Commun. Math. Phys. 186, 1-59 (1997)
Communications in
Mathematical Physics
(~) Springer-Verlag1997
Meanders and the Temperley-Lieb Algebra E Di Francesco, O. Golinelli, E. Guitter* Service de PhysiqueTh6odque, C.E.A. Saclay,F-91191 Gif sur YvetteCedex, France Received: 25 February 1996/Accepted: 12 August 1996
Abstract: The statistics of meanders is studied in connection with the Temperley-Lieb algebra. Each (multi-component) meander corresponds to a pair of reduced elements of the algebra. The assignment of a weight q per connected component of meander translates into a bilinear form on the algebra, with a Gram matrix encoding the fine structure of meander numbers. Here, we calculate the associated Gram determinant as a function of q, and make use of the orthogonalization process to derive alternative expressions for meander numbers as sums over correlated random walks.
1. Introduction The meander problem is one of these fundamental combinatorial problems with a simple formulation, which resist the repeated attempts to solve them. The problem is to count the number Mn of meanders of order n, i.e. of ineqnivalent 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. Two configurations are considered as equivalent if they are smooth deformations of one another. Apparently, the meander problem dates back the work of Poincar6 about differential geometry. Since then, it arose in various domains such as mathematics, physics, computer science [1] and fine arts [2]. In the late 80's, Arnold reactualized this problem in relation with Hilbert's 16th problem, concerning the enumeration of ovals of planar algebraic curves [3]. Meanders also emerged in the classification of 3-manifolds [4]. More recently, random matrix model techniques, borrowed from quantum field theory, were applied to this problem [5, 6]. As such, the meander problem seems to belong to the same class as large N QCD [7]. * E-mail: philippe,golinel,
[email protected]
2
P. Di Francesco,O. Golinelli, E. Guitter
In a previous paper [6], we made our first incursion into the meander problem, in trying to solve the compact folding problem of a polymer chain. Considering indeed a long closed polymer chain of say 2n identical monomers, we ask the question of counting the inequivalent ways of folding the whole chain onto itself, forbidding interpenetration of monomers. By compact folding, we mean that all the monomers are packed on top of each other. Accordingly, folding is a simple realization of objects with self-avoiding constraints. The reader may bear in mind the simple image of the folding of a closed strip of 2n stamps, with all stamps piled up on top of each other [8, 9].
~
~
bridge (a)
road
river (b)
Fig. 1. A compactlyfoldedpolymer(a) with 2n = 6 monomers,and the associatedmeander(b), obtainedby drawing a line (river) horizontallythroughthe monomers.Eachmonomerbecomesa bridge, and each hinge a segmentof road betweentwo bridges. The equivalence between this folding problem and the meander problem may be seen as follows. As illustrated in Fig. 1, drawing a line (river) across the 2n constituents (bridges) of the folded polymer, and pulling them apart, produces a meander of order n. The folding of a closed polymer chain and the meander problem are therefore completely identical. By analogy, we were led to define the meander counterpart of the folding problem of an open polymer chain: the semi-meanders. The latter are defined in the same way as meanders, except that the river is now semi-infinite, i.e. it has a source, around which the semi-meander is allowed to wind freely. We denote by ~rn the number of semi-meanders of order n, namely with n bridges. In this paper, we reconsider the meander and semi-meander problems in the framework of the Temperley-Lieb algebra [10]. This is based on a one-to-one correspondence between (multicomponent) semi-meanders and reduced elements of the Temperley-Lieb algebra. Similarly, (multicomponent) meanders are associated topairs of such elements. More precisely, the Tempedey-Lieb algebra is endowed with a bilinear structure out of which a Gram matrix can be constructed. In our language, the bilinear form associates to each pair of elements of the algebra a weight qC, where r denotes the number of connected components of the associated meander. In particular, the Gram matrix, as a polynomial of q, encodes all the relevant information about meander and semi-meander numbers. Here we obtain as a main result an exact compact expression for the determinant of the Gram matrix, referred to as the meander determinant. Far from solving the question of enumerating meanders, this gives however some partial information on the problem, and produces an exact solution to a meander-flavored issue. This result is summarized in Eq. (5.6), and proved by explicit orthogonalization of the Gram matrix. In a second step, we make use of the precise form of the change of basis in the orthogonalization process to derive various expressions for the semi-meander (Eq. (6.62)) and meander
Meanders and the Temperley-Lieb Algebra
3
(Eq. (6.63)) numbers as statistical sums over paths, with an interpretation as Solid On Solid (SOS) model partition functions. The paper is organized as follows. We start in Sect. 2 by giving basic definitions of (multi-component) meander (Eq. (2.1)) and semi-meander (Eq. (2.3)) numbers and associated polynomials in which a weight q is assigned to each connected component. The relation between (semi-)meanders and both arch configurations and walk diagrams is then discussed, and known results for q = • are reviewed (Eqs. (2.6)-(2.8)). Various conjectured and/or numerical asymptotic behaviors for large n are given (Eqs. (2.11)(2.18)). In Sect. 3, we introduce the Temperley-Lieb algebra TLn(q), and discuss its relation with walk diagrams and arch configurations, in one-to-one correspondence with reduced elements of the algebra. These reduced elements form a natural basis (basis 1) of TLr~(q). The contact with meanders is made through the introduction of a trace and a bilinear form on TL,~ (q) (Eqs. (3.11) and (3.14)). When evaluated on pairs of reduced elements (of the basis 1), this form generates the Gram matrix (Eq. (3.15)), which encodes the fine structure of meander numbers. In Sect. 4, we make a change from basis 1 to a new basis 2, in which the Gram matrix is diagonal. This allows for the calculation of the Gram determinant as a function of q (Eq. (5.6)), and the identification of its zeros (Eq. (5.10)) and their multiplicities (Eq. (5.23)). These results, together with a complete combinatorial proof are detailed in Sect. 5. The matrix for the change of basis 1 ~ 2 is studied in great detail in Sect. 6, where it is shown to obey a simple recursion relation (Eq. (6.29)). This equation is explicitly solved, in the form of matrix elements between two walk diagrams, factorized into a selection rule (with value 0 or 1, see Eq. (6.38)) multiplied by some weight, with a local dependence on the heights of the walk diagrams (Eq. (6.43)). This leads to expressions for the meander and semi-meander polynomials as sums over selected walk diagrams (Eqs. (6.62) and (6.63)). Analogous formulas are derived within the framework of SOS models (Eq. (6.90)), leading to various conjectures as to the asymptotic form of the meander and semi-meander polynomials for q _> 2. Section 7 is devoted to a refinement of the meander determinant for semi-meanders with fixed number of windings around the source of the river (Eq. (7.5)). A few concluding remarks are gathered in Sect. 8. Some technical ingredients are detailed in Appendices A,B and C.
2. Definitions
2.1. Meanders. A meander of order n is a planar configuration of a closed non-selfintersecting loop (road) crossing an infinite oriented line (river flowing from east to west) through 2n points (bridges). We denote by M,~ the number of topologically inequivalent meanders of order n. We extend the definition to a set of k roads (i.e., a meander with k possibly interlocking connected components). The number of meanders with k connected components is denoted by M~k). Note that necessarily 1 _< k < n. These numbers are summarized in the meander polynomial n
ran(q) = ~ M (k) qk k=l
(2.1)
The various meanders corresponding to n = 2 are depicted in Fig. 2. They correspond to the polynomial mz(q) = 2q + 2q 2. (2.2)
P. Di Francesco, O. Golinelli, E. Guitter
@00 k=l
k=2
Fig. 2, The four meanders of order n = 2, i.e. with 2n = 4 bridges. The two first ones have k = 1 connected component, the two other have k = 2 connected components
The numbers M~ k) are listed in [6] for 1 < k < n < 12.
2.2. Semi-meanders.
k=l
k=2
k=3
Fig. 3. The five semi-meandersof order n = 3, arranged according to their numbers k = 1,2, 3 of connected components.
A semi-meander of order n is a planar configuration of a closed non-self-intersecting loop (road) crossing a semi-infinite line (river with a source) through n points (bridges). Note that, in a semi-meander, the road may wind around the source of the river. We denote by )~/n the number of topologically inequivalent semi-meanders of order n, and by 37/~k) the number of semi-meanders with k connected components, 1 < k < n. We also have the semi-meander polynomial ran(q) =
M,~ ~ .
(2.3)
k=l
The various semi-meanders corresponding to n = 3 are depicted in Fig. 3. They correspond to the polynomial m3(q) = 2q + 2q 2 + q3. (2.4) The numbers ,~/~k) are listed in [6] for 1 < k < n < 14.
2.3. Arch configurations and (semi) meanders. A multicomponent meander may be viewed as the superimposition of two (top and bottom) arch configurations of order n, corresponding respectively to the configurations of the road on both sides of the river, as shown in Fig. 4. An arch configuration is simply a configuration of n planar non-intersecting arches (lying, say, above the river) linking the 2n bridges by pairs. The number of arch configurations o f order n is given by the Catalan number (2n)! cn ( n + 1)!n! (2.5) The set o f arch configurations o f order n is denoted by An. As an immediate consequence, as arbitrary multicomponent meanders are obtained by superimpositions of arbitrary top and bottom arch configurations, we have
Meanders and the Temperley-Lieb Algebra
5
...av i:iii:!ii
Fig. 4. Any meander is obtained as the superimposition of a top (a) and bottom (b) arch configurationsof same order (n = 5 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).
4
Fig. 5. Any semi-meander may be viewed as a particular meander by opening the semi-infinite river as indicated by the arrows. This doubles the number of bridges in the resulting meander, hence the order is conserved (n = 5 here). By construction, the lower arch configuration of the meander is always a rainbow arch configuration of same order. The number of connected components (k = 3 here) is conserved in the transformation.
m,~(1) = (c,~) 2
(2.6)
As for semi-meanders, upon o p e n i n g the semi-infinite river and dedoubling the bridges (cf. Fig. 5), they can also be v i e w e d as the superimposition of a top arch configuration of order n, and of a particular bottom "rainbow" arch configuration (namely that linking the ith bridge to the ( 2 n + 1 - i) th one, i = 1,2, ..., n). Therefore arbitrary m u l t i c o m p o n e n t semi-meanders m a y be obtained by s u p e r i m p o s i n g an arbitrary arch configuration with a rainbow of order n, leading to ~ n ( 1 ) = on.
(2.7)
In ref.[6], we have also proved the following results m~(-1) = {
O(cp)2
mn(-1) =
-(Cp)
{o
i f n = 2p ifn=2p+
1 ' (2.8)
i f n = 2p i f n = 2p + 1 "
Note that the o n e - c o m p o n e n t m e a n d e r and semi-meander n u m b e r s are recovered in the q ---+ 0 limit of respectively mn(q)/q and ~n(q)/q.
2.4. Walk diagrams.
A n arch configuration of order n m a y be viewed as a closed r a n d o m walk o f 2 n steps on a semi-infinite line, or equivalently its t w o - d i m e n s i o n a l
6
E Di Francesco, O. Golinelli, E. Guitter
extent, which we call a w a l k diagram, defined as follows 1. Let us first label the segments of river between consecutive bridges, namely the segment i lies between the i th and the (i + l ) th bridge, for i = 1,2, ..., 2n - 1. Let us also label by 0 and 2n, the semi-infinite portions of fiver respectively to the left of the first bridge and to the fight of the last one. To each portion of river i, we attach a height gi equal to the number of arches passing at the vertical of i. The nonnegative integers gi satisfy the following conditions go -- g2n -- 0, g i + l - g i C {4-1} i = 0 , 1 , . . . , 2 n - 1 .
(2.9)
The diagram formed by the broken line joining the successive points (i, gi), i = 0, 1,..., 2n, is the walk diagram corresponding to the initial arch configuration. This diagram represents the two-dimensional extent of a walk of 2n steps on the semi-infinite line g _> 0 starting and ending at its origin.
i
0
:
1 2
i i !
3
:
4 5 6 7 8
:
i
:
9 1011 12 1314 1 5 1 6 1718
Fig. 6. 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
Conversely, any walk diagram of 2n steps, characterized by integer heights g~ >_ 0, i = 0, ..., 2n, satisfying (2.9), corresponds to a unique arch configuration of order n. To construct the arch configuration corresponding to a walk diagram, notice that, going from left to fight along the river, whenever g~ - gi- 1 = 1, a new arch originates from the bridge i, whereas when gi - gi-1 ---- - - 1 , an arch terminates at the bridge i. We denote by Wn the set of walk diagrams of 2n steps. We have the identification Wn -
An.
(2.10)
In this paper, we will alternatively use the arch configuration and walk diagram pictures. Earlier numerical work [9, 5, 6] suggests that the (one-component) meander and semi-meander numbers behave in the large n limit as respectively 2.5. Asymptotics.
Rn
Mn
'~
nC` ,
~n J~n
~
n T ~
with 1 The walk diagrams are usually referred to as Dyck paths in the combinatorial literature.
(2.11)
Meanders and the Temperley-Lieb Algebra
7
/~ ~ 3.5...
R = R 2,
a = 7/2
"1' = 2.
(2.12)
The values of the exponents a and "7 are conjectured to be exact. The relation R =/~2 is a consequence of the polymer folding interpretation [6]: the entropy per monomer is the same for the open and closed polymer folding problems. Note however that the configuration exponents a and 2/depend on the boundary conditions (open or closed). A natural quantity of interest for the study of semi-meanders is the winding, namely the number of times the road winds around the source of the river in the river/road picture of the semi-meander. In the arch configuration picture, the winding of a semi-meander is the number of arches of the upper configuration passing at the vertical of the middle point; representing the upper arch configuration as a walk diagram a, the winding of the semi-meander is simply g~. Denoting by c(a) the number of connected components of the superimposition of the arch configuration a and of a rainbow configuration of order n, the average winding in semi-meanders of order n reads Wn(q) = Y'~aeW,~ g~ qc(a) ~ n~'(q), ~-~aEW,~ qc(a) "*---~oo
(2.13)
where we have identified a winding exponent v(q) E [0, 1]. In this paper, we give strong analytical evidence that v(q) = 1 for all q > 2. For 0 < q < 2, numerical work seems to indicate that 1/2 <_ v(q) < 1. More generally, we expect the meander and semi-meander polynomials to behave for large n as R(q) n ha(q),
rnn(q) ~
/~(q)n ~(q)
"
(2.14)
n'Y(q) '
where R(q) =/~(q)2 like in the q = 0 case, but only for q < 22 As an element of comparison, by using Stirling's formula for factorials, we have 4'* ~,dl)
= c,~ ,-~ / / 3 / 2 '
42 n rn,~(1) = (c~) 2 ,-~ n3 ,
(2.15)
hence/~(1) = 4 and a(1) = 3
"/(1) = 3/2.
(2.16)
We also have the obvious large q asymptotics ~'*(q)
~
qn, (4q) n
(2.17)
m'*(q) ~ c'* q~ ,'~ - -
723/2
2 This relation is only expected insofar as v ( q ) < 1. Indeed, in this case, comparing the numbers ran(q) and ~2~(q) of respectively meanders and semi-meanders with (2n) bridges, we see that the semi-meanders with significative winding (i.e., W n ~ n ) are negligible, hence we expect the two numbers to be of the same order, namely r a n ( q ) ~ R ( q ) n ~ rh2n(q) ~ /~(q)2n, hence R ( q ) =/~(q)2. According to the previous discussion, this fails for q > 2, where v ( q ) = 1. Indeed, it is easy to see that, for large q, R ( q ) / R ( q ) 2 ~ 4 / q ~ O, as r a n ( q ) ~ c n q n ~ (4q) n and rh2n(q) ~ q2n.
P. Di Francesco, O. Golinelli, E. Guitter
8
hence R(q) ,-~ 4q and [~(q) ~ q, whereas
~(~)
3
7(c~) = O.
=
(2.18)
3. Temperley-Lieb Algebra and Meanders 3.1. The Temperley-Lieb algebra and arch configurations. The Temperley-Lieb algebra of order n and parameter q, denoted by TL,~(q), is defined through its n generators 1, el, e2, ..., e n - l subject to the relations (i)
i = 1,2,...,n-
e~=qe~
[e~,ej] = 0
if ]i - j[ > 1,
(iii) eiei+l e~ = ei
i = 1,2,...,n-
(ii)
1, (3.1) 1.
This definition becomes clear in the "braid" pictorial representation, where the generators act on n parallel strings as follows:
1=
i i+l
ei =
i+l
(3.2)
n
and a product of elements is represented by the juxtaposition of the corresponding braid diagrams. 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)
2 ei =
(iii) ei ei+l ei = D
i
= q
+ti [ i+l =
9 ). ( - -
) (--
i i+l = q ei,
i
(3.3)
i+l = e~.
In the relation (i), the loop has been erased, but affected the weight q. The relation (iii) is simply obtained by stretching the (i + 2) th string. 3.2. The basis 1. The algebra T L n ( q ) is built out of arbitrary products of generators ei. Up to numerical factors depending on q, any such product can be reduced by using the relations (i)-(iii). The algebra TLn(q), as a real vector space, is therefore naturally endowed with the basis formed by all the distinct reduced elements of the algebra. This basis will be referred to as basis 1 in the following (as opposed to the basis 2, defined in Sect. 4 below). For illustration, the reduced elements of TL3(q) read
(3.4) cle2_-
e2o,_-
Meanders and the Temperley-Lieb Algebra
17 16 15 14 13 12 U 10
1 2 3
4 $ 6 7 8 9
10 U 12 1314 1516 1718
Fig. 7. The transformation of a reduced element of TL9 (q) into an arch configuration of order 9. The reduced element reads e3e4e 2 e5e 3 e 1e6e4e2
Let us now show that the reduced elements of TLn(q) are in one to one correspondence with arch configurations of order n. This is most clearly seen by considering the braid pictorial representation of a reduced element. Such a diagram has no internal loop (by virtue of (i)), and all its strings are stretched (using (iii)). As shown in Fig. 7, one can construct a unique arch configuration of order n by deforming the diagram so as to bring the (2n) ends of the strings on a line. This deformation is invertible, and we conclude that, as a vector space, T L n ( q ) has dimension
dim(TLn(q)) = Cn.
(3.5)
The basis 1 is best expressed in the language of walk diagrams. The walk diagrams of 2n steps are arranged .according to their middle height 2n = h, where h = n - 2p, 0 < p < n / 2 . For each value of h, the basic reduced element
f(n) = ele3es...e2p-1
f~n) = 1
(3.6)
corresponds to the lowest walk diagram ~A;~hn) with middle height h, namely
)'V~hn)
=
/N/~
.
/.
.
~
.
~.
.
(3.7)
/N/N
0 2 4 ... 2p ... n--.2(n-p)
...
2n
with 20 = 22 . . . . .
22p = 0,
21 : 23 . . . . .
22p--1 = 1,
(3.8)
22p+j = j
j = 1,2,...,h, 22n-j=g 9 j=0,1,2,...,n.
i
n
n
Fig. 8. An example of allowed left multiplication by ei. The initial walk diagram must have a minimum at the vertical of the point i. This operation adds a box to the walk diagram at the vertical of the point i < n.
It is then easy to see that any reduced element corresponding to a walk diagram with middle height g,, = h is obtained by repeated appropriate multiplications to the left or to the right of Jr h with e's. The walk diagrams of middle height h are constructed
10
E Di Francesco, O. Golinelli, E. Guitter
univocally b y adding "boxes" to the diagram W~n). As illustrated on Fig. 8, adding a box to a diagram )4; at the vertical of the point i is allowed only if i is a minimum of W , namely g~+t = gi-1 = gi + 1, in which case the new diagram, with the box added, has gi ~ gi + 2. For the associated basis 1 elements, this addition of a box corresponds to the left (resp. tight) multiplication by ei (resp. e 2 n - i ) when i < n (resp. i > n). This does not affect the middle height gn = h. For illustration, we list the elements o f the basis 1 for TL3 (q) together with the corresponding walk diagram (the middle height g3 takes only the values 1 (in 4 diagrams) and 3 (in 1 diagram)) el = f ~ 3 ) = _ ~ v
e2E1 = c 2
w
= w
ele2 = f~3)e2 =
_ _
e2
=
,
w
v
~
w
~
.
, -
(3.9)
e2
1 = f3~3) = _ ~
~
To avoid later confusion (with the basis 2), we will denote by (a)l the basis 1 element 3 corresponding to the walk diagram (or arch configuration) a E W~ ( ~ An).
3.3. Scalar product and meanders.
! L/Li,i' L ~. . . . . . . . . . . /,:, ::
Fig. 9. The trace of an element e E TL6(q) is obtained by identifying the left and right ends of its strings (dashed lines). In the arch configurationpicture, this amounts to closing the upper configurationby a rainbow of order 6. The corresponding semi-meanderhas 3 connected components,hence Tr(e) = q3 The standard scalar product on TLn(q) is defined as follows. First one introduces a trace over TLn(q). From the relation (i) o f (3.1), we see that in any element e of TLn (q) each closed loop may be erased and replaced by a prefactor q. Taking the trace of a basis 1 element e corresponds to identifying the left and right ends of each string as in Fig. 9, and assigning an analogous factor to each closed loop, which results in a factor Tr(e) = qC(~)
(3.10)
3 This notation will become clear when we introduce the basis 2. Indeed, the basis 2 elements will be indexed by the same walk diagrams ( ( a ) 2 ) , but will represent different combinations of products of e's, hence (a)z 5t(a)l in general.
Meanders and the Tempedey-LiebAlgebra
11
where c(e) is the number of connected components of the closure of e. The definition of the trace is extended to any linear combination of basis elements by linearity. Note that, with this definition, the trace is cyclic, namely Tr(ef) = Tr(fe). In the arch configuration picture, e(e) is easily identified as the number of connected components of the semimeander obtained by superimposing the arch configuration a corresponding to e and the rainbow of order n: indeed, the rainbow connects the ith bridge to the (2n + 1 - i)th which exactly corresponds to the above identification of string ends. In particular, this permits to identify the semi-meander polynomial (2.3) as
!~n(q ) =
~ eE basis 1
qC(e) = ~ Tr((a)l) aCWn
(3.11)
We also define the transposition on TL,~(q), by its action on the generators e~ = ei, and the relation (el) t = fret for any e, f E TLn(q). The definition extends to real linear combinations by (Ae + # f ) t = )~et + #ft. In the arch configuration picture, this corresponds to the reflection i --~ (2n + 1 - i) of the bridges. In the walk diagram picture, this is the reflection i ~ (2n - i).
Fig. 10. The scalar product (e, f) is obtained by first multiplying e with ft, and then identifying the left and right ends of the strings (by the dashed lines). Here we have (e, f) = q3. The correspondingmeander is obtainedby superimpositionof the upperarch configurationa correspondingto e and lowerarch configurationb correspondingto f (the transposition of f is crucial to recoverb as lowerarch configuration).Here the meander has c(a, b) = c(e, f) = 3 connectedcomponents For any two elements e and f E TLn(q), the scalar product is defined as (e, f ) = Tr(e ft).
(3.12)
This has a simple interpretation in terms of meanders. We have indeed
(e, f ) = qC(e,f) = qC(a,b),
(3.13)
where c(e, f ) = c(a, b) is the number of connected components of the meander obtained by superimposing the a and b arch configurations corresponding respectively to e and f (see Fig. 0 for an example). This permits to identify the meander polynomial as
Iron(q) = ~-~a,bEA,~ qC(a,b)= Ea,bcWn ((a)l,(b)l) J
(3.14)
Note that (e, 1) = Tr(e), hence the semi-meander expression (3.11) corresponds to taking (b)l = 1 in the above and summing over a E Wn only. This agrees with the abovementioned fact that the semi-meanders are particular meanders, namely with lower arch configuration fixed to be a rainbow. Indeed, the unit i E TL~(q) corresponds in the arch configuration picture to the rainbow of order n, (rn)l = 1.
3.4. Gram matrix. The Gram matrix G~(q) of the basis 1 of TLn(q) is the cn x Cn symmetric matrix with entries equal to the scalar products of the basis elements, namely
12
E Di Francesco, O. Golinelli, E. Guitter [ [~n(q)] a,b = ( (a)l'
(5)1) = qC(a,b)
For instance, G3(q)reads, in the basis 1 (3.9): q3 q2 q2 q3 G3(q) = q2 q q q2 q2 q
q2 q q3 q2 q
V a, b E An =- Wn
q2 q2 q3 q2 q2 q3 )
(3.15)
(3.16)
The meander and semi-meander polynomials are easily expressed in terms of the Gram matrix. Arranging the elements of basis 1 by growing middle height of the walk diagrams (in particular, the unit 1 is the last element), and defining the c,~-dimensional vectors g = (1, 1, 1 , . . . , 1)
~ = ( 0 , 0 , . . . , 0 , 1)
(3.17)
we have m~(q) = if- G,~(q)ff, ~hn(q) = ~" ~n(q)g,
(3.18)
where ~ . ff denotes the ordinary Euclidean scalar product of ~c~. Moreover, we also have rr~n(q 2) = tr (~n(q)2). (3.19) The Gram matrix Gn(q) contains therefore all the information we need about meanders. The remainder of the paper is devoted to a thorough study of this matrix and of the consequences on meanders.
4. The Basis 2
The multiplication of elements of the basis 1 involves many reductions, and therefore is quite complicated. In this section, we describe another basis for TLn(q), which we refer to as basis 2, in which the products of basis elements are trivialized, namely the product of any two basis 2 elements is either 0 or equal to another basis element. This second basis, described in detail in [11], will be instrumental in writing alternative expressions of the meander and semi-meander polynomials. 4.1. Definition of the basis 2. We need a few preliminary definitions. The Chebishev polynomials of the second kind are defined by the initial data Uo(x) = 1 and Ul(x) = x and the recursion relation
Un+l(X) =
x U n ( X ) -- U n - I ( X )
(4.1)
or equivalently by Un(z
-I- 1 ) =
Z n+l __ Z--n--1 z - 2; - 1
(4.2)
We also introduce the fractions On- j (q) Un-
subject to the recursion relation
Un(q)
(4.3)
Meanders and the Temperley-LiebAlgebra
13 1
- -
]~n+l
1 -
#n.
(4.4)
]Zl
To describe the basis 2, we use a walk diagram picture analogous to that for basis 1. Each basis element will be attached to a walk diagram of 2n steps. As in the case of basis 1, we start from the definition of the fundamental element ~_(n) h ' corresponding to ),V(h'~), the lowest walk diagram with middle height gn = h = n - 2p (3.7), namely
h •p(n)
=
(4.5)
(# 1)pele3"'" e2p-lEh(e2p+l, e2p+2, ..., en-1),
where the elements E h are defined recursively by
Eo=E1 =1 Eh+l(Ci,
(4.6)
Ci+l ~ ...~ e i + h - 1 ) ~
= E h ( e i , ei+l,..., ei+h--2)(1 -- IZhei+h-l)Eh(ei, ei+l,..., el+h-2).
9For instance, we have
E2(ei) = 1 - IZlei,
(4.7)
E3(ei, ei+l) = (1 - # l e 0 ( 1 - #2el+l)(1 - # l e i ) = 1 - Iz2(ei + ei+l) + Itl/Z2(eiei+l + ei+lei).
Note that E h is a projector 4 (E~ = Eh), and that the normalization factor in (4.5) ensures that ~(hn) is a projector too. In a second step, we construct the other basis elements corresponding to walk diagrams with middle height h. The latter are obtained by repeated left and right additions of boxes on the basic diagram W(hn). To define the corresponding basis 2 elements, it is sufficient to give the multiplication rule corresponding to a box addition (see Fig. 8). The rule reads as follows. I f a box is added on a minimum (gi+l = g i - l = gi + 1) of the walk diagram at the vertical of the point i < n (resp. 2n - i > n), the corresponding basis element is multiplied to the left (resp. right) by the quantity
~(
(4.8)
e i - - /-s
Applying these rules in the case of T L 3 ( q ) , we find the following basis 2 elements 4 This is easily proved by recursion on h, by simultaneously proving that E 2 ( E h (ei , ..., ez+h -- 2)e~+h -- 1)2 = # h I E h ( e l , . . . , ei+h-- 2 ) e l + h - 1.
=
Eh
and
14
P. Di Francesco, O. Golinelli, E. Guitter
.
-
-
= ~/-~-~2(e 2 _ / . t l ) p l e 1 V Pl = px/-~~(e2el -- #1el) =/Zlel ~ / ~ 2 (e2 -- ]A1)
.
-
.
V #1
= ~V/-~(ele2
.
-
-- /Alel)
V /-~1
(4.9) V ]~1
= m(e2 - ,ul(ele: + e2el) + / ~ e l ) ~
= ~(3)
3 = E3(el, e2)
v
v =
1 -- #2(el
+ e2) + #l#2(ele2
+
ezel)
4.2. Properties of the basis 2. The construction of the basis 2 basic elements ~h ^(nl is entirely dictated by the requirement that
ejEh(ei,ei+l,...,ei§
forj=i,i+l,...,i+h-1.
(4.10)
These relations were indeed used in [11] as a defining property for the Eh'S. The multiplication rule (4.8) ensures that whenever the multiplication by e~ acts on a slope of the corresponding walk diagram (i.e., when f i + l + g i - 1 -- 2gi -- 0), the result vanishes. In other words, ei (a)2 = 0
whenever gi+l + gia--1 -- 2g~ = 0.
(4.11)
These rules are also responsible for the following main property of the basis 2 elements. To write it explicitly, we need a more detailed notation for the walk diagrams of middle height gn = h, and the associated basis 2 elements. Such a diagram will be denoted a = lr, where I (resp. r) denotes the left (resp. right) half of the walk diagram, with i = 0, 1, ..., n (resp. i = 2n, 2n - 1, ..., n), namely l = {(i,gi)}
r = {(i,g2n-0}
(4.12)
for i = 0, 1,2..., n. Note that I is read from left to right on a and that r is read from right to left. Moreover, (/r) t = (rl). (4.13) Both half-walks start at height go = g2n = 0 and end at height g~ = h. To avoid confusion, we will denote the corresponding basis 1,2 elements by (lrh, (lr)2 respectively. The main property satisfied by the basis 2 elements reads, for any elements (a)2, (at)2 of the basis 2, a = Ir and a' = / ' r ' : (/r)2 (/'r')2 = ~r,l, (/r')2.
(4.14)
On this relation, we learn that all the self-transposed elements (i.e., with (a)2 = (a)t), namely those attached to symmetric walk diagrams (i.e., with l = r), are projectors. In particular, we recover the fact that qo(h~) = (l,V(h~))2 is a projector. As we shall see in the next section, the relation (4.14) implies also that the basis 2 is orthogonal with respect to the scalar product (3.12).
Meanders and the Temperley-Lieb Algebra
15
5. The Meander Determinant
5.1. The Gram matrix for basis 2. Thanks to the main property (4.14), the Gram matrix Fn(q) of the basis 2 elements takes a particularly simple diagonal form. Its cn x cn entries read
[rn(q)]o,a, = ((a)2,(a')2).
(5.1)
Let us compute the scalar product ((a)2, (a')2) = Tr((lr)2(l'r')tg) = Tr((Ir)2(r'l')2) = (ir,r' Tr((ll')2)
= Tr((l'r')t~(lr)2) = Tr((r'l')2(Ir)2) = 3t,t, Tr((rr')2)
(5.2)
= 3a,~, Tr((a)2(a) t) by direct application of (4.14) and use of the cyclicity of the trace and of (4.13). Hence the matrix F~(q) is diagonal. Moreover Tr((a)2(a) t) = Tr((rr)2) = Tr((ll)2)
(5.3)
for any r, l, does not depend on the half-path r of final height gn = h. It may be evaluated on the left half-path Ph corresponding to the walk diagram ),V(h~) of (3.7). A simple calculation shows that
Tr((phPh)2) = Tr(~ (n)) = Uh(q),
(5.4)
where U denotes the Chebishev polynomial (4.1). Hence Fn(q) is simply the diagonal matrix with the c,~ entries INn(q)] a , a = Uga (q), (5.5) where g~ denotes the middle height of the walk diagram a. We conclude that the basis 2 is orthogonal with respect to the scalar product ( , ) .
5.2. Main result. This remarkable property of the basis 2 will enable us to compute the determinant Dn(q) of the Gram matrix Gn(q) for the basis 1, also referred to as meander determinant. The result reads 5 Dn(q)
=
det (G,~(q)) = I~I Ui(q) an'i i=1
(5.6) 2n
where Ui(q) are the Chebishev polynomials (4.1), and we use the convention that (~) = 0 if j < 0. For instance, the determinant of the matrix ~3(q) (3.16) reads Da(q) = Ul(q) 4 U2(q) 4 U3(q) = q5 (q2 _ 1)4 (q2 _ 2).
(5.7)
As a nontrivial check, let us first compute the degree of D,~(q) as a polynomial in q 5 Ref. [4] presents a recursive algorithm for computing this determinant, which relies on direct manipulations of lines and columns of ~n. The main result of [4] is the identification of the zeros of Dn(q). Here we also give their multiplicities.
16
E Di Francesco,O. Golinelli,E. Guitter deg(Dn(q)) = E
ian'i =
= nc~,
n - 1
(5.8)
i=l
which is in agreement with the definition of the Gram matrix 6n: the term with highest degree in the expansion of the determinant comes from the product of the diagonal elements of Gn, namely
H
qC(a,a) = H
aEW~,
q'~ = qn~.
(5.9)
aEW,.,
as all the meanders with identical top and bottom arch configurations have the maximal number n of connected components.
5.3. The zeros of the meander determinant and their multiplicities. Before going into the proof of the formula (5.6), let us describe a few consequences of this result. The zeros zk,z of the polynomial Dn(q) are those of the Uk(q), for k = 1, ..., n, namely, using (4.2) zk,~ = 2 cos 7r k +l 1
1< I< k< n
(5.10)
hence we may rewrite
n
l
This yields the multiplicity d,~(zk,1) of each zero Zk,l, when (k + 1) and I are coprime integers, and 1 < (k + 1)/2 (Zk,k+l-t = --Zk,z has the same multiplicity as zk,z) [(n+l)/(k+l)]
d,~(zk,z) =
an,m(k+l)-I
E
(5.12)
m=l
For k = 1, l = 1 this yields the multiplicity of the zero q = 0,
[(n+l)/2] dn(O) =
(2:)
(2n) -
an,2m--1 =
n-
1
=
cn.
(5.13)
m=l
The fact that the zero q = 0 of dn(q) has multiplicity cn enables us to write, in the limit q-+0, , Dn(q) ~ qC, Dn(O), (5.14) where D~(0) 5/0 is the determinant of the matrix G ' ( 0 ) with entries
' 0
[~n(
{1
0
)]a,b----
ifc(a,b)=l otherwise
(5.15)
hence G~(0) is the one-connected component piece of the Gram matrix Gn(q). For instance,
0 01 )
G~(0)=
0 1 0
1 0 0
0 0 0
1
1
0
D~(0)=-2.
(5.16)
Meanders and the Temperley-Lieb Algebra
17
Noting that in the limit q ~ 0, U2i(q) --~ ( - 1 ) i
U2i-l(q)/q
~
(5.17)
i ( - 1 ) i-1,
the limit q ~ 0 of (5.6) yields [~/2] D~(0) =
l(n+l)/2]
I~(--l)ian'2~ H i=l
[i(--1)i-1] a'~':~-I
i=l
(5.18)
[(n+1)/2]
= (-1)~
1-I i~'~'-~" i=1
Therefore
loglD'(0)l=
~
n-2i
n-2i-1
+ n-2i-2
log/
i=1
(5.19)
4 2n
2
Cn ,
where the asymptotic estimate results from a saddle-point approximation to the sum. If I 0 ) were of the same order most of the eigenvalues ,k of the matrix Gn( t ~ (Dn(0))
1/c,~
(5.20)
~ eC~/2
we would have a meander polynomial, expressed through (3.19), of the order m~(q2) ~ q2 ~--~(,k)2 ~ q 2 e C ~ C n
(5.21)
which clearly is incompatible with the numerical estimate /~2n
mn(q2)/q 2
,-, M n ~
q---)o
(5.22)
n7/2
We conclude that the eigenvalues A of G~(0) do not have a localized distribution when n becomes large. This is also the case when q = 1. Indeed, the matrix ~,~(q = 1) is simply the cn • cn matrix with all entries equal to 1. It has the eigenvalue 0, with degeneracy (cn - 1), and the nondegenerate eigenvalue c~. This permits to recover the sum rules (2.6)-(2.7) easily. In this case, the distribution of eigenvalues of the Gram matrix is certainly not localized when n --- oc, as the only eigenvalue which matters diverges while all the other eigenvalues remain 0. More generally, the expression (5.12) can be resummed to yield 1 Cn - d,~(zk,t) = 2(k + 1) ~
k
7rm .2 ,
(2 sin k ~ i - ' /
71"m ~2n
{2 cos ~-~--i-)
(5.23)
ra=l
(see Appendix A for a detailed proof). The result is independent of l, under the requirement that 1 and k be coprime. For instance, for k = 2, 3, 4, 5 and n, we find
18
P. Di Francesco, O. Golinelli, E. Guitter
d~(• d~(•
2
= cn - 1 = c~ - 2 ~-1,
)=c,,--~
d~(+v/3) = cn d,~(2 cos
7rl )= 1 n+l
+
(5.24)
3 n-1 + 1 - - , 2 for 1 and (n + l) coprime.
The r.h.s, of (5.23) appears to be an integer in the following interpretation. Let ,Ak be the k • k symmetric matrix, with entries [Ak],,8 = 6s,,§
+ 6 .... 1
(5.25)
for r, s = 1,2,...,k. This matrix diagonalizes in the orthonormal basis {v, , r = 1, 2, ..., k}, where [vr]8 =
V/
-k~--f 7r?~8 2 sin f~+]
(5.26)
are the entries of the eigenvector vr of ,Ak, for the eigenvalue fl~ = 2 cos 7rr/(k + 1). Hence, the r.h.s, of (5.23) is nothing but k
Z
[vm] l( m)k 2,
1=
,,1
(5.27)
m=l
This expression is clearly an integer, as a matrix element of the (2n) th power of an integral matrix. Moreover, this permits to interpret the number cn - dn(Zk,Z) as counting the number of distinct closed walks of (2n) steps on a segment of size k, which start and end up at a fixed end of the segment. Indeed, .Ak is the adjacency matrix of a chain of k vertices, labeled 1,2, ..., k. The quantity [(.Ak)2n] 1,1 counts the number of distinct paths of length (2n) on the chain which start and end up at the vertex 1. In the language of walk diagrams, this is the number of walk diagrams w c W n whose heights do not exceed (k - 1). Denoting by W n , j = {a E W n I ta <- J for i = 0, 1,..., 2n},
(5.28)
Eq. (5.27) may be rephrased into Ca -- dn ( zk,t ) = card(W,~,k-1).
(5.29)
This interpretation permits to write a very simple generating function for the multiplicities dn(Zk,l). Indeed, for k >_ 1, let Gk(X)-'~ ~ X n (Cn - - d n ( z k , 1 ) ) . n=O
(5.30)
When k = l, we set Gl(X) = l, deciding by convention that d0(0) = 0, whereas co = 1. When k = 2, we are simply counting the only path of length 2n, going back and forth
Meanders and the Temperley-Lieb A l g e b r a
1
19
9-........ 9 9 ........
9 ........ 0 ........ -O. . . . . . . . .
9 9 9
9 9 9
2
3
4
k+l
A 2
A k
A k+l Fig. 11. A n y w a l k on .A~:+I m a y be viewed as the prolongation of a walk on ..42, by one or several w a l k s on .Ak, at each visit o f the vertex 2. Here we have represented a w a l k on ..42, of length 6, corresponding to the t e r m x 3 in G2(x). E a c h of its three visits of the vertex 2 m a y be arbitrarily p r o l o n g a t e d b y walks on ~4k (vertices 3 2, 3 ..... k + 1 on the figure), to generate all the walks on .A/~+I, resulting in the substitution x 3 --~ ( x G k ( x ) ) in the c o r r e s p o n d i n g generating function.
between the origin and the other vertex of the chain. Each such come and go picks a factor of x, resulting in the generating function 1
G2(x) = 1 + x + x 2 + . . . .
(5.31)
1-x
in agreement with the first line of (5.24). To compute Gk+l(X), knowing Gk(x), we may view all the walks on r as an arbitrary insertion of walks on .Ak at each visit of the vertex 2 by arbitrary walks on .A2, as indicated in Fig. 11. This leads to the following composition of generating functions: 1 Gk+l(X) = G2(xGk(x)) - 1 - x G k ( x )
1 -
1 -- t
x
,
(5.32)
""'l-x
where the fraction is iterated k times. Together with the initial condition Gl(x) = 1, eq.(5.32) completely determines G k ( x ) for all k. In fact, we find the following simple expression in terms of the Chebishev polynomials (4.2) and the function #k(q) (4.3): Ul(1/v~)Uk_ffl/v'~)
Uk(l/v~)
Gk(x)=
1
1
= v ~#k(~)'~/x"
(5.33)
where we have identified the recursion relation (5.32) with (4.4) upon a mutliplicative redefinition of # (which also gives G t ( x ) = 1) and the change of variable q = 1/v'~. Note that the expression (5.33) is valid as a series expansion in powers of x for small enough x. This in turn translates into the following expression for the generating function for the multiplicities d,~(zk,z):
(5.34) n=O
where 1 - v/1 - 4x C(x)
= ~ n=O
x n cn -
2x
(5.35)
20
E Di Francesco, O. Golinelli, E. Guitter
denotes the generating function of the Catalan numbers (2.5). The results (5.24) may be easily recovered from the expression (5.34), for k = 2, 3, 4, 5. Note that the series Fk(x) has the valuation k, namely Fk(x) ~ x k when x ~ 0, as dn(zk,D = 0, for n < k - 1, and dk(zk,t) = 1 for n = k. Hence limk__+~ Fk(x) = 0 uniformly for small enough x, which means that Gk(x) converges to C(x) uniformly when k ---+cxD:this establishes a link between the Chebishev polynomials and the Catalan numbers. Remarkably, the formula (5.23), together with the above interpretation (5.27), suggest a relation between the multiplicities dn(zk,t) of the zeros of D,~(q) and the rank r~(zk,1) = dim Im Gn(zk,z) of the matrix ~n(q = Zk,l), namely that
d~(zk,t) + rn(zk,z) = c~ = dim(TLn(q))
(5.36)
dim Ker G~(Zk,l) = d,~(zk,l).
(5.37)
or in other words that Indeed, the matrix ~ ( 0 ) = 0 has rank 0, whereas G,~(1) has rank 1 as all its lines are identical and non-vanishing. We also checked that 6 n ( v ~ ) has rank 2 n-1 for n = 1, 2, 3,4, 5. Equations. (5.27) and (5.36) would imply in general that the rank of the matrix Gn(zk,t) (for l and (k + 1) coprime) is equal to the number of walk diagrams w E Wn, whose heights are less or equal to (k - 1), i.e., card(W,~,k-l). Assuming that (5.36) is true, it is tempting to conjecture that the lines of G~(zk,z) corresponding to the diagrams a E Wn,k-1 form a collection of r,~(zk,l) independent vectors, of which any other line of G~(Zk,t) is a linear combination. In particular, the last line of G~(zk,l), corresponding to the diagram W (n), should be a linear combination (with coefficients A~) of the lines of Gn(Zk,t) pertaining to the diagrams a E W,~,k-l. This would result in a relation ~ xcO/V(nn) b) zk,z) ' = E ~an (ZkJ) c(a'b)" (5.38) aEWn,k-1
Summing this over b E W,~ would give a new expression for the semi-meander polynomial at q = Zkj, as a linear combination of the polynomials corresponding to the diagrams a E Wn,k-b namely
~n(zk,t) =
E
)~ ~ ( a , Zk,l),
(5.39)
aEWn,k-I
where
~(a,q) = E
q~(a,b).
(5.40)
bE W n
This conjecture is illustrated in appendix B, for q = ~
(k = 3, l = 1).
5.4. Proof of the main result. We now turn to the proof of the formula (5.6) expressing the meander determinant. Since the Gram matrix (5.5) is trivial in basis 2, we simply have to compute the determinant of the matrix of the change of basis 1 to 2. This is done by first showing that this matrix can be put in an upper triangular form and computing the product of its diagonal entries. The result, combined with (5.5), is identified with the desired expression (5.6) through a subtle mapping of walk diagrams. Preliminaries. Let Pn(q) denote the matrix of the change of basis 1 to 2, made of the column vectors of the basis 2 expressed in the basis 1. It satisfies (b)2= E aEWn
[P~(q)]a,b (ah"
(5.41)
Meanders and the Temperley-LiebAlgebra
21
Let us show that the walk diagrams indexing the vectors of both bases can be ordered in such a way that the matrix 7~n(q) is upper triangular. The basic element ~(hh) is, according to (4.5)-(4.6), a linear combination of the basis 1 elements of TLh(q) of the form
~(hh) = ~
"~a(a)l(el,.-., Ch-1),
(5.42)
aEWh
where the sum extends over all the diagrams of 2h steps, which are all walk W(hh). By inclusion of diagrams a, b E W,~, we mean a C b
iff
l~ < Ib k/i = 0, 1,..., 2n.
included in the (5.43)
Similarly, the basic element ~(h'~ is equal to the linear combination h = •fl(n)
~
"~a(IZl)P e l e 3 ' ' - e2p-1 (a)l(e2p+l, ..., en-1)
(5.44)
aEWh
which corresponds only to walk diagrams of 2n steps, included in W(hn). The other basis 2 elements with middle height h are obtained by repeated box additions on )/V(hn) (see Fig. 8), with the corresponding multiplication rule (4.8). It is then easy to prove recursively that any basis 2 element with middle height h, of the form (b)2, is a linear combination of basis 1 elements whose walk diagrams are included in b, namely [T'~(q)] a,b :/0 =* a C b. (5.45) Arranging the walk diagrams by growing middle height, we see that ~(hn) is expressed only in terms of lower basis 1 elements: this gives only upper triangular entries in the matrix T'n(q). More generally, the walk diagrams can be ordered for each fixed middle height h is such a way that all the diagrams included in a come before a: it is sufficient, for instance, to order the diagrams by growing number of boxes added to l/Y(hn). With such an ordering of the bases 1 and 2, the matrix T'n(q) is upper triangular (with nonzero terms on the diagonal). For instance, with the ordering of (3.9) and (4.9), we get the upper triangular matrix
~3(q) =
0
0 0
0
--/Zl/A2 --/Al~2
~l/A2 ]Zl~2
0 0
#2 0
-#2 1
~
(5.46)
Let us decompose the upper triangular matrix 7~n(q) into the product 3~
= Q,~(q).N'n(q),
(5.47)
where N'n(q) is a diagonal normalization matrix and Q,~(q) an upper triangular matrix, with diagonal entries equal to 1. This separates the redefinition of basis elements (which does not affect the Gram determinant), through the matrix Qn(q), from the change of overall normalization of the basis vectors (which affects the Gram determinant), through N',~(q). For n = 3, these matrices read
22
R Di Francesco, O. Golinelli, E. Guitter
Pl .hf3 (q) =
Q3(q) =
0
o o
o
0 0
0 0
0 0 0 0
1 0 0 0
0 1 0 0
0
00~
o
o o
0 0
#2 0
-~1 -~1 1 0
~1~2 ~1~2 -~2 1
,
)
(5.48)
The change of basis 1 --~ 2 translates into the matrix identity Fn(q) = Pn(q) t 6n(q) Pn(q)
9N'n(q) Qn(q) t 6n(q) Qn(q).hfn(q) hence, as det
Q~(q) =
(5.49)
1, we have the relation between determinants
(5.5o)
det[Fn(q)] = det[N'~(q)] 2 D~(q) with, according to ( 5 . 5 ) , det[Fn(q)] = H Ulg(q) aCWn [n/2]
b
= H [Un-2P(q)] ( n,~ 2,)
2
(5.51)
p--O
where bn,n-Zp is the number of half-walks of n steps with final height h = n - 2p, and constrained by gi _> 0, for i = 0, 1, ..., n. The walk diagrams of middle height h = n - 2p are simply obtained by taking arbitrary left and right halves of final height h, hence their number is (bn,,~_Zp) e. The number bn,n-2p is obtained by subtracting from (p), the total number of unconstrained walks with go = 0 and g~ = h, the number of those which touch the line g = - 1 , namely (p~_1)" Indeed, by a simple reflection (mirror image) with respect to the line g = - 1 of the portion of walk between its origin and the first encounter with g = - 1, we get a one-to-one mapping with unconstrained walks such that g~ = - 2 and g~ = h; the number of such walks is (pnl). Hence we have
The normalization Ar~(q). To get D~(q) from (5.50), we are left with the calculation of det[N'n(q)]. The diagonal entries of N'n(q) are computed as follows. For the diagram W(h~/, the entry reduces to the global normalization of the vector ~hn), namely [.N'n(q)] Wr),W 2, = (/Zl)p.
(5.53)
The entries corresponding to other walk diagrams of middle height h are simply the product of this factor by the product over all the box additions to l,V(h~) of the
Meanders and the Temperley-Lieb Algebra
23
Fig. 12. The left and right strip decomposition of a diagram of middle height h. The strip lengths are given by the numbers ~a corresponding to the maxima i of a, i ~ n
normalization factors V/#~+2/#e~+l which enter the multiplication rule (4.8). In other words
[J~fn(q)]aa
= (#1)P
'
H ~/+2 . box additions i V #ei+l
(5.54)
To make this formula more explicit, let us arrange the box additions needed to generate a from l/Y~hn) into p left and p right strips of consecutive boxes, oriented respectively to the right and left as indicated on Fig. 12. This is called the strip decomposition of a. Each strip ends at a local maximum of a, namely at the vertical of a point i with s = g i - - 1 = g i - - 1. The length of the corresponding strip is defined to be gi (there are actually gi - 1 boxes in a strip of length gi; a strip with no box has indeed gi = 1). The expression (5.54) becomes
['/~fn(q)]a,a
= ( # I ) P H V ~11 strips
(5.55)
where g denotes the length of each strip. As there are p left and p fight strips, the factors #t cancel out, and we are left with (5.56)
[J~fn(q)]a,a = H V ~ strips
Hence the prefactor in (5.50) reads n
det [.A/'~(q)]2 = H
H
/ze = H ( # ~ ) s~,',
aCWn strips of a
(5.57)
i=l
where sn,i denotes the total number of strips of length i in the strip decompositions of all the walk diagrams of W~, or equivalently the number of distinct diagrams of W,~, with a marked top of strip of length i. Using the relation Ui = 1/(#1#2... #~), we can rewrite det[Fn(q)] (5.51) as n
det[F~(q)] = H ( # ~ ) -h~,~,
(5.58)
i=l
where
hn,i = E
(b~,k)2
(5.59)
z_
k=n mod2 is the total number of walk diagrams of 2n steps with middle height larger or equal to i. We finally get n
Dn(q)
=
H ( P i ) -s'~'~-hn'~.
i=l
(5.60)
24
E Di Francesco, O. Golinelli, E. Guitter
Let us now prove that (5.61)
Sn# + hn,i = b2n,2i,
namely that the total number of walk diagrams of 2n steps with final height 2i is equal to the total number of strips of length i plus the total number of walk diagrams in Wn with middle height larger or equal to i. To prove this, we establish a map between the walk diagrams of length 2n and final height 2i and (i) the walk diagrams of W,~ with a marked top of strip of height i or (ii) the diagrams of W,~ with middle height gn _ i. The mapping of walk diagrams9 Starting from a given walk diagram w of 2n steps from g0 = 0 to g2n = 2i, with gk > O, for all k, the construction proceeds in three steps. b
2i
0 0
j
2n
Step 1
0
2n-j
2~
Step 2
Fig.
13. Reflection-translation of the diagram w. The rightmost crossing point between w and the line of constant height g = i, at an ascending slope, is marked by a black dot. The dot separates w into a left piece a and a right piece b. The reflection-translation consists in a reflection of a --+ a t, followed by a translation of a t in order to glue the two walks b and a t. The gluing point is indicated by a black dot on the second diagram.
Step 1. Let j be the largest point 0 < j < 2n of height gj = i on w, and such that g j - i = gj - 1 = gj+l - 2. As shown on Fig. 13, this separates the walk diagram w into a left piece a, with j steps and final height i (namely g~ = 0, g~ = i and g~ _> 0 for all k), and a right piece b, with (2n - j ) steps, initial height i and final height 2i. Note that the heights of b remain above the g = i line, by definition of j, hence b can be considered as a walk diagram of (2n - j) steps, with extremal heights gb = 0, g2b,~_j = i, and subject
to the constraint g~ _> 0, for all k. Step 2. Let us perform the following reflection-translation on w, shown in Fig. 13: reflect the a diagram (a --* a t) and translate it so that its height i (left) end is glued to the height i (right) end of b. Note that the resulting diagram w t is an element of Wn, as all its heights lie above the g = 0 line9 Let us mark this gluing point on the resulting diagram w' c Wn. This procedure maps the diagram w onto a marked diagram w ~ E W n . Step 3. Only two possibilities may occur for the marked point, denoted by j in the w t w p following: it is either (1) a m a x i m u m of w' (gj+l = gj-1 = g~' - 1) or (2) a descending w' + 2) of w '. Indeed, the point (j + 1) on w ~ has the height slope ( s w' = g ~ / + l = gj+l w /
a
~__
g3' -- 1. Case 1. When j is a maximum of w ~, the marked point (3, gj ) corresponds to the top end of a strip in the strip decomposition of w ~ unless j = n. Therefore we have the two subcases gj+l = t j - - 1
9
w
t
(1) (a): If the marked point is a m a x i m u m of w ~, not in the middle of w ~ (i.e. j ~ n), w ~ is a walk diagram of Wn, with a marked (right or left) top of strip at height i. (1) (b): If j = n, the diagram w ~ has a middle height i (hence enters the category of walk diagrams of Wn with middle height _> i). 9
/
Case 2. When the walk has a descending slope at j, the marked point (j, gjw ) corresponds to the top end of a (left) strip in the strip decomposition of w ~ only i f j < n. Therefore we have the three subcases
Meanders and the Temperley-Lieb Algebra
n
j'
25
j
(c~)
n
j
([~)
Fig. 14. The cases (2)(b)(c~) and (/3). We indicate the migration of the marked dot in the (c~)case. In the (/3) case, the diagram w has a middle height _> i.
(2) (a): If the marked point has j < n, w ' is a diagram with marked top of (left) strip. (2) (b): If j > n, we move the marked point to the left along a line of fixed height g = g~', until we reach a top of (right) strip (see Fig. 14 (a)). Of course, one may reach the middle of the diagram before crossing any top o f strip (see Fig. 14 (/3)). This leads to two more possibilities (2) (b)(a): The line of constant height g = g~o' crosses an ascending slope of w' at j ' ( g j , - 1 = gj, - 1 = gj,+l - 2), such that n < j ' < j. Taking for jl the largest such integer, we move the mark from j to f , and end up with a diagram w ' E Wn with a marked top of (right) strip. (2) (b)(/3): The line of constant height g = g~o' does not cross any ascending slope of w' between n and j. The diagram w ~ E Wn has therefore a middle height _> i. More precisely, we have either possibility: (2) (b)(/3)(i): The middle height is > i. (2) (b)(/3)(ii): The middle height is = i. (2) (c): The marked point is at j = n. The diagram w ~ E Wn has middle height i (hence enters the category of walk diagrams of Wn with middle height _> i). This exhausts all the diagrams with marked top of strips, according to whether - the top is a maximum (1)(a) the top is on a left descending slope (2)(a) - the top is on a right ascending slope (2)(b)(a) -
and all the diagrams with middle height _> i, according to whether - the middle - the middle the middle the middle -
-
height height height height
is > i (2)(b)(/3) is = i and is a maximum (1)(b) is = i and is either an ascending slope or a minimum (2)(b)(/3)(ii) is = i and is a descending slope (2)(c).
T h e i n v e r s e m a p . Conversely, any walk diagram w ~ E Wn with a marked top of strip at height i can be mapped onto a walk w of 2n steps with g~ = 0, g~',~ = 2i and g~o > 0 for all k as follows. The marked top of strip (j, g~o' = i) can be either (i) a maximum, (ii) a descending slope in the left half of w ~ (j < n) or (iii) an ascending slope in the right part of w' (j > n). In the cases (i) and (ii), the marked point separates the walk w ~ into a left piece a (with j steps and g~ = 0, g~ = i, g~ > 0 for all k), and a right piece b (with (2n - j) steps and gob = i, gzbn_j = 0, gb _> 0 for all k). The diagram w is built by the inverse
26
P. Di Francesco, O. Golinelli, E. Guitter
of the reflection-translation of Fig. 13, namely by first reflecting b --~ bt, and then by translating it and gluing its right end to the left end of a. After this transformation, the gluing point, now at position (2n - j) has an ascending slope on w at height i, and is the largest such point. In the case (iii), the marked point is first moved to the right until the first crossing of the line of constant height g = i with a descending slope is reached: such a point always exists, because the height g2w~ = 0 must be eventually reached. One then applies the above inverse reflection-translation to this new marked diagram. This produces again a diagram w E Wn where the gluing point is the largest point on w with ascending slope and height i. Finally, any walk diagram w ~ E Wn with middle height > i may first be marked as follows. Mark the first crossing j > n between the line of constant height g = i and the walk w' at a descending slope. Then apply the above inverse reflection-translation. In all cases, we have associated a walk diagram w to each diagram w ~ with either a marked end of strip of height i or a middle height ___ i. This concludes the proof of (5.61). Conclusion. Equation (5.60) implies that /z
D~(q) =
II(#i) -b2n,2~
(5.62)
i=l
or, reexpressed in terms of Ui through #i = Ui-1/U~,
Dn(q) = ~ I
[Ui(q)](b2~'z~-bz~'2i+2)'
(5.63)
i=1
which takes the desired form (5.6) with an,i = b2n,2i -- b2n,2i+2
=
(n-i)_2(n-i-1) 2n 2n
+
(n--i-2). 2n
(5.64)
6. Effective Meander Theory In this section, we study various properties of the matrix 79n(q) and its inverse, in relation with the meander and semi-meander polynomials through (3.18). Indeed, rewriting (5.49) as ~n(q) = (Pn(q)t) -1 F n ( q ) (79n(q)) -1 (6.1) the relations (3.18) become ?~n(q) = i f " G n ( q ) f f = ( ~ n ( q ) - l f f ) " Fn(q) 79n(q)-1ff,
ran(q) = if-Gn(q)u = ('Pn(q) -1~) 9Fn(q) 79~(q)-1~,
(6.2)
where the vectors g and g are defined in (3.17).
6.1. The matrix 79n(q)-k By definition, the matrix 79n(q) -1 describes the change of basis 2 --* 1, through
Meanders and the Tempedey-LiebAlgebra
27
(a)l = E
[Pn(q)-l]b,~ (b)2.
(6.3)
bCW~ Multiplying both sides to the right by (c) t, for some c E W,~, and taking the trace, we get
Tr((a)l(C)~) = E [Vn(q)-l]b,a Tr((b)2(c)~) beW~ , = [P~(q)-']~,Tr((c)2(c) t)
(6.4)
where we have used the orthogonality of the basis 2 elements. According to (5.3) (5.4), we have Tr((c)2(c)~) = Ue~(q), where g~ is the middle height of the diagram c, and we finally get
[79n(q)_l]ca - Tr ((a)l(c)~) - Ut~(q)Tr((ah(c)t).. 1 , Tr ((c)2(c) t )
(6.5)
6.2. Properties ofTen(q) -l. The formula (6.5) can be used to derive many properties of the matrix Pn(q) -1. Let us take a = W~n) (i.e., (ah = f~'~) = 1) in (6.5). This yields [Pn(q)-l]c'W~n'-
Tr((c)~)
Ugh(q)
(6.6)
Writing c t = Ir as a juxtaposition of a left and right half-walk, and using (4.14), we compute Tr(lr)2 = Tr ((lr)2(rr)2) = Tr((rr)2(Ir)2) = 5z,r Tr(rr)2. (6.7) Hence the trace of (ct)2 vanishes, unless c t is a symmetric diagram, i.e. with l = r, in which case the trace takes the value (5.3), (5.4) Tr(ct)2 = Tr(rr)2 = Tr(pe~ptg)2 = Ue~(q).
(6.8)
Putting (6.6) and (6.8) together, we simply find that
['Pn(q)- 1] c,W~~) =
1 if cis symmetric 0 otherwise ~c,symmetric
(6.9)
With the definition (3.17) of the vector ~7, this translates into
79n(q) -1 ff = g,
(6.10)
8a = ~a,symmetric.
(6.11)
where the vector g has the entries
Comparing with (3.17), this permits to rewrite the semi-meander polynomial as
~ n ( q ) = g" Fn(q)[Pn(q)] - l f f
= E a,bEWn a
symmetric
whereas the meander polynomial reads
[Pn(q)-l]a,b Ue•(q)'
(6.12)
28
P. Di Francesco, O. Golinelli, E. Guitter
ran(q) =
79n(q)-1U
"
I'n(q)~n(q)-lu (6.13)
aEWn
bEW~
Another interesting particular case of formula (6.5) is obtained by taking c = W ~ ), where e,~ is the smallest possible middle height in W,~, namely e~ = (1 - ( - 1 ) ' * ) / 2 = 5~,odd. The heights of a read g2i = 0 and g2i-1 = 1, for all i. This diagram is the smallest of all the diagrams in W,~, in the sense that it is included in all of them. It corresponds to the first entry of the bases 1 and 2, hence to the vector v7 = ( 1 , 0 , 0 , - . . ,0).
(6.14)
The corresponding basis 1 and 2 elements read respectively f ~ ) and ,_,,~,(n).By the definitions (4.5) and (3.6) taken at h = e~ (in which case E,~ = Eo or Ea, hence E,~ = 1), we find the following relation between them: D n) ---- (#l)[n/Zlf~:),
(6.15)
or equivalently
(]/~ en( n ) ]/ 2
(6.16)
= ( P l ) [ n / 2 ] (~/~(n)~
\
en ] l ~
where we have identified en = n - 2p, hence p = [n/2]. For the choice c = W ~ ), (6.5) reads
['Pn(q)-l]w•:)a
= Tr((a)t(W~: ))2)
,
Urn(q) = (]d,1) [n/2]
T r ( ( a h ( W ~ : ))1)
= (#,)[(n+l)/2]
U,,(q) [G,ffq)] a,W~:,
(6.17)
= (i,Zl)[(n+l)/2]-c(a,~'V~2 ))
In the second line, we have used the relation (6.16), whereas in the third line, we have used the fact that U~n(q) = q~" = (/q)-~" and that en + I n / 2 ] = [(n + 1)/2]. The last expression uses the definition of the Gram matrix (3.15): the quantity e(a, W~] )) is, in the arch configuration picture, the number of connected components of the meander obtained by superimposing the upper configuration a and the lower configuration b - W~: ), made of a sequence of n consecutive single arches, linking the bridges (2i - 1) and (2i), for i = 1,2, ..., n. The (meander) polynomial corresponding to the closings of W ~ ) was computed in [6] and reads 6 in(q)=v7
c(aw~')
~,~(q)~7= ~ "
q
aEWn
'
~l(k)(k-1) =
n
n
n
qk
(6.18)
k=l
6 In ref.[6], it has been shownthat the numberof closingsof ~4),(n~) with k connectedcomponentsis identical to that of arch configurationsof order n with k interior arches (i.e.. arches linking two neighboring bridges i and (i + 1)). In turn, this is nothing but the number of walk diagrams in W,~ with exactly k maxima (the notion of interior arch in an arch configurationis equivalent to that of a maximum in the correspondingwalk diagram). This number is (kn)(k~ 1)/n.
Meanders and the Temperley-Lieb Algebra
29
with the vectors g and C defined respectively in (3.17) and (6.14). Note that the polynomial in(q) is reciprocal, i.e. qni~(1/q) = i,~(q). Hence, from (6.17), we get a sum rule for the first line of the matrix ~~ E
[• n ( q ) - l ]
~Ew.
lA1(n) a = (~tl)[(n+l)/2] i n ( L )
"~'
/~1
(6.19)
= in(#l)/(l~)[n/2] by using the reciprocality of in(q).
6.3. Recursion relation for the matrix a similar way as P,~(q), as the matrix
Q~(q)-l. The matrix Qn(q) is constructed in of a redefinition of basis 1, except that all the normalization factors are dropped, namely the prefactor (#1) p in the definition (4.5) of p(,0 h is dropped, as well as the prefactor g//Zei+2/].tgi+l in the multiplication rule (4.8). This results in a diagonal of l's for Q,~(q). Qn(q) is the matrix of change of basis 1 to the unnormalized basis 2 (denoted by basis U), with elements (a)2, = (a)2/.Afa,a. Let us now derive recursion relations for constructing the inverse matrix Qn(q) -1. This matrix sends the unnormalized basis 2 ~into the basis 1, according to the identity (b)l = E
[Qn(q)-l]a,b (a)2'"
(6.20)
aE W~
Recall that the basis 1 elements are constructed by box additions (Fig. 8) on the basic elements f(hn), each box addition corresponding to the multiplication by some ei.
+
+
(ii)
(iii)
(i)
Fig. 15. The three possibilities for the multiplication ei(a)2,, represented as a box addition at the vertical of the point i on a diagram a E Wn. The latter may be above (i) a slope of a, (ii) a maximum of a or (iii) a minimum of a.
Let us study the consequences of a left box addition on b, at a minimum i < n of b. Let us denote by b + o the resulting diagram. Multiplying accordingly (6.20) to the left by e~, we find a recursion relation for the matrix elements of Qn(q) - t . Indeed
(b+O)l-- Z [Qo(q)-t]o,b+o(a)2' aew. =ei(b)l = E
(6.21)
[Qn(q)-l]a,b ei(a)2'
a E W,~
gives a relation between [Qn(q)-'],,b+o and elements of the f o r m [~n(q)-l]a,,b by identifying the coefficients of the basis 2' elements. Three situations may occur for ei(a)2,, as depicted in Fig. 15.
30
P. Di Francesco, O. Golinelli, E. Guitter
(i)
a = 2g~). Due to the The box addition is performed on a slope of a (gi~+l + gi-1 vanishing property (4.11), we find that the resulting element vanishes, namely [ e~ (a)2, = 0]
(6.22)
a = s ~ - 1). For (ii) The box addition is performed on a maximum of a (gi~+l = gi-a (a)2,, this maximum is itself the result of an (unnormalized) box addition with the roles of basis 2, hence a factor (ei - #k), where k = g~ - 1, according to (4.8). The multiplication by ei results in
1
x(ei--#k)+ ei(ei - #k) = ( 1 _ i.zk)e i = #1 #k+l
#k X 1, #k+l
(6.23)
where we have used the recursion relation (4.4) for the #'s. The first term in (6.23) restores the box of (a)2,, while in the second term the box is removed, yielding (a - <>)2,,where a - o denotes the walk diagram a with the box below the maximum removed. Hence 1 ei(a)2, = ((a)2, + I~k(a -- <>)2') (6.24) #k+l with k = g~ - 1. (iii) The box addition is performed on a minimum of a (g~+l = gi~-I = g~ + 1). We are left with the multiplication of (a):, by (6.25)
ei = (ei -- pk) + #k • 1, where k - gi + 1. Hence _
a
ei(a)2, = (a + <>)2' -I-]Ak(a)2, f
(6.26)
Substituting (6.22), (6.24),(6.26), in (6.21), we get [Qn(q)-lla,b+(>( a)2'=
~
aCW~
[Qn(q)-l]a,b
aCW~ X
{1
- - ~ a max(i) ((a)2, + # q - l ( a #e~ '
+ 6a,min(i) ((a + <>)2' + ]zg~+l(a)2')
- 0)2,),
(6.27)
)
where we use the notation da'max(i) = ~a,min(i) =
{o (1
-o-o if ~i~+l -~i--I -- ~i -- 1 otherwise ' if gia+l -o-o -- gi--1 -- gi + 1 otherwise
(6.28)
The identification of coefficients of (a)2, yields the relation
[Q'(q)-q.,b+o
(Ja,rnax(,)(~-~ [~'a(q)-l]a,b + [~n(q)-l]a_c,,b) +~ rQ " - 1 7 , +(~a,min(i)(fl'g:+l [~n(q)-l]a,b #~+2 k n(q) Ja+<>,b) (6.29)
Meanders and the Temperley-Lieb Algebra
31
where we have used (~a,max(i) = (~a--o,min(i) ~ 5a,min(i) = (~a+o,max(i), .
(6.30)
ga+~' = g~ + 2 Z
Together with the initial condition [ Q n ( q ) - l ] a w("' = a, w(-),
(6.31)
Equation (6.29) is an actual recursion relation, yielding all the entries of Q - l , column by column starting from the left. A first remark is in order: the entries of Q~(q)-I satisfy the property
[Qn(q)-l]a,b g 0 ~
a C b,
(6.32)
easily proved by recursion using (6.29). This last condition has been previously derived for the entries of ;O(q) (cf. (5.45)), but holds as well for the inverse matrix. Note that (6.29) also implies that [Qn(q)-l,]a,a'= 1 (6.33) in agreement with the normalization of Q.
6.4. The matrix Q~(q)-l. The recursion relation (6.29) will be solved in two steps. The idea is to treat separately the question of finding when [Q,~(q)- 1] a,b vanishes or not, and that of determining its precise value when it does not vanish. This suggests to separate the matrix element [Qn(q) -1 ] a,b into a product [ ~ n ( q ) - I ] a,b = Wa,b fa,b,
(6.34)
where f~,b is subject to the recursion relation
fa,b+o
=
( a,max(i) (fa,b + fa--o,b) + 6a,n n.)(A,b + A+o,b)
(6.35)
and
fa,w(,~' = d,,,,v,~("'"
(6.36)
Solving for f. From (6.35), (6.36), it is clear that the f ' s are nonnegative integers. In fact, the f ' s may only take the values 0 or 1, and act as selection rules on the couples of diagrams a C b. To describe the solution of (6.35), (6.36) we need one more definition. We will need a mixed representation of a couple a C b of walk diagrams in Wn, namely a E Wn is represented as a walk diagram, but b E A,~ -- Wn is represented as an arch configuration of order n. The diagram b is therefore represented by the permutation ab of the bridges, with a~ = 1, describing the arches (namely ab(i) = j iffthe bridges / and j are linked by an arch). The diagram a C b is said to be b-symmetric iff it satisfies g~b(~) - ea~,b(~)-I = -(g~ - gig_l)
(6.37)
In other words, we may represent on the same figure the arch configuration b and the walk diagram a, as illustrated in Fig. 16. Each bridge i of b sits at the vertical of the link (i - 1, i) of a. Then a is b-symmetric iff the links of a are pairwise symmetrical under
32
E Di Francesco, O. Golinelli, E. Guitter
:
:
:
:
:
;
:
:
:
:
:
:
Fig. 16. An example of walks a C b, where a is b-symmetric, b is represented in the arch configuration picture, and a in the walk diagram picture. The dotted lines continuing the arches of b indicate the links of a which have to be symmetrical: the two links connected to the same arch must be mirror image of each other
the pairs of bridges linked by an arch on b. In particular, if a is b-symmetric, then, b e l o w an interior arch of b (i.e., an arch linking two consecutive bridges i, (i + 1)), a must have a m a x i m u m or a m i n i m u m (the only two left-right symmetrical link configurations around i). Note also that a d i a g r a m a is symmetric iff it is w ~ n ) - s y m m e t r i c , and that the d i a g r a m / 4 2 Cn (n) is b-symmetric for all b E Wn. With this definition, the solution of the recursion relation (6.35), (6.36) reads
fa,b=(
1 0
if a is b - symmetric I otherwise
(6.38)
Hence, in (6.34), f selects the couples of diagrams a C b such that a is b-symmetric 7. With fi,,b as in (6.38) let us now c h e c k (6.35), (6.36). T h e relation (6.36) amounts to the fact that a is a - s y m m e t r i c . Indeed, an arch of a always starts (say, at the b r i d g e i) above an ascending link of a (g~ = gi~l + 1) and ends (say, at the bridge j = ~ra(i)) over a d e s c e n d i n g link of a (g~ - g~_ 1 = - 1 ) ; these two links are therefore symmetrical. To c h e c k (6.35), let us consider a d i a g r a m a C b+o, which is b+o-symmetric. N o t i n g that b + o has an interior arch linking the bridges i and (i + 1) (this is equivalent to a m a x i m u m above i on the corresponding walk diagram), by virtue of the a b o v e m e n t i o n e d property, the b + <>-symmetric d i a g r a m a must have either a m a x i m u m or a m i n i m u m above i. These two possibilities c o r r e s p o n d to the two lines o f (6.35). To complete the c h e c k of (6.35), we must prove that in either case one and only one of the two diagrams a and a -4- o is b-symmetric (then (6.35) simply reads 1 = 1). M o r e precisely, the b o x addition on b ~ b + o is interpreted in the arch configuration picture as the bridge move illustrated in Fig. 17. Before the b o x addition, b has a m i n i m u m at the vertical of i. This means that an arch (starting, say, at the bridge il < i) ends at the bridge i2 = i, and that another starts from the bridge i3 = (i + 1) ( and ends, say, at the bridge i4 > (i + 1)). The bridge m o v e of Fig. 17 replaces these two arches by an arch connecting the bridges it and i4, and an interior arch connecting i2 and i3. The creation of an interior arch corresponds to that o f a m a x i m u m (the top of the box) on b. Let us denote b y A , B , C, A r (like in Fig. 17), the regions of b lying respectively to the left of 7 Note, with the above definition, that fa,b 5t0 =~ a C b. Indeed, if fa,b 510, a cannot cross b, otherwise one would have g~ = g~ and gai+l = /?za + 1, ebi+l = 17'3~- 1, for some i. Take the smallest such i, this means that an arch of b ends at the bridge i. Let i ~ < i be the bridge where it starts, then by b-symmetry, we must h a v e ~ t + l = /?bi,+land g~, = gi'+Ia+ 1, gb = /?ai,+l-- 1, which contradicts the fact that i is the first crossing between a and b.
Meanders and the Temperley-Lieb Algebra
il
iz ia
33
14
I1
b
i2 ia
14
b+<>
Fig. 17. The bridge move b ---* b + <>on the corresponding arch configurations, b has a minimum at i = i2, hence an arch ends at the bridge i = i2, and another starts at the bridge i 3 = (i + 1). In b + o, this minimum has been changed into a maximum, hence the bridges (i~, i4) and (i2, i3) are connected. All the other parts A, /3, C and A ~ of b are unchanged.
between il and i2, between i3 and i4 and to the right of i4. Note that the regions A and A ~ may be connected to each other by arches passing above the (i], i2) and (/3, i4) arches, but/3 and C are only connected to themselves. il,
b+~
b
A
A' il
(i)
i2 i3
14
il
i2 i3
14
a= ((~1= %=4-])
(ii) a= ~
a-O
a
(el= - - ~ ] )
a-~
~
Fig. 18. Example of a walk a, which is b + <>-symmetric. The two possibilities (i) al = o2 = 1 and (ii) 0"1 = --0" 2 = - - I are represented. In both cases, one and only one of the two diagrams a and a - 0"2~ is b-symmetric.
Let us consider a walk diagram a which is b+o-symmetric (cf. Fig. 18). The portions a,/3, % a ~ of the walk a lying respectively below A , / 3 , C, A ~ satisfy the following properties: fl is B-symmetric, "~ is C-symmetric, and a a ~ is AA'-symmetric 8. All these portions of a remain untouched in a + o. Only the two links (i2 - 1, i2) and (i3 - 1, i3) of a will be affected. The b + <>-symmetry of a implies that (eia - - e ~ _ l ) --
= ( e ai4--1 _ s a ) ~ O'1 -= ( e ai 3 - - [ - e
ai3 ) ~
+l,
(6.39)
a2 = •
as the bridges (i], i4) and (i2, i3) are connected in b + 0. Two situations may now occur, according to the relative values of (xl and or2. (i) (71 = or2: a is not b-symmetric, because the links (i] - 1, il) and (i2 - l, i2) of a are not symmetrical (the same holds for the links (i3 - 1, i3) and (i4 - 1, i4)). On the s Here we extend slightly the notion of respective symmetry to walks c C d, with initial and final heights not necessarily equal to 0, by still imposing the condition (6.37
34
E Di Francesco, O. Golinelli, E. Guitter contrary, a - o2<>is b-symmetric, because both links (i2 - 1, iz) and (i3 - 1, i3) are flipped by the box addition/subtraction. This is illustrated on Fig. 18-(i).
(ii) cr1 = - a 2 : a is b-symmetric, but a - o-2<>is not, as the situation of the previous case is reversed. This is illustrated in Fig. 18-(ii). Hence, we have shown that, when a is b + o-symmetric, one and only one of the two diagrams a and a - o2<>appearing on the rhs of (6.35) is b-symmetric. This completes the check of the recursion relation (6.35) (which reduces in both cases crz = -4-1to i = 1). Equation (6.38) is the unique solution to (6.35), (6.36).
1
2
Fig. 19. A particular folding of the walk diagram b C Wn, leading to an a C Wn, such that a is b-symmetric. The solid horizontal lines represent the unfoldedfolding lines, while the horizontal dashed lines represent the lines along which b is effecti,lelyfolded (lines number 3,5,6). The total number of folding lines is n, the order of the diagrams (n = 6 here). In addition to their defining recursion relation, the f ' s satisfy a number of interesting properties, which will prove crucial in the study of meander and semi-meander polynomials. A m o n g the many interpretations of the condition f~,b = 1, the set o f a ' s such that f~,b = 1 for a given b E W~, may be obtained as shown in Fig. 19. First represent b as a walk diagram of 2n steps. Then draw horizontal lines joining the couples of points (of (j,9 gjb =_ gi), the f o r m ( i , g ~ ) b z,9 j >_ 1) corresponding to the beginning and end of all arches o f b (the arch starts at the bridge (i + 1) and ends at the bridge j). It is easy to see that there are exactly n such lines. The set of admissible a's is simply obtained by f o l d i n g the path b arbitrarily along these lines (see Fig. 19). Indeed, the folding operation preserves the b-symmetry of a, by simply reversing all the quantities (g~+l - g~) along the folding line. If no additional constraint was imposed on the a's, we would get 2 n possible foldings for each diagram b. However, a is further constrained to have nonnegative heights, which reduces this number, but we expect it to still behave as 2 '~ for most b's, in the large n limit. Conversely, here is an algorithm to generate, for fixed a E W~, all the walks b E Wn such that f~,b = 1. The path b = a is always admissible. Let us represent it by the sequence o f signs ti(a) = g~ - g~-l, i = 1,2, ..., 2n, and consider the modified sequence -
-
9 ~ri (a) --- ( - 1 )~ - 1 t~ (a) = ( - 1 )i - 1 (e~ -
gia_ 1 )"
(6.40)
Interpreting these indices i as bridge numbers (from 1 to 2n), the set of b's such that f~,b = 1 is simply the set of arch configurations linking these 2n bridges, such that each arch connects two bridges with the same value of the sign ~r~(a). An example is displayed in Fig. 20. The number of admissible b's for fixed a seems to depend strongly on a.
Meanders and the Temperley-Lieb Algebra
35
ti(a ) + + - b . - - + . - - - q- . . . . + - ~,(a) + i - - + i + : + + - - . - - -- + + + i
Fig. 20. For fixed a, the b's such that fa,b = 1 are the arch configurations connecting bridges with the same value of a~(a) = ( - 1 ) i - l t i ( a ) , where ti(a) = g~ - / ~ - l " for i = 1,2, ..,2n. Here n = 6, and we have represented one of the admissible b's.
Let us finally mention the following sum rule, proved in detail in Appendix C: 3 2n-l(__2n)!
E fa,b = a,bEWn
2n-I
n!(n + 2)! - 3 ~
(6.41)
cn
expressing the total number of couples (a, b) c W~ • W~, where a is b-symmetric. By Stirling's formula, we see that 3 8n fa,b ~ 2V/-~ n5/2.
E
(6.42)
a,bGWn
The leading behavior 8 '~ agrees with the expectation that the number of admissible a ' s for fixed b behave like 2 '~ for most b's (whose number is of the order of 4'~). Solving for w. To complete the solution of (6.29), we have to compute the weight Wa,b = [Q --1 ] a,b when a is b-symmetric. The form of Wa,b is entirely dictated by the coefficients of the recursion relation (6.29). The result reads 2n--1
Wa, b
"~
1
=
a
(6.43)
i=1
w(k, g, m)
b
(w(ea 1,e a
I~e+l (Izel.ze<) ~(k+m)-e ,ue
In order to check that this is compatible with (6.29), we note that, with the form (6.34), and when a is b + <>-symmetric, one and only one of the four terms in the r.h.s, of (6.29) is non-zero. Assuming for instance that a has a maximum at i, it is sufficient to check that 1
Wa,b+o _ Wa, b
[,tea
(6.44)
Wa,b+~ _ 1 Wa-o,b
irrespectively of which term survives. Equation(6.40 )follow directly from (6.43), and are exactly what is needed to absorb the coefficients in (6.29). Similarly, if a has a minimum at i, one easily checks that the sufficient conditions
36
P. Di Francesco, O. Golinelli, E. Guitter
Wa, b+o = Pea+l, 213a,b Wa,b+0 -- ~e~+l Wa+o,b ~e~+2
-
(6.45)
are fulfilled. Note also that wa,a = 1 as required.
b=
0
3
5678
12
Fig. 21. An example of computation of wa,b, for a C b. b is obtained from a by six box additions. The box weights are computed using the rules (i)-(ii)-(iii). Here we have wa,b = (l~ll~z)3/z.
Practically, for a C b irrespectively of whether a is b-symmetric, the weight (6.43) can be computed as a product of box factors over all the boxes which must be added to a to build b. (i)
ff the box is added at the vertical of a maximum of a, the weight w is multiplied by 1/#e~. (ii) If the box is added at the vertical of a minimum of a, the weight w is multiplied by
#e~+l(iii) If the box is added at the vertical of a slope of a, the weight w is multiplied by
i #e~+l/m~. This is a direct consequence of the expression of w(k, ~, m) in (6.43), where the power 89 + m ) - e distinguishes between maxima (value - 1 ) , minima (value 1) and slopes (value 0). An explicit example is given in Fig. 21.
6.5. The normalization matrix N'n(q). Let us reexpress the (diagonal) matrix elements of N'n(q) (5.56) in the language of weights w~,b (6.43). The result reads simply
I [./V'n(q)]a,a = Wa,~,y~) ]
(6.46)
where ],V(~n) is the largest walk diagram of W,~, i.e. containing all the others, with heights ~max = rain(i, 2n - i).
(6.47)
This is easily proved as follows. Let us first consider a symmetric diagram a. As mentioned before, this diagram is also I'V(~'~ hence f~,w,,~ = 1. By (6.9, we find that
Meanders and the Temperley-Lieb Algebra
1=[
37
~(q)]
a,l/y~ )
= [~rn(q)-l]a
a
, [Q~(q)
-1
]o,~,.
,*
(6.48)
hence [Hn(q)] o o =
[Qn(q) - 1 ]a,l/VO) o ---- W a~ ..,,n) VV n
(6.49)
as f~,w(~) = 1. This proves (6.46) for any symmetric diagram a. However, the expression (5.56) for arbitrary a is clearly factorized into products pertaining to the left and right halves of a = Ir, namely
[xn(q)]o,o: II
II
left strips on I
= n(l) n(r).
(6.50)
right strips on
Analogously, the quantity Wa,l/y(n) factorizes into two products pertaining respectively to the left and right halves of a = lr, namely W ~ , w ~ , = w(1) w(r),
where
(6.51)
n--1 w(1) ---- (x(g~_l,g~)) ~
H (W(~i--1' z ~i'z ~i+l z )) 88(g~ax__g~)
(6.52)
i=1
and x(k, ~) = ""(~tmax(k,O) k-g
(6.53) ---- (~t(k+g+l)/2) k--e
In the above, we have used the formula max(k, g) = ([k - g[ + k +g)/2 = (k +g+ 1)/2, as k - g = +1. The weights x separate the middle box factors in (6.43) into left and right halves. More precisely, each box factor
[ w(k, e, m) = x(k, e) x(,n, e)]
(6.54)
is factorized into a left half (k, g) (described from left to right on a) and a right half (m, g) (described from right to left on a). Equation (6.54) results from the identity /~g+l (iZglZg+l) 89
/ze
_ _
# kmax(k,g) -~ m-e Pmax(m,O'
(6.55)
easily proved by inspection. Now the result for symmetric diagrams a = It, r = l, reads [N'n(q)] a,a = n(/)2 = w(/)2,
(6.56)
hence n(1) = w(1). For arbitrary diagrams a = Ir, we have [.Afn(q)]a,a = n ( 1 ) n ( r ) = w ( 1 ) w ( r ) = W a,w~n,
(6.57)
which completes the proof of (6.46). As a by-product of the previous analysis, the local factorization property (6.54) and the obvious relation x ( k , g)x(g, k) = l, enable us to rewrite the general expression (6.43) as
38
R D i F r a n c e s c o , O . Golinelli, E . G u i t t e r
2n-- 1
Wa, b
1
b
a
H [x(er-~,e7)~(eT+,,e7)]~(~-~')
=
i=l 2n--I
= H [x(e~,e~+,)]~(~'`'-~>-(~'~'-~r>)
(6.58)
i=O 2n--I
H
=
i--O
1 1
a
a
8
8
("('+':+':+,>/')~(-"'"-'~'<"+'-"))
This last expression reduces directly to (5.56) in the case b = l/V(")..n, providing us with an alternative proof of (6.46). Indeed, gib+l -- gb = 1 if i < n, and - 1 if i >_ n, and the factor 89 - (g~+l - gi~)(gi+lb _g~)) takes the value 1 on a's (left and right) tops of strips, and 0 everywhere else, while (1 + g~ + g~+1)/2 is the corresponding length of strip.
6.6. The path formulation of(semi)-meanders. We now have all the elements to write alternative expressions for the meander and semi-meander polynomials, as weighted sums over paths. The entries of the matrix 79~(q) - l read, using (6.43) [79n(q)-l]a,b = [./~fn(q)-l]a,a [ Qn(q)--l]a,b "~ fa,b
Wa,b W a ,~/y(nn) 2n--1
= fa,b H
1
b
(6.59)
max
[w(~-l'~'~'eia+l)] 7)(g*-el >
i=1 2n--1
b
max
~z
a
a
.
= fa,b e 88~'=~ (e~-e~)~(e~_,,g~ ,e,+,.q), where gm~x is defined in (6.47), and
/ #e+,'~
k + m - 2g log (pgpe+l). 2
c~(k, g, m; q) = log l \ - ~ e / +
(6.60)
Using the alternative expression of Wa,b (6.58), we may also write
l
.1
[79,~(q)-l],~,b = fa,b e ~ ~ : o
[
[(e~-e~)-(ei\,-e~)] (e~L,-e?)log ~,(e~+,~,.,,/2
(6.61)
The passage from (6.59) to (6.61) may be viewed as a discrete integration by parts in the sum over i. Substituting the expression (6.61) in (6.12) and (6.13), we get the following expressions for the semi-meander and meander polynomials
r~nn(q)
L,b fa,w~-, ue~(q)
=
(6.62)
a,bG Wn
• ~ 1 EL-' ran(q) =
~
[(~,+,-~, >-",+,- ~>](~,+,-,> lo~..~+,= +,,. max
max
b
~b
a
ga
f,~,bf~,b' Ue~ (q)
a,b,b' cWn max max b ~b I ~b ! a ~a x ~ 1, = o 2r~ -- ] [~(e~+l-e, >-~,+,.>-( ~b,+,-, >](~.+,.>~og.,,:+~r+,+,,.
(6.63)
Meanders and the Temperley-Lieb Algebra
39
Note that the semi-meander expression (6.62) may be viewed as (6.63) in which U is fixed to be W(~n) - r,~, the walk diagram corresponding to the rainbow arch configuration of order n, which restricts the sum to symmetric walk diagrams a. The expressions (6.62), (6.63) should permit a detailed asymptotic study of the semi-meander and meander polynomials for large n.
6. 7. Connected components in meanders. For any b c A,~ -- W,~, let v~ be the vector with entries (V~)a = 5o,6. The matrix elements of G,~(q) can be expressed as [~n(q)] b,b' = V~, 9~n(q)~
(T'n(q)-lv~ ') 9 Fn(q)T'~(q)-lz~ = qc(b,b'),
=
(6.64)
where c(b, U). Equation (3.13) is the number of connected components of the meander obtained by superimposing the arch configurations b and U. Hence we can write a refined version of 6.63 for fixed b and b~ E A~
q~(b,b') = E
fo,b fo,b' Ue~(q) (6.65)
a E W~
I max max >( e ~ E : ~ -1 [2(ei+,-g , ) - ( s
b
b
b/
b!
a
a
)](gi+l--g,)log,te~+,a+l+l)/Z
Note that the highly non-local quantity c(b, b~) is expressed as a sum of local weights. However, the non-locality reemerges in a weaker form through the selection factors f , which induce mutually non-local constraints on the walks summed over. This formula gives an interesting expression for c(b, U) in the limit of large q. Indeed, we have, for q ~ oc,
Ue(q) ~ qe
#e ~ -1, q
(6.66)
hence (6.65) becomes
qc(b,b')~
E aE
f~,bf~188176
2n -- 1
b
b
b!
bt
a
a
((e,+~-e,)+(e,+,-e~)(t,.~-e,),
(6.67)
W~
where we see the contributions of the ~max,s and that of the Chebishev polynomial have cancelled each other, thanks to the identity g~_l
2n--1
~
a (gim+• - gimax )(gi+l - g~) = O.
(6.68)
i=O
For large q's, the sum in the rhs of (6.67) is dominated by some a C Wn for which the exponent of q is maximal. Such a maximum is unique, as the coefficient of qC(6,6') is 1. This yields the following formula for the number of connected components c(b, bt):
c(b, U) = ~1
max o~wn,
[ A..r
-- ~ i ) -b (gi+l -- gi )] (gi+l -- ~i )
b and b ~-symmetric
(6.69) A particular case corresponding to semi-meanders consists in taking br = }V~n) = rn the rainbow configuration of order n. Using (6.68), we find
40
E Di Francesco, O. Golinelli, E. Guitter
1
c(b) = c(b, ],V(~'~)) =
2n--1 max and b--symmetric
a ~ Wr~ ~ symmetric
i
2e~ + y~.
b a (~ib+l -- e i )(ei+ 1 -- e~)
} (6.70)
i-0
Another interesting consequence of the expression (6.65) is obtained if we take b = U, in which case c(b, b) = n. It takes the form of a sum rule for fa,b, namely, for any b E Wn,
q"=
1
[(e,.m~_e.~%_(gb+_g})](g~%l_eg)logl.(eg+~,~.l+,)/="(6.71)
2~.--1
aE W~ In particular, for b = 14~(~l, hence gb = gm~x for all i, we find, with
q~= E
f~,w~2~=
5~,symm~uic:
Ue,~(q)
a E Wf~ a
symmetric
In/21
(5.72)
-- ~ bn,n-2~ Un-2p(q) 1)-0 which is easily proved by recursion on n (the coefficient b,~,,~-2p, computed in (5.52), is indeed the number of symmetric diagrams with middle height h = n - 2p).
