¯ 1+t1+k − e¯ tk − e¯ 1+tk , ¯ 1+tk ,t1+k m → m + 2E
(A.23)
in the last term of (A.22) to arrive at ¯ 1,1+t1+k T −¯et1+k − E Ik = q O1,t1+k n+(m1+t1+k +nt1+k )θ(k odd) −¯e1+t1+k ¯ T −E1,1+tk + q O1,tk n+m1+tk θ(k even) −¯e1+tk ¯ 1,1+tk −¯etk − E OT 1,tk n+m1+tk θ(k even)+(mtk −1)θ(k odd) −q −¯etk − e¯ 1+tk k+1 X T ¯ 1,1+ta eta − E O1,ta n+(nta +m1+ta )θ(a even) −¯ q , = −¯eta − e¯ 1+ta
(A.24)
a=k
where to get the last line (3.20) or (3.21) has been used for k even or odd, respectively. Thus recalling (A.12) 2,8 we have for 1 ≤ k ≤ µ − 1, Ik =
k+1 X
q (1−δa,1 )(
ν0 −θ(a odd) L−1 ) 2 − 4
(t) fs (L − 1, eta −1 − E1,a + u0 (jµ )).
(A.25)
a=k
It remains to process Iµ . To this end we perform the change of variables n → n − e¯ l−1 + e¯ l + e¯ jµ −1 − e¯ jµ ¯ l,jµ −1 m → m − 2E in the lth term (l 6= dµ ) appearing in the sum (A.18) with k = µ. This yields
(A.26)
Rogers–Schur–Ramanujan Identities for Minimal Models of CFT
Iµ = q
OT 1,jµ −1 n+mdk θ(µ even)
jµ −1
+
X
q
¯ 1,dµ −E −¯edµ
385
T OT 1,jµ −1 n+Ol+1,jµ −1 (n+ejµ −1 )+(ml +mjµ −mjµ −1 )θ(µ even)
l=dµ
¯ l,jµ −2 ¯ 1,jµ − E −E , −¯el + e¯ jµ −1 − e¯ jµ
(A.27) where we used the empty sum convention. Using the telescopic expansion (A.16) we find Iµ = q +q
OT 1,jµ −1 n+mdk θ(µ even)
¯ 1,dµ −E −¯edµ
¯ 1,jµ −1 + e¯ jµ −1 − e¯ jµ −E e¯ jµ −1 − e¯ jµ ¯ ¯ OT ejµ −1 ) −E1,jµ −2 − E1+tµ ,jµ 1+tµ ,jµ −1 (n+¯ . −q e¯ jµ −1 − e¯ jµ
OT 1,jµ −1 n+(mjµ −mjµ −1 +1)θ(µ even)
(A.28)
For µ = 0 (A.28) yields
−¯e1 I0 = q + θ(j0 > 1)q mj0 −mj0 −1 +1 −¯e1 ¯ 1,j0 −1 + e¯ j0 −1 − e¯ j0 ¯ 1,j0 −1 + e¯ j0 −1 − e¯ j0 −2E −E − × e¯ j0 −1 − e¯ j0 e¯ j0 −1 − e¯ j0 m1
(A.29)
= fs (L − 1, u−1 (j0 )) + (q L−1 − 1)fs (L − 2, u0 (j0 )). where we used (A.9), (A.11), (A.12) 4,5. For µ > 0 we need to change variables again n → n − e¯ tµ − e¯ 1+tµ − e¯ jµ −1 + e¯ jµ , ¯ 1+tµ ,jµ −1 m → m + 2E
(A.30)
in the last term of the right-hand side of (A.28) to get ¯ 1,1+tµ −E Iµ = q −¯e1+tµ ¯ ¯ jµ −1 − e¯ jµ OT n+(mjµ +mjµ −1 +1)θ(µ even) −E1,jµ −1 + e + q 1,jµ −1 e¯ jµ −1 − e¯ jµ ¯ 1,1+tµ −¯etµ − E OT 1,jµ −1 n+m1+tµ θ(µ even)+(mtµ −1)θ(µ odd) −q −¯etµ − e¯ 1+tµ ¯ 1,jµ −1 + e¯ jµ −1 − e¯ jµ −E OT 1,jµ −1 n+(mjµ −mjµ −1 +1)θ(µ even) =q e¯ jµ −1 − e¯ jµ ¯ T etµ − E1,1+tµ O n+(m1+tµ +ntµ )θ(µ even) −¯ + q 1,jµ −1 , −¯e1+tµ OT 1,jµ −1 n+m1+tµ θ(µ even)
(A.31)
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A. Berkovich, B. M. McCoy, A. Schilling
where we used the elementary binomial recurrences (3.20) and (3.21). Recalling Eqs. 7, 8 of (A.12) we finally obtain for µ > 0, Iµ = q +q
ν0 −θ(µ odd) L−1 2 − 4
(t) fs (L − 1, u0 (jµ ) + etµ −1 − E1,µ )
ν0 −θ(µ even) L−1 2 − 4
(t) fs (L − 1, u−1 (jµ ) − E1,µ ).
(A.32)
The desired results (5.3) (for 2 ≤ j0 ≤ ν0 − 1) and (5.5) follow by combining formulas (A.17), (A.21), (A.25), (A.29) and (A.32).
Appendix B. Proof of the Initial Conditions for Propositions 1 and 2 In this appendix we prove the initial conditions for Propositions 1 and 2 as discussed in µ a Sect. 8. These proofs explain the origin of the factors q 2 (−1) in Propositions 1 and 2 and also demonstrate that at the dissynchronization point the form (3.40) does not hold. Proof of Proposition 1 with a = −1 and µ = 2. To prove Proposition 1 with a = −1, µ = 2 −2 : 2 we will use the following decomposition of the flow −→y−1 y2-2 -1
= y1-2 0
1
2
(8.2)b (8.5)
y1-2 0
2
y1-2 0
2
y1-2
(B.1)
0
The decomposition of the flow of length y2-1 used to prove Proposition 1 and 2 with µ=2, a=-1 whose proof is elementary and will be omitted. We recall that u20 is any 1 + tn+1 –dimensional vector with (u20 )j = 0 for j ≤ t2 . Then first using Proposition 1 with µ = 1, a = 0 and then using (5.3) with j0 = ν0 we obtain {fs (L, −et1 − et2 + u20 ), fs (L, e1 − et1 − et2 + u20 )} y1 −2
−→ {fs (L, e−1+t1 − et1 − et2 + u20 ), fs (L, −et2 + u20 )} 0 L 1 −→ fs (L, −et2 + u20 ), q − 2 fs (L, e1+t1 − et2 + u20 )
(B.2)
(8.2)b
L 1 + q 2 − L fs (L − 1, −et2 + u20 ) . q2
It may be seen that the second polynomial in the last pair of (B.2) for which b is at the dissynchronization value b2,−1 is not of the form (3.40). We may further evolve using flow −→2(8.5) to find
Rogers–Schur–Ramanujan Identities for Minimal Models of CFT
L u20 ), q − 2 fs (L, e1+t1
u20 )
387
L 2
1
u20 )
− et2 + + (q − L )fs (L − 1, −et2 + fs (L, −et2 + q2 ν0 −2 1 2 −→ q − 2 q − 4 {fs (L, −et1 + e2+t1 − et2 + u20 ), fs (L, e1 − et1 + e2+t1 − et2 + u20 )} (8.5) + {P0 (L, 1, e1+t1 − et2 + u20 ), P1 (L, 1, e1+t1 − et2 + u20 ), } . (B.3) where we used (5.4) with j1 = 1 + t1 and the first equation in (5.3) as well as the definitions (3.35), (8.27), (8.28). Now applying Propositions 1 and 2 with a = 0 and µ = 1 for −→0y1 −2 and lemma 2.1 for =⇒2 according to (B.1), we finally obtain upon recalling the definition (8.26), n
o y −2 1 2 (t) (t) + u20 ), fs (L, −E1,2 − et2 + u20 ) −→ q − 2 q c(t2 ) {P1 (L, 2, u20 ), fs (L, u20 )} . fs (L, −E1,2 −1
(B.4) − 21 is Therefore Proposition 1 for a = −1 and µ = 2 is proven. We note that the factor q µ a the origin of the factor q 2 (−1) in Proposition 1 with a = −1. Proof of proposition 2 with a = −1 and µ = 2. For the proof of Proposition 2 with a = −1, µ = 2 we shall again use the decomposition (B.1). However, in the present case it is convenient to rewrite the piece −→1(8.2)b −→2(8.5) in the bottom graph of (B.1) using the identity 1
2
2
1
−→ −→ = =⇒ −→ .
(8.2)b (8.5)
(8.2)a
(B.5)
To simplify our analysis, let us further assume that ν1 ≥ 3, ν0 ≥ 2. We start by applying Propositions 1 and 2 with a = 0 and µ = 1, {P0 (L, 2, u20 ), P1 (L, 2, u20 )} n o ν0 (t) (t) =q − 4 fs (L, e−1+t2 − E1,2 + u20 ), fs (L, e1 + e−1+t2 − E1,2 + u20 ) + {P0 (L, 1, u20 ), P1 (L, 1, u20 )} n ν0 −2 L y1 −2 − ν0 −→ q 4 P−1 (L, 1, e−1+t2 − et2 + u20 ) + q 2 − 4 fs (L, −et1 + u20 ) ,
(B.6)
0
fs (L, e−1+t2 − et2 + u20 )} . Evolving this once more according to =⇒2 and using Lemma 2.2 with µ = 1, j1 = 1 + t1 we find from (B.6), y1
{P0 (L, 2, u20 ), P1 (L, 2, u20 )} −→ o n ν0 (t) (t) + u20 ), fs (L, e1 + e−2+t2 − E1,2 + u20 ) q − 2 fs (L, e−2+t2 − E1,2 +q
ν − 40
(B.7)
{P0 (L, 1, e−1+t2 − et2 + u20 ), P1 (L, 1, e−1+t2 − et2 + u20 )} .
Evolving now according to −→1(8.2)a and using the fermionic recurrence (5.3) for fs and (6.1)for f˜s as well as definitions (8.27), (8.28), we derive for ν0 ≥ 2, ν1 ≥ 3,
388
A. Berkovich, B. M. McCoy, A. Schilling y1 +1
{P0 (L, 2, u20 ), P1 (L, 2, u20 )} −→ (B.1) n o ν (t) (t) − 20 fs (L, e1 + e−2+t2 − E1,2 q + u20 ), fs (L, e2 + e−2+t2 − E1,2 + u20 ) o n ν0 (t) (t) + q − 4 f˜s (L, e−2+t1 + e−1+t2 − E1,2 + u20 ), f˜s (L, e−3+t1 + e−1+t2 − E1,2 + u20 ) , (B.8) 2 −2 where the arrow in (B.8) represents the flow −→y−1 restricted to the first 1 + y1 steps. We now note that the polynomials in the rhs of (B.8) and in the rhs of (8.41) differ 2 −2 only by the overall factor of q −1/2 . Furthermore, according to (8.23) the flow −→y−1 and the flow −→y0 2 −2 have the same steps starting with the step y1 +2. These observations along with Proposition 2 for a = 0, µ = 2 clearly imply that y2 −2
{P0 (L, 2, u20 ), P1 (L, 2, u20 )} −→ q − 2 −1
1
n
q c(t2 )+
L−ν0 2
(t) fs (L, −E1,2 + u20 ), 0
o (B.9)
which concludes the proof of Proposition 2 with a = −1, µ = 2, ν0 ≥ 2, ν1 ≥ 3. The proof of the remaining cases with ν1 = 2 and ν1 ≥ 3, ν0 = 1 can be carried out in a similar fashion and is left to the reader. Finally, we note that the factor q −1/2 in (B.9) is (−)µ the origin of the factor q a 2 in Proposition 2 with a = −1, µ. Proof of Proposition 1 with a = −1 and µ = 3. To prove Proposition 1 with a = −1, µ = 3 3 −2 we shall use a decomposition of −→y−1 obtained from (8.24) with µ = 2, a = −1 by 2 replacing the second arrow =⇒ from the left by the composite arrow −→1(8.2)a −→1(8.2)b . This replacement is necessary because (8.24) is not valid as it stands for µ = 2. In what y3 −2 follows the notation −→b−2 −1 , y2 < b ≤ y3 is understood as the flow −→−1 restricted to the first b − 2 steps. For convenience we consider ν0 ≥ 2 and we start by applying Proposition 1 with µ = 2, a = 0 and subsequently using (8.43) with µ = 2, u20 = −et3 + u30 and lemma 2.1 with µ = 1, j1 = t2 to obtain n
o y 2 (t) (t) + u30 ), fs (L, e1 − E1,3 + u30 ) −→ fs (L, −E1,3 −1
q
+
ν0 −1 4
{fs (L, −et1 + e1+t2 − et3 + u30 ), fs (L, e1 q c(t2 ) {P0 (L, 1, −et3 + u30 ), P1 (L, 1, −et3 + u30 )} .
c(t2 )−
− et1 + e1+t2 − et3 + u30 )}
(B.10)
Application of Propositions 1 and 2 with a = 0 and µ = 1 to the rhs of (B.10) yields n
o y +y −2 2 1 (t) (t) fs (L, −E1,3 + u30 ), fs (L, e1 − E1,3 + u30 ) −→ −1 n ν0 −1 ν0 −1 L q c(t2 )− 4 fs (L, e−1+t1 − et1 + e1+t2 − et3 + u30 ) + q 2 − 4 fs (L, −et1 − et3 + u30 ), fs (L, e1+t2 − et3 + u30 )} . (B.11) Next evolving the rhs of (B.11) according to −→1(8.2)a and making use of (5.4) with j2 = 1 + t2 we derive
Rogers–Schur–Ramanujan Identities for Minimal Models of CFT
389
o (t) (t) + u30 ), fs (L, e1 − E1,3 + u30 ) fs (L, −E1,3 n ν0 +1 L y2 +y1 −1 c(t2 )− ν0 −1 (t) 4 fs (L, e1+t2 − et3 + u30 ), q 2 − 4 fs (L, e2+t2 − E1,3 −→ q + u30 ) n
−1
o ν0 L (t) (t) + u30 ) + fs (L, e−1+t1 + et2 + e1+t2 − E1,3 + u30 ) . +q 2 − 4 fs (L, e−1+t2 + e1+t2 − E1,3 (B.12) To proceed further we observe that the first two steps of −→1(8.2)b −→y1 2 −2 are the same as =⇒2 . The straightforward use of properties (5.3), (6.1) and definition (3.35) then gives n
oy
(t) (t) + u30 ), fs (L, e1 − E1,3 + u30 ) fs (L, −E1,3
2 +y1 +1
−→ q 2 {z1 (L, u30 ), z2 (L, u30 )} 1
−1
(B.13)
where za (L, u30 ) = q c(t2 )−
ν0 −1 4
q−
ν0 +1 4
(t) fs (L, ea + e2+t2 − E1,3 + u30 )+
ν0 (t) (t) q − 4 fs (L, ea + e−1+t2 + e1+t2 − E1,3 + u30 ) + f˜s (L, et1 −1−a + et2 + e1+t2 − E1,3 + u30 ) (B.14) with a = 1, 2. On the other hand, as can be seen from case 3 of (7.18) with j2 = (t) t2 + 1, j0 = 1, 2 and −E4,n replaced by u30 , n
oy
(t) (t) + u30 ), fs (L, e1 − E1,3 + u30 ) fs (L, −E1,3
2 +y1 +1
−→ {z1 (L, u30 ), z2 (L, u30 )} .
(B.15)
Comparing (B.13) and (B.15) we see that the expressions in the rhs of these expressions y3 −2 differ only by a factor of q 1/2 . According to (8.23) with µ = 3, a = −1 the flow −→−1 and the flow −→0y3 −2 have identical steps starting with the step y2 + y1 + 2. This fact together with (B.13), (B.15) and Proposition 1 with µ = 3, a = 0 imply that n
o y −2 1 3 (t) (t) fs (L, −E1,3 + u30 ), fs (L, e1 − E1,3 + u30 ) −→ q c(t3 )+ 2 {P−1 (L, 3, u30 ), fs (L, u30 )} , −1
(B.16) which concludes the proof of Proposition 1 with a = −1, µ = 3, ν0 ≥ 2. The proof of the special case ν0 = 1 follows along similar lines and will be omitted.
Appendix C. Proof of 10.10 and 10.15 We sketch the proof of (10.10) and (10.15) in several stages. First we use the results of Sect. 8 to prove (C.12) and (C.13). From Theorem 1 of Sect. 8 and (C.13) we then prove (10.15) for the case β = α + 1 and after that we use both (C.12) and (C.13) to prove the general case of (10.15) with 1 ≤ α ≤ β − 2 by induction. We conclude by proving (10.10). It is useful to introduce the notation j˜µ = jµ − tµ , 1 + tµ ≤ jµ ≤ tµ+1 + δµ,n . In 0 addition, we define the flow −→ab −2 with a = 0, −1 and yµ < b0 ≤ yµ+1 as part of the y −2 flow −→aµ+1 restricted to the first b0 − 2 steps. We also note the equality of flows
390
A. Berkovich, B. M. McCoy, A. Schilling j˜ µ yµ −2 2
j˜ µ yµ
−→ = −→ =⇒, 2 ≤ j˜µ ≤ νµ + δµ,n , 2 − |a| ≤ µ ≤ n a
a
and yµ
yµ −2 2
−→ = −→ =⇒, 2 − |a| ≤ µ ≤ n a
(C.1)
(C.2)
0
with a = 0, −1. These follow from (3.12) for a = 0 if we note the Takahashi decomposition ( (µ) (µ−1) lj µ + l tµ for 2 ≤ j˜µ ≤ νµ + δµ,n , 1 ≤ µ ≤ n (C.3) j˜µ yµ = (µ−1) l1+tµ for j˜µ = 1, 1 ≤ µ ≤ n. The identical argument works for a = −1 if we replace jµ by 1 + jµ . Next we use (C.1) and (C.2) along with (8.50)-(8.52) and Lemma 2.2 of Sect. 8 to obtain j˜ µ yµ
0 0 ), P1 (L, µ + 1, uµ+1 )} −→ {P0 (L, µ + 1, uµ+1 a
q
c(jµ )−c(tµ−1 )−
ν0 −θ(µ even) 1 + 2 a(−1)µ+1 4
µ+1
1
+q c(jµ )−c(tµ−1 )+ 2 a(−1)
(t) {fs (L, −E1,µ+1 + et1+µ −j˜ µ −1 + (t) fs (L, e1 − E1,µ+1 + et1+µ −j˜ µ −1
0 uµ+1 ), 0 + uµ+1 )}
(C.4)
0 {P0 (L, µ, et1+µ −j˜ µ − et1+µ + uµ+1 ), 0 )} P1 (L, µ, et1+µ −j˜ µ − et1+µ + uµ+1
with 1 + |a| ≤ j˜µ ≤ νµ − 2, 2 ≤ µ ≤ n − 1, a = 0, −1. Furthermore, from (8.23) and (8.25) we derive (µ) l1+j −2
j˜ µ yµ yµ−1 −2
−→ = −→ −→ µ
a
a
0
for 2 ≤ µ ≤ n, 1 ≤ j˜µ ≤ νµ − 1 + 2δµ,n , a = 0, −1. (C.5)
Then combining (C.5) with (C.4), Proposition 1 from Sect. 8 and (8.50) with a = 0 and µ replaced by µ − 1 gives (µ) l1+j −2
0 0 ),P1 (L, µ + 1, uµ+1 )} −→ {P0 (L, µ + 1, uµ+1 µ
a
q c(jµ )−
ν0 −θ(µ even) 1 + 2 a(−1)µ+1 4
(t) 0 0 {Y (L, µ, jµ , uµ+1 ), fs (L, −Eµ,µ+1 + et1+µ −j˜ µ −1 + uµ+1 )
+ q−
(−1)µ 4
(t) 0 fs (L, e−1+tµ − Eµ,µ+1 + et1+µ −j˜ µ + uµ+1 )} (C.6) with 1 + |a| ≤ j˜µ ≤ νµ − 2, 2 ≤ µ ≤ n − 1, a = 0, −1 and (t) 0 0 Y (L, µ, jµ , uµ+1 ) =P−1 (L, µ − 1, −Eµ,µ+1 + et1+µ −j˜ µ −1 + uµ+1 )
+q
(−1)µ 4
+q2− L
(t) 0 P−1 (L, µ − 1, e−1+tµ − Eµ,µ+1 + et1+µ −j˜ µ + uµ+1 ) (C.7)
ν0 −θ(µ even) 4
(t) 0 fs (L, −E1,µ−1 + et1+µ −j˜ µ + uµ+1 ).
The analogues of (C.6) and (C.7) for µ = 1 are
Rogers–Schur–Ramanujan Identities for Minimal Models of CFT
391
(1) l1+j −2
1 {P0 (L, 2, u20 ), P1 (L, 2, u20 )} −→
a
q c(j1 )−
ν0 4
+ a2
(C.8)
ν0
(t) {Y (L, 1, j1 , u20 ), q 2 − 4 fs (L, et2 −j˜ 1 −1 − E1,2 + u20 )+ L
(t) fs (L, e−1+t1 + et2 −j˜ 1 − E1,2 + u20 )}
and
Y (L, 1, j1 , u20 ) = fs (L, et2 −j˜ 1 − et2 + u20 )
(C.9)
with 1+|a| ≤ j˜1 ≤ ν1 −2, a = 0, −1. To derive (C.8)-(C.9) for a = 0 we compare case 3 of (7.18) with j0 = 0, 1 with case 2 of (7.18) with j0 = ν0 −1, and j1 replaced by j1 −1 and (t) by u20 . By comparing case 4 of (7.18) to find the desired result after replacing ej2 − E3,n (B.7)-(B.8) and (C.8) with j1 = 2, a = 0 and recalling that according to (8.23) the 2 −2 flows −→y0 2 −2 and −→y−1 have the same steps, starting with the step 1 + y0 + y1 we easily verify (C.8) for a = −1, 2 ≤ j˜1 ≤ ν1 − 2. (µ−1) l2+j −2
Next we apply the flow −→−1 µ−1 to the right-hand side (rhs) of (C.4) with a = 0. Taking into account (8.47) with a = −1, µ replaced by µ−1 and jµ replaced by jµ−1 +1 and (C.6) with a = −1, µ replaced by µ − 1, µ ≥ 3 or (C.8) with a = −1 with j1 replaced by 1 + j1 for µ = 2 we derive (µ−1) l2+j −2
ν0 −θ(µ odd)
1
µ+1
+ 2 (−1) 4 rhs of (C.4) −→ q c(jµ )+c(1+jµ−1 )− −1 0 ), {Y (L, µ − 1, 1 + jµ−1 , et1+µ −j˜ µ − et1+µ + uµ+1 µ−1
q δµ,2 ( 2 − L
+q +q
ν0 4
)
×
(t) 0 fs (L, −Eµ−1,µ+1 + etµ −j˜ µ−1 −2 + et1+µ −j˜ µ + uµ+1 )
(−1)µ −δµ,2 4
fs (L, e−1+tµ−1 −
(t) Eµ−1,µ+1
+ etµ −j˜ µ−1 −1 + et1+µ −j˜ µ +
(C.10) 0 uµ+1 )}
(−1)µ 4
0 {X(L, µ − 1, 1 + jµ−1 , et1+µ −j˜ µ −1 − et1+µ + uµ+1 ), (t) 0 + e1+jµ−1 + et1+µ −j˜ µ −1 + uµ+1 )} fs (L, −Eµ,µ+1
with 1 + tµ−1 ≤ jµ−1 ≤ tµ − 3, 1 + tµ ≤ jµ ≤ tµ+1 − 2, 2 ≤ µ ≤ n − 1 and 0 0 ) = θ(jµ > y1 ) P−1 (L, µ − δ1+tµ ,jµ , ejµ − et1+µ + uµ+1 )+ X(L, µ, jµ , uµ+1 q2− L
ν0 −θ(µ even) 4
(t) 0 fs (L, e−1+jµ − E1,1+µ + u1+µ )
(C.11)
+ δµ,1 δj1 ,y1 fs (L, −et2 + u20 ) for 1 ≤ µ ≤ n. To proceed further we note the following equality of flows: j˜ µ yµ
−→
(µ−1) l2+j −2
−→ µ−1
−1
=
(µ) (µ−1) l1+j +l1+j −2 µ
−→
µ−1
=
(µ−1) (µ) −2 l1+j −2 2 l1+j µ µ−1
−→ =⇒
−→
(C.12)
with µ ≥ 2, 1 ≤ j˜µ ≤ νµ − 2 + 2δµ,n , 1 ≤ j˜µ−1 ≤ νµ−1 − 1. The first equation in (C.12) follows from (8.25) with a = 0. The second equation in (C.12) is just properties
392
A. Berkovich, B. M. McCoy, A. Schilling
(µ) (3.12)–(3.13) of the map b → r with b = l1+j . Equation (C.12) together with (C.6) with µ a = 0 and (C.10) imply the flow (t) 0 0 {Y (L, µ, jµ , uµ+1 ), fs (L, −Eµ,µ+1 + et1+µ −j˜ µ −1 + uµ+1 )
+ q−
(−1)µ 4
(µ−1) l1+j −2
2
(t) 0 fs (L, e−1+tµ − Eµ,µ+1 + et1+µ −j˜ µ + uµ+1 )} ν0 +1
3
=⇒ −→ q c(jµ−1 )− 4 + 4 θ(µ odd) × 0 {Y (L, µ − 1, 1 + jµ−1 , et1+µ −j˜ µ − et1+µ + uµ+1 ), µ−1
q δµ,2 ( 2 − L
+q +q
(−1)µ 4
ν0 4
−
)
(C.13)
(t) 0 fs (L, −Eµ−1,µ+1 + etµ −j˜ µ−1 −2 + et1+µ −j˜ µ + u1+µ )}
δµ,2 4
(t) 0 fs (L, e−1+tµ−1 − Eµ−1,µ+1 + etµ −j˜ µ−1 −1 + et1+µ −j˜ µ + uµ+1 )}
(−1)µ 4
0 {X(L, µ − 1, 1 + jµ−1 , et1+µ −j˜ µ −1 − et1+µ + uµ+1 ), (t) 0 + e1+j˜ µ−1 + et1+µ −j˜ µ −1 + uµ+1 )} fs (L, −Eµ,µ+1
with 1 + tµ−1 ≤ jµ−1 ≤ tµ − 3, 1 + tµ ≤ jµ ≤ tµ+1 − 2, 2 ≤ µ ≤ n − 1. Next we use the recursive properties of Sect. 5 along with (8.47) and (C.6) with a = 0, µ replaced by µ − 1 and (C.8) with a = 0 for µ = 2 to obtain 0 ), fs (L, ejµ {X(L, µ, jµ , uµ+1
− et1+µ +
{X(L, µ − 1, jµ−1 , e1+jµ − et1+µ +
+q
(−1)µ 4
2 0 u1+µ )} =⇒
(µ−1) l1+j −2
−→ µ−1
(t) 0 uµ+1 ), fs (L, −Eµ,µ+1
q c(jµ−1 )−
ν0 +1 3 4 + 4 θ(µ
+ ejµ−1 + e1+jµ +
odd)
×
0 u1+µ )}
0 {Y (L, µ − 1, jµ−1 , ejµ − et1+µ + uµ+1 ),
q δµ,2 ( 2 − L
+q
(−1)µ 4
ν0 4
−
)
(t) 0 fs (L, −Eµ−1,µ+1 + etµ −j˜ µ−1 −1 + ejµ + uµ+1 )
δµ,2 4
(t) 0 fs (L, e−1+tµ−1 − Eµ−1,µ+1 + etµ −j˜ µ−1 + ejµ + uµ+1 )}
(C.14) with 1 + tµ−1 ≤ jµ−1 ≤ tµ − 2, 1 + tµ ≤ jµ ≤ t1+µ − 1 + δµ,n , 2 ≤ µ ≤ n. Theorem (t) 0 specified to be −Eµ+2,n along with 1 (of Sect. 8) with P = 0 and (C.14) with uµ+1 (µ) (µ−1) l1+j +l1+j −2 µ
−→
µ−1
(µ−1) (µ) −2 l1+j −2 2 l1+j µ µ−1
= −→ =⇒
−→
(C.15)
imply (µ) (µ−1) l1+j +l1+j −2 µ
ν0 +1
3
(t) (t) ), fs (L, e1 − E1,n )} −→ q c(jµ )+c(jµ−1 )− 4 + 4 θ(µ odd) × {fs (L, −E1,n (t) (t) ), fs (L, ejµ−1 + e1+jµ − Eµ,n )} {X(L, µ − 1, jµ−1 , e1+jµ − E1+µ,n
+q
(−1)µ 4
µ−1
(C.16)
(t) {Y (L, µ − 1, jµ−1 , ejµ − E1+µ,n ),
q δµ,2 ( 2 − L
+q
(−1)µ 4
ν0 4
−
)
(t) fs (L, etµ −j˜ µ−1 −1 + ejµ − Eµ−1,n )
δµ,2 4
(t) fs (L, e−1+tµ−1 + etµ −j˜ µ−1 + ejµ − Eµ−1,n )}
Rogers–Schur–Ramanujan Identities for Minimal Models of CFT
393
with 1 + tµ−1 ≤ jµ−1 ≤ tµ − 2, 1 + tµ ≤ jµ ≤ tµ+1 + δµ,n − 1, 2 ≤ µ ≤ n. The second terms of the right-hand side of (C.16) agrees with the right-hand side of (10.15) when β = α + 1 = µ which thus completes the proof of this special case. To complete the proof of (10.15) in the general case 1 ≤ α ≤ β − 2 we use the equality of flows (α) b(α+1,β)−2 2 l1+jα −2
−→
where b(α, β) =
β X
b(α,β)−2
=⇒ −→ = −→ ,
(C.17)
(µ) l1+j ; 1 ≤ α, 1 + α ≤ β ≤ n, µ
µ=α
1 + tµ ≤ jµ ≤ tµ+1 − 3, α ≤ µ ≤ β − 2, 1 + tβ−1 ≤ jβ−1 ≤ tβ − 2, 1 + tβ ≤ jβ ≤ tβ+1 − 1 + δβ,n .
(C.18)
Equation (C.17) is just properties (3.12)-(3.13) of the b → r map with b = b(α + 1, β). Then by starting with (C.16) and by use of (C.13) and (C.14) we may prove by induction (α + 1 → α) that b(α,β)−2
(t) (t) {fs (L, −E1,n ), fs (L, e1 − E1,n )} −→ X c(4) (j)+rf(4,1) (i) q {X(L, α, jα + iα+1 , u(4) (i, j) + et1+α ), fs (L, u(4,1) (i, j))} iα+1 ,···,iβ−1 =0,1 iα =0
X
+
q c(4) (j)+rf(4,2) (i) {Y (L, α, jα + iα+1 , u(4) (i, j) + et1 +α ),
iα+1 ,···,iβ−1 =0,1 iα =1
q δ1,α ( 2 − L
ν0 4
)
fs (L, u(4,2) (i, j)) + q −
(−1)α +δ1,α 4
fs (L, u(4,3) (i, j))},
(C.19) where c(4) (j), rf(4,1) (i), rf(4,2) (i), u(4) (i, j), u(4,1) (i, j), u(4,2) (i, j). and u(4,3) (i, j) are defined in Sect. 10.3. The second terms on the right-hand side of (C.19)agree with the right-hand side of (10.15) which thus completes the proof. It remains to prove (10.10). To do this we first compare cases 1 and 2 of (7.16) with (t) E3,n replaced by u20 to infer that (0) 2 l1+j −2
0 {X(L, 1, j1 , u20 ), fs (L, ej1 − et2 + u20 )} =⇒ −→
{q − q−
ν0 −2 4
ν0 −2 4
(t) (t) fs (L, ej0 −1 + ej1 +1 − E1,2 + u20 ) + f˜s (L, eν0 −j0 + ej1 − E1,2 + u20 ),
(C.20)
(t) (t) fs (L, ej0 + ej1 +1 − E1,2 + u20 ) + f˜s (L, eν0 −j0 −1 + ej1 − E1,2 + u20 )}
with 1 ≤ j0 ≤ ν0 − 1, 1 + t1 ≤ j1 ≤ t2 − 1. Furthermore we have {X(L, 1, j1 , u20 ), fs (L, ej1 − et2 + u20 )} −→ 1
(8.2) b
{fs (L, ej1 −et2 + u20 ), q −
ν0 −2 4
(t) (t) fs (L, ej1 +1 − E1,2 + u20 ) + f˜s (L, eν0 −1 + ej1 − E1,2 + u20 )} (C.21)
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with 1 + t1 ≤ j1 ≤ t2 − 1. Analogously by comparing cases 2 and 4 of (7.18) with (t) replaced by u20 we find ej2 − E3,n ν0
(t) {Y (L, 1, j1 , u20 ), q 2 − 4 fs (L, et2 −j˜ 1 −1 − E1,2 + u20 ) L
(0) 2 l1+j −2
0 (t) + u20 )} =⇒ −→ + fs (L, e−1+t1 + et2 −j˜ 1 − E1,2
{q − q−
ν0 −2 4
1 (t) fs (L, ej0 + et2 −j˜ 1 −1 − E1,2 + u20 ) + q 2 f˜s (L, eν0 −j0 −1 + et2 −j˜ 1 + u20 ),
ν0 −2 4
1 (t) (t) fs (L, ej0 +1 + et2 −j˜ 1 −1 − E1,2 + u20 ) + q 2 f˜s (L, eν0 −j0 −2 + et2 −j˜ 1 − E1,2 + u20 )} (C.22) with 1 ≤ j0 ≤ ν0 − 1, 1 + t1 ≤ j1 ≤ t2 − 2 and ν0
(t) + u20 ) {Y (L, 1, j1 , u20 ), q 2 − 4 fs (L, et2 −j˜ 1 −1 − E1,2 L
(t) + fs (L, e−1+t1 + et2 −j˜ 1 − E1,2 + u20 )} −→ 1
(8.2) b
ν0
(t) (t) {q 2 − 4 fs (L, et2 −j˜ 1 −1 − E1,2 + u20 ) + fs (L, e−1+t1 + et2 −j˜ 1 − E1,2 + u20 ), L
q−
ν0 −2 4
1 (t) (t) fs (L, e1 + et2 −j˜ 1 −1 − E1,2 + u20 ) + q 2 f˜s (L, eν0 −2 + et2 −j˜ 1 − E1,2 + u20 )} (C.23) with 1 + t1 ≤ j1 ≤ t2 − 2. The result (10.10) follows from combining (C.19) with α = 1 with (C.20)–(C.23).
Acknowledgement. The authors are grateful to G.E. Andrews for his interest and encouragement, to K. Voss and S.O. Warnaar for discussions and careful reading of the manuscript, and to T. Miwa for many helpful comments. One of us (AB) is pleased to acknowledge the hospitality of the ITP of SUNY Stony Brook where part of this work was done. This work is supported in part by the NSF under DMR9404747.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
Rogers, L.J.: Proc. Lond. Math. Soc. 25, 318 (1894) Schur, I., Preuss, S.-B.: Akad. Wiss. Phys.–Math. Kl, 302 (1917) Rogers, L.J. and Ramanujan, S.: Proc. Camb. Phil. Soc. 19, 214 (1919) MacMahon, P.A.: Combinatory Analysis. Vol. 2, Cambridge: Cambridge University Press, 1916 Andrews, G.E.: The Theory of Partitions, Encyclopedia of Mathematics and its Applications, Vol 2, G.-C. Rota ed. Reading, MA: Addison Wesley 1976 Slater, L.J.: Proc. Lond. Math. Soc. (2) 54, 147 (1951–52) Baxter, R.J.: J. Stat. Phys. 26, 427 (1981) Andrews, G.E., Baxter, R.J. and Forrester, P. J.: J. Stat. Phys. 35, 193 (1984) Date, E., Jimbo, M., Miwa, T. and Okado, M.: Phys. Rev. B35, 2105 (1987); Date, E., Jimbo, M., Kuniba, A., Miwa, T. and Okado, M.: Nucl. Phys B 290, 231 (1987) and Adv. Stud. in Pure Math. 16, 17 (1988) Belavin, A.A., Polyakov, A.M. and Zamolodchikov, A.B.: J. Stat. Phys. 34, 763 (1984) and Nucl. Phys. B 241, 333 (1984) Feigin, B. and Fuchs, D.B.: Funct. Anal. Appl. 17, 241 (1983) Feigin, B. and Fuchs, D.B.: Lecture Notes in Math. 1060, Springer-Verlag 1984 ed. L.D. Faddeev and A.A. Malcev) 230 Lepowsky, J. and Primc, M.: Structure of the standard modules for the affine Lie algebra A(1) 1 , Contemporary Mathematics, Vol. 46, Providence, RI: AMS, 1985
Rogers–Schur–Ramanujan Identities for Minimal Models of CFT
14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46.
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Kedem, R. and McCoy, B.M.: J. Stat. Phys. 71, 865 (1993) Dasmahapatra, S., Kedem, R., McCoy, B.M. and Melzer, E.: J. Stat. Phys. 74, 239 (1994) Kedem, R., Klassen, T.R., McCoy, B.M. and Melzer, E.: Phys. Letts. B 304, 263 (1993) Kedem, R., Klassen, T.R., McCoy, B.M. and Melzer, E.: Phys. Letts. B 307, 68 (1993) Dasmahapatra, S., Kedem, R., Klassen, T.R., McCoy, B.M. and Melzer, E.: Int. J. Mod. Phys. B 7, 3617 (1993) Rocha-Caridi, A.: In: Vertex Operators in Mathematics and Physics, ed. J. Lepowsky, S. Mandelstam and I.M. Singer, Berlin: Springer, 1985 Andrews, G.E.: Proc. Nat. Sci. USA 71, 4082 (1974) Andrews, G.E.: Pac. J. Math. 114, 267 (1984) Berkovich, A., McCoy, B.M. and Orrick, W.: J. Stat. Phys. 83, 795 (1996) Berkovich, A.: Nucl. Phys. B 431, 315 (1994) Foda, O. and Warnaar, S.O.: Lett. Math. Phys. 36, 145 (1996) Warnaar, S.O.: J. Stat. Phys. 82, 657 (1996) Foda, O. and Quano, Y-H.: Int. J. Mod. Phys. A 12, 1651 (1997) Berkovich, A. and McCoy, B.M.Lett. Math. Phys. 37 49 (1996) Melzer, E.: Int. J. Mod. Phys. A 9, 1115 (1994) Schilling, A.: Nucl. Phys. B 459, 393 (1996) and Nucl. Phys. B 467, 247 (1996) Andrews, G.E.: Scripta Math. 28, 297 (1970) Foda, O. and Quano, Y.-H.: Int. J. Mod. Phys. A 10, 2291 (1995) Kirillov, A.N.: Prog. Theor. Phys. Suppl. 118, 61 (1995) Warnaar, S.O.: Commun. Math. Phys. 184, 203 (1997) Forrester, P.J. and Baxter, R.J.: J. Stat. Phys. 38, 435 (1985) Andrews, G.E., Baxter, R.J., Bressoud, D.M., Burge, W.H., Forrester, P.J. and Viennot, G.: Europ. J. Combinatorics 8, 341 (1987) Warnaar, S.O., Pearce, P.A., Seaton, K.A. and Nienhuis, B.: J. Stat. Phys. 74, 469 (1994) Takahashi, M. and Suzuki, M.: Prog. of Theo. Phys. 48, 2187 (1972) Gaspar, G. and Rahman, M.: Basic Hypergeometric Series. Cambridge: Cambridge Univ. Press, 1990, Appendix I Berkovich, A. and McCoy, B.M.: Int. J. of Math. and Comp. Modeling (in press), hepth/9508110 Baver, E. and Gepner, D.: Phys. Lett. B 372, 231 (1996) Berkovich, A., Gomez, C. and Sierra, G.: Nucl. Phys. B 415, 681 (1994) Bailey, W.N.: Proc. Lond. Math. Soc. (2) 50, 1 (1949) Agarwal, A.K., Andrews, G.E. and Bressoud, D.M.: J. Indian Math. Soc. 51, 57 (1987) Berkovich, A., McCoy, B.M. and Schilling, A.: Physica A 228, 33 (1996) Schilling, A. and Warnaar, S.O.: Int. J. Mod. Phys. B 11, 189 (1997) and A Higher–Level Bailey Lemma: Proof and Application, To appear in The Ramanujan Journal (q-alg/9607014) Berkovich, A., McCoy, B.M., Schilling, A. and Warnaar, S.O.: Bailey flows and Bose-Fermi identities (1) (1) for the conformal coset models (A(1) 1 )N × (A1 )N 0 /(A1 )N +N 0 ,. Nucl. Phys. B 499, 621 (1997)
Communicated by T. Miwa
Commun. Math. Phys. 191, 397 – 407 (1998)
Communications in
Mathematical Physics c Springer-Verlag 1998
Box Dimensions and Topological Pressure for some Expanding Maps? Huyi Hu Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA. E-mail: [email protected] Received: 6 November 1996 / Accepted: 16 May 1997
Abstract: We consider an expanding map defined on an open region of the plane and study the box dimensions of its invariant sets. Under the condition that the map leaves invariant a “strong unstable foliation” F, we prove that the box dimension of an invariant set is given by δF + δT , where δT is its dimension transverse to F and δF is the root of a certain function involving topological pressure.
0. Introduction Self-similar sets, or fractals, can often be realized as invariant sets of expanding maps. When the map is conformal, i.e. when its derivative expands by the same amount in all directions, Bowen’s formula gives a relation between fractal dimension and topological pressure. More precisely, Bowen’s formula says that the Hausdorff or box dimension of a set 3 is the unique number δ such that the topological pressure P (f |3 , δφ) = 0, where φ(x) = − log |Df (x)|. (See [B1], also see [R].) In this paper we generalize Bowen’s result to a class of expanding maps that are not conformal. Let f be a C 2 map from an open set U ⊂ R2 to R2 and let 3 ⊂ U be a compact invariant set of f on which f is expanding. Suppose that f leaves invariant a foliation F of U along which it expands more strongly than in complementary directions. We project 3 onto a curve that is transversal to the foliation to obtain a set γ. We will show that the box dimension of γ exists and call it the transverse box dimension of 3, denoted by δT . The Lipschitzness of F guarantees that δT is independent of the choice of | det Df (x)| . the transversal curve. Define φF (x) = − log |Df (x)|F and φT (x) = − log |Df (x)|F It is easy to see that there exists a unique real number δF such that the topological ? This work was done while I was in the University of Maryland and was supported by NSF under grants DMS-8802593 and DMS-9116391, and by DOE (Office of Scientific Computing.)
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pressure P (f |3 , δF φF + δT φT ) = 0. We will show that the box dimension of 3 is given by δF + δT . We mention an earlier result of Bedford ([Be], also see [BU]) in which he studied a class of self-similar sets that are graphs of continuous functions. Bedford proved that the box dimension of such a graph is equal to 1 + t, where t is the root of a pressure function similar to ours. (The setting in [Be] is actually quite different. See the Remark in Sect. 1 for more detail.) In recent years there have been a number of results on the Hausdorff and box dimensions of invariant sets of “nonconformal” maps of the plane. (See, e.g. [AJ,F,GL,KP,Mc, PU,PW].) In most (though certainly not all, see e.g. [PW]) of these results, the maps are assumed to be piecewise affine; sometimes they are assumed to have fixed rates of expansion in horizontal and vertical directions. Our setting can be viewed as slightly more general than these. Our result, however, is only for box dimension, which is less subtle than Hausdorff dimension. It would be interesting to know if a similar formula for Hausdorff dimension holds.
1. Assumptions and Statement of Results Let U ⊂ R2 be an open set and let f : U → R2 be a C 2 map. A closed subset 3 ⊂ U is called invariant if f 3 = 3. A closed invariant set 3 is called locally maximal if there exists an open neighborhood ⊂ U of 3 such that any invariant set 30 , 3 ⊂ 30 ⊂ , coincides with 3. It is easy to see that if 3 is locally maximal, then f −1 3 ∩ = 3. The map f is called expanding on 3 if there exist constants κ > 1 and C > 0 such that for all x ∈ 3, |Df n (x)v| ≥ Cκn |v| v ∈ R2 . Without loss of generality, we may assume C = 1. In this paper we always assume that 3 is a locally maximal f -invariant set and f is expanding on 3. It can be proved under this setting that ∀β > 0, ∃ > 0 such that for any -pseudo orbit in 3, there exists a point in 3 β-shadowing it. Therefore 3 has Markov partitions with diameter less than . That is, there are finite number of subsets 31 , · · · , 3S with S [ 3i such that int 3i ∩ int 3j = ∅ if i 6= j, and each f 3i is a union of sets 3j . 3= i=1
We refer the reader to [B2] for more details. We also assume that f satisfies the following. Assumption A. (i) There exists a continuous family of cones C(x) such that Dfx C(x) ⊂ int C(f x) ∀x ∈ ∩ f −1 . (ii) There exists a line bundle E, E(x) ⊂ int C(x), such that Dfx E(x) = E(f x) ∀x ∈ ∩ f −1 . In this assumption, (i) implies that f has different expanding rates (see Lemma 4.1 ∞ \ [ for a proof), while (ii) implies that E(x) = Dfyn C(y) is the strong expanding n=1 y∈f −n x
direction (see the Remark after Lemma 4.1) and the direction is independent of preimages of x. Therefore, the strong unstable foliation F is well defined, which is a family of C 2 curves tangent to E. We can use F to state an equivalent assumption.
Box Dimensions for Expanding Maps
399
Assumption A’. There exists a foliation F of C 2 on such that (i) ∃C ≥ 1, λ < 1 such that ∀n ≥ 0, | det Df n (x)| ≤ Cλn |Df n (x)|F |2 (ii) f F (x) ⊃ F(f x) ∀x ∈
T
∀x ∈
n \
f −i ;
i=0
f −1 .
We say that F is Lipschitz, if for any smooth curve 0 ⊂ transversal to F -leaves, the map π : x → 0 ∩ F(x) is Lipschitz whenever it is defined. It is easy to see that Df induces a contract map F which sends the set of all line bundle L = {L(x) ∈ C(x) : x ∈ } to itself, and E is the unique fixed point of F . Moreover, by the same argument as in [HP] or [HPS], we can get that {E(x)} is C 1 and therefore, the strong unstable foliation F is C 1 . However, we only need that the foliation F is Lipschitz. We state it as the following. Fact. Under above assumptions, F is Lipschitz. Let i ⊂ be an open set containing a small neighborhood of 3i for each 3i . We may assume that the diameter of 3i is small enough such that restricted to i f is injective. Since E(x) is Lipschitz, we may also assume that ∀x, y ∈ i , the angle between E(x) and E(y) is small. So up to a coordinate system change, restricted to each i we can regard F (x) as a foliation given by parallel lines. For each i , take a C 1 curve 0i in the convex hull of i transversal to F-leaves. Let π : i → 0i be a continuous map defined by sliding along the F -leaves, i.e. for x ∈ i , π(x) = 0i ∩ F(x). The Lipschitzness means that π is Lipschitz. We assume that 0i is taken in such a way S S [ [ that π(x) is defined for all x ∈ i . Put γi = π3i , πi = π|i , 0 = 0i and γ = γi . i=1
i=1
The upper and lower box dimension of a bounded set A in a metric space are defined by dimB (A) = lim sup β→0
log N (A, β) − log β
and
dimB (A) = lim inf β→0
log N (A, β) − log β
respectively, where N (A, β) denotes the minimal number of balls of radius β that cover A. If dimB (A) = dimB (A), then we call the common value the box dimension of A and denote it by dimB (A). Since the projection π is Lipschitz, it is easy to see that both dimB (γ) and dimB (γ) are independent of the choice of 0. We will prove in Sect. 3 that dimB (γ) = dimB (γ). Therefore it makes sense to write δT = dimB (γ) and call it the transverse box dimension of 3, Denote |Df (x)|F = |Df (x)|E(x) | and |Df (x)|T = |det Df (x)| |Df (x)|F , where the subscripts F and T refer to “Foliation” and “Transversion” respectively. Put φF (x) = − log |Df (x)|F
and
φT (x) = − log |Df (x)|T .
Both are Lipschitz functions. Let P (f, φ) denote the topological pressure of f for the function φ (see e.g. [W] for a definition). P (f |3 , tφF + δT φT ) decreases as t is increased and goes to ±∞ as t goes to ∓∞. Now we state our result.
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Theorem. Let f be a C 2 map from an open set U ⊂ R2 to R2 and let 3 ⊂ U be a locally maximal compact invariant set of f on which f is expanding and topologically transitive. Suppose Assumption A or A’ is satisfied. Then the box dimension dimB (3) of 3 is equal to δF + δT , where δT is the transverse box dimension of 3, and δF = t is the unique real number such that the topological pressure P (f |3 , tφF + δT φT ) = 0. Remark. We mention some differences between Bedford’s setting [Be] and ours. In [Be] the sets are assumed to be graphs of continuous functions and the maps are assumed to preserve the weak unstable foliation rather than the strong unstable foliation. Bedford’s assumption is natural in the setting of fractal graphs. For general invariant sets, however, it seems more natural to work from the strong unstable directions than from the weak ones. See e.g. [LY]. We introduce some more notations. Suppose f 3i ⊃ 3j for some i and j. We denote by 3ij the component of the preimage of 3j contained in 3i , i.e. 3ij = 3i ∩ f −1 3j . Generally, if f 3ij ⊃ 3ij+1 for j = 0, · · · , k − 1, then we write 3i0 i1 ···ik =
k \
f −l 3ij ,
j=0
i0 i1 ···ik =
k \
f −l ij .
j=0
Since f is expanding on 3, we may regard that restricted to i0 i1 ···ik , f k : i0 i1 ···ik → ik is a diffeomorphism. Also we write 0i0 i1 ···ik = f −k 0ik ∩ i0 i1 ···ik ,
γi0 i1 ···ik = f −k γik ∩ i0 i1 ···ik .
Note that we have 3i0 i1 ···ik ⊂ 3i0 and i0 i1 ···ik ⊂ i0 , but in general 0i0 i1 ···ik 6⊂ 0i0 and γi0 i1 ···ik 6⊂ γi0 . Instead, the relations are π0i0 i1 ···ik ⊂ 0i0 and πγi0 i1 ···ik ⊂ γi0 . A word i0 i1 · · · ik , k ≤ ∞, is called admissible if f 3ij ⊃ 3ij+1 for j = 0, · · · , k − 1. In our proof we will use “≈” for the meaning “up to a constant factor”, where the constant depends only on f and 3. 2. Examples Example 2.1. Let R be a unit square in R2 . Take real numbers b > a > 1. Put n rectangles R1 , · · · , Rn of width a−1 and height b−1 in R, aligned with the axis of R. We require that these n small rectangles are disjoint and that their projections onto a horizontal line either coincide or are disjoint. Therefore these small rectangles form p columns, p ≤ n. Define a C 2 map f on R2 such that f maps each of Ri affinely onto R. Write fi = f |Ri : Ri → R, i = 1, · · · , n. The n inverses fi−1 determine an f -invariant set n S 3 ⊂ Ri in a usual way (see [H]). It is easy to see that 3 is a locally maximal invariant i=1
set of f , and f is expanding and topologically mixing on 3. Moreover, restricted to each small rectangle, f preserves vertical lines and expands faster along the vertical direction than the horizontal direction.
Box Dimensions for Expanding Maps
401
log p . Note that ∀x ∈ 3, log a φT (x) = − log a and φF (x) = − log b. Since both functions are constants, we have (see [W] Sect. 9.2) Clearly, the transverse box dimension of 3 is δT =
P (f |3 , tφF + δT φT ) = htop (f |3 ) − t · log b −
log p · log a = log n − t log b − log p. log a
Put P (f |3 , tφF + δT φT ) = 0 and then solve for t. We get t = result, dimB (3) =
log n − log p log p + . log b log a
log n − log p . So by our log b
log 2 . log 2 + = log 5 log 3.8 2 4 T S f −k Ri .) 0.431 + 0.519 = 0.950. (In the figure the black rectangles form the set In Fig. 1, a = 3.8, b = 5, n = 4 and p = 2. We have dimB (3) =
k=0
Figure 1
i=1
Figure 2
Example 2.2. We perturb the position of the small rectangles in Example 1. The perturbation is not necessarily small, as long as the rectangles stay in the unit square and disjoint. Let π be a projection from the unit square into a horizontal line, and let πi = π|Ri . Then γ = π3 is the invariant set for the contracting maps πi f −1 π −1 , i = 1, · · · , n. By Falconer’s result ([F], Theorem 5.4 and 5.3), for almost every such perturbation, the log n }. It is equal to 1 if transverse box dimension δT = dimB (γ) is equal to min{1, log a n ≥ a. Let 3 be such a set. Then we have 0 = P (f |3 , tφF + δT φT ) = htop (f |3 ) − t · log b − 1 · log a = log n − t log b − log a. Therefore t =
log n − log a log n − log a and the box dimension of 3 is + 1. log b log b
In Fig. 2 we take a = 3.8, b = 5 and n = 4. The box dimension of a set obtained in log 4 − log 3.8 . + 1 = 0.032 + 1 = 1.032. the above way is log 5
402
H. Hu
Remark. By the theorem stated in last section, we can see that the box dimensions of the invariant sets in the above examples do not change if we move the small rectangles along the vertical direction, as long as these rectangles are separated from each other.
3. Transverse Box Dimension Lemma 3.1 (Distortion estimates). There exists a constant J > 1 such that for any admissible word i0 i1 · · · ik , J −1 ≤
|Df k (z)|F ≤J |Df k (y)|F
and
J −1 ≤
|Df k (z)|T ≤ J, |Df k (y)|T
provided both y, z ∈ i0 i1 ···ik . Proof. Note that both |Df (x)|F and |det Df (x)| are Lipschitz, and so is |Df (x)|T . Also, f is uniformly expanding. With these facts, the arguments are standard. (For the arguments, see e.g. [M] pp. 173–174, [G] p. 71, or [MS] p. 353.) Lemma 3.2. There exists a constant K > 1 such that for any bounded subset A ⊂ R2 and for any constants β > 0, a > 1, a2 KN (A, aβ) ≥ N (A, β) ≥ N (A, aβ). Proof. This is clear.
Lemma 3.3. There exists a constant L > 1 such that for any i = 1, · · · , S, LN (γi , β) ≥ N (γ, β). Proof. Since f is topologically transitive on 3, we can find k > 0 such that ∀i = 1, · · · , S, f k i ⊃ . Put i0 = i. For each 1 ≤ j ≤ S, we fix one admissible word i0 i1 · · · ik with ik = j. Let γi0 i1 ···ik denote the preimage of γj under f k |i0 i1 ···ik . Note S [ that the projection πi : γi0 i1 ···ik → γi is an at most S to 1 map. We have ik =1
SN (γi , β) ≥
S X
N (γi0 i1 ···ik , β).
ik =1
By Lemma 3.2, N (γi0 i1 ···ik , β) ≥ N (γj , β|Df k |) ≥ (K|Df k | )−1 N (γj , β), 2
k
k
where |Df | = max|Df (x)|. Since N (γ, β) = x∈
S X
N (γj , β), the result follows.
j=1
Lemma 3.4. There exists b > 0 such that for any β > 0 small, for any m, n > 0, N (γ, β m+n ) ≥ bN (γ, β m )N (γ, β n ).
Box Dimensions for Expanding Maps
403
Proof. Since N (γ, β n ) denote the minimal number of β n -balls covering γ, we can find yj ∈ γ, j ≥ 1, · · · , 21 N (γ, β n ), such that each B(yj , 21 β n ) is contained in some i and B(yj 0 , 21 β n ) ∩ B(yj 00 , 21 β n ) = ∅ if j 0 6= j 00 . ˜ j , 1 β n ) = π −1 B(yj , 1 β n ), Suppose B(yj , 21 β n ) ∈ i0 for some i0 . Denote B(y i0 2 2 where πi : i → γi is the projection defined in Sect. 1. We can find an admissible word i0 i1 · · · ik such that ˜ j , 1 βn) i0 ···ik ⊂ B(y 2
but
˜ j , 1 β n ). i0 ···ik−1 6⊂ B(y 2
Recall that f k : i0 ···ik → ik is a diffeomorphism. By the distortion estimates, it is easy to check that for any α < 21 β n , z ∈ γi0 ···ik , B f k z, αJ −1 |Df k (x)| ¯ T ∩γik ⊂ f k B(z, α)∩γi0 ···ik ⊂ B f k z, αJ|Df k (x)| ¯ T ∩ γ ik , where x¯ = x¯ i0 ···ik ∈ i0 ···ik . That is, under the map f k , the image of an α-ball in γi0 ···ik is a subset of γik with diameter between 2αJ −1 |Df k (x)| ¯ T and 2αJ|Df k (x)| ¯ T. Therefore, N (γi0 ···ik , α) ≈ N (γik , α|Df k (x)| ¯ T ). n ¯ −1 By the distortion estimates, we have |Df k (x)| T ≈ diam(πi0 i0 ···ik ) ≈ β . Hence,
N (γi0 ···ik , α) ≈ N (γik , αβ −n ). Note that πi0 : γi0 ···ik → B(yj , 21 β n ) ∩ γi0 is an injective Lipschitz map. We have 1 N B(yj , β n ) ∩ γ, α ≥ N πi0 γi0 ···ik , α ≈ N (γi0 ···ik , α). 2 So there exist a constant c > 0 independent of the choice β, n, and yj such that 1 N B(yj , β n ) ∩ γ, α ≥ cN γik , αβ −n . 2 By Lemma 3.3 and this inequality, n 1 2 N (γ,β )
N (γ, α) ≥
X j=1
1 1 N B(yj , β n ) ∩ γ, α ≥ N (γ, β n ) · cL−1 N γ, αβ −n 2 2
1 ≥ cL−1 N (3, β n )N γ, αβ −n . 2 In particular, if we take α = β m+n , then the result follows by putting b =
1 −1 cL . 2
Proposition 3.5. dimB (γ) = dimB (γ). Proof. By Lemma 3.4, the sequence {log bN (γ, β n )} is superadditive. So the limit log bN (γ, β n ) n→∞ − log β n lim
exists. This implies the result of the proposition.
404
H. Hu
4. Proof of the Theorem Lemma 4.1. Under Assumption A, there exist C ≥ 1 and λ < 1 such that ∀n ≥ 0, | det Df n (x)| |Df n (x)|T = ≤ Cλn |Df n (x)|F |Df n (x)|2F
∀x ∈
n \
f −i .
i=0
Proof. For each x ∈ , take a coordinate system such that the cone C(x) is the union of the first and the third quadrants. Since C(x) is continuous, we may require that the change of the coordinate is continuous as x varies. a(x) b(x) Let denote the matrix of the map Df (x) under the coordinate systems c(x) d(x) and let 1(x) = |a(x)d(x) − b(x)c(x)|. The fact that Dfx C(x) ⊂ int C(f x) implies that for each x, a(x), b(x), c(x) and d(x) have the same sign. By continuity, we have b0 = min{|b(x)| : x ∈ ∩ f −1 } > 0 and c0 = min{|c(x)| : x ∈ ∩ f −1 } > 0. Also write 10 = max{1(x) : x ∈ ∩ f −1 }. 0 0 vf i x vx = Dfxi vx . We have ∈ E(x) and denote vf i x = Take a vector vx = vx00 vf00i x 2 2 vf0 x vf00x = a(x)d(x) + b(x)c(x) vx0 vx00 + a(x)c(x)vx0 + b(x)d(x)vx00 > 1(x) 1 +
2b0 c0 0 00 v x vx . 10
Therefore, vf0 n x vf00n x > vx0 vx00
n−1 Y
1(f i x) 1 +
i=0
2b0 c0 . 10
Since vx is away from the boundary of C(x), |vx |2 ≈ vx0 vx00 ∀x ∈ . Also | det Df n (x)| ≈ n−1 Y 1(f i x). So the above inequality gives i=0
|Df n (x)vx |2 > C −1 | det Df n (x)| 1 +
2b0 c0 n |vx |2 10
(1)
for some C > 1 independent of x and n. By the definition of | · |F and | · |T , we get | det Df n (x)| 2b0 c0 −n |Df n (x)|T =
Box Dimensions for Expanding Maps
405
Proof of the Theorem. Let β be small. We curtail each admissible infinite sequence i0 i1 · · · after the first term ik such that for any x ∈ i0 ···ik , the length of F(x) ∩ i0 ···ik is less than β and such that k is the smallest integer for the sequence with this property. Clearly, all such i0 ···ik form a cover of 3, denoted by C. By the distortion estimates we know that for any xi0 ···ik ∈ i0 ···ik , −1 k β ≈ diam ik · |Df k (xi0 ···ik )|−1 F ≈ |Df (xi0 ···ik )|F ,
(2)
and the width of i0 ···ik is proportional to |Df k (x)|−1 T . Since the height of i0 ···ik is less than β, we have N (3i0 ···ik , β) ≈ N (γi0 ···ik , β). Note that f k : γi0 ···ik → γik is a diffeomorphism. Lemma 3.1 and 3.3 imply N (γi0 ···ik , β) ≈ N (γik , β|Df k (xi0 ···ik )|T ) ≈ N (γ, β|Df k (xi0 ···ik )|T ), where xi0 ···ik ∈ i0 ···ik . So we get X N (3i0 ···ik , β) ≈ N (3, β) ≈ i0 ···ik ∈C
X
N (γ, β|Df k (xi0 ···ik )|T ).
(3)
i0 ···ik ∈C
By (2), β|Df k (xi0 ···ik )|T ≈
|Df k (xi0 ···ik )|T . |Df k (xi0 ···ik )|F
(4)
Lemma 4.1 implies that β|Df k (xi0 ···ik )|T → 0 as β → 0. Using the definition of box dimension, for any α > 0, we can find a constant C1 > 1 such that ∀β > 0, C1−1
|Df k (x
i0 ···ik )|F
δT −α
|Df k (xi0 ···ik )|T
≤ N (γ, β|Df k (xi0 ···ik )|T ) ≤ C1
For a function φ, denote Sk φ(x) = We have |Df k (x)| δT F
|Df k (x)|T
k−1 X
|Df k (x
i0 ···ik )|F
|Df k (xi0 ···ik )|T
δT +α
.
(5)
φ(f i x). Recall the definition of φT and φF .
i=o
n o = exp δT Sk φT (x) − δT Sk φF (x) n o n o = exp δT Sk φT (x) + δF Sk φF (x) exp −δT − δF Sk φF (x) .
Let µ be the Gibbs state of the function δT φT +δF φF . The fact that P (f, δT φT +δF φF ) = 0 implies that n o exp δT Sk φT (xi0 ···ik ) + δF Sk φF (xi0 ···ik ) ≈ µ3i0 ···ik for any xi0 ···ik ∈ 3i0 ···ik (see e.g. [B2]). Also, by (2) o n exp −δT − δF Sk φF (xi0 ···ik ) = |Df k (xi0 ···ik )|δFT +δF ≈ β −(δT +δF ) ,
406
H. Hu
and by (4), |Df k (x
i0 ···ik )|F
|Df k (x
α
i0 ···ik )|T
≈ β|Df k (xi0 ···ik )|T
−α
< β −α .
So the second inequality of (5) can be written as N (γ, β|Df k (xi0 ···ik )|T ) ≤ C2 µ3i0 ···ik β −(δT +δF +α) for some C2X > 1. Note µ3i0 ···ik = 1. Therefore by (3) we obtain that i0 ···ik ∈C
N (3, β) ≤ C
X
N (γ, β|Df k (xi0 ···ik )|T ) ≤ CC2 β −(δT +δF +α) .
i0 ···ik ∈C
Consequently, dimB (3) = lim sup β→0
log N (3, β) ≤ δT + δF + α. − log β
Similarly, by using the first inequality of (5) we can get dimB (3) = lim inf β→0
log N (3, β) ≥ δT + δF − α. − log β
Since these two inequalities are true for any α > 0, we know that dimB (3) = δT + δF . Acknowledgement. It is my pleasure to thank Professor Lai-Sang Young for her valuable help in forming the assumptions. I thank Professor M. Boyle and Professor J.Yorke for stimulating conversations. I would also thank the referee for carefully reading the manuscript and for suggesting various improvements.
References [AJ] [B1]
Alexander, J.C., Yorke, J.A.: Fat baker’s transformation. Ergodic Theory Dyn. Syst. 4, 1–23 (1984) ´ Bowen, R.: Hausdorff dimension of quisi-circles. Inst. Hautes. Etudes Sci. Publ. Math. 50, 11–25 (1979) [B2] Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lect. Notes Math. 470, New York: Springer-Verlag, 1975 [Be] Bedford, T.: The box dimension of self-affine graphs and repellers. Nonlinearity 2, 53–71 (1989) [BU] Bedford, T., Urba´nski, M.: The box and Hausdorff dimension of self-affine sets. Ergodic Theory Dyn. Syst. 10, 627–644 (1990) [F] Falconer, K.: The Hausdorff dimension of self-affine fractals. Math. Proc. Camb. Phil. Soc. 103, 339–350 (1988) [G] Godbillon, C.: Dynamical Systems on Surfaces. Berlin–Heidelberg–New York: Springer-Verlag, 1983 [GL] Gatzouras, D., Lalley, S.: Hausdorff and box dimensions of certain self-affine fractals. Indiana Univ. Math. J. 41, 533–568 (1992) [H] Hutchinson, J.E.: Fractals and self similarity. Indiana Univ. Math. J. 30, 713–747 (1981) [HP] Hirsch, M., Puch, C.: Stable manifolds and hyperbolic sets. In: Proc. Symp. in Pure Math. Vol 14, Providence RI: AMS, 1970, pp. 133–164 [HPS] Hirsch, M., Puch, C., Shub, M.: Invariant manifolds. Lect. Notes Math. 470, Berlin–Heidelberg–New York: Springer-Verlag, 1977
Box Dimensions for Expanding Maps
[KP] [LY] [M]
407
Kenyon, R., Peres, Y.: Hausdorff dimensions of affine-invariant sets. Preprint. Ledrappier, F., Young, L-S.: The metric entropy of diffeomorphisms. Ann. Math. 122, 509–574 (1985) Ma˜ne´ , R.: Ergodic Theory and Differentiable Dynamics. Berlin–Heidelberg–New York: SpringerVerlag, 1987 [Mc] McMullen, C.: The Hausdorff dimension of general Sierpinski carpets. Nagoya Math. J. 96, 1–9 (1984) [MS] De Melo, W., Van Strien, S.: One-Dimensional Dynamics. Berlin–Heidelberg–New York: SpringerVerlag, 1993 [PU] Przytycki, F., Urbanski, M.: On the Hausdorff dimension of some fractal sets. Studia Math. 93, 155–186 (1989) [PW] Pesin, Y., Weiss. H.: On the dimension of a general class of geometrically defined deterministic and random cantor-like sets. Preprint [R] Ruelle, D.: Bowen’s formula for the Hausdorff dimension of self-similar sets. In: Scaling and Selfsimilarity in Physics, Progress in Physics. Vol. 7, Boston: Birkh¨auser, 1983, pp. 351–358 [W] Walters, P.: An Introduction to Ergodic Theory. New York–Heidelberg–Berlin: Springer-Verlag, 1981 Communicated by Ya. G. Sinai
Commun. Math. Phys. 191, 409 – 466 (1998)
Communications in
Mathematical Physics c Springer-Verlag 1998
Dyonic Sectors and Intertwiner Connections in 2+1-Dimensional Lattice ZN -Higgs Models Jo˜ao C. A. Barata1 , Florian Nill2 1 Instituto de F´ısica da Universidade de S˜ ao Paulo, P. O. Box 66318, S˜ao Paulo 05315 970, SP, Brasil. E-mail: [email protected] 2 Institut f¨ ur Theoretische Physik der Freien Universit¨at Berlin, Arnimallee 14, Berlin 14195, Germany. E-mail: [email protected]
Received: 13 December 1996 / Accepted: 19 May 1997
Abstract: We construct dyonic states ωρ in 2+1-dimensional lattice ZN -Higgs models, i.e. states which are both, electrically and magnetically charged. These states are parametrized by ρ = (ε, µ), where ε and µ are ZN -valued electric and magnetic charge distributions, respectively, living on the spatial lattice Z2 . The associated Hilbert spaces Hρ carry charged representations πρ of the observable algebra A, the global transfer matrix t and a unitary implementation of the group Z2 of spatial lattice ! translations. We X X ε(x), µ(p) ∈ ZN × ZN these prove that for coinciding total charges qρ = x
p
representations are dynamically equivalent and we construct a local intertwiner connection U (0) : Hρ → Hρ0 , where 0 : ρ → ρ0 is a path in the space of charge distributions Dq = {ρ : qρ = q}. The holonomy of this connection is given by ZN -valued phases. This will be the starting point for a construction of scattering states with anyon statistics in a subsequent paper. Contents 1 2 2.1 2.2 2.3 2.4 3 3.1 3.2 3.3 3.4 4
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 The Basic Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 The local algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 Local transfer matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 Ground States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 External charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 The Construction of Dyonic Sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 Dyonic states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 Global transfer matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 Global charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 The dyonic self energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 State Bundles and Intertwiner Connections . . . . . . . . . . . . . . . . . . . . . . . 437
410
J. C. A. Barata, F. Nill
4.1 4.2 4.3 4.4 A A.1 A.2 B B.1 B.2 B.3 B.4 B.5 B.6 B.7
The local intertwiner algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 The intertwiner connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 The representation of translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 A Brief Sketch of the Polymer and Cluster Expansions . . . . . . . . . . . . . 447 Expansions for the vacuum sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 Expansions for the dyonic sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 The Remaining Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 Proof of Proposition 3.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 Proof of Propositions 3.1.2 and 3.1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 Proof of Theorem 3.1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 Proof of Proposition 3.4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 Completing the proof of Proposition 3.3.2 . . . . . . . . . . . . . . . . . . . . . . . 460 Proof of Theorem 4.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 Proof of Proposition 4.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
1. Introduction In this paper we continue our project initiated in [1] of a constructive analysis of states with anyonic statistics in a 2+1 dimensional lattice gauge theory. We investigate a general class of models with the discrete Abelian gauge group ZN for arbitrary N ∈ N, N ≥ 2, and with discrete Higgs fields. The vacuum expectations in this theory are represented by classical expectations of an (euclidean) statistical mechanics model given by the thermodynamic limit of finite volume expectations of the form R dαdϕ Bcl (α, ϕ) e−S3 R , (1.1) hBcl i3 := dαdϕ e−S3 with a generalized Wilson action: X X Sg (dα(p)) + Sh (dϕ(b) − α(b)) . S3 (ϕ, α) := p
(1.2)
b
Here ϕ and α are ZN -valued Higgs and gauge fields, respectively, on a euclidean spacetime lattice Z3 . Hence ϕ lives on sites, α lives on bonds, d denotes the lattice exterior derivative and the above sums go over all elementary positively oriented bonds b and plaquettes p in a finite space-time volume 3 of our lattice. Bcl is some classical observable, i.e. a gauge invariant function of the gauge field α and of the Higgs field ϕ with finite support. Here, gauge invariance means invariance under the simultaneous transformations ϕ → ϕ + λ and α → α + dλ for arbitrary λ : Z3 7→ ZN , with finite support. Above, the integrations over α and ϕ are actually finite sums, since these variables are discrete (to be precise, the discrete Haar measure on ZN is employed). In order to have charge conjugation symmetry and reflection positivity (i.e. a positive transfer matrix) the actions Sg and Sh will be chosen as even functions on ZN ≡ {0, . . . , N − 1} taking their minimal value at 0 ∈ ZN . Thus we have a general Fourier expansion N −1 1 X 2πmn , (1.3) βg/h (m) cos Sg/h (n) = − √ N N m=0
Dyonic Sectors and Intertwiner Connections in 2+1-D Lattice ZN -Higgs Models
411
n ∈ ZN , where we call βg (m) and βh (m), m ∈ ZN , the gauge and Higgs coupling constants, respectively1 . We also require βg/h (m) = βg/h (N − m), i.e. βg/h and −Sg/h are in fact Fourier transforms of each other. This model describes a ZN -Higgs model where the radial degree of freedom of the Higgs field is frozen, i.e. |φ| = 1 and φ(x) = 2πi e N ϕ(x) . We will be interested in the so-called “free charge phase” of this model, which, roughly speaking, is obtained whenever, for all 0 6= n ∈ ZN , Sg (n) − Sg (0) ≥ cg 0, Sh (n) − Sh (0) ≤
c−1 h
1,
(1.4) (1.5)
for large enough positive constants cg and ch . In this region convergent polymer and cluster expansions are available and have been analyzed in detail in [1], see also Appendix A for a short review. The first analysis of the structure of the charged states in this phase had been performed for the case of the group Z2 by Fredenhagen and Marcu in [2]. In that work it had been shown that electrically charged states exist in d + 1 dimensions, d ≥ 2 in the “free charge phase” of the model. The ideas employed by the authors involved a wide combination of methods from Algebraic Quantum Field Theory and Classical Statistical Mechanics. Later on the existence of electrically charged particles in the same model had been shown in [3] and the existence of multi-particle scattering states of these particles had been proven in [4], combining methods and results of [2] and of [5]. In [1] we extended some of these results to the ZN -Higgs model mentioned above and showed, after previous results of Gaebler on the Z2 case [6], the existence of magnetically charged states in 2 + 1 dimensions. In [1] we also proved the existence of electrically and of magnetically charged particles in this model. Our intention here is to show the existence of dyonic states ωρ in the “free charge phase” of our ZN -Higgs model, i.e. states carrying simultaneously electric and magnetic charges, ρ = (ε, µ). This had been performed in the Z2 case in [6]. We construct the associated charged representations of the observable algebra A as the GNS-triples (πρ , Hρ , ρ ) obtained from ωρ . These representations ! fall into equivalence classes laX X beled by the total charges qρ = ε(x), µ(p) ∈ ZN × ZN . We show that for x
p
each choice of q ∈ ZN × ZN the state bundle [ (ρ, Hρ ) Bq := ρ∈Dq
over the discrete base space Dq := {ρ : qρ = q} is equipped with a non-flat connection, i.e. a collection of unitary parallel transporters Uρ0 , ρ : Hρ → Hρ0 depending on paths 0 from ρ to ρ0 in Dq such that πρ0 = Ad Uρ0 ,ρ ◦ πρ . The holonomy of this connection is given by ZN -valued phases which appear as winding numbers between “electric” Wilson loops and “magnetic” vortex loops in the euclidean functional integral picture. In an upcoming paper [7] this construction will be the starting point for an analysis of the anyonic statistics of scattering states in these models. At this point we should also mention the previous work of J. Fr¨ohlich and P. A. Marchetti on the construction of dyonic and anyonic states in the framework of euclidean 1 The values of β (0) and β (0) only determine additive constants to the action and will be fixed by g h convenient normalization conditions.
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lattice field theories ([8, 9] and [10]). In their approach the analogue of our Hilbert spaces Hρ are obtained by Osterwalder-Schrader reconstruction methods which makes it difficult to discuss the representation theoretic background of the observable algebra. In difference with their approach we work with the Hamiltonian description. In particular, we construct our states as functionals on the quasi-local observable algebra generated by the time-zero fields. As usual, in this approach the euclidean description reappears in the form of local transfer matrices TV , whose ground state expectation values are given by functional integrals of the type (1.1). Still, we interpret and motivate our constructions mostly algebraically and consider the functional integral techniques only as a technical tool. We now describe the plan of this paper in some more detail. In Sect. 2 we introduce our basic setting. We follow the standard canonical quantization prescription of gauge theories in A0 = 0 gauge to define the local algebras generated by the time-zero fields. We then define an “euclidean dynamics” in terms of local transfer matrices whose ground states give rise to finite volume euclidean functional integral expectations with actions (1.2)-(1.3). We also review the notion of external charges (related to a violation of Gauss’ law) in order to distinguish them from the dynamical charges (“superselection sectors”) that we are interested in this work. Section 3 is devoted to the construction of dyonic sectors. We start with generalizing the Fredenhagen-Marcu prescription to obtain dyonic states ωρ on our observable algebra A. We then construct the associated GNS-representations (πρ , Hρ , ρ ) of A and implement the euclidean dynamics by a global transfer matrix on Hρ . We show ˆ ⊃ A (i.e. the ∗-algebra generated that as representations of the “dynamic closure” A by A and the global transfer matrix) these representations are irreducible and pairwise equivalent provided their total charges qρ ∈ ZN ×ZN coincide. (We also conjecture that they are dynamically inequivalent, if their total charges disagree.) Finally we show that the infimum of the energy spectrum in the dyonic sectors is uniquely fixed by requiring charge conjugation symmetry and cluster properties of correlation functions for infinite space-like separation. M Hρ by defining electric In Sect. 4 we construct an intertwiner algebra on Hq = ρ∈Dq
and magnetic “charge transporters” Eq (b), Mq (b) ∈ B(Hq ) living on bonds b in Z2 and fulfilling local Weyl commutation relations. In terms of these intertwiners we obtain a unitary connection U (0) : Hρ → Hρ0 intertwining πρ and πρ0 for any path 0 : ρ → ρ0 in Dq . The holonomy of this connection is given by ZN -valued phases. We conclude by applying our connection to construct a unitary implementation of the translation group in the dyonic sectors. We remark at this point that we do not touch the question of the existence of dyonic particles in these models, i.e. of particles in the dyonic sectors carrying simultaneously electric and magnetic charges. To study the existence of such particles requires adaptation of the known Bethe-Salpeter kernels methods for situations involving charged particles in lattice models. This will be performed elsewhere. Clearly, if these particles exist they should be expected to show anyonic statistics among themselves. A good part of the methods and results used here has been extracted from [1] and we will often refer to this paper when necessary. In particular, we will not repeat the proof of the convergence of the polymer expansion we are going to use since this point has been discussed in detail in [1], see however Appendix A for a short review. In fact, although polymer expansions are the main technical tool of this work (as well as of all other works on ZN -Higgs models cited above), our aim here is to formulate theorems
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and present results in a way which can be followed without any detailed knowledge on cluster expansions. Following this strategy, we abandon all statistical mechanics aspects of our proofs to Appendix B and reserve the main body of this work to exploit algebraic and quantum field theoretical concepts. Remarks on the notation.. Due to a different focus our notation will differ in many points from that of [1]. We will change our notation according to our needs of emphasis and clarity. The symbol indicates “end of proof”. Products of operators run from the left n Y Aa means A1 · · · An . For an invertible operator B, Ad B denotes the to the right, i.e. a=1
automorphism B · B −1 . If A ⊂ B(H) is an algebra acting on a Hilbert space H then we denote by A0 the commutant of A, i.e. the set of all operators of B(H) which commute with all elements of A. Here B(H) is the algebra of all bounded operators acting on H. 2. The Basic Setting We will always consider the lattice Zd , d = 2, 3 as a chain complex and denote by (Zd )p the elementary positively oriented p-cells in Zd . We also use the standard terminology “sites”, “bonds” and “plaquettes” for 0-, 1- and 2-cells, respectively. By a (finite) volume V ⊂ Zd we mean the closed chain sub-complex generated by a (finite) union of elementary d-cells in (Zd )d . We denote by Vp the set of elementary oriented p-cells in V where, by definition, a p-cell is contained in Vp ⊂ (Zd )p if and only if it lies in the boundary of some (p+1)-cell contained in Vp+1 . We denote by Cp (V ) ≡ ZVp the set of p-chains in V and by C p (V ) ≡ C p (V, ZN ) the set of ZN -valued cochains with support in V (i.e. group homomorphisms α : Cp (V ) → ZN ). As usual we identify C p (V ) with the group of ZN -valued functions on (Zd )p with support in Vp . Hence, for V ⊂ W we have the natural inclusion C p (V ) ⊂ C p (W ). We also denote C p := C p (Zd ) and define p ⊂ C p as the set of p-cochains with finite support. Often we will identify an elemenCloc tary p-cell c ∈ (Zd )p with its characteristic p-cochain (i.e. taking the value 1 ∈ ZN on c and 0 ∈ ZN else). |V | Considered as a finite Abelian group C p (V ) ∼ = ZN p is self-dual for all finite V , the pairing C p × C p → U (1) being given by the homomorphism X 2πi α(c)β(c) . (2.1) (α, β) 7→ eihα, βi := exp N c∈Vp
We denote by d : C p → C p+1 and d∗ : C p → C p−1 the exterior derivative and its adjoint, such that ∗ eihα, dβi = eihd α, βi p p−1 for all α ∈ Cloc and all β ∈ Cloc . In the main body of this paper we will be working with “time-zero” fields, i.e. cochains defined on the spatial lattice Z2 . The translation to the euclidean functional integral formalism will bring us to a space-time lattice Z3 . To unload the notation we will use the same symbols for both pictures as long as the meaning becomes obvious from the context. Hence, in both pictures ϕ ∈ C 0 will denote the Higgs field, α ∈ C 1 will denote the gauge field and a gauge transformation consists of a mapping (ϕ, α) 7→ (ϕ+λ, α+dλ) 0 . with λ ∈ Cloc
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2.1. The local algebras. As is well known, a d + 1-dimensional lattice system described by (1.1,1.2) can typically also be described as a quantum spin system, using the transfer matrix formalism. By this we mean an operator algebra living on a d-dimensional spatial lattice, together with discrete “euclidean time” translations given by e−tH , t ∈ Z, where T ≡ e−H is the transfer matrix. In this formulation expectations like (1.1) represent the vacuum or ground state of the “euclidean” dynamics defined by the transfer matrix. We have described in detail the quantum spin system of our model in [1] (see also [2]). It corresponds to the Weyl form of the usual canonical quantization prescription in temporal (α0 = 0) gauge. Let us recall here its main ingredients. On the spatial lattice Z2 we introduce the local algebra of time-zero Higgs and gauge fields in the following way. To each x ∈ (Z2 )0 we associate the unitary ZN -fields PH (x) and QH (x) and to each b ∈ (Z2 )1 we associate the unitary ZN -fields PG (b) and QG (b) (the subscripts G and H stand for “gauge” and “Higgs”, respectively) satisfying the relations: (2.2) PH (x)N = QH (x)N = PG (b)N = QG (b)N = 1l, and the ZN -Weyl algebra relations PG (α)QG (β) = e−ihα, βi QG (β)PG (α),
(2.3)
PH (γ)QH (δ) = e−ihγ, δi QH (δ)PH (γ),
(2.4)
1 ∈ Cloc γ(x)
where Y α, β PH (x)
0 Cloc
and γ, δ ∈ play the rˆole of test functions, i.e. PH (γ) := , etc. Operators localized at different sites and bonds commute and the
x∈(Z2 )0
G-operators commute with the H-operators. We denote [δQH ](α) := QH (d∗ α), [δ ∗ PG ](β) := PG (dβ), etc.,2 where d is the exterior derivative on cochains and d∗ is its adjoint. We will realize these operators by attaching to each lattice point x a Hilbert space Hx and to each lattice bond b a Hilbert space Hb , where Hx ∼ = Hb ∼ = CN . The operators QH (x), PH (x), QG (b) and PG (b) are given on Hx , and Hb , respectively, as matrices with matrix elements: PH (x)a, b = PG (b)a, b = δa, b+1(mod N ) and QH (x)a, b = QG (b)a, b = δa, b e
2πi N a
(2.5)
,
(2.6)
for a, b ∈ {0, . . . , N − 1}. The operators QH and QG have to be interpreted as the ZN versions of the Higgs field 2πi 2πi and gauge field, respectively: QH (x) = e N ϕ(x) , QG (x) = e N α(b) , with ϕ and α taking values in ZN . The operators PH and PG are their respective canonically conjugated “exponentiated momenta”, i.e. shift operators by one ZN -unit. Hence these operators indeed provide the Weyl form of the canonical quantization in α0 = 0 gauge, see also Eq. (2.11) below. We denote by Floc the ∗-algebra generated by these operators. Denoting by F(V ) the by QH (x), PH (x), QG (b) and PG (b) for x ∈ O V0 , b ∈ V C∗ -sub-algebra generated [ 1, V ⊂ O F(V ). The algebra F(V ) acts on HV := Hx Hb . Z2 finite, one has Floc = |V |<∞
x∈V0
b∈V1
We will denote by F the unique C∗ -algebra generated by Floc . Without mentioning 2
There was a misprint in these definitions in [1].
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explicitly we will frequently use that by continuity states ω on F or ∗-automorphisms γ of F are uniquely determined by their definition on Floc . Let S denote the group of spatial lattice translations by a ∈ Z2 and rotations by multiples of π/2. S acts naturally on F as a group of ∗-automorphisms τg , g ∈ S, given by τg (QH (x)) = QH (gx), etc. 0 We also have the group of local (time independent) gauge transformations G ≡ Cloc acting as ∗-automorphisms on F by QH (α) 7→ e−ihα, λi QH (α) ,
QG (γ) 7→ e−ihγ, dλi QG (γ),
(2.7)
and leaving all operators PH (x) and PY G (b) invariant. These gauge transformations are G(x)λ(x) , where3 implemented by the unitaries G(λ) := x
G(x) := PH (x) δ ∗ PG (x).
(2.8)
Note that G(x)∗ = G(x)−1 = G(x)N −1 . The operator G(x) is the generator of a ZN gauge transformation at the point x, as one can easily checks. It can be interpreted as the lattice analog of exp −2πi(div E − ρ)/N . We denote the gauge group algebra G ⊂ F as the Abelian C ∗ -sub-algebra generated by {G(x) | x ∈ Z2 } and put Gloc := G ∩ Floc and G(V ) := G ∩ F(V ). The algebra of observables A is defined as the gauge invariant sub-algebra of F : A := {A ∈ F : G(x)AG(x)∗ = A for all x ∈ Z2 }, i.e. the commutant of G in F. We call A(V ) := F(V ) ∩ A and Aloc := Aloc is the norm dense sub-algebra generated by PG (b), PH (x) and QGH (b) := QG (b) [δQH ] (b)∗
(2.9) [
A(V ). Then
V
(2.10)
for all x ∈ (Z2 )0 and b ∈ (Z2 )1 . We now provide a convenient “ket vector” notation for the local Hilbert spaces HV . Identifying Hx ∼ = Hb ∼ = CN ∼ = `2 (ZN ) we may naturally denote ON-basis elements of Hx/b by |ai, a ∈ ZN , i.e. the characteristic functions on ZN . Correspondingly, ON-basis elements of HV are labeled by O |ϕ, αi = |ϕ(x)i ⊗ |α(b)i , x∈V0 ,b∈V1
where (ϕ, α) run through all “classical configurations”, i.e. ZN -valued 0- and 1-cochains, respectively, with support in V. With this notation the representation of our local field algebras F (V ) is immediately recognized as the Weyl form of the canonical quantization in α0 = 0 gauge, i.e. QH (χ) |ϕ, αi QG (β) |ϕ, αi PH (χ) |ϕ, αi PG (β) |ϕ, αi
= eihϕ, χi |ϕ, αi, = eihα, βi |ϕ, αi, = |ϕ + χ, αi, = |ϕ, α + βi,
(2.11)
3 Here we replace our notation Q(x) of [1] by the more common one G(x) and also correct a misprint in equation. (2.9) of [1].
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where χ and β denote ZN -valued 0- and 1-cochains, respectively, with support in V. The action of the gauge transformations on HV now takes the usual form G(λ) |ϕ, αi = |ϕ + λ, α + dλi. It is well known that for a single site x ∈ Z2 the operators QH (x) and PH (x) generate End Hx and similarly for QG (b) and PG (b). Hence F(V ) ∼ = End (HV ) for all V implying Floc and F to be simple [11]. On the other hand, the observable algebra A is not simple since it contains G in its center. However, since eventually we are only interested in representations π of A satisfying π(G(x)) = 1l for all x ∈ Z2 (i.e. representations without external charges, see Sect. 2 below) we might as well consider B := A/J, as our “essential” observable algebra, where J ⊂ A is the two-sided closed ideal generated by [ {G(x) − 1l}x∈Z2 . In the obvious way we also define B(V ) = A(V )/J(V ) and B(V ). Then B is the C ∗ -closure of Bloc and by the same argument as Bloc = V ⊂Z2
above B is simple since we have Lemma 2.1.1. For all finite closed boxes V ⊂ Z2 the algebras B(V ) are isomorphic to full matrix algebras. Proof. We transform to “unitary gauge” by defining a unitary U : HV → HV0 ⊗ HV1 according to U |ϕ, αi := |ϕi ⊗ |α − dϕi . This gives
U G(χ)U −1 (|ϕi ⊗ |αi) = |ϕ + χi ⊗ |αi,
and therefore U G(V )U ∗ = M(V ) ⊗ 1l, whereM(V ) consists of all shift operators |V | |ϕi 7→ |ϕ + χi on HV0 = `2 (C 0 (V )) ≡ `2 ZN 0 . Now M(V ) being maximal Abelian in End HV0 and A(V ) being the commutant of G(V ) in F(V ) ≡ End HV we conclude U A(V )U ∗ = (M(V )0 ∩ End HV0 ) ⊗ End HV1 = M(V ) ⊗ End HV1 from which B(V ) ∼ = End HV1 follows.
Note that the above transformation also gives U PG (β)U ∗ |ϕi ⊗ |αi = |ϕi ⊗ |α + βi, U QGH (β)U ∗ |ϕi ⊗ |αi = |ϕi ⊗ eihα, βi |αi, showing that B is generated by the field operators PG + J and QGH + J, the later being defined in (2.10). We conclude this subsection with introducing the concept of charge conjugation. On Floc we define the charge conjugation iC to be the involutive ∗-automorphism given by iC (QH/G (χ)) = QH/G (−χ), iC (PH/G (α)) = PH/G (−α),
(2.12) (2.13)
Then iC extends to F by continuity. The generators of gauge transformations satisfy iC (G(x)) = G(x)∗ and therefore the observable algebra is invariant under iC :
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iC (A) = A. Also note that the restriction of iC to F(V ) is unitarily implemented on HV by |ϕ, αi → | − ϕ, −αi. 2.2. Local transfer matrices. Let us now introduce the dynamics by defining suitable finite volume transfer matrices. Infinite volume transfer matrices will be defined later. The form of the transfer matrix is justified by its finite volume ground state giving rise to the classical expectation associated with the euclidean ZN -Higgs model (1.1). We will consider local transfer matrices TV ∈ Aloc , for finite V ⊂ Z2 , defined by: TV = eAV /2 eBV eAV /2 ,
(2.14)
with N −1 N −1 1 XX 1 X X AV := √ βg (n) [δQG ] (p)n + √ βh (n) QGH (b)n , N p∈V2 n=0 N b∈V1 n=0
(2.15)
and N −1 N −1 1 XX 1 X X BV := √ γg (n) PG (b)n + √ γh (n) PH (x)n . N b∈V1 n=0 N x∈V0 n=0
(2.16)
Here βg/h and γg/h are even and real valued functions on ZN , such that TV is in fact positive, has positive matrix elements and is invertible. Moreover, this also implies that TV is charge conjugation invariant, iC (TV ) = TV . In the basis |ϕ, αi ∈ HV these operators have matrix elements given by X X Sg (dα(p)) − Sh (dϕ(b) − α(b)) |ϕ, αi (2.17) eAV |ϕ, αi = exp − p∈V2
and hϕ0 , α0 | eBV |ϕ, αi =
Y
b∈V1
Y
ρg (α(b) − α0 (b))
ρh (ϕ(x) − ϕ0 (x)),
(2.18)
x∈V0
b∈V1
where Sg and Sh are the euclidean actions (1.3) and where ρg and ρh are positive functions on ZN determined by " # X X n n 1 ρg (n)PG (b) = exp √N γg (n)PG (b) , n
X n
n
" ρh (n)PH (x)n = exp
√1 N
X
# γh (n)PH (x)n
.
n
With a suitable choice of the choice of the γg, h ’s as functions of the βg, h ’s one can arrange (see [1]) ρg,h = e−Sg,h
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and hence Tr TVn equals the partition function appearing in the denominator of (1.1) for a volume 3 ≡ Vn := V × {0, . . . , n − 1} ⊂ Z3 with periodic boundary conditions in “time” direction (free boundary conditions are also possible, see [2, 1] or below). The euclidean dynamics is given by the strong limit of the non-∗ automorphisms of F generated by the local transfer matrices: α(A) := lim α(A)V , V ↑Z2
A ∈ F,
(2.19)
where α(·)V is the automorphism of F defined through α(A)V := TV ATV−1 ,
A ∈ F.
(2.20)
For A ∈ Floc the limit in (2.19) is already reached at finite V . In [1] and [2] the notation αi in place of α was used, due to the interpretation of αi ≡ α as the generator of a translation by one unit in imaginary (euclidean) time direction. Clearly, α commutes with the action of lattice translations and rotations and with the charge conjugation iC . In this work we will consider the classical expectations (1.1) in the so called “free charge phase” of the ZN -gauge Higgs model. Let us introduce the functions g, h : ZN 7→ R+ defined by (see [1]) g(n) := exp(−Sg (n)), h(n) := F exp(−Sh ) (n),
(2.21) (2.22)
for n ∈ ZN , where F is the Fourier transform of functions on ZN . From now on we will fix the additive constants βg (0) and βh (0) through the conditions g(0) = h(0) = 1. In [1] the infinite volume limits of the classical expectations of local observables (1.1) have been shown to be analytic functions of the couplings g(1), . . . , g(N − 1), h(1), . . . , h(N − 1) whenever gc := max {|g(1)|, . . . , |g(N − 1)|} ≤ e−kg , and −kh
hc := max {|h(1)|, . . . , |h(N − 1)|} ≤ e
,
(2.23) (2.24)
with kg , kh > 0, large enough. For this a polymer and cluster expansion has been used. The above region of analyticity (for real couplings) is contained in the “free charge region” of the phase diagram. This phase is characterized by the absence of screening and confinement. All results of our present work, specifically those concerned with the existence and the properties of the charged states are valid for gc and hc sufficiently small. 2.3. Ground States. In this subsection we recall the important definition of a ground state and discuss some of its basic features. We start with introducing two useful concepts. First, the adjoint γ ∗ of an automorphism γ of a unital ∗-algebra C is defined through ∗ γ (A) := (γ(A∗ ))∗ , A ∈ C. We have γ ∗∗ = γ and γ is a ∗-automorphism iff γ = γ ∗ . For the composition of automorphisms one has (α ◦ β)∗ = α∗ ◦ β ∗ and consequently α∗ −1 = α−1 ∗ . For an invertible element A ∈ C one also has (Ad A)∗ = Ad A∗ −1 . Finally, if ω is a γ-invariant state on C then trivially it is also γ ∗ -invariant. Second, we say that a state ω on a ∗-algebra C has the cluster property for the automorphism γ if, for all A, B ∈ C, one has lim ω(Aγ n (B)) = ω(A)ω(B).
n→∞
We now come to a central definition which has first been introduced in [2].
(2.25)
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Definition 2.3.1. Let γ be a (not necessarily ∗-preserving) automorphism of a unital ∗-algebra C. A state ω on C is called a “ground state” with respect to γ and C if it is γ-invariant and if 0 ≤ ω(A∗ γ(A)) ≤ ω(A∗ A),
∀A ∈ C.
Actually γ-invariance follows from (2.26) (see, e.g. [1]).
(2.26)
We will motivate this abstract definition below when we discuss the ground state of the finite volume transfer matrix. The following lemma will be very useful for proving that certain states are ground states with respect to our euclidean dynamics α (or suitable modifications of α to be introduced in Sect. 3.1). This lemma was already implicitly used in [2]. Lemma 2.3.2. Let γ be an automorphism on a ∗-algebra C satisfying γ ∗ = γ −1 and let ω be a γ-invariant state on C which has the cluster property for γ (actually one just needs that, for each A ∈ C, the sequence ω(A∗ γ a (A)), a ∈ N, is bounded). Then, for all A ∈ C, (2.27) |ω(A∗ γ(A))| ≤ ω(A∗ A). The proof is easy. See [1]. We will now exhibit translation invariant ground states of the automorphism α. First, let us explain in this context the heuristic motivation of Definition 2.3.1. If V ∈ HV is the Frobenius eigenvector of the finite volume transfer matrix TV with eigenvalue kTV kHV then, in face of the positivity of TV , the inequalities 0 ≤ V , A∗ TV ATV−1 V ≤ V , A∗ AV (2.28) obviously hold for any A ∈ F(V ). This motivates to consider the relation (2.26) with γ = α as a characterization of a state replacing the vector states V for infinite volumes. One should notice here that the usual characterization of a ground state of a quantum spin system as a state ω for which lim ω(A∗ [HV , A]) ≥ 0 for all local A is inadequate for transfer matrix systems, due V ↑Z2
to the highly non-local character of the Hamilton operator HV ≡ − ln TV (at least in more than two space-time dimensions). Following [2], a translation invariant ground state for α with respect to the algebra F can be obtained by TrHV TVn BTVn EV0 , (2.29) ω0 (B) = lim lim V ↑Z2 n→∞ TrHV TV2n EV0 for B ∈ Floc , where EV0 is an in principle arbitrary operator with positive matrix elements. Before the limits V ↑ Z2 and n → ∞ are taken the expression in the righthand side of (2.29) is identical to the classical expectation (1.1) in a volume V × {−n, . . . , n} ⊂ Z3 of a suitable classical observable Bcl associated with B. A convenient choice for EV0 giving free boundary conditions in time-direction is (see [2]) EV0 := eAV /2 FV0 eAV /2 where FV0 :=
X (ϕ, α), (ϕ0 , α0 )
|ϕ, αihϕ0 , α0 |
(2.30) (2.31)
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Here the sum in (2.31) goes over all classical time-zero configurations with support in V , i.e. 0-cochains ϕ, ϕ0 ∈ C 0 (V ) and 1-cochains α, α0 ∈ C 1 (V ) . The existence of the thermodynamic limit in (2.29) can be established using Griffiths’ inequalities or the cluster expansions. The cluster expansions also provide a way to prove the translation and rotation invariance of the limit state ω0 . Another important fact derived from the cluster expansion is that the restriction of ω0 to A has the cluster property for the automorphism α. One of the most useful aspects of Definition 2.3.1 of a ground state is the possibility to define infinite volume transfer matrices. Indeed, if ω0 is a ground state of α with respect to F and (π0 (F), 0 , H0 ) is the GNS-triple associated with ω0 , we define, following [2], the infinite volume transfer matrix T0 as the element of B(H0 ) given on the dense set π0 (F)0 by T0 π0 (A)0 := π0 (α(A))0 ,
(2.32)
A ∈ F. One checks that this is a well-defined positive operator with 0 ≤ T0 ≤ 1. If moreover ω0 satisfies the cluster property (2.25), then 0 ∈ H0 is the unique (up to a phase) eigenvector of T0 with eigenvalue 1. Hence we call π0 the vacuum representation of (F, α). Since ω0 is translation invariant we can also define a unitary representation of the translation group in the vacuum sector through U0 (x)π0 (A)0 := π0 (τx (A))0 ,
(2.33)
A ∈ F, x ∈ Z2 . The momenta in the vacuum sector are therefore defined by U0 (x) = exp(iP · x), with sp P ∈ [−π, π]2 and the vacuum 0 is also the unique (up to a phase) translation invariant vector in H0 . Next we remark that ω0 is also charge conjugation invariant. This can be seen from (2.29) by using iC (TV ) = TV , iC (EV0 ) = EV0 and the fact that iC F(V ) is unitarily implemented on HV . Hence, charge conjugation is implemented as a symmetry of the vacuum sector by the unitary operator C0 ∈ B(H0 ) given on π0 (F)0 by C0 π0 (A)0 := π0 (iC (A))0
(2.34)
When constructing charged states ωρ in Sect. 3 this will no longer hold, i.e. there we will have ωρ ◦ iC = ω−ρ . 2.4. External charges . Let π be a representation of the field algebra F on some separable Hilbert space Hπ and for q ∈ C 0 let Hπq ⊂ Hπ be the subspace of vectors ψ ∈ Hπ satisfying 2πi (2.35) π(G(x))ψ = e N q(x) ψ for all x ∈ Z2 . According to common terminology we call Hπq the subspace with external electric charge q, since the operators π(G(x)) implement the gauge transformations on Hπ . Clearly, since A commutes with G we have π(A) Hπq ⊂ Hπq .
(2.36)
Moreover, using QH (x)G(x) = e2πi/N G(x)QH (x) for all x ∈ Z2 we have π(QH (q 0 ))Hπq = Hπq−q 0 for all q 0 ∈ Cloc .
0
(2.37)
Dyonic Sectors and Intertwiner Connections in 2+1-D Lattice ZN -Higgs Models 0
421
0
The representations of A on Hπq and on Hπq−q , π q and π q−q , respectively, cannot 2πi be unitarily equivalent for q 0 6= 0 with finite support since, by (2.35), G(x) − e N q(x) 1l ∈ 0 Ker π q and, hence Ker π q 6= Ker π q−q for q 0 6= 0. However, since QH (q)AQH (q)∗ = A one has for any ψq ∈ Hπq and A ∈ A π q (A)ψq = π(A)ψq = π(QH (q 0 ))∗ π ◦ Ad QH (q 0 )(A) π(QH (q 0 ))ψq 0 = π(QH (q 0 ))∗ π q−q ◦ Ad QH (q 0 )(A) π(QH (q 0 )ψq , (2.38) 0
thus showing that π q and π q−q ◦ Ad QH (q 0 ) are unitarily equivalent.4 Now Hπq might be zero for general representations π and general external charge distributions q (e.g. for the vacuum representation π0 and q with infinite support). In this work we are only interested in representations π containing a non-trivial subspace Hπ0 6= 0 of zero external charge (and therefore, by (2.37), also Hπq 6= 0 for all external charges with finite support). Moreover, Hπ0 will always be cyclic under the action of π(Floc ) and therefore we will always have5 Hπ =
M
Hπq
(2.39)
0 q∈Cloc
In fact, such a decomposition is always obtained for GNS-representations (πω , ω , Hω ) 0 or, equivalently, associated with states ω on F, provided ω ∈ Hωq for some q ∈ Cloc ω(F G(x)) = e2πi q(x)/N ω(F )
(2.40)
for all F ∈ F and all x ∈ (Z2 )0 . In this case we call ω an eigenstate (with external 0 ) of the gauge group algebra G. Note that ω is an eigenstate of G charge q ∈ Cloc with external charge q if and only if ω ◦ Ad QH (q 0 ) is an eigenstate with external charge 0 q + q 0 . Correspondingly, πω (QH (q 0 )∗ ) ω ∈ Hωq+q will also be a cyclic vector for πω (F). Hence, without loss, we may restrict ourselves to eigenstates ω of G with zero external charge. In fact, we will only be studying the restrictions of representations π(A) to Hπ0 as the “physical” subspace of Hπ , i.e. the subspace on which “Gauss’ Law”, π(G(x)) = 1l, holds. We emphasize that this notion of external (or “background”) charge is not to be confused with the concept of (dynamically) charged states and the associated charged representations of A. By this we mean representations of A with zero external charge, which are inequivalent to the vacuum representation (at least when extended to a suitable ˆ ⊃ A), e.g. by the appearance of different mass spectra or the “dynamic closure” A absence of a time and space translation invariant “vacuum” vector. By analogy with the terminology of quantum field theory we call an equivalence class of such charged representations a superselection sector. They are the main interest of our work. Hence, from now on by charged states we will always mean dynamically charged states in this later sense. 4 Note, however, that this equivalence does not respect the dynamics, since Ad Q (q) does not commute H with α. 0 5 Use that F loc = ⊕q Aloc QH (q) is a grading labeled by q ∈ Cloc , i.e. the irreducible representations of G in Floc .
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3. The Construction of Dyonic Sectors In [1] with the help of the cluster expansions (see Appendix A) we were able to show the existence of electrically and of magnetically charged sectors in the “free charges” phase of the ZN -Higgs model. We also proved the existence of massive charged 1-particle states in these sectors. This section is devoted to the construction of states and the associated sectors which are at the same time electrically and magnetically charged. Following the common use we call them dyonic states. Hereby we generalize and improve ideas from [6], where such states have been first constructed for the Z2 -Higgs model. In [7] we will continue this analysis by constructing scattering states in the dyonic sectors and identifying these to constitute an “anyonic Fock space” over the above mentioned 1-particle states. We start in Sect. 3.1 with reformulating the Fredenhagen-Marcu construction for charged states ωρ by defining them as the thermodynamic limit of ground states of modified local transfer matrices TV (ρ). For ρ = (ε, µ) these modified transfer matrices correspond to modified Hamiltonians, where the kinetic term in the Higgs fields is replaced by 1 1 (πH , πH ) → (πH + ε, πH + ε) 2 2 and the magnetic self energy is replaced by 1 1 (dα, dα) → (dα + µ, dα + µ). 2 2 In the functional integral these states are represented by euclidean expectations in the background of infinitely long vertical Wilson and vortex lines sitting above sites x ∈ (Z2 )0 and plaquettes p ∈ (Z2 )2 , respectively, of the time-zero plane Z2 . Their ZN values are given by values of the electric charges ε(x) and the magnetic charges µ(p), respectively. In Sect. 3.2 we construct the charged representations πρ of A associated with the states ωρ and the global transfer matrices implementing the euclidean dynamics in πρ . In Sect. 3.3 we prove that πρ and πρ0 are dynamically equivalent whenever their total charges coincide, qρ = qρ0 . We also conjecture that otherwise they are dynamically inequivalent and give some criteria for a proof. In Sect. 3.4 we show that the infimum of the energy spectrum in the dyonic sectors may be uniquely normalized by imposing charge conjugation symmetry and the requirement of decaying interaction energies between two charge distributions in the limit of infinite spatial separation. We recall once more that, without mentioning explicitly, all results of this section are valid in the free charge phase, i.e. for gc and hc (defined in (2.23), (2.24)) sufficiently small. 3.1. Dyonic states. We first recall the idea behind the Fredenhagen-Marcu (FM) string operator and its use in the construction of electrically charged states [2]. Starting from the usual method to create localized electric dipole states by applying a Mandelstam string operator to the vacuum x, y := φ(x)φ(y)∗ e
i
Ry x
Ai dz i
,
(3.1)
Dyonic Sectors and Intertwiner Connections in 2+1-D Lattice ZN -Higgs Models
423
Fredenhagen and Marcu [2] proposed a modification so as to keep the energy of the dipole state x, y finite as y → ∞. Using our lattice notation the FM-proposal reads6 M ∗ n F x,y := lim cn QH (x)QH (y) T0 QG (sxy ). n→∞
(3.2)
Here sx,y is an arbitrary path connecting x to y in our spatial lattice, T0 = e−H is the global transfer matrix in the vacuum sector and cn > 0 is a normalization constant M FM to get kF x,y k = 1. Note that x,y still lies in the vacuum sector and has zero external charge. In a second step one may then send one of the charges to infinity7 to obtain dynamically charged states as expectation values on Aloc , M FM (3.3) A ∈ Aloc . ωx (A) := lim F x,y , Ax,y , y→∞
Note that as a limit of eigenstates of G with zero external charge, ωx is also an eigenstate of G with zero external charge. Using duality transformations, an analogous procedure for the construction of magnetically charged states has been given in [1]. In order to be able to discuss dyonic states within a common formalism, we now pick up an observation of [1] to reformulate the above construction as follows. First we recall from [2] that in our range of couplings we have M ∗ F x,y = QH (x)QH (y) ψx, y ,
(3.4)
δ −δ
where ψx,y ∈ H0x y is the unique (up to a phase) ground state vector of the restricted δ −δ δ −δ global transfer matrix T0 H0x y , where H0x y ⊂ H0 is the subspace of an external electric charge-anticharge pair sitting at x and y, respectively. In fact, by looking at (3.2) we have (3.5) ψx, y = s − lim cn T0n QG (sx, y ), n→∞
and using our cluster expansion it is easy to check that the limit exists independently of the chosen string sx,y connecting x to y. To avoid the use of external charges (for which we do not have a magnetic analogue) M we now equivalently reformulate this by saying that F x,y is the unique (up to a phase) ground state vector of the modified transfer matrix T0 (δx − δy ) defined by T0 (δx − δy ) := QH (x)QH (y)∗ T0 QH (y)QH (x)∗ H00 ,
(3.6)
where H00 ⊂ H0 is the subspace without external charges. Since this modified transfer matrix generates a modified dynamics given by α0 = Ad (Q(x)Q(y)∗ ) ◦ α ◦ Ad (Q(y)Q(x)∗ ), M FM 0 we conclude that the state A 7→ (F x,y , A x,y ) is a ground state of α when restricted to the observable algebra, A ∈ A. A similar statement holds for magnetic dipole states and magnetically modified transfer matrices [1], see also below. To treat electric and magnetic charges simultaneously we now generalize this con0 2 and DM ≡ Cloc denote the set of ZN -valued struction as follows. Let DE ≡ Cloc 0-cochains (≡ electric charge distributions) and 2-cochains (≡ magnetic charge distributions), respectively, with finite support in our spatial lattice Z2 , and let D = DE ×DM . 6 By a convenient abuse of notation we often drop the symbol π when referring to the vacuum 0 representation. 7 Actually, in [2] the two limits, (3.2) and (3.3), were performed simultaneously.
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For ρ = (ε, µ) ∈ D, supp ρ ⊂ V0 × V2 , we generalize (3.6) and define the modified local transfer matrices in a finite volume V ⊂ Z2 as the element of A(V ) given by TV (ρ) := Z(µ)1/2 QH (ε)TV QH (ε)∗ Z(µ)1/2 , where we have used the notations
Y
QH (ε) :=
QH (x)ε(x)
(3.7)
(3.8)
x∈(Z2 )0
Y
and Z(µ) :=
Z (µ(p)) (p).
(3.9)
p∈(Z2 )2
Here the gauge invariant operator Z (n) (p), n ∈ {0, . . . , N − 1}, is defined by (see also [1]) −1 1 N X 2πi jn − 1 (δQG (p))j . βg (j) exp (3.10) Z (n) (p) = exp √ N N j=0
On our local Hilbert spaces HV it acts by Z (n) (p) |ϕ, αi =
e−Sg (dα(p)+n) |ϕ, αi . e−Sg (dα(p))
(3.11)
Hence, this operator can be interpreted as the operator creating a vortex with magnetic charge n at the plaquette p. The definition (3.7) is also motivated by the fact that under duality transformations we roughly have TV (ε, 0) ↔ TV ∗ (0, ε∗ ), see [1] for the precise statement. Also note that iC (TV (ρ)) = TV (−ρ) and that TV (ρ) is still gauge invariant, i.e. TV (ρ) ∈ Aloc . In a formal continuum notation, these modified transfer matrices correspond to modified Hamiltonians, where the kinetic term in the Higgs fields is replaced by 1 1 (πH , πH ) → (πH + ε, πH + ε) 2 2
(3.12)
and the magnetic self energy is replaced by 1 1 (dα, dα) → (dα + µ, dα + µ). 2 2
(3.13)
Together with these modified transfer matrices we also have a modified euclidean dynamics αρ given by the automorphism of F, αρ (A) := lim TV (ρ)ATV (ρ)−1 , V ↑Z2
A ∈ F,
(3.14)
such that α0 ≡ α. As in the case of α0 , for each A ∈ Floc the limit above is already reached at finite V . Moreover αρ (A) = A and iC ◦ αρ = α−ρ ◦ iC .
(3.15)
We emphasize that we introduce these modifications not as a substitute of our original “true” dynamics, but for technical reasons only. From (3.7) we conclude
Dyonic Sectors and Intertwiner Connections in 2+1-D Lattice ZN -Higgs Models
αρ = Ad Kρ ◦ α0 ◦ Ad Kρ∗ = Ad Lρ ◦ α0 , where
∈ Floc
Kρ := Z(µ)1/2 QH () and
Lρ := Kρ α0 (Kρ∗ ) = TV (ρ)TV (0)−1
425
(3.16) (3.17)
∈ Aloc ,
(3.18)
the last equality holding for V large enough. Note also that for ρ = (µ, ) and ρ0 = (µ0 , 0 ) such that supp µ ∩ supp µ0 = ∅ one has Kρ+ρ0 = Kρ Kρ0 .
(3.19)
If the distance between supp ρ and supp ρ0 is large enough one also has Lρ+ρ0 = Lρ Lρ0 .
(3.20)
Next, we define the total charge, qρ ∈ ZN × ZN , of a distribution ρ = (, µ) ∈ D as ! X X ε(x), µ(p) (3.21) qρ = e , mµ := x
and put
Dq := {ρ ∈ D |
p
qρ = q}.
(3.22)
Then, for all ρ = (ε, µ) ∈ D0 , (i.e. with vanishing total charge), there exist 1-cochains, sε and sµ , with finite support, such that d∗ sε = −ε
and
dsµ = −µ.
(3.23)
This allows to generalize the FM-construction and define, for ρ = (ε, µ) ∈ D0 , the family of states ωρ on A as ground states of the modified dynamics αρ by the following method. As already observed in [2] and [1], ground states for the modified transfer matrices can be obtained by taking the thermodynamic limit of the state defined by the formula n ωV, ρ (A) = lim ωV, ρ (A),
(3.24)
n→∞
where n ωV, ρ (A)
T rHV TV (ρ)n ATV (ρ)n EVρ = , T rHV TV (ρ)2n EVρ
A ∈ F(V ),
(3.25)
and where EVρ is some suitably chosen matrix to adjust the boundary conditions. There are two possibilities we will discuss. The first one is EVρ = 1l, thus getting periodic boundary conditions in euclidean time direction for the classical expectations associated n with ωV, ρ (A). In order to understand what happens algebraically one checks that, for µ = 0, the desired ground state simply becomes ω0 ◦ Ad QH (), which is a state with external electric charge. Hence, the resulting representation of A would be equivalent to the vacuum representation. A similar statement holds for µ 6= 0. Since this is not the kind of state we are interested in let us look at the second case. There we choose EVρ in such a way that we get free boundary conditions in the n euclidean time direction for the classical expectations associated with ωV, ρ (A), together with horizontal electric and magnetic strings “conducting” the charges among the points
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of the support of and µ, respectively, and located in the highest and in the lowest timeslices of Vn := V × {−n, . . . , n}. Hence we choose 1-cochains s and sµ obeying (3.23) and put h h i i∗ (3.26) EVρ := Z(µ)−1/2 M (sµ )QGH (sε ) EV0 Z(µ)−1/2 M (sµ )QGH (sε ) . Here EV0 has been defined in (2.30), QGH (b) are the gauge invariant link operators (2.10) and (3.27) M (sµ ) := e−AV /2 e−BV PG (sµ )eBV eAV /2 ≡ e−AV /2 PG (sµ )eAV /2 . S Note that M (sµ ) ∈ A(V ) for all V containing supp µ supp sµ and that M (sµ ) is actually independent of V provided the distance between supp sµ and the boundary ∂V is ≥ 2. The motivation for this definition comes from its effect in the euclidean functional integral, where QGH (sε ) produces a superposition of electric Mandelstam strings connecting the charges described by ε along the support of sε . Similarly, the operator Z(µ)−1/2 M (sµ ) creates magnetic Mandelstam strings joining the plaquettes of supp µ via the support of sµ . In the functional integral this magnetic string will appear in the form of shifted (or frustrated) vertical plaquettes placed between the first two and the last two time-slices of the space-time volume Vn . This can be seen by looking at the product TV (ρ)Z(µ)−1/2 M (sµ ) appearing in (3.25) due to the definition (3.26). There, the factor Z(µ)−1/2 e−AV /2 e−BV gets canceled by a corresponding factor in TV (ρ) and the matrix elements of PG (sµ )eBV (coming next according to (3.27)) are given by hϕ0 , α0 | PG (sµ )eBV |ϕ, αi # " X X 0 0 Sg (α(b) − α (b) + sµ (b)) − Sh (ϕ(x) − ϕ (x)) . = exp −
(3.28)
x∈V0
b∈V1
After transforming the functional integral expression for (3.25) to unitary gauge the shift sµ (b) in (3.28) appears on the vertical plaquette spanned by the horizontal bond b and the time-like bond htn−1 , tn i (and similarly, but with opposite sign at ht−n , t−n+1 i). The above construction of charged states can be understood as analogous to the construction of Fredenhagen and Marcu, except that, at finite volume, the horizontal Mandelstam strings are already located at the highest (respectively lowest) time-slices. Given the definition (3.26), the limit (3.24) is actually independent of the choice of the horizontal strings sε and sµ satisfying (3.23) and we can take the thermodynamic limit to obtain (3.29) ωρ (A) := lim ωV, ρ (A). V ↑Z2
EV0 implies G(x)EVρ
= EVρ for all x ∈ V and therefore ωρ provides Note that G(x)EV0 = an eigenstate of G with zero external charge. Also note that the boundary conditions (3.26)-(3.27) now imply ωρ ◦ iC = ω−ρ . (3.30) Rewriting (3.29) in terms of euclidean expectation values in the unitary gauge we get an expectation of a classical function Acl, ρ associated with A in the presence of both, “electric” Wilson loops LE (sε , n) living on bonds and “magnetic” vortex loops LM (sµ , n) living on plaquettes, i.e.