6.8. Asymptoticsfor
q > 2. In this section, we use the expressions (6.62), (6.63) to derive asymptotic formulas for the semi-meander and meander polynomials for large n. Such formulas can only be inferred when all the terms in the sums (6.62), (6.63) over walk diagrams are positive. This is the case for all q > 2, for which Urn(q) > 0 and #m > 0 for all m. q=2. As a preliminary exercise, let us start by taking the limit q ---* 2 of the sum rule (6.71). Due to the definition (4.2), we have
g U/(2) = (g + 1)
#e(2) = g + 1'
(6.73)
therefore, when q ~ 2, (6.71) becomes
2n
E
1
fa'b(e~+l)e~2:;-'
max
max
b
b
]
(e,.,-e,)-(e,,-e,)(e,§
a
la+g a"+1 +1 ~ g.~§e. +3
a
YL- -"I A ~
aCWn Note that, summing (6.74) over b c Wn we get the result
E
1_
fa,b(e~ + 1 ) e 2 E : 7 '
max
max
b
b
a
a
~,
t+l
[(gi+l-g' )-(g~+l-gi)](gi+l-gi)l~ ~?+~;~+,.3 e%~q" +1 = 2nc,~
(6.75)
a,bE Wn
which behaves, for large n, like n
n3/~--~ ~ n
E fa,b a,bEWn
(6.76)
by making use of the asymptotics (6.42). Comparing (6.75) and (6.76), we are led to the following scaling hypothesis for the values of g/5 and ga dominating the sum (6.75):
Meanders and the Temperley-Lieb Algebra
41
g~ ,~ n~g~(x)
gb ~ n,gb(x),
(6.77)
where x = i / n and u E [0, 1] is an exponent characterizing the average height o f the walk diagrams a, b. For this hypothesis to be compatible with (6.76), we m u s t necessarily have u = 1, in which case the exponential in (6.75) tends to a constant 9 (the s u m over i is of order n, but the logarithm is o f order l/n), and the factor (g~ + 1) tends to const, x n, which yields (6.76). This is an example of use of a scaling hypothesis o n the g's d o m i n a t i n g the sum (6.75), l e a d i n g to large n asymptotics. Analogously, if we make the same scaling hypothesis (6.77), with u = 1, on the g's d o m i n a t i n g the sums (6.62), (6.63), for q = 2, we find the asymptotic relations, valid for large n rhn(2)
~
n
~
f~,b
a,bEWn
o ',"~"~'
,-~ n
m,~(2)
(6.78)
fa,b fa,b'
Z a,b,b~ E W n
This expresses the asymptotics o f the m e a n d e r and s e m i - m e a n d e r polynomials at q = 2 in terms off~,b only. In going from (6.76) to (6.78), we have a s s u m e d that configurations of the same order of m a g n i t u d e d o m i n a t e both sums. In fact, we have m a d e a scaling hypothesis o n the matrix elements of p ~ - l ( q = 2) and F n ( q = 2), n a m e l y that the configurations with [7~,7~(2)]~,b "
[rn(2)]~,. = (g~ + 1)
J'~,b
,-, n "
(6.79)
dominate the three sums
Tr(
o(2))
~ n
fo,b,
a,bEW,~ 17"gn(2)Z7 ~ n ~
Z
fa,b,
a,bE Wn a symmetric
if" 9n(2)ff ~ n "
y~
(6.80)
f~,o f~,b'
a,b,b~ E W n
9 To see why, note that for large n and g's the sum in the exponential may be approximated by 2n-- 1
2-
max _ i
_ (ei+1
i+l - gi e ~ +[
i---O
i=0
where we have performed a discrete integration by parts. Hence the exponential of this sum is equivalent to Hi
min. of b (ca + 1)
( g ~ + l ) • 1-Ii . . . . . fb (~a +1) ~ const. The products extend respectively over the i's which are minima and maxima of the walk b and as there is always one more maximum than minima, the above ratio is exactly balanced, hence is of order 1 for large e~'s.
42
P. Di F r a n c e s c o , O. GolinelS, E. Guitter
with the same value of u = 1. Let us stress, however, that the scaling hypothesis (6.79) leads to a wrong result for the meander determinant, D~(2), for large n. Indeed, from (6.79), we would conclude that
]-[ 2 ~ f~,a n
D,~(2)
~
(6.81)
n ~'c'~
whereas, from the exact result (5.6) for D,~(2), we extract the large n asymptotics n
log D~(2) = Z a n , j
(6.82)
l o g o + 1) ~ x/-d-nc~
j=l
by the standard saddle point technique (note that we find exactly twice the previous result (5.19) for the large n asymptotics of log det D~ (0)). The correct asymptotics (6.82) contradict (6.81). This simply means that the configurations of a C 1 ~ dominating the meander determinant are very different from those dominating the trace of the Gram matrix or the (semi-)meander polynomial. q > 2 . We start again from the sum rule (6.71), with q = e ~ + c - ~ 0 > 0. We again make the hypothesis that, when summed over b E W,~, the sum (6.71) is dominated by large g's for large n. Noting that Urn(c~ + e - ~
~
emO 1 - e -2~
Pm ~
(6.83)
e -~
for large m, this gives the asymptotic formula cn(ca+e-~ n ~
c~
~
A,bl_e-2~c
Drnax~'-,igb- , - gb~,J~e,~+,-e~) ~] o ~:2o-' [tprnax L,~,+,--,
a,bC W n 1
-
~
1 -- C- 2 ~
0 ~"~2r~--llob
flb'~dpa
~Oa~
~ fa,be'~ A.~i=o , ~ + 1 - ~ . . . . i+1-- ~, a,b~ Wn
4n e_O)n n3/2 ( cO +
(6.84) where we have used (6.68). This gives an asymptotic sum rule involving the f~,b'S and q. Assuming that the same scaling hypothesis holds for the sums (6.62), (6.63), we find the following asymptotic formulas
~(e o+c-0)
~
~
A,bc o[eo+~
27-, b
b o
a,bG W n a symmetric 0
m,~(e ~ + e - ~
~
~
f~,bf~,b,e ~ s
2n--I
b
b
bt
b!
a
a
[,e,+,-eO+(~,,-e~ )],e~+,-e,)
a,b,b' E W~
(6.85) where we have dropped the prefactor 1/(1 - e-2~ subleading for 0 > 0. Indeed, the limits 0 -+ 0 and n --~ cc do not commute, hence (6.85) is only valid for 0 > 0. On the other hand, in the limit 0 ~ co, we recover the large q asymptotics
Meanders and the Temperley-Lieb Algebra
43
~ n ( q ) ~ q'* ~ e n~ (4e0)n
ran(q) ~ Cn qn ~
(6.86)
n 3/2
by using the two formulas (6.70), (6.69). As before," we can test the scaling hypothesis used above against the large n asymptotics of the meander determinant for q > 2. This hypothesis amounts to writing o [_2g~+X-,2n-bab _obve= _god [ ~ n l ( e 0 -t- e-0)]a,b "~ fa,b e 41- n /--~i=0 ,~,+l ",,' i+1 i ' ] , (6.87)
[Fn(e ~ + e - ~
"~ eOe:.
The corresponding large n estimate of the meander determinant reads
Dn(e~ + e-~
~" H f2,,y~o ~ e ~ o , aEWn
(6.88)
whereas the exact formula (5.6) leads to the asymptotics n
log D,~(e ~ + e - ~ = 0 E
an,j log
sinh(j + 1)0
~,, nc,~O
sinh 0
j=l
(6.89)
by the standard saddle point method. The agreement between the two estimates (6.88)(6.89) is a confirmation a posteriori that the scaling hypothesis (6.87) holds for a very large class of properties of the gram matrix G~(q), for q > 2 and large n. Finally, in view of the assumed q = 2 value u(2) = 1, and the exact q ~ c~ value u(cx~) = 1 (the semi-meander polynomial (6.86) is indeed dominated by the single diagram b = kV(n'~), with winding gb = n ~ n~(~176 it is reasonable to infer that u(q) is identically equal to 1 for all q _> 2.
6. 9. Meander and semi-meander polynomials as SOS partition functions. The asymptotic formulas (6.85) are to be compared with the following exact formulas rh~(e ~ + e - ~
=
E
0
a
1
2n--1
fa'be$[gn+~ Ei=~
b
b
~a
a
(~i+l--~i)(
i+l--~i)]
b +
b!
aE P n ,bE W n a symmetric 0
2r~--I
b
b!
a
a
A,bA,b,e~ E~:o [(~.,-e,)(~.+,-e~ )](~,+,-e,)
m~(e ~ + e -~
(6.90)
aEPn b,bt E W n
where a runs now over the set Pn of all closed paths of (2n) steps (with g~ = g~n = 0) not subject to the constraint g~ >_ O. The relations (6.90) may indeed be obtained as consequences of the following alternative formula for qC(b,b'), b, bt E Wn (to be compared with (6.65))
( e~ + e-~ c(b'b') = ~ a e P , fa,bf~,b 'e~ ~2"2~ [(e,,-eO+(e,+,-t, )](t,+,-e,)
(6.91)
Let us now prove (6.91). On the one hand, as a is both b and U-symmetric, the values of ti(a) = (s - g~) are fixed, up to an overall sign, along each connected component of the meander (b, U), and alternate on successive bridges along the connected component.
44
E Di Francesco, O. Golinelli, E. Guitter
3
6 \
-1
+1
0
0
Fig. 22. The four possible local environments of the (i + 1)th bridge together with the corresponding value si(b, b t) = =kl, 0
On the other hand, the quantity s i ( b , b') = [(eibl _ ~i)+(~i+lb b' _ gb')]/2 may only take the three values - 1,0 and + 1, corresponding to the four possibilities of local environment of the (i + 1) th bridge of the meander (b, U), depicted in Fig. 22. Along any connected component of (b, b'), the variable s i ( b , b') alternates as long as it remains nonzero, and discarding all the zeros leaves us with an alternating sign.
t"'x 1 23
10
Fig. 23. An oriented connected component K with 10 bridges. Starting from bridge 1, the sequence of visited bridges is 1, 8, 9, 10, 3, 4, 7, 6, 5, 2.
For illustration, with the connected component depicted in Fig. 23, this gives the sequence, starting from the bridge 1 bridgei
1
8
9
10
3
4
7
6
5
2
ti(a) s~(b,b') ti(a)s~(b,b')
+ + +
0 0
+ 0 0
+
+ + +
0 0
+ -
0 0
+ 0 0
0 0
turn
R
R
R
-
L
where we also indicated the type of turn (right=R, left=L) taken on the corresponding bridge. The global sign t i ( a ) s i ( b , U ) is thus constant between two zeros and is reversed through each zero. Since a zero indicates a transition from turning left to right and vice versa along the meander, the quantity 1
a
b
b~
(6.92)
i along K
summed along any connected component K of the meander (b, U), is simply equal, up to a sign, to the total number of right turns minus that of left turns ( n n - n r ) , taken on the bridges along K . As on any closed loop we have ( n n - n L ) = 5:2, we compute
Meanders and the Temperley-Lieb Algebra
f(K) = E
45
fa'bfa,b'e~ E , a,oogK t~(a)s,(a)
ti(a)=d=l i along K
(6.93)
= E eOe(nR--nL)/2 e=4-l = e 0 + e-O~
where the sum over e = -4-1 corresponds to the only overall sign ambiguity left on the t~(a) after taking into account the b and U-symmetry of a on K . The final result (6.91) is simply the product over all the connected components K of (b, b~) of the weight f ( K ) above, which completes the proof of the result. More generally, the above analysis can be carried over to q = z + 1/z, for any complex number z, resulting in Z
1
+ 1/z)C(b'5') = E A,bA,b'ZZ aEP~
2n--1 ~-~,--o
b
bt
bI
[(e'b+l-e')+(ei§
(ela+l-~)
(6.94)
This yields the following general expressions for semi-meander and meander polynomials at q = z + 1/z for arbitrary complex z
~n(Z + l/z)
=
E
fa,b zl [~a+l ~2n-1
,b
b
a
a
aEPn,bEWn a symmetric
m~(z + 1/z)
=
E
f~
zl
2n--I E,=O
b bp br [ ( ~ b + l - ~ / ) q - ( ' i + ' - ' i ) ] (~ia+l-~)
(6.95)
aE Pn b,bt E Wr~
Fig. 24. An example of SOS configuration attached to a meander. We display the value of the height L Note that it is entirely dictated by the choices of orientation of the connected components of the meander, and the fact that s = 0 at infinity
This analysis suggests to interpret the quantity qc(b,b') as the Boltzmann weight of a particular configuration, formed by the meander (b, b'), of a suitably defined SOS model. Indeed, the b and b/-symmetry of a E Pn implies that the variable/?a takes identical values on all segments of river which can be connected to each other without crossing any arch of b or b'. Therefore, the variable ~ may be considered as a height variable in the plane, constant on each connected component delimited by one ore several roads, and undergoing a jump discontinuity of-4-1 across each road (see Fig. 24 for an example), and continuous across the river. In particular, ~ = 0 at infinity, due to the boundary condition g0 = g2n = 0. Such an height configuration induces a unique orientation of the various
46
R Di Francesco, O. Golinelli, E. Guitter
connected components of(b, b'), by taking the convention that g --~ g+ 1 (resp. g + g - 1) across a road pointing to the right (resp. left). Conversely, a choice of orientation of the connected components of (b, b') specifies uniquely the height configuration, by further demanding that g = 0 at infinity. The Boltzmann weight b
b
b!
b!
z88 ~2~-1 [(gi+_el)+(g~+l_g* )](g~+l_g~)
(6.96)
corresponds to attaching to each bridge of (b, b') one of the following Boltzmann weights
!
!
Z 2
Z 2
1
1
_! Z
_! 2
Z
1
2
(6.97)
1
according to the local environment of the bridge, and taking the product over all the bridge weights. Again, summing over the two orientations of each connected component K of (b, b') results in a total weight per connected component
Z
z~('~R-nO/2 = z + 1/z = q,
(6.98)
e=4-1
where nR (resp. nL) is the number of right (resp. left) tums of the road on the bridges of K, and e = +1 accounts for the global orientation of K. In the language of SOS models, the expression (6.65) corresponds to a Restricted SOS version, in which the height variable is further restricted to be non-negative (in particular the configuration of Fig. 24 is ruled out). As a first element of comparison with the results of the previous section, if we write (6.95) at z = 1, hence q = 2, we see that ~n(2)
L,b
= a~Pn,bEW~.
a sy. . . . .
m,~(2)
(6.99)
fo,b fa,b,
= aE Pn b,bt E W n
to be compared with the asymptotic estimates (6.78): this gives a relation between sums over Pn and over Wn, involving the same combinations of f . Note that the same type of relation links the cardinals of the two sets over which a is summed, namely
card(Pn)=
n ) = (n 2n
+
1)en
(n + 1)card(Wn),
(6.100)
and also, using (6.41)
fa,b = 2 n aE Pn ,bE W n
Cn -~
2 -~(n + 2) ~
fa,b.
(6.101)
a,bE W n
The reader could wonder in what the restricted expressions (6.62), (6.63) of the previous section are really different from the simple SOS expressions (6.90) obtained above.
Meanders and the Temperley-Lieb Algebra
47
Actually, the considerations of the previous section on the heights g dominating the expressions (6.62), (6.63) for the meander and semi-meander polynomials, eventually leading to an exponent v = 1 for q = 2, could not be carried over here, because of the lack of an explicit prefactor proportional to (g + 1). Hence, in some sense, the formulas (6.62), (6.63) (at least for q = 2) give us access to more precise details on the path formulation. More generally, it is interesting to compare the q > 2 formulas (6.90) and (6.85). We see that these are identical, except for the range of summation over a ( W n in (6.85) and Pn in (6.90)). We conclude that the restriction condition that g~ _> 0 in (6.85) is not important in the large n limit, for q > 2.
7. Generalization: the Semi-meander Determinant In this section, we consider a possible generalization of the meander determinant to semi-meanders in the following way.
12345
Fig. 25. Any semi-meander may be viewed as the superimposition of an upper and a lower open arch configurations. Here the initial semi-meander has winding 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 number 5,4,1 of the upper configuration are respectively connected to the arches number 5,4,3 of the lower configuration).
Going back to the original fiver/road formulation of semi-meanders, we see on Fig. 25 that any given semi-meander, with winding number h, is obtained as the superimposition of two (upper and lower) open arch configurations of order n, with h open arches. By this, we mean that h semi-infinite vertical roads originate from h of the n bridges, otherwise connected by pairs through (n - h ) / 2 nonintersecting arches (the winding h has always the same parity as the order n in the semi-meanders). The semi-meander is re-built in a unique way by connecting the upper and lower open arches from the right to the left. In particular, only open arch configurations with the same number of open arches may be superimposed to yield a semi-meander. Let A ~ ~ denote the set of open arch configurations of order n with h open arches. It is a simple exercise to show that card(A~ ))
= bn,h
=
n~h
,,--~-/
--
n-h
~
- 1
"
(7.1)
Indeed, the open arch configurations of order n with h open arches are in one-to-one correspondence with the half-walk diagrams of n steps, with final height h, namely with Q = 0, g~ > 0 and gn = h. Let W~h) -= A ~ ) denote the set of half-walks of order n with final height h. The number of such half-walks has been derived in Eq. (5.52) above. We now define the semi-meander determinant of order n and winding h, as the determinant D~)(q) of the matrix G~)(q) with entries
48
P. Di Francesco, O. Golinelli, E. Guitter
[G(h)(q)] t,z' = qr
l, l' E W (h) = A (h),
(7.2)
where c(l, l') denotes the number of connected components of the semi-meander obtained by superimposing the open arch configurations 1 and l ~ and connecting their h open arches. For illustration, we list below the matrices corresponding to n = 4, h = 0, 2, 4, G4(~
= ( q2 q
q2) q
G4(2)(q)=
q3 q2
q2 q3
G4(4)(q)=q4
(7.3)
with the following ordering of open arch configurations h=0
: ~ ~-,, ~
h=2
: ~11,1~1
h=4
I1~
:
(7.4)
Note also that G~(q) = G2n(1) l(q) = Gn(q), hence the formula (5.6) applies to the winding zero and one cases. More generally, we conjecture that -~+1 TT
.~
otfh)
D~)(q) = det G~h)(q) = 1-I ujtq) "~,J
(7.5)
j=l (h) where the numbers c~,~,j read, in terms of the an,j of (5.6)
O•(2h) 2n,j (2h+l) r 1,j
= =
an,j+h + 2h an,j+h-1 an,j+h + 2h (an-l,j+h + an--l,j+h-l)
(7.6)
We checked the validity of this conjecture up to n = 9. For instance, for n = 8, 9, we have
= u ~' u~ 3
u6
u4
D(2) 8 n =8
= u ? 9 u~ 2 u~ 3 u4~ D~4) = u,~8 u~ ~ u 4 u6 D~6~ = u ~ D~ 8)
=u[ (7.7)
n : 9
D(91) (3) 9 D(95) D(97) D(9) 9
= U115 U240 U26 = U~2 U ~ u~ 2 = V 102
U 36
= U15~ = U9
U7
v~
U8
U5
u2
in agreement with (7.5), (7.6). We have performed various checks on the numbers a ~ } (7.6). In particular, the term of highest degree of D~)(q), as a polynomial of q, is given by the product of the diagonal terms in ~h)(q), namely qdeg(D~)) =
II
lEW~h)
n+h q--r-,
(7.8)
Meanders and the Temperley-LiebAlgebra
49
hence deg(D~) ) = n + h bn,h.
(7.9)
This can actually be derived from (7.6). We expect that (7.5), (7.6) can be proved by diagonalizing the matrix G(~h)(q). This matrix has again a simple interpretation as the Gram matrix of a certain subspace of TL,~(q), generated by some particular basis 1 elements. Inspired by the one-to-one correspondence between walk diagrams of order n and the elements of the basis 1, we attach to any half-walk I of n steps and final height h in W(~h) the basis 1 element (a)l corresponding to the walk diagram a = lr E W,~, where we have completed the half-walk 1 with a particular choice of right half-walk r of final height h, namely with g~ = [1 + ( - 1 ) i ] / 2 , i = 0, 1, ..., n - h, and g~ = i + h - n for i = n - h+ 1, n - h+2, ..., n. This corresponds to only retaining basis 1 elements which are obtained by acting on f('~) h (defined in (3.6)) through left multiplications by ei. In this new basis, the scalar product between two elements reads
(lr, l'r) = Tr((lr)l(l'r)~) = qC(tr,z'r) = q.V_~ qC(Z,l')
(7.10)
which coincides with (7.2) up to an overall prefactor of q(n-h)/2 due to the addition of (n - h)/2 trivial loops to the semi-meander ll r. A proof of (7.5), (7.6) should follow the lines of that of (5.6), by writing a change of basis which diagonalizes the Gram matrix (7.2). Note also that like in the meander case, the formula (7.5), (7.6) gives the multiplicities of the zeros of D ~ )(q), Finally, the product over all the possible windings of the semi-meander determinants takes the simple form
[gn(q) =
~I
D~)(q) = ~ I UJ (q)~n'j
h=O r ~ - h=O rood 2
(7.11)
j=l
where
\ n -- j / ~2n--l,j
=
\ n -- j -- 1
\n-j/
\n-j~
(7.12)
n-j-1
as a direct consequence of (7.6), with fln,j = Z h ~ (h) Equation (7.11) may be viewed as the semi-meander counterpart of (5.6). The semi-meander gram matrix (7.2) also gives access to refined properties of the semi-meanders. Indeed, we may compute = Tr n
=Z
29/~k)(h) qSk,
(7.13)
k=l
where ]f/I(~k)(h) denotes the total number of semi-meanders of order n with winding h and k connected components. An asymptotic study of these numbers should be made possible by the explicit diagonalization of ~(h)(q).
50
R Di Francesco, O. Golinelli, E. Guitter
8. Conclusion
In this paper, we have extensively studied the representation of the meander and semimeander enumeration problems within the framework of the Temperley-Lieb algebra TLn(q). This representation is induced by the existence of a map between the reduced elements of TLn(q) and the arch configurations of order n used to build meanders and semi-meanders. Moreover, we have seen that the standard trace over TLn(q) provides a tool for counting the number of connected components of meandric objects. The first result of this paper is a direct computation of the meander determinant (5.6), interpreted as the Gram determinant of the basis of reduced elements of TLn(q), and the exact study of its zeros (5.11) and associated multiplicities (5.23)-(5.24). Beyond the meander determinant, we have been able to rewrite the change of basis diagonalizing the Gram matrix in terms of local height variables defining a restricted SOS model (see (6.65)). We also derived an unrestricted SOS model interpretation (see (6.94)) of the Gram matrix elements. These lead to various expressions for the meander and semi-meander polynomials, as weighted sums over discrete paths (walk diagrams). It is tempting to try to approximate these sums by continuous path integrals, in the limit of large number of bridges. In the case q > 2, where all the SOS Boltzmann weights are positive, this path integral might even be dominated by a simple subset of configurations, obtained for instance through a saddle point approximation. A generalization of this approach to the semi-meanders with fixed winding (number of times the roads wind around the source of the river) should be possible, in view of the conjectured form (7.5) for the corresponding (fixed winding) semi-meander determinants. A proof of (7.5) should be at hand, by a simple adaptation of the proof of (5.6) presented here. This will be addressed elsewhere.
Appendix A. Proof of the formula (5.23) for the multiplicities of the zeros of the meander determinant
In order to prove (5.23), we note that
1
k
Oj+l,O mod (k+l) ---- k -t- 1 ~2--~(Cdk+l)m(j+l)' m=O
where Wk+l = e 2irr/(k+l), and rewrite
(A.1)
Meanders and the Temperley-Lieb Algebra |
k
d,~(zk,t) - k + 1 ~
51
n
Z(OOk+l)m(j+l)an'j
m=O j = l
- k+ 1E
n - j
(5dk+l)m(j+l) --
2(r
+ (COk+l)m(j-1)
m=O j = l
- (n271) _
-
1 ~--~(2sin rrm)2 g V i j=l k + 1 m=0
~,n-J) (oak+l) _
1 k 7rm )2 [ 1 2 ( k + l ) Z ( e s i n k + 1 (w v / 7 ~ + l + ~
2n
(A.2)
n-I
)2n
(2:)] -
m---0
- (n221) k
1 x--"2 sin rrm .2.~ rrm -2n 2(k+ 1) 2--at ~-~--i-)tzcos ~-f]-)
=cn
m=l
which is equivalent to (5.23). In the second line of (A.2), we have performed two discrete 2n integrations by parts, which have produced the boundary term (n-l)" In the fourth line of (A.2,)we have used the reality of dn(zk,t) to express the sum over j as
~-.~( 2n )wJ+w -' j=l
n-j
- - 2
_ 1 [ + 1 ]2n_ (2:)] 2 (x/~ ~ j
.
(A.3)
In the last line of (A.2), we have used the sum rule 1 k 2(k + 1) ~ / 2
/
.
7rm \2
sm k-7-i)
= 1
(A.4)
m=O
and recombined (2~)
-- ( n2,~ - - l ) = Cn"
Appendix B. The Gram matrix at q = v/2
Let us illustrate the conjecture (5.39) in the case k = 3, l = 1, namely q = z3,1 = v'~. For n = 3, 4 we have the following identities relating the last line of Gn(V/2) to those corresponding to diagrams of maximal height 2 (B.1) where each line vector is represented by its labeling diagram. In turn, the labeling diagram represents a basis 1 element for TLn(q = v~). Equations (B.1) translate into the fact that the element
52
P. Di Francesco, O. Golinelli, E. Guitter E3(el, e2) = 1 - v/-2(el + e2) + (e2el + ele2)
(B.2)
is orthogonal (with respect to the scalar product (3.12)) to all the elements of respectively TL3(v"2) and T L 4 ( v ~ ) . This is a direct consequence of the following identities:
el E3(el, e2) = e2 E3(el, e2) = 0, Tr(1 E3(e~, e2)) = ~/U3(V~) = 0,
,
(B.3)
Tr(e3E3(el, e2)) = v ~ U3(x/2) = 0 where the first and second lines are valid in both T L 3 ( v ~ ) (7/ = 1) and T L 4 ( v ~ ) (7/= v~), and the third line holds only in T L a ( v ~ ) . More generally, the element (B.2) is orthogonal to all the elements of TL,~(x/~) for any n > 5 as the ei commute with E3(el,e2) for i >__ 4. For n > 5 however, all the linear combinations we get involve diagrams with some heights > 3. For instance, for n = 5, the first combination reads
Z ~
= V'2 ( ~
+/'/~)
- (Z'r
+ ~"N,~)
(B.4)
Fig. 26. The enhancement transformationof a walk diagram. The walk diagram a = ABE Wn is enhanced at the point marked by a dot, by simply inserting a maximum at this point. Here A = l and B = r t, as the marked point lies in the middle of the diagram. The enhanced diagrambelongs to Wn+l. Note that going from TL4 to TL5 (as well as going from TL3 to TL4) amounts simply to enhancing the middle part of the diagrams, as depicted in Fig. 26, which results in a middle height 64 = 2 ~ 6L = 3 for the four diagrams on the r.h.s, of (B.4). To reexpress the combination (B.4) in terms of diagrams of W5,2, we note that the four diagrams appearing in the r.h.s, of (B.4) contain a middle sequence of heights of the f o r m (6 3 = 1 , 6 4 = 2, 65 = 3, 6 6 = 2, 6 7 = 1), as the result of two successive enhancements. Using the first line of (B.1), we may rewrite this central part as a linear combination o f four diagrams with central height _< 2, which results in the four combinations
(B.5) ~_~
= v~ (~..,,~
+ ~
)
- (z~x,.~
+ ~
)
= v~ ( , ~
+~
)
- (
+~
)
~
which, upon substitution into (B.4), yield the desired expression of the last line o f Gs(v'~) as a linear combination of the 25-1 = 16 lines corresponding to the elements of W5,2. Note that all these diagrams have middle height 1. For general n, we have the following recursive algorithm to generate the desired linear combination expressing the last line of Gn(x/2) in terms of the lines a E Wn,2, denotedby K,~ = Y~aew,,: AT(a). T h e combinations K3, K4 and K5 have been constructed above. Suppose we have constructed K , . Two situations may occur for K,~+I.
Meanders and the Temperley-Lieb Algebra
53
(i) If n = 2p - 1, the combination K2p is simply obtained by enhancing (see Fig. 26) the middle of all the diagrams of W2p-l,2 appearing in K2p-1, and keeping the coefficients of the combination fixed. But as the middle heights always satisfy gn = n mod 2, for all n, the diagrams of W2p-l,2 have all middle height g2p-1 = 1. Therefore, the combination Kzp only contains elements of W2p,2, with middle height g2p = 2. (ii) If n = 2p, the combination K2p+l is obtained in two steps. First enhance the middle of all the diagrams in K2p to get another linear combination Lzp+l. According to the previous discussion, the enhanced diagrams in L2p+l have all middle height equal to 3. But they actually arise from the diagrams appearing in Kzp_ 1, after two successive enhancements. This means that they all contain a middle sequence of heights of the form (gn-1 = 1,gn = 2, g~+1 = 3,gn+z = 2, gn+3 = 1). The second step consists in using the first line of (B. 1) to reexpress this middle piece as a linear combination of diagrams with middle height 1 < 2. This yields Kzp+l after substitution in L2p+I. By carefully following the above algorithm, we find the following compact expression for the linear combination K2p+l. P
(1A)( 2p+I )'~ _
9 "2p+l ' - K 2 p +1 = E
(- 1)J(v~)P-J
j---o
E (a) ae I~
(B.6)
where the sets Ij C W2p+l,2 are constructed recursively as follows. I0 is the set of symmetric diagrams of W2p+l,2. lk is the set of diagrams of W2p+1,2 which may be obtained from diagrams in Ik-1 by one box addition, and which are not already elements of some 1k-l, 1 > 1. One can easily show that card(Ij) = 2P(P). The reader will easily check (B.6) for n = 3, 4, 5, with the previous expressions (B. 1), (B.4), (B.5). The expression for K2p+Z is easily obtained by enhancing K2~1 (case (i) above). This leads to the relation (5.39) linking the semi-meander polynomial of degree (2p + 1) at q = x/2 to the polynomials (5.40) corresponding to the closures of all a E W2p+1,2, at the same value of q P
r~2p+,(vf2) = ~-~(-1)J(v~) p-j ~ j=O
~(a, V~)
(B.7)
aCIj
This proves the conjectured relation (5.39) in the case k = 3,1 = 1. Note also that changing x/~ ~ - x / ~ in (B.7) gives an analogous relation in the case k = 3,1 = 2. More generally, the element ~(n) = E,~(el, ..., en-1) (4.5) is orthogonal to all the elements of TLn(q = 2 cos 7r/(n + 1)), as a consequence of the identities e~ q0~n) = 0
for i = 1 , 2 , . . . , n -
Y r ~ n) = Un(q = 2 cos
7r n+l )=0,
1, (B.8)
This permits to express the last line of Gn(q = 2 cos 7r/(n + 1)) (corresponding to the diagram },V~'~) or equivalently to the element (w~n))l = 1) as a linear combination of the (Ca - 1) other lines, corresponding to diagrams with heights < (n - 1), and middle height (n - 2). This implies in particular that rn (2 cos 7r/(n + 1)) < c,~ - 1, and agrees with the conjectured relation (5.36), which reads here
54
E Di Francesco, O. Golinelli, E. Guitter
dn(2 cos
71"1 ) = 1,
r n ( 2 cos
n+
71" ) = Cn -- 1.
n+l
(B.9)
For m > n, E n ( e l , . . . , e,~_ l) remains orthogonal to all the e l e m e n t s of T L m ( 2 cos 7el(n+ 1)). This results in an expression o f the last line o f Gm(2COS~/(n + 1)) as a linear c o m b i n a t i o n o f the (Cn - l) r e p e a t e d ( m - n times) e n h a n c e m e n t s o f the e l e m e n t s o f W n , n - l , which belong to Win,m-1. For m = n + l, the elements o f the e n h a n c e d linear c o m b i n a t i o n still lie in W,~,n-1 as only the middle heights h a v e been affected, and c h a n g e d from ( n - 2) to ( n - 1). H e n c e all the linear c o m b i n a t i o n s corresponding to m = k n + 1 are the trivial e n h a n c e m e n t s of the linear c o m b i n a t i o n at m = kn. In all the o t h e r cases, m a n y reductions must be applied to the d i a g r a m s to eventually get a linear c o m b i n a t i o n o f elements o f W m , n - 1 only. We will not discuss the details o f this m e c h a n i s m here.
Appendix C. Proof of the sum rule (6.41)
b2 1
2j+2 al
a
I"
a2
II k ............ al
a
~1
Il
a2
~~k-1 ...... l//lI k 0 1
2j+l 2j+2
2n+2
Fig. 27. The recursion for .(k) , t n + 1 The diagram b C Wn+l is represented as an arch configuration, and we have represented its leftmost arch, separating its interior piece bl C Wj from its exterior p!ece b2 E Wn .4. The a's C P~+I (-k) which are b-symmetric are of either form depicted. In the first case, g~ -_ ga 2a+l = 1. The piece al 9
of a between these two points is bi-symmetric, and has its restriction condition lowered by 1: al G P3( - k - l ) (the dashed line represents the g = 0 line in the ai's). There are ~1~k+l) such couples (al, bl ). In the second case, g~ = gzad+l = --1. am is bl-symmetric, but now its restriction condition is raised by I: al E P3~.-k+l). There are r/? -1) such couples (al, bl). The piece a 2 is bz-symmetric and has its restriction condition unchanged in c--k)9There are r/(k) both cases: a2 C P~_j n - j couples (a2, b2) W e w i s h to establish the f o l l o w i n g result
Z fa'b=2ncn--12n+lcn+l a,bEWn
(C.1)
valid for n _> 1 (we set the n u m b e r on the lhs o f (C.1) to be 1 w h e n n = 0). B y a simple r e a r r a n g e m e n t of factorials, this is readily seen to b e equivalent to (6.41). Our
Meanders and the Temperley-Lieb Algebra
55
strategy will be the following. First we write a system of recursion relations linking the numbers (C. 1) to other numbers, to be defined below. We proceed and show that this set completely determines all the numbers, provided we take some suitable boundary conditions. Finally, we solve the system explicitly, and extract back the exact value (C. 1). Like in Sect. 6.9, we denote by Pr~ the set of unrestricted walks a, such that g~ = g ~ = 0, without the positivity constraint on the ga,s. Let p ( - k ) denote the set of walks a E Pn, whose (possibly negative) heights are bounded from below by - k , k a given nonnegative integer,
p(-k) = {a E Pn, s.t. g,'~ = g"~n= 0andg~ _> - k , Vi}.
(C.2)
In particular, p(O) = Wn. Note also that if k _> n, the above restriction amounts to no restriction at all, hence p ( - k ) = Pn. We define ~7~) to be the total number of couples (a, b), a E p ( - k ) and b E W,~, such that a is b-symmetric
7l~) =
E
f~,b
(C.3)
aEP(~-k), bEW,~
and Ek the generating function oo
Ek(x) = E
~7~) x~"
(C.4)
f~,b = 2 n C,~
(C.5)
n=0
Again, whenever k > n, we simply have ~(k)
=
E aE P~ , bE Wn
as shown in (6.101). The desired result (C. 1) amounts to writing that 1 (C(2x) E o ( x ) = C ( 2 x ) - 8xx
1 - 2x),
(C.6)
where C(x) denotes the generating function (5.35)of the Catalan numbers (the subtractions in the second term are ad hoc to yield the initial value %(0) = 1). Let us now derive a system of recursion relations for the numbers ~(nk). Let us count the pairs of walk diagrams (a E W~+l, b E Wn+~) such that a is b-symmetric. Representing b in the arch configuration picture as in Fig. 27, let us concentrate on its leftmost arch, connecting the first bridge (1) to, say, the bridge (2j + 2) (the bridge number must clearly be even). This arch isolates its interior, corresponding to the bridges 2, 3 ..... (2j + 1) from its exterior, corresponding to the bridges (2j + 3) ..... (2n + 2): these two sets of bridges cannot be connected to each other. Let us now count the a's which are b-symmetric, and consider an a E Wn+l, such that fa,b = 1. The part al of a corresponding to the interior bl of the leftmost arch of b is symmetric w.r.t, this piece of b. The same holds for the part a2 of a corresponding to the exterior b2 of this arch, which may be simply seen as a walk with 2(n - j) steps, i.e. an element of W n - j . In addition, we also have ~ - g~) = 1 = g~j+l -- g~j+2 by symmetry w.r.t, the leftmost arch of b, which implies that g~ = g~j+l = 1, while g~ _> 0 for i = 1 , 2 , . . , 2j + 1. Therefore, by a trivial translation of the heights and bridge numbers g~ = gi+l - 1, the part of a corresponding to the interior of the arch may be seen as a walk of (2j) steps with g~' = gzaj' = 0, but with the constraint
56
E Di Francesco,O. Golinelli,E. Guitter
that g~' _> - 1 for i = 0, 1, ..., 2j, hence as an element of pj(-1). Conversely, we may build any a which is b-symmetric by the juxtaposition of a walk in p)-l) and one in Wn-j, with the respective conditions that they are b-symmetric w.r.t, the corresponding portions of b, and elevating the interior portion by shifting the ga's of p ! - l ) by +1, and adding g~ = g~j+2 = 0. This is summarized in the following recursion relation: n ~(0) _-- ~ "tln+l ~
~(1)~(0) llj 'lln_ j 9
(C.7)
j--o (k) , More generally, the same reasoning applies to r/n+1 with the result (see Fig. 27)
r/(k) n+l
~
k+l) ~(k- 1)~ ~(k)
=
(r/5
+ "j
(c.8)
) "n--j,
j---0
where two situations may now occur for the part of a corresponding to the interior of the arch: either g~ = g2~+l = 1, in which case the restriction condition on a is lowered by 1 (term r/~k+l)), or g~ =
gEaj+l ---
--1, which may occur as soon as k >_ 1, in which case the
restriction condition is raised by 1 (term r/~k-1)). The exterior part of a is unaffected and keeps the restriction condition at level - k (term r/,~(k)_j).We may take (C.8) as generic recursion relation, also valid for k = 0, provided we define 77(--1) - - 0 for all n _> 0. In addition to this boundary condition, we set r/(k) = 1 for all k (there is exactly one walk diagram of 0 steps, with go = 0, whatever the restriction k). The recursion relations (C.8) together with the boundary conditions ?7(0k) =
r/(--1) -~ 0
(C.9)
1
determine all the numbers r/~) completely. Indeed, (C.8) expresses r/n+1 in terms of r/j, j < n, hence by repeated applications, we may express all the numbers ~(k) t/n in terms of the collection of numbers r/(0k). This establishes the uniqueness of the solution to (C.8), (C.9) provided it exists. To show the existence, we next exhibit the solution explicitly. It is best expressed in terms of the generating functions Ek(x) (C.4), in terms of the variable _
_
y = C(2x)2 - 1 = ~
2 n-I
cn
x~
(C.10)
n=l
easily invertible as x - - -Y (2y + 1)2
(C.11)
by use of (5.35). The general solution reads E2k(x)
=
2y+l--
E2k+l(x)
=
2y+l--
y+l
Uk(1/y)Ua+I( I /y)
(2y + 1)(y + 1)
y(Uk(1/y) + Uk+l(1/y) ) (Uk+l(1/y) + Uk+2(1/y)) (C.12)
Meanders and the Temperley-Lieb Algebra
57
where Uk(z) denote the Chebishev polynomials (4.2). Note in particular that for k = 0, we recover Eo(x) = 1 + 2y - y(y + 1) = 1 + y - y2, which yields the desired result (C.6), and therefore proves (C. 1). The first few generating functions read
Eo(x) = 1 + y - y2, El(x) =
(1 - y)(2y + 1) 2 l +y-y 2 1 + y - 2y 2 - y3
(C.13)
E2(x) =
1-y (1 - 2y2)(2y + l) 2 E3(x) = (1 + y - y2)(1 + y - 2y 2 - y3)" Note also that the expressions (C.12) make it clear that the E k ( x ) converge uniformly towards (2y + 1) = C(2x) when k ~ oc, for small enough x (indeed, when expanded at small y, (C.12) reads Ek(x) = 2y + 1 + O(y k+l) ~ 2y + 1 when k ~ c~). This is not surprising, as letting k tend to infinity amounts to progressively removing the constraints on the counted paths, whose numbers tend to 2ncn (they are actually exactly equal to this for all n _< k), and 2y + 1 = C(2x) is precisely the generating function for unconstrained paths. To prove (C. 12), let us rephrase the recursion relations (C.8) in terms of generating functions. We have E k(x) - 1 = x E k (x) (Ek+ l(X) -I- E k_ l(X)),
(C.14)
where we have used the boundary condition ~7rk) = 1 ~ Ek (0) = 1. The remainder of (C.9) implies that E - I ( x ) = O. (C.15) It is now a straightforward but tedious exercise to check that (C. 14) is satisfied by (C. 12). For odd k = 2p + 1, we have 1 -
x(E2p+2(z)+E2p(z))
2y y(y + 1) Up+2(1/y) + Up(l/y) - - + - 2y + 1 (2y + l) 2 Up(1/y)Up+l(1/y)Up+z(1/y) , 1 y+l + 2y + 1 ( 2 y + 1)2Up(1/y)Up+2(1/y)
=1 --
(C.16)
(2y + 1)Up(1/y)Up+2(1/y) + y + 1 (2y + 1)2Up(1/y)Up+2(1/y) where, in the second line, we have used the recursion relation (4.1). On the other hand, we compute
1 E2p+l(X)
y(Up + up+l)(u., + up+2) (2y + 1 ) ( y ( U p + U p + , ) ( U p + l + U p + 2 ) - f f - 1 )
Using the multiplication rule
(C.17)
58
P. Di Francesco, O. Golinelli, E. Guitter
m+k Uk(t) U.~(t) = E Uj(t) j=lm-kb
(C.18)
j = m + k mod 2
easily proved by recursion, and implying in particular that U 2 I = UpUp+2 + 1, we reexpress (Up(t) + Up+l(t)) (Vp+l(t) + Vp+2(t)) = Vp+l(Vp + Up+2) + Vp2+l + VpUp+2 = (t + I)U2+I + UpUp+2
(C.19)
= (t + l)(UpUp+2 + 1) + UpUp+2 = (t + 2)Up(t)Up+2(t) + t + 1 by various applications of (4.18) Substituting this into (C.17), with t = l / y , this gives exactly (C.16), thus proving (C.14) for k = 2p + 1. For even k = 2p, we have 1 -- X (E2p+l (x)+E2p_ l(X))
1
2y+ 1 1
2y + 1
y+1 up_~ + up + up+l + up+2 2y + 1 (Up-1 + Up)(U v + Up+t)(Uv+t + Uv+2) y+l y(2y + 1)(Up_l + Up)(Up+l + Up+2)
(C.20)
-- (Up-1 + Up)(gp+l + Up+2) + (y + 1)/y (2y + l ) ( g p - 1 + gp)(Up+l + Up+2) We then compute (Uv_l(t)+Uv(t))(Up+l(t) + Uv+2(t)) + t + 1
= (Up_lUp+l +UpUp+2) +Up-lUp+2+UpUp+l + t + 1 = (tUpUv+I - 1) + (UpUp+1 - U1) + UpUp+t + t + 1
(C.21)
= (t + 2)Uv(t)Up+l(t)
Finally, we write 1
E2p(x)
Up(1/y)Uv+l(1/y) (2y + 1)Up(1/y)Up+,(1/y) - y - 1
(C.22)
which, upon the substitution of (C.21), with t = i / y , is equal to (C.20). This completes the proof of (C.14) for k = 2p. Acknowledgement. We thank A. Zvonkin for bringing Ref.[4] to our knowledge, R. Balian for helpful discussions, S. Legendre for interesting historical remarks and J.-B. Zuber for a careful reading of the manuscript.
Meanders and the Temperley-Lieb Algebra
59
References 1. Hoffman, K., Mehlhorn, K., Rosenstiehl, E, Tarjan, R.: Sorting Jordan sequences in linear time using level-linked search trees. Information and Control 68, 170-184 (1986) 2. Phillips, A.: La topologia dei labirinti. In M. Emmet, ed. L' occhio di Horus: Itinerario nell'immaginario matematico, Istituto della Enciclopedia Italia, Roma, 1989 pp. 57-.67 3. Arnold, V.: The branched covering o f C P 2 ~ $4, hyperbolicity and projective topology. Siberian Math. Jour. 29, 717-726 (1988) 4. Ko, K.H., Smolinsky, L.: A combinatorial matrix in 3-manifold theory. Pacific. J. Math 149, 319-336 (1991) 5. Lando, S., Zvonkin, A.: Plane and Projective Meanders. Theor. Comp. Science 117, 227-241 (1993) and Meanders. Selecta Math. Sov. 11, 117-144 (1992) 6. Di Francesco, E, Golinelli, O. Guitter, E.: Meander, folding and arch statistics. J. Math. and Computer Modelling 144, (1996) 7. Makeenko, 5(.: Strings, Matrix Models and Meanders. Proceedings of the 29th Inter. Ahrenshoop Symp., Germany (1995) 8. Touchard, J.: Contributions ~tl'rtude du probl6me des timbres poste. Canad. J. Math. 2, 385-398 (1950) 9. Lunnon, W.: A map-folding problem. Math. of Computation 22, 193-199 (1968) 10. Temperley, H. 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) 1I. Martin, P.: Potts models and related problems in statistical mechanics. Singapore: World Scientific, 1991 Communicated by G. Felder
Commun. Math. Phys. 186, 61-85 (1997)
Communications in
Mathematical
Physics
9 Springer-Verlag1997
Unitarity of Rational N = 2 Superconformal Theories W. Eholzer, M. R. Gaberdiei Department of Applied Mathematics and Theoretical Physics, Universityof Cambridge, Silver Street, Cambridge, CB3 9EW, UK. E-mail:
[email protected],
[email protected] Received: 12 February 1996/ Accepted: 15 August 1996
Abstract: We demonstrate that all rational models of the N = 2 super Virasoro algebra are unitary. Our arguments are based on three different methods: we determine Zhu's algebra A(7-/0) (for which we give a physically motivated derivation) explicitly for certain theories, we analyse the modular properties of some of the vacuum characters, and we use the coset realisation of the algebra in terms of su(2) and two free fermions. Some of our arguments generalise to the Kazama-Suzuki models indicating that all rational N = 2 supersymmetric models might be unitary.
1. Introduction Among the various conformal field theories, the supersymmetric field theories play a special rrle as they are important for the construction of realistic string theories which involve fermions. There exist different classes of superconformal field theories which are parametrised by N , the number of fermionic (Grassmann) variables of the underlying space. For realistic string theories with N = 1 space-time supersymmetry, the worldsheet conformal field theory is believed to require N = 2 supersymmetry. In contrast to the N = 1 super Virasoro algebra which is rather similar to the nonsupersymmetric ( N = 0) algebra, the N = 2 algebra seems to be structurally different. For example the Neveu-Schwarz and Ramond sector of the N = 2 algebra are connected by the spectral flow [39], and the embedding structure of its Verma modules is much more complicated [9, 10]. In this paper another special feature of the N = 2 superconformal field theory is analysed in detail: the property that all rational theories are unitary. Here we call a theory rational if it has only finitely many irreducible highest weight representations, and if the highest weight space of each of them is finite dimensional. We shall use three different methods to analyse this problem which we briefly describe in tuna.
62
W. Eholzer, M. R. Gaberdiel
It was shown by Zhu [42] that a theory is rational in this sense if a certain quotient A(7-/0) of the vacuum representation 7-10 is finite-dimensional. This space also forms an associative algebra, and the irreducible representations of this algebra, the so-called Zhu algebra, are in one-to-one correspondence with the irreducible representations of the meromorphic conformal field theory 7-(0. For the case of the N = 2 superconformal theory, the algebra has always the structure of a finitely generated quotient of a polynomial algebra in two variables, and this implies that A(7-10) is finite dimensional for every rational theory. In this paper we give a physically motivated definition for A(~0). We then show, using the embedding diagrams of the vacuum representations of the N = 2 algebra [9, 10], that A(~0) is infinite dimensional for a certain class of non-unitary theories, thereby proving that these theories are not rational. In addition, we also calculate A(7-(0) explicitly for a few special values of the central charge. We find that A(7-/0) is indeed infinite dimensional for the non-unitary cases we consider (and finite dimensional in the unitary cases). In order to be able to determine the dimension of A(7-/0) for arbitrary central charge one would need to know all vacuum Verma module embedding diagrams and explicit formulae for certain singular vectors. The embedding diagrams are known [9, 10], but sufficiently simple explicit formulae for the singular vectors do not exist so far in general. It was also shown by Zhu [42] that the space of torus amplitudes (which is invariant under the modular group) is finite dimensional for a rational superconformal field theory 1. For such theories, this implies in particular that the orbit of the vacuum character under the modular group is a finite dimensional vector space. If this is not the case, on the other hand, the theory cannot be rational. We determine the vacuum characters using the embedding diagrams, and analyse the action of the modular group on it. We then show that the relevant space is infinite dimensional for c > 3, and for the class of non-unitary theories with c < 3 which was already analysed by the previous method. As a non-trivial check, we also show that this space is finite dimensional in the unitary minimal cases, where c < 3. The only cases which remain can be analysed using the coset realisation of the N = 2 super Virasoro algebra (see e.g. [30])
su(2-"~)k9
(~et) 2 ,
(1)
u(1) which is known to preserve unitarity [20]. Because of this property, a non-unitary (rational) N = 2 theory must correspond to non-integer level for the su(2) k. The only remaining cases correspond to admissible level k r N for which s u ( 2 ) k always has at least one (admissible) representation whose highest weight space is infinite dimensional. Following a simple counting argument due to Ahn et al. [1] we then show that this gives rise to infinitely many inequivalent representations of the N = 2 theory, thus proving that the theory cannot be rational. The reasoning should be contrasted with the situation for N = 0 and N = 1, where the corresponding counting argument does not work: there the admissible representations of su(2) k give rise to the non-unitary minimal models [27, 31]. The last method is well amenable to generalisation. Apart from some mathematical subtleties which we discuss, it can also be applied to the large class of Kazama-Suzuki A
1 Here we use again that for the case of the N = 2 theory,A(7-10)is finite dimensionalfor every rational theory.
Unitarity of Rational N = 2 SuperconformalTheories
63
models, and we therefore formulate it in this setup. As this class already provides most of the known N = 2 models, our arguments seem to indicate that actually all rational N = 2 superconformal field theories might be unitary. All three methods rely to varying degrees on the (conjectured) embedding diagrams for the vacuum representations of the N = 2 super Virasoro algebra which we shall discuss in some detail. For example we use the embedding diagrams to obtain a formula for the vacuum character, whose modular properties we analyse. In the calculation of A(7-/0) we conclude from the embedding diagrams that there exist no further relations, and finally, we check that the coset (1) actually realises the N = 2 algebra by comparing the coset vacuum character with the one obtained from the embedding diagrams. The paper is organised as follows. In Sect. 2 we fix our notations and describe the embedding diagrams for the vacuum Verma modules of the N = 2 super Virasoro algebra following DSrrzapf [9, 10]. In Sect. 3 we give a physically motivated derivation for A(7-(0), and calculate it for certain cases. In Sect. 4, we use the coset realisation of the algebra (1) to analyse the theories which correspond to an admissible level. Furthermore, we indicate how the arguments generalise to the Kazama-Suzuki N = 2 superconformal theories. Finally, we remark in Sect. 5 how the construction can be even further generalised and give some prospective rgmarks. In Appendix A we derive the vacuum character using the embedding diagrams of Sect. 2 as well as the coset realisation. In Appendix B, we analyse the modular properties of these characters, thereby showing that certain classes cannot be rational.
2. Preliminaries and Embedding Diagrams Let us first fix some notations and conventions. Throughout this paper we will consider the Neveu-Schwarz sector of the N = 2 super Virasoro algebra which is the infinite dimensional Lie super algebra with basis Ln, Tn, G~, C (n, r + 89 E Z) and (anti)commutation relations given by C 3 [Lm, Ln] = (m - n)Lm+n + i ~ ( m - m)gm+n,O, 1
+
[Lm, G~v ] = ( ~ m - r)Gm+r, [Lm, Tn] = - n T m + m 1
(~
,
[Tin, Tn] = ~ C m m+n o, [T.~, G~]
= + G . ~~= +~
C 2 1 {G~+, G~-} = 2L~+~ + (r - s)T~+~ + ~ ( r - -~)6~+~,o, [Lm, C] = [Tn, C] = [Gv=t=,C] = 0,
{a;, a : }
= {a;, a2} = o
for all m , n E Z and r , s E Z + 89 We denote the Verrna module generated from a highest weight state [h, q, c > with L0 eigenvalue h, To eigenvalue q and central charge C = c~ by )2h,q,c. An element of a highest weight representation of the N = 2 super Virasoro algebra which is not proportional to the highest weight itself will be called a 'singular vector' if it is annihilated
64
W. Eholzer, M. R. Gaberdiel
by all positive modes Ln, Tn, G~ (n, r + ~l E N := { 1 , 2 . . . }) and is an eigenvector of L0 and To. A singular vector is called uncharged if its To eigenvalue is equal to the To eigenvalue of the highest weight state and charged otherwise. The character Xv of a highest weight representation "1)is defined by
Xv(q)
:=
qh-c/24
~
dim02n)q,~
1 1 nE~N--~
where Vn is the Lo eigenspace with eigenvalue h + n, c is the central charge and h the conformal dimension of Y. The character of the Verma module ])h,q,c is for example given by XYh,q,c =
qh-c/24 1-I n=l
(1 + q n + l ) 2 (1 -- qn )2
and is called the 'generic' character. We call a meromorphic conformal field theory (MCFT) [19] rational if it possesses only finitely many irreducible M C F r representations2, and if the highest weight space of each of them is finite dimensional. We should stress that this definition of rationality differs from the definition used in the mathematical literature, where it is not assumed that the highest weight spaces are finite dimensional, but where in addition all representations are required to be completely reducible. It was shown by Dong et al. [12] that the mathematical definition of rationality implies the one used in this paper. On the other hand the converse is not true as there exists a counterexample [17]. For bosonic rational theories it has been shown by Zhu [42] that the space of toms amplitudes which is invariant under the natural action of the modular group is finite dimensional 3. The generalisation to the fermionic case has been studied in [24, Satz 1.4.6]. If all representations are completely reducible, the space of toms amplitudes is generated by the (finitely) many characters of the irreducible representations, In this case, the central charge and the conformal dimensions of the highest weight states are all rational numbers [2, 40]. We parametrise the central charge c as c(p, p') = 3(1 - 2P---~), P where p and pr will be chosen positive for e < 3. The well-known series of unitary minimal models then corresponds to the central charges being given as c(p, 1), where p > 2 [5, 7]. Finally, let us denote by Xp,p' the vacuum character of the model with central charge c(p, pr), i.e. the character of the irreducible quotient of V0,0,c(p,p, ). One of the main points realised in [8, 9] is that there can be up to two linearly independent uncharged singular vectors at the same level. Indeed, this happens for example for the Verma modules related to the unitary minimal models of the N = 2 super Virasoro algebra4. In [9] a complete list of all embedding diagrams of the N = 2 super Virasoro algebra has been conjectured. 5 2 A MCFT representation is a representation which is compatible with the vacuum representation, i.e. the null-fields of the MCFT act trivially. We do not assume that the graded components of a MCFT representation are finite dimensional. 3 Actually, Zhu showed that this is true if A(T/0) is finite dimensional. In the case under consideration, A ( ~ 0 ) is a finitely generated quotient of the polynomial algebra in two variables (as we shall show in Sect. 3), and thus, the theory is rational if and only if A(7-/0) is finite dimensional. 4 The embedding diagrams conjectured in [11, 32, 34] for the unitary case are not correct. 5 However some of them are still not correct [10].
Uniturity of Rational N = 2 Superconforrnal Theories
65
In contrast to the case of the Virasoro algebra it is not directly clear how to define embedding diagrams for the Verma modules of the N = 2 super Virasoro algebra. This is due to the fermionic nature of the N = 2 algebra: suppose that there is a singular vector ~b,~,p = O~,plh, q, c > of energy h + n and charge q + p in lPh,q,c and that ~t ,,p, = O%,p, Ih + n, q + p, c > is singular in )')h+n,q+p,c. T h e n Otn,,p, On,plh, q, e > might be identically zero in Vh,q,c. Our definition of embedding diagrams of Verma modules of the N = 2 super Virasoro algebra follows [9] and includes only those Verma modules which are actually embedded in the original Verma module. To be more specific, the embedding diagram of a Verma module of the N = 2 super Virasoro algebra shows the highest weight vector and all nontrivial singular vectors contained in it up to proportionality. These vectors are connected by a line to a singular vector if there exists an operator mapping the singular vector of lower level (or the highest weight vector) onto the singular vector of higher level. As in the case of the embedding diagrams of the Virasoro algebra we shall omit lines between two vectors if these vectors are already indirectly connected. We also want to include in the embedding diagrams information about the type of the singular vectors. To this end we use the following notation: the highest weight vector is denoted by a square and the singular vectors by circles. These circles are filled for singular vectors corresponding to Kac-determinant formula vanishings and unfilled for descendent singular vectors (for the explicit form of the Kac-determinant see [5, Eq. (6)]). Furthermore, uncharged singular vectors which have no singular descendents of positive or negative charge, respectively, are denoted by surrounding triangles pointing to the left or right, respectively. (These singular vectors are of type A (1,0) or A (0, 1) in the notation of [91). It has been shown in [9] that all singular vectors in Vh,q,~ have charge 0 or :t:1. Therefore we indicate the charge of a singular vector relative to the highest weight vector by drawing the uncharged vectors vertically underneath the highest weight vector, the - 1 charged vectors in a strip to the left of the highest weight vector and finally the +1 charged singular vectors in a strip to the right of the highest weight vector. Let us now consider all embedding diagrams of the Verma modules Vo,o,~r with p, ff > 0. For these values of the central charge there exist three types of embedding diagrams corresponding to p = l , f f ~ Q or p = 1,ff E N or 2 < p E N,ff E N, (.p, pP) = 1, whose respective embedding diagrams are shown in Fig. 2.1, Fig. 2.2 and Fig. 2.3. [1016. q:
--1
0
+l
Fig. 2.1. Embedding diagram for Vo,o,cO,vt) with pt ~ Q
G•
The Verma modules ],20,0,c( 1,pt) withp' r Q contain only two singular vectors, namely f2, where f2 = ]0, 0, c > is the vacuum vector (cf Fig. 2.1). The Verma modules 2
6 Note that in [9] the embedding diagram of type I I I ~ A +B - , I I 1 ~ A - B + (corresponding in our notation to Vo,0,~p,p,~ with coprime p, pt E N and p, pt > 2) is not correct. The correct embedding diagram is the same as the embedding diagram for the case I I I ~ A +- B + , I I I ~ A + B + - (corresponding in our notation to 12o,0,c~p,1) with 2 <_ p E N)[10].
66
W. Eholzer, M. R. Gaberdiel
corresponding to the embedding diagram in Figs. 2.2 and 2.3 contain infinitely many singular vectors whose conformal dimensions are given by pn(p~n i 1) (n = 1,2, 3 , . . . ) for the uncharged singular vectors and by pn(pt n + 1) + p~n + 89and p n ( p ' n - 1) - p i n - 89 (n = 0, 1 , 2 , . . . ) for the charged singular vectors (for the cases corresponding to Fig. 2.2 set p = 1). The main difference between these two diagrams is that in Fig. 2.2 all singular vectors are descendents of the two singular vectors G _~ 89f2, which is not true in Fig. 2.3. q:
--1
0
+1
E
Fig. 2.2.
q:
Embedding diagram for lffo,o,c(1,p,) with pt C N
--1
0
9
+1
~
Fig. 2.3. Embedding diagram for V0,0,c(p,p, ) with coprime p, pt and p _> 2
The line from the first uncharged singular vector to the two dimensional space of singular vectors at level p~(p + 1) in Fig. 2.3 deserves comment: it means that the first uncharged singular vector at level pt(p _ 1) has exactly one descendent uncharged
Unitarity of Rational N = 2 Superconformal Theories
67
singular vector at level p'(p + 1) which is contained in the two-dimensional space of singular vectors at this level. The module generated from this descendent singular vector contains singular vectors both of charge 0 and + 1. Let us end with two comments on the embedding diagrams of the vacuum Verma modules for c _> 3 which we will need in the appendix (for details see [9]). For c = 3 there are infinitely many singular vectors which are all embedded in the two generic singular vectors G • , f2. For c > 3 all embedding diagrams terminate, i.e. there are 2
only finitely many singular vectors contained in the vacuum Verma modules.
3. Zhu's Algebra In this section we shall first give a physically motivated derivation of Zhu's algebra; we shall then use this formulation to determine A(7-/0) for a certain class of theories of the N = 2 super Virasoro algebra, and for some special cases. A conceptually interesting way to determine all irreducible representations of a (bosonic) meromorphic conformal field theory is the method introduced by Zhu [42], whereby one associates an associative algebra, usually denoted by A(7-/0), to the vacuum representation 7-[0 of a conformal field theory. It was shown by Zhu that the irreducible representations of this associative algebra are in one-to-one correspondence with the irreducible representations of the meromorphic field theory 7-/0. To define this algebra, a certain product structure was introduced by means of some rather complicated formulae, and it was not clear how this construction could be understood from the more traditional point of view of conformal field theory. Here we shall give a different derivation for this algebra, from which it will be immediate that all representations of 7-[0 have to be representations of A(7-/0) (this derivation follows in the spirit [14] and [41]); to show the converse direction, a similar argument as the one given in [42] would be sufficient. Another virtue of our derivation is that the Neveu-Schwarz fermionic case (which has by now been independently worked out by Kac and Wang in [29]) can essentially be treated on the same footing. To fix notation, let us denote the modes of a holomorphic field S(z) of conformal weight h by
S(z) = ~
S-t z l-h 9
(2)
lEZ
Given two representations of the chiral symmetry algebra .At, 7-ll and 7-/2, and two points zl, z2 c C in the complex plane, the fusion tensor product can be defined by the following construction [15]. First we consider the product space (7-/1 | 7Y2) on which two different actions of the chiral algebra are given by the two comultiplication formulae [16] (n+h-lm+h_l)Z~ m ( S , ~ |
/x z,,~(sn)= /~ z,,z~(S~)= m=l-h
+ej
\ l+h l=l--h
1
z2
(3)
68
W. Eholzer, M. R. Gaberdiel
m=l-h
+el
n-h
(-z2)Z-n(ll|
'
(4)
l=n
A z,,z2(S_n) =
n - h
( - z O m - n ( S - m | n)
m=n
+el
~
(n+l-1) n -- h
(-
l)l+h_lz-(n+O(ll| 2
,
(5)
l=l--h
where in (3) we have n > 1 - h, in (4,5) n > h, and el is :q=l according to whether the left-hand vector in the tensor product and the field S are both fermionic or not. 7 The fusion tensor product is then defined as the quotient of the product space by all relations which come from the equality of A z1,~2 and A ~1,z2, (7-/1 | 7-/2)S := (7-/1 | 7-/2)/( A z,,z2 - A ~,,z2).
(6)
It has been shown for a number of examples that this definition reproduces the known restrictions for the fusion rules [15, 16]. To analyse the possible representations of the meromorphic field theory 7-/0, let us consider the fusion product of a given representation 7-I at z2 = 0 with the vacuum representation 7-/0 at zl = z. We shall be interested in the quotient of the fusion product by all states of the form A ~,0(.A_)(7-/0 | 7-/) s , (7) where .A_ is the algebra generated by all negative modes. (In the conventional approach to fusion in terms of 3-point functions, all such states vanish if there is a highest weight vector at infinity.) Using the comultiplication A z,0, it is clear that we can identify this quotient space with a certain subspace of (7"/0 | ~ ) f / A ~,o(A-) (7"/0 | ~ ) S C (7-/0 | ~(o)) ,
(8)
where 7-/(~ is the highest weight space of the representation 7-/. The idea is now to analyse this quotient space for the universal highest weight representation 7-[ = 7-tuniv, (o) i.e. to use no property of ~b E Tlun w, other than that it is a highest weight state. We can then identify this quotient space with a certain quotient of the vacuum representation 7-/0, thus defining A(~0), [
(o) \
/ A Z,0(~--) (no | n n v)s 9
t A(n0) | nun v) = (no |
(9)
In order to do this analysis without using any information about ~b, we have to find a formula for (~ | So)(7-/o | ~ )
rood
A z,o(.A-)(7-/o | ?-tuniv),
(10)
in terms of modes acting on the left-hand factor in the tensor product, where S is any bosonic field, i.e. S has integral conformal dimension h. 7 The second formula differs from the one given in [ 16] by a different e factor. There the two comultiplication formulae were evaluated on different branches; this is corrected here.
Unitarity of Rational N = 2 Superconformal Theories
69
The crucial ingredient we shall be using is the observation
(11)
" A O , - z ( S - h ) = A z,o(eZL-' S _ h e - z L - ' ) E A z , O ( A _ ) .
Hence we have (for h > 2) --1
o ~ So,-z(S-h) + ~
z -(h+z) n ~,o(S~)
l=l--h
\ h-h
--(s_men)+ l=l--h
+
-s I z -'h+'~ (~ | st)+ z -(~+'~ Z l {l+h-1 ) \,~ + h - 1 l=l--h k
m=l-h
_- (s_h | ~) - z-~ 0 | -1
+ Z
ZZ_m(,_qm@fl)}
So) + ( n |
/
Z
z-(h+m)(
l=l--h m=l--h
l+h-1 re+h-1
)
(Sm@ll)
where ~ is the algebra generated by the positive modes, and ~' denotes equality up to terms in the quotient. Evaluated on (~0 @ r where ~ is a highest weight, we then have (ll|176174
+
-1
l
Z
[" l+h- 1 )
Z z-'~\m+h-1
(S~|
.
(12)
l=l--h m=l--h
In particular, we can use this result to obtain a formula for the action of A ~,0(So) on (r | ~9) modulo vectors in the quotient, where r c A(7-/o). (It is clear that A z,o(So) is well-defined on the quotient.) We calculate A z,o(So) ~ z~(S-h | ~1)+ (So | n) + Z
re+h-1 +l=m \re+h-1
z-'~(Sm@ll)
m=l--h h rn=0
where we have used the identity
Z ( a+kk
=
a+/+ll
(13)
k=O
e.g.
(see [21, p. 174]) to rewrite the sum in curly brackets. This reproduces precisely the product formula of Zhu for z = 1 [42], h
S.r176162 m--O
~
(14)
70
W. Eholzer, M. R. Gaberdiel
If S is the field corresponding to a state in the subspace of the vacuum representation 7-/0 by which we quotient to obtain A(7-/0), then its zero mode vanishes by definition on all highest weight states. This implies that the product structure defined by the action of A z,0(S0) gives rise to a well-defined product on A(7-/0). We have now achieved our first goal, namely to express the zero modes of the holomorphic fields on a highest weight state in terms of modes acting in the vacuum representation, modulo terms which vanish if there is a highest weight vector at infinity. In the next step we want to derive the relations by which the vacuum representation has to be divided in order to give A(7-/0). In particular, we shall see that we can express all states of the form (S_n | R)(7-(0 | ~b) with n > h by corresponding states with n < h. (Again this can be done without using any property of ~b other than that it is a highest weight vector.) We shall do the calculation for the bosonic case first, i.e. for h E N; we shall explain later, what modifications arise in the fermionic case. As before we have
+ Z (n+l-l"] \ n - h / (--1)h-'~z -(n+t)A z,o(St)
0 ~ A'o,-z(S-n)
l=l-h ( nn- - h ) (-1)h-n z - n ( ' Q So) + (ll Q ,A+)
= (S_n|
+
~ I=l--h
n+l - 1 n-h
zl-rn(sm @ ]l).
~ \re+h-1 m=l-h
Using (12) we can rewrite the (ll @ So) term, and find after a short calculation (S-n |
~) ~
(:-') h
(--1)h-n
{
zh-n(S-h | ~) q-
z-(m+n)Cm(Sm | n) m=l-h
}
,
(15) where
-I
C.~=z(l+h-1 ) ( (n+l-1),(h-1),~ l=,~ re+h-1 1-(nZ-l~.(l+h-1)!]
"
(16)
For completeness we should also give the result for h = 1, where the analysis simplifies to
(T-n | l[) ~ --(--z)--n(]l ~ TO) ~ (-z)-n+l(T_l @ ~).
(17)
Taking n = h + 1, we note that (15) and (17) become
h+fi( h + l
0~
h
-m
)zm_h_l(S_m|
'
(18)
m=l
which, for z = 1, just reproduces the formula of Zhu [42]. Here we have used
m f h-l-l) Z
/=1
(see
e.g. [21, p.
176]).
lth-l-m
( --
h h+l-m
)
(19)
'
Unitarity o f Rational N = 2 Superconformal Theories
71
From our definition of A(7-/0) it is clear that every highest weight representation of gives rise to a representation of A(7-(0) with respect to the product structure induced by A z,0(S0). As our space is at most as large as the space of Zhu, it is then clear that our definition has to agree with the one of Zhu. The fermionic case is slightly simpler, as there are no zero modes, and thus there is no relation corresponding to ( 1 2 ) . W e therefore only have to calculate for n _> h, 7%
'/'
( S _ n @ ]l) ------AO,_z(S_n)--gll=_ -
(--1)Z+h-I(--Z)-(n+Z) (11| Sz)
_! =
el(-1)h-n
n-
h
l=l--h For
l =
1 -
h,
999
, - 3, 1 we have furthermore 1
, , (11 | & ) = A z,o(&) -
z
l)z
rn=l--h Hence we find for n _> h, l
(s_n | n)
2
(-1) h-n" E
(Sin | n),
(21)
m=l --h
where
_!
Din=
l=m
[,m+h-1
n-h
"
Using (13) this formula simplifies for n = h to h-~
m=O which, for z = 1, just reproduces the formula of Kac and Wang [29]. By the same reasoning as before, it is then clear that our definition agrees with the one of Kac and Wang. In the case of the N = 2 algebra, the only fermionic fields are G + of conformal weight h = 3/2. Because of (21), all negative modes of G_im with ra >_ 3/2 can be eliminated in the quotient. On the other hand, G+-m vanishes on the vacuum for m = 89 and thus all G • modes can be removed. Furthermore, using (15) and (17), all (negative) modes of L and T can be eliminated, except for T_l and L - 2 , L - 1 . On the other hand, L-1 vanishes on the vacuum, and thus can be removed by commuting it through to the right. The space A(7-/0) is therefore a certain quotient space of the space generated by L_2 and T-1. We can then equally well describe A(7-/0) as a quotient space of the space generated by h = L - 2 + L _ I and q = T_l; this formulation has the advantage that the two generators commute, and that they can be directly identified with the eigenvalues of the highest weight with respect to L0 and To (by (12)), since
72
W. Eholzer, M. R. Gaberdiel (ll | L0) ~ (h | 1),
(ll @ To) ~ (q | 11).
(24)
Thus A(7-/0) is a quotient space of the space of polynomials in h and q. Generically, this space is infinite dimensional, and to obtain some restrictions, we have to use singular vectors in the vacuum representation. It is clear that all bosonic descendents of singular vectors do not give new information for the quotient, as we can always replace negative modes of L and T by h, q, L - 1 and some non-negative modes. Apart from the L _ 1 contributions, these give only restriction which contain, as a factor, restrictions from the original singular vector. The L_1 contributions simply correspond to an infinitesimal shift in the insertion point z, and thus do not give new restrictions either. For fermionic descendents, a similar argument implies that the only descendents of potential interest are
G~_89
,
G+_89G~89
,
(25)
where .M is a singular vector. It is clear that all three are trivial for the first generic singular vectors of the vacuum representation, G • , ~2 = 0, but in general, they need 2
not be trivial. It follows from the embedding structure for the cases corresponding to Fig. 2.1 and Fig. 2.2 that all singular vectors are descendents of the generic singular vectors of the vacuum representation. We can thus conclude that A(7-/0) is isomorphic to a polynomial ring in two independent variables, and thus, in particular, infinite dimensional. This shows that the corresponding theories are not rational. In the case of the diagram of Fig. 2.3, there exists an additional independent bosonic singular vector .Ms . The relations of the generic singular vectors have already been taken into account in the above derivation, and A(7-/0) is therefore only finite dimensional (and the corresponding theory rational) if A/" gives rise to two independent relations. For a bosonic singular vector.IV', only the third descendent in (25) can contribute, as the other two have odd fermion number and thus are equivalent to zero in the quotient. We conclude from this that the potential rational models all have a non-trivial bosonic singular vector in the vacuum representation. Whether the model is actually rational depends then on whether the G + 89G - 89descendent gives an independent relation or not. We have calculated the relations coming from.N" and the G +_~ G - ~ descendent for a few examples explicitly. The first bosonic singular vector is given in each case as J~c=l =
(2L_2
3T-1T-t)f2
,
A/'c=3 =
(10T_3 - 3L_3 + 3G+3/zG-3/2 - 12L_2T-1 + 8 T - I T - I T - I ) if2 ,
A/'c=-6 = ( - 1 0 T _ 3 - 6 L - 3 + 6G+_3/gGS3/2 + 6 L - z T - 1 + T - I T - I T - I ) ~ , Arc=-1 = (42T_4 + 2 4 L - 4 + 2 7 T - z T - 2 - 84T-3T-1 - 6G+3/zG-5/2
+6G+_5/2G-3/2 - 3 2 L - z L _ 2 - 3 6 L - 3 T - 1 + 36T-tG+3/zG-3/2 +12L_zT-1T-1 + 9 T - a T - I T - I T - I ) f2 , .N'c=-12 = ( - 2 4 0 L _ 5 + 360G+3/2G-v/z + 120G+_5/zG-5/2 + 840L_zT-3 +360G+_v/zG-_3/2 + 6 0 0 L _ 3 L - 2 + 1 2 0 L - 3 T - 2 + 180L-4T-1 8 To avoid confusion, we shouldpoint out that L-1 ~ is a descendentof the two generic singular vectors. We should also note that we implicitly assume here, that the vacuum representation does not possess any subsingular vectors which might give additional relations.
Unitarity of Rational N = 2 SuperconformalTheories
73
- 6 0 T _ 1G+_3/2G-5/2 + 60T_ 1G+_5/2G-3/2 - 600L-2G+-3/2G~.3/2 - 3 0 0 L - 2 L - 2 T - 1 + 6 0 L - 3 T _ I T _ I - 60T_4T-1 + 30T-2T-2T_1 -60L-2T-1T-1T-1 -3T-IT-IT-IT-IT-1
+ 1 8 0 T - 3 T _ I T _ I - 60T-1T-1G+_3/2GZ3/2 - 1992T_s) ~ .
The singular vector and its descendent give rise to the polynomial relations (in h and q) Pl, and P2, respectively. The algebra A(7-/0) is then given by A(7"[o) = C[h, q]/ < pl(h, q),p2(h, q) > 9
Our results for the five cases above are contained in Table 3.1. Table 3.1. Polynomials determining A(7-(0)for certain values ofc c 1 3/2 -6 -1 -12
p~ (h, q) (2h - 3q 2) q(1 - 12h + 8q 2) q(2 + 6h + q2) ( - 4 h + 3 q 2 ) ( 1 + 8 h + 3 q 2) q(4+ 1 0 h + q 2 ) ( 6 + 1 0 h + q 2)
p2( h, q) ( - 2 h + q)(1 + 3q) (2h - q)(1 - 2h + 5q + 8q 2) (2h - q)(2 + 6h + q2) (2h - q)(2 + 3q)(1 + 8h + 3q 2) ( - 2 h + q)(4 + 10h + q2)(6 + 10h + qZ)
We note that for the unitary models c = 1 and c = 3, A(7-/o) is finite-dimensional, as the two relations are independent. In the other three cases, however, the two relations contain a common factor, and thus A(7-/0) is infinite dimensional.
4. The Coset Argument
We have shown in the last section that the N = 2 super Virasoro algebra is not rational for certain non-unitary cases. In this section we will analyse the remaining cases with c < 3. The analysis for c > 3, using the modular properties of the vacuum character, is contained in Appendix B. We shall use the coset realisation (1) of the N = 2 super Virasoro algebra to show A that certain admissible representations of su(2) k give rise to infinitely many irreducible representations in the non-unitary cases. The basic idea of this argument is due to Ahn et al. [ 1]. We shall present the argument in the more general setting of the Kazama-Suzuki models as the counting argument generalises. Recall that the Kazama-Suzuki models can be constructed from hermitian symmetric spaces [30]. More precisely, it has been shown in ref. [30] that if G / H is a hermitian symmetric space, the coset ~k @ (~et) 2n ~k ' (26) where n = rank(G) = rank(H), contains the N = 2 super Virasoro algebra and the explicit form of the N = 2 super Virasoro generators T, G • L in terms of the 2n free fermions and the ~k currents has been given. A complete list of hermitian symmetric space can, for example, be found in ref. [30, Table 1]. Note that for all hermitian symmetric spaces rank(G) = rank(H), g is simple and that 0 is of the form 0 = u(J.) | 0~_, where 01 is semisimple. The case of the N = 2 super Virasoro algebra corresponds to g = su(2) and 0 = u(1).
74
W. Eholzer, M. R. Gaberdiel
Before proceeding, we should note that it is in general rather difficult to determine the actual coset algebra of a given coset. In particular, even if the coset algebra is correctly identified for generic level (which is usually a tractable problem, for example by comparing generic characters), it is a priori not clear that this identification remains correct at arbitrary level. However, for the case of 13 = su(2), b = u(1), the character calculations of Appendix A show that the coset is indeed the N = 2 super Virasoro algebra for arbitrary level h 9. Using the explicit form of the u(1) current T and the Virasoro field L in the coset [30, Eq. (4.5)] it is easy to obtain a formula for the eigenvalues q and h of To and L0, respectively, acting on the subspace of the 9k highest weight space with t~-weight )~ of a ~k | ~et 2'~ highest weight representation h-
C2(~1) 2(k +9)
q-
(A, A + 2pb) 2(k +9) '
(27)
2 k+g (p~-p0'A)'
Here C2(9) denotes the second order Casimir of the 9 representation on the ~k highest weight space, 9 the dual Coxeter number of 1~, and p~ and po are half the sum of the positive roots of 9 and b, respectively. For the case of 9 = su(2), the formulae become h-
J0" + 1) (k+2)
m2 (k+2)'
q-
2m k+2'
(28)
where j and m label the spin and the magnetic quantum number of the corresponding su(2) representation. If for admissible k ~ N the admissible representations of su(2) k are MCFT representations, it follows directly from these formulae that there are infinitely many highest weight states, as was already observed by Ahn et al. [1]. (For more details see below.) This implies then directly that the corresponding theory__.._is not rational. It has now been shown that the admissible representations of su(2) are indeed MCFT representations [13, Corollary 2.11]. For general g, the corresponding result is not yet known, but we believe it to be true. Assuming this for the general case, the argument can be generalised as follows: we note that 0 has one simple root less than g which we denote by c~, and, that the Dynkin index of a is 1. For an admissible but non-integer level k there always exists an admissible representation of 9k whose Dynkin label corresponding to the fundamental weight dual to a is fractional ([28, Theorem 2.1 (c)] see also [35, p. 236]). This representation has in particular an infinite dimensional highest weight space. Let A be the q-weight of the highest weight vector VA of this representation, and denote by A~ the b-weight of VA. Furthermore, let E,~ be the step operator corresponding to a. Then the vectors E2~ VA are highest weight vectors of b, and their 0-weights are given as A,~ = AO + n#, where # is a non-zero b-weight and n + 1 E 1~. This implies that the expression (An, A,~ + 2p~) is unbounded for n ~ ~ and hence, that there are infinitely many different values for the conformal weight h in (27). Thus the coset (26) has infinitely many inequivalent representations. To relate the arguments for p = su(2) to the results of Sect. 3, let us parametrise the admissible level as k = p / p t - 2, where p, p~ are coprime positive integers and p > 2. The admissible representations are given by the .su(2)k weights [27, p. 4958] 9 This argumentrelies on the conjecturedembeddingdiagramsof the N = 2 algebra.
Unitarityof Rational N = 2 SuperconformalTheories
75
An,z = (k - n + l(k + 2))A0 + (n - l(k + 2))A1,
(29)
where n and l run through n = 0 , . . . ,p - 2, 1 = 0 , . . . ,p~ - 1, and A0, A1 are the A fundamental weights of su(2). We note that the spin j of the su(2) representation on A the highest weight space of the su(2) k representation corresponding to An,t is given by j = 89 - l(k + 2)). In particular, for k E N which corresponds to the unitary case, the spin j is always half-integral. If k r N, the admissible representations corresponding to the weights An,z with l 5/0 have an infinite dimensional highest weight space as 2j + 1 r N. These representations give rise to infinitely many MCFq" representations of the coset algebra, thus showing that it cannot be rational. Indeed, Eq. (28) implies that k+2
2
h+~--q-
j(j+l) - k-+ 2,
(30)
where j is the spin of the su(2) representation. This equation gives precisely the common factors in Table 3.1 for c = - 6 , - 1 and - 12, i.e. k + 2 = p / p ' = 2, 3 and ~0, where the A
spin j corresponds to the admissible representations of su(2) k with infinite dimensional highest weight space which are given by weights A~,z with 1 5/0. The additional discrete A
representations of A(7-/0) correspond to the su(2) k representations with the weights An,0 (n = 0 , . . . , p - 2): for example in the second case c -- - 1, (h, q) = (89 - ~) comes from n = 1 and (h, q) = (0, 0) from n = 0. We know that all irreducible representations satisfying the conditions given by the polynomials in Table 3.1 are MCFT representations, so in particular there exists a continuum of MCFT representations in the non-unitary cases. The above argument, however, only shows that those representations satisfying (28) can be obtained from the coset construction. Furthermore, it is clear that only countab/.~ly many representations of the coset MCFI" can be constructed from the admissible su(2)k representations. It therefore seems that the remaining representations cannot be constructed using the coset realisation. Finally, let us mention that our findings are in perfect agreement with the results obtained in ref. [4]. The authors of loc. cit. have investigated the representation theory of several exceptional N = 2 super W-algebras from a completely different point of view. The only rational models they found weie unitary and even contained in the unitary minimal series of the N = 2 super Virasoro algebra.
5. Conclusion In this paper we have analysed systematically the question whether the rational theories 9f the N = 2 superconformal algebra are always unitary. We have used three independent ~rguments to exclude the existence of rational non-unitary theories. Where possible we aave checked that the different methods lead to consistent conclusions. One of the methods is based on the coset realisation of the N = 2 algebra, and ~e have already indicated in section 4 how this argument can be generalised to the Kazama-Suzuki models. Apart from the (aforementioned) problem that the admissible 9epresentations are not yet known to be MCFF representations in general, this argument toes not exclude that the theories corresponding to non-admissible affine theories are 9ational. However, we expect that there should be 'fewer' singular vectors than in the ldmissible case, and thus that the corresponding theories should also not be rational. It
76
W. Eholzer, M. R. Gaberdiel
should be possible to settle both of these problems as soon as the representation theory of the Kac-Moody algebras at non-integer level is understood in detail. The coset argument is even more general, as it does not involve the fermions. Indeed, ignoring the fermions where applicable, it can be applied to cosets of the form ~(1)
~_(n)
kl G . , .
~ (1) 11 G where the ~
@~k~
~,(m) ' . . . G 'Jl,~
(31)
and ~I~) are simple affine Kac-Moody algebras at admissible level, and the
sum of the numbers of simple roots t,, -~ ~k, _(0 with ki ~ N, is bigger than the corresponding sum for the denominator. For the arguments to work for general (admissible) ki, we also have to assume that one of the ~(i) with ki i / N contains a simple root of Dynkin index 1 which is not contained in the denominator. In particular, the arguments apply to the diagonal coset
~k, 9 ~k2
~kl+k2 indicating that they only give rise to rational models if at least one of the levels kl or k2 is a positive integer (c.f. the conjecture in ref. [3, p. 2421]). This is for example the case for the coset realisation of the (non-unitary) minimal models of the Virasoro algebra where one has ~ = su(2) and k2 = 1. There are certain purely bosonic coset MCFTs which are of the general form (31), e.g. those corresponding to the unifying W-algebras associated to the unitary series of the Casimir W-algebras W.An or W D n [3, Table 7]. Our arguments confirm in these cases the conjecture of ref. [3, p. 2422] that all rational models of the unifying W-algebras are also rational models of the Casimir W-algebras (which are, in these cases, all unitary). Let us close by mentioning some open problems. It would be interesting to know under which conditions all representations of a coset MCF'I" ~/b can be obtained from MCFI" representations of ~ - - as we have seen in Sect. 4, this is not the case for certain non-rational N = 2 models, where there exists a continuum of representations. In the same spirit it would be interesting to describe all representations of the N = 2 super Virasoro algebra that can be obtained from the admissible su(2) k representations for given k and to investigate whether they define quasi-rational theories in the sense of [36]. It would also be important to have a more general criterion for determining whether a coset MFCT is rational or not. Finally, in order to complete the arguments for the general case, it would be necessary to have a better understanding of the representation theory of affine Kac-Moody algebras, in particular at admissible level.
A. Calculation of Vacuum Characters In this appendix we want to calculate the vacuum characters of the N = 2 super Virasoro algebra from the embedding diagrams of Sect. 2 and from the coset realisation described in Sect. 4. Let us first calculate the vacuum characters of the N = 2 super Virasoro algebra in the cases corresponding to the embedding diagrams in Figs. 2.1-2.3. In order to be able to determine the characters from the embedding diagrams we have to assume that there do not exist subsingular vectors in the vacuum representation, i.e. vectors which
Unitarity of Rational N = 2 Superconformal Theories
77
are not singular in the vacuum Verma module but become singular in the quotient of the vacuum Verma module by its maximal proper submodule 1~ We also want to assume that the character of a submodule of a Verma module generated from a level n charged singular vector is given by qn/(1 + qn-~') times the generic character where n ' < n is the level of the uncharged singular vector (or highest weight vector) connected by a line to the charged singular vector in the embedding diagram. This means for example that the character of the submodule generated from G +- ~ f2 or G - ~ ~ is just given by !
oo
(l+qn+ 89
....
q2/(1 + q89 Hn=l ~ , wnlcn is obvious in this case. In the case of the embedding diagram shown in Figs. 2.1 and 2.2 all singular vectors are embedded in the two submodules generated from G • ~ f2. Moreover, the embedding 2
diagram implies that the intersection of the two submodules generated by G +--31 ~'~ and G - 1 f2 is trivial. Therefore, the vacuum character of the N = 2 super Virasoro algebra with
c(1,p')
= 3(1 -
2p') ( f
Xl,p'(q) =
9~ Q o r p ' c N) is given by
q-C(1,p')/24 H n=l
('-l----q--~)2
1 --
1 + q~
The case corresponding to the embedding diagram in Fig. 2.3 is more interesting. Here we have to subtract and add successively the characters of the modules generated by the corresponding singular vectors. Using the two assumptions above we obtain that the vacuum character of the N = 2 super Virasoro algebra with c(p, f ) = 3(1 - 2 ~-); p, p' E N; ( p , / ) = 1; p > 2 is given by ~o
Xp,p'(q) = q-C(p'P')/24 H (1 + qn+ 89 n=l ('--1-S "~--~ X oo 61pn(ptn+l)+p,n+89 1 -- E qp'(n+l)(p(n+l)--t) + 2 ~ qpn+89
n--O
1+
oo qpn(p'n-- l)--p'n--89 ) + E qp'n(pn+l) + 2 -1+----~ " n=l
In particular, for / = 1 the above formula gives the vacuum character of the unitary minimal model with central charge c = 3(1 - 8) it Finally, note that the last expression for the vacuum character )~p,p, c a n be rewritten as
Xp,p,(q)=q-C(p,p,)/24 ( f i ( l + q n - 8 9 ' 1--qpn+89 n=l -(f:-~'))2 ] E qpn(pn+l) - - - - . nez 1 + qpn+89
(33)
In the second part of this appendix we use the coset realisation of the N = 2 super Virasoro algebra (1) for the calculation of the vacuum characters Xp,pt. Recall that the 10In [18] certain representations of the N = 2 superconformal algebra have been found which possess subsingular vectors. However, these representations are rather special and do not include the vacuum representation. ll Althoughthe multiplicities in the embedding diagrams of ref. [11, 34, 32] are not correct, the authors of Ioc.cit.obtainedthe correct characters in the unitary case.
78
W. Eholzer, M. R. Gaberdiel
central charge of the N = 2 algebra is given as c = ~
(k g 0, - 2 ) , and that the vacuum A
representation is given by the space of all uncharged u(1) highest weight states in the vacuum representation of su(2--""-)k | a c t 2. The character •p,p, is therefore the u(1 t ' ) uncharged part of X~(2-"---)Xa'r2 divided by the u(1) character
Xp,p'(q) =
~
1
Res,
(!
~
)
X~(2)(q,z)xar
,
(34)
)~m(q) where X~(1)(q)
1/~(q) = q - ~ / ~ ( q ) A
and r/(q)
e-~-r/(q)
e ~ - I-In=l( - qn). (Here
A
we have used the su(2) characters -Xk,~(2),_ kt/, z) = trqL~ 2J3o which also take the zero mode of j3 into account.) There exist two types of embedding diagrams for su(2)k vacuum representations with level k > - 2 (corresponding to c < 3) [33, L e m m a 4.1]: either the level k can be written as k = p i p ' - 2, where p, p~ c N, (/9, i f ) = 1 and p __ 2, or the vacuum representation is generic, i.e. all singular vectors are descendents of the level zero singular vector. In the former case the representation is admissible and the vacuum character is given by [27] A
XSU(2~)(q,k
z) = LqP"PP'(T, z i p I) -- O_p, ,pp, (T, z i p I) 01,2(% z) -- ~)-l,2(r, z)
where
'
(35)
qkn2z2kn "
~)k,X('r, Z) :=
nCZ+~ (Note that our z corresponds to qZ/2 in [27].) In the latter case the vacuum character is generic, and is given by q - Nc
su(2)~_ z) =
Xk
ktl~
oo [In=l(
l
-- qn)( 1 - qnz2)( 1 -- q n z - 2 )
Let us first consider the generic (i.e. not admissible) case with c < 3. To evaluate the above residue (34) we want to use the following expression for the fermionic character:
X:Se~2(q'z) = q-"~ Y I ( 1 + qn- 89 n>_l
(1+ qn- 89
which follows from the product formula for the
1 x--', =-7-7 L q~'m s z 2m , rltq) mcz
(36)
~)3 function (see e.g. [23, p. 164])
(DO
0 3 ( r / 2 , z) = ~
q89
= I I ( 1 - q~)(1 - z2qn-l/2)(1 -- z--2qn--1/2).
nCZ
(37)
n=l
Furthermore, we shall also use the following identity for the denominator of the su(2) character (see e.g. [26, p. 262 Eq. (5.26)])
1 ac ( 1 __ qnz2)( 1 _ qnz-2 ) I'In=l
where
-
1
~
~](q)2 leZ
r
21
(38)
Unitarity of Rational N = 2 Superconformal Theories
79
oo
r
= ~-~(- 1yqtr§ Y=O
An important property of ez(q) is that r = qlr The vacuum character of the N = 2 model is then (up to the
-Resz
~
E
q89162
q-C(1,p')/24 term)
- z-l))
'
m,ICZ where p / C N or ff ~ Q. The evaluation of the residue gives 1
/
,z2
1
x--,
1
.--., 1/2
lEZ
~z2
IcE
-
~.~q'=
Cz(q)(1 - q 89
~(q)3 fez oo
=
1 (1 ~/(q)3
- q 89
E(
-
1)rq89
)
lCZ r=O oo
=
lq ~( )3 (1 - q 8 9
E(_l)rq~qfl ~ ''~'
icZ r=O where 1 = l + r. We can now do the sums over l and r, and obtain the N = 2 vacuum character
Xl,p'(q) =
c"'P'~ (fi(l+qn-1/2)2~ ( _2 ql ~ q- 24 n=l (1--qn)-----i ] 1 l +q~ ]
where we have used that l-q3 = 1 - 2 q89 This is indeed the generic N = 2 vacuum l+q2
l+q~
character, where the only null vectors are G • , ~ (cf. Eq. (32)). Finally, consider the case where the su(2)k vacuum character is admissible, i.e. 2 with p, ff E N, (p, if) = 1 and p > 2. In this case we find, using (35) and the well-known denominator formula,
k = p/p~ -
q
. s u ( 2 ) t _ Z)
Xaxu(l~")(q) ~t/,
c(p,pZ)_l 24
/ ~ 2 p n + l __
_ IJn=l( oo 1 - q n z ~ ( 1 -- q n z - 2 ) E -
nEZ
qnp'(l+pn)'~"
Using (34), the N = 2 vacuum character is then (up to the we suppress for the moment) the residue -Res~
~(q)3
E n,rn,IEZ
~--2pn--l'~ '.
Z2 --~ 1
q-C(p,p')/24 term
qnp'(l+pn)q89162 Z21Z2m(Z2pn+l-- Z--2pn--1)
which
= (*)
"
The z-dependent part is Resz (z -l+2q§ - z l§247 , whose residue is easily obtained. By expressing m in terms of n and I, the sum then becomes
80
W. Eholzer, M. R. Gaberdiel
( * ) - ~/(q)31 ~
{q,W,(,+p,~)r189
_ q,W,O+pn)r189
n,lEg
- ~ ( q 1) 3 Z
[qnp'(l+pn)q89
n,lEZ
_qnp'(l+pn) ql (t:+p2nZ+l+2t+Zpn+2pnOr [qnp,(l+pn)q89
- ~(q)31 ~
21pn)(r
) _ q89162
]
n,lEZ
- f / ( q1) 3 Z
[qnp'(l+pn)q 89162
n,ICZ
where we have replaced 1 by - I in the second sum of the penultimate line, and used the previously mentioned symmetry of r in the last equation. Next we use the explicit expression for Ct to obtain oo
(*) = ~/(q)31 Z
qnP'(l+Pm( 1 _ qPn+89 Z ~-~(- 1)~q89
nEZ
lEZ r--'O
The last exponent of q can be rewritten as
-re+r+IZ+p2n2+2lr-2lpn = + r + 2rpn , 2 where l = 1 - pn + r. We then replace the sum over l by a sum over 1 which 1~2 00 1 + qm- 89): . The sum over r is the geometric series gives ~_Mczq -~ = rl(q) l-[m=l( ~_o(-1)~q 89 (r(l+2pn)) = 1/(1 + qpn+89 and we thus arrive at (compare for example [38, 25, 1])
Xp,p'(q) = q
24
~=1 (1
- q~)2 /] ~ez
1 +q pn+l "
This expression equals the one derived from the embedding diagram (33).
B. Modular Properties of the Vacuum Characters As already mentioned in Sect. 2, it was shown in [42] that for bosonic rational conformal field theories the space of torus amplitudes which is invariant under the natural action of the modular group is finite dimensional. We expect therefore that the dimension of the space spanned by the functions X[A(T) = x(A'r) (A c SL(2, Z)), where X is the vacuum character, is finite if and only if the N = 2 super Virasoro algebra is rational for the corresponding value of c. We shall show, using the following two lemmas, that the dimension of this space is infinite for c _> 3 and for the non-unitary models corresponding to the embedding diagrams in Figs. 2.1 and 2.2, i.e. c = c(1, p') with p = 1, p' ~ Q or p = I, p' E N. On the other hand, it is finite for the unitary models with c = c(p, 1) andff = 1,p > 3. (Forff = 1,p = 2 the dimension is clearly 1 since Xl,z = 1.)
Unitarity of Rational N = 2 SuperconformalTheories
81
L e m m a B.1. For k = p - 2 E N the vacuum characters Xk(r) := Xp,l(q) (q = e2~i~) of the N = 2 super Virasoro algebra are modular functions on I" (24k(k + 2)). More explicitly, they are given by
1
Xk(T) ----" ~ 3 ( T )
E
69L,(2~l+2,,~)('r)Om(k+2)'k(k+2)(r/2)'
mmod2k
(2kn+.k) 2
where the O;~,k = ~ n 6 Z q ~k are Riemann-Jacobi theta functions and the ~L,~ are Hecke indefinite modular forms (of weight one) associated to the lattice L = Z | Z and the quadratic form Q(7) = 2(k + 2)'72 - 2k7 2.
Proof We first recall the definition of a Hecke indefinite modular form (see [22] or [26, pp. 254] for more details). Let L C R 2 be a lattice of rank two and Q : L --+ 2Z an indefinite quadratic form such that Q(a:) = 0, x E L implies x = 0. Denote by L~ the lattice dual to L, L~ = {x E ]RZlB(x,g) c Z for y E L}, where B ( 7 , " / ) = 1 ( Q ( 7 + 7 ' ) - Q(7) - Q(7')) is the bilinear form associated to Q. Let Go be the subgroup of the identity component of the orthogonal group of (B, R 2) which preserves L and fixes all elements of L ~/L. Fix a factorisation Q(7) = l l (7)12('7), where I i and 12 are real linear and set sign(7) = sign(/1(7)). Then 69L,i*(r) :=
E
sign(7)qQ(7)/2
" / E L+/~ B('y,'/)>O -y raod G 0
is called a Hecke indefinite modular form associated to/z and L. It is a modular form of weight one on P (N), where N E N satisfies N Q ( 7 ) c 2Z for all "7 E L ~. The case we are interested in has been studied in [26, pp. 256]. We have L = g @ Z and Q(7) = 2(k + 2)72 - 2k72 =/1(7)/2(7), where ll(7) = ~ + 2)71 + x/~'72 and 12(7) = ~ ' 7 1 for 7 c L implies 7 = 0. Then B is given by
+ x/~"/2 so that Q('7) = 0
B(T, T') = 2(k + 2)717tl - 2k727~, implying that L ~ equals 2(~+~Z @ 2~Z. We observe that A, given by A(Tt, 72) = ((k + 1)71 + k72, (k + 2)7t + (k
+ 1)72) ,
satisfies Q(7) = Q(A'7), and that the group generated by A is the identity component of the orthogonal group of (B, IR2) which leaves L invariant. Furthermore, A 2 generates Go. Hence the functions 69L ~ , ~ (ra E Z) are modular forms of weight one on ~ 2(k+2) ~ 2k /
F (4k(k + 2)). Moreover, one can show that the Hecke indefinite modular forms (9 i ~ 1 ,,, are given by [26, p. 258]
82
W. Eholzer, M. R. Gaberdiel
~Lr l ~__~(T) = ~x 2 ( k + 2 } ~ 2 k ]
(
~
_
s> ,n>_O s
<0,~< )
0
(_l)Sq(k+2)(~+n~~ ~2 k~s+m~2 2,k§ - ,~ T, s (k+2)(~+n-
s_>0,n>0
1 ~2 kr
m~2
s<0,n<0
We thus find
~3 mm~od2kl~ L,( 2r ,~ )(T)om(k+2)'k(k+2)(T /2)
s>,n>0
?]3
~ s>0,n>0
=q
s<0,n<0
__ ~
(_l)sq(k+2)(~+n--~)--k(g+-~ -) 2k
s<0,n<0
3~ ~ ~]~(l+q'~+l/2) 2 4~k.2,- - i l - q - - ~ •
(39)
n=l
~+2,, ~ n=l
(1 + q,~+1/2)2
q(k§ nEZ
where we have used the definition of the Riemann-Jacobi theta functions ~m,k = ~m+2kZ,k and ~gL,~ = 69L,~+f0,Z)in the second equality, and the well-known product formula for the ~3 function (37) for 7-//2 and z = 1 in the third. The last expression equals (33) for p = k + 2, p' = 1. The property that the Xk are modular functions on F (24k(k+2)) follows now directly from the well-known modular properties of the Riemann-Jacobi theta functions and the function. [] It is then clear that the space spanned by Xp,11A (A E SL(2, Z)) is finite dimensional, as F (24p(p - 2)) has finite index in SL(2, Z) for p _~ 3. In order to prove that the space spanned by the functions X1 ,p'(At) ( A E SL(2, Z)) is infinite dimensional, we need the following lemma: LemmaB.2. Let f : C ---+ C be a function of the form f(7-) = q~P(q)/Q(q), where ~x C Q, P and Q polynomials and q = e 27fir. Then for any N > 0 and k E Z, the space spanned by the functions flk,A (A c F (N)) is infinite dimensional if f is not constant. Here flk,A is defined as flk,A(r) = (C7- + d ) - k f(AT-) , where A = ( a
db ) and AT- = ~+b c'r+d"
Unitarity of Rational N = 2 Superconformal Theories
83
Proof Assume that the space spanned by the flk,A (A C SL(2, Z)) is n dimensional, where n < c~, and that f is not constant. Let ~ be 7-Is where s is the denominator of a = ~, t~ = e 2 ~ , and let P, O be the polynomials given by
ff9(7t) = q~P(q), /5(~) =
p(q),
~)(71) =
Q(q) O(q) = q-~Q(q)
for a _> 0 f o r a < 0.
Then there exist n matrices Ai (i = 1, . . . , n) such that the functions flk,A~ (i = 1 , . . . , n) are linearly dependent over C. (Without loss of generality we can assume that A T 1Aj are not of the form ( * \o
* ] as we are interested in abasis over C.)Hence the polynomials /
P(qJ) I]i4j O(qi) (j = 1 , . . . , n) with qi = e 2~ia~ are linearly dependent over C[~], and thus the ~j are algebraically dependent over C[~]. Applying A11 to ~ we can assume that Al is the identity. Looking at the asymptotic behaviour of the qi for T ~ --icx~ we observe that there cannot be a term containing ql. By induction on n we find that the ~ are algebraically independent. This gives the desired contradiction. [] The proof of Lemma B.2 is due to J. Nekovar [37]. The last lemma proves that the space spanned by the functions XI,p,(AT) (A @ F (48)) is infinite dimensional since the function ~((~-§ r/(,'r) 4
X l,p' (T) satisfies the assump-
tions of the lemma and ~7(('r+1)/2)2 is invariant under the ]-I,A action for A G F (48). ~('r)4 Therefore, the space spanned by the functions XI,p,(A~-) (A E SL(2, Z)) is infinite dimensional. We also expect that the dimension of the corresponding space is infinite for e = e(p, p') with coprime integers p~, p _> 2. Finally the embedding diagrams of the vacuum Verma modules for e > 3 (cf the end of Sect. 2) imply that the corresponding vacuum characters are given by the product of the generic Verma module character and a rational function of q89 We can therefore again apply Lemma B.2 to conclude that the space obtained from the SL(2, Z) action on such a vacuum character is infinite dimensional. This shows that all theories with e _> 3 are not rational. Acknowledgement. We would like to thank M. D6rrzapf, H. Kansch, E Goddard, A. Kent, J. Nekovar and G. Watts for discussions, and A. Honecker and M. RSsgen for comments on a draft version of this paper. We are grateful to G. Watts for pointing out an error in an earlier version. We also thank the referee for pointing out the relevance of ref. [ 1]. W. E. is supported by the EPSRC, and M. R. G. is supported by a Research Fellowship of Jesus College, Cambridge. We also acknowledge partial support from PPARC and EPSRC, grant GR/J73322.
References 1. Ahn, C., Chung, S.,Tye, S.-H.: New Parafermion, su(2) Coset and N = 2 Superconformal Field Theories, Nucl. Phys. B365, 191-240 (1991) 2. Anderson, G., Moore, G.: Rationality in Conformal Field Theory. Commun. Math. Phys. 117, 441-450 (1988) 3. Blumenhagen, R., Eholzer, W., Honecker, A., Hornfeck, K., Hfibel, R.: Coset Realization of Unifying "W-Algebras. Int. J. Mod. Phys. A10, 2367-2430 (1995) 4. Blumenhagen, R., Hiibel, R.: A Note on Representations of N = 2 SVC-Algebras. Mod. Phys. Lett. A9, 3193-3204 (1994) 5. Boucher, W., Friedan, D., Kent, A.: Determinant Formulae and Unitarity for the N = 2 Superconformal Algebras in Two Dimensions or Exact Results on String Compactification. Phys. Lett. B172, 316-322 (1986)
84
W. Eholzer, M. R. Gaberdiel
6. Bouwknegt, E, Schoutens, K.: W-Symmetry in Conformal Field Theory. Phys. Rep. 223, 183-276 (1993) 7. Di Vecchia, P., Petersen, J.L., Yu, M., Zheng, H.B.: Explicit Construction of the N = 2 Superconformal Algebra. Phys. Lett. B174, 280-284 (1986) 8. D6rrzapf, M.: Analytic Expressions for Singular Vectors of the N = 2 Superconformal Algebra. Commun. Math. Phys. 180, 195-232 (1996)
9. D6rrzapf• M.: Superc•nf•rmal FieM The•ries and their Representati•ns. Cambridge: PhD thesis ( • 995) 10. Dtrrzapf, M.: Embedding Diagrams for Verma Modules of the N = 2 Superconformal Algebra. (in preparation) 11. Dobrev, V.K.: Characters of Unitarizable Highest Weight Modules over the N = 2 Superconformal Algebras. Phys. Lett. B186, 43-51 (1987) 12. Dong, C., Li, H., Mason, G.: Twisted Representations of Vertex Operator Algebras. Preprint, qalg/9509025 13. Dong, C., Li, H., Mason, G.: Vertex Operator Algebras Associated to Admissible Representations of s~2. Commun. Math. Phys. 184, 65-93 (1997) 14. Feigin, B.L., Fuchs, D.B.: Cohomology of some nilpotent subalgebras of the Virasoro and Kac-Moody Lie algebras. J. Geom. Phys. 5, 209-235 (1988) 15. Gaberdiel, M.R.: Fusion in Conformal Field Theory as the Tensor Product of the Symmetry Algebra. Int. J. Mod. Phys. A9, 4619--4636 (1993) 16. Gaberdiel, M.R.: Fusion Rules of Chiral Algebras. Nucl. Phys. B417, 130-150(1994) 17. Gaberdiel, M.R., Kausch, H.G.: A Rational Logarithmic Conformal Field Theory. Phys. Lett. B386, 131-137 (1996) 18. Gato-Rivera, B., Rosado, J.I.: New Interpretation for the Determinant Formulae of the N = 2 Superconformal Algebras. Preprint IMAFF-96-38, hep-th/9602166. 19. Goddard, P.. Meromorphic Conformal Field Theory. In: V.G. Kac (ed.), "Infinite Dimensional Lie Algebras and Lie Groups", Proceedings of the CIRM-Lumminy Conference 1988, Singapore: World Scientific, (1989) 20. Goddard, P., Kent, P., Olive, D.: Unitary Representations of the Virasoro and Super-Virasoro Algebras. Commun. Math. Phys. 103, 105-119 (1986) 21. Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics, New York: Addison-Wesley, (1992) 22. Hecke, E.: 13-bereinen Zusammenhang zwischen elliptischen Modulfunktionen und indefiniten quadratischen Formen. Nachrichten der K. Gesellschaft der Wissenschaften zu Gtttingen, Mathematischphysikalische Klasse 1925, pp. 35-44 23. Hille, E.: Analytic Function Theory II, London: Blaisdell Publishing Company, (1962) 24. Hthn, G.: Selbstduale Vertexoperator-Superalgebren unddasBabymonster, Bonn: PhD thesis, (1995) 25. Huitu, K., Nemeschansky, D., Yankielowicz, S.: N = 2 Supersymmetry, Coset Models and Characters. Phys. Lett. B246, 105-113 (1990) 26. Kac, V.G., Peterson, D.H.: Infinite-Dimensional Lie Algebras, Theta Functions and Modular Forms. Adv. in Math. 53, 125-264 (1984) 27. Kac, V.G., Wakimoto, M.: Modular Invariant Representations of Infinite-Dimensional Lie Algebras and Superalgebras. Proc. Natl. Acad. Sci. USA 85, 4956-4960 (1988) 28. Kac, V.G., Wakimoto, M.: Classification of Modular Invariant Representations of Affine Algebras. Adv. Series Math. Phys. 7, 138-177 (1989) 29. Kac, V.G., Wang, W.: Vertex Operator Superalgebras and Their Representations. Contemp. Math. 175, 161-191 (1994), (hep-th/9312065) 30. Kazama, Y., Suzuki, H.: New N = 2 Superconformal Field Theories and Superstring Compactification. Nucl. Phys. B321, 232-268 (1989) 31. Kent, A.: Infinite DimensionalAlgebras and the Conformal Bootstrap, Cambridge: PhD thesis, (1986) 32. Kiritsis, E.B.: Character Formulae and Structure of the Representations of the N = 1, N = 2 Superconformal Algebras. Int. J. Mod. Phys. A3, 1871-1906 (1988) 33. Malikov, EG.: Verma Modules over Kac-Moody Algebras of Rank 2. Leningrad Math. J. 2, 269-286 (1991) 34. Matsuo,Y.: Character Formula of G' < 1 Unitary Representation of N = 2 Superconformal Algebra. Prog. Theor. Phys. 77, 793-797 (1987) 35. Mathieu, P., Walten, M.A.: Fractional Level Kac-Moody Algebras and Non-Unitary Coset Conformal Theories. Prog. Theor. Phys. 102, 229-254 (1990) 36. Nahm, W.: Quasi-Rational Fusion Products. Int. J. Mod. Phys. BS, 3693-3702 (1994) 37. Nekovar, J.: Private communication
Unitarity of Rational N = 2 Superconformal Theories
85
38. Ravanini, E, Yang, S.-K.: Modular Invariance in N = 2 Superconformal Field Theories. Phys. Lett. B195, 202-208 (1987) 39. Schwimmer, A., Seiberg, N.: Comments on the N = 2, 3, 4 Superconformal Algebras in Two Dimensions. Phys. Lett. B184, 191-196 (1987) 40. Vafa, C.: Toward Classification of Conformal Theories. Phys. Lett. B206, 421-426 (1988) 41. Watts, G.M.T.: Fusion in the W3 Algebra. Commun. Math. Phys. 171, 87-98 (1995) 42. Zhu,Y.: Vertex Operator Algebras, Elliptic Functions, and Modular Forms, PhD thesis, Yale University (1990), Modular Invariance of Characters of Vertex Operator Algebras, Journal AMS 9, 237-302 (1996) Communicated by R.H. Dijkgraaf
Commun. Math. Phys. 186, 87-94 (1997)
Communications in
Mathematical
Physics
@ Springer-Verlag 1997
Representations of Lie Superalgebras and Generalized Boson-Fermion Equivalence in Quantum Stochastic Calculus* T.M.W. Eyre**, R.L. Hudson Mathematics Department, University of Nottingham, University Park, Nottingham NG7 2RD, Great Britain. E-mails:
[email protected];
[email protected] Received: 6 August 1996/Accepted: 15 September 1996
Abstract: The boson-fermion equivalence scheme of [5] can be generalized to N dimensions with r boson and N - r fermion creation and annihilation fields. The same stochastic integral prescription replaces the N 2 generalized number processes A}, which form representations of the Lie algebra 91(N), by processes forming representations of the Lie superalgebra gl(N, r).
1. Introduction The integrator processes of N-dimensional quantum stochastic calculus are conventiently denoted [2] as AT, a,/3 = 0, 1 , . . . N. The processes A~ consist of operators AT(t), t E R§ acting in the (boson) Fock space F(L2(R+, cN)) whose matrix elements between exponential vectors are given by
@(f),A~(t)e(g)} =fO<s
(1.1)
Here for f = ( f l . . . , fN) @ L2(R+, c N ) we define f0 = 1 and f3 = f3, /3 = 0, 1 , . . . , N. Then the process A0~ is time, A~ and A~, j = 1 , . . . , N are respectively the component creation and annihilation fields in the direction of the canonical basis vector ej of C N, and the A}, i,j = 1,..., N are the generalized number processes corresponding to the Dirac dyads e} = lej) (ei l. The latter processes form a time-indexed family of representations of the complex Lie algebra 91(N) of all N x N matrices, in so far as we have * Work supported by University of Nottingham Key Research Staff Initiative, and by British CouncilAlliance Anglo-French Research Project PAR/984/52. ** TMWE is supported by an EPSRC research studentship.
88
T.M.W. Eyre, R.L.Hudson
(Aj(t)te(f), A~te(g) ) - ( A~(t)te(f), Ajte(g) I = (e(f), (5~A~(t) - 5~ A~(t))e(g) ) . (1.2) Here and elsewhere in this paper all operators are defined on the exponential domain of finite linear combinations of exponential vectors, t denotes the restriction to the exponential domain of the adjoint (so that (A~(t))~f = A~(t)), and the weak product T = R S of two such operators is defined through its exponential matrix elements (Rte(f), Se(g)); thus (1.2) is a rigorous statement of the formal commutation relation
[A~(t),A~(t)l=(i~A~(t) - (ijk At(t). i
(1.3)
When time and creation and annihilation fields are included, (1.2) becomes a rigorous form of [A~(t), A~(t)] = ~A~(t) - ~ A ~ ( t ) , (1.4) wjere 83 is the Evans [2] (f-function which is 1 when c~ =/3 5/0 and otherwise 0. Thus the A~ form a representation of the Lie algebra glo(N) of ( N + l) x (N + 1) matrices in which the Lie bracket is given by
[M, N]A = M A N - N A M ,
A~ = ~ .
(1.5)
In the case N = 1, the boson-fermion equivalence scheme of [5] may be described in the present language as follows. Let us first note that the associative algebra fl, = M0(1) of 2 x 2 matrices in which matrix multiplication M N has been replaced by M A N is Z2-graded by making off-diagonal matrices odd and diagonal matrices even. That is to say, the linear decomposition r = .A+ @.4_ into diagonal and off-diagonal parts satisfies
.A+A+,A_A_ c_ fit+,
A+A_,A_.A+ c_ A_.
(1.6)
We may thus form the corresponding Lie superalgebra in which fit is equipped with the bracket operation {M, N} = M A N
- (--1)~(M)~(N)NAM,
(1.7)
where M and N are assumed of definite parity and a is 0 on even elements and 1 on odd. We denote this Lie superalgebra by g/0(1, 1). Let us now define processes ~ , c~,/3 = 0, 1 by the stochastic integral prescription -- [ -
~ ( 0 ) = 0,
I'~~
(1.8)
where we write =
Thus in the even cases we have . ~ = A], -0=~ = A0. However in the odd cases the processes ~o and ~d are fermionic creation and annihilation processes, indeed [5] they generate the Fock representation of the CAR over L 2(~+) according to the prescription bt(f) = f f(s)d~~
f f(s)(-1)A',(~)dA~
(1.9)
Their relation to the gauge and time processes may be summarized by the statement that, for each t E ~+, the map forms a representation of the Lie superalgebra g/0(1, l) on the Z2-graded Hilbert space
Lie Superalgebras in Quantum Stochastic Calculus
89
F(L2(R+)) = F+ 9 F_,
(1.10)
where F+ are the eigenspaces corresponding to eigenvalues 4-1 of the self-adjoint unitary grading operator ( - 1 ) A~(t). Thus the operators ~~ anticommute with the grading operator, whereas ~~ and E~ (t) commute with it, and we have ~ , - =~ 6 ( 0 - 3='~ (0 - ( - 1 ) '~'~'~"
~(t)~(t)
= ~(t)~(t)-
( - 1 ) ~%~'~ , d~_='~ 6 (t),
(1.11)
where products of unbounded operators are in the weak sense above9 It is not difficult to see (and it follows from the main theorem below) that this formulation of boson-fermion equivalence can be generalized to N dimensions by Z2grading the associative algebra A = M o ( N ) of ( N + 1) • ( N + 1) matrices equipped with the multiplication A A B according to the scheme
{[2~ 9
A+=
"'"
0
.
9
,
A_=
(1.12)
.
{[i 9
"'"
0
where the entries * can be arbitrary. The grading process ( - 1 ) A] is replaced by ~--,N
Aj
(-1)z--~ =' ~ in (1.8) with a corresponding modification of the grading operator defining the decomposition (1.10); in this way we construct the Fock representation of the CAR over L2(•+; c N ) . Again the fermionic creation and annihilation processes together with the component gauge and time processes satisfy the respresentation property (1.11) in the weak sense. Our purpose in the present paper is to show that (1.11) continues to hold if the Z2-grading (1.12) is replaced by r
A+ =
r
", 9
* *
....-
* *
0 0
... ...
0 0
:
:
"..
:
:
"..
:
9
*
...
*
0
...
0
0
0
...
0
*
...
*
9
:
"..
:
:
"..
:
0
...
0
*
...
*
0
r
0 0 A+ =
r
0
0 0 0
9 ..
0
*
.-.
*"
9 ..
0
*
...
*
9 ..
:
:
-..
:
9 ..
0
*
...
*
9..
*
0
...
0
9 ..
:
:
"..
:
9..
*
0
...
0.
/ / / /
(1.13)
where r is chosen arbitrarily between 1 and N , and the grading process is taken to be (-1)
~N+
J=
1 Z~
.
90
T.M.W. Eyre, R.L.Hudson
In particular, restricting our attention to the processes -=i j, i,j = 1,...,N (1.11) becomes : j ( t ) : 1 (t) - ( - 1 ) J z : k ( t ) : ~ ( t ) = 5~(t)5.](t) -- (--1) J t 5k:~(t),
so that
(1.14)
we obtain a representation of the Lie superalgebra gl(N, r). We use the notations s A t = rain{s, t}, s V t = max{s, t} for s, t E ]~+.
2. The theorem and its Proof From now on N and r are fixed natural numbers with 1 < r < N, ,,4 is the associative algebra of ( N + 1) • ( N + 1) matrices equipped with the usual involution and the multiplication A A B , where A = diag(0, 1 , . . . , 1). ,,4 is Z2-graded by the scheme (1.13). The grading process G is defined by
i
N a~(t), G(t) = ( _ I)Ej=~+I
or equivalently as the process of second quantizations G(t) = F(x[0,tl d i a g ( 1 , . . . , l, - 1 , . . . ,
- 1 ) + X(t,~)l),
where +l occurs r times and - 1 occurs N - r times in the diagonal matrix d i a g ( 1 , . . . , l, - 1 , . . . , - 1 ) . The processes ~ , c~,/3 = O, 1 , . . . , N are defined by
z (t)
=[
JO< s < l G ( s ) ~ d A ~ (s),
other notations being as in Sect. 1. Theorem. For all s, t E •+ the processes ~
~(s)Z~(t) - (-1)~,~ ~(t)~(s)
satisfy the commutation relations
= 5~ Z~(s A t) -- ( - - 1 ) ~ , ~ ~ = $ ( s A t), (2.1)
in the weak sense. The proof depends on four lemmas. First we prove that the grading process is commutative: L e m m a 1. For all s, t E ]~+, G(s)G(t) = G(t)G(s).
Proof of Lemma 1. Since G(s) is a self-adjoint unitary (e(f),G(s)G(t)e(g)}
= (G(s)e(f), G(t)e(g)) =
where
exp
/5
h(u)du,
IN f f (u)gy(u), h(u) =
u < sAt, u > sVt,
j:l ~ H(u)gj(u), j=lf J ( u ) g A u ) - j=r+l
sAt
Lie Superalgebras in Quantum Stochastic Calculus
Evidently (e(f),
G(t)G(s)e(9)) is given by the
Next we prove that the processes ~ ularly: L e m m a 2.
91
same expression.
[]
inherit the grades of the dyads e~ more partic-
For s < t E ~+ and 0 < ~, /3 < N, .U~(s)G(t) =
G(t)'~(s). quantization G(t)
( - 1)~
Proof of Lemma 2.
(2.2)
Notice that the second leaves the exponential domain invariant so that weak products may be indeed replaced by operator products as in (2.2). When c~ = 0, 2 2 = A~ and, from (1.1), we have
(e(f), ~(s)G(t)e(g)) = (7:h f<s,<s fz(sl)g~(s,)dsl (e(f), G(t)e(g) } , where (7:)~ is - 1 if a > r and +1 if a _< r. On the other hand, since
G(t) is self-adjoint,
G(t)Z~(s)e(g)} = (G(t)e(f), ~(s)e(9)} (e(f),
= (7:)2/
f~(sl)g~(Sl)dSl (G(t)e(f), e(9))
Jo <sl<s
= (7:)2 /"
f~(sl)g~(sl)dSl
(e(f),
G(t)e(g)),
J0 < s l < s
where (7:)2 is - 1 if 3 > r and +1 if 3 _< r. But since cr~ = 0, c~ and 3 are either both less than r or both greater than or equal to r so that (7:)1 = (7:)2 and (2.2) is proved in the even case. In the odd case we use the first fundamental formula of quantum stochastic calculus [6, 11] to write
(e(f), -~ (s)G( t)e(g) ) = (e(f), _v./n<s,<~G(sl )dA3a(sl)G(t)e(g)) = (7:)1
/
f3(sl)gC~(Sl)(e(f),G(sl)G(t)e(g)ds,
(2.3)
J0 < s l < s
and on the other hand
(e(f), G(t)~(s)e(9)) = (G(t)e(f), /
G(sa)dA~a(sOe(g))
dO < s l < s
= (7:)2 /
f~(sx)g~(sl)(G(t)e(f), G(sl)e(g)ds~,
(2.4)
Jo <sl<s
where (:V)I, (7:)2, defined as before, are now opposite in sign. For 0 < sl < s, the second quantizations G(sl) and G(t) commute by L e m m a 1, so that
(G(t)e(f), G(s~)e(g)) =
(e(f),
G(t)G(sl)e(9)) = (e(f), G(sl)G(t)e(g)).
Comparing (2.3) and (2.4) using (2.5) we establish (2.2) in the odd case also. Next we prove that the increments of the processes ~ in the weak sense, specifically that
(2.5) []
super-commute with the past
92
T.M.W. Eyre, R.L.Hudson
Lemma 3. For s <_t E ~+ we have
(~(s)te(f), (,~(t) - ~,'~(s))e(9)) = ( - 1 ) ~
ot
"r
((~r(t) - Z~(s))te(f), ~ ( s ) e ( 9 ) ) . (2.6)
Proof (of Lemma 3). when a~r = 0, (2.6) follows from the commutativity with the past of the increments of the process A~ = ~ .
When ~ = 1 we have
(~(s)t e(f), ( ~ (t) - . ~ (s))e(g)) ( ~ (s)ie(f), ~.t G(uldA'~(u)e(g)) = (~(s)te(f),
G(s)
/)
G~(u)dA~(u)e(g))
where we write G(u) = G(s)G~(u), where for u > s,
G~(u) =
,
,.~-~N
(A'~(u)--AJ(s))
t-~)~,::+~
,
.~
commutes with the past, in particular with A~(s). Since ~ ( s ) G(s) by Lemma l, we may write this as (-1)~
(~(
-- ( - 1 ) ~
-- ( - 1 ) ~
super-commutes with
s)~G(s)e ( f ) ), ( s t G~(u )dA~( u)e(g )}
IE; II;
1 1
G~(u) dA~(u)
G(s)e(f),Z.~(s)e(g)
a(u)dA~(u) e(f), ~(s)e(g)
= ( - 1 ) ~ ((Z~(t) - Z~(s)) e(f),ZX(s)e(9) )
)
I
[]
as required. L e m m a 4.
Proof (of Lemma 4). This is clear from the definition of ~ ( t ) , the self-adjointness of the parity processes G and the corresponding property of the processes A~.
[]
Proof of the Theorem. In view of lemma 3 it is sufficient to consider the case s = t. In view of Lemma 1 we have
( ~ ( t ) t e(f), ~ (t)e(9) ) =
(:~(t)~(f), z ~ ( t ) e ( 9 ) )
Lie Superalgebras in Quantum Stochastic Calculus
93
+ G(v)a~ G(u)cr~dA'~(u)dA~ (v) } + fo<~
((-1)~ ~ G(v) ~, G(u)~ dA~(u)dA~(v)
e(f),
+G(v) G(u) dAs(u)dAz(v ) +
/o
~
(~G(u)"~ +% dA~(u) e(g)
,
(2.7)
where we use Lemma 2 and the second fundamental formula of quantum stochastic calculus. Similarly we have
( Z~ (t) +e(f), Z~(t)e(g) ) = (e(f), ~ , Jfo < . < v < t ((-1)'~~176176
+G(v)~'~G(u) '~d~ua'~dA~(u)dA'~(v)) + fo<~
(2.8)
Subtracting (-1)"3~'~ • (2.8) from (2.7). we see that the four iterated integrals cancel, whereas the two "correction" terms, the first order integrals, are just
(e(f), (85'~_f~ ='Y -
( - 1)(r~(r~8~)e(g)),
where to obtain the latter terms we use the relations { a~(mod2) +
=
(mod 2)
if/3=7 if.
=
'
which are consequences of (2.7).
3. Discussion
By restricting our attention to the N 2 processes S j3, i, j = 1 , . . . , N we obtain representations (in the weak sense) of the Lie superalgebra gl(N, r). These may be decomposed into irreducibles and an analysis made of the super-commutative processes generated by the super-centre ff(gl(N, r)) of the enveloping algebra H(gl(N, r)) along the same lines as [7]. Details will be published elsewhere. The simultaneous existence of representations of a Lie superalgebra by processes (in the weak sense) and by their stochastic differentials may likewise be used to prove the existence of an abstract chaotic expansion for elements of H(gl(N, r)), and a concrete formula for this case can be found, parallel to [9]. Note that in this context it is essential that the differentials, "pointing into the future", super commute with the processes, allowing the co-algebra structure of Ll(gl(N, r)) to be brought into play. Again details will be published elsewhere [3]. Finally we note that the chaotic expansion for Ll(gl(N)) considered in [8] and [9] is closely related to the Driver-Gross isometry [1, 10] which is used to provide probabilistic
94
T.M.W. Eyre, R.L.Hudson
proofs of the Hall generalization [4] of the Bargmann-Segal isomorphism to Lie groups of compact type. The use of the corresponding super-theory to generalize in a similar way the fermionic Segal isomorphism [13] is under active investigation. Acknowledgement. The authors express their gratitude to the referee for correcting a number of minor errors and inaccuracies.
References 1. Driver, B.K.: On the Kakutani-It6-Segal-Gross and the Segal-Bargmann-Hall isomorphisms. J. Funct. Anal. 133, 69-128 (1995) 2. Evans, M.P.: Existence of quantum diffusions. PTRF 81, 473-482 (1989) 3. Eyre, T.M.W.: Chaos expansions in Lie superalgebras associated with a quantum stochastic calculus. Nottingham. Preprint. To appear in Commun. Math. Phys. 4. Hall, B.: The Segal-Bargmann "coherent state" transform for compact Lie groups. J. Funct. Anal. 122, 103-151 (1994) 5. Hudson, R.L., Parthasarathy, K.R.: Unification of Boson and Fermion stochastic calculus. Commtm. Math. Phys. 104, 457-470 (1986) 6. Hudson, R.L., Parthasarathy, K.R.: Quantum It6's formula and stochastic evolutions. Commun. Math. Phys. 93. 301-323 (1984) 7. Hudson, R.L., Parthasarathy, K.R.: Casimir chaos in a Boson Fock space. J. Funct. Anal. 119, 319-339 (1994) 8. Hudson, R.L., Parthasarathy, K.R.: Chaos map for the universal enveloping algebra of U(N), Math. Proc. Camb. Phil. Soc. 117, 21-30 (1995) 9. Hudson, R.L., Pulmannovfi, S.: Chaotic expansions of elements of the universal enveloping algebra of a Lie algebra associated with a quantum stochastic calculus. Nottingham, Preprint. To appear in Jour. London Math. Soc. 10. Gross, L., Malliavin, P.: Hall's transform and the Segall-Bargmann map. To appear in Taniguchi Proceedings, Warwick, 1995 11. Parthasarathy, K.R.: An introduction to quantum stochastic calculus. Basel: Birkhauser (1992) 12. Scheunert, M.: The theory of Lie superalgebras, springer LNM 716. Berlin-Heidelberg-New York: Springer-Verlag, 1979 13. Segal, I.E.: Tensor algebras over Hilbert spaces II. Ann. Math. 63, 160-175 (1956) Communicated by H. Araki
Commun. Math. Phys. 186, 95-107 (1997)
Communications in
Mathematical Physics
Q Spfinge~Vedag 1997
Optimal Heat Kernel Estimates for Schr6dinger Operators with Magnetic Fields in Two Dimensions* Michael Loss 1'**, Bernd Thaller 2 1 School of Mathematics, GeorgiaInstitute of Technology,Atlanta, GA 30332, USA 2 Institut fiir Mathematik, Universit~itGraz, A-8010 Graz, ?~usttia Received: 5 September 1996/Accepted: 3 October 1996
Abstract: Sharp smoothing estimates are proven for magnetic Schr6dinger semigroups in two dimensions under the assumption that the magnetic field is bounded below by some positive constant/30. As a consequence the L ~ norm of the associated integral kernel is bounded by the L ~ norm of the Mehler kernel of the Schr6dinger semigroup with the constant magnetic field B0.
1. Introduction A recurring problem in the theory of semigroups is to prove smoothing estimates, i.e., estimates of the semigroup acting as an operator from L p ----* L q, q > p (see [$82] for a general review on semigroups). In rare cases it is possible to render these estimates in their sharp form. Among the examples are Nelson's hypercontractive estimate [N66] and the sharp smoothing properties of the heat kernel (which follow from the sharp form of Young's inequality [B75, BL76]). A more recent example is furnished by the Mehler kernel associated with the S chr6dinger operator of a charged particle in a constant magnetic field which essentially is a result of Lieb [L90] although not explicitly stated there (see Theorem 1.1). An especially flexible approach to such problems, due to Gross [G76] (see also Federbush [F69]) and relevant for our paper, is the use of logarithmic Sobolev inequalities. In fact there is complete equivalence between Gross' logarithmic Sobolev inequality and Nelson's hypercontractive estimate. In a similar fashion, Weissler [W78] proved the equivalence of the sharp smoothing estimates for the heat semigroup and logarithmic Sobolev inequalities. * C)1996 by the authors. Reproductionof this article, in its entirety, by any means is permitted for noncommercial purposes. ** Work supported by N.S.E grant DMS-95-00840and the Erwin Schr6dingerInstitute
96
M. Loss, B. Thaller
On a more abstract level it was realized in Simon and Davies [DS84] (see also [D89]) that the technique of logarithmic Sobolev inequalities can be used to prove ultracontractivity for Markov semigroups. It is the aim of this note to implement Gross' method for magnetic SchrSdinger operators and to prove sharp smoothing estimates for the associated semigroup. The idea is similar to the one in [CL96] where it was realized that Gross' logarithmic Sobolev inequality when viewed as a family of sharp inequalities on ]Rn can be used to obtain sharp smoothing estimates of solutions of diffusion equations with a volume preserving drift. An example is the two dimensional Navier-Stokes equations in the vorticity formulation. Consider a single charged quantum particle in a magnetic field, i.e., consider the Hamiltonian H ( B ) = 2(V + iA) 2 ,
(1.1)
where A is a vector potential for the magnetic field B, i.e., curl A = B. Note that the semigroup generated by (1.1) is not Markovian, in fact not even positivity preserving. Usually, this difficulty is overcome by the important diamagnetic inequality ([K72, $77, 79, HSU77, see also AHS77 and CFKS87])
I(e"u)(x)l _< (e'=/21ul)(x),
(1.2)
which relates estimates on the magnetic Schr6dinger operator semigroup to estimates on the heat semigroup. The obvious disadvantage of using (1.2) is that all effects due to the magnetic field are completely eliminated. How, then, does the magnetic field affect the behavior of the semigroup? The first paper that addressed this question is presumably [M86]. It is proved in [M86] that for a magnetic Schr6dinger operator in 1R3, lim -1 In IletH(B)IIL,__+L~ < - 4 ) ( C ) B o ,
(1.3)
t--+oo t
for all curl-free magnetic fields B that satisfy 0 < 13o <__
[BI <_ CBo. Here
IBI = v / B 2 + B 2 + B 2 with Bl, B2, B3 being the components and the function ~ is
explicitly given in [M86]. It is noteworthy that the estimates were obtained by probabilistic techniques. This work was subsequently improved in [E94]. The estimate (1.3) holds in any dimension and without the condition that/3 be curl-free. In particular the following stronger estimate is true lim -1 In t--~
t
Ile~H(LB)IIL,_+L~ <- -
--CL min xE~ ~
IB(x)IL.
Here the constant C L -- 1 + O ( L ~ In L) as L --+ oc for some explicitly given a < 0. Thus, this result is in a certain sense optimal since by considering a constant magnetic field in two dimensions one cannot improve the exponent minxeR~ I/3(x)l. Again the results in [E94] were obtained by probabilistic techniques. Similar results can be found in [U94] where techniques from differential geometry were used. In this paper we consider the simpler problem for a magnetic field of constant direction, or what amounts to the same, a magnetic field in two dimensions. However, we try to retain as much information as possible about the magnetic field in the estimate on the semigroup.
Optimal Heat Kernel Estimates for SchrSdinger Operators
97
A magnetic field in two dimensions is a scalar function B which can be expressed in terms of a (non-unique) vector potential A,
B ( x ) - OAz(x) Oxl
OAl(x______) Ox2 '
A(x) = (A~(x), Az(x)) 9
For example, if B(x) = t3o is a constant magnetic field then Bo
A(x) = ~ - ( - x 2 , x l ) .
(1.4)
We want to estimate (for suitable p and q with l _< p < q < c~)
C(t;p,q) := sup Hu(t)[Iq u~Lp Ilullp
where
u(t)=etHu,
which is just the norm of the Schr6dinger semigroup exp(tH) as a mapping from LP(]R 2) to Lq(R2). This norm is finite, because (1.2) implies that
IIetHIILP-~L. <_ Ilet~nllL,-~Lq < ec. In case of a constant magnetic field B(x) = B0 > 0 these calculations can be done using the explicitly known integral kernel of the time evolution operator. If the vector potential A is chosen as in Eq. (1.4) above, the time evolution is given by
u(t, x) = f G(t, x, y) u(y) d2y , where G(t, x, y) is the Mehler kernel [$79, p. 168]
G(t, x, y) = Bo
1 4 7r sinh ( ~ )
exp
-
coth
( x - y)2 + i -~- (xly2 - x2Yl)
,
which is a complex, degenerate, centered Gaussian kernel (in the terminology of [L90]). In particular, we need the time evolution of the initial function
uo(x) = No exp
-4-~ao z2
'
ao > 0 .
(1.5)
The solution at time t is again a Gaussian which can be written as
uo(t, x) = N(t) exp
- ~
x2
where
N(t) =
No ao ao cosh ( - ~ ) + sinh (~-~) '
a(t) =
. o , + sinh - ao cosh (--~-)
ao sinh ( ~ )
The function a(t) is the unique solution of the initial value problem a(t) = ~Bo - (1 - a(t)2)
,
a(0) = a o .
* cosh ( - ~ ) "
98
M. Loss, B. Thaller
Writing ao = tanh(0) (for ao < 1), or a0 = coth(0) (for a0 > 1), we obtain / ~ a n h (0 + ~ - ~ ) f o r a0 < 1,
a(t) = l, coth (0 + Both2 J forf~a0a~=>1,1.
(1.6)
For a0 = 1, the function uo(x) = No exp(-Box2/4) is the eigenfunction of H corresponding to the eigenvalue - B o / 2 . The following theorem sets the stage for our investigation. It is essentially a corollary of a theorem in [L90]. We shall give the details in Sect. 2. Theorem 1.1. Let B(x) = Bo > O, all x E ]~2. For 1 < p < q < c~, there is a
centered Gaussian function Uo which maximizes the expression IIu(t) l[q/ll u lip over all u E LP(]~2). The function uo is of the form (1.5), where ao depends on p, q, t, and t3o. More explicitly, with $
ao = dt,p,q + ~/ ~,p,q + p - 1 ,
(1.7)
dt,p,q- 2etst
ct = cosh
,
we have Co(t;p,q) =- SUPueLP =
~
st = slnn ~
,
LI~H(BO)UlIq Ilu0(~)llq II~llp --Ilu011p
~4~aoTr]
4a(t)
"
Remark 1. The condition p < q in Theorem 1.1 is necessary as the following example shows. Consider the function u(x) = No ~
exp(-B0(x - a)2//4) ,
aEG
where G is a finite square lattice in ]~2 whose lattice spacing is large compared to 1//B0. Thus the Gaussian functions in the sum have essentially disjoint support and hence
,,Ullp
No ( 4-~op)lip IGI I/p ,
where Ia I denotes the number of lattice points. The function u is an eigenfuction of the magnetic semigroup with eigenvalue e -B~ Thus,
Hu(t)llq ~ Noe-B~
( 4~oq) l/q IGI 1/q ,
and
II (t)llq e_Bo /2 ( Bo Ilull ~ \ ]
p /PlCll/ _l/ ql/q
The last expression tends to infinity with [G[ i f p > q.
"
Optimal Heat Kernel Estimates for Schrtidinger Operators
99
Remark 2. Instead of using the results of [L90] one might approach Theorem 1.1 a la Gross [G76], i.e., trying to reduce the problem to an integral over infinitesimal time steps, and estimating the semigroup over these time steps by employing logarithmic Sobolev inequalities. It is clear that one would get some smoothing estimates, but can one obtain them in the sharp form? That there are some obstructions to reach this goal by this method can be seen as follows. Equations (1.5) and (1.7) determine the Gaussian function Uo that yields the norm of the magnetic heat kernel as an operator from L p to L q. With this Gaussian we may write Co(t;p,q) = [[u~ -Ilu0(s)ll~ Iluo(t)llq
Ilu01lp
[l~011p Ilu0(s)llr '
for some r and s with p < r _< q, 0 < s < t. Hence it is obvious that Co(t;p,q) <_ C o ( s ; p , r ) C o ( t -
s;r,q).
Clearly, the proposed method can only work, if there exists a number r with p < r < q such that the above formula is an equality, i.e., Co(t;p, q) = min {Co(s;p, r) Co(t - s; r, q)}. p<_r
(1.9)
However, it is easy to see that Eq. (1.9) cannot hold in general. For example, it is wrong in the specific case q = p > 2, 0 < s < t. From the explicit form of Co(t;p, q) given in Theorem 1.1 we find that Co(t;p, q) = Co(s;p, r(s)) Co(t - s; r(s), q)
(1.10)
holds for some r(s) E [p, q] if and only if the optimizer u0 giving C0(t; p, q) is also the optimizer for Co(s;p, r(s)) (i.e., Co(s;p, r(s)) = Ilu(s)ll~(8)/lIu011p). Equation (1.10) can only hold for all s E (0, t) if the function r(s) is monotonic nondecreasing, with r(0) = p and r(t) = q. Given u0 (and hence ao) as the optimizer for C0(t; p, q), we can easily determine r(s) using Theorem 1.1 by solving ao
=
ds,p,r(s)
+
V/2ds,p,r(s) + p -
1.
This gives the function P r(s) = [aocs + (1 - p)ss][cs/ao + s~] '
(1.11)
which is monotonic nondecreasing in the following cases: q-p q~p~)
>(st) 2
if2
q-p p~q)
>(st):
if2_>q>p.
I f p < 2 < q the function is always monotonic increasing. Thus, the formula (1.9) holds whenever the above inequalities are satisfied and in these cases Eq. (1.10) holds for all s c [0, t] with r(s) given as above. The identity a(s) = dt-s,r(s),q + v/d2t_s,r(~),q + r(8) - 1
100
M. Loss, B. Thaller
(with a(s) as in (1.6)) shows that the optimizer u0 for Co(s;p, r(s)) is mapped by the time evolution onto the optimizer Uo(S) for Co(t - s; r(s), q), if r(s) is chosen as in Eq. (1.11). The approach discussed at the beginning of Remark 2 has the great advantage of being quite flexible and we shall use it to obtain smoothing estimates in the cases where the magnetic field is not constant. Certainly, one cannot expect to get such detailed information as in Theorem 1.1 in this case, however, we have the following result which is proved in Sect. 4. Theorem 1.2. Assume that B(x) > Bo > 0 is continuous. Then the estimate
C(t;p, q) =
IIdH(B)IILp_~Lq <_ Co(t;p, q)
(with Co(t; p, q) given by Eq. (1.8)) holds for a) allt > O, ifp <_ 2, q >_ 2, b) t > Oand2 < p <
q < oc, if q ~ o Z ~ ) ->
e)t>Oandl
sinh
~
(1.12)
,
< _ p < q < 2 , if 2
Remark 1. In the case where neither a), b) nor c) holds b u t p < q, we get a nontrivial bound but it is not sharp. Remark 2. A simple consequence of Theorem 1.2 is the pointwise bound on the heat kernel
t30 IletmZ)llL,~L~ _< 47r s i n h ( - ~ )
'
which is the best possible. In fact there is equality if B(x) =_ 13o as can be seen fromthe Mehler kernel. Finally, also in Sect. 4, we extend our result on the "diagonal" of the magnetic heat kernel to the "off-diagonal" using a technique due to Davies [D89]. Theorem 1.3. Assume that B(x) > 13o > 0 is continuous. Then the magnetic heat
kernel satisfies the bound BO
[etH(x,y)l < 47r sinh(_~)e
_ <~-v)2
2,
(1.14)
Remark. The Ganssian decay on the right side of (1.14) is the one of the heat kernel, which is considerably weaker than the decay of the Mehler kernel. Is it true that [etH(x, Y)I is bounded by the Mehler kernel IG(t, x, U)[? The truth of this estimate would reveal a robust dependence of the magnetic heat kernel on the magnetic field. This is an open problem.
Optimal Heat Kernel Estimates for SchrSdinger Operators
101
2. Proof of Theorem 1.1 Let Gt be the integral operator with the Mehler kernel G(t, x, y). We want to determine = Co(t; p, q),
sup
which is either finite or infinite. By a result of E. Lieb (see [L90], Theorem 4.1) it is sufficient to take the supremum over centered Gaussian functions of the form e x p ( - x 9 J~x), where J~ is a complex-valued matrix, symmetric with respect to the scalar product in 1~2, and with a strictly positive real part. Even for this case the calculation is still somewhat tedious. Instead it is convenient to rework the proof of Theorem 4.1 in [L90]. Let t > 0 be arbitrary. We approximate Gt by the operator G~ with kernel
C~(x, y) = e -~x~ G(t, x, y), which for e > 0 is a non-degenerate, centered Gaussian kernel. According to [L90], Theorem 3.4, there is a unique (up to a multiplicative constant) centered Ganssian function u~ which yields the maximum of IlGfU]lq/llUllp over all u E LP(R2). The function u~ is of the form u~(x) = e x p ( - x . J~x), where Je is a (possibly complex-valued) matrix which is symmetric with respect to the scalar product in N 2 and has a strictly positive real part. But since the integral operator Gf commutes with rotations, the unique maximum u~ must also be rotationally invariant. Hence Je = a ~ l + i/3~1, where ae > 0, and fl~ is real. Since the integrals of Gaussian functions can be evaluated explicitly, we can evaluate the quotient IIG~ullq/llullp for u(x) = e x p [ - ( a + i/3)x 2] and maximize this expression over all a and/~. The maximum is obtained for (a,/3) = (a~, fl~), with ac > 0 and fl~ = 0. Since e x p ( - e x 2) < 1 we find for any Gaussian function u
IIG%llq ~ I[a~ullq, and hence
c~
IIG~u~llq
-[lu~[Ip
where u~ = e x p ( - a ~ x z) is the unique Gaussian maximizer for G~. By an explicit calculation one sees that lira C ~ =- C O -
~o
IIGtuollq IluoI[p '
with uo(x) = e x p ( - ~ x 2 ) , a = lim~--,o c~. Of course, we have C O < Co(t;p, q). Finally, for any Gaussian function u, the limit lim Ila~ultq
,-.0
Ilullp
IIa,ullq Ilul[p
exists by Lebesgue's dominated convergence, and from
IIG~ullq < c "
II~lfp we find immediately that
for all e > 0
102
M. Loss, B. Thaller
IlGtul[q ~ lim C ~ = C O for all Gaussian functions u. Hence, also sup IlGtu[[q/l[ull p = Co(t; p, q) <_ C ~ This proves C O = Co(t; p, q). A little calculation easily gives the explicit value of this constant.
3. A Differential Inequality In the following we assume that the magnetic field B is given by a differentiable vector field A as in Eq. (1.1) and satisfies B ( x ) _> t30 > 0, for all x 9 IR2. In particular we choose the vector potential to be in L4oc(1R2) and then by the Leinfelder-Simader Theorem [LS81] the formal expression (1.1) defines a selfadjoint operator on some domain D ( H ) with C~(1R 2) as a core. If we set u(s) = e ' * u o , then u(s) is a solution of
d-~U=Hu=
(V+iA)Zu.
(3.1)
If u0 E D ( H ) N L 1(lI~2) n ZCX~(I~2), then we have that u(s) 9 D ( H ) N L 1(R 2) n L ~ (R2). This follows from the diamagnetic inequality (1.2) for each s 9 (0, t] and the explicit form of the heat kernel. T h e o r e m 3.1. Let r : (0, t] --+ ]R be a twice differentiable function with r(s) > 2 and i'(s) > Ofor all s E (0, t]. Then
d ds In Ilu(s)ll~(s) < - Z ( r ( s ) , §
(3.2)
for all s E (0, t], where L(r, 7') = ~
f , 47ra(r, i') "l a(r, i~)Bo [2 + m BoT0r ?) + - - r
(3.3)
and l
"2
Equality holds in Eq. (3.2) for B ( x ) = Bo and u ( s , x ) = exp
4a(r(s),~(s)) z2
"
Proof. Pick uo 9 D ( H ) n L~(R 2) C) C~(R2). Since ~ In does not depend on k > 0, we assume during the following calculation, without loss of generality, that at the time r = s > 0 the solution is normalized such that Ilu(s)llr(s) = 1. For the derivative at r = s we obtain therefore d In 4-7
Ilu(s)l[r(s)
= ~~(s) f lu(s, x)l r(~) In lu(s, x)l ~(~) dZx
a- [ U ( 8 , X)] 2 d2x q- ~1 f ]U(8, x)[(~(,)-2) -ds
(3.4)
The integrals have to be taken over IR2. The formal computation can be easily justified by an approximation argument. For simplicity, the arguments s and x in the integrand on the right side will be omitted from now on.
Optimal Heat Kernel Estimates for SchrSdinger Operators
103
Using (3.1) we obtain, after a partial integration 1 f lu1r
ad lUl 2 = -- 89 -- 2 ) f lul (r-2} (VluD 2
- ~1 f ]u[(r-2) [(V + i A ) u[ 2 9
(3.5)
The integration by parts can be justified as follows. Since u E D ( H ) , by the LeinfelderSimader Theorem [LS81] we can pick a sequence un E C ~ ( I R 2) such that Un --+ u, H u n --'+ H u in L 2. Inspecting the proof of the Leinfelder-Simader Theorem one sees that the sequence u,~ can be chosen to converge to u in L t and to have a uniform bound on the L ~~ norm. Thus u,~ converges to u in L p for all 1 < p < c~. In particular, all the following computations can be justified in the same fashion and we can assume without restriction that u E C~(IR2). If we set u = f + ig then lul = r
+ 92. We find
[(V + iA) U[2 =- (V]U[) 2 -b IA + v S l 2 lul 2 = s 2 + y 2 ,
(3.6)
where we have introduced two real vector fields X and Y over R 2, X=V•
(~x2'
OXlO),ul '
Y = (A + V S ) lul. Here the symbol V S denotes the expression fVg - gVf
f2 + 92
'
which is defined wherever f2 + 92 > O. For any c > 0 we may estimate c2X2 + y 2 > 2 e X 9Y
(3.7)
with equality if and only if c X = Y. If B = Bo is a constant magnetic field we can choose a Gaussian function u = N exp { - / 3 0 X2 ;
4c
(3.8)
J
such that equality holds everywhere in (3.7), because (with A as in Eq. (1.2))
cX = c V •
B0
~ - ( - x 2 , x l ) u = Au= Y .
Let us now insert (3.6) and (3.7) into (3.5). Since r - 1 > 0 we can add and subtract a positive constant c 2 with 0 < c < v/r - 1 in order to obtain 1 f lU[(r--2) dssd[~t[2 = -- 89 -- 1 -- c 2) f I@ `-2) (Vlul) 2
_ !2 f lul(r-2)ca(Vrul)2 _ 89f lul(~-2)(A + VS) 2 lul 2 < - 89 - 1 - ca) f lul
104
M. Loss, B. Thaller
-~ f (V•
_--
9( A + V S )
- - rc
_
f { V • "([ul,(d + VS)) + BluV}
CfBlul
The only thing to observe is that V • - V S = 0, wherever lul > 0. Using B(x) > Bo and f ]u[ ~ = 1 these estimates lead to the following family of inequalities for each c E (0, v/7 - 1): d-~ d lnllull~ <_ ~~ f lul ~ In [ulr - 89
(3.9)
- 1 - c 2 ) f lUI(r-2)(VlU[) 2
~B0,
Equality holds in case of a constant magnetic field B = B0 and u a Gaussian function given by Eq. (3.8). We also note that in the special case ?(s) = 0 the above calculation can be simplified to give d In Ilul]~ < _ ~ z - ] -
ds
B0
-
(3.10)
r
Now we apply the following family of logarithmic Sobolev inequalities
[g21ng2-A j
-ln)~,
all,k>0,
(3.11)
7F J
where we assumed that f g2 = 1. Equality holds if g is a normalized Gaussian,
g(x) = ~
1
exp
--~
x2
..
This family is essentially Gross' logarithmic Sobolev inequality rewritten as an inequality on ]R2 (see, e.g., [CL96]). We insert (3.11) into (3.9) with
g
= 27r(r. - 1 - C 2 ) ,
= ['a] r / 2
(3.12)
.
Here we have to assume that ~ > 0. With this choice of ), we obtain d In ,,u,,~ < ~ { 27r(r_ 1_c2)} d--s --~-7 2+ln ~ This result holds for all 0 < c < ~ for c = a, where
cBo r
(3.13)
1. The expression on the right side has a minimum
B~r2(r - 1 ) - ~ ) = a(r, f) .
a = B--~~ (1 r
(3.14)
Note that for § = 0 this formula is consistent with Eq. (3.10). In order to determine the case of equality in (3.13) we determine ,k according to (3.12) for the optimal value a of c. This gives A = 4rca/Bor so that the Gaussian function which optimizes the logarithmic Sobolev inequality is given by g
ur,2N{ =
exp
- ~
X2
}
.
Optimal Heat Kernel Estimates for SchrSdinger Operators
105
As shown above, the corresponding function u at the same time optimizes Eq. (3.7) in case of a constant magnetic field. Hence we find that equality holds in the estimate (3.13) (with c = a) provided that B = B0 is constant and u is the Ganssian function defined above. For the optimal choice of c the inequality (3.13) becomes d d--~ In IIu(s)llr(s ) < - L ( r ( s ) , ~ ( s ) ) , where (with a = a(r, i~) as defined in (3.14))
f~ { 47ra ~ aBo L(r, ~) = ~-~ 2 + In Bor J + - - r This estimate remains true for ~(s) = 0 and a = v/r - 1 with L(r, O) = aBo/r, cf. Eq. (3.10). []
Remark. We note that if for some s E [0, t], the derivative ~ is negative, then d/ds In ]]u][r is unbounded. This can be seen by choosing a constant magnetic field A(x, y) = ( - y , x) and u(x, y) = e i~u g(x) h(y) , where g and h are positive with f g~ = f h r = 1. The integrals in (3.4) and (3.5) can be evaluated explicitly for Gaussian functions. Setting
(1) 1/2r ( X--2r) -
(~---~
exp
( Y--2r) -a
,
we find d In [lull, d--'s
= ~# (In a - 2 In Ir - 2) - -r - -7 - (1 1 + a ) - 1 ,
which for ~ < 0 tends to +cr as a ---+O.
4. P r o o f s o f T h e o r e m 1.2 a n d 1.3
Proof of Theorem 1.2. By duality, the semigroup e Ht : L p ~ Lq has the same norm as the one from L q' --+ L p' , where p~, q' are the dual indices. Thus c) follows from b). To prove b) for q < c~ we integrate the inequality (3.2) and obtain Ilu(t)[I--q < exp { - f o ' L ( r ( s ) , i ~ ( s ) ) d s } . Ilu(0)llp -
(4.1)
Recall that the only assumption on r(s) is that r(0) = p, r(t) = q and that ~(s) be nonnegative. Thus, we can optimize the right side of (4.1) over r(s) which leads to an interesting problem in the calculus of variations. The associated differential equation is
~ = 2 - - + Bgr(r - 2). r
(4.2)
It is easily verified that r(s) defined by Eq. (1.11) is a solution of (4.2) and satisfies r(0) = p. Moreover, if we choose a0 as in (1.7) we see that r(t) = q. Using the function r(s)
106
M. Loss, B. Thaller
as a trial function in (4.1) we obtain b) after an exceedingly tedious but straightforward computation. Next we consider b) with q = oe. We remark that the definition ( l A D of the trial function r(s) still makes sense in this case, the condition for monotonicity follows from (1.12). We have lims__+tr(s) = c~ and 2 < r(s) < cxD for s < t. Choose T with 0 < T < t and write u(T) = eTHuo. Then
tIetHuoIl~ IIuollp
IleTnuoll~T) Ildt-T)nu(T)[[~ IIu011p Ilu(T)lI~(r)
Here the second factor is bounded by some constant K(T) which tends to 1, as T ~ t, because the semigroup e sH is bounded as an operator from L ~ to L ~ for all r > 1. The first factor can be estimated as above by integrating (3.2), so that for all 0 < T < t,
IletHu~176176 p -< K(T) exp { - foTL(r(s)'f~(s)) ds} 9 For the trial function r(s) with q = oe one can see that ffs)/r(s) 2 as well as a(r(s), f~(s)) remain bounded for s ~ t. Therefore the function L(r(s), f~(s)) given by (3.3) has only a logarithmic and hence integrable singularity, as s --+ t. Hence the estimate above makes sense for T ~ t and we find that (4.1) remains valid even for q = cx~. To prove a) we again consider the function r(s) defined by Eq. (1.11). Again r(0) = p _ 2 and if we choose a0 as in (1.7), r(t) = q > 2. Denote by 0 the time where r(O) = 2. Such a time exists since r(s) is strictly increasing. Now
[[u(t)[[q
Ilu(t)llq [[u(o)ll2
Ilu(o)ll~
11~(o)112 Ilu(O)ll~ '
and the second factor on the right can be estimated according to part b) by the constant C(O;p, 2) whose value is given by the right side of (1.8) with q = 2 and t =/9. The first factor is estimated by writing
Ilu(t)llq = Ilemt-~ 11~(o)112 11~(o)112
< sup Ilu(t - O)llq Ilu(O)lI2 '
which is bounded by C(t -/9; 2, q). Combining these estimates and using formula (1.10) yields the result. []
Proof of Theorem 1.3. Following [D89] we consider the semigroup given by the kernel Qt(x, y) = e-'~x etH(B)(x, y)e ~y , which is generated by the operator 1
~ ( V + a + iA) 2 . Retracing the steps of the proof of Theorem 3.1 for u(t)(.) = f the estimate d d-'s In Ilu(s)llr(~ ) < - L(r(s), i~(s)) + c~2 , with
Qt(', y)uo(y)dy yields
L(r(s), ~(s)) given by (3.3). From the proof of Theorem 1.2 it now follows that
Optimal Heat Kernel Estimates for Schr6dinger Operators
IQt(x, y)l ~
107
Bo
e~2t,
47r sinh(-~)
or
leH(B)t( x, Y)I <
Bo
e~r
47r sinh(-~) [] Optimizing with respect to a yields the result. Acknowledgement. We would like to thank Eric Carlen,
Laszlo Erdts and Vitali Vugalter for many helpful
discussions.
References [AHS77]
Avron, J.E., Herbst, I., Simon, B.: SchrSdinger operators with magnetic fields, I. General interactions. Duke Math. J. 45, 847-883 (1978) [B75] Beckner, W. : Inequalities in Fourier Analysis. Ann. of Math. 102, 159-182 (1975) [BL76] Brascamp, H.J., Lieb, E.H.: Best constants in Young's inequality, its converse and its generalization to more than three functions. Adv. in Math. 20, 151-172 (1976) [CFKS87] Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: SchrSdinger operators (with application to Quantum Mechanics and Global Geometry), Berlin-Heidelberg-New York: Springer, 1987 [CL96] Carlen, E., Loss, M.: Optimal smoothing and decay estimates for viscously damped conservation laws, with application to the 2-D Navier Stokes equation. Duke Math. J. 81, 135-157 (1996) Davies, E.B.: Heat kernels and spectral theory. Cambridge tracts in Mathematics, Vol. 92, Cam[D891 bridge: Cambridge University Press, 1989 [DS84] Davies, E.B., Simon, B.: Ultracontractivity and the heat kernel for Schrtdinger operators and Dirichlet Laplacians. J. Funct. Anal. 59, 335-395 (1984) ErdSs, L.: Estimates on stochastic oscillatory integrals and on the heat kernel of the magnetic [E943 Schrtdinger operator. Duke Math. J. 76, 541-566 (1994) [F69] Federbush, P. : A partially ahernate derivation of a result of Nelson. J. Math. Phys. 10, 50-52 (1969) Gross, L.: Logarithmic Sobolev inequalities. Amer.J. Math. 97, 1061-1083 (1976) [G76] [HSU77] Hess, H., Schrader, R., Uhlenbrock, D.A.: Domination of semigroups and generalizations of Kato's inequality. Duke Math. J. 44, 893-904 (1977) Kato, T.: SchrSdinger operators with singular potentials. Isr. J. Math. 13, 135-148 (1972) [K72] Lieb, E.: Gaussian kernels have Gaussian maximizers. Invent. Math. 102, 179-208 (1990) [L90] Leinfelder, H. and Simader, C.: SchrSdinger operators with singular magnetic vector potentials. [LS81] Math. Z. 176, 1-19 (1981) Malliavin, P.: Minoration de l'etat fondamental de l'equation de Schrtdinger du magnetisme et [M86] calcul des variations stochastiques. C.R.Acad.Sci.Paris Ser.I.Math. 302, 481--486 (1986) Nelson, E.: A quartic interaction in two dimensions. In Mathematical Theory of Elementary [N66] Particles, R. Goodman and I. Segal, eds., Cambridge, MA: M.I.T. Press, 1966 [$77,79] Simon, B.: An abstract Kato inequality for generators of positivity preserving semigroups. Ind. Math. J. 26, 1067-1073 (1977). Kato's inequality and the comparison of semigroups. J. Funct. Anal. 32, 97-101 (1979) Simon, B.: Functional integration and Quantum Physics. New York: Academic Press, 1979 [S79] Simon, B.: Schrtdinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7, 447-526 (1982) [S82] Ueki, N.: Lower bounds on the spectra of SchrSdinger operators with magnetic fields. J. Funct. [U94] Anal. 120, 344-379 (1994) and 127, 257-258 (1995) [w781 Weissler, E: Logarithmic Sobolev inequalities for the heat-diffusion semigroup. Trans. A.M.S. 237, 255-269 (1978) Communicated by B. Simon
Commun. Math. Phys. 186, 109-136 (1997)
Communications in
Mathematical Physics
@ Springer-Verlag 1997
String Center of Mass Operator and its Effect on BRST Cohomology Alexander Astashkevich 1, Alexander Belopolsky 2,* 1 Departmentof Mathematics, MIT, Cambridge, MA 02139, USA. E-mail:
[email protected] 2 Center for Theoretical Physics, MIT,Cambridge, MA 02139, USA. E-mail:
[email protected] Received: 14 December 1995/Accepted: 8 October 1996
Abstract: We consider the theory ofbosonic closed strings on the flat background lR25'1. We show how the BRST complex can be extended to a complex where the string center of mass operator, x~, is well defined. We investigate the cohomology of the extended complex. We demonstrate that this cohomology has a number of interesting features. Unlike in the standard BRST cohomology, there is no doubling of physical states in the extended complex. The cohomology of the extended complex is more physical in a number of aspects related to the zero-momentum states. In particular, we show that the ghost number one zero-momentum cohomology states are in one to one correspondence with the generators of the global symmetries of the background i.e., the Poincar6 algebra.
1. Introduction and Summary As it was argued in ref. [1], the gauge parameters that include the zero mode of the X # operator have to be considered in order to prove the complete dilaton theorems. If we allow X ~' to appear in gauge parameters, it is natural to allow it to appear in the physical states as well. This calls for an extended version of the BRST complex where the zero mode of X ~' or, in other words, the string center of mass operator z0~ is well defined. In this paper we will consider the closed bosonic string with a fiat non-compact target space ~25,1. We will define the extended complex simply as a tensor product of the BRST complex with the space of polynomials of D = 26 variables. Our main objective is to calculate the semi-relative cohomology of this complex. When we add new vectors to a complex, two phenomena may occur in cohomology. First, some vectors that used to represent nontrivial cohomology classes may become trivial, and second, some new cohomology states may appear. Our original motivation [1] to use the extended complex was that the graviton trace ~, BRST-physical state, was * Supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under cooperative agreement #DF-FC02-94ER40818
110
A. Astashkevich, A. Belopolsky
trivial in the extended complex. As we will see, the extended complex provides many more examples of this kind. We will show that only one out of D 2 + 1 ghost number two zero-momentum BRST-physical states remains non-trivial in the extended complex. In the BRST complex, the spectrum of non-zero momentum physical states is doubled due to the presence of the ghost zero modes. We will see that there is no such doubling in the extended complex: only ghost number two states survive and all the ghost number three states become trivial I . Returning to the second phenomenon, the appearance of new physical states, we will be able to show that all such states can be obtained from the old ones by differentiation with respect to the continuous momentum parameter along the corresponding mass shells. In this sense no new physical state will appear. Zero momentum states will require special attention and the calculation of the cohomology of the zero momentum extended complex is technically the most difficult part of this work. We will find that the ghost number one discrete states at zero momentum correctly describe the global symmetries (Poincar6 group) of the background. The analysis presented in this paper shows that the semi-relative cohomology of the extended complex correctly describes the physics of the closed bosonic string around the flat D -- 26 background. The arguments that lead to this conclusion are the following: 1. Ghost number two non-discrete physical states are the same as in the BRST complex up to infnitesimal Lorentz transformations. In this sense the extended complex is as good as BRST (Sect. 5). 2. There is no doubling of the physical states, - ghost number three BRST-physical states are trivial in the extended complex (Sect. 4). 3. There is only one zero-momentum physical state at ghost number two which can be represented by the ghost dilaton (Sect. 4 and ref. [1]). 4. Ghost number one discrete states are in one to one correspondence with the generators of the Poincar6 group (Sect. 4). The paper is organized as follows. In Sect. 2 we start by describing the extended complex and the nilpotent operator Q. We define a cohomology problem for the extended complex and explain what we are going to learn about its structure. In Sect. 3 we formulate a simplified version of the problem in which we replace the closed string BRST complex by its chiral part. Section 3 is devoted to a detailed analysis of the cohomology of this complex. In Sect. 4 we investigate the (semi-relative) cohomology of the full extended complex using the same methods as in Sect. 3. At the end of Sect. 3 and Sect. 4 we will formulate two theorems which summarize our results on the structure of the cohomology of the chiral and the full extended complexes. We present a detailed analysis of the Lorentz group action on the cohomology of the extended complex in Sect. 5. For the sake of completeness we add two appendices: Appendix A, where we review some basic algebraic facts which we use to calculate the cohomology and Appendix B, where we prove the results which are necessary to calculate the cohomology at zero momentum.
2. Extended Complex When we describe a string propagating in flat uncompactified background, it seems natural to let the string center of mass operator x0 act on the state space of the theory. I The observation that ghost number three states are physical only in the case of finite space time volume was made in the book by Green, Schwarz, and Witten [2]. In the finite space time volume the momentum is discrete and x~ = -iO/Op~cannot be defined.
String Center of Mass Operator and BRST Cohomology
111
This operator must satisfy the Heisenberg commutation relation #
u
[a,~, x 0 ] = - i 5n,0 ~u~,
(2.1)
where ~?u" is the Minkowski metric. In order to incorporate an operator with such properties we define an extended Fock space as a tensor product of an ordinary Fock space with the space of polynomials of D variables:
xD-1l |
.Tp(a, ~) = C [ x ~
$'v(a, ~).
(2.2)
All operators except aft act only on the second factor which is an ordinary Fock space, x~ operators act by multiplying the polynomials by the corresponding x t' and, finally, the action of a0 is defined by a~ = 1 | pg - i~7~'v~-Sz~| 1.
(2.3)
Note that in the extended Fock module the action of c~ does not reduce to the multiplication by pU. The extended Fock module is also a module over the Virasoro algebra or, strictly speaking, over a tensor product of two Virasoro algebras corresponding to the left and right moving modes. The generators are given by the usual formulae Ln -~ ~ E rll~u : a ~ O~n_ ~' rn ."~ m --
1
. --#
(2.4)
--u
Ln = ~ ~ . ~ 0 ~ . 9a m OLn_m :
9
The central charge of the extended Fock module is 26, the same as that of .Tp(a, ~), and we can use it to construct a complex with a nilpotent operator Q (see refs. [2, 3]). Following the standard procedure we define the extended complex as a tensor product of the extended Fock module with the ghost module .T(b, c, b, ~)
~p = f ( b , c, ~, ~) | .~p(~, ~),
(2.5)
and introduce the nilpotent operator Q as
1
~) = E
c n Z - n - -~ E ( Tn --Tt):C_mC_nbm+n: + a . h . , Tt
(2.6)
m~%
where we put a hat over Virasoro generators in order to emphasize that they are acting on the extended Fock space. We can alternatively describe the extended complex as a tensor product of the BRST complex Vp with the space of polynomials C[x ~ x ~ Vp = C[x ~
x D - l ] | Vp,
(2.7)
and express the nilpotent operator (~ in terms of the BRST operator Q as follows Q) = 1 | Q - i~xu@O E ( c n a ~ _ n + ~n~U_n) -- [] @ C~, n
where c~ = (co + ~o)/2. and [] = z?"u o x .0o2x . "
(2.8)
112
A. Astashkevich,A. Belopolsky
So far we have constructed an extended complex Vp = C[x ~ x D- t] | V, equipped with a nilpotent operator ~) given by Eq. (2.8). One can easily check that Eq. (2.8) does define a nilpotent operator using Q2 = 0 and commutation relations between a u operators.
2.1. Cohomology of the extended complex. Our major goal is to calculate the cohomology of the operator ~) acting on the extended complex Vp for different values of the momentum p. More precisely, we will be looking for the so-called semi-relative cohomology, which is the space of vectors annihilated by ~) and bo = b0 - b0 modulo the image of ~) acting on the vectors annihilated by bo (see refs. [4-7]). In mathematical literature it is called the semi-infinite cohomology of the Vir x Vir algebra relative to the sub-algebra 12o generated by the central charge and L o -- L0 - L0 with the values in the extended Fock module 5~p, and is denoted by 7-[(Vir • Vir, E o, f p ) (see refs. [8, 9]). We denote this cohomology by H s ( Q , Vp). Before we start the calculations, let us describe what kind of information about the cohomology we want to obtain. Ordinarily, we are looking for the dimensions of the cohomology spaces at each ghost number. In the presence of the x0 operator these spaces are likely to be infinite dimensional (since multiplication by x does not change the ghost number). In order to extract reasonable information about the cohomology we have to use an additional grading by the degree of the polynomials in x. The operator Q mixes vectors of different degrees and we can not a priori expect the cohomology states to be represented by homogeneous polynomials. 2 Instead of the grading on V we have to use a decreasing filtration 9.. ~ F - k V 3 F - k + I V 3 . . .
3 F0V =-V,
(2.1.1)
where F - k V is the subspace of V consisting of the vectors with z-degree less or equal then k. The operator Q respects this filtration in a sense that it maps each subspace F r V to itself: Q, : Fr~" --* F~V. (2.1.2) This allows us to define a filtration on the cohomology of (~. By definition F_kH(Q, V) consists of the cohomology classes which can be represented by vectors with the xdegree less or equal then k. Although as we mentioned above there is no x-grading on the cohomology space, we can define a graded space which is closely related to it. We define Gr~H(Q, V) = FrH(Q, V)/Fr+~H(Q, V). (2.1.3) h
A
A
i
~
A
By definition Gr~H(Q, V) (for r _< 0) consists of the cohomology classes which can be represented by a vector with x-degree - r but not - r + 1. These spaces carry a lot of information about the cohomology structure. For example if we know the dimensions of Gr~ for r = 0, - 1 , . . . , - k we can find the dimension of F_k as their sum. On the other hand the knowledge of representatives of Grr states is not enough to find the representatives ofcohomology classes. This is so because Gr~ spaces contain information only about the leading in x terms of the cocycles of Q. To find the graded space G r H = ~ G r r H we will use the machinery known as the method of spectral sequences. The idea is to build a sequence of complexes E,~ with 2 For the p = 0 extended complexand only for this case we will be able to show that the cohomologycan be representedby homogeneouspolynomials,but this will come out as a non-trivialresult.
String Centerof Mass Operatorand BRSTCohomology
113
the differentials dn such that E n + 1 = H(dn, E,d, which converges to the graded space GrH. In our case we will be able to show that all differentials dn for n > 2 vanish and thus E3 = E4 . . . . . E~r Therefore, we will never have to calculate higher than the third terms in the spectral sequence. We will give more details on the application of the method of spectral sequences to our case in Appendix A.
3. Chiral Extended Complex Before attempting a calculation of the cohomology of the full extended complex let us consider its chiral version. This is a warm up problem which, nevertheless, captures the major features. We replace the Fock space ~p(a,-5) by its chiral version, )Vp(C0 which is generated from the vacuum by the left moving modes c~n only. Repeating the arguments of the previous section we conclude that the chiral version of Q is given by
O=l|
0
| E
Cne~-n - ~1 v" , [] |
(3.1)
n
We will calculate the cohomology of the chiral extended complex Vp for three different cases: case p: ~ 0, which describes the massive spectrum; case p2 = 0 - the massless one; and case p - 0, which besides the particular states from the massless spectrum describes a number of discrete states.
3.1. Massive states. Let us start the calculation of the cohomology of the chiral extended complex Vp by considering the case o f p 2 5~0. For this case the cohomology of the BRST complex is non-zero only for ghost number one and ghost number two. The cohomotogy contains the same number of ghost number one and two states which can be written in terms of dimension one primary matter states. Let Iv, p) E Vp be a dimension one primary state with no ghost excitations; then the following states, cocllv,p)
and
cllv,p),
(3.1.1)
represent nontrivial cohomology classes and, moreover, each cohomology class has a representative of this kind (see ref. [10]). We will calculate the cohomology of the extended complex in two steps. First, we extend the BRST complex by adding polynomials of one variable :~ = (p 9 x). The resulting space, Vp = C[~] | Vp, (3.1.2) is a subcomplex of Vp and we define its cohomology as H(~), Vp), where (~ is the restriction of ~) on Vp. Calculation of GrH(Q, Vp) is the objective of the first step. Second, we obtain the full extended space as a tensor product of Vp with the polynomials of the transverse variables ~p = C[:~1..., ~:D-1] | Fgp'
(3.1.3)
where
~i = X i
pi(p. X) p2
(3.1.4)
114
A. Astashkevich, A. Belopolsky
I
I
'..
s=4
'. 9
9149
s=3
.l
6 -'>.. ". 0 2
s=l
0
r=--4
r=-3
Gh# = 4
......
Gh# = 3
......
Gh# = 2
......
Gh# = 1
......
Gh# = 0
o0..
s=2
s
......
r=-2
r=--I
r=O
Fig. 1. A n a t o m y of a double complex
N
~
A
Using G r H ( Q , Vp) found in the first step, we will calculate GrH(QLVp). Let us calculate GrH(Q, Vp). Beside the ghost number, complex Vp has an additional grading - the x-degree. According to these two gradings we can write Vp as a double sum
-- 9
~r~8 E0 (p),
(3.1.5)
7"~8
where E~ r0, s = 5 ~ - - r | V, (pr + s ) is the space of ghost number r + s states with - r factors of ~. Note that in our notations r < 0. It will be convenient to represent a double graded complex like Vp graphically by a lattice (see Fig. 1) where each cell represents a space E 0 , columns represent the spaces with definite z-degrees and the diagonals represent the spaces with definite ghost numbers. The action of Q) on 7/p can be easily derived from the general formula (3.1). Any vector f r o m / ~ o k's can be represented as ~k | iv,p), where Iv,p) ~ Vv(8-k)is a vector from the BRST complex Vp with ghost number s - k. Applying ~9 to this state we obtain ~) ~k | iv, p } = ~k
|
- ikYck-1 | Z cn(p" ~-'~)I v,p)
(3.1.6)
n
k(k
- 1) p 2 : ~ k _ 2 |
2
CO I v , p ) .
According to Eq. (3.1.6) we decompose ~) in the sum of operators with a definite xdegree
String Center of Mass Operator and B R S T C o h o m o l o g y
s
[
":4
E1
]
115
E2 = E ~
s
" " " : .. . ~ ' . " 9"." i ~.." '.. i
:
O..-._ ':);
1
I -2
-1
9g ~ t ~
.:
I
0
, r
-2
--1
I
0
~r
Fig. 2. Spectralsequencefor Vp
~) = 00 -I- 01 -I- ~2,
(3.1.7)
where each On reduces the x-degree, or increases r, by n (see Fig. 1). Now we start building the spectral sequence of the complex (Vp, Q). For a short review of the method see Appendix A. The first step, the calculation of E~ '8 = GrrHr+8(00, V), reduces to the calculation of the cohomology of the BRST complex. Indeed, according to Eq. (3.1.6), 00 = 1 | Q, and therefore
E,~'~ = ~c-~ | H(~+~)(Q, Vp).
(3.1.8)
As we mentioned above, the BRST complex has nontrivial cohomology only at ghost numbers one and two. Thus the space/~1 = (~/~7 '~ looks as shown in Fig. 2 (left), where shaded cells correspond to non-zero spaces. The differential dl is induced on El by 01, and acts f r o m / ~ , s t o E1 +i's as shown in Fig. 2 (left). Since there are no states below the ghost number one and above ghost number two, the cohomology of dl at ghost number one is given by its kernel: / ~ ; , 1--r =
ker dl,
(3.1.9)
and at ghost number two by the quotient of/~2~'2-r by the image of dl:
~l,2--r
~r,2--r / I m dl. = E~
(3.1.10)
We are going to show that dl establishes an isomorphism of the corresponding spaces and, therefore, the only non-empty component of/~2 is/~o,1 _~ Hi(Q, Vp) as shown in Fig. 2 (right). Consider an operator B0 = ~ | b0. This operator is well defined o n / ~ 1 i.e., it maps cohomology classes to cohomology classes. On the other hand, its anticommutator with dl is given by {dl, Bo} =p2~, (3.1.11)
116
A. Astashkevich, A. Belopolsky
where k the x-degree operator. The last equation shows that if p2 5i 0, nontrivial cohomology of dl may exist only in k = 0 subspace of/~2. Moreover, if we apply {dl,/30} = dl B0 + B0 dl to ghost number one states only the second term will survive because there are no ghost number zero states i n / ~ . Thus we conclude that up to a ~ r , l - r and ~1 ~,r+l,l--r have the diagonal matrix Bo is an inverse operator to dl. Since ~l ~?r,l--r . ~--~r+l 1--r same dimension and dl is invertible it is an isomorphism between ~2 ana/5 2 ' for any r < 0. As shown in Fig. 2 (right),/~2 contains only one non-empty component. This means that the second differential d2 and all higher are necessarily zero and the spectral sequence collapses at/~2 = / ~ . Therefore, we conclude that GrH(Q, Vp) =
Hi(Q, Vp).
(3.1.12)
The second step in our program is trivial because the spectral sequence {E~ } of the full complex Vp = C [ ~ l , . . . , Y~D-1l | Vp, (3.1.13) stabilizes at J~l and J~l = J~oc = C [ ; ~ I , . . -
,XD--1] | GrH(Q, Vv)"
(3.1.14)
This happens simply because, according to Eq. (3.1.12), GrH(~), Vp), and thus /~1, contains only ghost number one states and therefore dl and all higher differentials must vanish. Combining Eqs. (3.1.14) and (3.1.12) we obtain GrH((~, Vp) = C[:~1,..., ~D-1I |
Hi(Q, Vp).
(3.1.15)
This completes our analysis of the cohomology of the chiral extended complex for p2 5~0.
3.2. Massless states. The analysis presented above can not be applied to the light-cone, p2 = 0. We could, in principle, repeat all the arguments using ~ 9x instead of ~, where is some vector for which ( . p 5/0, to build V and this would work everywhere except at the origin of the momentum space, p = 0. Yet it is instructive to make a covariant calculation in this case. Since there is no covariant way to choose a vector ~ we can not apply our two step program. Instead we will start from scratch and build a spectral sequence for the whole module Vv = C[x~ "'" , xD-1]
@ Vp,
(3.2.1)
graded by the total x-degree. According to Eq. (3.1), we can decompose ~) into a sum of operators of definite x-degree ~) = 00 + 01 + 02, (3.2.2) where 0o = I | i 0 o2 =
1
n [] |
(3.2.3)
String Center of Mass Operator and BRST Cohomology
117
Table 1. Chiral BRST cohomology at p = 0 Ghost # 3
Representatives Cl COC--1 IO>
1
2
CoOt~110) o~_ll0) IO)
D D 1
1 o
Dimension
The first step is to find cohomology of 00, which is just the tensor product of the BRST cohomology H(Q, Vp) with the space of polynomials
E1 = H(Oo, Vp) = C[x ~
X D-1 ] ~
H(Q, Vp).
(3.2.4)
Multiplying the representatives of H(Q, Vp) by arbitrary polynomials in x we obtain the following representatives of E1 cohomology classes
P u(x) ~ ClOL~ 1 IP),
Qu(x) |
c,0cloJ~ 11P),
(3.2.5)
where P~(x) and Qu(x) are polynomials in x that satisfy the transversality condition, p~Qu(x) = puPu(x) = 0, and are not proportional to pu. These transversality conditions come from the same conditions on BRST cohomology classes at p2 = 0. The first differential acts non-trivially from ghost number one to ghost number two states according to the following formula
dl :
P~ -+ Q~ = - i f f ' o ~ P~,.
(3.2.6)
It is easy to check that the map (3.2.6) is surjective and therefore E2 '8 = 0 for r + s = 2. As expected the cohomology of the massless complex has a similar structure to that of the massive one. There are no cohomology states with ghost number two and there is an infinite tower of ghost number one states with different x-degree.
3.3. Cohomology of the zero momentum chiral complex. The zero momentum complex is exceptional. Already in the BRST cohomology we encounter additional "discrete" states at exotic ghost numbers (see refs. [10, 11]). The cohomology is one dimensional at ghost numbers zero and three and D-dimensional at ghost numbers one and two. Explicit representatives for these classes can be written as given in Table 1. Let us denote the direct sum of spaces E~ '8 with the same ghost number m = r + s by E(~): E~m) - ~ : ) E~ ' m - r . (3.3.1) r_<0
As usual, the cohomology of 00 can be written in terms of the BRST cohomology as
El m) = Hm(Oo, Vo) = C [ x ~
x D-I] @ nm(Q, Vo).
According to Table l, H~ Vo) ~ H3(Q, V0) -~ C. Therefore, E~~ and El 3) are isomorphic to the space of polynomials C[x']. Similarly, E} 1) and El 2) are isomorphic to the space of polynomial vector fields CD[xu]. Evaluating the action of 01 on the representatives (see Table 1), we get the following sequence
118
A. Astashkevich, A. Belopolsky
i_
d _~ O. ~- 5 9, / .
,, " ~
di
3
"
"
t-
2
!"..%.1
-1
....
:
-::1
"..]
1
:i:i:i 0
-3
-2
[
-1
I
0
II- - - ~ / ,
-3
-2
-1
---~T
8
i:::!:i:!:i:i:i:L i:i::::::::::::.: .:.:.:.:.:.:.;.:.:. i:i:!:!:!:i:~:~:i.