Dyonic Sectors and Intertwiner Connections in 2+1-D Lattice ZN -Higgs Models
ωρ (A) = hAcl, ρ iρ := lim lim ZL−1E , LM V
n
Z
da Acl, ρ (a)e−SG (da+LM )−SH (a) e
427 2πi N (a, LE )
,
(3.31) where a denotes the euclidean lattice ZN -gauge field a := dϕ − α (unitary gauge) and ZLE , LM is the normalization such that ωρ (1l) = 1. Here the vertical parts of the loops LE and LM run from euclidean time t = −n to t = +n and are spatially located at the supports of ε and µ, respectively. The horizontal parts of LE and LM are given by the 1-cochains ±sε/µ (3.23), shifted to the euclidean time slice ±n, respectively. To be more precise, LE (sε , n) ∈ C 1 (Vn ) is the unique 1-cochain on the euclidean space-time lattice Z3 satisfying d∗ LE (sε , n) = 0 together with the condition that its horizontal part is nonzero only on the time slices ±n, where it coincides with ±sε . Similarly, LM (sµ , n) ∈ C 2 (Vn ) is the unique 2-cochain on Z3 satisfying dLM (sµ , n) = 0 plus the condition that the horizontal part of its dual 1-cochain is non-vanishing only on the time slices ±(n − 1/2), where it coincides with the dual of ±sµ . We will refer to such expectations by the following symbolic picture: . ............................................................................. ... C . .. C .. .. C ... .. . .. C .. .. VnC . . .. C ... .. C .. Acl .. . C . . . C .................................................................................. C euclid. . ωρ (A) = lim n, V .. C . ............................................................................ .. ... C .. .. C . . . .. . .. .. C .. .. Vn C .. C .. ... C ... .. C ... . . C ................................................................................ C euclid.
(3.32)
The dotted box indicates the space-time volume Vn . The shaded region in the numerator indicates the support of the classical function Acl associated with an observable A. The boldface vertical lines indicate the vertical part of the loops LM , i.e. the stacks of horizontal plaquettes whose projection onto the time-zero plane is given by µ. They are the euclidean realization of magnetic vortices located at µ. The dashed vertical lines indicate the vertical part of the loops LE . Their projection onto the time-zero plane is given by ε. They are the euclidean realization of electric charges located at ε. For finite n all vertical lines actually close to loops via the strings ±sε/µ running in (or just inside of) the horizontal boundary of Vn . In the limit n → ∞ the choice of these horizontal parts becomes irrelevant. In fact, applying our cluster expansion techniques [1] we have Proposition 3.1.1. For all ρ ∈ D0 the limit (3.29) provides a well defined state ωρ on F which is independent of the choice of 1-chains (sε , sµ ) satisfying (3.23). Moreover, ωρ A provides a ground state with respect to the modified dynamics αρ fulfilling the cluster property.
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Generalizing Eq. (3.2) we will show in Theorem 3.3.2 below, that for all ρ ∈ D0 , i.e. with vanishing total charge, the states ωρ are actually induced by vectors in the vacuum sector H00 . To get charged states ωρ for ρ ∈ Dq , q 6= 0, we now have to move a counter-charge to infinity. To this end let ρ · a denote the translate of ρ by a ∈ Z2 . Then ρ − ρ0 · a ∈ D0 for all ρ, ρ0 ∈ Dq . Proposition 3.1.2. For A ∈ Floc , 0 6= q ∈ ZN × ZN arbitrary and ρ, ρ0 ∈ Dq let ωρ (A) := lim ωρ−ρ0 ·a (A).
(3.33)
a→∞
Then the limit exists independently of the chosen sequence a → ∞ and it is independent of ρ0 . Moreover ωρ A provides a ground state with respect to the modified dynamics αρ fulfilling the cluster property. Clearly, (3.33) implies that now (3.30) also holds for qρ 6= 0. Next, we note that the symmetry group S of our spatial square lattice (i.e. consisting of translations by a ∈ Z2 and rotations by 21 nπ) acts naturally from the right on Dq by (ρ · g)(x) := ρ(g · x) and we obviously have ωV, ρ ◦ τg = ωg−1 V, ρ·g for all g ∈ S. Hence we get the Corollary 3.1.3. For all g ∈ S and all ρ ∈ D we have ωρ·g = ωρ ◦ τg , where τg denotes the natural action of S on F. We now give an interpretation of these states as charged states in the following sense. For V ⊂ Z2 , finite, define the charge measuring operators 8E (V ) :=
Y
δ ∗ PG (x)
and
x∈V
8M (V2 ) :=
Y
δQG (p).
(3.34)
p∈V2
R TheseR operators are lattice analogues of the continuum expressions exp −i ∇E and exp i dA, respectively, i.e. they measure the total electric charge inside V and the total magnetic flux through V , respectively. As in Theorem 6.2 of [1] we have Proposition 3.1.4. If V ⊂ Z2 is e.g. a square centered at the origin, one has, under the conditions of Proposition 3.1.2: ωρ (8E (V )) 2πie , = exp N V ↑Z2 ω0 (8E (V )) ωρ (8M (V2 )) 2πimµ lim , = exp N V ↑Z2 ω0 (8M (V2 )) lim
(3.35) (3.36)
where e and mµ were defined in (3.21). We omit the proof here, since it is analogous to the equivalent one found in [1]. Finally, we check that our charges have Abelian composition rules. Since, as opposed to the DHR-theory of superselection sectors [12], the states ωρ are not given in terms of localized endomorphisms, we take the following statement as a substitute for this terminology.
Dyonic Sectors and Intertwiner Connections in 2+1-D Lattice ZN -Higgs Models
429
Proposition 3.1.5. Let ρ, ρ0 ∈ D and for each a ∈ Z2 let ωρ and ωρ+ρ0 ·a be given by Propositions 3.1.1 or 3.1.2, respectively. Then lim ωρ+ρ0 ·a = ωρ .
a→∞
Proposition 3.1.5 is actually a special case of the following more general factorization property, which will be important in the construction of scattering states in [7]. Theorem 3.1.6. Let ρ1 , ρ2 ∈ D and for a ∈ Z2 put ρ(a) := ρ1 + ρ2 · a. Then, for all A, A0 , B and B 0 ∈ Aloc and all n ∈ N0 , n (A0 τa−1 (B 0 )) = lim ωρ(a) τa−1 (B)Aαρ(a) |a|→∞ ωρ1 Aαρn1 (A0 ) ωρ2 Bαρn2 (B 0 ) . (3.37)
A sketch of the proofs of Propositions 3.1.1, 3.1.2 and Theorem 3.1.6 is given in Appendices B.1–B.3. 3.2. Global transfer matrices. Given the family of dyonic states ωρ on A we denote by (πρ , Hρ , ρ ) the associated GNS representation πρ of A on the Hilbert space Hρ with cyclic vector ρ ∈ Hρ . (From now on we will no longer consider external charges, and therefore we simplify the notation by putting Hρ ≡ Hρ0 .) On Hρ we define a modified global transfer matrix Tρ by putting for A ∈ A, Tρ πρ (A)ρ = e−Eρ πρ (αρ (A))ρ ,
(3.38)
where Eρ ∈ R is an additive normalization constant, which will be determined later to appropriately adjust the self energies of the charge distributions ρ relative to each other. Theorem 3.2.1. For all ρ ∈ D Eq. (3.38) uniquely defines a positive bounded operator on Hρ , 0 ≤ Tρ ≤ e−Eρ , (3.39) which is invertible on πρ (A)ρ and implements the modified dynamics, i.e. Ad Tρ ◦ πρ = πρ ◦ αρ . Moreover, C ρ is the unique eigenspace of Tρ with maximal eigenvalue e−Eρ ≡ kTρ k. As a warning we recall that the charged states ωρ are ground states only with respect to the modified euclidean dynamics αρ and only when restricted to the observable algebra A ⊂ F. Hence, although πρ and Tρ could easily be extended to F and πρ (F)ρ , respectively, ρ would not necessarily be a ground state of Tρ in this enlarged Hilbert space containing external charges. However, since we will never look at external charges we will not be bothered by such a possibility. Proof of Theorem 3.2.1. First Tρ is well defined on πρ (A)ρ , since πρ (A)ρ = 0 implies ωρ (BA) = 0 for all B ∈ A and therefore e2Eρ kTρ πρ (A)k2 = ωρ αρ (A)∗ αρ (A) = ωρ αρ−1 (αρ (A)∗ )A = 0, by αρ -invariance of ωρ . Since αρ is invertible, Tρ is invertible on πρ (A)ρ . Moreover, by the ground state property we have
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πρ (A)ρ , Tρ πρ (A)ρ = e−Eρ ωρ (A∗ αρ (A)) ≤ e−Eρ ωρ (A∗ A),
(3.40)
which implies 0 ≤ Tρ ≤ e−Eρ . The identity Ad Tρ ◦ πρ = πρ ◦ αρ is obvious. Finally, let Pρ ∈ B(Hρ ) be the projection onto the eigenspace of Tρ with eigenvalue e−Eρ and ◦
P ρ = Pρ − |ρ ihρ |. Then, for all A, B ∈ A, lim ωρ (A∗ αρn (B)) = lim enEρ (πρ (A)ρ , Tρn πρ (B)ρ ) n
n
= (πρ (A)ρ , Pρ πρ (B)ρ ) ◦
= ωρ (A∗ )ωρ (B) + (πρ (A)ρ , P ρ πρ (B)ρ ). ◦
Hence the cluster property implies P ρ = 0.
(3.41)
Given the modified global transfer matrices Tρ we now use relation (3.16) to define on Hρ the family of transfer matrices Tρ (ρ0 ) := πρ (Lρ0 L−1 ρ )Tρ ,
(3.42)
where Lρ has been given in (3.18). Hence, Tρ ≡ Tρ (ρ) and we have Corollary 3.2.2. For all ρ, ρ0 ∈ D the operators Tρ (ρ0 ) are positive and bounded on Hρ and (3.43) Ad Tρ (ρ0 ) ◦ πρ = πρ ◦ αρ0 . Proof of Corollary 3.2.2. For the purpose of this proof we consider temporarily the enlarged GNS-triple (πρ (F), ρ , Hρ ), where Hρ now is the closure of πρ (F)ρ and contains vector states with external electric charges. Equation (3.43) follows immediately from (3.16). To see that Tρ (ρ0 ) is positive we use Lρ = Kρ α0 (Kρ∗ ), for Kρ defined in (3.17), to conclude Tρ (ρ0 ) = πρ (Kρ0 )Tρ (0)πρ (Kρ∗0 ) = πρ (Kρ0 Kρ−1 )Tρ (ρ)πρ (Kρ∗ −1 Kρ∗0 ), which is positive since Tρ (ρ) ≡ Tρ is positive.
(3.44)
In view of this result we call Tρ (0) the unmodified (“true”) global transfer matrix, since it implements the original dynamics α ≡ α0 . We also have the following important Corollary 3.2.3. Let Aρ ⊂ B(Hρ ) be the ∗-algebra generated by πρ (A) and Tρ (0). Then the commutant of Aρ is trivial: A0ρ = C 1l. Proof of Corollary 3.2.3. Since Tρ (ρ) ∈ Aρ we have [B, Tρ (ρ)] = 0 for all B ∈ A0ρ which implies Bρ = λρ for some λ ∈ C by the uniqueness of the ground state vector ρ of Tρ (ρ). Hence Bπρ (A)ρ = πρ (A)Bρ = λπρ (A)ρ for all A ∈ A and therefore B = λ1l. In view of Corollary 3.2.3 we may consider πρ as an irreducible representation of ˆ Algebraically A ˆ is defined to be the crossed ˆ ⊃ A such that Aρ = πρ (A). an extension A product of A with the semi-group N0 acting by n 7→ αn ∈ Aut A. Described in terms of ˆ is the ∗-algebra generated by A and a new selfadjoint element generators and relations A t ≡ t(0) with commutation relation tA = α(A)t for all A ∈ A. Note that the construction
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(2.32) shows that any GNS-representation πω of A from an α-invariant state ω naturally ˆ Moreover, Tω = πω (t) is positive provided the first extends to a representation of A. inequality in (2.26) holds. Using (3.18) we may also define ˆ t(ρ) := Lρ t ∈ A.
(3.45)
Then t(ρ) = t(ρ)∗ (since α(L∗ρ ) = Lρ )) and t(ρ) A = αρ (A) t(ρ) for all A ∈ A. By (3.42) ˆ by defining for fixed ρ and all this implies that also πρ extends to a representation of A ρ0 πρ (t(ρ0 )) := cρ Tρ (ρ0 ), (3.46) where cρ > 0 may be chosen arbitrarily. 3.3. Global charges. In this section we show that two charged representations of A, πρ and πρ0 , are dynamically equivalent if their total charges coincide, qρ1 = qρ2 . Here we take as an appropriate notion of equivalence the following Definition 3.3.1. Two representations, π and π 0 , of A are called dynamically equivalent, ˆ and if there exists a unitary intertwiner if they both extend to representations of A U : Hπ → Hπ0 such that U π(A) = π 0 (A) U for all A ∈ A and U π(t) = c π 0 (t) U for some constant c > 0. The use of the constant c is to allow for the possibility of different zero-point normalizations of the energy. Clearly, by rescaling the global transfer matrices one can always achieve c = 1. To prove that for qρ = qρ0 the representations πρ and πρ0 are dynamically equivalent in this sense we first show Proposition 3.3.2. Let q ∈ ZN × ZN be fixed and let ρ, ρ0 ∈ Dq . Then there exists a unique (up to a phase) unit vector 8ρ,ρ0 ∈ Hρ such that Tρ (ρ0 ) 8ρ,ρ0 = kTρ (ρ0 )k 8ρ,ρ0 . Moreover, 8ρ,ρ0 ∈ Hρ is cyclic for πρ (A) and it induces the state ωρ0 , i.e. (8ρ,ρ0 , πρ (A)8ρ,ρ0 ) = ωρ0 (A),
∀A ∈ A.
(3.47)
Proof. We adapt the proof of Theorem 6.4 of [2] to our setting. Hence we pick 1-cochains 1 such that (d∗ `e , d`m ) = ρ − ρ0 ∈ D0 and define `ε , `m ∈ Cloc 8nρ,ρ0 :=
Tρ (ρ0 )n ρ,ρ0 , kTρ (ρ0 )n ρ,ρ0 k
(3.48)
where ρ,ρ0 := πρ (Aρ,ρ0 (`e , `m )) ρ , 0 −1/2
Aρ,ρ0 (`e , `m ) := Z(µ )
(3.49) 1/2
M (`m )QGH (`e )Z(µ)
,
(3.50)
and where µ, µ0 denote the magnetic components in ρ, ρ0 . Using the definitions (3.243.29) we check in Appendix B.5 that the sequence of states (8nρ,ρ0 , πρ (A)8nρ,ρ0 ) converges to ωρ0 (A) for all A ∈ Aloc . We also show, that 8nρ,ρ0 is actually a Cauchy sequence in Hρ and therefore 8ρ,ρ0 := lim 8nρ,ρ0 ∈ Hρ (3.51) n
exists. By construction it is an eigenvector of Tρ (ρ0 ) with eigenvalue
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λρ0 = lim n
kTρ (ρ0 )n+1 ρ,ρ0 k = lim(8nρ,ρ0 , Tρ (ρ0 )2 8nρ,ρ0 )1/2 . n kTρ (ρ0 )n ρ,ρ0 k
(3.52)
Let now Hρ,ρ0 := πρ (A)8ρ,ρ0 ⊂ Hρ . Then Tρ (ρ0 )Hρ,ρ0 ⊂ Hρ,ρ0 and therefore ˆ ρ,ρ0 ⊂ Hρ,ρ0 by (3.46). Hence Hρ,ρ0 = Hρ by Corollary 3.2.3. By the clusπρ (A)H ter property of ωρ0 with respect to αρ0 we conclude λρ0 = kTρ (ρ0 )k and similarly as in (3.41) the associated eigenspace must be 1-dimensional. For later purposes we emphasize that Proposition 3.3.2 in particular implies that the choice of (`e , `m ) in (3.49) only influences the phase of the limit vector 8ρ,ρ0 in (3.51). Using Proposition 3.3.2 we now get the equivalence of charged representations whenever their total charge coincides. Theorem 3.3.3. Let ρ, ρ0 ∈ D and qρ = qρ0 . Then πρ and πρ0 are dynamically equivalent and kTρ (ρ00 )k kTρ (0)k = ∀ρ00 ∈ D. (3.53) kTρ0 (0)k kTρ0 (ρ00 )k Proof. Let U : Hρ0 → Hρ be given on πρ0 (A)ρ0 by U πρ0 (A)ρ0 := πρ (A)8ρ,ρ0 .
(3.54)
Then U extends to a unitary intertwining πρ0 and πρ . Moreover, since Tρ0 (ρ0 ) and Tρ (ρ0 ) implement αρ0 on Hρ0 and Hρ , respectively, one immediately concludes U Tρ0 (ρ0 )kTρ0 (ρ0 )k−1 = Tρ (ρ0 )kTρ (ρ0 )k−1 U.
(3.55)
Using (3.42) this implies U Tρ0 (ρ00 )kTρ0 (ρ0 )k−1 = Tρ (ρ00 )kTρ (ρ0 )k−1 U
∀ρ00 ∈ D
(3.56)
and therefore πρ and πρ0 are dynamically equivalent in the sense of Definition 3.3.1. Finally, (3.56) immediately implies (3.53). We now recall that fixing kTρ (ρ)k ≡ e−Eρ for a given ρ ∈ D amounts to fixing kTρ (ρ00 )k for any other ρ00 ∈ D. Hence, Theorem 3.3.3 shows that the energy normalization Eρ , ρ ∈ Dq , in (3.38) may be fixed up to a constant depending only on the total charge q by requiring kTρ (ρ00 )k = kTρ0 (ρ00 )k for all ρ, ρ0 ∈ Dq and some (and hence all) ρ00 ∈ D. In particular we now take ρ00 = 0 and require kTρ (0)k = e−(q) ,
∀ρ ∈ Dq
(3.57)
for some - as yet undetermined - function (q) ∈ R describing the “minimal self-energy” in the sector with total charge q ∈ ZN ×ZN . In the next subsection we will appropriately fix (q) such that 2(q) ≡ (q) + (−q) gives the minimal energy needed to create a pair of charge-anticharge configurations of total charge ±q from the vacuum and separate them apart to infinite distance. We close this subsection by remarking that we expect conversely (at least for a generic choice of couplings in the free charge phase) that the representations πρ and πρ0 are dynamically inequivalent if qρ 6= qρ0 . Inequivalence to the vacuum sector for qρ 6= 0 can presumably be shown by analogue methods as in [2], i.e. by proving the absence of a translation invariant vector in Hρ (see Sect. 4.3 for the implementations of translations
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in the charged sectors). More generally, one could try to prove that kTρ (ρ0 )k is not an eigenvalue of Tρ (ρ0 ) if qρ 6= qρ0 . To this end one would have to show that qρ 6= qρ0 implies s − lim n
Tρ (ρ0 )n πρ (A)ρ = 0 kTρ (ρ0 )kn
(3.58)
for all A ∈ Aloc . As in Proposition 5.5 of [2] a sufficient condition for this would be (πρ (A)ρ , Tρ (ρ0 )n πρ (A)ρ )2 = 0. n→∞ (πρ (A)ρ , Tρ (ρ0 )2n πρ (A)ρ ) lim
(3.59)
Using (3.42) this would amount to finding the asymptotics of the vertical string expectation value ! n−1 Y ∗ k −1 n αρ (Lρ0 Lρ )αρ (A) . (3.60) ωρ A k=0
Similarly as in the Bricmont-Fr¨ohlich criterion for the existence of free charges [13] Eq. (3.59) would follow provided (3.60) would decay like n−α e−n·const for some α > 0 (in [13] α = d/2). Since we have not tried to prove this we only formulate the Conjecture 3.3.4. If qρ 6= qρ0 then kTρ (ρ0 )k is not an eigenvalue of Tρ (ρ0 ). Together with Proposition 3.3.2 and Theorem 3.3.3 the Conjecture 3.3.4 would imply that πρ and πρ0 are dynamically equivalent if and only if their total charges coincide. 3.4. The dyonic self energies. In this subsection we determine the “dyonic selfenergies" (q) introduced in (3.57). In order to get charge conjugation invariant self energies it will be useful to start with implementing the charge conjugation as an intertwiner Cρ : Hρ → H−ρ by putting for A ∈ A and ρ ∈ D, Cρ πρ (A)ρ := π−ρ (iC (A))−ρ .
(3.61)
Lemma 3.4.1. Cρ extends to a well defined unitary Hρ → H−ρ such that i) Ad Cρ ◦ πρ = π−ρ ◦ iC . ii) Cρ∗ = C−ρ . iii) Cρ Tρ (ρ0 )eEρ = T−ρ (−ρ0 )eE−ρ Cρ . Proof. By Eq. (3.30) Cρ is well defined, unitary and obeys i). Part ii) follows from i2C = id. Finally, in the case ρ0 = ρ part iii) follows from Eq. (3.15). By (3.42) the case ρ0 6= ρ follows from iC (Lρ ) = L−ρ By charge conjugation symmetry we would of course like to have Eρ = E−ρ . We now fix the self-energies (q) in (3.57) of the sectors q ∈ ZN × ZN by first putting the vacuum energy to zero, (0) = 0, i.e. kTρ (0)k = 1,
∀ρ ∈ D0 .
(3.62)
This fixes e−Eρ ≡ kTρ (ρ)k = kπρ (Lρ )Tρ (0)k for all ρ ∈ D0 and in particular implies Eρ = E−ρ , ∀ρ ∈ D0 . In fact, we have Lemma 3.4.2. For fixed q ∈ ZN × ZN we have (q) = (−q) if and only if Eρ = E−ρ , ∀ρ ∈ Dq .
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Proof. Putting ρ0 = 0 in Lemma 3.4.1 iii) we see that kTρ (0)k = kT−ρ (0)k, ∀ρ ∈ Dq , is equivalent to Eρ = E−ρ , ∀ρ ∈ Dq , since Cρ is unitary. We now show that under the conditions (0) = 0 and (q) = (−q) the self-energies of the charged sectors are completely determined if we adjust them such that 2(q) ≡ (q) + (−q) becomes equal to the minimal energy needed to create a pair of chargeanticharge configurations of total charge ±q from the vacuum and separate them apart to infinity. Physically this means that for infinite separation we can consistently normalize the interaction energy between two charges to zero. Mathematically this is expressed by a factorization formula similar to Theorem 3.1.6 for matrix elements of the global transfer matrix . As it will turn out, this is further equivalent to an additivity property of the modified ground state energies, i.e. Eρ+ρ0 ·a → Eρ + Eρ0 as |a| → ∞. Theorem 3.4.3. There exists precisely one assignment of self-energies ZN ×ZN 3 q → (q) ∈ R such that kTρ (0)k = e−(q) , ∀ρ ∈ Dq and such that the following conditions hold i) (0) = 0, ii) (q) = (−q), iii) The factorization formula (3.37) extends to transfer matrices, i.e. πρ(a) (Aτ−a (B))ρ(a) , Tρ(a) (0)n πρ(a) (A0 τ−a (B 0 ))ρ(a) lim |a|→∞ = πρ1 (A)ρ1 , Tρ1 (0)n πρ1 (A0 )ρ1 πρ2 (B)ρ2 , Tρ2 (0)n πρ2 (B 0 )ρ2 , (3.63) where we used the notation of Theorem 3.1.6. To prove Theorem 3.4.3 and in particular the cluster property (3.63) we have to control norm ratios of (modified) transfer matrices, for which we have to invoke our polymer expansions. First we have Lemma 3.4.4. Let ρ, ρ0 ∈ Dq for some q ∈ ZN × ZN and choose 1-cochains `e , `m ∈ 1 such that (d∗ `e , d`m ) = ρ − ρ0 . Let Aρ,ρ0 ≡ Aρ,ρ0 (`e , `m ) ∈ Aloc be defined as in Cloc (3.50) and put # "ν−1 Y (ν) ∗ k −1 αρ (Lρ0 Lρ ) αρν (Aρ,ρ0 ) . (3.64) Xρ,ρ0 := Aρ,ρ0 k=0
Then we have (2ν+1)
ωρ (Xρ,ρ0 ) kTρ (ρ0 )k . = lim (2ν) ν→∞ kTρ (ρ)k ωρ (Xρ,ρ 0)
(3.65)
Proof. By Proposition 3.3.2 and Eqs. (3.48)–(3.51) we have for all q ∈ ZN × ZN and all ρ, ρ0 ∈ Dq , (ρ,ρ0 , Tρ (ρ0 )2ν+1 ρ,ρ0 ) , ν→∞ (ρ,ρ0 , Tρ (ρ0 )2ν ρ,ρ0 )
kTρ (ρ0 )k = (8ρ,ρ0 , Tρ (ρ0 )8ρ,ρ0 ) = lim where
ρ,ρ0 := πρ (Aρ,ρ0 (`e , `m )) ρ has been given in (3.49). The lemma follows by using (3.42) to rewrite
(3.66)
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Tρ (ρ ) =
"ν−1 Y
435
# αρk (Lρ0 L−1 ρ )
Tρ (ρ)ν ,
k=0
and recalling Tρ (ρ)ρ = kTρ (ρ)kρ by Theorem 3.2.1.
Note that the norm ratio (3.65) is obviously independent of the choice of the scale eEρ ≡ kTρ (ρ)k. We also recall from Proposition 3.3.2 that the choice of (`e , `m ) only influences the phase of 8ρ,ρ0 and hence the limit ν → ∞ in (3.65) is in fact independent of this choice. When expressing the r.h.s. of (3.65) in terms of euclidean path integrals we get a ratio of expectations of the form (3.31), where the classical function corresponding (ν) to Xρ,ρ 0 is itself a superposition of Wilson loops and vortex loops. These are built by the same recipes as before, i.e. with horizontal parts given by ±(`e , `m ) in the time slices t = ν and t = 0, respectively, and with vertical parts joining the endpoints of these strings. Hence we get the ratio of euclidean path integral expectations of two such loop configurations with time-like extension ν + 1 and ν, respectively, in the limit ν → ∞. Here these expectations have to be taken in the background determined by ρ, i.e. in the presence of the vertical parts of LE and LM described in (3.31), where there the limit n → ∞ and V → ∞ has to be taken first.8 Next we have to control certain factorization properties of the above norm ratios in the limit of the charge distributions being separated to infinity. Proposition 3.4.5. Let ρ1 , ρ2 ∈ D and for a ∈ Z2 put ρ1 (a) = ρ1 − ρ1 · a, ρ2 (a) = ρ2 − ρ2 · a. Then i) The limit kTρ1 (a) (ρ2 (a))k cρ1 , ρ2 := lim >0 (3.67) |a|→∞ kTρ1 (a) (ρ1 (a))k exists independently of the sequence a → ∞ and satisfies cρ1 ·g, ρ2 ·g = cρ1 , ρ2 , ∀g ∈ S. Moreover, for b ∈ Z2 we have the factorization property lim cρ1 +ρ01 ·b, ρ2 +ρ02 ·b = cρ1 , ρ2 cρ01 , ρ02 .
|b|→∞
ii) If qρ1 = qρ2 then c ρ1 , ρ 2 =
kTρ1 (ρ2 )k2 . kTρ1 (ρ1 )k2
(3.68)
(3.69)
Proof. Part i) is proven in detail in Appendix B. Here we just remark that it can roughly be understood from the perimeter law of Wilson and vortex loop expectations, which guarantees that ratios of loop expectations converge for infinitely large loops if the difference of their perimeters stays finite (in the above case this difference is given by two lattice units, due to the difference by one unit in time extension, see (3.65)). To prove ii) we pick in (3.66) the choice (3.70) ρ1 (a), ρ2 (a) = πρ1 (a) Aρ1 (a), ρ2 (a) (`e (a), `m (a)) ρ1 (a) , where we take (`e (a), `m (a)) = (`e − `e · a, `m − `m · a) This is why in our polymer expansion only the vertical parts of LE and LM matter, since the horizontal parts run in the time-like boundary of Vn and since the contributions from polymers reaching from the support of a classical function Acl to this boundary decay to zero for n → ∞. 8
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for some fixed configuration (`e , `m ) satisfying (d∗ `e , d`m ) = ρ1 − ρ2 . With this choice it is not difficult to check that for |a| large enough, Aρ1 (a), ρ2 (a) (`e (a), `m (a)) = Aρ1 , ρ2 (`e , `m )τ−a (Aρ1 , ρ2 (`e , `m )).
(3.71)
Similarly, for fixed ν and |a| large enough, = Xρ(ν) τ (Xρ(ν) ). Xρ(ν) 1 , ρ2 −a 1 , ρ2 1 (a), ρ2 (a)
(3.72)
When proving part i) in Appendix B.4 we show that plugging (3.72) into (3.65) the limit ν → ∞ is uniform in a and hence we may take the limit |a| → ∞ first. However, this takes us into the setting of the factorization formula (3.37) implying (ν) (ν) 2 lim ωρ(a) (Xρ(a), ρ0 (a) ) = ωρ1 (Xρ1 , ρ2 ) ,
|a|→∞
which proves part ii).
(3.73)
Proof of Theorem 3.4.3. The idea of the proof consists of translating the physically motivated conditions i)-iii) into conditions on the family of constants Eρ , ρ ∈ D. We have already remarked in (3.62) that i) may be obtained by suitably fixing Eρ for all ρ ∈ D0 . In Lemma 3.4.2 we have noticed that ii) is equivalent to Eρ = E−ρ for all ρ. We now show that iii) holds if and only if for all ρ1 , ρ2 ∈ D, lim
|a|→∞
To this end we use
Eρ1 +ρ2 ·a = Eρ1 + Eρ2 .
n Tρ (ρ)n = πρ (Lρ )Tρ (0) = πρ (Yρ(n) )Tρ (0)n ,
where Yρ(n) :=
n−1 Y
α0k (Lρ ).
(3.74)
(3.75)
(3.76)
k=0
Furthermore, for |a| large enough and ρ(a) = ρ1 + ρ2 · a we have Lρ(a) = Lρ1 τ−a (Lρ2 ) = τ−a (Lρ2 )Lρ1 .
(3.77)
Using that α0 commutes with τ−a we get for large enough |a| and A, A0 , B and B 0 ∈ Aloc , πρ(a) (Aτ−a (B))ρ(a) , Tρ(a) (0)n πρ(a) (A0 τ−a (B 0 ))ρ(a) n = ωρ(a) τ−a (B ∗ [Yρ(n) ]−1 )A∗ [Yρ(n) ]−1 αρ(a) (A0 τ−a (B 0 )) e−nEρ (a) . 2 1 Here we have chosen |a| large enough (depending on n) such that A∗ commutes with −1 τ−a Yρ(n) . On the other hand, we get for i = 1, 2 and all A, A0 ∈ Aloc , 2 πρi (A)ρi , Tρi (0)n πρi (A0 )ρi = ωρi A∗ [Yρ(n) ]−1 αρni (A0 ) e−nEρi . (3.78) i Thus, by Theorem 3.1.6 the factorization formula (3.63) is equivalent to (3.74). Hence, there can be at most one solution of the conditions i)–iii) above. Indeed, Eqs. (3.74) together with Eρ = E−ρ imply Eρ =
1 lim Eρ−ρa 2 |a|→∞
(3.79)
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for all ρ ∈ D. Since ρ − ρ · a ∈ D0 and since condition i) completely fixes Eρ = E−ρ for all ρ ∈ D0 , Eq. (3.79) fixes Eρ = E−ρ uniquely for all ρ ∈ D. We are left to prove that given Eρ = E−ρ for all ρ ∈ D0 via condition i) the limit (3.79) indeed exists for all ρ ∈ D and satisfies (3.74). However this is an immediate consequence of Proposition 3.4.5. Indeed, putting ρ2 = 0 in (3.67) and using kTρ1 (a) (0)k = 1 by condition i) we get for all ρ ∈ D, lim eEρ−ρ·a = cρ, 0 ,
|a|→∞
(3.80)
and hence (3.79) implies 1 ln cρ, 0 . (3.81) 2 Note that for qρ = 0 this is consistent with (3.69). The cluster property (3.74) now is an immediate consequence of (3.68). This concludes the proof of Theorem 3.4.3. Eρ =
kT (0)k
We remark that we have not worked out an analytic expression for the ratio kTρρ(ρ)k in the case qρ 6= 0, which is why we do not have further analytic knowledge of the dyonic self-energies (q). In particular we have not tried to confirm the natural “stability conjecture” (q) > 0 for all q 6= 0. In the purely electric (or purely magnetic) sectors the existence of 1-particle states [1] implies that e−(qρ ) ≡ kTρ (0)k lies in the spectrum of Tρ (0), i.e. (qρ ) is precisely the 1-particle self-energy at zero momentum. 4. State Bundles and Intertwiner Connections In this section we consider an appropriate analogue of what in the DHR-theory of super selection sectors would be called the state bundle. In our setting, by this we mean the ˆ obtained from collection of all GNS triples (Hρ , πρ , ρ ) of cyclic representation of A the family of states ωρ , ρ ∈ D. Modulo Conjecture 3.3.4, these representations fall into equivalence classes labelled by their total charges qρ . Hence we obtain, for each value of the global charge q ∈ ZN × ZN , a Hilbert-bundle Bq over the discrete base space Dq , which is simply given as the disjoint union [ ˙ Hρ . (4.1) Bq := ρ∈Dq
ˆ The fibers Hρ of this bundle are all naturally isomorphic as A-modules by Theorem ˆ acts irreducibly on Hρ by Corollary 3.2.3, these isomorphisms are 3.3.3. Since πρ (A) all uniquely determined up to a phase. When trying to fix this phase ambiguity one is naturally led to the problem of constructing an intertwiner connection on Bq , i.e. a family of intertwiners U (0) : Hρ → Hρ0 satisfying πρ0 = Ad U (0) ◦ πρ and depending on paths 0 in Dq from ρ to ρ0 . Here, by a path in Dq we mean a finite sequence (ρ0 , ρ1 , . . . , ρn ) of charge distributions ρi ∈ Dq , such that ρi and ρi+1 are “nearest neighbours” in Dq in a suitable sense to be explained below. In order to be able to formulate a concept of locality for these intertwiners we will also consider the Hilbert direct sum M Hρ (4.2) Hq := ρ∈Dq
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ˆ be represented by on which we let A ∈ A
M
Πq (A) :=
πρ (A) .
(4.3)
ρ∈Dq
The above mentioned connection will then be given in terms of an intertwiner ZN ˆ Locality in this framework Weyl algebra Wq acting on Hq and commuting with Πq (A). is formulated by the statement that Wq is generated by elementary “electric string operators” Eq (b) and elementary “magnetic string operators” Mq (b) satisfying for all oriented bonds b, b0 ∈ (Z2 )1 , Eq (b)N = Mq (b)N = 1l,
(4.4)
and the local ZN -Weyl commutation relations Eq (b)Eq (b0 )
= Eq (b0 )Eq (b),
Mq (b)Mq (b0 ) = Mq (b0 )Mq (b), Eq (b)Mq (b0 ) = e
2πi N δb, b0
(4.5)
Mq (b0 )Eq (b).
These operators will map each fiber Hρ , ρ ∈ Dq , isomorphically onto a “neighbouring one”, i.e. (4.6) Eq (b)Hρ = Hρ+(d∗ δb , 0) , Mq (b)Hρ = Hρ+(0, dδb ) ,
(4.7)
1 denotes the 1-cochain taking value one on b and zero otherwise. Hence, where δb ∈ Cloc one might think of Eq (b) (Mq (b)) as creating an electric (magnetic) charge-anticharge pair sitting in the boundary (coboundary) of the bond b. Defining charge distributions in Dq to be nearest neighbours if they differ by either such an elementary electric or magnetic dipole, the connection in Bq along a path (ρ0 , ρ1 , . . . , ρn ) in Dq is now given in the obvious way as an associated product of Eq (b)’s and Mq (b0 )’s, mapping each fiber Hρi isomorphically onto its successor Hρi+1 according to (4.6)-(4.7). We remark that in a DHR-framework one would expect these intertwiners to be given on the dense subspace πρ (A)ρ ⊂ Hρ in terms of unitary localized charge transporters Sel (b) and Smag (b) ∈ Aloc by
Eq (b)πρ (A)ρ = πρ+(d∗ δb , 0) (A Sel (b))ρ+(d∗ δb , 0) , Mq (b)πρ (A)ρ = πρ+(0, dδb ) (A Smag (b))ρ+(0, dδb ) ,
(4.8) (4.9)
which would be consistent and well defined if ωρ+(d∗ δb , 0) = ωρ ◦ Ad Sel (b), ωρ+(0, dδb ) = ωρ ◦ Ad Smag (b).
(4.10) (4.11)
Intuitively one could think of Sel as a kind of “electric” Mandelstam string operator as in (3.1) and of Smag (b) as its dual “magnetic” analogue. Unfortunately, due to the non-local energy regularization in (3.2) (and more generally, in (3.25)), the existence of such localized charge transporters Sel and Smag in Aloc satisfying (4.10)-(4.11) is very questionable. We also remark that, as opposed to the DHR-framework, in our lattice model the states ωρ are not given in the form ωρ = ω0 ◦ γρ for some localized automorphism γρ on
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A. This is why we do not have a field algebra extension of A carrying a global ZN × ZN symmetry, i.e. we are not able to define charged fields intertwining πρ ◦ γρ0 and πρ+ρ0 . It is therefore astonishing and, in our opinion, asks for further conceptual explanation that in this model one is nevertheless able to construct an intertwiner algebra with local commutation relations as given in (4.5). We will come back to this question in [7], where the existence of the local intertwiner algebra (4.4)-(4.5) will be the basic input for the construction of Haag-Ruelle scattering states in the charged sectors. 4.1. The local intertwiner algebra. We now proceed to the construction of the intertwiner algebra Wq obeying (4.4)-(4.5). For ρ, ρ0 ∈ Dq , let Pρ (ρ0 ) : Hρ → Hρ be the one-dimensional projection onto the ground state of Tρ (ρ0 ). Using the notation of (3.49)-(3.50) we then define for arbitrary 1 , 1-cochains l ∈ Cloc Pρ (ρ − (d∗ l, 0))πρ (Aρ, ρ−(d∗ l, 0) (l, 0))ρ , kPρ (ρ − (d∗ l, 0))πρ (Aρ, ρ−(d∗ l, 0) (l, 0))ρ k Pρ (ρ − (0, dl))πρ (Aρ, ρ−(0, dl) (0, l))ρ φmag , (l) = ρ kPρ (ρ − (0, dl))πρ (Aρ, ρ−(0, dl) (0, l))ρ k φel ρ (l) =
(4.12) (4.13)
which are well defined unit vectors in Hρ , by Proposition 3.3.2. We may then define ˆ by putting on the dense ˆ q (l) on Hq commuting with Πq (A) unitary operators Eˆq (l), M set πρ (A)ρ , A ∈ A, ρ ∈ Dq , Eˆq (l)πρ (A)ρ := πρ+(d∗ l, 0) (A)φel ρ+(d∗ l, 0) (l), mag ˆ Mq (l)πρ (A)ρ := πρ+(0, dl) (A)φρ+(0, dl) (l),
(4.14) (4.15)
which are well defined as in (3.54). It turns out that these intertwiners do not yet obey local commutation relations, however the violation of locality can be described by a kind of “coboundary” equation. Theorem 4.1.1. There exists an assignment of phases zρel (l), zρmag (l) ∈ U (1) such that 1 and all ρ ∈ Dq , for all l1 and l2 ∈ Cloc i) ii) iii)
el el zρ+(d ∗ l , 0) (l1 ) zρ (l2 ) 2 . Eˆq (l1 + l2 )ρ , Eˆq (l1 )Eˆq (l2 )ρ = zρel (l1 + l2 )
(4.16)
z mag dl2 ) (l1 ) zρmag (l2 ) ˆ q (l1 + l2 )ρ , M ˆ q (l1 )M ˆ q (l2 )ρ = ρ+(0, mag M . zρ (l1 + l2 )
(4.17)
z el dl1 ) (l1 ) zρmag (l1 ) ˆ q (l1 )ρ = eihl1 , l2 i ρ+(0, ˆ q (l1 )Eˆq (l2 )ρ , Eˆq (l2 )M (4.18) . M mag el zρ+(d ∗ l , 0) (l1 ) zρ (l2 ) 2
iv) If d∗ l = 0 then zρel (l) = 1. If dl = 0 then zρmag (l) = 1. el/mag
v) zρel/mag (l) = zρ·g
(l · g), ∀g ∈ S.
(4.19) (4.20) (4.21)
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The proof of this theorem and the precise definition of the phases zρel (l) and zρmag (l) will be given in Appendix B.6. 1 , We now use these phases to define on Hq the unitaries Zqel (l) and Zqmag (l), l ∈ Cloc by Zqel (l) Hρ := zρel (l)1lHρ , Zqmag (l) Hρ := zρmag (l)1lHρ .
(4.22) (4.23)
ˆ and we may define new Then Zqel (l) and Zqmag (l) clearly also commute with Πq (A) intertwiners Eq (l) := Eˆq (l)Zqel (l)−1 , ˆ q (l)Zqmag (l)−1 , Mq (l) := M
(4.24) (4.25)
which now obey local commutation relations. 1 the operators Eq (l) and Mq (l) are unitaries in B(Hq ) Theorem 4.1.2. For all l ∈ Cloc ˆ They satisfy the Weyl algebra relations commuting with Πq (A).
Eq (l1 )Eq (l2 ) = Eq (l1 + l2 ), Mq (l1 )Mq (l2 ) = Mq (l1 + l2 ), Eq (l1 )Mq (l2 ) = eihl1 , l2 i Mq (l2 )Eq (l1 ),
(4.26) (4.27) (4.28)
and restricted to each subspace Hρ ⊂ Hq , ρ ∈ Dq , we have Eq (l)Hρ = Hρ+(d∗ l, 0) , Mq (l)Hρ = Hρ+(0, dl) .
(4.29) (4.30)
Proof. It remains to prove the identities (4.26)–(4.28), for which it is enough to check them when applied to ρ ∈ Hρ , ∀ρ ∈ Dq . First we get Eq (l1 + l2 )ρ = zρel (l1 + l2 )−1 φel ρ+(d∗ l1 +d∗ l2 , 0) ,
(4.31)
which is a unit vector in the image of the one dimensional projection Pρ+(d∗ l1 +d∗ l2 , 0) (ρ). On the other hand we have Eq (l1 )Eq (l2 )ρ = zρel (l2 )−1 Eq (l1 )φel ρ+(d∗ l2 , 0) (l2 ).
(4.32)
ˆ and πρ0 +(d∗ l1 , 0) (A) ˆ we have Since Eq (l1 ) Hρ0 intertwines πρ0 (A) Eq (l1 )Tρ0 (ρ) Hρ0 = Tρ0 +(d∗ l1 , 0) (ρ)Eq (l1 ) Hρ0 , implying
(4.33)
(4.34) Eq (l1 )Pρ0 (ρ) Hρ0 = Pρ0 +(d∗ l1 , 0) (ρ)Eq (l1 ) Hρ0 . Hence (4.32) implies (4.35) Eq (l1 )Eq (l2 )ρ ∈ Pρ00 (ρ)Hρ , 00 ∗ ∗ where ρ = ρ + (d l1 + d l2 , 0). Comparing (4.31) and (4.35) we conclude that Eq (l1 )Eq (l2 )ρ and Eq (l1 + l2 )ρ only differ by a phase. This phase must be one since, by (4.16) and (4.24)–(4.25), (4.36) Eq (l1 + l2 )ρ , Eq (l1 )Eq (l2 )ρ = 1. Equations (4.27)-(4.28) are proven similarly.
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Putting Eq (b) ≡ Eq (δb ) and Mq (b) ≡ Mq (δb ), Theorem 4.1.2 provides the local intertwiner algebra Wq announced in (4.4)–(4.5). 4.2. The intertwiner connection. We now turn to the bundle theoretic point of view, where we consider the above intertwiners as a connection on Bq . As already explained, we call a pair ρ, ρ0 ∈ Dq nearest neighbours if they differ by an elementary electric or magnetic dipole, i.e. if there exists a bond b ∈ (Z2 )1 such that ρ0 − ρ = (±d∗ δb , 0) or ρ0 − ρ = (0, ±d δb ). In the first case we put Uρ0 , ρ = Eq (b)±1 :
H ρ → Hρ 0 ,
(4.37)
and in the second case we put Uρ0 , ρ = Mq (b)±1 :
H ρ → Hρ 0 .
(4.38)
If 0 = (ρ0 , . . . , ρn ) is a path in Dq , i.e. a finite sequence of nearest neighbour pairs, then we put U(0) := Uρn ρn−1 · · · Uρ1 ρ0 :
Hρ 0 → Hρ n
(4.39)
as the associated “parallel transport”. If 0 is a closed path, i.e. ρn = ρ0 , then U (0) must ˆ be a phase by the irreducibility of πρ0 (A). In order to determine these “holonomy phases” one may use the commutation relation (4.28) which allows to restrict ourselves to purely electric or purely magnetic loops 0 (i.e. where U(0) only consists of a product of Eq (b)’s or Mq (b)’s, respectively). Using (4.26)–(4.27) such loop operators U(0) are always of the form U (0) = Eq (le ) Hρ or U (0) = Mq (lm ) Hρ , where the loop condition on 0 implies d∗ le = 0 or dlm = 0, respectively. Note that this is consistent with (4.29)–(4.30), i.e. these loop operators must map Hρ onto itself. To compute the holonomy phases we now use the fact that d∗ le = 0 and dlm = 0 2 0 and k ∈ Cloc such that implies that there exist uniquely determined cochains s ∈ Cloc d ∗ s = le ,
dk = lm .
(4.40)
We also recall that in these cases Theorem 4.1.1 iv) implies Eq (le ) = Eˆq (le ) and Mq (lm ) = ˆ q (lm ). Hence, being a phase when restricted to Hρ , it is enough to apply these operators M to ρ where, by the definitions (4.22)–(4.23) and (4.24)–(4.25), they yield φel ρ (le ) and (lm ), respectively. φmag ρ Thus, we get our holonomy phases from 2 0 Proposition 4.2.3. Let s ∈ Cloc and k ∈ Cloc such that d∗ s = le and dk = lm . Then, for ρ = (, µ) we have ihµ, si ρ , φel ρ (le ) = e
(lm ) φmag ρ
=e
−ih, ki
ρ .
(4.41) (4.42)
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J. C. A. Barata, F. Nill
By the above arguments Proposition 4.2.3 implies Eq (le ) Hρ = eihµ, si , Mq (lm ) Hρ = eih, ki .
(4.43)
If we think of s being the characteristic function of a surface encircled by an electric loop le , then the holonomy phase (4.41) is just the magnetic flux through this surface. Interchanging electric and magnetic (and passing to the dual lattice), Eq. (4.42) may be stated analogously. This state of affairs will be the origin of anyon statistics of scattering states in [7]. Proposition 4.2.3 will be proven in Appendix B.7. 4.3. The representation of translations. Using our local intertwiner algebra Wq we are now in the position to define on each fiber Hρ a unitary representation Dρ (a), a ∈ Z2 , of the group of the lattice translation, such that Ad Dρ (a) ◦ πρ = πρ ◦ τa and such that Dρ (a) commutes with Tρ (0), ∀a ∈ Z2 . Moreover, for fixed q, these representations are all equivalent, i.e. U(0)Dρ (a) = Dρ0 (a)U(0) for all paths 0 : ρ → ρ0 . First we need a lift of the natural action of translations on Dq to bundle automorphisms on Bq . For the sake of generality let us formulate this by including also the lattice rotations. Lemma 4.3.4. For g ∈ S let Vρ (g) : Hρ → Hρ·g be given by Vρ (g)πρ (A)ρ := πρ·g (τg−1 (A))ρ·g .
(4.44)
Then Vρ (g) is a well defined unitary intertwining πρ ◦ τg with πρ·g 9 , i.e. Vρ (g)πρ (τg (A)) = πρ·g (A)Vρ (g), Vρ (g)Tρ (ρ0 ) = Tρ·g (ρ0 · g)Vρ (g),
(4.45) (4.46)
for all A ∈ A. Proof. Vρ (g) is well defined and unitary since ωρ ◦ τg = ωρ·g . Equations (4.45)-(4.46) follow from αρ ◦τρ·g = αρ·g and Eρ·g = Eρ (by (3.81) and S-invariance of cρ, ρ0 ). Note that the definition (4.44) implies the obvious identity Vρ·g (h)Vρ (g) = Vρ (gh).
(4.47)
for all g, h ∈ S, which means that the family of fiber isomorphisms Vρ (g), ρ ∈ Dq , may indeed be viewed as a lift of the natural right action of S on Dq to an action by unitary bundle automorphism on Bq . Equivalently, we now consider M Vq (g) := Vρ (g) (4.48) ρ∈Dq
as a unitary representation of S on Hq , satisfying Ad Vq (g) ◦ Πq = Πq ◦ τg−1 .
(4.49)
Moreover, we have 9 Note that the ∗-automorphism τ ∈ Aut A may be extended to a ∗-automorphism on A ˆ by putting g τg (t) = t.
Dyonic Sectors and Intertwiner Connections in 2+1-D Lattice ZN -Higgs Models
443
Lemma 4.3.5. Vq (g)Eq (l) = Eq (l · g)Vq (g), Vq (g)Mq (l) = Mq (l · g)Vq (g).
(4.50) (4.51)
Proof. By Theorem 4.1.1 v) we have Vq (g)Zqel/mag (l) = Zqel/mag (l · g)Vq (g).
(4.52)
ˆ q . However this Hence it is enough to prove the claim with Eq , Mq replaced by Eˆq , M is a straight forward consequence of the definitions and the fact that by (4.46), Vρ (g)Pρ (ρ0 ) = Pρ·g (ρ0 · g)Vρ (g),
(4.53)
and, therefore, el/mag
Vρ (g)φel/mag (l) = φρ·g ρ
(l · g).
(4.54)
We remark that Lemma 4.3.5 implies that the connection U (0) is S-invariant, i.e. Vρ0 (g)U(0) = U(0 · g)Vρ (g)
(4.55)
for all paths 0 : ρ → ρ0 and all g ∈ S. In order to arrive at an implementation of the translation group Z2 ⊂ S mapping each fiber Hρ onto itself we now have to compose Vρ (a)∗ , a ∈ Z2 , with a parallel transporter Uρ (a) : Hρ → Hρ·a along a suitable path 0 : ρ → ρ · a. To this end it is enough to consider the cases a = ei , i = 1, 2, where e1 = (1, 0) and e2 = (0, 1) denote the generators of Z2 . 0 there A minute’s thought shows that for any electric charge distribution ∈ Cloc 1 exists a unique 1-cochain `e (, i) ∈ Cloc with support only on bonds in direction i, such that the translated image · ei of by one unit in direction i satisfies · ei = + d∗ `e (, i).
(4.56)
2 there exists a unique 1-cochain Similarly, for a magnetic charge distribution µ ∈ Cloc `m (µ, i) with support only on the bonds perpendicular to the direction i, such that
µ · ei = µ + d`m (µ, i).
(4.57)
For ρ = (, µ) ∈ Dq we now define Uρ (ei ) : Hρ → Hρ·ei by putting Uρ (ei ) := Eq (`e (, i))Mq (`m (µ, i)) Hρ = Mq (`m (µ, i))Eq (`e (, i)) Hρ ,
(4.58)
where the second equality follows since, by construction, `m (µ, i) and `e (, i) always have disjoint support. With this construction we now define the unitaries Dρ (ei ) : Hρ → Hρ , i = 1, 2, by Dρ (ei ) := Vρ (ei )∗ Uρ (ei ).
(4.59)
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Theorem 4.3.6. The unitaries Dρ (ei ), i = 1, 2, generate a representation of Z2 impleˆ i.e. menting the translation automorphism on A, Dρ (e1 )Dρ (e2 ) = Dρ (e2 )Dρ (e1 ), Dρ (ei )πρ (A) = πρ (τei (A))Dρ (ei ),
(4.60) (4.61)
ˆ for all A ∈ A. Proof. Equation (4.61) immediately follows from (4.58) and (4.45)-(4.46). To prove (4.60) we first use Lemma 4.3.5 to conclude that Dρ (e2 )Dρ (e1 ) = Vρ (e2 )∗ Vρ·e2 (e1 )∗ Uρ·e1 (e2 )Uρ (e1 ), Dρ (e1 )Dρ (e2 ) = Vρ (e1 )∗ Vρ·e1 (e2 )∗ Uρ·e2 (e1 )Uρ (e2 ).
(4.62) (4.63)
Since (4.47) implies Vρ·e2 (e1 )Vρ (e2 ) = Vρ·e1 (e2 )Vρ (e1 ) = Vρ (e1 + e2 ),
(4.64)
we are left to check Uρ (e2 )−1 Uρ·e2 (e1 )−1 Uρ·e1 (e2 )Uρ (e1 ) = 1lHρ .
(4.65)
Using the definition (4.58) and the Weyl algebra relations (4.26)-(4.28), Eq. (4.65) is equivalent to (4.66) Mq (Lm (µ)) Eq (Le ()) Hρ = (u1 u2 u3 )−1 , where Lm (µ) = `m (µ, 1) + `m (µ · e1 , 2) − `m (µ · e2 , 1) − `m (µ, 2), Le () = `e (, 1) + `e ( · e1 , 2) − `e ( · e2 , 1) − `e (, 2),
(4.67) (4.68)
and where ui ∈ U (1) are the phases obtained by commuting in (4.65) all factors of Eq ’s to the right of Mq ’s, i.e. u1 (, µ) = exp −ih`e (, 2), `m (µ, 1) − `m (µ · e2 , 1)i , (4.69) u2 (, µ) = exp −ih`e ( · e2 , 1), `m (µ · e1 , 2)i , (4.70) u3 (, µ) = exp −ih`e ( · e1 , 2), `m (µ, 1)i . (4.71) To verify (4.66) we decompose and µ into a sum over “monopoles”, i.e. cochains supported on a single site or a single plaquette, respectively. Using the obvious fact that `e and `m are ZN -module maps, i.e. `e (n1 + 2 , i) = n`e (1 , i) + `e (2 , i), `m (nµ1 + µ2 , i) = n`m (µ1 , i) + `m (µ2 , i),
(4.72) (4.73)
0 2 and µ1, 2 ∈ Cloc , we conclude that (4.66) holds if and only for all n ∈ ZN , 1, 2 ∈ Cloc if it holds for all pairs of monopole distributions ρ = (, µ) = (δx , δp ), x ∈ (Z2 )0 , p ∈ (Z2 )2 . In fact, we have Y Y ui (δx , δp )(x)µ(p) , ui (, µ) = x∈(Z2 )0 p∈(Z2 )2
and similarly
Dyonic Sectors and Intertwiner Connections in 2+1-D Lattice ZN -Higgs Models
(, µ) , Mq (`m (µ))Eq (`e ())(, µ) Y
Y
445
=
(δx , δp ) , Mq (`m (δp ))Eq (`e (δx ))(δx , δp )
(x)µ(p)
.
(4.74)
x∈(Z2 )0 p∈(Z2 )2
Now, for = δx one easily verifies `e (δx , j) = −δhx−ej , xi ,
(4.75)
implying `e (δx · ei , j) = −δhx−ei −ej , x−ei i . For µ = δp and p = hy, y + e1 , y + e1 + e2 , y + e2 i we get `m (δp , 1) = δhy, y+e2 i , `m (δp , 2) = −δhy, y+e1 i . Plugging this into (4.69)-(4.71) we get 2πi δx, y+e2 − δx, y , u1 = exp N 2πi u2 = exp − δx, y+e2 , N 2πi u3 = exp − δx, y+e1 +e2 , N implying (u1 u2 u3 )−1 = exp
2πi δx, y + δx, y+e1 +e2 . N
(4.76) (4.77)
(4.78) (4.79) (4.80)
(4.81)
Next we look at (4.67)–(4.68) to compute Lm (δp ) = δhy, y+e2 i − δhy−e1 , yi − δhy−e2 , yi + δhy, y+e1 i = −dδy , (4.82) Le (δx ) = −δhx−e1 , xi − δhx−e1 −e2 , x−e1 i + δhx−e1 −e2 , x−e2 i + δhx−e2 , xi = −d∗ δq , (4.83) where q is the oriented plaquette q = hx − e1 − e2 , x − e2 , x, x − e1 i. This implies 2πi Mq (Lm (δp ))Eq (Le (δx ))ρ = exp δx−e1 −e2 , y + δx, y ρ . (4.84) N Comparing (4.81) with (4.84) we have proven (4.66) and therefore Theorem 4.3.6.
We remark that our definition (4.59) is consistent with the translation invariance of the vacuum 0 , since ρ = 0 implies U0 (ei ) = 1l and V0 (ei )0 = 0 . Next we show that our intertwiner connection also intertwines the representations Dρ of the translation group Z2 in Hρ , ∀ρ ∈ Dq . This is formulated most economically by putting M Dq (ei ) := Dρ (ei ), (4.85) ρ∈Dq
implying ˆ for all A ∈ A.
Dq (ei )Πq (A) = Πq (τei (A))Dq (ei ),
(4.86)
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Proposition 4.3.7. For i = 1, 2 the intertwiner algebra Wq commutes with Dq (ei ). Proof. Putting Uq (ei ) := Vq (ei )Dq (ei ) and using Lemma 4.3.5 we have to show Uq (ei )Eq (`0 ) = Eq (`0 · ei )Uq (ei ), Uq (ei )Mq (`0 ) = Mq (`0 · ei )Uq (ei ),
(4.87) (4.88)
1 . Since Uq (ei ) provides an intertwiner mapping Hρ → Hρ·ei it is again for all `0 ∈ Cloc enough to check these identities on ρ , ∀ρ ∈ Dq . Using (4.58), (4.59) and commuting Mq (`m (µ, i)) to the left, Eq. (4.87) is equivalent to 0
Eq (`e ( + d∗ `0 , i))Eq (`0 )(, µ) = eih` ·ei , `m (µ, i)i Eq (`0 · ei )Eq (`e (, i))(, µ) . (4.89) To prove (4.89) we compute L(`0 , i) := `e ( + d∗ `0 , i) + `0 − `0 · ei − `e (, i) = `e (d∗ `0 , i) + `0 − `0 · ei ,
(4.90)
1 yielding d∗ L(`0 , i) = 0. Let S(`0 , i) ∈ Cloc be such that d∗ S(`0 , i) = L(`0 , i). Then, by (4.43), Eq. (4.89) is equivalent to 0
0
eih` ·ei , `m (µ, i)i = eihS(` , i),
µi
.
(4.91)
Similarly as in the proof of Theorem 4.3.6 it is enough to check (4.91) for all `0 = δb , b ∈ (Z2 )1 , and all µ = δp , p ∈ (Z2 )2 . Hence, let b = hx, x + ej i and p = hy, y + e1 , y + e1 + e2 , y + e2 i. Then d∗ δb = δx+ej − δx and (4.75) gives L(δhx, x+ej i , i) = `e (δx+ej , i) − `e (δx , i) + δhx, x+ej i − δhx−ei , x+ej −ei i = δhx, x+ej i − δhx+ej −ei , x+ej i + δhx−ei , xi − δhx−ei , x+ej −ei i(4.92) . Thus, if j = i then L(δhx, x+ej i , i) = 0 implying (4.91), since in this case `m (µ, i) is perpendicular to the direction j. We are left to check the case j 6= i, which gives L(δhx, x+ej i , i) = (−1)j d∗ δq−ei , where q = hx, x + e1 , x + e1 + e2 , x + e2 i. Thus we get for i 6= j, 2πi (−1)j δx−ei , y . (4.93) eihS(δhx, x+ej i ), δp i = exp N On the other hand, Eqs. (4.76)–(4.77) give for i 6= j, `m (δp , i) = (−1)j δhy, y+ej i ,
(4.94)
and hence (4.93)–(4.94) imply (4.91) in the case i 6= j. Thus we have proven (4.87). Equation (4.88) is proven by similar methods. 4.4. Conclusions. In this work we have investigated the dyonic sector structure of 2 + 1dimensional lattice ZN -Higgs models described by the Euclidean action (1.2) in a range of couplings (2.23)-(2.24) corresponding to the “free charge phase” of the Euclidean statistical mechanics model (1.1). We have worked in the Hamiltonian picture by formulating the model in terms of its observable algebra A generated by the time-zero fields. A Euclidean (modified) dynamics αρ = lim Ad TV (ρ) has been defined on A in terms of local (modified) transfer V
Dyonic Sectors and Intertwiner Connections in 2+1-D Lattice ZN -Higgs Models
447
matrices TV (ρ), where ρ = (ε, µ) is a superposition of electric and magnetic ZN -charge distributions with finite support on the spatial lattice Z2 . Dyonic states ωρ have been constructed as ground states of the modified dynamics αρ on A. The associated charged representations (πρ , Hρ , ρ ) of A extend to irreducible representations of the “dynamic closure” Aˆ ⊃ A, where Aˆ = hA, ti is the abstract ∗-algebra generated by A and a global positive transfer matrix t implementing the “true” (i.e. unmodified) dynamics α0 . πρ and πρ0 are equivalent as representations of ˆ provided their total charges coincide, qρ = qρ0 . We have conjectured that the total A, charges qρ ∈ ZN × ZN indeed label the sectors of the model, i.e. πρ 6∼ πρ0 if qρ 6= qρ0 . The infimum of the energy spectrum (q) of each sector has been shown to be uniquely fixed by the conditions (0) = 0 and (q) = (−q) and the requirement of decaying interaction energies for infinite spatial separation. of the state bundle Bq = [In Sect. 4 we have analyzed structural algebraic aspectsM (ρ, Hρ ), Dq := {ρ | qρ = q}, by constructing on Hq = Hρ a local intertwiner ρ∈Dq
algebra Wq commuting with
M
ρ∈Dq
ˆ The generators of Wq are given by electric πρ (A).
ρ∈Dq
and magnetic “charge transporters”, Eq (b) and Mq (b), localized on bonds b in Z2 and fulfilling local Weyl commutation relations (Theorem 4.1.2). In terms of these charge transporters we have obtained a unitary connection U (0) : Hρ → Hρ0 intertwining πρ and πρ0 for any path 0 : ρ → ρ0 in Dq . The holonomy of this connection is given by ZN -valued phases related to the electric and magnetic charges enclosed by 0 (see (4.43)). Finally, the connection 0 7→ U (0) has been used to construct on each Hρ a unitary ˆ representation of the group of spatial lattice translations acting covariantly on πρ (A) and being intertwined by U (0). We remark that the existence of such representations in the charged sectors of our model is by no means an obvious feature. In [7] the holonomy phases of our connection 0 will be the main ingredient for establishing the anyonic nature of multiparticle scattering states of electrically and magnetically charged particles whose existence has been shown in [1]. Appendix A. A Brief Sketch of the Polymer and Cluster Expansions A.1. Expansions for the vacuum sector. In this Appendix we present the basics of the polymer and cluster expansion developed in [1]. We intend to present here only the most relevant facts to make the main ideas in the proofs of this Appendix understandable. For more details see [1]. Definition A.1.1. For a 1-cochains E with d∗ E = 0 and a 2-cochain D with dD = 0, both with finite support, we define the “winding number of E around D” as 2πi D hu , Ei . (A.1) [D : E] := exp N One easily sees that this definition does not depend of the choice of the particular configuration uD .
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Let us now prepare the definition of our polymers and their activities. Define the sets n P = P ∈ (Z3 )+2 : P is finite, co-connected and P = supp D, o for some D ∈ (Z3 )2 , dD = 0, D 6= 0 , n B = M ∈ (Z3 )+1 : M is finite, connected and M =
supp E,
o for some E ∈ (Z3 )1 , d∗ E = 0, E 6= 0 ,
where (Z3 )+1 (respectively (Z3 )+2 ) refers to the set of positively oriented bonds (plaquettes) of Z3 and the sets Ptotal = P ∈ (Z3 )+2 finite, so that P = supp D, for some D ∈ (Z3 )2 , dD = 0 , Btotal = M ∈ (Z3 )+1 finite, so that M = supp E, for some E ∈ (Z3 )1 , d∗ E = 0 . Note that the sets Ptotal and Btotal contain the empty set and that the non-empty elements of Ptotal and of Btotal are built up by unions of co-disjoint elements of P, respectively, by unions of disjoint elements of B. One has naturally P ⊂ Ptotal and B ⊂ Btotal . Each non-empty set P ∈ Ptotal and M ∈ Btotal can uniquely be decomposed into disjoint unions P = P1 + · · · + PAP , M = M1 + · · · + MBM (the symbol “+” indicates here disjoint union) where Pi ∈ P and Mj ∈ B. Then, if D ∈ (Z3 )2 is such that supp D = P , there is a unique decomposition D = D1 + · · · + DAP with Di ∈ (Z3 )2 , supp Di = Pi . Moreover if E ∈ (Z3 )1 is such that supp E = M , then there is a unique decomposition E = E1 + · · · + EBM with Ei ∈ (Z3 )1 , supp Ei = Mi . One can also decompose u = uD1 + · · · + uDAP with uDi ∈ (Z3 )1 , duDi = Di . For P ∈ Ptotal and M ∈ Btotal we define the sets D(P ) := {D ∈ (Z3 )2 so that
supp D
= P and dD = 0},
E(M ) := {E ∈ (Z ) so that
supp E
= M and d∗ E = 0}.
3 1
We consider now pairs (P, D) with P ∈ Ptotal and D ∈ D(P ) and pairs (M, E) with M ∈ Btotal and E ∈ D(M ) and define w((P, D), (M, E)) = w((M, E), (P, D)) ∈ {0, . . . , N − 1} as the “ZN -winding number” of (M, E) around (P, D): w((P, D), (M, E)) = w((M, E), (P, D)) := [D : E].
(A.2)
The pairs with P ∈ P and M ∈ B will be the building blocks of our polymers. With the help of w we can establish a connectivity relation between pairs (P, D) with P ∈ P, D ∈ D(P ) and pairs (M, E) with M ∈ B, E ∈ E(M ): we say that (P, D) and (M, E) are “w-connected” if w((P, D), (M, E)) 6= 1 and “w-disconnected” otherwise. We arrive then at the following Definition A.1.2. A polymer γ is formed by two pairs {(P γ , Dγ ), (M γ , E γ )} , with P γ ∈ Ptotal (V ), M γ ∈ Btotal (V ) and Dγ ∈ D(P γ ), E γ ∈ E(M γ ), so that the set
Dyonic Sectors and Intertwiner Connections in 2+1-D Lattice ZN -Higgs Models
449
γ γ {(P1γ , D1γ ), . . . , (PAγ γ , DA ), (M1γ , E1γ ), . . . , (MBγ γ , EB )} γ γ
(A.3)
formed by the decompositions P γ = P1γ + · · · + PAγ γ , M γ = M1γ + · · · + MBγ γ with γ γ , E γ = E1γ + · · · + EB with Piγ ∈ P(V ), Mjγ ∈ B(V ) and Dγ = D1γ + · · · + DA γ γ γ γ γ γ Di ∈ D(Pi ), Ej ∈ E(Mj ) is a w-connected set. Below, when we write (M, E) ∈ γ and (P, D) ∈ γ we are intrinsically assuming that M ∈ B with E ∈ E(M ) and that P ∈ P with D ∈ D(P ). For a polymer γ = ((P γ , Dγ ), (M γ , E γ )) we call the pair γg := (P γ , M γ ) the geometrical part of γ and the pair γc := (Dγ , E γ ) is the “coloring” of γ. Of course the coloring determines uniquely the geometric part. Each pair (D, E), D ∈ D(P ), E ∈ E(M ) with P ∈ P, M ∈ B is a color for (P, M ). Another important definition is the “size” of a polymer. We define the size of γ by |γ| = |γg | := |P γ | + |M γ |, where |P γ | (respectively |M γ |) is the number of plaquettes (respectively bonds) making up P γ (respectively M γ ). The activity µ(γ) ∈ C of a polymer γ is defined to be Aγ Bγ Y Y Y Y γ g(Diγ (p)) h(Ejγ (b)) , (A.4) µ(γ) := D : E γ γ γ i=1
j=1
p∈Pi
b∈Mj
with µ(∅) = 1. For a polymer model we need the notions of “compatibility” and “incompatibility” between pairs of polymers. This is defined in the following way. Two polymers γ and γ 0 are said to be incompatible, γ 6∼ γ 0 , if at least one of the following conditions hold: i)
0
0
There exist Maγ ∈ γg and Mbγ ∈ γg0 , so that Maγ and Mbγ are connected (i.e. there exists at least one lattice point x so that x ∈ ∂b and x ∈ ∂b0 for some bonds b ∈ Maγ 0 and b0 ∈ Mbγ ); 0
0
ii) There exists Paγ ∈ γg and Pbγ ∈ γg0 , so that Paγ and Pbγ are co-connected (i.e. there exists at least one cube c in the lattice so that p ∈ ∂c and p0 ∈ ∂c for some 0 plaquettes p ∈ Paγ and p0 ∈ Pbγ ); 0
0
0
0
iii) There exists (Maγ , Eaγ ) ∈ γ and (Pbγ , Dbγ ) ∈ γ 0 , so that (Maγ , Eaγ ) and (Pbγ , Dbγ ) are w-connected, or the same with γ and γ 0 interchanged. They are said to be compatible, γ ∼ γ 0 , otherwise. We will denote by G(V ) the set of all polymers in V ⊂ Z3 and by Gcom (V ) the set of all finite sets of compatible polymers. We want to express the vacuum expectation of classical observables in terms of our polymer expansion. We consider the following Definition A.1.3. Let α be a 1-cochain and β 2-cochain, both with finite support. Define the classical observable Y g((du + β)(p)) 2πi hα, ui . (A.5) B(α, β) := exp − N g(du(p)) p Any classical observable can be written as a linear combination of such functions.
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For a finite volume V ⊂ Z3 (say, a cube) we have the following10 hB(α, β)iV =
1 ZV1
X
X
[D − β : E − α]
Y
g(D(p))
p∈suppD
D∈V 2 E∈V 1 d(D−β)=0 d∗ (E−α)=0
Y
h(E(b)) .
b∈suppE
(A.6) Here the normalization factor ZV1 is given by X ZV1 = µ0 , Y
in multi-index notation, i.e. µ0 :=
(A.7)
0∈Gcom (V )
µ(γ). We will often identify the elements of Gcom
γ∈0
with their characteristic functions. The cochains D appearing in the sums in (A.6) can uniquely be decomposed in such a way that D = D0 + D1 with d(D0 − β) = 0 and dD1 = 0 and so that supp D0 is co-connected and co-disconnected from supp D1 . If dβ = 0 we choose D0 = 0. Analogously, the cochains E appearing in the sums in (A.6) can be decomposed uniquely in such a way that E = E0 + E1 with d∗ (E0 − α) = 0 and d∗ E1 = 0 and so that supp E0 is connected and disconnected from supp E1 . If d∗ α = 0 we choose E0 = 0. We denote by C1 (α) the set of the supports of all such E0 ’s, for a given α and by C2 (β) the set of the supports of all such D0 ’s, for a given β. For d∗ α = 0 we have C1 (α) = ∅ and for dβ = 0 we have C2 (β) = ∅. We define the sets of pairs Conn1 (α, V ) := (M, E), so that M ∈ C1 (α) and E ∈ V 1 , (A.8) with supp E = M and d∗ E = d∗ α} , Conn2 (β, V ) := (P, D), so that P ∈ C2 (β) and D ∈ V 2 , (A.9) with supp D = P and dD = dβ} . We then write
X
hB(α, β)iV =
[D − β : E − α]
Conn1 (α, V ) (P, D)∈Conn2 (β, V )
(M, E)∈
×
Y p∈P
" g(D(p))
Y
# h(E(b))
X 0∈Gcom
a0(M, E), α b0(P, D), β µ0 X
b∈M
µ0
,
0∈Gcom
(A.10) for
a(M, E), α (γ) :=
0, if M γ is connected with M, , γ [D : E − α] , otherwise
and 10
For simplicity we will neglect some boundary terms, which can be controlled with more work.
(A.11)
Dyonic Sectors and Intertwiner Connections in 2+1-D Lattice ZN -Higgs Models
b(P, D), β (γ) :=
451
0, if P γ is co-connected with P . γ [D − β : E ] , otherwise
(A.12)
It is for many purposes useful to write the normalization factor ZV1 in the form X c 0 µ0 . (A.13) ZV1 = exp 0∈Gclus (V )
Let us explain the symbols used above. Our notation is close to that of [2]. Gclus (V ) is the set of all finite clusters of polymers in V , i.e. an element 0 ∈ Gclus is a finite set of (not necessarily distinct) polymers building a connected “incompatibility graph”. An incompatibility graph is a graph which has polymers as vertices and where two vertices are connected by a line if the corresponding polymers are incompatible. We will often identify elements 0 ∈ Gclus with functions 0: G → N, where 0(γ) is the multiplicity of γ in 0 ∈ Gclus . The coefficients c0 are the “Ursell functions” and are of purely combinatorial nature. They are defined (see [2] and [14]) by c0 :=
∞ X (−1)n+1 n=1
n
Nn (0),
(A.14)
where Nn (0) is the number of ways of writing 0 in the form 0 = 01 + · · · + 0n where 0 6= 0i ∈ Gcom , i = 1, . . . , n. Relation (A.13) makes sense provided the sum over clusters is convergent. As discussed in [2] and [1] a sufficient condition for this is kµk ≤ kµkc , where kµk := sup |µ(γ)|1/|γ| , and kµkc is a constant defined in [2]. By (A.4), γ∈G
|γ| |µ(γ)| ≤ max{g(1), . . . , g(N − 1), h(1), . . . , h(N − 1)} ,
(A.15)
which justifies the conditions (2.23)–(2.24). Calling Conn1 (α) := Conn1 (α, Z3 ), Conn2 (β) := Conn2 (β, Z3 ) and Gclus := Gclus (Z3 ), we can also write the thermodynamic limit of hB(α, β)iV as
hB(α, β)i =
X
[D − β : E − α]
Conn1 (α) Conn2 (β)
(M, E)∈ (P, D)∈
× exp
X
Y
"
Y
g(D(p))
p∈P
# h(E(b))
b∈M
c0 a0(M, E), α b0(P, D), β − 1 µ0
! .
(A.16)
0∈Gclus
The presence of the phases [D − β : E − α] is an important feature of this last expression and is responsible for the emergence of the anyonic statistics. Note that for α and β such that dβ = 0 and d∗ α = 0 the expectation hB(α, β)i is proportional to [β : α], i.e. to the winding number of β around α. Let 0 be a cluster of polymers. We say that a polymer γ is incompatible with 0, i.e. γ 6∼ 0, if there is at least one γ 0 ∈ 0 with γ 6∼ γ 0 . For two clusters 0, 00 we have 0 6∼ 00 if there is at least one γ ∈ 0 with γ 6∼ 00 . For the polymer system discussed in this work we have the following result:
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Theorem A.1.4. There is a convex, differentiable, monotonically decreasing function F0 : (a0 , ∞) → R+ , for some a0 ≥ 0, with lima→∞ F0 (a) = 0 such that, for all sets of polymers 0, and for all a > a0 , X
e−a|γ| ≤ F0 (a) k0k,
(A.17)
γ6∼0
P where k0k = 0(γ 0 )|γ 0 |, 0(γ 0 ) being the multiplicity of γ 0 in 0. Once inequality (A.17) has been established, it has been proven in [2], Appendix A.1, that the two following results hold: X
|c0 | |µ0 | ≤ F1 (− ln kµk) k00 k,
(A.18)
0∈Gclus 06∼00
X
|c0 | |µ0 | ≤
0∈Gclus 06∼00 k0k≥n
kµk kµc k
n k00 kF0 (ac ),
(A.19)
where ac and kµc k > 0 are constants defined in [2], F1 : (ac + F0 (ac ), ∞) → R+ is the solution of F1 (a + F0 (a)) = F0 (a) and kµk := supγ |µ(γ)|1/|γ| . For a proof we refer the reader to [1] and [2]. The inequalities (A.18) and (A.19) are of central importance in the theory of cluster expansions and Pare often used for proving theorems. For instance, (A.19) tells us that the sums like 0 |c0 ||µ0 | involving only clusters with size larger than a certain n (and which are incompatible with some 00 fixed) decay exponentially with n. A.2. Expansions for the dyonic sectors. Let us now present the corresponding expansions for the states ωρ for ρ ∈ D0 . To each B ∈ F0 we can associate a classical observable Bcl, ρ = Bcl, ρ (dϕ − A) (see (3.31)). A possible but non-unique choice is (see [1] and [2]) Bcl, ρ
T rHV F(ϕ(0), A(0)), (ϕ(1), A(1)) B TV (ρ) , = T rHV F(ϕ(0), A(0)), (ϕ(1), A(1)) TV (ρ)
(A.20)
where F(ϕ(0), A(0)), (ϕ(1), A(1)) =
X
|ϕ(0), A(0)ihϕ(1), A(1)|,
(A.21)
(ϕ(0), A(0)), (ϕ(1), A(1))
and where ϕ(k), A(k) refers to the variables in the k th euclidean time plane. Since any such classical observable can be written as a finite linear combination of the functions B(α, β) previously introduced (with coefficients eventually depending on ρ) we concentrate on expectations of such functions. Proceeding as in the previous sections we can express hB(α, β)iρ defined in (3.31) as
Dyonic Sectors and Intertwiner Connections in 2+1-D Lattice ZN -Higgs Models
hB(α, β)iρ = X Conn1 (α) Conn2 (β)
[D − β : E − α] [D − β : −˜] [−µ˜ : E − α]
"
×
Y
#
X
h(E(b)) exp
b∈M
Y
g(D(p))
p∈P
(M, E)∈ (P, D)∈
453
c0 a0(M, E), α b0(P, D), β
−1
! a0 b0µ µ0
,
(A.22)
0∈Gclus
˜ n)) := µ(x) for all x ∈ (Z2 )2 and n ∈ Z where µ˜ is the 2-cochain on Z3 defined by µ((x, 3 and ˜ in the 1-cochain on Z defined by ˜((y, n + 1/2)) := (y), for all y ∈ Z2 and n ∈ Z. Here (y, n + 1/2) indicates the vertical bond in (Z3 )+1 whose projection onto Z2 is y and is located between the euclidean time-planes n and n + 1. Moreover, we defined a (γ) := [Dγ : −˜], bµ (γ) := [−µ˜ : E γ ].
(A.23) (A.24)
The cochains µ˜ and ˜ do not have finite support but, since the polymers are finite, the right hand side of the last two expressions can be defined using some limit procedure, for instance, by closing µ˜ and ˜ at infinity by adding, before the thermodynamic limit is taken, the cochains sµ and s to them. Since the polymers are finite, the limit does not depend on the particular sµ and s chosen.. The same can be said about the winding numbers [D − β : −˜] and [−µ˜ : E − α] in (A.22). Concerning the sum over clusters in (A.22), the following estimate can be established: Proposition A.2.1. For (M, E) ∈ Conn1 (α) and (P, D) ∈ Conn2 (β), one has X 0 0 0 0 0 c0 a(M, E), α b(P, D), β −1 a bµ µ ≤ c0 (|M |+|supp α|) + (|P |+|supp β|) , 0∈Gclus
(A.25) where c0 is a positive constant. Proof. One has X 0 0 0 0 0 c0 a(M, E), α b(P, D), β −1 a bµ µ ≤ 0∈Gclus
≤
X
|c0 | a0(M, E), α b0(P, D), β −1 |µ0 |
0∈Gclus
X
0∈Gclus 06∼γ1
|c0 | |µ0 | +
X
|c0 | |µ0 |
0∈Gclus 06∼γ2
≤ F1 (− ln kµk)(kγ1 k + kγ2 k),
(A.26)
by (A.18), where γ1 = (supp (E + α), E + α) and γ2 = (supp (D + β), D + β). Clearly, kγ1 k ≤ |M | + |sup α| and kγ2 k ≤ |P | + |sup β|. Proposition A.2.1 has a simple corollary
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Proposition A.2.2. If gc and hc are small enough and if min{αl , βl } ≥ n, where αl := inf{|M |, (M, E) ∈ Conn1 (α)}, βl := inf{|P |, (P, D) ∈ Conn2 (β)},
(A.27) (A.28)
|hB(α, β)iρ | ≤ ca e−cb n ,
(A.29)
then
for positive constants ca and cb . Proof. Using the representation (A.22) of hB(α, β)iρ in terms of cluster expansions and the estimate (A.25), one gets X | gc|P | h|M exp c0 (|M | + |supp α|) + (|P | + |supp β|) . |hB(α, β)iρ | ≤ c Conn1 (α) Conn2 (β)
(M, E)∈ (P, D)∈
(A.30) By standard arguments one has X
(hc ec0 )|M | ≤ const. e−ca n/2 ,
(A.31)
(M, E)∈Conn1 (α)
for some positive ca , provided hc is small enough and, analogously, X (gc ec0 )|P | ≤ const. e−ca n/2 ,
(A.32)
(P, D)∈Conn2 (β)
provided gc is small enough. This proves the proposition.
An important particular case of (A.22) occurs when dβ = d∗ α = 0. In this case we get simply ! X 0 0 0 0 0 c0 a(∅, 0), α b(∅, 0), β − 1 a bµ µ . hB(α, β)iρ = [β : α] [β : ˜] [µ˜ : α] exp 0∈Gclus
(A.33) Notice the presence of the ZN -factors [β : α] [β : ˜] [µ˜ : α] related to winding numbers involving α, β and the background charges ρ.
B. The Remaining Proofs B.1. Proof of Proposition 3.1.1. Let EG be the projection Floc → Aloc . Since ωV, ρ is gauge invariant, it is enough to prove the existence of lim ωV, ρ Aloc . Expectations V ↑Z2
like ωV, ρ (A) for A ∈ Aloc can be written as finite linear combinations of the previously introduced classical expectations hB(α, β)iV, ρ , whose thermodynamic limit was described in Subsect. A.2. To show that ωρ Aloc is a ground state with respect to αρ we first notice that, by the representation of ωV, ρ (A), A ∈ Aloc in terms of cluster expansions we can write, in analogy to (3.25),
Dyonic Sectors and Intertwiner Connections in 2+1-D Lattice ZN -Higgs Models
T rHV TV (ρ)n ATV (ρ)n+1 EVρ ωV, ρ (A) = lim , n→∞ T rHV TV (ρ)2n+1 EVρ
455
A ∈ F(V ),
(B.1)
and, hence, for V large enough, one has T rHV TV (ρ)n A∗ ATV (ρ)TV (ρ)n EVρ , n→∞ T rHV TV (ρ)2n+1 EVρ
ωV, ρ (A∗ αρ (A)) = lim
A ∈ F(V ).
(B.2) Now, by (3.26) and (2.30)-(2.31), EVρ is a positive operator and so, the numerator in (B.2) is clearly positive. This proves that ωρ (A∗ αρ (A)) ≥ 0. To show that ωρ (A∗ αρ (A)) ≤ ωρ (A∗ A) we can make use of Lemma 2.3.2 and show that ωρ fulfills the cluster property with respect to αρ . We can represent ωρ (A∗ αρn (A)) in terms of classical expectations of the classical functions associated to the operator A∗ αρn (A). These classical expectations can be written as a finite linear combination of expectations like hB(αn , βn )iρ , where the local cochains αn and βn can be written, for n large enough, as sums αn = α(0) + α(n) and βn = β(0) + β(n), where the local cochain α(n) (respectively β(n)) is the complex conjugate of the translate of α(0) (respectively, of β(0)) by n units in the euclidean time direction. Recalling now the representation (A.22) of hB(α, β)iρ in terms of cluster expansions we notice that, by Proposition A.2.2, the contributions of sets (M, E) ∈ Conn1 (α) connecting the support of α(0) to the support of α(n) decay exponentially with n, the same happening with the contribution of the sets (P, D) ∈ Conn2 (β) connecting the support of β(0) to the support of β(n). The only surviving terms, after taking the limit n → ∞ correspond to sets (M, E) ∈ Conn1 (α) and sets (P, D) ∈ Conn2 (β) connecting the supports of α(0), α(n), β(0) and β(n) with themselves. The contributions of these last terms converges to the product hB(α(0), β(0))iρ hB(α(n), β(n))iρ . This implies that ωρ (A∗ αρn (A)) → ωρ (A∗ )ωρ (A), n → ∞, thus proving the ground state property. The general cluster property ωρ (Aαρn (B)) → ωρ (A)ωρ (B), A, B ∈ Aloc , follows from the same arguments. B.2. Proof of Propositions 3.1.2 and 3.1.5 . In order to prove Proposition 3.1.2 we have to study lim hB(α, β)iρ−ρ0 a . We recall the representation (A.22) of hB(α, β)iρ−ρ0 a a→∞ and notice that, since c0 a0(M, E), α b0(P, D), β − 1 a0−0 a b0µ−µ0 a µ0 ≤ c0 a0(M, E), α b0(P, D), β − 1 µ0 , (B.3) which is summable, we can write X c0 a0(M, E), α b0(P, D), β − 1 a0−0 a b0µ−µ0 a µ0 = lim a→∞
X 0∈Gclus
0∈Gclus
c0 a0(M, E), α b0(P, D), β − 1 ( lim a0−0 a b0µ−µ0 a )µ0 . a→∞
(B.4)
But, clearly, lim a0−0 a b0µ−µ0 a = a0 b0µ for every cluster 0, since the polymers are a→∞ finite. The limit does not depend on the particular way as a → ∞. This shows that the representation (A.22) holds also for ρ ∈ Dq , q 6= 0, and can be used to describe ωρ (A), A ∈ Aloc with ρ ∈ Dq , q 6= 0. The cluster property, and consequently the ground state property for A ∈ Aloc , can be proven in the same way as in the previous case. The proof of Proposition 3.1.5 is analogous to the proof of Proposition 3.1.2 and does not need to
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be repeated, but in the next subsection we present the proof of the more general Theorem 3.1.6. n (A0 τa−1 (B 0 )) for B.3. Proof of Theorem 3.1.6 . Let us consider ωρ(a) τa−1 (B)Aαρ(a) n (A0 ) = A, B, A0 and B 0 ∈ Aloc . According to (3.16) one has, for |a| large enough, αρ(a) n αρn1 (A0 ) and αρ(a) (τa−1 (B 0 )) = αρn2 a (τa−1 (B 0 )) = τa−1 (αρn2 (B 0 )). Hence, for |a| large enough, n (A0 τa−1 (B 0 )) = ωρ(a) Aαρn1 (A0 )τa−1 (Bαρn2 (B 0 )) . ωρ(a) τa−1 (B)Aαρ(a)
(B.5)
The representation of the last expectation in terms of classical expectations is given by finite sums of classical expectations like hB(α(a), β(a))iρ(a) , where α(a) = α1 + α2 a and β(a) = β1 + β2 a, for local cochains α1, 2 and β1, 2 , where the cochains α1 and β1 are related to the operators Aαρn1 (A0 ) and where the cochains α2 and β2 are related to the operators Bαρn2 (B 0 ). Let us now consider the representation of hB(α(a), β(a))iρ(a) in terms of cluster expansions. It is given by hB(α(a), β(a))iρ(a) = X
˜ [D − β(a) : E − α(a)] D − β(a) : −(a)
Conn1 (α(a)) Conn2 (β(a))
(M, E)∈ (P, D)∈
× [−µ(a) ˜ : E − α(a)] × exp
X
hQ p∈P
i Q
g(D(p))
b∈M
h(E(b))
c0 a0(M, E), α(a) b0(P, D), β(a) − 1 a (a)0 bµ (a)0 µ0
(B.6)
! ,
0∈Gclus
Let us assume |a| so large as to include the set sup α1 ∪ sup β1 ∪ sup α2 ∪ sup β2 in the ball of radius |a|/8 centered at the origin and let us consider two infinite cylinders C1 and C2 = C1 + a with radius |a|/4, parallel to the euclidean time axis and extending from +∞ to −∞. The cylinder C1 contains the set sup α1 ∪ sup β1 and C2 contains the set (sup α2 ∪ sup β2 ) + a. By construction, the sets Conn1 (α(a)) and Conn2 (β(a)) will contain some elements which are entirely contained in C1 ∪ C2 and some which are not. These last ones must have a size larger than |a|/4 and, therefore, by arguments analogous to those used in the proof of Proposition A.2.2, their contribution to (B.6) decay exponentially with |a|. So, up to an exponentially falling error, we can restrict the sums over Conn1 (α(a)) and Conn2 (β(a)) to elements contained only in C1 ∪ C2 . The next question is, what happens to the sums over clusters, provided the elements M and P are now contained in C1 ∪ C2 ? Let us denote Ma := M ∩ Ca and Pa := M ∩ Ca , for a = 1, 2, with M = M1 ∪ M2 and P = P1 ∪ P2 , as disjoint unions and let E = E1 + E2 and D = D1 + D2 with sup Ea = Ma and sup Da = Pa for a = 1, 2. We claim that the difference
Dyonic Sectors and Intertwiner Connections in 2+1-D Lattice ZN -Higgs Models
X
457
c0 a0(M, E), α(a) b0(P, D), β(a) − 1 a (a)0 bµ (a)0 µ0 −
0∈Gclus
X
c0 a0(M1 , E1 ), α1 b0(P1 , D1 ), β1 − 1 a01 b0µ1 µ0
(B.7)
0∈Gclus
+
P
0 0 0 0 0 a c b − 1 a b µ 0 2 a µ 2 a (M2 , E2 ), α2 a (P2 , D2 ), β2 a 0∈Gclus
decays exponentially to zero with |a|. For, notice that the difference above is given by sums over clusters connecting C1 to C2 , having thus a size larger than |a|/2. Therefore, by (A.19), their contribution decay exponentially with |a|. Using now the exact factorization ˜ [D − β(a) : E − α(a)] D − β(a) : −(a) " # Y Y g(D(p)) h(E(b)) = ˜ : E − α(a)] [−µ(a) p∈P
b∈M
[D1 − β1 : E1 − α1 ] [D1 − β1 : −˜1 ] " # Y Y g(D(p)) h(E(b)) × [−µ˜ 1 : E1 − α1 ] p∈P1
(B.8)
b∈M1
[D2 − β2 a : E2 − α2 a] [D2 − β2 a : −˜2 a] " # Y Y g(D(p)) h(E(b)) , [−µ˜ 2 a : E2 − α2 a] p∈P2
b∈M2
valid in C1 ∪ C2 , we get using the translation invariance of the cluster expansions and taking |a| → ∞,
lim hB(α(a), β(a))iρ(a) = hB(α1 , β1 )iρ1 hB(α2 , β2 )iρ2 .
|a|→∞
With this, the proof of Theorem 3.1.6 is complete.
(B.9)
B.4. Proof of Proposition 3.4.5. Here we establish part i of Proposition 3.4.5. Let ρ = (, µ) and ρ0 = (0 , µ0 ) and define ρ0 = (0 , µ0 ) = ( − 0 , µ − µ0 ). We need (2ν+1) (2ν) first an expression in terms of the cluster expansions for the ratio ωρ (Xρ, ρ0 )/ωρ (Xρ, ρ0 ). Using a pictorial notation, this ratio can be written in terms of classical expectations as
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........................................................................... . .. .. .. .. `e or `m .. .. .. .. 6 .. .. .. .. . .. .. 2ν + 1... .. .. CC .. .. .. .. C .. .. C .. .. ? C .. .. 0 .. ρ−ρ C .. C ............................................................................... C ρ
C C
........................................................................... . . .. .. .. . `e or `m .. .. .. .. 6 .. .. . .. .. .. .. 2ν ... . .. .. CC .. .. .. C .. .. .. C .. . ? C . .. 0 . ρ−ρ C .. . C ................................................................................ C ρ
C
(2ν+1) ωρ (Xρ, ρ0 ) (2ν) ωρ (Xρ, ρ0 )
=
C
C
C
C CC
.
(B.10)
C
C
C C
C
CC
The infinite vertical lines indicate the background charges ρ and the finite loops are constructed over the charge distribution ρ − ρ0 . Their horizontal lines represent the strings `e and/or `m used in the definition (3.50) and their vertical lines have length 2ν + 1 in the numerator and 2ν in the denominator, respectively. Notice that ρ and ρ − ρ0 may have a non-empty overlap, a circumstance not shown in the figure for reasons of clarity. The next step is to find an expansion for the last expression in terms of our cluster expansions. The result is ( ) (2ν+1) X ωρ (Xρ, 0 0 0 ρ0 ) 0 0 0 0 = exp c0 a0 , 0, 2ν+1 bµ0 , 0, 2ν+1 − a0 , 0, 2ν bµ0 , 0, 2ν a bµ µ , (2ν) ωρ (Xρ, ρ0 ) 0 (B.11) where, in an almost self-explanatory notation, a0 , α, β (γ), α < β, represents the winding number of the magnetic part of the polymer γ with the electric loop built by the horizontal strings `e located at euclidean times α and β ∈ Z and by the vertical electric lines located over the support of 0 with length β − α. The quantity bµ0 , α, β is defined analogously. The right-hand side of (B.11) can be written as ) ( X 0 0 0 0 0 0 0 c0 a0 , 0, 1 bµ0 , 0, 1 − 1 a0 , −2ν, 0 bµ0 , −2ν, 0 a bµ µ , (B.12) exp 0
where, above, we used the factorization properties
Dyonic Sectors and Intertwiner Connections in 2+1-D Lattice ZN -Higgs Models
a0 , α, β (γ)a0 , β, δ (γ) = a0 , α, δ (γ), bµ0 , α, β (γ)bµ0 , β, δ (γ) = bµ0 , α, δ (γ),
459
(B.13) (B.14)
for α < β < δ ∈ Z and we used translation invariance. Taking the limit ν → ∞ of expression (B.12) is easy and gives ( ) X kTρ (ρ0 )k = exp c0 a00 , 0, 1 b0µ0 , 0, 1 − 1 a00 , −∞, 0 b0µ0 , −∞, 0 a0 b0µ µ0 , kTρ (ρ)k 0 (B.15) where a0 , −∞, 0 (γ) = lim a0 , −j, 0 (γ), etc., which is a well defined limit for each γ, j→∞
since the polymers are finite (for each γ, the limit is reached at finite j). kTρ1 (a) (ρ2 (a))k using its repreNext, we are interested in studying the limit lim |a|→∞ kTρ2 (a) (ρ2 (a))k sentation in terms of cluster expansions. The main technical problem we have to confront is the fact that, if ρ1 − ρ2 have a non-zero total charge, the strings `e and `m have to connect elements of the support of ρ1 − ρ2 with elements of the support of (ρ1 − ρ2 ) · a and have, hence, a length which increases with |a|. The crucial observation is, however, that the left hand side of (B.15) does not depend on the strings `e and `m , although this independence cannot apparently be seen from the representation in terms of cluster expansions. Let us consider two cylinders C1 (r) and C2 (r) = C1 (r)+a, such that C1 (r) is centered on the euclidean time axis, extending from −∞ to ∞ and has a radius r. Denote by r0 the largest distance from the set sup (ρ1 ) ∪ sup (ρ2 ) to the origin of the lattice and consider |a| large enough so that sup (ρ1 ) ∪ sup (ρ2 ) is contained in C1 (|a|/8) (by taking, say, |a| > 16r0 ). We first observe that the sum over clusters contained in Z3 \ (C1 (|a|/8) ∪ C2 (|a|/8)) does not contribute to (B.12). This can be seen at best in (B.11) by noticing that: 1) clusters contained in Z3 \(C1 (|a|/8)∪C2 (|a|/8)) crossing the t = 0 euclidean plane and having a side smaller than 2ν have a zero contribution (for them, one has a00 , 0, 2ν+1 b0µ0 , 0, 2ν+1 = a00 , 0, 2ν b0µ0 , 0, 2ν ); 2) clusters contained in Z3 \ (C1 (|a|/8) ∪ C2 (|a|/8)) crossing the t = 2ν euclidean plane, having a side smaller than 2ν and having a non-zero contribution cancel with their translates by one unit in euclidean time direction; 3) the only surviving clusters in Z3 \ (C1 (|a|/8) ∪ C2 (|a|/8)) must cross the planes t = 0 and t = 2ν, and therefore, their side is larger than 2ν and their contribution decays exponentially when the limit ν → ∞ is taken. It remains to consider two classes of clusters: a) those entirely contained in C1 (|a|/4) ∪ C2 (|a|/4) and having a non-empty intersection with C1 (d0 ) ∪ C2 (d0 ) for a fixed d0 with r0 < d0 < a/8 and b) those having a non-empty intersection with both C1 (d0 ) ∪ C2 (d0 ) and Z3 \ (C1 (|a|/4) ∪ C2 (|a|/4)). The contribution to the clusters belonging to class b decays exponentially with |a|. For, note that the clusters which give a non-zero contribution to (B.12) must either cross the t = 0 plane or the t = 1 plane (or eventually both). The clusters of this sort having a non-empty intersection with both C1 (d0 ) ∪ C2 (d0 ) and Z3 \ (C1 (|a|/4) ∪ C2 (|a|/4)) must have a size larger that |a|/8 and , hence, by (A.19), their contribution decays exponentially with |a|. It remains now to consider the clusters belonging to class a above. They are entirely contained inside one of the cylinders C1 (|a|/4) or C2 (|a|/4). Since we have freedom to choose the strings `e and `m at will, we choose them depending on a such that, inside of C1 (|a|/4) \ C1 (d0 ) and C2 (|a|/4) \ C2 (d0 ) they run parallel to a fixed direction, say
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to the positive x-axis of Z2 . Now, taking the limit |a| → ∞ is straightforward and gives cρ1 , ρ2 independent on the way the sequence a goes to infinity. The result is that cρ1 , ρ2 = dρ1 , ρ2 d−ρ1 , −ρ2 , where X dρ1 , ρ2 := exp c0 a012 , 0, 1; ∞ b0µ12 , 0, 1; ∞ − 1 0 0∩C1 (d0 )6=∅
a012 , −∞, 0; ∞ b0µ12 , −∞, 0; ∞ a01 b0µ1 µ0
(B.16)
,
for any sufficiently large d0 , where ρ1 − ρ2 = (12 , µ12 ) and where a12 , α, β; ∞ (γ) = lim a(1 −2 )−(1 −2 )·a, α, β (γ), |a|→∞
(B.17)
etc., where α < β ∈ Z and in a(1 −2 )−(1 −2 )·a, α, β (γ) the strings `e depend on a in the way described above, i.e. such that, inside of C1 (|a|/4) \ C1 (d0 ) and C2 (|a|/4) \ C2 (d0 ) they point parallel to the positive x-axis of Z2 . Note that, for each γ, the limit above is reached at finite values of |a|. The condition 0 ∩ C1 (d0 ) 6= ∅ means that the geometrical part of the cluster 0 must have a non-empty intersection with the cylinder C1 (d0 ). Note also that the convergence of the sum over clusters in (B.16) can be shown using the fact that the contributing clusters have a non-empty intersection with C1 (d0 ) and with the t = 0 and/or t = 1 euclidean time slices together with the exponential decay provided by (A.19). We can say, for instance, that the sum over clusters in (B.16) can be bounded by const.
∞ X
X
t=−∞
0
0∩(C1 (d0 )∩Tt )6=∅ k0k≥t
|c0 | |µ0 | ≤ const.
∞ X
e−ca |t| < ∞
(B.18)
t=−∞
with some positive constant ca , where Tt is the euclidean time-plane at euclidean time t. Using the representation above in terms of cluster expansions one can also easily show that d−ρ1 , −ρ2 = dρ1 , ρ2 . The next problem is to prove the factorization property (3.68). The arguments used are analogous to those leading to (B.16). We can namely prove that lim dρ1 +ρ01 ·b, ρ2 +ρ02 ·b = b→∞
dρ1 , ρ2 dρ01 , ρ02 . This can be obtained using the representation (B.16) with d0 depending on b such that C1 (d0 ) contains sup ρ1 ∪ sup ρ2 ∪ sup (ρ01 · b) ∪ sup (ρ02 · b). We consider again cylinders D1 (|b|/4) = C1 (|b|/4) and D2 (|b|/4) = D1 (|b|/4) + b, both contained in C1 (d0 (b)), with D1 (|b|/4) containing the set sup ρ1 ∪ sup ρ2 and D2 (|b|/4) containing the set sup (ρ01 · b) ∪ sup (ρ02 · b) for some |b| large enough. Repeating the previous arguments, we can neglect contributions from clusters contained outside of Z3 \ (D1 (|b|/4) ∪ D2 (|b|/4)) and take the pieces of the strings `e and `m which join the supports of ρ1 and ρ2 with the supports of ρ01 · b and ρ02 · b so that they again run parallel to the x-axis at sufficiently large distances. The desired relation will follow again from the usual clustering properties of the cluster expansions established above. B.5. Completing the proof of Proposition 3.3.2. We will here complete some missing points in the proof of Proposition 3.3.2. The ideas are actually contained in [2] and therefore we will concentrate only on the more relevant details.
Dyonic Sectors and Intertwiner Connections in 2+1-D Lattice ZN -Higgs Models
461
6 l1
m
6
`e or `m
n
n
l1
`e or `m
− 21
?? 66
− 21 0
`e or `m
? m
l2
? 1 0 0 0 Fig. 1. Schematic representation of the expression a0 l1 bl1 − 2 al1 , l2 bl1 , l2 − l2 (below)
1 2
and the loops l1 (above) and
.. .. .. `e or `m
ρ
6 b
? a6 ?
ρ0
0 `e or `m .. .. .. ..
Fig. 2. Schematic representation of the partial replacement of the infinite vertical line representing the background charge distribution ρ by ρ0 . The connections are performed at euclidean time planes b and −a, 0 ≤ a < b, through the strings `e and/or `m
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J. C. A. Barata, F. Nill
To show that the sequence of unit vectors 8nρ, ρ0 , n ∈ N is a Cauchy sequence it is enough to show that, for any > 0, one has |(8nρ, ρ0 , 8m ρ, ρ0 ) − 1| < , provided n is large enough, with m > n. The scalar product (8nρ, ρ0 , 8m ρ, ρ0 ) can be expressed as the exponential of a sum over clusters and, hence, it is enough to show that this sum is small enough provided n is large enough, with m > n. This sum can be written as X 1 0 1 0 0 0 a0m, m b0m, m a0ρ b0ρ µ0 . c 0 a l 1 bl 1 − a l 1 , l 2 b l 1 , l 2 − (B.19) 2 2 0 Above a0l1 , b0l1 , a0l1 , l2 and b0l1 , l2 are the electric and magnetic winding numbers on the loops l1 and l1 ∪ l2 schematically represented in Fig. 1 (where l2 = θl1 , θ meaning reflection on the t = 0 euclidean time plane). Also above a0a, b and b0a, b (with 0 ≤ a < b) are the electric and magnetic winding numbers around the infinite vertical lines schematically represented in Fig. 2. By a straightforward inspection we can verify that a cluster 0 with a size smaller than n with a non-trivial winding number with, say, the loop l1 are canceled in the sum (B.19) by the contribution of the reflected cluster θ0. The contribution of the clusters entirely contained between the time-slices n and m and the contribution of the clusters entirely contained between the time-slices −n and −m also cancel mutually. The only surviving clusters must have non-trivial winding numbers with both l1 and l2 simultaneously and must cross both planes at time n and −n. Therefore, they must have a size which increases with n. By estimate (2.31) their contributions disappear when n → ∞ uniformly m, completing thus the proof. B.6. Proof of Theorem 4.1.1. Let us start proving i). We will first consider the case where d∗ l1 6= 0 and d∗ l2 6= 0. Without loss, we will take l1 and l2 as having support on single lattice links. According to the definitions we have Eˆq (l1 + l2 )ρ = φel ρ1, 2 (l1 + l2 )
πρ1, 2 αρn Aρ1, 2 , ρ ((l1 + l2 ), 0) n→∞ N1 (n)
= lim with
ρ1, 2
N1 (n) := πρ1, 2 αρn Aρ1, 2 , ρ ((l1 + l2 ), 0) ρ1, 2
and
Eˆq (l1 )Eˆq (l2 )ρ = lim Eˆq (l1 )
πρ2 αρp Aρ2 , ρ (l2 , 0)
πρ2
where ρi = ρ + (d∗ li , 0), i = 1, 2 and ρ1, 2 = ρ + (d∗ (l1 + l2 ), 0). The vector in the right hand-side can be written as πρ1, 2 αρp Aρ2 , ρ (l2 , 0) αρq 2 Aρ1, 2 , ρ2 (l1 , 0) ρ1, 2 , lim lim p→∞ q→∞ N2 (p)N3 (q) where N1 (p) and N2 (q) are the normalization factors
(B.20)
(B.21)
ρ2
Aρ2 , ρ (l2 , 0) ρ2 πρ1, 2 αρp Aρ2 , ρ (l2 , 0) φel ρ (l1 ) 1, 2 , = lim p
p→∞ πρ2 αρ Aρ2 , ρ (l2 , 0) ρ2 p→∞
αρp
,
(B.22)
(B.23)
Dyonic Sectors and Intertwiner Connections in 2+1-D Lattice ZN -Higgs Models
463
N2 (p) := πρ2 αρp Aρ2 , ρ (l2 , 0) ρ2 ,
(B.24)
N3 (q) := πρ1, 2 αρq 2 Aρ1, 2 , ρ2 (l1 , 0) ρ1, 2 .
(B.25)
and
The scalar product in (4.16) can now be written as ρ1, 2 , πρ1, 2 (A) ρ1, 2 , N1 (n)N2 (p)N3 (q)
lim lim lim
n→∞ p→∞ q→∞
(B.26)
with ∗
A := αρn Aρ1, 2 , ρ ((l1 + l2 ), 0)
αρp Aρ2 , ρ (l2 , 0) αρq 2 Aρ1, 2 , ρ2 (l1 , 0) .
(B.27)
After expressing the expectation values above in terms of classical expectations (which involve only closed loops) and these in terms of cluster expansions, we arrive at the following expression: ! X 0 1 0 0 0 0 0 0 0 a + a3 + a4 − a5 aρ1, 2 bρ1, 2 , c 0 µ a1 − lim lim lim exp n→∞ p→∞ q→∞ 2 2 0 (B.28) where ai (γ) represent winding numbers of γ with respect to the loops successively presented in Fig. 3. The quantities aρ1, 2 (γ) and bρ1, 2 (γ) are electric and magnetic winding numbers with respect to the background charge ρ1, 2 . q p n
− 21
l1 l 2
− 21
+ 21
− 21
0
−n l1 l 2
−p l2 l1 l1
l1
−q
1 0 0 0 0 Fig. 3. Pictorial representation of the expression a0 1 − 2 a2 + a3 + a4 − a5 appearing in (B.38). The ai ’s are winding numbers with respect to the sets of loops presented in the picture (counted from the left to the right and separated by the associated factor ±1/2). The vertical lines are parallel to the euclidean time-axis. The open loops cross d∗ l1 and extend to the euclidean time infinity. At the right we indicate the different time planes
We have to perform a detailed analysis of the sum over clusters appearing in (B.28). For the sake of brevity we will sketch the main arguments.
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Define the time planes Ha := {(x, x0 ) ∈ Z3 with x0 = a} and denote by GB the set of all clusters 0 not crossing any of the planes H±n , H±p and H±q .It is easy to verify that for a cluster 0 ∈ GB one either has a01 − 21 a02 + a03 + a04 − a05 = 0 or it happens that its contribution cancels that of the other cluster in GB obtained by translating 0 in the time direction. This is, for instance, what happens for clusters located between Hp and Hq and translated clusters located between H−q and H−p . On the other hand, the size of clusters which cross at least two of the planes H±n , H±p or H±q is at least min{2n, p − n, q − p} (assuming q > p > n). After the limits q → ∞, p → ∞ and n → ∞ are taken the total contribution of such clusters is zero, which can be shown using the exponential decay given in (A.19) and noticing that the size of the loops of Fig. 3 grows only linearly in n, p or q. The remaining terms belong to clusters crossing one and only one of the planes H±n , H±p or H±q . Using the translation invariance of the sum over clusters, we may express these remaining terms (after the limits are taken) in the following form: 1 X c0 µ0 fl01 − (flt1 )0 a0ρ1, 2 b0ρ1, 2 + 2 06∼H0 P 1 0 fl02 − (flt2 )0 a0(d∗ l1 , 0) a0ρ1, 2 b0ρ1, 2 + 06∼H0 c0 µ 2
(B.29)
1 X c0 µ0 (flt1 )0 (flt2 )0 − fl01 fl02 a0ρ1, 2 b0ρ1, 2 , 2 06∼H0
where, with some abuse of notation, 0 6∼ H0 indicates that the geometric part of at least one polymer composing 0 crosses the plane H0 . Above, fl (γ) (respectively, flt (γ)) represents the winding number of the polymer γ with respect to the semi-infinite loops formed by l and by vertical lines starting at d∗ l and extending to the negative (positive) euclidean time infinity. See Fig. 4. Note that the second sum in (B.29) can be simplified, since a0(d∗ l1 , 0) a0ρ1, 2 b0ρ1, 2 = a0ρ2 b0ρ2 . .. .. .. ..
.. .. .. ..
flt :
l
fl :
l
.. .. .. ..
.. .. .. ..
Fig. 4. The semi-infinite loops for which fl (γ) and flt (γ) are defined. The horizontal lines represent the link l located at H0 . The vertical lines are parallel to the euclidean time axis and extend to the negative (left) or positive (right) euclidean time infinity
It is easy to show that each of the sums over clusters in (B.29) is absolutely convergent. Analogously, sums like
Dyonic Sectors and Intertwiner Connections in 2+1-D Lattice ZN -Higgs Models
X
c0 µ0 fl01 − 1 a0ρ1, 2 b0ρ1, 2
465
(B.30)
06∼H0
are also absolutely convergent. It can be seen by reflecting polymers on the plane H0 X that the last expression is the complex conjugate of c0 µ0 (flt1 )0 − 1 a0ρ1, 2 b0ρ1, 2 . 06∼H0
This means that each of the sums over clusters in (B.29) is purely imaginary. Defining 1 X c0 µ0 fl02 − (flt2 )0 a0ρ2 b0ρ2 , (B.31) zρel (l2 ) := exp 2 06∼H0
which is a pure phase, we conclude from (B.29) the proof of part i) of Theorem 4.1.1. Part ii) can be proven analogously, and we do not need to show the details. The proof of part iii) is also analogous but with an important difference. Since in this case l1 is a magnetic link and l2 an electric one, the closed loops formed by l1 and by l2 , appearing in the left Fig. 3, can have a nontrivial winding number, which can contribute to the classical expectations in the numerator of (B.26) with an additional ZN phase factor, as the phase factor [β : α] emerging from (A.33). This phase equals eihl1 , l2 i . In order to prove iv), consider that the support that l2 is, say, an elementary plaquette at H0 . Following the same steps of the proof of i) we would arrive at relations like (B.29) and (B.31), where both fl2 (γ) and flt2 (γ) represent the winding number of γ around this plaquette. Therefore, for any polymer γ, fl2 (γ) = flt2 (γ) and hence zρel (l2 ) = 1. The proof of v) is analogous. B.7. Proof of Proposition 4.2.3. We will prove only (4.41) since (4.42) is analogous. ihµ, si ρ k = 0, and to prove one has only to show Relation (4.41) means kφel ρ (le ) − e el ihµ, si that ρ , φρ (le ) = e . According to the definitions ρ , Pρ (ρ − (d∗ le , 0))πρ (Aρ, ρ−(d∗ le , 0) (le , 0))ρ el
. (B.32) ρ , φρ (le ) = Pρ (ρ − (d∗ le , 0))πρ Aρ, ρ−(d∗ le , 0) (le , 0) ρ Under the hypothesis d∗ le = 0 and, hence, we can write the right-hand side of (B.32) as ρ , πρ αρn (Aρ, ρ (le , 0)) ρ ρ , Tρ (ρ)n πρ Aρ, ρ (le , 0) ρ
= lim lim
Tρ (ρ)n πρ Aρ, ρ (le , 0) ρ
πρ αρn (Aρ, ρ (le , 0)) ρ . n→∞ n→∞ (B.33) We now expand the right hand side of (B.33) in terms of our cluster expansions and treat it with the same methods used in the proof of Theorem 4.1.1 above. We get el ihµ, si , (B.34) ρ , φel ρ (le ) = zρ (le ) e where the ZN phase factor eihµ, si emerges in this expression as the factor [µ˜ : α] emerges from (A.33): it represents the winding number of d∗ s in the background charge ρ. Actually eihµ, si = [µ˜ : s]. Since d∗ le = 0, one has zρel (le ) = 1 and the proposition is proven. Acknowledgement. We would like to thank K. Fredenhagen for stimulating interest and several useful discussions.
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References 1. Barata, J.C.A. and Nill, F.: Electrically and Magnetically Charged States and Particles in the 2+1 Dimensional ZN -Higgs Gauge Model. Commun. Math. Phys. 171, 27–86 (1995) 2. Fredenhagen, K. and Marcu, M.: Charged States in Z2 Gauge Theories. Commun. Math. Phys. 92, 81–119 (1983) 3. Barata, J.C.A. and Fredenhagen, K.: Charged Particles in Z2 Gauge Theories. Commun. Math. Phys. 113, 403–417 (1987) 4. Barata, J.C.A.: Scattering States of Charged Particles in the Z2 Gauge Theories. Commun. Math. Phys. 138, 175–191 (1991) 5. Barata, J.C.A. and Fredenhagen, K.: Particle Scattering for Euclidean Lattice Field Theories. Commun. Math. Phys. 138, 507–519 (1991) 6. Gaebler, F.: Quasiteilchen mit anomaler Statistik in zwei- und dreidimensionalen Gittertheorien. Diplomarbeit (1990). Freie Universit¨at Berlin 7. Barata, J.C.A. and Nill,F.: Anyon Statistics of Scattering States in the 2+1-Dimensional ZN -Higgs Model. In Preparation 8. Fr¨ohlich, J. and Marchetti, P.A.: Quantum Field Theory of Vortices and Anyons. Commun. Math. Phys. 121, 177–223 (1989) 9. Fr¨ohlich, J. and Marchetti, P.A.: Quantum Field Theory of Anyons. Lett. Math. Phys. 16, 347 (1988) 10. Fr¨ohlich, J. and Marchetti, P.A.: Soliton Quantization in Lattice Field Theories. Commun. Math. Phys. 112, 343–383 (1987) 11. Murphy, G.J.: C ∗ -Algebras and Operator Theory. New York: Academic Press, 1990 12. Doplicher, S., Haag, R. and Roberts, J.E.: Local Observables and Particle Statistics II. Commun. Math. Phys. 35 49–85 (1974) 13. Bricmont, J. and Fr¨ohlich, J.: An Order Parameter Distinguishing Between Different Phases of Lattice Gauge Theories with Matter Fields. Phys. Lett. 122B, 73–77 (1983) 14. Erhard Seiler: Gauge Theory as a Problem of Constuctive Quantum Field Theory and Statistical Mechanics. Lecture Notes in Physics 159, Berlin–Heidelberg–New York: Springer Verlag, 1982 Communicated by D. C. Brydges
Commun. Math. Phys. 191, 467 – 492 (1998)
Communications in
Mathematical Physics c Springer-Verlag 1998
Coalgebra Bundles? ´ Tomasz Brzezinski, Shahn Majid?? Department of Applied Mathematics & Theoretical Physics, University of Cambridge, Cambridge CB3 9EW, United Kingdom Received: 22 February 1996 / Accepted: 29 May 1997
Abstract: We develop a generalised theory of bundles and connections on them in which the role of gauge group is played by a coalgebra and the role of principal bundle by an algebra. The theory provides a unifying point of view which includes quantum group gauge theory, embeddable quantum homogeneous spaces and braided group gauge theory, the latter being introduced now by these means. Examples include ones in which the gauge groups are the braided line and the quantum plane.
1. Introduction In a recent paper [Brz96b] it was shown by the first author that a generalisation of the quantum group principal bundles introduced in [BM93] is needed if one wants to include certain “embeddable” quantum homogeneous spaces, such as the full family of quantum two-spheres of Podle´s [Pod87]. A one-parameter specialisation of this family was used in [BM93] in construction of the q-monopole, but the general members of the family do not have the required canonical fibering. The required generalised notion of quantum principal bundles proposed in [Brz96b], also termed a C-Galois extension (cf. [Sch92]), consists of an algebra P , a coalgebra C with a distinguished element e and a right action of P on P ⊗ C satisfying certain conditions. In the present paper we develop a version of such “coalgebra principal bundles” based on a map ψ : C ⊗ P → P ⊗ C and e ∈ C, and giving now a theory of connections on them. Another motivation for the paper is the search for a generalisation of gauge theory powerful enough to include braided groups [Maj91, Maj93b, Maj93a] as the gauge group. Although not quantum groups, braided groups do have at least a coalgebra and hence can be covered in our theory. We describe the main elements of such a braided ?
Research supported by the EPSRC grant GR/K02244 Royal Society University Research Fellow and Fellow of Pembroke College, Cambridge. On leave 1995 and 1996 at the Department of Mathematics, Harvard University, Cambridge MA 02138, USA ??