3 2
!iiiiiiilili::iiiill I
1 ::::::;:::::::::;: =====================
0
"::::::::::::::::::"
-3
-2
-1
0
---~r
Fig. 3. Spectral sequence for V0
0 ~
C[X/z] - ~
CD[x Iz] A CD[x ,u] V.~ C [ X # ] ---+ 0,
(3.3.2)
where the first nabla-operator is the gradient, which maps scalars to vectors and the second is the divergence, which does the opposite (and we have dropped an insignificant factor - i ) . It is convenient to interpret C[x u] = $2~ and CD[x u] = ~21 as a space of polynomial zero and one-forms on the Minkowski space. The first nabla-operator in Eq. (3.3.2) will be interpreted as an exterior derivative d and the second as its Hodge conjugate 5. With new notation we rewrite the sequence in Eq. (3.3.2) as
O ~ g2~ ~ g21 o s~l ~ s2o ~ 0 .
(3.3.3)
This is illustrated by Fig. 3 (left). Note that individual cells in Fig. 3 correspond to the subspaces of homogeneous polynomials rather than whole Ol or ~2~ One can easily calculate cohomology of this complex and obtain E (0) E!I) = = C , .Q1/d~Q~
E!2) = ker(5, D1),
(3.3.4)
E ~ 3) = 0.
Thus the second term in the spectral sequence has the structure presented by Fig. 3 (middle).
String Center of Mass Operator and BRST Cohomology
119
Now we have to calculate d2. This differential acts as shown in Fig. 3 (middle) and can be found from the following formula (see Appendix A): d2 = 02 -- 0100101 -
(3.3.5)
Let P = Pudx ~ E Y21 be a polynomial one form. The corresponding state representing a E~ 1~ cohomology class is given by Pu | OaU-tl0) 9
(3.3.6)
Applying 0t to (3.3.6) we obtain
oP.
O1Pl~ @ oJz_lCllO) = - i - -
Ox u
1 oP.
N C_lQ]O) = OO i - - -
| colO),
20x~
(3.3.7)
where we use the metric rlu" to raise and lower indices. From Eq. (3.3.7) we derive that Ot0~
i OF u
1 02Pt,
P " | c~-1c110)=at 20~Xp @ ('4)10)- 20x,Ox~,
|
~,lclcolO)" _
(3.3.8)
With the definition of 02 (see Eq. (3.2.3)) we immediately get 1
02 P , | o~u_lcl lO) = ~ [] PuclcolO).
(3.3.9)
By adding the last two equations, and dropping an insignificant factor 1/2, we see that the second differential acts on E~ 1) = C D [ x u ] / V C [ x u] as the following matrix differential operator: /.I /J (d2)~ = []6 u - Ot,Ov, (3.3.10) or, using the differential form interpretation E~21) = Y21/dY2 ~ d2 = [ ] - d ~ = ~d.
(3.3.11)
In order to calculate /?;3 = H(d2, F,z) we have to calculate the cohomology of the following complex:
0 ~ C o y21/dg?O ~ ker(6, Y2~) ~ 0,
(3.3.12)
0 --~ g2~ d j21 ~ F2t ~ /2 0 --~ 0.
(3.3.13)
or equivalently The later sequence is known to be exact in the last two terms 3 and thus E3 is given by E~~ = C, E~ t) = ker(Sd, Y21)/d~ ~
(3.3.14)
E~ 2) -- E~ 3) = 0.
Looking at Fig. 3 (right) one can easily deduce that all differentials dk for k _> 3 vanish. This allows us to conclude that ^ V0) ^ = E ~ s = E 3r,8 , Gr~Hr+8(Q,
(3.3.15)
and G r H has the structure shown in Fig. 3 (right). Since cohomology is non-trivial only at orie ghost number for every x-degree, we can easily find the dimensions of different G r r H spaces. We present the summary of our results on the cohomology of the chiral extended complex in the following theorem. 3 We would like to thank Jeffrey Goldstone for pointing this out to us.
120
A. Astashkevich,A. Belopolsky
Theorem 1. The cohomology of the chiral extended complex Vp admits a natural filtration (by the minus x-degree) :F~H ( Vp, Q ). The cohomology can be described using the associated graded spaces Gr~H(Vp, Q) = J%H(Vp, Q)IJ%+IH(Vp, Q) as follows: 1. Non-zero momentum 4 (I9 ~ 0): cohomology is trivial unless p2 = n - 1, where n is a non-negative integer. In the latter case dimGrrHl(~rp,~,) = ( 2 4 - r )
\
24 ) d ~ ,
dim HZ(Vp, ~)) = 0
for l r 1,
where dn denotes the number of the BRST states at mass level n. These numbers are generated by O~
0r
I - I ( 1 - zk)-24-~ Z k=l
dnzn ,
n=l
see ref [2].
. Zero momentum (p ~ 0): the cohomology appears at ghost number zero (the vacuum) dim H~
Q) = l,
and ghost number one (physical states, recall that r is minus x-degree).
0
/fr = 0,
D(D - 1) 2 dim GrrHl(Vp, ~)) = { D(D - 2)(D + 2) 3 Xr
for r = - 1 , for r = - 2 , for r <_ -3,
where
The cohomology is trivial for all the other ghost numbers
dim Ht(Vp, ~)) = 0 a We are usingthe normalizationof ref. [2] with a' = 1.
f o r l 4 0 , 1.
String Center of Mass Operator and BRST Cohomology
121
4. Cohomology of Full Extended Complex In this section we will calculate the semi-relative cohomology of the full extended complex. We will use the same technique as for the chiral complex and obtain similar results.
4.1. Review of semi-relative BRST cohomology. By definition, the semi-relative cohomology 7-/s consists of Q invariant states annihilated by b0 - b0, modulo QIA) where IA) is annihilated by b0 - b0. For future convenience we set b0~ = b0 • b0 and Co i = (1/2)(co • ~o). Semi-relative cohomology ~ s of the BRST complex can be easily expressed in terms of relative cohomology :HR- The latter consists of the Q invariant states annihilated both by bo and bo modulo QIA), where [A) is also annihilated by bo and bo separately. We can express ~ s in terms of ~ R using the following exact sequence (see ref. [12]): 9 -. --+ 7-/~ ~ - > 7-/~ b~> ~t./~--I {Q,~} ~t.~+l
i ) ~.~+1 ~
.-,.
(4.1.1)
The map i is induced in the cohomology by the natural embedding of the relative complex into the absolute complex. The relative cohomology can be found as a tensor product of the relative cohomologies of the left and right chiral sectors. This is particularly easy for the case p ~ 0 because in this case the chiral relative cohomology is non-zero only at ghost number one (see [10]) and, therefore, 7-(R consists of ghost number two states only. The exact sequence (4.1.1) reduces in this case to 0 ~ 7-I2 b ; 7_12 ~ 0
and
0 ---*7-[2
i
7_12 ---+0,
(4.1.2)
which means that 7-I2 is isomorphic to 7-/~ which in turn is isomorphic to 7-[2 and there are no semi-relative cohomology states at ghost numbers other than two or three. We can even write explicit representatives in terms of dimension (1, 1) primary matter states. Let Iv, p) denote such a state. Representatives of semi-relative cohomology classes can be written as C1C1 IV,p) and C~Cl-C 1Iv, p). (4.1.3)
4.2. Semi-relative cohomology of p ~ 0 extended complex. The calculation of the extended cohomology for the semi-relative complex in the case of p ~ 0 is line by line parallel to that of the chiral one. When we add polynomials of :~ = (p 9 x) the resulting cohomology is given by ghost number two BRST states only, and the semirelative cohomology of the full extended complex is obtained by adding polynomials of transverse components of x: Or 7-/s(Q, Vp) = C[Y~l, ... , YcD-1 ] | 7-/2.
(4.2.4)
The case of the massless (p2 = 0) complex may require some special consideration because there is no straightforward way to choose the transverse variables ~i but the answer is the same. In our opinion, an important result is that the extended cohomology does not contain ghost number three states. Let us explain in more detail what happens to the ghost number three semi-relative states when we extend the complex by x0. According to the results on extended cohomology for every state C~ClE1Iv, p) we should be able to
122
A. Astashkevich,A. Belopolsky
Table 2. Cohomologyof the relative BRST complex at p = 0 Ghost # 0
Representatives 10)
Dimension 1 ClC~Z110), ~1~ 110) 2D elO~l~l~Vl]0), cle_ll0), ~l~_t)10) D2+2 Cla~l~l~_l[0), tic l~l~Ull0) 2D ClC-l~lC ll0) 1
1
2
3 4
find a vector ]w,p) in the extended complex such that c~cl~l Iv,p) = Q,lw,p). We have found the leading part of Iw, t9) when we calculated the spectral sequence. It is given by 5 ~ c 1 ~ 1 Iv,p). Applying Q to this state we get
~
ClCIIV, p) = C~ClC1Iv, P) + ~ PlzC1-GI(C--nOZ~+ -C-n-~)I v, P)"
(4.2.5)
n
Thus in order to prove that c~ Cl~l Iv, P) is trivial we have to prove that the sum in the righthand side represents a ~) exact state. The latter is a direct consequence of the absence of relative cohomology of ghost number three. Indeed, this sum is annihilated by b0, b0, and Q and if it were nontrivial it would have represented a non-zero cohomology class of ghost number three in the relative complex.
4.3. Semi-relative cohomology of p = 0 extended complex. The calculation of the cohomology of the zero-momentum extended complex is based on the same ideas that we used for the chiral case but is technically more difficult. T h e major complication is due to a much larger BRST cohomology. We can find the BRST cohomology of the zero momentum semi-relative complex using the long exact sequence we mentioned above (see Eq. (4.1.1)) which in this case breaks in to the following exact sequences: 0 ____~7_/o
i
+
no
i
+
n4
, n4
b0 + 0,
"
b0
(4.3.1)
+
0,
The cohomology of the relative complex can be found as a tensor product of the left and right relative cohomologies. The later can be found, for example, in ref. [10] and consists of ghost number zero, one, and two states. Therefore, the relative complex has non-trivial cohomologies at ghost numbers from zero through four and their representatives and dimensions are listed in Table 2. Using the exact sequences (4.3.1) we can easily find the dimensions and representatives of the semi-relative cohomology as shown in Table 3. The extended module V0 is given by 5 This is not applicable to the massless case p2 = 0 which has to be analyzed separately.
String Center of Mass Operator and BRST Cohomology
123
Table 3. BRST cohomology of the semi-relative complex at p = 0 Ghost # 0
Representatives ]0)
1 2 3
4
ClC~_l [0), ~1~_1 [0) clc~_l~l~ll0),
(c~C_l - ~ _ l ) 1 0 ) ~l~-,)]0) ), C~ClC_l~l~_l]O)
C~C10~I~I~U_110>,C~(ClC--1 c~r
5
Dimension 1 2D D ~+ 1 D2+ 1 2D
C~ClC--ICl'C--I[O)
Vo = C [ x ~ . . . , x
1
D-l] @Yo,
(4.3.2)
and the operator Q has the following z-degree decomposition (4.3.3)
~) = 00 + 01 "k"02,
where Oo = l @ Q , 0 01 = -- i - h ~ , @ ~(CnC~_n + -~n'~_n), n
(4.3.4)
1
02 = --'~ [] @(Co + ~0) = -- [] @ C+.
As it was the case for the chiral complex, the cohomology of O0 coincides with the cohomology of the semirelative BRST module, 7-/s tensored with C [ x ~ x D-l] and the first term in the spectral sequence is given by E1 = Hs(Oo, Vo) = C[x~
x D-l] @ 7-/s.
(4.3.5)
Using information from Table 3, we can parameterize the space E1 = C [x 0 . . . x D- 1] @~s as follows: (R t~ = R E~ | 10), (~'~
-t't, ) =
@
+Pt, @Cl~-ll0)'
((9t2~ -~ttu~ RE2j) = QI21 tzu | ClOd' -- ~1~"-- 110) + R E21| ( o c - ~ - ~1~-1)10), (o~3~ R % = Q ~ | ~ 1 ~ " 1~1~~ 1IO) + R ~ | c~(q~_t - ~ - ~ ) 1 o ) , (/9[4] ~[4]. @ / _ . ) = p#[4] @ C;ClOlg_1~1~_110 ) + /-~[4] -,
(R ts~) = R ~~ @ c~c~c_~1~_110),
C~ClC_l_~l.~__l]O), (4.3.6)
where Qt,~, p , and Pu and R are correspondingly tensor-, vector-, and scalar-valued polynomials of D variables x". Applying 01 to the representatives given by (4.3.6) and dropping Q-trivial states we can find that the first differential acts according to the following diagram
A. Astashkevich, A. Belopolsky
124
t_
~:i:i:i:!:i:i:i:!:
4
::::::::::::' ..........-.....
:~:~:i:i:~:i:i:!:! iiiiiilili!!!!!!!!
3
::::::::::::::5::
2
: :iI D.~ 9 9 :.::
1
-
~,iiiii,i:i:
i~ii~iiiiiili!iiii i.!.;i.!.!.i.i.i.
--
0 -4
-3
-2
-1
iiiiii iiill -4
0 ---~-r
--3
--2
--1
0 --~r
Fig. 4. Spectral sequence for the zero momentum closed string states.
0:
(R t~ @ (0) -~[t].
1 : (p;I], --/z ) ~ 1
2 : (0[2 . . . ~ . ],
N R t2]) @
(0/~ R[0}, 0#/~[0]) ._ ---~[l]- Ovp~ll ' O~P~ --[11 - OuP~1]) to~Y~ (4.3.7)
N
3: ,(O[31 ~ u - , R[3])
(0, 0)
4 : ,(/O[4] u ' ~[4]. /~u ) ~ l
(~utO[3] , v -~uv +
0uR[3] ' -~u(~[3] -~v~ + 0t~Rt3])
"X 5 :
( R [5])
( 0 t t P t41 - 0/.tP [4])
Consider the cohomology of dl. From the explicit formulae (4.3.7) it is clear that E2 = H(dl, Et) contains no ghost number five states and the only state that survives at ghost number zero is given by a constant R t~ and is the vacuum. It is less trivial to show that E~4) = 0, and that E~ 1) contains only x-degree one and two states. We will prove these two results in Appendix B (Lemmas B.3 and B.1). This shows that the space E2 looks as shown in Fig. 4 (left). From the structure of E2 we conclude that d2 may only act from E~2) to E(23). In order to find -[21 R[21) = (02 - 0 1 0 0 1 0 1 ) (Q~]u, R[2]), d2 ((o~ttu, we have to keep the 00 trivial terms when we apply 01 to (otzl dropped in Eq. (4.3.7). Using (4.3.6) and (4.3.4) we infer
R [21) which were
[2] + O~,Rtz]) | c$el~-ll~ tot2] Rt2]) = 00 ((0 u Qt,. 01 , - ~ , ,
- ( 0 • O tat + 0 ~ R [2]) | c~cta" 1[0)).
(4.3.8)
N o w we can apply 01 to the argument of 00 above and add the image of 02 to obtain [2] , R [2]) = (t~O d2(Qu~, , v , ~ ,[3] ,
where
5R[31),
50[3] ~[21 _ CQX0v'~t')' (-)[21 + 20~'0~'R[21' -*t,v = [7(7)[2] ~t,u - 0 k 0~),~, _ ,..Q..k ,~p/-) [2] .a.. R[21. 5R[3]= '-" ~ ' ~ . X p - 2 []
(4.3.9)
(4.3.10)
String Center of Mass Operator and BRST Cohomology
125
As we will prove in Appendix B, this map is surjective and therefore the cohomology at ghost number three is trivial. The structure of E3 is presented in Fig. 4 (right). It is obvious that the spectral sequence collapses at E3 and E ~ = E3. Therefore, we obtain the result that is similar to that in the chiral case (see Eq. (3.3.15)): A
A ~'~8
Gr~Hr+S(Q, V0) = E/L~ = E 3 9
(4.3.11)
Note that d2 acts non-trivially only between ghost number two and ghost number three states, at the point where dl vanishes. This observation allows us to combine the calculation of E2 and E3 into one cohomology problem for the following complex 0 --+ V (0) ~
V (1) d,) V(2)
d2) V(3)
d, V(4) d,
V(5) --+ O,
(4.3.12)
pr,k-r are the ghost number k subspaces of Et. This is very similar where V (k) = , , ~ ~1 to what we did for the chiral complex (Eq. (3.3.12) and Eq. (3.3.13)), but, unfortunately, we lack a simple geometrical interpretation for the complex above. Since dl and d2 have x-degree - 1 and - 2 correspondingly, we can decompose the complex above into the sum of the following subcomplexes 0 ---~ 17(0) " n+2
d, ) 17(1) 9 n+l
d, ) V(2)
d2 ) "rr(3) dl V(4) Vn--2 ~ n--3
dl
IF(5) O, ~ "n--4 --'+
(4.3.13)
where the subscripts refer to x-degree. Using this decomposition and the results of Appendix B one can easily calculate the dimensions of Gr~Hs spaces. Let us summarize our results on the semi-relative cohomology of the extended complex. Theorem 2. The semi-relative cohomology of the extended complex Vp, admits a natural filtration (by the minus x-degree) ~ H s( Vp , Q ). The cohomology can be described using the associated graded spaces A
A
A
A
A
A
Grr Hs(VB, Q) = ~r Hs(Vp, Q ) /-
24
dim H t (Vp, Q,) = 0
2 dn'
for 1 5~2,
where dn are generated by the following partition function C<3
O0
1--~(1-- Zk)-24= Z dnzn , k=l n=l see ref [21.
126
A. Astashkevich,A. Belopolsky
2. Zero momentum case (p =- 0): the cohomology appears at ghost number zero (the vacuum) dim Hs~(Vp, ~)) = 1, ghost number one (global symmetries of the background, the Poincard algebra), D
ifr = O;
dim Gr~H~(Vp, Q) = { D ( D - 1) 2 0
ifr = - 1 ; otherwise;
and ghost number two (physical states),
dimGroH)(Vp,~" Q,) = 1, dim G r - I H 2'G,stvpQ,) = D (D: - 3 D + 8) 6 dim Gr - 2 H 2"#,stvpQ,) = D(D - 2)(D + 2) 3 Or-3 s2"~,stvp Q) = (D + 2) (5 D 3 - 16 D 2 + 15 D - 12) dim 24 dimGrrH~(Vp,s = xr for r < - 4 , where Xr = ( D Z + I ) [ ( D D ! I T ) - ( D D 3 1 r ) ]
+2D [ ( D o _ r 1 4) - ( O -
~)]
(D-r+I)-(DDrl5). +\ D-1 There are no non-trivial cohomology states at any other ghost number,
dim H~(Vp, s
=0
for 1 4 O, 1,2.
There is no evident structure among the ghost number two cohomology states. In the next section we are going to make some order in this zoo of zero momentum physical states using their transformation properties under the Lorentz group. 5. Lorentz Group and Extended Complex Since the cohomology of the extended complex is designed to describe the string theory near the flat background, the Lorentz group should act on it naturally, mapping the cohomology states to the cohomology states. In this section we investigate the action of the generators j u , of the infinitesimal Lorentz transformations. 5.1. Lorentz group acting on non-zero momentum extended complex. The generators of the infinitesimal Lorentz transformations, or the angular momentum operators of the closed string can be written as [2]
String Center of Mass Operator and BRST Cohomology
127
j u v = lUV + Evv + ~ u v
(5.1.1)
where i/zv
/z v v /z = X 00L 0 - - X 00~ 0 , o~
E vv = - i Z n=l
n (OL--nOLn --
OL--nOLn)' (5.1.2)
o~
-"
=
-"
-- OL nOLn). n=l
Applying the zero mode part l ~'" to Vp, we obtain
=x v|174
v~x. |
0
Xv
| 1.
(5.1.3)
One can easily check that the J ~ " operators commute with 0 and therefore are well defined on the cohomology. Note that for p ~ 0 these operators mix states of different x-degree. Consider the massive case, p2 ~ 0. Recall {hat any physical state in the extended complex can be represented by p(~l,...
, ~ T D - 1 ) @ Cl'Cl JV, p ) + ' ' "
,
(5.1.4)
where Iv, p) is a dimension (1, 1) primary matter state with the momentum p and dots stand for the lower x-degree terms. Suppose the transverse components are chosen as 5Ci = X i
pi(p. x) p2
(5.1.5)
We can obtain a state with the same leading term by applying the following operator to r p): p (p.jlu \ I )2
,"
,
pujD-lu'~ -~ ] r
)=P(~l,...,ScD-1)|
" . (5.1.6)
Together with the results on the cohomology of the extended complex obtained in the previous section, Eq. (4.2.4), this shows that the semi-relative cohomology of a p2 5r 0 extended complex is spanned by the states which are generated from the standard physical states by the infinitesimal Lorentz boosts. Although the analysis above fails for the massless states p2 = 0 (39 ~ 0), the result is still the same. One can check it using the explicit formulae for the cohomology states at p2 = 0. We conclude that the ghost number two cohomology of the extended complex at p ~ 0 has the same physical contents as that of the BRST complex. 5.2. Lorentz group and the zero momentum states. The Lorentz generators, J~", act on the zero-momentum states as linear operators of zero x-degree:
J~=x " ~ | Consider the complexes
0
@ 1 + 1 @ (E ~'" + ~F,~).
(5.2.1)
128
A. Astashkevich, A. Belopolsky
Table 4. Decomposition of the complex (5.2.2) into irreducible representations of SO(D -- 1, 1) for n > 2
Hn+ 2
2 Hn+ 2
Hn+ 2
Hn
4 Hn +2 Vn
5 Hn +Sn +An +2 Vn
Hn
Hn-2
4Hn-2+2Vn-2
Hn-4
4 H n - 4 + 2 Vn-4
6Hn-2+Sn-2 +An_2+4 Vn_ 2 6Hn-4+Sn-4 +An-4 +4 V n _ 4
5Hn-2+Sn-2 +An_2+2 Vn_ 2 6Hn-4+Sn-4 +An-4 +4 Vn-4
Hn-6
4 H n - 6 + 2 Vn--6
6Hn-6+Sn_ 6 +An_6+4Vn_ 6
6Hn-6+Sn_ 6 +An-6+4 V n - 6
0 ~ Lr(0) ~ - ~ V(1) 9 n+2
9 n+l
dl
> w(2)
d2> TT(3) dl Vr~--2 ~ If(a)" n-3
2Hn-2
dl
2 H n - 4 + 2 Vn--4
Hn-4
2Hn_6+2Vn_ 6
Hn-6
> ~I(5)" n - 4 ~ 0,
(5.2.2)
which calculate GrHs(V0, ~)). By definition V~~ and V~5) are the spaces of homogeneous polynomials of degree n. These spaces are reducible under the Lorentz group because the subspaces of the polynomials of the form (xUx~,)khn_2k are invariant under S O ( D 1, 1). Furthermore, if hn-2k are harmonic, []hn-2k = 0, these subspaces form irreducible representations of S O ( D - 1, 1). We will denote these irreducible representations by Hn. These representations can be alternatively described by Young tableaux as n #,
Hn---i I I " I i,
(5.2.3)
Now we can write the decomposition of V~~ or V~5) into irreducible representations as V~~ = V~5) = H,~ + I-In_2 + H,~-4 + . - . 9
(5.2.4)
At the ghost numbers one and four spaces we find another kind of irreducible representations: n
vn=i I I II
I] t,
(5.2.5)
and, finally in the decompositions of V(2) and V~3) we will encounter n
An=U I'"1 ] and Sn--
n
(5.2.6)
Suppose n _> 2. Table 4 shows the decomposition of the whole complex (5.2.2) into irreducible representations. From the series of lemmas presented in Appendix B, we know that the complex (5.2.2) has cohomology only in V~2). Using Table 4 we conclude that Gr-nH2(V0, ~)) = Ha + An + Sn, (5.2.7) which we obtain by "subtracting" the odd columns from column two and "adding" the even columns. A more detailed analysis shows that if we choose the representatives of
String Center of Mass Operator and BRST Cohomology
129
Gr_~H2(V0, ~)) so that they belong to Hn, An, or Sn, they also will represent cohomology classes of Q, no lower z-degree corrections required. Two exceptional cases, n = 0 and n = 1, have to be treated separately. The decompositions of the complex (5.2.2) into irreducible representations for these the first case is presented in Table 5. Using the results of Appendix B we can infer from Table 5 (left) Table 5. Decomposition of the complex (5.2.2) into irreducible representations for n = 0 (left) and n = 1 (right) V(0)
V(1)
Vo(2)
V3(0)
V2(1)
V(2)
H2 Ho
2 H2 2Ho + 2Vo
H2 Ho+Vo
H3 H1
2 H3 4Hi +2V1
H3 4H1 +A1 +2V1
that Gr_lH~(~), V0) = V0
and
Gr0H~(O, V0)= H0,
(5.2.8)
and from Table 5 (right) that G r - l H 2 ( Q), V0) = Ht + A1;
(5.2.9)
and again, if we pick the representatives of GrHs from the irreducible representations, they will be annihilated by Q and therefore represent the cohomology classes without lower x-degree corrections. For this case this can be easily checked by explicit calculation (see Appendix B). It is tempting to interpret the irreducible representations I-In, Sn, and A,~ as the dilaton, graviton, and antisymmetric tensor. If we do so, it is not quite clear why we have infinitely many irreducible representations for each field, and not just one. We speculate that these representations are related by infinitesimal shifts (the translational part of the Poincar6 algebra), which acts on the spaces of polynomials by differentiation with respect to x ~.
Appendix A. Spectral Sequence In this section we review some basic facts about a particular type of the spectral sequence which we use in our analysis of the extended complex. This is not intended to be a complete introduction to the method. Our only goal is to introduce the spaces E0, E1 and E2 equipped with differentials do, dl and d2 acting on them. We will prove that these differentials have zero square and provide some motivations to why their cohomologies are related to G r H . For a more detailed analysis of the first three terms of a spectral sequence, the reader is referred to the book by Dubrovin, Fomenko and Novikov [13]. A general introduction to the spectral sequences from the physicist's point of view and further references can be found in refs. [14, 15]. Let (C, d) be a complex with additional grading C = ~ Cr such that the differential d can be written as d = 00 + 01 + 02, (A.1) where On maps Cr to C r+,,. Since d mixes vectors from different gradings we can not define grading on cohomology H(d, C), but we can still define a decreasing filtration. By filtration of the element x E H(d, GrC) we will mean the smallest (negative) integer s such that x is representable by a cocycle
130
A. Astashkevich, A. Belopolsky
3 = x~ +xr+l + ' " ,
(A.2)
where xr c C~. We will denote the space of such vectors by FrH(d, C). Using the filtration we can define a graded space associated to the cohomology H(d, C) GrH = GGrsH'
where
G r s H = F~H/F~+IH.
(A.3)
8
The investigation of the spaces Gr~H '~ is carried out using the method of "successive approximations" based on what is called the "spectral sequence". The idea is to construct a sequence of complexes (E~, d~) such that En+l = H(d,~, En) which converges to GrH, 8~n--8 GrsH n = Eoo .
(A.4)
Differentials dn are acting on the spaces E~ '~ as follows: dn :
l~r+n,s-- n+ l
E~ '~ ~ --n
-
(1.5)
For a complete description of the spectral sequence and the proof of the theorem which states that the spectral sequence converges to G r H we refer to [13, 16]. Let us describe the first few terms of the spectral sequence. Suppose 3 given in Eq. (A.2) represent a cohomology class x in FsH(d, C). Applying d = 00 + 01 + 02 to 3 we obtain 6 X ----003~o + ( 0 1 3 r + 0 0 3 r + l ) + ( 0 2 3 r + 0 1 3 r + 1 + 003r+2) -I-(02Xr+ 1 -I- 013r+2 -1- 00Xr+3) -I- 9 9 9 ,
(A.6)
where we enclosed in braces the terms from the same Cr space. It follows that 003~ = 0,
013~ = -003~+1,
023~ = -013~+l - 003~+2,...,
(A.7)
from which we conclude that 3~ is a 00 cocycle and a 01 cocycle modulo image of 00. This suggests that the first approximation in the spectral sequence should be Ej = H(Oo, C) and do. = 00, the second approximation is 1772 = H(dl, El), where dl is induced on E1 = H(O0, C) by 01. Usin~ the second equation of (A.7) we can formally find 37.+1 in terms o f ~ as 3~+1 = - 0 o 013~ and rewrite the last equation of (A.7) as (02 -- O l O 0 1 0 1 ) 3 r ---- --00X~,+2.
(A.S)
The last formulae suggests that d2 is induced on E2 = H(dl, El) by 02 - 0100 101 and the third approximation is E3 = H(d2, E2). Let us show that these differentials are well defined and square to zero. For do the first is obvious since it acts on the same space as 0o and 0o2 = 0 follows from d 2 - 0 which is equivalent to 02 ---- {00, 01} ~. 02 -I- {00, 02} ----0~ -----0.
(A.9)
In order to show that dl is well defined we have to show that 01 maps 0o-closed vectors to 0o-closed vectors and 0o-trivial to 0o-trivial. This easily follows from the anticommutation relation {00, 01} = 0. Let us prove that d~ = 0. Suppose x E E1 can be represented by a cocycle 3 c C, 00~ = 0. Then applying 012 = - { 0 0 , 02} to 3 we obtain a trivial cocycle 023 = -00023. In cohomology this implies that dZx = O. Before we consider the differential dz, let us describe the space E 2 o n which it acts in greater detail. By definition E2 = H(dl, El), but E1 in turn is the cohomology of the original complex with respect to 0o. Therefore, in order to find a dl cocycle we should
String Center of Mass Operator and BRST Cohornology
131
start with a Oo cocycle N and require that its image under 01 is O0 exact. Two cocycles and ~ represent the same dl cohomology class if U - ~ E I m Oo + O1ker 00. Let ~ be a dl cocycle, which means that 00~ = 0
and
01~ = 0o~.
(A.10)
We define d2x as dzN = 02~ -- 0 1 y = (02 -- 0 1 0 o l 0 1 ) x .
(A.11)
Let us show that the result is again a dl cocycle. Indeed, using the properties of On listed in Eq. (A.9) we obtain
Oodz-x =
0002x + 0001y = 0002X -- 0100y = (00, 02}X -- 0~lX ----0
and 01d2~ = 0102~ - 02~ = 0o02~.
Similarly we can prove that if ~ and ~' belong to the same dl cohomology class then their d2 images belong to the same class as well. This will finally establish the correctness of the definition of d2 as an operator on E2. In conclusion let us show that d 2 = 0. With ~ as above we have d~2x-- 02(02~ - 01y ) - 0100101(02 ~ - 01y )
= 0102y + 0 1 0 o 1 0 2 0 1 ~
- 010oOo02y - 0~0o O20o~
= 0102y -- 0102y ----0,
which completes our analysis of d2.
Appendix B. Three Lemmas In this appendix we will calculate the cohomology of the following complex 0 --+ V (0) d(~ V (1) d(l)) V (2) a~2)) V (3) d~ V (4) d(4)) V (5) --~ 0,
(B.1)
where V (5) -~ V (1) -~ C[x 4. " ' x D - - 1 ] , V (2) "" V (3) "~ c 2 D [ x 0" " ' x D - l ] and V (~ --~ V (0) ~ cD2+I [Z 0"" 937.D - 1]. Following the notations of Sect. 4 we represent the elements (O[21 R[21~ of V (~ and V (5) as (R f~ and (RD]), elements of V (2) and V (3) as ,-~uu, , and Or31 RD]), and elements of V (1) and V (4) by (P~, Pu)- Differentials d (n) act as follows (R[~
(0)
tp[l] '. # ~ ~[1]~ # / ~ aA("~tO '" '. # R[0] ~ 0 # R [01~ / ( Q ~ ] , /~[2]).d(2~'~(0/~P[1] - our [I,, Op.p[llz, - 0,aN[l]) (O[31 R[3]'1 j")/~ (t~/O[31 ~R[31"I tD[4]
-~[4]~ ~(4--'~3-~t,qut')[3]+ ,0
(Risl) where
R[31 0~O[31 + 0•RI3I)
-.....~(Ot~p[41_ oupt
1),
(B.2)
132
A. Astashkevich, A. Belopolsky 6~013]_F10121 _ .qA,q g)[21 "*/*u----"-,~p,v
,a
.qA.q /3121
~,l.,,,,~,~u--,..*
t.,u,~p,A
+20.0~,R[21
(8.3)
~/:?[31 [21 + 2 [-1 R[21. . . . _ ._ .,qA Do r,~Ap
(B.4)
It is obvious that d (4) is surjective and the kernel of d (~ contains only constant polynomials, Thus we conclude that H ~ = C and H 5 = 0. The other cohomology spaces are described by the following lemmas
H 1 is finite dimensional and dim H 1 ---- D(D+I) H 1 cohomology classes 2 " can be represented by polynomials of degree no bigger than one. L e m m a B.1.
Proof According to (B.2) H 1 is a quotient of the space S of solutions to the system of first order differential equations
OuP~, = OvPj, OuPu = Ou~.
(B.5)
by the space T of trivial solutions Pu = P u = OuR" Note that both S and T naturally decompose into a direct sum of the spaces of homogeneous polynomials and so does the quotient
S=(~S
'~,
T=t~T
n,
H'=S/T=t~S~/T
n.
(B.6)
I
We want to prove that S n = T '~ for n > 1. Let Pu and Pu be homogeneous polynomials of degree n > 1 that satisfy Eq. (B.5). First, it is obvious that Pu = 0 for every # requires P u = 0. Indeed, if P~, = 0 for every # then according to the first equation in Eq. (B.5) O,,P u = 0 for every/z and v and since by assumption deg P u > 1 this means P u = 0. Second, using the first equation of (B.5) twice we obtain
O~O~Pu = O~OuP, = OuO~P. = OuO,P~.
(B.7)
Therefore for any a , / z and v, o ~ ( o ~ P u - O u P ~ ) = o.
(B.8)
And since deg(0,~Pu - OuP~) = n - 1 > 0 we conclude that O~Pu - OuP~ = 0 and there exist R such that Pu = OuR. Subtracting a trivial solution (OUR, OuR) from (Pu, P u ) we get another solution (Ps = 0, --'P, = flu - OUR)" According to our first observation Ps = 0 requires PJu = 0 and thus P u = Pu = OuR. It is easy to see that there are exactly D(_~- ~____non-trivial ) solutions of degree one and D non-trivial constant solutions which can be written as Pu = - P ,
= ~[,.lx ",
and
where ~I~-] is an antisymmetric tensor.
[]
L e m m a B . 2 . H 3 = 0.
Pu = - P ~ = const,
(B.9)
String Center of Mass Operator and BRST Cohomology
133
Proof This is the most difficult lemma in this work. We have to show that any solution of the system
OvtO[3] + 0 ~ R [3] = 0 7 (B.10)
v(9[31 + OUR[3] = O, "~ t.'/z can be represented in the form
~O[3] ()[2] -~uv = DO[2] -*tzu _ OAOizQ~] -- 0A0 u'~ u)~ + 20tz Ov R[2] , 5 R [31 = -O)'OpQ~lp + 2 [] R E21.
(B.1 1)
We will start from an arbitrary solution of the system (B.IO) and will be modifying it step by step by adding the trivial solutions of the form (B. 1 1) in order to get zero. We can use the same arguments as in Lemma B. 1 to consider only homogeneous polynomials of some degree m. Suppose m _> 1. We will describe an iterative procedure which will allow us to modify "'~u~'(O[31R[31) so that Q ~ will depend only on one variable, say x ~ and its only nonzero components be '3[3] '~ 0 and r313] "%0v" If this is the case, the cocycle condition (E-~q. (B. 10)) tells us that
O,R = OoQu,o = OoQo,u,
(B.12)
moreover, Qm0 = Cux~ and Qo,u = Cux~. Furthermore, using Eq. (B. 12) we conclude that C u = Cu. Integrating Eq. (B.12) we obtain
D--1 R
=
mCixix
g -1
+ Coxg 6.
(B.I3)
i=l One can check that such solution ((9 [31 R 13]) can be written in the form (B.1 1) with Q[2] = 0 and
~ 1 ( m - - ~ l xix~+l + R[21 = 21 i=1
Co
(m + 1)(m +
2) x~n+2 )
"
Now let us describe the procedure which reduces any solution to the abovementioned form. Our first objective is to get rid of xi dependence for i = 1..D - 1. Let us pick i. The following four step algorithm will make (Qua, R) independent of xi. We will see that when we apply the procedure to (Qua, R) which does not depend on some other xk it will not introduce xk dependence in the output. This observation will allow us to apply the algorithm D - 1 times and make ( Q ~ , R) depend only on x0.
Step 1. Let us introduce some notations. For a polynomial P we will denote the minimal degree ofx~ among all the monomials in P by hi(P). For a zero polynomial we formally set ni(O) = +oe. Given a matrix of polynomials Q~,., let N i ( Q ~ ) = nfafn ni(Qm,).
(B.14)
v~ Since (9131 _ ~ , are homogeneous polynomials of degree m we can write them as (9131 ..~ l,Z l l
Z
=
mo+.
. .+m
Cmo'"mD-l,~t]Xo D _ l =m
m0 . "'XD ~D--1 1
(B.15)
[34
A. Astashkevich, A. Belopolsky
Let us show that it is possible to add a trivial solution to (Q~,., R) and increase Ni(Qu~) by one. Indeed, let m0
~mo".mD_l,tzu
(0[2] ---"~ #~'
Z-
X0
mz+2
"''32 i
~/2D
...37D_
1
I
i ~ - i 7 ~{~rt/-7----~
mo+'.'+mD- imm
0
for #, v g i, otherwise,
and R [21 = 0. It is easy to see that Nitr3[3] ~r~[3l >-- Ni(Q[3]~) ~'~v + Otg~v) # + 1, where 6Q[3~ comes from the trivial solution generated by ( 0 [21 , R i21 ) according to (B. 1 1). Repeating this procedure, we will increase N~(tr -.--.[3] at least by one every time. Since .,~,~ O [31 are homogeneous polynomials of degree m, Ni is either less than m + 1 or equal +oc. Therefore after a finite number of steps we will make Ni = +oc which means that all Q~] are zero for # ~ i and v : / i .
Step 2. Since ",(c)I3] "V ,//,/J R[31) is a solution to the system (B.10) we can write OiQ[~3)
=
f~[31 O . R [3] = Oi~dip,
# ~t i,
[31 = Oi R[31 = 0 v Q i[3] 0 v Qvi ..
Suppose R [3] =
Z
Dmo...mD_lX~O
too+'" "+roD
" ' ' X D _m D 1 t
--1=77~
then we choose R [21 = 0,
Q[2[ ii =
~
Dmo. . . .
D-1
m,+2
3gr~o
(mi + 1)(mi + 2)
""xi
mD-,
"" ' X D - I
'
mo+...+mD_l=m
alia" (og>u"~[2l= 0 for all the other # a n d u. It is easy to see that (~[3l.~/,v= (013]-~/,v+ 5(0131-~pv
have the following properties: -
-
all "~[31 ~ / A v except Qii do not depend on zi (~[31 ~,/zb' =
0 for all # • i and v ~r i.
Step 3. Suppose to) [31 /~[31~ is of the form we obtained at the end of Step 2. Since OiQ~ = 0 for # ~ i, then a ~ R [3] = 0 for # ~ i and therefore, R [3] depends only on xi. Thus there exists R [21 such that R [3] = - 2 0 2 R [2~ and R ~2l depends only on xi. Adding a trivial solution generated by (0, R ~2]) we can make R [3] = 0.
Step 4. Using R [3J = 0 we can rewrite the system (B.10) as follows. O.g)[3] -z"~ii
S -~
-- ~
O.Q~Ii "
.4i Therefore, QI~1 = Q[3lii,0T xiQl3].i,1, where QI~I,0 and Q[31ii,1 do not depend on xi. For every Q~l,l we can find a polynomial P which depends on the same set of
String Center of Mass Operator and BRST Cohomology
135
variables and [Np = _Q[3I 0 [2] = 0 for ii, 1" Choose 0[2] "vii = Xi P , R [2] = 0 and -~u~ (p, u) ~ (i, i). Adding the corresponding trivial solution we achieve that 0"~/zv [3] and R [31 do not depend on xi. Repeating this program D - 1 times for each value of i = 1, . . . , D - 1, we make 0 31 R [3]) depend only on xo. Now we can repeat the first step once again with i = 0 and make Q[3] kj = 0 for k , j = 1, .. D - 1. We have already proven that such solution is trivial. Recall that in the very beginning of our analysis we have made an assumption that the polynomials have non-zero degree (m _> 1). Therefore we have to consider this last case separately. If polynomials 0 [3] and R [31 are constant, they trivially satisfy the system (B. 10). To show that any such constant solution can be represented in the form (B. 1 1), it is sufficient to take Q[2] # u = quu,o (xo) 2/2 + quu, l(Xl) 2/2, which generates (,~[31 ~4u~ R [3]) if qua,0 and qu~,l are chosen so that
qoo,, - qoo,o=Q[~ , _f~[3] q l l , 0 -- t / l l , 0 - - t ~ l l
qoo,l + qoo,o + q11,1 + qll,0 =R[3], q0u, 1=~d0u t-~[3] for v 5/0, qu0,~=Q~l for # 7( 0, q lv,O=tOdlv ,q[31 for v 7? 1, q/zl,O--=~/zl ,.~[3] for # ~' 1 [31 for p, u > 1. qu~,,o "t- q~,,~,l=Qu,,
This completes the proof of Lemma B.2. Lemma
B.3.
H 4 =
[]
0.
Proof It is almost obvious that the image of d ~3) covers the whole kernel of d (4) in V (4) because the space V (3) is much bigger than V (4) at every degree. Indeed we will show that it is sufficient to consider a subspace of V (3) spanned by the zeroth row and the zeroth column of the matrix Qm" Loosely speaking the row will cover Pu and the column will cover Pt,. Suppose (P~, flu) E kerd (4) o r equivalently d(4)(P~, P ~ ) = (OuP~ - 0~T~) = 0.
(B.16)
We want to show that subtracting vectors of the form d(3)(Q,,, 0) from (Pu, T , ) we can reduce it to zero. First of all we can easily get rid of the spatial components Pi and Pi for i = 1 9 .. D - 1 using Q ~ which has the only non-zero components given by
Qoi=fPidx~ This will reduce according to Eq. x ~ Thus varying we have a vector
and
Qio=fff~dx~
(B.17)
P~, and P u to the form Pu = aSo,, and P u = gS0,u. Furthermore, (B.16) polynomials a and g have the same derivative with respect to say Cl(x 1 9 99x D-l) in Eq. (B.17) we can achieve that a = g. Finally if given by P~ = P ~ = aSo,u we can use Q00 to reduce it to zero. []
136
A. Astashkevich, A. Belopolsky
Acknowledgement. We would like to thank Barton Zwiebach for reading the manuscript and giving many valuable suggestions. This work much benefited from discussions with Alexander Gorohovsky and Barton Zwiebach. One of us (A.B.) would also like to acknowledge conversations with Jeffrey Goldstone, Kenneth Johnson and Paul Mende.
References 1. Belopolsky, A. and Zwiebach, B.: Who changes the coupling constant? Nucl. Phys., B472, 109-138 (1996). e-Print Archive: hep-th/9511077 2. Green, M.B., Schwarz, J.H. and Witten E.: Superstring theory. Cambridge: Cambridge University Press, 1987 3. Friedan, D., Martinec, E. and Shenker, S.: Conformal invariance, supersymmetry and string theory. Nucl. Phys. B271, 93 (1986) 4. Nelson, P.: Covariant insertion of general vertex operators. Phys. Rev. Lett. 62, 993 (1989) 5. Distler, J. and Nelson, P.: Topological couplings and contact terms in 2-D field theory. Commun. Math. Phys. 138, 273-290 (1991) 6. Zwiebach, B.: Closed string field theory: Quantum action and the B-V master equation. Nucl. Phys. B390, 33-152 (1993). e-Print Archive: h e p - t h / 9 2 0 6 0 8 4 7. Becchi, C.M., Collina, R. and Imbimbo, C.: On the semirelative condition for closed (topological) strings. Phys. Lett. B322, 79-83 (1994). e-Print Archive: h e p - t h / 9 3 1 1 0 9 7 8. Feigin, B.: The semi-infinite homology of Kac-Moody and Virasoro Lie algebras. Russian Math. Surveys 39, 155-156 (1984) 9. Fuks, D.B.: Cohomology of infinite-dimensional Lie algebras. New York: Consultants Bureau, 1986 10. Frenkel, I.B., Garland, H. and Zuckerman, G.J.: Semi-infinite cohomology and string theory. Proc. Nat. Acad. Sci. USA $3, 8442-8446 (1986) 11. Distler, J. and Nelson, P.: New discrete states of strings near a black hole. Nucl. Phys. B374, 123-155 (1992) 12. Witten, E. and Zwiebach, B.: Algebraic structures and differential geometry in 2d string theory. Nucl. Phys. B377, 55-112 (1992). e-Print Archive: h e p - t h / 9 2 0 1 0 5 6 13. Dubrovin, B.A., Fomenko, A.T. and Novikov, S.P.: Modem Geometry - Methods and Applications. Part IlL Berlin-Heidelberg-New York: Springer-Verlag, 1984 14. Figueroa-O'Farrill, J.M. and Kimura, T.: The BRST cohomology of the NSR string: Vanishing and "No-Ghost" theorems. Commun. Math. Phys. 124, 105-132 (1989) 15. Dixon, J.A.: Calculation of BRS cohomology with spectral sequences. Commun. Math. Phys. 139, 495-526 (1991) 16. McCleary, J.: User's Guide to Spectral Sequences. Berkeley, CA: Publish or Perish, Inc., 1985 Communicated by R.H. Dijkgraaf
Comnmn. Math. Phys. 186, 137-165 (1997)
Communications in
Mathematical
Physics
@ Spfinger-Veflag1997
Characteristic Cohomology of p-Form Gauge Theories Marc H e n n e a u x 1'2, Bernard K n a e p e n 1,*, Christiane S c h o m b l o n d 1 1 Facult~des Sciences, Universit6Libre de Bruxelles, Campus Plaine C.P. 231, B-1050 Bruxelles, Belgium 2 Centro de Estudios Cientificos de Santiago, Casilla 16443, Santiago 9, Chile Received:4 July 1996 / Accepted: 8 October 1996
Abstract: The characteristic cohomology H~ha,.(d) for an arbitrary set of free p-form gauge fields is explicitly worked out in all form degrees k < n - 1, where n is the spacetime dimension. It is shown that this cohomology is finite-dimensional and completely generated by the forms dual to the field strengths. The gauge invariant characteristic cohomology is also computed. The results are extended to interacting p-form gauge theories with gauge invariant interactions. Implications for the BRST cohomology are mentioned.
1. Introduction The characteristic cohomology [ 1] plays a central role in the analysis of any local field theory. The easiest way to define this cohomology, which is contained in the so-called Vinogradov C-spectral sequence [2, 3, 4], is to start with the familiar notion of conserved current. Consider a dynamical theory with field variables r (i = 1 , . . . , M ) and Lagrangian E ( r i, 0 ~ r 0 m..4,k r The field equations read Z;i = 0,
(1.1)
with ~E OF_, Ci - 8r i - 0 r i
0
" OE # ~ )
,
..4,k (
+... + (-1)k0m
0s 0(0m...~kr
).
(1.2)
A (local) conserved current j~' is a vector-density which involves the fields and their derivatives up to some finite order and which is conserved modulo the field equations, i.e., which fulfills cg.j ~' ,-~ O. (1.3)
* Aspirant du Fonds National de la Recherche Scientifique (Belgium)
138
M. Henneaux, B. Knaepen, C. Schomblond
Here and in the sequel, ~ means "equal when the equations of motion hold" or, as one also says equal "on-shell". Thus, (1.3) is equivalent to
O,j" = ; ~ i + ,~'o~,s
+ . . . + ,V~,..~sO#,...~sz:~
(1.4)
for some )~im...~j, j = 0 , . . . , s. A conserved current is said to be trivial if it can be written as
jr ~ OS#~
(1.5)
for some local antisymmetric tensor density S "~" = - S ~ . The terminology does not mean that trivial currents are devoid of physical interest, but rather, that they are easy to construct and that they are trivially conserved. Two conserved currents are said to be equivalent if they differ by a trivial one. The characteristic cohomology in degree n - 1 is defined to be the quotient space of equivalence classes of conserved currents. One assigns the degree n - I because Eqs. (1.3) and (1.5) can be rewritten as dco ..~ 0 and co ~ d~b in terms of the (n - 1)-form co and (n - 2)-form ~b respectively dual to j r and S~. One defines the characteristic cohomology in degree k (k < n) along exactly the same lines, by simply considering other values of the form degree. So, one says that a local k-form co is a cocycle of the characteristic cohomology in degree k if it is weakly closed, d~ ~ 0; "cocycle condition" (1.6) and that it is a coboundary if it is weakly exact, co ~ d~p, "coboundary condition,"
(1.7)
just as it is done for k = n - 1. For instance, the characteristic cohomology in form degree n - 2 is defined, in dual notations, as the quotient space of equivalence classes of weakly conserved antisymmetric tensors, O~S ~ ~ O, S ~ = S I ~ ,
(1.8)
where two such tensors are regarded as equivalent iff S I~u -- S 'l~u ~, O p R p#u, R pItu = R [p#u] .
(1.9)
We shall denote the characteristic cohomological groups by H~ha,.(d). Higher order conservation laws involving antisymmetric tensors of degree 2 or higher are quite interesting in their own right. In particular, conservation laws of the form (1.8), involving an antisymmetric tensor S u~ have attracted a great deal of interest in the past [5] as well as recently [6, 7] in the context of the mechanism of"charge without charge" of Wheeler [8]. But the characteristic cohomology is also important for another reason: it appears as an auxiliary cohomology in the calculation of the local BRST cohomology [9]. This local BRST cohomology, in turn, is quite useful in the determination of the structure of the counterterms [ 10, 11 ] and the anomalies [ 12] in the quantum theory. It plays also a central role classically, in constraining the form of the consistent deformations of the action [13]. It is by establishing vanishing theorems for the characteristic cohomology that the problem of consistent deformations and of candidate anomalies has been completely solved in the cases of Yang-Mills gauge theories and of gravity [14, 6[. For this reason, it is an important question to determine the characteristic cohomological groups for any given theory.
Characteristic Cohomology of p-Form Gauge Theories
139
The purpose of this paper is to carry out this task for a system of free antisymmetric a tensor fields Bm...~p, a = 1, 9 N, with Lagrangian - 1
/2 = E
( 2 ( p a + 1)!
Ha HaU,...zpa+,), m...Upa+,
(1.10)
a
where the H a's are the "field strengths" or "curvatures", 1
H a
-
Ba
_
(pa + 1)!
H a
m...,,o+,
d x t''
""
. d x U ' o § = d B a,
1
a ...u~a d x #1 . . d. x U. ~ Pa ! Bu~
(1.11) (1.12)
The equations of motion, obtained by varying the fields B~,...up, are given by O p H ap#I'''pp~ = 0.
(1.13)
We consider simultaneously antisymmetric tensors of different degrees, but we assume 1 <_ Pa. We also assume n > Pa + 1 for each a so that the fields B~l...u~a all carry local degrees of freedom. Modifications of the Lagrangian by gauge invariant interactions are treated at the end of the paper. We give complete results for the characteristic cohomology in degree < n - 1, that is, we determine all the solutions to the equation O ~ S ~ ' ' ' ~ ' ~ ~ 0 with s > 0. Although we do not solve the characteristic cohomology in degree n - 1, we comment on the gauge invariance properties of the conserved currents and provide an infinite number of them, generalizing earlier results of the Maxwell case [15, 16, 17] a. The results of this paper will be used in [18] to compute the BRST cohomology of free, antisymmetric tensor fields. This is a necessary step not only for determining the possible consistent interactions that can be added to the free Lagrangian, but also for analyzing completely the BRST cohomology in the interacting case. Our results have already been used and partly announced in [19] to show the uniqueness of the Freedman-Townsend deformation of the gauge symmetries of a system of antisymmetric tensors of degree 2 in four dimensions. Antisymmetric tensor fields - or, as one also says, p-form gauge fields - have been much studied in the past [20, 21, 22, 23, 24] and are crucial ingredients of string theory and of various supergravity models [25]. The main feature of theories involving p-form gauge fields is that their gauge symmetries are r e d u c i b l e . More precisely, in the present case, the Lagrangian (1.10) is invariant under the gauge transformations B a --+ B 'a = B a + d A %
(1.14)
where A a are arbitrary (Pa -- 1)-fOrlTIS. NOW, if A a = d e a, then, the variation of B a vanishes identically. Thus, the gauge parameters A a do not all provide independent gauge symmetries: the gauge transformations (1.14) are reducible. In the same way, if I The determination of all the conserved currents is of course also an interesting question, but it is not sysn 1 tematically pursued here for two reasons. First the characteristic cohomology Hc~h-a,.(d) is infinite-dimensional for the free theories considered here and does not appear to be completely known even in the Maxwell case in an arbitrary number of dimensions. By contrast, the cohomological groups Hkchar(d), k < n - l, are all n l finite-dimensional and can be explicitly computed. Second, the group Hc~a,~(d) plays no role in the analysis of the consistent interactions of antisymmetric tensor fields of degree > l, as well as in the analysis of candidate anomalies if the antisymmetfic tensor fields all have degree > 2 [18].
140
M. Henneaux,B. Knaepen,C. Schomblond
e a is equal to dl,~ a, then, it yields a vanishing A a. There is "reducibility of reducibility" unless e a is a zero form. If ea is not a zero form, the process keeps going until one reaches 0-forms. For the theory with Lagrangian (1.10), there are thus P M -- 1 stages of reducibility of the gauge transformations (A a is a (Pa - 1)-form), where pM is the degree of the form of highest degree occurring in (1.10) [26, 27, 28, 29]. One says that the theory is a reducible gauge theory of reducibility order P M -- 1. General vanishing theorems have been established in [1, 2, 3, 9] showing that the characteristic cohomology of reducible theories of reducibility order p - 1 vanishes in form degree strictly smaller than n p 1. Accordingly, in the case of p-form gauge theories, there can be a priori non-vanishing characteristic cohomology only in form degree n - P M -- 1, n -- P M , etc., up to form degree n - 1 (conserved currents). In the 1-form case, these are the best vanishing theorems one can prove, since a set of free gauge fields A~ has characteristic cohomology both in form degree n - 1 and n - 2 [9]. Representatives of the cohomology classes in form degree n - 2 are given by the duals to the field strengths, which are indeed closed on-shell due to Maxwell equations. Our main result is that the general vanishing theorems of [1, 2, 3, 9] can be considerably strengthened when p > 1. For instance, if there is a single p-form gauge field and if n - p - 1 is odd, there is only one non-vanishing group of the characteristic cohomology n--p--1 in degree < n - 1. This is H;har (d), which is one-dimensional. All the other groups H ~ h a r ( d ) of the characteristic cohomology with n - p - 1 < k < n - 1 are zero, even though the general theorems of [1, 2, 3, 9] leave open the possibility that they do not vanish. As we shall show in [18], it is the presence of these additional zeros that give p-form gauge fields and gauge transformations their strong rigidity. Besides the standard characteristic cohomology, one may consider the invariant characteristic cohomology, in which the local forms w and g, occurring in (1.6) and (1.7) are required to be invariant under the gauge transformations (1.14). We also completely determine in this paper the invariant characteristic cohomology in form degree < n - 1. Our method for computing the characteristic cohomology is based on the reformulation performed in [9] of the characteristic cohomology in form degree k in terms of the cohomology H un_ k ((lid) of the Koszul-Tate differential (i modulo the spacetime exterior derivative d. Here, n is the form degree and n - k is the antighost number. This approach is strongly motivated by the BRST construction and appears to be particularly attractive and powerful. Our paper is organized as follows. In the next section, we formulate precisely our main results, which are (i) that the characteristic cohomology H k h a r ( d ) with k < n - 1 is generated (in the exterior product) by the exterior forms ~ a dual to the field strengths Ha; these are forms of degree n - Pa - 1; and (ii) that the invariant characteristic k,inv cohomology gchar (d) with k < n - 1 is generated (again in the exterior product) by the exterior forms H a and ~ a . We then review, in Sects. 3 and 4, the definition and properties of the Koszul-Tate complex. Section 5 is of a more technical nature and relates the characteristic cohomology to the cohomology of the differential (i + d, where (i is the Koszul-Tate differential. Section 6 analyses the gauge invariance properties of (f-boundaries modulo d. In Sect. 7, we determine the characteristic cohomology for a single p-form gauge field. The results are then extended to an arbitrary system of pform gauge fields in Sect. 8. The invariant cohomology is analyzed in Sect. 9. Section l0 discusses in detail the cohomological groups H * ( 5 1 d ) , which play a key role in the calculation of the local BRST cohomological groups H * ( s ] d ) . In Sect. 1 l, we show that the existence of representatives expressible in terms of the H a ' s does not extend to the characteristic cohomology in form degree n - 1, by exhibiting an infinite number
Characteristic Cohomologyof p-FormGaugeTheories
141
of (inequivalent) conserved currents which are not of that form. We show next in Sect. 12 that the results on the free characteristic cohomology in degree < n - 1 can be generalized straightforwardly if one adds to the free Lagrangian (1.10) gauge invariant interaction terms that involve the fields B~l...#p~ and their derivatives only through the gauge invariant field strength components and their derivatives (which are in general the only consistent interactions that one can add). We conclude in Sect. 13 by summarizing our results and indicating future lines of research. We assume throughout this paper that spacetime is the n-dimensional Minkowski space, so that the indices in (1.10) are raised with the inverse ~/~' of the fiat Minkowski metric ~ v - However, because of their geometrical character, our results generalize straightforwardly to curved backgrounds.
2. Results 2.1. Characteristic cohomologyo The equations of motion (1.13) can be rewritten as dH ~ ~ 0
(2.1)
in terms of the (n - p a - 1)-forms ~ a dual to the field strengths. It then follows that any polynomial in the H~'s is closed on-shell and thus defines a cocycle of the characteristic cohomology. The remarkable feature is that these polynomials are not only inequivalent in cohomology, but also completely exhaust the characteristic cohomology in form degree strictly smaller than n - 1. Indeed, one has: Theorem 2.1. Let ~ be the algebra generated by the -~a's and let V be the subspace containing the polynomials in the -H~ 's with no term of form degree exceeding n - 2. The subspace V is isomorphic to the characteristic cohomology inform degree < n - 1.
We stress again that the theorem does not hold in degree n - 1 because there exist conserved currents not expressible in terms of the ~a,s. Since the form degree is limited by the spacetime dimension n, and since ~ a has form degree n - Pa - 1, which is strictly positive (as explained in the introduction, we assume n - pa - 1 > 0 for each a), the algebra 7-/is finite-dimensional. In that algebra, the ~ a with even n - Pa - 1 commute with all the other generators, while the H~ with odd n - Pa - 1 are anticommuting objects. 2.2. Invariant characteristic cohomology. While the cocycles of Theorem 2.1 are all gauge invariant, there exists coboundaries of the characteristic cohomology that are gauge invariant, i.e., that involve only the field strength components and their derivatives, but which cannot, nevertheless, be written as coboundaries of gauge invariant local forms, even weakly. Examples are given by the field strengths H a = d B a themselves. For this reason, the invariant characteristic cohomology and the characteristic cohomology do not coincide. We shall denote by 7-/the finite-dimensional algebra generated by the (Pa + 1)forms H a, and by ,.7 the finite-dimensional algebra generated by the field strengths H a and their duals ~ a . One has
142
M. Henneaux, B. Knaepen, C. Schomblond
Theorem 2.2. Let kV be the subspace o f J containing the polynomials in the H ~ 's and the ~ a 's with no term o f f o r m degree exceeding n - 2. The subspace 14; is isomorphic to the invariant characteristic cohomology i n f o r m degree < n - 1.
Our paper is devoted to proving these theorems. 2.3. Cohomologies in algebra o f x-independent forms. The previous theorems hold as
they are formulated in the algebra of local forms that are allowed to have an explicit xdependence. The explicit x-dependence enables one to remove the constant k-forms (k > 0) from the cohomology, since these are exact, ci, i2...ik dxi' dxi2 9 9 dxik = d(ci, ~2...i~x ~ dx ~ . . . d x ik). If one restricts one's attention to the algebra of local forms with no explicit dependence on the spacetime coordinates, then, one must replace in the above theorems the polynomials in the curvatures and their duals with coefficients that are numbers by the polynomials in the curvatures and their duals with coefficients that are constant exterior forms.
Note that the constant exterior forms can be alternatively gotten rid of without introducing an explicit x-dependence, by imposing Lorentz invariance (there is no Lorentzinvariant constant k-form for 0 < k < n).
3. Koszul-Tate Complex The definition of the cocycles of the characteristic cohomology H ~ h ~ ( d ) involves "weak" equations holding only on-shell. It is convenient to replace them by "strong" equations holding everywhere in field space, and not just when the equations of motion are satisfied. The reason is that the coefficients of the equations of motion in the conservation laws are not arbitrary, but are subject to restrictions whose analysis yields useful insight on the conservation laws themselves. From this point of view, Eq. (1.4) involving the coefficients Aim~*j is a more interesting starting point than Eq. ( 1.3). One useful way to replace weak equations by strong equations is to introduce the Koszul-Tate resolution associated with the equations of motion (1.13). The details of the construction of the Koszul-Tate differential 5 can be found in [30]. Because the present theory is reducible, we must introduce the following set of BV-antifields [31]: B'am...#,=, B ' a m .-.upa-,,..., B*~u,, B *~.
(3.1)
The Grassmann parity and the antighost number of the antifietds B *am'f*v~ associated with the fields B I~1...IZpa a are equal to 1. The Grassmann parity and the antighost number of the other antifields is determined according to the following rule. As one moves from one term to the next one to its right in (3.1), the Grassmann parity changes and the antighost number increases by one unit. Therefore the parity and the antighost number of a given antifield B * a m u p - , are respectively j + 1 modulo 2 and j + 1. The Koszul-Tate differential acts in the algebra 7) of local exterior forms. By deftnition, a local exterior form co reads CO = E
COgi ""t* j d x l X l " " " d x l X Y :
(3.2)
where the coefficients cat,l...t,j are smooth functions of the coordinates x u, the fields Ba the antifields (3.1), and their derivatives up to a finite order. Although this is
Characteristic Cohomologyof p-FormGaugeTheories
143
not strictly necessary, we shall actually assume polynomiality in the fields, the antifields and their derivatives, as this is the situation encountered in field theory. The Koszul-Tate differential is defined by its action on the fields and the antifields as follows:
5B~,...u~o 5B *am'''upa 5B*at~,...t~p~-,
5B *au' 6B a*
= O,
(3.3)
= =
(3.4) (3.5)
OpH apul"''up~', OpB *apm'''up~-',
= OpB *apm, = OpB *ap.
(3.6) (3.7)
Furthermore we have,
5x u = O, 5(dx t~) = O.
(3.8)
The action of 5 is extended to an arbitrary element in 79 by using the rule
50~ = 0~6,
(3.9)
and the fact that 5 is an odd derivation which we take here to act from the left,
5(ab) = (~a)b + (-)C~
(3.10)
In 3.10, ea is the Grassmann parity of the (homogeneous) element a. These rules make 5 a differential and one has the following important property [32, 33, 30, 34]: Theorem 3.1. Hi(5) = Ofor i > 0, where i is the antighost number, i.e, the cohomology of 5 is empty in antighost number strictly greater than zero. One can also show that in degree zero, the cohomology of 5 is the algebra of "onshell functions" [32, 33, 30, 34]. Thus, the Koszul-Tate complex provides a resolution of that algebra. For the reader unaware of the BRST developments, one may view this property as the motivation for the definitions (3.3) through (3.7). One has a similar theorem for the cohomology of the exterior derivative d (for which we also take a left action, d(ab) = (da)b + (-)C~ Theorem 3.2. The cohomology of d in the algebra of local forms is given by,
H~ "~ R, Hk(d) = Of o r k 5~O, k 4 n, Hn(d) "~ space of equivalence classes of local forms,
(3.11) (3.12) (3.13)
where k is the form degree and n the spacetime dimension. In (3.13), two local forms are said to be equivalent if and only if they have identical Euler-Lagrange derivatives with respect to all the fields and the antifields.
144
M. Henneaux, B. Knaepen, C. Schomblond
Proof This theorem is known as the algebraic Poincar6 Lemma. For various proofs, see
[2, 35, 36, 37]. It should be mentioned that the theorem holds as such because we allow for an explicit x-dependence of the local exterior forms (3.2). If the local forms had no explicit x-dependence, then (3.12) would have to be amended as H k ( d ) ~ {constant forms} for k ~/0, k ~ n,
(3.14)
where the constant forms are by definition the local exterior forms (3.2) with constant coefficients. We shall denote in the sequel the algebra of constant forms by A* and the subspace of constants forms of degree k by A k. The following formulation of the Poincar~ lemma is also useful. []
Theorem 3.3. Let a be a local, closed k-form ( k < n) that vanishes when the fields and the antifields are set equal to zero. Then, a is d-exact. Proof The condition that a vanishes when the fields and the antifields are set equal to
zero eliminates the constants. This form of the Poincar6 lemma holds in both the algebras of x-dependent and x-independent local exterior forms. []
4, Characteristic Cohomology and Koszul-Tate Complex Our analysis of the characteristic cohomology relies upon the isomorphism established in [9] between H~ha~(d) and the cohomology H*(d[d) of 3 modulo d. The cohomology H#(rld) in form degree k and antighost number i is obtained by solving in the algebra of local exterior forms the equation, 6a ki + dbik_-~ = 0,
(4.1)
and by identifying solutions which differ by &exact and d-exact terms, i.e, Ik
k-1
k k ai ~,o ai -_ ai + oni-+l + drn i
.
(4.2)
One has
Theorem 4.1. Hkh~r(d)
~
H~_h(~ld), 0 < k < n,
(4.3)
H~
~
H~(6[d),
(4.4)
~
H~+k(61d), k > 0.
(4.5)
R 0
Proof Although the proof is standard and can be found in [37, 9], we shall repeat it
explicitly here because it involves ingredients which will be needed below. Let c~ be a class of H)har(d) (k < n) and let a0k be a representative of c~, c~ = [a0k]. One has 6a k+' + da~ = 0
(4.6)
for some a~ +1 since any antifield-independent form that is zero on-shell can be written as the ~ of something. By acting with d on this equation, one finds that da k§ is &closed and thus, by Theorem 3.1, that it is &exact, 6a k+2 + da k+l = 0 for some a k+2. One can repeat the procedure until one reaches degree n, the last term a~_~ fulfilling
Characteristic Cohomology of p-Form Gauge Theories
145
5a~_ k + d a n _ l _
= O,
(4.7)
and, of course, dan_ k = 0 (it is a n-form). For future reference we collect all the terms appearing in this tower of equations as ak _ n n-1 -- a n _ k + a n _ l _
k +...
-4- alk+l
+
a k.
(4.8)
Equation (4.7) shows that an k is a cocycle of the cohomology of 6 modulo d, in form-degree n and antighost number n - k. Now, given the cohomological class a of H~ha,.(d), it is easy to see, using again Theorem 3.1, that the corresponding element a n _ k is well-defined in H~_k(JId). Consequently, the above procedure defines an nonambiguous map m from H)har(d) to H~_k(Jld). This map is surjective. Indeed, let an_ k be a cocycle of Hg_k(Jld). By acting with d on Eq. (4.7) and using the second form of the Poincar6 lemma (Theorem 3.3), one finds that ann-l-k1 is also J-closed modulo d. Repeating the procedure all the way down to antighost number zero, one sees that there exists a cocycle a0k of the characteristic cohomology such that m([a0k]) = [a n_k ]" The map m is not quite injective, however, because of the constants. Assume that a0k is mapped on zero. This means that the corresponding an_ k is trivial in Hn_k(JId), i.e., an_ k n = Jbn_k+ln + ~'4bn-ln-k"Using the Poincar6 lemma (in the second form) one then finds successively that a n - Ik - 1 . . . up to a~ +1 are all trivial. The last term a0k fulfills da~ + Jdb~ = 0 and thus, by the Poincar6 lemma (Theorem 3.2), a0k = Jb~ + db~ -1 + c k. In the algebra of x-dependent local forms, the constant k-form c k is present only if k = 0. This establishes (49 and (49 That H,~(J[d) vanishes for ra > n is proved in [34]. [] The proof of the theorem shows also that (4.3) holds as such because one allows for an explicit x-dependence of the local forms. Otherwise, one must take into account the constant forms c k which appear in the analysis of injectivity and which are no longer exact even when k > 0, so that (4.3) becomes k
H~ha"(d) ~ H~_k(J[d), Ak -
(4.9)
while (4.4) and (4.5) remain unchanged.
5. Characteristic C o h o m o l o g y and C o h o m o l o g y of .4 = 6 + d
It is convenient to rewrite the Koszul-Tate differential in form notations9 Denoting the duals with an overline to avoid confusion with the antifield *-notation, and redefining the antifields by appropriate multiplicative constants, one finds that Eqs. (3.4) through (3.7) become simply I~r
a
~B 1
+
d H a = O,
6B 2
+
d B I = 0,
(5.1) ~Bp,~+ 1
+
dBp~
= O.
146
M. Henneaux,
B. Knaepen,
C. Schomblond
--$a ~ " - E l * a / y . | . I[~p ~-I j TheformB j dual to the antisymmetrlc tensor aenslty D "" ~ - (j = 1, . . . , p a + l ) has (i) form degree equal to n - pa - 1 + j ; and (ii) antighost number equal to j. Since B*~U~...t,,~+,-j has Grassmann parity j and since the product of (n - Pa - 1 + j ) dx's has Grassmann parity n - p~ - 1 + j , each B ; a has same Grassmann parity n - Pa - 1
(modulo 2), irrespective o f j . This is the same parity as that of the n - p~ - 1-form H " dual to the field strengths. Equation (5.1) can be rewritten as A/7/a = 0
(5.2)
A=~+d
(5.3)
with and
pa+l ~
(5.4)
j=l
The parity of the exterior form/~a is equal to n - p a - 1. The regrouping of physical fields with ghost-like variables is quite standard in BRST theory [38]. Expressions similar (but not identical) to (5.4) have appeared in the analysis of the Freedman-Townsend model and of string field theory [39, 40], as well as in the context of topological models [41, 42]. Note that for a one-form, expression (5.4) reduces to Ey. (9.8) of [14]. Quite generally, it should be noted that the dual ~ a to the field strength H a is the term of lowest form degree in/7/% It is also the term of lowest antighost number, namely, zero. At the other end, the term of highest form degree i n / J ~ is B;~+I, which has form degree n and antighost number p~ + 1. If we call the difference between the form degree and the antighost number the "A-degree", all the terms present in the expansion of H a have the same A-degree, namely n - pa - 1. The differential A = ~ + d enables one to reformulate the characteristic cohomology as the cohomology of A. Indeed one has T h e o r e m 5.1. The cohomology o f A is isomorphic to the characteristic cohomology, H k ( A ) ~_ Hkhar(d), 0 < k < n
(5.5)
where k in H k ( A ) is the A-degree, and in Hkchar(d) is the form degree. Proof. Let aok (k < n) be a cocycle of the characteristic cohomology. Construct a k as in the proof of Theorem 4. l, formula (4.8). The form a k is easily seen to be a cocycle of A, A a k = 0, and furthermore, to be uniquely defined in cohomology given the class of a0k. We leave it to the reader to check that the map so defined is both injective and surjective. This proves the theorem for k < n. For k = n, the isomorphism of H ~ ( A ) and H ~ , ~ ( d ) is even more direct (da~ = 0 is equivalent to Aa~ = 0 and a~ = db~-1 +Sb~ is equivalent to a~ = A(b~~-1 + b~)). [] Our discussion has also established the following useful rule: the term of lowest form degree in a A-cocycle a is a cocycle of the characteristic cohomology. Its form degree is equal to the A-degree k of a. For a = [ta, this reproduces the rule discussed above Theorem 5.1. Similarly, the term of highest form degree in a has always form degree equal to n if a is not a A-coboundary (up to a constant), and defines an element of g ~ k(Sld).
Characteristic Cohomologyof p-FormGauge Theories
147
Because A is a derivation, its cocycles form an algebra. Therefore, any polynomial in t h e / ~ is also a A-cocycle. Since the form degree is limited by the spacetime dimension n, and since the term ~ a with minimum form degree in H a has form degree n - p~ - 1, which is strictly positive, the algebra generated by the/_)a is finite-dimensional. We shall show below that these A-cocycles are not exact and that any cocycle of form degree < n - 1 is a polynomial in the ~ a modulo trivial terms. According to the isomorphism expressed by Theorem 5.1, this is equivalent to proving Theorem 2.1.
Remarks.
(i) The A-cocycle associated with a conserved current contains only two
terms, a = a~ + a~ -1,
(5.6)
where a 0n-1 is the dual to the conserved current in question. The product of such a Acocycle with a A-cocycle of A-degree k has A-degree n - 1 + k and therefore vanishes unless k = 0 or 1. (ii) It will be useful below to introduce another degree N as follows. One assigns N degree 0 to the undifferentiated fields and N-degree 1 to all the antifields irrespective of their antighost number. One then extends the N-degree to the differentiated variables according to the rule N(Ou~) = N ( ~ ) + 1. Thus, N counts the number of derivatives and of antifields. Explicitly,
N = Z Na
(5.7)
a
with
Na = ~-~ [(IJl ~-~ OjB~ O - ~ +(IJl+ l) ~ Od~*~aOj~a], J
i
J
(5.8)
i
where (i) the sum over J is a sum over all possible derivatives including the zeroth order one; (ii) IJI is the differential order of the derivative Oj (i.e., ]JI = k for 0m...uk); (iii) the sum over i stands for the sum over the independent components of Ba; and (iv) the sum over c~ is a sum over the independent components of all the antifields appearing in the tower associated with B ~ (but there is no sum over the p-form species a in (5.8)). The differential 5 increases N by one unit. The differentials d and A have in addition an inhomogeneous piece not changing the N-degree, namely dx~'(Oe~pUc~t/OxU),where O~pu~it/Ox~ sees only the explicit xU-dependence. The forms/1~ have N-degree equal to one.
6. Acyclicity and Gauge Invariance
6.1. Preliminaryresults. Under the
gauge transformations (1.14) of the p-form gauge fields, the field strengths and their derivatives are gauge invariant. These are the only invariant objects that can be formed out of the "potentials" B Idl a " " ~ P a and their derivarives. We shall denote by ZSmaU the algebra of local exterior forms with coefficients wm...t,J that depend only on the field strength components and their derivatives (and possibly x~). The algebras 7-/, 7-/and f f respectively generated by the (pa + 1)-forms H a, (n - Pa - 1)-forms H~ and ( H a, H~) are subalgebras of ~-'Small. Since the field equations are gauge invariant and since d maps Zsm~u on ZSmatt, one can consider the cohomological problem (1.6), (1.7) in the algebra ZSmaU.This defines the invariant * inv characteristic cohomology H c ~ r (d).
148
M . H e n n e a u x , B . K n a e p e n , C. S c h o m b l o n d
It is natural to decree that the antifields and their derivatives are also invariant. This can be more fully justified within the BRST context, using the property that the gauge transformations are abelian, but here, it can simply be taken as a useful, consistent postulate. With these conventions, the differentials 6, d and A map the algebra Z of invariant polynomials in the field strength components, the antifield components and their derivatives on itself. Clearly, Zs,~au C Z. The invariant cohomologies H*'inv(z~) and are defined by considering only local exterior forms that belong to Z. In order to analyze the invariant characteristic cohomology and to prove the nontriviality of the cocycles listed in Theorem 2.1, we shall need some preliminary results on the invariant cohomologies of the Koszul-Tate differential 6 and of d. The variables generating the algebra T' of local forms are, together with x ~' and dx ~,
H~'inv(51d)
B a l , t...I,~,~ , OpBat~, . . . , ~ , = , 9 9 9 , B *ap' . . . , p = _ m , OpB*al~l...,p~ - ~ ,
. . . , B'a,
O p B *a ' . . . .
These variables can be conveniently split into two subsets. The first subset of generators will be collectively denoted by the letter X- They are given by the field strengths (Ham ...u,=+, ) and their derivatives, the antifields and their derivatives. The field strengths and their derivatives are not independent, since they are constrained by the identity d H ~ = 0 and its differential consequences, but this is not a difficulty for the considerations of this section. The X'S are invariant under the gauge transformations and they generate the algebra 27 of invariant polynomials. In order to generate the full algebra we need to add to the X'S some extra variables that will be collectively denoted g'. The ~ ' s contain the field components B a~*l'''**,~ and their appropriate derivatives not present in the X'S. The explicit form of the ~P's is not needed here. All we need to know is that the ~ ' s are algebraically independent from the X's and that, in conjunction with the X'S, they generate 5D. Theorem 6.1. Let a be a polynomial in the X: a = a(x). If a = 5b, then we can choose b such that b = b(x). In particular, H~nv(5) ~- Of o r j > 0.
(6.1)
Proof We can decompose b into two parts: b = b + ~, with b = b(x) = b(~ = 0) and ~ = ~ , ~ R m ( x ) S m ( ~ ) , where Sm(k~) contains at least one ~P. Because 6~P = 0, we have, 5(b + b) = 5-b(x) + ~ , ~ 5 R m ( X ) S m ( ~ ) . Furthermore if M = M(X), then 6 M ( x ) = (SM)(x). We thus get, a(x ) = (Sb)(x) + ~ ( ~ R m ) ( X ) S m ( k ~ ) . m
The above equation has to be satisfied for all the values of the ~P's and in particular for = 0. This means that a(x) = (Sb)(x) = ~b(x). [] Theorem 6.2. Let ~ k be the subspace ofform degree k of the finite dimensional algebra 7-I of polynomials in the curvature (Pa + 1)-forms H a, TI = Gk T-Ik. One has
and
H~k , i n v (d) = 0, k < n, j > 0
(6.2)
H~k , i n v (d) = 7-/k, k < n.
(6.3)
CharacteristicCohomology of p-Form Gauge Theories
149
Thus, in particular, if a = a(x) with da = O, antighost a > 0 and deg a < n, then a = db with b = b(x). And if a has antighost number zero, then a = P ( H a) + db, where P ( H a) is a polynomial in the curvature forms and where b c ZSmaU. Proof The theorem has been proved in [36, 43] for 1-forms. It can be extended straightforwardly to the case of p-forms of odd degree. The even degree case is slightly different because the curvatures (p + D-forms H a are then anticommuting. It is fully treated in Appendix A. If the local forms are not taken to be explicitly x-dependent, Eq. (6.3) must be replaced by k,inv
H0
(d) = (A | H ) k.
(6.4) []
6.2. Gauge invariant 6-boundaries modulo d. We assume in this section that the antisymmetric tensors B a case of a single p-form.
have all the same degree p. This covers, in particular, the
Theorem 6.3. (Valid when the B a 's have all the same f o r m degree p). Let a~ = aq(X) E Z be an invariant local n-form of antighost number q > O. If aq is 5-exact n modulo d, aqn = 5Pq+t + dpqn - - 1 , then one can assume that #q+t and # q - t only depend on the X'S, i.e., are invariant (#qn+t and pq-1 C Z). Proof The p r o o f goes along exactly the same lines as the proof o f a similar statement made in [14] (Theorem 6. l) for 1-form gauge fields. Accordingly, it will not be repeated here 2 . []
Remark. The theorem does not hold if the forms have various formdegrees (see Theorem 10.1 below).
7. Characteristic Cohomology for a Single p-Form Gauge Field Our strategy for computing the characteristic cohomology is as follows. First, we compute H,~(~ld) (cocycle condition, coboundary condition) for a single p-form. We then use the isomorphism theorems to infer H'hat(d). Finally, we solve the case of a system involving an arbitrary (but finite) number o f p-forms of various form degrees.
7.1. General theorems. Before we compute H~.(61 d) for a single abelian p-form gauge field B m ..up, we will recall some general results which will be needed in the sequel. These theorems hold for an arbitrary linear theory of reducibility order p - 1.
Theorem 7.1. For a linear gauge theory of reducibility order p-1, one has, H~(~ld) =0,
j >p+
1.
(7.1)
2 We shall just mention a minor point that has been overlookedin the proof of Theorem 6.1 of [14], namely, that when p = 1 in Eq. (6.4) of [14], the form Z ~ need not vanish (in the notations of [14]). However,this does not invalidate the fact that one can replace Z ~. X~, etc. by invariantpolynomials as the recurrenceused in the proof of [14] and the absence of invariant cohomology for d in form degree one indicate. This is just what is needed for establishing the theorem.
150
M. Henneaux, B. Knaepen, C. Schomblond
Proof See [9], Theorem 9.1. See also [1, 2, 3].
[]
Theorem 7.1 is particularly useful because it limits the number of potentially nonvanishing cohomologies. The calculation of the characteristic cohomology is further simplified by the following theorem: T h e o r e m 7.2. Any solution of ~a + Opbp = 0 that is at least bilinear in the antifields is necessarily trivial.
Proof See [9], Theorem 11.2.
[]
Both theorems hold whether the local forms are assumed to have an explicit xdependence or not.
7.2. Cocycles of H~+1(3[d). We have just seen that the first potentially non-vanishing cohomological group is Hp+l(3ld). We show in this section that this group is onedimensional and provide explicit representatives. We systematically use the dual notations involving divergences of antisymmetric tensor densities.
H~+~(81d)
Theorem 7.3. is one-dimensional. One can take as representatives of the cohomological classes a = kB*, where B* is the last antifield, of antighost number p + 1 and where k is a number.
Proof Any polynomial of antighost number p + 1 can be written a = f B * + fPOp13* + ... + #, where f does not involve the antifields and where # is at least bilinear in the antifields. By adding a divergence to a, one can remove the derivatives of B*, i.e., one can assume fP = fo~ . . . . . 0. The cocycle condition da + Opbp = 0 implies then - O p f B*P + ~iz + Op(bp + f B *o) = O. By taking the Euter-Lagrange derivative of this equation with respect to B *p, one gets p 5LP - Opf+~((-1)
~-2-~) = 0.
(7.2)
This shows that f is a cocycle of the characteristic cohomology in degree zero since d(anything of antighost number one) m 0. Furthermore, if f is trivial in H~ then a can be redefined so as to be at least bilinear in the antifields and thus is also trivial in the cohomology of d modulo d. Now, the isomorphism of H~ with H~(~[d) implies f = k + ~g with k a constant (Hn~(3td) = 0 because n > p + 1). As we pointed out, the second term can be removed by adding a trivial term, so we may assume f = k. Writing a = kB* + #, we see that p has to be a solution of d# + Opb~p = 0 by itself and is therefore trivial by Theorem 7.2. So Hp+ 1(dl d) can indeed be represented by a = kB*. In form notations, this is just the n-form kB;+ l . Note that the calculations are true both in the x-dependent and x-independent cases. To complete the proof of the theorem, it remains to show that the cocycles a = L'B*, which belong to the invariant algebra 27 and which contain the undifferentiated antifields, are non-trivial. If they were trivial, one would have according to Theorem 6.3, that B;+I = (Su + dv for some u, v also in 2-. But this is impossible, because both 3 and d bring in one derivative of the invariant generators X while B;+l does not contain derivatives of X. [This derivative counting argument is direct if u and v do not involve explicitly the spacetime coordinates x ~. If they do, one must expand u, v and the equation B;+i = c~u + dv according to the number of derivatives of the fields in order to reach
Characteristic Cohomologyof p-Form Gauge Theories
151
the conclusion. Explicitly, one sets u = uo + . . . + uk, v = vo + . . . + vk, where k counts the number of derivatives of the H~,,...u~§ and of the antifields. The condition -Bp+1 = 5u + dv implies in degree k + 1 in the derivatives that 5Uk + d'vk = 0, where d' does not differentiate with respect to the explicit dependence on x t'. This relation implies in turn that uk is 5-trivial modulo d' since there is no cohomology in antighost number p + 2. Thus, one can remove uk by adding trivial terms. Repeating the argument for u k - 1, and then for Uk-2, etc., leads to the desired conclusion.] [] 7.3. Cocycles o f H~(51d) with i < p. We now solve the cocycle condition for the remaining degrees. First we prove
T h e o r e m 7.4. Let K be the greatest integer such that n - K ( n - p - 1) > 1. The cohomological groups H~(Sld) (j > 1) vanish unless j = n - k ( n - p - 1), k = 1 , 2 , . . . , K . Furthermore, f o r those values o f j , Hy(Sld) is at most one-dimensional. Proof We already know that Hn(5ld) is zero for j > p + 1 and that H~+l(51d ) is onedimensional. Assume thus that the theorem has been proved for all 3 s strtctly greater than J < p + 1 and let us extend it to J. In a manner analogous to what we did in the proof of Theorem 7.3, we can assume that the cocycles of n~(51d) take the form
f~l...~p§
J B*~~§
+/z,
(7.3)
where f.,...up+,_j does not involve the antifields and defines an element of H p ~ I S ( d ) . Furthermore, if f.,....p+~_ j is trivial, then the cocycle (7.3) is also trivial. Now, using the isomorphism Hv~s J (d) ~- H ~ _ v_ ~+g(51d) (p+ 1 - J > 0), we see that f is trivial unless j ' = n - p - 1+.J, which is strictly greater than J and is of the form j ' = n - k ( n - p 1). In this case, H ~ is at most one-dimensional. Since J = j ' - ( n - p - 1) = n - (k+ 1 ) ( n - p - 1) is of the required form, the property extends to J. This proves the theorem. [] Because we explicitly used the isomorphism HP~h~d(d) ~ n~n_p_~+j(5ld), which holds only if the local forms are allowed to involve explicitly the coordinates x~', the theorem must be amended for x-independent local forms. This will be done in Sect. 7.5. Theorem 7.4 goes beyond the vanishing theorems of [1, 2, 3, 9] since it sets further cohomological groups equal to zero, in antighost number smaller than p + 1. This is done by viewing the cohomological group H~(51d) as a subset of H~_v_l+~(51d) at a higher value of the antighost number, through the form (7.3) of the cocycle and the isomorphism between H P ~ i ( d ) and Hn_p_l+i(5ld). In that manner, the known zeros at values of the ghost number greater than p + 1 are "propagated" down to values of the ghost number smaller than p + 1. To proceed with the analysis, we have to consider two cases: (i) Case I: n - p - 1 is even. (ii) Case II: n - p - 1 is odd. We start with the simplest case, namely, Case I. In that case,/:/is a commuting object and we can consider its various powers (/~)k, k = 1 , 2 , . . . , K with K as in Theorem 7.4. These powers have A-degree k ( n - p - 1). By Theorem 5.1, the term of form degree n in (/:/)k defines a cocycle of H,~_ k(n-p- 1)(SId), which is non-trivial as the same invariance argument used in the previous subsection indicates. Thus, H~_ k(n-p-~)(5ld), which we know is at most one-dimensional, is actually exactly one-dimensional and one may take
152
M. Henneaux,
B. Knaepen,
C. Schomblond
as representative the term of form degree n in ( ~ ) k . This settles the case when n - p - 1 is even. In the case when n - p - 1 is odd, t7t is an anticommuting object and its powers (/7/)k, k > 0 all vanish unless k = 1. We want to show that H 2 _ k(n--p- 1)(~1d) similarly vanishes unless k = 1. To that end, it is enough to prove that H.'~_2(~_p_l)(Sld) = H~p+2_,~(~ld) = 0 as the proof o f Theorem 7.4 indicates (we assume, as before, that 2p + 2 - n > 1 since we only investigate here the cohomological groups H~(3ld) with i > 1). Now, as we have seen, the most general cocycle in H~p+2_,~(61d) may be assumed to take the form a = fl~p+2...l~,~B*"p§ + It, where # is at least quadratic in the antifields and where fu~+2...~.~ does not involve the antifields and defines an element of H "~h~p--1(d). But ~ r rhn=- ~P - I (d) _~ Hff+l(~ld) is one-dimensional and one may take as representative of H~h~P~-l(d) the dual k % , . . . # H m'''"~§ necessarily of the form, a = k~#l., .~n H#I
of the field strength. This means that a is
4s
Bg<#P+2""lLn "~ ]~3
(7.4)
and the question to be answered is: for which value of k can one adjust # in (7.4) so that
kem...un Hm'"u,+' aB*U~§ t*'~ + @ + Opb~ = 0?
(7.5)
In (7.5), # does not contain B *~,+2'''"~ and is at least quadratic in the antifields. Without loss of generality, we can assume that it is exactly quadratic in the antifields and that it does not contain derivatives, since ~ and 0 are both linear and bring in one derivative. [That p can be assumed to be quadratic is obvious. That it can also be assumed not to contain the derivatives of the antifields is a little bit less obvious since we allow for explicit x-dependence, but can be easily checked by expanding p and bp according to the number of derivatives of the variables and using the triviality o f the cohomology of modulo d in the space of local forms that are at least quadratic in the fields and the antifields.] Thus, we can write R~r
-" . P n ~(2p+l--n) + /At
"" " U p j ~ * ~ p + l
/.L = O'p, 1 . . . # n ~ ( 1 )
where # ' involves neither B ( ~ . ~ : ~ n nor B(*~.+l' ' ~ n We have explicitly indicated the tP tP -' . . . . antighost number in parentheses m order to keep track of it. Inserting this form of/* in (7.5) one finds that crm . ~ , is equal to ket, 1...~. if 2p + 1 - n > 1 (if 2p + l - n = l, 9 9 * 9 up.-..) n , R*up-~...un see below). One can then successwely ehmlnate B(2p_,~ --(ep-,~-1) ' etc " f r o m p , so that the question ultimately boils down to: is ~[~*fLl .. , p j ,~ j ~ * P j + I .--p2j ]s162 .. "P23 ~ ( p + l - - j ) " ~ ( p + l - - j )
(n even = 2 j ) or ,g
r ~ * / ~ l .../z3+l r r~*/~3+2..-/~2j+l IE/zj .../z2j+l D ( p _ j ) OD(p+l_j)
(n odd = 23 + 1) &exact modulo d, i.e., o f the form cSv + OpeV, where v does not involve the antifields B* for s > p + 1 - j (n even) or s > p - j (n odd)? That the answer to this question is negative unless k = 0 and a accordingly trivial, which is the desired result, is easily seen by trying to construct explicitly v. We treat for definiteness the case n even (n = 2j). One has // _-- /~/z 1
---
/j,2j R * # l ' " / z J ~(p+l--j)
R */z3+1'''/~23 ~(p+l--j)
Characteristic Cohomology of p-Form Gauge Theories
153
where )~/Zl...t`2j is antisymmetric (respectively symmetric) for the exchange of (/Z 1 . . . ~ j ) with (#j+l -. 9#2j) i f j is even (respectively odd) (the j - f o r m B~p+l-j) is odd by assumption and this can happen only if the components ~(p+l-j)n*mmare odd for j even, or even for j odd). From the equation /:~*/*1...t`4 .g j~*t`a+1 ...t`a.j ]g,:~,/Z1"',[~23~(p+l--j) ~(p+l--j) = ~b' + OpCP~
(7.6)
one gets ~ * ~ l . . . t ` j ~ jET~*Pt`34-1"*t`2j=2At`, 1,2jB *t`l"''t`~ /:~*pt`j+l...t`2j +OqpCp" ket`l ...t`23 ~(p+l--j) ~ P ~ ( p - - j ) ... (p+l--j) ~P~(,p--j)
(7.7)
Taking the Euler-Lagrange derivative of this equation with respect to -r~*u, ( p + l ...m - j ) yields next ~''t`l.. "t`23] ~ P ~ ( p - - j )
= O~
which implies k%l...mj = 2Am...mj. This contradicts the symmetry properties of At,,...m~, unless k = 0, as we wanted to prove.