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principal bundle theory as arising in this way. This is a first step towards a theory of braided-Lie algebra valued gauge fields, Chern-Simons and Yang-Mills actions, to be considered elsewhere. As well as providing a unifying point of view which includes our previous quantum group gauge theory [BM93], the theory of embeddable homogeneous spaces [Brz96b] and braided group gauge theory, our coalgebra bundles have their own characteristic properties. In particular, the axioms obeyed by ψ involve the algebra and coalgebra in a symmetrical way, opening up the possibility of an interesting self-duality of the construction. This becomes manifest when we are given a character κ on P ; then we have also the possibility of a dual “algebra principal bundle”, corresponding in the finitedimensional case to a coalgebra principal bundle with the fibre P ∗ , total space C ∗ and the structure map ψ ∗ . This is a new phenomenon which is not possible within the realm of ordinary (non-Abelian) gauge theory. Moreover, the axioms obeyed by ψ correspond in the finite-dimensional case to the factorisation of an algebra into P op C ∗ , which is a common situation [Maj90]. Indeed, all bicrossproduct quantum groups [Maj90] provide a dual pair of examples. Finally, we note that some steps towards a theory of fibrations based on algebra factorisations have appeared independently in [CKM94], including topological considerations which may be useful in further work. However, we really need the present coalgebra treatment for our infinite-dimensional algebraic examples, for our treatment of differential calculus and in order to include quantum and braided group gauge theories. We demonstrate the various stages of our formalism on some concrete examples based on the braided line and quantum plane. Preliminaries. All vector spaces are taken over a field k of generic characteristic and all algebras have the unit denoted by 1. C denotes a coalgebra with the coproduct 1 : C → C ⊗ C and the counit : C → k which satisfy the standard axioms. For the coproduct we use the Sweedler notation 1c = c(1) ⊗ c(2) ,
12 c = (1 ⊗ id) ◦ 1c = c(1) ⊗ c(2) ⊗ c(3) ,
etc.,
where c ∈ C, and the summation sign and the indices are suppressed. A vector space P is a right C-comodule if there exists a map 1R : P → P ⊗ C, such that (1R ⊗ id) ◦ 1R = (id ⊗ 1) ◦ 1R , and (id ⊗ ) ◦ 1R = id. For 1R we use the explicit notation ¯ ¯ 1R u = u(0) ⊗ u(1) , ¯
¯
where u ∈ P and u(0) ⊗ u(1) ∈ P ⊗ C (summation understood). For e ∈ C, we denote by PecoC the vector subspace of P of all elements u ∈ P such that 1R u = u ⊗ e. H denotes a Hopf algebra with product µ : H ⊗ H → H, unit 1, coproduct 1 : H → H ⊗ H, counit : H → k and antipode S : H → H. We use Sweedler’s sigma notation as before. Similarly as for a coalgebra, we can define right H-comodules. We say that a right H-comodule P is a right H-comodule algebra if P is an algebra and 1R is an algebra map. If P is an algebra then by n P we denote the P -bimodule of universal n-forms on P , which is defined as n P = 1 P ⊗P · · · ⊗P 1 P (n-fold tensor product over P ). By the natural identification P ⊗P P = P we have [Con85, Kar87] n P = {ω ∈ P ⊗n+1 : ∀i ∈ {1, . . . , n}, µi ω = 0}, where µi denotes a multiplication in P acting on the i and i + 1 factors in P ⊗n+1 . P = L n 0 n=0 P , where P = P , is a differential algebra with the universal differential
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d : P → 1 P , du = 1 ⊗ u − u ⊗ 1. When extended to n P ⊂ P ⊗ n+1 , d explicitly reads: n+1 X (−1)k u0 ⊗ . . . ⊗ uk−1 ⊗ 1 ⊗ uk ⊗ . . . ⊗ un . (1) d u 0 ⊗ u1 ⊗ . . . ⊗ u n = k=0 n
(P ) denotes a bimodule of n-forms on P obtained from n P as an appropriate quotient. Finally, if C is a coalgebra and P is an algebra then we define a convolution product ∗ in the space of linear maps C → P by f ∗ g(c) = f (c(1) )g(c(2) ), where f, g : C → P and c ∈ C. The map f : C → P is said to be convolution invertible if there is a map f −1 : C → P such that f ∗ f −1 = f −1 ∗ f = η ◦ , where η : k → P is given by η : α 7→ α1. In addition, we will also discuss examples based on the theory of braided groups [Maj91, Maj93b, Maj93a] and the theory of bicrossproduct and double cross product and Hopf algebras [Maj90, Maj94b], due to the second author. Chapters 6.2,7.2,9 and 10 of the text [Maj95] contain full details on these topics. 2. Coalgebra ψ-Principal Bundles In this paper we will be dealing with a particular formulation of C-Galois extensions or generalised quantum principal bundles. This formulation is more tractable than the one in [Brz96b], allowing us to develop a theory of connections for it in the next section. Yet, it is general enough to include all our main examples of interest. Our data is the following: Definition 2.1. We say that a coalgebra C and an algebra P are entwined if there is a map ψ : C ⊗ P → P ⊗ C such that ψ ◦ (id ⊗ µ) = (µ ⊗ id) ◦ ψ23 ◦ ψ12 ,
ψ(c ⊗ 1) = 1 ⊗ c,
∀c ∈ C
(2)
(id ⊗ 1) ◦ ψ = ψ12 ◦ ψ23 ◦ (1 ⊗ id), (id ⊗ ) ◦ ψ = ⊗ id, (3) where µ denotes multiplication in P , and ψ23 = id ⊗ ψ etc. Explicitly, we require that the following diagrams commute: C ⊗P ⊗P
id ⊗ µ C ⊗P
ψ ⊗ id ψ ? ? P ⊗C ⊗P P ⊗C HH * id ⊗ ψ j H µ ⊗ id P ⊗P ⊗C id ⊗ 1 P ⊗C ⊗C 6 ψ ⊗ id
C ⊗k
id ⊗ η C ⊗P
k⊗C
ψ ? - P ⊗C
η ⊗ id
(4) id ⊗
P ⊗C 6 ψ
P ⊗k
C ⊗P ⊗C C ⊗P YHid ⊗ ψ H 1 ⊗ id H C ⊗C ⊗P
k⊗P
⊗ id
P ⊗C 6 ψ C ⊗P (5)
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where η is the unit map η : α 7→ α1. In the finite-dimensional case this is exactly equivalent by partial dualisation to the requirement that ψ˜ : C ∗ ⊗ P op → P op ⊗ C ∗ is an algebra factorisation structure (which is part of the theory of Hopf algebra double cross products [Maj90]). This is made precise at the end of the section, where it provides a natural way to obtain examples of such ψ. Proposition 2.2. Let C, P be entwined by ψ. For every group-like element e ∈ C we have the following: 1. For any positive n, P ⊗n is a right C-comodule with the coaction 1nR = ψnn+1 ◦ ← −n ψn−1n ◦ . . . ◦ ψ12 ◦ (ηC ⊗ idn ) ≡ ψ ◦ (ηC ⊗ idn ), where ηC : k → C, α 7→ αe. n 2. The coaction 1n+1 R restricts to a coaction on P . 3. M = PecoC = {u ∈ P ; 11R u = u ⊗ e} is a subalgebra of P . 4. The linear map χM : P ⊗M P → P ⊗ C, u ⊗M v 7→ uψ(e ⊗ v) is well-defined. If χM is a bijection we say that we have a ψ-principal bundle P (M, C, ψ, e). P Proof. We write ψ(c ⊗ u) = α uα ⊗ cα and henceforth we omit the summation sign. In this notation, the conditions (4) and (5) are (uv)α ⊗ cα = uα vβ ⊗ cαβ ,
1α ⊗ cα = 1 ⊗ c,
uα ⊗ cα (1) ⊗ cα (2) = uαβ ⊗ c(1) β ⊗ c(2) α ,
(cα )uα = (c)u,
(6) (7)
for all u, v ∈ P and c ∈ C. 1. The map 1nR is given explicitly by 1nR (u1 ⊗ . . . ⊗ un ) = u1α1 ⊗ . . . ⊗ unαn ⊗ eα1 ...αn . Hence (1nR ⊗ id)1nR (u1 ⊗ . . . ⊗ un ) = u1α1 β1 ⊗ . . . ⊗ unαn βn ⊗ eβ1 ...βn ⊗ eα1 ...αn = u1α1 β1 ⊗ . . . ⊗ unαn βn ⊗ e(1) β1 ...βn ⊗ e(2) α1 ...αn = u1α1 ⊗ u2α2 β2 ⊗ . . . ⊗ unαn βn ⊗ eα1 (1) β2 ...βn ⊗ eα1 (2) α2 ...αn = ... = u1α1 ⊗ . . . ⊗ unαn ⊗ eα1 ...αn (1) ⊗ eα1 ...αn (2) = (idn ⊗ 1)1nR (u1 ⊗ . . . ⊗ un ), where we used the group-like property of e to derive the second equality and then we used the condition (5) n times to obtain the penultimate one. We also have (idn ⊗ )1nR (u1 ⊗ . . . ⊗ un ) = u1α1 ⊗ . . . ⊗ unαn (eα1 ...αn ) n α1 ...αn−1 = u1α1 ⊗ . . . ⊗ un−1 ) αn−1 ⊗ u (e
= . . . = (e)u1 ⊗ . . . ⊗ un = u 1 ⊗ . . . ⊗ un , where we have first used the condition (5) n-times and then the group-like property of e. Hence 1nR is a coaction.
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P P P P 2. If i ui ⊗v i ∈ ker µ then (µ⊗id) i 12R (ui ⊗v i ) = i uiα vβi ⊗eαβ = i (ui v i )α ⊗ eα = 0, using (6). Hence the coaction preserves 1 P . Similarly for n P . 3. Here M = {u ∈ P |uα ⊗ eα = u ⊗ e}, and if u, v ∈ M , then (uv)α ⊗ eα = uα vβ ⊗ eαβ = uvβ ⊗ eβ = uv ⊗ e as well, using (6). 4. It is easy to see that χM is well-defined as a map from P ⊗M P . Thus, if x ∈ M we have χM (u, xv) = u(xv)α ⊗ eα = uxα vβ ⊗ eαβ = uxvβ ⊗ eβ = χM (ux, v), using (6). We remark that parts 3 and 4 also follow from the theory of C-Galois extensions of [Brz96b], for P (M, C, ψ, e) is such an extension. The required right action of P on P ⊗ C is given by (µ ⊗ id) ◦ ψ23 : P ⊗ C ⊗ P → P ⊗ C. Example 2.3. Let H be a Hopf algebra and P be a right H-comodule algebra. The linear ¯ ¯ map ψ : H ⊗ P → P ⊗ H defined by ψ : c ⊗ u → u(0) ⊗ cu(1) entwines H, P . Therefore a quantum group principal bundle P (M, H) with universal differential structure as in [BM93] is a ψ-principal bundle P (M, H, ψ, 1). Proof. For any c ∈ H and u ∈ P we have uα ⊗cα = u(0) ⊗cu(1) . Clearly 1α ⊗cα = 1⊗c. We compute ¯
¯
uα vβ ⊗ cαβ = u(0) vβ ⊗ (cu(1) )β = u(0) v (0) ⊗ cu(1) v (1) = (uv)(0) ⊗ c(uv)(1) = (uv)α ⊗ cα , ¯
¯
¯
¯
¯
¯
¯
¯
hence the condition (4) is satisfied. Furthermore, (cα )uα = (cu(1) )u(0) = (c)u and ¯
¯
uαβ ⊗ c(1) β ⊗ c(2) α = u(0) β ⊗ c(1) β ⊗ c(2) u(1) = u(0)(0) ⊗ c(1) u(0)(1) ⊗ c(2) u(1) ¯
¯
¯ ¯
¯ ¯
¯
= u(0) ⊗ (cu(1) )(1) ⊗ (cu(1) )(2) = uα ⊗ cα (1) ⊗ cα (2) , ¯
¯
¯
so that the condition (5) is also satisfied. Clearly the induced coaction in Proposition 2.2 coincides with the given coaction of H. We can easily replace H here by one of the braided groups introduced in [Maj91, Maj93b]. To be concrete, we suppose that our braided group B lives in a k-linear braided category with well-behaved direct sums, such as that of modules over a quasitriangular Hopf algebra or comodules over a dual-quasitriangular Hopf algebra. This background quantum group does not enter directly into the braided group formulae but rather via the braiding 9 which it induces between any objects in the category. We refer to [Maj93a] for an introduction to the theory and for further details. In particular, a right braided B-module algebra P means a coaction P → P ⊗B in the category which is an algebra homomorphism to the braided tensor product algebra [Maj91] (u ⊗ b)(v ⊗ c) = u9(b ⊗ v)c.
(8)
The coproduct 1 : B → B⊗B of a braided group is itself a homomorphism to such a braided tensor product. Example 2.4. Let B be a braided group with braiding 9 and P a right braided Bcomodule algebra. The linear map ψ : B ⊗ P → P ⊗ B defined by ψ : c ⊗ u → ¯ ¯ 9(c ⊗ u(0) )u(1) entwines B, P . If the induced map χM is a bijection we say that the associated ψ-principal bundle P (M, B, ψ, 1) is a braided group principal bundle, and denote it by P (M, B, 9).
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Proof. This is best done diagrammatically by the technique introduced in [Maj91]. Thus, we write 9 = and products by µ = . We denote coactions and coproducts by . The proof of the main part of (4) is then the diagram:
=
= P B
P B
B P P
B P P
B P P
B P P
=
P B
P B ,
where box is ψ as stated, in diagrammatic form. The first equality is the assumed homomorphism property of the braided coaction . The second equality is associativity of the product in B, and the third is functoriality of the braiding, which we use to push the diagram into the right form. The minor condition is immediate from the axioms of a braided comodule algebra and the properties of the unit map η : 1 → P . Here 1 denotes the trivial object for our tensor product and necessarily commutes with the braiding in an obvious way (such that 1 is denoted consistently by omission). For the proof of (5) we ask the reader to reflect the diagram in a mirror about a horizontal axis (i.e. view it up-side-down and from behind) and then reverse all braid crossings (restoring them all to ). The result is the diagrammatic proof for the main part of (5) if we relabel the product of P as the coproduct of B and relabel the product of B as the right coaction of B on P . The minor part of (5) is immediate from properties of the braided counit. Example 2.5. Let H be a Hopf algebra and π : H → C a coalgebra surjection. If ker π is a minimal right ideal containing {u − (u)|u ∈ M } then ψ : C ⊗ H → H ⊗ C defined by ψ(c ⊗ u) = u(1) ⊗ π(vu(2) ) entwines C, H, where u ∈ H, c ∈ C and v ∈ π −1 (c), and we have a ψ-principal bundle H(M, C, ψ, π(1)) in the setting of Proposition 2.2, denoted H(M, C, ψ, π). Hence the generalised bundles over embeddable quantum homogeneous spaces in [Brz96b] are examples of ψ-principal bundles. Proof. In this case uα ⊗ cα = u(1) ⊗ π(wu(2) ), for any u ∈ H, c ∈ C and w ∈ π −1 (c). Clearly 1α ⊗ cα = 1 ⊗ c. We compute uα vβ ⊗ cαβ = u(1) vα ⊗ π(wu(2) )α = u(1) v (1) ⊗ π(wu(2) v (2) ) = (uv)α ⊗ cα , where w ∈ π −1 (c). Hence condition (4) is satisfied. Furthermore, we have (cα )uα = (π(wu(2) ))u(1) = (c)u and uαβ ⊗ c(1) β ⊗ c(2) α = u(1) α ⊗ π(w(1) u(2) ) ⊗ c(2) α = u(1) ⊗ π(w(1) u(2) ) ⊗ π(w(2) u(3) ) = u(1) ⊗ π(wu(2) )(1) ⊗ π(wu(2) )(2) = uα ⊗ cα (1) ⊗ cα (2) , where again w(1) ∈ π −1 (c(1) ) and w(2) ∈ π −1 (c(2) ). Therefore condition (5) is also satisfied. Some concrete examples of coalgebra bundles over quantum embeddable homogeneous spaces may be found in [Brz96b] (cf. [DK94]).
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We note that diagrams (4) and (5) are dual to each other in the following sense. The diagrams (5) may be obtained from diagrams (4) by interchanging µ with 1, η with and P with C, and by reversing the arrows. With respect to this duality property the axioms for the map ψ are self-dual. Therefore we can dualise Proposition 2.2 to obtain the following: Proposition 2.6. Let C, P be entwined by ψ : C ⊗ P → P ⊗ C. For every algebra character κ : P → k we have the following: 1. For any positive integer n, C ⊗n is a right P -module with the action /n = (κ ⊗ idn ) ◦ − →n ψ12 ◦ ψ23 ◦ . . . ◦ ψnn+1 = (κ ⊗ idn ) ◦ ψ . 2. The action /n maps 1n (C) to itself. 3. The subspace Iκ = span{c/1 u−cκ(u)|c ∈ C, u ∈ P } is a coideal. Hence M = C/Iκ is a coalgebra. We denote the canonical surjection by πκ : C → M . 4. There is a map ζ M : C ⊗ P → C ⊗M C defined by ζ M (c ⊗ u) = c(1) ⊗M c(2) /1 u, where C ⊗M C = span{c ⊗ d ∈ C ⊗ C|c(1) ⊗ πκ (c(2) ) ⊗ d = c ⊗ πκ (d(1) ) ⊗ d(2) } is the cotensor product under M . If ζ M is a bijection, we say that C(M, P, ψ, κ) is a dual ψ-principal bundle. Proof. 1. The explicit action is α1 n (cn ⊗ · · · ⊗ c1 )/n u = cα n ⊗ · · · ⊗ c1 κ(uα1 ···αn ).
Then clearly 1 β1 n βn ((cn ⊗ · · · ⊗ c1 ) /n u)/n v = cα ⊗ · · · ⊗ cα κ(uα1 ···αn vβ1 ···βn ) n 1
α
β
n−1 n−1 1 β1 n = cα ⊗ · · · ⊗ cα κ((uα1 ···αn−1 vβ1 ···βn−1 )αn ) n ⊗ cn−1 1 α1 n n ⊗ · · · ⊗ c κ((uv) = · · · = cα α ···α n 1 n ) = (cn ⊗ · · · ⊗ c1 )/ (uv) 1
for all ci ∈ C and u, v ∈ P . We used (6) repeatedly. 2. We have (c(1) ⊗ c(2) )/2 u = c(1) β ⊗ c(2) α κ(uαβ ) = cα (1) ⊗ cα (2) κ(uα ) = 1(c/1 u) by (7), and similarly for higher 1n (C). 3. Explicitly, Iκ = span{cα κ(uα ) − cκ(u)|c ∈ C, u ∈ P }. But using (7) we have 1(cα κ(uα ) − cκ(u)) = c(1) β ⊗ c(2) α κ(uαβ ) − c(1) ⊗ c(2) κ(u) = c(1) ⊗(c(2) α κ(uα ) − c(2) κ(u))+(c(1) β κ(uαβ )−c(1) κ(uα )) ⊗ c(2) α ∈ C ⊗ Iκ +Iκ ⊗ C. Hence Iκ is a coideal. 4. The stated map ζ M (c ⊗ u) = c(1) ⊗ c(2) α κ(uα ) has its image in C ⊗M C since c(1) ⊗ πκ (c(2) ) ⊗ c(3) α κ(uα ) = c(1) ⊗ πκ (c(2) β )κ(uαβ ) ⊗ c(3) α using (7) and πκ (Iκ ) = 0. By dimensions in the finite-dimensional case, it is natural to require that this is an isomorphism. This is also an example of a dual version of the theory of C-Galois extensions. The proposition is dual to Proposition 2.2 in the sense that all arrows are reversed. In concrete terms, if P, C are finite-dimensional then ψ ∗ : P ∗ ⊗ C ∗ → C ∗ ⊗ P ∗ and κ ∈ P ∗ make C ∗ (M ∗ , P ∗ , ψ ∗ , κ) a ψ ∗ -principal bundle. Here M ∗ = {f ∈ C ∗ |(κ ⊗ f ) ◦ ψ = f ⊗ κ}. If C, P are entwined and we have both e ∈ C and κ : P → k, we can have both a ψprincipal bundle and a dual one at the same time. An obvious example, in the setting of Example 2.3, is P = C = H a Hopf algebra and ψ(c ⊗ u) = u(1) ⊗ cu(2) by the coproduct.
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Then Proposition 2.2 with e = 1 gives a quantum principal bundle with M = k and right coaction given by the coproduct. On the other hand, Proposition 2.6 with κ = gives a dual bundle with action by right multiplication. Finally, we note that there is a close connection with the theory of factorisation of (augmented) algebras introduced in [Maj90, Maj94b] as part of a factorisation theory of Hopf algebras. According to this theory, a factorisation of an algebra X into subalgebras A, B (so that the product A ⊗ B → X is a linear isomorphism) is equivalent to a factorisation structure ψ˜ : B ⊗ A → A ⊗ B with certain properties. It was also shown that when A, B are augmented by algebra characters then the factorisation structure induces a right action of A on B and a left action of B on A, respectively. Proposition 2.7. Let C be finite-dimensional. Then an entwining structure ψ : C ⊗ P → P ⊗ C is equivalent by partial dualisation to a factorisation structure ψ˜ : C ∗ ⊗ P op → P op ⊗ C ∗ . In the augmented case, the induced coaction 11R and action /1 in Propositions 2.2 and 2.6 are the dualisations of the actions induced by the factorisation. ˜ ⊗ u) = ui ⊗ f i say, for f ∈ C ∗ and u ∈ P . The Proof. We use the notation ψ(f equivalence with ψ is by ui hf i , ci = uα hf, cα i, where h , i denotes the evaluation pairing. It is easy to see that ψ entwines C, P iff ψ˜ obeys [Maj94b]cf. [Maj90] ψ˜ ◦ (µ ⊗ id) = (id ⊗ µ) ◦ ψ˜ 12 ◦ ψ˜ 23 ,
˜ ⊗ 1) = 1 ⊗ f, ψ(f
(9)
ψ˜ ◦ (id ⊗ µ) = (µ ⊗ id) ◦ ψ˜ 23 ◦ ψ˜ 12 ,
˜ ⊗ u) = u ⊗ 1 ψ(1
(10)
for all f ∈ C ∗ and u ∈ P op . Thus, the first of these is ui hc, (f g)i i = uα hcα , f gi = uα hcα (1) , f ihcα (2) , gi = uαβ hc(1) β , f ihc(2) α , gi = uαi hc(1) , f i ihc(2) α , gi = uji hc(1) , f i i hc(2) , g j i using (7). Similarly for (10) using (6), provided we remember to use the opposite product on P . Such data ψ˜ is equivalent by [Maj94b, Maj90] to the existence of an algebra ˜ ⊗ u) in X, and X factorising into P op C ∗ . Given such X we recover ψ˜ by uc = µ ◦ ψ(c op ∗ ˜ ˜ conversely, given ψ we define X = P ⊗C as in (8), but with ψ. Also from this theory, if we have κ an algebra character on P op (or on P ) then / = (κ ⊗ id) ◦ ψ˜ is a right action of P op on C ∗ , which clearly dualises to the right action of P on C in Proposition 2.6. Similarly, if e is a character on C ∗ then . = (id ⊗ e) ◦ ψ˜ is a left action of C ∗ on P op (or on P ) which clearly dualises to the right coaction of C in Proposition 2.2. An obvious setting in which factorisations arise is the braided tensor product (8) of algebras in braided categories [Maj91, Maj93b, Maj93a], with ψ˜ = 9 the braiding. Thus if A⊗B is a braided tensor product of algebras (e.g. of module algebras under a background quantum group) we can look for a suitable dual coalgebra B ∗ in the category and the corresponding entwining ψ of B ∗ , Aop . This provides a large class of entwining structures. Another source is the theory of double cross products G ./ H of Hopf algebras in [Maj90]. These factorise as Hopf algebras and hence, in particular, as algebras. In this context, Proposition 2.7 can be combined with the result in [Maj90, Sect. 3.2] that the double cross product is equivalent by partial dualisation to a bicrossproduct H ∗ I/G. These bicrossproduct Hopf algebras (also due to the second author) provided one of the first general constructions for non-commutative and non-cocommutative Hopf algebras, and many examples are known. Proposition 2.8. Let CI/P op be a bicrossproduct bialgebra [Maj90, Sect. 3.1], where P op , C are bialgebras suitably (co)acting on each other. Then C, P are entwined by
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¯
ψ(c ⊗ u) = u(1) (0) ⊗ u(1) (1) (u(2) .c). ¯
¯
Here . is the left action of P op on C and u(0) ⊗ u(1) is the right coaction of C on P op , as part of the bicrossproduct construction. Proof. We derive this result under the temporary assumption that C is finite-dimensional. Thus the bicrossproduct is equivalent to a double cross product P op ./ C ∗ with actions ¯ ¯ ., / defined by f .u = u(0) hf, u(1) i and hu.c, f i = hc, f /ui for all f ∈ C ∗ . Then ψ˜ ˜ ⊗ u) = f (1) .u(1) ⊗ f (2) /u(2) according to [Maj90, Maj94b]. for this factorisation is ψ(f The correspondence in Proposition 2.7 then gives ψ as stated. Once the formula for ψ is known, one may verify directly that it entwines C, P given the compatibility conditions between the action and coaction of a bicrossproduct in [Maj90, Sect. 3.1]. Now we describe trivial ψ-principal bundles and gauge transformations in them. Proposition 2.9. Let P and C be entwined by ψ as in Definition 2.1 and let e be a group-like element in C. Assume the following data: 1. A map ψ C : C ⊗ C → C ⊗ C such that C C ◦ ψ23 ◦ (1 ⊗ id), (id ⊗ 1) ◦ ψ C = ψ12
(id ⊗ ) ◦ ψ C = ⊗ id,
(11)
and ψ C (e ⊗ c) = 1c, for any c ∈ C; 2. A convolution invertible map 8 : C → P such that 8(e) = 1 and ψ ◦ (id ⊗ 8) = (8 ⊗ id) ◦ ψ C .
(12)
Then there is a ψ-principal bundle over M = PecoC with structure coalgebra C and total space P . We call it the trivial ψ-principal bundle P (M, C, 8, ψ, ψ C , e) associated to our data, with trivialisation 8. Proof. The proof of the proposition is similar to the proof that the trivial quantum principal bundle in [BM93, Example 4.2] is in fact a quantum principal bundle. First we observe that the map 2 : M ⊗ C → P,
x ⊗ c 7→ x8(c)
is an isomorphism of linear spaces. Explicitly the inverse is given by 2−1 : u 7→ u(0) 8−1 (u(1) (1) ) ⊗ u(1) (2) , ¯
¯
¯
where 8−1 : C → P is a convolution inverse of 8, i.e. 8−1 (c(1) )8(c(2) ) = 8(c(1) )8−1 (c(2) ) = (c)1. To see that the image of the above map is in M ⊗ C we first notice that (12) implies that 11R ◦ 8 = (8 ⊗ id) ◦ 1 and that ψ(c(1) ⊗ 8−1 (c(2) )) = 8−1 (c) ⊗ e.
(13)
Therefore for any u ∈ P , 11R (u(0) 8−1 (u(1) )) = u(0) ψ(u(1) (1) ⊗ 8−1 (u(1) (2) )) = u(0) 8−1 (u(1) ) ⊗ e, ¯
¯
¯
¯
¯
¯
¯
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and thus u(0) 8−1 (u(1) ) ∈ M . Then it is easy to prove that the above maps are inverses to each other. We remark that 2 is in fact a left M -module and a right C-comodule map, where the coaction in M ⊗ C is given by x ⊗ c 7→ x ⊗ c(1) ⊗ c(2) . Moreover ψ ◦ (id ⊗ 2) = C ◦ ψ12 . (2 ⊗ id) ◦ ψ23 The proof that χM in this case is a bijection follows exactly the method used in the proof of [BM93, Example 4.2] and thus we do not repeat it here. ¯
¯
Next, we consider gauge transformations. Definition 2.10. Let P (M, C, 8, ψ, ψ C , e) be a trivial ψ-principal bundle as in Proposition 2.9. We say that a convolution invertible map γ : C → M such that γ(e) = 1 is a gauge transformation if C ◦ ψ12 ◦ (id ⊗ γ ⊗ id) ◦ (id ⊗ 1) = (γ ⊗ id ⊗ id) ◦ (1 ⊗ id) ◦ ψ C . ψ23
(14)
Proposition 2.11. If γ : C → M is a gauge transformation in P (M, C, 8, ψ, ψ C , e) then 80 = γ ∗ 8, where ∗ denotes the convolution product is a trivialisation of P (M, C, 8, ψ, ψ C , e). The set of all gauge transformations in P (M, C, 8, ψ, ψ C , e) is a group with respect to the convolution product. We say that two trivialisations 8 and 80 are gauge equivalent if there exists a gauge transformation γ such that 80 = γ ∗ 8. Proof. Clearly 80 is a convolution invertible map such that 80 (e) = 1. To prove that it satisfies (12) we first introduce the notation ψ C (b ⊗ c) = cA ⊗ bA
(summation assumed),
in which the condition (14) reads explicitly γ(c(1) )α ⊗ c(2) A ⊗ bαA = γ(cA(1) ) ⊗ cA(2) ⊗ bA , and then compute ψ ◦ (id ⊗ 80 )(b ⊗ c) = ψ(b ⊗ γ(c(1) )8(c(2) )) = γ(c(1) )α 8(c(2) )β ⊗ bαβ = γ(c(1) )α 8(c(2) A ) ⊗ bαA = γ(cA(1) )8(cA(2) ) ⊗ bA = (80 ⊗ id) ◦ ψ C (b ⊗ c).
This proves the first part of the proposition. Assume now that γ1 , γ2 are gauge transformations. Then (γ1 (c(1) )γ2 (c(2) ))α ⊗ c(3) A ⊗ bαA = γ1 (c(1) )α γ2 (c(2) )β ⊗ c(3) A ⊗ bαβA
= γ1 (c(1) )α γ2 (c(2) A (1) ) ⊗ c(2) A (2) ⊗ bαA = γ1 (cA(1) )γ2 (cA(2) ) ⊗ cA(3) ⊗ bA .
Therefore γ1 ∗ γ2 is a gauge transformation too. Clearly is a gauge transformation and thus provides the unit. Finally, to prove that if γ is a gauge transformation then so is γ −1 , we observe that if γ3 = γ1 ∗ γ2 and γ2 are gauge transformations then so is γ1 . Indeed, if γ1 ∗ γ2 is a gauge transformation then (γ1 (c(1) )γ2 (c(2) ))α ⊗ c(3) A ⊗ bαA = γ1 (cA(1) )γ2 (cA(2) ) ⊗ cA(3) ⊗ bA ,
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but since γ2 is a gauge transformation, we obtain γ1 (c(1) )α γ2 (c(2) A (1) ) ⊗ c(2) A (2) ⊗ bαA = γ1 (cA(1) )γ2 (cA(2) ) ⊗ cA(3) ⊗ bA . Applying γ2−1 to the second factor in the tensor product and then multiplying the first two factors we obtain γ1 (c(1) )α ⊗ c(2) A ⊗ bαA = γ1 (cA(1) ) ⊗ cA(2) ⊗ bA , i.e. γ1 is a gauge transformation as stated. Now applying this result to γ3 = and γ2 = γ we deduce that γ −1 is a gauge transformation as required. This completes the proof of the proposition. Although the existence of the map ψ C as in Proposition 2.9 is not guaranteed for all coalgebras, the map ψ C exists in most of the examples discussed in this section: Example 2.12. For a quantum principal bundle P (M, H) as in Example 2.3, we define ψ H (b ⊗ c) = c(1) ⊗ bc(2) , for all b, c ∈ H. Then (2.9)–(2.11) reduces to the theory of trivial quantum principal bundles and their gauge transformations in [BM93]. Proof. It is easy to see by standard Hopf algebra calculations that (11) is satisfied by the bialgebra axiom for H = C in this case. Moreover, (12) reduces to 8 being an intertwiner of 1R with 1. The condition (14) is empty. This recovers the setting introduced in [BM93]. In the braided case we use the above theory to arrive at a natural definition of trivial braided principal bundle: Example 2.13. For a braided principal bundle P (M, B, 9) as in Example 2.4, we define a trivialisation as a convolution-invertible unital morphism 8 : B → P in the braided category such that 1R ◦ 8 = (8 ⊗ id) ◦ 1, where 1R is the braided right coaction of B on P . We define a gauge transformation as a convolution-invertible unital morphism γ : B → M , acting on trivialisations by the convolution product ∗. This is a trivial ψ-principal bundle with ψ B (b ⊗ c) = 9(b ⊗ c(1) )c(2) , where 1c = c(1) ⊗ c(2) is the braided group coproduct. Proof. This time, (11) follows from the braided-coproduct homomorphism property of a braided group [Maj91]. From this and the form of ψ, we see that (12) becomes 1R ◦ 8(c) = ((8 ⊗ id) ◦ 9(b ⊗ c(1) ))c(2) . Setting b = e gives the condition stated on 8 because the braiding with e = 1 is always trivial. Assuming the stated condition, (12) then becomes 8(c(1) ) ⊗ bc(2) = ((8 ⊗ id) ◦ 9(b ⊗ c(1) ))c(2) , which is equivalent (by replacing c(2) by c(2) ⊗ Sc(3) and multiplying, where S is the braided antipode) to (8 ⊗ id) ◦ 9 = 9 ◦ (id ⊗ 8). When all our constructions take place in a braided category, this is the functoriality property implied by requiring that 8 is a morphism in the category. The theory of trivial ψ-bundles only requires this functoriality condition itself. Similarly, we compute the gauge condition (14) using ψ(b ⊗ γ(c)) = 9(b ⊗ γ(c)) because γ(c) ∈ M , and operate
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on it by replacing c(2) by c(2) ⊗ Sc(3) and multiplying. Then it reduces to 923 ◦ 912 ◦ (id ⊗ γ ◦ 1) = (γ ◦ 1 ⊗ id) ◦ 9. Since 1 is a morphism, we see (by applying the braided group counit) that the gauge condition (14) is equivalent to (γ ⊗ id) ◦ 9 = 9 ◦ (id ⊗ γ). As before, this is naturally implied by requiring that γ is a morphism in our braided category. It is clear that the convolution product ∗ preserves the property of being a morphism since 1 and 1R are assumed to be morphisms. For a ψ-principal bundle over a quantum homogeneous space as in Example 2.5, we can define a trivialisation if, for example, the map ψ C (b ⊗ c) = π(v (1) ) ⊗ π(uv (2) ),
(15)
where u ∈ π −1 (b), v ∈ π −1 (c) is well-defined. Then a trivialisation of the bundle is a convolution-invertible map 8 : C → H obeying 8 ◦ π(1) = 1 and 8(c)(1) ⊗ π(u8(c)(2) ) = 8 ◦ π(v (1) ) ⊗ π(uv (2) )
(16)
for all c ∈ C, u ∈ H, and v ∈ π −1 (c). Taking u = 1 requires, in particular, the natural intertwiner condition (8 ⊗ id) ◦ 1 = 1R ◦ 8. There is, similarly, a condition on gauge transformations γ obtained from (14). Hence our formulation of trivial ψ-principal bundles covers all the main sources of ψ-principal bundles discussed in this section. We conclude this section with some explicit examples of ψ-principal bundles. Example 2.14. Let H be a quantum cylinder Aq [x−1 ], i.e. a free associative algebra generated by x, x−1 and y subject to the relations yx = qxy, xx−1 = x−1 x = 1, with a natural Hopf algebra structure: 2|0
1x±1 = x±1 ⊗ x±1 ,
1y = 1 ⊗ y + y ⊗ x,
etc.
(17)
Consider a right ideal J in H generated by x − 1 and x−1 − 1. Clearly, J is a coideal and 2|0 therefore C = Aq [x−1 ]/J is a coalgebra and a canonical epimorphism π : H → C is a coalgebra map. C is spanned by the elements cn = π(y n ), n ∈ Z≥0 , and the coproduct and the counit are given by n X n c ⊗ cn−k , (cn ) = 0. (18) 1cn = k q k k=0
We are in the situation of Example 2.5 and thus we have the entwining structure ψ : C ⊗ H → H ⊗ C, which explicitly computed comes out as n X n m n q l(k+m) xm y k ⊗ cn+l−k , (19) ψ(cl ⊗ x y ) = k q k=0
where
and
[n]q ! n , = k q [n − k]q ![k]q ! [n]q ! = [n]q · · · [2]q [1]q ,
[0]q ! = 1,
[n]q = 1 + q + . . . + q n−1 .
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From this definition of ψ one easily computes the right coaction of C on H as well as the fixed point subalgebra M = k[x, x−1 ], i.e. the algebra of functions on a circle. By Example 2.5 we have just constructed a generalised quantum principal bundle 2|0 Aq [x−1 ](k[x, x−1 ], C, ψ, c0 ). Finally we note that the above bundle is trivial in the sense of Proposition 2.9. The 2|0 trivialisation 8 : C → Aq [x−1 ] and its inverse 8−1 are defined by 8(cn ) = y n ,
8−1 (cn ) = (−1)n q n(n−1)/2 y n .
(20)
One can easily check that the map 8 satisfies the required conditions. Explicitly, the map ψ C : C ⊗ C → C ⊗ C reads n X n q km ck ⊗ cm+n−k . ψ (cm ⊗ cn ) = k q C
k=0
Therefore n X n q km y k ⊗ cm+n−k ψ ◦ (id ⊗ 8)(cm ⊗ cn ) = ψ(cm ⊗ y ) = k q n
k=0
= (8 ⊗ id) ◦ ψ C (cm ⊗ cn ). Since the bundle discussed in this example is trivial, we can compute its gauge group. One easily finds that a convolution invertible map γ : C → k[x, x−1 ] satisfies condition (14) if and only if γ(cn ) = 0n xn (no summation), where n ∈ Z≥0 , 0n ∈ k and 00 = 1. Therefore the gauge group is equivalent to the group of sequences 0 = (1, 01 , 02 , ...) with the product given by n X n (0 · 0 )n = 0 00 . k q k n−k 0
k=0
For the simplest example of a braided principal bundle, one can simply take any braided group B and any algebra M in the same braided category. Then the braided tensor product algebra P = M ⊗B, along with the definitions 1R = id ⊗ 1,
8(b) = 1 ⊗ b,
8−1 (b) = 1 ⊗ Sb
(21)
put us in the setting of Examples 2.4 and 2.13. Note first that 1R is a coaction (the tensor product of the trivial coaction and the right coregular coaction) and makes P into a braided comodule algebra. Moreover, the induced map χM (m ⊗ b ⊗ n ⊗ c) = m9(b ⊗ n)c(1) ⊗ c(2) for m, n ∈ M , b, c ∈ B, is an isomorphism P ⊗M P → P ⊗ P ; it has inverse χ−1 M (m ⊗ b ⊗ c) = m ⊗ bSc(1) ⊗ 1 ⊗ c(2) . It is also clear that 8 is a trivialisation. This is truly a trivial braided principal bundle because P is just a (braided) tensor product algebra.
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Example 2.15. Let B = k[c] be the braided line generated by c with braiding 9(c ⊗ c) = qc ⊗ c and the linear coproduct 1c = c ⊗ 1 + 1 ⊗ c. It lives in the braided category Vecq of Z-graded vector spaces with braiding q deg( ) deg( ) times the usual transposition. Here deg(c) = 1. Let M = k[x, x−1 ] be viewed as a Z-graded algebra as well, with deg(x) = 1. Then P = k[x, x−1 ]⊗k[c] is a trivial braided principal bundle with the coaction and trivialisation n X n m n xm ⊗ ck ⊗ cn−k , 8(cn ) = 1 ⊗ cn . (22) 1R (x ⊗ c ) = k q k=0
As a ψ-principal bundle, this example clearly coincides with the preceding one, albeit constructed quite differently: we identify cn = cn and y = 1 ⊗ c, and note that in the braided tensor product algebra k[x, x−1 ]⊗k[c] we have the product (1 ⊗ c)(x ⊗ 1) = 2|0 9(c ⊗ x) = q(x ⊗ 1)(1 ⊗ c), i.e. P = Aq [x−1 ]. It is also clear that the coproduct deduced in (18) can be identified with the braided line coproduct which is part of our initial data here. This particular braided tensor product algebra P is actually the algebra part of the bosonisation of B = k[c] viewed as living in the category of comodules over k[x, x−1 ] as a dual-quasitriangular Hopf algebra (see [Maj95, p. 510]), and becomes in this way a Hopf algebra. This bosonisation is the Hopf algebra H which was part of the initial data in the preceding example. Finally, gauge transformations γ from the braided point of view are arbitrary degree-preserving unital maps k[c] → k[x, x−1 ], i.e. given by the group of sequences 0 as found before. This example demonstrates the strength of braided group gauge theory; even the most trivial braided quantum principal bundles may be quite complicated when constructed by more usual Hopf algebraic means. On the other hand, the following embeddable quantum homogeneous space does not appear to be of the braided type, nor (as far as we know) a trivial bundle. Example 2.16. Let P be the algebra of functions on the quantum group GLq (2). This is generated by elements α, β, γ, δ and D subject to the relations αβ = qβα,
αγ = qγα,
βδ = qδβ,
γδ = qδγ,
αδ = δα + (q − q −1 )βγ,
βγ = γβ,
(αδ − qβγ)D = D(αδ − qβγ) = 1.
Let C be a vector space spanned by cm,n , m ∈ Z>0 , n ∈ Z with the coalgebra structure m X m q k(m−k) c ⊗ cm−k,n+k , (cm,n ) = δm0 . 1(ci,j ) = k q−2 k,n k=0
Let the linear map ψ : C ⊗ P → P ⊗ C be given by ψ(ci,j ⊗ αk γ l β m δ n Dr ) n m X X m n q (m−s)(s+t−l)+(n−t)t−i(k+l−t−s) × = s q−2 t q−2
(23)
s=0 t=0 k+m−s l+n−t s t
α
β δ Dr ⊗ ci+m+n−s−t,j−r+t+s .
γ
Then ψ entwines P with C. Furthermore if we take e = c0,0 then the fixed point sub2|0 algebra PeC is generated by 1, α, γ and hence it is isomorphic to A1/q and there is a 2|0
ψ-principal bundle P (A1/q , C, ψ, e).
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Proof. The algebra GLq (2) can be equipped with the standard Hopf algebra structure 1
α γ
β δ
=
α γ
β α ⊗ δ γ
β , δ
S
α γ
β δ
=D
δ −qγ
−q −1 β , α
(α) = (δ) = 1, (β) = (γ) = 0. We define a surjection π : GLq (2) → C by π(αk β l γ m δ n Dr ) = δm0 q ln cl,n−r . In Sect. 5 of [Brz96b] it is shown that π is a coalgebra map and that the data H = GLq (2), π, C satisfy requirements of Example 2.5. Therefore we have a ψ-principal bundle with ψ as in Example 2.5. Written explicitly this ψ is exactly as in Eq. (23). In [Brz96b] it is also noted that the coalgebra C can be equipped with the algebra 2|0 structure of Aq−2 [x−1 ] by setting cm,n = q −mn xn y m . The coproduct in C is then the same as in Example 2.14, Eq. (17).
3. Connections in the Universal Differential Calculus Case From the first assertion of Proposition 2.2 we know that the natural coaction 1R = 11R of C on P extends to the coaction of C on the tensor product algebra P ⊗n for any positive integer n. Still most importantly this coaction can be restricted to n P by the second assertion of Proposition 2.2. Therefore the coalgebra C coacts on the algebra of universal forms on P . The universal differential structure on P is covariant with respect to the coaction 1nR in the following sense: Proposition 3.1. Let P , C, ψ and e be as in Proposition 2.2. Let d : P → 1 P be the universal differential, du = 1 ⊗ u − u ⊗ 1 extended to the whole of P as in the ← −n−1 ← −n for any integer n > 1. Therefore Preliminaries. Then ψ ◦ (id ⊗ d) = (d ⊗ id) ◦ ψ 1nR ◦ (id ⊗ d) = (d ⊗ id) ◦ 1n−1 R . P Proof. We take υ = i u0,i ⊗ u1,i ⊗ . . . ⊗ un,i ∈ n P (i.e., any adjacent product vanishes). Using conditions (4), and the explicit form of dυ (1), for any c ∈ C we compute ← −n+2 (c ⊗ dυ) ψ =
n+1 X
(−1)k
i
k=0
=
n+1 X k=0
X
(−1)k
X i
k−1,i k,i n,i α0 ...αk−1 βαk ...αn u0,i α0 ⊗ . . . ⊗ uαk−1 ⊗ 1β ⊗ uαk ⊗ . . . ⊗ uαn ⊗ c
k−1,i k,i n,i α0 ......αn u0,i α0 ⊗ . . . ⊗ uαk−1 ⊗ 1 ⊗ uαk ⊗ . . . ⊗ uαn ⊗ c
← −n+1 =(d ⊗ id) ◦ ψ (c ⊗ υ).
To discuss a theory of connections in P (M, C, ψ, e) it is important that the horizontal ← −2 one forms P 1 M P be covariant under the action of 12R or, more properly, ψ . The following lemma gives a criterion for the covariance of horizontal one-forms.
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Lemma 3.2. For a ψ-principal bundle P (M, C, ψ, e) assume that ψ(C ⊗M ) ⊂ M ⊗C. ← −2 Then ψ (C ⊗ P 1 M P ) ⊂ P 1 M P ⊗ C. Proof. Using (6) one easily finds that for any u, v ∈ P , x, y ∈ M and c ∈ C, ← −2 ψ (c ⊗ ux ⊗ yv) = uα xβ ⊗ yγ vδ ⊗ cαβγδ . If we assume further that xβ , yγ ∈ M then the result follows.
We will see later that the hypothesis of Lemma 3.2 is automatically satisfied for braided principal bundles of Example 2.4. In contrast, it is not necessarily satisfied for ψ-bundles on quantum embeddable homogeneous spaces of Example 2.5. For example, one can easily check that it is satisfied for the bundle discussed in Example 2.16. On the other hand the ψ-principal bundle over the quantum hyperboloid, which is an embeddable homogeneous space of Eq (2) [BCGST96] fails to fulfil requirements of Lemma 3.2. The covariance of P and P 1 M P enables us to define a connection in P (M, C, ψ, e) in a way similar to the definition of a connection in a quantum principal bundle P (M, H) (compare [BM93]). Definition 3.3. Let P (M, C, ψ, e) be a generalised quantum principal bundle such that ψ(C ⊗ M ) ⊂ M ⊗ C. A connection in P (M, C, ψ, e) is a left P -module projection ← −2 ← −2 Π : 1 P → 1 P such that ker Π = P 1 M P and ψ (id ⊗ Π) = (Π ⊗ id) ψ . It is clear that for a usual quantum principal bundle P (M, H), Definition 3.3 coincides with the definition of a connection given in [BM93]. Thus, the condition in ← −2 ← −2 Lemma 3.2 always holds for ψ as in Example 2.3, while ψ (id ⊗ Π) = (Π ⊗ id) ψ if and only if 12R Π = (Π ⊗ id)12R , which was the condition in [BM93]. In what follows we assume that the condition in Lemma 3.2 is satisfied. A connection Π in P (M, C, ψ, e) can be equivalently described as follows. First we define a map φ : C ⊗ P ⊗ ker → P ⊗ ker ⊗ C by the commutative diagram ← −2 ψ
C ⊗ 1 P id ⊗ χ ? C ⊗ P ⊗ ker
- 1 P ⊗ C χ ⊗ id
φ
? - P ⊗ ker ⊗ C
where χ(u ⊗ v) = uψ(e ⊗ v). The map φ is clearly well-defined. Indeed, because χM ← −2 is a bijection, ker χ = P 1 M P and then ψ (C ⊗ ker χ) ⊂ ker χ ⊗ C, by Lemma 3.2. Therefore φ(0) = 0. By definition of P (M, C, ψ, e) we have a short exact sequence of left P -module maps χ (24) 0 → P 1 M P → 1 P → P ⊗ ker → 0.
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The exactness of the above sequence is clear since the fact that χM is bijective implies ← −2 that χ is surjective and ker χ = P 1 M P . By definition, χ intertwines ψ with φ. Proposition 3.4. The existence of a connection Π in P (M, C, ψ, e) is equivalent to the existence of a left P -module splitting σ : P ⊗ ker → 1 P of the above sequence such ← −2 that ψ ◦ (id ⊗ σ) = (σ ⊗ id) ◦ φ. Proof. Clearly the existence of a left P -module projection is equivalent to the existence of a left P -module splitting. It remains to check the required covariance properties. Assume that σ has the required properties, then ← −2 ← −2 ψ ◦ (id ⊗ Π) = ψ ◦ (id ⊗ σ) ◦ (id ⊗ χ) ← −2 ← −2 = (σ ⊗ id) ◦ φ ◦ (id ⊗ χ) = (σ ◦ χ ⊗ id) ◦ ψ = (Π ⊗ id) ◦ ψ . Conversely, if Π has the required properties, then one easily finds that ← −2 ψ ◦ (id ⊗ σ ◦ χ) = (σ ⊗ id) ◦ φ ◦ (id ⊗ χ).
Since χ is a surjection the required property of σ follows.
To each connection we can associate its connection one-form ω : ker → 1 P by setting ω(c) = σ(1 ⊗ c). 1 Similarly to the quantum bundle case of [BM93] we have Proposition 3.5. Let Π be a connection on P (M, C, ψ, e). Then, for all c ∈ ker , the connection 1-form ω : ker → 1 P has the following properties: 1. χ ◦ ω(c) = 1 ⊗ c, ← −2 2. For any b ∈ C, ψ (b ⊗ ω(c)) = c(1) α c(2) βγ ω(eγ ) ⊗ bαβ , where c(1) ⊗M c(2) (summation understood) denotes the translation map τ (c) = χ−1 M (1 ⊗ c) in P (M, C, ψ, e). Conversely, if ω is any linear map ω : ker → 1 P obeying conditions 1-2, then there is a unique connection Π = µ ◦ (id ⊗ ω) ◦ χ in P (M, C, ψ, e) such that ω is its connection 1-form. Proof. For any b ⊗ u ⊗ c ∈ C ⊗ P ⊗ ker the map φ is explicitly given by φ(b ⊗ u ⊗ c) = uα c(1) β c(2) γδ ⊗ eδ ⊗ bαβγ . Therefore if ω is a connection one-form then ← −2 ← −2 ψ (b ⊗ ω(c)) = ψ ◦ (id ⊗ σ)(b ⊗ 1 ⊗ c) = σ(c(1) α c(2) βγ ⊗ eγ ) ⊗ bαβ = c(1) α c(2) βγ ω(eγ ) ⊗ bαβ . Conversely, if ω : ker → 1 P satisfies condition 1 then σ = (µ ⊗ id) ◦ (id ⊗ ω) gives a left P -module splitting of (24). Furthermore, Condition 2 implies We can equivalently think of a connection 1-form as a map C → P given by ω(c − e(c)). This was the point of view adopted in [BM93]. 1
1
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(σ ⊗ id) ◦ φ(b ⊗ u ⊗ c) = σ(uα c(1) β c(2) γδ ⊗ eδ ) ⊗ cαβγ = uα c(1) β c(2) γδ ω(eδ ) ⊗ cαβγ ← −3 ← −2 = uα ψ (bα ⊗ ω(c)) = (µ ⊗ id) ◦ ψ (b ⊗ u ⊗ ω(c)) ← −2 ← −2 = ψ (b ⊗ uω(c)) = ψ ◦ (id ⊗ σ)(b ⊗ u ⊗ c). Example 3.6. For a quantum principal bundle P (M, H), Condition 2 in Proposition 3.5 is equivalent to the AdR -covariance of ω. Proof. Using the definition of ψ in Example 2.3 one finds c(1) α c(2) βγ ⊗ eγ ⊗ bαβ = c(1) α c(2) β (0) ⊗ c(2) β (1) ⊗ bαβ ¯
¯
= c(1)(0) c(2) β (0) ⊗ c(2) β (1) ⊗ bβ c(1)(1) ¯
¯
¯
¯ ¯
¯
¯
¯ ¯
¯
¯
= c(1)(0) c(2)(0)(0) ⊗ c(2)(0)(1) ⊗ bc(1)(1) c(2)(1) ¯ ¯ ¯ ¯ = χM (c(1)(0) ⊗M c(2)(0) ) ⊗ bc(1)(1) c(2)(1) . From the covariance properties of the translation map [Brz96a] it then follows that c(1) α c(2) βγ ⊗ eγ ⊗ bαβ = χM (τ (c(2) )) ⊗ b(Sc(1) )c(3) = 1 ⊗ c(2) ⊗ bS(c(1) )c(3) . This also follows from covariance of χM as intertwining 12R projected to P ⊗M P with the tensor product coaction 11R ⊗ AdR on P ⊗ H. Hence Condition 2 may be written as ← −2 ψ (b ⊗ ω(c)) = ω(c(2) ) ⊗ b(Sc(1) )c(3) which is equivalent to 12R ◦ ω = (ω ⊗ id) ◦ AdR .
Example 3.7. For a braided group principal bundle P (M, B, 9) in Example 2.4, Lemma 3.2 holds. Moreover, Condition 2 in Proposition 3.5 is equivalent to AdR -covariance of ω, where AdR is the braided adjoint coaction as in [Maj94a]. Proof. The braided group adjoint action is studied extensively in [Maj94a] as the basis of a theory of braided Lie algebras; we turn the diagrams up-side-down for the braided adjoint coaction and its properties (or see earlier works by the second author). Firstly, ψ(B ⊗ M ) ⊂ M ⊗ B is immediate since by properties of e = 1, 9(B ⊗ M ) ⊂ M ⊗ B. Also clear is that 11R coincides with the given braided coaction of B on P and 12R coincides with the braided tensor product coaction on P ⊗ P . 12R projects to a coaction on P ⊗M P by Lemma 3.2. We show first that χM : P ⊗M P → P ⊗ B intertwines this coaction with the braided tensor product coaction 11R ⊗ AdR . We work with representatives in P ⊗ P and use the notation [Maj93a] as in the proof of Example 2.4. Branches labelled 1 are the coproduct of B; otherwise they are the given coaction of B on P . Thus, P
P
P
P
P
P
P ∆
∆ ∆
P P B
P P B
P
=
=
S
S
P ∆
∆
=
=
P
S
P P B
P P B
P P B
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where the upper box on the left is χM and the lower box is the braided adjoint coaction AdR . S denotes the braided antipode of B. The tensor product 11R ⊗ AdR uses the braiding and the product of B according to the theory of braided groups [Maj93b]. The first equality uses the homomorphism property of the given coaction of B on P . The second uses the comodule axiom. The third identifies an “antipode loop” and cancels it (using associativity and coassociativity, and the braided antipode axioms). The fourth equality uses the comodule axiom in reverse and also pushes the diagram into the form where we recognise the braided tensor product coaction 12R followed by χM . Using this intertwining property of χM , we write the right hand side of Condition 2 in Proposition 3.5 as B
B
B
B
B
B
τ
τ =
Ad
χ ω P P
P B
B
B
χ
=
ω
B
τ =
Ad
P B
ω
? =
ω
ω P
B
P P
P B
P B
P B
← −2 where τ = χ−1 M (1 ⊗( )). The left hand side ψ (b ⊗ ω(c)) is shown on the right hand side of the diagram (using associativity of the product in B). Hence equality is equivalent to 12R ◦ ω = (ω ⊗ id) ◦ AdR . We remark that in the framework with C ∗ in place of C as explained in Proposition 2.7, we can use for C ∗ braided groups of enveloping algebra type, in particular U (L) associated to a braided-Lie algebra L in [Maj94a] with braided-Lie bracket based on the properties of the braided adjoint action. In this case one could take ω ∈ L ⊗ 1 P with the corresponding covariance properties. Using the braided Killing form also in [Maj94a] one has the possibility (for the first time) to write down scalar Lagrangians built functorially from ω and its curvature. On the other hand, for a theory of trivial bundles (in order to have familiar formulae for gauge fields on the base) one needs to restrict trivialisations and gauge transforms in such a way that ω retains its values in L. This aspect requires further work, to be developed elsewhere. Example 3.8. Consider H(M, C, π), the ψ-principal bundle associated to an embeddable quantum homogeneous space in Example 2.5. Assume that ψ(C ⊗ M ) ⊂ M ⊗ C. Condition 2 in Proposition 3.5 is equivalent to 12R ◦ ω ◦ π = (ω ⊗ id) ◦ (π ⊗ π) ◦ AdR .
(25)
In particular, this implies that any linear inclusion i : M → H such that π ◦ i = id and (c) = ◦ i(c) gives rise to the canonical connection 1-form ω(c) = (Si(c)(1) )di(c)(2) , provided that (id ⊗ π) ◦ AdR ◦ i = (i ⊗ id) ◦ (π ⊗ π) ◦ AdR ◦ i. Proof. In this case ψ(c ⊗ v) = v (1) ⊗ π(uv (2) ), and τ (c) = Su(1) ⊗M u(2) , for any c ∈ C, v ∈ H and u ∈ π −1 (c). Also e = π(1). The transformation property of ω now reads
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← −2 ψ (b ⊗ ω(c)) = (Su(1) )α u(2) βγ ω(π(1)γ ) ⊗ π(v)αβ = (Su(2) )u(3) βγ ω(π(1)γ ) ⊗ π(vSu(1) )β
= (Su(2) )u(3) γ ω(π(1)γ ) ⊗ π(v(Su(1) )u(4) ) = (Su(2) )u(3) ω(π(u(4) )) ⊗ π(v(Su(1) )u(5) ) = ω(π(u(2) )) ⊗ π(v(Su(1) )u(3) ), where v ∈ π −1 (b). Choosing v = 1 we obtain property (25). The converse is obviously true. Before we describe some concrete examples of connections we construct connections in the trivial ψ-bundles of Proposition 2.9. Proposition 3.9. Let P (M, C, 8, ψ, ψ C , e) be a trivial coalgebra ψ-principal bundle such that ψ(C ⊗ M ) ⊂ M ⊗ C. Let β : C → 1 M be a linear map, β(e) = 0 and such that C ◦ ψ23 ◦ ψ12 ◦ (id ⊗ β ⊗ id) ◦ (id ⊗ 1) = (β ⊗ id ⊗ id) ◦ (1 ⊗ id) ◦ ψ C . ψ34
(26)
Then the map ω : ker → 1 P , ω = 8−1 ∗ d8 + 8−1 ∗ β ∗ 8
(27)
is a connection one-form in P (M, C, 8, ψ, ψ C , e). In particular for β = 0 we have a trivial connection in P (M, C, 8, ψ, ψ C , e). Proof. To prove the proposition we will show that ω satisfies conditions specified in Proposition 3.5. Firstly, however, we observe that the translation map in P (M, C, 8, ψ, ψ C , e) is given by (28) τ (c) = 8−1 (c(1) ) ⊗M 8(c(2) ). Indeed, a trivial computation shows that χM (τ (c)) = 1 ⊗ c, as required. The same computation shows that for any c ∈ ker , χ(8−1 (c(1) )d8(c(2) ) + 8−1 (c(1) )β(c(2) )8(c(3) )) = χ(8−1 (c(1) ) ⊗ 8(c(2) )) = 1 ⊗ c, and therefore Condition 1 of Proposition 3.5 is satisfied by ω. Now we prove that Condition 2 of Proposition 3.5 holds for 8−1 ∗d8 and 8−1 ∗β ∗8 separately. For the former the left hand side of Condition 2 reads ← −2 LHS = ψ (b ⊗ 8−1 (c(1) ) ⊗ 8(c(2) )) = 8−1 (c(1) )α ⊗ 8(c(2) )β ⊗ bαβ = 8−1 (c(1) )α ⊗ 8(c(2) A ) ⊗ bαA On the other hand we use the definition of τ (28) and the properties of 8 to write the right-hand side of condition 2 as follows: RHS = 8−1 (c(1) )α 8(c(2) )βγ 8−1 (eγ (1) ) ⊗ 8(eγ (2) ) ⊗ bαβ = 8−1 (c(1) )α 8(c(2) )βγδ 8−1 (eδ ) ⊗ 8(eγ ) ⊗ bαβ = 8−1 (c(1) )α 8(c(2) A )γδ 8−1 (eδ ) ⊗ 8(eγ ) ⊗ bαA
= 8−1 (c(1) )α 8(c(2) A (1) )8−1 (c(2) A (2) ) ⊗ 8(c(2) A (3) ) ⊗ bαA = 8−1 (c(1) )α ⊗ 8(c(2) A ) ⊗ bαA = LHS.
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← −2 To compute the action of ψ on the second part of ω we will use the shorthand notation ← −2 ψ (b ⊗ ρ) = ρα ⊗ bα , for any b ∈ C and ρ ∈ 1 P . In this notation Eq. (26) explicitly reads β(c(1) )α ⊗ c(2) A ⊗ bαA = β(cA(1) ) ⊗ cA(2) ⊗ bA . ← −2 Using the similar steps as in computation of the action of ψ on the first part of ω we find that the right hand side of Condition 2 reads 8−1 (c(1) )α β(c(2) A (1) )8(c(2) A (2) ) ⊗ bαA , while the left hand side is 8−1 (c(1) )α β(c(2) )α 8(c(3) A ) ⊗ bααA = 8−1 (c(1) )α β(c(2) A (1) )8(c(2) A (2) ) ⊗ bαA . From Proposition 3.5 we now deduce that ω is a connection one-form as stated.
Using similar arguments as in [BM93] we can easily show that the behaviour of β under gauge transformations is exactly the same as in the case of quantum principal bundles. For example, if we make a gauge transformation of 8, 8 7→ γ ∗ 8 and then view ω in this new trivialisation then the local connection one-from β will undergo the gauge transformation β 7→ γ −1 ∗ dγ + γ −1 ∗ β ∗ γ.
(29)
As before, we can specialise this theory to our various sources of ψ-principal bundles. For quantum principal bundles we recover the formalism in [BM93]. For braided principal bundles we make a computation similar to the one for γ in Example 2.13, finding that (26) is naturally ensured by requiring that β : B → 1 M is a morphism in our braided category. Then the same formulae (27) and transformation law (29) etc. apply in the braided case. Indeed, they do not involve any braiding directly. Now we construct explicit examples of connections in one of the bundles described at the end of Sect. 2. Example 3.10. Consider the quantum cylinder bundle Aq [x−1 ](k[x, x−1 ], k[c], ψ, 1) in Example 2.14. Then ψ(k[c] ⊗ k[x, x−1 ]) ⊂ k[x, x−1 ] ⊗ k[c]. The most general connection of the type described in Proposition 3.9 has the form 2|0
n q k(k−1)/2 y k dy n−k k q k=0 m n X XX n m k k((k−1)/2+i) + (−1) q 0 xi y k (dxm−k−i )y n−m(30) , m q k q i,m−k
ω(cn )=
i
n−1 X
(−1)k
m=0 k=0
where for all i ∈ Z, n ∈ Z≥0 , 0n,i ∈ k, 00,i = 0.
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Proof. If we set n = 0 in formula (19) then we find ψ(cl ⊗ xm ) = q lm xm ⊗ cl and the first assertion holds. This assertion also follows from Example 3.7. We identify C = k[c] by cn = cn , as a certain (braided) coalgebra. It is an easy exercise to check that a map β : k[c] → 1 k[x, x−1 ] satisfies condition (26) if and only if X 0n,i xi dxn−i , (31) β(cn ) = i
where i ∈ Z, 0n,i ∈ k, 00,i = 0. Now writing the explicit definition of trivialisation 8 (20), and the coproduct of cn (18) we see that ω in (30) is as in (27) with β given by (31). From the braided bundle point of view in Example 2.15 on the same bundle, we work in the braided category of Z-graded spaces and are allowed for β any degree-preserving that vanishes on 1. This immediately fixes it in the form (31), and hence ω from (27).
4. Bundles with General Differential Structures Let P (M, C, ψ, e) be a ψ-principal bundle as in Proposition 2.2. Let N be a subbimodule ← −2 ← −2 ← −2 of 1 P such that ψ (C ⊗ N ) ⊂ N ⊗ C. The map ψ induces a map ψ N : C ⊗ 1 P/N → 1 P/N ⊗ C and N defines a right-covariant differential structure 1 (P ) = 1 P/N on P . We say that 1 (P ) is a differential structure on P (M, C, ψ, e). Definition 4.1. Let P (M, C, ψ, e) be a coalgebra ψ-principal bundle and let ψ(C ⊗ M ) ⊂ M ⊗ C. Assume that N ⊂ 1 P defines a differential structure 1 (P ) on P (M, C, ψ, e). A connection in P (M, C, ψ, e) is a left P -module projection Π : ← −2 ← −2 1 (P ) → 1 (P ) such that ker Π = P 1 (M )P and ψ N ◦ (id ⊗ Π) = (Π ⊗ id) ◦ ψ N . Similarly as for the universal differential calculus case, a connection in P (M, C, ψ, e) can be described by its connection one-form. First we consider the vector space M = (P ⊗ ker )/χ(N ) with a canonical surjection πM : P ⊗ ker → M. Since χ is a left P -module map, χ(N ) is a left P -sub-bimodule of P ⊗ ker . Therefore M is a left P -module and πM is a left P -module map. The action of P on M is defined by X πM (uvi ⊗ ci ), u·υ = i
P −1 (υ). We denote 3 = πM (1 ⊗ ker ). The left where u ∈ P , υ ∈ M and i ui ⊗ ci ∈ πM P -module structure ofP M implies that for every element υ ∈ M, there exist ui ∈ P and λi ∈ 3 such that υ = i ui · λi . Therefore there is a natural surjection P ⊗ 3 → M. We assume that ψ(C ⊗ M ) ⊂ M ⊗ C, and hence the map φ can be defined. For any u ∈ P , c ∈ M and b ∈ C we have ← −2 φ(b ⊗ u ⊗ c) = φ(b ⊗ χ(n)) = (χ ⊗ id) ◦ ψ (b ⊗ n) ∈ χ(N ) ⊗ C, ← −2 where n ∈ N is such that χ(n) = u ⊗ c. We used the fact that ψ (C ⊗ N ) ⊂ N ⊗ C. Therefore we can define a map φN : C ⊗ M → M ⊗ C by the diagram
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C ⊗ P ⊗ ker
id ⊗ πM
-
- 0
C ⊗M φN
φ ? P ⊗ ker ⊗ C
πM ⊗ id
? M⊗C
-
- 0
The map χ induces a map χN : 1 (P ) → M by the commutative diagram πN
1 P
-
- 0
1 (P )
χ
χN
? P ⊗ ker
πM
-
? 0
? M
- 0
? 0
Clearly, χN is a left P -module map, i.e., χN (udv) = u · χN (dv). We can use the map χN to obtain another description of φN . Lemma 4.2. The following diagram
C ⊗ 1 (P )
← −2 ψN
- 1 (P ) ⊗ C
id ⊗ χN ? C ⊗M
χN ⊗ id φN
-
? M⊗C
is commutative. −1 (υ) and compute Proof. We take any υ ∈ 1 (P ), c ∈ C and υ˜ ∈ πN
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φN ◦ (id ⊗ χN )(c ⊗ υ) = φN ◦ (id ⊗ πM )(c ⊗ χ(υ)) ˜ = (πM ⊗ id) ◦ φ(c ⊗ χ(υ)) ˜ 2 ← − = (πM ⊗ id) ◦ (χ ⊗ id) ◦ ψ (c ⊗ υ) ˜ ← −2 ← −2 = (χN ⊗ id) ◦ (πN ⊗ id) ◦ ψ (c ⊗ υ) ˜ = (χN ⊗ id) ◦ ψ N (c ⊗ υ). Using arguments similar to the proof of Example 4.11 of [BM93] and the definition of a coalgebra ψ-principal bundle P (M, C, ψ, e) we deduce that χN
0 → P 1 (M )P → 1 (P ) → M → 0
(32)
is a short exact sequence of left P -module maps. Proposition 4.3. A connection in P (M, C, ψ, e) with differential structure induced by N is equivalent to a left P -module splitting σN of the sequence (32), such that ← −2 (σN ⊗ id) ◦ φN = ψ N ◦ (id ⊗ σN ). Proof. We use Lemma 4.2 to deduce the covariance properties of χN and then preform calculation similar to the proof of Proposition 3.4. To each connection Π we can associate its connection one form ω : 3 → 1 (P ) by ω(λ) = σN (λ). Similarly to the case of universal differential structure, one proves Proposition 4.4. Let Π be a connection in P (M, C, ψ, e) with differential structure defined by N ⊂ 1 P . Then, for all λ ∈ 3 the connection 1-form ω : 3 → 1 (P ) has the following properties: 1. χN ◦ ω(λ) = λ, ← −2 2. For any b ∈ C, ψ N (b ⊗ ω(λ)) = c˜(1) α c˜(2) βδ ω(πM (1 ⊗ eδ )) ⊗ bαβ , where c˜(1) ⊗M c˜(2) denotes the translation map χ−1 M (1 ⊗ c˜), and c˜ ∈ ker is such that πM (1 ⊗ c˜) = λ. Conversely, if M is isomorphic to P ⊗ 3 as a left P -module and ω is any linear map ω : 3 → 1 (P ) obeying Conditions 1–2, then there is a unique connection Π = µ ◦ (id ⊗ ω) ◦ χN in P (M, C, ψ, e) such that ω is its connection 1-form. In the setting of [BM93] the condition P ⊗ 3 = M is always satisfied for quantum principal bundles, and 3 = ker /Q, where Q is an AdR -invariant right ideal in ker that generates the bicovariant differential structure on the structure quantum group H as in [Wor89]. The detailed analysis of braided group principal bundles with general differential structures will be presented elsewhere. Here we remark only that it seems natural to assume that M = P ⊗3 and then choose 3 to be the space dual to the braided Lie algebra L as discussed in Sect. 3. This choice of 3 is justified by the fact that from the properties of the maps φ and φN it follows that the space 3 is invariant under the braided adjoint coaction (cf. Example 3.7). We complete this section with an explicit example of differential structures and connections on the quantum cylinder bundle in Example 2.14 (cf. Example 3.10). Example 4.5. We consider the quantum cylinder bundle of Example 2.14 (cf. Exam2|0 ple 2.15) and we work with differential structures on Aq classified in [BDR92]. Using the definition of ψ (19) one easily finds that there are two differential structures for which
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− →2 the covariance condition ψ (k[c] ⊗ N ) ⊂ N ⊗ k[c] is satisfied. The subbimodules N are generated by (1 + s)x ⊗ x − x2 ⊗ 1 − 1 ⊗ x2 , y ⊗ x − qxy ⊗ 1 − q1 ⊗ xy + qx ⊗ y (1 + q)y ⊗ y − y 2 ⊗ 1 − 1 ⊗ y 2 , where s ∈ k is a free parameter, in the first case, and by (1 + q)x ⊗ x − x2 ⊗ 1 − 1 ⊗ x2 , y ⊗ x − xy ⊗ 1 − q1 ⊗ xy + x ⊗ y, (1 + q)y ⊗ y − y 2 ⊗ 1 − 1 ⊗ y 2 , 2|0
in the second case. In both cases the modules of 1-forms 1 (Aq ) are generated by the exact one-forms dx and dy. Definitions of the N imply the following relations in 2|0 1 (Aq ) xdx = sdxx,
xdy = q −1 dyx,
ydx = qdxy,
ydy = qdyy,
in the first case, and xdx = qdxx,
ydx = qdxy + (q − 1)dxy,
xdy = dyx,
ydy = qdyy,
in the second one. In both cases (Aq [x−1 ] ⊗ ker )/χ(N ) = Aq [x−1 ] ⊗ 3, where 3 is a one-dimensional vector space spanned by λ = πM (1 ⊗ c) and can be therefore identified with a subspace of k[c] spanned by c. Also in both cases the most general connection is given by 2|0
Π(dx) = 0,
2|0
Π(dy) = dy + αdx,
where α ∈ k, and extended to the whole of 1 (Aq [x−1 ]) as a left Aq [x−1 ]-module map. The corresponding connection one form reads 2|0
2|0
ω(λ) = dy + αdx. The bundle is trivial and this connection can be described by the map β : k[c] → (k[x, x−1 ]) as in Proposition 3.9 (cf. Eq. (31)) with β(cn ) = 0 if n 6= 1 and β(c) = αdx. References [BCGST96] Bonechi, F., Ciccoli, N., Giachetti, R., Sorace, E. and Tarlini, M.: Free q-Schr¨odinger equation from homogeneous spaces of the 2-dim Euclidean quantum group. Commun. Math. Phys. 175, 161–176 (1996) [BDR92] Brzezi`nski, T., Da¸browski, H. and Rembieli´nski, J.: On the quantum differential calculus and the quantum holomorphicity. J. Math. Phys. 33, 19–24 (1992) [BM93] Brzezi´nski, T. and Majid, S.: Quantum group gauge theory on quantum spaces. Commun. Math. Phys. 157, 591–638 (1993) Erratum 167, 235 (1995) [Brz96a] Brzezi´nski, T.: Translation map in quantum principal bundles. J. Geom. Phys. 20, 349–370 (1996) [Brz96b] Brzezi´nski, T.: Quantum homogeneous spaces as quantum quotient spaces. J. Math. Phys. 37, 2388–2399 (1996)
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[CKM94] [Con85] [DK94] [Kar87] [Maj90] [Maj91] [Maj93a]
[Maj93b] [Maj94a] [Maj94b] [Maj95] [Pod87] [Sch92] [Wor89]
T. Brzezi´nski, S. Majid
Cap, A., Karoubi, M. and Michor, P.: The Chern-Weil homomorphism in non commutative differential geometry. Preprint 1994 Connes, A.: Noncommutative differential geometry. Publ. IHEs 62, 257–360 (1985) Dijkhuizen, M.S. and Koornwinder, T.: Quantum homogeneous spaces, duality and quantum 2-spheres. Geom. Dedicata 52, 291–315 (1994) Karoubi, M.: Homologie cyclique et K-th´eorie. Ast´erisque, 149 (1987) Majid, S.: Physics for algebraists: Non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction. J. Algebra 130, 17–64 (1990) /From PhD Thesis, Harvard, 1988) Majid, S.: Braided groups and algebraic quantum field theories. Lett. Math. Phys. 22, 167–176 (1991) Majid, S.: Beyond supersymmetry and quantum symmetry (an introduction to braided groups and braided matrices). In: M-L. Ge and H.J. de Vega, editors, Quantum Groups, Integrable Statistical Models and Knot Theory, Singapore: World Sci., 1993. pp. 231–282 Majid, S.: Braided groups. J. Pure and Applied Algebra 86, 187–221 (1993) Majid, S.: Quantum and braided Lie algebras. J. Geom. Phys. 13, 307–356 (1994) Majid, S.: Some remarks on the quantum double. Czech. J. Phys. 44, 1059–1071 (1994) Majid, S. Foundations of Quantum Group Theory. Cambridge: Cambridge University Press, 1995 Podle´s, P.: Quantum spheres. Lett. Math. Phys. 14, 193–202 (1987) Schneider, H.-J.: Normal basis and transitivity of crossed products for Hopf algebras. J. Algebra 152, 289–312 (1992) Woronowicz, S.L.: Differential calculus on compact matrix pseudogroups (quantum groups). Commun. Math. Phys., 122, 125–170 (1989)
Communicated by A. Connes
Commun. Math. Phys. 191, 493 – 500 (1998)
Communications in
Mathematical Physics c Springer-Verlag 1998
Decoherence Functionals for von Neumann Quantum Histories: Boundedness and Countable Additivity J. D. Maitland Wright Analysis and Combinatorics Research Centre, Mathematics Department PO Box 220, University of Reading, Whiteknights, Reading, RG6 6AX, United Kingdom Received: 6 December 1996 / Accepted: 18 May 1997
Abstract: Gell–Mann and Hartle have proposed a significant generalisation of quantum theory in which decoherence functionals perform a key role. Verifying a conjecture of Isham–Linden–Schreckenberg, the author analysed the structure of bounded, finitely additive, decoherence functionals for a general von Neumann algebra A (where A has no Type I2 direct summand). Isham et al. had already given a penetrating analysis for the situation where A is finite dimensional. The assumption of countable additivity for a decoherence functional may seem more plausible, physically, than that of boundedness. The results of this note are obtained much more generally but, when specialised to L(H), the algebra of all bounded linear operators on a separable Hilbert space H, give: Let d be a countably additive decoherence functional defined on all pairs of projections in L(H). If H is infinite dimensional then d must be bounded. By contrast, when H is finite dimensional, unbounded (countably additive) decoherence functionals always exist for L(H). 1. Introduction As in Wright [15], one of the key tools applied here is the solution of the Mackey–Gleason Problem, see [1–3] and the references given there, particularly [4,10]. In addition, deep results of Dorofeev and Shertsnev are needed [5–7]. Isham, Linden and Schreckenberg [11] determined the structure of bounded (or, equivalently, continuous) decoherence functionals (see below for definitions) on L(H), where H is a finite dimensional Hilbert space of dimension greater than two. They point out that Gell–Mann and Hartle have proposed a significant generalisation of quantum theory with a scheme whose basic ingredients are “histories” and decoherence functionals. For an excellent account of the physical significance of decoherence functionals – and much more – the reader should consult [11–13] and the references given there. The key importance of boundedness for decoherence functionals was observed by Isham et al. [11] in their penetrating analysis of the situation in finite dimensions. In
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[15] the author extended their analysis to infinite dimensions, showing, in particular, that whenever d is a bounded, finitely additive, decoherence functional associated with a von Neumann algebra A (where A has no direct summand of Type I2 ) then d can be represented as the difference of two norm-continuous semi-innerproducts on A. It could be argued that the assumption of boundedness is not immediately plausible physically. However, as remarked by Isham et al. in [11], it is physically reasonable to suppose that d is countably additive (see below for precise definitions). Our first aim here is to clarify the relationship between countable additivity and boundedness for decoherence functionals. It is shown in Theorem 3.1 that whenever A is a von Neumann algebra which is not commutative there exists a decoherence functional d, taking finite values on each pair of projections from A, such that d is unbounded. When A is finite dimensional, each family of non-zero, pairwise orthogonal projections is finite. Thus, when A is finite dimensional it is always possible to associate a “countably additive” unbounded decoherence functional with it. It follows from the work of Isham et al. [11] that such a d fails to be continuous. By contrast, let A be a von Neumann algebra which has no direct summand of Type In (where 2 ≤ n < ∞) and let d be a countably additive decoherence functional defined on all pairs of projections from A. Further assume that A can be embedded as a (weakly closed) subalgebra of L(H) where H is separable. Then we shall see in Theorem 4.1 that d is necessarily bounded. The hypothesis that A “lives” on a separable space is physically reasonable and simplifies some of the arguments. However this hypothesis can be dispensed with (at the price of some dull complications) if instead of requiring d to be countably additive it is required to be completely additive.
2. Decoherence Functionals In all that follows A shall be a von Neumann algebra and P (A) the lattice of projections in A. Definition. A decoherence functional associated with A is a function d : P (A) × P (A) → C with the following properties: 1. Hermiticity. For each p and q in P (A) d(p, q) = d(q, p)∗ (Here * denotes complex conjugation.) 2. Additivity. Whenever p1 is perpendicular to p2 and q is an arbitrary projection d(p1 + p2 , q) = d(p1 , q) + d(p2 , q) 3. Positivity d(p, p) ≥ 0 for each p in P (A).
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4. Normalisation. d(1, 1) = 1. If d satisfies only the properties of Hermiticity and Additivity then d is said to be an hermitian quantum bimeasure. 2# . Countable Additivity. A decoherence functional (or hermitian quantum bimeasure) is said to be countably additive if, whenever {pi : i = 1, 2, . . .} is a countable collection of pairwise orthogonal projections, then, for each q in P (A), X X d(pi , q). d( pi , q) = Here the series on the right hand side is rearrangement invariant and hence is absolutely convergent. ## 2 . Complete Additivity. A decoherence functional (or hermitian quantum bimeasure) is said to be completely additive if, whenever {pi : i ∈ I} is an infinite collection of pairwise orthogonal projections, X X d(pi , q) d( pi , q) = for each q in P (A). Here all but countably many of the terms d(pi , q) are zero and the convergence is absolute. 3. Unbounded Decoherence Functionals In this section we shall see that for “almost every” von Neumann algebra A, it is possible to produce an unbounded decoherence functional d which takes only finite values on each pair of projections from A. Theorem 3.1. Let A be a von Neumann algebra which is either not commutative or which has an infinite set of non-zero pairwise orthogonal projections. Then there exists a finite valued decoherence functional d : P (A) × P (A) → C such that d is unbounded. Proof. Let us consider the self-adjoint part of A, Asa , as a vector space over Q, where Q is the field of rational numbers. Let P0 = I and let us assume, for the moment, that there exists a countably infinite family of projections {Pn : n = 0, 1, 2, . . .} which is linearly independent over Q, that is, each non-empty finite subset is linearly independent over Q. Using Zorn’s Lemma, we can extend this countable set to a maximal set {Xλ : λ ∈ 3} of self adjoint elements of A which are linearly independent over Q (i.e. a Hamel base for Asa over Q). Here we assume that ω ⊂ 3 and that Xn = Pn for each n ∈ ω. Then every self-adjoint element S in A has a unique representation X S= sλ Xλ , where each sλ is rational and all but finitely many of the sλ are zero.
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P It follows that each element of A has a unique representation of the form zλ Xλ where, for each λ, zλ is complex rational and all but finitely many of the zλ are zero. Let us define a “pseudo-innerproduct” on A by defining X X X wλ Xλ i to be zλ wλ∗ . h zλ Xλ , Let us also define a map T from A to A, where T is linear when A is regarded as a vector space over the complex rationals, by setting T Pn = (n + 1)Pn for each n ∈ w, and for all other λ, T Xλ = 0 and extending T in the obvious way. Let D(z, w) = hT z, T wi for each z and w in A. Then D is additive in both variables and D(z, w) = D(w, z)∗ . Let P x be an arbitrary element of A. Then x has a unique representation of the form each λ, zλ is complex rational and all but finitely many of the zλ zλ Xλ , where, for P are zero. Then T x = zn (n + 1)Pn . So X D(x, x) = |zn |2 (n + 1)2 ≥ 0. On putting d(p, q) = D(p, q) for each p and q in P (A) we see that d is a quantum bimeasure where d(p, p) ≥ 0 for every projection p. Also d(Pn , Pn ) = hT Pn , T Pn i = (n + 1)2 . Since P0 = I, it follows that d is a decoherence functional. Clearly d is unbounded. To complete this argument we must establish the initial assumption. If A has a countably infinite sequence of non-zero orthogonal projections (Pn )(n = 1, 2, . . .) then {I, P1 , P2 , . . .} is linearly independent over the real numbers and hence also over Q. Otherwise, we may suppose that A is not commutative and hence contains a pair of noncommuting projections. Thus, see Takesaki [p. 306, 14], there is a projection e in A such that M2 (C) is a subalgebra of eAe. It now suffices to establish our initial assumption for M2 (C). Let √ t t(1 − t) . P (t) = √ t(1 − t) 1 − t Then P (t) is a projection for 0 ≤ t ≤ 1. Let θ be a transcendental number in the interval (0, 1), e.g. e−1 . Then {θn : n = 1, 2, · · ·} is linearly independent over Q for, otherwise, q would satisfy an algebraic equation with non-zero rational coefficients. Let P0 = I and let Pn = P (θn ) for n = 1, 2, · · · . Clearly these projections are linearly independent over Q. The proof is now complete. Corollary 3.2. Let V be a finite dimensional Hilbert space of dimension two or greater. Then there exists a countably additive, finite valued decoherence functional d associated with L(V ) where d is unbounded. Proof. Let d be obtained as in the theorem above. Then, since each family of non-zero pairwise orthogonal projections is finite, d is countably additive.
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4. Countably Additive Decoherence Functionals Unless stated otherwise we shall assume that the von Neumann algebra A has a faithful normal representation on a separable Hilbert space. Thus each family of non-zero pairwise orthogonal projections is countable. Hence each countably additive decoherence functional on A is, in fact, completely additive. Theorem 4.1. Let A be a von Neumann algebra with no Type In direct summand (2 ≤ n < ∞). Let A have a faithful normal representation on a separable Hilbert space. Let d be a countably additive decoherence functional associated with A (or, more generally, a countably additive hermitian quantum bimeasure). Then d is bounded. Furthermore, there exists D : A × A → C such that D(p, q) = d(p, q) for each p and q in P (A), where D is a bounded sesquilinear hermitian form and for each y, x → D(x, y) is a normal functional on A. Proof. For each fixed p in P (A), the map q → d(p, q) is a countably additive (and so, as remarked above, completely additive) quantum measure on the projections. By hypothesis, A = A1 ⊕ A2 , where A1 is abelian and where A2 has no direct summand of Type In for n < ∞. By classical measure theory, see Lemma III. 4.4 [p. 127, 8] the measure is bounded when restricted to the projections in A1 , indeed on any maximal abelian *-subalgebra. By deep results of Dorofeev and of Dorofeev and Shertsnev, see Theorem 2 [5] and [6,7], see also Theorem 3.2.20 [9], the measure is bounded on the projections of A2 . Thus this measure is bounded on P (A). Hence, by the Generalised Gleason Theorem for complex valued quantum measures, see [1–3], there exists a bounded linear functional φp such that φp (q) = d(p, q) for all q in P (A). Moreover φp is completely additive on orthogonal projections because of the hypotheses on d. Hence, see Corollary III.3.11 [14], φp is a normal functional on A. Thus φp may be identified with an element of A∗ , the predual of A. For each p ∈ P (A), let M (p) = φp . Whenever p1 and p2 are perpendicular we have M (p1 + p2 )(q) = d(p1 + p2 , q) = d(p1 , q) + d(p2 , q) = M (p1 )(q) + M (p2 )(q). Since finite linear combinations of projections are norm dense in A, M (p1 + p2 ) = M (p1 ) + M (p2 ) In order to be able to apply the vector valued Generalised Gleason Theorem we must show that M maps P (A) into a norm bounded subset of A∗ . By the Uniform Boundedness Theorem it suffices to show that, for each x in A, {φp (x) : p ∈ P (A)} is a bounded set. Clearly it is enough to establish the boundedness of this set when x = x∗ and 0 ≤ x ≤ I. Let us fix such an x and then let B be the von Neumann subalgebra of A generated by x and I. Then B is abelian and generated by its projections and may be identified with L∞ (µ) for an appropriate probability measure µ.
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Since x=
∞ X
2−n qn ,
1
where each qn is a projection in B, it suffices to show that 8 = {φp : p ∈ P (A)} is uniformly bounded on the projections of B. Suppose that this is false. Then, for each n, we can find fn ∈ 8 and en ∈ P (B) such that |fn (en )| ≥ n2 . Let ψn = n−1 fn so that |ψn (en )| ≥ n for each n. For each fixed q in P (B), |ψn (q)| = n−1 |fn (q)| = n−1 |φp(n) (q)| for some p(n) in P (A). Since, for each p in P (A), |φp (q)| = |d(p, q)| = |d(q, p)| = |φq (p)| ≤ kφq k, we have |ψn (q)| = n−1 |φq (p(n))| ≤ n−1 kφq k. So, on letting n → ∞, lim ψn (q) = 0 for each q. For each q in P (B) let F (q) = (ψr (q))(r = 1, 2 · · ·) and let Fn (q) = (ψ1 (q), · · · ψn (q), o, · · ·). Then F and Fn take their values in c0 and kF (q)−Fn (q)k → 0. On restricting ψj to B = L∞ (µ) = L∞ (X, B, µ), say, we obtain a countably additive measure on (X, B) which vanishes on µ-null sets and hence, by Lemma III.4.13 [8], is µ-continuous. It follows that each Fn corresponds to a µ-continuous (vector valued) measure on (X, B). So, by the Vitali–Hahn–Saks Theorem, see Corollary III.7.3 [8], F can be identified with a countably additive vector measure. Hence, by Corollary III.4.5 [8], its range is a bounded subset of c0 . But |ψn (en )| ≥ n for each n. This is a contradiction. So {φp : p ∈ P (A)} is uniformly bounded on P (B). Hence {φp (x) : p ∈ P (A)} is bounded. Thus M maps P (A) into a bounded subset of A∗ . Since A has no Type I2 direct summand we now apply the vectorial form of the Generalised Gleason Theorem [1,2] to deduce that M extends to a bounded linear operator (also denoted by “M ”) from A into A∗ . Following [15], let us define D : A × A → C by D(x, y) = M (x)(y ∗ ). Then D is a bounded sesquilinear form on A × A. Since the map (x, y) → M (x)(y) is a bilinear extension of d, it follows from Theorem 3 [15] that it is unique and that M (x)(y ∗ ) = M (y)(x∗ )∗ . Hence D is hermitian. We now fix y and set φ(x) = D(x, y) = D(y, x)∗ . Then φ(x∗ ) = M (y)(x)∗ . Since M (y) is normal so, also, is φ.
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5. Countably Additive and Bounded Decoherence Functionals Throughout this section the von Neumann algebra A will be assumed to be embedded as a (weakly closed) subalgebra of L(K), where K is a separable Hilbert space. Furthermore, we shall also assume that A has no Type I2 direct summand. Let d : P (A) × P (A) → C be a countably additive decoherence functional which is also bounded. It follows from Theorem 4.1 that if A has no Type In direct summand ( for 2 ≤ n < ∞) then the countable additivity of d automatically implies boundedness. However, as we have also seen, boundedness is not a consequence of countable additivity for Type In algebras (2 ≤ n < ∞). Lemma 5.1. There exists a unique bounded sesquilinear form D : A × A → C such that D(p, q) = d(p, q) for each p and q in P (A). Furthermore D is hermitian and the map x → D(x, y) is normal for each y in A. Proof. Since d is bounded, Corollary 4 [15] and its proof gives the existence and hermitian property of D. The uniqueness of D follows from its norm continuity and the observation that finite linear combinations of projections are dense in A. The normality of the map x → D(x, y) can be established by arguing as in the proof of Theorem 4.1. Remark. The Haagerup–Pisier–Grothendieck inequality easily implies that D is jointly strong continuous on the unit ball of A. See, for example, Lemma 8 [16]. Theorem 5.2. Let d : P (A)×P (A) → C be a bounded, countably additive decoherence functional. Let A have no direct summand of Type I2 . Then d can be extended to a unique normal Hermitian form D on A. There exist normal functionals (φn )(n = 1, 2, . . .) and (ψn )(n = 1, 2, . . .) such that, for each x and y in A, D(x, y) =
∞ X
φn (x)φn (y)∗ −
n=1
where, for same constant K, both k 2 ||x||2 .
∞ X
ψn (x)ψn (y)∗ ,
n=1
P∞
n=1
|φn (x)|2 and
P∞
n=1
|ψn (x)|2 are bounded by
Proof. The existence and uniqueness of a normal hermitian form D which extends d follows from Lemma 5.1. The existence of (φn )(n = 1, 2, . . .) and (ψn )(n = 1, 2, . . .) with the required properties then follows from Corollary 11 [16]. See, also, the elegant representation theorem of Ylinen [17]. Corollary 5.3. Let d be a countably additive decoherence functional on A, where A has no Type I2 direct summand. Then there exist semi-innerproducts h , i1 and h , i2 on A which are jointly strong* continuous on the unit ball of A and such that d(p, q) = hp, qi1 − hp, qi2 for each p and each q in P (A). P∞ P∞ Proof. Let hx, yi1 = n=1 φn (x)φn (y)∗ and let hx, yi2 = n=1 ψn (x)ψn (y)∗ .
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References 1. Bunce, L.J. and Wright, J.D.M.: The Mackey–Gleason Problem. Bull. Am. Math. Soc. 26, 288–293 (1992) 2. Bunce, L.J. and Wright, J.D.M.: The Mackey–Gleason Problem for vector measures on projections in von Neumann algebras. J. London Math. Soc. 49, 133–149 (1994) 3. Bunce, L.J. and Wright, J.D.M.: Complex measures on projections in von Neumann algebras. J. London Math. Soc. 46, 269–279 (1992) 4. Christensen, E.: Measures on projections and physical states. Commun. Math. Phys. 86, 529–538 (1982) 5. Dorofeev, S.: On the problem of boundedness of a signed measure on projections of a von Neumann algebra. J. Funct. Anal. 103, 209–216 (1992) 6. Dorofeev, S.: On the problem of boundedness of a signed measure on projections of a Type I von Neumann algebra. Proceedings of Higher Educational Institutions, Mathematics 3, 67–69, 1990 7. Dorofeev, S. and Shertsnev, A.N.: Frame type functions and their applications. Proceedings of Higher Educational Institutions, Mathematics 4, 23–29, 1990 8. Dunford, N., and Schwartz, J.T.: Linear operators I. New York: Interscience, 1967 9. Dvurechenskij, A.: Gleason’s theorem and its applications. London: Kluwer, 1993 10. Gleason, A.M.: Measures on the closed subspaces of a Hilbert space. J. Math. Mech. 6, 885–893 (1957) 11. Isham, C.J., Linden, N., and Schreckenberg, S.: The classification of decoherence functionals: An analogue of Gleason’s theorem. J. Math. Phys. 35, 6360–6370 (1994) 12. Isham, C.J., Linden, N.: Quantum temporal logic and decoherence functionals in the histories approach to generalised quantum theory. J. Math. Phys. 35, 5452–5476 (1994) 13. Isham, C.J.: Quantum logic and the histories approach to quantum theory. J. Math. Phys. 35, 2157–2185 (1994) 14. Takesaki, M.: Theory of operator algebras. New York: Springer 1979 15. Wright, J.D.M.: The structure of decoherence functionals for von Neumann quantum histories. J. Math. Phys. 36, 5409–5413 (1995) 16. Wright, J.D.M.: Linear representations of bilinear forms on operator algebras. Expos. Math. (to appear) 17. Ylinen, K.: The structure of bounded bilinear forms on products of C∗ -algebras. Proc. Am. Math. Soc. 102, 599–601 (1988) Communicated by A. Araki
Commun. Math. Phys. 191, 501 – 541 (1998)
Communications in
Mathematical Physics c Springer-Verlag 1998
Algebra of Screening Operators for the Deformed Wn Algebra Boris Feigin1 , Michio Jimbo2 , Tetsuji Miwa3,? , Alexandr Odesskii1 , Yaroslav Pugai1 1 2 3
L. D. Landau Institute for Theoretical Sciences, Chernogolovka, 142432, Russia Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606, Japan Institut Henri Poincar´e and Ecole Normale Superieure, France
Received: 3 March 1997 / Accepted: 20 May 1997
Abstract: We construct a family of intertwining operators (screening operators) between various Fock space modules over the deformed Wn algebra. They are given as integrals involving a product of screening currents and elliptic theta functions. We derive a set of quadratic relations among the screening operators, and use them to construct a Felder-type complex in the case of the deformed W3 algebra.
1. Introduction The method of bosonization is known to be the most effective way of calculating the conformal blocks in conformal field theory. The basic idea in this approach is to realize the commutation relations for the symmetry algebra (such as the Virasoro or affine Lie algebras) and the chiral primary fields in terms of operators acting on some bosonic Fock spaces. Quite often, the physical Hilbert space of the theory is not the total Fock space itself, but only a subquotient of it. In this case, it is necessary to ‘project out’ the physical space from the Fock spaces by a cohomological method. In the case of the Virasoro minimal models, Felder [2] introduced a two-sided complex d
d
d
d
d
· · · −→F (−1) −→F (0) −→F (1) −→F (2) −→ · · · ,
(1.1)
consisting of Fock spaces F (i) . As it turns out, the cohomology of this complex vanishes except at the 0th degree, and the remaining non-trivial cohomology affords the irreducible representation of the Virasoro algebra. The primary fields realized on the Fock spaces commute with the coboundary operator d, and hence make sense as operators on the cohomology space. Similar resolutions have been described for representations of affine Lie algebras [3–5]. For an extensive review on this subject, the reader is referred to [6]. ?
On leave from Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, Japan.
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It has been recognized that the idea of bosonization is quite fruitful also in non-critical lattice models [7, 8] and massive field theory [9]. The present work is motivated by recent progress along this line, on the restricted solid-on-solid models [10, 11]. In [10], the Andrews-Baxter-Forrester (ABF) model was studied. Here the counterpart of the conformal primary fields are the vertex operators (half transfer matrices) which appear in the corner transfer matrix method. Just as in conformal field theory, the bosonization discussed in [10] consists of two steps. The first step is to introduce a family of bosonic Fock spaces and realize the commutation relations of the vertex operators in terms of bosons. The second step is to realize the physical space of states of the model as the 0th cohomology of a complex of the type (1.1). In fact, the analogy with conformal field theory goes further. Each Fock space has the structure of a module over the deformed Virasoro algebra (DVA) discovered in [12], where the deformation parameter x (0 < x < 1) is the one which enters the Boltzmann weights of the models. As was shown in [10, 13], the above complex is actually that of DVA modules, i.e., the operator d commutes with the action of DVA. Felder’s complex (1.1) is recovered in the limit x → 1 (we shall refer to this as the conformal limit). The ABF models have sln generalizations, the former being the case n = 2. In [11], the first step of the bosonization was carried over to the case of general n. However the second step was not addressed there. The aim of the present paper is to construct an analog of the complex (1.1) in the case n = 3. In this situation the role of DVA is played by the deformed W3 algebra introduced in [14, 15]. We shall also construct for general n a family of intertwiners of deformed Wn algebras (DWA), which we expect to be sufficient to construct the complex in the general case. b 3 was constructed in [5]. In the conformal case, such a Felder-type complex for sl Strictly speaking, [5] discusses representations of affine Lie algebras, while our case corresponds (in the limit) to those of Wn algebras. In other words we are dealing with a coset theory rather than a Wess-Zumino-Witten theory. However the construction of the complex is practically the same for both cases. In the case n ≥ 3, each component F (i) of the complex is itself a direct sum of an infinite number of Fock spaces. The coboundary operator d can be viewed as a collection of maps between various Fock spaces. We call these maps the screening operators. They are given in the form of an integral of a product of screening currents, multiplied by a certain kernel function expressed in terms of elliptic theta functions. The main result of this paper is the explicit construction of these screening operators. In comparison with the conformal case, a simplifying feature is that the screening operators can be expressed as products of more basic, mutually commuting operators. In the conformal case, such a multiplicative structure exists only “inside the contour integral” (see [5] and Sect. 6 below). It has been pointed out [15] that the screening currents satisfy the commutation relations of the elliptic algebra studied in [16]. This connection turns out to be quite helpful in finding the basic operators referred to above and their commutation relations. Let us mention some questions that remain open. In order to ensure the nilpotency property d2 = 0, the signs of the screening operators have to be chosen carefully. We have verified that this is possible for n = 3. In the general case there are additional complications which we have not settled yet. More importantly, in this paper we do not discuss the cohomology of the complex, though we expect the same result persists as in the conformal case. The construction of the complex in [5] is based on the one-to-one b n , and the singular correspondence between intertwiners of Fock space modules over sl vectors in the Verma modules of Uq (sln ) with q a root of unity. It would be interesting to search for an analog of the latter in the deformed situation.
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The outline of this paper is as follows. In Sect. 2 we prepare the notation and the setting. Also the form of the complex to be constructed is briefly explained. The construction of the screening operators for general n is rather technical. To ease the reading, we first discuss in Sect. 3 the case n = 3 in detail. In Sect. 4 we introduce the screening operators in general, and state their commutation relations. In Sect. 5 we show that they commute with the action of DWA. In Sect. 6 we briefly discuss the CFT limit of the basic operators. The text is followed by 3 appendices. In Appendix A we discuss the condition when we construct intertwiners between two Fock modules. In Appendix B we list the commutation relations for the basic operators that will be used to derive the quadratic relations of the screening operators. In Appendix C we outline the proof that the screening operators commute with DWA.
2. Preliminaries In this section we prepare the notation to be used in the text. Throughout this paper, we fix a positive integer r ≥ n + 2 and a real number x with 0 < x < 1. 2.1. Lie algebra sln . Let us fix the notation concerning the Lie algebra sln . Let εi (1 ≤ i ≤ n) be an orthonormal in Rn relative to the standard inner Pbasis n product ( , ). We set ε¯i = εi − ε,ε = (1/n) j=1 εj . We shall denote by: Pi • ωi = j=1 ε¯j the fundamental weights, • αi = εi − εi+1 the simple roots, Pn−1 • θ = i=1 αi the maximal root, Pn • P = i=1 Zε¯i the weight lattice, Pn−1 • Q = i=1 Zαi the root lattice, • 1+ = {εi − εj | 1 ≤ i < j ≤ n − 1} the set of positive roots, • W ' Sn the classical Weyl group, • W ' W |× Q the affine Weyl group. For an element γ ∈ Q, we set |γ| =
n−1 X i=1
ci
for γ =
n−1 X
ci αi .
i=1
For a root α = εi − εj , rα signifies the corresponding reflection (often identified with the transposition (ij) ∈ Sn ). We also write si = rαi . j 2.2. Bosons. We recall from [11] our convention about the bosons. Let βm be the oscillators (1 ≤ j ≤ n − 1, m ∈ Z\{0}) with the commutation relations
[(n − 1)m]x [(r − 1)m]x δm+m0 ,0 (j = k), [nm]x [rm]x [m]x [(r − 1)m]x δm+m0 ,0 (j 6= k). = −mxsgn(j−k)nm [nm]x [rm]x
j k , βm [βm 0] = m
(2.1)
(2.2)
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n Here the symbol [a]x stands for (xa − x−a )/(x − x−1 ). Define βm by n X
j x−2jm βm = 0.
(2.3)
j=1
Then the commutation relations (2.1),(2.2) are valid for all 1 ≤ j, k ≤ n. We also introduce the zero mode operators Pλ , Qλ indexed by λ ∈ P . By definition they are Z-linear in λ and satisfy [iPλ , Qµ ] = (λ, µ)
(λ, µ ∈ P ).
j We shall deal with the bosonic Fock spaces Fl,k (l, k ∈ P ) generated by β−m (m > 0) over the vacuum vectors |l, ki: j j , β−2 , · · ·}1≤j≤n ]|l, ki, Fl,k = C[{β−1
where j |l, ki = 0, βm
(m > 0), r r r r−1 l− k)|l, ki, Pα |l, ki = (α, r−1 r √ r−1 √ r |l, ki = ei r−1 Ql −i r Qk |0, 0i. In what follows we set πˆ i =
p r(r − 1)Pαi .
It acts on Fl,k as an integer, πˆ i |Fl,k = (αi , rl − (r − 1)k). def
In this paper we work on Fλ = Fl,λ (λ ∈ P ) with a fixed value of l ∈ P . 2.3. Screening currents. We define the screening currents ξj (u) (j = 1, · · · , n − 1) by P √ r−1 1 r−1 1 (β j −β j+1 )(xj z)−m ξj (u) ≡ Uj (z) = ei r Qαj z r πˆ j + r : e m6=0 m m m :, (2.4) where the variable u is related to z via z = x2u . We shall need the following commutation relations between them. [u − v − 1] ξj (v)ξj (u), [u − v + 1] [u − v + 21 ] ξj±1 (v)ξj (u), ξj (u)ξj±1 (v) = − [u − v − 21 ] ξi (u)ξj (v) = ξj (v)ξi (u) (|i − j| > 1). ξj (u)ξj (v) =
Here the symbol [u] stands for the theta function satisfying [u + r] = −[u] = [−u], τ
[u + τ ] = −e2πi(u+ 2 )/r [u] where τ =
πi log x .
(2.5) (2.6) (2.7)
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505
Explicitly it is given by 2
[u] = xu /r−u 2x2r (x2u ), 2q (z) = (z; q)∞ (qz −1 ; q)∞ (q; q)∞ , ∞ Y (1 − zq i ). (z; q)∞ =
(2.8) (2.9) (2.10)
i=0
Quite generally we say that an operator X has weight ν if XFλ ⊂ Fλ+ν for any λ. Then ξj (u) has weight −αj . This implies (2.11) πˆ i ξj (u) = ξj (u) πˆ i − (αi , αj )(1 − r) . 2.4. The complex. In this section we describe the form of the complex we are going to construct. Fix an integral weight 3 ∈ P satisfying (3, αi ) > 0
(i = 1, · · · , n − 1),
(3, θ) < r.
(2.12)
Note that 0 < (3, α) < r
(2.13)
for any positive root α. Consider the orbit of 3 under the action of the affine Weyl group W . An element of W 3 can be written uniquely as λ = tγ σ3 = σ3 + rγ,
(2.14)
where σ ∈ W and γ ∈ Q. We assign a degree deg(λ) ∈ Z to (2.14) by setting deg(λ) = l(σ) − 2|γ|.
(2.15)
Here l(σ) denotes the length of σ ∈ W . (The right hand side of (2.15) is known as the modified length of w = tγ σ ∈ W , see e.g. [4, 5].) We shall construct a complex of the form d d d d d (2.16) · · · −→F3(−1) −→F3(0) −→F3(1) −→F3(2) −→ · · · , where
F3(i) =
M
Fλ
(i ∈ Z).
(2.17)
λ∈W 3 deg(λ)=i
Except for n = 2, F3(i) is a direct sum of an infinite number of Fock spaces. The coboundary map d : F3(i) → F3(i+1) can be viewed as a collection of operators dλ0 ,λ : Fλ(i) → Fλ(i+1) 0
(2.18)
associated with each pair λ, λ0 ∈ W 3 satisfying deg(λ0 ) = deg(λ) + 1. We shall impose a restriction on the possible pair λ, λ0 as explained below. For a positive root α and an element λ ∈ W 3, we define an integer mα (λ) by 0 < mα (λ) < r, mα (λ) ≡ (λ, α) mod r.
(2.19) (2.20)
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B. Feigin, M. Jimbo, T. Miwa, A. Odesskii, Ya. Pugai
In other words, if λ = tγ σ3, then we have (σ3, α) mα (λ) = (σ3, α) + r
if (σ3, α) > 0; if (σ3, α) < 0.
(2.21)
Set λ = λ − mα (λ)α = α
tγ rα σ3 tγ−α rα σ3
if (σ3, α) > 0; if (σ3, α) < 0.
(2.22)
Clearly the weight λα also belongs to the orbit W 3. Definition 2.1. We say that an ordered pair (λ, λ0 ) is admissible if the following hold for some positive root α. λ 0 = λα , deg(λα ) = deg(λ) + 1.
(2.23) (2.24)
We set dλ0 ,λ = 0 if λ, λ0 is not admissible. Otherwise, write λ0 = λα and dλα ,λ = Xα (λ) : Fλ −→ Fλα .
(2.25)
The construction of the complex is equivalent to finding an operator (2.25) for each admissible pair λ, λα , so that we have d2 = 0. We shall refer to (2.25) as a screening operator. We also require that the screening operators commute with the DWA generators. In practice, we find it convenient to construct the screening operators in the form Xα (λ) = sα (λ)X α (λ), where sα (λ) = ±1 is a sign factor. In Section 4 we give both sα (λ) and X α (λ) so that we have d2 = 0 for the case n = 3. The general case is incomplete because we could not find a proper choice of the signs sα (λ). The construction of the screening operators (2.25) is based on the screening currents (2.4). Let us consider the case where α in (2.25) is a simple root αj . It turns out that the pair λ, λαj is admissible for any λ ∈ W 3 (see Lemma A.3). In this case the operator (2.25) can be found as follows: X αj (λ) = Xja (a = mαj (λ)), I [u + 21 − πˆ j ] dz ξj (u) Xj = (z = x2u ). 2πiz [u − 21 ]
(2.26) (2.27)
Here the integration is taken over the contour |z| = 1. Notice that the kernel function F (u) = [u + 21 − πˆ j ]/[u − 21 ] has the quasi-periodicity F (u + τ ) = e2πi(1−πˆ j )/r F (u),
(2.28)
which ensures that the integrand of (2.27) is a single valued function in z. For n = 2, (2.26) exhausts the possible screening operators. For n ≥ 3 we must also construct operators corresponding to non-simple roots. As we shall see, they are given by similar (but more complicated) integrals over products of the screening currents.
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3. Case n = 3 Before embarking upon the construction of the complex in general, let us first elaborate on the case n = 3. Hopefully this will make clear the main points of the construction. The following figure shows the configuration of the weights in the orbit W 3 for n = 3.
A
C
B
Fig. 1. The orbit W 3 for n = 3. It forms a hexagonal lattice, consisting of three types of basic hexagons A, B, C and their translates by r times the root lattice Q
In the figure, each vertex represents a weight λ ∈ W 3. An arrow from λ to λ0 indicates that the pair (λ, λ0 ) is admissible. As was mentioned before, (λ, λα ) is always admissible for a simple root α = α1 , α2 . For n = 3 there is also the ‘third root’ θ = α1 +α2 . It turns out that (λ, λθ ) is admissible if and only if λ = tγ σ3
with σ = s1 , s2 , s1 s2 s1 .
The nilpotency d2 = 0 leads to two types of relations for the screening operators. The first type involves only screening operators corresponding to one simple root αj , and has the form Xja Xjr−a = 0. For n = 2, this relation was proved in [13]. The same argument applies to show Xjr = 0 for any j. The second type involves the root θ = α1 + α2 , and occurs for each square inside a (a) (a = mθ (λ)), indicating hexagon (see Fig.1). Let us write the operator X θ (λ) as X12 that it has weight −aθ. Let ai = (3, αi ) (i = 1, 2), a0 = r − a1 − a2 , and suppose λ = tγ σ3. If σ = s1 , then mθ (λ) = a2 , and the following relations must be satisfied: (a2 ) a0 X2r−a1 X1a2 ± X12 X2 = 0,
X2a2 X1r−a1 X1r−a0 X2a2 X1a2 X2r−a0
± ± ±
(a2 ) X1a0 X12 (a2 ) a1 X12 X1 a1 (a2 ) X2 X12
(3.1)
= 0,
(3.2)
= 0,
(3.3)
= 0.
(3.4)
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(a) The operator X12 and the signs ± in the commutation relations will be given later. The relations in the other cases σ = s2 , s1 s2 s1 are obtained by permuting the upper indices (a0 , a1 , a2 ) → (a2 , a0 , a1 ) successively. In the conformal case x = 1 (and n = 3), the ‘third’ screening operator Xθ (λ) satisfying the relations (3.1)-(3.4) was found in [5]. The construction in [5] is based on the observation that the screening operators Xj for the simple roots satisfy the same Serre relations as do the Chevalley generators of the quantum group Uq (sln ), with q being a root of unity (q 2r = 1 in the present notation). With the aid of these relations, the operator Xθ (λ) was expressed as a (non-commutative) polynomial in X1 and X2 . This method does not easily generalize to the deformed case, since it appears that there is no analog of the Serre relations between X1 and X2 . Nevertheless there exists a family of operators, in terms of which the third screening operator can be written in a simple factorized form. Consider an operator of weight −θ of the form II dz1 dz2 ξ1 (u1 )ξ2 (u2 )F (u1 , u2 ), 2πiz1 2πiz2
with some function F (u1 , u2 ) which is periodic in ui with period r. The integration is taken over |z1 | = |z2 | = 1. Recall (2.11) and (2.28). In order that the integrand be single valued in zi , we demand that F (u1 + τ, u2 ) = e−2πiπˆ 1 /r F (u1 , u2 ),
F (u1 , u2 + τ ) = e−2πi(πˆ 2 −1)/r F (u1 , u2 ),
where τ = πi/ log x. Assume further that F (u1 , u2 ) is holomorphic except for possible simple poles at ui = 1/2 and u1 − u2 + 1/2 = 0. (As for the last pole, we have taken into account the commutation relation (2.6).) If we regard the πˆ i ’s as constants, then the space of functions satisfying these conditions is 3 dimensional. It is straightforward to find a spanning set of such functions. This motivates us to introduce the following family of operators parameterized by k: II dz1 dz2 k ξ1 (u1 )ξ2 (u2 ) X12 (k) = (−1) 2πiz1 2πiz2 [u1 + k + 21 − πˆ 1 ] [u2 − k − 21 − πˆ 2 ] [u1 − u2 − k − 21 ] . (3.5) × [u1 − 21 ] [u2 − 21 ] [u1 − u2 + 21 ] Proposition 3.1. [X12 (k), X12 (l)] = 0 for any k, l, X1 X2 = X12 (−1), X2 X1 = X12 (0), X1 X12 (k) = X12 (k − 1)X1 , X2 X12 (k − 1) = X12 (k)X2 .
(3.6) (3.7) (3.8) (3.9)
The proof of these statements will be given later in the context of general n. Notice the periodicity relation (3.10) X12 (k + r) = (−1)r−1 X12 (k).
Algebra of Screening Operators for Deformed Wn Algebra
Set (a) (k) = X12
a Y
509
X12 (k − b + 1).
(3.11)
b=1
Then, for any non-negative integers a, b, we have (b) X1a+b X2b = X12 (−a − 1)X1a ,
(3.12)
(a) X1a X2a+b = X2b X12 (−b − 1), a a+b b (a) X2 X1 = X1 X12 (a + b − 1), (b) (a + b − 1)X2a . X2a+b X1b = X12
(3.13) (3.14) (3.15)
As an example, let us verify the first relation. Consider first the case b = 1. From (3.7) and (3.8) we have X1a+1 X2 = X1a X12 (−1) = X1a−1 X12 (−2)X1 = ··· = X12 (−a − 1)X1a . The case of general b follows immediately from this and the definition (3.11). The other relations are derived in a similar manner. Now set (a) X α1 +α2 (λ) = X12 (k),
a = mα1 +α2 (λ), k = (λ, α1 ) − 1.
Comparing (3.12)-(3.15) with (3.1)-(3.4) and taking (3.10) into account, we see that the desired relations are satisfied up to sign. It remains to settle the issue about the signs sα (λ). From the definition, the screening operators have the periodicity if β ∈ rZα1 + 2rZα2 .
X α (λ + β) = X α (λ)
Thus the signs can also be chosen according to the same periodicity. A direct verification shows that the following is one possible solution for the sα (λ). B
A +
+
+
- -
C +
-a -a +
+ +
+
+ -b
D +
+
+
- -
-b
b
E
b +
+
+
+
+
-a -a a
+ k
+ k
F + + -b -b
+ a
+
+
+ +
+
Fig. 2. The choice of the signs sα (λ). We set a = εr 1 , b = εr 2 , εr = (−1)r−1 . The vertices at the top row correspond to the weights A = 3, B = r1 r2 3 + α1 , C = r2 r1 3 + α1 , D = 3 + α1 − α2 , E = r1 r2 3 + 2α1 − α2 , F = r2 r1 3 + 2α1 − α2 , with rj = rαj
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4. Screening Operators 4.1. Basic operators. We are now in a position to introduce the operators which will play a basic role in the construction of screening operators for general n. Let α = αi + · · · + αi+m be a positive root. We often write it as αi···i+m . Define I Xα (k1 , . . . , km ) = ×
···
I i+m Y dzj ξi (ui ) ξi+m (ui+m ) ··· 1 2πizj [ui − 2 ] [ui+m − 21 ] j=i
i+m Y
[uj−1 − uj ] fα(k1 ,...,km ) (ui , . . . , ui+m ; πˆ i , . . . , πˆ i+m ). 1 [u − u + ] j−1 j 2 j=i+1
(4.1)
Here fα(k1 ,...,km ) (ui , . . . , ui+m ; πˆ i , . . . , πˆ i+m ) = (−1)k1 +···+km m m Y [ui+l−1 − ui+l − kl − 21 ] Y 1 [ui+l − kl + kl+1 − − πˆ i+l ], [ui+l−1 − ui+l ] 2 l=1
(4.2)
l=0
and k0 = −1, km+1 = 0 is implied. The integrand of (4.1) is a single-valued function in zj (i ≤ j ≤ i + m). To see this note that i+m Y
ξi (ui ) · · · ξi+m (ui+m )
− r1 πˆ j
zj
− r−1
zi+mr
(4.3)
j=i
is single-valued. When α = αj is a simple root, (4.1) reduces to (2.26). The basic property of (4.1) is the following commutativity. Theorem 4.1. For any k1 , · · · , km , p we have [Xα (k1 , . . . , km ), Xα (k1 + p, . . . , km + p)] = 0.
(4.4)
In view of this, we define for a non-negative integer a Definition 4.2. Xα(a) (k1 , . . . , km ) =
a Y
Xα (k1 − b + 1, . . . , km − b + 1).
(4.5)
b=1 (a) Sometimes, we abbreviate Xα(a)i···i+m (k1 , . . . , km ) to Xi···i+m (k1 , . . . , km ).
Proof of Theorem 4.1. Using the commutation relations (2.5)-(2.7) and (2.11), we can write the product Xα (k1 , . . . , km )Xα (k1 + p, . . . , km + p) in the form I
I i+m Y dzj dwj ξi (ui ) ξi (vi ) ξi+m (ui+m ) ξi+m (vi+m ) ··· ··· 1 1 2πiz 2πiw [ui+m − 21 ] [vi+m − 21 ] j j [ui − 2 ] [vi − 2 ] j=i ×F (ui , vi , · · · , ui+m , vi+m )
(zk = x2uk , wk = x2vk ),
and likewise for the product in the opposite order. Symmetrizing with respect to the integration variables and equating the integrand, we are led to prove the following equality:
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S fα(k1 ,...,km ) (ui , . . . , ui+m ; πˆ i − (1 − r), πˆ i+1 , . . . , πˆ i+m−1 , πˆ i+m − (1 − r))fα(k1 +p,...,km +p) (vi , . . . , vi+m ; πˆ i , . . . , πˆ i+m ) ×
i+m Y j=i
i+m−1 i+m [uj − vj−1 + 21 ] [uj − vj − 1] Y [uj − vj+1 + 21 ] Y (−1) [uj − vj ] [uj − vj+1 ] [uj − vj−1 ] j=i j=i+1
= S fα(k1 +p,...,km +p) (vi , . . . , vi+m ; πˆ i − (1 − r), πˆ i+1 , . . . , πˆ i+m−1 , πˆ i+m − (1 − r))fα(k1 ,...,km ) (ui , . . . , ui+m ; πˆ i , . . . , πˆ i+m ) ×
i+m Y j=i
i+m−1 i+m [uj − vj+1 − 21 ] Y [uj − vj−1 − 21 ] [uj − vj + 1] Y . (−1) [uj − vj ] [uj − vj+1 ] [uj − vj−1 ] j=i j=i+1
(4.6) Here the symbol S means the symmetrization of (uj , vj ) for each j = i, . . . , i + m. This is equivalent to Y m 1 3 [ui+l−1 − ui+l − kl − ][ui + k1 + − πˆ i ] A 2 2 l=1 m−1 Y 1 [ui+l − kl + kl+1 − − πˆ i+l ] × 2 l=1
Y 1 1 1 − πˆ i+m ] [vi+l−1 − vi+l − kl − p − ][vi + k1 + p + − πˆ i ] 2 2 2 l=1 m−1 Y 1 1 [vi+l − kl + kl+1 − − πˆ i+l ] [vi+m − km − p − − πˆ i+m ] 2 2 l=1 i+m i+m−1 i+m Y Y 1 Y 1 [uj − vj − 1] [uj − vj+1 + ] [uj − vj−1 + ] × 2 2 j=i j=i j=i+1 Y m 1 1 =A [ui+l−1 − ui+l − kl − ][ui + k1 + − πˆ i ] 2 2 l=1 m−1 Y 1 [ui+l − kl + kl+1 − − πˆ i+l ] × 2 m
×[ui+m − km +
l=1
Y 1 1 3 − πˆ i+m ] [vi+l−1 − vi+l − kl − p − ][vi + k1 + p + − πˆ i ] 2 2 2 l=1 m−1 Y 1 1 [vi+l − kl + kl+1 − − πˆ i+l ] [vi+m − km − p + − πˆ i+m ] × 2 2 l=1 i+m i+m−1 i+m Y Y 1 Y 1 [uj − vj + 1] [uj − vj+1 − ] [uj − vj−1 − ] . × 2 2 j=i j=i m
×[ui+m − km −
j=i+1
(4.7)
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Here the symbol A means the anti-symmetrization of (ui+j , vi+j ) for each j = 0, . . . , m. We prove this equality by induction on m. The (LHS) − (RHS) is a theta function of order 4 in ui . Using the induction hypothesis, one can check that it vanishes at ui = vi , vi ± 1. Taking into account the quasi-periodicity, we conclude that it must have a factor 1 [ui − vi ][ui − vi − 1][ui − vi + 1][ui + 2vi − ui+1 − vi+1 + − πˆ i ]. (4.8) 2 This is a contradiction unless (LHS) − (RHS) = 0. 4.2. Definition of screening operators. Let us come to the definition of the screening operators. Let (λ, λα ) be an admissible pair, with α = αi···i+m . We define X α (λ) : Fλ → Fλα by the formula X α (λ) = Xα(a) (k1 , . . . , km ), (4.9) where a = mα (λ), kj = (λ, αi···i+j−1 ) − 1.
(4.10) (4.11)
Note that on Fλ the operator πˆ j has the fixed value πˆ j |Fλ = (rl + (1 − r)λ, αj ) = (λ, αj ) mod r.
(4.12)
More explicitly the operator (4.9) is given as follows. Proposition 4.3. Notations being as in (4.2) and (4.9)–(4.12), we set (a) (1) (a) fα(a) (u(1) i , . . . , ui , . . . , ui+m , . . . , ui+m ) Y a (b) fα(k1 −b+1,...,km −b+1) (u(b) ˆ i − (a − b)(1 − r), πˆ i+1 , . . . , =S i , . . . , ui+m ; π b=1
Y
. . . , πˆ i+m−1 , πˆ i+m − (a − b)(1 − r))
1≤b
Y
×
(c) 1 [u(b) j − uj+1 + 2 ]
1≤b
=S
Y a
(c) [u(b) j − uj+1 ]
Y
(−1)
1≤b
(c) [u(b) j − uj − 1] (c) [u(b) j − uj ] (c) 1 [u(b) j − uj−1 + 2 ]
(c) [u(b) j − uj−1 ]
(4.13) (b) fα(k1 −a+b,...,km −a+b) (u(b) ˆ i − (a − b)(1 − r), πˆ i+1 , . . . , i , . . . , ui+m ; π
b=1
Y
. . . , πˆ i+m−1 , πˆ i+m − (a − b)(1 − r))
1≤b
×
Y 1≤b
(c) 1 [u(b) j − uj+1 + 2 ] (c) [u(b) j − uj+1 ]
Y 1≤b
(−1)
(c) [u(b) j − uj − 1] (c) [u(b) j − uj ] (c) 1 [u(b) j − uj−1 + 2 ] (c) [u(b) j − uj−1 ]
, (4.14)
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513
(a) where S means the symmetrization of (u(1) j , . . . , uj ) for each j = i, . . . , i + m. Then we have
I X α (λ) = × ×
I ···
Y
dzj(b)
1≤b≤a i≤j≤i+m
ξi (u(1) i ) 1 [u(1) i − 2]
Y 1≤b
···
2πizj(b)
ξi (u(a) i ) 1 [u(a) i − 2]
···
ξi+m (u(1) i+m ) 1 [u(1) i+m − 2 ]
Y
(c) [u(b) j − uj ] (c) [u(b) j − uj − 1]
···
ξi+m (u(a) i+m )
1 [u(a) i+m − 2 ]
(c) [u(b) j − uj+1 ]
1≤b,c≤a i≤j≤i+m−1
(c) 1 [u(b) j − uj+1 + 2 ]
(a) ×fα(a) (u(1) i , . . . , ui+m ).
(4.15)
(a) (1) (a) The function fα(a) (u(1) i , . . . , ui , . . . , ui+m , . . . , ui+m ) has the following zeros:
(i)
u(b) j =
1 2
(1 ≤ b ≤ a, i ≤ j ≤ i + m).
(c) (ii) u(b) j = uj+1 −
(d) 1 1 2 = uj+1 + 2 (1 ≤ b, c, d ≤ a, i ≤ j ≤ i + m − 1). (d) 1 u(c) j + 2 = uj+1 (1 ≤ b, c, d ≤ a, i ≤ j ≤ i + m − 1).
1 (iii) u(b) j − 2 = (iv) The function
(a) fα(a) (u(1) i , . . . , ui+m ) Q (b) 1 1≤b≤a [u j − 2] i≤j≤i+m
(b) is invariant under the simultaneous shift u(b) j 7→ uj + c. The function (a) fα(a) (u(1) i , . . . , ui+m )
Y
(c) [u(b) j − uj+1 ]
1≤b,c≤a i≤j≤i+m−1
is holomorphic everywhere. Proof. The equality of (4.13) and (4.14) follows from the proof of Theorem 4.1. Substituting (4.10)-(4.12) into (4.13) and comparing with the definition (4.2), we see that Qi+m (a) the function fα(k1 −b+1,...,km −b+1) in (4.13) has a factor j=i+1 [u(b) j − 1/2]. Therefore fα 1 has a zero at u(b) j = 2 for 1 ≤ b ≤ a and i + 1 ≤ j ≤ i + m. A similar argument applied to (4.14) shows that it is true also for j = i. For the proof of the rest of Proposition 4.3, we recall the definition of the elliptic algebra in [16]. (See also Appendix B.) The adaptation to our context goes as follows. Pn−1 (πˆ ,...,πˆ n−1 ) For πˆ i ∈ Z/rZ (1 ≤ i ≤ n − 1) and γ = i=1 ai αi (ai ∈ Z≥0 ) define Fγ 1 Pn−1 to be the set of functions in |γ| = i=1 ai variables u(b) i
(1 ≤ i ≤ n − 1, 1 ≤ b ≤ ai ) (a
)
n−1 satisfying the following properties: f (u(1) 1 , . . . , un−1 ) is
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(P 1)
meromorphic in u(b) i ∈ C,
(P 2)
(ai ) symmetric in each set of variables (u(1) i , . . . , ui ),
(P 3)
quasi-periodic in each variable u(b) i in the following sense: (b) f (u(b) i + r) = −f (ui ), (b)
2πi(ui f (u(b) i + τ ) = −e
+ τ2 +
(γ,αi )−1 −πˆ i )/r 2
f (u(b) i ),
(c) holomorphic except for (at most) simple poles at u(b) i = ui+1 , zero at one of the following:
(P 4) (P 5)
(c) • u(b) i = ui+1 − (c) • u(b) i+1 = ui −
1 2 1 2
1 = u(d) i+1 + 2 , 1 = u(d) i + 2.
Note that if ai = 0 for some i then πˆ i does not appear in the condition for f . If so, (πˆ ,...,πˆ n−1 ) we sometimes abbreviate πˆ i from the notation Fγ 1 . Let n−1 n−1 X (1) X (2) ai αi , γ (2) = ai α i , γ (1) = i=1
i=1
and let πˆ i0 be given by πˆ i0 = πˆ i − (γ (2) , αi )(1 − r).
(4.16)
Then one can show [16] that there exists an associative mapping (the ∗-product) 0 (πˆ 0 ,...,πˆ n−1 )
Fγ (1)1
(πˆ ,...,πˆ n−1 )
⊗ Fγ (2)1
(πˆ ,...,πˆ
)
→ Fγ (1)1+γ (2) n−1 ,
which sends f ⊗ g to f ∗ g, where (a(1) )
(a(1) )
(a(2) )
(1) n−1 1 (f ∗ g)(u(1) , v1(1) , . . . , v1 1 , . . . , u(1) 1 , . . . , u1 n−1 , . . . , un−1 , vn−1 , . . . , (a(2) ) (a(1) ) (a(2) ) (1) n−1 n−1 n−1 . . . , vn−1 ) = S f (u(1) , . . . , u )g(v , . . . , v n−1 n−1 ) 1 1
×
Y 1≤i≤n−1 1≤b≤a(1) i 1≤c≤a(2) i
×
Y 2≤i≤n−1 1≤b≤a(1) i 1≤c≤a(2) i−1
Y
(c) [u(b) i − vi − 1] (c) [u(b) i − vi ]
(−1)
(c) 1 [u(b) i − vi+1 + 2 ]
1≤i≤n−2 1≤b≤a(1) i 1≤c≤a(2) i+1
(c) 1 [u(b) i − vi−1 + 2 ] (c) [u(b) i − vi−1 ]
(c) [u(b) i − vi+1 ]
.
(1)
(a(2) )
(ai ) (1) , vi , . . . , vi i ) for each i = Here S means the symmetrization of (u(1) i ,...,u (c) 1, . . . , n − 1. Because of the symmetrization the pole at u(b) i = vi is canceled. The property (P5) is preserved by the ∗-product.
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(a(1) )
n−1 Suppose that F (u(1) ˆ 1 , . . . , πˆ n−1 ) is a function of the variables 1 , . . . , un−1 ; π
(a(1) )
(πˆ ,...,πˆ
)
n−1 n−1 ˆ 1 , . . . , πˆ n−1 ), and belongs to Fγ (1)1 for any choice of (u(1) 1 , . . . , un−1 ) and (π (πˆ 1 , . . . , πˆ n−1 ) ∈ Zn−1 . (πˆ ,...,πˆ n−1 ) is considered as an element Note that (πˆ 1 , . . . , πˆ n−1 ) in the expression Fγ (1)1 of (Z/rZ)n−1 . Therefore, the factor 1 − r in the shift (4.16) could be simply 1. However, the function F may have dependence on (πˆ 1 , . . . , πˆ n−1 ) ∈ Zn−1 . In fact, we use (4.16) in this form in (4.18) below. We define an operator X(F ) by
I
I
(a(1) )
n−1 dzn−1
(a(1) )
n−1 ) ξn−1 (un−1
ξ1 (u(1) 1 )
··· (1) 1 (a(1) ) (a(1) ) n−1 [u n−1 1 − 2] 2πizn−1 [un−1 − 21 ] (c) (c) Y [u(b) [u(b) i − ui+1 ] i − ui ]
X(F ) = ×
dz1(1)
2πiz1(1)
Y 1≤i≤n−1 (1) 1≤b
···
(c) [u(b) i − ui − 1]
(c) 1 [u(b) i − ui+1 + 2 ]
1≤i≤n−1 1≤b≤a(1) i 1≤c≤a(1) i+1
(a(1) )
n−1 ˆ 1 , . . . , πˆ n−1 ). ×F (u(1) 1 , . . . , un−1 ; π
(4.17)
(a(2) )
n−1 Similarly we define X(G) from G(v1(1) , . . . , vn−1 ; πˆ 1 , . . . , πˆ n−1 ). Set
(a(1) )
(a(1) )
(1) n−1 n−1 0 f (u(1) ˆ 10 , . . . , πˆ n−1 ), 1 , . . . , un−1 ) = F (u1 , . . . , un−1 ; π (a(2) ) n−1 g(v1(1) , . . . , vn−1 )
=
(4.18)
(a(1) ) n−1 G(va(1) , . . . , vn−1 ; πˆ 1 , . . . , πˆ n−1 ).
(4.19) Then we have X(F )X(G) = X(f ∗ g). From this follows (4.15). Then, (ii) and (iii) are nothing but the condition (P5). (1) 1 Let us prove (iv). Assume that f has zeros at u(b) i = 2 (1 ≤ i ≤ n − 1, 1 ≤ b ≤ ai ). Set (1)
(an−1 ) (a(1) ) f (u(1) ) a ,...,u n−1 , . . . , u ) = f¯(u(1) . Q n−1 1 (b) 1 1≤i≤n−1 [u i − 2]
(4.20)
(1) i
1≤b≤a
From (P3) we have the following periodicity with respect to each u(b) i : ¯ (b) f¯(u(b) i + r) = f (ui ), 2πi( f¯(u(b) i + τ) = e
(γ,αi ) −πˆ i )/r 2
f¯(u(b) i ).
(a(1) )
n−1 ¯ Fix generic (u(1) 1 , . . . , un−1 ) and consider a simultaneous shift of f , (1)
(an−1 ) f¯s (c) = f¯(u(1) 1 + c, . . . , un−1 + c).
(4.21)
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If (γ, γ) X ≡ ai πˆ i mod r, 2 n−1
(4.22)
i=1
then f¯s (c) is doubly periodic in c. In Proposition 4.3 we choose γ = (λ, α)α, and then (4.22) is valid. From (P4) f¯s has no pole in c, and therefore it is constant. This completes the proof of Proposition 4.3. Note that the statement (iv) of Proposition 4.3 corresponds to the closeness of Felder’s contour in the CFT limit. 4.3. Quadratic relations. With the screening operators introduced above, let us examine the nilpotency property d2 = 0. Unfortunately, we could not find a complete solution to this problem in the general case n ≥ 4. In Appendix A we show the following. Theorem 4.4. Suppose that α, β ∈ 1+ and (λ, λα , λα,β )
(4.23)
is admissible, i.e., both (λ, λα ) and (λα , λα,β ) are admissible. (i) If (α, β) = 2, then α = β and it is a simple root. (ii) Otherwise, there exists α0 (6= α), β 0 ∈ 1+ such that 0
0
0
0
0
(λ, λα , λα ,β )
(4.24)
is admissible and λα,β = λα ,β .
(4.25)
The pair (α0 , β 0 ) is uniquely determined by this condition. In the second case, we say that the set of admissible weights (4.23) and (4.24) satisfying (4.25) form a commutative square. Consider the case (i). If α = αj is a simple root and a = mα (λ), then we have mα (λα ) = r − a. Hence d2 = 0 is ensured by the relation Xjr−a Xja = Xjr = 0. This has been proved in [13]. In the case (ii), we need the following. Theorem 4.5. For each commuting square, we have the identity of screening operators 0
X β (λα )X α (λ) = sλ (α, β; α0 β 0 )X β 0 (λα )X α0 (λ), where sλ (α, β; α0 β 0 ) = ±1.
(4.26)
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517
The sign factor arises here because of the (anti-)periodicity property of the operator Xα (k1 , . . . . . . , km ), (εr = (−1)r+1 ).
Xα (k1 , . . . , km )|ki →ki +r = εr Xα (k1 , . . . , km ),
(4.27)
The precise formula for sλ (α, β; α0 β 0 ) = ±1 will be given below. To have d2 = 0 we must choose the signs sα (λ) appropriately. Theorem 4.5 reduces the problem to finding sα (λ) satisfying 0
sβ (λα )sα (λ) = −sλ (α, β; α0 , β 0 )sβ 0 (λα )sα0 (λ). We have no solution except for the special cases n = 2, 3. The assertion (4.26) amounts to a number of identities of theta functions. These are derived in Appendix B. Below we shall indicate which identities are used in each case. Case (α, β) = 0. In this case, α0 = β, β 0 = α hold (see Appendix A). The assertion (4.26) is nothing but (B.24) and (B.25). We can show that sλ (α, β; α0 , β 0 ) = 1. Case (α, β) = ±1. From the case-by-case analysis of Appendix A, we see that altogether there are 8 cases to consider. In the following we set γ1 = αi···i+l and γ2 = αi+l+1···i+l+m : Case A+ : α = γ1 , β = γ2 , mα (λ) > mβ (λα ), Case B+ : α = γ1 , β = γ2 , mα (λ) < mβ (λα ), Case C+ : α = γ2 , β = γ1 , mα (λ) > mβ (λα ), Case D+ : α = γ2 , β = γ1 , mα (λ) < mβ (λα ), Case A− : α = γ1 + γ2 , β = γ1 , Case B− : α = γ2 , β = γ1 + γ2 , Case C− : α = γ1 + γ2 , β = γ2 , Case D− : α = γ1 , β = γ1 + γ2 , The cases X+ and X− (X = A, B, C, D) form a commutative square. We will prove this statement and show the corresponding equality of the screening operators. Set κα (λ) =
mα (λ) − (σ3, α) . r
(4.28)
To get (4.26) for α = β 0 = γ1 , α0 = γ1 + γ2 , β = γ2 , we use (B.30) with a = mα0 (λ), b = mα (λ) − mα0 (λ), kj = (λ, αi···i+j−1 ) − 1 (1 ≤ j ≤ l or l + 2 ≤ j ≤ l + m). The signature in (B.24) is given by sλ (α, β; α0 , β 0 ) = εκr α (λ)|β|mα0 (λ) . To get (4.26) for α = γ1 , β = α0 = γ2 , β 0 = γ1 + γ2 , we use (B.31) with a = mα0 (λ), b = mα (λ), kj = (λ, αi···i+j−1 ) − 1
(1 ≤ j ≤ l or l + 2 ≤ j ≤ l + m).
(4.29)
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The signature in (B.24) is given by 0
sλ (α, β; α0 , β 0 ) = εκr α (λ)|α |mα (λ) . 0
(4.30)
0
To get (4.26) for α = γ2 , β = α = γ1 , β = γ1 + γ2 , we use (B.28) with a = mα0 (λ), b = mα (λ), kj = (λ, αi···i+j−1 ) − 1 (1 ≤ j ≤ l), kj = (λ, αi+l+1···i+j−1 ) − 1 (l + 2 ≤ j ≤ l + m). The signature in (B.24) is given by sλ (α, β; α0 , β 0 ) = εκr α0 (λ)|α|mα (λ) . 0
(4.31)
0
To get (4.26) for α = β = γ2 , β = γ1 , α = γ1 + γ2 , we use (B.29) with a = mα0 (λ), b = mα (λ) − mα0 (λ), kj = (λ, αi···i+j−1 ) − 1 (1 ≤ j ≤ l), kj = (λ, αi+l+1···i+j−1 ) − 1 (l + 2 ≤ j ≤ l + m). The signature in (B.24) is given by sλ (α, β; α0 , β 0 ) = εr(κα (λ)+κα0 (λ))|α|mα0 (λ) .
(4.32)
5. Operators X α (λ) as Intertwiners of DWA In this section we demonstrate that the screening operators introduced above are the intertwining operators for the DWA. A special case of the statement has been proved [14, 15] for the screening operators acting in the vacuum module, where the theta function factor becomes unit and the screening operator has a particularly simple form: I dz Uj (z). X αj = (5.1) 2πiz The proof of [14, 15] was based on the fact that the screening currents commute with DWA generators up to a total difference. As it was explained [13] on the example of the sl2 case, in dealing with general (α, λ) one needs to be sure that the additional theta function terms do not lead to nonvanishing contributions to the commutator of the DWA generators and X α (λ). We prove that this property is guaranteed by the relations of the elliptic algebra of screening operators. 5.1. Deformed W algebra. The DWA can be constructed as the subalgebra in the universal enveloping algebra of the Heisenberg algebra of operators βn and Pα . Let us introduce the local bosonic fields [14, 15] 1 1 The difference between our notations and those in [14, 15] is that we are working with the zero mode Pn−1 P shifted as P −→ P − √ 1 ω j . In the conformal limit x → 1 this corresponds to the j=1 r(r−1)
transform from the complex plane to the annulus. The parameters p, t, q in [14] are related with x and r as q = x−2r , t = x−2(r−1) , p = x−2 .
Algebra of Screening Operators for Deformed Wn Algebra
3j (z) = x−2
√
r(r−1)Pε¯ j
: exp (
519
X (xrm − x−rm ) m6=0
m
j −m βm z ): .
(5.2)
The explicit realization of the generators of DWA in terms of these fields is given by means of the non-linear transformation: : (x−2z∂z − 31 (z))(x−2z∂z − 32 (zx2 )) · · · (x−2z∂z − 3n (zx2n−2 )) : n X (−1)j W (j) (zxj−1 )x−2(n−j)z∂z , =
(5.3)
j=0
where W (0) (z) ≡ 1. In the limit x → 1 this formula leads to the quantum Miura transformation describing the bosonic realization of the Wn -algebra with the Virasoro subalgebra central charge n(n + 1) . c = (n − 1) 1 − r(r − 1) The bosonic fields W (j) (z) defined via the transform (5.3) constitute an associative algebra [14, 15]. For instance, the commutation relations between the currents W (j) (z) and W (1) (z) W (1) (z) =
n X
3s (z)
(5.4)
s=1
are described by w z f (j) W (1) (z)W (j) (w) − W (j) (w)W (1) (z)f (j) = −(x − x−1 )2 × z w w w − W (j+1) (x−1 w)δ x−(j+1) }. ×[r]x [r − 1]x {W (j+1) (xw)δ xj+1 z z (5.5) Here δ(z) =
P j∈Z
z j and
X zm [(n − j)m]x [rm]x [(r − 1)m]x . f (j) (z) = exp −(x − x−1 )2 m [nm]x m>0
It is important for us that the relation (5.5) leads [15] to Lemma 5.1. The Fourier modes Wt(1) , t ∈ Z of the currents W (1) (z), I dz t (1) (1) z W (z) Wt = 2πiz
(5.6)
generate the whole DWA. 5.2. Intertwining property of X α (λ).. The main statement in this section is that the screening operators defined by (4.9) satisfy the basic property of the intertwining operators for DWA:
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Proposition 5.2. [W (j) (z), X α (λ)] = 0 .
(5.7)
According to Lemma 5.1 the proof of Proposition 5.2 follows from Lemma 5.3. [W (1) (z), X α (λ)] = 0 .
(5.8)
We will prove this fact in Appendix C. Though we have no proof, we believe that Conjecture. The screening operators (4.9) exhaust all the intertwiners for DWA. We expect further that the irreducible representations of DWA arise as the cohomologies of the BRST complex (2.16). An important remark is that the definition of the DWA is invariant under the change r → 1 − r [14, 15]. For this reason there exists another set of intertwiners given by the second type screening operators. The construction for them is fairly obvious and we will not consider this case any more. Finally, let us stress that property iv) of Theorem 4.3 which is satisfied for X α (λ) turns out to be essential in the proof of Lemma 5.3. For this reason arbitrary products of basic operators or basic operators themselves do not commute with DWA generators and can not be treated as screening operators. It has been noted at the end of Sect. 2.4 and 6 that a similar situation takes place in the CFT. The general conditions for the existence of intertwining operators are discussed in Appendix A. 6. CFT Limit The problem of finding the intertwining operators of W -algebras [17] has been studied for the sl3 case (i.e., n = 3) in [5, 6]. However, it was not clear how to generalize the result for arbitrary n. In this section we take the CFT limit x → 1 , i log(z) ∼ 1 of the basic operators for DWA, and introduce the notion of basic operators for te W-algebra associated to the sln algebra with n ≥ 2. In the case n = 3 we recover the results of [5]. We restrict our discussion to a formal level in the sense that the well-definedness of integrals in the operators is not considered. In the deformed case, there is no such problem. The integrals are well-defined because the integrands are single-valued and the contours are on the unit circle. Our main goal is to find the formal limit of the operators (4.1) corresponding to the positive root α = αi + · · · + αi+m . Recall that these basic operators have the form: I I i+m Y dzj Ui (zi ) · · · Ui+m (zi+m ) Xα (k1 , . . . , km ) = · · · 2πizj j=i ×F (ui , . . . , ui+m ),
(6.1)
where the function F (ui , . . . , ui+m ) is given by F (ui , . . . , ui+m ) = (−1)k1 +···+km ×
m Y [ui+j−1 − ui+j − kj − 21 ] [ui+j−1 − ui+j + 21 ] j=1
m Y [ui+j − kj + kj+1 − 21 − πˆ i+j ] , [ui+j − 21 ] j=0
(6.2)
Algebra of Screening Operators for Deformed Wn Algebra
521
and again k0 = −1, km+1 = 0 is implied. Let us rewrite this expression into the sum of “elementary” integrals with the specific ordering of the variables on the unit circle: Z
X
Xα (k1 , . . . , km ) =
σ∈Sm+1
Z
i+m Y
··· 0<arg zi <···<arg zi+m <2π
j=i
dzj 2πizj
×Uσ(i) (zi ) · · · Uσ(i+m) (zi+m )Fσ (ui , . . . , ui+m ) ,
(6.3)
where Sm+1 is the permutation group of numbers (i, . . . , i + m) and the function Fσ is determined by the condition Ui (zσ−1 (i) ) · · · Ui+m (zσ−1 (i+m) )F (zσ−1 (i) , . . . , zσ−1 (i+m) ) = Uσ(i) (zi ) · · · Uσ(i+m) (zi+m )Fσ (zi , . . . , zi+m ),
(6.4)
or equivalently by Fσ (ui , . . . , ui+m ) = F (uσ−1 (i) , . . . , uσ−1 (i+m) )
Y j 0 <j σ(j 0 )−σ(j)=1
(−)
[uj − uj 0 + 21 ] . (6.5) [uj − uj 0 − 21 ]
Now we are able to work out the conformal limit of the operators Xα (k1 , . . . , km ). In what follows in this section, we will use the same notations for the CFT limits of the bosons, screening operators, etc., and mention only the changes in the definitions. First let us discuss the limit of the commutation relations for bosons and the screening current. The first one can be found directly by setting x → 1 in the formulae (2.1)-(2.3). j (1 ≤ j ≤ n−1, m ∈ Z\{0}) are defined by the commutation Namely, the oscillators βm relations (n − 1) (r − 1) δm+m0 ,0 (j = k), n r 1 (r − 1) mδm+m0 ,0 (j 6= k), =− n r
j k , βm [βm 0] = m
(6.6)
(6.7) n while βm is determined via the equation n X
j βm = 0.
(6.8)
j=1
Note that the definitions of the bosonic Fock spaces and the zero mode operators Pλ , Qλ remain the same as in Sect. 2.2. The prescription for taking the limit of the operator Uj (z) (j = 1, . . . , n − 1) is also very simple. We demand that together with x → 1, the parameter u tends to a limiting value in such a way that z = x2u is fixed. Therefore the screening currents of [14, 15] become the screening currents of [17]: P √ r−1 1 r−1 1 (β j −β j+1 )z −m :, (6.9) Uj (z) = ei r Qαj z r πˆ αj + r : e m6=0 m m m where z ∈ C. Note that the product of operators U (z)U (ζ) is defined apriori when |z| >> |ζ|, and to be understood as an analytic continuation. When we compare U (z)U (ζ) with U (ζ)U (z)
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in the deformed case, there is a common domain of convergence (a neighborhood of the unit circle |z/ζ| = 1). However, in the CFT limit, this will shrink because poles accumulate to z/ζ = 1. Therefore, in order to compare U (z)U (ζ) with U (ζ)U (z) in the CFT limit, we must specify the path of analytic continuation. In fact, we have −(α ,α ) q i j Uj (ζ)Ui (z), if arg z < arg ζ . (6.10) Ui (z)Uj (ζ) = if arg z > arg ζ q (αi ,αj ) Uj (ζ)Ui (z), Here the complex number q is q = eiπ
r−1 r
,
(6.11)
and the left hand side means the analytic continuation from the region, arg z = arg ζ and |z| > |ζ|, while the right hand side means the analytic continuation from the region, arg z = arg ζ and |z| < |ζ|. To work out the conformal limit of the theta depending part of (6.1) we parametrize the variables x, u as follows: x = e− v u= 2i
( > 0), (0 < Re v < 2π).
(6.12)
Now the limit is given by → 0, while z = eiv remains to be fixed. According to such a prescription the function (6.2) is changed to be the Pα -depending factor iπ
F (ui , . . . , ui+m ) → (−1)m q −m−k1 −···−km e r
(1−πˆ i −···−πˆ i+m )
(6.13)
if the condition arg(zj ) < arg(zj+1 ) holds for any j = i, . . . , i + m − 1 . For a non-trivial transposition σ ∈ Sm+1 , i.e., when some of the screening currents Uj+1 stands to the left of Uj the limit has the form iπ
Fσ (ui , . . . , ui+m ) → q f (σ) (−1)m q −m−k1 −···−km e r
(1−πˆ i −···−πˆ i+m )
(6.14)
where the function f (σ) is f (σ) =
X
(2kσ(j) + 1) .
(6.15)
j 0 <j σ(j 0 )−σ(j)=1
Now it is convenient to introduce the definition of an “elementary” integral Ii1 ···im+1 (1 ≤ i1 , . . . , im+1 ≤ n − 1) with the ordering of the variables as follows : Z Ii1 ···im+1 =
Z
i+m Y
··· 0<argzi <···<argzi+m <2π
j=i
dzj 2πizj
×Ui1 (zi ) · · · Uim+1 (zi+m ) .
(6.16)
In terms of these objects the formal limit of the basic operators (4.1) is given by the expression X q f (σ) Iσ(i)···σ(i+m) . (6.17) Xα (k1 , . . . , km ) = q −m−k1 −···−km σ∈Sm+1
Here, for notational convenience we have omitted the irrelevant common factor
Algebra of Screening Operators for Deformed Wn Algebra iπ
(−1)m e r
523
(1−πˆ i −···−πˆ i+m )
.
Indeed, whereas the operator P itself does not commute with screening operators, the important properties of (6.17) such as commutations with other basic operators and W -algebra generators do not depend on this factor. Note that the quasi-periodicity (3.10) of basic operators follows from q r = (−1)r−1 . The product of two integrals of the form Ii1 ···im , Iim+1 ···im+s is given according to the definition (6.16) as follows. Let Sm+s be the permutation group of numbers {1, . . . , m + s}. Denote by Sm,s the set of elements σ ∈ Sm+s such that σ −1 (j) < σ −1 (j + 1) for each j 6= m, m + s. Then X q gm,s (σ) Iiσ(1) ···iσ(m+s) , (6.18) Ii1 ···im Iim+1 ···im+s = σ∈Sm,s
where gm,s (σ) =
X
(αiσ(j) , αiσ(l) ).
(6.19)
jσ(l)
For instance, let us work out the decomposition of the product of two basic operators (see also [2, 5]): X12 (k) = q −k−1 I12 + q k I21 , X 1 = I1 ,
(6.20) (6.21)
into the “elementary” integrals. In this example we will also show the validity of (3.8). The formal product of these operators is the operator of weight −2α1 − α2 . It can be derived from the definition (6.18), Ii1 i2 Ii3 = Ii1 i2 i3 + q (αi3 ,αi2 ) Ii1 i3 i2 + q (αi3 ,αi1 )+(αi3 ,αi2 ) Ii3 i1 i2 , I i 1 I i 2 i 3 = I i1 i 2 i 3 + q
(αi1 ,αi2 )
I i2 i1 i3 + q
(αi1 ,αi3 )+(αi1 ,αi2 )
I i2 i 3 i1 .
(6.22) (6.23)
In particular, I12 I1 I21 I1 I1 I12 I1 I21
= (q + q −1 )I112 + I121 , = qI121 + (1 + q 2 )I211 , = (1 + q 2 )I112 + qI121 , = I121 + (q + q −1 )I211 .
(6.24)
Using these formulae we find that X1 X12 (k) = q −k (q + q −1 )I112 + (q k + q −k )I121 + +q k (q + q −1 )I211 = X12 (k − 1)X1 .
(6.25)
This confirms that (3.8) still holds in the CFT limit. Similarly, one can check (3.6), (3.7), (3.9) and similar properties of the basic operators in the n > 3 case. It would be useful, however, to express the basic operators in the “conventional” form, i.e., as non-commutative polynomials of Xj . Let us first prepare the necessary notations. Introduce the bracket
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{A, B}k ≡ −[k]q AB + [k + 1]q BA , {A, B}0 = BA , {A, B}−1 = AB , {{A, B}k1 , C}k2 = {A, {B, C}k2 }k1 , if [A, C] = 0.
(6.26) (6.27) (6.28) (6.29)
Now one finds that the following lemma holds: Lemma 6.1. q −m−k1 −···−km
X
q f (σ) Iσ(i)···σ(i+m)
σ∈Sm+1
=q
1−m−k1 −···−km−1
X
q f (σ) {Iσ(i)···σ(i+m−1) , Xm }km .
(6.30)
σ∈Sm
The proof follows from a straightforward decomposition of the right-hand side into “elementary integrals” Ii1 ···im+1 . Applying Eq. (6.30) we arrive at Proposition 6.2. Xi···i+m (k1 , . . . , km ) = {. . . {{Xi , Xi+1 }k1 , Xi+2 }k2 , . . . , Xi+m }km .
(6.31)
In particular, a conformal analogue of the operators (3.5) is X12 (k) = −[k]q X1 X2 + [k + 1]q X2 X1 .
(6.32)
Note that using this representation the equality of two decompositions (6.25) and CFT analogue for (3.9) can be rewritten as q-Serre relations [5] [k]q Xj2 Xj+1 − ([k + 1]q + [k − 1]q )Xj Xj+1 Xj + [k]q Xj+1 Xj2 = 0, 2 2 [k]q Xj+1 Xj − ([k + 1]q + [k − 1]q )Xj+1 Xj Xj+1 + [k]q Xj Xj+1 = 0.
These equations together with the commutativity of screening currents Uj , Ul for |j−l| 6= 1 imply that in the CFT limit, the screening operators corresponding to the simple roots satisfy the relations for the nilpotent half of the quantum group Uq (sln ). Using the properties of the bracket (6.26)–(6.29) one can check that the operators (6.31) satisfy the basic relations (B.14)–(B.18) of Lemma B.4, which now have the form: Xi···i+l−1 (k1 , . . . , kl−1 )Xi+l···i+l+m (kl+1 , . . . , kl+m ) = Xi···i+l+m (k1 , . . . , kl−1 , −1, kl+1 , . . . , kl+m ), (6.33) Xi+l···i+l+m (kl+1 , . . . , kl+m )Xi···i+l−1 (k1 , . . . , kl−1 ) = Xi···i+l+m (k1 , . . . , kl−1 , 0, kl+1 , . . . , kl+m ), (6.34) Xi Xi···i+m (k1 , k2 , . . . , km ) = Xi···i+m (k1 − 1, k2 , . . . , km )Xi , (6.35) Xi+m Xi···i+m (k1 , . . . , km−1 , km − 1) = Xi···i+m (k1 , . . . , km−1 , km )Xi+m , (6.36) 0 0 0 0 Xi···i+m (k1 , . . . , km )Xj···j+l (k1 , . . . , kl ) = Xj···j+l (k1 , . . . , kl ) if i + m + 1 < j. ×Xi···i+m (k1 , . . . , km ), (6.37)
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In the case n = 3 the procedure given in Sect. 3 may be followed in CFT limit to obtain the intertwining operators for W3 algebra starting from the basic operators (6.32). It can be easily verified that the expression for X α (λ) constructed in such a way coincides with the result of [5]. We also inspect directly in the CFT limit the commutativity property of basic operators for the n = 4. We believe that for general n, the construction of the intertwining operators X α (λ) for the W algebra is identical for those for the DWA (4.5) with the only difference in the definition of the basic operators (6.31). Our discussion was rather formal since we did not examine the well-definedness of the operators. To treat X α (λ) as operators on the Fock space one must fix the contour prescription [5]. We do not discuss this problem in this paper. 7. Discussion In this paper, we constructed the intertwining operators which commute with the action of the deformed Wn -algebra on the bosonic Fock spaces [14, 15]. In the conformal limit, the case we have discussed in this paper corresponds to representations of the Wn -algebra with the central charge n(n + 1) (7.1) c = (n − 1) 1 − r(r − 1) where r is an integer such that r ≥ n + 2. In the language of solvable lattice models, they correspond to the sln RSOS models [1]. The main difference in our construction compared to the corresponding conformal limit, is that we need to construct basic operators which change the weight of Fock spaces by a positive root. In the conformal case, the basic operators can be expressed in terms of the operators corresponding to the simple roots. In other words, the case n ≥ 3 contains a new feature which was not seen in the n = 2 case considered in [10]. One can look at this situation in the following way. It is well-known that the algebra of the screening operators in the conformal field theory is isomorphic to the algebra Uq (b+ ) with q = eπi(r−1)/r . An elliptic deformation of this algebra was considered in [16]. In this paper, we identified it with the algebra of the screening operators of the deformed Wn -algebra, and we derived a set of quadratic relations among the generators of that algebra. These relations can be considered as the elliptic deformation of the Serre relations. One of our original aims was to construct the Felder-type complex for the irreducible Wn modules. This was not achieved for two reasons. First of all, except for the case of n = 3, we cannot fix the signs so that we have d2 = 0 for the coboundary operators. Secondly, even for the case n = 3, we have no result on the cohomology of the complex. We considered the CFT limit of our construction. Our discussion stays at a formal level because we only considered the limit of the integrands of the screening operators. A. Admissibility and Commuting Squares A.1. Admissible pairs. Here we derive the condition for the admissibility of a pair (λ, λα ). We follow the notation of Sect. 2. In particular, we suppose λ = tγ σ3 throughout this section.
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> Lemma A.1. The condition (σ3, α) > < 0 is equivalent to l(rα σ) < l(σ). Proof. Suppose that rα = (ij) is the transposition and σ = σ(1), . . . , i, . . . , j, . . . , σ(n) .
(A.1)
rα σ = σ(1), . . . , j, . . . , i, . . . , σ(n) .
(A.2)
We have
< Therefore l(rα σ) > < l(σ) is equivalent to i > j. Since α is positive and rα = (ij), we have α = εi − εj if i < j and α = εj − εi if i > j. Noting that σ −1 εi = εσ−1 (i) −1 > and σ −1 (i) < σ −1 (j), we conclude that l(rα σ) > < l(σ) is equivalent to σ α < 0, and therefore to (σ3, α) = (3, σ −1 α) > < 0. Lemma A.2. l(rα ) = 2|α| − 1. Proof. If α = αi···i+m , a reduced expression of rα is given by rα = si · · · si+m−1 si+m si+m−1 · · · si .
We consider an operator Xα (λ) if and only if def
dα (λ) = deg(λα ) − deg(λ) = 1.
(A.3)
Note that dα (λ) =
l(rα σ) − l(σ) l(rα σ) − l(σ) + 2|α|
if (σ3, α) > 0; if (σ3, α) < 0.
(A.4)
In particular, we have 0 < dα (λ) < 2|α|.
(A.5)
Lemma A.3. A pair (λ, λα ) is admissible if and only if one of the following holds: (i) (σ3, α) > 0, and (σ3, β) < 0 or (σ3, γ) < 0 for any partition α = β + γ (β, γ ∈ 1+ ). (ii) (σ3, α) < 0, and (σ3, β) < 0 and (σ3, γ) < 0 for any partition α = β + γ (β, γ ∈ 1+ ). In particular, (λ, λα ) is always admissible for a simple root α = αj . Proof. We follow the argument in the proof of Lemma A.1. If l(rα σ) > l(σ), we set β = εi − εk and γ = εk − εj for k such that i < k < j. The condition dα (λ) = l(rα σ) − l(σ) = 1 is equivalent to σ −1 (k) < σ −1 (i) or σ −1 (j) < σ −1 (k) for any such k. This is equivalent to (σ3, β) < 0 or (σ3, γ) < 0, respectively. If l(rα σ) < l(σ), we set β = εj − εk and γ = εk − εi for k such that j < k < i. The condition dα (λ) = l(rα σ) − l(σ) + 2|α| = 1 is equivalent to σ −1 (i) < σ −1 (k) < σ −1 (j) for any such k. This is equivalent to (σ3, β) < 0 and (σ3, γ) < 0.
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A.2. Commuting squares. In this subsection we prove Theorem 4.4. The assertion (i) follows immediately from dα (λ) + dα (λα ) = 2|α|.
(A.6)
Below we shall prove the assertion (ii) case-by-case. Case (α, β) = 0. Set m = mα (λ) and m0 = mβ (λ). Recall (2.22). We have (rα σ3, β) = (σ3, β).
(A.7)
This implies mβ (λα ) = m0 . Therefore, we have λ − λα,β = mα + m0 β.
(A.8)
λ − λβ,α = mα + m0 β.
(A.9)
Similarly, we have
This implies dβ (λ) + dα (λβ ) = deg(λβ,α ) − deg(λ) = deg(λα,β ) − deg(λ) = 2.
(A.10)
Since dβ (λ), dα (λβ ) > 0, we have dβ (λ) = dα (λβ ) = 1.
(A.11)
Namely, (λ, λβ , λβ,α ) is admissible. Let us show the uniqueness of α0 , β 0 . Suppose that α = εi − εj and β = εk − εl . We consider only the case when k < i < j < l and set γ1 = εk − εi and γ2 = εj − εl . The other cases are similar. If (λ, λγ , λ − mα − m0 β)
(A.12)
is admissible and γ 6= α, β, then we have γ = εk − εj = γ1 + α or γ = εi − εl = α + γ2
(A.13)
m = m0 = mγ1 +α (λ) = mα+γ2 (λ).
(A.14)
and
Since mα (λ) ≡ (σ3, α) and mγ1 +α (λ) ≡ (σ3, γ1 + α) modr, we have (σ3, γ1 ) ≡ 0 mod r. This is a contradiction. Case (α, β) = 1. Set m = mα (λ) and m0 = mβ (λα ). We have (A.8). The only way other than (A.8) to write λ − λα,β as a positive linear combination of two positive roots is m(α − β) + (m + m0 )β if α − β ∈ 1+ ; λ − λα,β = (A.15) (m + m0 )α + m0 (β − α) if β − α ∈ 1+ . The uniqueness is then obvious from (A.15). Note that
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B. Feigin, M. Jimbo, T. Miwa, A. Odesskii, Ya. Pugai
(rβ σ3, α − β) = (σ3, α) = (rα σ3, β) = (σ3, β − α) =
m m−r
if (σ3, α) > 0; if (σ3, α) < 0,
(A.16)
m0 m0 − r
if (σ3, α − β) < 0; if (σ3, α − β) > 0.
(A.17)
Let us show that (λ, λβ , λβ,α−β ) is admissible if α − β ∈ 1+ . From (A.16) follows mα−β (λβ ) = m. Let us prove mβ (λ) = m + m0 .
(A.18)
If (σ3, α) > 0 and (σ3, α − β) < 0, the statement (A.18) follows from (A.16) and (A.17). The case (σ3, α) < 0 and (σ3, α − β) > 0 contradicts Lemma A.3 because (λ, λα ) is admissible. In the remaining cases, we have (σ3, β) = m + m0 − r. From Lemma A.3 (applied to (λ, λα )), we have m + m0 − r < 0, and therefore (A.18). Now, we have λβ,α−β = λα,β . This implies (see (A.11)) dα (λ) = dα−β (λβ ) = 1.
(A.19)
Thus we proved the admissibility of (λ, λβ , λβ,α−β ) if α − β ∈ 1+ . Next, we show that (λ, λβ−α , λβ−α,α ) is admissible if β − α ∈ 1+ . From (A.17) follows mβ−α (λ) = m0 . Let us prove mα (λβ−α ) = m + m0 .
(A.20)
Note that (rβ−α σ3, α) = (σ3, β). If (σ3, α) > 0 and (σ3, β − α) > 0, the statement (A.20) follows from (A.16) and (A.17). The case (σ3, α) < 0 and (rα σ3, β) = (σ3, β− α) < 0 contradicts with Lemma A.3 because (λα , λα,β ) is admissible and (rα σ3, α) = −(σ3, α) > 0. In the remaining cases, we have (σ3, β) = m + m0 − r. From Lemma A.3 (applied to (λα , λα,β )) we have (rα σ3, β −α) = (σ3, β) < 0, and therefore (A.20). Thus we proved (ii) when (α, β) = 1. Case (α, β) = −1. Set m = mα (λ) and m0 = mβ (λα ). Because of (2.13) we have (σ3, β) 6≡ 0 mod r, and therefore m 6= m0 . We have (A.8). The only way other than (A.8) to write λ − λα,β as a positive linear combination of two positive roots is (m − m0 )α + m0 (α + β) if m > m0 ; α,β = (A.21) λ−λ m(α + β) + (m0 − m)β if m < m0 . Again the uniqueness is obvious from (A.21). Note that m if (σ3, α) > 0; (rβ σ3, α + β) = (σ3, α) = m − r if (σ3, α) < 0, if (σ3, α + β) > 0; m0 (rα σ3, β) = (σ3, α + β) = m0 − r if (σ3, α + β) < 0.
(A.22) (A.23)
Let us show that (λ, λα+β , λα+β,α ) is admissible if m > m0 . From (A.23) follows mα+β (λ) = m0 . Let us prove mα (λα+β ) = m − m0 .
(A.24)
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> Note that (rα+β σ3, α) = −(σ3, β). If (σ3, α) > < 0 and (σ3, α + β) < 0, we have −(σ3, β) = m − m0 > 0, and therefore (A.24). If (σ3, α) < 0 and (σ3, α + β) > 0, we have −(σ3, β) = m − m0 − r < 0, and therefore (A.24). If (σ3, α) > 0 and (σ3, α + β) < 0, we have −(σ3, β) = m − m0 + r > r. This is a contradiction. Let us show that (λ, λβ , λβ,α+β ) is admissible if m < m0 . From (A.23) we have mα+β (λβ ) = m. Let us prove mβ (λ) = m0 − m.
(A.25)
> 0 If (σ3, α) > < 0 and (σ3, α + β) < 0, we have (σ3, β) = m − m > 0, and therefore (A.25). If (σ3, α) < 0 and (σ3, α + β) > 0, we have (σ3, β) = m0 − m + r > r. This is a contradiction. If (σ3, α) > 0 and (σ3, α+β) < 0, we have (σ3, β) = m0 −m−r < 0, and therefore (A.25). We have completed the proof of (ii) when (α, β) = −1. B. Generalized Serre Relations We modify the relations (2.5), (2.6) and (2.11) (keeping (2.7)) as follows. [u − v − δ] ξi (v)ξi (u), [u − v + δ] [u − v + δ2 ] ξj (v)ξi (u) if |i − j| = 1, ξi (u)ξj (v) = [u − v − δ2 ] πˆ i ξj (u) = ξj (u) πˆ i − (αi , αj )δ . ξi (u)ξi (v) =
(B.1) (B.2) (B.3)
Here, δ is a parameter. Note that if we set δ = 0 we have a commutative algebra. Fix (a1 , . . . , an−1 )
(ai ∈ Z≥0 ).
(B.4)
Consider a function f of the variables u(b) j (1 ≤ j ≤ n − 1, 1 ≤ b ≤ aj ) and (a )
j κj (1 ≤ j ≤ n − 1). We assume that f is symmetric in (u(1) j , . . . , uj ) for each 1 ≤ j ≤ n − 1. We call f a function of type (a1 , . . . , an−1 ). Suppose that f is of type (a1 , . . . , an−1 ) and g is of type (b1 , . . . , bn−1 ). We define the ∗-product f ∗ g of f and g to be the function of type (a1 + b1 , . . . , an−1 + bn−1 ) given by
(a
)
(b
)
(a1 ) (1) (b1 ) (1) (1) n−1 n−1 (f ∗ g)(u(1) 1 , . . . , u1 , v1 , . . . , v1 , . . . , un−1 , . . . , un−1 , vn−1 , . . . , vn−1 ; (an−1 ) (a1 ) (1) κ1 , . . . , κn−1 ) = S f (u(1) 1 , . . . , u1 , . . . , un−1 , . . . , un−1 ; κ1 + (−2b1 + b2 )δ,
κ2 + (b1 − 2b2 + b3 )δ, . . . , κn−1 + (bn−2 − 2bn−1 )δ) (b
)
(1) n−1 , . . . , vn−1 ; κ1 , κ2 , . . . , κn−1 ) ×g(v1(1) , . . . , v1(b1 ) , . . . , vn−1
Y 1≤j≤n−1 1≤a≤aj 1≤b≤bj
(b) [u(a) j − vj − δ] (b) [u(a) j − vj ]
Y 1≤j≤n−2 1≤a≤aj 1≤b≤bj+1
(b) δ [u(a) j − vj+1 + 2 ] (b) [u(a) j − vj+1 ]
Y 2≤j≤n−1 1≤a≤aj 1≤b≤bj−1
(b) δ [u(a) j − vj−1 + 2 ] (b) [u(a) j − vj−1 ]
.
(B.5)
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B. Feigin, M. Jimbo, T. Miwa, A. Odesskii, Ya. Pugai (a )
(b )
(1) j j Here the symbol S means the symmetrization of (u(1) j , . . . , uj , vj , . . . , vj ) for each 1 ≤ j ≤ n − 1. (k1 ,...,km ) be the following function of type (a1 , . . . , an−1 ) with Let fi···i+m
n aj =
1 0
if i ≤ j ≤ i + m; otherwise,
(k1 ,...,km ) (ui , . . . , ui+m ; κi , . . . , κi+m ) = fi···i+m
m Y [ui+j−1 − ui+j − (kj + 21 )δ] [ui+j−1 − ui+j ] j=1
×
m Y j=0
1 [ui+j − (kj − kj+1 + )δ − κi+j ] 2
(k0 = −1, km+1 = 0).
(B.6)
i ∨
δ If m = 0, we understand the function fi of type (0, . . . , 0, 1 , 0, . . . , 0), to be [u(1) i + 2 −κi ].
Theorem B.1. (k1 ,...,km ) (k1 +p,...,km +p) (k1 +p,...,km +p) (k1 ,...,km ) ∗ fi···i+m = fi···i+m ∗ fi···i+m . fi···i+m
Proof. This is similar to Theorem 4.4.
(B.7)
Set (k1 ,...,km ) , fi...i+m [k1 , . . . , km ] = fi···i+m a Y (a) [k1 . . . , km ] = ∗ fi···i+m [k1 − b + 1, . . . , km − b + 1], fi···i+m
(B.8) (B.9)
b=1
where the symbol ∗ in front of the usual product symbol means that this is a ∗-product. (a) [k1 , . . . , km ] satisfy a set of quadratic relations in ∗-product. They The functions fi···i+m are given below. By specialization δ = 1, we get the relations for the screening operators (a) (4.5) Xi···i+m (k1 , . . . , km ). For the proof of the quadratic relations we prepare a lemma. Let F be the algebra over C with the ∗-product, that is generated by elements fi···i+m [k1 , . . . , km ]. The algebra F is graded F = ⊕(a1 ,...,an−1 )∈Zn−1 Fa1 ,...,an−1 , ≥0
(B.10)
where Fa1 ,...,an−1 consists of the functions of type (a1 , . . . , an−1 ). Lemma B.2. If f, g ∈ F and f ∗ g = 0, then f = 0 or g = 0. Proof. Suppose that f is of type (a1 , . . . , an−1 ) and g is of type (b1 , . . . , bn−1 ). We expand f and g in power series of δ. If both f and g are non-zero, we have f = f0 δ m1 + o(δ m1 ), g = g0 δ m2 + o(δ m2 ),
f0 = 6 0, g0 6= 0,
for some m1 and m2 . From f ∗ g = 0 it follows that
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531
(an−1 ) (a1 ) (1) S f0 (u(1) 1 , . . . , u1 , . . . , un−1 , . . . , un−1 ; κ1 , . . . , κn−1 ) (bn−1 ) (1) ·g0 (v1(1) , . . . , v1(b1 ) , . . . , vn−1 , . . . , vn−1 ; κ1 , . . . , κn−1 ) = 0. (B.11) We will show that f0 = 0 or g0 = 0. Choose w1 , . . . , wn−1 ∈ C so that f0 and g0 (b) are holomorphic at u(b) j = wj and vj = wj , respectively. Power series expansion in (b) u(b) j − wj and vj − wj reduces the problem to the case when f0 and g0 are symmetric (a )
(a )
(1) j j polynomials in (u(1) j , . . . , uj ) and (vj , . . . , vj ), respectively. Finally the following lemma reduces the problem to the case of the polynomial ring.
Lemma B.3. Let Ga1 ,...,an−1 be the C-linear space of polynomials in the variables (a
)
(a1 ) (1) n−1 u(1) 1 , . . . , u1 , . . . , un−1 , . . . , un−1 , (a )
j that are symmetric in (u(1) j , . . . , uj ) for each 1 ≤ j ≤ n − 1. Set
G = ⊕(a1 ,...,an−1 )∈Zn−1 Ga1 ,...,an−1 .
(B.12)
≥0
Define the ∗-product in G by f ∈ Ga1 ,...,an−1 , g ∈ Gb1 ,...,bn−1 → f ∗ g ∈ Ga1 +b1 ,...,an−1 +bn−1 , where (a
)
(b
)
(a1 ) (1) (b1 ) (1) (1) n−1 n−1 (f ∗ g)(u(1) 1 , . . . , u1 , v1 , . . . , v1 , . . . , un−1 , . . . , un−1 , vn−1 , . . . , vn−1 ) (an−1 ) (a1 ) (1) (1) (b1 ) (1) = S f (u(1) 1 , . . . , u1 , . . . , un−1 , . . . , un−1 )g(v1 , . . . , v1 , . . . , vn−1 , . . . , (bn−1 ) . . . , vn−1 ) .
There is a ring homomorphism between G and the polynomial ring of the variables (1) (2) (0) (1) (2) (x(0) 1 , x1 , x1 , . . . , ; . . . , ; xn−1 , xn−1 , xn−1 , . . .).
Proof. For simplicity, we consider the case n = 2. The isomorphism is such that the subspace Ga of G corresponds to the space of degree a homogeneous polynomials in (1) (2) (x(0) 1 , x1 , x1 , . . .). The isomorphism is given by m1 ma 1) a) · · · x(m 7→ S (u(1) · · · (u(a) . x(m 1 1 1 ) 1 ) The basic relations are
(B.13)
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Lemma B.4. fi···i+l−1 [k1 , . . . , kl−1 ] ∗ fi+l···i+l+m [kl+1 , . . . , kl+m ] = fi···i+l+m [k1 , . . . , kl−1 , −1, kl+1 , . . . , kl+m ], (B.14) fi+l···i+l+m [kl+1 , . . . , kl+m ] ∗ fi···i+l−1 [k1 , . . . , kl−1 ] = fi···i+l+m [k1 , . . . , kl−1 , 0, kl+1 , . . . , kl+m ], (B.15) fi ∗ fi···i+m [k1 , k2 , . . . , km ] = fi···i+m [k1 − 1, k2 , . . . , km ] ∗ fi , (B.16) fi+m ∗ fi···i+m [k1 , . . . , km−1 , km − 1] = fi···i+m [k1 , . . . , km−1 , km ] ∗ fi+m , (B.17) 0 0 0 0 fi···i+m [k1 , . . . , km ] ∗ fj···j+l [k1 , . . . , kl ] = fj···j+l [k1 , . . . , kl ] if i + m + 1 < j. ∗fi···i+m [k1 , . . . , km ], (B.18) The proof is straightforward. Lemma B.5. fi···i+l [k1 , . . . , kl ] ∗ fi···i+l+m [k1 + kl+1 , . . . , kl + kl+1 , kl+1 , kl+2 , . . . , kl+m ] = fi···i+l+m [k1 + kl+1 , . . . , kl + kl+1 , kl+1 − 1, kl+2 , . . . , kl+m ] ∗fi···i+l [k1 , . . . , kl ], (B.19) fi+l···i+l+m [kl+1 , . . . , kl+m ] ∗ fi···i+l+m [k1 , . . . , kl−1 , kl − 1, kl+1 + kl , . . . , . . . , kl+m + kl ] = fi···i+l+m [k1 , . . . , kl−1 , kl , kl+1 + kl , . . . , . . . , kl+m + kl ] ∗ fi+l···i+l+m [kl+1 , . . . , kl+m ], (B.20) fi···i+l+m [k1 , . . . , kl−1 , kl , kl+1 + kl , . . . , kl+m + kl ] ∗fi+l···i+l+m+p [kl+1 , . . . , kl+m , −kl , kl+m+2 , . . . , kl+m+p ] = fi+l···i+l+m+p [kl+1 , . . . , kl+m , −kl − 1, kl+m+2 , . . . , kl+m+p ] ∗fi···i+l+m [k1 , . . . , kl−1 , kl − 1, kl+1 + kl , . . . , kl+m + kl ]. (B.21) The proof will be given later. The following are simple consequences of the above. Proposition B.6. (a) (b) [k1 , . . . , kl ] ∗ fi···i+l+m [k1 + kl+1 , . . . , kl + kl+1 , kl+1 + a − 1, kl+2 , . . . , fi···i+l (b) . . . , kl+m ] = fi···i+l+m [k1 + kl+1 , . . . , kl + kl+1 , kl+1 − 1, kl+2 , . . . , kl+m ] (a) [k1 , . . . , kl ], ∗fi···i+l
(B.22) (a) [kl+1 , . . . , kl+m ] fi+l···i+l+m
∗
(b) fi···i+l+m [k1 , . . . , kl−1 , kl
− 1, kl+1 + kl , . . . ,
Algebra of Screening Operators for Deformed Wn Algebra
533
(b) . . . , kl+m + kl ] = fi···i+l+m [k1 , . . . , kl−1 , kl + a − 1, kl+1 + kl , . . . , kl+m + kl ] (a) [kl+1 , . . . , kl+m ], ∗fi+l···i+l+m
(B.23) (a) fi···i+l+m [k1 , . . . , kl−1 , kl + b − 1, kl+1 + kl , . . . , kl+m + kl ] (b) ∗fi+l···i+l+m+p [kl+1 , . . . , kl+m , −kl + a − 1, kl+m+2 , . . . , kl+m+p ] (b) = fi+l···i+l+m+p [kl+1 , . . . , kl+m , −kl − 1, kl+m+2 , . . . , kl+m+p ] (a) [k1 , . . . , kl−1 , kl − 1, kl+1 + kl , . . . , kl+m + kl ]. ∗fi···i+l+m
(B.24) The following is also valid. However, the general case for a, b does not follow from the special case a = b = 1. Proposition B.7. (a) (b) [kl+1 , . . . , kl+m ] ∗ fi···i+l+m+p [k1 , . . . , fi+l···i+l+m
. . . , kl−1 , kl − 1, kl+1 + kl , . . . , kl+m + kl , kl + a − 1, kl+m+2 , . . . , kl+m+p ] (b) [k1 , . . . , kl−1 , kl + a − 1, kl+1 + kl , . . . , kl+m + kl , = fi···i+l+m+p (a) [kl+1 , . . . , kl+m ] kl − 1, kl+m+2 , . . . , kl+m+p ] ∗ fi+l···i+l+m
(B.25) We use these relations for integer parameters ki . However, they are valid without this restriction because the general case follows from the integer case. Let us derive (B.19). The other cases follow similarly. Without loss of generality one can assume that m = 1. Using (4.4) and (B.15) we have fi+l ∗ fi···i+l−1 [k1 , . . . , kl−1 ] ∗ fi···i+l [k1 + kl , . . . , kl−1 + kl , kl ] = fi···i+l [k1 + kl , . . . , kl−1 + kl , kl ] ∗ fi+l ∗ fi···i+l−1 [k1 , . . . , kl−1 ]. (B.26) Using (B.17) we have fi+l ∗ fi···i+l−1 [k1 , . . . , kl−1 ] ∗ fi···i+l [k1 + kl , . . . , kl−1 + kl , kl ] −fi···i+l [k1 + kl , . . . , kl−1 + kl , kl − 1] ∗ fi···i+l−1 [k1 , . . . , kl−1 ] = 0. (B.27) Since fi+l is not a zero divisor, we have (B.19). Combining all these, in particular (B.22) and (B.23), we arrive at the formulas which we need for the quadratic relations of the screening operators. Proposition B.8. (a+b) (b) [k1 , . . . , kl ] ∗ fi+l+1···i+l+m [kl+2 , . . . , kl+m ] fi···i+l (b) (a) = fi···i+l+m [k1 − a, . . . , kl − a, −a − 1, kl+2 , . . . , kl+m ] ∗ fi···i+l [k1 , . . . , kl ], (B.28) (a) (a+b) [k1 , . . . , kl ] ∗ fi+l+1···i+l+m [kl+2 , . . . , kl+m ] fi···i+l
534
B. Feigin, M. Jimbo, T. Miwa, A. Odesskii, Ya. Pugai (b) (a) = fi+l+1···i+l+m [kl+2 , . . . , kl+m ] ∗ fi···i+l+m [k1 , . . . , kl , −b − 1, kl+2 − b, . . . , . . . , kl+m − b],
(B.29) (a) (a+b) [kl+2 , . . . , kl+m ] ∗ fi···i+l [k1 , . . . , kl ] fi+l+1···i+l+m (b) (a) = fi···i+l [k1 − a, . . . , kl − a] ∗ fi···i+l+m [k1 , . . . , kl , a + b − 1, kl+2 , . . . , kl+m ], (B.30) (a+b) (b) [kl+2 , . . . , kl+m ] ∗ fi···i+l [k1 , . . . , kl ] fi+l+1···i+l+m (b) (a) = fi···i+l+m [k1 , . . . , kl , a + b − 1, kl+2 , . . . , kl+m ] ∗ fi+l+1···i+l+m [kl+2 − b, . . . , . . . , kl+m − b]. (B.31)
Proof. Let us derive (B.28). The other cases can be proven similarly. Using (4.5), (B.14), (B.22), we have (a+b) (b) [k1 , . . . , kl ] ∗ fi+l+1···i+l+m [kl+2 , . . . , kl+m ] fi···i+l (a+b−1) = fi···i+l [k1 , . . . , kl ] ∗ fi···i+l [k1 − a − b + 1, . . . , kl − a − b + 1] (b−1) ∗fi+l+1···i+l+m [kl+2 − b + 1, . . . , kl+m − b + 1] ∗ fi+l+1···i+l+m [kl+2 , . . . , kl+m ] (a+b−1) = fi···i+l [k1 , . . . , kl ] ∗ fi···i+l+m [k1 − a − b + 1, . . . , kl − a − b + 1, −1, (b−1) kl+2 − b + 1, . . . , kl+m − b + 1] ∗ fi+l+1···i+l+m [kl+2 , . . . , kl+m ] = fi···i+l+m [k1 − a − b + 1, . . . , kl − a − b + 1, −a − b, kl+2 − b + 1, . . . , (a+b−1) (b−1) . . . , kl+m − b + 1] ∗ fi···i+l [k1 , . . . , kl ] ∗ fi+l+1···i+l+m [kl+2 , . . . , kl+m ] (b) (a) = fi···i+l+m [k1 − a, . . . , kl − a, −a − 1, kl+2 , . . . , kl+m ] ∗ fi···i+l [k1 , . . . , kl ]. (B.32)
C. Commutativity with DWA In this section we prove Lemma 5.3. It would be more convenient for us to use here the “multiplicative” variable z instead of u. For this reason, let us define the theta function [[z]] ≡ [u]
z = x2u
,
(C.1)
having the periodicity property [[zx2r ]] = −[[z]]. Abusing the notations, let us use (a) (a) (b) the same symbol fα(a) (zi(1) , · · · , zi+m ) for the function fα(a) (u(1) i , · · · , ui+m ), where zj = (b)
x2uj . The screening operator X α (λ) in the notations (4.9)–(4.12) is given by (4.15), i.e., in the multiplicative variables, I X α (λ) = ×
I ···
Y 1≤b≤a i≤j≤i+m
Ui (zi(1) ) [[zi(1) /x]]
···
dzj(b) 2πizj(b)
Ui (zi(a) ) [[zi(a) /x]]
···
(1) ) Ui+m (zi+m (1) [[zi+m /x]]
···
(a) Ui+m (zi+m ) (a) [[zi+m /x]]
Algebra of Screening Operators for Deformed Wn Algebra
×
Y
[[zj(b) /zj(c) ]]
1≤b
[[zj(b) /x2 zj(c) ]]
Y 1≤b,c≤a i≤j≤i+m−1
535 (c) [[zj(b) /zj+1 ]]
(c) [[zj(b) x/zj+1 ]]
(1) (a) ×fα(a) (zi(1) , . . . , zi(a) , . . . , zi+m , . . . , zi+m ).
(C.2)
The non-trivial couplings between the screening currents with each other and with the field 3j (z) are: r−1
Uj (z)Uj+1 (w) = z − r s(w/z) : Uj (z)Uj+1 (w) : , r−1 Uj+1 (z)Uj (w) = z − r s(w/z) : Uj+1 (z)Uj (w) : , r−1 Uj (z)Uj (w) = z 2 r t(w/z) : Uj (z)Uj (w) : , 3j (z)Uj (w) = x−2(r−1) τj (w/z) : 3j (z)Uj (w) : , Uj (w)3j (z) = x−2(r−1) τj (w/z) : 3j (z)Uj (w) : , Uj (z)3j+1 (w) = τˆj (w/z) : 3j+1 (w)Uj (z) : , 3j+1 (w)Uj (z) = τˆj (w/z) : 3j+1 (w)Uj (z) : ,
(C.3)
where (x2r−1 z; x2r )∞ , (xz; x2r )∞ 2 2r (x z; x )∞ , t(z) = (1 − z) 2r−2 (x z; x2r )∞ r+j−2 1 − zx , τj (z) = 1 − zxj−r r−j−2 1 − zx τˆj (z) = . 1 − zx−r−j s(z) =
(C.4)
To be precise, the τj (w/z) (resp. τˆj (w/z)) in the formula for Uj (w)3j (z) (resp. 3j+1 (w)Uj (z)) must be understood as the expansion in z/w. The screening currents commute with the generator Wt(1) up to a total difference [14, 15]. Lemma C.1. [Wt(1) , Uj (z)] = (x−2r+2 − 1)(zxj−r )t : 3j (zxj−r )Uj (z) : −{z → zx2r } .
(C.5)
According to this lemma we have (a) )] (x−2r+2 − 1)−1 [Wt(1) , Ui (zi(1) ) · · · Ui+m (zi+m X (a) = (zj(f ) xj−r )t Ui (zi(1) ) · · · : 3j (zj(f ) xj−r )Uj (zj(f ) ) : · · · Ui+m (zi+m ) i≤j≤i+m 1≤f ≤a
−{zj(f ) → x2r zj(f ) } .
(C.6)
By normal-ordering the operator part we obtain an expression involving various functions and operators. For our purpose it is useful to introduce the following objects. Let 0 ≤ s ≤ m, 1 ≤ f0 , · · · , fs ≤ a and i ≤ k ≤ i + m − s. (The case of our immediate interest above is s = 0. However, the general case will be necessary as we proceed.) We define
536
B. Feigin, M. Jimbo, T. Miwa, A. Odesskii, Ya. Pugai (f0 ,...,fs ) Ik,...,k+s I I = − |w|=1
! |w|=x2r−s
dw (f0 ) (1) (a) Jk (zi , . . . , zi+m ) (f ) 2πiw z p =xs−p w
, (C.7) (0≤p≤s)
k+p
where (a) (a) ) = Uk(f0 ) (zi(1) , . . . , zi+m ) Jk(f0 ) (zi(1) , . . . , zi+m (a) (a) (a) )Tk(f0 ) (zi(1) , . . . , zi+m )Sk(f0 ) (zi(1) , . . . , zi+m ) × F (zi(1) , . . . , zi+m
and Uk(f0 ) , F , Tk(f0 ) and Sk(f0 ) are given below.
Y
(a) ) = (zk(f0 ) xk−r )t : 3k (zk(f0 ) xk−r ) Uk(f0 ) (zi(1) , . . . , zi+m
Uj (zj(b) ) :,
1≤b≤a i≤j≤i+m
(a) ) F (zi(1) , . . . , zi+m
Y
=
1≤b
(C.8)
Y
[[zj(b) /zj(c) ]]
(c) [[zj(b) /zj+1 ]]
1≤b,c≤a i≤j≤i+m
(a) ×fα(a) (zi(1) , . . . , zi+m )
Y
1
1≤b≤a i≤j≤i+m
(a) ) Tk(f0 ) (zi(1) , . . . , zi+m
Y
=
(zj(b) )
Y
,
t(zj(c) /zj(b) )
[[zj(b) /x2 zj(c) ]]
1≤b
×
2(r−1) r
[[zj(b) /x]]
x−2(r−1) τk (zk(b) xr−k /zk(f0 ) ),
(C.9)
1≤b≤a b6=f0
(a) ) Sk(f0 ) (zi(1) , . . . , zi+m
Y
=
(zj(b) )−
1≤b,c≤a i≤j≤i+m−1
×
Y
r−1 r
(c) s(zj+1 /zj(b) )
(c) [[xzj(b) /zj+1 ]]
(b) τˆk−1 (zk(f0 ) /xr−k zk−1 ).
(C.10)
1≤b≤a (f )
p = xs−p w (0 ≤ p ≤ s) is regular except for the functions The restriction zk+p
(f
)
(f )
p+1 p /zk+p ). For these singular terms we use the convention s(zk+p+1 (x2(r−1) ; x2r )∞ dz def −1 s(z ) = = Res z=x s(z −1 ) . 2πiz (x2r ; x2r )∞ z=x
(C.11)
Set I(s) =
i+m X
a X
I
k=i f0 ,...,fs =1
The contour for zj(b) is |zj(b) | = 1.
I ···
Y 1≤b≤a
i≤j≤i+m (b,j)6=(f0 ,k),...,(fs ,k+s)
dzj(b)
I (f0 ,...,fs ) . (b) k,...,k+s 2πizj
(C.12)
Algebra of Screening Operators for Deformed Wn Algebra
537
The induction goes as follows. Step 1. [Wt(1) , X α (λ)] = I(0). Step 2. I(s) = I(s + 1) (0 ≤ s ≤ m − 1). Step 3. I(m) = 0. Step 1 follows from (C.2), (C.3) and (C.6) We will show that the only poles of
(a) ) Jk(f0 ) (zi(1) , . . . , zi+m
tween the contours |w| = 1 and |w| = x2r−s are (A) (B)
(b) w = xzk+s+1
w=
(f )
(0≤p≤s)
(i ≤ k ≤ i + m − s − 1, 1 ≤ b ≤ a),
(b) x2r−s−1 zk−1
In particular, Step 3 follows. Let us abbreviate
bep zk+p =xs−p w
(i + 1 ≤ k ≤ i + m − s, 1 ≤ b ≤ a).
(a) ) Uk(f0 ) (zi(1) , . . . , zi+m
, etc., by p zk+p =xs−p w (f )
(0≤p≤s)
(C.13)
Uk(f0 )
. s
• Uk(f0 ) has no poles. This is because the term is normal-ordered. s
• F has no poles. This is proved in Proposition 4.3. s • Tk(f0 ) has no poles in the region s
x2r−s < |w| < 1. (a) Consider the poles of
[[zj(b) /x2 zj(c) ]]−1
(C.14)
. They are situated at w = zj(b) =xp w
x2rν+2−p zj(c) (ν ∈ Z). However the factor t(zj(c) /zj(b) )
cancels the poles for zj(b) =xp w
ν ≥ 0. Therefore, there are no such poles in the region (C.14). (b) Consider the poles of [[zj(b) /x2 zj(c) ]]−1 . The possible poles in (C.14) (c) zj =xp w (b) x2r−s−1 zk+1 .
are w = x2r−s−2 zk(b) (b < f0 ) τk (zk(b) xr−k /xs w), and the latter
or w = The former is canceled by is canceled by the property (P5) of fα(a) . (c) Consider the poles of t(zj(c) /zj(b) ) . The possible poles in (C.14) are w = (b) zj =xp w 2r−s−1 (c) 2r−s−2 (c) zk+1 or w = x zk+1 ] (f0 < c). The former is canceled x (a) (P5) of fα and the latter is canceled by τk (zk(c) xr−k /xs w).
(d) There is no pole of t(zj(c) /zj(b) )
in (C.14). zj(c) =xp w
(e) There is no pole of τk (zk(b) xr−k /xs w) in (C.14).
by the property
538
•
B. Feigin, M. Jimbo, T. Miwa, A. Odesskii, Ya. Pugai
Sk(f0 )
has two types of poles (A) and (B) of (C.13) arising from the factors (fs ) (f0 ) (c) r−k (b) and τˆk−1 (zk /x zk−1 ) , respectively. s(zk+s+1 /zk+s ) s
(f )
(fs ) zk+s =w
(c) (b) (a) Consider the poles of s(zj+1 /zj )
zk 0 =xs w
. They are situated at w
=
zj(b) =xp w
(c) (c) (ν ∈ Z≥0 ). The only poles in (C.14) are w = xzk+s+1 (1 ≤ c ≤ a). xp−s+1+2rν zk+p+1 (b) (b) (b) The pole of τˆk−1 (zk(f0 ) /xr−k zk−1 ) is at w = x2r−s−1 zk−1 . (f )
zk 0 =xs w
(c) Consider the poles of
(c) −1 [[xzj(b) /zj+1 ]]
. They are situated at w = zj(b) =xs−p w
(c) (ν ∈ Z). The only pole in (C.14) is w = x2r−s−1 zk(c) . This is canx2r+p−s−1 zk+p celed by the property (P5) of fα(a) . (c) −1 ]] . They are situated at w = (d) Consider the poles of [[xzj(b) /zj+1 (c) zj+1 =z s−p w
(b) (ν ∈ Z). The only pole in (C.14) is w = xzk+s−1 . This is canceled by the property (P5) of fα(a) . (b) x2rν+p−s+1 zk+p−1
We finished checking (C.13). We will conclude the induction by showing Step 2. We take the residues of (f0 ) (1) (a) Jk (zi , . . . , zi+m ) at (A) and (B) of (C.13). (fp ) zk+p =xs−p w
(0≤p≤s)
(fs+1 ) (fs+1 ) (A): We take the residue at w = xzk+s+1 , and then rename zk+s+1 to w in or (a) der to compare it with Jk(f0 ) (zi(1) , . . . , zi+m ) . Except for the (fp ) zk+p =xs+1−p w
(0≤p≤s+1)
(fs+1 ) (fs ) s(zk+s+1 /zk+s ), s−p
factor (fp ) zk+p = x
this procedure is equivalent to changing the substitution rule (fp ) w (0 ≤ p ≤ s) to zk+p = xs+1−p w (0 ≤ p ≤ s + 1). The residue of
(fs+1 ) (fs+1 ) dw s(zk+s+1 /w) 2πiw at w = xzk+s+1 gives s(x−1 ) in the convention of (C.11). Thus we get f0 ,...,fs+1 the term in Ik,...,k+s+1 that is corresponding to the cycle |w| = 1. (f1 ) (1) (a) (zi , . . . , zi+m ) (B): It is convenient to consider the residue of Jk+1
p ≤ s) at w = x2r−s−1 zk(f0 ) . We take the residue and then We will compare (f1 ) (1) (a) (I) Jk+1 (zi , . . . , zi+m ) (fp+1 ) (f0 ) zk+1+p =xs−p w
and (II)
, (0 ≤
)
zk =x−2r+s+1 w
(a) ) Jk(f0 ) (zi(1) , . . . , zi+m
taking care of the pole of
(0≤p≤s),
(f
p+1 zk+1+p =xs−p w (f0 ) rename zk to x−2r+s+1 w.
p zk+p =xs+1−p w (f )
(0≤p≤s+1)
(f1 ) τˆk (zk+1 /xr−k−1 zk(f0 ) )
(f0 )
at w = x2r−s−1zk . (f1 ) zk+1 =xs w
Algebra of Screening Operators for Deformed Wn Algebra
539
Let us abbreviate the restrictions (I) and (II) by and , respectively. We will I
compute the ratios of the corresponding terms in (I) and (II). (f1 ) (f0 ) • (U): Uk+1 /Uk = x−2(r−1) . I
II
II
This follows from the identity x2(r−1) : 3k+1 (xr+k z)Uk (z) :=: 3k (xr+k z)Uk (x2r z) : . Set T0 =
Y
[[zj(b) /x2 zj(c) ]]
1≤b
and S0 =
Y
(c) [[xzj(b) /zj+1 ]].
1≤b,c≤a i≤j≤i+m
• (F): F (T0 S0 )−1 /F (T0 S0 )−1 I
= 1. II
The signs arising from the periodicity [[x2r z]] = −[[z]] cancel out. (f1 ) • (T): Tk+1 T0 /Tk(f0 ) T0 = I
II
Y 1 − x2r−s−2 z (b) /w Y k+1 b6=f1
(b) 1 − x−s zk+1 /w
x−4(r−1)
b6=f0
1 − x2r−s−1 zk(b) /w 1 − x−s+1 zk(b) /w
.
We used t(x2r z) (1 − x2(r−1) z)(1 − x2r z) = . t(z) (1 − z)(1 − x2 z) • (S1): Resw=x2r−s−1 z(f0 ) τˆk (xs+k+1−r w/zk(f0 ) ) k
dw = x2(r−1) − 1. 2πiw
• (S2): Q
(f1 ) r−k−1 (b) zk ) b6=f0 τˆk (zk+1 /x
Q (f0 ) (b) r−k zk−1 ) b τˆk−1 (zk /x =
Y b6=f0
x2(r−1)
(f )
1 =xs w zk+1
(f )
zk 0 =xs+1 w
(b) 1 − x−s+1 zk(b) /w Y 1 − x−2r+s+2 w/zk−1
1 − x2r−s−1 zk(b) /w
b
(b) 1 − xs w/zk−1
.
(C.15)
540
B. Feigin, M. Jimbo, T. Miwa, A. Odesskii, Ya. Pugai r−1
In the restriction (I) we must pay a special attention to the factor (zk(f0 ) )− r (f1 ) ×s(zk+1 /zk(f0 ) ). We have r−1 (f0 ) − r−1 (f1 ) (f0 ) r s(zk+1 /zk ) z(f1 ) =xs w = x2(r−1) (xs+1 w)− r s(x2r−1 ) (zk ) z
k+1 (f0 ) −2r+s+1 =x w k
x2(r−1) (xs+1 w)− = 1 − x2(r−1)
r−1 r
s(x−1 ),
where s(x−1 ) is in the sense of (C.11). Taking this into account and collecting (S1) and (S2) we have (f1 ) (f0 ) • (S): Sk+1 S0 /Sk S0 = I
−x2(r−1)
II
1 − x−s+1 zk(b) /w Y 1 − x2r−s−1 zk(b) /w
x2(r−1)
b6=f1
(b) 1 − x−s zk+1 /w (b) 1 − x2r−s−2 zk+1 /w
.
From (U), (F), (T) and (S) we can conclude that the residues of (f1 ) (1) (a) (0 ≤ p ≤ s) Jk+1 (zi , . . . , zi+m ) (f
)
p+1 zk+p+1 =xs−p w
(f0 ,...,fs+1 ) at w = x2r−s−1 zk(f0 ) gives the term in Ik,...,k+s+1 that is corresponding to the cycle 2r−s−1 . This completes the proof of the commutativity [Wt(1) , X α (λ)] = 0. |w| = x
Acknowledgement. B. F. and A. O. would like to thank the RIMS for hospitality throughout their stay, during which this work was done. T.M. thanks IHP and ENS where he stayed in the last stage of the work. This work is partially supported by Grant-in-Aid for Scientific Research on Priority Areas 231, the Ministry of Education, Science and Culture.
References 1. Jimbo, M., Miwa, T. and Okado, M.: Solvable lattice models whose states are dominant integral weights of A(1) n−1 . Lett. Math. Phys. 14, 123–131 (1987) 2. Felder, G.: BRST approach to minimal models. Nucl. Phys. B 317, 215–236 (1989) 3. Bernard, D. and Felder, G.: BRST Fock representations and BRST cohomology in SL(2) current algebra. Commun. Math. Phys. 127, 145 (1990) 4. Feigin, B.L. and Frenkel, E.V.: Affine Kac-Moody algebras and semi-infinite flag manifold. Commun. Math. Phys. 128, 161–189 (1990) 5. Bouwknegt, P., McCarthy, J. and Pilch, K.: Quantum group structure in the Fock space resolution of sbl(n) representations. Commun. Math. Phys. 131, 125–155 (1990) 6. Bouwknegt, P., McCarthy, J. and Pilch, K.: Free field approach to 2-dimensional conformal field theories. Prog. Theoret. Phys. Supplement 102, 67–135 (1990) 7. Jimbo, M., Miki, M., Miwa, T. and Nakayashiki, A.: Correlation functions of the XXZ model for 1 < −1. Phys. Lett. A 168, 256–263 (1992) 8. Jimbo, M. and Miwa, T.: Algebraic Analysis of Solvable Lattice Models. CBMS Regional Conference Series in Mathematics, vol. 85, Providence, RI: AMS, 1994 9. Lukyanov, S.. Free field representation for massive integrable models. Commun. Math. Phys. 167, 183– 226 (1995)
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10. Lukyanov, S. and Pugai, Y.: Multi-point local height probabilities in the integrable RSOS model. Nucl. Phys. B [FS] 473, 631–658 (1996) 11. Asai, Y., Jimbo, M., Miwa, T. and Pugai, Y.: Bosonization of vertex operators for the A(1) n−1 face model. J. Phys. A 29, 6595–6616 (1996) 12. Shiraishi, J., Kubo, H., Awata, H. and Odake, S.: A quantum deformation of the Virasoro algebra and the Macdonald symmetric functions. Lett. Math. Phys. 38, 33–51 (1996) 13. Jimbo, M., Lashkevich, M., Miwa, T. and Pugai, Y.: Lukyanov’s screening operators for the deformed Virasoro algebra. Phys. Lett. A 229, 285–292 (1997) 14. Awata, H., Kubo, H., Odake, S. and Shiraishi, J.: Quantum WN algebras and Macdonald polynomials. 1995. q-alg/9508011 15. Feigin, B.L. and Frenkel, E.V.: Quantum W-algebras and elliptic algebras. 1995. q-alg/9508009 16. Feigin, B.L. and Odesskii, A.V.: Vector bundles on elliptic curve and Sklyanin algebras. 1995. RIMS1032, q-alg/9509021 17. Fateev, V.A., Lukyanov, S.L.: The models of two-dimensional conformal quantum field theory with Zn symmetry. Int. Jour. Mod. Phys. A. 3, 507–520 (1988) Communicated by G. Felder
Commun. Math. Phys. 191, 543 – 583 (1998)
Communications in
Mathematical Physics c Springer-Verlag 1998
Meander Determinants P. Di Francesco Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, N.C. 27599-3250, USA, Service de Physique Th´eorique, C.E.A. Saclay, F-91191 Gif sur Yvette Cedex, France. E-mail: [email protected] Received: 13 December 1996 / Accepted: 21 May 1997
Abstract: We prove a determinantal formula for quantities related to the problem of enumeration of (semi-) meanders, namely the topologically inequivalent planar configurations of non-self-intersecting loops crossing a given (half-) line through a given number of points. This is done by the explicit Gram-Schmidt orthogonalization of certain bases of subspaces of the Temperley-Lieb algebra. 1. Introduction The meander problem consists in counting the number Mn of meanders of order 2n, i.e. of inequivalent configurations of a closed non-self-intersecting loop crossing an infinite line through 2n points. The infinite line may be viewed as a river flowing from east to west, and the loop as a closed circuit crossing this river through 2n bridges, hence the name “meander”, although here the river and the road play symmetric roles. Two configurations are considered as equivalent if they are smooth deformations of one another. The meander problem probably first arose in the work of Poincar´e about differential geometry. Since then, it has emerged in many different contexts, such as mathematics, physics, computer science [1–3] and even fine arts [3]. The problem was recently reactualized by Arnold, in relation with Hilbert’s 16th problem [4]. Meanders also emerged in the classification of 3-manifolds [5]. More recently, random matrix model techniques, borrowed from quantum field theory, were applied to this problem [7–9]. In the present paper, we rather adopt the purely algebraic approach advocated in [9], based on a pictorial representation of the elements of the Temperley-Lieb algebra [10] (see also P. Martin’s book [11] for an elementary introduction) using strings (each element is a sort of domino with string-ends on its boundary, and elements are multiplied by connecting the string-ends of the corresponding dominos), by means of which the meanders are constructed. A particular basis (set of basic dominos) of the TemperleyLieb algebra will provide us with the building blocks for the construction of meanders, or
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some of their generalizations, the semi-meanders, introduced in [7]. Roughly speaking, a (multi-component, i.e. made of possibly several non-intersecting roads) meander is obtained as the concatenation of two dominos, and the identification of their free stringends: this is exactly the manipulation involved when evaluating the standard bilinear form of the Temperley-Lieb algebra on these two dominos. Moreover, the value of the bilinear form is simply q L , q a given complex number, and L the total number of loops formed by the connection of the strings, namely the number of loops in the corresponding meander. Theorem 1 below is a formula expressing the determinant of the Gram matrix of this basis of the Temperley-Lieb algebra, and was first derived in [9] (an algorithm for its computation was also given in [5]). By choosing a particular subset of dominos (hence a particular basis of some subspace of the Temperley-Lieb algebra), we may obtain meanders with more specific details: the semi-meanders with fixed winding numbers are some of these. To obtain the latter, consider the following semi-meander problem: enumerate the inequivalent planar configurations of a loop crossing a half-line through a given number of points n. The loop of such a semi-meander may freely wind around the origin of the half-line (interpreted as the source of the river), and we can define a winding number associated to this. The (multi-component, i.e. with possibly several non-intersecting roads) semi-meanders with fixed winding number w may be obtained as the concatenation of particular elements (dominos) of the Temperley-Lieb algebra, namely those with exactly w of the n strings going across the domino. The Theorem 2 below is the generalization to semi-meanders of the abovementioned meander determinant formula. The latter was only conjectured in [9], in a slightly different form. The paper is organized as follows. In Sect. 2, we recall a few definitions and facts on (semi-) meanders, in particular their various formulations as (i) superpositions of two (open) arch configurations (ii) superposition of two (open) walk diagrams, and we state the main results of the paper (Theorems 1 and 2), in the form of determinant formulas. Section 3 is devoted to the proof of the formula for the meander determinant. The proof relies on the interpretation of any meander as the product of two elements of the Temperley-Lieb algebra. After displaying the various mappings between arch configurations, walk diagrams and reduced elements of the Temperley-Lieb algebra, we reformulate the meander determinant as the Gram determinant of a particular basis of the Temperley-Lieb algebra, or rather of one of its ideals. The proof is then carried out, by performing the explicit Gram-Schmidt orthogonalization of this basis (Proposition 1). This appears in fact as the consequence of a stronger statement regarding the orthogonalization of all the products of any two basis elements (Lemma 1). The computation of the meander determinant is then a combinatorial exercise (Proposition 2) in rearranging all the normalization factors introduced in the orthonormalization process, which we carry out by performing some mapping of decorated walk diagrams. In Sect. 4, we turn to the semi-meander generalization. We follow the same strategy as in Sect. 3, with a number of complications, due to the fact we now deal with a subspace of the Temperley-Lieb algebra, which is not an ideal, i.e. has no good multiplication properties between its elements. Nevertheless, we are still able to perform the explicit Gram-Schmidt orthogonalization of our initial basis (Proposition 3 and Lemmas 2, 3, 4). The determinant formula then follows from a rearrangement (Propositions 4, 5 and 6) of the normalization factors introduced in the orthonormalization process. We gather in Sect. 5 a few concluding remarks.
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2. Meander determinants: the results 2.1. Arch configurations and meanders. A meander of order1 2n is a planar configuration of a closed non-selfintersecting loop (road) crossing a line (river) through 2n distinct points (bridges), considered up to smooth deformations preserving the topology of the configuration (i.e., preserving the succession of bridges).
a
b Fig. 1. Any meander is obtained as the superimposition of a top (a) and bottom (b) arch configurations of same order (2n = 10 here). An arch configuration is a planar pairing of the (2n) bridges through n non-intersecting arches lying above the river (by convention, we represent the lower configuration b reflected with respect to the river: this will actually be denoted by bt in the following)
The river separates the meander into an upper and a lower planar configuration of n non-intersecting pieces of road (arches) joining the 2n bridges by pairs (see Fig. 1 for an example), respectively contained in the upper and lower half-planes defined by the river. Such a configuration, considered up to the abovementioned equivalence, is called an arch configuration of order 2n. Let A2n denote the set of all arch configurations of order 2n. Let us label the bridges from left to right 1, 2, ..., 2n. The total number cn of arch configurations of order 2n is obtained by considering the leftmost arch, joining say the bridge 1 to the bridge 2j, 1 ≤ j ≤ n. This arch separates the arch configuration into two pieces: the portion below the leftmost arch, and that to the right of this arch. These are two arbitrary P arch configurations of respective orders 2(j − 1) and 2(n − j). Hence we have cn = 1≤j≤n cj−1 cn−j . With c0 = 1, we get |A2n | = cn =
(2n)! (n + 1)!n!
(2.1)
which is the nth Catalan number. Superposing two arbitrary arch configurations a, b ∈ A2n (after a reflection of b w.r.t. the river) will in general lead to a multi-component meander, made of several non-intersecting roads. We denote by κ(a|b) the corresponding number of connected components. 2.2. Open arch configurations and semi-meanders. A semi-meander of order n is a planar configuration of a non-selfintersecting loop (road) crossing a half-line (river with a source) through n distinct points (bridges), up to smooth deformations of the road preserving the topology of the configuration. The main difference with a meander is that 1 In this paper, the order will always refer to the total number of bridges in the configuration. A different convention was adopted in refs. [8, 10], where the order of a meander is rather half its number of bridges.
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the road may now freely wind around the source of the river, therefore the number of bridges needs not be an even integer. We define the winding number of a semi-meander to be the number of pairs of bridges linked by an arch encircling the source of the river (each such arch contributes 1 to the total winding number). Note that a semi-meander of order n may only have a winding number h = n mod 2.
1 2 3 4 5
Fig. 2. Any semi-meander may be viewed as the superimposition of an upper and a lower open arch configuration. Here the initial semi-meander has order n = 5 and winding h = 3. The two open arch configurations on the right have h = 3 open arches. To recover the initial semi-meander, these open arches must be connected two by two, from the right to the left (the arches numbered 5, 4, 3 of the upper configuration are respectively connected to the arches numbered 3, 2, 1 of the lower configuration)
In any given semi-meander with winding number h, the river still separates the configuration into an upper and a lower one (see Fig. 2 for an example), corresponding to the portion lying respectively above and below the river, but these are linked by h arches encircling the source of the river, and connecting h upper parts of bridges to h lower parts of bridges. Let us cut these h arches and extend them so as to form h vertical half-lines on the upper configuration and h vertical half-lines on the lower configuration. The resulting objects are called open arch configurations of order n with h open arches2 . Such an open arch configuration is formed by a line with n distinct points (upper halfbridges) either connected by pairs through arches in the upper half-plane (there are (n − h)/2 such arches), or connected to “infinity” through a vertical half-line (there are h such open arches), with a total of (n + h)/2 arches. We denote by A(h) n the set of open arch configurations of order n with h open arches. Note again that h = n mod 2. In particular, A(0) 2n = A2n . To compute the cardinal cn,h of A(h) n , we concentrate again on the leftmost arch of a given configuration. Two cases may occur: (i) This arch is open. The configuration lying on the right of this arch is an arbitrary open arch configuration of order n − 1 with h − 1 open arches. (ii) This arch connects the bridges 1 and say 2j, thus separating the configuration into two parts: the one below the leftmost arch is an arbitrary arch configuration of order 2(j − 1), whereas the one to the right of bridge 2j is an arbitrary open arch configuration of order n − 2j with h open arches. These are summarized in the following recursion relation: X
[n/2]
cn,h = cn−1,h−1 +
cj−1 cn−2j,h ,
(2.2)
j=1
where cn denotes the Catalan number (2.1), and [x] is the largest integer smaller or equal to x. With the initial condition c2n,0 = cn for all n ≥ 0, this determines the numbers cn,h completely, and we have 2
As before, the order refers to the total number of bridges in the configuration.
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|A(h) n |
= cn,h =
n
−
n−h 2
n , n−h 2 −1
(2.3)
where cn,h are some generalized Catalan numbers. (Note again that A(h) n is only defined if n = h mod 2.) Like in the meander case, given two arbitrary open arch configurations a, b ∈ A(h) n , we may consider their superposition (after reflecting b w.r.t. the river) obtained by gluing their half-bridges, and connecting the upper and lower open arches starting from the rightmost one so as to form h arches encircling the source of the river. This leads in general to a multi-component semi-meander formed of possibly many non-intersecting roads crossing the river, and possibly winding around its source, with a total winding number h. By analogy with the meander case, we still denote by κ(a|b) the resulting number of connected components. 2.3. Meander and semi-meander determinants. With the above definitions, let us introduce, for any given complex number q, the meander and semi-meander matrices G2n (q) and Gn(h) (q) of respective sizes cn × cn and cn,h × cn,h , with entries a, b ∈ A2n , G2n (q) a,b = q κ(a|b) (2.4) (h) κ(a|b) Gn (q) a,b = q a, b ∈ A(h) n . (0) As A(0) 2n = A2n , we have G2n (q) = G2n (q), hence the meander matrix is just a particular case of a semi-meander matrix with h = 0 winding number. Nevertheless, for a clearer exposition, we will distinguish between the two cases. Let us denote by Um (q) the Chebyshev polynomials of the second kind, namely such that sin(m + 1)θ (2.5) Um (2 cos θ) = sin θ for all m ≥ 0. With this definition, and the integer numbers cn,m defined in (2.3), we have the following compact formulas for the determinants of the matrices G2n (q) and Gn(h) (q)
Theorem 1.
n Y a2n,2m det G2n (q) = , Um (q) m=1
(2.6)
a2n,2m = c2n,2m − c2n,2m+2 . Theorem 2. det
Gn(h) (q)
n−h 2 +1
=
Y
a(h) n,m
Um (q)
,
m=1
(2.7)
a(h) n,m = cn,2m+h − cn,2m+2+h + h(cn,2m+h−2 − cn,2m+h ), Theorem 1 was proved in [9], whereas formula (2.7) was only conjectured there (in a slightly different, but equivalent form). In the following, for pedagogical reasons, we will first give a simplified proof of Theorem 1, in the same spirit as [9]. We will then show how to generalize this proof to that of Theorem 2. Clearly, Theorem 2 contains Theorem 1 as the particular case h = 0. Before turning to the proofs of Theorems 1 and
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2 above, we wish to provide the reader with an alternative picture for (open or closed) arch configurations, which will prove useful in the following. The idea is to view an (open or closed) arch configuration of order n as a walk of n steps on a half-line.
2.4. Arch configurations and closed walk diagrams. There is a bijection between the arch configurations of order 2n and the closed paths of 2n steps on a half-line, or rather their two-dimensional extent, which we call a walk diagram of 2n steps. The mapping goes as follows. Let us index by i the portion of river in between two consecutive bridges i and i + 1, 1 ≤ i ≤ 2n − 1, by 0 the portion to the left of the first bridge, and by 2n the portion to the right of the last bridge.
0
1 2
3
4 5 6 7 8
9 10 11 12 13 14 15 16 17 18
Fig. 3. A walk diagram of 18 steps, and the corresponding arch configuration. Each dot corresponds to a segment of river. The height on the walk diagram is given by the number of arches intersected by the vertical dotted line
To each of these we associate the height h(i) equal to the number of arches passing at the vertical of the corresponding portion of river. With this definition, we have h(0) = h(2n) = 0, h(i)− h(i − 1) = ±1, according to whether an arch originates or terminates at the bridge i, and h(i) ≥ 0 for all i. The function h(i) can be thought of as the coordinate of a walker on the half-line after i steps. The two-dimensional extent of the trajectory is simply obtained by joining the consecutive points (i, h(i)) i.e., by plotting the graph of the function h, as illustrated in Fig. 3. We denote by W2n the set of such walk diagrams of 2n steps, with h(0) = h(2n) = 0. In particular, the bijection implies that |W2n | = |A2n | = cn for all n ≥ 0. In the following, we will denote indifferently by the same letter a ∈ A2n or W2n an arch configuration or the corresponding (closed) walk diagram.
2.5. Open arch configurations and open walk diagrams. For all h ≤ n, h = n mod 2, there is a bijection between the set A(h) n of open arch configurations of order n with h open arches and the set of open walk diagrams on a half-line, starting at the origin and ending at height h after n steps.
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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Fig. 4. An open walk diagram of n = 14 steps with final height h = 4, and the corresponding open arch configuration. The height on the walk diagram is given by the number of arches intersected by the dotted lines, plus that of open arches lying on the left of the point considered
Starting from some open arch configuration a ∈ A(h) n , let us label as before by 0, 1, ..., n the portions of river in between consecutive bridges of a (including that to the left of the first bridge, 0 and to the right of the last bridge, n). To each of these, we associate the height h(i) equal to the number of arches passing at the vertical of the corresponding portion of river, plus the total number of open arches originating from the bridges number 1, 2, ..., i, namely the total number of open arches lying to the left of the portion i of river. With this definition, the function i → h(i) satisfies h(0) = 0 ,
h(n) = h ,
h(i) ≥ 0 ,
and h(i + 1) − h(i) = ±1
(2.8)
according to whether an (open or closed) arch originates from the bridge i or a (closed) arch terminates at the bridge i. The function i → h(i), satisfying the properties (2.8), defines a unique walk on the half-line, starting at the origin (height h(0) = 0), and ending after n steps of ±1 in height at height h(n) = h. The graph of the function, (i, h(i)), with consecutive points linked by segments of line, is the two-dimensional extent of such a walk, which we call an open walk diagram of n steps with final height h. We denote by Wn(h) the set of open walk diagrams of n steps with final height h (note that this is only defined for h = n mod 2). In particular, the above bijection implies |Wn(h) | = |A(h) n | = cn,h .
(2.9)
In the following, we will also use indifferently the same letter a to denote an element of Wn(h) or A(h) n whichever picture is more convenient. 3. The Meander Determinant: Proof of Theorem 1 In this section, we give a detailed proof of Theorem 1. We first recall the equivalence between arch configurations and reduced elements of the Temperley-Lieb algebra T Ln (q), or rather a certain left ideal In (q) of T L2n (q), isomorphic to T Ln (q). In the latter language, the meander matrix G2n (q) (2.4) is interpreted as the Gram matrix of a basis (called basis 1) of In (q) with respect to the standard bilinear form. The determinant of G2n (q) will be a by-product of the orthogonalization of this matrix.
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3.1. Temperley-Lieb algebra and arch configurations. The arch configurations of order 2n have a direct interpretation in terms of reduced elements of the Temperley-Lieb algebra T Ln (q), for a given complex number q. The latter is best expressed in its pictorial form, as acting on a “comb” of n strings, with the n generators 1, e1 , e2 , ..., en−1 defined as 1
.. ..
1 =
i i+1
ei =
.. ..
i i+1 .
(3.1)
n
The most general element e of T Ln (q) is obtained by composing the generators (3.1) like dominos. The algebra is defined through the following relations between the generators (i) e2i = q ei i = 1, 2, ..., n − 1, (ii) [ei , ej ] = 0 if |i − j| > 1, (iii) ei ei±1 ei = ei i = 1, 2, ..., n − 1.
(3.2)
The relation (ii) expresses the locality of the e’s, namely that the e’s commute whenever they involve distant strings. The relations (i) and (iii) read respectively
(i)
e2i
.. ..
=
.. ..
(iii) ei ei+1 ei =
i = q
i i+1 =
.. . .. .
i i+1 = q ei , (3.3)
i i+1 = ei .
In (i), we have replaced a closed loop by a factor q. Therefore we can think of q as being a weight per connected component of string. In (iii), we have simply “pulled the string” number i + 2. An element e ∈ T Ln (q) is said to be reduced if all its strings Q have been pulled and all its loops removed, and if it is further normalized so as to read i∈I ei for some minimal finite set of indices I. A reduced element is formed of exactly n strings. 1 2 3 4 5 6 7 8 9
18 17 16 15 14 13 12 11 10
1 2 3
4 5 6 7 8 9
10 11 12 13 14 15 16 17 18
Fig. 5. The transformation of a reduced element of T L9 (q) into an arch configuration of order 18. The reduced element reads e3 e4 e2 e5 e3 e1 e6 e4 e2
There is a bijection between the reduced elements of the Temperley-Lieb algebra T Ln (q) and the arch configurations of order 2n. Starting from a reduced element of T Ln (q), we index the left ends of the n strings by 1, 2, ..., n, and the right ends of the
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551
strings 2n, 2n−1, ..., n+1 from top to bottom (see Fig. 5 for an illustration). Interpreting these ends as bridges, and aligning them on a line, we obtain a planar pairing of bridges by means of non-intersecting strings (arches), hence an arch configuration of order 2n. Conversely, we can deform the arches of any arch configuration of order 2n to form a reduced element of T Ln (q). As a consequence, we have dim(T Ln (q)) = cn , as vector space with a basis formed by all the reduced elements. In the following, we will rather use the identification between T Ln (q) and the left ideal In (q) of T L2n (q) generated by the element un = e1 e3 ...e2n−1 , which goes over to reduced elements.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Fig. 6. The arch configuration of order 18 of Fig. 5 is immediately interpreted as an element of the ideal I9 (q) of T L18 (q), by adding a succession of n = 9 strings linking the consecutive upper ends of strings by pairs (for simplicity, the element of T L18 (q) is now read from bottom to top). The corresponding reduced element of I9 (q) reads, from bottom to top, (e3 e9 )(e2 e4 e8 e10 e12 e14 e16 )(e1 e3 e5 e7 e9 e11 e13 e17 ).
Indeed any reduced element of In (q) has a pictorial representation as a set of 2n strings linking by pairs the 2n left and 2n right ends of strings, as illustrated in Fig. 6. The pairing of the right ends of strings is very simple, and represents the right factor e1 e3 ...e2n−1 . It consists of n arches connecting the n pairs of successive right ends of strings. Therefore the 2n left ends of strings are connected among themselves through the n remaining strings. This gives exactly an arch configuration of order 2n.
Fig. 7. Example of a walk diagram in W8 , expressed as the result of four box additions on the fundamental walk a4
The converse construction is best expressed in the walk diagram representation of arch configurations. Let us construct a map ρ from W2n to the set of reduced elements of the ideal In (q) of T L2n (q). Let an denote the fundamental walk diagram of W2n , such that an : h(0) = h(2) = ... = h(2n) = 0
and
h(1) = h(3) = ... = h(2n − 1) = 1. (3.4)
To this diagram, we associate the reduced element un = ρ(an ) = e1 e3 ...e2n−1 .
(3.5)
Now any walk diagram in W2n is obtained from an by successive box additions, illustrated in Fig. 7. A box addition at a minimum i of a ∈ W2n , i.e. where h(i + 1) = h(i−1) = h(i)+1, simply consists in shifting h(i) → h(i)+2, which amounts to formally add a square box which fills the minimum at i and transforms it into a maximum. We
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will denote by a → a + i this operation on a ∈ W2n (this notation keeps track of the point i at the vertical of which the box is added). This enables us to define the length of a walk diagram a ∈ W2n as the number of box additions which have to be performed on the fundamental an to build a. We set |a| = #boxes in a
for all a ∈ W2n .
(3.6)
In particular, we have |a + i | = |a| + 1. The mapping ρ is then defined as ρ(a + i ) = ei ρ(a),
(3.7)
where the box addition is made at the point i (necessarily a minimum of a).
e1
e2
e3 e3
e4
e5
e6
e7
Fig. 8. Example of the mapping µ between an element a ∈ W8 and e = ρ(a) ∈ I4 (q). We read the element e = ρ(a) from the various layers of box additions, using the formula (3.7). Here we get e = (e3 )(e2 e4 e6 )(e1 e3 e5 e7 ) (the parentheses correspond to the successive layers of boxes added.
The most general reduced element e = ρ(a) in In (q), a ∈ W2n , is therefore written as the product over all box additions leading from the fundamental walk an to a, of the corresponding ei ’s. This is illustrated in Fig. 8. The reduced elements of In (q) form a basis (which we call basis 1 from now on) of the corresponding vector space over the complex numbers. We have established that |W2n | = dim(In (q)) = cn . For simplicity, we will adopt the following notation for the basis 1 elements: we write (a)1 = ρ(a)
for any a ∈ W2n .
(3.8)
As an example, the basis 1 for I3 (q) is formed by the c3 = 5 following elements = e1 e 3 e 5 1 = e2 e 1 e 3 e 5 1 = e4 e 1 e 3 e 5 (3.9) 1 = e2 e 4 e 1 e 3 e 5 1 = e3 e 2 e 4 e 1 e 3 e 5 1
indexed by the 5 walk diagrams of W6 . We now show how to reconstruct the string-domino pictorial representation attached to an element of In (q), from the box decomposition of the corresponding walk a ∈ W2n . The idea is to represent the ei ’s forming the fundamental element un (3.5) by boxes as well. Actually each box will have the meaning of a left multiplication by ei , this time acting on 1. Starting from some walk diagram a ∈ W2n , we write it as the result of box
Meander Determinants
553
additions on the empty diagram. To go to the string-domino picture, we have to draw “arches” using the box configurations. This is done by marking each box with a pair of strings as follows →
(3.10)
and by continuing each string with a vertical line ending at some string-end on the border of the corresponding domino.
Fig. 9. The box decomposition of an element a ∈ I5 (q), and the corresponding string-domino picture. Note that the domino is now read from top to bottom, rather than from bottom to top as it used to be in Fig. 6
This is illustrated in Fig. 9, where the strings are represented in thick black lines. 3.2. Gram matrix for the basis 1 of In (q).
Fig. 10. Computing the trace of a reduced element e of T L6 (q): (i)connect the left and right ends of strings (dashed lines) (ii) count the number of connected components of string: we find κ(e) = 3 here. This leads to the trace Tr(e) = q 3
The Temperley-Lieb algebra is endowed with a standard trace, defined on the reduced elements as the number (3.11) Tr(e) = q κ(e) , where κ(e) is the number of connected components of strings after the identification of the left ends of strings with the right ones, as depicted in Fig. 10. This definition extends by linearity to any element of T Ln (q). Given a reduced element e ∈ T Ln (q), we may consider the adjoint et , obtained by reflecting the corresponding arch configuration w.r.t. the river. The corresponding operation on T Ln (q) satisfies eti = ei (the generators are self-adjoint), and (ef )t = f t et for any reduced elements e, f . Taking the adjoint simply
554
P. Di Francesco
reflects the string-domino picture of the corresponding reduced element, and exchanges the left and right ends of strings. Again, this extends to any element of T Ln (q) by linearity. We can now introduce the bilinear form for any e, f ∈ T Ln (q).
(e, f ) = Tr(ef t )
(3.12)
The above definitions extend by restriction to any ideal of the Temperley-Lieb algebra. Let us now concentrate on the ideal In (q) of T L2n (q). Let us consider the Gram matrix of the basis 1, with respect to the bilinear form (3.12), namely the matrix 02n (q), with entries (3.13) 02n (q) a,b = (a)1 , (b)1 = Tr (a)1 (b)t1 , where a, b run over the walk diagrams of W2n which are used to index the corresponding basis 1 elements (a)1 , (b)1 . 1 2 3 4 5 6 7 8 9 10
}b
ft e
t
}a 1 2 3 4 5 6 7 8 9 10
Fig. 11. The bilinear form (e, f ) is obtained by first multiplying e with f t , and then identifying the upper and lower ends of the strings (The bridges numbered 1, 2, ..., 10 are identified), and counting the number of connected components of strings. Here we have created n = 5 simple loops at the connection between the two dominos, and κ(a|b) = 3 other loops, from the superposition of the arch configurations a and b of order 10, corresponding respectively to e and f . Note that b → bt is reflected w.r.t. the river, corresponding to the adjoint in f t . Finally we have Tr(ef t ) = q n+κ(a|b) = q 8 here
But evaluating the matrix element (3.13) just amounts, as illustrated in Fig. 11, to connecting the domino corresponding to (a)1 and the reflection of the domino corresponding to (b)1 , and identifying the left ends of strings (on the bottom of the figure) with the right ones (on top of the figure), and counting the number of connected components of strings. Because of the particular form of the elements of In (q) (see Fig. 6), this procedure will create n loops along the connection line between the two dominos (these loops are formed by the arches connecting consecutive ends of strings) plus an extra κ(a|b) loops, namely those appearing in the superposition of the arch configurations a and (the reflection of) b. Therefore we have (3.14) 02n (q) a,b = q n+κ(a|b) = q n G2n (q) a,b by comparison with the previous definition (2.4) of the meander matrix G2n (q). Hence the meander determinant is simply related to the Gram determinant of the basis 1, through (3.15) det(02n (q)) = q ncn det G2n (q). The remaining subsections of this section will be devoted to the computation of the Gram determinant of the basis 1 of In (q). 3.3. Orthonormalization of the basis 1. In the following, we will compute the Gram determinant (3.15) by performing an explicit Gram-Schmidt orthonormalization of the basis 1 w.r.t. the bilinear form (3.12). The orthonormalization process consists in a change of basis from the basis 1 to another basis, which we call basis 2, satisfying the following properties:
Meander Determinants
555
The basis 2 elements are still indexed by the walk diagrams of W2n , we denote them by (a)2 , a ∈ W2n . (ii) The basis 2 is orthogonal w.r.t. the bilinear form (3.12), namely (a)2 , (b)2 = 0 whenever a 6= b. (iii) The basis 2 elements have all the same norm 1, namely (3.16) for any a ∈ W2n . (a)2 , (a)2 = 1 (i)
The basis 2 elements are constructed as follows. We start from the fundamental element (an )2 , indexed by the fundamental walk of W2n (3.4), and defined as (an )2 = q −n e1 e3 ...e2n−1 .
(3.17)
The normalization factor ensures that the property (iii) above holds for the norm of this element, namely that (3.18) (an )2 , (an )2 = 1. (Indeed, (an )2 (an )t2 = (an )2 , and Tr(an )2 = q −n q n = 1.) As any walk diagram a ∈ W2n is obtained from the fundamental one an by successive box additions, we define the other basis 2 elements by the following box addition rule, which amounts to a recursion. Suppose we have constructed (a)2 for some a ∈ W2n . The following rule gives the element (a + i,` )2 , where a box addition has been performed on a minimum i of a, with h(i + 1) = h(i − 1) = h(i) + 1 = `, the height of the box addition, r µ`+1 (3.19) ei − µ` (a)2 , (a + i,` )2 = µ` where we have used the notation µ` =
U`−1 (q) U` (q)
for ` = 1, 2, 3...
(3.20)
in terms of the Chebyshev polynomials (2.5). Note that the recursion relation Um+1 (q) = qUm (q) − Um−1 (q) translates into the relation 1 1 − µm = µ1 µm+1
for all m ≥ 1.
(3.21)
The rule (3.19) may be viewed as a deformation of the rule (3.7) used to construct the basis 1. However, two new ingredients have appeared: (i) the box addition now depends on the height ` at which it is performed (hence the notation p i,` , to keep track of this height) and (ii) there is an overall change of normalization µ`+1 /µ` . Together with the initial point (3.17), the recursive rule (3.19) determines the basis 2 elements completely. By construction, these elements all have the right factor e1 e3 ...e2n−1 , hence belong to the ideal In (q). Moreover, when expressed on basis 1 elements, they read X Pb,a (b)1 , (3.22) (a)2 = b⊂a b∈W2n
where the matrix elements of P are products of factors involving the µm ’s, and the sum extends only on the walk diagrams b included in a, i.e., such that a is obtained from b by some box additions (this includes the case b = a with no box addition). The change of basis 1 → 2 is therefore triangular, as the walk diagrams a ∈ W2n may be arranged
556
P. Di Francesco
by growing length |a| (3.6), thus making the matrix P of the change of basis upper triangular (this implies that the basis 2 is indeed a basis of In (q)). To distinguish between the two different box additions (3.7) for basis 1 and (3.19) for basis 2, which could both be performed on a given walk diagram a ∈ W2n , representing either a basis 1 or a basis 2 element, we decide to represent (3.7) by grey boxes, whereas (3.19) is represented by white boxes, namely r ei =
and
m
µm+1 e i − µm = µm
(3.23)
i
i
(We have indicated the position i and the height m at which the box acts.) In this pictorial representation, the basis 2 elements of I3 (q) read )2 = µ31 = µ31 2
(
5/2 1/2
= µ1 µ2 (e2 − µ1 )e1 e3 e5 5/2 1/2
2
= µ31
= µ1 µ2 (e4 − µ1 )e1 e3 e5
2
= µ31
= µ21 µ2 (e2 − µ1 )(e4 − µ1 )e1 e3 e5
= µ31
= µ21 µ2 µ3 (e3 − µ2 )(e2 − µ1 )(e4 − µ1 )e1 e3 e5
= µ31 e1 e3 e5
2
1/2 1/2
(3.24) With these definitions, we have the Proposition 1. The basis 2 is orthonormal w.r.t. the bilinear form (3.12), namely for all a, b ∈ W2n . (3.25) (a)2 , (b)2 = δa,b We will first prove by recursion, using box additions, the following Lemma 1. (a)t2 (b)2 = 0
for all a, b ∈ W2n such that |a| ≤ |b| and a 6= b.
(3.26)
Note that this result is stronger than the one for the trace of (a)t2 (b)2 , which is implied in Proposition 1. We learn from Lemma 1 that the product of any two distinct basis 2 elements vanishes. This stronger result is linked to the property that In (q) is an ideal. This remark will take its full strength when we study the semi-meander determinant. For an element (a)2 of basis 2, the length of a, |a|, represents its number of white boxes. We will therefore prove the Lemma 1 by recursion on the white box addition. Suppose that the property P (P) : (a)t2 (b)2 = 0
for all b such that |b| ≥ |a| and b 6= a
(3.27)
holds for some a ∈ W2n . Let us prove that P holds for a + , for any white box addition on a. Pick any walk diagram b ∈ W2n such that |b| ≥ |a + | = |a| + 1.
(3.28)
Meander Determinants
557
We wish to evaluate the quantity (a + i,` )t2 (b)2
(3.29)
and show that it vanishes. The idea is simply to transfer the box addition from a to b, namely use the commutation of ei with ej , |j − i| > 1, to let the white box i,` act on (b)2 (the white box is self-adjoint, and multiplies (a)t2 to the right, hence we can let it act on (b)2 by left multiplication). There are however two problems associated with this transfer: (i) b may not have a minimum at i or (ii) if b has a minimum at i, it may lie at a different height m 6= `. We therefore have to distinguish between the following three possible configurations of b at i (maximum, slope, minimum) (1) b has a maximum at i, namely with h(i + 1) = h(i − 1) = h(i) − 1 = m. This means that b itself is the result of a box addition at i on the walk b0 = b − i,m with h(i + 1) = h(i − 1) = h(i) + 1 = m, and all other h(j) identical to those of b. Hence the white box addition on (b)2 reads r r µ`+1 µ`+1 µm+1 (ei − µ` )(ei − µm )(b − i,m )2 ei − µ` (b)2 = µ` µ` µm r µ`+1 µm+1 = (µ−1 1 − µ` − µm )(ei − µm ) µ` µm (3.30) + (µ−1 − µ )µ ) (b − m m i,m 2 1 r r µ`+1 −1 µ`+1 µm = (µm+1 − µ` )(b)2 + (b − i,m )2 , µ` µ` µm+1 where we have used the relation (3.21), and the property e2i = qei = µ−1 1 ei . In the second line of (3.30), we have reconstructed a white box addition on the minimum of b − i,m at i (first term) up to an additive constant (second term), resulting respectively in the terms (b)2 and (b − i,m )2 of the result. (2) b has an ascending slope (resp. a descending slope) at i, namely with h(i+1)−1 = h(i) = h(i − 1) + 1 = m (resp. h(i + 1) + 1 = h(i) = h(i − 1) − 1 = m). In either case, let us write the white box addition on (b)2 as r µ`+1 √ ei − µ` µ`+1 , (3.31) µ` namely as a term proportional to a grey box addition (multiplication by ei ) plus a constant. But the grey box addition on a white slope of (b)2 has a zero result. Indeed, the slope is itself the result of prior white box additions, hence, in the case of an ascending slope, r
m
..