7.4. Characteristic Cohomology. By means of the isomorphism theorem of section 4, our results on H2(51d) can be translated in terms of the characteristic cohomology as follows: (i) If n - p - 1 is odd, the only non-vanishing group of the characteristic cohomology n--p--1 in form degree < n - 1 is Hchar (d), which is one-dimensional. All the other groups vanish. One may take as representatives for H~h-~-l(d) the cocycles k H . Similarly, the only non-vanishing group Hi(A) with j < n - 1 is H ' ~ - P - a ( A ) with representatives k/7/and the only non-vanishing group H~(5]d) with i > 1 is H~+1(61d) with representatives kB~+ 1. (ii) If n - p - 1 is even, there is further cohomology. The degrees in which there is non-trivial cohomology are multiples of n - p - 1 (considering again values of the form degree strictly smaller than n - 1). Thus, there is characteristic cohomology only in degrees n - p - 1, 2(n - p - 1), 3(n - p - 1), etc. The corresponding groups are one-dimensional and one may take as representatives kH, k(-H) z, k ( H ) 3, etc. There is also non-vanishing ALcohomology for the same values of the A-degree, with representative cocycles given by k H , k(/]) 2, k(/]) 3, etc. By expanding these cocycles according to the form degree and keeping the terms of form degree n, one gets representatives for the only non-vanishing groups n~(51d) ( with i > 1), which are respectively Hp+ 1(S Id), Hp+ l_(n_p_ 1), Hp+1-2(n-p-I), etc. An immediate consequence of our analysis is the following useful theorem: T h e o r e m 7.5. If the polynomial Pk( H) of form degree k < n in the curvature (p + 1)form H is &exact modulo d in the invariant algebra Z, then pa( H) = O.
Proof The theorem is straightforward in the algebra of x-independent local forms, as a direct derivative counting argument shows. To prove it when an explicit x-dependence is allowed, one proceeds as follows. Ifpk(H) = 5akl+da~ -1 where alk and a0k - l E Z, then day +rak2+~ = 0 for some invariant a 2k+l . Using the results on the cohomology of 5 modulo d that we have just established, this implies that a~ differs from the component of form degree k and antighost number 1 of a polynomial Q ( / t ) by a term of the form 5p + da,
154
M. Henneaux, B. Knaepen, C. Schomblond
where p and cr are both invariant. But then, 5a~ has the form d ([Q(/7/)]ok-' + &r), which implies p k (H) = d ( - [Q(H)]0k- 1 _ 5a + a0k- ~), i.e., p k (H) = d(invariant). According to the theorem on the invariant cohomology of d, this can occur only if P k ( H ) = O. [] 7.5. Characteristic cohomology in the algebra of x-independent local forms. Let us denote ([/)m by Pm (ra = 0 , . . . , K). We have just shown (i) that the most general cocycles of the A-cohomology are given, up to trivial terms, by the linear combinations AmPm with Am real or complex numbers; and (ii) that if AmPm is b-exact, then the Am are all zero. In establishing these results, we allowed for an explicit x-dependence of the local forms (see comments after the proof of Theorem 7.4). How are our results affected if we work exclusively with local forms with no explicit x-dependence? In the above analysis, it is in calculating the cocycles that arise in antighost number < p + 1 that we used the x-dependence of the local forms, through the isomorphism H P + I - J ( ,4~ char ~l ~ - Hn-p-l+j(rSld) 9 If the local exterior forms are not allowed to depend explicitly on x, one must take the constant k-forms (k > 0) into account. The derivation goes otherwise unchanged and one finds that the cohomology of A in the space of xindependent local forms is given by the polynomials in the P,~ with coefficients A~,~that are constant forms, Am = Am(dx). In addition, if AmP,~ is b-exact, then, A,mP ~ = 0 for each m. One cannot infer from this equation that Am vanishes, because it is an exterior form. One can simply assert that the components of A~ of form degree n - m(n - p - 1) or lower are zero (when multiplied by Pro, the other components of Am yields forms of degree > n that identically vanish, no matter what these other components are). It will be also useful in the sequel to know the cohomology of A t, where A t is the part of A that acts only on the fields and antifields, and not on the explicit x-dependence. One has A = A' + dx, where dx =- O~PliCit/Ox~ sees only the explicit x-dependence. By the above result, the cohomology of A t is clearly given by the polynomials in the Pm with coefficients Am that are now arbitrary spacetime forms, Am = Am(X, dx).
8. Characteristic Cohomology in the General Case To compute the cohomology H~(Sld) for an arbitrary set of p-forms, one proceeds along the lines of the Kunneth theorem. Let us illustrate explicitly the procedure for two fields B~I ...~p, and B 2 .--~2 . One may split the differential A as a sum of terms with definite No-degrees, A = z~ 1 -I- A 2 "1- d x (8.1) (see (5.8)). In (8.1), dx leaves both N1 and Na unchanged. By contrast, Al increases Nl by one unit without changing N2, while A2 increases Na by one unit without changing N1. The differential Al acts only on the fields B 1 and its associated antifields ("fields and antifields of the first set"), whereas the differential A2 acts only on the fields B 2 and its associated antifields ("fields and antifields of the second set"). Note that A1 + A2 = A'. Let a be a cocycle of A with b-degree < n - 1. Expand a according to the Nl-degree, a = ao + a l
+ a2
+ . . . + am, N l ( a j ) = j.
(8.2)
The equation A a = 0 implies A l a m = 0 for the term a,,~ of highest Nl-degree. Our analysis of the At-cohomology for a single p-form yields then am = c,~(/:/1) k + Al(something), where cm involves only the fields and antifields of the second set, as
Characteristic Cohomology of p-Form Gauge Theories
155
well as d x ~ and possibly x u. There can be no conserved current in am since we assume the A-degree of a - and thus of each aj - to be strictly smaller than n - 1. Now, the exact term in am can be absorbed by adding to am a A-exact term, through a redefinition of a m - 1 . Once this is done, one finds that the next equation for am and am-1 following f r o m / 1 a = 0 reads [(/12 + dx)cm](ffI1) k + Alam-1 = 0.
(8.3)
But we have seen that Am(H1) k cannot be exact unless it is zero, and thus this last equation implies both [(A2 + dx)cm](/~l) k = 0
(8.4)
~11am-1 = O.
(8.5)
and
Since (~1)k has independent form components in degrees k ( n - p 1), k ( n - p 1)+ 1 up to degree n, we infer from (8.4) that the form components of (/12 + dx)cm of degrees 0 up to degree n - k ( n - p - 1) are zero. If we expand Cm itself according to the form degree, c,,~ = ~ c~, this gives the equations ~eim +dcim-1 = O, i = 1 , . . . , n -
k(n-p-
1),
(8.6)
and ~c~ = 0.
(8.7)
Our analysis of the relationship between the Zl-cohomology and the cohomology of modulo d indicates then that one can redefine the terms of form degree > n - k ( n - p - 1) of Cm in such a way that Acre = 0. This does not affect the product C m ( f t l ) k. We shall assume that the (irrelevant) higher order terms in Cm have been chosen in that manner. With that choice, cm is given, up to trivial terms that can be reabsorbed, by )~m(/12) t, with )~,,~ a number, so that am = Am(/7/2)t(H1) k is a / 1 - c o c y c l e by itself. One next repeats successively the analysis for a m - 1 , a m - 2 to reach the desired conclusion that a may indeed be assumed to be a polynomial in the ~ a , s , as claimed above. The non-triviality of the polynomials in the/~ra,s is also easy to prove. If p(/7/) = Ap, with p = Po + Pl + . . . + Pro, N l ( p k ) = k, then one gets at Nl-degree m + 1 the condition ( P ( H ) ),~ +I = A I Pro, which implies ( P ( H ) )m +I = 0 and A 1pro. = O, since no polynomial i n / ~ l is Al-trivial, except zero. It follows that P m = u(/7/1) m up to trivial terms that play no role, where u is a function of the variables of the second set as well as of x u and d x ~. The equation of order m implies then ( P ( H ) ) m = ((A2 + dx)u) ([tl)m + A l P m - 1. The non-triviality of the polynomials in t7/1 in A v c o h o m o l o g y yields next A l p , ~ _ l = 0 and (P(/q))m = ((/12 + d x ) u ) ( f 1 1 ) m. Since the coefficient of (/7/1)m in (P(-f-1))m is a polynomial in/it2, which cannot be (//2 + dx)-exact, one gets in fact ( P ( H ) ) m = 0 and (/12 + d x ) u = 0. It follows that Pm fulfills ~1Pro = 0 and can be dropped. The analysis goes on in the same way at the lower values of the /11-degree, until one reaches the desired conclusion that the exact polynomial P(/~) indeed vanishes. In view of the isomorphism between the characteristic cohomology and H* (A), this completes the proof of Theorem 2.1 in the case of two p-forms. The case of more p-forms is treated similarly and left to the reader.
156
M. Henneaux, B. Knaepen, C. Schomblond
9. Invariant Characteristic Cohomology 9.1. Isomorphism theorems for the invariant cohomologies. To compute the invariant characteristic cohomology, we proceed as follows. First, we establish isomorphism thek,i~ (d), Hn_ n # k~ (5ld) and H k # ~ ( A ) . Then, we compute Hk'~n~(A) orems between H~h~ for a single p-form. Finally, we extend the calculation to an arbitrary systems of p-forms. Theorem 9.1. k,inv Hchar (d)
~k
0
n,inv
~-- H~_ k (~ld), 0 _< k < n,
(9.1)
~
(9.2)
H~;~(5ld), k > 0.
Theorem 9.2. The invariant cohomology of A is isomorphic to the invariant characteristic cohomology, k,inv Hk'inv(z~) "~ Hchar (d), 0 < k < n.
(9.3)
Proof First we prove (9.1). To that end we observe that the map m introduced in the k inv n inv demonstration of Theorem 4.1 maps Hc~a~ (d) on H~'_ k ((~Id). Indeed, in the expansion (4.8) for a, all the terms can be assumed to be invariant on account of Theorem 6.1. The surjectivity of m is also direct, provided that the polynomials in the curvature P ( H ) are not trivial in H*(fild), which is certainly the case if there is a single p-form (Theorem 7.5). We shall thus use Theorem 9.1 first only in the case of a single p-form. We shall then prove that Theorem 7.5 extends to an arbitrary system of forms of various form degrees, so that the proof of Theorem 9.1 will be completed. To compute the kernel of m, consider an element a0k E Z such that the corresponding n,inv a~_ k is trivial in Hn_ k (5[d). Then, again as in the proof of Theorem 4.1, one finds that all the terms in the expansion (4.8) are trivial, except perhaps a0k, which fulfills da k + 6db~ = 0, where bk E 2" is the k-form appearing in the equation expressing the triviality of a~ +1, a~+l = db~ +Sbk2+l. This implies d(a~ - 6b~) = 0, and thus, by Theorem 6.2, a k = P + db~-l + 5b~ with P E 7-/k and b~-1 E Z. This proves (9.1), since P is not trivial in H*(5[d) (Theorem 7.5). [Again, we are entitled to use this fact only for a single p-form until we have proved the non-triviality of P in the general case.] The proof of (9.2) is a direct consequence of Theorem 6.1 and parallels step by step the proof of a similar statement demonstrated for 1-forms in [14] (Lemma 6.1). It will not be repeated here. Finally, the proof of Theorem 9.2 amounts to observing that the map m ~ that sends [a0k] on [a] (Eq. (4.8)) is indeed well defined in cohomology, and is injective as well as surjective (independently of whether P ( H ) is trivial in the invariant cohomology of 3 modulo d). Note that if the forms do not depend explicitly on x, one must replace (9.1) by k ir~v d
(A | 7-t)k - H~:-k (~ld).
[]
(9.4)
9.2. Case of a single p-form gauge field. Theorem 6.3 enables one to compute also the invariant characteristic cohomology for a single p-form gauge field. Indeed, this n,inv theorem implies that H~_ k ( 5 ]d) and H~_k(51d ) actually coincide since the cocycles of
Characteristic Cohomology of p-Form Gauge Theories
157
H n - k (~ld) are invariant and the coboundary conditions are equivalent. The isomorphism of Theorem 9.1 shows then that the invariant characteristic cohomology for a single pform gauge field in form degree < n - 1 is isomorphic to the subspace__of form degree < n - 1 of the direct sum 7-/G 7-/. Since the product H A H has form degree n, which exceeds n - 1, this is the same as the subspace W of Theorem 2.2. The invariant characteristic cohomology in form degree k < n - 1 is thus given by (7"t | ~ ) k , i.e., by the invariant polynomials in the curvature H and its dual H with form degree < n - 1. Similarly, by the isomorphism of Theorem 9.2, the invariant cohomology H k #n~(A) of A is given by the polynomials in t7/and H with A-degree smaller than n - 1. 9.3. Invariant cohomology o f A in the general case. The invariant A-cohomology for an
arbitrary system of p-form gauge fields follows again from a straightforward application of the Kunneth formula and is thus given by the polynomials in the ~ a , s and H a's with b-degree smaller than n - 1. The explicit proof of this statement works as in the noninvariant case (for that matter, it is actually more convenient to use as degrees not N1 and N2, but rather, degrees counting the number of derivatives of the invariant variables X's. These degrees have the advantage that the cohomology is entirely in degree zero). In particular, none of the polynomials in the 9 a , s and H a's is trivial. The isomorphism of Theorem 9.2 implies next that the invariant characteristic COhomology nk,inv char (d) (k < n - l) is given by the polynomials in the curvatures H a and their duals ~ a , restricted to form degree smaller than n - 1. Among these, those that involve the curvatures H ~ are weakly exact, but not invariantly so. The property of Theorem 7.5 thus extends as announced to an arbitrary system of dynamical gauge forms of various form degrees. Because the forms have now different form degrees, one may have elements in k,inv (d) (k < n - l) that involve both the curvatures and their duals. For instance, if Hchar B 1 is a 2-form and B 2 is a 4-form, the cocycle H 1 A ~ 2 is a (n -- 2)-form. It is trivial in H~h~(d), k but not in H ~k inv (d).
10. Invariant Cohomology of 6 mod d
n,inv
The easiest way to work out explicitly H~_ k (~]d) in the general case is to use the above isomorphism theorems, which we are now entitled to do. Thus, one starts from H k # '~" (A) and one works out the component of form degree n in the associated cocycles. Because one has elements in H k # n v ( A ) that involve simultaneously both the curvan,inv d and H,~_k(~ld ) coincide ture and its A-invariant dual/~, the property that H~_ k (~1) may no longer hold. In the previous example, one would find that H ( ~ v B *(2)~, which has antighost number two, is a ~-cocycle modulo d, but it cannot be written invariantly n#nv so. An important case where the isomorphism H ~ _ k (~ld) ~ H~_k(~ld) (k > l) does hold, however, is when the forms have all the same degrees. To write down the generalization of Theorem 6.3 in the case of p-forms of different degrees, let P ( H ~, ~ a ) be a polynomial in the curvatures (Pa + 1)-forms H a and their A-invariant duals/7/% One has A P = 0. We shall be interested in polynomials of Adegree < n that are of degree > 0 in both H a and ~ a . The condition that P be of degree > 0 in H ~ implies that it is trivial (but not invariantly so), while the condition that it be of degree > 0 i n / / a guarantees that when expanded according to the antighost number, P has non-vanishing components of antighost number > 0,
158
M. Henneaux, B. Knaepen, C. Schomblond n
P = ~_~[P]~-k"
(10.1)
j=k
From A P = 0, one has 8 [ P ] ~ - k + d[p]n--~-I = O. There is no polynomial in H a and/Qa with the required properties if all the antisymmetric tensors B /*l--->pa a have the same form degree (Pa = P for all a's) since the product H a l l b has necessarily A-degree n. When there are tensors of different form degrees, one can construct, however, polynomials P with the given features. The analysis of the previous subsection implies straightforwardly. T h e o r e m 10.1. Let aq = aq (X) E I be an invariant local n-form o f antighost number
q > O. Ifa~ is 5-exact modulo d, aqn = 5#qnl + dlZqn--1 , then one has 'n 1 + d # ~ - I aqn = [P]q + al~q+
(10.2)
for some polynomial P ( H ~, [I ~) o f degree at least one in H a and at least one in [1 a, and where #'qn1 and # ~ -1 can be assumed to depend only on the X 's, i.e., to be invarianr In particular, if all the p-form gauge fields have the same form degree, [P];~ is absent and one has ' '~-I aqn = alZqn+l + d #
(10.3)
where one can assume that #'qn+l and lzq -~ are invariant (p'qn+l and p ~ - I E Z).
11. R e m a r k s on Conserved Currents
That the characteristic cohomology is finite-dimensional and entirely generated by the duals ~ a , s to the field strengths holds only in form degree k < n - 1. This property is not true in form degree equal to n - 1, where there are conserved currents that cannot be expressed in terms of the forms H a , even up to trivial terms. An infinite number of conserved currents that cannot be expressible in terms of the forms ~ a are given by T~s163 1 ... O~~/~l ... ~ T
=
L ( 1p!H ~ p l . . .pp,oq O~sHlj~I'''pB ,fll...~r ...
-2
1 H* H *p2''pn-v-1 ) (n - p - 2)!--~P2""Pn-P-I"~'"a~ ~ ,3,...~,-"
(11.1)
These quantities are easily checked to be conserved T vvq t* ...c~/51 ...3r,# = 0
(11.2)
and generalize the conserved currents given in [15, 16, 17] for free electromagnetism. They are symmetric for the exchange of # and u and are duality invafiant in the critical dimension n = 2p + 2, where the field strength and its dual have the same form degree p + 1. In this critical dimension, there are further conserved currents which generalize the "zilches", Zt~Ucq...c~fll...3s
=
H~Zal...ap,cq...~ H , V
~1 ...O'p
_H.#al
...ap,cq
,~1 ' " ~
..... H~ , .... p ,/~1...~,
(11.3)
Characteristic Cohomology of p-Form Gauge Theories
159
Let us prove that the conserved currents (11.1) which contain an even total number of derivatives are not trivial in the space of z-independent local forms. To avoid cumbersome notations we will only look at the currents with no/~ indices. One may reexpress (11.1) in terms of the field strengths as TI*~,,~,...,~
1
_
2p!
( H n`~, . . . . p,al...OZm HUal
1 - - H +rf*~2(p+ 1)!
~ .... ~§
"
,~ + H~*o~...O~, H ~' ~ ~1 . . . .
H ~' "''~
,,~, . . . . . )
'~ . . . . .
(11.4)
If one takes the divergence of this expression one gets, Tt*~'~' . . . . .
,t*
=
5 K~"~'""~,
(11.5)
where K ~ . . . . . differs from k H % ~ . . . ~ , , ,'~I'"'~'"B*~I"'~, by a divergence. It is easy to see that T ~/jO/1*~ is trivial if and only if H%~ .... p ,~'''~m B * " ' " ' ~ is trivial. So the question is: can we write, H ~,
(9"1 . . . O p
,,~, . . . . . B . a , . . . o - p = 6 M U a l . . . . .
+ O p N P U a l ...a,~
(11.6)
for some M ~ l ' ' ' ~ m and N puc~ . . . . . .9 Without loss of generality, one can assume that M and N have the Lorentz transformation properties indicated by their indices (the parts of M and N transforming in other representations would cancel by themselves). Moreover, by Theorem 6.3, one can also assume that M and N are gauge invariant, i.e., belong to I . If one takes into account all the symmetries of the left-hand side and use the identity d H = 0, the problem reduces to the determination of the constant c in, H u o . , . . . a v , a , . . . . . . B *al'''crp
=
5(cH~,,~, . . . . p-~(~,,~2 . . . . . )B*~' .... "-~)
+ O p N ~ , ~ l . . . e , m + terms that vanish on-shell.
(11.7)
If one takes the Euler-Lagrange derivative of this equation with respect to B *~'''~p one gets, (11.8) HmTl...c~p,oq...o~m ~ ( - - ) p+l CH v[~l...~p-ll(m,~=.-.~,,)l~l' where the right-hand side is symmetrized in c q . . . c ~ and antisyinmetrized in al .. 9 ap. The symmetry properties of the two sides of this equation are not compatible unless c = 0. This proves that T #v~l . . . . . (with m even) is not trivial in the algebra of xindependent local forms. It then follows, by a mere counting of derivative argument, that the T u " ~ ' ' ' " " define independent cohomological classes and cannot be expressed as polynomials in the undifferentiated dual to the field strengths H with coefficients that are constant forms. The fact that the conserved currents are not always expressible in terms of the forms ~ a makes the validity of this property for higher order conservation laws more striking. In that respect, it should be indicated that the computation of the characteristic cohomology in the algebra generated by the H a is clearly a trivial question. The non-trivial issue is to demonstrate that this computation does not miss other cohomological classes in degree k < n - 1. Finally, we point out that the conserved currents can all be redefined so as to be strictly gauge-invariant, apart from a few of them whose complete list can be systematically determined for each given system of p-forms. This point will be fully established in [ 18], and extends to higher degree antisymmetric tensors a property established in [44] for one-forms (see also [45] in this context).
160
M. Henneaux, B. Knaepen, C. Schomblond
12. Introduction of Gauge Invariant Interactions The analysis of the characteristic cohomology proceeds in the same fashion if one adds to the Lagrangian (1.10) interactions that involve higher dimensionality gauge invariant terms. As we shall show in [18], these are in general the only consistent interactions. These interactions may increase the derivative order of the field equations. The resulting theories should be regarded as effective theories and can be handled through a systematic perturbation expansion [46]. The new equations of motion read
OuE ~m''m'''~'p" = 0,
(12.1)
where/2 am'la2"''"pa are the Euler-Lagrange derivatives of the Lagrangian with respect to the field strengths (by gauge invariance, E involves only the field strength components and their derivatives). These equations can be rewritten as d ~ a ~ 0,
(12.2)
where Z a is the (n - pa - l)-form dual to the Euler-Lagrange derivatives. The Euler-Lagrange equations obey the same Noether identities as in the free case, so that the Koszul-Tate differential takes the same form, with H~ replaced everywhere by Z ~. It then follows that p+l
E~ = Z~ + Z
--g,(z
Bj
(12.3)
j=l
fulfills AZ~a = 0.
(12.4)
This implies, in turn, that any polynomial in the s is A-closed. It is also clear that any polynomial in the Z a is weakly d-closed. By making the regularity assumptions on the higher order terms in the Lagrangian explained in [9], one easily verifies that these are the only cocycles in form degree < n - 1, and that they are non-trivial. The characteristic cohomology of the free theory possesses therefore some amount of"robustness" since it survives deformations. By contrast, the infinite number of non-trivial conserved currents is not expected to survive interactions (even gauge-invariant ones). [In certain dimensions, one may add Chern-Simons terms to the Lagrangian. These interactions are not strictly gauge invariant, but only gauge-invariant up to a surface term. The equations of motion still take the form d(something) ~ 0, but now, that "something" is not gauge invariant. Accordingly, with such interactions, some of the cocycles of the characteristic cohomology are no longer gauge invariant. These cocycles are removed from the invariant cohomology, but the discussion proceeds otherwise almost unchanged and is left to the reader.]
13. Summary of Results and Conclusions In this paper, we have completely worked out the characteristic cohomology//~ha,.(d) in form degree k < n - 1 for an arbitrary collection of free, antisymmetric tensor theories. We have shown in particular that the cohomological groups H~ha,.(d) are finite-dimensional and take a simple form, in sharp contrast with H~h~l~(d), which is
Characteristic Cohomologyof p-FormGaugeTheories
161
infinite-dimensional and appears to be quite complex. Thus, even though one is dealing with free theories, which have an infinite number of conserved local currents, the existence of higher degree local conservation laws is quite constrained. For instance, in ten dimensions, there is one and only one (non-trivial) higher degree conservation law for a single 2-, 3-, 4-, 6-, or 8-form gauge field, in respective form degrees 7, 6, 5, 3 and 1. It is d H ~ O. For a 5-form, there are two higher degree conservation laws, namely d H ~ 0 and d(H) 2 ~ 0, in form deg_rees 4 and 8. For a 7-form, there are four higher degree conservation laws, namely d H ~ 0, d ( n ) 2 ~ 0, d ( n ) 3 ~ 0 and d(H) 4 ~ 0, in form degrees 2, 4, 6 and 8. Our results provide at the same time the complete list of the isomorphic groups Hk(A), as well as of H~_k(51d). We have also worked out the invariant characteristic cohomology, which is central in the investigation of the BRST cohomology since it controls the antifield dependence of BRST cohomological classes [14]. An interesting feature of the characteristic cohomology in form degree < n - 1 is its "robustness" to the introduction of gauge invariant interactions, in contrast to the conserved currents. As we pointed out in the introduction, the characteristic cohomology is interesting for its own sake since it provides higher degree local conservation laws. But it is also useful in the analysis of the BRST cohomology. The consequences of our study will be fully investigated in a forthcoming paper [18], where consistent interactions and anomalies will be studied (see [47] for the 2-form case in this context). In particular, it will be pointed out how rigid the gauge symmetries are. We will also apply our results to compute the BRST cohomology of the coupled Yang-Mills-two-form system, where the field strength of the 2-form is modified by the addition of the Chem-Simons 3-form of the Yang-Mills field [48]. This computation will use both the present results and the analysis of [50, 36, 49, 14]. Acknowledgement. M.H. is gratefulto LPTHE(Universit6sParis VI and Pads VII) for kind hospitality.This workhas beenpartly supportedby researchfunds fromF.N.R.S. and a researchcontractwith the Commission of the,EuropeanCommunity.
A. Proof of Theorem 6.2
To prove Theorem 6.2, it is convenient to follow the lines of the BRST formalism. In that approach, gauge invariance is controlled by the so-called longitudinal exterior derivative operator "7, which acts on the fields and further variables called ghosts. The construction of "7 can be found in [31, 30]. For simplicity, we consider throughout this appendix the case of a single p-form; the general case is covered by means of the Kunneth formula. The important point here is the reducibility of the gauge transformations. Because of this, we need to introduce p ghost fields: Cm...~p_l,... , Cm...~p_j,... , C.
(A.1)
These ghosts carry a degree called the pure ghost number. The pure ghost number of Cm...t~p_ , is equal to 1 and increases by one unit up to p as one moves from the left to the right of (A. 1). The action of '7 on the fields and the ghosts is given by, "TB
=
dC1,
(A.2)
"7C1
=
dC2,
(A.3)
162
M. Henneaux, B. Knaepen, C. Schomblond
"TCp_ 1
=
"TCp = "7(anti field) =
dVp,
(A.4)
0,
(A.5)
0.
(A.6)
In the above equations, Cj is the p - j form whose components a r e C~l ...l~p_j. For p even, Cp is a commuting object. One extends '7 such that it is a differential that acts from the left and anticommutes with d. The motivation behind the above definition is essentially contained in the following theorem:
Theorem A.1. The cohomology of'7 is given by,
H('7) = Z | Cp,
(A.7)
where Cp is the algebra generated by the last, undifferentiated ghost Cp. In particular, in antighost and pure ghost numbers equal to zero, one can take as representatives of the cohomological class the gauge invariant functions, i.e, the functions which depend solely on the field strengths and their derivatives. [It is in that sense that the differential "7 incorporates gauge invariance.] The proof of this theorem follows the lines given in [43], by redefining the generators of the algebra so that "7 takes the standard form 'Txi = Yi, 'Tyi = 0, 'Tz~ = 0 in terms of the new generators xi, yi, z~. The paired variables xi, Yi disappear from the cohomology, which is entirely generated by the unpaired variables z,~. In the present case, one easily convinces oneself that the generators of Z | Cp are precisely of the z~-type, while the other generators come in pairs. The derivatives of the last ghost Cp are paired with the symmetrized derivatives of the next-to-last ghost Cp_lu, the other derivatives of the next-to-last ghost Cp_lu which may be expressed as derivatives of the "curvatures" 0uCp_ 1~ - O~Cp_ l u, are paired with the derivatives of the previous ghost Oat .... k C(p-2),u~ involving a symmetrization, say on cq and #, etc. The details present no difficulty and are left to the reader. According to the theorem, any solution of the equation "Ta = 0 can be written,
a=E
c~z(x)CZ + "Tb.
(A.8)
l
Furthermore, if a is 7-exact, then one has c~t ~ 0 since the various powers of C are linearly independent. The previous theorem holds independently of whether p is even or odd. We now assume that p is even, so that the curvature (p + 1)-form H is anticommuting and the last ghost Cp is commuting, and prove Theorem 6.2 in that case (the case when p is odd parallels the 1-form case and so need not be treated here). Assume that da~) = 0 with a 0k a polynomial in the field strengths and their derivatives. By the Poincar6 lemma we have a~ = da~ -1, but there is no guarantee that a0k- 1 is also in 7?s,~au. Acting with "7 on this equation we get, again using the Poincar6 lemma, "Tako-1 + da~ -2 = 0. One can thus construct a tower of equations which take the form,
Characteristic Cohomology of p-Form Gauge Theories
163
a~
=
da k - l ,
"ya~- l +dak1-2
=
O,
(A.IO)
=
O,
(A.11)
=
0.
(A.12)
"yakq-l-q
- k-2-q
+aaq+ 1
k--2-q 'yaq+ 1
Let
r = k -
2 -
q and
q + 1 = m.
If m
=
(A.9)
pl then the last equation o f the tower implies,
r = CplP + "Yam-1, am
(A.13)
with P C 2-s,~,al. If m ~ pl then we simply have arm = "ya~_ 1- In that case, an allowed redefinition o f the tower allows one to suppose that the tower stops earlier with "yam,r = 0 and m' = pl. A n allowed redefinition of the tower simply adds to a0k a term of the form dbko- 1, where b0k - 1 is gauge invariant. So from now, we shall assume that indeed m = pl. If we substitute (A.13) in (A.11) we get, t ~§ 1 + 1Cp-lCtp - ' ) + CZpdP = 0 (A.14) "y~am_ (the trivial term ' y a ~ _ 1 is absorbed in an allowed redefinition of the tower). Since the action of d is well defined in ~small this implies d P = O. The form degree of P is strictly smaller than the form degree of a0k, so let us make the recurrence hypothesis that the theorem holds for P . Because we treat the case p even, H is odd and P = d H + c + dQ, where c and c I are constants and Q E ZsmaU. We thus have,
a r =cClp +c'CZpH +dQelp.
(A.15)
The last two terms in (A. 15) are trivial. For the first one we have,
dBCZp = d(QCZp) - "y(QtCp_,c~).
(A.16)
Then we note that,
-
1
l+--~ (d(
E
Ci, Ci2""Ci~.,)
(A.17)
..+il+l=pl
i l +. 0_
--'Y(
E C{, Ci2... Cil+l)). il+...+il+l=Pl--I
(A.18)
0 < i 1 < p , . . . ,0__
To prove the above identity one sets Co = B and Cp+l - 0 since the ghost of highest pureghost is Cp. One also uses the fact that 'yCi -- dCi§ This shows that we only need to consider the bottom a fm cC~. Let us now prove that if I > 1 then c = O. We can write,
a pl o =c
~ il+...+il=Pl
Ci, C~:... Ci,.
0 < i I < p , . . . , 0 < i , + 1 _
It is easy to show that for p(l - 1) + 1 < k < pl we have,
(A.19)
164
M. Henneaux, B. Knaepen, C. Schomblond
Ci, Ci: 9 9Ci~). il+...*il=te o~_zl~_p,...,O~_il~_p
(A.20)
il+...+il=k--1 o<~ 1 _
This implies that, 1-1
aP(l-1) = (--)Pc p
(A.21)
i l +. ..+il=p(l--1) O--
where Q E 2-~m~lz. If we substitute this in .~,~p(Z-1)-I + daP(Z_l) = 0 we get, I~p+l
(-1)PcIHCtp -1 +dOClp-t +~((-)Pc
Z
CilCi2""Ciz
zl+...+il=P(l--1) 1 0 < i l <_p,... ,O~iz+ l _
+Q(l -
l )Cp_lCIp -2 + r
= O.
(A.22)
The above equation implies,
dQ + (-1)PcIH = 0.
(A.23)
Because the form degree of H is strictly smaller than the form degree of a0k, the above recurrence hypothesis tells us that this equation is impossible unless c = 0. To show that c = 0 we had to lift the bottom a~ 1 p + 1 times. This is only possible when the tower has p + 1 steps, which is the case when I > 1. If I = 1 then the bottom is ato = cCp. This bottom can be lifted to the top of the tower and yields a0p+I = c H + dN, N E Zsmau. To validate the recurrence hypothesis we observe that if a0k is of form degree 0 then necessarily a0~ = k. This ends the proof of the theorem and shows that forp even we have a = e+ctH+dN, N E Zsmall, as desired.
References 1. Bryant, R.L., Griffiths, E: Characteristic Cohomology of Differential Systems (I): General Theory. Duke University Mathematics Preprint Series, volume 1993 no 1, January 1993 2. Vinogradov, A.M.: Sov. Math. Dokl. 18, 1200 (1977); Sov. Math. Dokl. 19, 144 (1978) 3. Vinogradov, A.M.: Soy. Math. Dokl. 20, 985 (1979); J. Math. Anal. Appl. 100, 1 (1984) 4. Tsujishita, %: Osaka J. Math. 19, 311 (1982); Diff. Geom. Appl. 1, 3 (1991) 5. Unruh, W.: Gen. Rel. Grav. 2, 27 (1971) 6. Barnich, G., Bran&, E, Henneanx, M.: Nucl. Phys. B455, 357 (1995) 7. Torte, C.G.: Class. Quant. Grav. 12, L43 (1995) 8. Misner, C.W., Wheeler, J.A.: Ann. Phys. (N.Y.) 2, 525 (1957) 9. Barnich, G., Bran&, E, Henneaux, M.: Commun. Math. Phys. 174, 57 (1995) 10. Voronov, B.L., Tyutin, I.V.: Theor. Math. Phys. 52, 628 (1982) 11. Gomis, J., Weinberg, S.: hep-tld9510087, to appear in Nucl. Phys. B469, No 3 (1996) 12. Stora, R.: In: New Developments in Quantum Field Theory and Statistical Mechanics. Carg~se 1976, Levy, M., Mitter, E (eds.), NATO ASI Series B26, New York: Plenum Press, 1977 13. Barnich, G., Hermeaux, M.: Phys. Lett. B311, 123 (1993) 14. Barnich, G., Brandt, E, Henneanx, M.: Commun. Math. Phys. 174, 93 (1995) 15. Lipkin, D.M.: J. Math. Phys. 5, 698 (1964) 16. Morgan, T.A.: J. Math. Phys. 5, 1659 (1964) 17. Kibble, T.W.: J. Math. Phys. 6, 1022 (1965); O'Connell, R.E, Tompkins, D.R.: J. Math. Phys. 6, 1952 (1965)
Characteristic Cohomology of p-Form Gauge Theories
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. 47. 48. 49. 50.
165
Henneaux, M., Knaepen, B., Schomblond, C.: In preparation Henneaux, M.: Phys. Lett. B368, 83 (1996) Ogievetsky, V.I., Polubarinov, I.V.: Sov. J. Nucl. Phys. 4, 156 (1967) Kalb, M., Ramond, P.: Phys. Rev. D9, 2273 (1974) Freedman, D.Z., Townsend, P.K.: Nucl. Phys. B177, 282 (1981) Nepomechie, R.I.: Phys. Rev. D31, 1921 (1984) Teitelboim, C.: Phys. Lett. B167, 63,69 (1986) Green, M., Schwarz, J., Witten, E.: Superstring Theory (2 vols). Cambridge: Cambridge University Press, 1987 Siegel, W.: Phys. Lett. B93, 275 (1980) Thierry-Mieg, J.: Nucl. Phys. B335, 334 (1990) Baulieu, L., Thierry-Mieg, J.: Nucl. Phys. B228, 259 (1983) Baulieu, L.: In: Perspectives in Particles and Fields. Carg6se 1983; Levy, M., Basdevant, J.-L., Speiser, D., Weyers, J., Jacob, M., Gastmans, R. (eds.), NATO ASI Series B126, New York: Plenum Press, 1983 Henneaux, M., Teitelboim, C.: Quantization of Gauge Systems. Princeton: Princeton University Press, 1992 Batalin, I.A., Vilkovisky, G.A.: Phys. Rev. D28, 2567 (1983); Phys. Rev. D30, 508 (1984) Fisch, J., Henneaux, M., Stasheff, J., Teitelboim, C.: Commun. Math. Phys. 120, 379 (1989) Fisch, J.M.L., Henneaux, M.: Commun. Math. Phys. 128, 627 (1990) Henneaux, M.: Commun. Math. Phys. 140, 1 (1991) Vinogradov, A.M.: Sov. Math. Dokl. 19, 1220 (1978); De Wilde, M.: Lett. Math. Phys. 5, 351 (1981); Tulczyjew, W.M.: Lecture Notes in Math. 836, 22 (1980); Dedecker, P., Tulczyjew, W.M.: Lecture Notes in Math. 836, 498 (1980); Tsujishita, T.: Osaka J. of Math. 19, 311 (1982); Bonora, L., Cotta-Ramusino, P.: Commun. Math. Phys. 87, 589 (1983); Olver, P.J.: Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics, Vol. 107, New York: Springer Veflag, 1986; Wald, R.M.: J. Math. Phys. 31, 2378 (1990); Dickey, L.A.: Contemp. Math. 132, 307 (1992) Brandt, E, Dragon, N., Kreuzer, M.: Nucl. Phys. B332, 224 (1990) Dubois-Violette, M., Henneaux, M., Talon, M, Viallet, C.M.: Phys. Lett. B267, 81 (1991) Stora, R.: In: Progress in Gauge Field Theory. 't Hooft, G., Jaffe, A., Lehmann, H., Mitter, P.K., Singer, I.M., Stora, R. (eds.), New York: Plenum Press, 1984 Thorn, C.B.: Phys. Rep. 175, 1 (1989) Baulieu, L., Bergshoeff, E., Sezgin, E.: Nucl. Phys. B307, 348 (1988) Carvalho, M., Vilar, L.C.Q., Sasaki, C.A.G., Sorella, S.P.: hep-th/9509047 Baulieu, L.: hep-tlg9512026 Dubois-Violette, M., Henneaux, M., Talon, M., Viallet, C.M.: Phys. Lett. B289, 361 (1992) Barnich, G., Brandt, F., Henneaux, M.: Phys. Lett. B346, 81 (1995) Torre, C.G.: J. Math. Phys. 36, 2113 (1995) Weinberg, S.: Physica 96A, 327 (1979) Bandelloni, G., Blasi, A.: Class. Quant. Grav. 10, 1249 (1993) Nicolai, H., Townsend, P.K.: Phys. Lett. 98B, 257 (1981); Chapline, M., Maton, N.: Phys. Lett. 120B, 105 (1983); Bergshoeff, E., de Roo, M., de Wit, B., van Nieuwenhuizen, P.: Nucl. Phys. B195, 97 (1982) Bran&, F., Dragon, N., Kreuzer, M.: Nucl. Phys. B332, 250 (1990) Dubois-Violette, M., Talon, M., Viallet, C.M.: Math. Phys. 102, 105 (1985)
Communicated by R. H. Dijkgraaf
Commun. Math. Phys. 186, 167-197 (1997)
Communications in
Mathematical
Physics
(~) Springer-Verlag 1997
Estimate of the Difference between the Kac Operator and the Schriidinger Semigroup Takashi Ichinose*, Satoshi Takanobu** Department of Mathematics, Faculty of Science, KanazawaUniversity, Kanazawa, 920-11, Japan Received: 1 March 1996/Accepted: 28 October 1996
Abstract: An operator norm estimate of the difference between the Kac operator and the Schr6dinger semigroup is proved and used to give a variant of the Trotter product formula for Schr6dinger operators in the L p operator norm. This extends Helffer's result in the L 2 operator norm to the case in the L p operator norm for more general scalar potentials and with vector potentials. The method of the proof is probabilistic based on the Feynman-Kac and Feynman-Kac-It6 formula.
1. Introduction It is well-known (e.g. [8]) that if A and B are selfadjoint operators in a Hilbert space bounded below and A + B is essentially selfadjoint, the Trotter product formula lim ( e - t B / n e - t A / n ) n
=
(1.1)
e -to
~---40G
and its variant lim ( e - - t A / 2 n e - - t B / n e - - t A / 2 n ) n
=
e -tC
(1.2)
n----~ ( ~
hold in the strong operator topology, where C is the unique selfadjoint extension of A + B. More generally, it is Kato's Trotter product formula that (1.1) and (1.2) are valid also for the form sum C of A and B which is selfadjoint. In [2, 3 and 4], B. Helffer investigated the Kac operator (the transfer matrix) e - t V / 2 e - t ( - 89A ) e - - t V / 2 for a Kac model in statistical mechanics, associated with a potential V(x). In order to understand better its properties, he related in some asymptotic * Research partially supportedby Grant-in-Aidfor Scientific Research No. 07454023, Ministry of Education, Science and Culture, Japanese Government. ** Research partially supportedby Grant-in-Aidfor Scientific Research No. 07640293, Ministry of Education, Science and Culture, Japanese Government.
168
T. Ichinose,S. Takanobu
limit the spectral properties of the Kac operator to those of the Schrtdinger operator - 89A + V. To this end he proved in [4] (cf. [3]), when V(x) is a real-valued C~ in IRd bounded below by a constant b and satisfying [O~V(x)l <_ C~(1 + x2)<2-1~1)+/2 for every multi-index c~, the following remarkable estimate of the difference between the Kac operator and the Schrtdinger semigroup:
Ile-tV/ae -t(-~-A)e-tv/a- e-t(-89
=
O(t2),
(1.3)
as t $ 0, where a+ = max{a, 0} and II 9 IIt(L=> stands for the operator norm of bounded linear operators on La(Rd). Hence as a by-product the variant of the Trotter product formula in operator norm
II(e-tV/2•e-t( - 89A)/ne--tV/2n)n __ e-t(- 89A+V>IIaL > = n-lO(t2),
n ~ oo
(1.4)
is also readily derived. Here, as - 8 9 + V is essentially selfadjoint on C~(]R~), so its unique selfadjoint extension is also denoted by the same - 89A + V. The aim of the present paper is to extend Helffer's result to the case for more general potentials V(x) and even in the L p operator norm for 1 < p < oc. It also turns out that the results are true with the operator - 89A replaced by the magnetic SchrSdinger operator 89 - A(x)) 2. The main results, Theorems 2.1, 2.2 and 2.3, are stated in Sect. 2. They are proved in Sect. 3, where two further theorems are given, Theorems 3.1 and 3.2, to estimate the integral kernel of the difference between the Kac operator and the Schr6dinger semigroup. They are used to prove Theorems 2.1 and 2.3. Sections 4 and 5 are devoted to the proof of Theorems 3.1 and 3.2.
2. T h e M a i n T h e o r e m s
To formulate our theorems we want to consider the SchrSdinger operator H(A) + V = 89 - A(x)) 2 + V(x) not only in L2(]~a) but also in LP(]~d), 1 < p < oz, and also in the Banach space Co~(R d) of the continuous functions in ~d vanishing at infinity, equipped with L ~ norm. Here A(x) and V(x) are the vector and scalar potentials which are respectively an ]~d-valued and a real-valued functions in ]~a. By the result of Leinfelder-Simader [7] (cf. Kato [6]), H(A) + V is essentially selfadjoint on C~(~d), if A(x) is in L~oc(lRd) with div A(x) in L~o~(]~d) and if V(x) is in L~oc(~ a) and bounded below by a constant b. So its unique selfadjoint extension is also denoted by the same H(A) + V. Then its semigroup e -t(1-l(A)+V) is represented by the Feynman-Kac-It6 formula (e.g. Simon [101)
(e-t(H(A)+V) f )(x) E~ [e-i~(t'A) e x p ( -
~(t, A) =
/o
/o
)
l
V(X(s))ds f(X(t)) ,
jo
A(X(s))dX(s) + 89
divA(X(s))ds,
f E LZ(Ra),
(2.1a)
(2.1b)
KacOperator
169
where Ex means the expectation or integral with respect to the Wiener measure/zx on the space of the continuous paths X : [0, oo) ---+ R d starting at X(0) = x and the first integral on the right-hand side of (2. lb) is It6's stochastic integral. We can see via (2.lab) that the operators e - t ( n ( A ) + V ) defined as bounded operators on L2(IRd) extend from LP(]Rd) fq Lz(]R d) to bounded operators on LP(1Rd) for 1 < p < oo (cf. Simon [11]). e - t ( n ( A ) + V ) is a strongly continuous semigroup obeying
Ile-t(H(A)+W)fl[ p
_< e-tbllfllp ,
(2.2)
for f E LP(1Rd), 1 < p < oo. We denote also the generator Hp of this semigroup in LP(IRd) by the same H ( A ) + V. When p = oo, e -t(n(a)+V) is defined on Loo(R d) as the dual of the L 1 operator. It is not strongly continuous, but (2.2) holds for the p = oo operator. The p = oo operator Hoo is the adjoint of the p = 1 operator H1. In addition, if A ( x ) is a Cl-function and V ( x ) is continuous, the F e y n m a n - K a c It5 formula (2. lab) defines as well the strongly continuous semigroup e -t(H(A)+V) on Coo(IRa ) obeying (2.2). Needless to say, the operator H ( A ) + V becomes - 8 9 + V, if A(x) vanishes. In the following, I1" II stands for the operator norm of bounded operators on LP(IRd), 1 _< p _< ~ , or on Coo(llU). T h e o r e m 2.1. Let 0 < ~ <_ 1. Let m be a nonnegative integer such that m~ < 1 and (m + 1)~ _> 1. Suppose that V ( z ) is a Cm-function in ]Ra bounded below by a constant b which satisfies that
IO"v(x)l
b+ 1) '-I~1~,
_< c ( v ( x ) -
with a constant C > O, and further that OaV(x),
0 < [a[ < m,
I~1 = m ,
IO"V(x)- O~V(y)l <_ CIx- yl ~,
(2.3)
are HSlder-continuous: x , y E IRd,
(2.4)
with constants C > 0 and 0 < t~ <_ 1. Then it holds in the operator norm II " II of bounded operators on LP(IRd), 1 <_ p < c~, and on Coo(R d) that, as t ~ O,
IIe - t v / z e - t ( =
k A) e-tV/2 _ e - t ( - k A+v) ii
~ O(tl+(m+~)/2),
re=O, 1,
[
m _> 2.
O(t1+26),
(2.5)
Note that the condition (2.4) with t~ = 1 is equivalent to that O~V(x), Ic~l = m + 1, are essentially bounded. T h e o r e m 2.2. For the same function V ( x ) as in Theorem 2.1, the variant of the Trotter product formula in the operator norm II" II ofbounded operators on Lp(Rd), 1 <_p <_ c~, and on Coo(]~ d) holds: as n ~ ~ , 1[(e-tV/Zn e - t ( - 89A)/n e--tV/2n)n __ e - t ( - 89za+V) ]i
f n-(m+m/20(tl+tm+~)/2), n-z60(tl+26),
m = 0, 1, m > 2,
(2.6)
so that the left-hand side of(2.6) converges to zero uniformly on every compact t-interval in [0, c~).
170
T. I c h i n o s e , S. T a k a n o b u
Example 1. The functions X 2 (harmonic oscillator potential) and X 4 - - X 2 (double well potential), Ix I satisfy the conditions (2.3) and (2.4) for V(x) in Theorem 2.1 with (6, m , ~ ) = ( 89 (~,3, l 1), (1,0, 1), respectively, while }xl 1/2, Ixl 3/2 satisfy the conditions (2.3) and (2.4) with (5, m, ~) = (1,0, 89 (2, 1, 1), respectively. Remark 1. Helffer [4] (cf. [2, 31) was the first to prove (1.3) by the techniques of pseudodifferential operators when V(x) is a C~-function bounded below by b and satisfying IOc~V(x)] < Ca(1 + x2)(2-1al)*/2 for every multi-index a. His result is included in the case p = 2, m = 1,5 = 1/2 and ~ = 1 in Theorem 2.1. In fact, his condition implies that IO~V(x)]
<_ C ( V ( x ) - b + 1)(1-1al/2)+
(2.7)
for the same a. There he also cited that F. Daumer proved a result weaker than (2.5) with O(t l+min{1/2,~)) in place of O(tl+2~), when V(x) is a C~-function satisfying that [O~V(x)[ < C~(V(x) - b + 1)(1-1~16)+ for every a. Note this condition implies the conditions (2.3) and (2.4) with n = 1. With condition (2.7) Dia and Schatzman [1] also have recently given a functionalanalytic proof of (1.3).
Remark 2. Rogava [9] proved (1.1) and (1.2) in operator norm with convergence rate O ( n - 1 / 2 Inn), when the domain D[A] of A is included in the domain D[B] of B and A + B is selfadjoint on D[A + B] = D[A] N D[B] = D[A], so that B is A-bounded. Notice that in our Theorems 2.1 and 2.2, neither V is ( - 8 9 nor - 8 9 is V-bounded. Remark 3. The method of the present paper can show that the statements of Theo1 - - t V e -t(-!A) rems 2.1 and 2.2 are valid with e-tV/2e-t(-89 -tv/2 replaced by ~(e 2 + e-t(- 89 However, as for e-tVe-t(-~ A) and e-t(- 89 in place of e-tV/2e -t(- 89 e -tv/2, it can only show that, as t ~ O, I l e - t V e - t ( - 8 9 A} -- e-t(-89 and Ile-t(-89A)e -tV -- e-t(-89 are O(tl+~/2), i f m = 0, and 0(tl/2+5), i f m _> 1. Next we shall see these theorems are valid with the operator - 89A replaced by the magnetic Schr6dinger operator H ( A ) = 89 - A(x)) 2 with vector potential A(x) including the case of constant magnetic fields. Theorem 2.3. a) Suppose that V(x) is H61der-continuous, i.e. satisfies (2.4) with m = O, and that A(x) is in L4oc(~ d) with divA(x) in L~oc(~ d) (resp. A(x) is a Cl-function). Then it holds in the operator norm l l " ll of bounded operators on LP(~d), 1 < p < ~ ,
(resp. on C~(]~d)) that, as t ~ O, Ile-tV/2e-tH(a)e-tV/2 -- e-t(H(A)+V)II
=
o(tl+~/2)"
(2.8)
b) Suppose that V(x) satisfies (2.3) and (2.4) with m >__ 1, and that A(x) = (A1 (x), ..., Ad(X)) is a Cl-function which satisfies that the magnetic fields are bounded:
IOk&(z) -
OjAk(x)l
<_ M,
x E ]~d
(2.9)
for 1 < j, k < d with a constant M > O, and further that OyAk(x), 1 < j, k < d, are HOlder-continuous: I O j A k ( x ) - OjAk(y)l
<_ M i x -
yl ~,
x , y 9 ]Rd,
(2.10)
Kac Operator
171
for 1 < j, k <_ d with constants M > 0 and 0 <_ 7 <- 1 (By 7 = 0 we understand OjAk(x) bounded). Then it holds in the operator norm I1" II of bounded operators on LP(~d), 1 <_ p < c~, and on C~(]~ a) that, as t ~ O, IIe-tV/2 e-tH(A) e-tV/2 -- e-t(H(A)+V)II =
f
O(tl+O+m/2),
m = 1,
[
0(t1+25),
m >_ 2.
(2.11)
Consequently, in the cases a) and b), the variant of the Trotter product formula in the operator norm I1" II of bounded operators on LP(Ra), 1 <_p <_ c~z, and on C ~ ( R a) holds: as n ~ cx~,
[l(e-tV/2ne--tH(A)/ne--tV/2n) n -- e-t(H(A)+V)[I =
~ n-(m+m/20(tl+(m+m/2),
rn = O, 1,
[
m > 2,
n-250(tl+25),
(2.12)
so that the left-hand side of (2.12) converges to zero uniformly on every compact tinterval in [0, e~). Remark 4. Notice that the condition (2.9) with (2.10) on A(x) includes the case of constant magnetic fields. Remark 5. As Theorems 2.1 and 2.2 extend to Theorem 2.3, so Remark 3 is valid with H ( A ) in place of - 8 9 The method of our proof is probabilistic based on the Feynman-Kac and FeynmanKac-It6 formula (e.g. [10]).
3. Proofs of the Main Theorems
In this section Theorems 2.1, 2.2 and 2.3 are proved. To prove Theorems 2.1 and 2.3 we give stronger results in Theorems 3.1 and 3.2 which estimate the integral kernel of the difference between the Kac operator and the Schr6dinger semigroup.
3.1. Proof of Theorem 2.1. Put K(t)
=
e-tV/2e-t(- 89
-- e -t(- 89
t > O.
(3.1)
Multiplying K ( t ) by e t ( b - 1) and then taking V ( x ) - b+ 1 as a new V(x), we may suppose that V ( x ) _> 1, and the condition (2.3) holds with b = 1. Since K ( t ) are uniformly bounded operators on LP(~ d) and C~(IIU) in t > 0, and since C ~ ( R d) is dense in Lp(Rd), 1 < p < cxD, and C ~ ( R d) , we have only to show that for f E C ~ ( ~ d)
p,
IIK(t)fllp
=
m = O, 1,
{O(tl+(m+'~)/2)llfI O(t,+2Z)llfllv '
m > 2.
(3.2)
172
T. Ichinose, S. Takanobu
Here note that the L C ~ ( ~ d) c a s e follows as the dual of t h e L1 (]l;[d) case. By the Feynman-Kac formula (2.lab) we have for f E C~(]Rd),
- exp ( -
fotV(X(s))ds))f(X(t))],
(3.3)
where Ex is the expectation with respect to the Wiener measure #~ on the space Cx([0, ~x~) ~ ]~d) of the continuous paths X : [0, cx~) ---+ ]Rd with X(0) = x. If F(xl, ...,Xn) is a bounded Borel measurable function on (~d)n, then for 0 = So < 81 < " ' " < 8 n = 8 , n
Z --r F(xl,...,x~)
E~[F(X(sl),...,X(sn))I . . . .
d
JRd
• p ( 8 1 , XO -- X l ) P ( 8 2 - - 81, Xl -- X 2 ) " " ' p ( S n - - S n - - 1 , X n - - l - -
xn)dxl"" dxn, (3.4a)
with x0 = x, where
p(s, x) is the heat kernel, the integral kernel of e-~(- 89 p(s,x) = (27rs)-d/2e-~2/2~.
(3.4b)
We have
f x~p(s,x)dx = s, 1 < j < d; f Ixlap(s,x)dx = C(a)s a/2, (3.4c) .]Rd JRd with a constant C(a) depending on a > 0. Precisely speaking, C(a) also depends on the dimension d, but, since it is fixed throughout this paper, we suppress the dependence on d. We use the conditional expectation first E~[ 9 IX(t) = y] and next E0[ 9 IX(t) = 0] to rewrite (3.3) as
(K(t)f)(x) : f~df(y)p(t,x-y)Ez[exp(-~(V(x)+V(y))) -exp(- fotV(X(s))ds) lX(t):y]dy = ~ f(y)p(,,x-y)Eo[exp(-t(V(x)+V(y))) -exp(-fo'V(x+~(y-x)+X(s))ds ) X(t)=O]dy = ~af(y)p(,,x-y)Eo[exp(-~(V(x)+V(y))) -exp(- fotV(x+ ~(y-x)+ X:'~ =- dRfaf(y)p(t, x - y)d(t, x, y)dy,
(3.5a)
Kac Operator
173
where
d(t,x,y)
=
(3.5b)
Eo[v(t,x,y)]
with
v(t, x, y)
= e x p ( - - ~ ( V ( x ) + V(y)) ) -exp(-
Xo'~
fotV(x+
~(y-x)+
(3.5c)
Xo'~
in the fourth member of (3.5a) or in (3.5c) is the Brownian bridge: Xo'~
=
X(s)
-
~x(t),
(3.6)
o < s < t.
The function
k(t,x, y)
(3.7)
= p ( t , x - y)d(t, x, y)
is the integral kernel of the operator K ( t ) in (3.1). By Taylor's theorem
e-a=
e -b = - ( 1 - ea-b)e -a ~,(a - b)Je -~ - ~ . ( a - b) m§
-
/o 1dO(1 -
O)me -(1-O)a-Ob.
j=l
This yields the following expansion of v(t, x, y) in (3.5c):
v(t, x, y)
-- - (~(v(x) § v ~ ) ) - for v(x § ~(y x)§ ~ ~
~v,x,~v,~,
~ ( ~ ( V ( x ) + V ( y ) ) - fo V ( x + ~ ( y - x) . ~o , o , , . o j ~ j=2
m! ~(V(x) + V(y)) -
/o
. .1/ol
Y ( x + t(Y - x) + X~' (s))ds)
dO(1 - O)m
Put t
w(t, x, y) and
=
~(V(x) + V(y)) -
fo
V ( x + ~(y - x) + X~'~
(3.9)
174
T. Ichinose, S. Takanobu 3
d(t, x, y) = ~
d~(t, x, y),
(3.10a)
dl(t, x, y) = -Eo[w(t, x,y)]e -~(V(~)+V(y)),
(3.lOb)
i=1
m
d2(t, x, y) = - ~
~.Eo[w(t, z, y)J]e -~-(V(~)+V(y)),
(3.10c)
3=2
d3(t, x, y) -
'[
m! Eo w(t, x, y)m+l
/o1dO(1 -
fo
xexp(-(~(V(x)+V(y))-O
O)m
V(x+ t(Y-x)+X~'~
)1,
(3.10d)
where we note from the expansion (3.8) that if m = 1, the term d2(t, x, y) is absent, and if m = 0, both the terms dl(t, x, y) and d2(t, x, y) are absent. Then we have the following estimate of k(t, x, y) in (3.7) in terms of p(t, x - y) times a finite sum of the powers of t and Ix - Yl.
Theorem 3.1. For m = O,
Ik(t,x,y)l
<_ p ( t , x - y)[d(t,x,y)l,
(3.11a)
Id(t,x,y)l
<_ C l ( t l x - yl ~ +t1+~/2),
(3.11b)
with a constant C1 depending on C and ~. F o r m = 1, [k(t,x,y)[
<
p ( t , x - y)(lda(t,x,y)[ + Id3(t,x,y)]),
Id~(t,x,y)l
<_ C 2 ( t ] x - yl ~+~ +tl+(x+~)/2),
[d3(t,x,Y)l
~
C21(t251x-
(3.12a) (3.12b)
yl 2 + t 1+25)
+Cz2((tlx - yll+~) 2 + tz(l+(1+m/2)),
(3.12c)
with constants (22 and C22 depending on C and ~, and C21 depending on C and 5. Form>2,
3 Ik(t,x,y)l
<_ p ( t , x -
y) ~-~ld~(t,x,y)l,
(3.13a)
i=1 m
Iddt, x,y)l
<
~[~ Cj((t261x-
ylZ)J/2 +t(a+a6)j/2)
j=2
+Cm+l(t]x - ylm+~ + tl+(m+m/2), m
(3.13b)
m
[d2(t,x,y) I <_ Z { Z C i k ( ( t : 6 1 x - - y 1 2 ) J k / 2 j=2
+t(l+26)Jk/2 )
k=l
+Cj,m+l((t[x _ y[m+,)j + t(l+fm+~)/2)j)~, )
(3.13c)
Kac Operator
175
Id3(t, x,
Y)l <- ~ Cm+l,k((t 261x -- yl2) (m+t)k/2 + $(l+25)(m+l)k/2) k=l
+Cm+l,m+l((tlx - ylm+~) "~+1 + t(l+(m+a)/2)(m+l)).
(3.13d)
Here the constants Cj, 2 < j <_ m, depend on C , j and 5, the constant Cm+l depends on C, m and t% the constants Cjk, 2 < j < m + 1, 1 < k < m, depend on C, j, k, m and 5, and the constants Cj,m+l , 2 < j < m + 1 depend on C, j, m and ~. The proof of Theorem 3.1 is postponed to Sect. 4. Now we are in a position to complete the proof of Theorem 2.1, using Theorem 3.1. To do so first note by (3.4c) that for a > 0,
II
f~.
pit,. -
y)l" -ylalf(y)ldyllp
<- C(a)ta/Zllfllp,
f C LP(]~d),
(3.14)
for 1 < p < c~. This is obvious for p = oc. For 1 < p < co, this can be seen, because, by the H61der inequality with 1/p + 1/q = 1, the left-hand side of (3.14) is smaller than or equal to
f dx(i p(~,x-y)lx- ylady)PlqL p(l~,x,-y)ix-ylalf(y)lPdy)l/P,
J~d
J~d
d
which is by (3.4c) equal to the right-hand side of (3.14). Then for m = 0 we have from (3.1 lb),
II s~fd Ik(t, .,y)llf(y)ldyllp
C1 (C(/'~) +
1)tl+'~/Zllfllp
= c~,,~tt+'~/21[fllp ,
(3.15)
with a constant CI,~ depending on C and ~, which yields (2.5) for m = 0. Next for rn > 1 put 3
k(t,x,y)
=
Z
(3.16a)
ki(~'x'Y)'
i=l
ki(t,x,y)
=
p(t,x-
y)di(t,x,y),
i = 1,2,3,
(3.16b)
where k2(t, x, y) is absent if m = 1. Then for m = 1 we have from (3.12bc) [[ L d Ikl(t' ",Y)llf(y)ldyllp
_< C2(C(1 + t~) +
1)tt+<~+~>/Zllfll p
89 [I L e [k3(t, ",y)l[f(y)ldyl[p
(3.17a)
{ 6 2 1 ( 6 ( 2 ) + l ) t 1+2~
+C2z(C(2(1 + ~)) + 1)t2(l+(l+~)/2)}llf] ]p I ~1,--, 2 , ~ .,1+(1+,~)/211~l t' liJllp,
(3.17b)
176
T. Ichinose, S. Takanobu
with a constant C2,,~ depending on C and ~, because if m = 1, then 1/2 _< 5 < 1, so that 1 + 23 _> 2. It follows by (3.16a) that
IIA~ Ik(t,., y)l If(y)ldyllp < I[ f~e Ik~(t'"Y)tlf(Y)ldyllP + II ~e lk3(t' "'Y)llf(Y)IdytlP < C2,~tl+(l+'~
(3.17c)
yielding (2.5) for m = 1. Finally, for m _> 2 we have from (3.13bcd) by (3.14) II
f~ Ikl( t, ", Y)IIf(Y)[dyIIp m
+ 1)t (1+25)j/2 +
Cm+l(C(m + e;) + 1)tl+(m+m/21llf IIp
j=2
_< ~cm,~t rPfll~, 1+25
1
(3.18a)
f
[[]~d Ik2(t' "'Y)]]f(Y)]dy]IP m
m
-< {Z(Zcjk(c
)+
j=2 k=l
+Cj,m+l(C((m + ~)j) + 1)t (l+(m+~)/2)j) }l[fl [p
_< ~,~,5~ Elfllp, II f~d Ik3(t, ", Y)IIf(y)[dyI[p 1 f~
(3.18b)
.d+25
m
_<
~
Cm+l,k(C((m + 1)k) +
1)t (l§
% - -
k=l
+Cm+l,m+l(C((m + t~)(m + ~--
1 : (1+25)(m+1)/2 ~Cm,ot
with a constant
/r a Ip,
(3.18C)
Cm,5 depending on C, m
IIf~ Ik(t,.,y)llf(y)ldyllp a
1)t(l+(m+m/2)(m+l)}lifIIp
1)) +
and 5. It follows by (3.16a) that 3
<_
~--~llf~[k~(t,.,Y)llf(y)ldyllp i=1
a
< Cm,stl+zsllfllp. This yields (2.5) for m > 2, ending the proof of Theorem 2.1.
(3.18d)
Kac Operator
177
3.2. Proof of Theorem 2.2. We may again assume V(x) _> 1. Then, since all the operators e-t(-89 A) , e - t v and e-t(-89 are contractions, we have
II( e - t v / 2 n
e - t ( - 89A)/n e--tV/2n) n -- e - t ( - 89A+V) II n
l[ ~-'~(e-tV/2n e-t(- 89
j-1
j=l X ( e - t V / 2 n e - t ( - 89A)/ne--tV/2n -- e t(- 89A+V)/n)e-(n-J)t(- 89A+V)/n ][
<_ nlle-tV/2n =
e-tV/2n
~ nO((t/n)l+(m+€
_ e-t(- 89 A+V)/nll
= n-(m+~)/20(tl+(m+~)/2),
L nO((t/n) t+2~) =
n-2aO(tl+2a),
m=0,1, m_>2,
ending the proof of Theorem 2.2.
3.3. Proof of Theorem 2.3. Put K(t,A)
= e-tV/2e-tH(A)e -tv/2 -- e -t(H(A)+V),
t > 0.
(3.19)
By the Feynman-Kac-It6 formula (2.lab) we have for f E C~(]~d),
(K(t,A)f)(x)
= Ex [e-~V(t'A) ( e x p ( - ~ (V(x) + V(X(t))) ) -exp(-fotV(X(s))ds))f(X(t))],
(3.20)
with ~(t, A) in (2.1b). By the same argument as used to get (3.5abc) we rewrite (3.20) fls (K(t,A)f)(x)
=
f ~ f(y)p(t,x - y)Eo[e-i~(t'z'Y)v(t,x,y)]dy
=
[ f(y)p(t, x - y)d(t,x, y;A)dy, J~ d
(3.21a)
where
d(t,x,y;A)
= Eo[e-iV(t'~'Y)v(t,x,y)]
(3.21b)
with v(t, x, y) in (3.5c) or (3.8), and
qS(t, x, y)
lfo'
= 7
+
(y - z)A(x + }(y - x) + Xo'~
fo
+89
t A ( x + ~ ( y - x)+ X ot,o (s))dX~)t,o (s) divA(x + ~(y - x) + xt'~
(3.21c)
178
T. Ichinose, S. Takanobu
Then the function k(t, x, y; A) =
p(t, x - y)d(t, x, y; A)
(3.22)
is the integral kernel of the operator K ( t , A) in (3.19). Similarly to (3.10abcd) put with w(t, x, y) in (3.9), 3
d(t, x, y; A)
-- E
dz(t, x, y; A),
(3.23a)
l=l t
dl (t, x, y; A)
= -Eo[e-ie(t#'y)w(t, x, y ) l e - g (V(~)§
d2(t, x, y; A)
= -
~
.
.
(3.23b) t
~Eo[e-~e(t'x'y)w(t,x, y)a]e-7(v(~)+v(Y)),
(3.23c)
j=2
d3(t,x,y;A)
--
t
[e-iqs(t,x,y)w(t~x~y)m+l
-~Eo
• exp,.-(~(V(x)
+ V(y)) - 0
/o
fO 1 d O ( 1
-
O) "~
V ( x + ~(y - x) +
,
(3.23d)
where i f m = 1, the term d2(t, x, y; A) is absent, and ifm = 0, both the terms dl (t, x, y; A) and d2(t, x, y; A) are absent. Then we have the following estimates of k(t, x, y; A) in (3.22). Theorem 3.2. a ) F o r m = O,
Ik(t,x,y;A)l
<
p(t,x-y)ld(t,x,y;A)l,
(3.24)
where d( t, x, y; A) has the same bound as d( t, x, y) in (3.11b). b ) F o r m > 1, 3
] k ( t , x , y ; A ) I <<_ p(t, x - y) ~
Idz(t, x,
y; A) I,
(3.25a)
l=l
I d , ( t , x , y ; A ) I <<_ ~--~ Cj((t251x - yl2)J/2 + t(1+25)3/2) j=2
+Cm+, (fix - yl ~+~ + t '+(~+~)/2) + C[t '+a Ix - y[
+c~(t'/2+alx -
yl 2+'r + tl+a+~/21x - yl)
+C~t 3/2+6 + C~(t U2+51x - yl'r + t3/2+5+'~/2) +cs(tl+alz - yl 1+7 + t3/2+a+-y/2),
(3.25b)
where the constants Cj, 2 <_ j < m, depend on C, j and 5, the constant Cm+l depends 9n C, m and n, the constants C~ and C~ depend on C, M and 5, and the constants C~, ~! and C~ depend on C, ~I, 5 and 7. On the right-hand side of (3.25b) the first term '~4 m ~j=2 is absent if m = 1. The other dz(t, x, y; A) and d3(t, x, y; A) have the same bounds' ~s dz(t, x, y) in (3.13c) and d3(t, x, y) in (3.12c), (3.13d).
Kac Operator
179
Now we use Theorem 3.2 to complete the proof Theorem 2.3. The proof of Theorem 3.2 is also postponed to Sect. 5. a) For m = 0, since d(t,x,y;A) has the same bound as d(t,x,y) in (3.11b) in Theorem 3.1, it is obvious that k(t, x, y; A) obeys the same inequality (3.15) as k(t, x, y). b) For m > 1 put 3
k(t, x, y;A)
= ~
ki(t, x, y; A),
(3.26a)
i=l
ki(t,z,y;A)
= p ( t , z - y)di(t,x,y;A),
i = 1,2,3,
(3.26b)
where k2(t, x, y; A) is absent if m = 1. For m > 1, since dz (t, x, y; A) and d3 (t, x, y; A) have the same bounds as d2(t, x, y) in (3.13c) and d3(t, x,y) in (3.12c), (3.13d) in Theorem 3.1, it is also obvious that k2(t, x, y; A) and k3(t, x, y; A) obey the same inequalities (3.18b) and (3.17b), (3.18c) as k2(t, x, y) and k3(t, x, y). Therefore we have only to show for kl(t, x, y; A) that
L
11
~ Ikl( t , ' , y ; A ) l l f ( y ) l d y f l p
<
Ct,~ta+(l+m/al[f[Ip, m = 1, ( cm,at,+2611fllp, m >_ 2,
(3.27)
with a constant Cl,t~ depending on C and n and one Cm,6 depending on C and ~. In fact, we have from (3.25b) by (3.14) II
~
[kl(t, ", y;
A)llf(y)ldyllp
m
<_ {~-~ Cj(C(j) + l)t (1+2~)j/2 + Cm+l(C(/7~ + tO)4- 1)t l+(m+n)/2
j=2
+C~C(1)t 3/2+6 + C~(C(2 + 7) + 1)t3/2+6+'r/2 + C~t3/2+6 +C~(C(7 ) + 1)t 3/2+~+7/2 + C~(C(1 + "7) + 1)t3/2+6+7/2}llf] [p, which is bounded by the right-hand side of (3.27), because if m = 1, then 1/2 _< ~ < 1, so that 3 / 2 + ~ > 1 +(l+n)/2, a n d i f m > 2, then ~ < 1/2, so that 3/2 + ~ _> 1 + 2 & Thus we have shown (2.11). Once (2.1 i) is obtained, the proof of (2.12) is analogous to that of Theorem 2.2. This ends the proof of Theorem 2.3.
4. Proof of Theorem 3.1
We first prove (3.1 lb) for m = 0, second (3.12b) and (3.13b), third (3.13c) and finally (3.12c) and (3.13d) for m > 1. We may suppose that V(z) > 1, and the condition (2.3) holds with b = 1. We begin with calculating some basic quantities concerning the Brownian bridge Xo'~ in (3.6) in the following lemma.
180
T. Ichinose, S. Takanobu
L e m m a 4.1. Let 0 < s < t.
Eo[IX~'~
2C(a)(t-
s) a/2,
-
2aC(a)(t
- s ) ~/2,
<_
2a+2 ~§ ~(-:(_,.=/z+l ,~,~ , -a72~,a)~ ,
~]
<
ds
0 < a < 1, (4.1a) a >_ 1.
4-~-('7(0~ta/2+1 f o t Eo[IX~'~
0 < a < l, (4. lb)
a > 1.
Eo[(iz - Yl + IX~'~ Ix - y[~ + 2C(a)(t -<
2a-I
fo t Eo[(Ix - yl +
t
- 8)a/2,
0 < a < 1,
(Ix - yl ~ + 2aC(a)(t - s)~/2),
a > 1.
[Xo'~
4 (7{~x.a/2+l, tlx - yl a + h-gg~,a)~ 2a-1 (tlx - yl a --'4" C("'~t~/2+1~,-,
0
2a+xa+2
<-
(4.2a)
j,
1,
(4.2b)
a _> 1.
Proof W e use (3.4abc). For 0 < a < 1,
Eo[IX~'%)l ~]
=
Eo[IX(8) -
X(t) +
~-~X(t)l ~]
Eo[IX(O - X(s)l ~ + (~-~)~ IX(t)l ~]
=
C(a)(1 +(L~)~/2)(t-
s) ~/2
<_ 2 C ( a ) ( t -
s) ~/:.
F o r a > 1,
Eo[IX~'~
l~1
=
Eo[IX(s)
<_
(&[IX(t)-
= ((s =
- X(t) +
~-~X(t)[ a]
x < 8 ) I ~ U ~ + L=iaEo[lX(t)l~]l/~) ~
Ixlap(t-- 8, x)dx)l/a+(~'~'(/Ndlxlap(t,x)dx)l/a)a
C(a)(1 + (~-~)U2)a(t-
s) ~/2
<_ 2 a C ( a ) ( t -
s) a/:.
This shows (4.1a). Then (4.1b) is an i m m e d i a t e consequence of (4.1a). For 0 < a < 1, we have by (4.1 a)
Eo[(Ix
For a > 1,
- yl + IX~'%)1) ~]
_
Eo[Ix - yl ~ + Ix~'~
a)
<
lx - yt ~ + 2 C ( a ) ( t - s) ~/2.
Kac Operator
181
Eo[(Ix - Yl + IXo'%)l) <']
-<
(Eo[Ix - yt~] '/~ + E o I I X o ' % ) I ~ ] V ~ ) ~
=
(Ix
-<
2a-1
Yl § 1 7 6
-
t - ~)ln)<<
(Ix - yl ~ + 2aC(a)( t -- s)al2) 9
This shows (4.2a). Then (4.2b) is an immediate consequence of (4.2a). This ends the proof of Lemma 4.1. Now we come to the proof of Theorem 3.1. Rewrite (3.9) as w(t, x, y)
f
=
tl2 (V(x + ~(y - x) + X~'~
- V ( x ) ) ds
--dO
f,12(v(y
+ (1 - ~)(x - y) + Xo'~
-
- Y(y))ds.
(4.3)
4.1. Proof of (3.11b) for d(t, x, y) with rn = O. We obtain directly from (4.3) by (2.4), t/2
Iw(t,x,y)l
<_
fdO
CIt(y-x)+Xo'~ CI(1 - t)(x - y) § xt'~
+ 2
_< C
/o'
+ IXo'~
(Ix-yl
Then we have by (4.2b)
Id(t, x, Y) I <_ Eo[lv(t,x,y)l] _< C
L'
_
Eo[Iw(t,x,y)l]
E0[(Ix - Yl + [X~'~
<_ C ( t l x - y l ~ §
4 t,metr162
ds ,,
(4.4)
yielding (3.1 lb) for m = 0.
4.2. Proof of (3.12b) and (3.13b) for dl(t, x, y) with m > 1. By Taylor's theorem we have
V(x + z) - V(x) =
m--1 1 E l(zO)kV(x) + ~ k=l
L 1
dT(1 -- T ) m - I ( z O ) m V ( X + VZ)
= ~ ~(zO)kY(x) k=l
+~
L 1d~(1 - ~)m-, ((zO)mV(x + Tz) - (zO)mY(x)).
182
T. Ichinose, S. Takanobu
Applying this to (4.3) yields w ( t, x, y)
~-
--
-
f
t/2
(~(y - x) + X~'~ 9 ft/2
s
~
k=2
9O]kV(x)d~
[(~(v - x) + x~,~
JO
,
(m--l)!
-
9O V ( x l d s
JO
f;
~
1
f
ft/2
( ( 1 - ~)( ~ x - y) + X o ' ~
9OV(y)ds
2
__
1
~
[((1 - ~ ) ( x - y) + X ~ ' ~
k=2
2
1 ~ (m--l)!
dT(1 -- T) "~-1
2
9OlkV(y)ds
[((1 -- t ) ( X -- y) + X o ' ~
9 O] "~
(4.5) m
where the second and fifth terms - ~k=2 on the right are absent if m = 1. It follows that w(t,x,y)
=
4 Zwi(t'x'Y)' i=l
(4.6a)
where wl(t,x,y)
=
w2(t,x,y)
=
w3(t,x,y)
=
(4.6b)
~ t ( x - y) . ( O V ( x ) - O V ( y ) ) ,
k=2
+
Z
[((1 - t ) ( x -
y)+ X;'~
2
1
W4(~;, X, y)
-
1 f o dr(1 (m--l)!
-
T) m - I
9O]kV(y)ds},
(4.6d)
Kac O p e r a t o r
183
{for~2(
x
+
[ ( ~s ( y - x ) +
[(0 -
X~'~
t)(x
- O],~
t o
2
x ( V ( y + r((1 - t , ( x -
y ) + xt'~
- V(y)))ds},
(4.6e)
where the first term wl(t, x, y) and second w2(t, x, y) come from the first and fourth m terms of (4.5), and the third term w3(t, x, y) = - ~k=2 " " " is absent if m = 1. According to the decomposition (4.6a) of w(t, x, y) rewrite dl(t, x, y) as 4
dl(t,x,y)
= E
(4.7a)
dli(t,x,y),
i=1
dli(t, x, y)
= -Eo[wi(t, x, y)]e-~(v(~)+v(Y)),
1 < i < 4,
(4.7b)
where dl3(t, x, y) is absent if m = 1. We estimate dli(t, x, y), 1 < i < 4, in the following. First, to treat du(t, x, y) for m > 2, we use Taylor's theorem to get m-1
/=1
11
dcr(1 - a)m-Z((x - y). O)m (V(y + cr(x - y)) - V(y)) }.
+ ~
(4.8)
Then we have by (2.3) and (2.4) IEo[wl(t, x, Y)]] --- ]Wl( t, x, Y)I m--1
1=i
lal=l+l
- y)m fro I d c r ( 1 - if)m-2( E
+~]x
)O'V(y + ff(x - y)) - O'V(y)12) 1/2 }
t(,l=~ m
_< ~ t { E a * /~2 c - - x
] ~ n c x--ylm+~}. y.~V(y) 1-16 + a(m-l)
(4.9)
1=2
To proceed note here that for a > O,
tae -t/2 Then we have from (4.9) by (4. I0)
<__
~--e~(2a~a ~
t >_ O.
(4.10)
184
T. I c h i n o s e , S. T a k a n o b u
t
Idt,(t, x, Y)I = lEo[wl(t,x,y)]le-~(v(z)+V(Y))
<
IEo[wi(t,x,y)]le-2
_tV
(y)
m
<
1Ct
dl/2 ( 2 - 2 1 J ' l l - 1 6 § (/--1)! t e "
X -I
yl l +
yl m+~
/=2 rti
-<
E 7lCttS, x - yl I + 89
- y l m+~,
(4.11a)
1=2
with constants Ct, 2 < l < m, depending on C, l and 5, and Cm§ depending on C and m, where if m = 1, the first term ~tm2 in the last member is absent. As for rill(t, x, y) for m = 1, we obtain directly from wl(t, x, y) in (4.6b) by (2.4) t
[d~dt, x , y ) i
=
IEo[w~(t,x,y)lle-~(v(:)+v(u)) < IEoi~,(t,z,u)]l
<
~ d ~ / 2 C t l x _ y]l+,~
(4.11b)
Second, for dl2(t, x, y), we see
dl2(t,x,y)
=
0,
(4.12)
because E0[X~'~ = Eo[X(s) - ~X(t)l = 0, so that Eo[w:(t, x, y)] = 0. Third, to estimate dl3(t, x, y) we first bound w3(t, x, y) by the aid of (2.3): m
Iw3(t,x,Y)l
~ f,/2 I~(y -
+ ~m ~
X~'~
IO"Y(~)12)'/: lal=k
i;,
k=2
<
x)+
JO
I(~ - g ) ( ~ -
y)+ X~'~
~--~ ~ i : c v
~ IO"U(u)l:) '1~ io~l=k
2
( x ) 1-k~ /0 t12 (I x - Yl +
IX~'~
k=2
x - , ---gr-vtY) ~i~c . . . . l-k~ +2_..,
'~ k=2
if/ 2 (ix
- y] +
IXo,O(s)l)kds.
Then we have for d13(t, x, y) by (4.10) and (4.2b), [dl3(t, x, Y)I <- Eo[IW3( t, x, y)l]e - ~(V(x)+V(y)) t
dk/2C g2-2kShl-kSC:-(1-kS) k! ~ e J v
Eo[([x - Yl + IX~'~
ds
k=2
~gV--(---7--)dkl2-2k~ 2c 1-k~ t -(1-k~)2k-l (t]x _ yik + ~2k+~C'kt'k/2+l~)t
)
k=2
= ~ 89 k=2
-- yl k +t(~+2~)k/2),
(4.13)
KacOperator
185
with constants Ck, 2 < k < m, depending on C, k and 5. Finally, to estimate d14(t, x, y) we first bound w4(t, x, y) by the aid of (2.4): Iw4(t, X, y)[
< --
fO' dr(1
1
(m--l)!
- ~_)m- 1{~'0t/2 If (Y - x) + X ~ ' ~
• ~ [O'~V(x + ~-(t(Y-x)+ Xt'~
-O~V(x)12)'/2ds
l-l=m
+
s;
1(1 - t ) ( x - y) + Xo'~
m
2
[O"V( y + ~-((1 - ~)(x - y)+ Xt'~
x( E
- O"V(y)lZ)l/Zds}
I~l=m
< --
1
+ < --
dT(1
(m--l)!
f;
2
dm/2c~-'~l~(y - x) + Xo'~
-- T) m-1
dm/2CT~l(1 -- t)(X -- y) + X~'~
I'
d'~/2C m!
(I x _ y[ + [Xo'O(s)l)m+~ds.
Then we have for dl4(t, x, y) by (4.2b),
[d14(t,x,y) I < < --
t
Eo[lwg(t,x,y)l]e- g(v(~)+V(Y)) <_ Eo[lwg(t,x,y)[] dm/=c m!
<_ 89
/o'
Eo[(Ix- Yl + IX~'~
ds
x _ ylm+~ + tl+(m+~)/2),
(4.14)
with a constant C.~+1 which depends on C, m and t~ and which is taken to be larger than the constant Cm+l in (4.11a). Thus, recalling (4.7ab) and summing (4.1lab), (4.12), (4.13) and (4.14) we have proved (3.12b) and (3.13b).
4.3. Proof of (3.13c) for d2(t, x, y ) f o r m _> 2. We have from (4.5), Iw(t, x,
_<
y)l
k=l
~
JO
+__L_l
(m--l)!
I~(y- x)+ x~,~
Io-v(x)12)'/2 lal=k
/odT(1 - ~_)m-1 V It(Y
- x) + X~'~
dO
X( E IO(~Y(x + T(~(y--X)+ xt'O(8))) -O~ l~l=r-
m
186
T. Ichinose, S. T a k a n o b u
+
h
I~
I(~ - ~)(~- y)+Xo'~
k=l
2
+
d'r(1
-- T) m-1
2 Z
•
:)'/~
Ic~l=k
IO~V( y+r((1
I(1 -
~)(x - y) + x t ' ~
t o (s))) - ~ ) ( x - y) + X o'
Oc~V(y)12)l/2ds,
which is by (2.3) and (2.4) smaller than or equal to
m k=~
et/2 dk/zc . . . . 1-k(5 I s k----~t~)
Jo
+JL__ (m-- 1)! fO dr(1 +
--
T)m-1 it~2
d~s
1 (m--l)!
I(1
- ~ ) ( x - y)+ X~'~
2
/01
dr(1
~ ~V(x) k=l
d ' ~ / 2 C r ' ~ l t ( y - x) + X~'~
JO
k=l +
xt,O(s)lkd8
ICu-x)+
--
T) m-1
d~/2eT~l(l - t)(x - y) + X~'~
1-kS ft/2(Ix-ylq-IX,~'~ JO
m
d~/2C .v.t. y. ) + ~-~ ~ _ --N--.
1-ka
; (Ix-
k=l
+dm/:Cr~!j 0
2
yl +
2
t (IX -- Yl +
Ixt'~
iZo,O(s)L)kds
m+~ds.
Then w e have for j > 2 by the HOlder inequality
(l/j + (j - 1)/j
= 1)
Iw(t, x, y)l j m
(Jo
(Ix-yl + Ix;'~
j
k=l rn
t
.t_V~(dk/2C)J~v(q~ k=l +(--~., ) <-
y)J(1-kS)[kL. f "(~x t/2 (
[
(1~-
yl + IX~'~
yl +
(2/rt + 1)J--1 { ~-'~(dk/2C'Jl/'(~'J(l-k6)(t-'J-1 /_.~, k~ , - w ,2,
k=l
j
f00t/2(Ix
- yl +
Ix~'~
jkds
187
Kac Operator
m
_,_V ~ ( dk/2C~Jv(a,]j(1-k6)[ t ~j-1 [ -z_.'
k~ .
-,~,
tg,
k=l
§ dm /2 c ~j ~J -1
~0"t(Ix -
Jt/2
(Ix-
Yl +
IX~,~(s)l)~(m§
yl +
IXo'~
jkd8 (4.15)
as
It follows by (4.10) and (4.2b) with (3.10c) that
Id2(t,x,y)l m
9
t
~ ~Eo[Iw(t,x,y)tqe-~r247 j=2 m
<
m
~
(2m+l)J-I
j=2
f~-'~(dk/2c~j(~j(1-k6)t-j(1-k6){t.lj-1 k=l x
(2m+l) j - I
-<
j!
j=2
/o
- Yl +
IX~'~
f~"~(dk/2c~j[~]j(1-k6)o-j+lt-l+jk6
I.A..,,---EV--.T, ,
~
,
-
~
k=l
3k+1(~,(4le~§ _ yljk + 2Jk+2 ~'~j~
• +(~)jtj--
• m
Eo[(lx
J
12J(m+~)-- 1 _
ylj ( m + ~ ) 2+j(m+'~)+l ~c(j(m .
§ I~))tJ(m+~)/2+l)}
m
{~-~ Cjk(tJk~lx-- yl ja + t('+26)Jk/2)
j=2
k=l
+Cj,m+l (t j Ix - y[(m+a)j + t(l+(m+~)/2)j)},
(4.16)
with constants Cjk,2 <_ j <_ m, 1 < k < m, depending on C , j , k , m and 5, and Cj,m+l, 2 < j < m, depending on C , j , m and n. This yields (3.13c).
4.4. Proof of (3.12c) and (3.13d)for d3(t, x, y)for m > 1. We have from (3.10d) and (4.15) w i t h j = m + 1 by (4.10),
Id3(t, x, Y)I <_ ~Eo 1 [ Iw(t,x, y)r,~+l fol dO(l-O) m • e x p \ - ( V~( x ) (
+ V(y))-0
/o V(x + ~(y- x)+ Xt,
~]'~
188
T. Ichinose, S. Takanobu
dk/2C m+l t m ) (~) fo' dO(1 m (---gV-.
(2m+l)m{~
_ <
r~
m 2(m+l)(1-k6) (m+l)(1-k6)
O) (
~
)
k=l x((1 - O)t)-(m+l)(1-k~)
+ 1 (dm/2C,m+ltm
~'~-f-"
"
Eo[(I ~c - Yl +
IX~'~
ds
f t E o [ ( [ x _ yl+lxt,O(s)l)(m+l)(.~+~)]ds}. Jo
Then we obtain by (4.2b),
Id3(t, x, Y)I --~
m (2m+l)m ~(m+l)! ~--~(~)dk/2C m+l 2 --m ( 2(m+l)(l--kS)e k
) (m+l)(1-kS) t --l+(m+l)k5
k=l
•
x
-
yl(m+l) k + ~2(m+l)k+l C((m
+ 1)k)t (~+l)k/2+l)
+ ~__~. ( T~ dm/ZC ) 1 m+l t m 2 (m+l)(m+n)-- 1(tla_~ yl(m+l)(m+n) 2(m+l)(m+tr
+( ~ ~
C((m + 1)(m + n))t (m+l)(m+~)/2+l)}
m
=
~_, Cm+l,k(t(m+l)knlx
--
yl(m+l)k + t(l+2$)(m+l)k/2)
k=l
+ C m+l,m+l(t m+l IX-- yl(m+~)(m+l) + t(l+(m+~)/2)(m+l)) .
(4.17)
Here Cm+l,k, 1 < k < m, are constants depending on C, k, m and 5, and Cm+l,m+l a constant depending on C, m and ~. This yields (3.12c) for m = 1 and (3.13d) for m > 2, completing the proof of Theorem 3.1.
5. Proof of Theorem 3.2
5.1. The case m = 0. We obtain directly from (3.21b) Id(t,x,y;A) I <
Eo[Iv(t,x,y)l]
<
Eo[Iw( t,x,y)l],
so that d(t, x, y; A) has the same bound as d(t, x, y) in (4.4), that is, (3.1 lb).
5.2. The case m > 1. We may again suppose that V(z) > 1, and the condition (2.3) holds with b = 1. We can use for d2(t, x, y; A) and d3(t, x, y; A) the results of Theorem 3.1, but have to make a new treatment for dl(t, x, y; A). In fact, we have
KacOperator
189
Id2(t,x,y;A)l
< ~_~Eo[l~(t,x, -
"'J"] e
Y)I
~
,
(5.1)
j=2
Ida(t,x,y',A) l < l e o [Llw(t,x,y)lm§
dO(1 - O)~
So we can give for d2(t, x, y; A) and d3(t, x, y; A) the same bounds as d2(t, x, y) and d3(t, x, y) in (3.13c), (3.12c) and (3.13d). Now we come to dl (t, x, y; A), which is rewritten based on the decomposition (4.6a) of w(t, x, y) as 4
dl(t,x,y;A)
= E
dll(t,x,y;A),
(5.3a)
l=l
d l l ( t , x , y ; A)
=
-Eo[e-i~(t'x'Y)wl(t,x, y)]e -~(v(x)+y(y)), 1 < l < 4,
(5.3b)
with wt(t, x, y), 1 < l < 4, in (4.6bcde), where d13(t, x, y; A) is absent if m = 1. Among dll(t, x, y; A), 1 < l < 4, we have to treat d12(t, x, y; A) anew, but can use for the others the previous results in Sect. 4. In fact, we have
[dlz(t,x,y;A)[
<_ Eo[lwt(t,x,y)l]e -~(V(~)+V(y)),
l = 1,3,4.