= ei
. i i+1
r
µm+1 (ei+1 − µm )(ei − µm−1 )... µm−1
µm+1 (1 − µ−1 1 µm + µm µm−1 )ei − µm−1 ei ei+1 ... µm−1 √ = − µm+1 µm−1 ei ei+1 ... , =
(3.32)
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P. Di Francesco
where we have used the relations e2i = µ−1 1 ei and ei ei+1 ei = ei , and where (3.21) has implied the vanishing of the coefficient of ei in the second line of (3.32). We are left with an expression involving the action of a grey box at the point i + 1 (factor ei+1 ) on a white slope of the rest of (b)2 (symbolized by the ... in (3.32)). We can therefore repeat the calculation (3.32) with i → i + 1 (and m → m − 1), and so on, until the “bottom” of the diagram is reached, namely the situation where the slope is formed by the piling up of a grey and a white box:
m
r = ei+m−1
µ2 (ei+m − µ1 )ei+m−1 ... , µ1 ,
(3.33)
i+m = 0 by using ei+m−1 ei+m ei+m−1 = ei+m−1 and e2i+m−1 = µ−1 1 ei+m−1 . The same reasoning applies for a descending slope: such a slope may indeed be viewed as the adjoint of an ascending slope, whereas the white box to be added is self-adjoint. The addition of a white box on any slope of b therefore reduces to the second term of (3.31), namely r µ`+1 √ (3.34) ei − µ` (b)2 = − µ`+1 µ` (b)2 µ` . (3) b has a minimum at i, namely with h(i + 1) = h(i) + 1 = h(i − 1) = m. Then, writing r r r r µ`+1 µ`+1 µm µm+1 µ`+1 × (ei − µm ) + (µm − µ` ), (3.35) e i − µ` = µ` µ` µm+1 µm µ` where we have reconstructed a white box addition at point i and height m in the first term, we simply get r r r µ`+1 µ`+1 µm µ`+1 (b + i,m )2 + (µm − µ` )(b)2 . (3.36) ei − µ` (b)2 = µ` µ` µm+1 µ` These three situations are summarized in the following recursion relation, according to the configuration of b at i, respectively denoted δb,max(i,m) (case (1)) and δb,min(i,m) (case (3)) r µ`+1 √ (ei − µ` )(b)2 = − µ`+1 µ` (b)2 µ` r r µ`+1 −1 µ`+1 µm + δb,max(i,m) µ (b)2 + (b − i,m )2 µ` m+1 µ` µm+1 r r µ`+1 µm µ`+1 + δb,min(i,m) (b + i,m )2 + µm (b)2 . µ` µm+1 µ` (3.37) In all cases (1-3), this enables us to reexpress r X µ`+1 (ei − µ` )(b)2 = λb0 (a)t2 (b0 )2 (3.38) (a + i,` )t2 (b)2 = (a)t2 µ` 0 b ∈W2n
Meander Determinants
559
as a linear combination of terms of the form (a)t2 (b0 )2 , where b0 = b, b + or b − , hence with |b0 | ≥ |b| − 1. But, by hypothesis (3.28), we have |b| ≥ |a| + 1, hence |b0 | ≥ |a|. We can therefore apply the recursion hypothesis P (3.27) to each of the products (a)t2 (b0 )2 in (3.38), which must then vanish, and we finally get a zero answer for (a + )t2 (b)2 . This establishes the property P (3.27) for a + , under the assumption that it is satisfied for a. To complete the recursion, we have to establish the property P (3.27) for the initial point a = an . It will then hold for any a ∈ W2n . Let us prove that for all b ∈ W2n such that b 6= an .
(an )t2 (b)2 = 0
(3.39)
(The condition that |b| ≥ |an | = 0 does not give any restriction on b.) We have to evaluate the product e1 e3 ...e2n−1 (b)2 , namely the addition of a row of n grey boxes to b. As those cover the whole width of b, there is at least one such grey box which acts on a white slope of b (otherwise, b should have no slope, hence would be equal to an ). But in Eqs. (3.32)-(3.33) above, we have proved that the addition of a grey box on a white slope of (b)2 yields a zero answer. This completes the proof of (3.39), and the Lemma 1 follows by recursion. To prove the Proposition 1, we note that by symmetry of the bilinear form (3.12) (a)2 , (b)2 = Tr (a)2 (b)t2 = Tr (b)t2 (a)2 (3.40) = (b)2 , (a)2 = Tr (a)t2 (b)2 . If a 6= b we immediately get (a)2 , (b)2 = 0 by applying Lemma 1 to (a, b) if |a| < |b| or (b, a) if |a| > |b|, and either of the two if |a| = |b|. This gives the orthogonality of the distinct basis 2 elements w.r.t. the bilinear form (3.12). The norm of (a)2 , a ∈ W2n , is easily computed by recursion. We have already seen ) , (a ) = 1. Suppose in (3.18) that the fundamental basis 2 element has norm (a n 2 n 2 that (a)2 , (a)2 = 1 for some a ∈ W2n . Let us compute the norm of the element (a + )2 , for some white box addition on a. We have µ`+1 (a + i,` )t2 (a + i,` )2 = (a)t2 (ei − µ` )2 (a)2 µ` r r µ`+1 −1 µ`+1 t = (a)2 1 + (µ − µ` ) (ei − µ` ) (a)2 µ` `+1 µ` (3.41) r µ `+1 = (a)t2 (a)2 + (µ−1 − µ` )(a)t2 (a + i,` )2 µ` `+1 = (a)t2 (a)2 , t where we have used e2i = µ−1 1 ei and the relation (3.21) in the first line, and (a)2 (a + by direct application of Lemma 2. Equation (3.41) implies that )2 = 0 in the second, (a + )2 , (a + )2 = (a)2 , (a)2 = 1, by the recursion hypothesis. Together with the initial point (3.18), this proves that (a)2 , (a)2 = 1, for all a ∈ W2n . Proposition 1 follows.
3.4. The meander determinant. The meander determinant (3.15) follows from the Gram determinant for the basis 1. Let us now compute the latter. The basis 2 being orthonormal, its Gram matrix is the cn × cn identity matrix I. The change of basis from basis 1 to 2 (with the upper triangular matrix P (3.22)) therefore reads P 02n (q)P t = I.
(3.42)
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P. Di Francesco
Hence we have det 02n (q) = (det P )−2 . As P is an upper triangular matrix, only the diagonal elements Pa,a enter the determinant formula. From the definition of the basis 2 elements by white box additions (3.19) on the fundamental (an )2 , we immediately get that the matrix elements Pa,a satisfy the recursion r µ`+1 Pa+i,` ,a+i,` = Pa,a (3.43) µ` With the initial condition Pan ,an = µn1 for the fundamental walk diagram, this determines the Pa,a completely. We have Y µ`+1 2 Pa,a = µ2n . (3.44) 1 µ` boxes of a
2 4 3 2 1 1
Fig. 12. The decomposition of a walk a ∈ W12 into strips of white boxes. There are n = 6 such strips, with respective lengths 2, 4, 3, 2, 1 and 1 (note that an empty strip has by definition length 1)
The boxes of any a ∈ W2n can be arranged into n strips, as illustrated in Fig. 12, namely n diagonal lines of boxes of increasing consecutive heights and positions. Each such line has an upper end, the top of the rightmost box in the line. Let us call the height of this end the length of the corresponding strip. For instance, the fundamental diagram is formed of n strips of length 1. With this definition, we simply get Y 2 = µn1 µ` , (3.45) Pa,a strips of a
where, in the product over the n strips of a, ` stands for the corresponding strip length. The Gram determinant of the basis 1 reads then Y Y −2 n det 02n (q) = Pa,a = µ−nc µ−1 (3.46) 1 ` . a∈W2n
strips of all a∈W2n
We also get a formula for the meander determinant, using (3.15) Y det G2n (q) = µ−1 ` .
(3.47)
strips of all walks ∈W2n
This can be rewritten as det G2n (q) =
n Y m=1
µm
−s2n,m
,
(3.48)
Meander Determinants
561
where s2n,m denotes the total number of strips of length m in all the walk diagrams of order 2n, W2n . The formula of Theorem 1 will follow from the explicit computation of the numbers s2n,m . We have the Proposition 2.
s2n,m = c2n,2m =
2n 2n − . n−m n−m−1
(3.49)
This will be proved by establishing a bijection between the walks a ∈ W2n with a marked end of strip at height m, and the walks of 2n steps on a half-line starting at the (2m) . The origin h(0) = 0 and ending at height h(2n) = 2m, namely the elements b ∈ W2n (2m) cardinal of the latter set being equal to |W2n | = c2n,2m (see eq.(2.9)), Proposition 2 will follow.
R
L
h=2m
a=
h=m h=0 i
R
L b=
h=m h=0 i
i’
(2m) Fig. 13. The map from any a ∈ W2n to a walk b ∈ W2n with a marked end of strip at height m. i is the rightmost intersection of the line h = m with a at an ascending slope. The walk a is cut into two parts: (m) . We have b = LR¯ ∈ W2n , with the left L ∈ Wi(m) , and the right R, such that its reflection R¯ ∈ W2n−i the marked point at i. If h(i + 1) = m − 1 (not the case in the present figure), this is the desired walk of W2n with a marked end of strip at height m. If h(i + 1) = m + 1 (the case of the present figure), we migrate i → i0 =min{j > i|h(j) = m = h(j + 1) + 1}, and mark i0 . The migration is indicated by an arrow. The corresponding strip of length m has also been represented
(2m) Let us consider any walk a ∈ W2n . As shown in Fig. 13, the line h = m intersects the walk a at least once along an ascending slope (at some point j, where h(j) = m and h(j − 1) = m − 1 on a). Let i denote the position of the rightmost3 such intersection, namely i =max{j|h(j) = m = h(j − 1) + 1}. Cutting the walk a at the point (i, h(i) = m) separates the walk into a left part L ∈ Wi(m) and a right part R, which may be viewed (m) as an element of W2n−i (see Fig. 13). Indeed, from the definition of i, the walk R stays above the line h = m until its end: subtracting m from all its heights, and counting its (m) . steps from 0 to 2n − i (instead of from i to 2n) expresses R as an element of W2n ¯ i.e. describing it in the opposite direction (R¯ is a walk on the half-line Reflecting R → R, 3
The fact that we take the rightmost intersection here is responsible for the bijectivity of the mapping.
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P. Di Francesco
starting at height m and ending at height 0 after 2n − i steps), and composing L and ¯ i.e. attaching the origin of R¯ to the end of L, we form a walk b = LR¯ ∈ W2n (see R, Fig. 13). In this walk, we have h(i) = m. If h(i + 1) = m − 1, i is an end of strip of height m, which we mark. If h(i + 1) = m + 1, i cannot be an end of strip. Nevertheless, we just have to consider the smallest point i0 > i such that h(i0 ) = m = h(i0 + 1) + 1, which always exists, as the walk a goes back to height 0 at position 2n. This point i0 is an end of strip at height m, which we mark. Conversely, let us start from some a ∈ W2n with a marked end of strip at position i and height m. By definition, this end of strip satisfies h(i) = m and h(i + 1) = m − 1. If i is a maximum of a, namely h(i − 1) = m − 1, it separates the walk a into a left part L and a right part R. The left part is a walk on the half-line, ending at height m after i steps, hence L ∈ Wi(m) . The right part R is a walk on the half-line starting at height m and ending at the origin, after 2n − i steps. The reflected walk R¯ is obtained by describing R in the opposite direction, namely starting from the origin, and ending at height m, after (m) . Now if we compose the walks L and 2n − i steps. Hence we can write that R¯ ∈ W2n−i (2m) R¯ (attach the origin of R¯ to the end of L), the resulting walk b = LR¯ ∈ W2n , and due to the fact that i was a maximum of a, we have h(2n − 1) = 2m − 1 and h(2n) = 2m in b. If i is not a maximum of a, we first migrate the marked point from i to the largest value i0 < i, such that h(i0 ) = m = h(i0 − 1) + 1 (the closest ascending slope at height m to the left of i). Then we apply the previous cutting, reflecting and pasting procedure (2m) , with the particular property that at the point i0 . This produces a walk b = LR¯ ∈ W2n h(2n − 1) = 2m + 1 and h(2n) = 2m on b. We have in fact established a more refined mapping between (i) the a ∈ W2n with a marked maximum, of height m (namely at a point i such that h(i) = m = h(i + 1) + 1 = (2m−1) , (ii) the a ∈ W2n with a marked descending h(i − 1) + 1) and the b ∈ W2n−1 slope at height m (i such that h(i) = m = h(i − 1) − 1 = h(i + 1) + 1) and the (2m+1) . This forms a bijection between the walks a ∈ W2n with a marked end of b ∈ W2n−1 (2m) strip (either a maximum or a descending slope) and the walks b ∈ W2n (with either h(2n − 1) = 2m − 1 or h(2n − 1) = 2m + 1). Hence we conclude that (2m) | = c2n,2m , s2n,m = |W2n
(3.50)
which proves Proposition 2. To translate the result (3.49) of Proposition 2 into the formula of Theorem 1, using (3.48), we simply have to reexpress the meander determinant in terms of the Chebyshev polynomials Um (q), using µm = Um−1 /Um . Equation (3.48) becomes det G2n (q) =
s2n,m n n Y Y s2n,m −s2n,m+1 Um = Um Um−1
m=1
by noting that s2n,n+1 =
2n −1
−
(3.51)
m=1
2n −2
= 0. This takes exactly the form of (2.6), with
a2n,2m = s2n,m − s2n,m+1 = c2n,2m − c2n,2m+2 , which completes the proof of Theorem 1.
(3.52)
Meander Determinants
563
4. The Semi-Meander Determinant: Proof of Theorem 2 The strategy of the proof of Theorem 2 is exactly the same as for Theorem 1. It is based on the representation of open arch configurations by a particular set of reduced elements of the Temperley-Lieb algebra, forming the basis (still called basis 1, but not to be confused with that of previous section) of a vector subspace thereof. The semi-meander determinant is then expressed in terms of the Gram determinant of this basis 1. The next step is the explicit Gram-Schmidt orthogonalization of this basis, defining another basis, called basis 2. The semi-meander determinant is then computed by using the change of basis 1 → 2. 4.1. Temperley-Lieb algebra and open arch configurations. The open arch configurations of A(h) n , with order n and with h open arches, can be represented by some particular reduced elements of the Temperley-Lieb algebra T Ln (q).
Fig. 14. The interpretation of an open arch configuration of order n = 15 and with h = 3 open arches (right diagram) as a reduced element of T L15 (q) (left diagram). Note that exactly h = 3 strings go across the domino, namely link three lower to (the three rightmost) upper ends. The linking of the upper ends of the domino is made through (n − h)/2 = 6 strings connecting consecutive ends by pairs
In Fig. 14, we have represented in the string-domino pictorial representation the domino corresponding to a reduced element of the Temperley-Lieb algebra, immediately interpretable as an open arch configuration. Starting from a ∈ A(h) n , let us construct an element, still denoted4 by (a)1 of T Ln (q): representing the corresponding domino as acting from bottom to top, the connection of its n lower ends of strings is realized through the closed arches of a, whereas the h open arches just go across the domino, and connect h of the lower ends to the h rightmost upper ends of strings. The remaining n − h ends are then connected by consecutive pairs like in the meander case. This construction establishes a bijection between A(h) n and the reduced elements of T Ln (q) with exactly h strings connecting lower ends to the h rightmost upper ends, and (n − h)/2 strings connecting the remaining n−h upper ends by consecutive pairs. Let us denote by In(h) (q) the vector space spanned by these reduced elements. From now on, we will refer to the basis {(a)1 |a ∈ A(h) n } as the basis 1. Like in the meander case, the basis 1 is best expressed in the equivalent language (h) of walk diagrams a ∈ Wn(h) . Let a(h) n be the fundamental element of Wn , with h(0) = h(2) = ... = h(n − h) = 0, h(1) = h(3) = ... = h(n − h − 1) = 1 and h(n − h + j) = j for j = 1, 2, ..., h. Any a ∈ Wn(h) may be viewed as the result of box additions on the (h) fundamental a(h) n . The construction of (a)1 , a ∈ Wn is performed recursively. We first set (4.1) (a(h) n )1 = e1 e3 ...en−h−1 , 4 Here we adopt the same notation for elements of T L (q) corresponding to open arch configurations as n that used before for closed arch configurations. These will correspond to another basis {(a)1 } for a ∈ A(h) n , which we will refer to again as the basis 1. This should not be confusing, as we are only dealing with the open arch case from now on.
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P. Di Francesco
and then for a box addition at position i, we set (a + i )1 = ei (a)1 .
(4.2)
As an example, the basis 1 elements for I4(2) read
= e1
= e2 e 1
1
1
(4.3)
= e3 e 2 e 1 1
where we have represented the boxes added on the walk diagrams.
Fig. 15. The string-domino picture corresponding to the box decomposition of an open walk diagram (3) . Note that exactly 3 strings join upper and lower ends. The domino is rather read from top to a ∈ W11 bottom, as opposed to the case of Fig. 14, where it is read from bottom to top
To make direct contact with the string-domino pictorial representation, we may attach to the box decomposition of any walk diagram a ∈ Wn(h) a domino using the same rule as in Sect. 3.1, namely represent all the boxes corresponding to left multiplications by ei (including those of the fundamental element a(h) n ), and decorate them by a horizontal double line (string), as in (3.10). The picture is then completed by drawing vertical strings joining the string ends on the upper and lower borders of the domino. This is illustrated in Fig. 15, where the strings are represented in thick black lines. The main and new difficulty here, in comparison with the former meander case, is that these reduced elements of T Ln (q) do not form an ideal5 . For instance, we have listed in (4.3) the basis 1 elements for I4(2) (q). If we multiply the first (fundamental) element by the third one, we find (e1 )(e3 e2 e1 ) = e1 e3 , which does not belong to the space I4(2) (q) (there is no string connecting lower and upper ends in e1 e3 , whereas there must be 2 such strings in any element of I4(2) (q)), which is therefore not an ideal. Nevertheless, we can still form the Gram matrix 0(h) n (q) for the basis 1, by using the restriction to In(h) (q) of the bilinear form (3.12). This reads
0(h) n (q)
a,b
= (a)1 , (b)1
for a, b ∈ A(h) n .
(4.4)
5 This will be responsible for the absence of a generalization of the Lemma 1 of Sect. 3.3 for the present case.
Meander Determinants
565
t
(b)1
a
(a) 1
b
t
Fig. 16. Computation of (a)1 , (b)1 . We put the reflected domino (b)t1 on top of the domino (a)1 (here, (3) ). The upper ends are then identified one by one to the lower ends of strings. Counting the loops a, b ∈ W11 formed yields: (n−h)/2 = 4 central loops formed at the connection between the two dominos, plus κ(a|b) = 3 loops coming from the superposition of the open arch configurations a and bt (reflected w.r.t. the river). This gives finally (a)1 , (b)1 = q 7
As illustrated in Fig. 16, to compute (a)1 , (b)1 , we glue the dominos (a)1 and the reflected (b)t1 , identify the upper and lower string ends, and count the number of resulting connected components. The connection of the two dominos creates (n − h)/2 loops, from the strings connecting the upper ends by consecutive pairs on (a)1 and (b)1 . The remaining part simply creates κ(a|b) loops, from the superposition of the open arch configurations a and bt (reflected w.r.t. the river), and the connection of their h open arches (see Fig. 16). Hence the Gram matrix for the basis 1 of In(h) (q) is simply related to the semi-meander matrix (2.4), through n−h n−h 2 +κ(a|b) = q 2 (4.5) Gn(h) (q) a,b . [0(h) n (q) a,b = q The semi-meander determinant is therefore related to the Gram determinant of the basis 1 through n−h 2 cn,h
det Gn(h) (q) = µ1
det 0(h) n (q).
(4.6)
4.2. Orthogonalization of the basis 1. In this section, we introduce a basis 2 of In(h) (q), still indexed by a ∈ Wn(h) , which will be orthonormal with respect to the bilinear form (3.12). Like in the meander case, the basis 2 will be defined recursively through box additions. We start from the basic element n/2
(h) (4.7) (a(h) n )2 = µ1 (an )1 , n+h (h) −n n−h where the normalization ensures that (a(h) q 2 q 2 = 1, where we n )2 , (an )2 = q have counted the contributions of the (n − h)/2 loops formed by the strings pairing upper ends by consecutive pairs on (a)1 , and that of the κ(a|a) = (n + h)/2 loops created by the superposition of a with its own reflection at . To proceed, we need to define the concept of floor of a walk diagram a ∈ Wn(h) . Let us denote by h(i), i = 0, 1, 2, ..., n the heights of a, with h(0) = 0 and h(n) = h. The floor of a is yet another diagram f (a) ∈ Wn(h) , such that f (a) ⊂ a, and with heights h0 (i), i = 0, 1, 2, ..., n, defined as follows. Let us denote by J the set of integers
J = {j ∈ {0, 1, ..., n} such that h(k) ≥ h(j) , ∀ k ≥ j}.
(4.8)
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P. Di Francesco
J3 J4
} } }
J2
}
J1
}
J0 0
4 5 6
12 13 14 16 17 19 20
(6) Fig. 17. A diagram a ∈ W20 (thick black line) and the construction of its floor f (a) ⊂ a. The segments J0 , J1 , ..., J4 of positions forming J are indicated by dotted lines. The floor f (a) is represented filled with grey boxes. The boxes in between f (a) and a are represented in white. The floor-ends have positions 0, 4, 6, 12, 14, 16, 17, 19, 20
As illustrated in Fig. 17, this set J is clearly the union of ordered segments of positions, of the form j = J0 ∪ J1 ∪ ... ∪ Jk , with Ji = {ji , ji + 1, ji + 2, ..., ji + ni }, for some integers ni and ji , i = 0, ..., k. These segments correspond to the ascending slopes of a such that no point on their right has a lower height. With these notations, the floor f (a) of a is defined to have the heights h0 (j), j = 0, 1, ..., n, according to the following rules: h0 (j) = h(j) 0
∀ j ∈ J,
0
h (ji + ni + 2r) = h (ji + ni ) 0
∀ r ≥ 0 with 2r ≤ ji+1 − ji − ni ,
0
h (ji + ni + 2r − 1) = h (ji + ni ) − 1
∀ r ≥ 1 with 2r − 1 ≤ (ji+1 − ji − ni ). (4.9) This is valid for all i = 1, 2, ..., k. For i = 0, we have to be more careful, as the leftmost floor piece has a different status. If J0 6= {0} (this leftmost floor piece is empty), then (4.9) is valid for i = 0 as well. If J0 = {0} (this leftmost floor piece is not empty: this is the case in Fig. 17), we have to add the values h0 (0) = h0 (2) = · · · = h0 (j1 ) = 0, h0 (1) = h0 (3) = · · · = h0 (j1 − 1) = 1.
(4.10)
The floor diagram is represented filled with grey boxes in Fig. 17. The floor diagram f (a) is in fact a succession of horizontal broken lines, with heights alternating h(ji +ni ) = `+1, h(ji + ni + 1) = `, h(ji + ni + 2) = ` + 1,..., h(ji+1 ) = ` + 1, on the intermediate positions in between the segments Ji and Ji+1 . These are separated by ascending slopes (along the segments Ji ). For each such intermediate floor Fi , we define the floor height to be the number ` = h(ji + ni − 1) = h(ji + ni + 1) = ... = h(ji+1 ) − 1, for i ≥ 1. The leftmost floor F0 , of height 0 if J0 = {0}, is a little different as we have ` = 0 = h(j0 = 0) = h(2) = ... = h(j1 ) from (4.10). We will also refer to these intermediate floors as simply the floors of a, for which this decomposition is implied. The endpoints with positions ji + ni and ji+1 (and equal height h(ji + ni ) = h(ji+1 ) except maybe for the rightmost floor-end) of each of these floors will be called floor-ends in the following. To define the basis 2 of In(h) (q), we will need a pictorial representation of the walk diagrams a ∈ Wn(h) in which the floor f (a) is also represented. As in Sect. 3, we adopt the representation (3.23) by grey and white boxes ofpthe left mutliplications of a reduced element of In(h) (q) by respectively ei at position i or µm+1 /µm (ei −µm ) on a minimum of height m and position i. The basis 2 elements then correspond to
Meander Determinants
567
(i) grey box additions for all the boxes forming the floor f (a), including the basic boxes forming a(h) n (see below) (ii) white box additions for all the superstructures of a above its floor f (a). There is however a final subtlety with the height of these white boxes, which is counted along strips, w.r.t. the grey floor. In the case (4.7) of the fundamental diagram, the representation is simply a(h) n
n/2
2
= µ1
....
(4.11)
n/2
= µ1 e1 e3 ...en−h−1 (h) as the floor of this element is simply f (a(h) n ) = an , and we have represented the basic grey boxes under the floor. The other elements of the basis 2 are obtained by white box additions on a(h) n 2 . The novelty, when compared to the case of Sect. 3, is that some box additions may create a new floor, namely change previously added white boxes into grey ones. In general, the best way to construct the basis 2 elements, is to first list all the walk diagrams a ∈ Wn(h) , represent them together with their floor f (a) ⊂ a, and then write the corresponding products of grey and white boxes. This is illustrated now in the case of W6(2) . = µ31 = µ31 e1 e3 , 2 5/2 1/2 = µ31 = µ1 µ2 (e2 − µ1 )e1 e3 , 2 5/2 1/2 = µ31 = µ1 µ2 (e4 − µ1 )e1 e3 , 2 = µ31 = µ21 µ2 (e2 − µ1 )(e4 − µ1 )e1 e3
2 5/2 1/2
= µ31
= µ1 µ2 (e5 − µ1 )e4 e1 e3 ,
= µ31
= µ21 µ2 (e2 − µ1 )(e5 − µ1 )e4 e1 e3 ,
2
2
= µ31 2 1/2 1/2
= µ21 µ2 µ3 (e3 − µ2 )(e2 − µ1 )(e4 − µ1 )e1 e3 ,
= µ21 µ2 (e3 − µ1 )(e5 − µ1 )e2 e4 e1 e3 ,
= µ31 2
= µ31
,
2 1/2 1/2
= µ21 µ2 µ3 (e4 − µ2 )(e3 − µ1 )(e5 − µ1 )e2 e4 e1 e3 ,
(4.12)
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P. Di Francesco
where we have represented the grey and white boxes corresponding to each walk diagram. Note e.g. for the last element of (4.12) that the rightmost white box is counted to have height 1 (instead of 2) because this height is the relative height w.r.t. the grey floor on the same strip, which is already at height 1. This construction results in the following change of basis 1 → 2 X Pb,a (b)1 (4.13) (a)2 = f (a)⊂b⊂a
with possibly non-vanishing matrix elements Pb,a only for the walks b ∈ Wn(h) such that b is above the floor of a (f (a) ⊂ b) and below a (b ⊂ a). Like in the meander case of Sect. 3, we can arrange the walk diagrams by growing length (number of boxes, grey and white), and make the matrix P upper triangular. With this definition, the basis 2 satisfies the following Proposition 3. The basis 2 elements are orthonormal with respect to the bilinear form (3.12), namely for all a, b ∈ Wn(h) . (4.14) (a)2 , (b)2 = δa,b This result will be proved in the remainder of this section. Note first, in comparison with the meander case (Proposition 1), that no stronger statement (generalizing Lemma 1) will hold here for the products of elements of In(h) (q). This is because, as mentioned earlier, In(h) (q) is no longer an ideal, hence we have no good control of what the product of two elements of In(h) (q) can be. Thus, instead of resorting to the multiplication of elements, we will directly consider the bilinear form (3.12). The main forthcoming results (Lemmas 2, 3 and 4 below) will deal with reexpressions and simplifications of this bilinear form, when evaluated on two elements of In(h) (q). In particular, Lemma 3 will give a reexpression in terms of the form (3.12), evaluated respectively on elements (h) (q) and Ip (q), which will enable us to use the results of Sect. 3, namely the of In−2p proposition 1, to eventually compute (4.14). To prove Proposition 3, we need a few more definitions. As we are basically dealing with elements of the basis 2, it will be useful to trade the usual notion of walk diagram a ∈ Wn(h) for that of bicolored box diagram, namely the corresponding pictorial representation using grey and white box addition, i.e. the arrangement of grey and white boxes forming (a)2 . For convenience, we still denote by (a)2 the bicolored box diagram corresponding to (a)2 , with a ∈ Wn(h) .
(a)2
s (a) s (a) s (a) s4(a) s (a) 1
2
3
5
Fig. 18. The bicolored box diagram corresponding to an element a ∈ Wn(n−10) for all n ≥ 14. The width of the diagram is w = 5. It is decomposed into 5 strips sj (a), j = 1, 2, ..., 5
Such a bicolored box diagram may be viewed as the succession of strips s1 (a), s2 (a), ..., sw (a), made of a succession of grey, then white boxes of consecutive positions and heights. The number w stands for the number of these strips, namely the width of the base of (a)2 , i.e. the number of grey boxes of height 0 in (a)2 . Note that for all a ∈ Wn(h) ,
Meander Determinants
569
the element (a)2 has width w = (n − h)/2. Moreover, we have the following identity between elements of In(h) (q): n/2
(a)2 = µ1 s1 (a)s2 (a)...sw (a)
(4.15)
by considering the strips (i.e. successions of grey and white boxes) as elements of the Temperley-Lieb algebra. To proceed with the proof of Proposition 3, we will compute the quantity (a)2 , (b)2 = Tr (a)t2 (b)2 . The strategy is the following. We will start by comparing the rightmost strips sw (a) and sw (b) of (a)2 and (b)2 . Both are a succession of grey boxes, topped by one white box, in the form r µ2 (e2w+j−2 − µ1 )e2w+j−3 e2w+j−4 ...e2w e2w−1 (4.16) s = µ1 with possibly different values of j = ja or jb , the total size (total number of boxes) of the strip. Note that if ja = 1, sw (a) is reduced to a single grey box, without white box on top (this is the case when (a)2 only has one floor of height 0). We have the first result Lemma 2. For all a, b ∈ Wn(h) , and w = (n − h)/2, if sw (a) 6= sw (b), then (a)2 , (b)2 = 0. If sw (a) 6= sw (b), have different size. Let us assume that ja < jb . then these strips Writing (a)2 , (b)2 = Tr (b)2 (a)t2 , and (b)2 = Bsw (b), (a)2 = Asw (a), we have (4.17) (a)2 , (b)2 = Tr Bsw (b)sw (a)t At . In this expression, we now transfer the boxes of sw (b) onto (a)t2 , starting from the lowest one, up to the top of sw (b). These boxes now act on sw (a)t from below. Thanks to the relation ei ei−1 ei = ei , the first ja − 2 grey boxes of the strip sw (a)t are annihilated by the action of the first ja − 1 grey boxes of sw (b), namely µ2 (e2w+jb −2 − µ1 )e2w+jb −3 ...e2w e22w−1 µ1 × e2w ...e2w+ja −3 (e2w+ja −2 − µ1 ) µ2 = 2 (e2w+jb −2 − µ1 )e2w+jb −3 ...e2w+1 e2w e2w+1 µ1 × ...e2w+ja −3 (e2w+ja −2 − µ1 ) µ2 = 2 (e2w+jb −2 − µ1 )e2w+jb −3 ...e2w+ja −3 (e2w+ja −2 − µ1 ). µ1
sw (b)sw (a)t =
(4.18)
(Note that the last factor (e2w+ja −2 − µ1 ) must be replaced by e2w+ja −2 = e2w−1 in the case ja = 1, but this does not alter the following discussion.) Let us now transfer in the same way all the boxes of sw−1 (b), sw−2 (b), ..., s1 (b) onto (a)t2 . But these occupy only positions k ≤ 2w + jb − 3, and the largest position k = 2w + jb − 3 may only be occupied by a white box. Hence, after the transfer of (b)2 onto (a)t2 is complete, the resulting element is a linear combination of the form (b)2 (a)t2 = α C 0 (e2w+jb −2 − µ1 )e2w+jb −3 C + β D0 e2w+jb −3 (e2w+jb −2 − µ1 )e2w+jb −3 D,
(4.19)
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P. Di Francesco
where C, D, C 0 , D0 are elements of the Temperley-Lieb algebra only involving the generators ek , k < 2w + jb − 3, and α and β two complex coefficients, coming from the various normalization factors. The second term in (4.19) vanishes identically, thanks to the identity ei (ei+1 − µ1 )ei = 0. We are therefore left with (a)2 , (b)2 = α Tr C 0 (e2w+jb −2 − µ1 )e2w+jb −3 C (4.20) = α Tr (e2w+jb −2 − µ1 )e2w+jb −3 CC 0 . To show that this expression vanishes, let us use the string representation of the ei , and the definition of the trace as computing q L , where L is the number of loops of the string representation of the element, after identification of the upper and lower ends of its strings. In this picture (setting i = 2w + jb − 2, CC 0 = E, and taking the adjoint of the expression in the trace, which does not change its value), we have Tr E(e1 , e2 , ..., ei−2 )ei−1 (ei − µ1 )
E
=
− µ1
E
(4.21)
=
X
i-1 i
αr (q − µ1 q L
i-1 i
L+1
)
r
= 0, where we have expanded E(e1 , e2 , ..., ei−2 ) as a linear combination of diagrams involving only grey boxes with positions k ≤ i − 2. In each of these diagrams, the second term has always one more loop than the first one, hence the cancellation, with the factor 2. µ1 = q −1 . This completes the proof of Lemma Lemma 2 guarantees that (a)2 , (b)2 = 0 as soon as the last strips sw (a) and sw (b) are distinct. In the latter case, Proposition 3 is therefore proved. Let us assume now that (a)2 and (b)2 have the same last strip, say with j boxes. Then both a and b have a rightmost floor of height H = j −1. Let pa and pb denote their respective widths, namely the respective numbers of grey boxes of height j − 1 forming this floor in a and b. Two situations may occur for these floors: (i) they have the same width pa = pb . In this case, we will show that (a)2 , (b)2 is factored into the bilinear form (3.12) evaluated on smaller diagrams, obtained by cutting (a)2 and (b)2 into two pieces (Lemma 3 below). (ii) The width of the rightmost floor of a is strictly smaller that that of the rightmost floor of b pa < pb . In this case, we will show that (a)2 , (b)2 = 0 (Lemma 4 below). Let us treat these cases separately. CASE (i). The two rightmost floors of a and b have the same width pa = pb = p. We will simply grind the j − 2 consecutive layers of grey boxes underlying the floor of height j − 1, and detach the corresponding portions of a and b, so that the quantity (a)2 , (b)2 will factorize into a product of analogous terms, for smaller diagrams (see Lemma 3 below). More precisely, let us compute the quantity S(a)S(b)t = sw−p+1 (a)sw−p+2 (a)...sw (a)sw (b)t ...sw−p+1 (b)t
(4.22)
Meander Determinants
571
involved in the computation of (a)2 (b)t2 . In S of (4.22), all the strips involved have a floor of height j − 1, i.e. have the form sw−m+1 = ... e2w−2m+j−1 e2w−2m+j−2 ...e2w−2m+1 = s˜w−m+1 e2w−2m+j−2 ...e2w−2m+1 ,
(4.23)
where each strip s˜ has a floor of only one grey box, topped by white boxes. The idea is to transfer the grey boxes from sw (a) to sw (b), from below, just like we did in (4.18), and do it again for sw−1 (a) and sw−1 (b), etc..., until we are left only with the amputated strips s. ˜ The final result simply reads ˜ S(b) ˜ t = s˜w−p+1 (a)s˜w−p+2 (a)...s˜w (a)s˜w (b)t ...s˜w−p+1 (b)t . (4.24) S(a)S(b)t = S(a) This result implies the following Lemma 3. If a and b ∈ Wn(h) have identical rightmost floors of width p, then (a)2 , (b)2 = (a0 )2 , (b0 )2 (a00 )2 , (b00 )2 , where
n−2p 2
(a0 )2 = µ1
n−2p 2
0
(b )2 = µ1
(4.25)
s1 (a)s2 (a)...sw−p (a), s1 (b)s2 (b)...sw−p (b),
(a00 )2 = µp1 s˜w−p+1 (a)...s˜w (a),
(4.26)
(b00 )2 = µp1 s˜w−p+1 (b)...s˜w (b). The normalizations in (4.26) are chosen to guarantee that all the elements (a0 )2 , (a00 )2 , ... have norm 1, as we will see below. Llemma 3 will follow from the application of (4.24) to the computation of (a)2 , (b)2 = Tr (a)2 (b)t2 . Indeed, we simply write 0 t 0 t (a)2 , (b)2 = µ2p 1 Tr (a )2 S(a)S(b) (b )2 0 ˜ ˜t 0 t = µ2p 1 Tr (a )2 S S (b )2 (4.27) = Tr (a00 )2 (b00 )t2 (b0 )t2 (a0 )2 = (a00 )2 , (b00 )2 × (a0 )2 , (b0 )2 In the last step, we have noted that (a00 )2 (b00 )t2 involves only generators ek with positions ˜ k ≥ 2w − 2p + j − 1 (position of the leftmost grey box in S(a) = (a00 )2 ), whereas 0 0 t (a )2 (b )2 involves only generators ek with k ≤ 2w − 2p + j − 3 (maximum position of the rightmost (white) box in (a0 )2 ). The last line of (4.27) follows then from the locality of the trace, namely that for any two sets of positions I, J, with i < j − 1 for all i ∈ I, j ∈ J, Y Y Y Y ej ) = Tr( ei ) Tr( ej ), (4.28) Tr( ei i∈I
j∈J
i∈I
j∈J
which follows from the definition of the trace (the loops arising from the two terms are independent). In (4.25), the bilinear forms (a)2 , (b)2 , (a0 )2 , (b0 )2 and (a00 )2 , (b00 )2 , are respec(h) (1) tively evaluated in the spaces In(h) (q), In−2p (q) and I2p+1 (q). Let us concentrate on the (1) 00 00 (q) and last term (a )2 , (b )2 . There is a simple morphism ϕ of algebras between I2p+1 (0) I2p+2 (q) = Ip (q),
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P. Di Francesco
ϕ(E) =
√ µ1 E e2p+1 ∈ Ip (q)
(1) ∀ E ∈ I2p+1 (q).
(4.29)
The morphism ϕ consists simply in adding the missing rightmost grey box (e2p+1 ) to complete the floor of E into that of the ideal Ip (q). Moreover, we have added an ad-hoc multiplicative normalization factor µ1 . With this normalization, we have the following simple correspondence between traces over the two spaces: t Tr ϕ(E)ϕ(F )t = µ1 µ−1 1 Tr E e2p+1 F
=
=
(4.30)
= Tr EF t , t where we have used e22p+1 = µ−1 1 e2p+1 , then transferred all the boxes of F onto E, and represented the result in the pictorial string-domino representation of the trace (see Fig. 15), to show that the presence of the grey box does not change the value of the trace (it does not affect the structure of the loops). Using this fact, we can now apply 00 00 ) , (b ) of (4.25), by the result of Proposition 1 of Sect. 3.3 above to the factor (a 2 2 00 00 simply interpreting ϕ (a )2 and ϕ (b )2 as elements of Ip (q). We conclude that 00 00 00 00 (a )2 , (b )2 = ϕ (a )2 , ϕ (b )2 (4.31) = δa00 ,b00 .
Hence, if a00 6= b00 , the bilinear form vanishes, and Proposition 3 follows. If a00 = b00 , we go back to the beginning of our study, with now (a)2 and (b)2 replaced with (a0 )2 and (h) (b0 )2 ∈ In−2p (q). If only the case (i) occurs, we will dispose successively of each portion of a and b above their common successive floors (as above), and get an expression Y δa00 ,b00 . (4.32) (a)2 , (b)2 = portions a00 ,b00 above successive floors
If case (ii) occurs, the result will vanish, as we will see now. CASE (ii). The diagrams (a)2 and (b)2 have a rightmost floor, of same height j − 1, but with different widths p = pa < pb . As in the case (i), we concentrate on the portions of a and b above this rightmost floor, over a width p, namely consider S(a) = sw−p+1 (a)sw−p+2 (a)...sw (a), (4.33) S(b) = sw−p+1 (b)sw−p+2 (b)...sw (b). To compute the quantity (a)2 , (b)2 , we now write (a)2 = (a0 )2 S(a) and (b)2 = (b0 )2 S(b), and get (a)2 , (b)2 = Tr (a0 )2 S(a)S(b)t (b0 )t2 (4.34) ˜ S(b) ˜ t . = Tr (b0 )t2 (a0 )2 S(a) Using the cyclicity of the trace and the symmetry of (3.12), we may also write
Meander Determinants
573
(a)2 , (b)2
˜ ˜ t (a0 )t2 (b0 )2 S(b) = Tr S(a) ˜ t (b0 )2 S(b) ˜ . = Tr (a0 )t2 S(a)
(4.35)
˜ We have used the commutation of S(a), which involves only boxes of positions ≥ α = 0 2w−2p+j−1, with (a )2 , which only involves boxes of positions ≤ α−2 (as its rightmost floor has now an height < j − 1). Let us now compute (4.35) by transferring the white ˜ ˜ ˜ t onto (b0 )2 S(b). Once this transfer is complete, (b0 )2 S(b) is replaced by boxes of S(a) ˜ with all possible box additions/subtractions a linear combination of diagrams (c0 )2 S(c) induced by the process of transfer. Note that the left portion (b0 )2 of (b)2 is also affected, ˜ For notational simplicity, we have used the as these boxes act on both (b0 )2 and S(b). denomination c0 for the left part of c, so that we still have (c)2 = (c0 )2 S(c). We are then ˜ t , namely those of height j −1. left with the transfer of the first layer of grey boxes of S(a) ˜ (the action To get a non-zero result, those must only hit minima or maxima on (c0 )2 S(c) of a grey box on a white slope vanishes, according to (3.32), (3.33)). Concentrating ˜ above the position α = 2w − 2p + j − 1 (namely the on the configuration of (c0 )2 S(c) ˜ only two situations may yield configuration of (c0 )2 above the leftmost grey box in S(c)), a non-zero answer (a) (c0 )2 has no white box above the position α. (b) (c0 )2 has a white maximum at α. In this case, this maximum is necessarily at height j + 2 (white box of height j + 1, hence of relative height 2 w.r.t. the floor), because no white slope is allowed at any of the positions α, α + 2, ..., α + 2p − 2 = 2w + j − 3, ˜ and the rightmost white box of S(c) has an height ≤ j, hence a relative height ˜ t= ˜ t (call it S(d) ≤ 1. Let us transfer this white box back onto what is left of S(a) eα eα+2 ...eα+2p−2 ). Actually, this diagram has now a grey maximum at the position α (this is the position of the leftmost The white p grey box in the floor of S(a)). √ box acts on this grey maximum as µ3 /µ2 (eα − µ2 )eα = eα / µ2 µ3 , hence is eliminated up to some multiplicative constant. We therefore end up in a situation where (c0 )2 → (c0 − α )2 has no white box above the position α hence in the case (a) above. ˜ In either case, we end up in a situation where α is a floor-end on both diagrams (a0 )2 S(d) 00 ˜ 00 0 00 0 and (c )2 S(c), where c = c in the case (a) and c = c − α in the case (b). So we can reexpress X ˜ S(d) ˜ t (a)2 , (b)2 = λc Tr (a0 )t2 (c00 )2 S(c) c
=
X
λc
c
(4.36) = µ1
X
λc
c
= µ1
X
˜ S(d) ˜ t , λc (a0 )2 , (c00 )2 Tr S(c)
c
where, by using the string-domino picture, we have removed the grey box linking the left and right parts of the operator in the trace, at the expense of creating a new loop,
574
P. Di Francesco
hence the extra factor of µ1 = q −1 . Note that in this argument it was crucial that there should be no white box above the grey box we have removed, further linking the left and right parts: this is why we had to go through case (b) above and modify c0 → c00 = c0 − to get back to the situation (a). Now the main feature of (c00 )2 is that it has still a rightmost grey floor of height j − 1, whereas by definition (a0 )2 has a rightmost grey floor of height < j − 1. Hence the rightmost strips in both diagrams are distinct: sw−p (a0 ) 6= sw−p (c00 ). We can therefore apply Lemma 2, to conclude that (4.37) (a)2 , (c00 )2 = 0 in (4.36), so that finally (a)2 , (b)2 = 0. Hence we deduce the Lemma 4. For any two bicolored box diagrams (a)2 and (b)2 , with rightmost floors of same height j − 1, but of differents widths pa < pb , we have (4.38) (a)2 , (b)2 = 0. The proof of Proposition 3 is now straightforward. We start with the two bicolored box diagrams (a)2 and (b)2 . If their rightmost strips are distinct, then (a)2 , (b)2 = 0 by Lemma 2. Otherwise, we focus our attention to their rightmost floors, which have the same height j − 1. If they have different widths, Lemma 4 above implies Q that (a)2 , (b)2 = 0. If they have the samewidth, Lemma 3 expresses (a)2 , (b)2 = δa00 ,b00 , hence we finally get that (a)2 , (b)2 = 0 unless a and b are identical, in which case (a)2 , (a)2 = 1. This completes the proof of Proposition 3. 4.3. The semi-meander determinant: A preliminary formula. The semi-meander determinant (4.6) follows from the Gram determinant of the basis 1. The latter is best expressed through the change of basis 1 → 2, in which the Gram matrix is sent to the cn,h × cn,h identity matrix I. With the upper triangular matrix P defined in (4.13), this reads t (4.39) P 0(h) n (q)P = I. −2 Hence det 0(h) n (q) = (det P ) . The diagonal elements of P are linked by the recursion relation r µ`+1 Pa+i,` ,a+i,` = Pa,a , (4.40) µ`
where the box addition i,` is performed at the point i, and at relative height `, with n/2 (4.7) for respect to the grey box floor in a. With the initial condition Pa(h) (h) = µ 1 n ,an the fundamental walk diagram of Wn(h) , this gives Y µ`+1 2 Pa,a = µn1 , (4.41) µ` white boxes of a
where ` denotes the height of the white box addition, relative to the grey floor in a.
Meander Determinants
575
Fig. 19. The strips of white boxes on a walk a ∈ Wn(h) with n = 20 and h = 6. The walk is represented in a thick black line. We have also represented the floor of grey boxes for this walk. We have (n − h)/2 = 7 strips of white boxes, of respective lengths 2, 1, 3, 2, 2, 2, 2
Like in the meander case, let us arrange the white boxes of any bicolored box diagram corresponding to an a ∈ Wn(h) into strips of white boxes, namely sequences of white boxes with consecutive positions and heights, added on top of the grey floor of a (see Fig. 19 for an illustration). There are exactly (n−h)/2 such white strips. The strip length is now defined as the relative height of the top of the white box sitting on top of the strip (hence an empty strip has length 1). With this definition, we simply have Y n+h 2 = µ1 2 µ` , (4.42) Pa,a white strips of a
where, in the product over the (n − h)/2 strips of a ∈ Wn(h) , ` stands for the strip length (all denominators have been cancelled along the strips, except for the µ1 ones, which have rebuilt the prefactor). This yields the determinant of the basis 1 Y Y −(n+h)cn,h /2 −2 Pa,a = µ1 µ−1 (4.43) det 0(h) n (q) = ` , white strips of (h) all a∈Wn
a∈Wn(h)
and thanks to (4.6), the semi-meander determinant Y
−hcn,h
det Gn(h) (q) = µ1
µ−1 ` .
(4.44)
white strips of (h) all a∈Wn
The latter can be recast into det Gn(h) (q)
=
−hc µ1 n,h
n−h 2 +1
Y
µm
−s(h) n,m
,
(4.45)
m=1
where s(h) n,m denotes the total number of white strips of length m in all the bicolored box diagrams corresponding to the walk diagrams of Wn(h) (the notation is such that s(0) 2n,m = s2n,m (3.50)). Note also that the strips all have length ≤ (n − h)/2 + 1, hence the upper bound in the product in (4.45). The formula of Theorem 2 will follow from the explicit computation of the numbers s(h) n,m . This will be done in two steps. The first step (Proposition 4, Sect. 4.4 below) consists in arranging the s(h) n,m walks above according to their floor configuration (namely their configuration of grey boxes). The second step (Proposition 5, Sect. 4.5 below) consists in enumerating the walks with minimal floor configurations (namely made of only one layer of grey boxes). Finally in Proposition 6, Sect. 4.6, the combination of these two results will eventually lead to a formula for s(h) n,m , which will complete the proof of Theorem 2. 4.4. Enumeration of the floor configurations. In this first step, we note that many different diagrams a ∈ Wn(h) have the same contribution to (4.44), namely those with identical white strips, but different floors of grey boxes. Assembling all these contributions leads to the following formula for s(h) n,m
576
P. Di Francesco
Proposition 4. s(h) n,m
=
X h + k − 1 k≥0
k
fn−h+2,m,k ,
(4.46)
where f2n,m,k denotes the total number of walk diagrams a ∈ W2n with k + 1 floors of height 0, and with a marked top of strip of length m.
Fig. 20. A typical walk diagram a ∈ Wn(h) is represented in thick black line on the upper diagram. We have also represented its floor of grey boxes, and the white boxes topping it. The floor of grey boxes in a is a succession of a number k + 1 of horizontal floors, F0 , F1 , ..., Fk , with respective heights H0 = 0, H1 , H2 , ..., Hk ≥ 0. The conjugates of a are obtained by varying these heights, without changing the white strips of a (this is done by letting the floors slide along the dashed lines separating them). The minimal conjugate aˆ ∈ Wn−h+2 of a is represented below it: it has H1 = H2 = ... = Hk = 0. The floor-ends are indicated by arrows
Indeed, as illustrated in Fig. 20, in any walk diagram a ∈ Wn(h) , the floor of grey boxes may be viewed as a succession of a number, say k + 1 of consecutive horizontal floors of grey boxes F0 , F1 , ..., Fk , with respective heights H0 = 0, H1 , ..., Hk , and Hj ≥ 0 for all j ≥ 1. The leftmost floor F0 , of height H0 = 0, is made of one layer of grey boxes of the form e1 e3 e5 ..., and occupies a segment I0 = {i0 = 0, 1, 2, ..., i1 − 1} of positions (we include here the case when F0 = ∅, i.e. I0 = {0}, corresponding to the case J0 6= {0} of (4.8)). It is topped by white strips of arbitrary lengths. Any horizontal floor Fj , j = 1, 2, ..., k, of height Hj ≥ 0, is a parallelogram made of Hj + 1 horizontal layers of grey boxes, whose base occupies a segment of positions Ij = {ij , ij +1, ij +2, ..., ij+1 }, with ik+1 = n −h +1. What distinguishes these floors from F0 is that they are necessarily topped with at least two layers of white boxes, resulting in white strips of lengths m ≥ 2 only. The separation between two consecutive floors of height Hj ≥ 0 is formed by the strips of length 2 (with one white box), as illustrated in the upper diagram of Fig. 20, where the floor separations are indicated by dashed lines. The various floor heights are subject to the constraint 0 ≤ H1 ≤ H2 ≤ ... ≤ Hk ≤ h − 1,
(4.47)
arising from the original definition of the floor of a walk a ∈ Wn(h) (the floor Fj is always of lesser or equal height than Fj+1 ).
Meander Determinants
577
By varying only the heights H1 , H2 , ..., Hk subject to (4.47), and by keeping the white strips fixed, we describe the set of all conjugates of a given walk diagram a ∈ Wn(h) . There are therefore h+k−1 (4.48) |{(H1 , ..., Hk ) ∈ N s.t. 0 ≤ H1 ≤ H2 ≤ ... ≤ Hk ≤ h − 1}| = k such conjugates for each diagram a ∈ Wn(h) with k + 1 floors. We now choose among the conjugates of a, the minimal one, namely that with H1 = H2 = ... = Hk = 0, which we denote by aˆ (the bottom diagram of Fig. 20). We may amputate this diagram from the final slope with positions n − h + 2, n − h + 3, ..., n, and view it as a diagram aˆ ∈ Wn−h+2 . Indeed, the diagram aˆ has h(n − h + 1) = 1, the height of the rightmost floor-end, hence we may complete it by h(n − h + 2) = 0 into an element of Wn−h+2 . Denoting by f2n,k,m the total number of walks of W2n with k + 1 floors of height 0, and with a marked top of strip of length m, Proposition 4 follows, by enumerating these fn−h+2,k,m walk diagrams with a marked top of strip of length m, and weighing each of them by the number of its conjugates (4.48). 4.5. The mapping of walk diagrams. The second step of the calculation of s(h) n,m is the computation of the numbers f2n,k,m appearing in (4.46). The result reads Proposition 5. The total number f2n+2,k,m of walks in W2n+2 , with k + 1 floors of height 0, and with a marked top of strip of length m reads for m ≥ 2, n ≥ 1, f2n+2,k,m = c2n−k,2m+k + kc2n−k,2m+k−4 for m = 1, n ≥ 1, f2n+2,k,1 = c2n−k,k+2
(4.49)
where the numbers cn,h are defined in (2.3). Here we have excluded the trivial case n = 0, for which no strip appears, hence f2,k,m = 0
for all k and m.
(4.50)
To prove this proposition, we will construct a bijection from the set of walk diagrams of W2n+2 with k + 1 floors of height 0, and with a marked top of strip of length m ≥ 2 (2m+k) (2m+k−4) to (i) the set W2n−k (ii) k copies of the set W2n−k , which will prove (4.49), as (h) |Wn | = cn,h . (In the case m = 1, only part (i) will apply, namely, we will construct a bijection between the walks of W2n+2 with a marked end of empty strip (length 1) and (k+2) W2n−k .) We start from a ∈ W2n+2 , with k + 1 floors, all of height 0, and with a marked top of strip of length m. Two cases may occur: (i) The marked top of strip lies above the leftmost floor (F0 ). In this case, we will con(2m+k) struct a walk b ∈ W2n−k by a cutting-reflecting-pasting procedure on a, analogous to that used in the meander case. This will produce the first term in (4.49). (ii) The marked top of strip lies above one of the k other floors (F1 , F2 , ..., Fk ). This is possible only if m ≥ 2, as there is no empty strip above these floors, by definition. By a circular permutation of the k floors, we can always bring the block containing the marked point to the right. We therefore have a k-to-one mapping to the situation where the marked strip is above the rightmost floor. This k-fold circular permutation symmetry is responsible for the factor k in the second term of (4.49). The diagrams with the marked top of strip of length m above the rightmost (Fk ) floor are then
578
P. Di Francesco
mapped to the walk diagrams with a marked top of strip at height m − 2 above (2(m−2)+k) the leftmost (F0 ) floor considered in the case (i), hence to the set W2n−k = (2m+k−4) W2n−k (the case m = 2 will have to be treated separately). Together with the multiplicity factor k this will produce the second term of (4.49). Let us now construct the maps for the cases (i) and (ii) above. CASE (i). We start from a walk diagram a ∈ W2n+2 , with k + 1 floors of height 0, and with a marked top of strip of length m above its leftmost floor F0 , say at position i. The point (i, h(i) = m) separates a into a left L and right R parts, respectively such (m) . Reflecting L and pasting it again at the left end of that L ∈ Wi(m) and R¯ ∈ W2n+2−i (2m) R, we create a walk diagram b0 , whose reflection b¯0 ∈ W2n+2 . To construct the eventual (2m+k) image b ∈ W2n−k of a, we perform the following amputations of the walk b0 . We will suppress some pieces of b0 at each separation of floor, according to the following rules: (1) (2) (3)
j
i
i i
j j
j
(h → h + 1, o → o − 1),
j
(h → h − 2, o → o − 2),
i
i
i j
(h → h + 3, o → o − 1), (4.51)
where we have represented the floor end by an empty circle, and where we indicate the change in final height (h) and in the order (o) resulting from the amputation. Considering that the rules in (4.51) apply respectively (1) to the first floor separation only (between F0 and F1 ), (2) to the k − 1 intermediate floor separations (between Fj and Fj+1 , j = 1, 2, ..., k − 1), and (3) to the rightmost floor end (right end of Fk ), we get an overall change from the initial values (h = 2m, o = 2n + 2) of the height and order of b0 to the amputed b00 with h → h + 3 + (k − 1) − 2 = 2m + k
o → o − 1 − (k − 1) − 2 = 2n − k. (4.52)
(2m+k) . Hence taking b = b¯00 , we get an element of W2n−k To prove that this mapping is bijective, let us compute its inverse. Starting from (2m+k) , let i be the position of the rightmost intersection between b and the b ∈ W2n−k line h = m + k at an ascending slope (h(i − 1) + 1 = h(i) = h(i + 1) − 1). This point separates the walk b into a left part L and a right part R. Let us reflect R and (k) . As before, if paste it again to the right end of L. This produces a walk a0 ∈ W2n−k h(i + 1) − 1 = h(i) = m + k, we mark the point i, which will be an end of strip (in the eventually reflected walk). Otherwise, h(i + 1) = h(i) − 1, and we migrate the mark to the point i0 = max{j < i|h(j + 1) = h(j) − 1 = m + k}). Let us now mark (by black dots) the rightmost intersections between a0 and the lines h = k, h = k − 1, ..., h = 1 at ascending slopes of a0 , and also the left end (i = 0, h = 0) of a0 . We reconstruct the k + 1 separations of floors using the following rules (inverse of (4.51)):
(1)
(h → h + 2, o → o + 2),
(2)
(h → h − 1, o → o + 1),
(3)
(h → h − 3, o → o + 1).
(4.53)
The corresponding separations have been represented by empty circles. They all lie at height h = 1 in the resulting final walk a00 . The three rules of (4.53) apply respectively
Meander Determinants
579
(1) to the left end of a0 , (2) to any of the k − 1 intermediate points of intersection with the lines h = 1, ..., h = k − 1, and (3) to the rightmost intersection with the line h = k. The rules (4.53) therefore result in a change of final height and order (h = k, o = 2n − k) → (h + 2 − (k − 1) − 3 = 0, o + 2 + (k − 1) + 1 = 2n + 2), hence a00 ∈ W2n+2 . The last step consists simply in reflecting a00 , to produce a = a¯00 , with a marked top of strip of height m + k − k = m above the leftmost (F0 ) floor, and a has a total of k + 1 floors, all of height 0. As before, the bijectivity of the map follows from the fact that we considered rightmost points of intersection, which makes the construction unique. This bijection yields the number c2n−k,2m+k of walks in W2n+2 with k + 1 floors of height 0, and with a marked top of strip of length m above F0 . This is the first term of (4.49). CASE (ii). We start from a walk a ∈ W2n+2 , with k + 1 floors F0 , F1 ,...Fk , all of height zero, and with a marked top of strip of length m above its rightmost floor Fk . By definition, we necessarily have m ≥ 2, and in fact there is one and only one strip of length 2 above the floor Fk (the one just above the right floor-end), and all other strips have length ≥ 3. We now construct a bijection between these walks and the b ∈ W2n+2 with k + 1 floors F00 , F10 , ..., Fk0 , all of height 0, and with a marked top of strip of length m − 2 above their leftmost floor F00 . If m = 2, the above remark shows that the number of walks a with k + 1 floors of height 0, and with a marked top of strip of length 2 above Fk is equal to the number of such walks, without marked top of strip (there is exactly one such strip of length 2 per walk). Skipping the cutting-reflecting-pasting procedure of case (i) (we have no more marked top of strip), we can still apply the amputation rules (4.51) on the walk (k) . Conversely, starting from any b0 = a¯ ∈ W2n+2 : this results in a walk b ∈ W2n−k (k) b ∈ W2n−k , let us apply to it the inverse of the amputation rules (4.53), after marking the rightmost intersections at ascending slopes with the lines h = k, h = k − 1, ..., h = 1. This produces a walk a0 ∈ W2n+2 , and finally a = a¯0 ∈ W2n+2 has k + 1 floors of height 0. This bijection yields the number c2n−k,k of walks in W2n+2 with k + 1 floors of height 0, and a marked top of strip of height m = 2 above Fk . Together with the k-fold cyclic degeneracy of the case (ii) this gives the second term of (4.49), for m = 2.
Fig. 21. The exchange map on walk diagrams of Wn(h) , maps the walks with a marked strip of length m above their rightmost floor onto those with a marked strip of length m−2 above the leftmost floor (the corresponding strip of length m = 3 is marked with a black dot on the figure). We have indicated by a thick broken line the portions exchanged. The double-layer of white boxes on the rightmost floor is adapted to fit the exchange
580
P. Di Francesco
If m ≥ 3, we simply exchange the floors F0 and Fk in the following way. The floor Fk is by definition topped by at least two layers of white boxes (see Fig. 21). Let ik , ik +1, ..., ik+1 denote the positions occupied by Fk , the ends ik and ik+1 being at height 1. Let us cut out the portion ak of a in between the positions ik + 2 and ik+1 − 2, both at height 3 (the level of the second layer of white boxes). Let us also cut the portion a0 of a above the leftmost floor F0 , in between the positions i0 = 0 and i1 − 1, both at height 0. We form a walk b ∈ W2n+2 by simply exchanging the portions a0 and ak in a, as depicted in Fig. 21. The marked top of strip on ak has been therefore transferred above the leftmost floor of b, but as two layers of white boxes have been suppressed, all the lengths of strips have been decreased by 2. Hence the walk b ∈ W2n+2 has k + 1 floors of height 0, and a marked top of strip of length m − 2 above its leftmost floor F00 . This construction is clearly invertible, by just exchanging again ak and a0 . From case (i) above, we learn that (2(m−2)+k) (2m+k−4) the walk b can be mapped onto an element of W2n−k = W2n−k , in a bijective way. This bijection yields the number c2n−k,2m+k−4 of walks a ∈ W2n+2 with k + 1 floors of height 0, and with a marked top of strip of length m above its rightmost floor Fk . With the overall k-fold cyclic degeneracy mentioned above, this gives the second term in (4.49) for all m ≥ 3. The mappings of the cases (i) and (ii) above complete the proof of Proposition 5, with the understanding that the case m = 1 only gives rise to case (i), hence the different answer. 4.6. The semi-meander determinant: The final formula. Combining the results of Propositions 4 and 5, namely Eqs. (4.46) and (4.49), we get the following formula for the (h) numbers s(h) n,m of walk diagrams in Wn with a marked top of strip of length m above its floor: X h + k − 1 = for m ≥ 2, (cn−h−k,2m+k + kcn−h−k,2m+k−4 ) s(h) n,m k k≥0 (4.54) X h + k − 1 (h) for m = 1. sn,1 = cn−h−k,k+2 k k≥0
This is valid for h ≤ n − 1. If h = n, (4.50) yields s(n) n,m = 0 for all m. By a direct calculation, we find Proposition 6. The numbers of walks in W2n+2 with k+1 floors of height 0 and a marked end of strip of length m read s(h) n,m = cn,h+2m + hcn,h+2m−2 s(h) n,1 (n) sn,m
= cn,h+2
for m ≥ 2, h ≤ n − 1,
for m = 1, h ≤ n − 1,
(4.55)
for h = n and all m ≥ 1.
= 0
The proof relies on the following classical identity for binomial coefficients: c−d X k+a c−k a+c+1 = b d b+d+1
(4.56)
k=b−a
for all integers a, b, c, d. This is easily proved by use of generating functions. We now simply have to apply (4.56) to the various sums appearing on the r.h.s. of (4.54):
Meander Determinants
581
X k + h − 1n − h − k
n n+h 2 +m
n
= = n−(h+2m) , n−h h−1 2 +m 2 (4.57) X k + h − 1 n − h − k n n = n+h = n−(h+2m) , n−h −1 h−1 2 +m+1 2 +m+1 2 k≥0
k≥0
X k + h − 1
hence
and, noting that k
k+h−1 k
cn−h−k,k+2m = cn,h+2m
k
k≥0
=h
(4.58)
k+h−1 k−1
, we also get X k+h−1 k cn−h−k,k+2m−4 = hcn,h+2m−2 . k
(4.59)
k≥0
Proposition 6 follows from (4.58) and (4.59). Substituting the result (4.55) above into (4.45), we finally get the semi-meander determinant n−h 2 +1 Y −(cn,h+2m +hcn,h+2m−2 ) (h) , (4.60) µm det Gn (q) = m=1 −hc
where we have absorbed the prefactor µ1 n,h of (4.45) into the m = 1 term of the product. Finally, using the fact that µm = Um−1 (q)/Um (q), for m ≥ 1, Theorem 2 follows. 5. Conclusion In this paper, we have proved two determinant formulas for meanders and semimeanders. This has been done by the Gram-Schmidt orthogonalization of the corresponding bases 1 of the Temperley-Lieb algebra or some of its subspaces. The main philosophy of the construction leading to the Gram-Schmidt orthogonalization of these bases 1 lies in the concept of box addition, the building block of the definition of the bases 2 elements. We believe that this type of construction should be much more general and apply to many other cases of algebra-related Gram-Schmidt orthogonalization. An important remark about Theorems+1 and+2 above is that they implicitly give the structure (including multiplicity) of the zeros of the Gram determinants, considered as functions of the variable q. Due to the definition of the Chebyshev polynomials, the zeros of the Gram determinant always take the form q = 2 cos(π
m ) p+1
(5.1)
with 1 ≤ m ≤ p ≤ n−h 2 + 1 in the semi-meander case. These zeros actually correspond to the cases when the corresponding subspace of the Temperley-Lieb algebra is reducible (there are linear combinations of the basis 1 elements which are orthogonal to all the basis 1 elements: i.e. there may be vanishing linear combinations of the basis 1 elements, the basis 1 being therefore no longer a basis at these values of q). The multiplicity of these zeros is linked to the degree of reducibility (namely to how many such independent linear combinations exist).
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P. Di Francesco
Unfortunately, the information we obtain from these determinant formulas on the meanders and the semi-meanders themselves is very difficult to exploit. Indeed, quantities such as asymptotics (for large order) of the meander and semi-meander numbers and distributions are only indirectly related to the Gram determinants, as they would rather involve the exact knowledge of the asymptotics of the Gram matrices, or at least of their eigenvalue spectra. However, the exact orthogonalization performed above is useful to derive new asymptotic formulas for the meander numbers, as sums over walk diagrams (see [9] for the meander example). We hope to return to this question in a later publication. Theorems 1 and 2 above can probably be generalized in many directions. A first possibility relies on the fact that there exists a canonical Temperley-Lieb algebra attached to any non-oriented, connected graph (see [12] and references therein), which may still be interpreted as the image of a walk diagram on that graph. The only constraint is that the number q must be an eigenvalue of the adjacency matrix of the graph (a matrix Ga,b made of 1’s and 0’s according to whether the couple (a, b) of vertices of the graph is joined by a link or not). In the examples treated here, this graph is simply the set of heights, namely the integer points on the (infinite) half-line, linked by segments between consecutive pairs (hence Gi,j = δj,i+1 + δj,i−1 for i, j > 0 and G0,j = δj,1 , with the eigenvalue q for the (infinite) eigenvector v = (U0 (q), U1 (q), U2 (q), ...)). But nothing prevents us from considering more complicated graphs. We believe that there exists a general determinant formula, associated to each such graph, expressing the result in terms of features of the graph only (with c2n,2m replaced by a corresponding number of paths of given length and given origin and end on the graph, and Um (q) replaced by the components of the eigenvector of the adjacency matrix for the eigenvalue q). Another direction of generalization has to do with replacing the Temperley-Lieb algebra by a more general quotient of the Hecke algebra. Indeed, recall that the TemperleyLieb algebra T Ln (q) is nothing but a simple quotient of the Hecke algebra, defined as follows. The Hecke algebra Hn (q) is defined by generators 1, e1 , e2 , ..., en−1 satisfying the following relations (i) e2i = q ei , (ii) [ei , ej ] = 0 if |i − j| > 1, (iii) ei ei+1 ei − ei = ei+1 ei ei+1 − ei+1 ,
(5.2)
hence the Temperley-Lieb algebra is the quotient of the Hecke algebra by the ideal generated by the elements ei ei±1 ei − ei . This quotient was identified as the commutant of the quantum enveloping algebra Uqˆ (sl2 ) acting on the fundamental representation of Hn (q), with q = qˆ + qˆ−1 . More quotients are found by considering the commutants of other quantum enveloping algebras (such as Uqˆ (slk ) for instance) [13]. These quotients await a good combinatorial interpretation, but should lead to a natural generalization of meanders and semi-meanders. We believe that many Gram determinants can still be computed exactly in this framework. References 1. 2. 3.
Hoffman, K., Mehlhorn, K., Rosenstiehl, P. and Tarjan, R.: Sorting Jordan sequences in linear time using level-linked search trees. Information and Control 68, 170–184 (1986) Touchard, J.: Contributions a` l’´etude du probl`eme des timbres poste. Canad. J. Math. 2, 385–398 (1950) Lunnon, W.: A map–folding problem. Math. of Computation 22, 193–199 (1968)
Meander Determinants
4. 5. 6. 7. 8. 9. 10. 11.
12. 13. 14.
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Phillips, A.: La topologia dei labirinti. In: M. Emmer, ed. L’ occhio di Horus: itinerario nell’immaginario matematico. Roma: Istituto della Enciclopedia Italia, 1989, pp. 57–67 Arnold, V.: The branched covering of CP2 → S4 , hyperbolicity and projective topology. Siberian Math. Jour. 29, 717–726 (1988) Ko, K.H., Smolinsky, L.: A combinatorial matrix in 3-manifold theory. Pacific. J. Math 149) 319–336 (1991) Lando, S. and Zvonkin, A.: Plane and Projective Meanders. Theor. Comp. Science 117, 227–241 (1993) and Meanders. Selecta Math. Sov. 11, 117–144 (1992) Di Francesco, P., Golinelli, O. and Guitter, E.: Meander, folding and arch statistics. To appear in Mathematical and Computer Modelling (1996), hep-th/950630 Makeenko, Y.: Strings, Matrix Models and Meanders. Proceedings of the 29th Inter. Ahrenshoop Symp., Germany (1995) Di Francesco, P., Golinelli, O. and Guitter, E.: Meanders and the Temperley-Lieb algebra. To appear in Commun. Math. Phys. (1996), hep-th/9602025 Temperley, H. and Lieb, E.: Relations between the percolation and coloring problem and other graphtheoretical problems associated with regular planar lattices: some exact results for the percolation problem. Proc. Roy. Soc. A322, 251–280 (1971) Martin, P.: Potts models and related problems in statistical mechanics. Singapore: World Scientific, 1991 Di Francesco, P.: Integrable lattice models, graphs and modular invariant conformal field theories. Int. Jour. Mod. Phys. A7, 407–500 (1992) Reshetikhin, N.: Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links 1 and 2. LOMI preprints E-4-87 and E-17-87 (1988)
Communicated by D. C. Brydges
Commun. Math. Phys. 191, 585 – 601 (1998)
Communications in
Mathematical Physics c Springer-Verlag 1998
The sh Lie Structure of Poisson Brackets in Field Theory G. Barnich1,4,? , R. Fulp2 , T. Lada2 , J. Stasheff 3,?? 1 Center for Gravitational Physics and Geometry, The Pennsylvania State University, 104 Davey Laboratory, University Park, PA 16802, USA 2 Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, USA 3 Department of Mathematics, The University of North Carolina at Chapel Hill, Phillips Hall, Chapel Hill, NC 27599-3250, USA 4 Freie Universit¨ at Berlin, Institut f¨ur Theoretische Physik, Arnimallee 14, D-14195 Berlin, Germany
Received: 5 March 1997 / Accepted: 21 May 1997
Abstract: A general construction of an sh Lie algebra (L∞ -algebra) from a homological resolution of a Lie algebra is given. It is applied to the space of local functionals equipped with a Poisson bracket, induced by a bracket for local functions along the lines suggested by Gel’fand, Dickey and Dorfman. In this way, higher order maps are constructed which combine to form an sh Lie algebra on the graded differential algebra of horizontal forms. The same construction applies for graded brackets in field theory such as the BatalinFradkin-Vilkovisky bracket of the Hamiltonian BRST theory or the Batalin-Vilkovisky antibracket.
1. Introduction In field theories, an important class of physically interesting quantities, such as the action or the Hamiltonian, are described by local functionals, which are the integral over some region of spacetime (or just of space) of local functions, i.e., functions which depend on the fields and a finite number of their derivatives. It is often more convenient to work with these integrands instead of the functionals, because they live on finite dimensional spaces. The price to pay is that one has to consider equivalence classes of such integrands modulo total divergences in order to have a one-to-one correspondence with the local functionals. The approach to Poisson brackets in this context, pioneered by Gel’fand, Dickey and Dorfman, is to consider the Poisson brackets for local functionals as being induced by brackets for local functions, which are not necessarily strictly Poisson. We will analyze here in detail the structure of the brackets for local functions corresponding to the Poisson ? Research supported by grants from the Fonds National Belge de la Recherche Scientifique and the Alexander-von-Humboldt foundation. New address: Universit´e Libre de Bruxelles, Facult´e des Sciences, Campus Plaine C.P. 231, B-1050 Bruxelles, Belgium. ?? Research supported in part by NSF grant DMS-9504871.
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brackets for local functionals. More precisely, we will show that these brackets will imply higher order brackets combining into a strong homotopy Lie algebra. The paper is organized as follows: In the case of a homological resolution of a Lie algebra, it is shown that a skewsymmetric bilinear map on the resolution space inducing the Lie bracket of the algebra extends to higher order “multi-brackets” on the resolution space which combine to form an L∞ -algebra (strong homotopy Lie algebra or sh Lie algebra). For completeness, the definition of these algebras is briefly recalled. For a highly connected complex which is not a resolution, the same procedure yields part of an sh Lie algebra on the complex with corresponding multi-brackets on the homology . This general construction is then applied in the following case. If the horizontal complex of the variational bicomplex is used as a resolution for local functionals equipped with a Poisson bracket, we can construct an sh Lie algebra (of order the dimension of the base space plus two) on the graded differential algebra of horizontal forms. The construction applies not only for brackets in Darboux coordinates as well as for non-canonical brackets such as those of the KDV equation, but also in the presence of Grassmann odd fields for graded even brackets, such as the extended Poisson bracket appearing in the Hamiltonian formulation of the BRST theory, and for graded odd brackets, such as the antibracket of the Batalin-Vilkovisky formalism. 2. Sh Lie Algebras from Homological Resolution of Lie Algebras 2.1. Construction. Let F be a vector space and (X∗ , l1 ) a homological resolution thereof, i.e., X∗ is a graded vector space, l1 is a differential and lowers the grading by one with F ' H0 (l1 ) and Hk (l1 ) = 0 for k > 0. The complex (X∗ , l1 ) is called the resolution space. (We are not using the term “resolution” in a categorical sense.) Let C∗ and B∗ denote the l1 cycles (respectively, boundaries) of X∗ . Recall that by convention X0 consists entirely of cycles, equivalently X−1 = 0. Hence, we have a decomposition X0 = B0 ⊕ K,
(1)
with K ' F. We may rephrase the above situation in terms of the existence of a contracting homotopy on (X∗ , l1 ) specifying a homotopy inverse for the canonical homomorphism η : X0 −→ H0 (X∗ ) ' F. We may regard F as a differential graded vector space F∗ with F0 = F and Fk = 0 for k > 0; the differential is given by the trivial map. We then consider the chain map η : X∗ −→ F∗ with homotopy inverse λ : F∗ −→ X∗ ; i.e., we have that η ◦ λ = 1F∗ and that λ ◦ η ∼ 1X∗ via a chain homotopy s : X∗ −→ X∗ with λ◦η −1X∗ = l1 ◦s+s◦l1 . Observe that this equation takes on the form λ◦η −1X∗ = l1 ◦s on X0 . We may summarize all of the above with the commutative diagram s ←− · · · −→ X2 −→ X1 x l1 x λyη λyη
s ←− −→ l1
· · · −→
−→ H0 = F .
0
−→
0
X0 x λyη
The sh Lie Structure of Poisson Brackets in Field Theory
587
It is clear that η(b) = 0 for b ∈ B0 . Let (−1)σ be the signature of a permutation σ and unsh(k, p) the set of permutations σ satisfying σ(1) < . . . < σ(k) {z } |
and
first σ hand
σ(k + 1) < . . . < σ(k + p). | {z } second σ hand
We will be concerned with skew-symmetric linear maps l˜2 : X0 ⊗ X0 −→ X0
(2)
that satisfy the properties (i) l˜2 (c, b1 ) = b2 , X (−1)σ l˜2 (l˜2 (cσ(1) , cσ(2) ), cσ(3) ) = b3 , (ii)
(3) (4)
σ∈unsh(2,1)
where c, ci ∈ X0 , bi ∈ B0 and with the additional structures on X∗ as well as on F that such maps will yield. We begin with N Lemma 1. The existence of a skew-symmetric linear map l˜2 : X0 X0 −→ X0 that satisfies N condition (i) is equivalent to the existence of a skew-symmetric linear map [·, ·] : F F −→ F. N F via the diagram Proof. l˜2 induces a linear mapping on F N l˜2 X0 X0 X0 −→ λ⊗λ↑ ↓η N [·,·] F F −→ F . The fact that l˜2 satisfies condition (i) guarantees that [·, ·] is well-defined on the homology classes. N Conversely, given [·, ·] : F F −→ F , define l˜2 = λ ◦ [·, ·] ◦ η ⊗ η. It is clear that l˜2 is skew-symmetric because [·, ·] is, and condition (i) is satisfied in the strong sense that l˜2 (c, b) = 0. N Lemma 2. Assume that l˜2 : X0 X0 −→ X0 satisfies condition (i). Then the induced bracket on F is a Lie bracket if and only if l˜2 satisfies condition (ii). Proof. Assume that the induced bracket on F is a Lie bracket; recall that the bracket is given by the composition η ◦ l˜2 ◦ λ ⊗ λ. The Jacobi identity takes on the form, for arbitrary fi ∈ F, X (−1)σ (η ◦ l˜2 ◦ λ ⊗ λ)(η ◦ (l˜2 ⊗ 1) ◦ (λ ⊗ λ ⊗ 1) σ∈unsh(2,1)
X
⇔
(fσ(1) ⊗ fσ(2) ⊗ fσ(3) ) = 0 σ
(−1) (η ◦ l˜2 ◦ λ ⊗ λ)(η ◦ l˜2 (λ(fσ(1) ) ⊗ λ(fσ(2) )) ⊗ fσ(3) ) = 0
σ∈unsh(2,1)
⇔
X
(−1)σ η ◦ l˜2 (λ ◦ η ◦ l˜2 (λ(fσ(1) ) ⊗ λ(fσ(2) )) ⊗ λ(fσ(3) )) = 0
σ∈unsh(2,1)
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G. Barnich, R. Fulp, T. Lada, J. Stasheff
X
⇔
(−1)σ η ◦ l˜2 ((1 + l1 ◦ s) ◦ l˜2 (λ(fσ(1) ) ⊗ λ(fσ(2) )) ⊗ λ(fσ(3) )) = 0
σ∈unsh(2,1)
X
⇔
σ∈unsh(2,1)
X
+
(−1)σ η ◦ l˜2 (l˜2 (λ(fσ(1) ) ⊗ λ(fσ(2) )) ⊗ λ(fσ(3) ))
(−1)σ η ◦ l˜2 (l1 ◦ s ◦ l˜2 (λ(fσ(1) ) ⊗ λ(fσ(2) )) ⊗ λ(fσ(3) )) = 0
σ∈unsh(2,1)
X
⇔ η(
(−1)σ l˜2 (l˜2 (λ(fσ(1) ) ⊗ λ(fσ(2) )) ⊗ λ(fσ(3) ))) + η(b) = 0.
σ∈unsh(2,1)
But η(b) = 0 and so X
(−1)σ l˜2 (l˜2 (λ(fσ(1) ) ⊗ λ(fσ(2) )) ⊗ λ(fσ(3) )) = b0 ∈ B0 .
(5)
σ∈unsh(2,1)
The converse follows from a similar calculation.
Remark. The interesting case here is when F is only known as X0 /B0 and the only characterization of the Lie bracket [·, ·] in F is as the bracket induced by l˜2 . An important particular case, to be considered elsewhere, occurs when X0 is a Lie algebra G with Lie bracket L2 and B0 a Lie ideal. The bracket l˜2 is defined by choosing a vector space complement K of the ideal B0 in X0 and then projecting the Lie bracket L2 onto K. Hence, L2 (c1 , c2 ) = l˜2 (c1 , c2 ) + b(c1 , c2 ), where b(c1 , c2 ) is a well-defined element of B0 . Indeed, by definition, property (i) holds with zero on the right hand side. Property (ii) follows from the Jacobi identity for L2 : 0= =
X
X
(−1)σ L2 (L2 (cσ(1) , cσ(2) ), cσ(3) )
σ∈unsh(2,1)
(−1)σ [l˜2 (L2 (cσ(1) , cσ(2) ), cσ(3) ) + b(L2 (cσ(1) , cσ(2) ), cσ(3) )]
σ∈unsh(2,1)
=
X
(−1)σ [l˜2 (l˜2 (cσ(1) , cσ(2) ), cσ(3) ) + b(L2 (cσ(1) , cσ(2) ), cσ(3) )].
(6)
σ∈unsh(2,1)
We now turn our attention to the maps that l˜2 induces on the complex X∗ . N Lemma 3. A skew-symmetric linear map l˜2 : X0 X0 −→ XN 0 that satisfies condition (i) extends to a degree zero skew-symmetric chain map l2 : X∗ X∗ −→ X∗ . N ˜ Proof. We first Nextend l2 to a linear map l2 : X1 X0 −→ X1 by the following: N let x1 ⊗ x0 ∈ X1 X0 . Then l1 (x1 ⊗ x0 ) = l1 x1 ⊗ x0 + x1 ⊗ l1 x0 = l1 x1 ⊗ x0 ∈ X0 X0 . So we have that l2 l1 (x1 ⊗ x0 ) = l˜2 (l1 x1 ⊗ x0 ) = b by condition N (i). Write b = l1 z1 for z1 ∈ X1 and define l2 (x1 ⊗ x0 ) = z1 . Also extend l2 to X0 X1 by skew-symmetry: l2 (x0 ⊗ x1 ) = −l2 (x1 ⊗ x0 ). Note that l2 is a chain map by construction. Now assume thatN l2 is defined and is a chainNmap on elements of degree less than or equal to n in X∗ X∗ . Let xp ⊗ xq ∈ Xp Xq , where p + q = n + 1. Because l1 (xp ⊗ xq ) has degree n, l2 l1 (xp ⊗ xq ) is defined. We have that
The sh Lie Structure of Poisson Brackets in Field Theory
589
l1 l2 l1 (xp ⊗ xq ) = = l1 l2 [l1 xp ⊗ xq + (−1)p xp ⊗ l1 xq ] = l2 l1 [l1 xp ⊗ xq + (−1)p xp ⊗ l1 xq ] because l2 is a chain map = l2 [l1 l1 xp ⊗ xq + (−1)p−1 l1 xp ⊗ l1 xq +(−1)p l1 xp ⊗ l1 xq + xp ⊗ l1 l1 xq ] = 0
(7)
because l12 = 0 and (−1)p and (−1)p−1 have opposite parity. Thus l2 l1 (xp ⊗ xq ) is a cycle in Xn and so there is an element zn+1 ∈ XN n+1 with l1 zn+1 = l2 l1 (xp ⊗ xq ). Define l2 (xp ⊗ xq ) = zn+1 . As before, extend l2 to Xq Xp by skew-symmetry and note that l2 is a chain map by construction. Nn X∗ −→ Remark. We will be concerned with (graded) skew-symmetric maps fn : Nn+k Nk X∗ −→ X∗ via the equation X∗ that have been extended to maps fn : fn (x1 ⊗ . . . ⊗ xn+k ) = X (−1)σ e(σ)fn (xσ(1) ⊗ . . . ⊗ xσ(n) ) ⊗ xσ(n+1) ⊗ . . . ⊗ xσ(n+k) ,
(8)
unsh(n,k)
where e(σ) is the Koszul sign (see e.g. [16]). This extension arises from the skewsymmetrization of the extension of a linear map as a skew coderivation on the tensor coalgebra on the graded vector space X∗ [15]. The extension of the differential l1 assumed in the previous lemma is equivalent to the one given by this construction. We assume for the remainder of this section that all maps have been extended in this fashion when necessary; moreover, we will use the uniqueness of such extensions when needed. When the original skew-symmetric map l˜2 satisfies both conditions (i) and (ii), there is a very rich algebraic structure on the complex X∗ . N Proposition 4. A skew-symmetric linear map l˜2 :N X0 X0 −→ X0 that satisfies conditions (i) and (ii) extendsNto a chain N map l2 : X∗ X∗ −→ X∗ ; moreover, there exists a degree one map l3 : X∗ X∗ X∗ −→ X∗ with the property that l1 l3 +l3 l1 +l2 l2 = 0. Here, we have suppressed the notation that is used to indicate the indexing of the summands over the appropriate unshuffles as well as the corresponding signs. They are given explicitly in Definition 5 below. N Proof. We extend l˜2 to l2 : X∗ X∗ −→ X∗ as in the previous lemma. In degree zero, l2 l2 (x1 ⊗ x2 ⊗ x3 ) is equal to a boundary b in X0 by condition (ii). There exists an element Nz ∈ X N1 with l1 z = b and so we define l3 (x1 ⊗ x2 ⊗ x3 ) = −z. Because l1 = 0 on X0 X0 X0 , we have that l1 l3 + l2 l2 + l3 l1 = 0. N N Now assume that l3 is defined up to degree p in X∗ X∗ X∗ and satisfies the relation l1 l3 + l2 l2 + l3 l1 = 0. Consider the map l2 l2 + l3 l1 which is inductively defined on N N elements of degree p+1 in X∗ X∗ X∗ . We have that l1 [l2 l2 +l3 l1 ] = l1 l2 l2 +l1 l3 l1 = l2 l1 l2 + l1 l3 l1 = l2 l2 l1 + l1 l3 l1 = [l2 l2 + l1 l3 ]l1 = −l3 l1 l1 = 0. Thus the image of l2 l2 + l3 l1 is a boundary in Xp which is then the image of an element, p+2 ∈ Xp+2 . Define now Nsay zN l3 applied to the original element of degree p + 1 in X∗ X∗ X∗ to be this element zp+1 . In the proof of the proposition above, we made repeated use of the relation l1 l2 −l2 l1 = 0 when extended to an arbitrary number of variables. We may justify this by observing that the map l1 l2 − l2 l1 is the commutator of the skew coderivations l1 and l2 and is thus
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a coderivation; it follows that the extension of this map must equal the extension of the 0 map. The relations among the maps li that were generated in the previous results are the first relations that one encounters in an sh Lie algebra (L∞ algebra). Let us recall the definition [17, 16]. Definition 5. An sh Lie structure on a graded vector space X∗ is a collection of linear, Nk X∗ −→ X∗ of degree k − 2 that satisfy the relation skew symmetric maps lk : X
X
e(σ)(−1)σ (−1)i(j−1) lj (li (xσ(1) , . . . , xσ(i) ), . . . , xσ(n) ) = 0,
i+j=n+1 unsh(i,n−i)
(9) where 1 ≤ i, j. Remark. Recall that the suspension of a graded vector space X∗ , denoted by ↑ X∗ , is the graded vector space defined by (↑ X∗ )n = Xn−1 while the desuspension of X∗ is given by (↓ X∗ )n = Xn+1 . It can be shown, [17, 16], Theorem 2.3, that the data in the definition is equivalent to V∗ ↑ X, the cocommutative coalgebra a) the existence of a degree −1 coderivation D on on the graded vector space ↑ X, with D2 = 0. and to V∗ ↑ X ∗ , the exterior algebra on the b) the existence of a degree +1 derivation δ on 2 suspension of the degree-wise dual of X∗ , with δ = 0. In this case, we require that X∗ be of finite type. Let us examine the relations in the above definition independently of the underlying vector space X∗ and write them in the form X
(−1)i(j−1) li lj = 0,
i+j=n+1
where we are assuming that the sums over the appropriate unshuffles with the corresponding signs are incorporated into the definition of the extended maps lk . We will require the fact that the map X
(−1)i(j−1) lj li :
n O
X∗ −→ X∗
i,j>1
is a chain map in the following sense: Lemma 6. Let {lk } be an sh Lie structure on the graded vector space X∗ . Then l1
X
(−1)(j−1)i lj li = (−1)n−1 (
i,j>1
where i + j = n + 1.
X
(−1)(j−1)i lj li )l1 ,
i,j>1
(10)
The sh Lie Structure of Poisson Brackets in Field Theory
591
Proof. Let us reindex the left hand side of the above equation and write it as l1
n−1 X
(−1)(n−i)i ln−i+1 li
i=2
which, after applying the sh Lie relation to the composition l1 ln−i+1 , is equal to n−i n−1 XX
(−1)(n−i)i (−1)(k−1)(n−i−k)+1 lk ln−i−k+2 li .
i=2 k=2
On the other hand, the right hand side may be written as (−1)n−1
n−1 X
(−1)(i−1)(n−i+1) li ln−i+1 l1
i=2
which in turn is equal to (−1)n−1
n−i n−1 XX
(−1)(i−1)(n−i+1) (−1)(n−i−k+1)k (−1)n−i+1 li ln−i+2−k lk .
i=2 k=2
It is clear that the two resulting expressions have identical summands and a straightforward calculation yields that the signs as well are identical. The argument in the previous proposition may be extended to construct higher order maps lk so that we have N Theorem 7. A skew-symmetric linear map l˜2 : X0 X0 −→ X0 that satisfies conditions (i) and (ii) extends to an sh Lie structure on the resolution space X∗ . Proof. We already have the required maps l1 , l2 and l3 from our previous work. We use induction to assume that we have the maps lk for 1 ≤ k < n that satisfy the relation in the definition of an sh Lie structure. To construct the map ln , we begin with the map X
(−1)(j−1)i lj li :
n O
X0 −→ Xn−3
i+j=n+1
with i, j > 1. Apply the differential l1 to this map to get X X (−1)(j−1)i lj li = (−1)n−1 (−1)(j−1)i lj li l1 = 0 l1 i+j=n+1
(11)
i+j=n+1
where the first equality follows from the lemma and the second equality from the fact Nn X0 . The acyclicity of the complex that l1 is 0 on Nn X∗ will then yield, with care X∗ and it will satisfy the sh Lie to preserve the desired symmetry, the map ln on relations by construction. Finally, assume that all of the maps lk for k < n have been constructed so as to satisfy the sh relations and that ln has been constructed in a similar fashion through NLie n X∗ . We have the map degree p in
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G. Barnich, R. Fulp, T. Lada, J. Stasheff
X
(−1)(j−1)i lj li : (
n O
X∗ )p −→ Xp−3
i+j=n+1
to which we may apply the differential l1 . This results in X X (−1)(j−1)i l1 lj li = (−1)(j−1)i l1 lj li + (−1)n−1 l1 ln l1 i+j=n+1
i,j>1
=
X
(−1)(j−1)i (−1)n−1 lj li l1 + (−1)n−1 l1 ln l1
i,j>1
=
X
(−1)(j−1)i (−1)n−1 lj li l1 + (−1)n−1 (−
i,j>1
X
(−1)(j−1)i lj li l1 ) = 0
i,j>1
and so again, we have the existence of ln together with the appropriate sh Lie relations. Remarks. (i) It may be the case in practice that the complex X∗ is truncated at height n, i.e. we have that X∗ is not a resolution but rather may have non-trivial homology in degree n as well as in degree 0. In such a case, our construction of the maps lk may be terminated by degree n obstructions. More precisely, we have that the vanishing of Hk X∗ for k different from 0 and n will then only guarantee the existence of the requisite Nk X∗ )p −→ Xp+k−2 for p + k − 2 ≤ n. maps lk : ( (ii) If property (i) holds with zero on the right hand side, i.e. so that l2 vanishes if one of the xi ’s is in B0 , then l2 can be extended trivially (to be a chain map) as zero on N2 ( X∗ )p for p > 0. It is easy to see that in the recursive construction, one can choose similarly trivial extensions of the maps lk for k > 2 to all of the resolution complex, i.e. they are defined to vanish whenever one of their arguments belongs to B0 or Xp Nk for p > 0. Hence they induce well-defined maps lˆk on F . With these choices, each of the defining equations of the sh Lie algebra on X∗ involves only two terms, namely l1 lk + lk−1 l2 = 0 for k ≥ 3. (iii) If Hk X∗ = 0 for 0 < k < n and property (i) holds with zero on the right hand side, Nn+2 we have defined a map on F which may be non-zero. If so, the only non-trivial defining equation of the induced sh Lie algebra on F reduces to lˆn+2 lˆ2 = 0. For example, if n = 1, Σ lˆ3 (lˆ2 (xi , xj ), xk , xm ) = 0 where the sum is over all permutations of (1234) such that i < j and k < m. In Sect. 3, we apply this construction in the context of Poisson brackets in field theory. 2.2. Generalization to the graded case. The above construction of an L∞ -structure on the resolution of a Lie algebra can be extended in a straightforward way to the graded case, i.e. when the Lie algebra is graded (either by Z or by Z/2, the super case) and the bracket is of a fixed degree, even or odd, satisfying the appropriate graded version of skew-symmetry and the Jacobi identity. We will refer to all of these possibilities as graded Lie algebras although the older mathematical literature uses that term only for the case of a degree 0 bracket. In these situations, the resolution X∗ is bigraded and the inductive steps proceed with respect to the resolution degree (see [20, 15] for carefully worked out examples).
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The graded case occurs in the Batalin-Fradkin-Vilkovisky approach to constrained Hamiltonian field theories [8, 2, 7, 14] where their bracket is of degree 0 and in the Batalin-Vilkovisky anti-field formalism for mechanical systems or field theories [3, 4, 14] with their anti-bracket of degree 1. In all these cases, one need only take care of the signs by the usual rule: when interchanging two things (operators, fields, ghosts, etc.), be sure to include the sign of the interchange. 3. Local Field Theory with a Poisson Bracket We first review the result that the cohomological resolution of local functionals is provided by the horizontal complex. Then, we give the definition of a Poisson bracket for local functionals. The existence of such a Poisson bracket will assure us that the conditions of the previous section hold. Hence, we show that to the Poisson bracket for local functionals corresponds an sh Lie algebra on the graded differential algebra of horizontal forms. 3.1. The horizontal complex as a resolution for local functionals. In this subsection, we introduce some basic elements from jet-bundles and the variational bicomplex relevant for our purpose. More details and references to the original literature can be found in [18, 19, 5, 1]. For the most part, we will follow the definitions and the notations of [18]. Although much of what we do is valid for general vector bundles, we will not be concerned with global properties. We will use local coordinates most of the time, though we will set things up initially in the global setting. Let M be an n-dimensional manifold and π : E → M a vector bundle of fiber ∞ : dimension k over M . Let J ∞ E denote the infinite jet bundle of E over M with πE ∞ ∞ ∞ J E → E and πM : J E → M the canonical projections. The vector space of smooth sections of E with compact support will be denoted 0E. For each (local) section φ of E, let j ∞ φ denote the induced (local) section of the infinite jet bundle J ∞ E. The restriction of the infinite jet bundle over an appropriate open U ⊂ M is trivial with fibre an infinite dimensional vector space V ∞ . The bundle π ∞ : J ∞ EU = U × V ∞ → U
(12)
then has induced coordinates given by (xi , ua , uai , uai1 i2 , . . . , ).
(13)
We use multi-index notation and the summation convention throughout the paper. If j ∞ φ is the section of J ∞ E induced by a section φ of the bundle E, then ua ◦ j ∞ φ = ua ◦ φ and uaI ◦ j ∞ φ = (∂i1 ∂i2 ...∂ir )(ua ◦ j ∞ φ), where r is the order of the symmetric multi-index I = {i1 , i2 , ..., ir }, with the convention that, for r = 0, there are no derivatives. The de Rham complex of differential forms ∗ (J ∞ E, d) on J ∞ E possesses a differential ideal, the ideal C of contact forms θ which satisfy (j ∞ φ)∗ θ = 0 for all sections φ with compact support. This ideal is generated by the contact one-forms, which in local coordinates assume the form θJa = duaJ − uaiJ dxi . Contact one-forms of order 0 satisfy (j 1 φ)∗ (θ) = 0. In local coordinates, contact forms of order zero assume the form θa = dua − uai dxi .
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Remarkably, using the contact forms, we see that the complex ∗ (J ∞ E, d) splits as a bicomplex (though the finite level complexes ∗ (J p E) do not). The bigrading is described by writing a differential p-form α as an element of r,s (J ∞ E), with p = r + s J (θJA ∧ dxI ), where when α = αIA dxI = dxi1 ∧ ... ∧ dxir
and
θJA = θJa11 ∧ ... ∧ θJass .
(14)
J be local We intend to restrict the complex ∗ by requiring that the functions αIA functions in the following sense.
Definition 8. A local function on J ∞ E is the pullback of a smooth function on some finite jet bundle J p E, i.e. a composite J ∞ E → J p E → R. In local coordinates, a local function L(x, u(p) ) is a smooth function in the coordinates xi and the coordinates uaI , where the order |I| = r of the multi-index I is less than or equal to some integer p. The space of local functions will be denoted Loc(E), while the subspace consisting ∞ ∗ ) f for f ∈ C ∞ M is denoted by LocM . of functions (πM Henceforth, the coefficients of all differential forms in the complex ∗ (J ∞ E, d) are required to be local functions, i.e., for each such form α there exists a positive integer p such that α is the pullback of a form of ∗ (J p E, d) under the canonical projection of J ∞ E onto J p E. In this context, the horizontal differential is obtained by noting that dα is in r+1,s ⊕ r,s+1 and then denoting the two pieces by, respectively, dH α and dV α. One can then write J J A θJA ∧ dxi ∧ dxI + αIA θJi ∧ dxi ∧ dxI }, dH α = (−1)s {Di αIA
where A = θJi
s X r=1
(15)
(θJa11 ∧ ...θJarr i ... ∧ θJass ),
and where Di =
∂ ∂ + uaiJ a ∂xi ∂uJ
(16)
is the total differential operator acting on local functions. We will work primarily with the dH subcomplex, the algebra of horizontal forms ∗,0 , which is the exterior algebra in the dxi with coefficients that are local functions. In this case we often use Olver’s notation D for the horizontal differential dH = dxi Di where Di is defined above. In addition to this notation, we also utilize the operation which is defined as follows. Given any differential r-form α on a manifold N and a vector field X on N , X α denotes that (r-1)-form whose value at any x ∈ N and (v1 , ..., vr−1 ) ∈ (Tx N × · · · × Tx N ) is αx (Xx , v1 , ..., vr−1 ). We will sometimes use the notation X(α) in place of X α. ∞ ∗ Let ν denote a fixed volume form on M and let ν also denote its pullback (πE ) (ν) to J ∞ E so that ν may be regarded either as a top form on M or as defining elements P ν of n,0 E for each P ∈ C ∞ (J ∞ E). We will almost invariably assume that ν = dn x = dx1 ∧ · · · ∧ dxn . It is useful to observe that for Ri ∈ Loc(E) and α = (−1)i−1 Ri (
∂ ∂xi
dn x),
(17)
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595
then dH α = (−1)i−1 Dj Ri [dxj ∧ (
∂ ∂xi
dn x)] = Dj Rj dn x.
(18)
Hence, a total divergence Dj Rj may be represented (up to the insertion of a volume dn x) as the horizontal differential of an element of n−1,0 (J ∞ E). It is easy to see that the converse is true so, that, in local coordinates, one has that total divergences are in one-to-one correspondence with D-exact n-forms. Definition 9. A local functional Z Z L(x, φ(p) (x))dvolM = (j ∞ φ)∗ L(x, u(p) )dvolM L[φ] = M
(19)
M
is the integral over M of a local function evaluated for sections φ of E of compact support. The space of local functionals F is the vector space of equivalence classes of local functionals, where two local functionals are equivalent if they agree for all sections of compact support. If one does not want to restrict oneself to the case where the base space is a subset of Rn , one has to take the transformation properties of the integrands under coordinate transformations into account and one has to integrate a horizontal n-form rather than a multiple of dxn by an element of Loc(E). Lemma 10. The vector space of local functionals F is isomorphic to the cohomology group H n (∗,0 , D). Proof. Recall that one has a natural mapping ηˆ from n,0 (J ∞ E) onto F defined by Z (j ∞ φ)∗ (P )ν ∀φ ∈ 0E. (20) η(P ˆ ν)(φ) = M
It is well-known (see e.g. [18]) that η(P ˆ ν)[φ] = 0 for all φ of compact support if and only if in coordinates P may be represented as a divergence, i.e., iff P = Di Ri for ˆ ν) = 0 if and only if there exists a form some set of local functions {Ri }. Hence, η(P n−1,0 β∈ such that the horizontal differential dH maps β to P ν. So the kernel of ηˆ is precisely dH n−1,0 and ηˆ induces an isomorphism from H n (dH ) = n,0 /dH n−1,0 onto the space F of local functionals. For later use, we also note that the kernel of ηˆ coincides with the kernel of the Euler-Lagrange operator: for 1 ≤ a ≤ m, let Ea denote the ath component of the Euler-Lagrange operator defined for P ∈ Loc(E) by Ea (P ) =
∂P ∂P ∂P ∂P − ∂i a + ∂i ∂j a − ... = (−D)I ( a ). a ∂u ∂ui ∂uij ∂uI
(21)
The set of components {Ea (P )} are in fact the components of a covector density with respect to the generating set {θa } for C0 , the ideal generated by the contact one-forms of order zero. Consequently, the Euler operator E(P dn x) = Ea (P )(θa ∧ dn x),
(22)
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for {θa } a basis of C0 , gives a well-defined element of n,1 . We have E(P ν) = 0 iff P ν = dH β. For convenience we will also extend the operator E to map local functions to n,1 , so that E(P ) is defined to be E(P dn x) for each P ∈ Loc(E). In Sect. 2, we have assumed that we have a resolution of F ' H n (∗,0 , dH ). In the case where M is contractible, such a resolution necessarily exists and is provided by the following (exact) extension of the horizontal complex ∗,0 (Loc(E), dH ): dH dH dH dH R −→ 0,0 −→ . . . −→ n−2,0 −→ n−1,0 −→ n,0 ↓η ↓η ↓η ↓η ↓η 0 −→ 0 −→ . . . −→ 0 −→ 0 −→ H0 = F Alternatively, we can achieve a resolution by taking out the constants: the space X∗ is given by Xi = n−i,0 , for 0 ≤ i < n, Xn = 0,0 /R, and Xi = 0, for i > n + 1. Either way, we have a resolution of F and can proceed with the construction of an sh Lie structure. (For general vector bundles E, the assumption that such a resolution exists imposes topological restrictions on E which can be shown to depend only on topological properties of M [1].) 3.2. Poisson brackets for local functionals. To begin to apply the results of Sect. 2, we must have a bilinear skew-symmetric mapping l˜2 from n,0 × n,0 to n,0 such that: (i) l˜2 (α, dH β) belongs to dH n−1,0 for all α ∈ n,0 and β ∈ n,0 , and P n−1,0 |σ| ˜ ˜ for all α1 , α2 , (ii) σ∈unsh(2,1) (−1) l2 (l2 (ασ(1) , ασ(2) ), ασ(3) ) belongs to dH n,0 α3 ∈ . To introduce a candidate l˜2 , we define additional concepts. We say that X is a generalized vector field over M iff X is a mapping which factors through the differential of the projection of J ∞ E to J r E for some non-negative integer r and which assigns to each ∞ (w). Similarly Y is a generalized vector field w ∈ J ∞ E a tangent vector to M at πM r over E iff Y also factors through J E for some r and assigns to each w ∈ J ∞ E a ∞ (w). In local coordinates one has vector tangent to E at πE X = X i ∂/∂xi ,
Y = Y j ∂/∂xj + Y a ∂/∂ua ,
(23)
where X i , Y j , Y a ∈ Loc(E). A generalized vector field Q on E is called an evolutionary vector field iff (dπ)(Qw ) = 0 for all w ∈ J ∞ E. In adapted coordinates, an evolutionary vector field assumes the form Q = Qa (w)∂/∂ua . Given a generalized vector field X on M , there exists a unique vector field denoted ∞ )(T ot(X)) = X and θ(T ot(X)) = 0 for every contact one-form θ. Tot(X) such that (dπM In the special case that X = X i ∂/∂xi , it is easy to show that T ot(X) = X i Di . We say that Z is a first order total differential operator iff there exists a generalized vector field X on M such that Z = T ot(X). More generally, a total differential operator Z is by definition the sum of a finite number of finite order operators Zα for which there exists functions ZαJ ∈ Loc(E) and first order total differential operators W1 , W2 , ...Wp on M such that Zα = ZαJ (Wj1 ◦ Wj2 ◦ ... ◦ W jp ),
(24)
where J = {j1 , j2 , ..., jp } is a fixed set of multi-indices depending on α (p = 0 is allowed).
The sh Lie Structure of Poisson Brackets in Field Theory
597
In particular, in adapted coordinates, a total differential operator assumes the form Z = Z J DJ , where Z J ∈ Loc(E) for each multi-index J, and the sum over the multiindex J is restricted to a finite number of terms. In an analogous manner, for every evolutionary vector field Q on E, there exists its ∞ )(pr(Q)) = prolongation, the unique vector field denoted pr(Q) on J ∞ E such that (dπE Q and Lpr(Q) (C) ⊆ C, where Lpr(Q) denotes the Lie derivative operator with respect to the vector field pr(Q) and C is the ideal of contact forms on J ∞ E. In local adapted coordinates, the prolongation of an evolutionary vector field Q = Qa ∂/∂ua assumes the form pr(Q) = (DJ Qa )∂/∂uaJ . The set of all total differential operators will be denoted by T DO(E) and the set of all evolutionary vector fields by Ev(E). Both T DO(E) and Ev(E) are left Loc(E) modules. One may define a new total differential operator Z + called the adjoint of Z by Z Z (j ∞ φ)∗ [SZ(T )]ν = (j ∞ φ)∗ [Z + (S)T ]ν (25) M
M
for all sections φ ∈ 0E and all S, T ∈ Loc(E). It follows that [SZ(T )]ν = [Z + (S)T ]ν + dH ζ
(26)
for some ζ ∈ n−1,0 (E). If Z = Z J DJ in local coordinates, then Z + (S) = (−D)J (Z J S). This follows from an integration by parts in (26) and the fact that (26) must hold for all T (see e.g. [18] Corollary 5.52). Assume that ω is a mapping from C0 × C0 to T DO(E) which is a module homomorphism in each variable separately. The adjoint of ω denoted ω + is the mapping from C0 × C0 to T DO(E) defined by ω + (θ1 , θ2 ) = ω(θ2 , θ1 )+ . In particular ω is skew-adjoint iff ω + = −ω. Using the module basis {θa } for C0 , we define the total differential operators ω ab = a b ω(θ , θ ). From these operators, we construct a bracket on the set of local functionals [9–13] (see e.g. [18, 5] for reviews) by Z ω(θa , θb )(Eb (P ))Ea (Q)dn x, (27) {P, Q} = M
n
n
where P = P d x and Q = Qd x for local functions P and Q. As in other formulas of this type, it is understood that the local functional {P, Q} is to be evaluated at a section φ of the bundle E → M and that the integrand is pulled back to M via j ∞ φ before being integrated. We find it useful to introduce the condensed notation ω(E(P ))E(Q) = ω(θa , θb ) (Eb (P ))Ea (Q) throughout the remainder of the paper. In order to express ω(E(P ))E(Q) in a coordinate invariant notation, note that pr( ∂u∂ a ) E(L) = Ea (L)dn x for each local function L. Consequently, if ∗ is the operator from n,0 E to Loc(E) defined by ∗(P ν) = P , then ω(E(P ))E(Q) = ω(θa , θb )(∗[pr(
∂ ∂ ) E(P )])(∗[pr( a ) E(Q)]). b ∂u ∂u
If coordinates on M are chosen such that ν = dn x, then it follows that Z ω(θa , θb )(∗[pr(Yb ) E(P )])(∗[pr(Ya ) E(Q)])ν, {P, Q} = M
(28)
(29)
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where {Ya } and {θb } are required to be local bases of Ev(E) and C0 , respectively, such that θb (Ya ) = δab . It is easy to show that the integral is independent of the choices of bases and consequently, one has a coordinate-invariant description of the bracket for local functionals. 3.3. Associated sh Lie algebra on the horizontal complex. This bracket for functionals provides us with some insight as to how l˜2 may be defined; namely for α1 = P1 ν and α2 = P2 ν ∈ n,0 , define l˜2 (α1 , α2 ) to be the skew-symmetrization of the integrand of {P1 , P2 } : 1 l˜2 (α1 , α2 ) = [ω(E(P1 ))E(P2 ) − ω(E(P2 ))E(P1 )]ν. 2
(30)
By construction, l˜2 is skew-symmetric and, from E(dH β) = 0 for β ∈ n−1,0 , it follows that l˜2 (α, dH β) = 0. Thus a strong form of property (i) required above for l˜2 holds. The symmetry properties of ω may be used to simplify the equation for l˜2 (α1 , α2 ). Skew-adjointness of ω implies ω(E(P1 ))E(P2 )ν = −ω(E(P2 ))E(P1 )ν + dH γ
(31)
n−1,0
, which depends on α1 = P1 ν and α2 = P2 ν. In fact, since for some γ ∈ E(dH β) = 0, the element γ depends only on the cohomology classes P1 , P2 of α1 and α2 . A specific formula for γ can be given by straightforward integrations by parts. Hence, from (30) and (31), we get 1 l˜2 (α1 , α2 ) = ω(E(P1 ))E(P2 )ν − dH γ(P1 , P2 ). 2 R ∞ ∗ Furthermore, since M (j φ) dH γ = 0 for all φ ∈ 0E, we see that Z Z ∞ ∗ {P1 , P2 }(φ) = (j φ) [ω(E(P1 ))E(P2 )]ν = (j ∞ φ)∗ l˜2 (α1 , α2 ). M
(32)
(33)
M
In order to explain the conditions necessary for l˜2 to satisfy the required “Jacobi” condition, we formulate the problem in terms of “Hamiltonian” vector fields (see e.g. [18] Chapter 7.1 or [5] Chapter 2.5) and their corresponding Lie brackets. Given a local function Q, one defines an evolutionary vector field vωEQ by vωEQ = ω ab (Eb (Q))∂/∂ua = ω(θa , θb )(∗[pr(∂/∂ub ) E(Q)])∂/∂ua .
(34)
Again, the vector field vωEQ depends only on the functional Q and not on which representative Q one chooses in the cohomology class Q ∼ Qν + dH n−1,0 . Thus, for a given functional Q, letR vˆ Q = vωEQ for any representative Q. Since {P1 , P2 } = M l˜2 (α1 , α2 ), we see that vˆ {P1 ,P2 } = vωE(l˜2 (α1 ,α2 )) .
(35)
ω(E(P1 ))E(P2 ) = ω ab (Eb (P1 ))Ea (P2 ) = pr[ω ab Eb (P1 )∂/∂ua ] E(P2 ) = pr(vωE(P1 ) ) E(P2 ) = pr(vˆ P1 ) E(P2 ).
(36)
Note also that
Moreover, integration by parts allows us to show that
The sh Lie Structure of Poisson Brackets in Field Theory
pr(Q)(P )ν = pr(Q) d(P ν) = pr(Q) E(P ν) + dH (pr(Q) σ),
599
(37)
for arbitrary evolutionary vector fields Q and local functions P , and for some form σ ∈ n−1,0 depending on P . For every such Q, let IQ denote a mapping from n,0 to n−1,0 such that pr(Q)(P )ν = pr(Q) E(P ) + dH (IQ (P ν))
(38)
for all P ν ∈ n,0 . Explicit coordinate expressions for IQ can be found in [18] chapter 5.4 or in [5] chapter 17.5 . It follows from the identities (32), (36) and (38) that 1 l˜2 (α1 , α2 ) = pr(vˆ P1 )(P2 )ν − dH Ivˆ P1 (P2 ν) − dH γ(P1 , P2 )). 2
(39)
Thus, for α1 , α2 , α3 ∈ n,0 , we see that l˜2 (l˜2 (α1 , α2 ), α3 ) = −l˜2 (α3 , l˜2 (α1 , α2 )) = = −l˜2 (α3 , pr(vˆ P1 )(P2 )ν − dH (·)) = −l˜2 (α3 , pr(vˆ P1 )(P2 )ν) (40) and l˜2 (l˜2 (α1 , α2 ), α3 ) = −pr(vˆ P3 )(pr(vˆ P1 )(P2 ))ν + dH ζ,
(41)
where ζ is given by 1 ζ(P1 , P2 , P3 ) = Ivˆ P3 (pr(vˆ P1 )(P2 )ν) + γ(P3 , {P1 , P2 }). 2
(42)
Rewriting the left hand side of the Jacobi identity in Leibnitz form and using (35), (39) and (41), we find X (−1)|σ| l˜2 (l˜2 (ασ(1) , ασ(2) ), ασ(3) ) = σ∈unsh(2,1)
= −l˜2 (α3 , l˜2 (α1 , α2 )) − l˜2 (l˜2 (α1 , α3 ), α2 ) + l˜2 (α1 , l˜2 (α3 , α2 )) = = [−pr(vˆ P3 )(pr(vˆ P1 )(P2 )) + pr(vˆ P1 )(pr(vˆ P3 )(P1 )) −pr(vˆ {P1 ,P3 } )(P2 )]ν + dH η,
(43)
with η(P1 , P2 , P3 ) = ζ(P1 , P2 , P3 ) − ζ(P3 , P2 , P1 ) 1 +Ivˆ {P1 ,P3 } (P2 ν) + dH γ({P1 , P3 }, P2 ). 2
(44)
Although ζ depends on the representative P2 and not its cohomology class, η depends only on the cohomology classes Pi because it is completely skew-symmetric. It follows from this identity that if pr(vˆ {P1 ,P2 } ) = [pr(vˆ P1 ), pr(vˆ P2 )]
(45)
for all P1 , P2 , then {·, ·} satisfies the Jacobi condition. Under these conditions, the mapping H : F −→ Ev(E) defined by H(P) = vˆ P is said to be Hamiltonian. Equivalent conditions on the mapping H alone for the bracket {·, ·} to be a Lie bracket can be found
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in [18, 5]. The derivation given here allows us to give, in local coordinates, an explicit form for the exact term (44) violating the Jacobi identity. If H is Hamiltonian, the bracket l˜2 satisfies condition (ii) and the construction of Sect. 2 applies. Because the resolution stops with the horizontal zero-forms, we get a possibly non-trivial L(n + 2) algebra on the horizontal complex. If we remove the constants, we can then extend to a full L∞ −algebra by defining the further li to be 0. In addition, because property (i) holds without any l1 exact term on the right hand side, remark (ii) at the end of Sect. 2.1 applies, i.e., we need only two terms in the defining equations of the sh Lie algebra and the maps lk induce well-defined higher order maps on the space of local functionals. On the other hand, if we do not remove the constants, the operation ln+2 takes values in Xn = 0,0 = Loc(E) and induces a multi-bracket on H n (∗,0 , dH ) ' F , the space of local functionals, with values in Hn (X∗ , l1 ) = H 0 (∗,0 , dH ) ' HDR (C ∞ (M )) = R. We have thus proved the following main theorem. Theorem 11. Suppose that the horizontal complex without the constants (∗,0 /R, dH ) is a resolution of the space of local functionals F equipped with a Poisson bracket as above. If the mapping H from F to evolutionary vector fields is Hamiltonian, then to the Lie algebra F equipped with the induced bracket lˆ2 = {·, ·}, there correspond maps li : (∗,0 /R)⊗i → ∗−i+2,0 /R for 1 ≤ i ≤ n + 2 satisfying the sh Lie identities l1 lk + lk−1 l2 = 0. The corresponding map lˆn+2 on F ⊗n+2 with values in H 0 (∗,0 , dH ) = R satisfies lˆn+2 lˆ2 = 0. Specific examples for n = 1 are worked out in careful detail by Dickey [6].
4. Conclusion The approach of Gel’fand, Dickey and Dorfman to functionals and Poisson brackets in field theory has the advantage of being completely algebraic. In this paper, we have kept explicitly the boundary terms violating the Jacobi identity for the bracket of functions, instead of throwing them away by going over to functionals at the end of the computation. In this way, we can work consistently with functions alone, at the price of deforming the Lie algebra into an sh Lie algebra. Our hope is that this approach will be useful for a completely algebraic study of deformations of Poisson brackets in field theory. Note: After this paper had been submitted for publication, it was pointed out to us by M. Markl, that if condition (i), (Eq.(3)), holds with zero on the right hand side, the maps lk with k > 3 can be chosen to vanish. This means in the case of the Gelfand-DickeyDorfman bracket that only the homotopy for the Jacobi identity l3 is in fact non-trivial. It will be interesting to explore its physical significance. Acknowledgement. The authors want to thank I. Anderson, L.A. Dickey and M. Henneaux for useful discussions.