(5.4)
So dll(t, x, y; A), 1 = 1,3, 4, have the same bounds as dll(t, x, y) in (4.1 lab), (4.13) and (4.14) in Sect. 4. Thus what remains to do is to estimate dl2(t, x, y; A) in (5.3b). To do so we write for simplicity the Brownian bridge X~'~ in (3.6) as Y(s) = (Yl(s), ..., Yd(S)) in the following. We have to investigate qS(t, x, y) - ~--~i3=1~ ( t , x, y) in (3.21c). By the mean value theorem we have for each j, 1 _< j _< d,
A j ( x + ~(y - x) + Y(s)) d
= Aj(x) + E ( t ( Y k
- xk) + Yk(s))OkAj(x)
k=l 1
d
eo
§
- xk) +
Zk = l
x)+ r(s))) -
-
Noting this and that t
d
71 fO E ( y j j,k=l =
-- X j ) ( ~ ( y k -- Xk) + Y k ( s ) ) d s O k A j ( x )
d t d 1 E (YJ -- xj)(yk - x k ) O k A j ( x ) + 7 yj -- xj)Yk(s)dsOkAj(x), j,k=l 0 j,k=l
lf E(
190
T. Ichinose, S. Takanobu
we have for the first term 4~ (t, x, y) of ~(t, x, y), d
qS'l(t,x,y )
d
Z ( y j -- x j ) A j ( x ) + 1 ~ (yj _ Xj)(Yk -- Xk)OkAj(x) j=l j,k=l
=
t d +y1 ~0 Z (YJ -- xJ)Yk(s)dsOkAj(x) j,k=l 1
t
d
+~l fo dOfo ~
(yj - - X j ) ( ~ ( y k - - X k ) +
Yk(8))
j,k=l
•
+O(t(y- x)+ Y(s))) -OkAj(x))ds.
(5.5)
Since
t d
fo ~-~ (~(Yk - xk) + Yk(s))dYj(s)OkAj(x) j,k=l t ----
d
--T1 fo
~-~ (Yk -- xk)Yj(s)dsOkAj(x) j,k=l t
Zfo
d
Yk(s)dYj(s)(OkAy(x) - OjAk(x)) - ~ ~ OjAj(x),
l<j
j=l
the second term 4~(t, x, y) of r
x, y) is rewritten as
4~(t, x, y) t
d
_-- --tl fO ~-~ (Yk -- Xk)Yj(s)dsOkAj(x) j,k=l t
+
d
Yk(s)dYj(s)(OkAj(x) - OjAk(x)) -- ~ ~ OjAj(x) l<j
j=l
t
+ ~ fo (t(Yk - xk) + Yk(s))dYj(s) j,k=l
• foldO(OkAj(x+O(~(y-x)+ Y(s)))-OaAj(x)). Here we have used
fot dYj(s) fo s d ~ ( s )
'
= Yj(t) - Yj(O) =
[sYAs)] t -
O,
/0 ' Yj(s)ds
=
-
/0' Yj(s)ds,
(5.6)
191
Kac O p e r a t o r
and ItS's formula
/o
/o
Yj(s)dYj(s) = 89 Y~(s)dYk(s) +
dYj(s) 2
/o
~1
-
Yk(s)dY3(s) =
ds = ~1 [ Y j ( s ) 2 t]o
t
_
~ -
t
-~,
d(Yj(s)Yk(s))
=
[Yj(s)Yk(s)]~
= o, j r k.
The third term 4~(t, x, y) of 4~(t, x, y) is rewritten as e ; (t, x, y) t
d
= ~divA(x) + 1 f
~-~(OjAj(x+ t(Y - x)+Y(s)) - OjAj(x))ds.
JU
(5.7)
j=t
Thus with (5.5), (5.6) and (5.7) we have obtained .5
~(t, x, y)
=
E
4~l(t, x, y),
(5.8a)
l=0
where d
9 o(t,x,y)
=
d
E(y j=l
xj)Aj(x) +
j -
j _ x j ) ( y k -- X k ) O k A j ( x ) ,
(5.8b)
j,k=l t
4)l(t,x,y)
l E(y
d
= ~ fo
E
(YJ - xj)Yk(s)ds(OkAj(x) - OjAk(X)),
(5.8c)
j,k=l,j~Zk
4~2(t,x,y)
=
E
Yk(s)dYj(s)(OkAy(x) - OjAk(z)),
(5.8d)
l<_j
q53(t,x,y)
,/oI/o
= ~
dO
E(yj--Xj)(t(Yk--Xk)+Yk(s)) j,k=l
•247
(5.8e)
t d
~s(t, x, y)
=
(t(Yk - xk) + Y k ( s ) ) d Y j ( s ) 3kl
0
Then, since qs0(t , x, y) does not depend on Y(s), dt2(t, x, y; A) can be written as
192
T. Ichinose, S. Takanobu
dl2(t, x, y; A)
= --e-~rP~
-z~t=l dt,x,y)
9
.
5
q5
--
1)w2(t,x,
y)]e-~(v(~)+Y(u)),
(5.9)
with w2(t, x, y) in (4.6c), because Eo[wz(t, x, y)] = O. Since we have by (2.3) and the Schwarz inequality
Iw2(t, x, y) le- ~(v(~)+v(u)) <
dll2c(V(x)I_s Lt/2
< (~-)l-~dl/2Ct-l§
IY(s)las +
t V(y) 1-5 fttl 2 IY(s)lds)e-~(v(x)+v("))
f',sO Ig(s)lds
< (2~)l-~dli2of~-llZ(i'ly(s)lZds) 112 --
-
-
(5.10)
" J 0
we have Idlz(t, x, y; A)I 5
_< ~
Eo[l~t(t, x, Ylllw2(t, x, y)[le-~ (v(~)+w(u))
/=1 5
t
x.,,(f ~
l=l
<--(2~e25)l-Sdl/2ctS-l/2 --
5
~
t "11/2 Eo[l~z(t, x, y)lelWzEo [fO IY(s)l~ds]
l=l 5
(2-~e2"--'~5)l-5(23c(2"---))l/2d1/2ctl/2+6 E Eo[l~t(t, x, Y11211/2, l=l
where in the last inequality we have used (4. lb). We have to estimate E0[i~Sl(t, x, y)lZ]W2, 1 < 1 < 5. First, for q51($,x, y) in (5.8c) we see by (2.9),
I~l(t, x, Y)I
<- y,io
< ~d
lyj-xJl[Yk(s)[dslOkAj(x)- OjAk(X)l
j,k=l,j:/'k t
~_ M d t - ~ l x - yl
L
so that by (4. lb) and the Schwarz inequality
IY(s)lds,
(5.11)
Kac O p e r a t o r
193
[(/o'IY(s)lds)2]'12 (io'
<_ Mdt-ilx- ylEo
Eo[l~,(t, x, y)121l/z
<- Mdt-'l:l x - yl
Eo[Ir(s)l:lds) l/2
< (~)'I2MdtU2Ix-yl-
(5.12)
Second, to treat ~2(t, x, y) in (5.8d) note by (3.6) that for j 5t k, t
L
t Yk(s)dYj(s)
1
-~Xj(t) =
/;
t
/o'Xk(s)ds +
(t)Xk(t)
/o'sds
Xk(s)dXj(s) - lxj(t)Xk(t)
+~(Xk(t) LtXj(s)ds - Xj(t)LtXk(s)ds). Then we have by (2.9), Eo[l~2(t, x, y)12]1/2
l<j
ft
\ 27 1/2
+IEo[(Xk(tL ) Xj(s)ds-X,(t)JoXk(s)ds) < M E
{
E~189
J
E~
l<j
=
2-1(d 2 -
d)M(2-112t + 2 - i t
+ 2- 3-i/2t)
<
3(d2 -
d)Mt,
where we have employed
Eo
E(/o'Xa(s)dXj(s) )21 = /o'Eo[Xk(s)2]ds = it2,
~o[(S~ ~.)~.) ]:'~,to~o~~176176 t
2
Lemma 4.1 based on (3.4abc), Third, for ~3(t, x, y) in (5.8e) we see by (2.10),
j 5~k,
(5.13)
194
T. Ichinose, S. Takanobu
I~/i3(t, x, Y)I d
<
MO~IYy -
X
s jllz(Yk - Xk) + Yk(s)l[~(Y - x) + Y(s)l ~&
j,k=l
<_ M d t - l l x - y l
I~(y-x)+Y(s)[l+'Yds t
<- M d t - l l x - Yl
fo
(I x - Y] + ]Y(s)]) l+~ds
<-- Mdt-1/21x - Yl ( ~ 0 "t (I x - Yl + IY(s)I) 2(1+'Otis)\ 1/2 , where in the last inequality we have used the Schwarz inequality. It follows by (4.2b) that E0[l~3(t, x, y)12]1/2
<- Mdt-l/Zlx -- Yl
Eo[(Ix - Yl + IY(s)l)Z(l+2()] ds) 1/2 23+2",(
<
Mdt-1/2]x _ y]20+z'Y)/z(tlx _ yl2(1+2()+ 4~sC(2(1 + 7))t2+2()1/2
--
<
23+2"vC(2( 1 +./)))1/2t1+2(/2 ) 21/2+~Mdt-1/21x _ y] (tl/Z]x _ y]l+7 + (4_4$_4~
<
2U2+2(Md(lx_ y]2+~ + (4-4~ Z3+2~ C(2(1 + ~/)))l/2tr
- y[).
(5.14)
Fourth, for ~b4(t, x, y) in (5.8f) we see by (2.10) and the Schwarz inequality 1r
x, y)l
a f0t l ~ ( y -
<_ ~ M d <
89
x)+ Y(s)[2(ds
1/2 (for ( ] x - y l +IY(s)]) 2~ds).1/2 ,
so that by (4.2b) Eo[l~4(t, x, y)i2] 112
<_ 89Malt1~2 <
//o'
Eo[(I z - Yl + IY(s)D22(I ds) ~/2
1Mdtl/2(max{1,222(-1})1/2(tlx--
<- m~{1'2"-l/2}Mdtl/2(t1/21 z 2
y122(+ max{4,2 ~ r+2"Y} C(2~)t 1+2() 1/2
-Y12( + ( l ~ C(2~/))l/2t(r+2()/2)"
(5.15)
Finally, to treat ~5(t, x, y) in (5.8g) we use another representation of the Brownian bridge Y(s) = Xo'~ in (3.6): it is identified with the solution Y(s) of the stochastic differential equation
K a c Operator
195
dY(s)
= dX(s)-
Y(O)
=
Yt(~_S)~ds,
O,
(5.16)
where X(s) is the Brownian motion starting at X(0) -- 0 (see Ikeda-Watanabe [5] p. 243). Then d
9 5(t,x,y)
=
t
E
fo (~(Yk -- Xk) + Yk(s))dXj(s)
j,k=l 1
• 1
f0
dO(OkAj(x + O(~(y - x) + Y(s))) - OkAj(x)) t
d j,k=l
• (OkAj(x + O(~(y -- x) + Y(s))) - OkAj(x)) --
~sl(t, x, y) + ~52(t, x, y).
(5.17)
For ~52(t, x, y) in (5.17) we see by (2.10), [q552(t, x, y)[ 1
t
d
3,k=l
<
M d fo t J-Y-~ t-~ ]~(y - x) + Y(s)ll+T ds
<
Md
< --
Md21/2tl/n
IY(s)[tl~ - vl + [Y(s)l)l+'Yds
t h . _ Yl + IY(s)l) 2(l+'~)ds
1/2
(t__s)3/2 \1 ~
,
where the last inequality is due to the Schwarz inequality. Then we have by (4.1a) and '4 2a), Eo[l~52(t, x, y)121
<_ 2(Md)2t 1/2
ff Eo ['r(~)'~'~ [(t_~)~:kl~ - Yl + Ig(s)l) z(l+~)]ds
< --
2(Md) 2tl/2
E~(t_s)3/2
z -- Yl + IY(s)l)40+7)] 1/eds
<_
2(Md)2tUzfot 2 2 C ( 4 ) 1 / 2 ( t
- s)-1/223/2+27
( I x - y14(l+'~)
+240+7)C(4(1 + 3'))(t - 8)2(1+~'))
1/2d8
196
T. Ichinose, S. Takanobu <_ 21+2+3/2+27C(4)I/2(Md)2tV2 fO t (t - 8)-1/2(]x - yl 2(~+~) +220+7)C(4(1 + 3,))1/2(t - s)l+7)ds =
29/2+Z~C(4)a/Z(Md)2tl/2
((t - s)-l/2]x - yl zO§ +220+7)C(4(1 + -,/))1/2(t _ s)l/2+7)ds
=
29/z+2~C(4)VZ(Md)2tUz (2t~/21x - Yl 2(~+~) + 22(1+'r C(4(1+',/)) 3+23" 1/2 2t3/2+-r
=
211/2+27C(4)l/2(Md)2(tlx
_
y[2(1+7) + 22(1+'~)C(4(1+-,/)) 3+23, 1/2t2+,),'~1"
Hence Eo[l~52(t, x, y)[211/2 < 21U4+'~C(4)l/4Md( tl/21 x - Yl 1+7 + 21+7 c(4(1+7))'/4 § __
(3+2.y)1/2
~
1"
(5.18)
For 4~51(t, x, y) in (5.17) we see by (2.10),
Eo[l~si(t, x, y)l a] d ~(yk - zk) + rk(s)) Eo [(j~_dl ~ot(j~ol ~-~.( k=l
-
d
t
1
(/o
d
x (OkAj(x + O(t(y - x) + Y(s))) - OkAj(x))dO <
Eo[E j=l
ds
MOTl~(yk -- x k ) + Y k ( s ) l l ~ ( y - - x ) + Y(s)lO'dO k=l d
t
<
<
M 2 d 2 ~0 t Eo[(Ix - y[ + ]Y(s)l)2(l+7)]ds
<_
21+27M2d2(tlx- -
Hence
)2]
23+27 2 ( | + 7)) t 2+7). yl2(l+'Y) + 4-~C(
ds
Kac Operator
197
E o [ l ~ 5 t (t , x, y) 12] 1/2
<- 21/2+TMd(tl/2lx
--
yll+
,- r23+= ,4-7-5-4,~l/2/ql,~[ ~"~"
t + 7))1/zt1+~/2)"
(5.19)
Thus we have obtained from (5.18) and (5.19) with (5.17), E0[lr
x, V)12]t/2
_< Eo[l#st(t,
<_ CM,
x,
y)12] 1/2 + E0[l#52(t, x, V)12] 1/2
(tl/2lz - yll§
§
(5.20)
with a constant CM, 7 depending on M and 7Thus recalling (5.11) with (5.12), (5.13), (5.14), (5.15) and (5.20), we conclude that Idtz(t, x, y; A)[
<
c t +alx
- yl + c~(tt/e+6lx
-
y] 2+'~ + tl§
-
Yl)
+C~t 3/2+~ + C~(t 3/2+aIx - yl + t 3/2+~+3'/2) §
tl+61z - yl 1+"/ § t3/2+6+')'/2),
(5.21)
where C~ and C~ are constants depending on C, M and 6, and C~, C~ and C~ constants depending on C, M, 6 and 7. This proves (3.25b), completing the proof of Theorem 3.2. Acknowledgement. One of the authors (T. I.) is grateful to Professor Bernard Helffer for stimulating his interest in this problem and valuable discussions during the preparation of this paper. He also wishes to thank Professor Rainer Hempel for a private communication on the Schrtdinger operator in Lp.
References 1. Dia, B.O., Schatzman, M.: An estimate on the transfer operator. To appear in J. Funct. Anal. 2. Helffer, B.: Spectral properties of the Kac operator in large dimension. CRM Proceedings and Lecture Notes 8, 179-211 (1995) 3. Helffer, B.: Correlation decay and gap of the transfer operator (in English). Algebra i Analiz (St. Petersburg Math. J.) 8, 192-210 (1996) 4. Helffer, B.: Around the transfer operator and the Trotter-Kato formula. Operator Theory: Advances and Appl. 78, 161-174 (1995) 5. Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, 2nd ed., Amsterdam / Tokyo: North-Holland / Kodansha, 1989 6. Kato, T.: Schr6dinger operators with singular potentials. Israel J. Math. 13, 135-148 (1972) 7. Leinfelder, H., Simader, C.: Schr6dinger operators with singular magnetic vector potentials. Math. Z. 176, 1-19 (1981) 8. Reed, M., Simon, B.: Methods of Modem Mathematical Physics, I: Functional Analysis, Revised and enlarged ed. New York: Academic Press, 1980 9. Rogava, Dzh. L.: Error bounds for Trotter-type formulas for self-adjoint operators. Funct. Anal. and Its Appl. 27, 217-219 (1993) 10. Simon, B.: Functional Integration and Quantum Physics. London: Academic Press, 1979 11. Simon, B.: Brownian motion, L p properties of Schr6dinger operators and the localization of binding. J. Funct. Anal. 35, 215-229 (1980)
Communicated by H. Araki
Commun. Math. Phys. 186, 199-218 (1997)
Communications in
Mathematical Physics
9 Springer-Verlag1997
The Relativistic Non-Abelian Chern-Simons Equations Yisong Yang * Department of Applied Mathematics and Physics, Polytechnic University,Brooklyn, New York 11201, USA Received: 24 September 1996/Accepted: 1 November 1996
Abstract: We study the r x r system of nonlinear elliptic equations r
r
7~
Na
Au~= -AEKabe~b+AE E eUbK~beU~Kb~+47cE6P~J' b=l
a=
1,2,-..,r,
b=l
c=l
j=l
x E R 2,
where A > 0 is a constant parameter, K = (Kab) is the Caftan matrix of a semi-simple Lie algebra, and 5pis the Dirac measure concentrated atp E R 2. This system of equations arises in the relativistic non-Abelian Chern-Simons theory and m a y be viewed as a nonintegrable deformation of the integrable Toda system. We establish the existence of a class of solutions known as topological multivortices. The crucial step in our method is the use of the decomposition theorem of Cholesky for positive definite matrices so that a variational principle can be formulated.
1. Introduction The interest in the study of the (2+ 1)-dimensional Chern-Simons type field theory models is that these models may reveal, at least partially, some unique properties promised by anyon physics [ 10, 11 ]. In particular, it is expected that these theoretical models may find applications in high-temperature superconductivity, the quantum Hall effect and other planar physical systems. The static soliton-like solutions are called vortices which are both electrically and magnetically charged [6, 25, 35, 36] and are known to be impossible in the classical Yang-Mills framework [22, 33]. Analytically, the static Chern-Simons equations are difficult to approach due to the dominance of the lower-order ChernSimons term and the nonlinearity in the way of coupling various fields together. It was * Research supported in part by the National Science Foundation under grant DMS-9596041
2~
~ Y~g
only after the independent work of Hong-Kim-Pac [ 16] and Jackiw-Weinberg [20] that people began to be able to find exact solutions [18, 19, 9, 38, 39, 5, 4, 7] in a certain critical limit, known as the Bogomol'nyi self-dual phase [3]. There are two categories of field-theoretical models, relativistic and non-relativistic, with different mathematical features. The non-relativistic models are based on imposing the gauged nonlinear Schrrdinger equation, governing the dynamics of the matter field, so that the gauge field obeys the Chern-Simons dynamics. The self-dual equations are characterized by a zeroenergy condition and may be reduced into the hierarchies of integrable equations, under the names Toda, affine Toda, sinh-Gordon, nonlinear sigma model, CP N model, etc. [9], under various reductions. In this sense, the non-relativistic Chern-Simons theory provides a unified framework for these diverse equations [9]. On the other hand, however, none of the self-dual relativistic models is integrable and one has to resort to the methods in nonlinear functional analysis to prove the existence of solutions. The levels of difficulty of these nonlinear problems are actually reflected by their gauge groups. When the gauge group is Abelian [16, 20], namely U(1), a nearly complete understanding is already achieved. First the rigorous work in [42, 38] proves the existence of topological solutions. In particular, the work in [38] provides a globally efficient method to approximate the most superconducting solutions. Then, in [39, 5], the existence of radially symmetric non-topological solutions is established and it is also shown [5] that the electric and magnetic charges, energy, and momentum, of a non-topological solution may take arbitrarily designated values in explicitly determined open intervals. Furthermore, the work in [4] proves the existence of doubly periodic solutions, resembling the Abrikosov condensates [1] in the Ginzburg-Landau theory [13] of superconductivity. This type of solutions are studied more thoroughly in [41], where it is shown that already at this compact level there are two solutions for each prescribed vortex location: One solution resembles the topological solution in the non-compact setting and another resembles the non-topological solution. When the gauge group is non-Abelian, such as S U ( N ) , a systematic mathematical analysis is yet to be carried out. In fact, the special case N = 2 leads to the same nonlinear elliptic equation [26, 27] as in the Abelian case and is well understood. For N = 3, the solutions are governed by a 2 x 2 system of equations [23] for which the results or methods in the existing literature [18, 19, 9, 38, 39, 5, 4, 41] already do not apply and new methods must be devised. In the recent unifying work of Dunne [8] the relativistic self-dual Chern-Simons equations in the presence of an arbitrary (compact) non-Abelian gauge group with a symmetric Cartan matrix are derived which reduce to the integrable Toda system in the non-relativistic limit c ---+e~ (c is the vacuum speed of light) and contain the various equations mentioned earlier as limiting cases. The present paper is aimed at a proof of existence of topological solutions for this general system of nonlinear elliptic equations. Here is an outline of the contents to follow. In the next section we review the problem to be studied and then state our main result concerning the existence of Chern-Simons vortices. In Sect. 3 we state our existence theorem for the equivalent system of elliptic equations and formulate a variational principle for the problem. In Sect. 4 we use the methods in nonlinear functional analysis to prove the existence of an energy minimizer for the variational problem. This minimizer is necessarily a solution to the original system of partial differential equations. In Sect. 5 we derive the asymptotic behavior at infinity for the solution obtained. Although our existence theorem is proved for arbitrary semisimple Lie algebra for which the Cartan matrix is not necessarily symmetric, stronger exponential decay estimates for the solution and its first derivatives are established for the system with a symmetric Cartan matrix. In Sect. 6 we obtain again for the symmetric
Relativistic Non-Abelian Chern-Simons Equations
201
Cartan matrix case some natural quantized integrals related to the calculation of electric and magnetic charges in the model, completing the proof of the main result. In Sect. 7 we consider a few examples. In Sect. 8 we make some remarks. Since we will be dealing with a general r x r system of nonlinear elliptic equations, there is a lack of maximum principle and the method of sub/supersolutions fails. The key step in our approach here is the use of the Cholesky decomposition theorem for positive definite matrices to obtain a powerful variational principle. In fact, since the coefficient matrix in the system of equations to be studied is the Cartan matrix determined by the inner products of the simple roots of the Lie algebra of the gauge group, we may rewrite it as the product of a diagonal matrix and a positive definite matrix, the latter can uniquely be written as the product of a lower triangular matrix with positive diagonal entries and its transposition according to the Cholesky theorem. In this way, although the original system of equations are not in a variational form, a suitable manipulation now makes the system variational. This important feature renders the subsequent study successful.
2. The Non-Abelian Chern-Simons Equations Following [7, 8, 9], the non-Abelian relativistic self-dual Chern-Simons model in (2 + 1) dimensions is described by the Lagrangian action density /
s = -tr([D,O]t[D"r
-
o
\
~e"~tr{OuA,,A~, + :A,A,,A,~
\
3
7
- V(r r
where Du = 0u + [Au, .] is the gauge co-variant derivative to be applied to the Higgs field r in the adjoint representation of the compact gauge group G for which the associated semi-simple Lie algebra is denoted G, [-,-] is the Lie bracket Over G, A~ is G-valued gauge field, n > 0 is the Chern-Simons coupling parameter, the symbol tr refers to the trace in the matrix representation of ~, the Minkowski spacetime metric is defined by 9u~, =diag{-1, 1, 1} which is used to raise or lower indices, and the potential energy density of the Higgs field is given by the special formula V(r ct)
_-
4_~tr{([[r ' r
r
_
,02r162
r
r __ ,02r
in order to write the energy density of the model as a sum of squares and a divergence term, and v > 0 is a constant which measures either the scale of the broken symmetry or the subcritical temperature of the system. In the static case, this special property of the energy density implies an explicit energy lower bound, expressed in terms of magnetic or electric charges, which may be attained by the solutions of the following self-dual Chem-Simons equations: D_r
0,
(1)
l
F+_ = ~ [ v 2 r - [[r r
r
where D_ = D1 - iD2 and F+_ = O_A+ - O+A_ + [A_, A+] with A+ = A1 • and 0+ = 01 + i02. Of course the solutions of (1) also solve the Euler-Lagrange equations of /2. For the above general system, the only known exact solutions so far in the literature are the zero-energy (trivial) solutions determined by the algebraic equation
202
Y. Yang
The purpose of the present paper is to establish an existence theorem for a large class of nontrivial solutions for the original self-dual Chern-Simons system (1), commonly called topological solitons or multivortices. Since ~ is semi-simple, it is most convenient to use the Cartan-Weyl basis or the root space decomposition to express the generators of G or ~ for which the mutually commuting set, generating the Cartan subalgebra of ~, is denoted by {H~}l<~
[Ha, Hb]=O,
a,b=l,2,...,r,
[Ha, E ~ n ] = "4-olanE-4-n, a = 1 , 2 , . . . , r ,
n= 1,2,...,s,
r
[E~,E_n] = E a~Ha, ~
n = 1,2,...,s,
a=l
n,n'=J=l,•177
[ E n , E n , ] = Nnn, En+n, ,
n~-n',
where a n = (c~, 1 9 a~) (n = =t=1,+2, 99 9 i s ) are the root vectors, Nnw's are suitable constants, and we' do not assume summation convention for repeated Lie algebra indices. To simplify (1), Dunne, in [8], proposes the following ansatz 7"
r
=
a/~?a,
~r a=l
(2) r
Au = i
AauHa a=l
in which the Ea's (a = 1 , 2 , . . . , r) now refer to the simple root ladder generators [15]. Under this ansatz, the system describes r Abelian Chern-Simons gauge fields A~ coupled to r complex scalar Higgs fields q5a, with the couplings determined by the Cartan matrix K = ( h a d of the Lie algebra ~. Here, with the Euclidean inner product r
(aa, O~b)= Z
~ abc
c=l
for the simple root vectors, the entries K~b are given by the expressions [15] Kab-
2((~, ab)
(a~,cea~'
a,b = 1 , 2 , - . . , r .
(3)
For commonly used gauge groups such as G = SU(N), one may renormalize the root vectors to make (c~a, a~) = 1 for all a which makes K symmetric. It is in this situation that Dunne [7, 8] reduces (1) into the system (5) to be studied in the next section. In this section alone we always understand that K is symmetric to work within the field theoretical framework in [7, 8]. However, for greater generality, our existence theorem for the governing elliptic system will not be confined to such a special case. In other
RelativisticNon-AbelianChern-SimonsEquations
203
words, for the existence part, we shall work in the full generality that K is the Caftan matrix of an arbitrary semi-simple Lie algebra as given in the above formulas. Note that, with the ansatz (2), the trivial solutions, also called the principal embedding vacuum [8], in the broken symmetry case, q~0) ~ 0 (Va), are characterized by r
)l 2 =
V
K
)ab,
a =
1,2,
9 .. , r,
(4)
b=l
for which the gauge fields all vanish [8]. These solutions define the symmetry-breaking vacuum states. On the other hand, nontrivial multivortex solutions of (1) under the ansatz (2) are characterized by the zeros of the complex scalar fields, (~a,s, and their asymptotic behavior at infinity. These zeros may be viewed as electrically and magnetically charged particles residing in equilibrium or defects in a superconductor so that the solution configuration interpolates the point normal state achieved at the defects and the superconducting vacuum at infinity. Our main existence theorem for such particle-like solutions is stated as follows. Theorem 1. For any given nonnegative integers Na and the corresponding prescribed
point sets Za = {Pal,Pa2,''" ,PaN=},
a = 1,2,...,r,
the relativistic self-dual Chern-Simons equations (1) have a solution satisfying (2) so that Za is the zero set o f t a, the magnetic charge induced from F~: takes the quantized value 9 ~=/"
F~:=27rNa,
a=l,2,...,
r~
JR 2
and there hold the decay estimates at infinity: IqS~l2 -[qS~)] 2 = O(e-XO~2(2v~o/2mO-~)l~l),
F~_ = O(e-)~~ [Dj ~'~ 12 =. O(e -)'~
2v/2~~ 2v'~~
j = 1,2,
a= 1,2,...,r,
where ~ E (0, 1) is arbitrary, A~ is the smallest positive eigenvalue of K R K , with the diagonal matrix R being defined by
and (to by
2~
Z Y~g
3. The Elliptic System and its Variational Principle Following Dunne [8], away from the zeros of Ca's, the Chern-Simons equations (1) under the ansatz (2) are equivalent to
b=l
b=l
a = 1,2,.. 9,r,
(5)
c=l
x E R 2.
With the prescribed data stated in Theorem 1 and the substitution 2) 4
If5~l2=v2e u",
a= l , 2 , . . ,
r,
A ---- -~ 2
~
the system (5) becomes r
r
r
Na
AUa = - / ~ E g a b e U b + / ~ E E e u b K b c e U C K a c + 4 7 r E ( ~ p a s , b=l b=l c = l j=l
a = 1,2,--. ,r,
(6)
x E R 2.
Due to a technical reason, in the rest of the paper we shall consider a variant of the above system, namely, r
r
r
Na
b=l
b=l
c=l
3=1
a= 1 , 2 , . . . , r ,
(7)
x E R 2,
It is easily seen that when the matrix K is symmetric as stated in Sect. 2 and assumed in [8], the systems (6) and (7) coincide. Thus, to prove Theorem 1, it suffices to solve (7), for which K is symmetric, with the corresponding boundary condition at infinity, because the original field configurations r A~'s may all be constructed as in [21 ] from ua's which is now a standard fact. Therefore it suffices to concentrate on (7).
Remarks.
(i) The extensively studied integrable Toda system is T
Aua=-AZ
K~be~b ,
a= l , 2 , . . . , r ,
b=l
which arises in the non-relativistic Chem-Simons models [91. The work of Kostant [24] and Leznov-Saveliev [28, 29, 30] leads to an explicit construction of solutions. The beautiful solution theory for the Toda system and its generalizations has been further advanced by various authors. See [12, 31, 32, 34]. In our context the system (7) may be viewed as a non-integrable deformation of the Toda equations. (ii) It may be worthwhile to point out that although the Cartan matrix K has the special property that its diagonal entries are all the same, Kaa = 2, a = 1, 2 , . . - , r and its off-diagonal entries can only assume values from the numbers - 3 , - 2 , - 1, 0 (cf. [15]), our method in solving (7) is not restricted to such a property. In fact, we will
Relativistic Non-AbelianChern-SimonsEquations
205
establish our existence theorem for (7) under the general condition that the coefficient matrix K has the decomposition K = PS, (8) where P is a diagonal matrix with positive diagonal entries and S is a symmetric positive definite matrix, both of size r x r. Such a condition is broad enough to contain the case where K is a Cartan matrix because, due to the representation (3), one may choose P = 2 diag {(a l, a l ) - l , (a2, ~2)-1, . . . , (at, a r ) - l } ,
S = ((O~a, ab)).
Throughout the rest of this paper, we always assume the validity of the hypothesis (8) on the matrix K in the system (7). Here is our main existence theorem for (7). T h e o r e m 2. Suppose that the coefficient matrix K in the system (7) satisfies r
~-~(K-1)ab > 0,
a = 1, 2 , . . . , r.
(9)
b=l
Then (7) has a classical solution verifying the asymptotic condition lim ua(x) = vao =- In (K-i)ab , I~1--+~ ~ b=l lim I V u ~ l = 0 , Jxl-*~
(10)
a=l,2,...,r.
Furthermore, if K is symmetric and A2 is the smallest positive eigenvalue of the matrix K R K , where R is the diagonal matrix defined by R = diag(R1, R 2 , ' - ' , R r ) ,
~-~( K -1 ) ab~
Ra-
a = l , 2 , . . . ,r,
b=l
then (10) can be strengthened to the following exponential decay estimate max l u ~ ( x ) - u a
l
+ max l
=
\
/
(11)
near infinity, where 0 < e < 1 is arbitrary and cro is the smallest member among the numbers R1, R 2 , " ', Rr. Besides, in this case, there are quantized integrals
2 =
2 b=l ~=1
(12)
a= 1,2,...,r. However, if K is not symmetric but the symmetrization of K R K , namely M = I(KRK
+ KrRKr
is positive definite, then the same decay estimate (11) and the quantization of integrals (12) hold for the solution, where A2 now denotes the smallest eigenvalue of M.
206
Y. Yang
Remark. If K is the Cartan matrix of an arbitrary finite-dimensional semi-simple Lie algebra, the assumption (9) is redundant. In fact, let A1, A2, 99-, Ar be the fundamental dominant weights relative to the given simple root vectors a l , c~2,. 9 9 a t . Then K is the transition matrix from the basis A1, ) , 2 , ' " , A~ to the basis cq, c~2, 99 9 c~ for the root space,
a= 1,2,.-.,r,
C~a=~KabAb, b=l
and K -1 defines the inverse of the transition whose entries are all nonnegative [17]. However, since K -1 exists, so the condition (9) is always valid as claimed. As mentioned earlier, we need to find a variational principle for (7). Our method depends crucially on the Cholesky decomposition theorem for positive definite matrices. We proceed as follows. Since the r • r matrix S is symmetric and positive definite, by the theorem of Cholesky [14, 40] there is a (unique) lower triangular r • r matrix L with positive diagonal entries so that (13)
S = L L r.
Here and in the sequel, we use the superscript ~ to denote transposition for an r • r matrix or an r-vector. It will be useful to relate L to S in an explicit way. In fact, let S = (Sab) and L = (Lab) (a, b = 1 , 2 , . . . , r). Then we have the inductive scheme ~al
Lll = V/-~11, Lal = L-~ll'
Laa =
Saa --
Lab ,
a = 2,...,r,
a = 2," " ,r,
b=l
S a b - Eb~=l 1 La~Lb~
a=b+l,...,r,
Lab --
b= 2,...,r.
Lbb
Write and
1 = ( 1 , 1 , . . . , 1 ) ~-
U=(e~',e~2,...,e~)
~.
Introduce the background functions ga
u~
= - ~ - ~ ln(1 + Ix -P~31-2), j=l
N~ 9a(X) = 4 E
j=l
1 (1 + Ix - pail2) 2'
a= 1,2,...,r,
x C R 2.
Using the substitution Ua = u 0a + Va, the system (7) takes the regular form
(14)
Relativistic Non-Abelian Chern-Simons Equations
A V a = --A
K ab e ~~
207
~ e 4 + ' ~ Kab e=~+~K5~ + ga,
+A
b=l
(15)
b=l c=l
a= 1,2,...,r,
x E R 2.
Now we apply the transformation w = (PL)-lv,
V = (Vl,V2,''',Vr)
h = (PL)-lg,
g = ( g l , g 2 , . . . , g , ) ~,
r, (16)
and consider the energy functional (this is a crucial step in the method) /.
U =
(
eu~
Wl
, eU~
gabWb
)"
~
(17)
grbWb
where we adopt the notation P L = (gab). Define then the diagonal matrix
{
0
r
]
o o o t ~ b w b , . . . , e U r + Z2 6=,~rbWb . U = diag e u~+e"w' . . . , eU~+~b=~
(18)
It is straightforward (but somewhat tedious) to verify that a critical point of the functional I, say w, satisfies the nonlinear elliptic equations Aw = -),L~'U + A L ~ P U L L ~ ' U + h.
(19)
Here it may be useful in the derivation of (19) to apply the result
dw (e'U)= (eL) uc, where e is a constant r-vector. Returning to the original variable v via (16) and noting that the product P U is commutative, we obtain from (19) the system Av = - A K U + A K U K U + g.
(20)
It can be examined that (20) is the matrix form of Eqs. (15) as desired. Thus the expected variational principle is established. Note that the form of the variational energy functional (17) imposes the boundary condition L~-U - ( P L ) - l l = 0 (21) at infinity. It is this condition that leads to our main assumption in Theorem 2 which ensures the existence of a unique positive constant solution, U > 0, to the above equation and will be employed in the next section9 In fact the above equation is identical to the one that defines the "principal embedding vacuum" (4) as stated in Sect9 2.
208
Y. Yang
4. The Existence of a M i n i m i z e r
We first derive suitable asymptotics for the unknown w of (19) viewed as a critical point of the functional (17). S i n c e u ~ (ul, 0 u2, o . - - , uT,) o~- vanishes at infinity, the finite-energy condition for the functional (17), namely (21), implies that v approaches a constant vector v ~ satisfying LT,V0 = ( p L ) _ l l '
vo =
0 v~0 . . . e ~o e vl,e
(22)
at infinity. As before, denote by K - l = ( ( K - 1 ) a D the inverse of the matrix K given in (8). The relation (22) leads to the following expressions for the asymptotic states of the unknown Va' S:
K-1)ab
v 0a = l n b=l
)
,
a=l,2,
o..
,r.
(23)
Withv~ = rvO o r . we now shift w according to the rule ~ 1, v 2o , ' " , vT,) w H w + (PL)-lv ~ Then the functional (18) becomes 7"
1 E [Vwa[2 + ~ t L . R U _ ( P L ) _ I I { 2 I(w)= fR :2~=1
+h-w +h" (PL)-lv ~
(24)
where U is as defined in (17) and the matrix R is given by R=diag
(o
o}
e ~l,ev~,. - . , e %
.
Due to (22), the finite-energy condition implies that the shifted unknown vector w satisfies the boundary condition u = 1
(25)
at infinity because R I = V ~ Or equivalently, w~ ~ 0 as Ix[ ~ oo. This is a convenient boundary condition to work with. We claim that there is a constant co > 0 so that
ILT,RU - ( P L ) - I l l 2 >_ colU - 112,
(26)
which is a stronger version of (25). In fact, by (22), we can rewrite the left-hand side of (26) as IL':R(U - 1)12. Thus (26) follows from the fact that U R is nonsingular. We are ready to use functional analysis to prove the existence of a minimizer of the energy (24) in the standard Sobolev spaces. For this purpose, let W k,p denote the space of scalar or vector-valued functions with distributional derivatives up to order k which are all lying in LV(R2). The norm o f L p will be written II " lipConsider the optimization problem min{I(w) [ w
E wl'2).
(27)
Because the matrix P L is nonsingular, we may switch to the variable v from w via (17) back and forth according to convenience. Using (26) and the variable v, we can find suitable constants cl, c2, c3 > 0 such that
Relativistic Non-Abelian Chem-Simons Equations
209
I(w) >_ Cl ~_,(IJVv~ll~ + lle"~ +'~ - lll~) - c2 ~_, ll lhl valll - c3. a=l
(28)
a=l
It is easily checked that
Isl
le s - 11 > 1 + Is---T'
Vs 9
R.
Consequently we have the lower hound o
o
/R
lie ~'o+~o - 111~ >
l U0a + V a [0 2
~ (l+luO+vOl)2 (29)
-> 2
~ (1 + I~~ + Iv~l) 2 -
~ (1 + luOl)2"
Note that since (14) implies u ~ =O(Ixl-2) for large Ix I, the last integral on the right-hand side of (29) is convergent. We recall the standard embedding inequality in two dimensions:
Ilvll] _< v~llvll21lVvll2,
v 9 w 1,2.
(30)
Thus, in view of (30), we have for any meaningful insertion u0 the upper bound
Ilvll4 = < 2
ivl)(1 + luol + I~1)1~1 2 (1 +
_< C(1 +
'v12
luol + Ivl)2
~ (lu~ + Iv12 + Iv'4)
Ilvll~ + Ilvll~llW@
s
IvF ~ (1 + luol + Ivl)2
1 4+C( ivl 2 4 < ~llvll2 [~2 (l+luol+lvl) 2] +llVvll8+
1).
Here and in the sequel, C denotes a generic positive constant. Thus we get
(
Ilvll~ _< c 1 + IlVvlt~ +
/. 2 (1 + luol + Ivl)~)
"
(31)
On the other hand, for any v E W 1'2, we have in view of (30) that
II Ihlvll,
< Ilhl14/311vl14
< ellvll2 + CIIvvll2 +
c
_< ~(llvl12 + IlVvll~) + ~. Eo
(32)
210
Y. Yang
Now insert (29), (31) and (32) into (28) with u ~ = u ~ v~ = v (a = 1 , 2 , . . . , r). We arrive at the following coercive inequality: Z(w) > ClllVliw,,~ - c2 (33) c31lwllw~,=
- c4,
where the constants C ' s above are all positive because of the fact that (16) defines an invertible transformation w ~ v from W ~,2 to itself. It is easily seen that (24) is finite everywhere on W 1,2. In fact, similar to (26), we have I L ' R U - ( P Z ) - l l l 2 5 cl[U -- 1[2 7"
_< 2 c , ~ ( l e u ~ (e vo - 1)12 + le ~~ - ll2) a=l ?-
< c:}-~(fe wo - it2+ re~~ _ 112) a=l
for some constants Cl, c2 > 0. Using the MacLaurin series (e f - 1 ) 2 = f 2 + ~
2k-2fk k=3
and (37) in the next section as in [44], we may find a bound C'w > 0 depending on w c W 1'2 so that f [L~RU - ( P L ) - I l l 2 < Cw, which proves the finiteness of (24) on
Wl, 2. In view of (33) we see that the functional (24) is bounded from below on W 1'2. Set r/0 = i n f { I ( w )
[w
E wl'2),
and let {w (n) } he a sequence in W 1,2 satisfying I ( w (m) --~ z/0 as n --~ oo. The inequality (33) says that {w (n) } is bounded in W 1,2. Without loss of generality, we can assume that {w (m } weakly converges to an element w E W 1'2. We now show that w is a solution to the problem (27). Of course, the finiteness of I(w) implies that for any e > 0, there is bounded domain so that the truncated energy (24) over ~ (in other words the integral in (24) is now taken over ~ instead of R2), Is?(w), satisfies I o ( w ) > I ( w ) - e. Recall the Trudinger-Moser inequality of the form
s
2 e y _< Cle c211f[lwl'2(Y2),
f E wl'2(~'~),
and the compact embedding W1,2(~) --~ LP(O) (p > 2). Thus the structure of I ~ implies that Is9 is weakly lower semi-continuous over W1,2(S2). Consequently, lim I a ( w (n)) > I o ( w ) ,
n--+OO
Relativistic Non-Abelian Chem-Simons Equations
211
where, without loss of generality, we have assumed the convergence of the sequence of numbers {Io(w(~))} because otherwise we can always focus on a convergent subsequence. Besides, we may also assume that the bounded domain f2 is so chosen that f < [
h . w (") + h 9 ( P L ) - l v 0 < e,
Vn.
2_~
Hence, we obtain I(w(n)) --> In(w(n)) + f n : - ~ h . W (n) + h . ( P L ) - l v 0
> I n ( w (n)) - e. Letting n --~ oo in the above, we have z/0 _> Is2(w) - e > I ( w ) - 2e. Since e > 0 can be arbitrarily small, we conclude that I ( w ) _< 70- This proves that w solves (27). Thus w is a weak solution of (19). Using the elliptic regularity theory, we see that w is a smooth solution. Therefore a solution o f (20) is obtained. We now turn to the study of the behavior of the solution at infinity.
5. Asymptotic Behavior We first rewrite (20), after taking the shift v ~-~ v ~ + v, in the form Av = A(KRU - KR)(KRU
-
Since the matrix K R is invertible and K R 1
IKRU
- KRI
<_clU
- 1],
1) - A K R ( K R U
-
1) + g.
(34)
= 1, we have IKRU
-
1[ < ciU - II,
(35)
where c > 0 is a suitable constant. Using (35) in (34), we see that there are constants C 1 , C2 > 0 SO that
]the right-hand side of (34) I < c, ~ ( e ~'~
-
le ~'~176 - I I + Ig[. (36)
1) 2 + c2
a=1
a=l
The existence proof carried out in the last section already showed e ~'~ - 1 E L 2 (a = 1 , 2 , . . . , r ) . We need to derive now e u~ - 1 E L 4 by using va E W 1'2 (a = 1 , 2 , . . . , r ) established earlier also in the existence proof part. Denote by u ~ and v any pair among u l , o . . . , u ro a n d v l , v 2 , . . . , v r , respectively. We proceed as follows. Since u ~ < 0, we have [e u~
-
1[ _< le v - 1[ + [ e u~ - - 1].
So it suffices to verify that e v - 1 E L 4 for v E W 1'2. For this purpose, we recall the following embedding inequality in two dimensions:
II/1[,-<
7r
I[fllw,,=,
p > 2.
(37)
212
Y. Yang
We then use the MacLaurin series oo 4 k _ 4 ( 3 k + l ) + 6 . 2 (ev -- 1)4 = E k! k--4
k Vk
and (37) to obtain a formal upper estimate, k--2
Ile~ - 1114 <
~4k--4(3k+l)+6"2k(k~._2)
kt
-r-
~
rlvll~,,~.
(38)
k--4
The series (38) is convergent. This proves the desired conclusion. Return now to (36). We see that the right-hand side of (34) lies in L 2. Thus v c W 2'2. Because we are in two dimensions, v must vanish at infinity. Besides, the embedding W z,2 ~ W l,p (p > 2) implies v E W I'p. The boundedness of v makes it direct to see that e ~ - 1 E L p for any p > 2. Therefore, by LP-estimates and (34), (36), we have v C W 2,p (p > 2). Hence Ojv also vanishes at infinity (j = 1,2). We now consider the decay rates for the original vector variable u. Away from the vortex points p ~ j ' s , the corresponding matrix form of the system (7) is Au = -AKU + AKUKU, U = dia
g{
e TM e "2,
e *'r
}
U =
(
(39)
e TM e ~2
e u~
Note that, in (39), the definitions of U and U are up-dated to simplify notation. = o o , eVOy Set Uo (eVl, e~2,... = K - 1 1 . Then (39) may be rewritten as Au = A(KUK)(U
(40)
- Uo).
To proceed further, we rewrite (40) as A ( u -- v ~ = A ( K R K R ) ( u
- v~ + A(KUKU~ 9
- KRKR)(u u ~
- v~
(41)
u ~
where the diagonal matrix U~ is defined by U~ =dlag{e 1, e 2 , . . . , e ~ } with U~a lying between Ua(X) and v a~ = 1,2,...,r). The eigenvalues a l , a 2 , . . . , a~ of the matrix K R K are arranged in an increasing order so that 0 < A2 - a l ~ a2 _< -. 9 _< err. Choose an r • r orthogonal matrix O to satisfy O ~ ( K R K ) O = d i a g { a l , c % . . . , a~}. Set f = (u - v~
- v ~ and define w by R ( u - v ~ = Ow. Then Eq. (41) implies Af
~ 2(u - v~
- v~
= 2A(R[u - v ~
- v~
+2A(u - v ) ~ - R ( K U K U ~ - K R K R ) ( u
- v)
(42) _> 2AAo2w~-w - b ( x ) f > 2AAo2(U- v~ _> (2AA2~ro - b ( x ) ) f ,
- v~ - b(x)f
Relativistic Non-Abelian Chern-Simons Equations
213 o
o
o
where ~ro is the smallest among the numbers e ~ , e v 2 , . . . , e "~ and b(x) ---* 0 as Ixl ~ c~. Using a suitable comparison function, standard techniques in elliptic inequalities, and the fact that f ~ 0 as Ixl ~ c~, we conclude that for any ~ C (0, 1) there is a constant C(~) > 0 so that
f = ~-'~ eV~
- V~ 2 < C(e)exp ( - A o ~ ( 1 -
e)lxl)
forlxllarge.
(43)
a=l
Similarly we can get the decay rate for the derivatives. In fact, for fixed j = 1,2, let v = 0ju. Here and in the sequel we only consider the problem away from the vortex points. Thus, differentiating (40), we have
A v = A ( K U K U ) v + A ( K U V K ) ( U - U0),
V = diag{vl, v 2 , . . . , v r ) .
(44)
Set h = C ' R v and define Rv = Ow as before. Then (44) yields
A h > 2v~-RAv = 2A(RC')(KRK)(Rv) + 2Av~R(KUKU - KRKR)v + 2 A v ~ R ( K U V K ) ( U - U0), which is analogous to the first half of (42). Thus we see that the estimate (43) is valid for h. Namely,
h=~-~e'ol0ju~l 2 <_C(e)exp(-Ao~(1-e)lxl)
forlxllarge.
(45)
a=l
The desired decay estimates stated in (11) follow from (43) and (45). We then consider the case where K is not symmetric. Eq. (41) takes the form A(u -- v ~ = ~ A ( K R K + K ~ R K T ) ( R [ u - v~
+ l A ( K R K - K~" RK~')(R[ u _ v~ +A(KUKU~ - K R K R ) ( u - v~ Use A2 to denote the smallest positive eigenvalue of the matrix K R K + K " R K ~. Since (R[u - v~ - K~RK~')(R[u - v~ = 0, we see that the scalar function f still satisfies the last part of (42). Hence (43) is valid. The same symmetrization method may be applied to show that the derivatives of u also obey (45) as expected. This completes the verification of the exponential decay estimates.
214
Y. Yang
6. Quantized Integrals Here, again, the matrix K is symmetric. We now turn back to the notation in Sect. 3. Since IVua I decays exponentially fast at infinity and u ~ is as defined in (14), the function va must satisfy IVva [ =O(]z I-3) asymptotically at infinity. Consequently the divergence theorem and the smoothness of Va over R 2 enable us to arrive at the conclusion
fa Ava= o-*lim ~ j(l~i = pOv~ ~ n ds =0' On the other hand, it is straightforward to check that the functions g~ (a = l, 2 , . . . , r) given in (14) satisfy
fR 2 g~=47rN~, a= l,2,...,r. In view of the above two results, we immediately obtain the expected conclusion concerning the quantized integrals stated in Theorem 2.
7. Some Examples We first consider the case when the gauge group is SU(N). The first non-Abelian member, N = 2, as well as the general SU(N) model in the semiunitary gauge, was studied by Lee in [26, 27]. The governing Chern-Simons vortex equation is identical to that in the Abelian theory [16, 20]. Hence its solutions are well understood [38, 39, 5, 4, 41]. The next member is N = 3. In [23], Kao and Lee studied self-dual SU(3) ChernSimons vortices. These soliton-like solutions carry fractional spins and non-Abelian charges. In their model the Higgs particle lies in the adjoint representation of the gauge group. The vortices are shown to be governed by the 2 x 2 system of nonlinear elliptic equations NI A U 1 = - - 2 e TM + e u2 + 4 e 2 u l - - 2 e 2u2 - - e ul+uz +
4rr~ dptj , j=l
(46)
N~
A U 2 = e TM - -
2e u2 -
2 e 2ul + 4 e 2u2 -
e ul+u2 +
4re~_, ap23 j=l
after a rescaling. The nonlinear terms in (46) may be rewritten in the matrix form -(
21 21 )
{ e'*~
21)(
02)(21
21)(
)
which says that (46) is covered as a special case of the general system (7) for which the Cartan matrix is (2-1) K= -1 2 with (g-1)ll + (K-1)12 = (K-l)21 + (K-1)22 = 1.
Relativistic Non-Abelian Chern-Simons Equations
215
Consequently v ~ = v ~ = 0 and the existence of a multivortex solution (~1, U2) of (46) so that UI(X),U2(Z)--+O, I•Ul[,IVu2l 4 0 aslxl ~cc exponentially fast according to the asymptotic rate
u~(x) + u2(x) + IVUl(X)l 2 + IVu2(x)l 2 =
O(e-~(1-r
near infinity is ensured in view of Theorem 2 because R =diag{ 1, 1 } and the smaller eigenvalue of K R K = / ( 2 is 1. Besides, there hold the following quantized integrals:
/ ~ 2eUl _ eU2 _ 4e2ul + 2e 2u2 +
e ul+u2
47rN1,
2
JR --eUl + 2eUZ + 2e2u1 -- 4e2U2 + eUl+U2 = 47rN2. 2
For the general gauge group S U ( N ) , N _> 3, the rank r is N - 1 and the governing system has N - 1 unknowns and N - 1 nonlinear vortex equations [7]. The structure of the system is completely determined by the ( N - 1) • ( N - 1) Cartan matrix which is tridiagonal:
K=
211 2--11 0 . 0. . 0 . . . . . . - 1 2 - 1 ... ......
0 0 9
(47)
9. - 1 2 -1 9 .. 0 -1 2 Inserting (47) into (7), it is straightforward as for the case N -- 3 of Kao and Lee [23] above to state a similar existence theorem. Here we omit the detail. Next, we consider the Lie algebra G2. The Cartan matrix is [15, 17] K=
-3
2
(48,
"
Thus the associated system (7) (with ~ = 1) is of the form Nl
A u I = --2e TM + e u2 + 4e 2ul - 2e 2u2 + e ul+u2 + 4rr E
(~lj
j=l
(49) N2 AU2 = 3e TM - 2e ~'2 - 6e 2ul + 4e 2u2 - 3e ul+u2 + 47r Z a2j. j=l
The property ( K - 1 ) l l + ( / ( - 1 ) 1 2 -- 3, ( K - l ) 2 1 + ( / ( - 1 ) 2 2 = 5 leads to the result v ~ = In 3 and v ~ = In 5. Consequently the statement in Theorem 2 says that (49) has a classical solution (Ul, u2) satisfying Ul(X) --~ 3, u2(x) ~ 5, and IVUl(X)] 2 + IVu2(z)l 2 ~ 0 as Ixl ~ c~. Using R =diag{3, 5}, we have
M=
( K R K + / ( ~ R K ~-)=
-32
29
'
which is not positive definite. So (11) and (12) stated in Theorem 2 are not directly applicable here.
216
Y. Yang
8. Concluding Remarks In conclusion, we obtained in this paper a class of solutions of (7), commonly referred to as topological solutions, which give rise to the multivortex solutions of the relativistic non-Abelian self-dual Chern-Simons equations (1). These solutions approach the socalled principal embedding vacuum at infinity. It will be interesting to establish existence of nontopological solutions of (7) (for r > 2) which are characterized by the boundary condition lim eU~(~)=O, a=l,2,...,r. Besides, the existence of doubly periodic solutions of (7) is also an important open problem. It is not clear whether the methods in [4, 41], for the simplest case r = 1, may be applied to (7) for r > 2. Technically, we have seen that the Cholesky decomposition is a powerful tool in the variational formulation of a system of nonlinear partial differential equations. It is expected that such a tool will be useful for the study of other problems involving systems of nonlinear equations. In fact we have been successful to obtain a complete resolution for the general system r
Na
AUa= EI(ab(eub--rb)+47rE~p~ b=l
j,
j=l
(50)
lim eU~(~)=r~, a = l , 2 , . . . , r , Ixl--,~ where K = (Ka6)is a positive definite symmetric matrix and r~ > 0 (a = 1, 2 , . . . , r) are constants. For (50), the case r = 1 is the self-dual vortex equation in superconductivity [21], the case r = 2 arises in the extended electroweak model of Bimonte and Lozano containing two Higgs doublets which lie in the fundamental representation of the gauge group [2, 43], and the general case, r = 1 , 2 , . . . , is derived by Schroers as the governing equations for gauged linear sigma model solitons [37]. In [44] existence and uniqueness results for arbitrary r are presented for the system defined either on a closed 2-surface or the full plane R 2. Acknowledgement. I would like to thank Gerald Dunne for helpful correspondence, whose talk at the AMS special session on gauge field theory held at the Courant Institute in April 1996 initiated this work.
References 1. Abrikosov, A.A.: On the magnetic properties of superconductors of the second group. Sov. Phys. JETP 5, 1174-1182 (1957) 2. Bimonte, G., Lozano, G.: Vortex solutions in two-Higgs-doublet systems. Phys. Lett. B326. 270-275 (1994) 3. Bogomol'nyi, E.: The stability of classical solutions. Sov. J. Nucl. Phys. 24, 449454 (1976) 4. CaffareUi, L.A., Yang, Y.: Vortex condensation in the Chern-Simons Higgs model: An existence theorem. Commun. Math. Phys. 168, 321-336 (1995) 5. Chen, X., Hastings, S., McLeod, J.B., Yang, Y.: A nonlinear elliptic equation arising from gauge field theory and cosmology. Proc. R. Soc. Lond. 446, 453-478 (1994) 6. de Vega, H.J., Schaposnik, E: Electrically charged vortices in non-Abelian gauge theories with ChernSimons term. Phys. Rev. Lett. 56, 2564-2566 (1986)
Relativistic Non-Abelian Chem-Simons Equations
217
7. Dunne, G.: Self-Dual Chern-Simons Theories. Lecture Notes in Physics, vol. m36, Berlin: Springer, 1995 8. Dunne, G.: Mass degeneracies in self-dual models. Phys. Lett. B345, 452-457 0995) 9. Dunne, G., Jackiw, R., Pi, S.-Y.,Tmgenberger, C.: Self-dual Chern-Simons solitons and two-dimensional nonlinear equation. Phys. Rev. D43, 1332-1345 (1991) 10. Frthlich, J., Marchetti, E: Quantum field theory of anyons. Lett. Math. Phys. 16, 347-358 (1988) 11. Frthlich, J., Marchetti, E: Quantum field theory of vortices and anyons. Commun. Math. Phys. 121, 177-223 (1989) 12. Ganoulis, N., Goddard, E, Olive, D.: Self-dual monopoles and Toda molecules. Nucl. Phys. B205, 601636 (1982) 13. Ginzburg, V.L., Landau, L.D.: On the theory of superconductivity. In: Collected Papers of L. D. Landau, edited by D. Ter Haar, New York: Pergamon, 1965, pp. 546-568 14. Golub, G.H., Ortega, J.M.: Scientific Computing and Differential Equations. San Diego: Academic, 1992 15. Greiner, W., Miiller, B.: Quantum Mechanics - Symmetries. 2nd ed., Berlin-New York: Springer, 1994 16. Hong, J., Kim, Y., Pac, E-Y.: Multivortex solutions of the Abelian Chern-Simons-Higgs theory. Phys. Rev. Lett. 64, 2330-2333 (1990) 17. Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory. New York and Heidelberg: Springer, 1972 18. Jackiw, R., Pi, S.-Y.: Soliton solutions to the gauged nonlinear Schr(idinger equation on the plane. Phys. Rev. Lett. 64, 2969-2972 (1990) 19. Jackiw, R., Pi, S.-Y.: Classical and quantum nonrelativistic Chem-Simons theory. Phys. Rev. D42, 35003513 (1990) 20. Jackiw, R., Weinberg, E.J.: Self-dual Chem-Simons vortices. Phys. Rev. Lett. 64, 2334-2337 (1990) 21. Jaffe, A., Taubes, C.H.: Vortices and Monopoles. Boston: Birkh~iuser, 1980 22. Julia, B., Zee, A.: Poles with both magnetic and electric charges in non-Abelian gauge theory. Phys. Rev. Dll, 2227-1232 (1975) 23. Kao, H.C., Lee, K.: Self-dual SU(3) Chem-Simons Higgs systems. Phys. Rev. D50, 662645632 (1994) 24. Kostant, B.: The solution to a generalized Toda lattice and representation theory. Adv. Math. 34, 195-338 (1979) 25. Kumar, C., Khare, A.: Charged vortex of finite energy in non-Abelian gauge theories with Chem-Simons tenn. Phys. Lett. B178, 395-399 (1986) 26. Lee, K.: Relativistic nonabelian self-dual Chem-Simons systems. Phys. Lett. B255, 381-384 (1991) 27. Lee, K.: Self-dual nonabelian Chem-Simons solitons. Phys. Rev. Lett. 66, 553-355 (1991) 28. Leznov, A.N.: On the complete integrability of a nonlinear system of partial differential equations in two-dimensional space. Theoret. Math. Phys. 42, 225-229 (1980) 29. Leznov, A.N., Saveliev, M.V.: Representation of zero curvature for the system of nonlinear partial differential equations x~,zz = exp(kx)= and its integrability. Lett. Math. Phys. 3, 489-494 (1979) 30. Leznov, A.N., Saveliev, M.V.: Representation theory and integration of nonlinear spherically symmetric equations to gauge theories. Commun. Math. Phys. 74, 111-118 (1980) 31. Mansfield, E: Solutions of Toda systems. Nucl. Phys. B208, 277-300(1982) 32. Mikhailov, A., Olshanetsky, M., Perelomov, A.: Two-dimensional generalized Toda lattice. Commun. Math. Phys. 79, 473-488 (1981) 33. Nielsen, H., Olesen, E: Vortex-line models for dual strings. Nucl. Phys. B61, 45-61 (1973) 34. Olive, D., Turok, N.: The symmetries of Dynkin diagrams and the reduction of Toda field equations. Nucl. Phys. B215, 4713--494 (1983) 35. Paul, S., Khare, A.: Charged vortices in an Abelian Higgs model with Chem-Simons tenn. Phys. Lett. B174, 420-422 (1986) 36. Paul, S., Khare, A.: Charged vortex of finite energy in nonabelian gauge theories with Chem-Simons tenn. Phys. Lett. B178, 395-399 (1986) 37. Schroers, B.: The spectrum of Bogomol'nyi solitons in gauged linear sigma models. Nucl. Phys. B475, 440-468 (1996) 38. Spruck, J., Yang, Y.: Topological solutions in the self-dual Chem-Simons theory: Existence and approximation. Ann. Inst. Henri Poincar6 -Anal. non lintalre; 12, 75-97 (1995) 39. Spruck, J., Yang, Y.: The existence of non-topological solitons in the self-dual Chern-Simons theory. Commun. Math. Phys. 149, 361-376 (1992) 40. Stoer, J., Burlirsch, R.: Introduction to Numerical Analysis. Berlin-New York: Springer, 1983 41. Tarantello, G.: Multiple condensate solutions for the Chem-Simons-Higgs theory. J. Math. Phys. 37, 3769-3796 (1996)
218
Y. Yang
42. Wang, R.: The existence of Chem-Simons vortices. Commun. Math. Phys. 137, 587-597 (1991) 43. Yang, Y.: Topological solitons in the Weinberg-Salam theory. Physica D 101, 55-94 (1997) 44. Yang, Y.: On a system of nonlinear elliptic equations arising in theoretical physics. Preprint, 1996 Communicated by T. Miwa
Commun. Math. Phys. 186, 219-231 (1997)
Communications in
Mathematical Physics
Q Spfinge~Vedag1997
A New N = 6 Superconformal Algebra Shun-Jen C h e n g 1,*, Victor G. Kac 2,** 1 Department of Mathematics, National Cheng-Kung University,Tainan, Taiwan. E-mall: [email protected] 2 Department of Mathematics, MIT, Cambridge, MA 02139, USA. E-maih [email protected] Received: 3 October 1996/ Accepted: 12 December 1996
Abstract: In this paper we construct a new N = 6 superconformal algebra which extends the Virasoro algebra by the S06 current algebra, by 6 odd primary fields of conformal weight 23-and by 10 odd primary fields of conformal weight i.1 The commutation relations of this algebra, which we will refer to as CK6, are represented by short distance operator product expansions (OPE). We construct CK6 as a subalgebra of the SO(6) superconformal algebra/(6, thus giving it a natural representation as first order differential operators on the circle with N = 6 extended symmetry. We show that CK6 has no nontrivial central extensions.
1. Introduction Superconformal algebras have been playing a fundamental role in string theory and conformal field theory. A rigorous mathematical definition of a superconformal algebra is as follows [3]. This is a simple Lie superalgebra over C spanned by the modes of a finite family ~ of pairwise local fields such that (i) ~ contains the Virasoro field L(z) = ~-:nEz Lnz-n-2, (ii) the coefficients of the OPE between the fields from ~ are linear combinations of the fields from ~r and their derivatives. One can show ([3] Corollary 4.7) that (ii) follows from the property that L(z) is an energy-momentum field, i.e. [ L - t , ~b(z)] = O~b(z),~(z) c ~. In the study of representations of a superconformal algebra g it is important to consider central extensions of g. This amounts to considering C-valued 2-cocycles on ~. Hence we can (and will) ignore central terms in the study of superconformal algebras and consider the calculation of 2-cocycles as a separate problem. * Partially supported by NSC grant 85-2121-M-006-019 of the ROC. ** Partially supported by NSF grant DMS-9622870.
220
s.-J. Cheng, V.G.Kac Recall the OPE for L(z):
OL(w) L(z) . L(w) ~ z - w -
-
3 I-
2L(w) (z - w) 2"
-
-
In particular the Virasoro algebra is a superconformal algebra with trivial odd part. Recall that a field r is called primary of conformal weight A C C if
Or L(z).r
z-w
Ar ( z - w ) 2"
Note that if all fields from ~ are primary, then L(z) is an energy-momentum field. A primary field of conformal weight 1 is called a current. A current subalgebra is a collection of currents {r such that r 9 ej(w) ,-~ ~-~kZ~j~k(w). (For a detailed --W discussion of all these notions see e.g. [3].) One may find a conjecture on classification of superconformal algebras in [3] (cf [5]). This conjecture states that the algebra CK6, constructed in the present paper, is the only new superconformal algebra. Towards the proof of this conjecture the following result has been established. Theorem 1.1. ([3, 4]) Let g be a superconformal algebra such that its even part 96 is
spanned by L( z ) and currents and its odd part is spanned by fields of conformal weights 12and 3. Then ~ is isomorphic to one of the following superconformal algebras: (i) KN for 0 < N < 3 (see [1, 2, 5] or Sect. 2) spanned by 2 N fields, (ii) N = 4 superconforrnal algebra (see [1, 5] or Sect. 3) spanned by 8fields, (iii) W(1,2) (see [2, 5] or Sect. 2) spanned by 12fields, (iv) CI4f6 (see Sect. 3) spanned by 32fields. At this point one should mention the paper by Ramond and Schwarz [6] which gives a classification of algebras satisfying conditions of Theorem 1.1 and some additional conditions. The additional conditions exclude algebras W(1,2) and CK6, so that the answer given in [6] is correct, but the proof is not quite correct. In the present paper we give an explicit construction of the superconformal algebra CK6 and prove that it has no non-trivial central extensions (in contrast to all other superconformal algebras appearing in the list of Theorem 1.1). The absence of central extensions seems to be disappointing as it prevents from construction of simple chiral algebras based on CK6. On the other hand this feature seems to be useful for string theory as it automatically excludes anomalies.
2. Preliminaries In this section we will set the notation to be used throughout this paper. We denote by C[t, t -1] the ring of Laurent polynomials in the indeterminate t over the complex numbers C. Let (1, ~2, 99 9 ~N be N odd indeterminates, and let A (N) be the Grassmann superalgebra in these indeterminates. We let A (1, N) stand for the (super) commutative associative superalgebra A (N) | C[t, t - l ] . Let o be the even differentiation with respect to t and let ~ be the odd differentiation with respect to ~i, for i = 1,2,. 99 N. The Lie superalgebra of derivations of A (1, N ) is denoted by W(1, N). It consists of elements of the form ([2, 5]):
New N = 6 SuperconformalAlgebra 0
N
221 0
D = aoN + ~= ~b-g~'
ao, a l . . . , a n E A(1,N).
The algebras KN ([2, 5]). Consider the differential form f2 = dt - ~ N 1 {id{i. The Lie superalgebra KN is the subalgebra of W(1, N) given by KN := {D E W(1,N)IDfa = Pg2, for some P E A(1,N)}. This Lie superalgebra is isomorphic to the algebra known in physics as the S O ( N ) superconformal algebra [ 1]. Equivalently KN is the subalgebra of W(1, N) consisting of elements of the form N DU := 2 u O + (--1)degu E D i ( u ) D i , i=l
where Oi = ~i o + o and u E A (1, N). From this we see that KN can be identified with A (1, N), where the Lie bracket in KN is identified with the contact bracket on A (1, N), which is given by ([2, 5]): N
N N Ou Ov Ou Ov [ u , v ] = ( 2 u - ~_ {i-~ii)-~ -- ~-(2V -- Ei=I{i0--~/) + (--1)dCg'* Ei=, O{i 0~,"
Ou Ov
Next we write down the commutation relations of KN in field notation. Let I := {il, 9 9 ik } be an ordered subset of the set { 1,. --, N} of positive integers from 1 to N. We let III= k and ~i := ~i, A . . . A ~ik E A (N), and define the field ~I(Z) := E ( ~ I n~Z
@ t n) Z - n - 1 .
Proposition 2.1. Thefields ~i(z) are pairwise local, and the operatorproduct expansion of the fields (i(z) and ~j(z), where I and d are ordered subsets o f { l , . . . , N}, is given
by 5z(z) 95Aw) ,-~ (1I[ + [J[
-
4) ~I(ZA--~j(w) W)2 +(1II - 2) 0(~I A ~j)(w) z--w
N O~ A O0~i(W)
+ (-1)lrl E
o~ z - w
i=l
In particular, KN is a superconformal algebra with the Virasoro feld - l ~o(z ) = - 31 ~ n E z (1 | tn)z-n-l' and the field ~i(z) is primary (with respect to this Virasoro field) of conformal weight 2 - ~2" Proof The proof of the proposition is straightforward using the definition of the contact bracket above. []
222
S.-J. Cheng, V.G. Kac
3. The Algebras K6 and CK6 The goal of this section is to construct a new N = 6 superconforrnal algebra as a subalgebra of/s In order to do so we will first prove a few simple lemmas that will facilitate the computations in this section significantly. First let us recall the notion of the Hodge dual. Let I = { i l , i 2 , ' ' ' , i k } be a subset of { 1 , 2 , . . . , N } . Let I* = {jl, J2, " " " , J N - k } be the complement of I in { 1,2, 999 N}. For ~ I = ~il /~ ~i2 A ' ' " / k ~ik we define the Hodge dual o f ~ to be ~ = d:~j I A (j~ A . . . A ~jN-k such that (t A ~ = (1 A (2 A . . . A {N. The Hodge dual ~ ( z ) of the field ~x(z) is defined in an obvious way. In what follows small letters i, j, k, etc. will denote elements in { 1 , 2 , . - . , N}, while capital letters I, J will denote subsets of {1,2,. 99 N}. Lemma 3.1.
~~v O~i-~ A 0 ~
(_l)la(~ A ~j)*.
=
Proof If i E J, then the right-hand side and the left-hand side are both 0. Hence we can assume that i r J. In this case there exists some J' such that ~ = ~ A ~j,. We have ~j A ~3 = ~j A ~ A ~y, = ~12-..N, which implies that (-1)lJl~i A ~a A ~a, = ~12...N. Hence (~i A ~a)* = (_l)lgl~j,. On the other hand ~--~, ~ A ~ = ~ , ~ A ~o~, = ~J'" [] The next two lemmas can be proved similarly and will be stated without proofs. Lemma 3.2. N
N
t=l
t=l
( Z O~ij O~J).
O~ij
.
O~j
Lemma 3.3. For N = 6 we have
@ A @r. = - ~
60~*j -~
A
O~;lm o~
t=l
Lemma 3.4. For N = 6 we have the foUowing equality of singular parts of the OPEs:
~*jk(z)" ~l-n(w) ~ @k(z). ~Zm.(W). Proof If I{i, j, k} N {l, m, n}l -> 2, then the right-hand side and the left-hand side both equal to O. So we can assume that either {i, j, k} O {l, m, n} = Oor {i, j, k} N {1, m, n} = i = l: Let us first consider the case {i, j, k} fq {l, m, n} = •. We can assume that {ijk A ~tmn = ~123456.The left-hand side equals 2 (~zm~ A - ~ j k ) ( w ) + O(~gr~ A --@k)(W) (Z -- W) 2
Z -- W
while the fight-hand side is
2(@k/~ ~,~,~)(w) + 0(@k/x ~tmn)(W) (z -- W) 2
z -- W
New N = 6 Superconformal Algebra
223
Clearly both sides are equal. Now suppose that {i, j, k} N {/, m, n} = i = 1. Let {i, j, k, m, n, s} = { 1,2, 3, 4, 5, 6}. , ~- 6~mn A ~s, where e = +1. It follows that ~%~ = e~jks. The left-hand We have ~ijk side then equals - (~mnh~Jk)(w) while the fight-hand side equals - ( ( ~ h ~ r a nZ)-(- wW) [] Z--W ~ As an immediate consequence we have Corollary 3.1. For N = 6 we have: ~*~(z)
Thus ~ i j k ( Z )
+
. ~tm,(W)
~ --~i~(Z)
" ~--,(W).
O~i*jk(Z) is a (super) commutative subalgebra of K6, where o~2 =
--1.
L e m m a 3.5.
N 0~
0~
t=l
Proof Both sides equal to 0 unless k = i or k = j. This is clear for the left-hand side. For the right-hand side note that if k 5t i and.k 4"j, then ~ contains both ~i and ~j. But always either contains ~i or ~j. OCt Assume that k = i. The left-hand side in this case equals ~ A ~*j. Now ~ A ~*j = ~2--.N. Hence ~ = ~j A ~i*j. So the right-hand side equals - ~ []
A ~ o~
= ~i A ~i*j.
Now we are in a position to write down some of the OPEs of K6 in a form from which it is much easier to deduce that it contains an N = 6 superconformal subalgebra. The proofs of these formulas are simple applications of the previous lemmas. We have
~ ~ij(z) . ~k(w) ~
~i(Z)" ~j(W) ~
~i(z)"
(z -
w) 2 + - - ' z - w
2~ij(w)
O~j(w)
5ij~O(w)
(Z -- W ) 2
Z -- W
Z -- W
~jkz(w) ~
0~t(w)
~(zo),
Z--W
Z--W
~jk~.~(w) + ~ ~ j ( z ) . ~klm(W) ~ ~_--~--~ ~i*j(z) " ~ k l m ( w ) ~ij* /x ~ k ( w ) ~i*j(z) " ~ k ( w ) ~
(z -
. ~;dw)
~
(Z -- W) 2
Z -- W
'
Z--W
+ _~ i ~_( w )
w) 2
25ik~jl~;(W) ~j(z)
( o~~" A ~ ) ( w )
~:~
~
'
z -
w
+ ~, t=l
+
20((i*j A (k)(W) Z--W
~",~o~t A ~o~t ~,"* (- w ) Z--W
S.-J. Cheng, V.G. Kac
224
~3(z). ~;(w) ~
(~)*(w)
~ 26~(w)
~(z).~(w)
Z--W
~i3(w) ~i~o~(w) ~ - - 7
(Z -- W) 2
~i(z)-G(w)~
Z -- W
Z -- W
~;(w). Z--W
Theorem 3.1. For each complex number a such that a 2 = - 1 the following fields form an N = 6 superconformM subalgebra of K6: (i) L ( z ) = - 8 9 -~0 ~ 3~o(z) 9 is the Virasoro field, (ii) ~ j ( z ) + aO~i*j(z) are currents, (iii) ~ ( z ) - aOZ~*(z) are odd primary fields of conformal weight 3, 1 (iv) ~jk(z) + a~*jk(z) are odd primary fields of conformal weight ~. The operator product expansions of this algebra are as follows (we omit the ones involving the Virasoro feld as usual): (~,~k(z) + a ~ k ( z ) ) . (~,~,(w) + a~;-,(w)) ~ 0, .o~14
Et6=l ( - ~ t )(W)z--w A ~176
( ~ i j ( z ) + O~O~i~(Z)) " ( ~ k l ( w ) + O~O~l(W)) ~'~
Z--~I)
((dz)
- aOZ~;(zl)
9 (~j(w)
- aoZ~2(wl)
O(&Aw) + aO~i~(w))
25ijL(w)
Z--W
Z--W
,.~ -
2(~ij(w) + oeO~i*3(w))
(z - w) 2
6
(~q(z) + aO~i~(z)) . (~klm(w) + a ~ ; m ( W ) ) ~ ~
[0_~ A ~)(W)
" 04, z --
t=l 6 0 [ 0 04 ~. ` A ~)*(W) + O~) Z--W t=l
(~ij (z) + ~O~,*j (z)) . ((k(w) - aO 2 ~ (w)) o~q tw~ - a O 2 ( ~ 04k ~
(Z -- W) 2
)*(w)
Z--W
(~i(z) - aO2~*(z)) 9 (~r
Z--W
+ a~kl(w)) ~
a(~i~ k; (w) + aO~i~ k; (w)) Z--W
9
New N = 6 SuperconformalAlgebra
225
Proof Thefactthatthefieldsofconformalweight 89form a commutative subalgebra was already mentioned in Corollary 3.1. The calculations of the other OPEs are straightforward, although somewhat tedious. However those OPEs of/s written down explicitly above and the lemmas greatly simplify the task. [] We denote the algebra constructed in Theorem 3.1 by CK6. Actually, Theorem 3.1 gives two different embeddings of CK6 into K6. It is immediate from the construction that CK6 is a superconformal algebra whose current subalgebra is the S06 current algebra, and that the odd part of CK6 is spanned by six primary fields of conformal weight 3 and ten primary fields of conformal weight 89 on which S06 acts as in the vector representation and as in one of two irreducible components of the exterior cube of the vector representation, respectively.
Remark. Incidentally the construction of CK6 inside K6 can be imitated to obtain the N = 4 superconformal algebra inside K4. Let f12 _ 1. Explicitly the N = 4 superconformal algebra can be realized inside/s as follows (cf [1]): (i)
L(z) = - l ~o(z) - ~O~(~(z) is a Virasoro field.
(ii) ( o ( z ) -/3(i*j(z) are currents. 3 (iii) ~ ( z ) + flO(*(z) are odd primary fields of conformal weight 2.
For completeness' sake let us also include the OPEs: ( ~ ( z ) + ~O~;(zl). ( ~ ( w ) + ~ O ~ ( w l ) ~
(z-~)~
--O(~ij(w) --/3~*j(w)) + 25ijL(w) Z--W
4 taOj A ~
t=l
(O(*j
_ s(O(*j
~)*(w)
Z--W
t=l 4
~(~
~ktt),(W)
Z--W
( ~ j ( z ) - fl~**j(z)) . (~k(w) + f l O ~ ( w ) )
~
o~k . . . . z - , o ~ , , o~;~ (w) + ~_~,o~_% , w _ f l o~k p u t o~ ) ( )
Z--W These OPEs can be easily calculated using Lemma 3.1, Lemma 3.2 and the fact that for N = 4 one has ~i A ~2 = --~* A ~j.
Remark. If we consider the subalgebra g of CK6 consisting of polynomial vector fields on C (i.e. generated by non-negative modes of the fields of CK6), we obtain a simple primitive Lie superalgebra (see [2] for definition). Let t have degree 2 and let ~i have degree 1. This induces a Z-gradation IJ = Oj~ with dimg_2 = 1, dimgl = 6. Furthermore, go = S06 (9 Ct ~ ~ GL(4) and gl as an SO6-module is isomorphic to S2(sl4) 0 so6 on which t ~ot acts as the scalar - 1 . This Z-graded superalgebra should be added to the list of Theorem l0 from [2].
226
s.-J. Cheng, V.G. Kac
Another Realization of CK6.
In the remaining part of this section we give another realization of K6 and hence of CK6 using 4 x 4 matrices. Although we will not use this realization, we believe that it is worth presenting it here, as it might be useful in the future. As before let us start by analyzing the algebra K6. The quadratic fields of K6 (i.e. ~I (z) such that Ill = 2) is an S06 current algebra over C. We will identify S06 with SL4. In particular SL4 can be represented by traceless 4 x 4 matrices over C. Hence we can identify the zero modes of the quadratic fields (i.e. the linear span of ~I @ t 0) with SL4. The fixed modes of the linear fields of K6 (i.e. ~i(z), i = 1 , . . - , 6) form the standard representation so6 of this Lie algebra S06, and it is isomorphic to A 2(8/4) when considered as an SL4-module, where 814 here denotes its standard representation. Thus the linear fields o f K6 can be represented by quadratic terms of A (4), on which a natural Hodge involution acts. Let's denote this Hodge involution by w. It should not be confused with the Hodge dual in K6, described above. Now the quadratic terms of A (4) can be represented by skewsymmetric 4 x 4 matrices, hence w acts on 4 • 4 skewsymmetric matrices. The action is as follows: Given a skewsymmetric 4 • 4 matrix/3 of the form
(0 a
B =
we have
-a -b -c
0 -d -e
d 0 -f
e f 0
o f -e d) w( B) =
- f e -d
0 -c b
c 0 -a
The action of SL4 on these skewsymmetric matrices is given b y J G _ + G_ j T (which is skewsymmetric), where here J C SL4 and G _ is a 4 x 4 skewsymmetric matrix representing an element of A 2(s/4). The fixed modes of the cubic fields of K6 is not an irreducible representation of SL4. Rather it consists of the representation $2(8/4) and its dual $2(814) *. We will represent both $2(8/4) and ,..q2(8/4)* by symmetric 4 x 4 matrices with the corresponding SL4actions being J F + + F +jT and _ j T F ++ - F ++J, respectively, where F + and F ++ are both symmetric 4 x 4 matrices. The quartic fields form the adjoint representation of SL4 and will also be represented by traceless 4 x 4 matrices with obvious SLn-action. The quintic fields form a representation isomorphic to the representation formed by the linear fields and will also be represented by 4 • 4 skewsymmetric matrices with the same action. Finally the highest degree field is the trivial representation, while the lowest degree field is the Virasoro field. With this in mind it is not surprising that we have the following realization o f / ( 6 :
Even felds. L(z) is a Virasoro field. J(z) are currents, where J is a 4 x 4 matrix with t r J = 0. J ( z ) are primary fields of conformal weight 0, where J is a 4 x 4 matrix with t r J = 0.
A(z) is a primary field of conformal weight - 1. Odd fields. G _ (z) are primary fields of conformal weight 23-,where G _ is a skewsymmetric 4 x 4 matrix.
New N = 6 Superconformal Algebra
227
F +(z) are primary fields of conformal weight 89 where F + is a 4 x 4 symmetric matrix. F ++(z) are primary fields of conformal weight 89 where F ++ is a 4 x 4 symmetric matrix. G _ _ (z) are primary fields of conformal weight - 1 , where G _ _ is a skewsymmetric 4 x 4 matrix. For an arbitrary 4 x 4 matrix A we define A0 := A - l(trA) 9 1, where I is the identity matrix. The fields of K6 satisfy the following operator product expansions (a, b, a,/3, i, j from now on are always indices): J a ( z ) . Jt'(w) ~ [J~, Jb](w) (z - w)
'
1 (JG--G-JT)+(w)+(JT~~176
J(z) . G_(w) ~
~
.
(z_w)Z
.+ ( J G - + G _ j T ) _ (w}
(z-w)
G ~_(z)" G ~_(w) ~
tr(G~- w(G~-))L(w) + O(G~w(G~_))o(w) _ (G ~_w(G~))o(w) (z - w)
G_(z).
J(z) .
r'+(w)~--
r+(w)~
(JF++
1 ( r +~(G_))(w)
I o ( r +~(G_))(w)
2
4
(z - w)
F+JT)+(w) (z - w)
Y(z)- r + ( w ) ~ ](z) . G_(w)
(z - w) 2
(z - w)
1 (JF+F+JT)__(w) 2 (z - w) 2 '
(Jr+-
r.2T)__(w). (z - w)
O ( J G _ +G _ ,l T ) _ _ (w) , 1 ( J G _ +O _ j T ) _ _ (w)
~ ~
- - "1" -~
(z_,,,)~
( J G - - - G - J T )+(w) "t- ( J T t~ G - )--t~ G - )J)++(w) , (z-w) (z-w)
J'~(z). Jb(w) ~ [ J % Jb](w) 1 tr(J'~jb)A(w) (z--w) +2 (z--w) 2 ' J(z).
G__(w)
G,~(z) . G ~ _ ( w ) ~ -
,.~
(JG__ + G_-JT)__(w) (z - w)
(O~J(a~--))0(~) --
(z-w)
z
1 tx(G~-w(Ge ))OA(w) +
4
z-w- -
1 tr(G~_oa(G o _ _ ) ) A ( w ) 2 (z-w) 2
G _ ( z ) . A(w),,~
G__(w)
- -
Z--W
,
N
Fi+(z). FJ++(w)"o
t (r'+r'+§
~ u ( r + r {+)A(~v)
1 tr(F + i F {+)OA(w)
2
4
8
z-w
(z-w) z
z-w
'
228
S.-J. Cheng, V.G. Kac
r ~(z). r ~(w) ~ o. Now we can describe the subalgebra CK6 of K6 as follows: 1
= L(z) - ~03A(z),
~_(z) = G_(z) + ~OZG__(z), 1
-
J(z) = J(z) + -~OJ(z), +(z) = r §
To be more precise it is the following algebra:
Even fields. L(z) is a Virasoro field. J(z) are currents, where J i s a 4 • 4 matrix with t r J = 0.
Odd fields. G_(z) are primary fields of conformal weight ~, where G _ is a skewsymmetric 4 • 4 matrix. +(z) are primary fields of conformal weight 1, where F + is a 4 • 4 symmetric matrix. The fields of CK6 satisfy the following OPEs: Ja(z) " Jb(w) ~ [ya,
ffb](w)
(z - w)
'
J(z) 9G_(w) ~ ( J ~ - + G - J'T)-(w) + ( J ~ - -- G - J'T)+(w) (z - w)
(z -
w) 2
'
Y(z). ? +(~)~ (y~§ + P+~)§ (z - w)
O~_(z). ~_(w) ~
tr(&-~(~-))~(~) + ~ ( ~ ( ~ _ ) ) o ( ~ ) (z - w)
_ (&_~(~_))o(~) (z -
w) 2
'
G_(z). ~ +(~) ~ I ( P § 2
(z - w)
Ai
r + ( z ) . P~(w) ~ o.
As before w(G_) denotes the Hodge dual of the matrix G_. All OPEs can be checked using those OPEs of/('6 given above.
Remark. The currents form, as for/s the SL4 loop algebra. Their zero modes therefore form an SL4. The fixed modes of the fields of weight ~ form the module A 2(s/4) and the fixed modes of the fields of weight 1 form the module S2(sl4).
New N = 6 SuperconformalAlgebra 4. Central Extensions of
229
CK6
In this section we will prove the following theorem: Theorem 4.1. C K6 has no nontrivial central extensions. Let C be a central (even) element of the algebra CK6. The OPEs of the mutually local fields of CK6 are equivalent to the commutation relations of CK6, viewed as an infinite A
A
~ i
dimensionalLie superalgebra with basis {Jna, G~, F k, Ln},wheren E Z a n d k E 89 Here a, a, i are used as indices of bases. We will write down the commutation relations of a central extension of CK6. The coefficients fCab,lZac~,c ~3 7"~,iPai,Jg~z, A~Z, r below are of course the structure constants of CK6 and can be calculated explicitly from the OPE given in Sect. 3. But since we will not need to know them explicitly, we will not do it here. g(., .) of course is the 2-cocycle of this central extension, while Cab is an invariant symmetric bilinear form on SL4. Using the same argument as in [5] we see that any central extension of CK6 has the following form: [Zn' Zm]
= (n -
m ) Z n + m 4- ~ n ' - - m ( n 3
-- n ) C ,
12
= f~bJ~+m + ne~b~,-mC, [Jna, G~n] ='f~,~a~~zn+m +n~-iaaFn+m, 7a
[
9 Aj
r m] = + 2Ac~z(n -
t a n , e~m] : ~ ~c~
~ i
~ i
r
m)Ja+m + g(O~, G~)C, ^i
a ~a
[G,~, F ml = ~oiJ~+m + g(G~, F m)C, n,
m
)c.
(Here we have used the convention of summing over repeated indices.) Let J d e n o t e the linear span of the zero modes of a basis of J~ over C. Furthermore we let Gk denote the linear span of the U h modes of a basis of Oa, and G - k denote the linear span of the - k th modes of a basis of G~, for a fixed k. Finally let F k denote the linear span of the k th modes of abasis of F , and F - k denote the linear span of the - k th modes of a basis of F , for a fixed k. ] is the adjoint representation of SL4 (denoted by adSL4), while Gk and G - k both are the SL4-module A 2(sl4). Similarly F k and - k are both the representation S2(sl4). The central element C can be viewed as the trivial representation of SL4. With this in mind we see from the Jacobi identity that the existence of a central extension implies the existence of the following SL4-module homomorphisms: A 2(814) @ A 2(sl4) ---+ C, adSL4 | adSL4 --* C, A 2(8/4) @ $2(814) --* C, $2(814) @ $2(814) --+ C. Since A 2(sl4) and S2(sl4) are not contragredient and S2(sl4) is not self-contragredient, we obtain immediately that g ( F i~, F ~ ) - - g(G~,^ ^iFm) = O.
230
S.-J. Cheng, V.G. Kac
Lemma 4.1. e is trivial.
Proof Consider the Jacobi identity Ai
= [[d~, G~ ], F 811 + [G~, [J~, ~ ~]]. We collect the central term and use the fact that g( F n, F m) = g(G~, ~o~ F m) = 0 to get (summation over repeated indices): b
n@c~iCab~n,-r-s = O. The equation is valid for all n, r, s. So we get b
~)~is
=0.
Suppose that e is nontrivial. Then, since e is an invariant symmetric bilinear form o n S L 4 , it is nondegenerate and we can assume that the set {J~, j b , . . . } is chosen so that they form an orthonormal basis. Since ~ is a non-zero homomorphism from A 2(sl4)@$2(814) onto adSL4, we can find c~ and i such that ~ is not zero for each fixed a. Thus c ~ = 0 and so ~ is trivial, which is a contradiction. [] Lemma 4.2. 9(G~, G ~ ) = 0.
Proof Let [G~, G ~ I = c ~ Ln+m ~ " ( n - m) J~+,~ "i', +9(G,~, ~ Gm)C. ^~ +2A~;~ As weight spaces GL4 = SL4 @ Lo it follows that 9(Gn, ~ G~) 4 0 only when a = - / 3 and n = - m . Let C a and G# belong to oppositeffeight spaces with respect to SL4. We can find an in the Cartan of SL4 such that [H, G '~] =~0. Consider the Jacobi identity of
We collect the central term and use Lemma 4.1 to get 9(G~+r, G ] ) = g(G ^ r , G~+8), A;~ for all r, s, n. Choose s = - n - r, we obtain g(d~_~, G~) = 9 ( d ~ , d ~ ) , for all r, s. Now we pick s = r and we get 9(G~, 0~-~) = 9(d~-~, d~), which implies that -
=Y~,
s~
-s)~
Vr~8.