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References 1. Anderson, I.: The Variational Bicomplex. Preprint, Utah State University, 1996 2. Batalin, I.A. and Vilkovisky, G.S.: Relativistic s-matrix of dynamical systems with boson and fermion constraints. Phys. Lett. 309–312 (1977) 3. Batalin, I.A. and Vilkovisky, G.S.: Gauge algebra and quantization. Phys. Lett. 102 B, 27–31 (1981) 4. Batalin, I.A. and Vilkovisky, G.S.: Quantization of gauge theories with linearly dependent generators. Phys. Rev. D 28, 2567–2582 (1983); Erratum: Phys. Rev. D30, 508 (1984) 5. Dickey, L.A.: Soliton Equations and Hamiltonian Systems. Advanced Series in Mathematical Physics, vol. 12, Singapore: World Scientific, 1991 6. Dickey, L.A.: Poisson brackets with divergence terms in field theories: two examples. Preprint, University of Oklahoma, 1997 7. Fradkin, E.S. and Fradkina, T.E.: Quantization of relativistic systems with boson and fermion first and second class constraints. Phys. Lett. 72B, 343–348 (1978) 8. Fradkin, E.S. and Vilkovisky, G.S.: Quantization of relativistic systems with constraints. Phys. Lett. 55B, 224–226 (1975) 9. Gel’fand, I.M. and Dickey, L.A.: Lie algebra structure in the formal variational calculus. Funkz. Anal. Priloz. 10 no 1, 18–25 (1976) 10. Gel’fand, I.M. and Dickey, L.A.: Fractional powers of operators and hamiltonian systems. Funkz. Anal. Priloz. 10 no 4, 13–29 (1976) 11. Gel’fand, I.M. and Dorfman, I.Ya.: Hamiltonain operators and associated algebraic structures. Funkz. Anal. Priloz. 13 no 3, 13–30 (1979) 12. Gel’fand, I.M. and Dorfman, I.Ya.: Schouten bracket and Hamiltonian operators. Funkz. Anal. Priloz. 14 no 3, 71–74 (1980) 13. Gel’fand, I.M. and Dorfman, I.Ya.: Hamiltonian operators and infinite dimensional Lie algebras. Funkz. Anal. Priloz. 15 no 3, 23–40 (1981) 14. Henneaux, M. and Teitelboim, C.: Quantization of Gauge Systems. Princeton, W: Princeton Univ. Press, 1992 15. Kjeseth, L. BRST cohomology and homotopy Lie-Rinehart pairs. Dissertation, UNC-CH, 1996 16. Lada, T. and Markl, M.: Strongly homotopy Lie algebras. Comm. in Algebra 2147–2161 (1995) 17. Lada, T. and Stasheff, J.D.: Introduction to sh Lie algebras for physicists. Intern’l J. Theor. Phys. 32, 1087–1103 (1993) 18. Olver, P.J.: Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics, vol. 107, Berlin–Heidelberg–New York: Springer-Verlag, 1986 19. Saunders, D.J.: The Geometry of Jet Bundles. London Mathematical Lecture Notes, vol. 142, Cambridge: Cambridge Univ. Press, 1989 20. Stasheff, J.D.: Homological reduction of constrained Poisson algebras. J. Diff. Geom. 45, 221–240 (1997) Communicated by T. Miwa
Commun. Math. Phys. 191, 603 – 611 (1998)
Communications in
Mathematical Physics c Springer-Verlag 1998
Differentiation of Operator Functions and Perturbation Bounds Rajendra Bhatia, Dinesh Singh, Kalyan B. Sinha Indian Statistical Institute, New Delhi-110016, India. E-mail: [email protected] Received: 4 April 1997 / Accepted: 28 May 1997
Abstract: Given a smooth real function f on the positive half line consider the induced map A → f (A) on the set of positive Hilbert space operators. Let f (k) be the k th derivative map f . We of the real function f and Dk f the k th Fr´echet
of the operator
derivative identify large classes of functions for which Dk f (A) = f (k) (A) , for k = 1, 2, .... This reduction of a noncommutative problem to a commutative one makes it easy to obtain perturbation bounds for several operator maps. Our techniques serve to illustrate the use of a formalism for “quantum analysis” that is like the one recently developed by M. Suzuki.
1. Introduction In several problems of quantum and statistical physics, one has to study various functions on the space of operators in a Hilbert space. Calculus of such functions, therefore, is of great interest. In a recent paper [12] Suzuki has developed a formalism for such a calculus and called it “quantum analysis”. Several references where such analysis is useful are given in this paper. Motivated by our interest in perturbation inequalities for operator functions, we have used a somewhat different, but essentially equivalent, formalism in some of our work. A summary of this approach is given in [2] ; see especially Sect. X.4. One attractive feature of this approach is the ease it affords in calculating norms of derivatives. In [4, 5, 6], it was observed that the norms of certain noncommutative derivatives are, rather surprisingly, equal to those of their commutative analogues. In the present paper this approach is taken further. We will evaluate precisely norms of higher order derivatives of some operator functions. At the same time this will illustrate the simplicity and the power of our methods which should be useful in other problems. Let B(H) denote the space of all bounded linear operators on a Hilbert space H. Let Bs (H) be the set of all self-adjoint operators and B+ (H) the set of all positive definite
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operators. Let f be a real valued measurable function on (0, ∞). This induces (via the spectral theorem) a map from B+ (H) to Bs (H), which we denote again by f. One of the central problems in perturbation theory is to find bounds for kf (A) − f (B)k in terms of kA − Bk. All functions considered here are smooth and have derivatives of all orders. Note that Bs (H) is a real linear space and B+ (H) is an open subset of it. For k = 1, 2, ..., we denote by D k f (A) the k th order Fr´echet derivative of f at A. This is a symmetric multilinear map from the k-fold product Bs (H) × · · · × Bs (H) into Bs (H). Its action can be described as [Dk f (A)](B1 , ..., Bk ) =
∂k f (A + t1 B1 + · · · + tk Bk ) |t1 =···=tk =0 . ∂t1 · · · ∂tk
(1.1)
For basic facts about the Fr´echet differential calculus, the reader may see [1, 2, 8 and 10]. The norm of Dk f (A) is defined as
k
k
D f (A) =
[D f (A)](B1 , ..., Bk ) . sup (1.2) kB1 k=···=kBk k=1
The Taylor theorem says that for all B sufficiently close to A, we have f (B) = f (A) + [Df (A)](B − A) + · · · +
1 [Dk f (A)](B − A, ..., B − A) + · · ·. (1.3) k!
From this we have kf (B) − f (A)k ≤
N X
1
Dk f (A) kB − Akk + O kB − AkN +1 . k!
(1.4)
k=1
We call this an N th order perturbation bound for f . First order perturbation bounds for several matrix and operator functions(not restricted to the class we have delimited above) have been obtained by many authors. See, e.g. [11] for several references on such works for the exponential function and [3] for works on various matrix factorisations. Higher order bounds are rarer to find. One reason for this is the relatively more complex nature of the expression (1.1) for the higher Fr´echet derivatives. In this paper, we will obtain such bounds for two large families of functions of positive operators. These include the exponential function and the power functions Ap , −∞ < p < ∞ . Let f (k) be the (ordinary) derivative of the real valued function f . Then for each A ∈ B+ (H),
(k) k
f (A) = [D f (A)](I, ..., I) . Consequently,
k
D f (A) ≥ f (k) (A) , for all A.
For k = 1, 2, ..., let
Dk = f : Dk f (A) = f (k) (A) for all A ∈ B+ (H) .
(1.5)
The classes Dk turn out to be nonempty, and have several unexpected properties, some of which have been studied in [4, 5, and 6]. In [4] it was shown that all operator monotone functions on (0, ∞) are in D1 and D2 . It is well known that the power function f (t) = tp is operator monotone if and only if 0 ≤ p ≤ 1. In [5] it was shown that every operator
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monotone function is in Dk for all k = 1, 2, ... . The major application of these results in 1 the two papers was to find nth order perturbation bounds for the function |A| = (A∗ A) 2 . This function has been of great interest to physicists as well as numerical analysts ; see [5] for references. It was also shown in [4] that the functions f (t) = tn , n = 2, 3, ... , and f (t) = exp t are also in D1 . None of these are operator monotone. In [6] it was shown p that the √ power function f (t) = t is in D1 if p is in (−∞, 1] or in [2, ∞), but not if p is in (1, 2). It will follow from results proved below that the functions f (t) = exp t and f (t) = tp , −∞ < p ≤ 1, are in the class Dk for all k = 1, 2, ..., and that for p > 1 the function f (t) = tp is in the class Dk for all k ≥ [p + 1] . In the earlier papers mentioned above we relied heavily on integral representations for the functions under consideration. The approach here is somewhat different: we use more of power series. A recent paper [7] follows the ideas in [6] to go up to second order derivatives. Much of this is subsumed here in our analysis. Of course once a function f is shown to be in the class Dk , the problem of finding the norm of Dk f (A) is reduced to that of finding the norm of f (k) (A). A noncommutative problem is thus reduced to its commutative version.
2. The Main Results We want to study the derivatives Dk f (A) for functions f represented by power series. First consider the function f (A) = An , A ∈ B(H) , where n is any natural number. For 1 ≤ k ≤ n , the derivative Dk f (A)(B1 , ..., Bk ) is given by an expression that is linear and symmetric in B1 , ..., Bk , and when dim H = 1 it reduces to the expression f (k) (x) = n(n − 1) · · · (n − k + 1)xn−k =
n! xn−k , (n − k)!
(2.1)
for the k th derivative of the function f (x) = xn . These three requirements dictate that X X Aj1 Bσ(1) · · · Ajk Bσ(k) Ajk+1 , (2.2) Dk f (A)(B1 , ....Bk ) = σ∈Sk
ji ≥0 j1 +···+jk+1 =n−k
n! terms where Sk is the set of permutations on {1, 2, ..., k}. Note that this is a sum of (n−k)! , each of which is a word of length n in which n − k letters are A and the remaining k letters are B1 , ..., Bk , each occurring exactly once. When k = 2, for instance, this reduces to X Aj1 B1 Aj2 B2 Aj3 + Aj1 B2 Aj2 B1 Aj3 . D2 f (A)(B1 , B2 ) = j1 +j2 +j3 =n−2
The reader may find it instructive to compare our formula (2.2) with (4.25b) in [12], and see how one can be derived from the other. Both are noncommutative analogues of (2.1) but the noncommutativity is expressed in different ways by them. Now if f is an analytic function with power series f (z) =
∞ X n=0
an (z − α)n ,
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then we have Dk f (A)(B1 , ..., Bk ) =
X
an Dk (A − α)n (B1 , ..., Bk ),
(2.3)
where Dk (A − α)n is the k th derivative of (A − α)n given by the expression (2.2) with (A − α) in place of A. The only point that needs a little elaboration here is whether the term by term differentiation is justified. An easy way to see this is via the Cauchy integral formula. Let f (z) be analytic in the open right half plane C + and let A be a positive operator whose spectrum is contained in the positive real line . Let γ be a simple closed curve in C + such that the spectrum of A is enclosed by γ in its interior. In that case, by the Riesz functional calculus Z f (z) 1 dz, f (A) = 2πi γ (z − A) so that using (1.1) we see Df (A)(B) =
1 2πi
Z f (z) γ
1 1 B dz, (z − A) (z − A)
and in general Dk f (A)(B1 , ..., Bk ) Z X 1 1 1 1 1 f (z) [ = Bσ(1) ··· Bσ(k) ]dz. (z − A) (z − A) (z − A) (z − A) 2πi γ σ∈Sk
Using this formula, it is easy to justify the term by term differentiation in (2.3). The formula (2.3) should be compared with TheoremVII in [12]. We are now in a position to state and prove our main results. P Theorem 2.1. Let f have a power series representation, f (t) = an tn with an ≥ 0 for all n. Then ∞ \ Dk . f∈ k=1
Proof. Let A ≥ 0. By what has been said above Dk f (A)(B1 , ..., Bk ) =
∞ X
an (Dk An )(B1 , ..., Bk ),
n=k
where the summands (Dk An )(B1 , ..., Bk ) are given by (2.2). So ∞
k
X
D f (A) ≤ an n=k
From (2.1) above f (k) (A) =
∞ X n=k
Since A ≥ 0 and an ≥ 0,
an
n! n−k kAk . (n − k)!
n! An−k . (n − k)!
Differentiation of Operator Functions and Perturbation Bounds
(k)
f (A) =
607
∞
X
n!
n−k A an
(n − k)! n=k
=
∞ X
an
n=k
Thus
n! n−k kAk . (n − k)!
k
D f (A) = f (k) (A) for all k.
(2.4)
Examples. Given below are some functions that satisfy the conditions of Theorem 2.1. Let f (t) be any polynomial all of whose coefficients are non-negative. Let f (t) = exp t. Let f (t) be sinh(t) or cosh(t). The argument presented in the proof of Theorem 2.1 works equally well if the power series representing the function has a finite circle of convergence. Of course one then considers only those operators A which have spectra inside the circle of convergence of the given power series. Thus f (t) = arcsin t, |t| ≤ 1 is in Dk for all k. (v) Hansen [9] has shown that functions that satisfy the hypothesis of Theorem 2.1 are exactly the ones that are monotone with respect to the entry-wise order relation on the space of Hermitian matrices with real entries. This makes Theorem 2.1 more interesting, since we already know [5] that operator monotone functions are in the class ∩∞ k=1 Dk . We are thankful to the referee for pointing this out. P Remark 2.2. If the function f (t) = an tn is such that an ≥ 0 for n ≥ k then the same proof shows that f is in the class Dn for all n ≥ k.
(i) (ii) (iii) (iv)
Our next theorem concerns functions whose Taylor series have real coefficients that are alternately positive and negative. Theorem 2.3. Let f be a map of the positive half line into R such that f has an analytic extension into the right half plane. Further, suppose that the Taylor series of f around each positive α is given by f (z) =
∞ X
an (z − α)n ,
(|z − α| < α)
n=0
with an an+1 < 0 for all n. Then f ∈ Dk for all k ≥ 1. Proof. Let A be a positive operator with inf σ(A) = a > 0 and sup σ(A) = b. Choose a positive real number α such that α > b and assume that f has the Taylor expansion around α as given by the statement of the theorem.We also assume, without loss of generality, that an > 0 for even n and an < 0 for odd n. Let an = b2m when n is even and an = −b2m+1 when n is odd, and where bm is positive for all m. Then f (z) =
∞ X n=0
so that,
b2n (z − α)
2n
−
∞ X n=0
b2n+1 (z − α)2n+1 ,
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f (A) =
∞ X
b2n (A − α)2n −
n=0
∞ X
b2n+1 (A − α)2n+1 .
n=0
Then, as in the proof of Theorem 2.1, we have
k
D f (A)(B1 , ..., Bk ) ≤
X 2n≥k
X (2n + 1)! (2n)! 2n−k 2n+1−k b2n kA − αk b2n+1 kA − αk + . (2n − k)! (2n + 1 − k)! 2n≥k
Now suppose that k is even and k = 2m (the argument for odd k is similar). Then the right hand side of the above inequality is ∞ X
∞ X (2n)! (2n + 1)! 2n−k b2n (a − α) b2n+1 (α − a)2n+1−k + = (2n − k)! (2n + 1 − k)! n=m n=m ∞ X (2n)! (2n + 1)! b2n (a − α)2n−k − b2n+1 (a − α)2n+1−k . (2n − k)! (2n + 1 − k)! n=m n=m
(k)
Since a − α < 0 , this is also equal to f (A) . ( See the definition of a and α at the beginning of the proof and use the spectral resolution of A.) This shows that
k
D f (A) ≤ f (k) (A) .
=
Thus
∞ X
k
D f (A) = f (k) (A) .
Corollary 2.4. Let f be a completely monotone function on (0, ∞) ; i.e., let Z ∞ f (t) = e−λt dµ (λ) , 0
where µ is a positive measure on (0, ∞). Then f ∈ Dk for all k ≥ 1. [For k = 1 this is Theorem 2.1 of [6] and for k = 2 this is Theorem 1 of [7]. Proof. From the nature of the function f , it is easy to check that in the Taylor series expansion about any positive number α, the Taylor coefficients satisfy an an+1 < 0 for all n. Hence f is in Dk for all k. Remark 2.5. It is known that any function that maps the positive real line into itself and satisfies f (n) (x)f (n+1) (x) < 0 for all n and all x is completely monotone. See Widder [13, p. 145, 161]. Remark 2.6. It must be mentioned that Theorem 2.3 is valid in a much more general situation than the one described above. In effect the same proof works to give the following: Let f map the positive real line into the set of real numbers and let it be analytic in C + . Further suppose that there exists a sequence of positive numbers {αn } such that αn → ∞ and such that for each n and for each positive integer k, f (k) (αn )f (k+1) (αn ) < 0. Then f is in Dk for all k. An example of a function that satisfies these criteria but does not satisfy the conditions of Theorem 2.3 is given by f (t) = e−t sin t.
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Remark 2.7. We also note that the same proof leads to the following: Let f map the positive half line into R and suppose that there exists a positive integer k such that f (n) (x)f (n+1) (x) < 0 for all n ≥ k and for all x ≥ δ > 0 for some δ. Then f is in Dn for all n ≥ k. Corollary 2.8. Let f and g be two functions whose Taylor expansions as given by Theorem 2.3 satisfy an an+1 < 0, for all n. Then f + g and f g are also in Dk for all k. Proof. This is a simple consequence of the fact that the Taylor series of the sum and the product is also of the type whose coefficients satisfy an an+1 < 0 for all n. Corollary 2.9. Let f (t) = tp , p ≤ 1. Then f ∈ Dk for all k. Proof. The proof is obvious when we look at the Taylor coefficients of tp .
Corollary 2.10. Let f (t) = tp , 1 < p. Then f ∈ Dk for all k ≥ [p + 1]. Proof. The Taylor coefficients of tp , for the above values of p, satisfy an an+1 < 0 for all n ≥ [p + 1]. So the assertion follows from Remark 2.7. Examples. Let f (t) = e−t . It is easy to see ( by looking at the Taylor series) that f is completely monotone. (ii) Let f (z) be analytic in the entire complex plane except in a disk in the open left half plane centred at a point a of the real line. Assume that its Laurent series representation is given by
(i)
f (z) =
∞ X n=1
αn , (z − a)n
where each αn ≥ 0. Then f (t) is in the class Dn for all n ≥ 0. This is again a consequence of the nature of the Taylor series of f (z) around any point α of the positive real line. It can be easily seen that the Taylor coefficients satisfy the condition of complete monotonicity around any point of the positive half line. 1
(iii) Let f (z) = e z . Though this function has a singularity at zero and not on the negative half line, the logic of the previous example applies to show that the Taylor coefficients of the function (around any point on the positive half line ) satisfy the condition of Theorem 2.3 and so it is in Dk for all k. (iv) It is known that the function f (t) = t + 1/t is not in D, see [6] . However it now follows from Remark 2.7 by looking at the Taylor series of f about any positive α, that the function is in Dk for all k ≥ 2. (v) As a final illustration we consider the function Z ∞ 1 1 − e−λ dλ. e−λt log(1 + ) = t λ 0 This function is in Dk , for all k, being completely monotone. It is interesting to note this because of its similarity to the function log(1 + t). This latter function is in Dk for all k since it is operator monotone.
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3. Perturbation Bounds We have already explained in Sect. 1 that once we know how to evaluate, or estimate,
the norms Dk f (A) , we can use the inequality (1.4) to find perturbation bounds for f in a neighbourhood of A. Thus, for all f in ∩∞ k=1 Dk we can find such bounds up to any desired order. In Sect. 2 we have also shown that certain functions are in the classes Dk for all k larger than some integer m. For such functions too we can obtain perturbation bounds to any desired order. This is explained below. Suppose f ∈ Dk for k ≥ m. Let p be the polynomial of degree m − 1 obtained by keeping the first m terms in the Taylor expansion of f . Let g = f − p. Then, it is easy to see that g is in Dk for all k ≥ 1. If B is close to A, we can write kf (B) − f (A)k ≤ kp(B) − p(A)k + kg(B) − g(A)k . The quantity kg(B) − g(A)k can be estimated by the method explained above. The quantity kp(B) − p(A)k is easy to estimate. Note that kAr − B r k =
r−1 X
B j (A − B)Ar−j−1 .
j=0
From this it follows that kAr − B r k ≤ rM r−1 kA − Bk , where M = max(kAk , kBk). Since p(B) − p(A) is a finite linear combination of such terms, its norm can be easily estimated. Acknowledgement. The second author is on leave from the University of Delhi and is grateful to the Indian Statistical Institute, New Delhi for a visiting appointment and to the University Grants Commission for a Career Award. K.B.S. acknowledges the support of Jawaharlal Nehru Center for Advanced Scientific Research, Bangalore, India.
References 1. Ambrosetti, A. and Prodi, G.: A Primer of Non-Linear Analysis. Cambridge: Cambridge Univ. Press, 1993 2. Bhatia, R.: Matrix Analysis. New York: Springer-Verlag, 1997 3. Bhatia, R.: Matrix factorisations and their perturbations. Linear Algebra Appl. 197/198, 245–276 (1994) 4. Bhatia, R.: First and second order perturbation bounds for the operator absolute value. Linear Algebra Appl. 208, 367–376 (1994) 5. Bhatia, R.: Perturbation bounds for the operator absolute value. Linear Algebra Appl. 226, 539–545 (1995) 6. Bhatia, R. and Sinha, K.B.: Variation of real powers of positive operators. Indiana Univ. Math. J. 43, 913–925 (1994) 7. Bist, V. and Vasudeva, H.L.: Second order perturbation bounds. Publ. RIMS Kyoto Univ (To appear) 8. Dieudonn´e, J.: Foundations of Modern Analysis. London–New York: Academic Press, 1969 9. Hansen, : Functions of matrices with nonnegative entries. Linear Algebra Appl. 166, 29–43 (1992) 10. Hille, E. and Phillips, R.S.: Functional Analysis and Semigroups. American Mathematical Society Colloquium Publ., 1975
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11. Najfeld, I. and Havel, T.F.: Derivatives of the matrix exponential and their computation. Advances Appl. Math. 16, 321–375 (1995) 12. Suzuki, M.: Quantum analysis – non-commutative differential and integral calculi. Commun. Math. Phys. 183, 339–363 (1997) 13. Widder, D.V.: The Laplace Transform. Princeton: Princeton Univ. Press, 1946 Communicated by H. Araki
Commun. Math. Phys. 191, 613 – 626 (1998)
Communications in
Mathematical Physics c Springer-Verlag 1998
On Zero-Mass Ground States in Super-Membrane Matrix Models ¨ Fr¨ohlich, Jens Hoppe? Jurg Theoretical Physics, ETH-H¨onggerberg, CH–8093 Z¨urich, Switzerland Received: 2 April 1997 / Accepted: 28 May 1997
Abstract: We recall a formulation of super-membrane theory in terms of certain matrix models. These models are known to have a mass spectrum given by the positive halfaxis. We show that, for the simplest such matrix model, a normalizable zero-mass ground state does not exist. 1. Introduction and Summary of Results Some time ago [1], super-membranes in D space-time dimensions were related to supersymmetric matrix models where, in a Hamiltonian light-cone formulation, the D − 2 transverse space coordinates appear as non-commuting matrices [2]. It has been proven in [3] that the mass spectrum of any one of these matrix models, which is given by the (energy) spectrum of some supersymmetric quantum-mechanical Hamilton operator [4], fills the positive half-axis of the real line. This property of the mass spectrum in super-membrane models is in contrast to the properties of mass spectra in bosonic membrane matrix models [2] which are purely discrete, see [5]. One of the important open questions concerning super-membrane matrix models is whether they have a normalizable zero-mass ground state. Such states would describe multiplets of zero-mass one-particle states, including the graviton, (see [1]). A new interpretation of the mass spectrum of super-membrane matrix models (in terms of multi-membrane configurations) has been proposed in [6]. A first step towards answering the question of whether there are normalizable zeromass ground states in super-membrane matrix models has been undertaken in [1]. In this note, we continue the line of thought described in [7] and show that, in the simplest matrix model, a normalizable zero-mass ground state does not exist. Let us recall the definition of super-membrane matrix models. The configuration space of the bosonic degrees of freedom in such models consists of D − 2 copies of ?
Heisenberg Fellow. On leave of absence from Karlsruhe University
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the Lie-algebra of SU (N ), for some N < ∞, where D is the dimension of space-time, with D = 4 (5,7) 11. A point in this configuration space is denoted by X = (Xj ) with Xj =
2 N −1 X
XjA TA , j = 1, · · · , D − 2,
(1.1)
A=1
where {TA } is a basis of su(N ), the Lie algebra of SU (N ). In order to describe the quantum-mechanical dynamics of these degrees of freedom, we make use of the Heisenberg algebra generated by the configuration space coordinates XjA and the canonically conjugate momenta PjA satisfying canonical commutation relations A (1.2) Xj , XkB = PjA , PkB = 0, A B AB δjk . Xj , Pk = i δ To describe the quantum dynamics of the fermionic degrees of freedom, wemake D−2
use of the Clifford algebra with generators ΘαA , A = 1, · · · , N 2 − 1, α = 1, · · · , 2 2 (times 2, if D = 5 or 7), and commutation relations A (1.3) Θα , ΘβB = δαβ δ AB . The generators ΘαA can be expressed in terms of fermionic creation- and annihilation operators: A bA α + cα √ , 2 A i bA α − cα A √ Θ2α = , 2 D−2 ∗ 1 A 2 = c , α = 1, · · · , 2 with bA , and α α 2 A B A B bα , bβ = cα , cβ = 0, A B bα , cβ = δαβ δ AB . A = Θ2α−1
(1.4)
(1.5)
The Hilbert space, H, of state vectors (in the Schr¨odinger representation) is a direct sum of subspaces Hk , k = 0, · · · , K := (N 2 − 1) 9 =
1 2
2
D−2 2
. A vector 9 ∈ Hk is given by
K X 1 A1 α1 ···αk k b · · · bA αk ψA1 ···Ak (X), k! α1
(1.6)
k=0
where X = XjA , j = 1, · · · , D − 2, A = 1, · · · , N 2 − 1. We require that cA α 9 = 0,
for all
9 ∈ H0 .
(1.7)
The scalar product of two vectors, 9 and 8, in H is given by h9, 8i =
K X 1 k! k=0
X Z Y α1 ···αk A1 ···Ak
j,A
α1 ···αk 1 ···αk dXjA ψA (X) × φα A1 ···Ak (X). 1 ···Ak
(1.8)
On Zero-Mass Ground States in Super-Membrane Matrix Models
615
The Hilbert space H carries unitary representations of the groups SU (N ) and SO(D−2). Let H(0) denote the subspace of H carrying the trivial representation of SU (N ). D−2
One can define supercharges, Qα and Q†α , α = 1, · · · , 21 2 2 , with the properties that, on the subspace H(0) , o n † † {Qα , Qβ } = Qα , Qβ =0, (1.9) H(0)
and
n
Qα , Q†β
o
H(0)
H(0)
= δαβ H
,
(1.10)
H(0)
where H = M 2 , and M is the mass operator of the super-membrane matrix model. Precise definitions of the supercharges and of the operator H can be found in [1] (formulas (4.7) through (4.12)). In [3] it is shown that the spectrum of H |H(0) consists of the positive half-axis [0, ∞). The problem addressed in this note is to determine whether O is an eigenvalue of H corresponding to a normalizable eigenvector 90 ∈ H(0) . Using Eqs. (1.9) and (1.10), one can show that 90 must be a solution of the equations Qα 9 = Q†α 9 = 0,
for some α, 9 ∈ H(0) .
(1.11)
If Eqs. (1.11) have a solution, 90 = 9α0 , for α = α0 , they have a solution for all values of α, (by SO(D − 2) covariance). The problem to determine whether Eqs. (1.11) have a solution, or not, can be understood as a problem about the cohomology groups determined by the supercharges Qα . We define (0) , H+ := ⊕ H2l l≥0
(0) H− := ⊕ H2l+1 . l≥0
We define the cohomology groups o n (0) , 9 | 9 = Qα 8, 8 ∈ H−σ Hσ,α := 9 ∈ Hσ(0) | Qα 9 = 0 σ = ±1. If Hσ,α is non-trivial, for some σ and some α, then Eqs. (1.11) have a solution. 2. The (D = 4, N = 2) Model The goal of this note is a very modest one: We show that, for D = 4 and N = 2, Eqs. (1.11) do not have any normalizable solutions. Our proof is not conceptual; it relies on explicit calculations and estimates and does therefore not extend to the general case in any straightforward way. When D = 4 and N = 2 we use the following notations: ~qj := Xj1 , Xj2 , Xj3 , j = 1, 2, ~λ = λ1 , λ2 , λ3 := b1 , b2 , b3 , (2.1) α
and
∂ = ∂~λ
∂ ∂ ∂ , , 1 2 ∂λ ∂λ ∂λ3
α
α
:= c1α , c2α , c3α ,
α = 1. The operators representing the generators of su(2) on H are given by
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J. Fr¨ohlich, J. Hoppe
~ = −i L
∂ ~ ~ ~ ~q1 ∧ ∇1 + ~q2 ∧ ∇2 + λ ∧ . ∂~λ
(2.2)
The supercharges are given by (see [1], Eq. (4.20)) ~ 2 · ∂ + i ~q · ~λ , ~ 1 − i∇ Q = ∇ ∂~λ and ~ 1 + i∇ ~ 2 · ~λ − i ~q · ∂ , Q† = − ∇ ∂~λ where
(2.3)
~q = ~q1 ∧ ~q2 ,
(2.4)
and ∧ denotes the vector product. We then have that ~ , (Q† )2 = i (~q1 + i ~q2 ) · L, ~ Q2 = i (~q1 − i ~q2 ) · L and H =
Q, Q†
.
(2.5)
A vector 9 ∈ H+ can be written as 1 ~ ~ ~ λ∧λ · ψ 2 1 = ψ + εABC λA λB ψ C . 2
9=ψ+
(2.6)
~ = 0), Eqs. (1.11) imply the following system (∗ ) For 9 ∈ H+(0) (i.e., 9 ∈ H+ with L9 of first-order differential equations: ~, ~ 1 − i∇ ~2 ∧ψ i ~q ψ = ∇ (2.7)
and
~ =0, ~q · ψ
(2.8)
~, ~ 2 ψ = i ~q ∧ ψ ~ 1 + i∇ ∇ ~ =0. ~ 1 + i∇ ~2 · ψ ∇
(2.9)
~ = 0 yields Moreover, the equation L9 ~ 1 + ~q2 ∧ ∇ ~2 ψ = 0, ~q1 ∧ ∇ X ~ 1 + ~q2 ∧ ∇ ~2 ~q1 ∧ ∇ ψB + εABC ψc = 0 , A
(2.10)
(2.11) ∀ A, B .
(2.12)
C
(0) It is straightforward to verify that, for 9 ∈ H− , Eqs. (1.11) imply a system of equation equivalent to (2.7) through (2.12). This can be interpreted as a consequence of Poincar´e duality. The formal expression for the Hamiltonian H = Q, Q† is given by
On Zero-Mass Ground States in Super-Membrane Matrix Models
617
H = HB + H F ,
(2.13)
where
~2 − ∇ ~ 2 + ~q 2 HB = − ∇ 1 2
~λ ∧ ~λ − (~q1 − i ~q2 ) · ∂ ∧ ∂ . ∂~λ ∂~λ As shown in [5], the spectrum of HB is discrete, with and
HF = (~q1 + i ~q2 ) ·
inf spec HB = E0 > 0 .
(2.14)
(2.15)
The representation of the group SO(D − 2) ' U (1), (D = 4) on H is generated by the operator ~ 2 − ~q2 · ∇ ~ 1 − 1 ~λ · ∂ . (2.16) J = − i ~q1 · ∇ 2 ∂~λ While J does not commute with Q or Q† , it does commute with QQ† and Q† Q and hence with H. It is therefore sufficient to look for solutions of Eqs. (2.7) through (2.12) transforming under an irreducible representation of U (1), i.e. solutions that are eigenvectors of J corresponding to eigenvalues j ∈ 21 Z . Thespectrum of the restriction of J to the subspace H+ is the integers, while spec J |H− consists of half-integers.
3. Analysis of Equations (∗ ) In this section, we assume that Q9 = Q† 9 = 0 has a solution 9 ∈ H+(0) and then show that 9 = 0. The assumption that Q9 = Q† 9 = 0 implies that hQ9, Q9i + hQ† 9, Q† 9i = 0 .
(3.1)
6 Let ξ := (~q1 , ~q2 ) ∈ R6 . Let p gn (ξ) ≡ gn (|ξ|), n = 1, 2, 3, · · · , be a function on R 2 2 only depending on |ξ| := ~q1 + ~q2 with the properties that gn is smooth, d monotonic decreasing, gn (|ξ|) = 1, for |ξ| ≤ n, gn (|ξ|) = 0 for |ξ| ≥ 3n, and dt gn (t) ≤ n1 . Let hk (ξ), k = 1, 2, 3, · · · R, be an approximate δ-function at ξ = 0 with the properties that hk is smooth, hk ≥ 0, hk (ξ)d6 ξ = 1, and 1 supp hk ⊆ ξ |ξ| ≤ 2 . (3.2) k
We define a bounded operator, Rn,k , on H by setting Z Rn,k 8 (ξ) = gn (ξ) hk ξ − ξ 0 8 ξ 0 d6 ξ 0 , for any 8 ∈ H. Clearly
s n→∞ − lim Rn,k 8 = 8 ,
(3.3)
(3.4)
k→∞
for any 8 ∈ H. Next, we note that, for a vector 8 in the domain of the operator Q, (3.5) Q, Rn,k 8 (ξ) = In,k (ξ) + IIn,k (ξ) ,
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J. Fr¨ohlich, J. Hoppe
where
Z ~ 1 − i∇ ~ 2 gn (ξ) · hk ξ − ξ 0 ∂ 8 ξ 0 d6 ξ 0 , ∇ ∂~λ
In,k (ξ) =
Z
and IIn,k (ξ) = i gn (ξ)
hk ξ − ξ 0
~q (ξ) − ~q ξ 0
· ~λ8 ξ 0 d6 ξ 0 .
(3.6)
(3.7)
bounded by 1. Since The operator norm of the operators ∂λ∂A and λA , A = 1, 2, 3, is d ~ 1 − i∇ ~ 2 gn (·) gn (t) ≤ 1 , the operator norm of the multiplication operator ∇ dt
n
1 is by 6n 0 . The operator norm of the convolution operator 8 (ξ) 7→ R bounded 0above 0 hn ξ − ξ 8 ξ d ξ is equal to 1. This implies
18 ∂ 6
kIn,k k ≤ k8k . (3.8) 8 ≤ n ∂~λ n
Next, we note that, for ξ in the support of the function gn , hk ξ − ξ 0 ~q (ξ) − ~q ξ 0 ≤ 7n hk ξ − ξ 0 . 2 k Thus, for k ≥ n, kIIn,k k ≤
21 k8k . n
(3.9)
In conclusion
40 k8k , (3.10) k Q, Rn,k 8k ≤ n for k ≥ n. A similar chain of arguments shows that, for 8 in the domain of Q† , 40 k8k , k Q† , Rn,k 8k ≤ n
(3.11)
for k ≥ n. Next, we suppose that 9 solves (3.1). We claim that, given ε > 0, there is some finite n(ε) such that, for 9n,k := Rn,k 9 ,
and
k9k ≥ k9n,k k ≥ (1 − ε) k9k ,
(3.12)
hQ9n,k , Q9n,k i + hQ† 9n,k , Q† 9n,k i ≤ εk9k2 ,
(3.13)
for all k ≥ n ≥ n(ε). Inequality (3.12) follows directly from (3.4) and the fact that the operator norm of Rn,k is = 1. To prove (3.13), we note that, for k ≥ n, 2 # # 40 # # h9, 9i , (3.14) hQ 9n,k , Q 9n,k i = h Q , Rn,k 9, Q , Rn,k 9i ≤ n where Q# = Q or Q† . This follows from the equations Q9 = Q† 9 = 0 and inequalities (3.10) and (3.11). We now observe that, by the definition of Rn,k , 9n,k = Rn,k 9 is a smooth function of compact support in R6 , for all n ≤ k < ∞. It therefore belongs to the domain of definition of the operators Q Q† and Q† Q. Thus, for all n ≤ k < ∞,
On Zero-Mass Ground States in Super-Membrane Matrix Models
619
hQ9n,k , Q9n,k i + hQ† 9n,k , Q† 9n,k i = h9n,k , Q, Q† 9n,k i = h9n,k , HB 9n,k i + h9n,k , HF 9n,k i ,
(3.15)
where HB and HF are given in Eq. (2.14), (and it is obvious from (2.14) that 9n,k belongs to the domains of definition of HB and HF ). As proven in [5], (3.16) h8, HB 8i ≥ E0 k8k2 , for some strictly positive constant E0 (= inf spec HB ), for all vectors 8 in the domain of HB . Thus, for k ≥ n ≥ n(ε), and using (3.13), we have that ε k9k2 ≥ h9n,k , HB 9n,k i + h9n,k , HF 9n,k i ≥ (1 − ε)2 E0 k9k2 + h9n,k , HF 9n,k i .
(3.17)
Our next task is to analyze h9n,k , HF 9n,k i. If 8 = (ϕ, ϕ ~ ) ∈ H+ belongs to the domain of definition of HF then Z ~ (ξ) d6 ξ + c.c. ϕ (ξ) (~q1 − i ~q2 ) · ϕ (3.18) h8, HF 8i = 2 Note that 9n := lim 9n,k , where 9n,k = Rn,k 9 and 9 solves (3.1), belongs to the k→∞ ~ solves the equations Q9 = Q† 9 = 0, domain of definition of HF . Since 9 = ψ, ψ ~ For ~q 6= 0, we find that see (3.1), we can use Eqs. (2.8) and (2.9) to eliminate ψ: ~ (ξ) = i ~q ∧ ∇ ~ 1 + i∇ ~ 2 ψ(ξ) ψ q2
(3.19)
(recall that ~q = ~q1 ∧ ~q2 ). Inserting (3.19) on the R.S. of (3.18), for 8 = 9n , we arrive at the equation Z (gn ψ) (ξ) (~q1 − i ~q2 ) h9n , HF 9n i = 2 i ~q ~ ~ × gn (ξ) ∧ ∇1 + i ∇2 ψ (ξ) d6 ξ + c.c. (3.20) q2
Let T := 2 (~q1 − i ~q2 ) ·
i ~q ~ ~ . ∧ ∇1 + i ∇ 2 q2
Then h9n , HF 9n i = h9n , T 9n i + c.c. Z − | ψ (ξ) |2 gn (ξ) [T, gn ] (ξ) d6 ξ + c.c.
(3.21)
Next, we make use of the fact that 9 must be SU(2)–invariant. This is expressed in Eq. (2.11), which implies that ψ(ξ) only depends on SU(2)–invariant combinations of the variables ~q1 and ~q2 , i.e., on r1 := |~q1 |, r2 := |~q2 |, x :=
~q1 · ~q2 . r1 r2
(3.22)
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J. Fr¨ohlich, J. Hoppe
Instead, we may use variables q, p and ϕ defined by 1 1 2 (~q1 + i ~q2 )2 = r1 − r22 + i r1 r2 x, 2 2
p eiϕ := with
0 ≤ p < ∞,
0 ≤ ϕ < 4π,
q := |~q1 ∧ ~q2 |
0≤q < ∞.
(3.23)
(3.24)
If F is an SU(2)–invariant function then Z∞
Z 6
d ξ F (ξ) = c
Z∞ dq
0
Z4π dp
0
0
qp dϕ p F (q, p, ϕ) , q 2 + p2
(3.25)
where c is some positive constant. If ϕ is SU(2)–invariant then p q 2 + p2 ∂ ϕ, Tϕ=c q ∂q 0
(3.26)
where c0 is a positive constant. Using (3.26) in (3.21), we find that h9n , HF 9n i = c
00
Z∞
Z4π dp
0
− c00
Z∞ 0
dϕ 0
Z4π dp
Z∞ dq p
∂ |ψn (q, p, ϕ) |2 ∂q
0
Z∞ dq p |ψ (q, p, ϕ) |2
dϕ 0
∂ ∂q
p 2 g n 2 q 2 + p2 , (3.27)
0
with c00 = c.c0 > 0. By the definition of gn , ∂ ∂q
p 2 2 2 gn 2 q + p ≤0,
pointwise. Therefore h9n , HF 9n i ≥ − c
00
Z∞
Z4π dϕ p |ψn (q = 0, p, ϕ) |2 .
dp 0
In passing from (3.27) to (3.28), we have used that with respect to the measure p dp dϕ dq and that Z∞ 0
∂ ∂q |ψn
(q, p, ϕ) |2 is an L1 –function
Z4π dϕ p |ψn (q, p, ϕ) |2
dp
(3.28)
0
0
is right-continuous at q = 0. These facts will be proven below. Combining Eqs. (3.15), (3.17) and (3.28), we conclude that
On Zero-Mass Ground States in Super-Membrane Matrix Models
εk9k2 ≥ lim
621
hQ9n,k , Q9n,k i + hQ† 9n,k , Q† 9n,k i
k→∞
≥ (1 − ε) E0 k9k2 Z∞ Z4π − c00 dp dϕ p |ψn (q = 0, p, ϕ) |2 , 0
(3.29)
0
for all n ≥ n(ε). Choosing ε sufficiently small, we conclude that either 9 = 0, or there is a constant β > 0 such that Z∞
Z4π dϕ p |ψn (q = 0, p, ϕ) |2 ≥ β ,
dp 0
(3.30)
0
for all sufficiently large n. Next, we explore the consequences of (3.30). Since 9 solves (3.1), we can use (3.19) to conclude that ~ 2 ∞ > k9k2 = kψk2 + kψk Z ~ 1 + i∇ ~ 2 ψ(ξ)|2 | ∇ = d6 ξ |ψ(ξ)|2 + . |~q|2 Using that 9 is SU(2)–invariant and passing to the variables q, p and ϕ, one finds that Z∞
Z∞ dp p
2 0
where ψ,x :=
dq q
Z4π
0
0
∂ψ ∂x
, and
iψ,ϕ dϕ ψ,p + p
Z∞
Z∞ dp
0
Z4π dq
0
0
! 2 2 e + | ψ,q | (q, p, ϕ) < K,
pq 2 e dϕ p |ψ (q, p, ϕ)| < K, 2 2 p +q
(3.31)
(3.32)
e = k9k < ∞ (with the constant c appearing in (3.25)). with K c Inequalities (3.31) and (3.32) also hold for ψn , instead of ψ, with a constant K that 2 ∂ is uniform in n → ∞. These inequalities prove that ∂q |ψn (q, p, ϕ)| is an L1 –function with respect to the measure p dp dϕ dq and that 2
Z∞
Z4π dϕ p |ψn (q, p, ϕ)|
dp
fn (q) := 0
2
0
is right-continuous at q = 0, properties that were used in our derivation of (3.28). By the Schwarz inequality and (3.31),
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J. Fr¨ohlich, J. Hoppe
Z∞
Z4π dp
0
dϕ 0
≤ 2
Z∞
∂ 2 dq p |ψn (q, p, ϕ)| ∂q
0
Z∞
Z∞
dq q
dp p 0
0
Z∞ 0
Z4π q dq
0
1/2 2 dϕ |ψn,q (q, p, ϕ)|
0
Z∞ dp p
Z4π
1/2
2 dϕ |ψn (q, p, ϕ)|
≤ K 0 n4 ,
0
for some finite constant K 0 ! To prove continuity of fn (q) in q, we note that, for q1 > q2 , Z∞ |fn (q1 ) − fn (q2 )| ≤
Z4π dp
0
Zq1 dϕ
∂ 2 dq p |ψn (q, p, ϕ)| ∂q
q2
0
2
∂ |ψn (q, p, ϕ)| is an L1 –function. which tends to 0, as (q1 − q2 ) → 0, because ∂q Next, we make use of the SO(D − 2) ' U (1) symmetry with generator J given in Eq. (2.16). We have noted below (2.16) that J commutes with QQ† and Q† Q, and hence that 9 ∈ H+ can be chosen to be an eigenvector of J corresponding to some eigenvalue m ∈ Z. In the variables q, p, ϕ,
J = −2i Hence we may write ψ (q, p, ϕ) = ei
m 2
∂ . ∂ϕ ϕ
m
p 2 φ (q, p) ,
(3.33)
for some function φ independent of ϕ. Eqs. (3.31) and (3.32) then simply Z∞
Z∞ dp p
α
0
and
dq 2 2 |φ,p | + |φ,q | (q, p) < ∞ q
(3.34)
pα q 2 dq p |φ (q, p)| < ∞, p2 + q 2
(3.35)
0
Z∞
Z∞ dp
0
0
where α = m + 1. Furthermore, inequality (3.30), in the limit as n → ∞, yields Z∞ 2
dp pα |φ (q = 0, p)| ≥
β . 4π
(3.36)
0
Let IN := N1 , N . Then inequality (3.34) implies that there exists a set ⊆ [0, ∞) with the property that ∩ [0, δ] has Lebesgue measures δ2 , for any δ > 0, and such that
On Zero-Mass Ground States in Super-Membrane Matrix Models
1 N
|α| Z
623
2
dp |φ,p (q, p)| ≤ Kδ
(3.37)
IN
for some constant Kδ independent of N and all q ∈ ∩ [0, δ]. Moreover, Z 2 lim dp |φ,p (q, p)| = 0, q→0 q∈
(3.38)
IN
for all N . It follows that, for q ∈ ∩ [0, δ] , p1 , p2 ∈ IN , N < ∞, p Z 2 dp φ,p (q, p) |φ (q, p1 ) − φ (q, p2 )| = |p1 − p2 | |p − p | 1 2 p1 1/2 1/2 ≤ |p1 − p2 | N |α| Kδ .
(3.39)
Thus, for q ∈ ∩ [0, δ] and p1 , p2 ∈ IN , φ (q, p) is uniformly H¨older–continuous with exponent 21 . Thus φ0 (p) := lim φ(q, p) is uniformly continuous in p ∈ IN , for all q→0 q∈
N < ∞, and it then follows from (3.38) that φ0 (p) = φ0 = const.
(3.40)
Inequality (3.36) then implies that |φ0 | must be positive! Without loss of generality, we may then assume that φ0 > 0. Thus the function φ introduced in (3.33) has the following properties: (A)
lim φ (q, p) = φ0 > 0, q→0 q∈
Z∞
Z∞ dp
(B) 0
0
Z∞
Z∞ dp
(C) 0
qpα 2 dq p |φ (q, p)| < ∞, p2 + q 2 dq
pα 2 2 |φ,p (q, p)| + |φ,q (q, p)| < ∞. q
0
We now show that such a function φ (q, p) does not exist. Let us first consider the case α ≥ 0. We choose an arbitrary, but fixed p ∈ (0, ∞). Using the Schwarz inequality, we find that, for 0 < q0 < ∞, Zq0 0
1 dq 2 |φ,q (q, p)| ≥ q q0
Zq0 dq |φ,q (q, p)| 0
1 ≥ q0
Zq0
2
2 dq |φ,q (q, p)|
0
2 q∗ (p) Z 1 ≥ dq φ,q (q, p) , q0 0
(3.41)
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J. Fr¨ohlich, J. Hoppe
where q ∗ (p) ∈ [0, q0 ] is the point at which |φ(q, p)| takes its minimum in the interval [0, q0 ]. Note that φ(q, p) is continuous in q ∈ [0, q0 ], for almost every p ∈ [0, ∞). The R.S. of (3.41) is equal to 2 1 ∗ . (p), p − φ φ q 0 q02 Thus
1 |χ (p) − φ0 | q0
2
Zq0 ≤
dq 2 |φ,q (q, p)| , q
(3.42)
0 ∗
where χ(p) = φ(q (p), p). By property (C), Z∞
Zq0 dp p
dq 2 |φ,q (q, p)| ≤ ε(q0 ), q
α
0
0
for some finite ε(q0 ), with ε(q0 ) → 0, as q0 → 0. Hence Z∞ 2
dp pα |χ(p) − φ0 | < q02 ε (q0 ) .
(3.43)
0
We define a subset Mδ ⊆ [0, ∞) by Mδ := p |χ (p)| ≤ φ0 − δ . Then
Z dp p
α
1 ≤ 2 δ
Mδ
Z∞ 2
dp pα |χ (p) − φ0 | <
q02 ε (q0 ) . δ2
0
By property (B), Z∞ ∞>
Z∞
0
0
Z∞
Zq0
≥
dp 0
q0/2
≥
It follows that
q02 4
q0 α 2 p
p + q0
Zq0 dp
Mδc
p2 + q 2
dq
Z ≥
qpα
dq p
dp
p + q0
q0/2
(φ0 − δ)
|φ (q, p)|
q0 α 2 p
dq
Z 2 Mδc
|φ (q, p)|
dp
2
|φ (q, p)| pα . p + q0
2
2
(3.44)
On Zero-Mass Ground States in Super-Membrane Matrix Models
Z dp
Z
pα + p + q0
dp Mδc
Mδ
pα 1 ≤ p + q0 q0
Z
625
Z dp pα +
dp Mδc
Mδ
pα < ∞. p + q0
R pα diverges. This is a contradiction, since Mδ ∪ Mδc = [0, ∞), and dp p+q 0 Next, we consider the case α ≤ −1. We change variables, k := p1 , dp = −
1 ∂ ∂ = − k2 . dk, 2 k ∂p ∂k
Then conditions (A)–(C) take the form (A0 )
lim φ (q, k) = φ0 > 0, q→0 q∈
Z∞
0
Z∞ dq q
dk
(B )
(C 0 )
0
0
Z∞
Z∞ dk
0
dq
q k γ−2 2 |φ (q, k)| < ∞, 1 2 + q2 k
k γ−2 4 2 2 k |φ,k (q, k)| + |φ,q (q, k)| < ∞, q
0
where γ := −α > 0. Repeating the same arguments as above, we get again
1 |χ (k) − φ0 | q0
2
Z∞ ≤
dq 2 |φ,q (q, k)| , q
(3.45)
0
where χ(k) is the value of φ(q, k) at the minimum of |φ(q, k)|, for q ∈ [0, q0 ]. By (C’), Z∞
Zq0 dk
0
dq
k γ−2 2 |φ,q (q, k)| < ε0 (q0 ) < ∞, q
0
0
with ε (q0 ) → 0, as q0 → 0. Hence Z∞
dk k γ−2 |χ(k) − φ0 | ≤ q02 ε0 (q0 ) . 2
(3.46)
0
Let Lδ ⊆ [0, ∞) be the set defined by Lδ := k |χ(k)| ≤ φ0 − δ . Then we have that Z dk k
γ−2
Lδ
Condition (B’) implies that
1 ≤ 2 δ
Z∞ 2
dk k γ−2 |χ(k) − φ0 | ≤ 0
q02 ε0 (q0 ) . δ2
(3.47)
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J. Fr¨ohlich, J. Hoppe
Z∞ ∞>
Z∞ dk k
0
Zq0 dk k
q0/2
γ−2
1 k
q0/2
0
Zq0
Z ≥
dk k Lcδ
≥
1 k
0
Z∞ ≥
0
2
(φ0 − δ)
+ q0
|φ (q, k)|
2
q0/2 (φ0 − δ)2 1 + k q0
γ−1 q0/2
q 2
q 2 |φ (q, k)| +q
γ−2
Z 2
dk Lcδ
k γ−1 . 1 + k q0
(3.48)
Combining (3.47) and (3.48) we find that Z Z k γ−1 k γ−1 dk + dk 1 + k q0 1 + k q0 Lδ
≤
1 q0
Z
Lcδ
Z
dk k γ−2 + Lδ
dk Lcδ
k γ−1 < ∞. 1 + k q0
R kγ−1 This is a contradiction, because Lδ ∪ Lcδ = [0, ∞) and dk 1+k q0 diverges for γ ≥ 1. This completes the proof that functions satisfying properties (A), (B) and (C) do not exist. We have thus proven that Eq. (3.1) only has the trivial solution 9 = 0. Acknowledgement. We would like to thank H. Kalf for useful discussions.
References 1. de Wit, B., Hoppe, J., Nicolai, H.: Nuclear Physics B 305, [FS23] 545 (1988) 2. Goldstone, J.: Unpublished. Hoppe, J.: MIT Ph.D. Thesis, 1982, and Quantum Theory of a Relativistic Surface. In: Constraint’s Theory and Relativistic Dynamics. (eds. G. Longhi, L. Lusanna) Arcetri, Florence 1986, World Scientific 1987 3. de Wit, B., L¨uscher, M., Nicolai, H.: Nuclear Physics B 320, 135 (1989) 4. Baake, M., Reinicke, P., Rittenberg, V.: J. Math. Physics 26, 1070 (1985) Claudson, M., Halpern, M.: Nucl. Phys. B 250, 689–715 (1985) Flume, R.: Annals of Physics 164, 189 (1985) 5. L¨uscher, M.: Nuclear Physics B 219, 233 (1983) Simon, B.: Annals of Physics 146, 209 (1983) 6. Banks, T., Fischler, W., Shenker, S.H., Susskind, L.: M Theory as a Matrix Model: A Conjecture. hep-th/9610043 7. Hoppe, J.: On Zero-Mass Bound-States in Super-Membrane Models. hep-th/9609232 Communicated by A. Jaffe
Commun. Math. Phys. 191, 627 – 639 (1998)
Communications in
Mathematical Physics c Springer-Verlag 1998
A Variational Principle Associated to Positive Tilt Maps Sen Hu Department of Mathematics, Fine Hall, Princeton University, Princeton, NJ 08544-1000, USA. E-mail: [email protected] Received: 14 May 1996 / Accepted: 30 May 1997
Abstract: In this paper we establish a variational principle for positive tilt maps. This implies that the Aubry–Mather set exists for positive tilt maps and most of the theory developed by Mather for twist maps now applies for positive tilt maps.
1. Introduction Let f be a C1 area preserving, exact, orientation preserving and end preserving diffeomorphism of the cylinder R/Z × R. For composition of positive twist maps Mather [7] established a variational principle h satisfying (H0 ) − (H6θ ). Then from Bangert’s work [1] for any h satisfying those conditions one can construct an Aubry–Mather set for each rotation number. One knows that compositions of twist maps are also positive tilt maps. It is not known whether those two sets coincide (cf. [14]). In this paper we establish a variational principle h for positive tilt maps satisfying (H0 ) − (H6θ ). Since most of Mather’s work on twist maps are based on a variational principle h satisfying (H0 ) − (H6θ ), this work implies that much of his work applies for positive tilt maps too. For example, Mather’s work on connecting orbits ([11, 13]), destruction of invariant curves ([8]), and differentiability of the average action functions ([10]), etc., are also true for positive tilt maps. We will not state all theorems. The interested reader may look at the cited papers. The work to establish such a variational principle for positive tilt maps relies on proving an elementary lemma suggested by Mather (Main Lemma in Sect. 4). It basically asserts that the tilt where the action function takes minimum for various points (x, y) with 0 fixed (x, x ) must be less than or equal to π. The action function h admits a geometrical interpretation as an area of some region (see Sect. 3). The proof of the main lemma involves counting areas for various situations. The proof is entirely elementary. It only uses the basic fact on the Jordan curve in plane topology to analyze positive tilt diagrams.
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Once we have the main lemma it is not difficult to show that h satisfies (H0 ) − (H6θ ). For completeness we also give the proof of those. It is interesting to note that if we don’t have the Main Lemma we even cannot prove h satisfies (H0 ), i.e. h is continuous. 2. Statement of the Theorem 2.1. Positive tilt maps. Let f : R/Z × R → R/Z × R be a diffeomorphism of the cylinder. Let γ be a C1 embedded curve defined on R in R/Z × R, such that, γ(t)2 → −∞ as t → −∞, γ(t)2 → +∞ as t → +∞, where γ(t)2 means the second component of the coordinates. Then we define the tilt of γ as the real valued function on R satisfying the following: 0
1) tilt(γ)(t) := angle which γ (t) makes with the vertical (mod2π); 2) If γ(t)2 > γ(s)2 for all s < t, then −π/2 ≤ tilt(γ)(t) ≤ π/2. A C1 area preserving, exact, orientation preserving and end preserving map f : R/Z × R → R/Z × R is said to be a positive tilt map if for each vertical line l, tilt(f ol) > 0. 0
0
Let f (x, y) = (x , y ). To say that f is area-preserving means that f preserves the 0 0 area form dxdy. This implies that the 1-form y dx − ydx is a closed form. To say that f is exact means that this 1-form is exact. Remark. A positive twist map is a map satisfying 0 < tilt(f ol) < π for every vertical l. Remark. If f and g are positive tilt maps, then so is gof . Notice that if f and g are positive twist maps, gof may not be a positive twist map. 2.2. Action function h and conditions (H0 ) − (H6θ ). Let f¯ be a lift of f to R2 , f¯ : R2 → R2 . Since f¯ is area-preserving we can define h0 as a function of R2 , such that 0
0
dh0 (x, y) = y dx − ydx. Now define
0 0 h(x, x ) = min{h0 (x, y)|π1 f¯(x, y) = x }.
y
Here π1 is the projection to the first factor. The minimum is taken over the set 0 Σx,x0 = {y|π1 f¯(x, y) = x }.
We say that h satisfies conditions (H0 ) − (H6θ ) if it satisfies the following. H0 : h is a continuous function. 0 0 0 H1 : h(x + 1, x + 1) = h(x, x ), for all x, x ∈ R. H2 : lim|ξ|→∞ h(x, x + ξ) = ∞, uniformly in x. 0
0
0
0
0
0
H3 : h(x, x ) + h(ξ, ξ ) < h(x, ξ ) + h(ξ, x ) if x < ξ, x < ξ .
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0
0
¯ x, ξ ) are both minimal segments, and are distinct, then (x− ¯ H4 : If (x, ¯ x, x ), (ξ, ¯ ξ)(x − 0 ξ ) < 0. 0
0
H5 : There exists a positive continuous function ρ on R2 , such that h(ξ, x ) + h(x, ξ ) − R ξ R ξ0 0 0 0 0 h(x, x ) − h(ξ, ξ ) ≥ x x0 ρ, if x < ξ and x < ξ . 0
0
0
0
0
H6θ : x → θx2 /2 − h(x, x ) is convex, for any x , and x → θx 2 /2 − h(x, x ) is convex for any x. Here θ = cot α, α is a lower bound of tilt of f . 2.3. Statement of the theorem. Theorem. Let f : R/Z × R → R/Z × R be a C1 area preserving, exact, orientation preserving and end preserving positive tilt diffeomorphism. Let h be the action function defined above. Then h satisfies (H0 ) − (H6θ ). Remark. Again from Bangert [1] we have Aubry–Mather sets that exist for h. ∞ h(xi , xi+1 ) be the funcLet x = (..., xi , ...) be a configuration and W (x) = Σi=−∞ tional. We say that (..., xi , ...) is minimal if for any j < k, (xj , ..., xk ) is minimal k−1 k−1 h(xi , xi+1 ) ≤ Σi=j h(yi , yi+1 ) for any subject to constraint yj = xj , yk = xk , i.e. Σi=j (yj , ..., yk ) satisfying yj = xj , yk = xk . In [1] (see also [5]) it is shown that a minimal configuration corresponds to an orbit of f and has a well defined rotation number, i.e. lim xi /i = ρ(x) exists. (xj , ..., xk ) is called a minimal segment.
Corollary. For a positive tilt diffeomorphism f there exists minimal configurations for each rotation number. Remark. Existence of Aubry-Mather sets can also be established by topological method, see [4, 3, 2]. 3. Geometrical Interpretation of h0 0
0
Let f (x, y) = (x , y ). The generating function h0 (x, y) can be obtained by integrating 0 0 the 1-form y dx − ydx along the vertical segment from (x, 0) to (x, y) and along the horizontal segment from (x0 , 0) to (x, 0). See also [2]. The integral along the vertical 0 segment is the algebraic area of the region bounded by the vertical line through (x , 0), the image of the vertical line segment joining (x, 0) and (x, y), the vertical line through f (x, 0) and the horizontal line y = 0. The integral along the horizontal segment is a function of x, say U (x), only. Since the mapping is exact, U (x) is periodic. We see for our concern U (x) does not contribute. For conditions (H3 ) − (H5 ), U (x) cancelled in the expression. For condition (H6θ ), U (x) does not contribute because it is periodic. Because of this we will drop the U (x) part and consider h0 (x, y) to be the area we described above. Notice that the image of the vertical line through (x, 0) may have several intersections 0 with the vertical line through (x , 0). In this case we take the image of the vertical line through (x, 0) up to its intersection whose image is (x, y). Here the points in the vertical line are given the same order as the order of the vertical line. See Fig. 1. In Fig. 2 we illustrate this fact by describing a special case where the vertical line 0 through (x , 0) intersect with the image of the vertical line through (x, 0) at three points.
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(x0,0)
(x, 0)
Fig. 1.
A B
Fig. 2.
We see that h0 (x, y2 ) = h0 (x, y1 )+ area (A) and h0 (x, y3 ) = h0 (x, y2 )− area (B). The sign before the areas depends on the orientation of the curve f (l) at points where the curve intersect with the vertical line. 4. The Main lemma 4.1. Approximate a diagram of infinite many intersections with a finite one. It may 0 happen that there are infinitely many number of intersections of f (l) with l . Since the mapping is smooth and we only restrict ourself to a compact region we can only have the following picture: either there are only finitely many transverse intersections or there are infinitely many intersections yet all intersections except finitely many are almost vertical (up to a given precision). This is true because the mapping is smooth. Because of the smoothness of f, f (l) locally looks like a line. And we restrict ourself to a compact region so we can divide the finite part with the part near the ends breaking down into several small pieces, each approximated by its tangent very well.
Fig. 3.
At the points of intersection if the tangent is quite different from the vertical direction, the curve and the line intersect transversely. If the tangent is almost parallel to the vertical
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line we will just replace this part by a vertical segment. Then we get a new diagram with finitely many transverse intersections plus some vertical segments. Our main lemma applies to those kinds of diagrams. The difference of actions of corresponding points is very small, i.e. by the given precision number times the length of the replaced segment. We will see that for our purpose, i.e. to prove (H0 ) − (H6θ ), this approximation is good enough. In the following we only consider approximated diagrams, i.e., diagrams with finitely many intersections or finitely many vertical segments. 4.2. The main lemma. 0
Main Lemma. If h(x, x ) = h0 (x, y), then 0 < tilt(x,y) (f ) ≤ π, 0
0
and f (x, .) crosses x ×R from left to right, i.e., the two unbounded components of f (l)−l 0 0 0 0 in R2 − l belong to two different components of R2 − l . Here l = x × R, l = x × R . This lemma follows from the following proposition. In the proposition we do not deal with the mapping of the cylinder directly. We will deal with a curve intersecting with a line. For a precise definition see Sect. 6. 0
Proposition. Let P be any point of intersection of f (l) with the line l . Suppose that 0 f (l) does not cross l from left to right, or that its tilt is greater than π at P , then h0 (P ) >
min
0
h0 (x, y).
And (x, y) runs for all points of intersections which are before P . We will prove the proposition by induction in the last section. 5. Proof of the Theorem 5.1. Proof of the theorem, (H0 ) − (H2 ). We first prove that h satisfies (H0 ), i.e. h is 0 0 continuous. We only prove that h(x, x ) is continuous as a function of x . The proof that h is continuous as a function of x is the same. There are several cases to be considered: 0 0 0 1) f (l) intersects l (the vertical line through (x , 0)) transversely. When x changes the points where h achieve minimum varies continuously. In this case it is clear that h is continuous. We also need to consider the point P where h achieves minimum jumps from one to another discontinuously. In this case the minimum value only changes slightly 0 because jumping to different points won’t change the minimum value. So h(x, x ) is still 0 continuous in x . 0 0 2) f (l) intersects with l tangentially. In this case f (x, .) always intersects l from 0 0 left to right from our main lemma. So if we move l slightly h(x, x ) changes slightly. 0 Hence h(x, x ) is also continuous. 0 0 It is obvious that h satisfies the condition (H1 ) that h(x + 1, x + 1) = h(x, x ) for all 0 x, x ∈ R. That h satisfies the condition (H2 ) that h(x, x + ξ) → ∞ as |ξ| → ∞ uniformly in x follows from the fact that f also satisfies end preserving property. If x + ξ stays in the the neighborhood of the end we then have uniform increase of the area corresponding
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to h(x, x + ξ). The end preserving property just ensures that when ξ goes to ∞, x + ξ stays in the neighborhood of an end. 5.2. Proof of the theorem, (H3 ) and (H5 ). To verify that h satisfies the conditions (H2 ) 0 0 and (H5 ) we observe that we only need to consider those pairs (x, ξ) and (x , ξ ) such 0 0 that x − x and ξ − ξ are very small. The reason is that the conditions (H2 ) and (H5 ) are additive with respect to subdivision of the square formed by those numbers. Hence if it were true for all small squares it would be true for all squares. 0 0 When x − x is very small, the images of the vertical lines through x and through x 0 are very close. Now ξ − ξ is also very small. We first consider the case that in the region 0 between the vertical lines through ξ and ξ and the images of the vertical lines through x 0 0 and x there is no point in the image of the vertical lines between x and x with infinity slope, i.e. the tangent of the image at that point is vertical. Recall that at points where h0 take minimum values we have 0 < tilt(x,y) f (l) < π, which is also true for nearby 0 0 0 0 (x, y)0 s. For pairs (x, ξ), (x, ξ ), (x , ξ), (x , ξ ), h0 takes a minimum at nearby points. Thus we see the sum 0
0
0
0
h(ξ, x ) + h(x, ξ ) − h(x, x ) − h(ξ, ξ ) is the shaded area in Fig. 4. Remember we have to drop the U (x) part in calculating h0 (x, y). Since the tilt is between 0 and π we have the area is always positive. Hence h satisfies conditions (H3 ) and (H5 ). We have to consider the case that the tilt where
Fig. 4.
h takes a minimum be π. From the Main lemma f (x, .) cross from left to right. Again for nearby lines and curves we have similar picture. We then draw the picture for the difference of actions 0
0
0
0
h(ξ, x ) + h(x, ξ ) − h(x, x ) − h(ξ, ξ ) we see that it is even more positive. Again the error of approximation is small compared 0 0 with the difference above. The error is less than times |ξ − x |× length of vertical segment. It is much smaller than the shaded area. See Fig. 5. 5.3. Proof of the theorem, (H6θ ) and (H4 ). Recall that (H6θ ) is the following: for θ > 0, 0
0
x → θx2 /2 − h(x, x ) is convex in x for all x , and 0
0
0
0
x → θx 2 /2 − h(x, x ) is convex in x for all x.
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Fig. 5.
This is equivalent to say that the slopes of the image of the vertical line through (x, 0) are bounded above at points where h0 take minimums. If α is the bound of the slope where h take minimum we can take θ = cot α. From the main lemma, there are two possibilities. One case is that at point of intersection the tilt is π. In this case cot of the slope is −∞, it is bounded above by 0. The other case is that at point of intersections the tilt is between zero and π. By compactness the tilt is bounded below by some α in a compact region. Hence h satisfies (H6θ ) for θ = cot α. From [7] it is known that (H4 ) follows from (H0 ) − (H2 ) and (H5 ) − (H6θ ). 6. Proof of the Main Lemma 6.1. Several lemmas. In this subsection we prove several lemmas leading to the proof of the main lemma. First we give a definition. 0
Definition. (Diagram) The intersection pattern of a curve γ (f (l) in application here) 0 with a line γ (l in application) is called a diagram. We say two patterns are the same if the points of intersection have the same order as the order of either the curve or the line. A diagram is called a positive tilt diagram if the pattern of the diagram can be realized so that the angle between the curve and the line is always positive. Remark. In the following we only consider diagrams with only finitely many transverse intersections and finitely many vertical segments. As we see before such diagrams approximate general diagrams very well. And it is enough to only consider such diagrams. Definition. A pair of points of intersections is called a positive contractible element if the two points are consecutive from either the order of the curve and the order of the line and they have the orientation, as in Fig. 6. From the geometrical interpretation in Sect. 3, the action function h0 at P2 is equal to the action function h0 at P1 + shaded area. Similar definition applies for negative contractible element. Lemma 1. For each positive tilt diagram there exists at least one contractible element. Proof. The crucial fact is that we have only finitely many intersections. For each pair of consecutive points on the curve look at the segment on the line bounded by those two points. Those two segments, i.e. the segment on the curve and the segment on the
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+
+
Fig. 6.
line whose end points are the given consecutive points, bound a region D. Suppose there were no other points of intersection on the line segment then this would give a contractible element. Supposing there are other points of intersection on the line segment, we claim that there must be two consecutive points contained in the line segment. For any point P 0 00 of intersection in the line segment one of the two points P , P proceeding and after P has to be in the line segment. This follows from the Jordan curve theorem. Suppose 0 that P is not on the line segment, then the segment on the curve whose end points are 0 0 00 P, P cannot be inside D. This cannot happen for both segments P, P and P, P on the curve. 0 Now we have a smaller domain D bounded by two segments in the line and in the curve bounded by one of those new consecutive points of intersection in the line segment. This process has to be terminated in finite steps because we have only finitely many points of intersection. Hence we always have a contractible element. Remark. We proved more that there exist a contractible element in each side of the line. Lemma 2. For any positive tilt diagram with three or more intersections there exists at least one positive contractible element and one negative contractible element. Proof. We prove it by induction. We only consider the case of the positive contractible element. The proof for the case of the negative contractible element is the same. If the number of intersections is equal to three the lemma is obviously true. Now the lemma is true for cases such that the number of intersections is less than or equal to 2k − 1. Let us consider the case that n = 2k + 1. Suppose the lemma is not true. Since there must exist a contractible element we have at least one negative contractible element. We now apply an operation to the diagram by pulling out the negative contractible element, i.e. we move the segment on the curve whose end points are points of the negative contractible element so that it won’t intersect the line and we get a new diagram. The new diagram is still a positive tilt diagram because the operation does not affect the tilt of the remaining part. We then apply the induction hypothesis. There exists a positive contractible element for the new diagram. The only case for which it fails to be a positive contractible element for the original diagram is that all positive contractible elements for the new diagram contain the negative contractible element we started with. We then have a picture as in Fig. 7.
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+
-
Fig. 7.
Fig. 8.
We then pull out the positive contractible element for the new diagram. Then the only case for which the positive contractible element fails to be a positive contractible element for the original diagram is shown in Fig. 8. For all other cases a positive contractible element of the new diagram is also a positive contractible element of the original diagram. We apply the same procedure and we have the same pattern. Since there are only finitely many intersection points this procedure has to stop after finitely many steps. Thus we obtain a positive contractible element for the original diagram. This completes the proof of Lemma 1. Definition (Doubled diagram). Take a positive tilt diagram L, we double the curve and connect the two curves at the end of the curve. The new diagram is called a doubled diagram. It is a positive tilt diagram. It has only one positive contractible element or only one negative contractible element depending on how the two curves are connected at the end of the curve. This is quite obvious from the construction, since it is not needed, we omit the proof. Lemma 3. Suppose that a diagram have only one positive contractible element or only one negative contractible element, then the diagram is a doubled diagram.
Fig. 9.
For the proof of the main proposition we will use the following corollary.
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Corollary. For diagrams with only one positive contractible element there must be a negative contractible element after the positive contractible element in the order of the curve. Proof of Corollary. We have to look at the doubling process more closely. We call the original diagram K. By doubling K and connecting two curves at one of the end points 0 of the original curve we get K . We have two ways of doubling depending on how we connect the end points of the curve. One of the diagrams gives a unique positive contractible element while the other gives a unique negative contractible element. In either way the orientations of the new curve and that of the original curve restricted to each segment are opposite. We apply Lemma 1 to diagram K so there exists a positive contractible element. When we double the curve the new segment has to be inside the domain bounded by the segment of the positive contractible element and the segment of the line with same end points of the positive contractible element. Otherwise the positive contractible element 0 would give another positive contractible element for K . We observe that the orientation of the new curve and the original curve are opposite, thus gives a negative contractible element, which is after the unique positive contractible element. Proof of Lemma 2. Let us say that there is only one positive contractible element. Now pull out this contractible element. Since there is only one positive contractible element for the original pattern, we can only have one of the following three possibilities as shown in Fig. 10.
Fig. 10.
Possibility three cannot happen by a Jordan curve argument. We have two parallel elements such that there is no other point in the line segment. Suppose that the segment of the curve on the right is ahead of the segment on the left. We then form a Jordan curve by the segment on the curve connecting the two segments and the line segment connecting the two segments. The other end of the curve cannot go out of the domain bounded by the Jordan curve. See Fig. 11. That is a contradiction. Similar proof applies for the other case. The other two possibilities can be obtained from the double procedure. Since there is only one positive contractible element we apply the pull out procedure again, and again we finally get the doubled diagram. We can squeeze the diagram to get another diagram. The original diagram can be obtained by doubling. Since the resulting new diagram after squeezing can be considered as part of the original diagram it is also positive tilt.
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Fig. 11.
6.2. Proof of the proposition. The proof is again by induction. We only consider the case that all intersections are transverse. If we had proved this case, we would be able to do it for all cases. For the cases with vertical segment, we can quotient vertical segments to get another diagram. The values of actions at points of intersection are the same. Without loss of generality we assume that there are only transverse intersections for the new diagram. We then have minimums only achieved at points whose tilt are between 0 and π. Some of those points may correspond to the vertical segment of the original diagram. By comparing actions with points before or 0 after the point we consider we see that f (x, .) must across x × R from left to right. Now let us do the induction on the number of points of intersection of the diagram K. It is obvious for cases of intersection number one and three. By Lemma 1 there exists at least one positive contractible element. We pull out this contractible element by strengthening the segment in the curve bounded by the two consecutive points, the 0 resulting diagram K is still a positive tilt diagram. We do it in a manner that we only kill the action of the shaded area. This means that this procedure does not affect actions for points before the contractible element and decrease actions by the amount of shaded area for points after the contractible element. 0
By the induction hypothesis the proposition is true for K . This implies that the proposition is true for K for points of intersection before and after the positive contractible element. If there were other positive contractible element we do the same procedure again. It is easy to see that the proposition is then true for all points of intersection. If there were only one positive contractible element by the corollary after Lemma 2 we would have a negative contractible element after the positive contractible element. We do the same procedure for the negative element, i.e. pull out this negative 00 contractible element and get a new diagram K . We then have that the proposition is true for the first point of the positive contractible element. Since for the second point of the positive contractible element the action is larger, so the proposition is also true for
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Fig. 12.
the second point. Hence we have shown that the proposition is true for all points. And the proposition is proved. Acknowledgement. I would like to thank John Mather for many helpful conversations and for his encouragements. Without discussing with him this paper won’t exist. I would like to thank P. Boyland and the anonymous referee for useful comments. I would like to thank Department of Mathematics at Princeton University and National Natural Science Foundation of China for their financial support.
References 1. Bangert, V.: Mather sets for twist maps and geodesics on tori. Dynamics reported, Volume 1. Edited by U. Kirchgraber and H. O. Walther, 1–57 (1988) 2. Boyland, P.: Rotation sets and Morse decomposition in twist maps. Ergodic Theory and Dynamical Systems, 8*, 33–61 (1988) 3. Franks, J.: Recurrence and fixed points of surface homeomorphisms. Ergodic Theory and Dynamical Systems, 8*, 99–107 (1988) 4. Hall, R.: A topological version of a theorem of Mather on twist maps. Ergodic Theory and Dynamical Systems, 4, 585–603 (1984) 5. Katok, A.: Some remarks on Birkhoff and Mather twist map theorems. Ergodic Theory and Dynamical Systems, 2, 185–192 (1982) 6. MacKay, R., Meiss, J. and Percival, I.: Transport in Hamiltonian systems. Physica 13D, 55–81 (1984) 7. Mather, J.N.: Modulus of continuity for Peierls’s barrier. In: Periodic Solution of Hamiltonian Systems and Related Topics, ed. P.H. Rabinowitz et al. NATO ASI Series C 209 Dordrecht: D. Reidel, pp. 177–202 1987 8. Mather, J.N.: Destruction of invariant curves. Ergodic Theory and Dynamical Systems 8, 199–214 (1988) 9. Mather, J.N.: Minimal measures. Comment. Math. Helvetici 64, 375–394 (1989) 10. Mather, J.N.: Differentiability of the minimal average action as a function of the rotation number. Bol. Soc. Bras. Mat., Vol 21 no 1, 59–70 (1990) 11. Mather, J.N.: Variational construction of orbits of twist diffeomorphisms. J. of AMS, 4, 207–263 (1991) 12. Mather, J.N.: Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z., 207, 169–207 (1991)
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13. Mather, J.N.: Variational construction of connecting orbit. Ann. Inst. Fourier, Grenoble, 43, 5, 1349–1386 (1993) 14. Moser, J.: Monotone twist mappings and the calculus of variations. Ergodic Theory and Dynamical Systems, 6, 401–413 (1986) Communicated by A. Jaffe
Commun. Math. Phys. 191, 641 – 662 (1998)
Communications in
Mathematical Physics c Springer-Verlag 1998
Regularisable and Minimal Orbits for Group Actions in Infinite Dimensions M. Arnaudon, S. Paycha Institut de Recherche Math´ematique Avanc´ee, Universit´e Louis Pasteur et CNRS, 7, rue Ren´e Descartes, 67084 Strasbourg Cedex, France Received: 27 November 1995 / Accepted: 30 May 1997
Abstract: We introduce a class of regularisable infinite dimensional principal fibre bundles which includes fibre bundles arising in gauge field theories like Yang-Mills and string theory and which generalise finite dimensional Riemannian principal fibre bundles induced by an isometric action. We show that the orbits of regularisable bundles have well defined, both heat-kernel and zeta function regularised volumes. We introduce a notion of µ-minimality (µ ∈ R) for these orbits which extend the finite dimensional one. Our approach uses heat-kernel methods and yields both “heat-kernel” (obtained via heat-kernel regularisation) and “zeta function” (obtained via zeta function regularisation) minimality for specific values of the parameter µ. For each of these notions, we give an infinite dimensional version of Hsiang’s theorem which extends the finite dimensional case, interpreting µ-minimal orbits as orbits with extremal (µ-regularised) volume.
0. Introduction This article is concerned with the notions of regularisability and minimality of orbits for an isometric action of an infinite dimensional Lie group G on an infinite dimensional manifold P. Our study is based on heat-kernel regularisation methods but it involves a larger class including zeta function regularisations. The ones we consider are parametrised by µ ∈ R; we recover the zeta function regularisation by setting µ = γ, the Euler constant and the heat-kernel regularisation by setting µ = 0. Notions of regularisability and minimality for actions of infinite dimensional Lie groups on infinite dimensional manifolds have already been studied by other authors (see [KT, MRT]) in a particular context and using zeta function regularisation methods. We recover these notions for µ = γ. We shall introduce a class of principal fibre bundles called (resp. pre-)regularisable fibre bundles which generalise to the infinite dimensional case finite dimensional Riemannian principal fibre bundles arising from a free isometric action. We show that the
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fibres of these (resp. pre-)regularisable bundles have a well defined regularised (resp. preregularised) volume which is Gˆateaux differentiable. This class of (pre-) regularisable fibre bundles includes some infinite dimensional principal bundles arising from gauge field theories such as Yang-Mills and string theory. We introduce the notion of strong minimality and µ-minimality using µ- regularisation, all of which extend the finite dimensional notion and coincide in the finite dimensional case. However, µ-minimality depends on the choice of the parameter µ, in particular zeta function minimality (µ = γ) does not in general coincide with heat-kernel minimality (µ = 0). We show that if the metric on the structure group is fixed, µ-(resp. strongly) minimal fibres of a (resp. pre-)regularisable principal fibre bundle coincide with the ones with extremal µ-(resp. pre-)regularised volume among orbits of the same type for the group action. This gives an infinite dimensional version of Hsiang’s theorem on (pre-) regularisable principal fibre bundles with structure group equipped with a fixed Riemannian metric, which we extend (adding a term which reflects the variation of the metric on the structure group) to any (pre-)regularisable principal bundle. Starting from a systematic review of the notions of µ- regularised determinants in Sect. 1, in Sect. 2 we introduce the notions of regularisable principal fibre bundle, ( resp. pre-) regularisability and µ- (resp. strong) minimality of orbits, relating (resp. strong) minimality with the Gˆateaux-differentiability of µ (resp. pre-)regularised determinants interpreted as volumes of fibres. In Sect. 3, we compare these notions for different values of µ. The relations we set up between the regularised mean curvature vector and the directional gradients of the regularised determinants yield an infinite dimensional version of Hsiang’s theorem. To avoid making this article any longer than it already is, we chose not to treat examples in detail here. Let us just however point out some examples the results in this article can be applied to. When applied to the coadjoint action of a loop group, one recovers some results concerning regularisability and minimality of fibres studied in [KT]. The notion of minimality investigated in this article also applies to the study of orbits of a Yang-Mills action (see e.g [FU, KR, MV] for the corresponding geometric setting). A notion of zeta function minimality in the Yang Mills context had already been suggested in [MRT1]. Our heat-kernel approach leads to a slightly different definition; when the underlying manifold is of dimension 4, only if the irreducible connections are Yang-Mills, do the different notions of minimality (in particular zeta function and heat-kernel) coincide. Let us stress that in both the examples mentioned here, the space P, resp. the group G are modelled on a space of sections of a vector bundle E, resp. F with finite dimensional fibres on a closed finite dimensional manifold M and G acts on P by isometries. The case of diffeomorphisms acting on metrics which has been investigated carefully in [MRT2] is also very interesting since it relates to string theory. One could show, in a similar way to the Yang-Mills case, that the bundle M−1 → M−1 /Diff0 (see [FT, T]) arising in bosonic string theory ( where M−1 is the manifold of smooth Riemannian metrics with curvature −1 on a compact boundaryless Riemannian surface of genus greater than 1 and Diff0 is the group of smooth diffeomorphisms of the surface which are homotopic to zero), is also a regularisable fibre bundle so that most results of this paper can be applied to this fibre bundle. Unlike the case of Yang-Mills theory, its structure group Diff0 is not equipped with a fixed Riemannian structure but with a family of Riemannian metrics which is parametrised by g ∈ M−1 ; this example was our initial motivation when considering the general case of a structure group equipped
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with a family of metrics indexed by P . Investigating carefully the geometry of the orbits in this particular example leads to interesting questions concerning the geometry of some associated determinant bundles [PR]. In this particular example, minimality of the fibres is still equivalent to extremality of the volumes of the fibres since the additional term arising from the varying metric on the group (see Proposition 2.2) vanishes. The geometric notions developed in this paper play a important role when projecting a class of semi-martingales defined on the total manifold onto the orbit space for a certain class of infinite dimensional group actions. The regularisation based on heatkernel methods used here yields natural links between the geometric and the stochastic picture, which we investigate in [AP2]. The stochastic picture described in [AP2] leads to a stochastic interpretation of the Faddeev-Popov procedure used in gauge field theory to reduce a formal volume measure on path space to a measure on the orbit space, the formal density of which is a regularised “Faddeev–Popov” determinant. 1. Regularised Determinants In this section, we recall some basic facts about regularised determinants, comparing different regularisation methods. Although the results presented here are in some way well known (see e.g [BGV]) and frequently used in the physics literature, the presentation we offer is maybe a little unusual, since it involves defining a one parameter family of regularised limits (parametrised by µ ∈ R). Zeta function regularisation corresponds to µ = γ, the Euler constant, heat-kernel regularisation to µ = 0. Let us first introduce some notations. For a function t 7→ f (t), defined on an interval PK j of R+∗ containing ]0, 1], we shall write f (t) '0 j=−J aj t m + blogt, aj ∈ R, b ∈ R, J, K ∈ N, m ∈ N∗ , if f (t) =
K X
j
aj t m + blogt + O(t
K+1 m
)
∀
0 < t < 1.
(1.1)
j=−J
Let us call C this class of functions and C0 the subclass of functions f ∈ C with no logarithmic divergence at zero (i.e. b = 0). In the following, we shall always assume that J ≥ m. Functions in the class C arise naturally as primitives of functions in the class C0 as shown in the following Lemma 1.1. If f is a differentiable function on an interval I of R containing ]0, 1] with K X j K+1 aj t m + O(t m ), then derivative f 0 ∈ C0 , then f ∈ C. More precisely, if f 0 (t) = j=−J
f (t) =
K+m X
j
αj t m + βlogt + O(t
K+m+1 m
) with αj =
m j aj−m
for j 6= 0 and β = a−m
j=−J+m
and for some α0 ∈ R. Proof. Let us set g(t) = f (t)−
K+m X j=−J+m,j6=0
j K+1 m aj−m t m −a−m logt. Since g 0 (t) = O(t m ), j k+1
for 0 < s < t ≤ 1 we can estimate |g(t) − g(s)| ≤ Ct m |t − s| ≤ Ct
K+1 m
and this yields
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the existence of the limit lim g(t) = α0 . Using the same estimate for s = 0 then yields the result.
t→0
In similar way to [BGV], we set the following: Definition. For f ∈ C, µ ∈ R, and with the notations of (1.1): K X j aj t m − blogt − µb, Limµ f (t) ≡ lim f (t) − t→0
t→0
(1.2)
j=−J
which we call the µ-regularised limit of f at point zero. Remark. Although the parameter µ might seem artificial at this stage, it will prove to be useful when comparing heat-kernel regularisations and zeta-function regularisations. A similar parameter µ arises in the work of Bismut and Freed on determinant bundles where similar regularisations are needed [BF]. Further analogies between the gauge orbit picture discussed here and the determinant bundles picture are discussed in [P]. Of course, for a function in the class C0 , this limit does not depend on the parameter µ. Let (At ), t ∈]0, 1] be a one parameter family of trace-class operators on a separable Hilbert space H (in particular tr(A1 ) is finite) such that t → trAt is a function in the class C, then for any µ ∈ R we can define the µ-regularised limit trace of A ≡ (At , t ∈]0, 1]) by µ A ≡ Limµt→0 trAt . (1.3) trreg This regularised limit trace depends of course on the whole one parameter family A and on the choice of the parameter t. Whenever the context we are working in allows no ambiguity on the choice of µ, we shall sometimes leave the explicit mention of µ out. We now introduce a family of heat-kernel operators which play a fundamental role in Zthis paper. For this we define for ε > 0 a function hε : R+∗ → R by hε (λ) ≡ ∞ −tλ e dt. Notice that hε is C ∞ , non decreasing and (hε )0 (λ) = λ−1 e−ελ . Writing − t ε R ∞ −t R ε −λt R1 hε (λ) − log ε = − ε e t dt − ελ e t dt + ε 1t dt, we find that the function ε 7→ hε (λ) lies in C for fixed λ. Moreover we have: Limµε→0 hε (λ) = logλ − µ + γ,
(1.4)
R 1 1−e−t R ∞ e−t dt− t dt 1 where γ = e 0 t is the Euler constant. For a strictly positive self-adjoint operator B on a Hilbert space H, we can define hε (B) which yields a one parameter family of operators (hε (B), ε > 0). Definition. Let B be a strictly positive self-adjoint operator on a separable Hilbert space. Whenever the one parameter family B ≡ (hε (B), ε ∈]0, 1]) has a regularized limit trace, for any µ ∈ R, we shall call µ-regularised determinant detµreg (B) of B the expression: µ µ (1.5) detµreg B ≡ etrreg (B) = eLimε→0 tr hε (B) . In the following, we give conditions under which we can define the heat-kernel regularized determinant of an operator B. But before that, let us state an easy lemma which will prove to be useful for what follows.
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Lemma 1.2. Let B be a strictly positive self-adjoint operator on a separable Hilbert space such that 1) e−εB is trace class for any ε > 0. 2) The function ε → tr(e−εB ) lies in the class C0 with (bj , j ≥ −J) as coefficients in the expansion (1.1). Then the operator B has a heat-kernel regularised determinant and we have for µ∈R µ
det µreg B = eLimε→0 trhε (B) R ∞ e−tB R Pm−1 mbj − tr t dt− 01 j − 1 j=−J,j6=0 =e
F (t) t dt
(1.6)
−µb0
with m−1 X
F (t) = tre−tB −
j
bj t m .
(1.7)
j=−J
Z
∞
e−tB dt is trace-class. Since t ε all the terms involved are positive, we can exchange the integral and sum symbols R ∞ −tB so that trAε = − ε tr e t dt. Let us check that the family (Aε , ε ∈]0, 1]) has a regularized limit trace. The map t → trAt is differentiable and from tr(e−tB ) = 0 X PK j j d K+1 m + O(t m ) follows that trAt '0 bj+m t m . Applying Lemma j=−J bj t dt Proof. One easily shows that Aε ≡ hε (B) = −
j=−J−m
1.1 to f (t) = tr(At ) shows that the one parameter family A = (Aε ) has a finite regularised limit trace trµreg (A) ≡ Limµε→0 trAε . By (1.5) this in turn yields that B has a µR ∞ e−tB µ regularised determinant detµ B = e trreg A . Since trA = − tr dt, integrating 1
reg
Ft t
1
t
between ε and 1 yields
trAε −
m−1 X j=−J,j6=0
mbj j ε m − b0 log ε = − j
m−1 X j=−J,j6=0
mbj − j
Z
1 ε
F (t) dt − t
ε→0
−1 X mbj j ε m − b0 log ε) = lim (trAε − ε→0 j
j=−J
∞
tr 1
e−tB dt. t (1.8)
Since m ≥ 1, we have lim (trAε −
Z
m−1 X j=−J,j6=0
which combined with (1.8) and using (1.5) yields (1.6).
mbj j ε m − b0 log ε) j
The following lemma gives a class of operators which fit in the framework described above. Lemma 1.3. Let B be a strictly positive self adjoint elliptic operator of order m > 0 on a compact boundaryless manifold. For any ε > 0, e−εB is trace class and B has a well defined µ-regularised determinant.
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Proof. We shall show that the assumptions of Lemma 1.2 are fulfilled. Condition 1) in Lemma 1.2 follows from the fact that a strictly positive s.a elliptic operator on a compact boundaryless manifold has purely discrete spectrum (λn )n∈N , α > 0 ( see e.g [G], Lemma 1.6.3). Indeed, from λn > 0, λn ' Cnα , for some C > 0,P this fact easily follows that tre−εB = n e−ελn is finite. Conditions 2) of Lemma 1.1 follow from the fact that for a s.a elliptic operator B of order m on a compact manifold of dimension d without boundary, tre−tB '0 PK j m for any K > 0 (this follows for example from Lemma 1.7.4 in [G]). j=−d aj t Applying Lemma 1.1, we can therefore define the heat-kernel regularised determinant of B. The above definition extends to a class of positive self-adjoint operators which satisfy requirements 1) and 2) of Lemma 1.2 and have possibly non zero kernel. Requirement 1) of the lemma implies that this kernel is finite dimensional. Let PB be the orthogonal projection onto the kernel of the operator B acting on H and let us set H ⊥ ≡ (I −PB )H. Let us consider the restriction B 0 ≡ B/H ⊥ . It is easily seen that the operator B 0 satisfies requirements of Lemma 1.2 with coefficients b0j = bj for j 6= 0 and b00 = b0 −dim(KerB). Formula (1.6) extends to B 0 with adapted changes in the coefficients. Let us at this stage see how the zeta-function regularised determinant fits into this picture. We refer the reader to [BGV, G] for a precise description of the zeta-function regularisation procedure and only describe the main lines of this procedure here. Recall that for a strictly positive self adjoint operator B acting on a separable Hilbert space with purely discrete spectrum given by the eigenvalues (λn , n ∈ N) with the property λn ≥ Cnα , C > 0, α > 0 for large enough n, we can define the zeta function of B by: X 1 λ−s s ∈ C, Res > . ζB (s) ≡ n , α n Furthermore, ζB (s) admits a meromorphic continuation on the whole plane (see e.g [G], Lemma 1.10.1) which is regular at s = 0 and one can define the zeta function regularised determinant of A by 0 (1.9) detζ (B) = e−ζB (0) . Remark. From the definition, easily follows that in the finite dimensional case the zetafunction regularised and the ordinary determinants coincide. The following lemma compares the zeta function and µ-regularisations. Lemma 1.4. Let B be a strictly positive self-adjoint densely defined operator on a Hilbert space H such that 1) B has purely discrete spectrum (λn )n∈N with λn ≥ Cnα ,C > 0, α > 0 for large enough n, 2) The function ε → tre−εB lies in C0 . Then for µ ∈ R, eζB (0)(γ−µ) detζ B = detµreg (B),
(1.10)
where γ = limn→∞ (1 + 21 + · · · + n1 − log n) is the Euler constant. In particular, detζ B = detγreg B.
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Remark. A proof of this result for µ = 0 and the Laplace operator on a compact Riemannian surface without boundary can be found in [AJPS]. Proof. Before starting the proof, let us recall that the function Gamma is defined by R ∞ −t 0(z) = 0 e t tz dt for 0 < Rez. Moreover 0(z)−1 is an entire function and we have ∞ Y z −z (1 + )e n , where γ is the Euler constant. From this follows that in 0(z)−1 = zeγz n n=1
−1 = s + γs2 + O(s3 ). a neighborhood of zero, we have the asymptotic expansion 0(s) Z +∞
Using the Mellin transform of the function λ−s = 0(s)−1
ts−1 e−tλ dt we can
0
write:
Z 0(s)ζB (s) =
1
ts−1 tre−tB dt +
Z
0
∞
ts−1 tre−tB dt.
(1.11)
1
Notice that the last expression on the r.h.s converges 0 for, setR ∞ for Res ≤ R, RR > 1 1 ∞ ting CR = supn supt≥1 tR−1 e− 2 tλn , we have 1 tR−1 e−tλn ≤ CR 1 e− 2 tλn = − 21 λn 2CR λ−1 , which is the general term of a convergent series. n e As before we set m−1 X j bj t m . (1.12) F (t) ≡ tre−tB − j=−J
Using (1.11 ) and (1.12), we can write for s ∈ C with large enough real part, Res > Z 1 Z ∞ m−1 X bj ζB (s) = 0(s)−1 ts−1 tre−tB dt + ts−1 F (t)dt . + j + s 1 0 j=−J m
J m:
This equality then extends to an equality of meromorphic functions on Res > 0 with poles s = −j m . Using the asymptotic expansion of the inverse of the Gamma function −1 0(s) around zero, we have Z 1 Z ∞ m−1 X bj 0 (s) = (s + γs2 + O(s3 )) ts−1 tre−tB dt + ts−1 F (t)dt , + ζB j + s 1 0 j=−J m which yields b0 = ζB (0). Moreover m−1 X 0 (s) = (1 + 2γs + O(s2 )) ζB
bj + +s
j j=−J m
m−1 X
+(s + γs2 + O(s3 )) −
j=−J
bj + j (m + s)2
Z
Z
∞
ts−1 tre−tB dt +
1
1
1 0
Z
∞
ts−1 F (t)dt
ln(t)ts−1 tre−tB dt .
ts−1 F (t)ln(t)dt+ 0
Z
1
Letting s tend to zero, s > 0, since the divergent terms bs0 and −s sb02 arising in each of the terms of this last sum compensate, we get Z ∞ −tB m−1 X mbj Z 1 F (t) tre 0 + dt + dt . (0) = b0 γ + ζB j t t 0 1 j=−J,j6=0
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Hence, comparing with the expression of detµreg B given in (1.5), for any µ ∈ R we find 0 log detζ (B) = −ζB (0) = −ζB (0)γ + log det µreg (B) + µζB (0) and hence the equality of the lemma. Remarks. 1) In the finite dimensional case with dimH = d, since limε→0 tre−εB = d = ζB (0), from the result of Lemma 1.4 and the fact that the zeta function regularised determinant coincides with the ordinary one, it follows that for µ ∈ R: detµreg B = ed(γ−µ) detζ B = ed(γ−µ) detB,
(1.13)
where detB denotes the ordinary determinant of B. For µ = γ, detγreg B = detB = detζ B. 2) Let M be a Riemannian manifold of dimension d and B a positive self-adjoint elliptic operator with smooth coefficients acting on sections of a vector bundle V on M with finite dimensional fibres of dimension k. We know by [G] Theorem 1.7.6 (a) that ζB (0) = 0 if n is odd. However, in general the coefficient ζB (0) is a complicated expression given in terms of the jets of the symbol of the operator B. In the following we shall be concerned with the dependence of ζB (0) on the geometric data given on that manifold.
2. Regularisable Principal Fibre Bundles The aim of this section is to describe a class of principal fibre bundles for which we can define a notion of regularised volume of the fibres and for which these regularised volumes have differentiability properties. Let P be a Hilbert manifold equipped with a (possibly weak) right invariant Riemannian structure. The scalar product induced on Tp P by this Riemannian structure will be denoted by < ·, · >p . We shall assume this Riemannian structure induces a Riemannian connection denoted by ∇ and an exponential map with the usual properties. In particular, for all p0 , expp0 yields a diffeomorphism of a neighborhood of 0 in the tangent space Tp0 P onto a neighborhood of p0 in the manifold P. Let G be a Hilbert Lie group ( in fact a Hilbert manifold with smooth right multiplication is enough here, see e.g. [T]) acting smoothly on P on the right by an isometric action Θ : G × P → P, (2.0) (g, p) → p · g. Let for p ∈ P,
τp : G → Tp P, d u 7→ (p · etu ) , dt t=0
(2.0bis)
where G denotes the Lie algebra of G. We shall assume that the action Θ is free (so that τp is injective on G) and that it induces a smooth manifold structure on the quotient space P/G and a smooth principal fibre bundle structure given by the canonical projection π : P → P/G. Let us furthermore equip the group G with a smooth family of equivalent (possibly weak) Adg invariant Riemannian metrics indexed by p ∈ P. The scalar product induced on G by the Riemannian metric on G indexed by p ∈ P will be denoted by (·, ·)p . Since
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the metrics are all equivalent, the closure of G w.r.t (·, ·)p does not depend on p and we shall denote it by H. Since G is dense in H, τp is a densily defined operator on H and we can define its adjoint operator τp∗ w.r. to the scalar products (·, ·)p and < ·, · >p . We shall assume that τp∗ τp has a self-adjoint extension on a dense domain D(τp∗ τp ) of H. Definition. The orbit of a point p0 is volume preregularisable if the following assumptions 1) and 2) on the operator τp∗ τp are satisfied: ∗
1) Assumption on the spectral properties of τp∗0 τp0 . The operator e−ετp0 τp0 is trace class for any ε > 0 and for any vector X at point p0 , there is a neighborhood I0 of p0 on ∗ the geodesic pκ = expp0 κX such that for all p ∈ I0 , e−ετp τp is trace class. 2) Regularity assumptions. We shall assume that the maps p 7→ τp and p 7→ τp∗ τp are ∗ Gˆateaux differentiable and that for any t > 0, the function p 7→ tre−tτp τp is Gˆateaux differentiable at point p0 . We furthermore assume that the Gˆateaux-differentials at point p0 in the direction X of these operators are related as follows: ∗
∗
δX (tre−ετp τp ) = −εtr(δX (τp∗ τp )e−ετp τp ).
(2.1)
Moreover, for any vector X at point p0 , there are constants C > 0, u > 0 and a neighborhood I0 of p0 on the geodesic pκ = expp0 κX such that for any p ∈ I0 : ∗
tre−tτp τp ≤ Ce−tu
(2.2)
and ∗
(τp∗ τp )e−tτp τp |||∞ MI0 (t) ≡ supp∈I0 |||δX(p) ¯
(2.3)
is finite and a decreasing function in t. Here ||| · |||∞ denotes the operator norm on G induced by (·, ·)p , X¯ is a local vector ¯ κ ) = expp ∗ (κX)(X). field defined in a neighborhood of p0 by X(p κ The orbit Op0 is called volume-regularisable if dim Kerτp∗ τp is constant on some neighborhood of p0 on any geodesic containing p0 and if the following assumption is satisfied: 3) Assumption on the asymptotic behavior of the heat-kernel traces. Both the functions ∗ ∗ t 7→ tre−tτp τp and t 7→ δX tre−tτp τp lie in the class C0 (see Sect. 1). There is an integer m > 0 and a family of maps p 7→ bj (p), j ∈ {−J, · · · , m − 1} which are Gˆateaux differentiable in the direction X at point p0 such that ∗
tre−ετp τp '0
m−1 X
j
bj (p)ε m
(2.4)
j=−J
in a neighborhood I0 of p0 on the geodesic p = expp0 κX, and ∗
δX tre−ετp τp '0
m−1 X j=−J
j
δX bj (p)ε m .
(2.5)
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Furthermore, setting Fp (t) ≡ tre
−tτp∗ τp
−
m−1 X
j
bj (p)t m , for any vector X at point
j=−J
p0 , there is a constant K > 0, and a neighborhood I0 of p0 on the geodesic κ → pκ = expp0 κX such that: supp∈I0 kδX(p) Fp (t)k∞ ≤ Kt. ¯
(2.5bis)
A principal bundle as described above with all its orbits volume-preregularisable (resp. volume- regularisable) will be called preregularisable (resp. regularisable). Remark. Since the Riemannian structure on P is right invariant and the one on G is Adg invariant, the above assumptions do not depend on the point chosen in the orbit for we have τp·g = Rg∗ τp Adg . Most fibre bundles we shall come across are not only preregularisable but also regularisable so that the notion of preregularisabiblity might seem somewhat artificial. However, in the case of the coadjoint action of loop groups mentioned in the introduction, it is sufficient to verify the conditions required for preregularisability in order to prove a certain minimality of the orbits, namely strong minimality, a notion which will be defined in the following and which implies minimality. Natural examples of regularisable fibre bundles arise in gauge field theories (YangMills, string theory). In gauge field theories, P and G are modelled on spaces of sections of vector bundles E and F based on a compact finite dimensional manifold M and the operators τp∗ τp arise as smooth families of Laplace operators on forms. As elliptic operators on a compact boundaryless manifold, they have purely discrete spectrum which satisfies condition 1) (see [G] Lemma 1.6.3) and (2.4) (see [G], Lemma 1.7.4.b)). By classical results concerning one parameter families of heat-kernel operators, they satisfy (2.1) (see [RS], Proposition 6.1) and (2.2) (see proof of Theorem 5.1 in [RS]). Since δX Bp is also a partial differential operator, by [G], Lemma 1.7.7, δX tre−εBp satisfies (2.5). Assumptions on the Gˆateaux-differentiability and assumptions (2.3 ), (2.5 bis) are fulfilled in applications. Indeed, the parameter p is a geometric object such as a connection, a metric on M and choosing these objects regular enough (of class H k for ∗ k large enough) ensures that the maps p 7→ τp , p 7→ τp∗ τp , p 7→ tre−tτp τp , etc., are regular enough for they involve these geometric quantities and their derivatives, but no derivative of higher order. Remark. In the context of gauge field theories, the underlying Riemannian structure w.r.to which the traces (arising in (2.2)-(2.5 bis)) are taken are weak L2 Riemannian structures, the ones that also underlie the theory of elliptic operators on compact manifolds. In [AP2], we discuss how far this weak Riemannian structure could be replaced by a strong Riemannian structure, in order to set up a link between this geometric picture and a stochastic one developed in [AP2]. Notation. We shall set with the notations of Sect. 1, for ε > 0 and p ∈ P detε (Bp ) = exptr(hε (Bp )). Proposition 2.1. Let Op0 be a volume-preregularisable orbit such that for any geodesic containing p0 , there is a neighborhood of p0 on this geodesic on which τp∗ τp is injective. 1) detε (τp∗ τp ) is well defined for any ε > 0 and for p in a neighborhood of p0 on any geodesic containing p0 .
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2) The map p 7→ detε (τp∗ τp ) is Gˆateaux-differentiable at point p0 , the operator Z +∞ ∗ δX (τp∗ τp )e−tτp τp dt is trace class for any p in a neighborhood of p0 on any ε
geodesic of p0 . For any tangent vector X at point p0 , we have: Z ∞ ∗ tr (δX τp∗ τp )e−tτp0 τp0 dt δX log detε (τp∗ τp ) = ε Z +∞ ∗ (δX τp∗ τp )e−tτp0 τp0 dt. = tr
(2.6a)
ε
3) If the orbit Op0 is moreover volume-regularisable, for any µ ∈ R, the map p 7→ detµreg (τp∗ τp ) is Gˆateaux differentiable in all directions at point p0 , and for p in a geodesic neighborhood of p0 , the map ε 7→ δX log det ε (τp∗ τp ) lies in the class C. For µ ∈ R, Limµε→0 δX log detε (τp∗ τp ) = δX log detµreg τp∗ τp ! ∗ Z 1 Z ∞ m−1 X m e−tτp τp δX Fp (t) δX bj − dt − dt − µδX b0 . =− δX tr j t t 1 0 j=−J,j6=0
(2.6b) Proof. We set Bp =
τp∗ τp
and as before, detε (Bp ) = exptrhε (Bp ).
1) By the first assumption for volume-preregularisable orbits, we know that e−εBp is p trace class so that by Lemma 1.1 so is Apε ≡ log hε (Bp ). Hence detε (Bp ) = etrAε is well defined. 2) Let us show the first equality in (2.6 a). Assumption 2) for volume-preregularisability t t Bp e−tBp )| ≤ CMI0 ( )e− 2 u . yields that for any p ∈ I0 and any t > ε > 0 |tr(δX(p) ¯ 2 Here, we have used the fact that |tr(U V )| ≤ |||U |||trV | for any bounded opt erator U and any trace class operator V applied to U = δX(p) Bp e− 2 Bp0 and ¯ t V = Re− 2 Bp . Hence, by the Lebesgue dominated convergence theorem, the map ∞ p 7→ ε t−1 tre−tBp dt is Gˆateaux-differentiable in the direction X at point p0 and Z ∞ Z ∞ −1 −tBp δX t tre dt = t−1 δX tre−tBp dt ε ε Z ∞ tr((δX Bp )e−tBp0 )dt, =− ε
Z
+∞
using (2.2). Using the fact that log detε (Bp ) = −
t−1 tre−tBp dt then yields the
ε
first equality in (2.6 a). The second equality in (2.6 a) and the fact that we can swap the trace and the integral follow from the estimate: |||δX Bp e−tBp0 |||1 ≤ |||δX Bp e
−ε 2 Bp 0
≤ C|||δX Bp e
|||∞ k|e− 2 tBp0 k|1
−ε 2 Bp 0
1
|||∞ e−tu ,
(∗)
valid for t ≥ ε, using Assumption Z +∞ Z +∞ (2.2). We finally obtain by dominated convergence: tr δX Bp e−tBp0 dt = trδX Bp e−tBp0 dt. ε
ε
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3) Let us first check that the map p 7→ detµreg Bp is Gˆateaux differentiable at point p0 in the direction X. By (1.5), we have m−1 X
log det reg Bp = −
j=−J,j6=0
bj (p) − j
Z
∞
tr 1
e−tBp dt − t
Z
1 0
Fp (t) dt. t
The first term on the r.h.s. is Gˆateaux differentiable in the direction X by the assumption on the maps p 7→ bj (p). The second term on the r.h.s. is Gˆateaux differentiable by the result (applied to ε = 1) of part 2 of this proposition which tells us that p 7→ detε (Bp ) is Gˆateaux differentiable. The Gˆateaux differentiability of the last term follows from the local uniform upper bound (2.5 bis). Pm−1 j mb We now check (2.6 b). The map p 7→ log detε (Bp ) − j=−J,j6=0 j j ε m − b0 log ε is Gˆateaux differentiable in the direction X and we can write δX ( log detε (Bp ) − ∞
= δX (−
tr ε
e−tBp dt − t
m−1 X
= δX −
j=−J,j6=0
=−
j
j=−J,j6=0
Z
m−1 X
m−1 X j=−J,j6=0
δ X bj
bj m − j m − j
Z
mbj ε m − b0 log ε) j m−1 X
j=−J,j6=0
Z
∞ 1
e−tBp dt − tr t
∞
δX tr 1
j
mbj ε m − b0 log ε) j Z
e−tBp dt − t
1 ε
Z
Fp (t) dt t 1
δX ε
as in (1.8)
Fp (t) dt, t
which tends to δX log detreg Bp by (1.6) and dominated convergence. Here we have used Z ∞ −tBp e dt = tr the results of point 2) of the proposition applied to ε = 1 to write δX t 1 Z ∞ R 1 δ F (t) R 1 F (t) δX tre−tBp dt and (2.5 bis) to write δX ε pt dt = ε X tp dt. 1
Remark. These results extend to the case when instead of assuming that τp∗ τp is injective locally around p0 , one considers orbits of an action at points p0 for which the dimension of the kernel of τp is constant on some neighborhood of p0 on each geodesic starting at point p0 . For this, one should replace detε τp∗ τp and detreg τp∗ τp by det0ε τp∗ τp and det0reg τp∗ τp . This extension is useful for the applications mentioned in the introduction. A naive generalisation of the finite dimensional notion of volume to volume of infinite dimensional orbits would give infinite quantities. But for volume-preregularisable or regularisable orbits, one can define a notion of preregularised or µ-regularised volume (µ ∈ R), which justifies a posteriori the term “volume-preregularisable or volumeregularisable orbits” for these orbits. Since τp·g = Rg∗ τp Adg and since the metric on G ∗ is Adg and that on P right invariant, for any ε > 0, we have detε (τp·g τp·g ) = detε (τp∗ τp ) so that it makes sense to set the following definitions:
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Definition. 1) Let Op be a volume-preregularisable orbit, then volε (Op ) ≡ p det ε (τp∗ τp )0 defines a one parameter family of preregularised volumes of Op . q 2) Let Op be a volume-regularisable orbit, then for µ ∈ R, volµreg (Op ) = det µreg (τp∗ τp )0 defines the µ-regularised volume of Op . q 3) Let Op be a volume-regularisable orbit, then volζ (Op ) = det ζ (τp∗ τp )0 defines the zeta function regularised volume of Op . From Lemma 1.4 it follows that 1
0
volζ (Op ) = e 2 (−γ+µ)b0 (p) volµreg (Op ),
(2.7)
where γ is the Euler constant and b00 (p) = ζτp∗ τp (0) − dim Ker(τp∗ τp ) is the coefficient arising from the heat-kernel asymptotic expansion of τp∗ τp given by (2.4). In finite dimensions, when dimH = d and G is a compact LieZ group equipped with the Haar µ (Op ) = e(µ−γ)d |detτp | measure dvol, this yields volreg
As a consequence of Proposition 2.1:
|detAdg dvol(g)|. G
Proposition 2.2. For any µ ∈ R, the µ-(resp. pre)-regularised volume of a volume(pre)regularisable orbit Op is Gˆateaux-differentiable at the point p. Let us now introduce a notion of extremality of orbits which generalises the corresponding finite dimensional notion [H]. Definition. A strongly extremal orbit is a volume-preregularisable orbit, the preregularised volume of which is extremal, i.e. Op is strongly extremal if δX volε (Op ) = 0 for any horizontal vector X at point p and any ε > 0. For a given µ ∈ R, a µ- extremal orbit of a preregularisable bundle is an orbit, the µ-regularised volume of which is extremal, i.e. δX volµreg (Op ) = 0 for any horizontal vector X at point p. Notice that whenever ζτp∗ τp (0) − dim(Ker(τp∗ τp )) does not depend on p, the extremality of the volume of an orbit does not depend on the parameter µ. From (2.7) it also follows that this notion generalises the finite dimensional notion of extremality of the volume of the fibre.
3. Minimal Orbits as Orbits with Extremal Volume We shall consider a preregularisable principal fibre bundle P → P/G. By assumption, the bundle is equipped with a Riemannian connection given by a family of horizontal spaces Hp , p ∈ P such that Tp P = Hp ⊕ Vp , where Vp is the tangent space to the orbit at point p and the sum is an orthogonal one. For a horizontal vector X at point p, we define the shape operator H X : V p → Vp ¯ v (p), Y 7→ −(∇Y X)
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where the subscript v denotes the orthogonal projection onto Vp and X¯ is a horizontal vector field with value X at p. Similarly, we define the second fundamental form: S p : V p × V p → Hp (Y, Y 0 ) 7→ (∇Y¯ Y¯ 0 )h (p), where Y¯ , Y¯ 0 are vertical vector fields such that Y¯ (p) = Y , Y¯ 0 (p) = Y 0 . These definitions are independent of the choice of the extensions of X,Y and Y 0 . An easy computation shows that the shape operator and the second fundamental form are related as follows: < HX (Y ), Y 0 >p =< S p (Y, Y 0 ), X >p .
(3.1)
Note that this explicitly shows that HX only depends on X and not on the extension X¯ of X. Since S p is symmetric, so is HX . As in the finite dimensional case, one can define the notion of totally geodesic orbit, an orbit Op being totally geodesic whenever the second fundamental form S p vanishes. Definition. The orbit Op of a point p ∈ P will be called preregularisable if for any horizontal vector X at p, ∀ε > 0, ∗
∗
ε ≡ e− 2 ετp τp HX e− 2 ετp τp HX 1
1
(3.2)
is trace class. A preregularisable orbit Op will be called strongly minimal if moreover ε = 0 ∀ε > 0. for any q ∈ Op and X a horizontal vector at point q, trHX ε trace class) is autoRemarks. 1) The preregularisability of the orbits ( namely HX matically satisfied if the manifold P is equipped with a strong smooth Riemannian structure, since in that case the second fundamental form is a bounded bilinear form and its weighted trace is well defined (see also [AP2] where this is discussed in further detail). 2) Since on a preregularisable bundle, the Riemannian structure on P is right invariant and the one on G is Adg invariant, the notion of (pre)regularisability and (strong) minimality of the orbit does not depend on the point chosen on the orbit. 3) Notice that if HX is trace class, as in the finite dimensional case, strong minimality implies that trHX = 0 and hence ordinary minimality. The fact that strong minimality implies minimality in the finite dimensional case motivates the choice of the adjective “strong”. ε and the second fundamental form are related 4) This preregularised shape operator HX as follows: ∗
∗
ε (Y ), Y 0 >p =< S p (e− 2 ετp τp Y, e− 2 ετp τp Y 0 ), X >p < HX 1
1
ε Since τp τp∗ is an isomorphism of the tangent space to the fibre Tp Op , HX vanishes whenever the second fundamental form vanishes and an orbit is totally geodesic whenever this regularised shape operator vanishes on the orbit for some ε > 0.
Definition. A preregularisable orbit Op will be said to be regularisable if furthermore, ε , ε ∈]0, 1] admits a regularised limit-trace (as defined in the one parameter family HX Sect. 1). For µ ∈ R, we denote by trµreg HX its µ-regularised limite trace.
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Definition. For a given µ ∈ R, a regularisable orbit Op will be called µ− minimal if trµreg HX = 0 for any horizontal vector X at point p. Remarks. 1) As we shall see later on, for different values of µ, the notions of µminimality do not coincide in general. ε admits a regularised limit 2) In the finite dimensional case, the one parameter family HX trace given by the ordinary trace trµreg HX = trHX and µ- minimality is equivalent to the finite dimensional notion of minimality. 3) A strongly minimal preregularisable orbit Op is µ-regularisable and µ-minimal for any µ ∈ R. The notion of minimality of orbits for group actions in the infinite dimensional case has been discussed in the literature before. King and Terng in [KT] introduced a notion of regularisability and minimality for submanifolds of path spaces using zeta-function regularisation methods. They show zeta function regularisability and minimality for the orbits of the coadjoint action of a (based) loop group on a space of loops in the corresponding Lie algebra. One can check that these orbits are also regularisable and strongly minimal (hence minimal) within our framework . A notion of zeta function regularisability and minimality was discussed by Maeda, Rosenberg and Tondeur in [MRT1] (see also [MRT2]) in the case of orbits of the gauge action in Yang-Mills theory. In fact, it can be seen as a particular example of µ-minimality for µ = γ, the Euler constant. Let us introduce some notations. Let P → P/G be a preregularisable principal fibre bundle and let (Tnp )n∈N be a set of eigenvectors of τp∗ τp in G corresponding to the eigenvalues (λpn )n∈N counted with multiplicity and in increasing order. Let p0 be a fixed point in P and let Ipp0 be the isometry from (G, (·, ·)p0 ) into (G, (·, ·)p ) which takes the orthonormal set (Tnp0 )n of eigenvectors of τp∗0 τp0 to the orthonormal set of eigenvectors (Tnp )n of τp∗ τp . Notice that Ipp00 = I. Lemma 3.1. Let P → P/G be a preregularisable principal fibre bundle. Let p0 ∈ P be a point at which the map p 7→ Ipp0 u is Gˆateaux-differentiable for any u ∈ G. Let X be a horizontal vector at p0 . We shall consider eigenvalues λpn that correspond to eigenvectors that do not belong to Ipp0 Kerτp∗0 τpo . p 1) The maps p → λpn are Gˆateaux-differentiable Z in the direction X at point p0 , δX λn = +∞
(δX (τp∗ τp )Tnp0 , Tnp0 )p0 and δX log hε (λpn ) =
∗
(δX (τp∗ τp )e−tτp0 τp0 Tnp0 , Tnp0 )p0 dt.