Hence g(G~, G~_~) is a constant function of r. Now we use the Jacobi identity
and compare the central terms of both sides. The left-hand side, if non-zero, must be a function in n (in fact a polynomial in n with zero constant term). The right-hand side, however, is a constant, if non-zero. This implies that the Virasoro subalgebra has trivial center and also g(G~, G ~ ) = 0. [] This completes the proof of Theorem 4.1.
New N = 6 Superconformal Algebra
231
References 1. 2. 3. 4.
Ademollo, M. et al: Dual strings with U(1) colour symmetry. Nucl. Phys. B l l l , 77-111 (1976) Kac, V. G.: Lie superalgebras. Adv. Math. 26, 8-96 (1977) Kac, V. G.: Vertex algebras for beginners. Providence: AMS, University Lecture Notes, vol. 9, 1996 Kac, V. G.: Superconformal algebras and transitive group actions on quadrics. Commun. Math. Phys. 186, 233-252 (1997) 5. Kac, V.G., van de Leur, J.W.: On classification of superconformal algebras. In: S.J. Gates et al. (eds.) Strings 88, Singapore: World Scientific, 1989 pp. 77-106 6. Ramond, E, Schwarz, J.H.: Classification of dual model gauge algebras. Phys. Lett. 64B, no. 1, 75-77 (1976) Communicated by G. Felder
Commun. Math. Phys. 186, 233-252 (1997)
Communications in
Mathematical
Physics
(~) Springer-Verlag1997
Superconformal Algebras and Transitive Group Actions on Quadrics Victor G. Kac* Departmentof Mathematics,MIT,Cambridge,MA 02139, USA Received: 16 October 1996/Accepted: 12 December1996 To Ernest Borisovich Vinberg on his 60 th birthday
Abstract: A classification of "physical" superconformal algebras is given. The list consists of seven algebras: the Virasoro algebra, the Neveu-Schwarz algebra, the N = 2, 3 and 4 algebras, the superalgebra of all vector fields on the N = 2 supercircle, and a new algebra C K 6 constructed in [3]. The proof relies heavily on the classification of all connected subgroups of S O N ( C ) which act transitively on the quadric (v, v) = 1.
O. Introduction Superconformal algebras have been playing a fundamental role in superstring theory and conformal field theory. It is an important and challenging problem to give their complete classification. My conjecture is that the examples given in Sect. 2 of this paper exhaust all possibilities (cf. [7 and 6]. The main theorem of the present paper (Theorem 4.1), which gives a classification of "physical" superconformal algebras, is the first step towards the proof of this conjecture 1. A superconformal algebra is a Lie superalgebra g over C spanned by the coefficients of a finite family F of pairwise local formal distributions (fields) such that the following three properties hold: 1) F contains a Virasoro formal distribution L(z) = ~ c z [L(z), L(w)] = OL(w)~5(z - w) + 2 L ( w ) ~ ( z - w));
L n z - ' ~ - 2 (recall that
2) the coefficients of the delta-function expansion of the commutators of formal distributions from F are from R := C[0]F; 3) g contains no non-trivial ideals spanned by coefficients of formal distributions from a C [0]-submodule of R. * Partially supportedby NSF grant DMS-9622870 1Added in proof. This conjecturehas been proved.
234
V.G. K a c
Recall that a formal distribution ~5 with coefficients in g is called primary of conforreal weight A if [L(z), qS(w)] = O~(w)~(z - w) + A ~ ( w ) ~ ( z -- w). A superconformal algebra (g, F ) is called physical if the set of even (resp. odd) formal distributions from F consists of L and primary distributions of conformal weight 1 (resp. primary distributions of conformal weight 3 and ~1 ). Theorem 4. l(see Sect. 4) states that there are seven physical superconformal algebras, namely, the N = 0 (Virasoro), N = 1 (Neveu-Schwarz), N = 2, 3 and 4 superconformal algebras, the superalgebra W2 of all vector fields on the N = 2 supercircle, and a new superalgebra CK6 constructed in [3]. The rank of R over C[0] (i.e. the minimal number of formal distributions generating g) is 1,2, 4, 8, 8, 12 and 32 respectively. The proof is based on the analysis of the conformal superalgebra R = C [ 0 ] F associated to (g, F). Apart from the C[0]-module structure, R carries a C-bilinear product a(w)(n)b(w) for each n E Z+, which is simply the coefficient of 5(~)(z - w)/n! in the commutator [a(z), b(w)]. The Lie superalgebra axioms are translated into some complicated identities for R ([6], Sect. 2.7). These identities show, in particular, that the 0 th product defines a Lie algebra structure over C on the space a of all primary distributions from R of conformal weight 1 and a faithful representation of a on the space V of all odd primary distributions from R of conformal weight ~ with a non-degenerate invariant symmetric bilinear form (., .). The basic observation of the paper is that the complex connected algebraic group A corresponding to the Lie algebra a acts transitively on the quadric (v, v) = 1, v c V. Let au denote the stabilizer of a point u on this quadric. If F contains no odd primary 1 then a~ = 0 ([6],Sect. 5.9) which immediately distributions of conformal weight ~, leads to a conclusion that the only possibilities for dim V are 0, 1,2 and 4. These cases correspond to N = 0, 1,2 and 4 superconformal algebras respectively. 1 Denote by E the space of all odd primary distributions in R of conformal weight ~. Another important observation is that the a-module E when restricted to au is isomorphic to the a~-module ad a~. In Sect. 3 1 classify all connected algebraic subgroups of the group SON(C) which act transitively on the quadric (Theorem 3.1). The above restriction on a~ rules out most of the cases if a~ ~ 0. ! avoid doing this inspection by using the numerical restriction (4.8), which leaves only six cases, of which three are ruled out by the above condition on a~, and the remaining three cases correspond to N = 3 superconformal algebra, to W2 and to the new superconformal algebra CK6 constructed in [3]. This is, in fact, how CK6 has been predicted (in the spring of 1996)! Ramond and Schwarz [ 13] gave a classification of Lie superalgebras satisfying conditions of Theorem 4.1 and some additional conditions. The additional conditions exclude algebras W2 and CK6, so that the answer given in [13] is correct. However the argumentation of [13] is incomplete and is not quite correct (like the "proof" of simplicity of the current algebra on p. 76 ). On the other hand, restrictions given by Lemma 4.2b and Lemma 4.3 are similar to their observations. This paper is dedicated to my teacher, E.B. Vinberg, who taught me invariant theory (among many other things ), which turned out to be indispensable for this paper. I am grateful to him and to A.L. Onishchik for consultations on transitive group actions.
1. Preliminaries on Lie Superalgebras of Formal Distributions
Let g = g~ + gT be a Lie superalgebra over C. A formal distribution (usually called a field by physicists) with coefficients in g is a generating series of the form :
Superconformal Algebras and Transitive Group Actions on Quadrics
a(z) = E
235
a('~)z-n-l'
n~Z
where a(~) E 9 have the same parity for all n E Z and z is an indeterminate. The formal distribution a(z) is called even (resp. odd) if all a(n) are even (resp. odd). Two formal distributions a(z) and b(z) are called (mutually) localiffor some N E Z+ one has: (z - w)N[a(z), b(w)] = 0. (1.1) Introducing the formal delta-function ~
nEZ
we may write a condition equivalent to (1.1) : N-1
[a(z), b(w)] = E (a(j)b) (w)o~(~5(z - w)/j! j=0
(1.2)
for some formal distributions (a@)b)(w) ([5],Theorem 2.3), which are uniquely determinedand given by the formula
(a(j)b) (w) = Res(z - w) j [a(z), b(w)].
(1.])
Z
Formula (1.3) defines a C-bilinear product a(j)b for each j E Z+ on the space of all formal distributions with coefficients in 9Note also that the space (over C) of all formal distributions with coefficients in 9 is a (left) module over C [Oz], where the action of Oz is defined in the obvious way:
so that Oza(z) = ~ ( 0 a ) ( ~ ) z - n - l , where (Oa)(~) = -na(~-l). The Lie superalgebra g is called a formal distributions Lie superalgebra if there exists a family F of pairwise local formal distributions whose coefficients span g. In such a case we say that the family F spans 9. We denote by Fg (resp. FT) the set of all even (resp. odd) distributions from F, so that F 6 (resp. FT) spans gg (rasp. ~tT)-We often will write (9, F ) to emphasize the dependence of F. The simplest example of a formal distributions Lie superalgebra is the current superalgebra g associated to a Lie superalgebra ~: 0 = C [t,t -1] | It is spanned by the following family of pairwise local formal distributions (a E 3) :
a(z) = E
(t~ | a)z-~-I"
nEZ
Indeed, it is immediate to check that [a(z), b(w)] = [a, bl(w)5(z - w).
236
V.G. Kac
The most fundamental for us example is the (centerless) Virasoro algebra, the Lie algebra with the basis L,~ (n E Z) and commutation relations
[L,~, Ln] = (m - n)Lm+n. It is spanned by the local formal distribution L(z) = ~-~nez L n z - n - 2 ' since one has : [L(z), L(w)] = O,,L(w)(~(z - w) + 2L(w)O~(z - w).
(1.4)
A formal distribution L(z) with coefficients in g satisfying (1.4) is called a Virasoro distribution. Note that C[0] F is a C [0]-submodule of the space of formal distributions which still consists ofpairwise local formal distributions. The formal distributions Lie superalgebra g spanned by F is called simple ifg contains no non-trivial ideals spanned by a submodule of C[0]F. The Virasoro algebra has no ideals at all, but the current algebra always has "evaluation" ideals. Nevertheless the current algebra g associated to Lie superalgebra ~ is simple in the above sense iff ~ is simple in the usual sense. I conjecture that the only simple formal distributions Lie algebras with a finite family F are either the quotients of current algebras associated to simple finite-dimensional Lie algebras and their twisted analogues or the Virasoro algebra. A solution of this conjecture under the assumption of existence of a gradation follows from a very difficult theorem of Mathieu [8]. The list of simple formal distributions Lie superalgebras (g, F ) with a finite family F is much larger. Apart from the quotients of current superalgebras associated to simple finite-dimensional Lie superalgebras, it includes three series: WN ( N >_0), SN (N > 2) and KN (N _> 0) (see [7] and Sect. 2 of the present paper), and the exceptional superalgebra CK6 (see [3]). One can also change modings, i.e. consider twisted superalgebras (see [7]). (For SN there is a 1-parameter family of modings; for all other examples of Sect. 2 there are two modings.) The conjecture is that this is a complete list (cf. [6],[7]). Definition 1.1. We say that a formal distribution Lie superalgebra (g, F) is finite if F is finite and the module C[0]F is closed under all products (1.3). The Virasoro algebra and the quotients of current superalgebras associated to finitedimensional Lie superalgebras are finite. All the superalgebras described in Sect. 2 are finite. The finiteness condition provides the choice of the "non-twisted" moding.
Example. Let ~ be a finite-dimensional Lie algebra and let ~ = ~o + ~1 be a Z/2Zgradation. Then g := C [t,t -1] |176 +t89 [t,t -1] | is a subalgebra of the Lie algebra C It89 t- 89 | ~0. This is a "twisted" current algebra. It is spanned by pairwise local formal distributions L
a(z) = E
(tn | a) z-n-1 for a r ~0
nEZ
and
a(z) = E ( tn+89• a)z -n-l nEZ
for a r
E 1.
Superconformal Algebras and Transitive Group Actions on Quadrics
237
However, [a(z), b(w)] = w[a, b](w)d(z - w) if a, b E ~1. Hence ~t is not a finite Lie algebra of formal distributions, at least if ~ = [~, ~]. Finite formal distributions Lie superalgebras can be studied via conformal superalgebras introduced in [6]. Definition 1.2. A conformal superalgebra R is a left Z/2Z-graded C[O]-module R = R~ G R~ with a C-bilinear product a(n)bfor each n E Z+ such that the following axioms hoM (a, b, c E R, m, n E Z+): (CO) (C1)
a(n)b = Ofor n >> 0, (Oa)(n)b = -na(n_l)b,
(C2)
a(n)b= ( - 1)p(a)p(b)~-'~j~o(- 1)J+n+l(OJ/J !)b(n+j)a,
(C3)
a(m) (b(n)C) = }-'~d~__0( 7 ) (a(j'b)(,~+n-2 c + (--1)P(a)P'b)b('~)(a(m)C).
It is shown in [6],Sec. 2.7 that if (G F) is a formal distributions Lie superalgebra such that R(g, F) := C[O]F is closed under all products (1.3), then R(g, F) is a conformal superalgebra with respect to these products. Note that, by definition, (~, F) is finite iff R( G F) is a finitely generated C[0]-module. Conversely, if R = @ieiC[O]a i is a free as the C[0]-module conformal superalgebra, we may associate to R a formal distributions Lie superalgebra g(R) with the basis 9 i --n--1 a~m ) (i E I, m E Z) and F = {ai(z) = ~-]n a(n)z }~er with the bracket (cf. (1.2)): [a'(z)'aJ(w)] = E
(a~k)aj) (W)O~5(z--w)/k!,
kEZ+
so that R(~(R), F) = R. Proposition 1.1. Every finite formal distributions Lie superalgebra with trivial center is a quotient of a Lie superalgebra I~(R), where R is a conformal superalgebra finitely and freely generated as a C[O]-module and with trivial center, by an ideal that does not contain all a(n), n E Z, for a non-zero element a E R. For such Lie superalgebras the C[O]-module R(g, F) is free. Proof Follows from the above discussion and [6], Proposition 2.7 which states that a finitely generated C[0]-module conformal superalgebra with a trivial center is a free C[0]-module. [] All notions concerning formal distributions Lie superalgebras of course are automatically translated into the language of conformal superalgebras [6]. The following simple observation is extremely useful. Lemma 1.1. Let R be a conformal superalgebra. Then, with respect to 0 th product, OR is a 2-sided ideal of R and R / O R is a Lie superalgebra over C. Moreover, the 0taproduct defines a structure of a left R/OR-module on R (over C[0]). Proof See e.g. [6],Corollary 2.3c.
[]
The simplest examples of conformal superalgebras are the current conformal superalgebras : R(g) C[0] | =
with the products defined by :
238
V.G. Kac
a(o)b = [a, b], a(j)b = 0 for j > 0, a, b E ~, and the Virasoro conformal algebra V = C[O]L with the products (cf. (1.4)):
L(o)L = OL, L(1)L = 2L, L(j)L = 0 for j > 1. More complicated (finite rank) conformal superalgebras are associated to (finite) Lie superalgebras of formal distributions constructed in Sect. 2. I conjecture [6] that all simple conformal superalgebras which are finitely generated C[0]-modules are either current conformal superalgebras associated to finitedimensional simple Lie superalgebras (classified in [5]) or conforrnal superalgebras associated to one of the superconformal algebras listed in Sect. 2. A proof of this conjecture is in sight now 2. The non "super" case has been recently settled. Let ~t be a Lie superalgebra and let L(z) be its Virasoro distribution. One says that a formal distribution a(z) with coefficients in g is an eigendistribution of conformal weight A~ E C with respect to L(z) if L(z) and a(z) are mutually local and [L(z), a(w)] = Owa(w)5(z - w) + Aaa(w)OwS(z - w) + . . . .
(1.5)
The formal eigendistribution a(z) is called a primary distribution with respect to L(z) if the coefficients of O~6(z - w) with j > 1 in (1.5) vanish. This is equivalent to the commutation relations: [Lm, a(z)] = z m (zOz + (m + 1)Aa) a(z).
(1.6)
(Equality (1.5) is equivalent to (1.6) for m = 0 and - 1 ) . A primary distribution of conformal weight 1 is usually called a current. The following proposition is a straightforward but useful bookkeeping device (see e.g. [6],Lemma 5.9).
Proposition 1.2. Let a(z) and b(z) be eigendistributions (with respect to L(z)) ofconformal weights Aa and Ab respectively. Then Oza( z ) is an eigendistribution of conformal weight Aa + 1, and ( a~j)b) ( z ) is an eigendistribution of conformal weight Aa + Ab-- j -- 1. Let (g, F ) be a formal distributions Lie superalgebra. A formal distribution L(z) E F is called an energy-momentum distribution if L(z) is a Virasoro distribution and all a(z) E F are its eigendistributions. Definition 1.3. A formal distributions Lie superalgebra (g, F ) is called a superconformal algebra if it is simple, finite and F contains a Virasoro distribution L. The most important examples of superconformal algebras g that appear in conformal field theory have the property that L is an energy-momentum distribution and that A,, = 1 if a E F~ is different from the energy-momentum distribution, and Aa = 3 or = 89if a E F T. We then say that (9, F ) is a physical superconformal algebra. It follows from Proposition 1.2 that for a physical superconformal algebra all formal distributions from F are primary and all currents from F span a current Lie algebra. In the study of representations of a superconformal algebra (g, F ) it is important to consider its central extensions. By definition, this amounts to adding to the righthand side of the bracket [a(z), b(w)] in (1.2) a linear combination of terms of the form ctj(a, b)OJ~5(z - w ) / j ! , where aj(a, b) E C is a C-valued C-bilinear form defined for
2 Addedinproof. This conjecturehas been proved
Superconformal Algebras and Transitive Group Actions on Quadrics
239
each j E Z§ on R = C[0]F. The usual properties of a 2-cocycle of a Lie superalgebra are then equivalent to the following identities for all m, n E Z+ ([6], Sec.2.7):
Olr~(Oa,b) = -nc~n_l(a, b), c~n(a, b) = ( - 1)n+l+p(a)p(b)an(b, a),
(1.7)
Olm (a,b(n)C) = ~-~ ( ~. ) Olm+n-j (a(j)b,c) +(-1)p(a)p(b)oLn (b,a(m)C) . j--o As usual, the trivial cocycle an(a, b) = f (a(n)b), where f : R ---+C is a C-linear map, defines a trivial central extension of g (isomorphic to the direct sum of g and C). Two cocycles that differ by a trivial cocycle are called equivalent; they define isomorphic central extensions. Following [7],[3] we list in Sect. 2 all, up to equivalence, central extensions of all superconformal algebras discussed there.
2. Examples of Supereonformal Algebras Denote by A(1, N) the tensor product of the algebra of Laurent polynomials C It,/~--1] over C in the indeterminate t and the Grassmann algebra A(N) over C in the indeterminates ~l,. 99 iN. This is an associative superalgebra with the parity
p(t) = -0, p (~i) = 1, i = 1,..., N. The Lie superalgebra of all derivations of the superalgebra A(1, N) is denoted by
WN. This is a simple Lie superalgebra [5, 7]. Introducing the even derivation 00 and odd derivations Oi for i = 1 , . . . , N by 0
0
we can write every element of W N in the form of a linear differential operator N
D = EPiOi,
where Pi C A(1,N).
(2.1)
i=O
Given an element A E A(N), we introduce for each j = 0, 1 , . . . , N the following formal distributions with coefficients in WN:
AJ(z) = E
(t~AOj) z-"-l"
nCZ
Note that W N is spanned by ( N + 1)2 N linearly independent over C[0z] formal distributions AJ(z), where A runs over all monomials in A(N) and j over the set 0, 1 , . . . , N. Of course, W0 is the Virasoro algebra. The commutation relations of the Lie superalgebra WN can be written in a compact form by making use of the formal delta-function as follows.
Proposition 2.1. For arbitrary A and B C A(N) one has:
240
V . G . Kac
(a) [Ai(z),BJ(w)] : ((A cOiB)j ( w ) + ( - 1 ) p(A) ((OjA) B ) ' (w)) 6(z - w) if i , j = l , . . . , N . (b) [Ai(z), B~ = (A OiB) ~ (w)5(z - w) - (-1)p(B)(AB)i(w)O~5(z - w) /f i = 1 , . . . , N . (c) [A~ B~ = - O ~ ( A B)~ - w) - 2(AB)~ - w). Proof is straightforward by using (1.2) and (1.3)
[]
Here are some important special cases of Proposition 1.1: [-l~
AJ(w)] = O~AJ(w)5(z - w) + (1 + 5jo) AJ(w)O~5(z - w),
(2.2)
[(~i) ~ (z), AJ(w)] = (5~A -- 5~j) AJ(w)5(z - w) if j = 1 , . . . , N ,
(2.3)
[(~i)i (z), A~
= 5iAA~
-- w) -- (-- l) p(A) (~iA) i (w)O~5(z - w).
(2.4)
In the last two formulas we assume that A is a monomial ~i~ ... ~i~ and we let (~iA
----
1 if OiA 5t 0, and = 0 if OiA = O.
Let now ), = (A1, 9 9 )~N) C C g and consider the following formal distribution: N
L~(z) = - l ~
+E
;~iOz (~i) i (z).
(2.5)
i=l
Proposition 2.2. (a) For each )~, the formal distribution L ~( z) is an energy-momentum formal distribution. (b) Let A = ~il ... ~i~ be a monomial. Then the formal distribution AJ(z), where j = 1 , . . . , N, isprimary with respect to L~(z) with conformal weight A= l+.~j - E)~k. k
(c) Let A = ~i, ... ~is be a monomial. Then the formal distribution A~ formal weight A =2- E
has con-
"~k
k
with respect to LA(z), but is not primary, the extra term in [LA(z), A~
(--1)P(A) E
hi (~iA) i (w)O2 ~(z - w). i
If A 5r l, then the corrected formal distribution A~
+
( - 1 )p(A) ----Z-~
E
)~i (~iA) i (w) i
is primary ( of conformal weight A ).
being
SuperconformalAlgebras and TransitiveGroup Actions on Quadrics
241
Proof Ffollows from (2.2)-(2.4) and the usual observations. First, if L(z) is a Virasoro formal distribution, and ~(z) is even primary of conformal weight 1 with respect to L(z) and [~(z), ~(w)] = 0, then L(z) + A0z~(z) is again a Virasoro formal distribution. Second, if [L(z), a(w)] = O~.a(w)5(z - w) + Aa(w)O~5(z -- w) + b(w)O2 ~(z - w), then a(w) - F-10b(w) is primary of conformal weight A with respect to L(z).
[]
The following corollary is immediate by Proposition 2.2. Corollary 2.1. (a) The Lie superalgebra WN is a superconformal algebra for all N >_ O. It is physical with respect to L ;~(z) iff A = ( 1 , . . . , 89 and N < 2. (b) The even part of W1 is spanned by L89 and a current (~1) 1 (z). The odd part of W1 is spanned by two primary formal distributions of conformal weight 3: l l(z) and ((1) ~ (z). (This is the N = 2 superconformal algebra.) (c) The even part of W2 is spanned by L(89189 and five currents:
(~)J (z) for i , j = 1, 2 and (~1~2)~ (z). The odd part of W2 is spanned by four primary formal distributions of conformal weight 3. li(z)for i = 1,2 and (~i)~ (z) for i-- 1,2, 1. and two primary formal distribution of conformal weight ~.
(~1~2)i (Z) for i = 1,2. Remark 2.1. It is shown in [7] that any central extension of WN is trivial unless N <_ 2; the superconformal algebras WN with N _< 2 have a unique (up to equivalence) central extension. For W0 it is the usual Virasoro central extension. For W1 it looks as follows (we write down only non-zero values of c~k's):
Oq ((~l) 1 ,(~l) l) =C/3, OL2 ((~l) l ,(1) 0) =C/3, OL2 ((1)1,(~1)0) = --C/3. For W2 it looks as follows:
OL1
((~1)2,(~2)l) =C/6,
0~1((~1)1,(~2)2)
---
--C/6,
oq ((~1~2)1 ,(1) 2) =c/6.
oq (>( ~ 1 ~ 2 ) 2 , ( 1" ) 1 ) = - c / 6 ,
Here and further the normalization is chosen in such a way that for the "physical" energy-momentum formal distribution L = L(89..... 89 we have ol3(L, L) = c/2. Recall that the divergence of a linear differential operator (2.1) is defined by [5]
N div D = OoPo + E(-1)P(POoiPi,
(2.6)
i=l and that its basic property is div [D1, D2] = D1 (div D2) - ( - 1)P(D')p(D2)D2(div D1).
(2.7)
242
V.G. Kac
It follows that SN := {D E WNI divD =0} is a subalgebra of the Lie superalgebra WN. This Lie superalgebra is not simple since =
9
...
However, the Lie superalgebra SN :=
is simple provided that N > 2 [5, 7]. The odd part of Sl is an ideal9 It is straightforward to check the next proposition. Proposition
9
N
2.3. (a) L~(z) lies in SN iff ~ = 1 Ai = 1.
(b) The following formal distributions span SN (A E A(N)): A~j(z) := (O~A)j (z) + (OjA) ~(z), i , j = 1,..., N, Aoj(z) := (OjA) ~ (z) - (--1)P(A)OzAJ(z), j = 1 , . . . , N. One can choose N 2 N linearly independent over C [Oz] among them that still span S~r (c) If A = ~i~ ... ~i~ is a monomial, then the formal distributions Aij(z), i, j = 1, . . . , N, are primary with respect to L;~(z) of conformal weight 1 +A~ +Aj - E A i k " k
(d) If A = ~il ... ~ is a monomial, then Aoj(z) with respect to L X(z) has conformal weight 2 + Aj - E
Aik.
k
Corollary 2.2. (a) The Lie superalgebra SN is a superconformal algebra iff N > 2. It is physical with respect to LX(z) iff N = 2 and ),1 = ),2 - ~. (b) The even part of $2 is spanned by L(z) := L(89189 and three currents: _
1
(~1) 2 (Z), (~2) 1 (Z), (~1) 1 (Z) -- (~l)2"(Z).
The odd part of $2 is spanned by four primary formal distributions of conformal weigth 3.
li(z), i = 1,2, (~1) 0 (Z) +
0 z (~1~2) 2 (Z), ~20(Z) -- Oz (~1~2) 2 (Z).
(This is the N = 4 superconformal algebra.) Remark 2.2. According to [7], any central extension of SN with N > 2 is trivial. The
only (up to equivalence) central extension of $2 looks as follows: a3(L,L)=c/2,
a,
o~1 ((~,), _ (~,)2 (r
((~1)2 , (~2)1) _ (~,)2)
=c/6,
= e/6, ]
a2 ((1) > r (~0~ + 0 ((~i~,)' + (~i~2)z)) -- - e / 3 , i = 1,2.
Superconformal Algebras and Transitive Group Actions on Quadrics
243
Consider the differential form N
w =dt-
Z~d~i i=1
and the following subalgebra of WN:
KN := {D E WN IDw = Pa~ for some P E A(1, N)}, introduced independently in [1] and [5] (see [71). This Lie superalgebra consists of linear differential operators of the form ( f E A(1, N)): N
D: := fOo +
I(-I)P(':)ED~(f)Di, i=l
where we let
Di=~iO0+Oi, i = 1 , . . . , N . (Note that Dfw = (Oof) w), It easily follows [5, 7] that KN is a simple Lie superalgebra for N > 0, unless N = 4 in which case we have: K4 = [1s K4] | CD t-1~2~3~4 and [K4,/s is simple. Using the vector space isomorphism A(N) -% KN given by f H D : , we identity KN with A(1, N). Then the bracket of differential operators on KN gets identified with the following bracket on A(1, N) [5],[7]:
If, y]= I - ~ 1 -Oof ( g - :
~0J
Oog
Ni~=l ~Oig ) ( _+l ) p ( D ~ O- i f O i g
(2.8)
i=l
Given A E A(N), define the following formal distribution with coefficients in A(1, N) = K N :
A(z) = Z
(tnA) z-n-l"
nEE
Note that KN is spanned by 2 N linearly independent over C [0z] formal distributions A(z), where A runs over all monomials in A(N). Proposition 2.4. (a) LX(z) lies in I(N iff A1 . . . . . AN = 89 One also has: 1 1 1 L(~'~ ..... ~)(z) = - l(z). (b) For arbitrary monomials A = ~ ... ~i. and B = ~jl ... ~J. one has:
[A(z),B(w)]
= ((~ - 1 ) OwAB(w)+(-1)~ 89 y~.N, (OiAOiB)(w)) 6(z - w) + (~-~ - 2) AB(w)O~5(z - w).
In particular, A(z) is a primary formal distribution of conformal weight 2 - ~ with respect to the energy-momentum distribution -1 (z). Proof Straightforward as that of Proposition 2.1, using (2.8).
[]
244
V.G. Kac
Corollary 2.3. The Lie superalgebra KN is a superconformal algebra for all N >_O. It is physical with respect to L(z) = - l ( z ) / f i N < 3.
Remark 2.3. It is proved in [7] that any central extension of KN is trivial unless N < 4. For KN with N < 3 the only non-trivial (up to equivalence) central extension look as follows:
ao (~j~k,~i~j~k) = C/12 (i 7~j •k), 0~1 (~i~j,~i~j) = c/12 (i r C~2(~i, ~i) = c/6, a3(1, 1) = c/2. It follows from [7] that, up to equivalence, the only "field-theoretic" central extension of K4 looks as follows, where we let u = ~1~2~3~4: OLl(1,/2 ) = C, Oq
(~i,Oil]) = C, Oq (~i~j,OiOjl)) = (3,
the Virasoro central charge being 0.
Remark 2.4. One has the following isomorphisms between superconformal algebras: W0 -~ K0 (Virasoro), W1 ~ K2 ( N = 2 superconformal algebra). Finally, the superconformal algebra CK6 is constructed as a subalgebra of/s [3] as follows. Let u = ~1~2~3~4~s~6 and for a monomial A = ~, ~i2 ... denote by A* the Hodge dual monomial: A* = 0i, 0i2 999u. Then CK6 is spanned by the energy-momentum distribution - l ( z ) + x/-Z]-03 u(z), by 15 currents ~i~j (z)+ V/-~Oz (~i~j)* (Z), by 6 primary distributions of conformal weight 3: ~ ( z ) - v/-L-]-02~* (z), and 10 primary distributions of conformal weight i.1, ~i~j~k(z) + ~ (~i~j~k) * (z).any central extension of CK6 is trivial [3].
3. Transitive Group Actions on Quadrics Let S O N be the complex orthogonal group, i.e. the group of all unimodular linear transformations of the N-dimensional complex vector space V preserving a non-degenerate invariant bilinear form (., .) on V. Then SON acts transitively on the complex quadric
Q(V) = {v E Vl(v,v)= 1}. Taking the group GLN of complex invertible matrices acting linearly on the Ndimensional complex vector space V1 and defining on the 2N-dimensional vector space V = 171 | VI* the symmetric bilinear form (u + u*, v + v*) = (u, v*) + (v, u*) (where (,) is the bilinear pairing between V and V*), we obtain an embedding of GLN in S02N. The subgroup SLN of this GLN still acts transitively on the complex quadric. The subgroup SLN is contained in a unique up to conjugacy maximal proper closed algebraic subgroup of S02N; this subgroup has the form GLN ~
GLN, SLN ~
(3.1)
Another important example is the subgroup SpN • SL2 (N even) of S02N which is obtained by looking at V = 171 | C 2 with the symmetric bilinear form on V being the
SuperconformalAlgebras and TransitiveGroup Actions on Quadrics
245
tensor product of symplectic forms on V1 and C 2. The subgroup SpN of SpN • SL2 still acts transitively on the complex quadric. We have an embedding of C x and C in SL2 respectively by: A~--~ 0
A- 1
and A H
0
'
and a decomposition UN = U~v x C such that SpN normalizes U~v and commutes with C. Since SpN acts irreducibly on Lie U~v and SpN is a maximal subgroup of SLN, we see that a complete list of connected algebraic subgroups strictly between Spy and S02N is
SpN •
x, SpN xC, (SpN• x) ~
SpN N UtN, SpN ~
(3.2)
Of course, all the groups from (3.1) and (3.2) act transitively on the complex quadric. Theorem 3.1. Let G be a connected closed subgroup of the group SON acting transitively on the complex quadric. Then G is conjugate to a subgroup from the following list:
1) 2) 3) 4) 5) 6)
SON, SLN C G c S02N, SpN ~ G C S02N (N even, N >_4), G = G2 C S07, G = Spiny C ~08, G = Spin9 C S016.
The proof of this theorem is based on the following simple lemma (whose proof is the same as that of Montgomery theorem [9] for real Lie groups [2]). L e m m a 3.1. Let G be an affine algebraic group acting regularly and transitively on an irreducible algebraic variety M of(complex) dimension d, and suppose that Hd( M, C) 5r
O. Then any maximal reductive subgroup K of G has a dense orbit on M. Proof. We have: M ~- G/H, where H is the stabiliser of a point m of M. We may choose the point m in such a way that L := H N K is a maximal reductive subgroup of H. We have the following two fibrations: H / L ---* G/L ~ G / H and K / L ---* G / L ---* G / K . Since both H / L and G / K are contractible (see e.g. [12]), we obtain that
H*(G/L, C) = H*(G/H, C) = H*(K/L, C). Hence Hd(K/L, C) r 0. It follows that K . m is a dense orbit in M (see Remark 3.3). []
Remark 3.1. (a) If under the assumptions and notation of Lemma 3.1 the variety M is affine, then K acts transitively on M . The following proof of this fact was supplied by A.L.Onishchik. Indeed, take a K-orbit O on M of minimal dimension. It is closed, hence affine, and therefore by the Matsushima-Onishchik theorem (see [4] for a simple proof of this theorem), the stabiliser F of a point of O is a reducfive subgroup. The subgroup
246
v.G. Kac
F is conjugate to a subgroup of L, hence dim O = dim K / F > dim K / L = dim M. Hence O is dense (and closed) in M, and therefore coincides with M. (b) The following example was supplied by E.B.Vinberg. Let G be the group of affine transformations of C 2 and let M be the (non-affine) variety of all complex straight lines in C 2. Then dim M = 2, H2(M, C) 5r 0, G acts transitively on M, but its maximal reductive subgroup GL2(C) does not (but has an open orbit).
Proof of Theorem 3.1. Let K be a maximal reductive subgroup of G. It follows from Lemma 3.1 that the algebra C[V] K of K-invariant polynomials on V is generated by a non-degenerate quadratic form Q. There are, clearly, two possibilities: (a) V is irreducible with respect to K, (b) V = V1 | Va*, where V1 is irreducible with respect to K. In case (a) K must be semisimple. In case (b) the semisimple group (K, K ) C K still has the above property (indeed, if P(x, y), where x E V1, y c VI*, lies in C[V] (K,K) but is not a polynomial in Q, then P(x, y)P*(x, y), where P* is the "contragredient" invariant, lies in C[V] K and is not a polynomial in Q). Thus we may assume that the algebraic group K is connected semisimple. In case (a) either (i) the group K is simple, or (ii) it is a tensor product of two subgroups of the orthogonal group or (iii) it is a tensor product of two subgroups of the symplectic group. Case (ii) is ruled out since SOre | SOn with m > n > 3 has at least two basic invariants. In case (iii) one of the symplectic subgroups should be SL2 since Spm | Spn with m >_ n > 4 has at least two basic invariants. Observing that K | C x has an open orbit in V, we derive (e.g. from [14]) that in case (i) we have SON, G2 c S07, Spin7 C S08 and Spin9 C S016 and in case (iii) we have only Spin | SL2 (m > 2). In case (b) the group K must be simple since for the action of GL~ x GL,~ on (C m @ C '~) @ (C m | C'~) * with m > n > 2, there are at least two basic invariants. We derive (e.g. from [4] observing that the action of K on V is polar) that K is either SLy or SpN acting on C N @ ( c N ) *. This completes the proof of the theorem. []
Remark 3.2. A.L.Onishchik pointed out that the classification of reductive subgroups K of SON (C) acting transitively on quadrics is equivalent to the classification of decompositions SON(C) = K . SON_I(C), and due to [10], Theorem 2.1, this is equivalent to classification of decompositions of compact Lie groups: SON = K- SON- 1. A solution to the latter problem is a classical result (see [11]).
Remark 3.3. Let G be an affine algebraic group and let H be its closed subgroup. Let K be a maximal compact subgroup of G and let L C K be a maximal compact subgroup of H . The same argument as in the proof of Lemma 3.1 shows that H*(G/H, C) = H*(K/L, C). It follows from this and Remark 3.1a that a homogeneous affine variety M = G/H of an algebraic group G is a homogeneous variety of a reductive group if and only if Ha(M, C) ~ 0 for d = dim M. One always has: H3(M, C) = 0 f o r j > d.
4. A Classification of Physical Superconformal Algebras We prove here the main result of the paper.
Theorem 4.1. Thefollowing is a complete list of physical superconformal algebras:
KN for 0 < N <_3, $2, W2, and CK6.
Superconformal Algebras and TransitiveGroup Actions on Quadrics
247
Let (9, 5L-)be a physical superconformal algebra. Consider the associated conformal superalgebra R = C[0].T. Due to [6], Proposition 2.7, R is a finitely-generated free C[0]-module. Hence, in view of Proposition 1.2, we have: / ~ = (C[O]L) | (C[O] | a), Ri = (C[O] | V) | (C[O] | F ) , where L is the energy-momentum distribution, a is the space of all even primary distributions of conformal weght 1, and V (resp. F ) is the space of all odd primary distributions of conformal weight 3 (resp. 1). It also follows from Proposition 1.2 that a is a finitedimensional Lie algebra over C and C [0] | a is the associated current conformal algebra. (In other words, ~ is the usual semidirect product of the centerless Virasoro algebra and the Lie algebra C [t, t - l ] @c a.) Furthermore, it follows from Proposition 1.2 that we have representations 7r of a on V and cr of a on F defined respectively by:
a(o)v = ~ ( a ) v + O p ( a |
for a C a, v @ V,
(4.1)
where p : a | V ---+ F is a homomorphism of a-modules;
a(o)v = cr(a)v for a E a, v C F.
(4.2)
The remaining non-trivial products, due again to Proposition 1.2, have the following form:
u(o)v u(1)v
= 2(u,v)L + O ~ ( u | for u,v C V, = ~)(U| for u,v E V,
u(o)V =
X(u|
for u E F ,
(4.3) (4.4)
vcV,
(4.5)
where (u, v) is a symmetric C-valued bilinear form on V, invariant with respect to the representation 7r of a on V, ~ and ~b are homomorphisms of n-modules V | V ) ad a, and X is a homomorphism of n-modules F | V ~ ad a. Due to the property (C3) of R, we have for u, v C V: L(2) (uco)v) = (Lco)U)(2) v + 2 (L0)u)0) v. It follows, using (C1), that L(2) (u(o)v) = U(l)V. Applying L(2) to both sides of (4.3), we deduce that ~ = 2~.
(4.6)
Lemma 4.1. (a) The bilinear form (., .) on V is non-degenerate. (b) The representation 7r is faithfuL
Proof Let V ~ C V be the kernel of the form (., .). Then I := C[0] |
(a + V ~ + F ) is an ideal of the conformal superalgebra R, since a(n)I C 1, V(n)I C I and F(n)I C I, proving (a) (since R is simple). Let a ~ = KerTr. Then J := C[0] | ( a ~ a~o)F) is an ideal of R since a(n)J C J, F ~ ) J C J (because F(n) (a~o)F) has non-positive conformal weight) and V0~)J C J (because V(n,a ~ = 0 and En_>o V(~)(a~o)F ) C En_>oa(~ (Vu~)F) C a~o)a C a~ proving (b). [] If V = 0, then R is the Virasoro conformal algebra K0 since C[0] | ideal in this case. Hence from now on we shall assume that V 5/0.
(a + F ) is an
248
v.G. Kac
Lemma 4.2. Let (g, 3L-) be a superconformal algebra such that ~0 contains, apart from L, only currents, and let R be the associated conformal superalgebra. Let a and a' be odd eigendistributions of R of conformal weights A and A' such that ,4 + A' = 3, so that a(0)a' = 2(a, at)L + O~(a, a'), where (a, a ~) E C and ~(a, a') is a current. Then (a) In the left R/OR-module R defined by Oth product (see Lemma 1.1) we have: [a, a'] = 2(a, a')OIn.
(4.7)
(b) If(a, a') :fi O, then dimc[o] RO = dimc[o] R i.
Proof (a) is clear by Lemma 1.1 and the definition of the energy-momentum distribution. (b) follows by taking the supertrace (see [5]) of both sides of (4.7). [] Lemma 4.3. Under the assumptions of Lemma 4.2, let V be a subspace over C of R consisting of odd eigendistributions ofconformal weight 3/2, and suppose that the bilinear form (., .) defined by (4.3) is non-degenerate on V. Then d := dimctoj R is divisible by 289(N-l) (resp. 21 N) if N := dim V is odd (resp. even).
Proof We have, by Lemma 4.2, in the left R/OR-module R: uv + vu = 2(u, v)O. Tensoring with C(0), we obtain a representation of a Clifford algebra over a vector space over C(0) of dimension N with a non-degenerate bilinear form 2(u, v)O in a vector space over C(0) of dimension d. This proves the lemma. [] Let N = dimc V (> 1), a = d i m c a, f = dimc F. We have by Lemma 4.1a, Lemma 4.2 and Lemma 4.3: a + 1 = N + f is divisible by 2 [89
(4.8)
Furthermore, let u E V be such that (u, u) = 1, and let u • be the orthogonal complement to Cu in V. Consider the following basis of R over C[0]: L, basis of a, u, basis of u • basis of F. The matrix Mu of u, viewed as an element of R / O R acting on R by 0 -th product, looks in this basis as follows (we use that u has conformal weight 3/2 and formulas (4.1), (4.3), (4.5)) :
M~,=
0 0
0 0
2 aO
0 AO
0 / /3
0/2 0 0
7 u #9
o 0 0
o 0 0
o 0 0
Herec~,/3, 7, A, #anduarematricesoverCofsizesax 1, a x f , 1 xa, a x ( N - 1 ) , f x a and (N - 1) • a respectively. But due to Lemma 4.2a, the square of the matrix M~ is OI. This is equivalent to c~ = 0, -y = 0 and the following relations:
vA = I N - l , tz/3 = Iy, vfl = O, #A = O, Au +/3lz = It.
(4.9a) (4.9b) (4.9c) (4.9d)
Superconformal Algebras and Transitive Group Actions on Quadrics
249
It follows from (4.9a) that rank v > N - 1. This implies
rc(a)u has codimension < 1 in V.
(4.10)
Let A be the connected linear algebraic subgroup of S O ( V ) whose Lie algebra is 7r(a) (recall that the bilinear form (., .) on V is a-invariant). It follows from (4.10) that A . is an open orbit on the quadric (u, u) = 1. Since this holds for any point of this quadric, we conclude that A acts transitively on the quadric (u, u) = 1.
(4.11)
Consider the following diagram of maps (we denote the map by the same letter as the corresponding matrix): # a
,
F
/]
U•
/3 P
a
Since rank u _> N - 1 and rank # _> f (due to (4.9b)), we conclude that the maps u and # are surjective. It follows from (4.9c) that fl(F) is annihilated by u. Finally the relation (4.9d) implies that both maps/3 and A are injective and their images have zero intersection. Since all four maps #, u, A and/3 are a~,-module homomorphisms, we conclude that the a~,-module a (with respect to ad) is isomorphic to the a~,-module F q~ u • But the au-modules a/a~, and u • are isomorphic. It follows that the a~,-modules F and ad a~, are isomorphic. We thus obtain the following condition a-module F restricted to au is isomorphic to ad a~,.
(4.12)
Now we are in a position to prove Theorem 4.1. First, due to condition (4.11), a is a Lie subalgebra of so(V) corresponding to one of the subgroups listed by Theorem 3.1. If N = 2n+ 1 is odd, we see that there are two possibilities: a = sou or a = Lie G2 c so7. The second case is ruled out immediately (14 + 1 is not divisible by 8). In the first case condition (4.8) gives: n(2n + 1) + 1 is divisible by 2 n which eliminates all cases except for (a) and (b) below: (a) N = 1, a = O, f = O;
(b) N = 3 ,
a=so3, f=l.
Let now N = 2n be even. First of all, the exceptional cases 5) and 6) of Theorem 3.1 are ruled out by (4.8). If a = SON (case 1 of Theorem 3.1), condition (4.8) gives: n(2n - 1) + 1 is divisible by 2 n-1 which permits only the following cases: (c) N = 2 ,
a=so2, f=0;
(d) N = 6, a = 806, f - - - 1 0 .
250
v.G. Kac
In case 2) of Theorem 3.1 there are the following 4 possibilities for a C so2n, n > 1 (see (3.1)): a = gln, a = sln ~ Un, a = gln ~4 Un, a = sln, where dim un = 89
- 1). Hence we have respectively:
a = n 2, a =
l, a = ~1n ( 3 n - 1 ) ,
n(3n-1)-
a = n2 - 1 .
Due to (4.8) this number should be divisible by 2 ~-1 . This permits only the following cases: (e) N = 4 ,
a=sl2,
f=0;
(f) N = 4, a = gl2 ~
(g) N = 6 ,
a = s l 3 ~
f=2; f=6;
(h) N = 8, a = 814, Case 3) of Theorem 3.1 remains. According to (3.2), there are 9 possibilities (including Sp~). Recall that n is even and we may assume that n > 4 (since sl: ~- sp2). The corresponding possibilities of values of a are these: n(n+l),
1 ~n(n+l)+l,
1 ~n(n+l)+2,
1 ~n(n+l)+3,
n 2 -- 1, n 2~ n 2 + 1. Condition (4.8) says that this number should be divisible by 2 n-1 , which permits only one possibility: N=8, a=sp4~
In all cases (a) - (f) there exists a unique, up to a constant factor, n-module homomorphism r : V | V ~ ad a. The constant factor is determined uniquely from the identity (C3) for a, b, c c V and m = 1, n = 0. Hence in the relations (4.3) and (4.4),
Superconformal Algebras and Transitive Group Actions on Quadrics
251
the maps ~ and ~ = 2~ (see (4.6)) are uniquely determined. It follows that in the cases when f = 0, i.e. (a),(c) and (e), the only possibility is K t , / ( 2 and $2 respectively. In order to prove uniqueness in the remaining three cases one has to look at the a-module F. It suffices to check the following condition: (,) the a-module homomorphisms p:(ada)|
and x : F Q V ~ a d a
are unique up to constant factors, say, a and/3. Indeed, both constants must be non-zero since otherwise our conformal superalgebra R is not simple. By rescaling we can fix the constant/3. After that the constant c~is uniquely determined from (C3) for a, c E V, b E a and m = n = 0. Condition (,) is obvious in case (b) since F is the trivial 1-dimensional s/2-module and V is the adjoint s/2-module in this case. Case (d) is more interesting. Recall that in this case the a-module V is isomorphic to the defining representation of so6, so that a~ is isomorphic to sos. Now we use the well-known isomorphisms: so6 ~-- sl4 and so5 ~- sp4. Then V becomes the s/4-module A2C 4 and the sp4-module ad a~ becomes $2C 4. Hence, by (4.12), the s/4-module F is isomorphic to $2C 4. Now (,) is easily checked. Finally, in case (f) the Lie algebra a is isomorphic to the direct sum of sl2 and the 2-dimensional non-commututive Lie algebra b, its module V is isomorphic to C 2 | ad b, and it is easy to see that au -~ b. Due to condition (4.12), it follows that the a-module F must be either the adjoint representation of b extended to the sl2 summand trivially, or the 2-dimensional irreducible representation of sl2 extended to b by a scalar. But the first case is inconsistent with p ~ 0, X 5/ 0, and it is easy to establish uniqueness in the second case as we did above. This completes the proof of the theorem. R e m a r k 4 . 1 . One can show, using the same type of arguments and representation theory of conformal superalgebras, that if the conditions of Theorem 4.1 on the even part of a superconformal algebra remain the same, but the conditions on the odd part are dropped, then the list remains the same.
References 1. Ademollo, M., Brink, L., D'Adda, A., D'Auria, R., Napolitano, E., Scinto, S., Del Guidice, E., Di Veccia, P., Ferrara, S., Gliozzi, E, Musto, Pettorino, R., Schwarz, J.H.: Dual strings with U(1) colour symmetry. Nucl. Phys. B 111, 77-111 (1976) 2. Borel, A.: Les bouts des espaces homogenes des groupes de Lie. Ann. Math., 58, 443-457 (1953) 3. Cheng, S.-J.,Kac, V.G.: A new N = 6 superconformal algebra. Commun. Math. Phys. (1977) 4. Dadok, J.,Kac, V.G.: Polar representations. J. Algebra, 92, 504-524 (1985) 5. Kac, V.G.: Lie superalgebras. Adv. Math., 26, 8-96 (1977) 6. Kac, V.G.: Vertex algebras for beginners. University lecture series, vol 10, Providence RI: AMS, 1996 7. Kac, V.G., Leur, J.W.: On classification of superconformal algebras. In: S.J.Gates et al. eds, Strings 88, Singapore: World Sci, 1989, pp. 77-106 8. Mathieu, O.: Classification des algrbres de Lie gradudes simple de croissance < 1. Invent. Math. 86, 371-426 (1986) 9. Montgomery, D.: Simply connected homogeneous spaces. Proc. AMS, l, 467-469 (1950) 10. Onishchik, A.L.: Decompositions of reductive Lie groups. Math. USSR~Sbornik 9, 515-554 (1969) 11. Onishchik, A.L.: Topology of transitive transformation groups. J. A. Barth Verlag, 1994 12. Onishchik, A.L., Vinberg, E.B.: Lie groups and algebraic groups. Berlin-Heidelberg-New York: Springer Verlag, 1990
252
V.G. Kac
13. Ramond, E, Schwarz, J.H.: Classification of dual model gauge algebras. Phys. Lett. B64B, No. 1, 75-77 (1976) 14. Sato, M., Kimura, T.: A classification of irreducible prehomogeneous vector spaces and their relative invariants. Nagoya Math. J. 65, 1-155 (1977) Communicated by A. Jaffe
Commun. Math. Phys. 186, 253 – 264 (1997)
Communications in
Mathematical Physics c Springer-Verlag 1997
Decomposition of the Adjoint Representation of the Small Quantum sl2 V. Ostrik Independent University of Moscow, P.O. Box 230, Moscow 11743, Russia. E-mail: [email protected] Received: 30 May 1996 / Accepted: 15 July 1996
Abstract: In this Paper we find all indecomposable components of adjoint representation of small quantum sl2 with multiplicities.
1. Introduction √ 1.1 . Given a finite type root datum and a primitive root of unity q = l 1, G. Lusztig has defined √ in [Lu] a remarkable finite dimensional Hopf algebra u over the cyclotomic field Q( l 1). It is called a restricted quantum universal enveloping algebra, or a small quantum group. Recall that for a Hopf algebra A with a coproduct 1, antipode S, and counit ε the adjoint representation is defined in the following way. A is an A-bimodule with respect to left and right multiplication. Using the antipode we can consider A as an A ⊗ Amodule. Combining this with the coproduct we get a new structure of A-module on A. It is called the adjoint representation and denoted by ad. 1.2 . In this note we study the adjoint representation of u in the simplest case of the root datum sl2 . The semisimple part of this representation is of big importance in the study of local ˆ 2 at level l − 2 in arbitrary genus. systems of conformal blocks in WZW model for sl The problem of distinguishing the semisimple part is closely related to the problem of integral representation of conformal blocks (see [BFS]). We find all the indecomposable direct summands of ad with multiplicities. To formulate the answer let us recall a few notations from the representation theory of u. The representation ad is naturally Z-graded, so we consider the category C of Z-graded umodules. The simple modules in this category are parametrized by their highest weights
254
V. Ostrik
which can assume arbitrary integer values. The simple module with a highest weight λ ∈ Z is denoted by L(λ), and its indecomposable projective cover is denoted by P (λ). 1.3 . Main Theorem. Let l ≥ 3 be an odd integer. The adjoint representation ad is isomorphic to the direct sum of the modules P (0), P (2), . . . , P (l − 3); L(−l + 1), L(−l + 3), . . . , L(2l − 4), L(2l − 2) with the following multiplicities: (i) The multiplicity of P (l − 1) ' L(l − 1) is l; let i ∈ [0, . . . , l−3 2 ]. (ii) The multiplicity of P (2i) is l+1 2 + i. (iii) The multiplicity of L(2i) is l − 1 − 2i. (iv) The multiplicities of L(2l − 2 − 2i) and L(−2 − 2i) are
l−1 2
− i.
1.4 . G. Lusztig has defined (see e.g. [Lu]) a quantum enveloping algebra with divided powers U containing u as a Hopf subalgebra. Its finite dimensional irreducible representations are parametrized by their highest weights which can assume arbitrary nonnegative integer values. The simple module with a highest weight λ ∈ N is denoted ˆ by L(λ), and its indecomposable projective cover is denoted by Pˆ (λ). There is a natural restriction functor from the category of finite dimensional U -modules to C. It appears that ad lies in its essential image. Corollary 1.1. The structure of Z-graded u-module on ad can be lifted to the structure of U -module isomorphic to the direct sum of the modules Pˆ (0), Pˆ (2), . . . , Pˆ (l − ˆ ˆ ˆ ˆ 3); L(0), L(2), . . . , L(2l − 4), L(2l − 2) with the following multiplicities: ˆ − 1) is l; (i) The multiplicity of Pˆ (l − 1) ' L(l l−3 let i ∈ [0, . . . , 2 ]. (ii) The multiplicity of Pˆ (2i) is l+1 + i. 2
ˆ (iii) The multiplicity of L(2i) is l − 1 − 2i. ˆ (iv) The multiplicity of L(2l − 2 − 2i) is l−1 2 − i. 1.5 . The author is very grateful to Michael Finkelberg; it is hard to overestimate his help with writing down the present work. Also I am happy to express my sincere gratitude to Thomas Kerler who taught me everything I know about ad. 2. Notations 2.1 . In this section we recall the necessary facts about u and its representations, following mainly [Lu]. Let l be an odd integer, l > 1. Let q ∈ C be a primitive l root of unity. Let i −q −i (i)q = qq−q −1 . Let u be an associative algebra over C with generators E, F, K, K −1 and relations: KK −1 = K −1 K = 1;
Decomposition of the Adjoint Representation of the Small Quantum sl2
255
KEK −1 = q 2 E, KF K −1 = q −2 F ; EF − F E =
K − K −1 ; q − q −1
E l = F l = 0, K l = 1. The algebra u is finite-dimensional and dim u = l3 . Let ω be an automorphism of the algebra u given on generators by the formulas: ω(E) = F, ω(F ) = E, ω(K) = K −1 . The algebra u is Hopf algebra with respect to coproduct 1, antipode S and counit ε given by the formulas: 1(E) = E ⊗ 1 + K ⊗ E, 1(F ) = F ⊗ K −1 + 1 ⊗ F, 1(K) = K ⊗ K; S(E) = −K −1 E, S(F ) = −F K, S(K) = K −1 ; ε(E) = ε(F ) = 0, ε(K) = 1. 2.2 . Let C be a category of finite-dimensional Z-graded u-modules V =
L i∈Z
V i such
that the following conditions hold: (a) E is operator of degree 2, i.e. E acts from V i to V i+2 ; (b) F is operator of degree −2; (c) K acts on V i by multiplication by q i . The morphisms in category C are morphisms of u-modules compatible with Zgrading. 2.3 . We introduce the duality D on category C. If V ∈ C, then D(V ) is V ∗ as a vector space. The action of x ∈ u on D(V ) is given by the formula (xf )(v) = f (ωS(x)v), where f ∈ D(V ), v ∈ V, S is the antipode. 2.4 . Let us define the adjoint representation ad ∈ C (see e.g. [LM]). Let x be an element of u. The adjoint action of generators is given by the folloing formulas: ad(E)x = Ex − KxK −1 E = K[K −1 E, x], ad(F )x = F xK − xF K = [F, x]K, ad(K)x = KxK −1 . 2.5 . Now we introduce Z-grading on adjoint representation. We put deg(E) = 2, deg(F ) = −2, deg(K) = 0 and deg(ab) = deg(a) + deg(b) for any a, b ∈ u such that deg(a), deg(b) are defined. Note that all the weights of ad are even integers in the interval [2 − 2l, . . . , 2l − 2]. 2.6 . It is known (see e.g. [LM] or [BFS]) that D(ad) ' ad.
256
V. Ostrik
3. u-Modules 3.1 . It is easy to check that an element X = EF +
qK + q −1 K −1 q −1 K + qK −1 = F E + (q − q −1 )2 (q − q −1 )2
(1)
lies in the center of algebra u (see e.g. [Ke]). The element X is called a Casimir element. It satisfies the following equation of degree l (see loc.cit.): Y (X − bj ) = 0, (2) P (X) := j∈Z/lZ j+1
−j−1
+q l j 0 where bj = q(q−q −1 )2 (q = 1, so q is well defined). In particular bj = bj 0 if j +j = l−2. The root b−1 = (q−q2−1 )2 of P has multiplicity 1, and the rest roots bj have multiplicity 2.
3.2 . Let j ∈ Z/lZ. Let Cj be a full subcategory of C such that (X − bj ) acts nilpotently on objects of Cj . In what follows we will identify Cj and Cj 0 if j + j 0 = l − 2. 3.3 . Let us fix H 0 — a maximal subset of Z/lZ with the following property: if {j, j 0 } is two-element subset of H 0 , then j + j 0 6= l − 2. We have H 0 = {−1} ∪ H, where H = H 0 − {−1}. 3.4 . For any j ∈ H we define the integers J, J 0 by the following properties: (1) 0 ≤ J < J 0 < l; (2) J + J 0 = l − 2; (3) (J − j)(J 0 − j) ≡ 0 (mod l). 3.5 . The category C is a direct sum of subcategories Cj where j runs through H 0 . 3.6 . We denote by u± ⊂ u the subalgebra generated by K, E (resp. K, F ). For λ ∈ Z ± denote by C± λ the one-dimensional Z−graded u −module of weight λ such that K acts λ as q and E (resp. F ) acts as zero on it. We denote by M ∓ (λ) the Z−graded u−module ± ± ± u⊗u± C± λ . The modules M (λ) are called Verma modules. Let M (λ) 3 v (λ) := 1⊗1. Let V ∈ C, v ∈ V λ and E ·v = 0 (resp. F v = 0). Then v is called an upper singular (resp. a lower singular) vector in V, and there exists a unique morphism φ : M ± (λ) → V such that φ(v ± (λ)) = v. 3.7 . For each λ ∈ Z there is a unique up to isomorphism simple module L(λ) ∈ C with highest weight λ. The modules L(λ1 ) and L(λ2 ) are isomorphic iff λ1 ≡ λ2 (mod l). Indecomposable projective cover of L(λ) will be denoted by P (λ). We have D(L(λ)) ' L(λ) and D(P (λ)) ' P (λ). In particular P (λ) is injective; dim Hom(L(λ), P (λ)) = 1. 3.8 . The set of isomorphism classes of simple objects in category C−1 is {L(λ), λ ≡ −1(mod l)}. As u-modules without grading all the L(λ) are isomorphic to one and
Decomposition of the Adjoint Representation of the Small Quantum sl2
257
the same u-module St (Steinberg module). It has dimension l. The category C−1 is semisimple. In particular P (λ) = L(λ). 3.9 . Let j ∈ H. The set of isomorphism classes of simple modules in Cj is {L(λ), λ ≡ J(mod l) or λ ≡ J 0 (mod l)}. The modules L(λ), λ ≡ J(mod l) (resp. λ ≡ J 0 (mod l)) are isomorphic as u-modules. Their dimension is J + 1 (resp. J 0 + 1). The projective module P (λ) admits a filtration P (λ) ⊃ W (λ) ⊃ L(λ) ⊃ 0 such that P (λ)/W (λ) ' L(λ), W (λ)/L(λ) ' L(λ0 ) ⊕ L(λ00 ), where λ0 6= λ00 , λ0 ≡ λ00 ≡ −2 − λ (mod l), |λ − λ0 | < 2l > |λ − λ00 |. In particular dim P (λ) = 2l. 3.10 . It is easy to see from 2, 3 and 3 that all the simple subquotients of adjoint representation have the type L(λ), where λ is an even integer from the interval [1 − l, . . . , 2l − 2]. Hence a projective module P (λ) can be a subquotient of ad only if λ ∈ 2Z∩[0, . . . , l−1]. In particular each subcategory Cj contains only one isomorphism class of such projectives. 3.11 . Lemma 3.1. Suppose V ∈ C is indecomposable and the action of Casimir element X on V is not semisimple. Then there exists λ 6≡ −1 (mod l) such that V ' P (λ). Proof. Casimir acts nonsemisimply on regular representation (see (2)). It follows that action on projective modules P (λ), λ 6≡ −1 (mod l) is not semisimple. It is easy to see that the space of eigenvectors of Casimir in P (λ) is W (λ). Let W ⊂ V be a maximal submodule of V such that X acts on W semisimply. Choose 0 6= ϕ ∈ Hom(V, L), where L = L(λ) for some λ ∈ Z such that Ker ϕ contains W. We have a morphism ψ ∈ Hom(P, V ), where P = P (λ) such that the diagram is commutative: P ψ
? ϕR - L V
-0
If Ker ψ 6= 0, then X acts on Im ψ semisimply. Therefore W is not maximal. We have a contradiction. If Ker ψ = 0, then we have injection P ,→ V. From 3.7. it follows that P is direct summand of V. The proof is complete. 4. The Blocks of Adjoint Representation In what follows we always will identify Sj and Sj 0 , where j + j 0 = l − 2 and S is an object, map, etc. Also we will identify SJ and Sj if J ∈ Z, J ≡ j (mod l). 4.1 . The regular action of Casimir X (by multiplication) is an endomorphism of adjoint representation. This gives a decomposition of adjoint representation into blocks ad = L adj , where (X − bj ) acts nilpotently on adj . Let prj denote a projection onto adj . j∈H 0
4.2 . Let j ∈ H. Let Mj = Ker(X − bj ), Nj = adj ∩ Im(X − bj ). Each adj admits a filtration adj ⊃ Mj ⊃ Nj ⊃ 0. The rest of this section is a computation
258
V. Ostrik
of associated graded factors of this filtration. Evidently adj /Mj ' Nj . It remains to compute Nj , Mj /Nj . It is convenient to put N−1 = ad−1 . 4.3 . Recall (see 2) that ad0 ⊂ ad denotes the zero weight space. Let ad0j = adj ∩ ad0 0 (for all j ∈ H 0 ), Nj0 = Nj ∩ ad0 , Mj0 = Mj ∩ ad0 (for j ∈ H) and N−1 = ad0−1 . We will compute the action of ad(X) on ad0j . We start with a computation of action of ad(X) on Nj0 . 4.4 . Lemma 4.1. We have (a) dim adj = 2l2 if j ∈ H and dim ad−1 = l2 ; (b) if j ∈ H then dim Nj = dim adj /Mj = (J + 1)2 + (J 0 + 1)2 and dim Mj /Nj = 4(J + 1)(J 0 + 1). (c) dim ad2m = l(l − |m|) for all m ∈ Z such that |m| < l; (d) dim ad0j = 2l for j ∈ H and dim Nj0 = l for all j ∈ H 0 ; ≥ 2(l − |m|) if j ∈ H and |m| ≥ l − 1 − J. (e) dim ad2m j Proof. (a), (b), (c), (d) are trivial. Let us prove (e). Suppose m > 0. It is easy to see from consideration of u-action on Verma modules M + (J) and M + (J 0 ) that E m acts nontrivially at least on 2(l − m) weights. By standard arguments with a Vandermond determinant we obtain that the desired dimension is at least 2(l − m). The proof for m < 0 is similar.
4.5 . The subspace ad0 ⊂ u is a subalgebra of u. It is generated as an algebra by K and X (see [Ke]). Moreover ad0 is a free module over a subalgebra generated by K (see loc. cit.). In particular we have Mj0 = Nj0 . We have ad(X)K i =
q 2i−1 + q 1−2i i K − (q i − q −i )2 XK i+1 + (i)q (i + 1)q K i+2 . (q − q −1 )2
(3)
4.6 . For j ∈ H the elements (X − bj )prj K i , i = 1, . . . , l (resp. pr−1 K i , i = 1, . . . , l for j = −1) form a basis of Nj0 . In this basis ad(X) acts as a lower-triangular matrix: A(j) =
b0 (q − q −1 )2 bj (1)q (2)q 0 .. .
0 b2 (q 2 − q −2 )2 bj (2)q (3)q .. .
0 0 b4 (q 3 − q −3 )2 bj .. .
... ... ... ... .. .
0 0 0 0 .. .
0
0
0
...
b0
(4)
Decomposition of the Adjoint Representation of the Small Quantum sl2
259
4.6.1. Remark 1. The vectors prj K i , i = 1, . . . , l; (X − bj )prj K i , i = l + 1, . . . , 2l form a basis of ad0j (j ∈ H). 4.7 . Let k ∈ 2Z ∩ [0, . . . , l − 1]. The eigenvalues of this matrix are bk (with multiplicity 2 if k 6= l − 1 and multiplicity 1 if k = l − 1). Let k 6= l − 1. It is obvious that there exists 1 or 2 eigenvectors of A(j) corresponding to eigenvalue bk . We have 2 eigenvectors iff the determinant d(j, k) of matrix (see Lemma 6.1) k/2+1 − q −k/2−1 )2 bj bk+2 − bk 0 ... (q (k/2 + 1)q (k/2 + 2)q (q k/2+2 − q −k/2−2 )2 bj bk+4 − bk . . . D(j, k) = . . . . . . . . . . . . ...
0
...
...
(5) is equal to zero. It is easy to see that this determinant is a polynomial in b2j of degree l−1−k . Since b2j = b2j 0 ⇒ bj = bj 0 , the polynomial d(j, k) vanishes for at most l−1−k 2 2 values of j ∈ H 0 . 4.8 . In this section we prove the following proposition: Proposition 4.1. For any j ∈ H we have the following decomposition: l−3
J 2 M M Nj ' (L(2i) ⊕ L(2i)) ⊕ P (2i) ⊕ L(l − 1). i=0
i=J+1
l−5 Proof. The proof proceeds by induction: we start from j = l−3 2 , then proceed to j = 2 , etc. It follows from 4 that for any j ∈ H 0 the module Nj contains as subquotients L(0), L(2), . . . , L(l − 3) with multiplicities 2 and L(l − 1) with multiplicity 1 (since only these modules have nontrivial zero weight space).
Lemma 4.2. Let j + 1 =
l−1 2 .
Then Nj ' L(0) ⊕ L(0) ⊕ . . . ⊕ L(l − 3) ⊕ L(l − 1).
l+1 2 l +1 2 Proof. Let us compute the dimensions. We have dim Nj = ( l−1 2 ) + ( 2 ) = 2 (see Lemma 4(b)). On the other hand by the above we have dim Nj ≥ 2 dim L(0) + 2 2 dim L(2)+. . .+2 dim L(l−3)+dim L(l−1) = 2·1+2·3+. . .+2·(l−2)+l = l 2+1 . It follows that in this case Nj is a direct sum of L(0), L(2), . . . , L(l − 3) with multiplicities 2 and L(l − 1) with multiplicity 1 (since Ext1 (L(λ), L(µ)) = 0 ∀λ, µ ∈ {0, 2, . . . , l − 1}). 2
The lemma implies that all eigenvalues of A( l−3 2 ) are semisimple. It follows from 4 that eigenvalue bl−3 is not semisimple for all the rest of j. Therefore for j 6= l−3 2 the corresponding Nj contains the projective submodule P (l − 3) (see 3). l2 +9 Now let j + 1 = l−3 2 . Then dim Nj = 2 . On the other hand dim Nj ≥ 2 · 1 + . . . + 2 2 · (l − 4) + 2l + l = l 2+9 . So in this case Nj is a direct sum of L(0), L(2), . . . , L(l − 5) with multiplicities 2, L(l − 1) with multiplicity 1, and P (l − 3). As above it follows that all the rest of Nj contains projective submodules P (l − 5) and P (l − 3), etc. The proposition is proved.
260
V. Ostrik
4.8.1. Corollary 4.1. We have: l−1
ad−1 = N−1 =
2 M
P (2i).
i=0
Proof. It follows from the proof of Proposition 4.1 that all the eigenvalues of the matrix A(−1), except for bl−1 , are not semisimple. Hence the result follows from Lemma 3.1 and computation of dimensions. 4.9 . Corollary 4.1 gives a decomposition of ad−1 . So in what follows we will assume that j 6= −1, i.e. j ∈ H. Lemma 4.3. The module Mj /Nj is a direct sum of modules L(λ) with multiplicity 2, where λ is even and satisfies one of the following conditions: (i) λ lies in the interval [2(l − J + 1), . . . , 2l − 2]; (ii) λ lies in the interval [−2J − 2, . . . , −2]. Proof. Recall that Mj0 = Nj0 . Hence Mj /Nj contains only subquotients L(λ), where either λ > l or λ < 0. By Lemma 4.1 (e), Proposition 4.1, and Corollary 4.1, we have l+1 ≥ 2(l − |m|) if j ∈ H and dim ad2m dim ad2m j −1 ≥ l − |m| for any m ≥ 2 . It follows = 2(l − |m|) for any j ∈ H and m ≥ l+1 from Lemma 4.1 (c) that dim ad2m j 2 . l+1 Hence for any m ≥ 2 we have exactly two upper (resp. lower) singular vectors of weight 2m (resp. −2m). It follows that adj has two simple subquotients with highest weight 2m and two simple subquotients with lowest weight −2m for any m ≥ l+1 2 . Thus ] with multiplicities 4; adj has the following subquotients: L(2i), where i ∈ [0, . . . , l−3 2 l−3 L(l − 1) with multiplicity 2; L(−2 − 2i) and L(2l − 2 − 2i), where i ∈ [0, . . . , 2 ], with multiplicities 2. The computation of dimensions shows that these modules are all the subquotients of adj . It follows from Proposition 4.1 that [Nj : L(λ)] = 1 if λ = −2 − 2i and λ = 2l − 2 − 2i, where i ∈ [J + 1, . . . , l−3 2 ]. Since adj /Mj ' Nj , any simple subquotient of Mj /Nj is of the type L(λ), where λ ∈ [2(l − J + 1), . . . , 2l − 2] ∪ [−2J − 2, . . . , −2] is even. But for any λ, µ satisfying such conditions we have Ext1 (L(λ), L(µ)) = 0. The result follows. 4.9.1. = 2(l − |m|) for any Remark. It follows from the proof of Lemma 4.3 that dim ad2m j j ∈ H and m ∈ Z, |m| < l. 4.10 . Let k be an even integer and k ∈ [0, l−1]. Let adj (k) be a summand corresponding to the subcategory Ck in adj . Let Mj (k) = Mj ∩ adj (k) and Nj (k) = Nj ∩ adj (k). Let us summarize the results of the present section. 4.10.1. ad−1 (k) = P (k).
Decomposition of the Adjoint Representation of the Small Quantum sl2
261
4.10.2. If j ∈ H then (a) adj (l − 1) = L(l − 1) ⊕ L(l − 1); (b) if k ≥ 2J + 2 then adj (k) = P (k) ⊕ P (k); (c) if k ≤ 2J then adj (k) admits a fitration adj (k) ⊃ Mj (k) ⊃ Nj (k) ⊃ 0 with the following associated graded factors: Nj (k) ' L(k) ⊕ L(k); Mj (k)/Nj (k) ' L(2l − 2 − k) ⊕ L(2l − 2 − k) ⊕ L(−2 − k) ⊕ L(−2 − k); adj (k)/Mj (k) ' L(k) ⊕ L(k). 5. The Proof of the Main Theorem 5.1 . Let us find the multiplicities of projective submodules in adj . Recall (see Remark in 4.6.1) that the vectors prj K i , i = 1, . . . , l; (X − bj )prj K i , i = l + 1, . . . , 2l form a basis of ad0j . In this basis ad(X) acts as a block matrix A(j) 0 A0 (j) = , B A(j) where A(j) is a matrix (4) and B is a matrix (see (3)) 0 0 0 0 0 −(q − q −1 )2 0 −(q 2 − q −2 )2 0 ... ... ...
... ... . ... ...
5.2 . Let bk be a nonsemisimple eigenvalue of A(j). Then a summand corresponding to the subcategory Ck in adj is a sum of 2 copies of projective P (k). 5.3 . Let bk (k 6= −1) be a semisimple eigenvalue of A(j), i.e. k ≤ 2J. 5.3.1. Lemma 5.1. In this case adj contains the projective module from category Ck . Proof. It is enough to prove that the matrix A0 (j) has exactly 3 eigenvectors with eigenvalue bk or, equivalently, that the matrix A0 (j) − bk has corank 3. Let us denote by A˜ 0 (j, k) the matrix A0 (j, k) − bk with the ith and (i + l)th columns divided by positive numbers −(q i − q −i )2 for any i ∈ [1, . . . , l − 1]. In order to apply Lemma 6.2 let us put A0 = A˜ 0 (j, k). Then the corresponding matrix D in notations of Lemma 6.2 is the matrix D(j, k) with columns divided by some positive numbers. In order to check the semisimplicity of the matrix D, put q = exp(πi l+1 l ). Then the conditions of Lemma 6.3 Indeed, off-diagonal entries of D(j, k) are either (t)q (t + 1)q or bk+2t − bk = (t)q (t + k + 1)q . In both cases this entry is (t1 )q (t2 )q where t1 , t2 ∈ [1, . . . , l − 1] and one of t1 , t2 is even and another is odd. But if q = exp(πi l+1 l ) and t ∈ [1, . . . , l − 1] then (t)q > 0 ⇔ t is odd. Finally note that the entries of D are the entries of D(j, k) divided by some positive numbers. The lemma is proved.
262
V. Ostrik
5.3.2. The above lemma implies that adj (k) is a sum of projective module and some module Y (k, j). It follows from 4.9.1 that Y (k, j) admits a filtration of length 3 with the following associated graded factors: L(k); L(−2 − k) ⊕ L(2l − 2 − k); L(k). 5.3.3. For any 0 ≤ s < l an element prj (K −1 E)s is an upper singular vector in adj . Let us −1 E)s 6= 0. Consider the action of this element prove that if s ≤ l−1 2 then (X −bj )prj (K on P (J 0 ). Then (X −bj ) is a surjection onto L(J 0 ) ⊂ P (j 0 ) and the desired result follows s from the fact that dim L(J 0 ) ≥ l+1 2 . Similarly the vector prj F is a lower singular vector l−1 k s in adj , and prj F 6∈ Mj if s ≤ 2 . For s = 2 we obtain that adj contains a submodule L(k) which does not lie in Mj . Indeed, the submodules generated by prj (K −1 E)s and prj F s coincide since adj (k)/Mj (k) ' L(k) ⊕ L(k) and adj (k) ⊃ P (k) 6⊂ Mj (k). 5.3.4. It follows that L Hence L Y (k, j) = Z(k, j)⊕L(k), where Z(k, j) contains a submodule L(k). adj (k) is a direct sum of a few copies of P (k) and Y (k) = Y (k, j) ad(k) = j∈H 0
j∈H 0
where all the subquotients L(k) of Y (k) are the submodules of Y (k). Now from the autoduality (see 2.6) we see that all the subquotients L(k) are direct summands of Y (k). Thus Y (k) is a direct sum of its simple subquotients. Hence in the case 4.10.2 (c) (i.e. if k ≤ 2J) we have that adj (k) = P (k) ⊕ L(k) ⊕ L(k) ⊕ L(2l − 2 − k) ⊕ L(−2 − k).
This completes the proof of the Main Theorem.
6. Three Matrix Lemmas The results of this section were used in the previous sections. 6.1 . Let A be a r × r lower-triangular matrix: A=
α1 β1 γ1 0 .. .
0 α2 β2 γ2 .. .
0 0 α3 β3 .. .
... ... ... ... .. .
0 0 0 0 .. .
0
0
...
βr−1
αr
.
(6)
Lemma 6.1. Let αi = αj = α for some i < j and αk 6= α for k 6= i, j. The matrix A has 2 different eigenvectors with eigenvalue α iff the determinant of (j − i) × (j − i) matrix
Decomposition of the Adjoint Representation of the Small Quantum sl2
D=
263
βi γi 0 0 .. .
αi+1 − α βi+1 γi+1 0 .. .
0 αi+2 − α βi+2 γi+2 .. .
... ... ... ... .. .
0 0 0 0 .. .
0
0
...
γj−2
βj−1
(7)
vanishes. Proof. Clear.
6.2 . Suppose αi = αj iff i + j = r + 1. Let A0 be the following matrix A0 = where
0 1 0 B = 0 ...
A B0
0 A
0 0 1 ...
0 0 0 ...
,
... ... . ... ...
Lemma 6.2. Suppose the matrix A above has 2 eigenvectors with eigenvalue αi . Suppose the matrix D is semisimple of corank 1. Then the matrix A0 − αi has corank 3. Proof. It suffices to consider the case i = 1, j = r, α1 = αr = 0. Deleting two rows and columns consisting of zeros we obtain a matrix 0
D =
D E
0 D
,
where E is a unit matrix. We have to prove that corank of D0 is equal to 1. Let Im(D) (resp. Im(D0 )) denote the linear space generated by columns of D (resp. D0 ). Let pr : Im(D0 ) → Cr−1 be a map forgetting the last r − 1 coordinates. Then pr(Im(D0 )) = Im(D) has dimension r − 2. Let us prove that Ker(pr) has dimension r − 1. Indeed, Ker(pr) ⊃ Im(D) and Ker(pr) contains the kernel of operator D. Since D is semisimple, Im(D) 6⊃ Ker(D). The result follows.
6.3 . Lemma 6.3. Suppose D is a real matrix, and all the off-diagonal entries are negative. Then D is semisimple. Proof. It is easy to see that conjugating matrix D by some diagonal matrix we can obtain a symmetric matrix.
264
V. Ostrik
References [BFS] Bezrukavnikov, R., Finkelberg, M., Schechtman, V.: Localization of u-modules. V. The modular structure on the category F S. Preprint, to appear [Ke] Kerler, T.: Mapping class group actions on quantum doubles. Comm. Math. Phys. 168, 353–388 (1995) [Lu] Lusztig, G.: Finite-dimensional Hopf algebras arising from quantized universal enveloping algebras. J. of AMS 3, 257–296 (1990) [LM] Lyubashenko, V., Majid, S.: Braided groups and quantum Fourier transform. Preprint DAMTP/91-26
Communicated by G. Felder
Commun. Math. Phys. 186, 265 – 293 (1997)
Communications in
Mathematical Physics c Springer-Verlag 1997
The Brjuno Functions and Their Regularity Properties S. Marmi1 , P. Moussa2 , J.-C. Yoccoz3 1 2 3
Dipartimento di Matematica “U. Dini”, Universit`a di Firenze, 50134 Firenze, Italy CEA, Service de Physique Th´eorique, CE-Saclay, F-91191 Gif-sur-Yvette Cedex, France Universit´e de Paris-Sud, Math´ematiques, Bˆat. 425, 91405 Orsay, France
Received: 21 March 1995 / Accepted: 8 August 1996
Abstract: We show that various possible versions of the Brjuno function, based on different kinds of continued fraction developments, are all equivalent and we study their regularity (Lp , BMO and H¨older) properties, through a systematic analysis of the functional equation which they fulfill.
0. Introduction When an irrational rotation is analytically perturbed, it is a natural question to ask whether or not there exists a neighborhood of the fixed point where the dynamics looks like the unperturbed case. More precisely, does there exist a local holomorphic coordinate for which the perturbed transformation is expressed as an ordinary rotation? When the rotation number satisfies the Brjuno condition [Br], such a coordinate exists in a domain called a Siegel disk. The Brjuno function tells more: it gives an estimate of minus the logarithm of the size of the Siegel disks as a function of the rotation number [Yo]. A similar result also holds in the simpler case of certain complex area–preserving maps [Ma, Da]. In [Yo], the relation between the logarithm of the size of the Siegel disks and the Brjuno function is established through a geometrical renormalisation argument, which uses both a change of variable, combined with the replacement of the original map with a suitably chosen first return map. If the original map has rotation number x (where 0 < x < 1), the replaced map has rotation number 1/x mod 1. The argument gives then the functional equation fulfilled by the Brjuno function, but only establishes inequalities between the Brjuno function and the logarithm of the radius of the Siegel disk (see [Pe] for a review). In this work, we will analyse the consequences of an additional assumption, which remains to be proven, that is we assume that the logarithm of the size of the Siegel disk fulfills the same equation as the logarithm of the Brjuno function, up to a remainder term
266
S. Marma, P. Moussa, J.-C. Yoccoz
having sufficient regularity conditions. We shall discover that a natural assumption is that this remainder be continuous and satisfying an H¨older condition of exponent 1/2. We first observe that regularity assumptions are quite common and usually implicitly made on remainder terms by physicists in their approach of renormalisation group type applied to dynamical systems. The assumption is also supported here by previous numerical calculations [Ma], which display the a priori surprising following result: the difference between the Brjuno function and the suitable measure of regularity adapted to the particular dynamical system, is not only bounded, as anticipated from Yoccoz’s result, but moreover continuous. This observation holds in several cases: 1) holomorphic maps with indifferent fixed points, where the Brjuno function is compared to some measure of the logarithm of the size of the Siegel disk (or Herman rings), as function of the rotation number, 2) complex area preserving maps (semistandard map, modulated singular maps) where the Brjuno function is compared to the logarithm of the critical function. It is remarkable also that the plot of the ratio of each of these functions (size of disks and rings, critical functions) to the exponential of minus the Brjuno function, displays the same general features, boundedness, continuity, and universal shape [Ma, see for instance Fig. 6, 13, and 16]. One of the purposes of this work is to explain that these numerical features become natural outcomes of the above mentioned regularity hypothesis of the renormalisation equation. In Sect. 1, we first analyze the relation between the Brjuno function and the various kind of continued fractions. This required a review of the properties of these various continued fraction expansions. A Brjuno function is associated in Sect. 2 to each kind of continued fraction expansion, so that we get a one parameter family of functions Bα (x), and the Brjuno condition on x is equivalent to Bα (x) bounded, for any fixed α. We shall see later that only two cases appear to be relevant to our purpose, that is the usual Gauss case α = 1 and the nearest integer case α = 1/2 which is used in [Yo]. We establish the functional equation fulfilled by each Brjuno function, and show that its solution requires the inversion of an operator T (α) (see Eqs. (2.5) and (2.7) below—usually, the superscript α is omitted in T (α) , when the value of α is clearly fixed). We estimate for any α the value of the spectral radius of the operator T (and of some of its generalisations) for all Lp norms. This shows that the Brjuno functions belong to all Lp spaces for finite p. In Sect. 3, we fix α = 1/2, and we estimate the norm of T (and of some of its generalisations) in the BMO–space (BMO meaning Bounded Mean Oscillation–see a short review on BMO norms in the Appendix); in this case, the Brjuno function B1/2 is obtained as the action of (1 − T )−1 on a logarithmic function which belongs to the BMO–space. Since the spectral radius of T in this space is shown to be smaller than one, the Brjuno function is also in this space. Noticing that the adjoint of T is nothing else than the generalized Ruelle-Frobenius-Perron operator associated to the dynamical system which generates the continued fraction, the identification of the space adapted to T seems to us promising for the study of dynamical properties. Another nice picture of the case α = 1/2 is that T also acts on the space of continuous functions. We study in Sect. 4 this action with respect to H¨older continuity properties. We show that regular perturbations (at least C 1/2 ) of the logarithmic term do modify the solution only by a C 1/2 contribution, so that the most singular part remains unchanged. This is where the above regularity assumption for the remainder of the renormalisation equation can be used: if these remainder perturbative terms to the renormalisation equation have regularity C β (β > 1/2), then the most singular part of minus the logarithm of the size of the stability domains as function of the rotation number is universally (that is modulo a C 1/2 function), described by the Brjuno function. One could infer from these results that
Brjuno Functions and Their Regularity Properties
267
the Brjuno function for the α = 1/2 case is the best candidate for applications; actually, we also found that the difference B1 −B1/2 is H¨older-1/2 continuous. Therefore both B1 and B1/2 are suitable. This is somewhat surprising, since the operator T (1) is not defined is the space of continuous functions (indeed it is not even defined in the BMO-space). However a specific character of B1/2 is that it is an even function. As an example, we find it natural to set up the following conjecture: The difference between the Brjuno function B1/2 (equivalently B1 ), and minus the logarithm of the conformal radius of the Siegel disk of the quadratic polynomial z exp(2iπx)+z 2 is a 1/2H¨older continuous function of the real variable x. Actually, Douady-Hubbard’s theory of quadratic-like maps [DH] suggest that this could even be true also for perturbations of the quadratic maps. Also we would like to replace the radius in the above conjecture by the critical function of holomorphic area preserving maps, of the kind considered in [Ma, Da]. A further motivation for the present study, and particularly the BMO-space results, is the problem of building a complex analytic extension of the Brjuno function. This will be the subject of a subsequent paper. One of us (S. M.) wishes to thank the Italian CNR for financial support, and A. Berretti and S. Isola for useful discussions. Part of this work was made during visits of the second author (P. M.), who thanks the Department of Mathematics ‘U. Dini’ of the University of Florence, the Italian Institutes INFN and INFM for hospitality and/or financial support. Support from the EC contract ERBCHRXCT94-0460 for the project “Stability and universality in classical mechanics” is also acknowledged.
1. On a Family of Continued Fraction transformations Let α ∈ [1/2, 1] and let x ∈ R. We define [x]α = min{p ∈ Z | x < α + p} ,
(1.1)
that is [x]α = p iff
α−1+p≤x<α+p.
Note that [x]α = [x − α + 1] , where [ ] = [ ]1 denotes the usual integer part of a real number. We will consider the iteration of (1.2) Aα : (0, α) 7→ [0, α] defined by
1 1 , Aα (x) = − x x α
with branches 1 −k x 1 Aα (x) = k − x Aα (x) =
Let
for for
1 1 <x≤ , k+α k 1 1 <x≤ . k k+α−1
(1.3)
268
S. Marma, P. Moussa, J.-C. Yoccoz
√ G=
5+1 , 2
0=
√ 2+1
√
and
√ 5−1 , γ = 0−1 = 2 − 1 . 2 Theorem 1.1. The dynamical system defined by the iteration of (1.3) preserves an absolutely continuous (w.r.t. Lebesgue) probability measure mα with density cα ρα (x): mα (dx) = cα ρα (x)dx. The density is given by : −1
g=G
=
(i) If g ≤ α ≤ 1, then 0 ≤ 1 − α ≤ Aα (α) = α−1 − 1 ≤ α , and 1 , log(1 + α) 1 1 χ(α−1 −1,α) (x) + χ(1−α,α−1 −1] (x) ρα (x) = 1 +x 2+x 1 1 + + χ(0,1−α] (x) . 2+x 2−x cα =
(1.4)
ii) If 1 − γ ≤ α < g, then 0 ≤ Aα (α) = 2 − α−1 < 1 − α ≤ Aα (α − 1) = (1 − α)−1 − 2 < α , and 1 , log G 1 1 −1 −1 χ χ (x) + (x) ρα (x) = G + x ((1−α) −2,α) 2 + x (1−α,(1−α) −2] 1 1 + + χ(2−α−1 ,1−α] (x) x+2 G+1−x 1 1 + + χ(0,2−α−1 ] (x) . x+2 2−x cα =
iii) If
1 2
(1.5)
≤ α < 1 − γ, then 0 ≤ Aα (α) = 2 − α−1 ≤ Aα (α − 1) = (1 − α)−1 − 2 < 1 − α ≤ α ,
and 1 , log G 1 1 1 ρα (x) = χ(1−α,α) (x) + + χ((1−α)−1 −2,1−α] (x) G+x G+x G+1−x 1 1 + + χ(2−α−1 ,(1−α)−1 −2] (x) x+2 G+1−x 1 1 + + χ(0,2−α−1 ] (x) , (1.6) x+2 2−x cα =
Brjuno Functions and Their Regularity Properties
269
where χ(a,b) (x) denotes the characteristic function of the interval (a, b). The theorem can be proven by direct calculations, which are very tedious for general values of α. It is much easier it from similar results of Nakada [Na], who to derive considers the map A˜ α = x1 − x1 α , instead of Aα = |A˜ α |. Note that if α = 1, one 1 recovers Gauss’ result: c1 ρ1 (x) = (1 + x) log 2 , and if α = 1/2, one finds c1/2 ρ1/2 (x) = 1 1 1 log G G + x + G + 1 − x . Remark 1.2. From the above theorem, it is obvious that the density of the measure mα is bounded above and below, so that mα and the Lebesgue measure are equivalent. More precisely, bounding each term of the type (b ± x)−1 by its values at 0 or 1, one easily shows that there exists a constant c0 such that for any α ∈ [1/2, 1], c−1 0 ≤ c α ρα ≤ c 0 .
(1.7)
c0 = 2/ log G = 4.156... ,
(1.8)
One finds which can be reduced to 1/ log G when α = 1/2. Remark 1.3. It is well known that the Kolmogorov-Sinai entropy of these maps is given by Z α
h(α) = −2
cα ρα (x) log x dx . 0
If α = 1 one obtains h(1) =
π2 , 6 log 2
for α = 1/2 one obtains, h(1/2) =
π2 , 6 log G
a result already given by Rieger [Ri]. For general α, we find that cα h(α) does not depend on α, thus π2 6 log(1 + α) if g ≤ α ≤ 1 , h(α) = 2 π if 1/2 ≤ α ≤ g , 6 log G as already shown by Nakada [Na]. Using the terminology of the thermodynamical formalism for dynamical systems, these results show the existence of a phase transition at α = g. To each x ∈ R \ Q we associate a continued fraction expansion by iterating Aα as follows. Let x0 = |x − [x]α |, a0 = [x]α ,
(1.9)
then one obviously has n x 0 = a0 + ε 0 x 0
where
ε0 =
+1 iff x ≥ [x]α −1 otherwise .
(1.10)
270
S. Marma, P. Moussa, J.-C. Yoccoz
We now define inductively for all n ≥ 0, xn+1 = Aα (xn ), 1 an+1 = ≥ 1, xn α
(1.11)
thus x−1 n
= an+1 + εn+1 xn+1
where εn+1 =
+1 iff x−1 n ≥ an+1 −1 otherwise .
(1.12)
Therefore we have ε0 = ... a 1 + ε1 x1 ε0 = a0 + ε1 a1 + .. εn−1 a2 + . + an + εn xn
x = a 0 + ε0 x 0 = a0 +
(1.13)
and we will write x = [(a0 , ε0 ), (a1 , ε1 ), . . . , (an , εn ), . . .] .