ε
2) Furthermore,we have p0
ε ˜p ˜p Un , Un >p0 +e−ελn (δX Ipp0 Tnp0 , Tnp0 )p0 = − < HX
1 δX log hε (λpn ), 2
(3.5)
where we have set U˜ np = kτp Tnp k−1 τp Tnp . 3) If the Riemannian structure on G is fixed (independent of p), then δX Ipp0 is antisymmetric and Z ∗ 1 +∞ ε ˜p ˜p U n , Un > p 0 (δX (τp∗ τp )e−tτp τp Tnp , Tnp )p0 dt = − < HX 2 ε (3.6) p0 1 1 −1 = δX log hε (λpn ) = λpn0 δX λpn e−ελn . 2 2
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Proof. As before, we shall set Bp = τp∗ τp . Since p0 is fixed, we drop the index p0 in Ipp0 and denote this isometry by I p . Notice that I p0 = I. As before, we denote by (Tnp )n∈N the orthonormal set of eigenvectors of τp∗ τp which correspond to the eigenvalues (λpn )n∈N in increasing order and counted with multiplicity. We shall set T˜np = τp Tnp , T¯np = τp Tnp0 . 1) Using the relations (I p ·, I p ·)p = (·, ·)p0 , I p (Tnp0 ) = Tnp , I p ∗ I p = I, we can write λpn = (Bp Tnp , Tnp )p = (Bp I p Tnp0 , I p Tnp0 )p0 and the map p 7→ λpn is Gˆateaux differentiable in all directions at point p0 since p 7→ Bp , p 7→ I p are Gˆateaux-differentiable by assumption on the bundle. Furthermore δX (Bp Tnp , Tnp )p = δX (I p ∗ Bp I p Tnp0 , Tnp0 )p0 = ((δX Bp )Tnp0 , Tnp0 )p0 + (δX (I p ∗ )Bp0 Tnp0 , Tnp0 )p0 + + (I p0 ∗ Bp0 (δX I p )Tnp0 , Tnp0 )p0 = ((δX Bp )Tnp0 , Tnp0 )p + λpn0 ([I p0 ∗ δX (I p ) + (δX I p ∗ )I p0 ]Tnp0 , Tnp0 )p0 . Since I p ∗ I p = I, we have δX I p ∗ I p0 + I p0 ∗ δX I p = 0 so that finally λpn is Gˆateauxdifferentiable and δX λpn = ((δX Bp )Tnp0 , Tnp0 )p0 . Using the local uniform estimate (2.3), and with the same notations, we have for t > ε p0 1 1 (Bp )e−tBp0 Tnp0 , Tnp0 )p0 k ≤ MI0 ( t)e− 2 tλn so that the map p 7→ log hε (λpn ) k(δX(p) ¯ 2 is Gˆateaux-differentiable at point p0 in the direction X and Z ∞ t−1 (e−tBp Tnp , Tnp )dt δX log hε (λpn ) = −δX ε Z +∞ δX (Bp )e−tBp0 Tnp0 , Tnp0 )p0 dt. =( ε
2) By definition of hε we have: δX log hε (λpn ) = (log hε )0 (λpn )δX λpn p0
= (λpn0 )−1 e−ελn δX λpn . On the other hand δX λpn = δX < T˜np , T˜np >p = 2 < δX (τp I p )Tnp0 , T¯np0 >p0 = 2 < δX T¯np , T¯np0 >p0 +2 < τp δX I p Tnp0 , T¯np0 >p0 ¯ T¯np0 >p0 +2 < τp δX I p Tnp0 , T¯np0 >p0 = −2 < ∇T¯np0 X, ¯ T˜np0 >p0 +2 < τp δX I p Tnp0 , T¯np0 >p0 = −2 < ∇T˜np0 X, = −2λpn0 < HX U˜ np0 , U˜ np0 >p0 +2λpn0 (δX I p Tnp0 , Tnp0 )p0 , where for the third equality, we have used the fact that, X¯ being right invarip ¯ np0 , U¯ np0 >p0 ¯ = 0. Hence δX loghε (λpn ) = −2e−ελn0 < HX(p ant, [T¯np , X] ¯ 0)U p0
+2e−ελn (δX I p Tnp0 , Tnp0 )p0 , which yields 2). 3) On one hand, since the scalar product on the Lie algebra is fixed, we have δX I p∗ ⊂ (δX I p )∗ . On the other hand, since I p∗ I p = I, we have −δX I p ⊂ δX I p∗ so that the second term in the l.h.s of (3.5) vanishes.
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Definition. We shall call an orbit Op0 of a preregularised bundle an orbit of type (T ) whenever the following conditions are satisfied: 1) The map p 7→ Ipp0 is Gˆateaux-differentiable at point p0 . ∗
2) The operator δX Ipp0 e−ετp0 τp0 is trace class for any p0 ∈ P and ε > 0. ∗ 3) For any p ∈ P, tr Ipp0 e−ετp0 τp0 is Gˆateaux-differentiable at point p0 ∈ P and ∗
∗
δX tr(Ipp0 e−ετp0 τp0 ) = tr(δX Ipp0 e−ετp0 τp0 ). Whenever the Riemannian structure on G is independent of p, any orbit satisfying condition 1) is of type (T ), for in that case the traces involved in 2) and 3) vanish, δX Ipp0 being an antisymmetric operator. Proposition 3.2. Let P → P/G be a preregularisable principal fibre bundle. Then 1) Any orbit of type (T ) is preregularisable. More precisely, if Op0 is an orbit of type ε is trace class, the map (T ), for any horizontal vector X at point p0 , the operator HX p 7→ volε (Op ) is Gˆateaux differentiable in the direction X at point p0 and ∗
ε trHX − δX tr(Ipp0 e−ετp0 τp0 ) = −δX log vol0ε (Op ) Z ∗ 1 +∞ 0 tr [δX (τp∗ τp )e−tτp0 τp0 ]dt. =− 2 ε
(3.7)
2) If the Riemannian structure on G is independent of p, the orbit of any point p0 is a preregularisable orbit and Z ∗ 1 +∞ 0 ε = −δX log vol0ε (Op ) = − tr [δX (τp∗ τp )e−tτp0 τp0 ]dt, (3.7bis) trHX 2 ε where tr0 means we have restricted to the orthogonal of the kernel of τp∗0 τp0 and volε0 means that we only consider eigenvalues λpn that correspond to eigenvectors that do not belong to Ipp0 Kerτp∗0 τpo . Remarks. 1) In finite dimensions, for a compact connected Lie group acting via isometries on a Riemannian manifold P of dimension d, we have for any ε > 0 and using the various definitions of the volumes, including the µ-volume, µ ∈ R: lim δX log volε (Op ) = δX log volµreg Op
ε→0
= δX log volOp . Hence going to the limit ε → 0 on either side of (3.7 bis) we find: trHX = −δX log volOp . If the Gˆateaux-differentiability involved is a C 1 - Gˆateaux-differentiability, this yields trS p = −grad log volOp This leads to a well known result, namely (Hsiang’s theorem [H]) that the orbits of G whose volume are extremal among nearby orbits is a minimal submanifold of M .
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2) Equality (3.7) tells us that whenever the Riemannian structure on G is independent of p (as in the case of Yang-Mills theory), strongly minimal orbits of a preregularisable principal fibre bundle are pre-extremal orbits. This gives a weak (in the sense that we only get a sufficient condition for strong minimality and not for minimality) infinite dimensional version of Hsiang’s [H] theorem. 3) If both the spectrum of τp∗ τp and the Riemannian structure on G are independent of p, as in the case of Yang-Mills theory in the abelian case (where the spectrum only depends on a fixed Riemannian structure on the manifold M ), the orbits are strongly minimal (see also [MRT 1] par.5). Proof of Proposition 3.2. We set Bp = τp∗ τp . For the sake of simplicity, we assume that Bp is injective on its domain, the general case then easily follows. 1) From the preregularisability of the principal bundle follows (see Proposition 2.1) that ) is Gˆateaux-differentiable in the direction X at point p0 and the map p 7→ detε (B Zp +∞
δX log det ε (Bp ) =
dttr(δX Bp e−tBp ). On the other hand, by Lemma 3.1
ε
Z +∞ p0 1 h dt(δX Bp e−tBp )Tnp , Tnp ip − e−ελn (δX I p Tnp0 , Tnp0 )p0 2 ε ε ˜p ˜p Un , Un ip . = −hHX
(∗)
The fibre bundle being preregularisable, by the results of Proposition 2.0, the first term on the left-hand side is the general term of an absolutely convergent series. On the other hand, the orbit being of type (T ), the series with general term given p0 by e−ελn (δX I p Tnp0 , Tnp0 )p0 is also absolutely convergent. Hence the right-hand side ε of (*) is absolutely convergent and HX is trace class since (U˜ n )n∈N is a complete orthonormal basis of Imτp , Z +∞ ε dttr(δX Bp e−εBp ) = trHX − δX log Volpε 0 (Bp ) = −δX log det ε (Bp ), − ε
which then yields (3.7). 2) This follows from the above and point 3) of Lemma 3.1 and holds for any orbit Op of a regularisable fibre bundle since it does not involve δX Ip . The following proposition gives an interpretation of trµreg HX in terms of the variation of the regularised volume of the orbit. Proposition 3.3. The fibres of a regularisable principal fibre bundle with structure group equipped with a fixed (p-independent) Riemannian metric are regularisable. 1) For a given µ ∈ R, orbits are µ- minimal whenever they are µ- extremal. More precisely, for any point p0 ∈ P and any horizontal vector X at point p0 , the ε has a limit trace trµreg HX and one parameter family HX trµreg HX = −δX log volµreg (Op ) Z 1 Z ∞ m−1 . ∗ 1 X bj (p) δX Fp (t) + dt + δX t−1 δX tre−tτp τp dt − µδX b00 = 2 j t 0 1 j=−J,j6=0
(3.8)
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For µ0 ∈ R, 0
trµ HX = trµ HX + γ(µ0 − µ)δX b00 ,
(3.9)
where as before b00 = b0 − dimKerB. 2) Orbits are µ-minimal whenever Z ∞ X p 1 0(s)−1 ts−1 e−tλn δX λpn dt + (s − 1)−1 δX b00 (p)dt lim − s→1 2 0 λn 6=0
exists for any horizontal field X at point p. Furthermore, setting µ = γ, the Euler constant, we have Z ∞ X p 1 γ HX = − lim 0(s)−1 ts−1 e−tλn δX λpn dt + (s − 1)−1 δX b00 (p) . trreg 2 s→1 0 λn 6=0
(3.10) If moreover δX b00 = 0 for any horizontal vector X at point p0 , if an orbit is µ-minimal for one value of µ, it is for any value of µ. Remarks. 1) From (3.9) follows that unless δX b0 = 0, µ-minimality depends on the choice of the parameter µ. 2) In the case of a compact connected Lie group acting via isometries on a finite dimensional Riemannian manifold P of dimension d, the various notions of minimality coincide since b0 = d, volµreg (Op ) = vol(Op ) (this being the ordinary volume) and (1.10) yields: trS p = −grad log vol(Op ), where S p is the second fundamental form. It tells us that the orbits of G, the volume of which are extremal among nearby orbits is a minimal submanifold of P. This proposition therefore gives an infinite dimensional version of Hsiang’s theorem [H]. 2) A zeta function formulation of Hsiang’s theorem in infinite dimensions was already discussed in [MRT1] in the context of Yang-Mill’s theory. However, there was an obstruction due to the factor b0 (p) in the zeta-function regularisation procedure which does not appear here (see also [MRT2]). A formula similar to (3.10) (but using zeta function regularisation) can be found in [GP] (see in [GP] formula (3.17) combined with formula (A.3)). Proof of Proposition 3.3. As before, we set Bp = τp∗ τp . and we shall assume for simplicity that Bp is injective; the proof then easily extends to the case when the dimension of the kernel is locally constant on each geodesic containing p0 . 1) Since the fibre bundle is regularisable, we know by Proposition 2.1 that the map p 7→ detreg (Bp ) is Gˆateaux-differentiable in the direction X. Let us now check that ε HX has a regularized limit trace, applying Lemma 1.1. For this, we first investigate ε . By the result of Proposition 3.2, we the differentiability of the map ε 7→ trHX Z ∞ −tBp 1 1 e ε have trHX = − δX log detε (Bp ). The differentiability in ε = dtδX tr 2 ε t 2 easily follows from the shape of the middle expression.
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Setting as before Fp (t) = tre−tBp −
Pm−1
j
j=−J
bj t m , we have furthermore
∂ 1 ε trHX = − ε−1 δX tre−εBp ∂ε 2 m−1 j−m 1 Fp (ε) 1 X − = − δX δ X bj ε m . 2 ε 2 j=−J
F (ε)
From the regularisability of the fibre bundle follows that |δX pε | ≤ K for some K > 0 and 0 < ε < 1 (see assumption (2.5 bis)) which in turn implies that 1 ∂ ε trHX '0 − ∂ε 2
−1 X
j
δX bj+m ε m .
j=−J−m
ε ) in Lemma 1.1, we can define the regularised limit trace Setting f (ε) ≡ tr(HX −1
1 1 X δ X bj j 1 ε trµreg HX + µδX b0 = lim (trHX ε m + δX b0 log ε) + m ε→0 2 2 j 2 j=−J −1 X 1 bj j δX log detε (Bp ) − mδX ε m − δX b0 log ε by (3.7 bis ) = lim − ε→0 2 j j=−J −1 X 1 mbj j = lim − δX log detε (Bp ) − ε m − b0 log ε ε→0 2 j j=−J
1 = − δX log det µreg (Bp ) by (1.5) 2 Z 1 Z ∞ m−1 ∗ δX Fp (t) 1 X mδX bj + + dt t−1 δX tre−tτp τp dt] by (2.6 b). = [ 2 j t 0 1 j=−J,j6=0
R∞ P 2) It is well known that the expression 0(s)−1 0 ts−1 n e−tλn is finite for Res large enough and that it has a meromorphic continuation to the whole plane. Since 0(s) = (s − 1)0(s − 1), we have for s with large enough real part:
0(s)−1
Z
∞ s−1
t
0
X p e−tλn δX λpn dt = (s − 1)−1 n
1 = −(s − 1)−1 0(s − 1)
Z
∞ s−2
t
1 0(s − 1)
Z
∞
ts−1
0
X
p
e−tλnδX λpn dt
n
δX tre−tBp dt
0
see assumption (2.2) and Lemma 3.1
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m−1 XZ 1 j 1 = −(s − 1)−1 t m +s−2 δX bj dt + 0(s − 1) 0 j=−J ! Z Z ∞
+ 1
δX Fp (t)ts−2 dt
by (2.5)
0
= −(s − 1) Z
1
ts−2 δX tre−tBp dt +
∞
+
−1
m−1 X 1 0(s − 1)
j j=−J m
ts−2 δX tr e−tBp dt +
1
Z
1 δ X bj +s−1
1
#
ts−2 δX Fp (t)dt , 0
Pm−1 j where we have set Fp (t) = tre−εBp − j=−J bj (p)t m . Hence, since 0(s)−1 = s + γs2 + O(s3 ) around s = 0, going to the limit s → 1, we find: Z ∞ X p −1 lim [0(s) ts−1 e−tλn δX λpn dt + (s − 1)−1 δX b0 (p)] = s→1
0
n
= lim (−1 − γs + O(s2 )) s→0
Z
m−1 X j=−J,j6=0
#
1
t
+
s−1
1 δ X bj + j m +s
Z
∞
ts−1 δX tr e−tBp dt
1
δX Fp (t)dt − γδX b0
0
= δX det0reg (Bp ) − γδX b0 = −2tr
0 reg HX
by formula (1.6) (with µ = 0) and (2.6 b)
− γδX b0 ,
R∞ R∞ where lims→0 1 ts−1 δX tr e−tBp dt = 1 t−1 δX tr e−tBp dt holds using estimate R1 (*) arising in the proof of Proposition 2.0 and lims→0 0 ts−1 δX Fp (t)dt+s−1 δX b0 = R 1 −1 t δX Fp (t)dt by (2.5 bis) and using dominated convergence. 0 The rest of the assertions of 2) then easily follow.
Acknowledgement. We would like to thank Steve Rosenberg most warmly for very valuable critical comments he made on a previous version of this paper. We would also like to thank David Elworthy for his generous hospitality at the Mathematics Department of Warwick University where part of this paper was completed.
References [AMT]
[AP1] [AP2]
Abraham, R., Marsden, J.E., Ratiu, T.: Manifolds, tensor analysis and applications, Global Analysis, Pure and Applied. Modern Methods for the study of non linear phenomena in engineering, Reading, MA: Addison Wesley, 1983 Arnaudon, M., Paycha, S.: Factorization of semi-martingales on infinite dimensional principal bundles. Stochastics and Stochastic Reports, Vol. 53, 81–107 (1995) Arnaudon, M., Paycha, S.: Stochastic tools on Hilbert manifolds, interplay with geometry and physics. Commun. Math. Phys. 187, 243–260 (1997)
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[AJPS]
M. Arnaudon, S. Paycha
Albeverio, S., Jost, J., Paycha, S., Scarlatti, S.: A mathematical introduction to string theoryvariational problems, geometric and probabilistic methods. To appear, Cambridge: Cambridge University Press [BF] Bismut, J.M. et Freed, D.S: The Analysis of elliptic families I. Commun.Math.Phys. 106, 159–176 (1986) [BGV] Berline, N., Getzler, E., Vergne, M.: Heat-Kernels and Dirac Operators. Second edition, Berlin– Heidelberg–New York: Springer Verlag 1996 [FT] Fischer, A.E., Tromba, A.J.: On a purely Riemannian proof of the structure and dimension of the ramified moduli space of a compact Riemann surface. Mathematische Annalen 267, 311–345 (1984) [FU] Freed, D.S, Uhlenbeck, K.K.: Instantons on four manifolds. Berlin–Heidelberg–New York: Springer Verlag (1984) [G] Gilkey, P.B.: Invariance Theory, The heat equation and the Atiyah-Singer index theorem. Wilmington, DE: Publish or Perish, 1984 [GP] Groisser, D., Parker, T.: Semi-classical Yang-Mills theory I:Instantons. Commun. Math. Phys. 135, 101–140 (1990) [H] Hsiang, W.Y.; On compact homogeneous minimal submanifolds. Proc.Nat. Acad. Sci. USA 56, 5–6 (1966) [KR] Kondracki, W., Rogulski, J.. On the stratification of the orbit space. Dissertatione Mathematicae Polish Acad. of Sci. 250, 1–62 (1986) [KT] King, C., Terng, C.L.: Volume and minimality of submanifolds in path space. In: Global Analysis and Modern Mathematics, ed. K. Uhlenbeck, Wilmington, DE: Publish or Perish ( 1994) [MRT1] Maeda, Y., Rosenberg, S., Tondeur, P.: The mean curvature of gauge orbits. In: Global Analysis and Modern Mathematics. Ed. K. Uhlenbeck, Wilmington, DE: Publish or Perish ( 1994) [MRT2] Maeda, Y., Rosenberg, S., Tondeur, P.: Minimal orbits of metrics and Elliptic Operators. To appear in J. Geom. Phys. [MV] Mitter, P.K., Viallet, C.M.: On the bundle of connections and the gauge orbit manifold in Yang-Mills theory. Commun. Math. Phys. 79, 457–472 (1981) [P] Paycha, S.: Gauge orbits and determinant bundles: Confronting two geometric approaches. Proceedings of conference on Infinite dimensional K¨ahler manifolds. Ed. A. Huckleberry, Bael, Boston: Birkh¨auser Verlag (to appear) [PR] Paycha, S., Rosenberg, S.: Work in progress [RS] Ray, D.S., Singer, I.M.: R-torsion and the Laplacian on Riemannian manifolds. Advances in Mathematics 7, 145–210 (1971) [T] Tromba, A.J.: Teichm¨uller theory in Riemannian geometry. Basel–Boston: Birkh¨auser Verlag, 1992 Communicated by A. Jaffe
Commun. Math. Phys. 191, 663 – 696 (1998)
Communications in
Mathematical Physics c Springer-Verlag 1998
Yangian Gelfand-Zetlin Bases, glN -Jack Polynomials and Computation of Dynamical Correlation Functions in the Spin Calogero-Sutherland Model Denis Uglov Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, Japan. E-mail: [email protected] Received: 1 April 1997 / Accepted: 1 June 1997
Abstract: We consider the glN -invariant Calogero-Sutherland Models with N = 1, 2, 3, . . . in the framework of Symmetric Polynomials. In this framework it becomes apparent that all these models are manifestations of the same entity, which is the commuting family of Macdonald Operators. Macdonald Operators depend on two parameters q and t. The Hamiltonian of the glN -invariant Calogero-Sutherland Model belongs to a degeneration of this family in the limit when both q and t approach an N th elementary root of unity. This is a generalization of the well-known situation in the case of the Scalar Calogero-Sutherland Model (N = 1). In the limit the commuting family of Macdonald Operators is identified with the maximal commutative sub-algebra in the Yangian action on the space of states of the glN -invariant Calogero-Sutherland Model. The limits of Macdonald Polynomials which we call glN Jack Polynomials are eigenvectors of this sub-algebra and form Yangian Gelfand-Zetlin bases in irreducible components of the Yangian action. The glN -Jack Polynomials describe the orthogonal eigenbasis of the glN -invariant Calogero-Sutherland Model in exactly the same way as Jack Polynomials describe the orthogonal eigenbasis of the Scalar Model (N = 1). For each known property of Macdonald Polynomials there is a corresponding property of glN -Jack Polynomials. As a simplest application of these properties we compute two-point Dynamical Spin-Density and Density Correlation Functions in the gl2 -invariant Calogero-Sutherland Model at integer values of the coupling constant. 1. Introduction In this paper we study the spin generalization of the Calogero-Sutherland Model [23] which was proposed in [4] and [1]. This model describes n quantum particles with coordinates y1 , . . . , yn moving along a circle of length L (0 ≤ yi ≤ L). Each particle carries a spin with N possible values, and the dynamics of the model are governed by the Hamiltonian
664
D. Uglov
1 X ∂2 π2 + 2 2 ∂yi 2L2 n
H β,N = −
i=1
X
β(β + Pij ) , π sin2 L (yi − yj ) 1≤i6=j≤n
(1.1)
where integer β > 0 is a coupling constant and the Pij is the spin exchange operator for particles i and j. As pointed Q out in [1] itπ is convenient to make a gauge transformation (yi − yj ) and defining the gauge-transformed of (1.1) by taking W = 1≤i<j≤n sin L Hamiltonian Hβ,N by Hβ,N =
L2 −β W H β,N W β . 2π 2
(1.2)
If we set zj = exp( 2πi L yj ), then Hβ,N acts on the vector space
FN,n := C[z1±1 , . . . , zn±1 ] ⊗ (⊗n CN )
antisymm
,
(1.3)
where antisymm means total antisymmetrization. In this paper we always will be working with the gauge-transformed model which has FN,n as its space of quantum states. This space of states is a Hilbert space with a β-dependent scalar product ( · , · )β,N which we describe in Sect. 2. It is well-known that the scalar version of the model (N = 1) is best understood in the framework of symmetric polynomials. In this case the space of states F1,n is naturally isomorphic – by multiplication with the Vandermonde determinant – to the vector space of symmetric Laurent polynomials, and the orthogonal eigenbasis of Hβ,1 1 is described by Jack Polynomials with parameter β+1 in notations of Macdonald’s book [17]. Properties of Jack Polynomials known in mathematical literature (e.g. [17, 22]) are of paramount importance for the scalar Calogero-Sutherland Model, for a rather straightforward application of these properties allows to compute two-point Dynamical Correlation Functions [9, 15, 14]. The main observation of the present paper is that, from our viewpoint, the Spin Model with arbitrary N is best understood in the framework of symmetric polynomials as well. At the first glance the model with N ≥ 2 seems to have little to do with symmetric polynomials since it has two disparate types of degrees of freedom: coordinates and spins. It turns out, however, that there is an isomorphism between the Hilbert space FN,n and the space of symmetric Laurent polynomials such that an orthogonal eigenbasis of Hβ,N is described by symmetric polynomials which are natural generalizations of Jack Polynomials, and which we call glN -Jack Polynomials. These symmetric polynomials are generalizations of Jack Polynomials in the sense that they are also limiting cases of Macdonald Polynomials [17]. The Jack Polynomial is a limit of the Macdonald Polynomial when both parameters in the latter approach 1 [17]. And the glN -Jack Polynomial is a limit of the Macdonald Polynomial when both parameters in the latter approach the N th root of unity ωN = exp( 2πi N ). It is clear that for the glN -Jack Polynomials one can derive analogues of all properties known for Macdonald polynomials just by taking the limit. In precisely the same way as in the case N = 1 these properties are straightforward to apply in order to compute two-point Dynamical Correlation Functions in the Spin Calogero-Sutherland Model with arbitrary N. As an example, in this paper we give a derivation of Spin-Density and Density two-point Dynamical Correlation Functions for the case of N = 2 and non-negative integer β. Let us now outline our approach and results in a more detailed manner.
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665
1.1. Main result. As we have already mentioned, we treat the Spin Calogero-Sutherland Model with non-negative integer β by mapping it isomorphically onto a problem formulated in the language of symmetric polynomials. ±1 ±1 Sn be the vector space of symmetric To be more precise, let 3± n = (C[x1 , . . . , xn ]) Laurent polynomials in variables x1 , . . . , xn with complex coefficients. For any Laurent polynomial f = f (x1 , . . . , xn ) we will denote by [f ]1 the constant term in f. And we will use the bar to denote the complex conjugation. The vector space 3± n is equipped with a scalar product which we denote by h · , · iβ,N . This scalar product is defined in terms of the weight function Y −N β (1 − xN ) (1 − xi x−1 (1.4) 1(β, N ; x1 , . . . , xn ) := i xj j ), 1≤i6=j≤n
and its value on any two symmetric Laurent polynomials f and g is given by the formula h f , g iβ,N =
i 1 h −1 f (x−1 1 , . . . , xn )1(β, N ; x1 , . . . , xn )g(x1 , . . . , xn ) . n! 1
(1.5)
Thus the Hilbert space structure on 3± n is defined. On the other hand the space of states FN,n is also a Hilbert space with the scalar product ( · , · )β,N which we define in Sect. 2. The Hilbert space 3± n has an orthogonal basis whose elements are parameterized by an integer number r and a partition λ with length less or equal to n − 1. The elements of this basis are (1.6) (x1 · · · xn )r Pλ(N β+1,N ) (x1 , . . . , xn ), where for any positive real number γ the symmetric polynomial Pλ(γ,N ) (x1 , . . . , xn ) is defined as the following limit of the Macdonald Polynomial Pλ (q, t; x1 , . . . , xn ) (cf. [17]): Pλ(γ,N ) = Pλ(γ,N ) (x1 , . . . , xn ) = lim Pλ (ωN p, ωN pγ ; x1 , . . . , xn ). (1.7) p→1
Here ωN is an N th elementary root of unity. We will call the polynomial Pλ(γ,N ) a glN Jack Polynomial since when N = 1 this polynomial is nothing but the Jack Polynomial −1 Pλ(γ ) in notations of Macdonald [17]. Here it may be useful to observe that with γ = N β + 1 the scalar product h · , · iβ,N is just the limit of the scalar product for Macdonald Polynomials [17]. Now we formulate our main result. Main result. In the Hilbert space of states FN,n of the Spin Calogero-Sutherland Model with β > 0 there exists an orthogonal eigenbasis of the Hamiltonian Hβ,N . The elements (β,N ) are parameterized by an integer r and a partition λ with length of this eigenbasis Xr,λ less or equal to n − 1. Moreover for integer positive β there exists an isomorphism of Hilbert spaces FN,n and 3± n such that (β,N ) ) = (x1 · · · xn )r Pλ(N β+1,N ) (x1 , . . . , xn ), (Xr,λ
where Pλ(N β+1,N ) (x1 , . . . , xn ) is the glN -Jack Polynomial.
(1.8)
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Let us make two comments. When N = 1 the statement above is well-known. In this case acting with the isomorphism amounts simply to dividing a skew-symmetric Laurent polynomial by the Vandermonde determinant so that the result is a symmetric Laurent polynomial. Another comment is about the severity of the restriction that β be an integer. We need this restriction in order to be able to carry out all our proofs in a completely algebraic manner – so as not to deal with questions of convergence of various integrals. We conjecture that all our results and formulas, in particular the result above, are valid for all real positive β as well, modulo some evident modifications. We do not however have necessary proofs which would require analytical considerations. 1.2. Yangian Gelfand-Zetlin bases in the spin Calogero-Sutherland Model. Let us now (β,N ) } in the previous section. explain how we specify the eigenbasis {Xr,λ It is known from the work [1] that the space of states FN,n admits an action of the algebra Y (glN ) – the Yangian of glN [6, 20], such that this action commutes with the Hamiltonian Hβ,N . The Yangian Y (glN ) has a maximal commutative sub-algebra A(glN ) [2, 19, 20] which is generated by centers of all sub-algebras in the chain Y (gl1 ) ⊂ Y (gl2 ) ⊂ . . . ⊂ Y (glN ),
(1.9)
where Yangians Y (gl1 ) ⊂ Y (gl2 ) ⊂ . . . ⊂ Y (glN −1 ) are realized inside Y (glN ) by the standard embeddings [20] (see Sect. 3). The Hamiltonian Hβ,N is known [1] to belong to the center of the Y (glN )-action on FN,n which is generated by the Quantum Determinant. This means that the Hamiltonian belongs to the commutative family of operators A(glN ; β) which give action of the subalgebra A(glN ) on FN,n . Because of this it is natural to concentrate our attention on this commutative family rather than on the Hamiltonian alone. Doing this has two advantages. The first is that the spectrum of the commutative family A(glN ; β) is simple [24] unlike the spectrum of the Hamiltonian which has degeneracy coming from the Yangian symmetry. Therefore an eigenbasis of A(glN ; β) is defined uniquely up to normalization (β,N ) }. of eigenvectors. It is precisely the eigenbasis {Xr,λ The second advantage is that since each operator from the family A(glN ; β) is selfadjoint relative to the scalar product ( · , · )β,N [24], the elements of the eigenbasis (β,N ) {Xr,λ } are mutually orthogonal automatically because of the simplicity of the spectrum. According to [20] Yangian representations where the action of the sub-algebra A(glN ) is diagonalizable are called tame, and A(glN )-eigenbases in irreducible tame representations are called Yangian Gelfand-Zetlin bases. As was established in [24] (see also Sects.3.3 and 3.4 of the present paper) the space of states FN,n is a tame and com(β,N ) } is just a direct sum of pletely reducible Yangian representation. The basis {Xr,λ Yangian Gelfand-Zetlin bases of the irreducible components of FN,n . From the main result given in Sect. 1.1 and the above discussion it is apparent that the image of the commutative family A(glN ; β) under the isomorphism is nothing but the degeneration of the commutative family of Macdonald Operators of which Macdonald Polynomials are eigenvectors [17]. 1.3. Computation of the spin-density and density dynamical correlation functions for N = 2 . One of the consequences of our main result is that we may compute SpinDensity and Density Dynamical Correlation Functions in the Spin Calogero-Sutherland
Yangian Gelfand-Zetlin Bases, Jack Polynomials and Correlation Functions
667
Model (N ≥ 2) in practically the same way as in the Scalar Calogero-Sutherland Model (N = 1) [9, 15, 18]. The only extra work which has to be done in the case N ≥ 2 is to identify the ground state, and identify the operators on the space of symmetric Laurent polynomials that are obtained when we twist the Spin-Density and Density operators with the isomorphism . The identification of the ground state is a painstaking process of finding the state with the lowest energy eigenvalue which, however, is easy to do when N = 2 and the number of particles in the Model n is even such that n/2 is odd. In this case we find that the ground state is a basis in a one-dimensional Yangian representation and is, in particular, a spin singlet. It is mapped by the into the Laurent polynomial 1. Images of the Spin-Density and Density operators under the isomorphism are just power-sums acting as multiplication operators on the space of symmetric Laurent polynomials. In particular, when N = 2 the Spin-Density is mapped into a sum of odd power sums, and the Density is mapped into a sum of even power sums. This being established, our computation of the Correlation Functions for N = 2 follows exactly the computation for the scalar case [9, 15] with the only difference that we use the gl2 -Jack Polynomials instead of Jack Polynomials. As in the scalar case our result for e.g. the Spin-Density Correlation Function is represented as a sum over all partitions λ of length less or equal to n such that for a non-negative integer β the summand vanishes if the diagram of λ contains the square with leg-colength 1 and arm-colength 2β + 1 [17]. In the present paper we do not consider the thermodynamic limit of these Correlation Functions. Another problem which we do not consider in this paper is a computation of Green’s functions [9, 15]. This problem, however, also appears to be tractable in our approach if we take into account the Cauchy formula for glN -Jack Polynomials: X λ
Pλ(γ,N ) (x1 , . . . , xn )Pλ(γ0
−1
,N )
(y1 , . . . , yn ) =
n Y
(1 + xi yj ),
(1.10)
i,j=1
where λ0 is the partition conjugated to λ. This Cauchy formula is obviously the limit of the corresponding formula for Macdonald Polynomials [17]. 1.4. Plan of the paper. Let us now outline the plan of the present paper. In Sect. 2.1 we summarize some notational conventions to be used troughout the paper. We also define here the wedge vectors, or simply, wedges which form a basis of the space of states FN,n . The isomorphism mentioned in Sect. 1.1 will map these wedges into Schur polynomials [17], or, more precisely, into natural extensions of Schur polynomials which form a basis in the space of symmetric Laurent polynomials. In Sect. 2.2 we define an appropriate scalar product on the space FN,n . In Sect. 2.3 we describe the gauge-transformed Hamiltonian Hβ,N and recall its relation to the Cherednik-Dunkl operators following [1]. In Sect. 2.4 we construct a certain eigenbasis of Hβ,N . This eigenbasis is not orthogonal, but it plays an important role in subsequent considerations. Sections 3.1–3.4 deal with the Yangian symmetry of the Spin Calogero-Sutherland Model and the Yangian Gelfand-Zetlin bases. In Sect. 3.1 we summarize properties of the Yangian algebra giving particular attention to the maximal commutative subalgebra A(glN ). In Sect. 3.2 we recall the definition of the Yangian action in the Spin CalogeroSutherland Model following [1].
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Section 3.3 is the summary of the results of the paper [24] concerning the decomposition of the space of states FN,n into irreducible Yangian subrepresentations. In Sect. 3.4 we describe the eigenbasis of the commutative family A(glN ; β) in the (β,N ) } mentioned in the Sect. 1.1 of this introduction. space of states. This is the basis {Xr,λ Sections 4.1–4.5 are concerned with formulation of the Spin Calogero-Sutherland Model in the language of symmetric polynomials. In Sect. 4.1 we summarize notations concerning partitions. In Sect. 4.2 we define the isomorphism which was discussed in Sect. 1.1. Actually we define an infinite family of isomorphisms {K | K ∈ Z} between the space of states FN,n and the space of symmetric Laurent polynomials in variables x1 , . . . , xn . The isomorphisms K are related to each other by the trivial shift: K = (x1 . . . xn )K+1 ,
(1.11)
and the isomorphism whose existence is claimed in Sect. 1.1 is K with an arbitrary integer K. The main part of this section is the proof that each of the K is an isomorphism of Hilbert spaces i.e. that it respects scalar products. In the brief Sect. 4.3 we describe the basis in the space of symmetric Laurent polynomials obtained from the A(glN ; β)-eigenbasis by the map with the isomorphism K for any fixed K. In Sect. 4.4 we define the glN -Jack Polynomials and discuss some of their properties. In Sect. 4.5 we establish the main result of this paper which was described in Sect. 1.1. Finally, in Sects. 5.1 and 5.2 we compute the Correlation Functions. The Appendix contains proofs of some of the statements in the main text.
2. Gauge-Transformed Hamiltonian of the Spin Calogero-Sutherland Model In this part of the paper we summarize some properties of the gauge-transformed Hamiltonian Hβ,N and give the definition of the Hilbert space of states on which it acts. This part mainly follows the paper [1] and our primary objective here is to introduce our notations and to formulate the definition of the model in a way suitable for our subsequent considerations. 2.1. Preliminary remarks and notations. Let N be a positive integer. In this paper N has the meaning of the number of spin degrees of freedom of each particle in the Spin Calogero-Sutherland Model. For any integer k define the unique k ∈ {1, . . . , N } and the unique k ∈ Z by setting k = k − N k. And for a k = (k1 , k2 , . . . , kn ) ∈ Zn set k = (k1 , k2 , . . . , kn ), k = (k1 , k2 , . . . , kn ). For any sequence k = (k1 , k2 , . . . , kn ) ∈ Zn let |k| be the weight: |k| = k1 +k2 +· · ·+kn , and define the partial ordering ( the natural or the dominance ordering [17] ) on Zn by setting for any two distinct k, l ∈ Zn : iff
|k| = |l|,
k>l and k1 + · · · + ki ≥ l1 + · · · + li
for all
i = 1, 2, . . . , n.
(2.1)
n For r ∈ N let L(r) n be a subset of Z defined as n L(r) n = {k = (k1 , k2 , . . . , kn ) ∈ Z | ki ≥ ki+1
and
∀s ∈ Z #{ki | ki = s} ≤ r}. (2.2)
Yangian Gelfand-Zetlin Bases, Jack Polynomials and Correlation Functions
669
(1) In particular the L(n) n is the set of non-increasing sequences of n integers and the Ln is the set of strictly decreasing sequences, i.e. such k = (k1 , k2 , . . . , kn ) ∈ Zn that ki > ki+1 . Let V = CN with the basis {v1 , v2 , . . . , vN } and let V (z) = C[z ±1 ]⊗V with the basis {uk | k ∈ Z}, where uk = z k ⊗ vk . For monomials in vector spaces C[z1±1 , . . . , zn±1 ], ⊗n V and ⊗n V (z) = C[z1±1 , . . . , zn±1 ] ⊗ (⊗n V ) we will use the convention of multiindices:
z t = z1t1 z2t2 · · · zntn , v(a) = va1 ⊗ va2 ⊗ · · · ⊗ van , u k = uk 1 ⊗ u k 2 ⊗ · · · ⊗ uk n ,
t = (t1 , t2 , . . . , tn ) ∈ Zn ; a = (a1 , a2 , . . . , an ) ∈ {1, . . . , N }n ; k = (k1 , k2 , . . . , kn ) ∈ Zn .
(2.3) (2.4) (2.5)
Let Kij be the permutation operator for variables zi and zj in C[z1±1 , . . . , zn±1 ] ( operator of coordinate permutation ), and let Pij be the operator exchanging ith and j th factors in the tensor product ⊗n V (operator of spin permutation ). Let An be the antisymmetrization operator in ⊗n V (z): X An (uk1 ⊗ uk2 ⊗ · · · ⊗ ukn ) = sign(w)ukw(1) ⊗ ukw(2) ⊗ · · · ⊗ ukw(n) , (2.6) w∈Sn
where Sn is the symmetric group of order n. We will use the notation uˆ k = uk1 ∧ uk2 ∧ · · · ∧ ukn for a vector of the form (2.6), and will call such a vector a wedge. A wedge uˆ k is normally ordered if k ∈ L(1) n , that is k1 > k2 > . . . > kn . Let FN,n be the image of the operator An in ⊗n V (z). Then FN,n is spanned by wedges and the normally ordered wedges form a basis in FN,n . Equivalently the vector space FN,n is defined as the linear span of all vectors f ∈ C[z1±1 , . . . , zn±1 ] ⊗ (⊗n V ) such that for all 1 ≤ i 6= j ≤ n: Kij f = −Pij f.
(2.7)
This is the definition of FN,n adopted in [1]. 2.2. Scalar product. Here we define a scalar product on the space of states FN,n . Our definition has three steps. First we define scalar products on the vector spaces ⊗n V and C[z1±1 , . . . , zn±1 ] separately. Then we define a scalar product on the tensor product ⊗n V (z) = C[z1±1 , . . . , zn±1 ] ⊗ (⊗n V ). Finally we define a scalar product on FN,n considered as a subspace of ⊗n V (z). On ⊗n V define a sesquilinear, i.e., anti-linear in the first argument and linear in the second argument, scalar product ( · , · )N by requiring that pure tensors in ⊗n V be orthonormal: ( v(a) , v(b) )N = δab , a, b ∈ {1, . . . , N }n . (2.8) For w = (w1 , w2 , . . . , wn ) ∈ Cn , |w1 | = |w2 | = . . . = |wn | = 1 and a non-negative real number δ let: Y 1(w; δ) = (1 − wi wj−1 )δ , (2.9) 1≤i6=j≤n
and define for all f (z), g(z) ∈
C[z1±1 , . . . , zn±1 ]
( f (z) , g(z) )0δ =
1 Y n! n
j=1
Z
a scalar product ( · , · )0δ by setting :
dwj 1(w; δ)f (w)g(w), 2πiwj
(2.10)
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D. Uglov
where the integration over each of the variables wj is taken along the unit circle in the complex plane, and the bar over f (w) means complex conjugation. On the linear space ⊗n V (z) = C[z1±1 , . . . , zn±1 ] ⊗ (⊗n V ) we define a scalar product ( · , · )0δ,N as the composition of the scalar products (2.10) and (2.8), i.e. for f (z), g(z) ∈ C[z1±1 , . . . , zn±1 ]; u, v ∈ ⊗n V we set: ( f (z) ⊗ u , g(z) ⊗ v )0δ,N = ( f (z) , g(z) )0δ ( u , v )N ,
(2.11)
and extend the definition on all vectors by requiring that ( · , · )0δ,N be sesquilinear. Finally, on the subspace FN,n ⊂ ⊗n V (z) a scalar product ( · , · )δ,N is defined as the restriction of the scalar product ( · , · )0δ,N . Note that the normally ordered wedges are orthonormal relative to this scalar product when δ = 0: ( uˆ k , uˆ l )0,N = δkl ,
k, l ∈ L(1) n .
(2.12)
2.3. The gauge-transformed Hamiltonian. Here we briefly recall the relationship between the gauge-transformed Hamiltonian and the Cherednik-Dunkl operators [3, 4, 1]. We will use the following convention: if B is an operator acting on C[z1±1 , . . . , zn±1 ] ( or ⊗n V ) then we will denote by the same letter B the operator B ⊗ id ( or id ⊗ B ) acting on the space C[z1±1 , . . . , zn±1 ] ⊗ (⊗n V ). Q π (yi − yj ) and set zj = exp( 2πi Let W = 1≤i<j≤n sin L L yj ). Then the gaugetransformed Hamiltonian Hβ,N is defined as follows [1]: L2 −β W H β,N W β 2π 2 n n X X 2 = Di + β (2i − n − 1)Di
Hβ,N =
i=1
+ 2β
X
(2.13)
i=1
1≤i<j≤n
β 2 n(n2 − 1) , θij Di − Dj + θji (Pij + 1) + 12
where we set: Di = zi ∂/∂zi and θij = zi /(zi − zj ). The Hamiltonian Hβ,N acts in the space FN,n [1]. To see this let us recall, that the Cherednik-Dunkl operators [3, 7]: X X θji (Kij − 1) − θij (Kij − 1), (i = 1, 2, . . . , n), (2.14) di (β) = β −1 Di − i + i<j
i>j
act on C[z1±1 , . . . , zn±1 ] and together with permutations Kij satisfy the defining relations of the degenerate affine Hecke algebra: Kii+1 di (β) − di+1 (β)Kii+1 = 1, Kii+1 dj (β) = dj (β)Kii+1 , (j 6= i, i + 1), di (β)dj (β) = dj (β)di (β).
(2.15) (2.16) (2.17)
These relations imply, in particular, that symmetric polynomials in Cherednik-Dunkl operators commute with permutations Kij and therefore are well-defined operators on the space FN,n . The action of the operator
Yangian Gelfand-Zetlin Bases, Jack Polynomials and Correlation Functions
β2
n X
di (β) +
i=1
+ 2β
X
n+1 2 θij
1≤i<j≤n
2 =
n X i=1
Di2 + β
n X
671
(2i − n − 1)Di
i=1
β 2 n(n2 − 1) Di − Dj − θji (Kij − 1) + 12
(2.18)
on any vector f ∈ FN,n is seen to coincide with the action of the Hamiltonian Hβ,N since Kij f = −Pij f . Thus we may write: 2 n X n+1 2 Hβ,N = β (2.19) di (β) + 2 i=1
as operators on FN,n . Note that from the last equation it follows that the Hβ,N is self-adjoint relative to the scalar product ( · , · )β,N since the Cherednik-Dunkl operators are self-adjoint relative to the scalar product ( · , · )0β as one can easily verify. 2.4. An eigenbasis of the Hamiltonian Hβ,N . In this section we construct a certain eigenbasis of the Hamiltonian Hβ,N . This eigenbasis is not orthogonal with respect to the scalar product ( · , · )β,N . However it has a triangular expansion in the basis of wedges. This is an important property to be used later in Sect. 3. Let k = (k1 , k2 , . . . , kn ) ∈ L(1) n . Note that k1 > k2 > . . . > kn implies, in particular, that k1 ≤ k2 ≤ . . . ≤ kn . And let us act with the Hamiltonian Hβ,N on the (normally ordered) wedge uˆ k = uk1 ∧ uk2 ∧ . . . ∧ ukn . Using the expression (2.18) we find: X hij uˆ k , (2.20) Hβ,N uˆ k = E(k; β)uˆ k + 2β 1≤i<j≤n
where
Pn Pn 2 2 2 −1) E(k; β) = i=1 ki + β i=1 (2i − n − 1)ki + β n(n , 12 and hij (uk1 ∧ . . . ∧ uki ∧ . . . ∧ ukj ∧ . . . ∧ ukn ) =
=
Pkj −ki −1 r=1
(2.21)
(kj − ki − r)(uk1 ∧. . .∧uki −N r ∧. . .∧ukj +N r ∧. . .∧ ukn ). (2.22)
Normally ordering the wedges in the right-hand side of (2.20) we find from (2.22) that: X h(β) ˆ l, (2.23) Hβ,N uˆ k = E(k; β)uˆ k + kl u l∈L(1) n , l>k, l
with certain real coefficients h(β) kl . Now recall from [17] that for positive β we have: E(k; β) 6= E(l; β) when l > k. Then (2.23) leads to: Proposition 1. (β) For any k ∈ L(1) n there is a unique eigenvector 9k of Hβ,N such that: (2.24) X (β) (β) ˆk + ψkl uˆ l (ψkl ∈ R). 9(β) k =u l∈L(1) n , l>k
Eigenvalue of Hβ,N for this eigenvector is E(k; β). The coefficient
(β) ψkl
vanishes unless l < k.
(2.25) (2.26)
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D. Uglov
Note that for any integer M the wedge: vac(M ) = uM ∧ uM −1 ∧ · · · ∧ uM −n+1
(2.27)
is an eigenvector of the Hamiltonian Hβ,N as implied by either (2.24) or (2.20, 2.22). We will call any vector of the form (2.27) a vacuum vector. As we have mentioned already, the eigenbasis {9(β) k } is not orthogonal. To construct an orthogonal eigenbasis we need to utilize the Yangian symmetry of the model as discussed in the next part of the paper. 3. Yangian Gelfand-Zetlin Bases and an Orthogonal Eigenbasis of the Spin Calogero-Sutherland Model Our objective in this part of the paper is to construct an orthogonal eigenbasis of the Spin Calogero-Sutherland Model. As was shown in the work [24] to do this it is natural to use the Yangian symmetry of the model. The orthogonal eigenbasis then is defined uniquely up to normalization as the eigenbasis of the commutative family of operators which give the action of the maximal commutative subalgebra in the Yangian action on the space of states FN,n . 3.1. The Yangian of glN and its maximal commutative subalgebra. The Yangian Y (glN ) (s) , where a, b ∈ [6] is an associative unital algebra with generators: the unit 1 and Tab {1, . . . , N } and s = 1, 2, . . . . In terms of the formal power series in variable u−1 : (1) (2) Tab (u) = δab 1 + u−1 Tab + u−2 Tab + ··· ,
(3.1)
the relations of Y (glN ) are written as follows (u − v)[Tab (u), Tcd (v)] = Tcb (v)Tad (u) − Tcb (u)Tad (v),
(3.2)
and the coproduct 1 : Y (glN ) → Y (glN ) ⊗ Y (glN ) is given by: 1(Tab (u)) = PN c=1 Tac (u) ⊗ Tcb (u). The center of Y (glN ) is generated by coefficients of the series X sign(w)T1w(1) (u)T2w(2) (u − 1) · · · TN w(N ) (u − N + 1) (3.3) AN (u) = w∈SN
called the Quantum Determinant of Y (glN ). The Yangian has a distinguished maximal commutative subalgebra A(glN ) [2, 19, 20]. This subalgebra is generated by coefficients of the series A1 (u), A2 (u), . . . , AN (u) defined as follows: X sign(w)T1w(1) (u)T2w(2) (u − 1) · · · Tmw(m) (u − m + 1), Am (u) = (3.4) w∈Sm m = 1, 2, . . . , N. That is to say by the centers of all algebras in the chain: Y (gl1 ) ⊂ Y (gl2 ) ⊂ . . . ⊂ Y (glN ),
(3.5)
where for m = 1, 2, . . . , N − 1 the Y (glm ) is realized inside Y (glN ) as the subalgebra generated by coefficients of the series Tab (u) with a, b = 1, 2, . . . , m.
Yangian Gelfand-Zetlin Bases, Jack Polynomials and Correlation Functions
673
Finite or infinite dimensional Y (glN )-modules with a semisimple action of the subalgebra A(glN ) are called tame, and eigenbases of A(glN ) in irreducible tame modules are called (Yangian) Gelfand-Zetlin bases [20]. Let us now consider certain tame Yangian modules which appear in the context of the Spin Calogero-Sutherland Model. Let f ∈ C and π(f ) : Y (glN ) → End(V ) be the Y (glN )-homomorphism defined by: π(f )(Tab (u)) = δab 1 +
Eba , u+f
(3.6)
where Eba ∈ End(V ) is the matrix unit: Eba vc = δac vb . Also for f = (f1 , f2 , . . . , fn ) ∈ Cn we will denote by π(f ) the tensor product of the homomorphisms (3.6): π(f ) = π(f1 ) ⊗ π(f2 ) ⊗ · · · ⊗ π(fn ) : ⊗n Y (glN ) → End(⊗n V ), so that π(f ) 1(n) (Tab (u)) , where 1(n) is the coproduct iterated n − 1-times, defines a Y (glN )-module structure on ⊗n V . Let now M be an integer ( unrelated to the M in (2.27)) and let p = (p1 , p2 , . . . , pM ) ∈ {1, . . . , N }M be a sequence of positive integers such that: n = p1 + p2 + · · · + pM .
(3.7)
With these (p1 , p2 , . . . , pM ) define integers q0 , q1 , . . . , qM by ps = qs − qs−1 ,
(q0 := 0),
(s = 1, 2, . . . , M ).
(3.8)
For 1 ≤ i < j ≤ n define the partial anti-symmetrization operator A(i,j) ∈ End(⊗n V ) by setting for a = (a1 , . . . , an ) ∈ {1, . . . , N }n : A(i,j) (va1 ⊗ va2 ⊗ · · · ⊗ van ) := X sign(w)va1 ⊗ va2 ⊗ · · · ⊗ vai ⊗ vai+w(1) ⊗ vai+w(2) ⊗
(3.9)
w∈Sj−i
· · · ⊗ vai+w(j−i) ⊗ vaj+1 ⊗ · · · ⊗ van . And let (⊗n V )p be the image in ⊗n V of the operator: Ap :=
M Y
A(qs−1 ,qs ) .
(3.10)
s=1
If we define the set Tp labeled by the sequence p by
then the set
Tp := {a = (a1 , . . . , an ) ∈ {1, . . . , N }n | ai < ai+1 when qs−1 < i < qs for each s = 1, . . . , M },
(3.11)
{ϕ(a) := Ap v(a) | a ∈ Tp }
(3.12)
n
is a basis of (⊗ V )p . Further, let f = (f1 , f2 , . . . , fn ) be a sequence of real numbers satisfying the following two conditions: fi = fi+1 + 1 when qs−1 < i < qs for each s = 1, 2, . . . , M ; and fqs > fqs +1 + 1 for s = 1, 2, . . . , M − 1. Then we have:
(C1) (C2)
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D. Uglov
Proposition 2. The coefficients of π(f )1(n) (Tab (u)) ∈ End(⊗n V )[[u−1 ]] leave the subspace (⊗n V )p ⊂ ⊗n V invariant. In (⊗n V )p there is a unique up to normalization of eigenvectors π(f )A(glN ) − eigenbasis: {χ(a) | a ∈ Tp }. X c(a, b)ϕ(b), c(a, b) ∈ R. χ(a) = ϕ(a) +
(3.13) (3.14) (3.15)
b>a
π(f )1(n) (Am (u))χ(a) = Am (u; f ; a)χ(a), m = 1, 2, . . . , N ; n Y u + fi + δ(ai ≤ m) where Am (u; f ; a) := . u + fi
(3.16)
i=1
The N -tuples of rational functions in u: A1 (u; f ; a), . . . , AN (u; f ; a) (3.17) are distinct for distinct a ∈ Tp . In other words: the spectrum of the π(f )A(glN ) on (⊗n V )p is simple. In the expression for the eigenvalue Am (u; f ; a) above we have used the convention that for a statement P, δ(P ) = 1 if P is true, and δ(P ) = 0 otherwise. We will often use this convention in this paper. We give a proof of this proposition in the Appendix, Sect. A. 3.2. Yangian in the Spin Calogero-Sutherland Model. The space FN,n admits a Y (glN )i−1 n−i (i) = 1⊗ ⊗ Eab ⊗ 1⊗ ∈ End(⊗n V ), and let action defined as follows [1]: let Eab L(i) ab (u; β) = δab 1 +
(i) Eab . u + di (β)
(3.18)
(n) (2) Set Tab (u; β) = L(1) ac1 (u; β)Lc1 c2 (u; β) · · · Lcn−1 b (u; β), where summation over the indices ci is assumed. The degenerate affine Hecke algebra relations satisfied by the Cherednik-Dunkl operators imply that coefficients of the series Tab (u; β) act on FN,n and satisfy the defining relations of the Yangian [1]. Denote by Y (glN ; β) the Y (glN )action on FN,n defined by the Tab (u; β) and denote by A(glN ; β) the corresponding action of the subalgebra A(glN ). In particular the Quantum Determinant of Tab (u; β) has the form [1]: n Y u + 1 + di (β) , (3.19) AN (u; β) = u + di (β) i=1
and therefore the Hamiltonian Hβ,N (2.3) is an element in the center of the action Y (glN ; β) and hence an element in A(glN ; β). For an operator B acting on FN,n let B ∗ be its adjoint relative to the scalar product ( · , · )β,N and for a series B(u) with operator-valued coefficients we will use B(u)∗ to denote the series whose coefficients are adjoints of coefficients of the series B(u). In [24] it is shown, that generators of the action Y (glN ; β) satisfy the following relations: (3.20) Tab (u; β)∗ = Tba (u; β). These relations imply, in particular, that the action of the subalgebra A(glN ; β) is selfadjoint. That is we have [24]: Am (u; β)∗ = Am (u; β),
m = 1, 2, . . . , N.
(3.21)
Yangian Gelfand-Zetlin Bases, Jack Polynomials and Correlation Functions
675
3.3. Decomposition of the space of states with respect to the Yangian action Y (glN ; β). Here we recall the decomposition of the space of states FN,n into irreducible Yangian representations with respect to the Yangian action Y (glN ; β). For details one may consult the work [24] which contains proofs of the statements made in this section. The Yangian decomposition of the space of states FN,n is constructed by using the Non-symmetric Jack Polynomials (see [21, 5, 16]). Non-symmetric Jack polynomials Et(β) (z) are labelled by t = (t1 , . . . , tn ) ∈ Zn and form a basis of C[z1±1 , . . . , zn±1 ]. For any t ∈ Zn let t+ denote the element of the set L(n) n (2.2) obtained by arranging parts of t in non-increasing order. The polynomials Et(β) (z) have the triangular expansion in the monomial basis: X (β) etr z r , (3.22) Et(β) (z) = z t + r≺t
where e(β) tr are real coefficients and + t > r+ or r≺t ⇔ t+ = r+ and the last non-zero difference ti − ri is negative. Moreover, polynomials Et(β) (z) are eigenvectors of the Cherednik-Dunkl operators: di (β)Et(β) (z) = fi (t; β)Et(β) (z),
(i = 1, 2, . . . , n),
−1
where fi (t; β) = β ti − ρi (t), and ρi (t) = #{j ≤ i | tj ≥ ti } + #{j > i | tj > ti }.
(3.23) (3.24) (3.25)
In [24] it is shown that the space FN,n splits into an infinite number of irreducible ) Yangian submodules Fs labelled by elements of the set L(N n (2.2): ) Fs . FN,n = ⊕s∈L(N n
(3.26)
) To describe the component Fs which corresponds to a given s ∈ L(N let M be the n number of distinct elements in the sequence s = (s1 , s2 , . . . , sn ). And let q0 , q1 , . . . , qM be defined by: q0 = 0, qM = n; sqj > sqj +1 , j = 1, 2, . . . , M − 1. As in (3.8) a sequence p(s) = (p1 , p2 , . . . , pM ) is defined by: pj = qj − qj−1 , j = 1, 2, . . . , M . Clearly we have p1 + p2 + · · · + pM = n. As a Y (glN )-module the space Fs is isomorphic to (⊗n V )p(s) (3.1) where the Yangian action is given by π(f (s))1(n) (Tab (u)) and f (s) = (f1 , f2 , . . . , fn ) with fi = fi (s; β). This isomorphism is explicitly given by the operator U (s; β) : (⊗n V )p(s) → Fs which is defined for any v ∈ (⊗n V )p(s) as follows: X (β) U (s; β)v = Et (z) ⊗ Rt(β) v, (3.27) t∼s
where the sum is taken over all distinct rearrangements t of s, and Rt(β) ∈ End(⊗n V ) is defined by the recursive realtions: Rs(β) = 1, (β) Rt(i,i+1)
= −Rˇ ii+1 (fi (t; β) − fi+1 (t; β))Rt(β)
(3.28) for
ti > ti+1 .
(3.29)
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D. Uglov
Here Rˇ ii+1 (u) = u−1 + Pii+1 and t(i, i + 1) denotes the element of Zn obtained by interchanging ti and ti+1 in t = (t1 , . . . , tn ). Observe now that, since β > 0, the conditions (C1,C2) are satisfied by the sequence f (s). Therefore setting for all a ∈ Tp(s) : 8(β) (s, a) := U (s; β)ϕ(a), X (β) (s, a) := U (s; β)χ(a);
(3.30) (3.31)
from (3.14 - 3.17) and the fact that Fs is isomorphic to (⊗n V )p(s) we obtain the following: Proposition 3. {X (β) (s, a) | a ∈ Tp(s) } is the unique up to normalization of eigenvectors (3.32) A(glN ; β) − eigenbasis of Fs . X X (β) (s, a) = 8(β) (s, a) + c(a, b)8(β) (s, b), c(a, b) ∈ R. (3.33) b>a
(s, a) = Am (u; f (s); a)X (β) (s, a), m = 1, 2, . . . , N ; n Y u + fi (s; β) + δ(ai ≤ m) Am (u; f (s); a) := . u + fi (s; β)
Am (u; β)X where
(β)
(3.34)
i=1
The N -tuples of rational functions in u:A1 (u; f (s); a), . . . , AN (u; f (s); a) (3.35) are distinct for distinct a ∈ Tp(s) . Thus the set {X (β) (s, a) | a ∈ Tp(s) } is a Yangian Gelfand-Zetlin basis of the irreducible Yangian representation Fs . 3.4. A(glN ; β)-eigenbasis of the space of states FN,n . Here we define the orthogonal eigenbasis of the commutative family A(glN ; β) in the entire space of states FN,n . This eigenbasis is just the union of bases {X (β) (s, a) | a ∈ Tp(s) } taken over all elements s ) from the set L(N n . We prefer, however, to choose a different parameterization of elements in this basis, such that a triangularity of these elements expanded in the basis of normally ordered wedges becomes manifest. For any k = (k1 , k2 , . . . , kn ) ∈ L(1) n the sequence s = (kn , kn−1 , . . . , k1 ) is an element ) of the set L(N . And the sequence a = (kn , kn−1 , . . . , k1 ) belongs to the set Tp(s) defined n by the sequence s as in the previous section. With these k, s and a we have: (−1)
n(n−1) 2
8(β) (s, a) = 9(β) k ,
(3.36)
where the 9(β) k is the eigenvector of the Hamiltonian Hβ,N defined in (2.24). A proof of the equality (3.36) is contained in the Appendix, Sect. B. Now with the same k, s and a as above let us define: Xk(β,N ) := (−1)
n(n−1) 2
X (β) (s, a).
Then we have the following statements about the vectors Xk(β,N )
(3.37)
Yangian Gelfand-Zetlin Bases, Jack Polynomials and Correlation Functions
677
Proposition 4. {Xk(β,N ) | k ∈ L(1) n } is the unique up to normalization of eigenvectors (3.38) A(glN ; β) − eigenbasis of FN,n . X x(β) ˆ l , x(β) (3.39) Xk(β,N ) = uˆ k + kl u kl ∈ R. l∈L(1) n , l
Am (u; β)Xk(β,N ) where
= Am (u; β; k)Xk(β,N ) , m = 1, 2, . . . , N ; n Y u + β −1 ki + i − n − 1 + δ(ki ≤ m) Am (u; β; k) := . u + β −1 ki + i − n − 1 i=1
(3.40)
The N -tuples of rational functions in u: A1 (u; β; k), . . . , AN (u; β; k) (3.41) are distinct for distinct k ∈ L(1) n . ( Xk(β,N ) , Xl(β,N ) )β,N = 0
for
k 6= l.
(3.42)
Proof. The statement (3.38) is a direct consequence of (3.32), of the decomposition (3.26) and of the easily verified fact that the correspondence between the set of all pairs ) (1) (s, a) where s ∈ L(N n , a ∈ Tp(s) , and the set of strictly decreasing sequences Ln given by (N ) k = (k1 , k2 , . . . , kn ) ∈ L(1) n → s = (kn , kn−1 , . . . , k1 ) ∈ Ln , (3.43) a = (kn , kn−1 , . . . , k1 ) ∈ Tp(s) is one-to-one. The triangularity of Xk(β,N ) expressed by (3.39) is established as follows. In view of (3.33) and (3.36) we have X (β) c(β) (c(β) (3.44) Xk(β,N ) = 9(β) k + kl 9l kl ∈ R). l∈L(1) n , l=k, l
Now observe that l = k, l < k imply l = k, l < k. And therefore (3.39) follows from the triangularity of eigenvectors 9(β) l expressed by (2.24) and (2.26). Equation (3.40) follows immediately from (3.34) and the explicit expression for the eigenvalues of the Cherednik-Dunkl operators (3.24). The simplicity of the spectrum of the A(glN ; β) expressed by the statement (3.41) is proved as follows. First of all, (3.35) shows that the spectrum of A(glN ; β) is simple on each irreducible component Fs of the Yangian action Y (glN ; β). Next, by a straightforward modification of the proof of the statement (3.17) which is contained in the Appendix, Sect. A, one shows that the spectrum of the Quantum Determinant AN (u; β) separates between these irreducible components. Finally, the orthogonality (3.42) is a consequence of (3.41) and the self-adjointness (3.21) of A(glN ; β) relative to the scalar product ( · , · )β,N . The orthogonal eigenbasis {Xk(β,N ) | k ∈ L(1) n } is precisely the eigenbasis of FN,n that was described in Sect. 1.1. To see this we simply have to observe that if we fix (n) an integer M and define for any k ∈ L(1) n a sequence σ = (σ1 , σ2 , . . . , σn ) ∈ Ln by ki = σi + M − i + 1, then the correspondence between this k and the pair r = σn , λ = (σ1 − σn , σ2 − σn , . . . , σn−1 − σn , 0) (3.45)
678
D. Uglov
is bijective. In other words we may label the elements of the basis {Xk(β,N ) | k ∈ L(1) n } by a pair which consists of an integer r and a partition λ with length less or equal to n − 1. This is the parameterization adopted in Sect. 1.1 of the Introduction. 4. Spin Calogero-Sutherland Model in the Language of Symmetric Polynomials Starting with this section we move into the realm of symmetric polynomials so that many objects we are about to encounter will be parameterized by partitions. Below we summarize partition-related notations we are going to use. Mainly we will conform to the notations of the book [17]. 4.1. Notations. For a partition λ = (λ1 , λ2 , . . . ) we will denote by l(λ) the length, i.e. the number of non-zero parts λ1 , λ2 , . . . in λ. All partitions that appear in this paper have length less or equal to n which is the number of particles in the Spin Calogero-Sutherland Model. We will identify a partition λ with its diagram defined as the set { (i, j) | 1 ≤ i ≤ l(λ), 1 ≤ j ≤ λi },
(4.1)
graphically represented as a collection of squares with coordinates (i, j), where i is increasing downward and j is increasing from left to right as on Fig. 1, which represents λ = (6, 4, 4, 3, 1).
Fig. 1. Partition λ = (6, 4, 4, 3, 1)
For a square s ∈ λ arm-length aλ (s), leg-length lλ (s), arm-colength a0 (s) and legcolength l0 (s) are defined as the number of squares in the diagram of λ to the east, south, west and north from s respectively. Also for a square s we define Content: c(s) = a0 (s) − l0 (s). Hook-length: hλ (s) = aλ (s) + lλ (s) + 1. And for a real number γ their refinements: h∗λ (s; γ)
c(s; γ) = a0 (s) − γl0 (s), = aλ (s) + γlλ (s) + 1, hλ∗ (s; γ) = aλ (s) + γlλ (s) + γ.
(4.2) (4.3)
For a positive integer N which as before has the meaning of the number of spin degrees of freedom in the Spin Calogero-Sutherland Model we define the following two subsets of λ:
Yangian Gelfand-Zetlin Bases, Jack Polynomials and Correlation Functions
CN (λ) = {s ∈ λ | c(s) ≡ 0 mod N }, HN (λ) = {s ∈ λ | hλ (s) ≡ 0 mod N }.
679
(4.4) (4.5)
And for any subset of squares S ⊂ λ we denote by |S| the number of squares in S. 4.2. An isomorphism between the space of states of the Spin Calogero-Sutherland Model and the space of symmetric Laurent polynomials. Here we discuss the main technical points of this paper – the definition and properties of the isomorphism between the space of states of the Spin Calogero-Sutherland Model and the space of symmetric Laurent polynomials. To avoid having to deal with questions of convergence of various integrals, we assume from now and until the end of this paper that β is a non-negative integer number. With this assumption it will be possible to keep our discussion completely algebraic. ±1 Let Ln = C[x±1 1 , . . . , xn ] be the C-algebra of Laurent polynomials in variables x1 , . . . , xn . If f = f (x1 , . . . , xn ) ∈ Ln let f (x1 , . . . , xn ) be the Laurent polynomial −1 with complex conjugated coefficients, let f ∗ = f (x−1 1 , . . . , xn ), and let [f ]1 denote the constant term in f . Let A± n be the subspace of skew-symmetric Laurent polynomials in Ln . For l = (l1 , . . . , ln ) ∈ Zn define the antisymmetric monomial aˆ l analogous to the wedge vector (2.6) as follows: X l l aˆ l = xl1 ∧ xl2 ∧ · · · ∧ xln = sign(w)x1w(1) x2w(2) · · · xlnw(n) . (4.6) w∈Sn ± Then {ˆal | l ∈ L(1) n } is a basis of An . Now let us introduce
˜ 1(β, N ) :=
Y
−N β (1 − xN ) . i xj
(4.7)
1≤i6=j≤n
˜ Since β is a non-negative integer, 1(β, N ) is a symmetric Laurent polynomial. We define a scalar product hh · , · iiβ,N on Ln by setting for f, g ∈ Ln : hh f , g iiβ,N =
1 ∗˜ [f 1(β, N )g]1 . n!
(4.8)
Let us use hh f , g ii0 as a short-hand notation for hh f , g ii0,N , then we have ˜ hh f , g iiβ,N = hh f , 1(β, N )g ii0 ,
(4.9)
˜ where we regard the 1(β, N ) as a multiplication operator on Ln . When β is a non-negative integer, the definition of the scalar product ( · , · )0β (2.10) may be formulated as follows: for f, g ∈ C[z1±1 , . . . , zn±1 ]: ( f , g )0β =
1 ∗ [f 1(β)g]1 , n!
(4.10)
Q where 1(β) = 1≤i6=j≤n (1 − zi zj−1 )β is now a symmetric Laurent polynomial. Then for the scalar product ( · , · )β,N on the space FN,n we have with f, g ∈ FN,n : ( f , g )β,N = ( f , 1(β)g )0,N ,
(4.11)
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where we consider the 1(β) as a multiplicaltion operator on the space FN,n . Note that in the last formula we have 1(β)g ∈ FN,n since 1(β) is a symmetric Laurent polynomial. n 0 ˆ k ) = aˆ k . Clearly Let now ω 0 : FN,n → A± n be a C-map defined for k ∈ Z by ω (u this map is an isomorphism of linear spaces. Moreover for any f, g ∈ FN,n we have: ( f , g )0,N = hh ω 0 (f ) , ω 0 (g) ii0 ,
(4.12)
( uˆ k , uˆ l )0,N = δkl = hh aˆ k , aˆ l ii0
(4.13)
for if k, l ∈ L(1) n then
which shows (4.12) since {uˆ k | k ∈ L(1) ak | k ∈ L(1) n } and {ˆ n } are bases of the spaces ± FN,n and An respectively. Now we are ready to formulate one of the crucial technical points of this paper, which is the following relation between the scalar product ( · , · )β,N (Sect. 2.2) on the space of states FN,n of the gauge-transformed Spin Calogero-Sutherland Model and the scalar product (4.8) on the space of skew-symmetric Laurent polynomials A± n: Proposition 5. ( f , g )β,N = hh ω 0 (f ) , ω 0 (g) iiβ,N .
For any f, g ∈ FN,n we have
(4.14)
Proof. Let P = P (z1 , . . . , zn ) ∈ C[z1±1 , . . . , zn±1 ] be a symmetric Laurent polynomial. Then for any f ∈ FN,n the following relation holds: −N 0 ω 0 (P (z1 , . . . , zn )f ) = P (x−N 1 , . . . , xn )ω (f ).
(4.15)
Indeed, the algebra (C[z1±1 , . . . , zn±1 ])Sn of symmetric Laurent polynomials in variables z1 , . . . , zn is generated by the power-sums: pr = pr (z1 , . . . , zn ) =
n X
zir ,
(r = 0, 1, 2, . . . )
(4.16)
i=1
and the element
z1−1 z2−1 · · · zn−1 .
(4.17)
So it is enough to show (4.15) when P equals to one of the Laurent polynomials (4.16), (4.17). Also, since uˆ k , k ∈ Zn span FN,n we may assume that in (4.15) f = uˆ k . Calculating: ! n X 0 0 ω (pr uˆ k ) = ω uk1 ∧ · · · ∧ uki −N r ∧ · · · ∧ ukn = (4.18) i=1
=
n X
−N xk1 ∧ · · · ∧ xki −N r ∧ · · · ∧ xkn = pr (x−N ak ; 1 , . . . , xn )ˆ
i=1
and =x
ω 0 (z1−1 z2−1 · · · zn−1 uˆ k ) = ω 0 (uk1 +N ∧ uk2 +N ∧ · · · ∧ ukn +N ) = k1 +N
∧x
k2 +N
∧ ··· ∧ x
kn +N
=
N xN 1 x2
(4.19)
· · · xN ˆ k, na
we confirm (4.15). Now since 1(β) is a symmetric Laurent polynomial, we have for any f, g ∈ FN,n :
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681
hh ω 0 (f ) , ω 0 (g) iiβ,N = (by 4.9) ˜ N )ω 0 (g) ii0 = (by 4.15) = = hh ω 0 (f ) , 1(β, = hh ω 0 (f ) , ω 0 (1(β)g) ii0 = (by 4.12) = ( f , 1(β)g )0,N = (by 4.11) = ( f , g )β,N .
±1 ±1 Sn be the linear space of symmetric Laurent polyNow let 3± n = (C[x1 , . . . , xn ]) nomials in variables x1 , . . . , xn and introduce on 3± n a scalar product h · , · iβ,N as ± follows: for f, g ∈ 3n set
1 ∗ [f 1(β, N )g]1 , n! Y ˜ where 1(β, N ) = 1(β, N) (1 − xi x−1 j ) h f , g iβ,N =
=
Y
(4.20)
1≤i6=j≤n −N β (1 − xN ) (1 − xi x−1 i xj j ).
(4.21)
1≤i6=j≤n ± 0 : A± Let K ∈ Z and ωK n → 3n be an isomorphism of linear spaces defined for ± f ∈ An by:
Q
where aˆ δ = A± n we have:
−K 0 ωK (f ) = x−K · · · x−K aδ , n f /ˆ 1 x2
1≤i<j≤n (xi
(4.22)
− xj ) is the Vandermonde determinant. Then for any f, g ∈
0 0 h ωK (f ) , ωK (g) iβ,N = hh f , g iiβ,N .
(4.23)
0 Now set K = ωK ω 0 . The map K for any integer K is an isomorphism of linear ± spaces FN,n and 3n . Moreover combining (4.14) and (4.23) we arrive at the following conclusion
Proposition 6. For any f, g ∈ FN,n we have
h K (f ) , K (g) iβ,N = ( f , g )β,N .
(4.24)
That is FN,n with the scalar product ( · , · )β,N and the space of symmetric Laurent polynomials 3± n with the scalar product h · , · iβ,N are isomorphic as Hilbert spaces with the isomorphism defined by the map K , where K is an arbitrary integer. 4.3. Basis of the space of symmetric Laurent polynomials defined by the A(glN ; β)eigenbasis of the space of states FN,n . Fix M ∈ Z and for any k = (k1 , . . . , kn ) ∈ (n) L(1) n define σ = (σ1 , . . . , σn ) ∈ Ln by ki = σi + M − i + 1. Then with the sequence δ = (n − 1, . . . , 1, 0) we have M −n+1 (uˆ k ) = (x1 · · · xn )−(M −n+1) aˆ k /ˆaδ = aˆ σ+δ /ˆaδ := sσ .
(4.25)
In particular, when σn ≥ 0, the sσ is just the Schur polynomial labelled by the partition σ. Note that M −n+1 (vac(M )) = 1. Denote now M −n+1 (Xk(β,N ) ) by Yσ(β,N ) . Then from (3.32) - (3.35) we obtain:
682
D. Uglov ± {Yσ(β,N ) | σ ∈ L(n) n } is a basis of 3n such that: X (β) (β) Yσ(β,N ) = sσ + vστ sτ , vστ ∈ R;
h Yσ(β,N ) ,
τ <σ Yτ(β,N ) iβ,N
=0
if
σ 6= τ.
(4.26) (4.27) (4.28)
Note that the basis {Yσ(β,N ) |σ ∈ L(n) n } with the properties (4.27) and (4.28) is unique. This follows from the standard argument [17] based on the Gram-Schmidt orthogonalization of the basis of the Laurent polynomials sσ . Note also that {Yλ(β) } where λ runs through all partitions of length less or equal to n, is a basis of the space of symmetric polynomials 3n = (C[x1 , . . . , xn ])Sn . 4.4. glN -Jack polynomials. Let q and t be parameters and let 3q,t n = (C(q, t)[x1 , . . . , xn ])Sn be the C(q, t)-algebra of symmetric polynomials in variables x1 , . . . , xn . For f = f (x1 , . . . , xn ), g = g(x1 , . . . , xn ) ∈ 3q,t n a scalar product is defined as follows [17]: h f , g iq,t = n Z 1 Y dwj = n! 2πiwj j=1
Y
(wi wj−1 ; q)∞
(twi wj−1 ; q)∞ 1≤i6=j≤n
f (w1 , . . . , wn )g(w1 , . . . , wn ),
(4.29)
where the integration in each Q of the variables wj is taken along the unit circle in the ∞ complex plane, and (x; q)∞ = r=0 (1 − xq r ). Then for each partition λ of length less or equal to n the Macdonald Polynomial Pλ (q, t) = Pλ (x1 , . . . , xn ; q, t) [17] is uniquely defined as the element of 3q,t n such that: X uλµ (q, t)mµ , (4.30) Pλ (q, t) = mλ + µ<λ
where mλ = mλ (x1 , . . . , xn ) is the monomial symmetric polynomial and uλµ (q, t) ∈ C(q, t); and h Pλ (q, t) , Pµ (q, t) iq,t = 0 if λ = 6 µ. (4.31) Let now γ be a positive real number, let ωN = exp( 2πi N ) and consider the limit: q = ωN p,
t = ω N pγ ,
p → 1.
(4.32)
In this limit the Macdonald Polynomial Pλ (q, t) degenerates into a symmetric polynomial which we will denote by Pλ(γ,N ) and call the glN -Jack Polynomial. In particular when −1
N = 1 the gl1 -Jack Polynomial is nothing but the usual Jack Polynomial: Pλ(γ,1) = Pλ(γ ) in the notation of [17]. First, let us verify that the polynomial Pλ(γ,N ) is well-defined. To do this we will take the limit (4.32) in the coefficients uλµ (q, t) of the expansion (4.30). For a partition λ a tableau T of shape λ [17] is a sequence of partitions: ∅ = λ(0) ⊂ λ(1) ⊂ . . . ⊂ λ(r) = λ
(4.33)
such that each skew diagram θ(i) = λ(i) − λ(i−1) (1 ≤ i ≤ r) is a horizontal strip. The sequence (|θ(1) |, . . . , |θ(r) |) is called the weight of T. The coefficient of mµ in the expansion of Pλ (q, t) is [17]:
Yangian Gelfand-Zetlin Bases, Jack Polynomials and Correlation Functions
uλµ (q, t) =
X
ψT (q, t)
683
(4.34)
T
summed over tableaux of shape λ and weight µ. To describe the ψT (q, t), for partitions λ and µ such that µ ⊂ λ let Cλ/µ (resp. Rλ/µ ) denote the union of the columns (resp. rows) that intersect λ − µ. Then: ψT (q, t) =
r Y
ψλ(i) /λ(i−1) (q, t),
(4.35)
bµ (s; q, t) , bλ (s; q, t)
(4.36)
i=1
where for µ ⊂ λ: Y
ψλ/µ (q, t) =
s∈Rλ/µ −Cλ/µ
(
and bλ (s; q, t) =
1 − q aλ (s) tlλ (s)+1 1 − q aλ (s)+1 tlλ (s) 1
if s ∈ λ,
(4.37)
otherwise.
Now taking the limit (4.32) in bλ (s; q, t) we obtain: ) b(γ,N (s) = lim bλ (s; ωN p, ωN pγ ) λ p→1 ( aλ (s) + γlλ (s) + γ = aλ (s) + γlλ (s) + 1 if s ∈ λ, hλ (s) ≡ 0 mod N ; 1 otherwise.
(4.38)
) ) Thus the coefficient uλµ (q, t) in (4.30) has a well-defined limit u(γ,N such that u(γ,N λµ λµ > 0 for γ > 0, and X (γ,N ) Pλ(γ,N ) = mλ + uλµ mµ . (4.39) µ<λ
Sometimes it is more convenient to use the expansion of the polynomial Pλ(γ,N ) in the basis of Schur polynomials sλ which is also unitriangular in the natural ordering of partitions: X (γ,N ) (γ,N ) vλµ sµ , vλµ ∈ R, (4.40) Pλ(γ,N ) = sλ + µ<λ
since the bases {mλ } and {sλ } are related by a unitriangular transformation [17]. Note also, that Pλ (q, q) = sλ [17] gives Pλ(1,N ) = sλ . Consider next the behaviour of the scalar product (4.29) in the limit (4.32). To avoid the question of convergence of the integral in (4.29) in this limit, we now assume that t = q k where k is a non-negative integer. Then for any f, g ∈ 3± n we have [17]: 1 ∗ [f 1(q, t)g]1 , (4.41) n! Y Y k−1 Y (xi x−1 j ; q)∞ = (1 − q r xi x−1 1(q, t) = j ). (4.42) −1 (tx x ; q) i j ∞ 1≤i6=j≤n 1≤i6=j≤n r=0
h f , g iq,t = where
Let now k = γ and let γ = N β + 1, where β is a non-negative integer. Then taking the limit (4.32) in (4.42) we obtain:
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D. Uglov
Y
1(β, N ) = lim 1(ωN p, ωN p ) = γ
p→1
=
Y
Nβ Y
r (1 − ωN xi x−1 j )
1≤i6=j≤n r=0
(1 −
−N β xN ) (1 i xj
−
(4.43)
xi x−1 j ).
1≤i6=j≤n
This is exactly the weight function of the scalar product h · , · iβ,N which was introduced in (4.21). Thus for γ = N β + 1 the scalar product for Macdonald Polynomials h · , · iq,t degenerates in the limit (4.32) into the scalar product h · , · iβ,N defined in (4.21), so that from (4.31) we get h Pλ(N β+1,N ) , Pµ(N β+1,N ) iβ,N = 0,
if λ 6= µ.
(4.44)
It is now clear that for the glN -Jack Polynomials we may establish analogues of all properties known for Macdonald Polynomials just by taking the limit (4.32). We will not go here into discussion of the Cauchy formulas, Duality etc. [17] all of which are straightforward to derive for the glN -Jack Polynomials. In this paper we will restrict ourselves to computing the normalization and the expansion of power-sums in terms of Pλ(γ,N ) because these are relevant to computation of Dynamical Spin-Density and Density Correlation Functions in the Spin Calogero-Sutherland Model. First of all taking the limit (4.32) in the norm formula for the Macdonald Polynomial [17]: h Pλ (q, t) , Pλ (q, t) iq,t = cn (q, t) cn (q, t) = h 1 , 1 iq,t =
Q
0
0
1−q a (s) tn−l (s) s∈λ 1−q a0 (s)+1 tn−l0 (s)−1
Q
Q
1−q aλ (s)+1 tlλ (s) s∈λ 1−q aλ (s) tlλ (s)+1 ,
(tj−i ;q)∞ (qtj−i ;q)∞ 1≤i<j≤n (tj−i+1 ;q)∞ (qtj−i−1 ;q)∞ ,
we obtain the norm formula for the glN -Jack Polynomial: ) h Pλ(γ,N ) , Pλ(γ,N ) iβ,N = c(γ,N n
Y s∈λ c(s)≡nmod N
×
a0 (s) + γ(n − l0 (s)) a0 (s) + 1 + γ(n − l0 (s) − 1)
Y
s∈λ hλ (s)≡0mod N
aλ (s) + γlλ (s) + 1 . aλ (s) + γlλ (s) + γ
In the last formula we have set γ = N β + 1 and Y ) c(γ,N = h 1 , 1 iβ,N = C (γ,N ) (j − i), where n
(4.45)
(4.46)
1≤i<j≤n
C (γ,N ) (k) = 0 =
0
γk−γ+1−a +1 0 γk+γ−1−a +1 N N 02 γk−a +1 N γk−γ+1 +1 0 γk+γ−1 +1 N N γk 0 γk N +1 0 N
when k ≡ a mod N and a = 1, . . . , N − 1; (4.47) when k ≡ 0 mod N.
In the paper [24] the norm formulas for eigenvectors of the commutative family A(glN ; β) were computed by another, rather cumbersome method. For the case N = 2
Yangian Gelfand-Zetlin Bases, Jack Polynomials and Correlation Functions
685
we have verified that the formulas in [24] give exactly the same result as the formula above. Consider now the formula for the expansion of the power-sums in the basis of Macdonald Polynomials [14]: Q 0 0 n lλ (s) X X − q aλ (s) s∈λ\(1,1) t m m Q xi = (1−q ) Pλ (q, t), (m = 0, 1, 2, . . . ). pm = lλ (s) q aλ (s)+1 s∈λ 1 − t i=1
λ,|λ|=m
(4.48) For any partition λ let us define the subsets: CN (λ) = {s ∈ λ |C(s) ≡ 0 mod N }, HN (λ) = {s ∈ λ |hλ (s) ≡ 0 mod N }.
(4.49) (4.50)
Then taking the limit (4.32) in the formula (4.48) we obtain the formula for expansion of the power-sums in the basis of glN -Jack Polynomials: X (γ,N ) (γ,N ) pm = χ λ Pλ , (4.51) λ|λ|=m ) are as follows: where the coefficients χ(γ,N λ Q c(s) Q 0 0 s∈λ\CN (λ) (1 − ωN ) s∈CN (λ)\(1,1) (a (s) − γl (s)) (γ,N ) n(λ) Q = |λ|ωN Q χλ hλ (s) ) s∈HN (λ) (aλ (s) + γlλ (s) + 1) s∈λ\HN (λ) (1 − ωN
when |λ| ≡ 0 mod N,
(4.52)
|CN (λ)| = |HN (λ)|; ) =0 χ(γ,N λ
(4.53)
|CN (λ)| > |HN (λ)|; Q c(s) Q 0 0 s∈λ\CN (λ) (1 − ωN ) s∈CN (λ)\(1,1) (a (s) − γl (s)) |λ| n(λ) (γ,N ) Q = (1 − ωN )ωN Q χλ hλ (s) ) s∈HN (λ) (aλ (s) + γlλ (s) + 1) s∈λ\HN (λ) (1 − ωN (4.54) when |λ| 6≡ 0 mod N, |CN (λ)| = |HN (λ)| + 1;
when |λ| ≡ 0 mod N,
) =0 χ(γ,N λ
(4.55) P
when |λ| 6≡ 0 mod N, |CN (λ)| > |HN (λ)| + 1. Here n(λ) = (i − 1)λi . Note that for any partition λ such that |λ| ≡ 0 mod N we have |CN (λ)| ≥ |HN (λ)|. And for any partition λ such that |λ| 6≡ 0 mod N we have |CN (λ)| ≥ |HN (λ)| + 1. 4.5. Identification of glN -Jack polynomials and polynomials Yλ(β,N ) . Here we identify the symmetric Laurent polynomials Yσ(β,N ) (4.26) with the glN -Jack polynomials multiplied by a certain power of the Laurent polynomial x1 x2 · · · xn . This is the final step in our construction, and it gives us the proof of the main result of this paper. n For σ ∈ L(n) n = {σ = (σ1 , . . . , σn ) ∈ Z | σ1 ≥ σ2 ≥ · · · ≥ σn } let us define a symmetric Laurent polynomial Pσ (q, t) by the relation: Pσ (q, t) = (x1 x2 · · · xN )σn P(σ−σn In ) (q, t).
(4.56)
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D. Uglov
Here (σ−σn In ) = (σ1 −σn , σ2 −σn , . . . , σn−1 −σn , 0) is a partition and P(σ−σn In ) (q, t) is the corresponding Macdonald Polynomial. If in σ all σi are non-negative, then σ is identified with a partition, say λ, of length less or equal to n and Pσ (q, t) = Pλ (q, t) since [17] (4.57) x1 x2 · · · xn Pλ (q, t) = P(λ+In ) (q, t) for any partition λ such that l(λ) ≤ n. Similarly for any σ ∈ L(n) n we define (γ,N ) Pσ(γ,N ) = (x1 x2 · · · xN )σn P(σ−σ , n In )
(4.58)
sσ = (x1 x2 · · · xN )
s(σ−σn In ) = aˆ σ+δ /ˆaδ .
(4.59)
(γ,N ) (vστ ∈ R),
(4.60)
σn
Now from (4.40, 4.44) we obtain: Pσ(γ,N ) = sσ + h Pσ(N β+1,N ) ,
X
(γ,N ) vστ sτ ,
τ <σ Pτ(N β+1,N )
iβ,N = 0
if σ 6= τ.
(4.61)
Comparing this with (4.26 - 4.28) and taking into account the uniqueness of Laurent polynomials staisfying (4.26 - 4.28), we establish the main result of this paper which implies the statement given in Sect. 1.1 of the Introduction. Theorem 7. For any β ∈ N and M ∈ Z the Laurent polynomials (4.26) obtained from the eigenvectors of the commutative family A(glN ; β) under the action of the isomorphism M −n+1 : FN,n → 3± n are identical with the Laurent polynomials (4.58), where γ = N β + 1, i.e: Yσ(β,N ) = Pσ(N β+1,N ) ,
σ ∈ L(n) n .
(4.62)
In particular for any partition λ of length less or equal to n the symmetric polynomial Yλ(β,N ) is identical with the glN -Jack Polynomial Pλ(N β+1,N ) . To end this section let us now note, that the Laurent polynomials Pσ (q, t) are eigenvectors of the Macdonald Operator [17]: D1 (q, t) =
n Y X txi − xj i=1 j6=i
xi − xj
q
∂ xi ∂x
i
,
D1 (q, t)Pσ (q, t) = Eσ (q, t)Pσ (q, t), n X where Eσ (q, t) = q σi tn−i .
so that
(4.63) (4.64) (4.65)
i=1 ± Let the map χ : 3± n → 3n be defined on any Laurent polynomial f (x1 , . . . , xn ) by −1 χf (x1 , . . . , xn ) = f (x , . . . , x−1 n ). Then we have:
χD1 (q, t)χ = tn−1 D1 (q −1 , t−1 ), χPσ (q, t) = Pσ∗ (q −1 , t−1 ), σ ∈ L(n) n .
(4.66) (4.67)
Here for σ = (σ1 , σ2 , . . . , σn ) we used the notation: σ ∗ = (−σn , −σn−1 , . . . , −σ1 ). From (4.67) it follows that ) . (4.68) χPσ(γ,N ) = Pσ(γ,N ∗
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687
5. Dynamical Correlation Functions for N = 2 In this section we will apply results obtained thus far in order to compute SpinDensity and Density two-point Dynamical Correlation Functions in the Spin CalogeroSutherland Model. So as to avoid cumbersome technical details we will consider only the case where the spin has two values, i.e. N = 2 and the number of particles in the model n is an even number such that n/2 is odd. 5.1. A parameterization of energy, momentum and spin eigenvalues. Let {Xk(β,2) | k ∈ L(1) n } be the A(gl2 ; β)-eigenbasis of the space F2,n constructed in Sect. 3.4. Pn 2 Hβ,2 (2.13), the momentum operator P = 2π The energy operator 2π i=1 zi ∂/∂zi L L2 P n and the spin operator S = i=1 σiz /2, where σiz are Pauli matrices, all belong to the commutative family A(gl2 ; β). Therefore their eigenvalues in the basis {Xk(β,2) | k ∈ L(1) n } can be read from the formulas (3.40) with the following result: 2π 2 2π 2 (β,2) H X = Eβ,2 (k)Xk(β,2) , β,2 k L2 L2 n n X X β 2 n(n2 − 1) 2 ; ki + β (2i − n − 1)ki + where Eβ,2 (k) = 12 P Xk(β,2) = where
i=1 P (k)Xk(β,2) , n X
P (k) =
2π L
(5.1) (5.2)
i=1
(5.3) ki ;
(5.4)
i
SXk(β,2) = S(k)Xk(β,2) , n 1X where S(k) = δ(ki = 1) − δ(ki = 2). 2
(5.5) (5.6)
i=1
A ground state of the model is defined as a state with the lowest energy eigenvalue among states with zero momentum. Under our assumption that n is even and n/2 is odd, the ground state is unique and is found to be Xk(β,2) 0
with
k 0 = (k10 , . . . , kn0 ) = (M, M − 1, . . . , M − n + 1), where M =
n + 1. 2 (5.7)
From the expansion (3.39) we find that = vac( Xk(β,2) 0
n + 1) = uk0 ∧ uk0 ∧ · · · ∧ ukn0 . 1 2 2
(5.8)
When the number of particles n is even, any state vac(M ) (2.27) with even M is a basis in a one-dimensional trivial representation of Y (gl2 ; β). In particular, the spin of any such state is zero. (n) 0 For k ∈ L(1) n define, as in Sect. 4.3, a sequence σ ∈ Ln by ki = σi + ki . Then the (β,2) (2β+1,2) . In particular, the isomorphism M −n+1 with M = n/2 + 1 maps Xσ+k0 into Pσ (β,2) ground state is mapped into 1 and a state Xλ+k0 , where λ is a partition is mapped into the gl2 -Jack Polynomial Pλ(2β+1,2) . For a state of this form we will find it convenient to express the corresponding eigenvalues of energy, momentum and spin in terms of the diagram associated with the partition λ.
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Fig. 2. Coloring of the partition λ = (6, 4, 4, 3, 1).
Let us color the diagram of λ by white and black colors in a checkerboard order so that the square (1, 1) is colored white. An example of this coloring for λ = (6, 4, 4, 3, 1) is given in Fig. 2. Then a square s ∈ λ is white(black) if and only if the content of this square c(s) is even(odd). Let Wλ (Bλ ) be the subset of all white(black) squares in λ. In the notation of Sect. 4 we have Wλ = C2 (λ). With this coloring we have for the eigenvalues 2 0 of spin, momentum, and normalized energy E(k) = 2π L2 (Eβ,2 (k) − Eβ,2 (k )) : S(λ + k 0 ) = |Wλ | − |Bλ |, P (λ + k 0 ) = 2π L |Wλ |, E(λ + k 0 ) =
2π 2 L2
nw (λ0 ) − (2β + 1)nw (λ) +
n 2 (2β
+ 1) +
1 2
|Wλ | .
(5.9) (5.10) (5.11)
0
In the last P formula λ is the partition conjugated to λ and nw (λ) which is analogous to n(λ) = (i − 1)λi is defined as follows: X X l0 (s) = (i − 1)wi (λ), (5.12) nw (λ) = s∈Wλ
where wi (λ) is the number of white squares in the ith row of λ. Note that we have X X X nw (λ0 ) = a0 (s) = wi (λ)2 − wi (λ) + wi (λ)2 . (5.13) s∈Wλ
i:even
i:odd ∗
Finally, from the expressions (5.2 - 5.6) we find that if λ = (−λn , −λn−1 , . . . , −λ1 ) then S(λ∗ + k 0 ) = − S(λ + k 0 ), P (λ∗ + k 0 ) = − P (λ + k 0 ), E(λ∗ + k 0 ) = E(λ + k 0 ).
(5.14) (5.15) (5.16)
5.2. Dynamical Correlation Functions. We will consider two-point correlation functions of the following operators: Spin-Density: Density:
s(0, 0) = ρ(0, 0) =
n X i=1 n X i=1
δ(yi )σiz /2 = δ(yi ) −
n 1 XX m z zi σi /2, L
n 1 = L L
m∈Z i=1 n ∞ X X m=1 i=1
(zim + zi−m ).
(5.17) (5.18)
Yangian Gelfand-Zetlin Bases, Jack Polynomials and Correlation Functions
689
Also, as an intermediate step in the computation we will need to consider correlation functions of the operators s± (0, 0) =
n X
δ(yi )σi± =
i=1
n 1 XX m ± zi σi . L
(5.19)
m∈Z i=1
Now let s be an integer and define Js+ =
so that s± (0, 0) = have:
1 L
Pn
−s−1 + σi , i=1 zi Js = Js+ + Js− ,
P s∈Z
Pn Js− = i=1 zi−s σi− , Pn Ks = i=1 zi−s ,
(5.20) (5.21)
Js± . Acting on a wedge in F2,n with operators Js and Ks we
J s u k 1 ∧ u k 2 ∧ · · · ∧ uk n =
n X
Ks uk1 ∧ uk2 ∧ · · · ∧ ukn =
uk1 ∧ uk2 ∧ · · · ∧ uki +2s+1 ∧ · · · ∧ ukn , (5.22)
i=1 n X
uk1 ∧ uk2 ∧ · · · ∧ uki +2s ∧ · · · ∧ ukn .
(5.23)
i=1
Let f and g be eigenvectors of the spin operator in F2,n such that Sf = s(f )f and Sg = s(g)g, where s(f ) and s(g) are corresponding eigenvalues. Since the operator S is self-adjoint relative to the scalar product ( · , · )β,2 we have: ( f , Js± g )β,2 = δ (s(f ) − s(g) = ±1) ( f , Js g )β,2 .
(5.24)
Now observe that (5.22, 5.23) imply that the isomorphism M −n+1 transforms the operators Js and Ks into odd and even power-sums respectively: M −n+1 Js −1 M −n+1 =
n X
M −n+1 Ks −1 M −n+1 =
x2s+1 = p2s+1 , i
i=1 n X
x2s i = p2s ,
(5.25) (5.26)
i=1
so that ( f , Js± g )β,2 = δ (s(f ) − s(g) = ±1) h M −n+1 (f ) , p2s+1 M −n+1 (g) iβ,2 , (5.27) ( f , Ks g )β,2 = δ (s(f ) − s(g) = 0) h M −n+1 (f ) , p2s M −n+1 (g) iβ,2 .
(5.28)
In virtue of these relations computation of matrix elements of the Spin-Density and Density operators is reduced to computation of matrix elements of power sums. Because of this from now on the logic of our computation follows exactly the logic of computation of the Density Correlation Function in the scalar Calogero-Sutherland Model [9, 14]. Let us denote by t and x the time and the space coordinates in a two-point Correlation Function. Taking formulas discussed so far in this section into account we have:
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D. Uglov
hs∓ (x, t)s± (0, 0)i := ( vac( n2 +1) , e
2 −it 2π2 L
(5.29) Hβ,2−ixP
s∓ (0, 0)e
2 it 2π2 L
Hβ,2+ixP
s± (0, 0)vac( n2 +1) )β,2
( vac( n2 + 1) , vac( n2 +1) )β,2 =
(β,2) (β,2) n X ( vac( n2 + 1) , s∓ (0, 0)Xσ+k 0 )β,2 ( Xσ+k 0 , s± (0, 0)vac( 2 + 1) )β,2 (β,2) (β,2) ( vac( n2 + 1) , vac( n2 + 1) )β,2 ( Xσ+k 0 , Xσ+k 0 ) β,2
σ∈L(1) n
×eitE(σ+k =
(5.30)
1 L2
X
0
)+ixP (σ+k0 )
h
(1) σ∈Ln S(σ+k0 )=±1
Pσ(2β+1,2)
=
(5.31)
P
2 m:odd (pm + p−m ) iβ,2 itE(σ+k0 )+ixP (σ+k0 ) e . Pσ(2β+1,2) , Pσ(2β+1,2) iβ,2
,
h 1 , 1 iβ,2 h
(5.32)
Now using (4.51), (4.68) and (5.14 - 5.16) we obtain for the Spin-Density Correlation Function:
=
1 2L2
(5.33) hs(x, t)s(0, 0)i = 41 hs− (x, t)s+ (0, 0)i + 41 hs+ (x, t)s− (0, 0)i = 2 h P (2β+1,2) , P (2β+1,2) i P 0 λ λ β,2 itE(λ+k ) χ(2β+1,2) e cos(xP (λ + k 0 )). λ, |λ|:odd λ h1,1i β,2
S(λ+k0 )=±1
In view of (4.45), (4.54, 4.55) and (5.9) the last expression may be written as follows hs(x, t)s(0, 0)i =
=
1 2L2
Y
X
Y
λ, |λ|:odd |Wλ |−|Bλ |=±1 |Wλ |=|H2 (λ)|+1
× Zλ (β, n) e
c(s; 2β + 1)2
s∈Wλ \(1,1)
h∗λ (s; 2β + 1)hλ∗ (s; 2β + 1)
(5.34)
s∈H2 (λ)
itE(λ+k0 )
cos(xP (λ + k 0 )),
where Y
Zλ (β, n) =
s∈Wλ
a0 (s) + (2β + 1)(n − l0 (s)) . + 1 + (2β + 1)(n − l0 (s) − 1)
a0 (s)
In complete analogy with the Spin-Density Correlation Function, we find for the Density Correlation Function: 2 hρ(x, t)ρ(0, 0)i = 2 L
X
χ(2β+1,2) λ
2 h P (2β+1,2) , P (2β+1,2) i λ λ β,2
λ, |λ|:evenS(λ+k0 )=0
h 1 , 1 iβ,2
0
× eitE(λ+k ) cos(xP (λ + k 0 )). (5.35) Which in view of (4.45), (4.52 , 4.53) , (5.6) we may write as
Yangian Gelfand-Zetlin Bases, Jack Polynomials and Correlation Functions
hρ(x, t)ρ(0, 0)i =
=
2 L2
X
Y |λ|2 Y
λ, |λ|:even |Wλ |−|Bλ |=0 |Wλ |=|H2 (λ)|
691
c(s; 2β + 1)2
s∈Wλ \(1,1)
h∗λ (s; 2β + 1)hλ∗ (s; 2β + 1)
(5.36)
s∈H2 (λ) 0
× Zλ (β, n) eitE(λ+k ) cos(xP (λ + k 0 )), where Zλ (β, n) =
Y s∈Wλ
a0 (s) + (2β + 1)(n − l0 (s)) . + 1 + (2β + 1)(n − l0 (s) − 1)
a0 (s)
Notice that according to definition (4.2) the summands in formulas for Correlation Functions vanish if the diagram of λ contains the (white for integer β) square (2, 2β + 2). This observation must facilitate a computation of the thermodynamic limit. In this paper we do not consider this problem. 6. Discussion In this paper we have constructed an isomorphism between the space of states of the glN invariant Calogero-Sutherland Model and the space of Symmetric Laurent Polynomials. With this isomorphism we have shown that the glN -invariant Calogero-Sutherland Models with N = 1, 2, 3, . . . when considred in an appropriate unified framework, which is the framework of Symmetric Polynomials, manifest themselves as limiting cases of the same entity, which is the commuting family of Macdonald Operators. Macdonald Operators depend on two parameters q and t. And the Hamiltonian of the glN -invariant Calogero-Sutherland Model belongs to a degeneration of this family in the limit when both q an t approach the N th elementary root of unity: q = ωN p,
t = ωN pN β+1 ,
p → 1,
and β is the coupling constant of the glN -invariant Calogerowhere ωN = Sutherland Model. This picture provides a generalization of the well-known situation in the case of the Scalar Calogero-Sutherland Model (N = 1). In the limit the commuting family of Macdonald Operators is identified with the maximal commutative subalgebra A(glN ) in the action of the Yangian algebra Y (glN ) on the space of states of the glN -invariant Calogero-Sutherland Model. The limits of Macdonald Polynomials which we call for obvious reasons glN -Jack Polynomials are eigenvectors of this subalgebra and, by definition, form Yangian Gelfand-Zetlin bases in irreducible components of the Yangian action. In view of the isomorphism between the space of states of the glN -invariant CalogeroSutherland Model and the space of Symmetric Laurent Polynomials the latter admits an action of the Yangian Y (glN ). This fact is not surprising for it is known that the b admits a Yangian action coming level-1 Fock space module of the affine Lie algebra gl N from that of the Yangian action of the Calogero-Sutherland Model in a certain projective b 2 case ) and the Fock space module is isomorphic by limit n → ∞ ( see [25] for the gl Fermion-Boson correspondence to the space of Symmetric Functions [11]. In fact the isomorphism of our present paper is to be regarded as nothing more than a finite-size version of the Fermion-Boson correspondence. exp( 2πi N )
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D. Uglov
The connection with the Macdonald Operators seems to be more mysterious. It provides another example of a situation where the degree of a root unity appears as the rank of a Lie algebra. To understand this phenomenon better it would be desirable to understand of which algebra the Yangian is the limit when q and t approach the root of unity. The glN -Jack Polynomials describe the orthogonal eigenbasis of glN -invariant Calogero-Sutherland Model in exactly the same way as Jack Polynomials describe the orthogonal eigenbasis of the Scalar Model (N = 1). For each known property of Macdonald Polynomials there is a corresponding property of glN -Jack Polynomials. As a simplest application of these properties we compute two-point Dynamical Spin-Density and Density Correlation Functions in the gl2 -invariant Calogero-Sutherland Model at integer values of the coupling constant. In this paper we did not consider the problem of computation of the Green’s Functions. However we expect that the Cauchy formula for the glN -Jack Polynomials: X
) (γ,N ) b(γ,N Pλ (x1 , . . . , xn )Pλ(γ,N ) (y1 , . . . , yn ) = λ
n Y
N β (1 − xN i yj ) (1 − xi yj ),
i,j=1
λ ) where γ = N β + 1 and (see 4.38) b(γ,N = λ
X
Pλ(γ,N ) (x1 , . . . , xn )Pλ(γ0
−1
Q
,N )
(γ,N ) (s) s∈λ bλ
and its dual
(y1 , . . . , yn ) =
n Y
(1 + xi yj )
i,j=1
λ
will play a role in a solution of this problem together with an appropriate evaluation formula for glN -Jack Polynomials. Appendix A. Proof of Proposition 2 The statement (3.13) follows from the well-known construction of Yangian representations by the fusion procedure (see e.g. [10]). So we will not discuss proof of this statement here. To prove the rest of the statements, for any two increasing sequences a = (a1 , a2 , . . . , am ), b = (b1 , b2 , . . . , bm ), where m, ai , bi ∈ {1, . . . , N } define (cf.[20]): X Qa,b (u) := sign(w)Ta1 ,bw(1) (u)Ta2 ,bw(2) (u − 1) · · · Tam ,bw(m) (u − m + 1). (A.1) w∈Sm
With this definition: Am (u) = Qm,m (u),
m = {1, 2, . . . , m},
(m = 1, 2, . . . , N ).
(A.2)
According to Proposition 1.11 in [20]: X Qm,a(1) (u) ⊗ Qa(1) ,a(2) (u) ⊗ · · · ⊗ Qa(n−1) ,m (u), 1(n) (Am (u)) =
(A.3)
where
a(1) ,...,a(n−1)
where the summation is taken over all increasing sequences of integers 1, 2, . . . , N of length m.
Yangian Gelfand-Zetlin Bases, Jack Polynomials and Correlation Functions
693
By considering the action of the operator E1,1 + 2E2,2 + . . . + N EN,N ∈ End(V ) we find for g ∈ R, a ∈ {1, . . . , N } that: π(g)(Qa,b (u))va ∈ R[[u−1 ]]va+|b|−|a| ,
(A.4)
and hence that for any a ∈ {1, . . . , N }n the expression (π(f1 ) ⊗ π(f2 ) ⊗ · · · ⊗ π(fn )) 1(n) (Am (u))v(a)
(A.5)
is a linear combination of v(b) such that the b ∈ {1, . . . , N }n has as its elements bi = ai + |a(i) | − |a(i−1) |,
(i = 1, 2, . . . , N ),
(A.6)
for certain a(1) , . . . , a(n−1) appearing in (A.3) and a(0) = a(n) = m. Since for any such b we have b1 + b2 + · · · + bi = a1 + a2 + · · · + ai + |a(i) | − |m|,
(i = 1, 2, . . . , n),
(A.7)
and since in the last equation |a(i) | − |m| > 0 unless a(i) = m, we find by taking (A.2) into account that (π(f1 ) ⊗ π(f2 ) ⊗ · · · ⊗ π(fn )) 1(n) (Am (u))v(a) = (π(f1 )(Am (u)) ⊗ π(f2 )(Am (u)) ⊗ · · · ⊗ π(fn )(Am (u))) v(a) + =
X
X
c(u; f ; a, b)v(b)
b>a
c(u; f ; a, b)v(b),
b≥a
(A.8)
where c(u; f ; a, b) ∈ R[[u−1 ]] and c(u; f ; a, a) =
n Y u + fi + δ(ai ≤ m) = Am (u; f ; a). u + fi
(A.9)
i=1
Now by definition (3.12) we have X ϕ(a) = v(a) + e(a, b)v(b),
(a ∈ Tp ,
e(a, b) ∈ {0, ±1}),
(A.10)
b>a
and hence: (π(f1 ) ⊗ π(f2 ) ⊗ · · · ⊗ π(fn )) 1(n) (Am (u))ϕ(a) = X = c(u; f ; a, a)v(a) + d(u; f ; a, b)v(b), (d(u; f ; a, b) ∈ R[[u−1 ]]).
(A.11)
b>a
On the other hand the subspace (⊗n V )p is left invariant by the Yangian action, so that
(π(f1 ) ⊗ π(f2 ) ⊗ · · · ⊗ π(fn )) 1(n) (Am (u))ϕ(a) = X = g(u; f ; a, b)ϕ(b), (g(u; f ; a, b) ∈ R[[u−1 ]]). b
Comparing this equation with (A.11) and using (A.10) we get
(A.12)
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D. Uglov
(π(f1 ) ⊗ π(f2 ) ⊗ · · · ⊗ π(fn )) 1(n) (Am (u))ϕ(a) = X = Am (u; f ; a)ϕ(a) + c(u; f ; a, b)ϕ(b), (c(u; f ; a, b) ∈ R[[u−1 ]]).
(A.13)
b>a
Now define the sequence of real numbers h(1) , h(2) , . . . , h(M ) by h(s) := fqs−1 +1 ,
(s = 1, 2, . . . , M ).
(A.14)
And define for any a ∈ Tp : qs X
(s) lm =
δ(ai ≤ m),
(m = 1, 2, . . . , N ; s = 1, 2, . . . , N ).
(A.15)
i=qs−1 +1
Then for the eigenvalue Am (u; f ; a) we have Am (u; f ; a) =
M Y
u + 1 + h(s)
s=1
(s) u + 1 + h(s) − lm
.
(A.16)
The correspondence (s) | s = 1, 2, . . . , M ; Tp → {lm
m = 1, 2, . . . , N }
(A.17)
given by the relation (A.15) is bijective. The conditions (C1, C2) in Sect. 3.1 and the (s) ≤ qs − qs−1 imply that inequalities 0 ≤ lm (s) (s+1) h(s) − lm − h(s+1) + lm > 0,
(s = 1, 2, . . . , M − 1;
m = 1, 2, . . . , N ), (A.18)
so that the N -tuple of rational functions in the variable u: A1 (u; f ; a), A2 (u; f ; a), . . . , (s) | s = 1, 2, . . . , M, m = 1, 2, . . . , N } uniquely. AN (u; f ; a) determines the set {lm This proves the statement (3.17). Now the statements (3.14),(3.15) and (3.16) follow from (A.13) and (3.17). Thus the proof is finished. B. Proof of Equality (3.36) Using the expression (3.27) for the operator U (s; β) we have X (β) Et (z) ⊗ Rt(β) ϕ(a), 8(β) (s, a) =
(B.1)
t∼s
where t ∼ s means that t belongs to the set of all distinct rearrangements of the sequence ) s ∈ L(N n . In view of the triangularity (3.22) of the Non-symmetric Jack Polynomial (β) Et (z) we may split the Et(β) (z) with t ∼ s as follows: Et(β) (z) = Et(β) (z)0 + Et(β) (z)00 , where Et(β) (z)0 =
X rt, r∼s
Both of the vectors
r e(β) tr z ,
and
Et(β) (z)00 =
(B.2) X
) m∈L(N n ,
m<s, r∼m
r e(β) tr z .
(B.3)
Yangian Gelfand-Zetlin Bases, Jack Polynomials and Correlation Functions
X
Et(β) (z)0 ⊗ Rt(β) ϕ(a)
X
and
t∼s
695
Et(β) (z)00 ⊗ Rt(β) ϕ(a)
(B.4)
t∼s
belong to the subspace FN,n since 8(β) (s, a) belongs to FN,n and monomials z r which appear in the decomposition of Et(β) (z)0 as a vector in C[z1±1 , . . . , zn±1 ] ⊗ (⊗n V ) are distinct from monomials z r which appear in the decomposition of Et(β) (z)00 . Taking into account that Rs(β) = 1 we may write: X
X
Et(β) (z)0 ⊗ Rt(β) ϕ(a) = z s ⊗ ϕ(a) +
t∼s
z t ⊗ ϕ¯ t (a),
(ϕ¯ t (a) ∈ ⊗n V ). (B.5)
t∼s, t≺s
Since the expression above is a vector in FN,n and by using the definition of the vector ϕ(a) in (3.12) we obtain X
Et(β) (z)0 ⊗ Rt(β) ϕ(a) = An (z s ⊗ v(a)) = (−1)
n(n−1) 2
uˆ k ,
(B.6)
t∼s
where An (2.6) is the operator of total antisymmetrization in the space C[z1±1 , . . . , zn±1 ]⊗ (⊗n V ), and as in the paragraph preceding Eq. (3.36) we have k = (k1 , k2 , . . . , kn ) ∈ L(1) n such that s = (kn , kn−1 , . . . , k1 ),
a = (kn , kn−1 , . . . , k1 ).
(B.7)
Furthermore, the expansion of the vector X
Et(β) (z)00 ⊗ Rt(β) ϕ(a)
(B.8)
t∼s
in C[z1±1 , . . . , zn±1 ] ⊗ (⊗n V ) contains only monomials z r such that r+ < s. Therefore expansion of (B.8) in the basis {uˆ l | l ∈ L(1) n } contains only normally ordered wedges uˆ l such that l > k. Taking this, and (B.6) into account we have (−1)
n(n−1) 2
8(β) (s, a) = uˆ k +
X l∈L(1) n,
ϕ(β) ˆl kl u
(ϕ(β) kl ∈ R).
(B.9)
l>k
However according to (2.24) an eigenvector of the Hamiltonian Hβ,N with the above (β) expansion in the basis {uˆ l | l ∈ L(1) n } is unique and equals to 9k . This proves (3.36). Acknowledgement. I am grateful to Kouichi Takemura with whom we collaborated on the paper [24] from which the present work has evolved and to Professor P.J. Forrester for encouragement and discussions during his stay at RIMS, Kyoto University in Summer 1996. I am also grateful to Professors T. Miwa and M. Kashiwara for discussions and support. After this work had been completed I have learned about the unpublished typescript of Y. Kato [12] in which he computes the Green Function of the Spin Calogero-Sutherland Model for N = 2 and β = 1 and conjectures expressions for the Green Function at all integer values of β. The method of Kato is completely different from the method of our paper and is in the spirit of the papers [13, 8]. I am grateful to T. Yamamoto for bringing the paper [12] to my attention.
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References 1. Bernard, D., Gaudin, M., Haldane, F.D.M. and Pasquier, V.: Yang-Baxter equation in long-range interacting systems. J. Phys., A26, 5219–5236 (1993) 2. Cherednik, I.V.: A new interpretation of Gelfand-Zetlin bases. Duke Math. J. 54, 563–577 (1987) 3. Cherednik, I.V.: A unification of the Knizhnik-Zamolodchikov and Dunkl operators via affine Hecke algebras. Inv. Math., 106, 411–432 (1991) 4. Cherednik, I.V.: Integration of quantum many-body problems by affine Knizhnik-Zamolodchikov equations. Preprint RIMS-776 (1991); Advances in Math. 106, 65–95 (1994) 5. Cherednik, I.V.: Double Affine Hecke algebras and Macdonald’s conjectures. Annals Math. 141, 191– 216 (1995); Non-symmetric Macdonald Polynomials IMRN 10, 483–515 (1995) 6. Drinfeld, V.G.: A new realization of Yangians and quantized affine algebras. Sov. Math. Dokl. 36, 212– 216 (1988); “Quantum Groups.” In: Proceedings of the International Congress of Mathematicians, Am. Math. Soc., Providence, RI. 1987 pp. 798–820 7. Dunkl, C.F.: Trans. Am. Math. Soc. 311, 167 (1989) 8. Forrester, P.J.: Int. J. Mod. Phys. B9, 1234 (1995) 9. Ha, Z.N.C.: Phys. Rev. Lett. 73, 1574 (1994); Nucl. Phys., B435[FS], 604 (1995) 10. Jimbo, M.: it Topics from representation theory of Uq (g) – an introductory guide to physicists. Nankai Lectures on Mathematical Physics, Singapore: World Scientific, 1992 11. Kac, V.G. and Raina, A.: Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras. Singapore: World Scientific, 1987 12. Kato, Y.: Green Function of the Sutherland Model with SU (2) internal symmetry. Preprint 13. Kato, Y. and Kuramoto, Y.: Phys. Rev. Lett. 74, 1222 (1995) 14. Konno, H.: Nucl. Phys. B473, 579 (1996) 15. Lesage, F., Pasquier, V. and Serban, D.: Nucl. Phys. 435[FS], 585 (1995) 16. Macdonald, I.G.: Affine Hecke algebra and Orthogonal Polynomials. S´eminaire Bourbaki, 47 No. 797, 1–18 (1995) 17. Macdonald, I.G.: Symmetric functions and Hall polynomials 2-nd ed., London: Clarendon Press, 1995 18. Minahan, J. and Polychronakos, A.P.: Phys. Lett. B326, 288 (1994); Phys. Rev., B50, 4236 (1994) 19. Molev, A.I.: Gelfand-Tsetlin Bases for Representations of Yangians. Lett. Math. Phys., 30, 53–60 (1994) 20. Nazarov, M. and Tarasov, V.: Representations of Yangians with Gelfand-Zetlin bases. Preprint UWSMRRS-94-148 (1994). to be published in J. f¨ur Reine und Angew. Math. 21. Opdam, E.: Harmonic analysis for certain representations of graded Hecke algebras. Acta Math. 175, 75–121 (1995) 22. Stanley, R.P.: Adv. Math. 77, 76 (1989) 23. Sutherland, B.: J. Math. Phys. 12, 246, 251 (1971); Phys. Rev. A4, 2019 (1971); ibid. A5, 1372 (1972) 24. Takemura K. and Uglov D.: The Orthogonal Eigenbasis and Norms of Eigenvectors in the Spin CalogeroSutherland Model. Preprint RIMS-1114 (solv-int/9611006) 25. Uglov, D.: Semi-infinite wedges and the conformal limit of the fermionic spin Calogero-Sutherland model of spin 21 . Nucl. Phys. B478, 401–430 (1996) Communicated by G. Felder
Commun. Math. Phys. 191, 697 – 721 (1998)
Communications in
Mathematical Physics c Springer-Verlag 1998
2-Cocycles and Twisting of Kac Algebras? Leonid Vainerman?? Lab. de Math. Fondamentales, Universit´e Pierre et Marie Curie, Case 191, Aile 46-0, 3-´eme e´ tage, 4 place Jussieu, F-75252 Paris Cedex 05, France. E-mail: [email protected] Received: 21 March 1997 / Accepted: 2 June 1997
Abstract: We describe the twisting construction with the help of 2-cocycles on Hopf– von Neumann and George Kac algebras; we show that twisted Kac algebras are again Kac algebras. Using this construction, we give a wide class of new quantizations of the Heisenberg group and describe several series of non-trivial finite- dimensional C∗ -Hopf algebras (Kac algebras) of dimensions 4n and 2n2 (n ∈ N) as twisting of finite groups. 1. Introduction 1.1. This paper can be considered as the development and generalization of [EVa]. There one can find a motivation and some results, notions and notations which we use here. We recall only the most important of them. 1.2. Let us recall that in ([L], §1 and 2), M. Landstad gave a construction of quantum groups, taking the von Neumann algebra L(G) generated by the left regular representation λG of a locally compact group G, and deforming the canonical coproduct 0G s of L(G) defined, for all g in G, by: 0G s (λG (g)) = λG (g) ⊗ λG (g), in order to obtain a new coproduct which can be written, for all x in L(G): ∗ 0 (x) = 0G s (x) ,
where is a “dual” 2-cocycle lifted from an abelian subgroup H of G. The idea of such a construction which is called “twisting” is due to V.G. Drinfeld [D]. ? This research was supported in part by Ukrainian Foundation for Fundamental Studies and by International Science Foundation ?? On leave of absence from: International Solomon University, Zabolotny Street, 38, apt. 61, 252187 Kiev, Ukraine. E-mail: [email protected]
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This procedure, deforming the coproduct of L(G), gives, on the predual A(G) of L(G), a deformation of the product; it is the point of view developed by M. Rieffel in ([R1], §1) or ([R2], §2). This leads to the dual situation, where one takes an algebra of functions on G, deforms the product in this algebra, and leaves untouched the canonical coproduct on this algebra. 1.3. In [EVa] it was shown that, under certain conditions, Landstad’s deformations of locally compact groups are Kac algebras (on the theory of Kac algebras we refer to [ES]), and the analogue of this construction for general Kac algebras was done. It is important to stress that the co-involution of the initial Kac algebra in that framework remained undeformed. The aim of the present paper is to give such a generalization of the twisting construction which includes a deformation of a co-involution, and to show that the result of such a deformation is again a Kac algebra. In this way we also get a much wider class of non-trivial Kac algebras. Notice that another construction of a deformation of a coproduct (or a multiplicative unitary), using a 2-cocycle, can be found in ([BS], 8.24, 8.26). 1.4. The paper is organized as follows: in the second section we give foundations of the theory of 2-cocycles, 2-pseudo-cocycles and the twisting of Hopf–von Neumann algebras. This section is inspired by ([M], 2.3). In the third section we deal with Woronowicz algebras; this is a class of objects introduced in [MN], which contains Kac algebras and also compact and discrete quantum groups. Then, the constructions of Sect. 2 allow us to consider the twisting of the multiplicative unitaries associated with Woronowicz algebras. In Sect. 4 we consider the twisting of Kac algebras and Woronowicz algebras by 2-cocycles lifted from 2-cocycles on their locally compact subgroups. Notice that in [EVa] we worked only with 2-cocycles lifted from special 2-cocycles associated with very special abelian subgroups. Then the main result of this paper is given: in the case of Kac algebras, when a 2-cocycle is of the above mentioned type, and the image of a subgroup belongs to the centralizer of the Haar weight, then the coinvolutive Hopf–von Neumann algebra obtained in Sect. 2, with undeformed Haar weight, is again the Kac algebra. We also describe the deformation of the corresponding multiplicative unitary. In Sect. 5 we give a class of new quantizations of the Heisenberg group, much wider than in ([EVa], 6.3). Some of them have a deformed antipode, and some have an undeformed one; however, all of them are unimodular Kac algebras. In Sect. 6 we describe several series of non-trivial finite-dimensional C∗ -Hopf algebras (Kac algebras) of dimensions 4n and 2n2 (n ∈ N) as twisting of finite groups. It should be noted, that finite-dimensional Kac algebras in the past few years found the important application in studying the irreducible inclusions of factors (see [EN, HS, IK] and the literature cited there).
2. 2-Cocycles and Twisting of Hopf–von Neumann Algebras Definition 2.1. Let (M, 0) be a Hopf–von Neumann algebra ([ES], 1.2.1). An element x ∈ M is said to be group-like if it is unitary: xx∗ = x∗ x = 1 and 0(x) = x ⊗ x. Obviously all group-like elements form a group - an intrinsic group of Hopf–von Neumann algebra (cf. [ES], 1.2.2).
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Definition 2.2. For all unitaries u in M (resp., in M ⊗ M ), where (M, 0) is a Hopf–von Neumann algebra, a notion of a coboundary ∂1 u in M ⊗ M (resp., ∂2 in M ⊗ M ⊗ M ), was defined in ([EVa], 2.2) by: ∂1 u = (u∗ ⊗ u∗ )0(u) (resp., ∂2 = (i ⊗ 0)(∗ )(1 ⊗ ∗ )( ⊗ 1)(0 ⊗ i)() ). A 1-cocycle of (M, 0) is a unitary u ∈ M such that ∂1 u = 1, or, equivalently: 0(u) = (u ⊗ u); A 2-cocycle of (M, 0) is a unitary in M ⊗ M such that ∂2 = 1, or, equivalently: ( ⊗ 1)(0 ⊗ i)() = (1 ⊗ )(i ⊗ 0)(). A 2-pseudo-cocycle of (M, 0) is a unitary in M ⊗ M such that ∂2 belongs to (0 ⊗ i)0(M )0 ([EVa], 2.3). Proposition 2.3. Let (M, 0) be a Hopf–von Neumann algebra, and a unitary in M ⊗ M ; then, let us put, for all x in M : 0 (x) = 0(x)∗ . Then, (M, 0 ) is a Hopf–von Neumann algebra, iff is a 2-pseudo-cocycle of (M, 0); and we shall say that (M, 0 ) (resp., 0 ) is twisted from (M, 0) (resp., from 0) by . Proof. The equality (0 ⊗ i)0 (x) = (i ⊗ 0 )(0 (x)) can be rewritten as ( ⊗ 1)(0 ⊗ i)()(0 ⊗ i)0(x)(0 ⊗ i)(∗ )(∗ ⊗ 1) = (1 ⊗ )(i ⊗ 0)()(i ⊗ 0)0(x)(i ⊗ 0)(∗ )(1 ⊗ ∗ ), or, using 2.2, as ∂2 (0 ⊗ i)0(x) = (i ⊗ 0)(0(x))∂2 , which gives the result.
Let us remark that in ([EVa], 2.5) it was shown that the condition above is sufficient. Proposition 2.4. Let (M, 0) be a Hopf–von Neumann algebra, a unitary in M ⊗ M , u a unitary in M . Let us put: u := (u∗ ⊗ u∗ )0(u). Then, u is a 2-pseudo-cocycle (resp., a 2-cocycle) for (M, 0), iff is. If both and u are 2-pseudo-cocycles, then ∂2 u = ∂2 .
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Proof. ∂2 u = (i ⊗ 0)(0(u∗ )∗ (u ⊗ u))× (1 ⊗ 0(u∗ )(u ⊗ u))((u∗ ⊗ u∗ )0(u) ⊗ 1)(0 ⊗ i)((u∗ ⊗ u∗ )0(u)) = (i ⊗ 0)0(u∗ )(i ⊗ 0)(∗ )(u ⊗ 0(u))(1 ⊗ 0(u∗ )(u ⊗ u))× ((u∗ ⊗ u∗ )0(u) ⊗ 1)(0(u∗ ) ⊗ u∗ )(0 ⊗ i)()(0 ⊗ i)0(u) = (i ⊗ 0)0(u∗ )(i ⊗ 0)(∗ )(1 ⊗ ∗ )( ⊗ 1)(0 ⊗ i)()(0 ⊗ i)0(u) = [(0 ⊗ i)0(u)]∗ ∂2 (0 ⊗ i)0(u). Now one can see that ∂2 u ∈ (0 ⊗ i)0(M )0 iff ∂2 ∈ (0 ⊗ i)0(M )0 , and that ∂2 u = 1 iff ∂2 = 1. Obviously, if both and u are 2-pseudo-cocycles, then ∂2 u = ∂2 . Definition 2.5. 2-pseudo-cocycles (resp., 2-cocycles) 1 and 2 are said to be pseudocohomologous (resp., cohomologous) if ∗1 (2 )u ∈ 0(M )0 (resp., 1 = (2 )u ) for some unitary u ∈ M . Obviously, the condition ∗1 (2 )u ∈ 0(M )0 is equivalent to any from the following conditions: [(2 )u ]∗ 1 ∈ 0(M )0 , [(1 )u ]∗ 2 ∈ 0(M )0 or ∗2 (1 )u ∈ 0(M )0 . Proposition 2.6. Let (M, 0) be a Hopf–von Neumann algebra, 1 a 2-pseudo- cocycle and (M, 01 ) the twisting of (M, 0) as in 2.3. Let 2 be a unitary in M ⊗ M and 02 (x) := 2 0(x)∗2 be a monomorphism from M to M ⊗ M . Then (M, 02 ) is the twisting of (M, 0) isomorphic to (M, 01 ) by means of an inner automorphism π(x) := uxu∗ of M (this means that 02 ◦ π = (π ⊗ π) ◦ 01 , u ∈ M is a unitary) iff 2 is a 2-pseudo-cocycle pseudo-cohomologous to 1 by means of u. Proof. Rewrite the relation 02 ◦ π = (π ⊗ π) ◦ 01 as 2 0(u)0(x)0(u∗ )(2 )∗ = (u ⊗ u)1 0(x)∗1 (u∗ ⊗ u∗ ) or as ∗1 (2 )u 0(x) = 0(x)∗1 (2 )u (∀x ∈ M ). After that the first statement is obvious. If the mentioned condition is satisfied, then (M, 02 ) is a Hopf–von Neumann algebra, from which we have, using 2.3, that 2 is a 2- pseudo-cocycle, pseudo-cohomologous to 1 . Obviously, in this case the twistings (M, 01 ) and (M, 02 ) are isomorphic. Corollary 2.7. Let 1 be a 2-pseudo-cocycle of a Hopf–von Neumann algebra (M, 0) and 2 be a unitary in M ⊗ M such that ∗1 (2 )u ∈ 0(M )0 for some unitary u ∈ M . Then 2 is a 2-pseudo-cocycle, pseudo-cohomologous to 1 . Proposition 2.8. Let (M, 0, κ) be a co-involutive Hopf–von Neumann algebra ([ES], 1.2.5), be a 2-pseudo-cocycle of (M, 0). Then a Hopf-von Neumann algebra (M, 0 ) posesses a co-involution of the form κ (x) = uκ(x)u∗ (u ∈ M is a unitary) iff ς(κ ⊗ κ)()()u ∈ 0(M )0 , where ς(x ⊗ y) = y ⊗ x, x, y ∈ M.
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Proof. In fact, the relation 0 (κ (x)) = ς(κ ⊗ κ )0 (x) (∀x ∈ M ) is equivalent to the relation 0(uκ(x)u∗ )∗ = ς(u ⊗ u)(κ ⊗ κ)(0(x)∗ )(u∗ ⊗ u∗ ), or 0(u)0(κ(x))0(u∗ )∗ = (u ⊗ u)ς(κ ⊗ κ)(∗ )0(κ(x))ς(κ ⊗ κ)()(u∗ ⊗ u∗ ), or
ς(κ ⊗ κ)()u 0(κ(x)) = 0(κ(x))ς(κ ⊗ κ)()u ,
from which the result is obvious.
Remark 2.9. Let (M, 0, κ) be a co-involutive Hopf–von Neumann algebra, and be a 2-(pseudo-)cocycle for (M, 0); then it is straightforward to check that (κ ⊗ κ)(∗ ) is a 2-(pseudo-)cocycle for (M, ς0), and, therefore, that ς(κ ⊗ κ)(∗ ) is a 2-(pseudo)cocycle for (M, 0). Due to Corollary 2.7, Proposition 2.8 can be reformulated: a twisting (M, 0 ) of a co-involutive Hopf–von Neumann algebra (M, 0, κ) by means of a 2pseudo-cocycle posesses a co-involution of the form κ (x) = uκ(x)u∗ iff a 2-pseudococycle ς(κ ⊗ κ)(∗ ) is pseudo-cohomologous to . Definition 2.10. Let (M, 0, κ) be a co-involutive Hopf–von Neumann algebra, be a 2(-pseudo)-cocycle for (M, 0). We shall say that is pseudo-co-involutive (resp., co-involutive) if a 2(-pseudo)-cocycle ς(κ ⊗ κ)(∗ ) is pseudo-cohomologous to (resp.,ς(κ ⊗ κ)(∗ ) = u ). We shall say that is strongly co-involutive, if ς(κ ⊗ κ)(∗ ) = . Notice that the paper [EVa] was devoted to studying strongly co-involutive 2(-pseudo)cocycles of Hopf–von Neumann and Kac algebras (see, in particular, ([EVa], 2.8, 2.10). Examples 2.11. (i) Let G be a locally compact group, and let us define, for all f in L∞ (G), κa (f ) by the equality, for all s in G: −1 κG a (f )(s) = f (s ). G Then, (L∞ (G), 0G a , κa ) is a co-involutive Hopf–von Neumann algebra ([ES], 1.2.9); let ω be a 2(-pseudo)-cocycle on G, as defined in 2.2. If ω(t, s) satisfies, for almost all s, t in G the equality: ω(s−1 , t−1 ) = ω(t, s),
then it is strongly co-involutive (see also ([EVa], 2.9)). On the other hand, let us suppose that the 2-cocycle ω(t, s) is continuous on G × G and ω(e, s) = ω(s, e) = 1 (∀s ∈ G). Put in the main 2-cocycle equality s1 = s3 = (s2 )−1 = s−1 , then we have: u(s) = ω(s−1 , s) = ω(s, s−1 ) = (κG a u)(s). G ∗ −1 −1 Let us verify that a 2-cocycle ς(κG a ⊗ κa )(ω )(t, s) = ω(s , t ) is always cohomologous to ω(t, s) by means of the mentioned u(s), i.e., that
ω(s−1 , t−1 ) = ω(t−1 , t)ω(s−1 , s)ω(t, s)ω(s−1 t−1 , ts).
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In fact, using twice the main 2-cocycle equality, one has: ω(s−1 , t−1 )ω(s−1 t−1 , ts)ω(t, s) = ω(s−1 , t−1 )ω(s−1 t−1 , t)ω(s−1 , s) = ω(t−1 , t)ω(s−1 , s). So that, any mentioned 2-cocycle on a locally compact group is co-involutive. (ii) In the special case G = Rn the main 2-cocycle equality has the following form: ω(x, y)ω(x + y, z) = ω(y, z)ω(x, y + z) (∀x, y, z ∈ Rn ), and one can easily verify that this equality is satisfied for ω(x, y) := exp(iB(x, y)), where B(x, y) is any real bilinear form on Rn . Obviously, the above 2-cocycle is always co-involutive, and it is strongly co-involutive iff the bilinear form B(x, y) is skewsymmetric.
3. 2-Cocycles of Kac Algebras and Woronowicz Algebras 3.1. One can find the notions of a Kac algebra K = (M, 0, κ, ϕ), its fundamental unitary ˆ = (M ˆ the dual Kac algebra K ˆ , 0, ˆ κ, W , the canonical involutive isometries J and J, ˆ ϕ) ˆ and the duality theory for Kac algebras in [ES]. Similar notions and facts concerning Woronowicz algebras W = (M, 0, κ, τ, ϕ) can be found in [MN]. There one can also find the examples of Kac and Woronowicz algebras. The following statements 3.2, 3.3, 3.4, 3.5 are the generalizations of the corresponding statements ([EVa], 3.6, 3.7, 3.8, 3.9) for the case when the co-involution is also deformed (i.e., u is not necessarily 1). Because of that, the deformation of the fundamental unitary of a Kac algebra is more complicated. Proposition 3.2. Let K = (M, 0, κ, ϕ) be a Kac algebra (or W = (M, 0, κ, τ, ϕ) a Woronowicz algebra), W its fundamental unitary. Let R be unitary from M ⊗ M , and a 2-pseudo- cocycle of (M, 0). With the notations of 3.1, let us put ˜ R˜ = (Jˆ ⊗ J)R∗ (Jˆ ⊗ J) and W,R = W R. Then, for all x in M , we have ∗ 0 (x) = W,R (1 ⊗ x)W,R .
Proof. By the definition of J, we get that R˜ belongs to M ⊗ M 0 ; therefore, the result is trivial. Lemma 3.3. With the hypothesis of 3.2, let us suppose that is a co-involutive 2cocycle for (M, 0, κ), u is a corresponding unitary in M , and R = u V (where V is a unitary from M ⊗ M such that (0 ⊗ i)(V ) = (i ⊗ 0)(V )). We have then: ∗ ˜ ˜ ˜ u )(σ ⊗ i), (0 ⊗ i)(R)( ⊗ 1) = (σ ⊗ i)(1 ⊗ W ∗ )(σ ⊗ i)(1 ⊗ R)(σ ⊗ i)(1 ⊗ W )(1 ⊗
˜ u = (Jˆ ⊗ J)(u )∗ (Jˆ ⊗ J). where σ means the flip on Hϕ ⊗ Hϕ ,
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Proof. Let us write j(x) = Jx∗ J; this way we define an anti-isomorphism from M to M 0 , or from M 0 to M . We have then R˜ = (κ ⊗ j)(R), and, therefore ˜ = (0 ⊗ i)(κ ⊗ j)(R) = (ς(κ ⊗ κ) ⊗ j)(0 ⊗ i)(R). (0 ⊗ i)(R) Using 2.10, we then get: ∗ ˜ ⊗ 1) = (ς(κ ⊗ κ) ⊗ j)[(u ⊗ 1)(0 ⊗ i)(u )(0 ⊗ i)(V )] = (0 ⊗ i)(R)( (ς(κ ⊗ κ) ⊗ j)[(1 ⊗ u )(i ⊗ 0)(u )(i ⊗ 0)(V )] = [(ς(κ ⊗ κ) ⊗ j)(i ⊗ 0)(R)](ς ⊗ i)(1 ⊗ ˜u ), which is equal, using the implementations of j and κ, to ˜ u )(σ ⊗ i), (σ ⊗ i)(Jˆ ⊗ Jˆ ⊗ J)(1 ⊗ W )(σ ⊗ i)(1 ⊗ R∗ )(σ ⊗ i)(1 ⊗ W ∗ )(Jˆ ⊗ Jˆ ⊗ J)(1 ⊗ ˆ is equal to and, using the property linking W , W ∗ and J, J, ˜ u )(σ ⊗ i), (σ ⊗ i)(1 ⊗ W ∗ )(Jˆ ⊗ Jˆ ⊗ J)(σ ⊗ i)(1 ⊗ R∗ )(σ ⊗ i)(Jˆ ⊗ Jˆ ⊗ J)(1 ⊗ W )(1 ⊗
which gives the result.
Proposition 3.4. With the hypothesis and notations of 3.3, we have: (0 ⊗ i)(W,R ) = (W,R )23 (W )13 , where W := W,R for R = u . Proof. Using the definition, we get ∗ ˜ ⊗ 1) = (0 ⊗ i)(W,R ) = ( ⊗ 1)(0 ⊗ i)()(0 ⊗ i)(W )(0 ⊗ i)(R)( ∗ ˜ ⊗ 1). = (1 ⊗ )(i ⊗ 0)()(0 ⊗ i)(W )(0 ⊗ i)(R)( Using then the fact that (0 ⊗ i)(W ) = (1 ⊗ W )(σ ⊗ i)(1 ⊗ W )(σ ⊗ i), we get, using ∗ ˜ ⊗ 1) is equal to 3.3, that (0 ⊗ i)(W )(0 ⊗ i)(R)( ˜ ˜ u )(σ ⊗ i). (1 ⊗ W )(1 ⊗ R)(σ ⊗ i)(1 ⊗ W )(1 ⊗
As (i ⊗ 0)() = (1 ⊗ W )(σ ⊗ i)(1 ⊗ )(σ ⊗ i)(1 ⊗ W ∗ ), we then get that ∗ ˜ ⊗ 1)(1 ⊗ W ∗ ) (i ⊗ 0)()(0 ⊗ i)(W )(0 ⊗ i)(R)( is equal to u
˜ ˜ )(σ ⊗ i) (1 ⊗ W )(σ ⊗ i)(1 ⊗ )(σ ⊗ i)(1 ⊗ R)(σ ⊗ i)(1 ⊗ W )(1 ⊗ ˜ commute, we finally get: and, as (σ ⊗ i)(1 ⊗ )(σ ⊗ i) and (1 ⊗ R) (0 ⊗ i)(W,R ) = (1 ⊗ W,R )(σ ⊗ i)(1 ⊗ W )(σ ⊗ i), which is the result.
Theorem 3.5. Let K = (M, 0, κ, ϕ) be a Kac algebra (or W = (M, 0, κ, τ, ϕ) a Woronowicz algebra), W its fundamental unitary, a co-involutive 2-cocycle of ˜ = (Jˆ ⊗ J)∗ (Jˆ ⊗ J), where J is the canonical involutive isom(M, 0, κ). Let us put etry Jϕ constructed by the Tomita- Takesaki theory, Jˆ is the canonical implementation on Hϕ of the anti-automorphism κ, and W = W ˜u ; then we get: (i) W∗ is a multiplicative unitary; ˜ Jˆ ⊗ J). (ii) W∗ = (Jˆ ⊗ J)u W ( Proof. By 3.2 and 3.4, we get (i); by the formula W ∗ = (Jˆ ⊗ J)W (Jˆ ⊗ J) and the ˜ (ii) is trivial. definitions of ˜u and ,
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4. Twisting of Kac Algebras The results of this section are the generalizations of the results of Sects. 4 and 5 of [EVa] for the case when the co-involution is also deformed (i.e, u is not necessarily 1) and for more general subgroups and 2-cocycles. Because of that, the corresponding calculations and arguments are more complicated. Lemma 4.1. Let (M, 0, κ, ϕ) be a Kac algebra (or W = (M, 0, κ, τ, ϕ) a Woronowicz algebra), W its fundamental unitary. Let K be a locally compact group, α a co-involutive K Hopf–von Neumann morphism from (L∞ (K), 0K a , κa ) to (M, 0, κ), ω(t, s) a continuous counital 2-cocycle from K ⊗ K to T and u(s) = ω(s−1 , s) a corresponding unitary in L∞ (K) as in 2.11, and let = (α⊗α)(ω). If ai , bi ∈ L∞ (K) are such that Σi (ai ⊗bi ) →i ω strongly, and of norm less or equal to 1, then: K ∗ ∗ ∗ u ∗ (i) Σi 0K a (κa (ai ))(bi ⊗ 1) is weakly converging to (u ⊗ 1)(ω ) , where −1 −1 −1 −1 ω u (t, s) := (u∗ ⊗ u∗ )ω0K a (u) = ω(t , t)ω(s , s)ω(t, s)ω(s t , ts); u ∗ ˆ ˜u ˆ (ii) Σi (α(bi )∗ ⊗ Jα(κK a (ai ))J is weakly converging to = (J ⊗ J)( ) (J ⊗ J). K ∗ ∗ Proof. (i) The finite sums Σi 0K a (κa (ai ))(bi ⊗ 1) are converging to the function G ω(s−1 t−1 , t), which equals (u∗ ⊗ 1)ς(κG a ⊗ κa )(ω)(t, s) (to see that, put in the main −1 −1 2-cocycle equality s1 = s , s2 = s3 = t). Since the 2-cocycle ω is co-involutive, the above function equals (u∗ ⊗ 1)(ω u )∗ . (ii) The application x → Jα(x∗ )J from L∞ (K) to M 0 is a homomorphism, and, therefore, all the finite sums Σi (α(bi )∗ ⊗ Jα(κK a (ai ))J are of norm less or equal to 1. Let ξ, ξ 0 , η, η 0 in Hϕ ; we then get 0 0 ((Σi α(bi )∗ ⊗ Jα(κK a (ai ))J)(ξ ⊗ ξ )|η ⊗ η ) = K 0 0 ˆ ((Jˆ ⊗ J)ς(κK a ⊗ κa )Σi (α(ai ) ⊗ α(bi ))(J ⊗ J)(ξ ⊗ ξ )|η ⊗ η ) → → ((Jˆ ⊗ J)ς(κ ⊗ κ)()(Jˆ ⊗ J)(ξ ⊗ ξ 0 )|η ⊗ η 0 ) =
((Jˆ ⊗ J)(u )∗ (Jˆ ⊗ J)(ξ ⊗ ξ 0 )|η ⊗ η 0 ) = ˜ u (ξ ⊗ ξ 0 )|η ⊗ η 0 ). ( Definition 4.2. With the hypothesis of 4.1, we shall say that is an abelian coinvolutive 2-cocycle of (M, 0, κ), constructed by (K, ω, α). As in ([EVa], 4.6), where the definition of an abelian cocycle was done for a more special situation, we can always suppose that α is injective. Remarks 4.3. (i) If W = (M, 0, κ, τ, ϕ) a Woronowicz algebra with ϕ finite (it comes then, using ([BS], §4), from a compact quantum group), or if K = (M, 0, κ, ϕ) is a compact type Kac algebra ([ES], 6.2), and is an abelian co-involutive 2-cocycle of (M, 0, κ) constructed by (K, ω, α); then, as ϕ ◦ α is a bounded left Haar measure on K, so K is compact. (ii) If W = (M, 0, κ, τ, ϕ) is a Woronowicz algebra such that the predual M∗ has a unit (it comes then from a discrete quantum group), or if K = (M, 0, κ, ϕ) is a discrete type Kac algebra ([ES], 6.3), and is an abelian co-involutive 2-cocycle of (M, 0, κ) constructed by (K, ω, α); then, as ◦ α is a unit of L1 (K), so K is discrete.
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Proposition 4.4. Let K = (M, 0, κ, ϕ) be a Kac algebra, W its fundamental unitary, K a locally compact group, α a co-involutive Hopf–von Neumann morphism from K (L∞ (K), 0K a , κa ) to (M, 0, κ), an abelian co-involutive 2-cocycle of (M, 0, κ) constructed by (K, ω, α) as in 4.2, u the corresponding unitary in (L∞ (K)). Let α(L∞ (K)) be included in M ϕ = {x ∈ M, σtϕ (x) = x, ∀t ∈ R}. Then ϕ is left-invariant with respect to 0 , i.e., we have, for all z in M + : (i ⊗ ϕ)0 (z) = ϕ(z)1. Proof. Let ai , bi in L∞ (K) such that Σi (ai ⊗ bi ) →i ω strongly, and all finite sums are of norm less or equal to 1. Let x, y in Nϕ ∩ N∗ϕ , and ξ 0 , η 0 be right bounded vectors with respect to ϕ; using 4.1(ii), we get that Σi (α(bi )∗ ⊗Jα(κK a (ai ))J) is weakly converging to u ∞ ϕ ˜ ˜ u (3ϕ (x) ⊗ 3ϕ (y)) , and, therefore, using that α(L (K)) lies in M , we get that W is the weak limit of W Σi (α(bi )∗ ⊗ Jα(κK a (ai ))J)(3ϕ (x) ⊗ 3ϕ (y)) = ∗ W Σi (3ϕ (α(bi )∗ x) ⊗ 3ϕ (yα(κK a (ai )) )) K ∗ ∗ = 3ϕ⊗ϕ (0(y)(α ⊗ α)(Σi 0K a (κa (ai ) )(bi ⊗ 1))(x ⊗ 1)),
˜ u (3ϕ (x) ⊗ 3ϕ (y)) is the weak limit of and therefore we get that (π 0 (ξ 0 ) ⊗ π 0 (η 0 ))W K ∗ ∗ ((π 0 (ξ 0 ) ⊗ π 0 (η 0 ))3ϕ⊗ϕ (0(y)(α ⊗ α)(Σi 0K a (κa (ai ) )(bi ⊗ 1))(x ⊗ 1)) = K ∗ ∗ 0 0 0(y)(α ⊗ α)(Σi 0K a (κa (ai ) )(bi ⊗ 1))(x ⊗ 1)(ξ ⊗ η )
and is, using 4.1(i), equal to 0(y)(u )∗ (α(u∗ )x ⊗ 1)(ξ 0 ⊗ η 0 ) = 0(yα(u∗ ))∗ (x ⊗ α(u))(ξ 0 ⊗ η 0 ). We then deduce that 0(yα(u∗ ))∗ (x ⊗ α(u)) belongs to Nϕ⊗ϕ and that u
˜ (3ϕ (x) ⊗ 3ϕ (y)) = 3ϕ⊗ϕ (0(yα(u∗ ))∗ (x ⊗ α(u))), W from which we get W (3ϕ (x) ⊗ 3ϕ (y)) = 3ϕ⊗ϕ (0 (yα(u∗ ))(x ⊗ α(u))), and then, taking into account that α(u) ∈ M ϕ , we have (ω3ϕ (x),3ϕ (x) ⊗ ϕ)[(1 ⊗ α(u∗ ))0 (α(u)y ∗ yα(u∗ ))(1 ⊗ α(u))] = (ϕ ⊗ ϕ)[(x∗ ⊗ α(u∗ ))0 (α(u)y ∗ yα(u∗ ))(x ⊗ α(u))] = (ϕ ⊗ ϕ)((x∗ ⊗ 1)0 (α(u)y ∗ yα(u∗ ))(x ⊗ 1)) = ϕ(x∗ x)ϕ(y ∗ y). Now the proof can be finished exactly as in ([EVa], 5.1).
4.5. In the situation described in 4.4 we had the formula: W (3ϕ (x) ⊗ 3ϕ (y)) = 3ϕ⊗ϕ (0(yα(u∗ ))∗ (x ⊗ α(u))) (∀x, y ∈ Nϕ ∩ N∗ϕ ). Let us introduce the following operator: Wu := (Jα(u)J ⊗ Jα(u)J)W (Jα(u∗ )J ⊗ Jα(u∗ )J). Obviously, Wu is equal to
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(1 ⊗ Jα(u)J)W (1 ⊗ Jα(u∗ )J) = (1 ⊗ Jα(u)J)W (Jˆ ⊗ J)0(α(u∗ ))∗ (α(u) ⊗ α(u))(Jˆ ⊗ J)(1 ⊗ Jα(u∗ )J) = (1 ⊗ Jα(u)J)W (Jˆ ⊗ J)0(α(u∗ ))(Jˆ ⊗ J)× ˆ Jˆ ⊗ Jα(u)J)(1 ⊗ Jα(u∗ )J) = (Jˆ ⊗ J)∗ (Jˆ ⊗ J)(Jα(u) ˆ ˆ ˜ Jα(u) ˜ Jα(u) Jˆ ⊗ 1) = W ( Jˆ ⊗ 1). (1 ⊗ Jα(u)J)(1 ⊗ Jα(u∗ )J)W ( We used here the inclusion of the commutative W∗ -algebra Jα(L∞ (K))J into M 0 and the relation W (Jˆ ⊗ J)0(a)(Jˆ ⊗ J) = (Jˆ ⊗ J)(1 ⊗ a)(Jˆ ⊗ J)W, which follows from the definition of W . We also have, using that α(L∞ (K)) ⊂ M ϕ and the above formula for W , that Wu (3ϕ (x) ⊗ 3ϕ (y)) = (1 ⊗ Jα(u)J)W (3ϕ (x) ⊗ 3ϕ (yα(u)) = (1 ⊗ Jα(u)J)3ϕ⊗ϕ (0 (y)(x ⊗ α(u))) = 3ϕ⊗ϕ (0 (x)(y ⊗ 1)). ∗ ˆ ) = α(u)Jˆ is the Lemma 4.6. (Wu )∗ = (Jˆu ⊗ J)Wu (Jˆu ⊗ J), where Jˆu := Jα(u ∗ implementation of κ (·) := uκ(·)u .
Proof. Using 4.5, we have: ˆ ˜ Jα(u) Jˆ ⊗ 1)∗ = (Wu )∗ = (W ( ∗ ˆ ˆ (Jα(u )J ⊗ 1)(Jˆ ⊗ J)(Jˆ ⊗ J)W ∗ (Jˆ ⊗ J)(Jˆ ⊗ J)∗ = ∗ ˆ ˆ ˆ Jˆ ⊗ 1)(Jα(u )J ⊗ 1)(Jˆ ⊗ J) = (Jˆu ⊗ J)W (Jˆ ⊗ J)∗ (Jˆ ⊗ J)(Jα(u) ∗ ˆ ˆ )J ⊗ 1)(Jˆu ⊗ J) = (Jˆu ⊗ J)Wu (Jˆu ⊗ J). (Jˆu ⊗ J)Wu (Jα(u
Corollary 4.7. (Wu )∗ is multiplicative unitary. Proof. The statement follows from 3.5 (i) and ([BS], 1.2).
Theorem 4.8. Let K = (M, 0, κ, ϕ) be a Kac algebra, W its fundamental unitary, K a locally compact group, α a co-involutive Hopf–von Neumann morphism from K ∞ ϕ (L∞ (K), 0K a , κa ) to (M, 0, κ) such that α(L (K)) ⊂ M , ω a co-involutive cocycle on K as in 4.1, and = (α ⊗ α)(ω). Then K = (M, 0 , κ , ϕ) is again a Kac algebra whose fundamental unitary is Wu . Proof. Using 4.4, 4.5 instead of ([EVa], 5.1) and 4.6 instead of ([EVa], 3.9 (ii)), one can repeat the proof of ([EVa], 5.2). Remarks 4.9. (i) If a 2-cocycle ω from K × K to T is strongly co-involutive (see 2.10), then u = 1, Theorem 4.8 is valid, even if ω is measurable, κ = κ, and we are in the situation, slightly more general than in ([EV], 5.2). (ii) If in the situation described in 4.8 a Kac algebra K is unimodular, i.e., ϕ = ϕ ◦ κ is the trace on M (see [ES], 6.1.3), then so is K . (iii) It is possible to get in the context of Woronowicz algebras a result similar to Theorem 4.8, if, in addition, α(L∞ (K)) ⊂ M τ = {x ∈ M, τt (x) = x, ∀t ∈ R} (see [MN] for the definition of τt ).
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5. Quantizations of the Heisenberg Group 5.1. The general way of getting examples of non-trivial Kac algebras is the deformation of a locally compact group G with the help of 2-(pseudo)-cocycles lifted from its abelian closed subgroup H such that 1G (s) = 1, for all s in H, where 1G is the modulus of G (see ([EVa], 6.1, 6.2), but now we can deal with more general subgroups and cocycles. This gives a possibility to get a much wider class of deformations 5.2. The Heisenberg group Hn (R) can be considered as the semi-direct product Rn+1 ×α Rn , where the action α of Rn on Rn+1 is given, for a, b in Rn , t in R, by αa (b, t) = (b, t + (a|b)). Therefore, the product rule in Hn (R) is given by (a, a0 , b, b0 in Rn , t, t0 in R): (b, t, a)(b0 , t0 , a0 ) = (b + b0 , t + t0 + (a|b0 ), a + a0 ) and, as the action α leaves the Lebesgue measure of Rn invariant, we are in both situations described in ([EVa], 6.2). The group Hn (R) is unimodular, and the Hilbert space L2 (Hn (R)) can be identified with L2 (Rn ) ⊗ L2 (R) ⊗ L2 (Rn ). So, we get that the left regular representation λ(b, t, a) of Hn (R) is defined by λ(b, t, a)f (v, u, w) = f (v − b, u − t − (a|v − b), w − a), where u belongs to R, v, w to Rn , f to L2 (Rn ) ⊗ L2 (R) ⊗ L2 (Rn ). Let us define a unitary U on that Hilbert space by (see ([EVa], 6.3)): Z 1/2 ˆ uˆ iuuˆ f (v, u, vˆ − w)ei(v|v) e dudv, U f (v, ˆ u, ˆ w) = (|u|/2) ˆ Rn ×R
where u, uˆ belong to R, v, w, vˆ to Rn , f to L2 (Rn ) ⊗ L2 (R) ⊗ L2 (Rn ) and we get that the left regular representation λ(b, t, a) of Hn (R) verifies: U λ(b, t, a)U ∗ f (u, v, w) = ei(b|v)u eitu f (u, v + a, w). Therefore, this representation is equivalent to the representation π on L2 (Rn+1 ) defined, for any φ in L2 (Rn+1 ), by: π(b, t, a)φ(u, v) = ei(b|v)u eitu φ(u, v + a). This representation generates the von Neumann algebra Ł∞ (R) ⊗ L(L2 (Rn )), which is therefore isomorphic to L(Hn (R)); by this isomorphism, there exits a symmetric Kac algebra (Ł∞ (R) ⊗ L(L2 (Rn )), 0s , κs , ϕs ), such that 0s (π(b, t, a)) = π(b, t, a) ⊗ π(b, t, a), κs (π(b, t, a)) = π(−b, −t + (a|b), −a). Since we have
π(b, t, 0) = ((u, v) → ei(b|v)u eitu ),
[ n+1 ) into Ł∞ (R)⊗L(L2 (Rn )) we get from the inclusion then the morphism β1 from L∞ (R Rn+1 ⊂ Hn (R), sends the function (u, v) → eitu ei(b|v) on π(b, t, 0), and, therefore, for [ n+1 ), β (f ) is the function (u, v) → f (u, uv). any f in L∞ (R 1
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As π(0, 0, a) = 1 ⊗ ρ(a), where ρ is the right regular representation of Rn , the cn ) into Ł∞ (R) ⊗ L(L2 (Rn )) we get from the inclusion Rn ⊂ morphism β2 from L∞ (R Hn (R), is given by: β2 (f ) = 1 ⊗ F f F ∗ , cn ) to L2 (Rn ). where F is the Fourier–Plancherel unitary from Ł2 (R [ n+1 by the formula: Using 2.11 (ii), we can construct a 2-cocycle on R ˆ v; ˆ uˆ 0 ,vˆ 0 ) ˆ v; ˆ uˆ0 , vˆ0 ) = eiB(u, , ωB (u,
[ n+1 . b v, cn , B(·, ·; ·, ·) is a real bilinear form on R where u, ˆ uˆ0 belong to R, ˆ vˆ0 belong to R So, B = (β1 ⊗ β1 )(ωB ) is a 2-cocycle for (Ł∞ (R) ⊗ L(L2 (Rn )), 0s , κs ), and B is the operator of multiplication on the fucnction defined on Rn+1 × Rn+1 by 0
0
0
B (u, v, u0 , v 0 ) = eiuu B(u,v;u ,v ) . A deformed coproduct 0 is defined by 0B (π(b, t, a)) = B (π(b, t, a) ⊗ π(b, t, a))∗B , and we obtain 0
0
0
0
0
0
0B (π(b, t, a))f (u, v; u0 , v 0 ) = eiuu [B(u,v;u ,v )−B(u,v+a;u ,v +a)] eit(u+u ) × 0
0
×ei[(v|b)u+(v |b)u ] f (u, v + a; u0 , v 0 + a), where u, u0 belong to R, v, v 0 to Rn , and f belongs to L2 (Rn+1 × Rn+1 ). The operator uB acting in L2 (Rn+1 ) is the operator of multiplication on the function defined on Rn+1 by 2 uB (u, v) = e−iu B(u,v;u,v) . A deformed co-involution κB is defined by κB (π(b, t, a)) = uB κs (π(b, t, a))u∗ B , and we obtain 0B (π(b, t, a))φ(u, v) = eiu
2
[B(u,v−a;u,v−a)−B(u,v;u,v)] iu((a|b)−t) i(v|b)u
e
e
φ(u, v − a),
cn ). for any φ in L2 (R × R According 4.8, this construction gives a series of new unimodular Kac algebras (Ł∞ (R) ⊗ L(L2 (Rn )), 0B , κB , ϕs ). We can consider the Lie algebra L of the Heisenberg group Hn (R), and define, where u belongs to R, v to Rn , vk is the k th component of v, and φ is in the Schwartz algebra S(Rn+1 ), its generators Pk , R, Qk (k ∈ {1, ..., n}) by Pk φ(u, v) =
∂ ∂φ [π(b, t, a)φ(u, v)]|a=0,b=0,t=0 = (u, v), ∂ak ∂vk
∂ [π(b, t, a)φ(u, v)]|a=0,b=0,t=0 = iuφ(u, v), ∂t ∂ [π(b, t, a)φ(u, v)]|a=0,b=0,t=0 = iuvk φ(u, v), Qk φ(u, v) = ∂bk Rφ(u, v) =
2-Cocycles and Twisting of Kac Algebras
709
which are the infinitesimal operators of the representation π. These operators are linked by the commutation relations [Pk , R] = [Qk , R] = 0 and [Pk , Qj ] = δjk R. The symmetric coproduct 0s of the enveloping algebra U (L) satisfies: 0s (Pk ) = Pk ⊗ 1 + 1 ⊗ Pk , 0s (R) = R ⊗ 1 + 1 ⊗ R, 0s (Qk ) = Qk ⊗ 1 + 1 ⊗ Qk ; the antipode κs and the involution ∗ verify κs (Pk ) = Pk∗ = −Pk , κs (R) = R∗ = −R, κs (Qk ) = Q∗k = −Qk . Then the above quantization leads to the following formulas on the level of quantized universal enveloping algebra: 0B (Pk ) = Pk ⊗ 1 + 1 ⊗ Pk + R ⊗
∂B 2 ∂B (R , iQ) + 0 (R2 , iQ) ⊗ R, ∂vk ∂v k
0B (R) = R ⊗ 1 + 1 ⊗ R, 0B (Qk ) = Qk ⊗ 1 + 1 ⊗ Qk , and also κB (Pk ) = −Pk + (
∂B ∂B + )(R2 , iQ), P ∗ k = −Pk , ∂vk ∂v 0 k
κB (R) = R∗ = −R, κB (Qk ) = Q∗k = −Qk , where k ∈ {1, ..., n}, Q is the vector with the components Qk . One can see from these formulae that the above deformation is symmetric iff the bilinear form B is symmetric. In particular, for B(u, ˆ v; ˆ uˆ0 , vˆ0 ) = qjk vˆ j vˆ0 k , (where n is more than 1, j, k ∈ ˆ {1, ..., n}, j 6= k, vˆ j (resp., v 0 j ) is the j th component of vˆ (resp., vˆ0 ), qjk ∈ R)), we have on the level of quantized universal enveloping algebra, for l, m ∈ {1, .., n}, l 6= k, l 6= j: 0jk (Pl ) = Pl ⊗ 1 + 1 ⊗ Pl , 0jk (Pj ) = Pj ⊗ 1 + 1 ⊗ Pj + iqjk (R ⊗ Qk ), 0jk (Pk ) = Pk ⊗ 1 + 1 ⊗ Pk + iqjk (Qj ⊗ R), 0jk (R) = R ⊗ 1 + 1 ⊗ R, 0jk (Qm ) = Qm ⊗ 1 + 1 ⊗ Qm , κjk (Pl ) = −Pl , P ∗ l = −Pl , κjk (Pj ) = −Pj + iqjk RQk , P ∗ j = −Pj , κjk (Pk ) = −Pk + iqjk RQj , P ∗ k = −Pk , κij (R) = R∗ = −R, κij (Qm ) = Q∗m = −Qm . cn by the formula: Using 2.11 (ii) again, we can construct a 2-cocycle ω B on R 0
ˆ vˆ ) ω B (v, ˆ vˆ 0 ) = eiB(v, ,
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cn , B(·, ·) is a real bilinear form on R cn . So, B = (β2 ⊗ β2 )(ω B ) is a where v, ˆ vˆ 0 ∈ R ∞ B 2 n 2-cocycle for (Ł (R) ⊗ L(L (R )), 0s , κs ), and as the operator can be written as follows: ˆ vˆ 0 ) (1 ⊗ F ∗ ⊗ 1 ⊗ F ∗ ). B = (1 ⊗ F ⊗ 1 ⊗ F )eiB(v, Let B˜ be an n by n matrix over R corresponding to the bilinear form B : B(v, ˆ vˆ 0 ) = (B˜ v| ˆ vˆ 0 ), and let B˜ t be the corresponding transposed matrix. Then one can calculate: ˆ ˆ = eitu ei(a|v) φ(u, vˆ − ub) (1 ⊗ F ∗ )π(b, t, a)(1 ⊗ F )φ(u, v)
cn ), from which we get: for any φ in L2 (R × R 0
0
ˆ vˆ )) (1 ⊗ F ∗ ⊗ 1 ⊗ F ∗ )0B (π(b, t, a))φ(u, v; ˆ u0 , vˆ 0 ) = eit(u+u ) ei(a|(v+ × 0
0
0
ˆ vˆ ) −iB(v−ub, ˆ vˆ −u b) ×eiB(v, e φ(u, vˆ − ub; u0 , vˆ 0 − u0 b)
cn × R × R cn ), and, using the Fourier transform again, we get for any φ in L2 (R × R 0
0
0
0
0B (π(b, t, a))f (u, v; u0 , v 0 ) = eit(u+u ) e−iuu B(b,b) e[u(v|b)+u (v |b)] × ˜ + a), f (u, v + u0 B˜ t b + a; u0 , v 0 + uBb where u, u0 ∈ R, v, v 0 ∈ Rn . In a similar way one can calculate a deformed co-involution κB : κB (π(b, t, a))f (u, v) = ˆ v) ˆ ˆ v) ˆ (1 ⊗ F )e−iB(v, (1 ⊗ F ∗ )(π(−b, (a|b) − t, −a))(1 ⊗ F )eiB(v, φ(u, v) ˆ = ˆ v) ˆ iu((a|b)−t) −i(a|v) ˆ iB(v+ub, v+ub) ˆ (1 ⊗ F )e−iB(v, e e e ˆ φ(u, vˆ + ub) =
eiu((a|b)−t) eiu
2
B(b,b)
ˆ iu[B(v,b)+B(b, ˆ v)] ˆ (1 ⊗ F )e−i(a|v) e φ(u, vˆ + ub) =
eiu((a|b)−t) e−iu
2
B(b,b) −iu(v|b)
e
f (u, v + u(B˜ + B˜ t )b − a).
This construction gives a series of new unimodular Kac algebras (Ł∞ (R) ⊗ L (L (Rn )), 0B , κB , ϕs ). On the level of a quantized universal enveloping algebra we have 0B (Pk ) = Pk ⊗ 1 + 1 ⊗ Pk , 2
0B (R) = R ⊗ 1 + 1 ⊗ R, 0B (Qk ) = Qk ⊗ 1 + 1 ⊗ Qk − iR ⊗
∂B ∂B (P ) − (P ) ⊗ iR, ∂v 0 k ∂vk
κB (Pk ) = P ∗ k = −Pk , κB (R) = R∗ = −R, κB (Qk ) = −Qk − R(
∂B ∂B + )(iP ); Q∗k = −Qk , ∂v 0 k ∂vk
where k ∈ {1, ..., n}, P is the vector with the components Pk . One can see from these formulae that the above deformation is symmetric iff the bilinear form B is symmetric.
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711
In particular, for B(v, ˆ vˆ0 ) = qjk vˆ j vˆ0 k , (where n more than 1 j, k ∈ {1, ..., n}, j 6= k, th 0 ˆ vˆ j (resp.,v j ) is the j component of vˆ (resp., vˆ0 ), qjk ∈ R)), we have on the level of a quantized universal enveloping algebra, for l, m ∈ {1, .., n}, l 6= k, l 6= j: 0jk (Pm ) = Pm ⊗ 1 + 1 ⊗ Pm , 0jk (R) = R ⊗ 1 + 1 ⊗ R, 0jk (Qk ) = Qk ⊗ 1 + 1 ⊗ Qk − iqjk (Pj ⊗ R), 0jk (Qj ) = Qj ⊗ 1 + 1 ⊗ Qj − iqjk (R ⊗ Pk ), 0jk (Ql ) = Ql ⊗ 1 + 1 ⊗ Ql , κjk (Pm ) = P ∗ m = −Pm , κij (R) = R∗ = −R, κjk (Qj ) = −Qj − iqjk RPk , Q∗ j = −Qj , κjk (Qk ) = −Qk − iqjk RPj , Q∗ k = −Qk , κjk (Ql ) = Ql ∗ = −Ql . Notice that the quantizations of the Heisenberg group of the above mentioned type corresponding to the simplest skew-symmetric forms B, were considered in ([EVa], 6.3). On the other hand, representing any real bilinear form B as the sum of its symmetric and skew symmetric components, one can show that, up to an isomorphism, the twisting is given essentially by the skew-symmetric part of B.
6. Finite Dimensional Kac Algebras 6.1. First of all, recall that a finite dimensional Kac algebra is nothing but a finite dimensional C∗ -Hopf algebra ([ES], 6.6). This means that M is a semisimple *-algebra over C such that xx∗ 6= o for any x ∈ M , and that 0 and κ commute with ∗. This means also that there exist a linear mapping m : M ⊗M → M which defines the multiplication in M , an ε ◦ κ = κ, (ε ⊗ i)0(x) = (i ⊗ ε)0(x) = x, m(κ ⊗ i)0(x) = ε(x)1, m(i ⊗ κ)0(x) = ε(x)1 for all x ∈ M . Remark that a compact quantum group in the sense of [W1], or a discrete quantum group in the sense of [ER], if it is finite dimensional, is nothing but a finite dimensional Kac algebra (see ([W1, A.2], [ER, Va]). Lemma 6.2. Any counital 2-cocycle of a finite dimensional Kac algebra (i.e., such that (ε ⊗ i) = (i ⊗ ε) = 1), satisfies the condition ς(κ ⊗ κ)(∗ ) = u with a unitary u = m(i⊗κ) = m◦ς(κ⊗i) = κ(u). This means that is co-involutive in the sense of 2.10. Also, ε(u) = 1.
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Proof. a) Rewrite the main 2-cocycle equality with the usual tensor notation = (1) ⊗ (2) ([Abe]): 0
0
0
0
((1) ⊗ (2) )0( (1) ) ⊗ (2) = (1) ⊗ ((1) ⊗ (2) )0( (2) ), 0
0
where (1) ⊗ (2) stands for another copy of . Apply i ⊗ κ ⊗ i to both sides of this equality: 0
0
((1) ⊗ 1 ⊗ 1)[(i ⊗ κ)0( (1) ) ⊗ (2) ](1 ⊗ κ((2) ) ⊗ 1) = 0
0
(1 ⊗ 1 ⊗ (2) )[ (1) ⊗ (κ ⊗ i)0( (2) )](1 ⊗ κ((1) ) ⊗ 1). Apply m ◦ (i ⊗ m) = m ◦ (m ⊗ i) to both sides of this equality, using that m[(a ⊗ 1)(b ⊗ c)(1 ⊗ d)] = a · m(b ⊗ c) · d ∀a, b, c, d ∈ M : 0
0
m(((1) ⊗ 1)[m(i ⊗ κ)0( (1) ) ⊗ (2) ](κ((2) ) ⊗ 1)) = 0
0
m((1 ⊗ (2) )[ (1) ⊗ m(κ ⊗ i)0( (2) )](1 ⊗ κ((1) ))), from where one can get, using the relations of 6.1 and the counitality of : u = (1) κ((2) ) = (2) κ((1) ), or u = m(i ⊗ κ) = m ◦ ς(κ ⊗ i) = κ(u). The property ε(u) = 1 follows from the counitality of . b) Let us show that uu∗ = 1. Indeed, we have 0
0
uu∗ = (1) κ((2) )κ( (1)∗ ) (2)∗ = 0
0
00
(1) κ( (1)∗ (2) ) (2)∗ = 00
00
(1)∗
(1)∗
ε(
00
(2)∗
0
0
0
)(1) κ( (1)∗ (2) ) (2)∗ =
(1) κ( (1)∗ (2) )ε(
00
(2)∗
0
) (2)∗ ,
00
where (1) ⊗ (2) stands for the third copy of . Let us show that the right-hand side of this equality is equal to 1. Rewrite the main 2-cocycle equality as follows: (0 ⊗ i)(∗ ) = (i ⊗ 0)(∗ )(1 ⊗ ∗ )( ⊗ 1) or 00
0((1)∗ ) ⊗ (2)∗ =
(1)∗
00
(1) ⊗ [0(
(2)∗
0
0
)( (1)∗ (2) ⊗ (2)∗ )].
From this we have 00
(i ⊗ κ)0((1)∗ )(1 ⊗ (2)∗ ) =
00
(1)∗
(1)∗
00
(1) ⊗ m(κ ⊗ i)[0( 0
(1) ⊗ κ( (1)∗ (2) )ε(
00
(2)∗
(2)∗
0
0
)( (1)∗ (2) ⊗ (2)∗ )] =
0
) (2)∗ .
Applying the map m to both sides of this equality and using the counitality of ∗ , one has: 00 0 00 0 1 = (1)∗ (1) κ( (1)∗ (2) )ε( (2)∗ ) (2)∗ , which gives uu∗ = 1. Let us show that u∗ u = 1. Indeed, we have 0
0
u∗ u = κ( (1)∗ ) (2)∗ (1) κ((2) ) = κ(
00
(1)
)ε(
00
(2)
0
0
)κ( (1)∗ ) (2)∗ (1) κ((2) ) =
2-Cocycles and Twisting of Kac Algebras 0
00
κ( (1)∗
713 (1)
0
) (2)∗ (1) ε(
00
(2)
)κ((2) ).
Let us show that the right-hand side of this equality is equal to 1. Rewrite the main 2-cocycle equality as follows: (0 ⊗ i)() = (∗ ⊗ 1)(1 ⊗ )(i ⊗ 0)() or 0
0
00
0((1) ) ⊗ (2) = ( (1)∗ ⊗ (2)∗ (1) ⊗ (2) )(
(1)
00
⊗ 0(
(2)
)).
From this we have 0
00
(κ ⊗ i)0((1) ) ⊗ (2) = κ( (1)∗
(1)
0
00
) ⊗ ( (2)∗ (1) ⊗ (2) )0(
(2)
),
and from this: 0
00
1 ⊗ 1 = (κ( (1)∗
(1)
0
) (2)∗ (1) ⊗ (2) )0(
00
(2)
).
Applying the map m ◦ (i ⊗ κ) to both sides of this equality we finally have 0
00
1 = κ( (1)∗
(1)
0
) (2)∗ (1) ε(
00
(2)
)κ((2) ),
which gives uu∗ = 1. Thus, u is unitary. c) Let us show that 0(u) = ∗ (u ⊗ u)ς(κ ⊗ κ)(∗ ), which is obviously equivalent to the main statement of the lemma. Using the main 2-cocycle equality in various forms and some of the above relations, we have 0(u) = 0(m ◦ (i ⊗ κ)()) = (m ⊗ m)(i ⊗ ς ⊗ i)(0 ⊗ 0)((i ⊗ κ)()) = (m ⊗ m)(i ⊗ ς ⊗ i)(i ⊗ i ⊗ ς(κ ⊗ κ))(i ⊗ i ⊗ 0)(0 ⊗ i)() = (m ⊗ m)(i ⊗ ς ⊗ i)(i ⊗ i ⊗ ς(κ ⊗ κ))(i ⊗ i ⊗ 0)(∗ ⊗ 1)(1 ⊗ )(i ⊗ 0)() = (m ⊗ m)(i ⊗ ς ⊗ i)(∗ ⊗ 1 ⊗ 1)(i ⊗ i ⊗ ς(κ ⊗ κ))[(1 ⊗ (i ⊗ 0)())(i ⊗ (i ⊗ 0)0)()] = ∗ (m⊗m)(i⊗ς ⊗i)(i⊗i⊗ς(κ⊗κ))[(1⊗(1⊗∗ )(⊗1)(0⊗i)())(i⊗(0⊗i)0)()] = ∗ (m ⊗ m)(i ⊗ ς ⊗ i)[(i ⊗ i ⊗ ς(κ ⊗ κ))((1 ⊗ ⊗ 1)((1) ⊗ (0 ⊗ i)(0((2) ))) (1 ⊗ 1 ⊗ ς(κ ⊗ κ)(∗ ))] = ∗ (m⊗m)(i⊗ς⊗i)(i⊗i⊗ς(κ⊗κ))[(1⊗⊗1)((1) ⊗(0⊗i)(0((2) )))]ς(κ⊗κ)(∗ ). Now we have to show only that (m ⊗ m)(i ⊗ ς ⊗ i)(i ⊗ i ⊗ ς(κ ⊗ κ))[(1 ⊗ ⊗ 1)((1) ⊗ (0 ⊗ i)(0((2) )))] = u ⊗ u. Indeed, (m ⊗ m)(i ⊗ ς ⊗ i)(i ⊗ i ⊗ ς(κ ⊗ κ))[(1 ⊗ ⊗ 1)((1) ⊗ (0 ⊗ i)(0((2) )))] = 0
(m ⊗ m)(i ⊗ ς ⊗ i)[(1 ⊗ (1) ⊗ 1 ⊗ 1)× 0
((1) ⊗ (i ⊗ ς(κ ⊗ κ))(0 ⊗ i)(0((2) )))(1 ⊗ 1 ⊗ 1 ⊗ κ( (2) ))] = 0
0
(1 ⊗ (1) )[(m ⊗ m)(i ⊗ ς ⊗ i)((1) ⊗ (i ⊗ ς(κ ⊗ κ))(0 ⊗ i)(0((2) )))](1 ⊗ κ( (2) )) = 0
0
((1) ⊗ (1) )[(m ⊗ m)(i ⊗ ς ⊗ i)(1 ⊗ (i ⊗ ς(κ ⊗ κ))(0 ⊗ i)(0((2) )))](1 ⊗ κ( (2) )). One can get the following equality for any A ∈ M ⊗ M : (m ⊗ m)[1 ⊗ (ς ⊗ i)(i ⊗ ς(κ ⊗ κ))(0 ⊗ i)(A)] =
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(m ⊗ m)[1 ⊗ (ς ⊗ i)(i ⊗ ς)((i ⊗ κ)0(A(1) ) ⊗ κ(A(2) ))] = (m ⊗ m)[1 ⊗ κ(A(2) ) ⊗ (i ⊗ κ)0(A(1) )] = κ(A(2) ) ⊗ ε(A(1) )1 = ς(1ε ⊗ κ)(A). In particular, we have (m ⊗ m)(i ⊗ ς ⊗ i)(1 ⊗ (i ⊗ ς(κ ⊗ κ))(0 ⊗ i)(0((2) ))) = ς(1ε ⊗ κ)(0((2) )) = κ((2) ) ⊗ 1. That is why we finally get: (m ⊗ m)(i ⊗ ς ⊗ i)(i ⊗ i ⊗ ς(κ ⊗ κ))[(1 ⊗ ⊗ 1)((1) ⊗ (0 ⊗ i)(0((2) )))] = 0
0
0
0
((1) ⊗ (1) )(κ((2) ) ⊗ 1)(1 ⊗ κ( (2) )) = (1) κ((2) ) ⊗ (1) κ( (2) ) = u ⊗ u. Remark 6.3. The similar statement was proved in ([M], 2.3.4) for quasitriangular Hopf algebras, but the quasitriangularity was used there in an essential way. 6.4. Commutative and symmetric (i.e., co-commutative) finite dimensional Kac algebras are associated with finite groups in the way explained in ([ES], 4.2.4, 4.2.5, 6.6). That is why it is interesting to study non-trivial (i.e., non commutative and non symmetric) finite dimensional Kac algebras. Let us specialize our construction for the case of finite group G and its abelian ˆ H > be the duality between subgroup H. Let Hˆ be the dual group for H and < H, them. Then there exists a family of selfadjoint idempotents Phˆ generating the abelian subalgebra C(H) of the group algebra C(G) such that: X X ˆ h > Pˆ , Pˆ = 1 ˆ h > λ(h), < h, < h, Phˆ Pgˆ = δh, ˆ gˆ Ph ˆ , λ(h) = h h ||H|| ˆ H ˆ h∈
0s (Phˆ ) =
X
h∈H
Pgˆ ⊗ Pgˆ −1 hˆ , κs (Phˆ ) = Phˆ −1 ,
ˆ g∈ ˆ H
ˆ ||H|| is the order of H. where g, h ∈ H, g, ˆ hˆ ∈ H, Using these idempotents, one can write the following formulae for , u and ∂2 : X X ω(x, y)(Px ⊗ Py ), u = m(i ⊗ κ) = ω(x, x−1 )Px , = ˆ x,y∈H
∂2 =
X
ˆ x∈H
ω(y, z)ω(x, yz)ω(x, y)ω(xy, z)(Px ⊗ Py ⊗ Pz ).
ˆ x,y,z∈H
Thus, in order to construct a concrete example of a non-trivial finite dimensional Kac algebra, one can choose a finite non commutative group G, its abelian subgroup H and a 2-(pseudo)-cocycle ω : Hˆ × Hˆ → T. Then, using the above formulae, one gets and ˆ u. In the case when ω is a 2-cocycle, also is, but if ω is only a 2-pseudo-cocycle on H, one should verify if is a 2-pseudo-cocycle on (L(G), 0s , κs ). If it is, and it is at least pseudo-coinvolutive (2.10), then we have already new finite dimensional coinvolutive Hopf–von Neumann algebra. If it has a counit, then it is a Kac algebra ([ES], 6.3.5). For this one should choose ω to be counital: ω(e, x) = ω(x, e) = 1 ∀x ∈ H which gives the counitality of : (ε ⊗ i) = (i ⊗ ε) = 1. Finally, if the new coproduct is not symmetric (i.e., non co-commutative), we have a non-trivial Kac algebra. The necessary and sufficient condition for this is, obviously, that the unitary ∗ ς() does not belong to 0(M )0 .
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Remark 6.5. The construction above and the examples below are valid in the framework of general semisimple Hopf algebras over any algebraically closed field of characteristic zero (to show this one needs some results from [Abe, LR, M], see also [N]). 6.6. We shall describe now two series of non-trivial Kac algebras by twisting two classical series of finite groups: dihedral and quasiquaternion. Let us start with the groups G1 = D2n = Z2n ×α Z2 (2 ≤ n ∈ N), with the following action α of Z2 = {1, b} on Z2n = {ak (k = 0, 1, ..., 2n − 1)}: αb (ak ) = a2n−k (k = 0, 1, ..., 2n − 1); and G2 = Qn having two generators: a of order 2n, (n ∈ N), and b of order 4 such that b2 = an and bab−1 = a−1 . So, any group G1 , G2 has 4n elements: {ak , bak (k = 0, 1, ..., 2n − 1)}, and the group algebra L(G) is in both cases isomorphic to C ⊕ C ⊕ C ⊕ C ⊕ M2 (C) ⊕ . . . ⊕ M2 (C) | {z } n−1
(see [HR], 27.61). Let e1 , e2 , e3 , e4 ,ej11 , ej12 , ej21 , ej22 (j = 1, ..., n−1), be the matrix units of this algebra; we can now write the left regular representation λ of G1 ([HR], 27.61): X j jk(2n−1) j (εjk e22 ), λ(ak ) = e1 + e2 + (−1)k (e3 + e4 ) + n e11 + εn j
λ(bak ) = e1 − e2 − (−1)k (e3 − e4 ) +
X
j (εjk(2n−1) ej12 + εjk n n e21 ),
j
where εn = eiπ/n . Consider the subgroup H = {e, an , b, ban } which is isomorphic to Z2 × Z2 . Since the dual group Hˆ is isomorphic to H, then, following 6.4, one can ˆ compute the orthoprojectors Phˆ , (hˆ ∈ H): Peˆ = e1 +
1 + (−1)n e 4 + q1 , 2
1 + (−1)n e 3 + q2 , 2 1 − (−1)n e 4 + p1 , Paˆ n = 2 1 − (−1)n e 3 + p2 , Pbˆ aˆ n = 2 where p1 , p2 , q1 , q2 are the orthogonal projectors defined by: X0 j (e11 + ej12 + ej21 + ej22 ), p1 = 1/2 Pbˆ = e2 +
j
p2 = 1/2
X0
q1 = 1/2
j
(ej11 − ej12 − ej21 + ej22 ),
X00 j
(ej11 + ej12 + ej21 + ej22 ),
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q2 = 1/2
X00 j
(ej11 − ej12 − ej21 + ej22 ),
P0 P00 where (resp., ) means that the corresponding index of summation takes only odd (resp., even) values. Respectively, one can write the left regular representation λ of G2 ([HR], 27.61): X j −jk j λ(ak ) = e1 + e2 + (−1)k (e3 + e4 ) + (εjk n e11 + εn e22 ), j
λ(ak b) = e1 − e2 + (−1)k (e3 − e4 ) +
X
j (εj(k−n) ej12 + ε−jk n n e21 ),
j
for even n, and λ(ak b) = e1 − e2 + i(−1)k (e3 − e4 ) +
X
j (εj(k−n) ej12 + ε−jk n n e21 ),
j
for odd n, where εn = eiπ/n . Consider the subgroup H = Z4 = {e, b, b2 , b3 }. As the dual group Hˆ is isomorphic ˆ to H, then, following 6.4, one can compute the orthoprojectors Phˆ , (hˆ ∈ H): Peˆ = e1 + e3 + q1 , Pbˆ = p1 , Pbˆ 2 = e2 + e4 + q2 , Pbˆ 3 = p2 , f or even n, Peˆ = e1 + q1 , Pbˆ = e4 + p1 , Pbˆ 2 = e2 + q2 , Pbˆ 3 = e3 + p2 , f or odd n, where p1 , p2 , q1 , q2 are the orthogonal projectors defined by: X0 j (e11 − iej12 + iej21 + ej22 ), p1 = 1/2 j
p2 = 1/2
X0 j
q1 = 1/2 q2 = 1/2
(ej11 + iej12 − iej21 + ej22 ),
X00 j
X00 j
(ej11 + ej12 + ej21 + ej22 ),
(ej11 − ej12 − ej21 + ej22 ),
P0 P00 where (resp., ) means that the corresponding index of summation takes only odd (resp., even) values. There exists a standard structure of a symmetric Kac algebra on L(G) with a symmetric coproduct 0s , involution ∗ and a co-involution κs defined, for all g in G by 0s (λ(g)) = λ(g) ⊗ λ(g), κs (λ(g)) = λ(g), (λs (g))∗ = λ(g −1 ). One can see that e1 is the projection given by the co-unit of L(G) ([ES], 6.3.5). The abelian subalgebra generated by λ(e), λ(an ), λ(b), λ(ban ) (resp., by λ(e), λ(b), λ(b2 ), λ(b3 )), is generated also by the mutually orthogonal orthoprojectors Peˆ , Paˆ n , Pbˆ , Pbˆ aˆ n (resp., Peˆ , Pbˆ , Pbˆ 2 , Pbˆ 3 ). The orthogonal projectors Peˆ + Pbˆ and Paˆ n + Pbˆ aˆ n (resp., Peˆ + Pbˆ 2 and Pbˆ + Pbˆ 3 ) are central.
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[ [2 2 c c Let us consider now the function ω on (Z 2 ) × (Z2 ) (resp., on Z4 × Z4 ) such that, [ 2 c for all u, h in (Z 2 ) (resp., in Z4 ): ω(e, ˆ u) = ω(u, u) = 1, ω(h, u) = ω(u, h) ˆ = ω(b, ˆ bˆ ˆ an ) = ω(bˆ ˆ an , aˆ n ) = i (resp., ω(b, ˆ bˆ 2 ) = ω(bˆ 2 , bˆ 3 ) = ω(bˆ 3 , b) ˆ = i). and ω(ˆan , b) [ 2 One can verify that this function is the 2-cocycle on (Z 2 ) (resp., the 2-pseudoc c cocycle on Z4 ; remark that all 2-cocycles on Z4 are symmetric: ω(h, u) = ω(u, h) and so give only symmetric twisting; that is why for getting a non-trivial twisting in this case we are forced to use 2-pseudo-cocycles). Let be the unitary element of L(G) ⊗ L(G) obtained by the lifting construction. For the group G1 we get = Peˆ ⊗ I + Paˆ n ⊗ (Peˆ + Paˆ n + i(Pbˆ − Pbˆ aˆ n )) + Pbˆ ⊗ (Peˆ + Pbˆ + i(Pbˆ aˆ n − Paˆ n )) + Pbˆ aˆ n ⊗ (Peˆ + Pbˆ aˆ n + i(Paˆ n − Pbˆ )). For the group G2 we respectively get = Peˆ ⊗ I + Pbˆ ⊗ (Peˆ + Pbˆ + i(Pbˆ 2 − Pbˆ 3 )) + Pbˆ 2 ⊗ (Peˆ + Pbˆ 2 + i(Pbˆ 3 − Pbˆ )) + Pbˆ 3 ⊗ (Peˆ + Pbˆ 3 + +i(Pbˆ − Pbˆ 2 )). It is useful to note that in both cases we can write = 1 + i2 . [ 2 Since ω is a 2-cocycle on (Z 2 ) , then is a 2-cocycle for (L(G1 ), 0s ). If we consider c c c on Z4 × Z4 × Z4 the function (s1 , s2 , s3 ) → ω(s2 , s3 )ω(s1 , s2 s3 )ω(s1 , s2 )ω(s1 s2 , s3 ), it takes value 1 except if s1 , s2 or s3 equals bˆ or bˆ 3 , for which it takes value -1; therefore, we get: ∂2 = I ⊗ I ⊗ I − 2(Pbˆ ⊗ Pbˆ ⊗ Pbˆ + Pbˆ ⊗ Pbˆ ⊗ Pbˆ 3 + Pbˆ ⊗ Pbˆ 3 ⊗ Pbˆ + Pbˆ 3 ⊗ Pbˆ ⊗ Pbˆ + Pbˆ ⊗ Pbˆ 3 ⊗ Pbˆ 3 + Pbˆ 3 ⊗ Pbˆ ⊗ Pbˆ 3 + Pbˆ 3 ⊗ Pbˆ 3 ⊗ Pbˆ + Pbˆ 3 ⊗ Pbˆ 3 ⊗ Pbˆ 3 ) = I ⊗ I ⊗ I− 2((Pbˆ + Pbˆ 3 ) ⊗ (Pbˆ + Pbˆ 3 ) ⊗ (Pbˆ + Pbˆ 3 )). Thus, ∂2 belongs to Z(L(G2 )) ⊗ Z(L(G2 )) ⊗ Z(L(G2 )) and is therefore a 2pseudo-cocycle for (L(G2 ), 0s ). So, in both cases (L(G), (0s ) ) is a Hopf–von Neumann algebra. It is clear that the above 2-cocycle is strongly co-involutive on (L(G1 ), 0s , κs ); let us show that the 2-pseudo-cocycle is pseudo-co-involutive on (L(G2 ), 0s , κs ) in the sense of 2.10, i.e., that ς(κs ⊗ κs )()(u∗ ⊗ u∗ )0s (u) belongs to 0s (L(G2 ))0 , where u = m(i ⊗ κs )() = Peˆ + Pbˆ 2 + i(Pbˆ 3 − Pbˆ ) (remark that κs (u) = u∗ !). Let us remark now that ς1 = (κs ⊗ κs )(1 ) = 1 ∗ = 1 , ς2 = (κs ⊗ κs )(2 ) = −2 ∗ = −2 , 1 2 = 2 1 , where 1 , 2 were introduced above. Using these relations, one can calculate that ς(κs ⊗κs )()(u∗ ⊗u∗ )0s (u) = (Peˆ +Pbˆ 2 −Pbˆ −Pbˆ 3 )⊗I which belongs to Z(L(G2 )) ⊗ Z(L(G2 )) ⊂ 0s (L(G2 ))0 . Thus, according to 2.8, 2.10 we get that (L(G1 ), (0s ) , κs ) (resp., (L(G2 ), (0s ) , uκs (·)u∗ )) is the co-involutive Hopf–von Neumann algebra. Moreover, in both cases we get that (e1 ⊗ x) = (e1 ⊗ x) = e1 ⊗ x, for all x in L(G); from which we conclude that e1 still gives a co-unit for the new structure; we also get that (0s ) (e1 ) = 0s (e1 ), and, by ([ES], 6.3.5), we get that this new structure is the Kac algebra.
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To prove that this new Kac algebra is non symmetric, it is enough to prove that ς(0s ) (λ(a)) 6= (0s ) (λ(a)), which is: ς(1 + i2 )(λ(a) ⊗ λ(a))(1 + i2 )∗ 6= (1 + i2 )(λ(a) ⊗ λ(a))(1 + i2 )∗ : Using the above relations for 1 , 2 , we see that this inequality is equivalent to: 1 (λ(a) ⊗ λ(a))2 6= 2 (λ(a) ⊗ λ(a))1 , and it is even enought to prove that (p ⊗ q)1 (λ(a) ⊗ λ(a))2 6= (p ⊗ q)2 (λ(a) ⊗ λ(a))1 , where p = p1 + p2 , q = q1 + q2 are in the center of L(G); one can obtain this last inequality, using the above expressions for 1 , 2 and λ(a), by direct calculations (for the case of the group G1 - only if 3 ≤ n, for n = 2 the deformation is co-commutative). Therefore, the Kac algebras in consideration are non symmetric and we obtain two series of non-trivial Kac algebras. Since, by construction, the coproducts 0s and (0s ) are equal on the subalgebra generated by λ(H), it is clear that the corresponding elements belong to the intrinsic group of this Kac algebra (i.e., they are group-like elements) (see 2.1). Therefore, the dual Kac algebra has at least 4 one-dimensional representations. The intrinsic group of the dual Kac algebra is equal to (Z2 )2 for the case of dihedral groups and to (Z2 )2 (if n is even) or to Z4 (if n is odd) for the case of quasiquaternion groups. That is why, at least for odd n, the above Kac algebras are non-isomorphic. Remark that some other description of Kac algebras with the above block-structure of an algebra is also given in ([IK], 5.5, 5.6). Remarks 6.7. (i) The above deformation for the case of dihedral groups was done in ([EVa], 6.5) for n = 3; it was observed in ([N], 5.3) that the construction is valid for general n. (ii) The above deformation for the case of quasiquaternion groups for n = 2 is nothing but the historical Kac-Paljutkin example of a non-trivial Kac algebra (see [KP3], 8.19, [KP2]), its description by means of twisting was done in ([EVa], 6.4); for n = 3, this deformation was done in ([Va1], 6.6); then it was observed in ([N], 5.4) that the construction is valid for general n. (iii) From [Wi, Mas2] one can deduce that the unique (up to an isomorphism) non-trivial semisimple Hopf algebra of dimension less or equal to 11 is the above example by Kac and Paljutkin. Since any finite dimensional semisimple Hopf algebra has no more than one Kac algebra structure (up to isomorphism) (see [And, Vays]), the same is true for Kac algebras of dimension less or equal to 11. On the other hand, using the theory of extensions of groups to Kac algebras (see [K1]) or its modification for finite dimensional semisimple Hopf algebras (see [Mas1] and references there), one can show (see [F]) that there exist exactly 2 (up to an isomorphism) non-trivial Kac algebras of dimension 12, both of them are selfdual and can be presented as extensions of the form 1 → CZ2 → K → CS3 → 1 ∗
and have the same C -algebra structure C ⊕ C ⊕ C ⊕ C ⊕ M2 (C) ⊕ M2 (C). c4 . Now it is clear that these Kac Their intrinsic groups are respectively (Z2 )2 and Z algebras are exactly isomorphic to the above mentioned ones, described by twisting.
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It was shown in [K2] that all Kac algebras of prime dimension are commutative and symmetric, all of them are associated with the corresponding cyclic group in the sense of 3.2 (i), (ii). Finally, one can conclude from [Mas1, LR] that all Kac algebras of dimensions 14, 15 are commutative or cocommutative. So, the three above described Kac algebras are all the non-trivial Kac algebras of dimension less or equal to 15. (iv) On the other hand, it was shown in ([N], 4.5, 4.6) that there exist non-trivial Kac algebras of dimensions 16 and 2m2 ( m ≥ 3 is prime) which cannot be obtained by twisting of any finite group. Indeed, in [S] a series of non-trivial Kac algebras 2 Km , m ≥ 3 with C∗ -algebra structure Cm ⊕ Mm (C) was constructed. Let now m be prime. We shall show that there is no group algebra with such a C∗ -algebra structure. Let G be any group of order 2m2 , where m ≥ 3 is prime. Then G should contain a subgroup H of order m2 (first Sylow theorem). The index of H is 2, so H is normal ([Kur, 9]). So, there exists a homomorphism α : G → G/H ' Z2 . Let χ1 be a non-trivial character of Z2 , then χ = χ1 ◦ α is a character of G and χ2 is a trivial character of G. The group of characters of G has an even order because it contains an element χ of order 2 (Lagrange’s theorem). Therefore, the number of one-dimensional representations of G is even and is not equal to m2 . So, Km cannot appear as an algebra of a deformation of a symmetric Kac algebra, when m ≥ 3 is a prime. However, below we shall see that the dual Kac algebras for Km can be obtained by twisting of certain finite groups. 6.8. We shall describe now a series of non-trivial Kac algebras by twisting the finite groups G = Zm 2 ×α Z2 (3 ≤ m ∈ N), with the following action α of Z2 = {1, s} on H = Zm 2 = {(a, b) (a, b = 0, 1, ..., m − 1)} : αs (a, b) = (b, a) (we use here the additive notation). So, G has 2m2 elements and any character of G is necessarily symmetric on H; since such a character takes the value 1 or -1 in s, G has exactly 2m characters. This group has an invariant abelian subgroup H of index 2, that is why all other irreducible representations of G are of dimension 2 ([CR], 53.18). Now it is clear that the group algebra L(G) is isomorphic to C ⊕ . . . ⊕ C ⊕ M2 (C) ⊕ . . . ⊕ M2 (C) . {z } | | {z } m(m−1) 2
2m
\2 \ 2 Let us consider now the following function ω on (Z m ) × (Zm ) : ˆ c)). ˆ cˆ, d) ˆ = exp( 2πi (ˆadˆ − bˆ ω(ˆa, b; m \ 2 One can verify that this function is the 2-cocycle on (Z m ) . Let be the unitary element of L(G) ⊗ L(G) obtained by the lifting construction: =
X ˆ H ˆ H×
exp(
2πi ˆ ˆ (ˆad − bˆc))P(a, ˆ ⊗ P(c, ˆ, ˆ b) ˆ d) m
where P(·,·) are orthoprojectors generating L(H) obtained as in 6.4. Since ω is a 2∗ \ 2 cocycle on (Z m ) , then is a 2-cocycle for (L(G), 0s ). One can also note that = ς,
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from where one can conclude that 0 is non-symmetric if and only if 2 (λ(s) ⊗ λ(s)) 6= (λ(s) ⊗ λ(s))2 . This inequality can be obtained by straightforward calculation. Thus, we have a series of non-trivial Kac algebras of dimension 2m2 . If m is odd, 2 then the dual Kac algebras have the above C∗ -algebra structure ([S], 3.1): Cm ⊕Mm (C). 2 Remark that we could use above for twisting the finite groups G = Zm ×α Z2 (3 ≤ \2 \ 2 m ∈ N) some other 2-cocycles ω on (Z m ) × (Zm ) : ˆ cˆ, d) ˆ = exp( ω(ˆa, b;
2πi ˆ cˆ, d)), ˆ B(ˆa, b; m
\ 2 where B(·; ·) is general bilinear form on (Z m) . Remark also that some other description of Kac algebras with the above blockstructure of algebra is also given in ([IK], 5.1). Remark 6.9. In the paper [N] the ring of representations (K0 -ring) of a finite-dimensional semisimple Hopf algebra (or Kac algebra) was considered and it was shown there that this ring is the twisting invariant. Using this fact, it was shown that the twisting of a simple finite group is a simple Hopf (or Kac) algebra (i.e., it does not have proper normal Hopf subalgebras). In this case the dual Hopf algebra does not have proper Hopf subalgebras. Using the lifting construction explained in 6.4, a series of nontrivial deformations of the symmetric groups Sn (n > 3) and the alternating groups An (n > 4) was described. The latter Hopf algebras are simple. Acknowledgement. I am deeply grateful to M. Enock for many valuable discussions without which this work could not appear. I am also grateful to V.G. Kac and H.-J. Schneider for important information on finitedimensional Hopf algebras. Finally, I am grateful to D. Nikshych for stimulating discussions.
References [Abe] Abe, E.: Hopf algebras. Cambridge: Cambridge University Press, 1977 [And] Andruskiewitsch, N.: Compact involutions of semisimple quantum groups. Czech. J. Phys. 44, 963– 972 (1994) [BS] Baaj, S. Skandalis, G.: Unitaires multiplicatifs et dualit´e pour les produits crois´es de C∗ -alg`ebres. Ann. Sci. ENS 26, 425–488 (1993) [CR] Curtis, C.W., Reiner, I.: Representation theory of finite groups and associative algebras New York. Interscience Publishers, 1962 [D] Drinfeld, V.G.: Quasi-Hopf algebras. Leningrad. Math. J. 1, 1419–1457 (1990) [ER] Effros, E.G. and Ruan, Z.-J.: Discrete quantum groups, I; the Haar measure. Int. J. Math. 5, 681–723 (1994) [ES] Enock, M. and Schwartz, J.-M.: Kac algebras and duality of locally compact groups, Berlin: Springer, 1992 [EN] Enock, M. and Nest, R.: Irreducible inclusions of Factors, Multiplicative Unitaries and Kac Algebras J. Funct. Anal. 137, 466–543 (1996) [EVa] Enock, M. and Vainerman, L.: Deformation of a Kac algebra by an Abelian subgroup Commun. Math. Phys. 178, 571–596 (1996) [F] Fukuda, N.: Semisimple Hopf algebras of dimension 12. Preprint [HR] Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis, v.II. Berlin: Springer-Verlag, 1970 [HS] Hong, J.H. and Szymanski, W.: Composition of Subfactors and Twisted Bicross Products. Preprint (1995) [IK] Izumi, M., Kosaki, H.: Finite dimensional Kac algebras arizing from certain group actions on factors. Preprint (1995) [K1] Kac, G.I.: Extensions of groups to ring-groups. Math. USSR Sbornik 5, 451–474 (1968)
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[K2] Kac, G.I.: Certain arithmetic properties of ring-groups. Funct. Anal. Appl. 6, 158–160 (1972) [KP1] Kac, G.I., Paljutkin: V.G.: Finite ring groups. Trans. Moscow Math. Soc. 15, 251–294 (1966) [KP2] Kac, G.I., Paljutkin, V.G.: An example of a ring group of order eight. (Russian), Soviet Math. Surveys 20 n.5, 268–269 (1965) [Kur] Kurosh, A.G.: it Group theory. Moscow: 1966 [L] Landstad, M.B.: Quantization arising from abelian subgroups. Int. J. Math. 5, 897–936 (1994) [LR] Larson, R.G. and Radford, D.E.: Semisimple Hopf algebras. J. Algebra 171, 5–35 (1995) [M] Majid, S.: Foundations of Quantum Group Theory. Cambridge: Cambridge University Press, 1995 [MN] Masuda, T. and Nakagami, Y.: A von Neumann Algebra Framework for the Duality of the Quantum Groups. Publ. RIMS, Kyoto Univ. 30 n.5, 799–850 (1994) [Mas1] Masuoka, A.: Semisimple Hopf algebras of dimension 2p. Preprint (1995) [Mas2] Masuoka, A.: Semisimple Hopf algebras of dimension 6 and 8. Preprint (1994) [N] Nikshych, D.: K0 -rings and twisting of finite dimensional Hopf Algebras. To appear in Commun. in Algebra (1997) [R1] Rieffel, M.A.: Compact Quantum groups associated with Toral subgroups. Contemp. Math. 145, 465–491 (1993) [R2] Rieffel, M.A.: Deformation Quantization for Actions of Rd . Memoirs A.M.S. 506 (1993) [S] Sekine, Y.: An example of finite dimensional Kac algebras of Kac-Paljutkin type. To appear in Proc. Amer. Math. Soc. [Va] Vainerman, L.I.: Relations between Compact Quantum Groups and Kac Algebras. Adv. in Soviet Math. 9, 151–160 (1992) [Va1] Vainerman, L.: 2-Cocycles and Twisting of Kac Algebras, Preprint-Nr. gk-mp-9610/41, Mathematisches Institut Ludwig-Maximilians-Universit¨at M¨unchen, (Oktober 1996) 36pp [Vays] Vaysleb, E.: Cosemisimple Bialgebras and Discrete Quantum Semigroups. Preprint, UCLA (1996) [Wi] Williams, R.: Finite dimensional Hopf algebras, Thesis, Florida State University, 1988 [W1] Woronowicz, S.L.: Compact matrix pseudogroups. Commun. Math. Phys. 11, 613–665 (1987) [W2] Woronowicz, S.L.: Compact quantum groups. Preprint (1993) Communicated by A. Connes
Commun. Math. Phys. 191, 723 – 734 (1998)
Communications in
Mathematical Physics c Springer-Verlag 1998
The Kowalevski Top: its r-Matrix Interpretation and Bihamiltonian Formulation I.D. Marshall? Centre des math´ematiques et leurs applications, Ecole normale sup´erieure de Cachan, 61 avenue du pr´esident Wilson, 94235 Cachan Cedex, France Received: 18 December 1995 / Accepted: 2 June 1997
Abstract: The Lax pair representation for the Kowalevski top and for the Kowalevski gyrostat are obtained directly via the r-matrix method. The construction of Reyman and Semenov–Tian-Shansky is then used to give a bihamiltonian description of the Kowalevski gyrostat 1. Introduction The Kowalevski top is the third integrable case of a heavy rigid body with a fixed point, rotating in a constant gravitational field. The other two cases are due respectively to Euler and to Lagrange. This third case was published in 1889 [6]. It has historical importance for the field of integrable systems and has been much studied since. It is remarkable that for so long this system resisted any attempts to give it a Lax representation. A significant milestone in the modern subject of integrable systems was the publication in 1987 of three different papers by three different groups of authors. These papers finally established the existence of a Lax representation for the Kowalevski top. In fact in each of the three papers a different Lax pair was found. The results of [4] and of [1] are important as they demonstrate interesting connections between the Kowalevski top and other well-known systems with Lax pairs. In both papers methods of algebraic geometry are used. Haine and Horozov in [4] showed that on any level surface of the invariants, the equations of motion for the Kowalevski top are equivalent to the Neumann system. Adler and van Moerbeke in [1] found connections between the Kowalevski top and the systems of Manakov and Henon-Heiles. A disadvantage with both of these results is that the entries of the matrices of the Lax pair depend on the physical variables via rational transformations. It should be underlined that both sets of results are distinguished by the fact that the Lax pairs can be said to ? Present address: Mathematics Department, University of Glasgow, Glasgow, G12 8QW, UK. E-mail: [email protected]
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have been constructed rather than guessed. This last fact represents a major achievement for each of the approaches of the respective pairs of authors of [7] and [1]. The result of interest for the present paper is that of Reyman and Semenov–TianShansky [8], in which they gave a Lax pair in the form of 4 × 4 matrices, and in which the variables of the system appear explicitly. The Lax pair given in [8] was arrived at by means of a process of symmetry reduction applied to a system of two interacting tops. From this point of view the Kowalevski top is a particular member of a whole family of integrable systems, which could therefore be considered to be a family of multi-dimensional generalisations of the Kowalevski top. The result presented here is supplementary to the result of Reyman and Semenov– Tian-Shansky in that it gives a direct description of the Lax pair in [8] by means of the r-matrix construction. The best reference for the background to the result in the present paper is the article [2] by Reyman and Semenov–Tian-Shansky with Bobenko. There it is explained additionally how with the help of their Lax representation one can study the geometry of the spectral curve and that the form of the solutions one obtains by using the machinery of finite-gap integration theory are simpler than those found by Kowalevski. This should not be understated, for in the work of Kowalevski the explicit solution of the system in terms of physical variables was extremely complicated and difficult to compute, and this part of the result was completed by K¨otter [5]. Another important reference is the chapter contributed by Reyman and Semenov–Tian-Shansky to Volume 16 of the Encyclopaedia of Mathematics series [9]. This is a good place to read about the r-matrix construction in the general context and it contains several other relevant references, as well as the Kowalevski top result. The book [3] is highly recommended: it describes the work of Kowalevski (concerning other results as well as the famous top result) in the context of nineteenth century mathematics and helps to makes it clear why this result was so highly thought of and how it represented such a significant advance at the time of its discovery.
Reyman and Semenov–Tian-Shansky defined a family of Hamiltonian systems on T ∗ SO(p, q) by considering the space ` so(p, q), τ of twisted loops based on so(p, q) for q ≤ p, where the involution τ is given by τ (X) = −X T , and using a construction originating in [7]. This construction gives, for any simple Lie group G with Cartan involution σ, a family of Poisson maps from T ∗ G with its canonical symplectic structure into `(g∗ , σ) with its r-matrix Poisson structure arising from the (standard) splitting corresponding to C [λ, λ−1 ] = C [λ] + λ−1 C [λ−1 ]. Restriction to a subfamily followed by the consequent Hamiltonian reduction by the action of SO(q) leads to a Poisson map from T ∗ SO(p) into ` so(p, q), τ . Let us denote this map 8. The r-matrix construction says that the family F of invariant functions on ` so(p, q), τ are in involution and hence the family 8∗ F are in involution on T ∗ SO(p). Due to the fact that sufficiently many of the elements in 8∗ F are independent, this procedure gives integrable systems on T ∗ SO(p). The image under d8 of the equations of motion takes the form of a Lax pair. For p = 3, q = 2 one of these systems turns out to be the Kowalevski top in the stationary reference frame. In physical variables the transformation from the stationary ∗ frame T ∗ SO(p) to the moving frame, where the phase space is so(p) n (Rp )q with its Lie Poisson bracket, is implemented in the loop space by a gauge transformation and this gauge transformation is not a Poisson map. What is important however is that it does preserve the property of the equations of motion that theyare prsented in Lax formg.
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∗ Let us denote by 9 the map thus obtained from so(p) n (Rp )q to ` so(p, q), τ . 9 is invertible whilst 8 in general is not invertible. Thus, there are two maps given in [2]: 8 : T ∗ SO(p) → ` so(p, q), τ , which ∗ is Poisson but noninvertible, and 9 : so(p) n (Rp )q → ` so(p, q), τ , which is invertible but not Poisson. The Poisson (non-Poisson) property of 8(9) is with respect to the standard r-matrix Poisson bracket on ` so(p, q), τ . For H(L) = 21 tr L2 |λ0 , 8∗ H gives the Hamiltonian for the generalised Kowalevski top in the stationary reference frame and 9∗ H gives the Hamiltonian for the generalised Kowalevski top in the moving reference frame. The map 8 gives an elegant proof of the integrability of the Kowalevski top as well as its generalisations, but as it is noninvertible it cannot be said to provide a true Lax representation. However the map 9 which, other than by long-hand computation in the moving frame phase space, cannot be used to prove the integrability of the Kowalevski top, is invertible and so does provide a true Lax representation. Indeed it can be used, as was demonstrated for the Kowalevski top case (as opposed to for all of its generalisations), in [2] via the finite-gap method, to solve the equations of motion explicitly. It would be nice to have a map from the physical phase space to a loop algebra which is both Poisson and invertible. In fact it is sufficient just to make 9 be a Poisson map and that is achieved in the present paper. Speaking in general terms, where g is any Lie algebra, if g can be split as a vector space direct sum into two subalgebras A and B, then a (nonstandard) Poisson structure can be defined on `(g∗ ) by splitting the ring `g into λg[λ] + A and λ−1 g[λ−1 ] + B and then specifying an r-matrix in the obvious way corresponding to this splitting of `g. The Lax pair representation of the Kowalevski top in the Euler-Poisson picture is shown to be a Poisson map with respect to a Poisson structure on `(so(p, q), τ ) of this kind. An unforseen advantage of using the new Poisson structure on the loop space is that it permits the use of a method of Reyman and Semenov–Tian-Shansky to find a bihamiltonian version of the integrability of the Kowalevski gyrostat. The most attractive example to treat from the point of view of this paper is the example of the Kowalevski top, or more particularly the Kowalevski gyrostat, for which p = 3 and q = 2. This is because for this case, when we adopt the procedure to be described in Sect. 3, we are led unavoidably to the example of the classical Kowalevski top. The more general (p, q) cases are found by the observation that one can restrict to certain Poisson subspaces. In the present context we can view the form of these subspaces as being suggested by the p = 3, q = 2 example, although of course we know what they are already following [8]. For this reason we devote Sect. 2 and Sect. 3 to the p = 3, q = 2 case for which we use the identification so(3, 2) = sp(4). Then we describe the general case in Sect. 4. In Sect. 5 when we discuss the bihamiltonian formalism for the Kowalevski top and its generalisations, we shall continue to artificially distinguish between the cases g = sp(4) and g = so(p, q) because of the simple form that the results take with the g = sp(4) notation, which then suggests the form that the results should take for the general case. The possibility to generalise the result of Sect. 3 to the general (p, q) case was pointed out to the author by A.G.Reyman [private communication 1996]. 1.1 Notation. Much use will be made of the Kronecker product of matrices. This has a standard notation, but in order to avoid confusion it is defined below. If A is a p × q matrix and B is an m × n matrix A ⊗ B is the pm × qn matrix,
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a11 B A ⊗ B = ...
a12 B .. .
...
ap1 B
ap2 B
...
a1q B .. . .
(1.1)
apq B
Use will also be made of the standard Pauli matrices σ1 , σ2 , σ3 . In fact we will use σ1 =
0 1
1 0
,
iσ2 =
0 1 −1 0
,
σ3 =
1 0
0 −1
,
(1.2)
as well as the 2 × 2 identity matrix which will be denoted by the symbol 1. The notation mat(p × q, R) means “the set of real p × q matrices”.
2. The Kowalevski Top: Some Known Results The content of this section can be found in [2] (and in [8] or [9]). In the moving reference frame the Kowalevski top is a hamiltonian system on the dual space e(3)∗ of the Lie algebra of the Euclidean group E(3) of rigid motions of three-dimensional Euclidean space. The phase space is six-dimensional and the standard dynamical variables are `1 , `2 , `3 , g1 , g2 and g3 , with Poisson bracket given by {`i , `j } = ijk `k ,
{`i , gj } = ijk gk ,
{gj , gj } = 0 .
(2.1)
In other words the phase space is the dual space of the semidirect product Lie algebra so(3) n R3 , with the Lie–Poisson bracket. There are two Casimir functions, I1 = g 2 and I2 = ` · g, and equating these to constants defines the generic symplectic leaves which are four dimensional. A hamiltonian system on this space will be integrable therefore if it has one non-trivial constant of motion. The Hamiltonian for the Kowalevski top is given by (2.2) H(`, g) = 21 (`21 + `22 + 2`23 ) − g1 . Reyman and Semenov–Tian-Shansky introduced a more general kind of system. They called their system the Kowalevski gyrostat in two fields. Its phase space is ninedimensional and can be thought of as the dual to the semidirect product Lie algebra so(3) n (R3 ⊕ R3 ). That is, its standard coordinates are `1 , `2 , `3 , g1 , g2 g3 , h1 , h2 and h3 and the Poisson bracket is given by {`i , `j } =ijk `k , {`i , gj } = ijk gk , {`i , hj } = ijk hk , {gi , gj } = {gi , hj } = {hi , hj } = 0 .
(2.3)
There are three Casimir functions, C1 = g 2 , C2 = h2 and C3 = g·h and generic symplectic leaves are 6 dimensional. The Hamiltonian function for the Kowalevski gyrostat is given by (2.4) H(`, g, h) = 21 (`21 + `22 + 2`23 ) + γ`3 − (g1 + h2 ) , where γ is a constant. The Kowalevski top is obtained from the Kowalevski gyrostat by imposing the invariant condition h = 0, i.e. C2 = 0, together with γ = 0. The following result is due to Reyman and Semenov–Tian-Shansky:
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Theorem [8]. The equations of motion for the Kowalevski gyrostat are equivalent to the Lax equation L˙ = [L, M ] where −`2 1 − `1 iσ2 −γiσ2 0 0 + 21 L(λ) = −λ `2 1 − `1 iσ2 −(2`3 + γ)iσ2 0 σ3 (2.5a) g 3 σ3 + h3 σ1 1 −1 (g1 − h2 )σ3 + (g2 + h1 )σ1 +2λ g 3 σ3 + h3 σ1 −(g1 + h2 )σ3 + (g2 − h1 )σ1 and
M (λ) = λ
0 0
0 σ3
+
1 2
−(2`3 + γ)iσ2 −`2 1 + `1 iσ2
`2 1 + `1 iσ2 (2`3 + γ)iσ2
.
(2.5b)
Two independent commuting integrals for the Kowalevski gyrostat are given by the functions and K2 = tr L4 |λ−2 , (2.6) K1 = tr L4 |λ0 where the subscripts mean respectively “take the λ0 component” and “take the λ−2 component” in the expansion in powers of λ. The fact that the Hamiltonian function of (2.4) (up to addition of a constant) is equal to H = − 21 tr L2 |λ0
(2.7)
is an indication that a straightforward r-matrix description might be possible. Note that the matrix L defined in (2.5a) differs from that in [8] by a factor of a half. This is convenient for present purposes, and of course makes no difference to the validity of the Lax pair.
3. The r-Matrix Construction Applied to the Kowalevski Gyrostat In this section it is shown that the Lax pair given by (2.5) can be obtained as a direct consequence of the r-matrix construction. Let g be sp(4). That is g = sp(4) = {X ∈ gl(4, R) | JXJ = X T } ,
(3.1)
where J = 1 ⊗ iσ2 , so that g = {X = A ⊗ 1 + S1 ⊗ σ1 + S2 ⊗ iσ2 + S3 ⊗ σ3 | AT = −A, SiT = Si } .
(3.2)
Define an involutive automorphism τ : g → g (i.e. τ satisfies τ 2 = id and [τ X, τ Y ] = τ [XY ] ∀X, Y ∈ g) by (3.3) τ (X) = −X T , then X = τ (X) ⇒ X = A ⊗ 1 + S ⊗ iσ2
(3.4)
X = −τ (X) ⇒ X = S1 ⊗ σ1 + S3 ⊗ σ3 .
(3.5)
and The twisted loop algebra associated with τ is given by
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`(g, τ ) = {X ∈ `g | X(−λ) = τ (X(λ))} n X = X= Xk λk ∈ `g | X2n is of the form A ⊗ 1 + S ⊗ iσ2 , k
o X2n+1 is of the form S1 ⊗ σ1 + S3 ⊗ σ3 .
(3.6)
In (3.4), (3.5) and in (3.6) the symbols S, S1 and S3 denote arbitrary symmetric 2 × 2 matrices, whilst A denotes an arbitrary antisymmetric 2 × 2 matrix. Defining the pairing h , i : `g × `g → R by hX, Y i = tr X(λ)Y (λ)|λ0 ,
(3.7)
we have Proposition 3.1. The restriction to `(g, τ ) × `(g, τ ) of the pairing given by (3.7) is nondegenerate and hence it can be used to identify `(g, τ )∗ with `(g, τ ). From now on we will identify `(g, τ )∗ = `(g, τ ) by means of h , i. For any element X ∈ `(g, τ ) we can write X = X+ + X0 + X−
(3.8)
with X+ (λ) a series in λ, X0 independent of λ and X− (λ) a series in λ−1 . Let P± and P0 be the projection operators on `(g, τ ) with respect to the decomposition of (3.8). Let `(g, τ )0 be the λ0 –component of `(g, τ ), i.e. `(g, τ )0 = P0 `(g, τ ) = {A⊗1+S ⊗ iσ2 }. Consider the splitting of `(g, τ )0 as a vector space direct sum of two subalgebras, `(g, τ )0 = A + B ,
(3.9)
with A = {A ⊗ 1 + S ⊗ iσ2 | tr S = 0},
B = {x(1 + σ3 ) ⊗ iσ2 | x ∈ R} .
(3.10)
Note that `(g, τ )0 is u(2) and A is su(2), while B is abelian. Defining ρ ∈ End (`(g, τ )0 ) by
ρ = PA − PB ,
(3.11)
where PA and PB are projection operators on `(g, τ )0 with respect to the splitting given by (3.9), we have the following Proposition 3.2. R ∈ End(`(g, τ )) given by R = P+ + ρ ◦ P0 − P−
(3.12)
Σ = 21 (σ3 − 1) ⊗ σ3 .
(3.13)
is a classical r-matrix. Let Then we have Proposition 3.3. The space K ⊂ `(g, τ ) (∼ `(g, τ )∗ ) given by K = {L(λ) = λΣ + α + λ−1 β} ,
(3.14) ∗
is a Poisson subspace with respect to the R-Lie-Poisson bracket on `(g, τ ) .
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Proof. The Proposition follows by a direct calculation and depends crucially on the fact that Σ commutes with B. In fact, if we suppose that C = S1 ⊗ σ1 + S3 ⊗ σ3 satisfies [C, (1 + σ3 ) ⊗ iσ2 ] = 0 (the condition that C be in the centraliser of B), then without loss of generality we can assume that S1 = 0 (otherwise we could make a similarity transformation which changes σ1 and σ3 and leaves 1 and iσ2 unchanged) and then this condition reads S3 (1 + σ3 ) + (1 + σ3 )S3 = 0 ,
(3.15)
i.e. S3 has the form x(1 − σ3 ), for some x ∈ R. If we choose as real coordinates on K the functions `i , gi , hi for i = 1, 2, 3 and γ, given by writing α = − 21 `1 σ1 ⊗ iσ2 − 21 `2 iσ2 ⊗ 1 − 21 `3 (1 − σ3 ) ⊗ iσ2 − 21 γ1 ⊗ iσ2
(3.16)
and β = 21 h1 σ3 ⊗σ1 − 21 h2 1⊗σ3 + 21 h3 σ1 ⊗σ1 + 21 g1 σ3 ⊗σ3 + 21 g2 1⊗σ1 + 21 g3 σ1 ⊗σ3 , (3.17) then we get Proposition 3.4. With respect to { , }R on the Poisson subspace K of `(g, τ )∗ , we have {γ, f } = 0, {`i , `j } = ijk `k ,
for any f ∈ C ∞ (K),
{`i , hj } = ijk hk ,
{`i , gj } = ijk gk ,
{hi , hj } = {gi , gj } = 0 .
(3.18) (3.19) (3.20)
Denoting by Kγ the subset of K given by fixing the value of γ arbitrarily, it follows that ∗ Kγ is isomorphic as a Poisson space to so(3) n (R3 ⊕ R3 ) , which is the phase space for the Kowalevski gyrostat. The r-matrix method guarantees that all coadjoint invariant functions on Kγ commute with respect to { , }R . It follows that the subset of C ∞ (K) generated by the functions Hk,j given by −2j 1 2k tr L(λ) λ (3.21) Hk,j (L) = 0 2k λ
commute amongst themselves. H = −H1,0 is the Hamiltonian function for the Kowalevski gyrostat. H2,0 and H2,−1 are two commuting second integrals. Remarks. (i) It has been shown that the Lax representation for the Kowalevski gyrostat can be obtained by direct application of the r-matrix approach for a particular choice of splitting of the twisted loop algebra based on sp(4). As such, the result is a minor but necessary comment on the result of Reyman and Semenov–Tian-Shansky. (ii) The use of the r-matrix of (3.12), with the operator ρ acting on the λ0 –component, amounts to the identification via the trace form of the dual space to su(2) not with su(2) itself (which is an obvious identification to make), but with the space span{iσ1 , iσ2 , i(1 − σ3 )} ∼ span{σ1 ⊗ iσ2 , iσ2 ⊗ 1, (1 − σ3 ) ⊗ iσ2 } .
(3.22)
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In effect, the choice of A and B in (3.10) was made in order to force this identification. However many other choices for B, such as span{1 ⊗ iσ2 + σ1 ⊗ iσ2 } with Σ = (1 − σ1 ) ⊗ σ3 , or
(3.23) span{1 ⊗ iσ2 + iσ2 ⊗ 1} with Σ = σ1 ⊗ σ1 + σ3 ⊗ σ3 ,
and so on, would eventually lead to precisely the same system, namely to the Kowalevski gyrostat. 4. The General Case In this section it is shown how to use the r-matrix method to obtain the generalisations of the Kowalevski top given in [8]. The author thanks A.G.Reyman for pointing out that the previous section can be extended in this way. The idea is to do just the same as was done in [8] by starting with the twisted loop algebra `(so(p, q), τ ), where τ : so(p, q) → so(p, q) is given by τ (X) = −X T , but to define on `(so(p, q), τ ) a nonstandard r-matrix analogous to the one used in Sect. 3. Indeed in the case p = 3, q = 2, using the identification so(3, 2) = sp(4), the two r-matrices are just the same. We assume that q ≤ p and we define the p × q matrix E by I (4.1) E= q , 0 where Iq is the q × q identity matrix. In the rest of this section we shall use the blockdecomposition of (p + q) × (p + q) matrices according to which the (1, 1) block is a p × p matrix and the (1, 2) block is a p × q matrix. Let g be any fixed element of the group SO(p). In an analogous fashion to that of Eq. (3.9), let us split `(so(p, q), τ )0 as a vector space direct sum of two subalgebras `(so(p, q), τ )0 = A + B, with
n A=
a 0
o 0 a ∈ so(p) , 0
n B=
gEbE T g T 0
(4.2) o 0 b ∈ so(q) . b
(4.3)
The freedom to choose g arbitrarily corresponds to the freedom in the choice of B mentioned at the end of the last section. We will suppose from now on with no loss of generality that g in (4.3) is the identity. We may note that `(so(p, q), τ )0 is so(p) ⊕ so(q), A is the same as so(p) whilst B is the same as so(q). As in the last section, we now define ρ ∈ End(`(so(p, q), τ )0 ) by ρ = PA − PB ,
(4.4)
where PA and PB are projection operators on `(so(p, q), τ )0 with respect to the splitting given by (4.2), and we go on as in the last section to define the r-matrix R ∈ End(`(so(p, q), τ )) by (4.5) R = P+ + ρ ◦ P0 − P− . We use the same formula as that in (3.7) to identify `(so(p, q), τ )∗ with `(so(p, q), τ ). We let
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0 E (4.6) ET 0 and we prove precisely the same proposition in this new context as was proved in the last section. Let us define the Poisson subspace K ⊂ `(so(p, q), τ ) by the same formula as that in (3.14) and consider a generic hamiltonian vector-field on K. If we write O 0 ω 0 0 E −1 + , (4.7) + λ K 3 L(λ) = λ 0T 0 0 ET 0 Σ=
with ω ∈ so(p), ∈ so(q), 0 ∈ mat(p × q, R), then for any function φ ∈ C ∞ (K) the hamiltonian vector-field generated by φ is given by ω˙ = [X − EY E T , ω] + 21 (50T − 05T ), ˙ = −E T [X, ω]E − [Y, ] + 1 E T (05T − 50T )E, 2
(4.8)
0˙ = (X − EY E T )0, where the components X ∈ so(p), Y ∈ so(q), 5 ∈ mat(p × q, R), of dφ are given by d φ(ω + tδω, 0 + tδ0, + tδ) = tr X δω + tr 5T δ0 + tr Y δ. (4.9) dt t=0
The constraint + E T ωE = γ, where γ is an arbitrary constant for q = 2 and γ = 0 for q 6= 2, is invariant under the flow generated by all hamiltonian vector-fields on K and hence the space P of all L ∈ K of the form P 3 L(λ) = L(λ)(ω, 0) = λΣ + α + λ−1 β ω 0 E + =λ T 0 0 E
0 γ − E T ωE
+λ
−1
O 0T
0 , 0
(4.10)
is itself a Poisson subspace of `(so(p, q), τ ). On P the R-Lie-Poisson bracket gives (4.11) {φ, ψ}(ω, 0) = tr ω[δω φ, δω ψ] + tr 0T δω φ δ0 ψ − δω ψ δ0 φ , or equivalently, the hamiltonian vector-field XH for H ∈ C ∞ (P) is given by ω˙ = [δω H, ω] + 21 δ0 H 0T − 0 δ0 H T , 0˙ = δω H 0.
(4.12)
In other words P is the dual of semidirect product Lie algebra so(p) n (Rp )q with its Lie-Poisson bracket. The functions on P of the form 1 tr L(λ)2k λ−2j (4.13) Hk,j = 2k λ0 are in involution and the generalised Kowalevski top on P has the Hamiltonian H1,0 = 21 tr ω 2 + 21 tr ωEE T ωEE T − tr ωEγE T + 2tr E T 0 + 21 γ 2 .
(4.14)
Thus the equations of motion for the generalised Kowalevski top are ω˙ = [EE T ωEE T − EγE T , ω] + (E0T − 0E T ), 0˙ = ω + EE T ωEE T − EγE T 0, where, as we recall, γ ∈ so(q) is zero unless q = 2.
(4.15)
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5. Bihamiltonian Formulation In this section it is shown how to extend the results of Sects. 3 and 4 in a very simple way, so as to give a bihamiltonian formulation of the Kowalevski gyrostat. We recall a construction of Reyman and Semenov–Tian-Shansky (see §4.2 in [9]): Let `(g) = g ⊗ C[λ, λ−1 ] and let `(g)+ = g ⊗ C[λ] and `(g)− = g ⊗ λ−1 C[λ−1 ]. Let R ∈ End(`(g)) be given by R|`(g)± = ±id and let Rk = R ◦ 3k , where 3 ∈ End(`(g)) is given by (3X)(λ) = λX(λ). Then all the Rk are classical r-matrices and the corresponding R-Lie-Poisson brackets on `(g)∗ are all compatible. If we wish to apply this construction to a twisted loop algebra `(g, τ ) then we must modify it to take into account the order of the involution τ . Thus we must let Rk = R ◦ 3ks , where s is the order of τ , i.e. s is the smallest positive integer for which τ s = id. Indeed for the r-matrices defined in (3.12) and in (4.5) it is easy to check that Rk = R ◦ 32k are all classical r-matrices and, by the construction of Reyman and Semenov–Tian-Shansky, that they give rise to compatible Poisson structures on `(sp(4), τ ) or on `(so(p, q), τ ). For the reader’s convenience the formula is now given for the two Poisson brackets on the whole space `(g, τ ) where, as is the case in thepresent examples, τ is an order 2 involution. For L ∈ `(g, τ ); for φ, ψ ∈ C ∞ `(g, τ ) let ξ = δL φ and η = δL ψ, i.e. φ(L+t1) = φ(L)+h1, ξi+O(t2 ) for all 1 ∈ `(g, τ ) and ψ(L+t1) = ψ(L)+h1, ηi+O(t2 ) for all 1 ∈ `(g, τ ). Then {φ, ψ}k (L) = h3−2k L, [(32k ξ)+ + (32k ξ)0A , (32k η)+ + (32k η)0A ] − [(32k ξ)0B + (32k ξ)− , (32k η)0B + (32k η)− ]i. (5.1) The next problem in applying the construction of Reyman and Semenov–TianShansky is to find convenient Poisson subspaces of the respective loop algebras. As is easily seen, whilst K in Sect. 3 and P ⊂ K in Sect. 4 are Poisson subspaces with respect to the R0 -Lie-Poisson bracket on `(g, τ ) for g = sp(4) or so(p, q) respectively, they are not Poisson subspaces with respect to the other Rk -Lie-Poisson brackets for k 6= 0. However one can check that a simple modification of these spaces gives what we want. The modification consists in adding a λ−2 -term which is to represent a Casimir with respect to the R0 -Lie-Poisson bracket encountered already, so that the extended space is invariant under all hamiltonian flows with respect to the R−1 -Lie-Poisson bracket. Thus for the g = sp(4) case we extend K ⊂ `(sp(4), τ ) and define the space ˆ (5.2) Kˆ = L(λ) = λΣ + α + λ−1 β + λ−2 x1 ⊗ iσ2 , where the matrices Σ, α, β are exactly the same as those in (3.14) and x ∈ R. For g = so(p, q) we extend P ⊂ K ⊂ `(so(p, q), τ ) and define n o 0 0 ˆ Pˆ = L(λ) = λΣ + α + λ−1 β + λ−2 , 0 B
(5.3)
where Σ, α, β are as they were in (4.10) and B ∈ so(q). It can be checked that Kˆ and Pˆ are Poisson subspaces of `(sp(4), τ ) and `(so(p, q), τ ) respectively with respect to the R0 -Lie-Poisson bracket and that the variables x and B respectively are Casimirs. It can also be checked that with respect to the R−1 -LiePoisson bracket both spaces are still Poisson subspaces. It follows that these spaces have compatible Poisson structures given by the R0 – and the R−1 –Lie-Poisson brackets.
The Kowalevski Top
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It is traditional for discussing bihamiltonian structures to write the compatible Poisson structures as hamiltonian vector-fields. Thus on Kˆ for the sp(4) case, we have ˙ `=ρ∧`+ξ∧g+η∧h g˙ = ρ ∧ g − xη X(0) H : h˙ = ρ ∧ h + xξ x˙ = 0; (5.4a) ˙1 = η3 + `2 k ` `˙2 = −ξ3 − `1 k `˙3 = ξ2 − η1 g˙1 = −(`3 + γ)η1 − `2 ξ3 + `3 ξ2 + (h1 + g2 )k g˙ = −ρ − (` + γ)η − ` ξ + ` ξ + (h − g )k 2 3 3 2 3 1 1 3 2 1 X(−1) : H g˙3 = ρ2 − (`3 + γ)η3 − `1 ξ2 + `2 ξ1 + h3 k ˙ h1 = ρ3 + (`3 + γ)ξ1 − `2 η3 + `3 η2 + (h2 − g1 )k h˙ 2 = (`3 + γ)ξ2 − `3 η1 + `1 η3 − (h1 + g2 )k h˙ 3 = −ρ1 + (`3 + γ)ξ3 − `1 η2 + `2 η1 − g3 k x˙ = (g · η − h · ξ) + h1 η2 − h2 η1 + g1 ξ2 − g2 ξ1 − ρ1 `2 + ρ2 `1 , (5.4b) where the components, ρ, ξ, η ∈ R3 and k ∈ R, of dH are given by d H(` + tδ`, g + tδg, h + tδh, x + tδx) = ρ · δ` + ξ · δg + η · δh + kδx. (5.4c) dt t=0 On Pˆ for the so(p, q) case, we have T 1 ω˙ = [X, ω] + 2 50 − 05 X(0) 0˙ = X0 + 21 5B H : ˙ B = 0; ω˙ = 21 E5T − 5E T + [ω, EY E T ] 0˙ = −XE + 21 ω5 + 5E T ωE + 0Y − EY E T 0 X(1) H : B˙ = E T [ω, X]E + [B, Y ] + 21 0T 5 − 5T 0 + E T (05T − 50T )E ,
(5.5a)
(5.5b)
where the components, X ∈ so(p), Y ∈ so(q), 5 ∈ mat(p × q, R), of dH are given by d H(ω + tδω, 0 + tδ0, B + tδB) = tr X δω + tr 5T δ0 + tr Y δB. (5.5c) dt t=0 It is now a simple exercise to compute, firstly for the sp(4) case: C = 21 x2 ⇒ X(0) = 0 = X(0) and ⇒ X(−1) C K ⇒
X(−1) K
=
X(0) H
for K = −`3 x − 21 (g 2 + h2 ) − γx for H =
1 2 2 (`1
+
`22
+
2`23 )
+ γ`3 − g1 − h2 ;
(5.6)
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and for the so(p, q) case C = 21 tr B 2 ⇒ X(0) = 0 = X(0) and ⇒ X(−1) C K ⇒
X(−1) K
=
X(0) H
for K = −tr EBE T ω + tr 0T 0 for H =
1 2 tr ω
T
ω + EE ωEE
T
(5.7) + 2tr 0E.
(−1) In both cases, X(0) gives the equations of motion of the Kowalevski top after H = XK setting x = 0 or B = 0. Thus we have a bihamiltonian version of the Kowalevski top.
Note. The results of this section were motivated by a talk of F.Magri. He had mentioned that it was possible to give a bihamiltonian version of the Kowalevski top and this gave the impetus to try and find an appropriate extension of the results of Sects. 3 and 4. It seems that the bihamiltonian formulation presented here differs in some way from that of which Magri spoke, which is due to G.Magnano. An account of the other version can be found in [Magnano 1996]. Acknowledgement. I would like to thank Franco Magri for lengthy discussions on the bihamiltonian formalism, especially concerning the Kowalevski top. I would also like to thank Alexei Reyman, who actually gave a material contribution to this paper, as is clear from §4. This paper was finally completed during a stay at the Institut Henri Poincar´e in Paris and the kind generosity of my hosts there is very much appreciated.
References 1. Adler, M., van Moerbeke, P.: The Kowalewski and H´enon-Heiles motions as Manakov Geodesic flows on SO(4) – a two-dimensional family of Lax pairs. Commun. Math. Phys. 113, 659–700 (1988) 2. Bobenko, A.I., Reyman, A.G., Semenov–Tian-Shansky, M.A.: The Kowalevski top 99 years later: A Lax pair, generalisations and explicit solutions. Commun. Math. Phys. 122, 321–354 (1989) 3. Cooke, R.: The Mathematics of Sonya Kowalewskaya. Berlin Heidelberg: Springer-Verlag, 1984 4. Haine, L., Horozov, E.: A Lax pair for Kowalewski’s top. Physica D 29, 173–180 (1987) 5. K¨otter, F.: Sur le cas trait´e par Mme Kowalewskie de rotation d’un solide autour d’un point fixe. Acta Math. 17, 209–264 (1893) 6. Kowalevski, S.: Sur le probl`eme de la rotation d’un corps solide autour d’un point fixe. Acta Math. 12, 177–232 (1889) 7. Reyman, A.G.: Integrable Hamiltonian systems connected with graded Lie algebras. Differential Geometry, Lie groups and mechanics II. Zap. Nauchn. Semin. LOMI 95, 3–54 (1980) [J.Soviet Math. 19, 1507–1545 (1982)] 8. Reyman, A.G., Semenov–Tian-Shansky, M.A.: Lax representation with a spectral parameter for the Kowalevski top and its generalisations. Lett. Math. Phys. 14, 55–62 (1987) 9. Reyman, A.G., Semenov–Tian-Shansky, M.A.: Group-theoretical methods in the theory of finite dimensional integrable systems. Part 2, Chapter 2. In: Encyclopaedia of Mathematical Sciences vol.16, eds. V.I.Arnold & S.P.Novikov, Berlin Heidelberg: Springer-Verlag 1994 Communicated by M.Jimbo This article was processed by the author using the TEX style file pjour1 from Springer-Verlag.
Commun. Math. Phys.191, 735 (1998)
Communications in
Mathematical Physics c Springer-Verlag 1998
Erratum Phase Space Bounds for Quantum Mechanics on a Compact Lie Group Brian C. Hall Department of Mathematics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada Received: 30 September 1997 / Accepted: 30 September 1997
Commun. Math. Phys. 184, no 1, 233–250 (1997) ˙ indicating the Three times on p. 238 the symbol γ¯ appears. This symbol should be γ, time derivative of γ. Communicated by D. Brydges