(1.14)
Note that for α = 1 we recover the standard continued fraction expansion defined through the iteration of Gauss’ map x 7→ x−1 mod 1, and all εn = +1. When α = 1/2 one has the so-called nearest integer continued fraction, and an ≥ 2 for all n ≥ 1. The nth -convergent is defined by pn = [(a0 , ε0 ), (a1 , ε1 ), . . . , (an , εn )] = a0 + qn a1 +
ε0 ε1
,
(1.15)
. εn−1 a2 + . . + an
and it is immediate to check that the numerators pn and denominators qn are recursively determined by (1.16) p−1 = q−2 = 1 , p−2 = q−1 = 0 , and for all n ≥ 0, pn = an pn−1 + εn−1 pn−2 , (1.17)
qn = an qn−1 + εn−1 qn−2 . Moreover pn + pn−1 εn xn qn + qn−1 εn xn qn x − pn xn = −εn qn−1 x − pn−1 qn pn−1 − pn qn−1 = (−1)n ε0 . . . εn−1 . x=
(1.18) (1.19) (1.20)
Let n xi = (−1)n ε0 . . . εn (qn x − pn ) βn = Πi=0
for n ≥ 0,
and β−1 = 1 .
(1.21)
Brjuno Functions and Their Regularity Properties
271
Then xn =
βn βn−1
βn−2 = San βn−1 + εn βn .
(1.22)
From the definitions given one easily proves by induction the following proposition Proposition 1.4. Given α ∈ [1/2, 1], for all x ∈ R \ Q and for all n ≥ 1 one has i) qn+1 > qn > 0; ii) pn > 0 when x > 0 and pn < 0 when x < 0; 1 1 1 iii) |qn x − pn | = qn+1 + εn+1 qn xn+1 , so that 1 + α < βn qn+1 < α ; iv) if α > g , βn ≤ αg n ; v) if α ≤ g , βn ≤ αγ n . Proof. One gets parts (i) and (ii) by recursion using (1.17), in fact it is obvious only when α = 1. When α 6= 1, one could alternatively use Lemma 1.8 below. Part (iii) is easily obtained from (1.18). The proof of (iv) is also easy: either xk ≤ g for all k = 0, . . . , n, or xk > g for some −1 − 1 = g, thus xk xk+1 = 1 − xk < 1 − g = g 2 . In k. Then xk+1 = x−1 k − 1 and xk+1 < g the sequence βn = x0 · · · xn one then isolates the pairs xk , xk+1 such that xk > g (since for each pair xk xk+1 < g 2 ). The other terms in βn are all smaller or equal to g, except for xn < α. The proof of (v) is more complicated: the result is obvious if among x0 , x1 , . . . , xn , there are less than two of them taking values greater than γ. Otherwise, let xk and xk+p+1 be two successive values greater than γ. Note that xk > γ implies xk+1 = |2 − x−1 k | < γ, therefore we must have p ≥ 1. Now statement (v) is an immediate consequence of the following assertion which we will then prove: k+p xi < γ p+1 . if xk > γ, p ≥ 1, xk+1 , . . . , xk+p < γ and xk+p+1 > γ, then Πi=k We divide the proof into some different cases. (1) (2)
2 If γ < xk ≤ 1/2, then xk+1 = x−1 k − 2 and xk xk+1 = 1 − 2xk < 1 − 2γ = γ , and the assertion holds. By the way this closes the proof in the case α = 1/2. 2 2 Let xk > 1/2, thus xk+1 = 2 − x−1 k < g . Now observe that the image of [1/3, g ] 2 is [0, g ]. If xk+1 ≤ 1/3 we let m = 1 , and if xk+1 > 1/3 we let m ≥ 2 such that
xk+1 , . . . , xk+m−1 ∈ (1/3, g 2 ] ,
xk+m ∈ [0, 1/3] .
Note that p ≥ m ≥ 1. (2.1) If m ≥ 4, then xk xk+1 · · · xk+m ≤ 13 g 2m−1 , since xk < g, xk+1 , . . . , xk+m−1 ≤ g 2 and xk+m ≤ 1/3. A numerical exercise shows that for m ≥ 4, 13 g 2m−1 < γ m+1 , and the assertion follows. We must now consider the three cases left: m = 1, m = 2 and m = 3. < 1 − γ, then xk xk+1 = 2xk − 1 < γ 2 ; otherwise xk ≥ 1 − γ, and (2.2) m = 1. If xk √ xk+1 ≥ 1 − 2/2 so that 2/7 < xk+1 ≤ 1/3 and xk+2 = x−1 k+1 − 3, xk+1 xk+2 = 1 − 3xk+1 and finally xk xk+1 xk+2 = 3 − 5xk < 5γ − 2 = γ 3 , which shows the assertion.
272
S. Marma, P. Moussa, J.-C. Yoccoz
3 (2.3) m = 2, then xk+2 = 3 − x−1 k+1 and xk xk+1 xk+2 = 5xk − 3. If this is smaller √ than γ , then the assertion follows. √ If not, which is equivalent to assume xk > 2 − 4/5, ∈ [2/7, 1/3]. then xk+1 > (48 − 25 2)/34 and xk+2 > 0.3111 . . ., so that xk+2 √ Thus xk+3 = x−1 − 3 and x x x x = 8 − 13x < 8 − 13( 2 − 4/5). A k k+1 k+2 k+3 k k+2 numerical exercise shows that this last number is smaller than γ 4 , which completes the assertion in this case. 4 (2.4) m = 3, then assume √ xk xk+1 xk+2 xk+3 = 13xk − 8 > γ . This would be equivalent to xk > (25−12 2)/13. However, from the definition of m, one gets xk+3 ≤ 1/3 which implies xk ≤ 21/34, and the two inequalities on xk are contradictory. Thus xk xk+1 xk+2 xk+3 ≤ γ 4 , and the proof of the assertion is completed.
Remark 1.5. From (iii) and (iv) one gets if α > g , qn ≥ α(11+ α) Gn−1 , and similarly, from (iii) and (v) if α ≤ g , qn ≥ α(11+ α) 0n−1 . Remark 1.6. From (iii) one gets 1 pn 1 1 1 ≤ < x − < < 2qn qn+1 qn (qn + qn+1 ) qn (αqn + qn+1 ) qn qn qn+1
(1.23)
if εn+1 = +1, whereas
1 pn 1 1 < < x − < qn qn+1 qn qn (qn+1 − (1 − α)qn ) αqn2
(1.24)
if εn+1 = −1. Note also that assertions (iv) and (v) remain valid for x ∈ Q, with the convention that βn = 0 as soon as one of the xk , k ≤ n, vanish (in which case the xk with larger order are undefined). Remark 1.7. Using the estimates of Remark 1.5, qk ≥ max(1, Gk−1 /2), and the ele1/2 mentary inequality log qk ≤ (2/e)qk , there exists two positive constants c1 and c2 such that ∞ X log qk k=0 ∞ X k=0
qk log 2 qk
≤ c1 =
2 e
3+
√ 2 √ G− G
= 5.214... ,
≤ c2 = 5 log 2 = 3.465... ,
(1.25) (1.26)
for all α ∈ [1/2, 1] and for all x ∈ (0, α). Let Pn /Qn denote the nth convergent to x according to the standard continued fraction expansions (i.e. obtained by the iteration of the Gauss map A1 ). Following the method of [Bo], we shall now relate the nth convergents of the α–continued fractions to Pn /Qn . In fact the following result is obtained through a repeated use of the identity: A−
1 1 =A−1+ . B+x 1 + B −11 + x
Brjuno Functions and Their Regularity Properties
273
Lemma 1.8. For fixed x ∈ R\Q, let k α : N → N be the arithmetic function inductively defined by k α (−1) = −1 and α
k (n + 1) =
k α (n) + 1 k α (n) + 2
if εn+1 = +1, if εn+1 = −1,
where εn+1 is defined as in (1.10) and (1.12). Then k α is strictly increasing and for all n∈N pn Pkα (n) = . qn Qkα (n) Moreover, when k α (n + 1) = k α (n) + 2, we have for the denominators of the convergent of Gauss’continued fraction Qkα (n+1) = Qkα (n)+2 = Qkα (n)+1 + Qkα (n) . In the remainder of this Section, we collect a few technical facts concerning the nearest integer continued fraction which will be systematically used in the various proofs of Sects. 3 and 4. The reader who is mainly interested in the results can skip the following lemmas. Let A = A1/2 . We say that x and x0 belong to the same branch of An , when xk and x0k belong to the same branch of A for 0 ≤ k ≤ n − 1. Then the coefficients ak , εk , of the expansion of x and the coefficients a0k , ε0k of the expansion of x0 do coincide for 0 ≤ k ≤ n. We now define an integer n(x, x0 ) which represents the number of iteration steps needed to separate the orbits of x x0 . Definition 1.9. Let x, x0 be two distinct irrationals in (0, 1/2). The splitting order n(x, x0 ) is the greatest integer m such that x, x0 belong to the same branch or to two adjacent branches of Am . We define also the integer δ(x, x0 ) such that m = n(x, x0 )−δ(x, x0 ) is the greatest integer such that x, x0 belong to the same branch of Am . We shall see in the sequel that δ(x, x0 ) is equal to 0, 1, or 2. Indeed there are four possible situations, provided we also include the cases where x and x0 are permuted (for brevity we will write n and δ for n(x, x0 ) and δ(x, x0 )): (A) x and x0 belong to the same branch of An . Then δ = 0. (B) x and x0 belong to the same branch of An−1 , and there exists k ≥ 3 such that 1 2 2 < xn−1 < < x0n−1 < . 2k + 1 k 2k − 1
(1.27)
Since x and x0 belong to the same branch of An−1 and to adjacent branches of An , in this case δ = 1. (C) x and x0 belong to the same branch of An−2 and there exists k ≥ 3 such that 2 5 5 < xn−2 < < x0n−2 < . 5k − 2 2k − 1 5k − 3
(1.28)
In this case, both xn and x0n belong to [2/5, 1/2]. x and x0 belong to adjacent branches of An−1 as well as of An , and δ = 2.
274
S. Marma, P. Moussa, J.-C. Yoccoz
(D) x and x0 belong to the same branch of An−1 and there exists k ≥ 3 such that 2 1 1 < xn−1 < < x0n−1 < . k 2k − 1 k−1
(1.29)
In this case δ = 1, as in case (B) above, but one must add the condition that one of the numbers xn , x0n (at least) does not belong to [2/5, 1/2], otherwise, one is in case (C). From the above definitions, for all l ≤ n − δ one has al = a0l , εl = ε0l , pl = p0l , ql = ql0 , |βl (x) − βl (x0 )| = ql |x − x0 | , |x − x0 | = |xl − x0l |βl−1 (x)βl−1 (x0 ) .
(1.30)
In the case (B) one has an = a0n = k , pn = p0n , qn = qn0 , εn = +1 , ε0n = −1 . Let
x00 =
pn ∈ (x, x0 ) . qn
(1.31) (1.32)
Then one has x00 n−1 = k −1 , βn−1 (x00 ) = qn−1 , x00 n = 0 and |x − x00 | = qn−1 βn−1 (x)xn = qn−1 βn (x), |x0 − x00 | = qn−1 βn−1 (x0 )x0n = qn−1 βn (x0 ) .
(1.33)
In the case (C) one has an−1 = k , εn−1 = −1 , an = 2 , εn = +1, a0n−1 = k − 1 , ε0n−1 = +1 , a0n = 2 , ε0n = +1, 0 0 + qn−2 , qn = qn0 = qn−1 + qn−1 , qn−1 = qn−1
(1.34)
pn−1 = p0n−1 + pn−2 , pn = p0n = pn−1 + p0n−1 . Let x00 =
pn pn−1 + p0n−1 = ∈ (x, x0 ) . 0 qn qn−1 + qn−1
(1.35)
Then one has x00 n−2 = 2/(2k − 1), βn−2 (x00 ) = 2qn−1 , x00 n−1 = 1/2, βn−1 (x00 ) = qn−1 , x00 n = 0 and |x − x00 | = qn−1 βn−1 (x)xn = qn−1 βn (x), |x0 − x00 | = qn−1 βn−1 (x0 )x0n = qn−1 βn (x0 ) . In the case (D) one has an = k , εn = −1 a0n = k − 1 , ε0n = +1 Let
qn = qn0 + qn−1 pn = p0n + pn−1 .
(1.36)
(1.37)
Brjuno Functions and Their Regularity Properties
x00 =
pn + p0n ∈ (x, x0 ) . qn + qn0
Then one has x00 n−1 = 2/(2k − 1), x00 n = 1/2, βn−1 (x00 ) = 2(qn + qn0 )−1 and |x − x00 | = 2(qn + qn0 )−1 βn−1 (x) 21 − xn , |x0 − x00 | = 2(qn + qn0 )−1 βn−1 (x0 ) 21 − x0n . We recall that in the case (D) one also has 1 1 1 0 − xn , − xn ≥ . max 2 2 10
275
(1.38)
(1.39)
(1.40)
We give now two lemmas which relate the separation between two numbers and their splitting orders. Lemma 1.10. There exists a positive constant c3 independent on x and x0 , such that for all l < n = n(x, x0 ), we have 0 0 c−1 3 βl (x ) < βl (x) < c3 βl (x ) .
(1.41)
Indeed one can take c3 = 9/2. Proof. For l < n − δ, this is just a consequence of (1.30) and Proposition 1.4 (iii), and the constant obtained in this case is 3. When δ = 1, one gets from (1.27) or (1.29) 2/3 ≤ xn−1 /x0n−1 ≤ 3/2, which leads to a constant 9/2. When δ = 2, one gets from (1.28), 12/13 ≤ xn−2 /x0n−2 ≤ 13/12, and 4/5 ≤ xn−1 /x0n−1 ≤ 5/4, so that we also get for the constant 65/16 < 9/2. Lemma 1.11. There exists a positive constant c4 > 0 such that for all x, x0 ∈ (0, 1/2), and n ≥ n(x, x0 ), one has max(βn (x), βn (x0 )) ≤ c4 |x − x0 |1/2 . √ Indeed one can take c4 = 9 15/2 = 17.42....
(1.42)
Proof. In the case (D) one has |x − x0 | = |x − x00 | + |x00 − x0 |, so that |x − x0 | ≥
1 1 −2 1 2 (qn + qn0 )−1 inf(βn−1 (x), βn−1 (x0 )) ≥ q ≥ β (x) , 5 15 n 60 n−1
(1.43)
since qn = qn0 + qn−1 > qn0 , and (2/3)qn−1 ≤ βn−1 (x) ≤ 2qn−1 . The previous lemma 2 then shows that |x − x0 | ≥ (c23 /60)βn−1 (x0 ). Since βn ≤ (1/2)βn−1 , the constant c4 is √ √ at most equal to c3 15 = 9 15/2. In the cases (B) and (C) one has |x − x0 | = qn−1 (βn (x) + βn (x0 )) ≥ qn−1 max(βn (x), βn (x0 )) .
(1.44)
0 0 However, qn−1 √ ≥ (1/2)βn−1 (x) ≥ (1/2c3 )βn−1 (x ) = (1/9)βn−1 (x ), and therefore we get c4 ≥ 3/ 2. 0 −1 0 Finally in the case (A) one has |x−1 n − x n | ≥ 1 since x, x do not belong to n+1 0 two adjacent branches of A , from which follows that |xn − xn | ≥ |xn ||x0n |. Suppose xn > x0n (the other case resulting by symmetry), then if x0n ≥ xn /2, we get |xn − x0n | ≥
276
S. Marma, P. Moussa, J.-C. Yoccoz
|xn |2 /2 >, and if x0n < xn /2, we get |xn − x0n | = xn (1 − x0n /xn ) > xn /2 > x2n /2. Therefore 1 (1.45) |xn − x0n | ≥ [max(x2n , x02 n )] 2 and |x − x0 | = βn−1 (x)βn−1 (x0 )|xn − x0n | ≥
βn−1 (x)βn−1 (x0 ) [max(x2n , x02 n )], 2
and using the previous lemma, |x − x0 | ≥
2 2 max(βn−1 (x), βn−1 (x0 )) [max(x2n , x02 n )], 2c3
so that one gets c4 ≥ 3 in this case.
(1.46)
Lemma 1.12. Let J be the interval of definition of one single branch of Am , and |J| its length. One of its end-points is equal to pm /qm . We have 1 1 ≤ |J| ≤ 2 , 2 3qm qm
and for x ∈ J ,
m dA (x) 9 2 1 2 ≤ q , q ≤ 4 m dx 4 m
(1.47)
so that dAm (x) 9 1 ≤ |J| ≤ . 12 dx 4
(1.48)
Proof. The end-points of J are obtained in setting xm = 0 or xm = ±1/2 in (1.18), that is pm /qm and (2pm ± pm−1 )/(2qm ± qm−1 ) respectively. Therefore we have |J|−1 = qm (2qm ± qm−1 ). On the other hand, one gets dx/dxm from (1.18), so that its inverse |dAm /dx| = (qm ± qm−1 xm )2 , and the lemma follows easily. Lemma 1.13. Let J and J 0 be the intervals of definition of two adjacent branches of Am , with respective lengths |J| and |J 0 |, then there exists a constant c5 (which can be taken c5 = 12), such that |J| c−1 ≤ c5 . (1.49) 5 ≤ |J 0 | Proof. Let x be the common end-point. If Am (x) = 0, then one has x = pm /qm , 0 and the other end-points are (2pm ± pm−1 )/(2qm ± qm−1 ). Therefore qm = qm , and |J|/|J 0 | = (2qm ± qm−1 )/(2qm ± qm−1 ) ≤ 3. If Am (x) = 1/2, then one of the two intervals has the form [pm /qm , (2pm ± pm−1 )/(2qm ± qm−1 )] and the same holds for 0 the other, but with pm and qm replaced by p0m and qm respectively. For the same reasons 0 ± qm−1 , as in case (C) above (see Eq. (1.34)), we have pm = p0m ± pm−1 , qm = qm 0 0 pm−1 = p0m−1 , qm−1 = qm−1 , so that qm /qm ≤ 3, and the length ratio |J|/|J 0 | = 2 0 qn (2qn ± qn−1 )/qn0 (2qn0 ± qn−1 ) ≤ 3qn2 /q 0 n ≤ 12. Note that in both cases we have qm 1 ≤ 0 ≤3. 3 qm
(1.50)
Brjuno Functions and Their Regularity Properties
277
2. The Brjuno Functions Following Yoccoz [Yo] we define a (generalized) Brjuno function: Definition 2.1. The α-Brjuno function Bα : R \ Q → R is defined by the formula Bα (x) = −
∞ X
βi−1 (x) log xi ,
(2.1)
i=0
where the xn follow x0 = x by repeated iterations of Aα , as defined in (1.10) and (1.11), and the βn ’s are given by (1.21). We have posed β−1 = 1. Remark 2.2. It is useful to extend the above definition x ∈ Q, by setting Bα (x) = +∞, or exp(−Bα (x)) = 0. The Brjuno function defined in [Yo] corresponds to B1/2 , the one defined by the nearest integer continued fraction map A1/2 . Proposition 2.3. Given α ∈ [1/2, 1] one has (i) Bα (x) = Bα (x + 1) for all x ∈ R; (ii) for all x ∈ (0, α)
1 ; Bα (x) = − log x + xBα x
(2.2)
(iii) if x ∈ [α − 1, 0) then Bα (−x) = Bα (x); (iv) there exists a constant C1 > 0 (independent of α) such that for all x ∈ R \ Q one has ∞ X log qj+1 Bα (x) − (2.3) ≤ C1 , qj j=0 where {qj }j≥0 denotes the sequence of the denominators of the convergents to x of the α-continued fraction expansion. Proof. Given x ∈ R \ Q, the sequences (xi )i≥0 and (βi )i≥0 associated to x and x + 1 are the same, which proves (i). The same is true for x and −x if x ∈ (α − 1, 0), which proves (iii). If x ∈ (0, α), let y = 1/x and denote by yi , ai (y), βi (y), and xi , ai (x), βi (x) the sequences (1.11) and (1.21) associated to y and to x respectively. From (1.9) and (1.10) it follows that x0 = x, a0 (y) = a1 (x), y0 = x1 and by induction for all n ≥ 0, yn = xn+1 and βn (y) = (βn+1 (x))/x. Thus Bα (y) = −
∞ X
βi−1 (y) log yi = − log y0 −
i=0 ∞
∞ X 1 βi (x) log xi+1 x i=1
1X 1 =− βi−1 (x) log xi = [Bα (x) + log x] , x x i=1
which proves (ii). To prove (iv) we first remark that (1.21) implies qi βi−1 + εi qi−1 βi = 1
278
S. Marma, P. Moussa, J.-C. Yoccoz
for all i ≥ 0. Then Bα (x) +
∞ X log qi+1
qi
i=0
=
∞ X i=0
=
∞ X
∞ X βi qi−1 βi−1 log + βi log qi+1 βi−1 + εi βi−1 qi i=0 ∞ X
βi−1 log βi qi+1 −
i=0
βi−1 log βi−1 +
i=0
∞ X i=0
εi
qi−1 βi log qi+1 , qi
but by (1.21), Proposition 1.4 (iii), and the estimates of Remark 1.7, ∞ ∞ X X log 2 βi−1 log βi qi+1 ≤ 2 ≤ 2c2 , qi i=0 i=0 ∞ ∞ X X log 2 + log qi βi−1 log βi−1 ≤ 2 ≤ 2(c1 + c2 ) , qi i=0 i=0 ∞ ∞ X X qi−1 log qi+1 εi βi log qi+1 ≤ 2 ≤ 2c1 , qi qi+1 i=0
i=0
from which it follows that ∞ X log qi+1 ≤ C1 = 4(c1 + c2 ) . Bα (x) − qi i=0
By means of Lemma 1.8 one can prove the following Proposition 2.4. There exists a positive constant C2 > 0 such that for all α ∈ [1/2, 1] and for all x ∈ R \ Q one has ∞ X log Q j+1 Bα (x) − (2.4) ≤ C2 . Qj j=0 P∞ Proof. Thanks to (iv), Proposition 2.3, it suffices to compare j=0 (1/qj ) log qj+1 with P∞ j=0 (1/Qj ) log Qj+1 . By Lemma 2.3, one has qj = Qk(j) for all j, where for brevity we write k(j) for k α (j). Thus ∞ X log qj+1 j=0
qj
=
X k(j+1)=k(j)+1
log Qk(j+1) + Qk(j)
X k(j+1)=k(j)+2
log Qk(j+1) . Qk(j)
Using the fact that Qk(j+1) = Qk(j)+2 = Qk(j)+1 + Qk(j) we have log Qk+2 log(Qk+1 + Qk ) log Qk+1 log 1 + Qk /Qk+1 = = + Qk Qk Qk Qk but
Brjuno Functions and Their Regularity Properties
279
log 1 + Qk /Qk+1 log 2 0≤ ≤ , Qk Qk By applying the estimates of Remark 1.7 one gets the result: ∞ ∞ X log qj+1 X log Qj+1 − ≤ 2c2 + c1 , qj Qj j=0
j=0
and one gets C2 = 6c2 + 5c1 .
Remark 2.5. The Brjuno numbers [Br] are usually defined by the Brjuno condition ∞ X log Qi+1 i=0
Qi
< +∞ .
Proposition 2.4 shows that the α-Brjuno functions Bα are finite at x if and only if x is a Brjuno number and that all the generalized Brjuno functions differ one from the other for a L∞ function. On the other hand, the advantage of the functions Bα with respect to the Brjuno condition is that they verify a nice functional equation (2.2) under the action of the modular group SL (2, Z). Another important characterization of the generalized Brjuno functions comes from their “uniqueness”, as it is an immediate consequence of Theorem 2.6 below. For fixed 1/2 ≤ α ≤ 1, let us consider the operator 1 ν , (Tν f )(x) = x f x
(2.5)
if x ∈ (0, α), where ν ≥ 0, defined for the moment on measurable functions of R which verify f (x) = f (x + 1) for almost every x ∈ R ,
(2.6)
f (−x) = f (x) for a.e. x ∈ (0, 1 − α) . It is understood that the function Tν f is completed outside (0, α) by imposing on Tν f the same parity and periodicity conditions which are expressed for f in (2.6). As noticed in the Introduction, one should write Tν(α) instead of Tν , however, we will omit the α dependance for shortness, since the value of α is usually obviously given by the context. The functional equation for the α-Brjuno function can be written in the form [(1 − T1 )Bα ](x) = − log x ,
(2.7)
for all x ∈ (0, α), complemented with the periodicity and symmetry conditions (2.6). This suggest to study the operator Tν on the Banach spaces Xα,p = {f : R → R | f verifies (2.6) , f ∈ Lp ((0, α), dmα (x)), }
(2.8)
p
endowed with the norm of L ((0, α), dmα (x)), where dmα (x) = cα ρα (x) dx is the invariant measure defined in Sect. 1, so that
(2.9)
280
S. Marma, P. Moussa, J.-C. Yoccoz
Z
1/p
α
||f ||α,p =
|f (x)|p dmα (x)
,
(2.10)
0
as α varies in (1/2, 1) and p ∈ [1, ∞]. Note that if p < p0 one has the obvious inclusion Xα,p0 ⊂ Xα,p and let Xα = ∩p≥1 Xα,p .
(2.11)
If (1 − T1 ) is invertible in the considered space, then (2.7) has a unique solution for Bα , provided that the right hand side also belongs to the space, which is easy to check. The invertibility property is given by the following theorem, which states in particular that the spectral radius of T1 is strictly smaller than 1. Theorem 2.6. Tν is a linear bounded operator from Xα,p into itself for all ν > 0, for all α ∈ [ 21 , 1] and for all p ∈ [1, ∞]. Indeed its spectral radius on Xα,p satisfies r(Tν ) ≤
gν , γν ,
if α > g if α ≤ g.
(2.12)
Proof. It is a simple calculation. We observe first that (Tνn f )(x) = (βn−1 (x))ν f (xn ) = (βn−1 (x))ν (f ◦ Anα )(x) ,
(2.13)
therefore Z
α 0
Z |Tνn f (x)|p mα (x)dx =
0
α
(βn−1 (x))νp |f (Anα (x))|p dmα (x)
≤ [αγαn−1 ]νp
Z
α
|f (x)|p dmα (x),
(2.14) (2.15)
0
where we have used Proposition 1.4 (iv) and (v) (therefore γα = g if α > g, γα = γ if α ≤ g) and the invariance of the measure dmα (x) w.r.t. Aα . From (2.14) it immediately follows that ||Tνn f ||α,p ≤ [αγαn−1 ]ν ||f ||α,p , and one gets (2.12) by taking the 1/nth root of both sides.
(2.16)
The use of the invariant measure in (2.8) makes the evaluation of the spectral radius remarkably simple. We of course get the same result if we replace the measure in (2.8) by the Lebesgue measure which is equivalent (see Remark 1.2). In particular, the above theorem implies that the spectral radius is also bounded by (2.12) for the operator Tν in the spaces Lp (T), naturally introduced using the periodicity property. It is more difficult to tell whether Tν is itself contracting (see [MMY] for some results in this direction), we only mention here that in the case α = 1/2, T1 is a contraction for all Lebesgue Lp -norms on [0, 1/2].
Brjuno Functions and Their Regularity Properties
281
3. The Brjuno Function and the BMO Space In the previous section, it has beenTshown that the Brjuno functions Bα belong to Lp (T) ∞ and therefore to the intersection p=1 Lp (T). The purpose of this section is to show a stronger result: the Brjuno functions Bα belong to BMO(T). We recall the definition and the main properties of BMO spaces in Appendix A. In fact, we already know that all Brjuno functions Bα differ by L∞ functions, and since L∞ ⊂ BMO, it will be enough to prove that Bα is in BMO(T) for a fixed value of α. In this section we fix α = 1/2 and denote A1/2 and B1/2 simply by A and B respectively. We will also write dm(x) = dm1/2 (x) =
1 log G
1 1 + G+x G+1−x
dx .
On the interval I, we define now the mean value fI of f Z Z 1 fI = f (x) dm(x) , mI = dm(x) , mI I I
(3.1)
(3.2)
and its quadratic oscillation OI (f ) OI (f ) = =
1 mI
Z f (x) − fI
2
1/2 dm(x)
I
1 2m2I
Z f (s) − f (t)
2
(3.3)
21 dm(s) dm(t)
.
I×I
We now consider the space X∗ ⊂ X1/2 = ∩∞ p=1 X1/2,p defined as: X∗ = {f ∈ BMO(R) | f (x + 1) = f (x) ∀x ∈ R ,
(3.4)
f (−x) = f (x) ∀x ∈ [0, 1/2]} , with the norm ||f ||∗ = |f |∗ + ||f ||2 ,
(3.5)
where ||f ||2 = ||f ||L2 ((0,1/2),dm) , and |f |∗ =
sup
OI (f ) .
(3.6)
I⊂[0,1/2]
Therefore we have ||f ||∗ =
sup I⊂[0,1/2]
Z
1/2
1 2m2I
Z
2
I×I
f (s) − f (t) ! 21
(f (x))2 dm(x)
+ 0
.
21 dm(s) dm(t) (3.7)
282
S. Marma, P. Moussa, J.-C. Yoccoz
Remark 3.1. Due to the equivalence between the measure m and the Lebesgue measure, one gets an equivalent norm in replacing m(I) by the length |I| in (3.7) We explain in the appendix why the norm we use here is equivalent to the usual BMO norm: it differs with the usual one [Gr,GCRF] in two respects, first, we use here the invariant measure dm instead of the Lebesgue measure, second, we use here a L2 -norm definition of BMO instead of the usual L1 -norm definition. The equivalence between the L1 and L2 definitions is a corollary of the John-Nirenberg Theorem which is far from obvious. Furthermore, the BMO-norm on [0, 1/2] is equivalent to the BMO-norm on T only for even functions, which is the case for B1/2 . We shall now prove the following theorem, Theorem 3.2. The Brjuno function B = B1/2 belongs to X∗ , and therefore to BMO(T). For 1/2 ≤ α ≤ 1, the functions Bα also belong to BMO(T). The proof follows immediately from the following Theorem 3.3, which states in particular that for α = 1/2, 1 − T1 is invertible in X∗ . And it is easy to show by direct computations that, for α = 1/2, the right hand side of (2.7), namely the even and periodic function equal to log x on (0, 1/2], is in X∗ . It results that B = B1/2 is also in X∗ , as well as all Bα for 1/2 ≤ α ≤ 1. Note that B 6⊂ L∞ because the logarithmic function is unbounded. Theorem 3.3. In the case α = 1/2, and for all ν > 0, Tν is a bounded linear √ operator from X∗ to X∗ . Indeed, its spectral radius in X∗ is at most equal to γ ν = ( 2 − 1)ν . Proof. In order to prove the theorem, we must estimate OI (Tνm f ) for m ≥ 0, I ⊂ [0, 1/2]. Let I = [x, x0 ], n = n(x, x0 ) the splitting order of x and x0 . We divide the proof into three cases. First case: One has m > n. Let Ib denote the union of the domains of the branches of Am b and from (1.49), 1 ≤ |I|/|I| b which meet the interval I. Then one has I ⊂ I, ≤ (1 + 2c5 ), since in this case I contains the full interval of at least one branch of Am . Setting g = Tνm f , one gets Z Z Z 1 1 1 (g − gI )2 dm ≤ g 2 dm ≤ c0 (1 + 2c5 ) g 2 dm , (3.8) OI2 (g) = b Ib mI I mI I |I| where we have used (1.7) and (1.49). For the domain J of any complete branch of Am , we take Am (t) instead of t as the integration variable. Then, it follows from (1.48) that Z Z m 2 2ν 2 (Tν f ) dm ≤ ||βm−1 ||C 0 (f ◦ Am )2 dm ≤ 12|J|c20 ||βm−1 ||2ν (3.9) C 0 ||f ||2 , J
where ||
J
||0C
denotes the sup-norm on [0, 1/2]. Therefore one gets Z 2 b (Tνm f )2 dm ≤ 12c20 ||βm−1 ||2ν C 0 ||f ||2 |I| , Ib
from which it follows that OI (Tνm f ) ≤ 2c0
p
3c0 (1 + 2c5 )||βm−1 ||νC 0 ||f ||2 .
(3.10)
(3.11)
Second case: One has m ≤ n − δ. Then I is contained in the domain J of a single branch of Am . Let I1 = Am (I) ⊂ [0, 1/2]. We have
Brjuno Functions and Their Regularity Properties
283
Z 2m2I OI2 (T m f ) = where
(Tνm f (s) − Tνm f (t))2 dm(s) dm(t) ≤ 2(M1 + M2 ) ,
(3.12)
2ν βm−1 (s)(f (Am s) − f (Am t))2 dm(s) dm(t) ,
(3.13)
ν ν (f (Am t))2 (βm−1 (s) − βm−1 (t))2 dm(s) dm(t) .
(3.14)
I×I
Z M1 = Z
I×I
M2 = I×I
Now, from the bound (1.47) on dAm /dx, we deduce upper and lower bounds on the ratio |I1 |/|I|, m −1 2 2 dA (x) |I1 | 9qm qm ≤ 4 ≤ 9 |I| ≤ 9c20 mI . ≤ ≤ , and, (3.15) 2 4 |I| 4 dx qm |I1 | mI1 Taking Am (s) and Am (t) as new integration variable, one gets m −2 ! Z dA 4 2ν M1 ≤ c0 ||βm−1 ||C 0 max (f (s0 ) − f (t0 ))2 dm(s0 ) dm(t0 ) x∈I1 dx I1 ×I1 (3.16) ≤ = √ 4 with c6 = 9 2c0 . On the other hand, from (1.30) one gets 2 2 81c80 ||βm−1 ||2ν C 0 2mI OI1 (f )
2 2 c26 ||βm−1 ||2ν C 0 mI OI1 (f )
,
|βm−1 (s) − βm−1 (t)| ≤ qm−1 |I| ≤ |I|1/2 ,
(3.17)
−2 −2 ≤ qm−1 . Then, using the obvious inequality since from (1.47), one has |I| ≤ |J| ≤ qm ν ν ν−1 ν−1 )|x − y|, and Lemma 1.10, we get |x − y | ≤ ν max(x , y Z 2ν−2 2ν−2 M2 ≤ ν 2 (f (Am t))2 max(βm−1 (s), βm−1 (t))(βm−1 (s) − βm−1 (t))2 dm(s)dm(t) I×I Z 2ν−2 (f (Am t))2 βm−1 (t) dm(s)dm(t) (3.18) ≤ ν 2 |I|c23 I×I
Z
M2 ≤ ν 2 |I|mI c0 c23 ||βm−1 ||2ν C0
I
−2 2 2 (f (Am (t)))2 βm−1 (t)dt ≤ c27 ||βm−1 ||2ν C 0 mI ||f ||2 .
(3.19) We have bounded βm−1 in the integral using Proposition 1.4 (iii) and taken Am (t) as a 3/2 new integration variable using (1.47), so that setting c7 = 3νc0 c3 , one obtains OI (Tνm f ) ≤ ||βm−1 ||νC 0 c26 OI21 (f ) + c27 ||f ||22
1/2
≤ ||βm−1 ||νC 0 c6 OI1 (f ) + c7 ||f ||2 .
(3.20)
Third case: One has n − δ < m ≤ n. Thus one is in one of the cases (B), (C) or (D), discussed above after Definition 1.9 : the interval I is contained in the union of two adjacent branches of Am and the point x00 is the common point of the two branches. Let I − = [x, x00 ], I + = [x00 , x0 ] and we still define M1 and M2 as in Eqs. (3.13) and (3.14),
284
S. Marma, P. Moussa, J.-C. Yoccoz
so that we get (3.12) as a bound of the oscillation on I. We first bound M2 as in the previous case, but now from (1.30), |βm−1 (s) − βm−1 (t)| ≤ |βm−1 (s) − βm−1 (x00 )| + |βm−1 (x00 ) − βm−1 (t)| 0 |I− | (3.21) ≤ qm−1 |I+ | + qm−1 √ 1/2 1/2 1/2 ≤ |I+ | + |I− | ≤ 2|I| , where the beforelast inequality is obtained as (3.17) above. Then Z Z Z −2 m 2 −2 f (A (t)) βm−1 (t)dt ≤ + (t)dt ≤ 2c27 ||f ||22 , (3.22) f (Am (t))2 βm−1 I−
I
I+
from which it follows that 2 2 M2 ≤ 4c27 ||βm−1 ||2ν C 0 mI ||f ||2 .
On the other hand
(3.23)
Z
M1 ≤
c20 ||βm−1 ||2ν C0
(f (Am (s)) − f (Am (t)))2 dsdt .
(3.24)
I×I
Let I1+ = Am (I + ), I1− = Am (I − ); one has, using (3.15), Z (f (Am (s)) − f (Am (t)))2 dsdt I ε ×I ε
|I ε |2 ≤ 81 ε 2 |I1 |
Z I1ε ×I1ε
(f (s0 ) − f (t0 ))2 ds0 dt0
(3.25)
≤ 81c40 2m2I OI2 ε (f ) . 1
Finally one gets Z I + ×I −
(f (Am (s)) − f (Am (t)))2 ds dt
|I + ||I − | ≤ 81 + − |I1 ||I1 |
Z I1+ ×I1−
(f (s0 ) − f (t0 ))2 ds0 dt0 .
(3.26)
Let I1 = I1+ ∪I1− . From the discussion of cases (B), (C) and (D) made after Definition 1.9, it follows that I1+ and I1− have a common end-point which is either 0 or 1/2. Therefore, either I1+ ⊂ I1− or I1− ⊂ I1+ , then Z (f (s0 ) − f (t0 ))2 ds0 dt0 ≤ c20 2m2I1 OI21 (f ) . (3.27) I1+ ×I1−
However, as read from (3.15), Then, from (1.50), one gets
|I − | 0 2 |I + | 2 q both belong to the interval [4/9, 4]. + qm , and |I1 | |I1− | m |I + ||I − | ≤ c28 |I1+ ||I1− |
|I| |I1 |
2 ,
(3.28)
Brjuno Functions and Their Regularity Properties
285
with c8 = 22 34 , from which it follows that Z (f (Am (s)) − f (Am (t)))2 dsdt ≤ c29 2m2I OI21 (f ) ,
(3.29)
I + ×I −
with c9 = 81c28 c40 . Putting all constants together, one therefore finds c10 and c11 such that 2 2 2 2 M1 ≤ c10 ||βm−1 ||2ν C 0 mI (OI + (f ) + OI − (f ) + OI1 (f )) , 1
OI (Tνm f )
≤
c11 ||βm−1 ||νC 0 (OI1+ (f )
1
(3.30)
+ OI − (f ) + OI1 (f ) + ||f ||2 ) . 1
In the three cases we have considered, we get a bound for the oscillation of Tνm (f ) as a linear combination with constant coefficients of the sup of the oscillation of f , the norm ||f ||2 , and an overall factor ||βm−1 ||νC 0 which contains the only dependence on m. Since (12.5) gives the same result for the L2 -norm, we easily deduce that, for the norm (3.5), there exist a positive constant c12 independant of m and f , such that ||Tνm f ||∗ ≤ c12 ||βm−1 ||νC 0 ||f ||∗ ≤ (γ ν )m (c12 /2γ)||f ||∗ .
(3.31)
The first consequence is that Tν is a bounded linear operator from X∗ to X∗ , the second is that its spectral radius r(Tν ) = lim ||(Tν )m ||1/m is bounded by γ ν . The spectral radius computation is rather general, and does not require a tight adjustment for the constants: more specific results on the norm of Tν itself depend on the peculiar norm taken. See [MMY] for some results of this kind. We observe that the computation of the spectral radius for the Lp -norm and the BMO-norm both come from the leading behaviour of βm−1 , as given by Theorem 1.4. The embedding of the BMO space in between the Lp spaces and L∞ makes this result very natural. We expect interesting consequences will follow for the complex extension of the Brjuno functions. 4. The Brjuno Function and its H¨older Stability Properties The functional equation (1.29) for the Brjuno function for α = 1/2 is [(1 − T1 )B1/2 ](x) = − log x ,
(4.1)
for all x ∈ (0, 1/2), complemented with the condition that B = B1/2 is even and periodic. In this section we will suppose that the right hand side of this equation is perturbed, by an additional term f , which is less singular than the logarithmic function. We want to study the singular properties of the perturbed solution. Since the equation is linear, we only need to consider the action on f of T1 and (1 − T1 )−1 , which will be conveniently called the Brjuno operator B. We will consider even and periodic functions f which are continuous. It is sufficient to know the value of f on [0, 1/2], so we assume 0 . One can check that T1 f (resp. Tν for ν > 0) is also continuous provided we f ∈ C[0,1/2] set (T1 f )(0) = 0 (resp (Tν f )(0) = 0). We need now the usual H¨older’s type semi-norms for continuous functions. 0 , Then we define the H¨older’s η-norm as Definition 4.1. Let f ∈ C[0,1/2]
|f |η =
|f (x) − f (y)| , |x − y|η 0≤x
(4.2)
286
S. Marma, P. Moussa, J.-C. Yoccoz
with 0 < η ≤ 1. This is a seminorm since it vanishes on constant functions, so that we introduce the norm: (4.3) ||f ||η = |f |η + ||f ||C 0 , η 0 , if f ∈ C[0,1/2] and |f |η is where ||f ||C 0 = max0≤x≤1/2 |f (x)|. We say that f ∈ C[0,1/2] finite.
We now have: η , 0 < η ≤ 1. Then: Theorem 4.2. Let f ∈ C[0,1/2]
(1) Tνm f is H¨older continuous, of exponent η = min(η, ν/2). (2) If ν > 2η (thus η = η), Tν is a bounded linear operator in C η , of spectral radius −1 smaller or equal to γ ν−2η P∞< 1.mThe operator Bν = (1 − Tν ) is defined in this space and fulfills Bν = m=0 Tν . (3) If ν = 2η (thus η = η = ν/2), there exists a positive constant c14 > 0 such that ||Tνm ||C ν/2 (0,1/2) ≤ c14 , for all m ≥ 0 .
(4.4)
Proof. Let x, x0 ∈ [0, 1/2], m ≥ 0, η ∈ (0, 1], ν > 0. We want to estimate |Tνm f (x) − Tνm f (x0 )| η under the assumption that f ∈ C[0,1/2] . We let n = n(x, x0 ).
First case: m > n. From (2.13), one has ν ν (x) + βm−1 (x0 )) |Tνm f (x) − Tνm f (x0 )| ≤ ||f ||C 0 (βm−1
≤
(4.5)
ν ν (xn+1 ) + βnν (x0 )βm−n−2 (x0n+1 )) ≤ ||f ||C 0 (βnν (x)βm−n−2 2cν4 ||f ||C 0 |x − x0 |ν/2 ||βm−n−2 ||νC 0 ,
(4.6) (4.7)
where we have used Lemma 1.11. Second case: m ≤ n − δ. One has |Tνm f (x) − Tνm f (x0 )| ≤ (βm−1 (x))ν |f (Am (x)) − f (Am (x0 ))| ν ν (x) − βm−1 (x0 )| . +|f (Am (x0 ))||βm−1
(4.8)
From (1.47) and Proposition 1.4 (iii), one has |Am (x) − Am (x0 )| ≤ 9(βm−1 (x))−2 |x − x0 | ,
(4.9)
and, using (1.30), and (1.41) ν−1 ν−1 ν ν |βm−1 (x) − βm−1 (x0 )| ≤ ν max(βm−1 (x), βm−1 (x0 ))|βm−1 (x) − βm−1 (x0 )|
≤ νcν−1 (βm−1 (x))ν−1 qm−1 |x − x0 | 3 ≤ Therefore
(βm−1 (x))ν−2 |x νcν−1 3
0
−x|.
(4.10)
Brjuno Functions and Their Regularity Properties
287
|x − x0 |−η |Tνm f (x) − Tνm f (x0 )|
||f ||C 0 (βm−1 (x))2η−2 |x − x0 |1−η , ≤ (βm−1 (x))ν−2η 9η |f |η + νcν−1 3 ||f ||C 0 (16/9)1−η ≤ (βm−1 (x))ν−2η 9η |f |η + νcν−1 3 = Kf (βm−1 (x))ν−2η ,
(4.11)
−1 where we have used |x−x0 | ≤ |J|, and (from 1.48) βm−1 (x) ≥ (3/4)qm ≥ (3/4)|J|1/2 , J being the domain of the branch of Am which contains x and x0 . Third case: One has n − δ < m ≤ n: one is then led to consider the cases (B), (C) or (D) defined above, after Definition 1.9. One then introduces the intermediate point x00 and one gets the same estimates of the second case. More precisely,
|Tνm f (x) − Tνm f (x0 )| ≤ |Tνm f (x) − Tνm f (x00 )| + |Tνm f (x0 ) − Tνm f (x00 )| ≤ 2Kf max (βm−1 (x))ν−2η |x − x00 |η , (βm−1 (x0 ))ν−2η )|x0 − x00 |η (βm−1 (x))ν−2η |x − x0 |η , ≤ 2Kf cν−2η 3
(4.12)
where we have used (1.41). One can summarize the possible cases, in view of Theorem 4.3, by stating that there exists always a constant c13 such that |Tνm f (x) − Tνm f (x0 )| ≤ c13 ||f ||η |βm−1 (x)|ν−2η |x − x0 |η .
(4.13)
The statements of the theorem result easily from Eqs. (4.3), (4.7), (4.11), (4.12), and from the obvious inequality for the C0 norm, ν (x)] ≤ |Tνm f (x)| ≤ ||f ||C 0 sup[βm−1
1 (m−1)ν γ ||f ||C 0 . 2
(4.14)
Remark 4.3. It will be useful to notice here that an estimate similar to (4.13) holds for |εm (x)Tνm f (x) − εm (x0 )Tνm f (x0 )| provided f (0) = f (1/2) = 0 ,
(4.15)
namely |εm (x)Tνm f (x) − εm (x0 )Tνm f (x0 )| ≤ c13 ||f ||η |βm−1 (x)|ν−2η |x − x0 |η .
(4.16)
In order to get (4.16), we follow the same argument as we used in the previous proof, and distinguish three cases. The proof is immediate in the first case where m > n, and the bound is obtained in the same way as (4.7). In the second case where m ≤ n − δ, we also get the result as in (4.8), since we now have εm (x) = εm (x0 ). In the third case, where n − δ < m ≤ n, one introduces once more the intermediate point x00 , and one observes that either x00m = 0, or x00m = 1/2, according to the discussion which follows Definition 1.9. It follows that Tνm (x00 ) = βm−1 (x00 )f (x00m ) = 0, due to condition (4.15). Therefore |εm (x)Tνm f (x) − εm (x0 )Tνm f (x0 )| ≤ |Tνm f (x) − Tνm f (x00 )| + |Tνm f (x0 ) − Tνm f (x00 )|, and the required estimate is obtained as in (4.12). This completes the proof of (4.16). η , with ν < 2η, we easily deduce from the proof of the preceding When f ∈ C[0,1/2]
theorem that Tν f is in C ν/2 , and that Bν f is in C θ , for any θ < ν/2. In fact, one has a slightly stronger result for Bν f .
288
S. Marma, P. Moussa, J.-C. Yoccoz
η Theorem 4.4. Let f ∈ C[0,1/2] ,
(1) If ν < 2η, then the function Bν f =
P m≥0
Tνm f is in C ν/2 .
(2) If ν = 2η, then Bν f is in C η for 0 ≤ η < ν/2. Indeed it admits xν/2 | log x| as continuity modulus. Proof. We have, using (4.13) and setting n = n(x, x0 ), 0
|Bν f (x) − Bν f (x )| ≤
n X
|Tνm f (x)
−
Tνm f (x0 )|
≤ c13 ||f ||η |x − x |
|Tνm f (x) − Tνm f (x0 )|
n+1
0 0 ν/2
+
∞ X
0 η−ν/2
|x − x |
n X 0
! ν−2η
(βm−1 (x))
∞ X + ||βm−n−2 ||νC 0
! .
n+1
(4.17) From the bound on the βn , one sees immediately that the second series in the previous inequality converges. Furthermore all xk belong to (0, 1/2], and one has 2η−ν n n X X 1 ν−2η ν−2η (βm−1 (x)) ≤ (βn−1 (x)) = c15 (qn (x))2η−ν , 2n−m 0
0
Pn with c15 = (3/2)2η−ν 0 2−(m(2η−ν)) , which is finite. Now, if x and x0 belong to the same branch of An , let K be its interval of definition. If they belong to two adjacent branches, let K be the union of their intervals of definition. In both cases one has |x − x0 | ≤ |K| and, using Lemmas 1.12 and 1.13, one easily sees that |K|qn2 (x) is bounded, above and below, so that finally, one can find c16 such that for
|Bν f (x) − Bν f (x0 )| ≤ c16 ||f ||η |x − x0 |ν/2 .
(4.18)
|Bν f (x) − Bν f (x0 )| ≤ c17 (1 + n(x, x0 ))||f ||η |x − x0 |ν/2 .
(4.19)
ν < 2η ,
When ν = 2η, we get
However, x and x0 belong to the same branch |J| of An−2 , so that, using (1.48), one gets −2 |x − x0 | ≤ |J| ≤ qn−2 , and following remark 1.5, |x − x0 | ≤ (3/4)γ n−3 . We therefore have, up to some constant c, log |x − x0 | ≥ n log γ + c, which means that n is bounded: n ≤ (log(|x − x0 |−1 ) − c)/ log 0. Therefore there exists a constant c18 such that ν = 2η ,
for
|Bν f (x) − Bν f (x0 )| ≤ c18 ||f ||η log(|x − x0 |−1 )|x − x0 |ν/2 . (4.20)
As a consequence of the above theorem, we observe that if we consider a perturbation of the functional equation (2.7) for the Brjuno function (in the case α = 1/2), that is for 0 < x ≤
1 , 2
[(1 − T1 )Bf ](x) = − log x + f (x) ,
the function Bf differs from B by 1/2-H¨older continuous function when f is analytic, or at least 1/2- H¨older continuous. Therefore the ‘most singular’ part of Bf does not depend on the perturbation when it is sufficiently regular. The present result could provide
Brjuno Functions and Their Regularity Properties
289
an explanation for the so-called ‘modular smoothing’ of critical functions observed in earlier works [BPV,MS]. The above result seems to give a special role to the Brjuno function B = B1/2 . However, the theorem below displays a somewhat surprising result, namely that the difference B1 − B1/2 is not only bounded, as we already know, but also 1/2-H¨older continuous. We first give a preparatory statement. Proposition 4.5. Let B1+ and B1− be the even and odd part of B1 . B1+ and B1− are periodic, so that they are determined by their values in [0, 1/2]. We have x 1−x − , (4.21) for x ∈ [0, 1/2] , B1 (x) = log 2 x 1 B1+ (x) = xB1+ + g(x) − log x , (4.22) x with, still for x ∈ [0, 1/2], x g(x) = − log 2
1−x x
+
xB1−
1 . x
(4.23)
Proof. For x ∈ [0, 1/2]∩[R\Q], we have B1 (−x) = B1 (1−x), and 1 < (1−x)−1 < 2, so that B1 (−x) = − log(1 − x) + (1 − x)B1 ((1 − x)−1 − 1). But x 1−x 1 x 1 − 1 = B1 = log + B1 , B1 1−x 1−x x 1−x x since 0 < x/(1 − x) < 1. Therefore B1 (−x) = −x log(1 − x) − (1 − x) log x + xB1 (x−1 ) . Since we also have
B1 (x) = − log x + xB1 (x−1 ) ,
we easily get (4.21) by subtraction. By addition, we get x 1−x 1 1 + − log x − log = xB1 − log x − B1− (x) , (4.24) B1 (x) = xB1 x 2 x x which leads to (4.22) since we already know B1− .
B1− ,
The odd part of B1 − B1/2 coincides with and (4.21) shows that B1− is in C η for 0 ≤ η < 1. The even part of B1 − B1/2 , namely 1(x) = B1+ (x) − B1/2 (x)
(4.25)
satisfies 1(x) = g(x)+x1(x−1 ). From (4.23) and Theorem 4.2, one deduces that g ∈ C η , for 0 ≤ η < 1/2. Then Theorem 4.4 tells that 1 ∈ C η for 0 ≤ η < 1/2, and the same holds for the difference B1 − B1/2 . It is easy to see that the continuity extends to x = 0 and x = 1/2, by setting 1(0) = 1(1/2) = 0. In fact, the following theorem gives a stronger result which includes the case η = 1/2. Theorem 4.6. The difference B1 − B1/2 may be extended from R \ Q to R as an 1/2H¨older continuous periodic function with period one.
290
S. Marma, P. Moussa, J.-C. Yoccoz
Proof. From (4.24) and (4.25), we get for 0 < x < 1/2, 1(x) = x1(x−1 ) + xB1− (x−1 ) − B1− (x) = x0 1(x1 ) + ε1 x0 B1− (x1 ) − B1− (x0 ) , (4.26) where we have used the first step of the continued fraction expansion of x. Solving (4.26) by iteration leads to the even and periodic functions 1, 11 , 12 defined for 0 < x < 1/2 as: 1(x) = 11 (x) + 12 (x) , ∞ X 11 (x) = − βn−1 (x)B1− (xn ) , 12 (x) =
n=0 ∞ X
εn (x)βn−1 (x)B1− (xn ) .
(4.27) (4.28) (4.29)
n=1
We deduce from Theorem 4.4 that 11 = −(1 − T1 )−1 B1− is an even periodic function in C 1/2 , since B1− is sufficiently regular. We now observe that B1− (0) = B1− (1/2) = 0, therefore the evaluation of |12 (x)−12 (x0 )| is made following the proof of Theorem 4.4, for ν = 1, using (4.16) instead of (4.13). The conclusion is the same as Theorem 4.4, part (i), that is that also 12 ∈ C 1/2 . Now, 1 is the even part of B1 − B1/2 , and since the odd part of B1 −B1/2 is nothing but B1− , we deduce that B1 −B1/2 is 1/2-H¨older-continuous. Appendix: B. M. O. Norms Let f ∈ L1loc (R). We define the mean value fI of f on the interval I as: Z 1 fI = f dx , |I| I where |I| is the length of the interval I. Then we define for any interval U , Z 1 |f − fI | dx . ||f ||∗,U = sup I⊂U |I| I
(A.1)
(A.2)
We then say f belongs to the space BMO(U ) if ||f ||∗,U < ∞, i.e. is finite. BMO is an abbreviation for ‘bounded mean oscillation’. ||f ||∗,U is a seminorm on BMO(U ), since for any constant c, we have ||f + c||∗,U = ||f ||∗,U . In particular ||f ||∗,U = 0 if f is constant on U . This applies to U = R and leads to the space BMO(R), abbreviated as BMO and the seminorm ||f ||∗,R on BMO(R) will simply be written ||f ||∗ . In fact this seminorm is a norm on the quotient space of function in L1loc (R) modulo the constant functions. With this norm, the quotient space is complete. We list now some more or less classical results and lemmas [Gr,GRCF]. Proposition A.1. The space L∞ (U ) is a subspace of BMO(U ), and ||f ||∗,U ≤ inf c ||f − c||∞,U . Proposition A.2. Let f in BMO(U ) and let I be an interval. Then for any λ > 0, the Lebesgue measure of the set of points t ∈ I such that |f (t) − fI | > λ is bounded by K1 exp(−K2 λ/||f ||∗,U ).
Brjuno Functions and Their Regularity Properties
291
This is the John-Nirenberg Theorem, see Garnett [Gr]. The proof there is for U = R, but a careful reading show that it works for any U . The constants K1 and K2 do not depend on U , λ, and f . Roughly speaking, this theorem says that where f is unbounded, it behaves at most as a logarithmic function. Proposition A.3. We have the following “magic reverse H¨older’s inequality´´: let f ∈ L1loc (R) and suppose that for some interval U ⊂ R, the seminorm ||f ||∗,U is finite, then for any bounded real p ≥ 1, there exists a constant Ap such that sup I⊂U
1 |I|
Z |f − fI | dx p
p1
≤ Ap ||f ||∗,U .
(A.3)
I
In fact it is an easy corollary of the John-Nirenberg theorem, see Garnett [Gr]. The constant Ap does not depend on U , and may be shown to be smaller than pC with an explicit constant C. Note that the inequality does not work in the limit p → ∞. The preceding proposition shows that replacing the L1 norm in the definition of the BM O norm ||f ||∗,U , by the analogous Lp norm (with p finite), leads to the same BM O space. More precisely using the usual Lp norm Z |f | dx
||f ||p,U =
p
p1 ,
(A.4)
U
we define ||f ||∗,p,U = sup
I⊂U
1 |I|
Z |f − fI | dx p
I
p1
1
= sup |I|− p ||f − fI ||p,I .
(A.5)
I⊂U
We then have Proposition A.4. The space BMO(U ), is a subspace of Lp (U ) when U is a bounded interval. p ∞ Thus, BMO(U ) is a subspace of ∩∞ p=1 L (U ), but not a subspace of L (U ). In fact, on BMO(U ), we have a family of equivalent norms, as shown in the following proposition.
Proposition A.5. On BMO(U ), where U is a bounded interval, define for any real a > 0, and b > 0, and for any integer p ≥ 1 (finite), the following family of norms N (f, a, b, p) = a||f ||∗,p,U + b||f ||p,U ,
(A.6)
then these norms are all equivalent for various a and b and p. In BMO(R), we will also define the seminorm ||f ||∗,T as follows Z 1 |f − fI | dx , ||f ||∗,T = sup |I| ≤ 1 |I| I
(A.7)
the supremum being taken over intervals I ⊂ R with length less or equal to 1. The seminorm ||f ||∗,T is convenient for periodic functions with period 1. For any f ∈ BMO(R) we obviously have: ||f ||∗,[−(1/2),+(1/2)] ≤ ||f ||∗,T ≤ ||f ||∗,R . This observation will be useful if we now consider functions f ∈ BMO(R) which are even and periodic with period 1: for such functions, we also have the following result
292
S. Marma, P. Moussa, J.-C. Yoccoz
Proposition A.6. There exist constants K3 > 1, K4 > 1, K5 > 1 such that for any f ∈ BMO(R), which is even and periodic with period 1, we have a) b) b)
||f ||∗,R ≤ K3 ||f ||∗,T , ||f ||∗,T ≤ K4 ||f ||∗,[0,1] , ||f ||∗,[0,1] ≤ K5 ||f ||∗,[0,1/2] ,
.
See [MMY] for a detailed proof. Note that parts b) and c) are not true if the periodic function f is not even. A non trivial, but immediate consequence of these results is the following corollary. Corollary A.7. Let f be a function defined in [0, 1/2], which belongs to BMO([0, 1/2]). The function g which is even and periodic with period 1, and which coincides with f on [0, 1/2] is in BMO(R). As indicated in Propositions A.5 and A.6, we have a wide choice among possible equivalent norms. The L2 -norm is especially useful, due to the following elementary identity: Z Z 1 1 2 (f − fI ) ds = (f (s) − f (t))2 dsdt . (A.8) |I| I 2|I|2 I×I Defining the oscillation of f in I with the L2 -norm, namely OI (f ) =
1 2|I|2
Z (f (s) − f (t)) dsdt 2
21 ,
(A.9)
I×I
the norm N (f, a, b, 2) can be rewritten as N (f, a, b, 2) = a sup OI (f ) + b||f ||2,U ,
(A.10)
I⊂U
In the above expression, (A.9) and (A.10), replacing the Lebesgue measure on U by any equivalent mesure, leads to an equivalent norm. We now end this appendix by quoting Fefferman’s theorem, which makes the link between analysis on the real line and harmonic or complex extension on the upper halfplane. Proposition A.8. The space BMO(R) is the dual space of the Hardy space H 1 on R. If f ∈ L1loc (R), f ∈ BMO(R) if and only if there exist a constant c, and functions φ and ψ in L∞ , such that f = c + φ + Hψ, where the Hilbert transform Hψ is the harmonic conjugate of ψ. Furthermore, φ and ψ can be chosen such that ||φ||∞ ≤ C||f ||∗ and ||ψ||∞ ≤ C||f ||∗ , with C a constant. References [Bo] [Br] [BPV] [Da] [DH]
Bosma, W.: Optimal continued fractions. Indag. Math. A90, 353–379 (1987) Brjuno, A. D.: Analytical form of differential equations. Trans. Moscow Math. Soc. 25, 131–288 (1971); 26, 199–239 (1972) Buric, N., Percival, I. C., Vivaldi, F.: Critical function and modular smoothing. Nonlinearity 3, 21–37 (1990) Davie, A. M.: The critical function for the semistandard map. Nonlinearity 7, 219–229, (1994). The same author has announced similar results for the standard map (private communication) ´ Norm. Douady, A., Hubbard, J. H.: On the dynamics of polynomial-like mappings. Ann. Scient. Ec. Sup. 4e` me S´erie 18, 287–343 (1985)
Brjuno Functions and Their Regularity Properties
293
[Ga] Gauss, E. F.: Collected Works. No. X1 , Leipzig: Teubner, 1917, p. 372 [Gr] Garnett, J. B.: Bounded Analytic Functions. New York: Academic Press, 1981 [GCRF] Garcia–Cuerva, J. and Rubio de Francia, J.L.: Weighted Norm Inequalities and Related Topics. North Holland Mathematical Studies, 116, Amsterdam: North Holland, 1985 [LM] Lasota, A., Mackey, M. C.: Probabilistic properties of deterministic systems. Cambridge: Cambridge University Press, 1985 [Ma] Marmi, S.: Critical functions for complex analytic maps. J. Phys. A: Math. Gen. 23, 3447–3474 (1990) [MMY] Marmi, S., Moussa, P., Yoccoz, J.-C.: Continued fraction transformations, Brjuno functions, and BMO spaces. Note CEA-N-2788, Feb. 1995. This internal report is in fact a preliminary version of the present work [MS] Marmi, S. Stark, J.: On the standard map critical function. Nonlinearity 5, 743–761 (1992) [Me1] Meyer, D. H.: On a ζ-function related to the continued fraction transformation. Bull. Soc. Math. France 104, 195–203 (1976) [Me2] Meyer, D. H.: On the Thermodynamic Formalism for the Gauss Map. Commun. Math. Phys. 130, 311–333 (1990) [Me3] Meyer, D. H.: Continued fractions and related transformations. In: Ergodic theory, symbolic dynamics and hyperbolic spaces, T. Bedford, M. Keane, C. Series, editors, Oxford: Oxford University Press, 1991 [Na] Nakada, H.: On the invariant measures and the entropies for continued fraction transformations. Keio Math. Rep. 5, 37–44 (1980) [Pe] P´erez-Marco, R.: Solution compl`ete au probl`eme de Siegel de lin´earisation d’une application holomorphe au voisinage d’un point fixe. S´eminaire Bourbaki nr. 753, Ast´erisque 206, 273–310 (1992) [Ri] Rieger, G. J.: Mischung und Ergodizit¨at bei Kettenbruchen nach n¨achsten genzen. J. Reine Angew. Math. 310, 171–181 (1979) [Yo] Yoccoz, J.-C.: Th´eor`eme de Siegel, polynˆomes quadratiques et nombres de Brjuno. Ast´erisque 231, 3–88 (1995) Communicated by M. Herman
Commun. Math. Phys. 186, 295 – 322 (1997)
Communications in
Mathematical Physics c Springer-Verlag 1997
Quantum Deformation of Lattice Gauge Theory D.V. Boulatov? International School for Advanced Studies (SISSA/ISAS), via Beirut 2-4, I-34014 Trieste, Italy. E-mail: [email protected] Received: 29 April 1996 / Accepted: 24 September 1996
Abstract: A quantum deformation of 3-dimensional lattice gauge theory is defined by applying the Reshetikhin-Turaev functor to a Heegaard diagram associated to a given cell complex. In the root-of-unity case, the construction is carried out with a modular Hopf algebra. In the topological (weak-coupling) limit, the gauge theory partition function gives a 3-fold invariant, coinciding in the simplicial case with the Turaev-Viro one. We discuss bounded manifolds as well as links in manifolds. By a dimensional reduction, we obtain a q-deformed gauge theory on Riemann surfaces and find a connection with the algebraic Alekseev-Grosse-Schomerus approach. 1. Introduction The lattice regularization of non-abelian gauge theory (LGT) proposed by K.Wilson in 1974 [19] plays a fundamental role of a non-perturbative definition of QCD. The main principle has been to give up the Poincar´e invariance of the theory and preserve the local gauge symmetry as more fundamental. In the weak coupling regime, the broken translational and rotational invariance restores dynamically. If the gauge coupling is strictly equal to 0, the gauge strength tensor vanishes identically and the theory becomes topological: one can take any finite lattice from a given equivalence class without changing the content of the model. The lattice formulation extends in a natural way the set of acceptable gauge groups for all compact groups while the continuous one is based on the notion of the Lie algebra thus excluding finite groups, for example. After the theory of quantum groups had appeared as a distinct mathematical subject [10, 21], the natural question arose whether the notion of gauge symmetry could be extended to incorporate quantum groups as well. This problem has not been only of academic interest. As was noticed in Ref. [5], the ? Address after 1 October 1996: Institute for Theoretical and Experimental Physics, B. Cheremushkinskaya 25, 117259 Moscow, Russian Federation
296
D.V. Boulatov
Ponzano-Regge model [14] (which coincides with the classical (q → 1) limit of the Turaev-Viro construction [17]) can be represented as LGT defined on lattices dual to simplicial complexes. Then it was natural to assume that the Turaev-Viro invariant could be related to a kind of LGT built on quantum group symmetry (for the sake of brevity, we shall call it qQCD3 ). This program was carried out in Ref. [6] explicitly (see also [7]). Technical difficulties originating from a complicated structure of the representation 2π ring of SLq (2) at a root of unity, q = ei k+2 , was avoided in [6] by establishing a direct connection with the ribbon graph invariants of Reshetikhin and Turaev [15]. The gauge invariance is implicit in this formulation (however, it does not mean that the model does not enjoy it). On the other hand, relative simplicity makes this explicit representation very convenient. The classical (q → 1) limit of the Turaev-Viro construction was discovered long ago by Ponzano and Regge in the framework of Regge calculus [14]. They argued that it can be regarded as a discretization of 3d gravity with the Einstein-Hilbert action. On the other hand, the Turaev-Viro invariant is related to Witten’s Chern-Simons invariant [20]. Witten has shown that, with the ISO(2,1) gauge group, the latter is connected with 3d quantum gravity. In the Euclidean regime for a negative cosmological constant, the gauge group becomes isomorphic to SO(4) = SU (2) × SU (2). It means that qQCD3 possessing SLq (2) gauge symmetry is interesting from the physical point of view (an exposition of the subject can be found in Ref. [7]). The structure of the answers for the partition function suggests that, in 2 dimensions, the corresponding q-deformed LGT (qQCD2 ) is related to the topological (G/G)k coset model (as was argued in Ref. [8] these two models are in some sense dual to each other). An alternative purely algebraic approach to qQCD2 was put forward in Ref. [1, 3, 4]. The starting point for them was a Poisson bracket on 2d lattice connections proposed by Fock and Rosly [11]. In 2 dimensions there is a cyclic order of links incident to a vertex. Demanding that variables performing gauge rotations at each vertex form a quasi-triangular Hopf algebra, Alekseev, Grosse and Schomerus have deduced an algebra of gauge fields. In contrast to this situation, in 3 dimensions there is a natural cyclic order of faces sharing the same link of a lattice, which suggests that one should start with gauge fields forming a Hopf algebra while gauge transformations are interpreted as changes of bases these fields act on. As in the quantum case there is no group manifold behind the construction, gauge fields and gauge transformations have clearly the different statuses. One could say that this is one more occurrence of the principle “Quantization removes degenerations”. Descending from 3 to 2 dimensions, one finds a model which seems, at first sight, to be different from the AGS one. However, as we shall show they are locally equivalent. The outline of the paper is the following. Chapter 2 is devoted to the construction of q-deformed LGT in three dimensions. In Sect. 2.1 we introduce classical LGT and make some general remarks on its quantum deformation. In Sect. 2.2 we collect facts from 3-manifold topology which are used in the sequel. In Sect. 2.3 we introduce the Reshetikhin-Turaev functor in the form adopted for our purposes. In Sect. 2.4 we define qQCD3 functor and discuss the notion of gauge invariance within our framework. In Sect. 2.5 we introduce the qQCD3 partition function in the case of Uq (su(n)) gauge group. Sect. 2.6 is devoted to the root-of-unity case: Uq (sl(n, R)), q ` = 1. In Sect. 2.7 we prove that the weak-coupling partition function introduced in the previous section is a topological invariant. In Sect. 2.8 we discuss the case of bounded manifolds and shortly outline the introduction of Wilson loop averages in our model. Chapter 3 is devoted to the 2-dimensional case. Here we derive Verlinde’s
Quantum Deformation of Lattice Gauge Theory
297
formula and discuss a connection between our approach and the AGS algebra. We conclude with a few general remarks.
2. qQCD3
2.1. Formulation of the problem. To introduce lattice gauge theory, one needs a cell decomposition of a manifold (in physicist usage, a lattice). A gauge field is a map from a set of oriented edges to a compact group: ` 7→ g` ∈ G. A change of an orientation corresponds to the conjugation: g` → g`−1 . One attaches to every vertex a G-module (usually the regular representation). Gauge transformations rotate bases of the modules independently at each vertex. The gauge field is interpreted as performing a parallel transport between vertices, thus relating bases at adjacent ones. If an oriented link, k, connects vertices v2 and v1 , the gauge transformation of the group element gk is gk → hv1 gk h−1 v2
(2.1)
A holonomy associated with a path {L} in the lattice is an ordered product of gauge field elements along {L}: Y gkk , (2.2) hL = k∈L
where k = +1, if the k edge is directed along the path, and k = −1, if their directions are opposite. Gauge invariant quantities are those taking values in the set of conjugacy classes of G. A trace of the holonomy along a closed loop in any representation of G is an example of such an invariant. The Boltzmann weights are functions of holonomies along boundaries, ∂f , of faces, f . One of the standard choices is the so-called group heat kernel X dR χR (h∂f )e−βCR , (2.3) Wβ (h∂f ) = th
R
P
In Eq. (2.3), R is the sum over all finite-dimensional irreps of a gauge group G; χR (x) is the character of an irrep R; dR = χR (I) is its dimension; CR is a second Casimir eigenvalue; β is a real parameter called a coupling constant. The construction makes sense for compact groups whose unitary finite dimensional irreps span the regular representation. The choice (2.3) ensures that Wβ (h∂f ) becomes the group δ-function in the weak coupling limit, β → 0: W0 (h∂f ) = δ(h∂f , I). We shall call this limit topological. The partition function is defined as the integral of the product of the Boltzmann weights over all faces: Z Y Y Y dg` Wβ ( gkk ), (2.4) Zβ = G `
f
k∈∂f
where dg` is the Haar measure on the group G, and the product
Q `
goes over all edges.
298
D.V. Boulatov
If the term “q-deformed” is to mean that gauge variables take values in a quantum group, any presentation of the model should be reducible to a form where the variables are represented in a standard fashion as matrices of non-commutative elements. The simplest and most famous example is SLq (2), which can be introduced as the set of matrices a b g= (2.5) c d the entries of which obey the commutation relations ba = qab db = qbd cb = bc, ca = qac dc = qcd da − ad = (q − q −1 )bc, −1 ad − q bc = 1.
(2.6)
The relations (2.6) imply the existence of the R-matrix
q 0 R= 0 0
0 1 q − q −1 0
0 0 1 0
0 0 0 q
(2.7)
and the RT T = T T R equation (g ⊗ 1)(1 ⊗ g)R = R(1 ⊗ g)(g ⊗ 1).
(2.8)
The R-matrix obeys the quantum Yang-Baxter equation R12 R13 R23 = R23 R13 R12 .
(2.9) 1
2
3
Indices show at which positions in the tensor cube of representation spaces, V ⊗ V ⊗ V , acts the R-matrix. SLq (2) has two real forms: SUq (2), for real q, and SLq (2, R), for |q| = 1. The matrices can be multiplied. If entries of both g and h obey Eq. (2.6) and are mutually commutative, the entries of the product gh obey (2.6) as well. Therefore, matrices on different links of a lattice have to co-commute with one another in the tensor product. The algebra of matrices (2.5) naturally extends to the quasi-triangular Hopf algebra Fq (SL(2)) of quantized functions on SL(2). Owing to the famous duality, its basis is provided by the matrix elements of finite-dimensional irreducible representations of the quantized universal envelopping (QUE) algebra Uq (sl(2)). Therefore, to construct qQCD, we have at hands co-multiplication, R-matrix, antipode and Clebsch-Gordan coefficients (CGC). The q-deformation of Eq. (2.4) is roughly speaking a way to write it down in terms of elements of the Hopf algebra. A priori, it is not unique. The guiding principle here can be to identify any transformation of the construction with some isometry of a base cell complex in a self-consistent way. Then all algebraic manipulations become geometrically meaningful. It is close in spirit to the Reshetikhin-Turaev functor from the category of ribbon tangles to the modular Hopf algebras [15]. Our presentation of qQCD3 is in many respects inspired by their work. To describe it, we need to look at LGT from a bit more general than the usual point of view.
Quantum Deformation of Lattice Gauge Theory
299
2.2. Topological background. For the reader’s convenience we collect in this section some definitions which we shall use in the sequel. A k-cell is a polyhedron homeomorphic to a k-dimensional ball. A cell complex is a union of a finite number of cells such that an intersection of any 2 k-cells is either empty or a finite number of less dimensional cells. A cell complex can be obtained starting with a finite set of points by attaching subsequently cells of higher dimensions, any cell being attached to a finite number of lower dimensional cells. A union of all cells of dimension ≤ n is called an n-skeleton. A cell complex is a manifold if and only if the neighbourhood of each vertex is a spherical ball. A complex is called simplicial if all cells are simplexes (i.e., points, links, triangles, tetrahedra, etc.). Physicists usually mean by a lattice a cell complex such that an intersection of any two k-cells either empty or consists of only one entire less dimensional cell. We adopt this notion. Simplicial complexes are lattices by definition. e is constructed by putting into correspondence its k-cells to cells A dual complex, C, of C having complimentary dimensions, n − k. To introduce LGT, we need a presentation of a cell complex, i.e., an effective way to describe it unambiguously. In the classical case, one needs to know only a 2-skeleton of a complex. From the topological point of view, the construction of LGT described at the beginning of the previous section is reminiscent of the definition of H 1 (C, G), the noncommutative first cohomology of C with coefficients in G. In the topological limit, all holonomies along contractible loops vanish and gauge fields obey the defining relations of π1 (C). Therefore, being properly normalized, the partition function Z0 counts the number of conjugacy classes of injective homomorphisms from π1 (C) into a gauge group G: Z0 = |Hom(π1 (C), G)/G|.
(2.10)
Of course, it makes sense only if G is finite. If G is a Lie group, one speaks about a moduli space of flat G-connections, which is defined as a set of fields modulo gauge transformations: MG := {Hom(π1 (C), G)/G}.
(2.11)
It is easy to see that classical topological LGT is completely determined by a homotopy type of a complex. The construction of qQCD3 requires a more precise presentation of a complex. It is known that any oriented 3-manifold can be obtained by gluing up two 3dimensional handlebodies along their boundaries. This operation is the Heegaard splitting. The minimal genus of the handlebodies is called the Heegaard genus of the manifold. We can obtain a Heegaard splitting for a given oriented manifold M from its cellular decomposition, C, as follows. We take a tubular neighbourhood, H, of the 1-skeleton of e = M \H, can be regarded as a tubular neighbourhood C. The complement of H in M , H e of the 1-skeleton of the dual complex C. Every 1-cell σi1 ∈ C determines a disk Di ⊂ H whose detachment destroys a handle of H. The boundaries of the disks ∂Di ⊂ ∂H give a system of cycles on the
300
D.V. Boulatov
boundary, ∂H, of the handlebody H. We shall call them the α-cycles: αi := ∂Di . Dual f1 ∈ C e determine analogously a system of α e of 1-cells σ e-cycles on the boundary, ∂ H, j e Images of the α H. e-cycles on ∂H produced by a gluing homomorphism h are called the characteristic curves (or γ-cycles) of the Heegaard diagram and define the manifold S e unambiguously. M =H hH b ⊂ {D} such that the detachment of them Let us fix a number of the disks {D} b ∼ makes the handlebody connected and simply-connected (i.e., H\{D} = B 3 ). We can b put into correspondence a generator ai of the fundamental group to each disk Di ∈ {D}. Defining relations are read off in an obvious way from a system of the characteristic curves {γ}. That is, if γj intersects disks Dj1 , Dj2 , . . . , Djk subsequently, then the j j j corresponding relator is Γj = aj1 1 aj2 2 . . . ajkk , where k = ±1 is the intersection number depending on a mutual orientation of γj and the k th disk at the intersection point. This set of relators is of course excessive. A minimal set can be fixed by choosing a number of be α e-cycles which span a disjoint collection of disks {D} in the complementary handlebody be e = M \H such that the detachment of all the disks from the set {D} e a 3-ball: H makes H b 3 ∼ e D} e =B . H\{ One can deform a Heegaard diagram by any 2-dimensional isomorphism of a boundary ∂H which extends to the whole handlebody H. A set of generators for such isomorphisms is called in the literature the Suzuki moves (see, e.g., refs. [12, 9] for an exposition accessible to a physicist). It can be shown that any class of isotopic diffeomorphisms of a genus g surface Mg2 onto itself has a representative which can be constructed as a composition of the Dehn twists, Tµ , where µ is one of the basic cycles on Mg2 and = ±1. One detaches from Mg2 a thin neighborhood Uµ ∼ = S 1 × [0, 1] of a cycle µ and then attaches it back after the full twist Uµ → Uµ : ϕ × t → (ϕ + 2πt) × t, (where t ∈ [0, 1] and φ ∈ [0, 2π] parametrizes S 1 ). In the sequel, we shall only need the following fact: all the Suzuki moves are combinations of Dehn twists on loops in ∂H which bound disks D ⊂ H, except for the handle slide defined in the following way. Imagine solid handles attached to a surface of a spherical ball. One drags one end of a handle up, along and down another handle. As a result of this operation, an α-cycle corresponding to the second handle slides around an α-cycle corresponding to the first one. It can be described as a multiplication of loops on ∂H defined in the standard fashion as in the definition of the fundamental group π1 (Mg2 ) e as well. The same operations can be applied to H It is a classical result that any two Heegaard diagrams representing the same manifold can be connected by a sequence of the following operations: e They do not change a presentation of 1. Dehn twists on loops contractible in H or H. π1 (C). 2. Cycle slide, which consists in the multiplication of a cycle by another one: γj → γj γk . It means that a relator Γj in a presentation of π1 (C) is substituted by Γj Γk . The same operation applied to the α-cycles, αj → αj αk , corresponds to the change of generators of π1 (C): aj is substituted for aj ak . 3. Stabilization, which consists in adding a new handle to H and extending a gluing diffeomorphism by the identity on its boundary. It means that one adds one character-
Quantum Deformation of Lattice Gauge Theory
301
istic curve and one α-cycle to a Heegaard diagram or, equivalently, a new generator ag+1 along with the trivial relation, Γg+1 = ag+1 = 1, to a presentation of π1 (C). It should be noted that an isotopy within a handlebody itself cannot necessarily take place for its embeddings in R3 . The obvious obstruction is that the characteristic curves can become linked in R3 . We shall need the operation of the connected sum of two manifolds: M = M1 #M2 . One constructs M by deleting spherical balls from M1 and M2 and then gluing the manifolds together along the boundaries. Obviously, M #S 3 ∼ = M , which can be represented as the attachment of a single 3-cell to the spherical boundary of the ball obtained from M . This operation introduces an abelian semi-group structure and any 3-fold invariant can be regarded as a representation of this semi-group. A manifold is called simple, if it cannot be represented as a connected sum of two nonspherical manifolds. Any compact oriented 3-manifold possesses a unique expansion into a connected sum of simple manifolds. By performing a Heegaard splitting, a manifold is constructed out of two handlebodies joined by some homeomorphism of their boundaries. Such a homeomorphism can be continued into small neighborhoods of the boundaries. It means that the characteristic curves can submerge a bit into the inside of H. If nij = (αi , γj ) is an intersection number of two cycles on the boundary, then γj will have after the deformation the same number as a linking coefficient with αi . The linking coefficient of two loops in R3 is equal, by definition, to the intersection number of the first with a disk spanned by the second. If αi and γj are linked, the corresponding 1-cell, σi1 , enters the boundary of the corresponding 2-cell, σj2 : σi1 ∈ ∂σj2 . And vice versa for co-boundaries: σj2 ∈ δσi1 . The boundary of a 2-cell defines a natural cyclic order of 1-cells belonging to it. A peculiarity of the dimension 3 is that 2-cells forming a co-boundary of a 1-cell are naturally ordered as well. It is a cyclic order of dual 1-cells forming a boundary of a dual 2-cell. 2.3. Reshetikhin-Turaev functor. The quantized function (QF) algebra Fq (SL(n)) is dual to the QUE algebra Uq (sl(n)), therefore the topological basis of Fq (SL(n)) is given by the matrix elements of irreducible representations of Uq (sl(n)). For example, the 2 × 2 matrix realization of SLq (2) given in Eqs. (2.5) and (2.6) exactly corresponds to the matrix elements of the 2-dimensional irreducible representation of Uq (sl(2)). In this paper, we shall deal with real forms of Uq (sl(n, C)) with respect to some ∗-structures, whose existence is always assumed. The discussion in the previous section suggests that we can construct qQCD3 with help of the Reshetikhin-Turaev functor from the category of colored ribbon tangles ctang to the category of representation rings of Uq (sl(n)), repUq . The basic geometric object is a tangle, which was defined in Ref. [15] as “a link of circles and segments in the 3-ball, where it is assumed that ends of segments lie on the boundary of the ball”. One puts into correspondence to every tangle a linear operator, f , acting on a tensor product of modules associated with segments (which have therefore to be oriented). f : Vi1 ⊗ . . . ⊗ Vin → Vj1 ⊗ . . . ⊗ Vjk or, graphically,
(2.12)
302
D.V. Boulatov
j1
...jk ∼ fij11...i = n
jk
66666 , f
(2.13)
66666
i1
in
where j1 , . . . , jk and i1 , . . . , in are some indices numerating the modules. The simplest example is the identity operator represented by a single segment: α δα,β ∼ =
.
(2.14)
β All modules considered in this paper are assumed to be irreducible. We shall draw linear operators acting on them as small boxes (coupons) with labels inside. The elementary building blocks are the matrix elements α
i (a) ∼ Dα,β =
a ,
(2.15)
i, β where a is an element of Uq . The arrows show a direction of the action of an operator. We use the Greek letters to numerate basis vectors of irreducible modules and the Latin ones, to numerate the modules. They will often be omitted. Uq possesses several ∗-structures. We shall draw conjugate objects as β
i (a∗ ) ∼ Dα,β =
a
i, α
β
=
a
.
(2.16)
i, α
The last equality takes place for a real form of Uq , where the ∗-structure matches basis vectors of a module: α → α. The operators form an algebra A. We can translate this property in pictures as
Quantum Deformation of Lattice Gauge Theory
a, b ∈ A → ab ∈ A
303
∼ =
a
ab
,
(2.17)
b
∃1 ∈ A : a1 = 1a = a, ∀a ∈ A
∼ =
a
a1
1a
.
(2.18) A is a ring, i.e., an Abelian group under some + operation, which we shall understand as a formal sum of pictures with the natural definition of the multiplication by an integer number. To have a bi-algebra structure on A, we need a co-multiplication ∆ : V → V ⊗ V and a co-unit ε. We introduce ∆ as
i ∆(a) ∼ =
∆(a)
X
=
a
a
a
.
(2.19)
i
In general, the last equality is simply a convenient pictorial representation and has to be given a precise meaning in every particular case. In Eq. (2.19), the 3-valent vertices,
j3 , α 3
j1 , α 1 j α ∼ = Cj13α13;j2 α2
j1 , α 1
j2 , α 2
j2 , α 2 j1 α1 ;j2 α2 ∼ = C j3 α3
j3 , α 3
(2.20)
304
D.V. Boulatov
are the quantum Clebsch-Gordan coefficients ejα11 ⊗ ejα22 =
X
Cjj13αα13;j2 α2 ejα33 (ejα is a
j3 ,α3
basis of Vj ). They obey the properties
X
;
=
i
=
.
(2.21)
i
j1
j2
j1
j2
j
j
which simply means that they are elements of a unitary matrix connecting bases in V and V ⊗ V : X iβ j 0 α0 ;j 0 α0 Cj1 α1 ;j2 α2 C iβ1 1 2 2 = δj1 ,j10 δj2 ,j20 δα1 ,α01 δα2 ,α02 . i,β
We can check the properties of the co-multiplication graphically
a
∆(ab) = ∆(a)∆(b) ∼ =
a
(2.22)
ab b b
and
∆(1) = 1 ⊗ 1
∼ =
=
.
(2.23)
In these formulas, the sum over intermediate states is assumed. In what follows, we shall often omit the sum sign in pictures. Thus, the co-associativity is coded in the properties of the Clebsch-Gordan coefficients. The co-unit is a homomorphism to an abelian group associated with a field over which A is defined,
Quantum Deformation of Lattice Gauge Theory
305
ε(ab) = ε(a)ε(b) ε(1) = 1. (2.24) We shall connect the co-unit with a projection on the trivial representation of a quantum group. In other words, with the group integration. To have the Hopf algebra structure on A, we introduce an antipode map: S : A → A:
S
a
=
=
a
a∗
(2.25)
obeying · (S ⊗ id) ◦ ∆ = ·(id ⊗ S) ◦ ∆ = 1 ◦ ε which looks graphically as
· (S ⊗ id) ◦
=
a
(2.26)
=
a
ε(a),
(2.27)
where a is an arbitrary element of A and ◦ means a composition of operations. This property shows that the antipode can serve as a q-analog of the inverse. However, in general, S 2 6= 1. The maps V ⊗ V → C and C → V ⊗ V are constructed with help of CGC: 0 j
:= j
p dj
j
;
j
j
j
p dj
,
j
:=
j
(2.28)
0
where dj is the quantum dimension of the module Vj . These objects become the ordinary δ-functions in the q → 1 limit. The self-consistency requires that
S(ab) = S(a)S(b)
∼ =
b ab
a
a
b
. (2.29)
306
D.V. Boulatov
For this property to hold, it is important that
a a
b
b
a
.
(2.30)
b
The other property of the antipode is ∆ ◦ Sa = τ ◦ (S ⊗ S) ◦ ∆a. It can be checked graphically:
∆◦
a
=
a
= τ ◦(S ⊗S)
a
, (2.31)
a
τ
where τ is the flip operator: Vj1 ⊗ Vj2 −→ Vj2 ⊗ Vj1 . Having equipped A with an R-matrix, we obtain a quasi-triangular Hopf algebra (A, R). The R-matrix obeys the Yang-Baxter equation (2.9). In our context, it will be ˇ more convenient to consider the R-matrix Rˇ := τ ◦ R
(2.32)
which can be represented graphically as X ˇ αi ⊗ βi ∼ R=τ◦ =
(2.33)
i
It is invertible X −1 ˇ βi ⊗ S(αi ) ∼ R =τ◦ = i
=
. (2.34)
A Hopf algebra is called triangular if Rˇ 2 = 1. ˇ The standard definition of the R-matrix requires that the following properties hold:
(∆ ⊗ id)Rˇ = Rˇ 12 Rˇ 23
∼ =
(∆ ⊗ id)
=
, (2.35)
Quantum Deformation of Lattice Gauge Theory
(id ⊗ ∆)Rˇ = Rˇ 12 Rˇ 23
∼ =
307
=
(id ⊗ ∆)
(2.36)
along with the general form of Eq. (2.8)
ˇ R(∆a) Rˇ −1 = ∆a, ∀a ∈ A ∼ =
a
a
=
a
a
.
(2.37)
.
(2.38)
,
(2.39)
These equations are equivalent to the Yang-Baxter one
Rˇ 12 Rˇ 23 Rˇ 12 = Rˇ 23 Rˇ 12 Rˇ 23
∼ =
=
The standard relations including the antipode are
−1 ∼ ˇ ˇ (S ⊗ id)R = R = (S ⊗ id)
−1 ˇ ˇ (id ⊗ S)R = R ∼ = (id ⊗ S)
= =
= =
, (2.40)
308
D.V. Boulatov
and those with the co-unit are
(ε ⊗ id)Rˇ = (id ⊗ ε)Rˇ = 1
∼ =
=
j
0
= 0
j
.
(2.41)
j
The standard way to introduce the ribbon Hopf algebra structure on A is to bring forward the element u ∈ A defined as [15] u :=
X
S(βi )αi
∼ =
.
(2.42)
i
The element v 2 = uS(u) lies in the center of A. ˇ v) is a quasi-triangular Hopf algebra (A, R) ˇ A ribbon Hopf algebra Uq = (A, R, equipped with a central invertible element v ∈ A,
v :=
v −1 :=
=
=
(2.43)
obeying the properties: ε(v) = 1,
uS(u) = v 2 ∼ = u◦
=
=
=
.
(2.44)
Quantum Deformation of Lattice Gauge Theory
S(v) = v
∼ =
∆(v) = (Rˇ 2 )−1 (v ⊗ v)
309
∼ =
,
=
(2.45)
=
=
. (2.46)
The element
uv −1 ∼ =
=
(2.47)
allows for defining the q-trace of an operator
qtr(a) := tr(auv −1 ) ∼ =
a
.
(2.48)
In the tensor square of spaces it takes the form
tr[a ⊗ b ◦ ∆(uv −1 )] = qtr(a) qtr(b)
∼ =
a
b
=
a
b
.
(2.49) Following Reshetikhin and Turaev, we shall call this operation the closing of a tangle. The quantum dimension of a module, Vj , is, by definition, the q-trace of the identity operator:
310
D.V. Boulatov
dj := qtr(1V j ) ∼ =
.
(2.50)
2.4. Algebra of fields and gauge invariance. In Sect. 2.2 we have described a Heegaard diagram as a handlebody with a given system of α-cycles and characteristic curves on its boundary. Every α-cycle span a disk D in a handlebody H. The disk can beTthickened to a plate P . In this way we obtain a collection of disjoint plates in H (Pi Pj = Ø, if i 6= j). Each plate corresponds to a 1-cell of a base cell complex C from which the Heegaard diagram has been read off. By detaching the plates, H reassembles into a collection of 3-balls {B}, each corresponding to a 0-cell of C. Definition 1. We construct qQCD3 functor in the following way: 1. A gauge variable taking values in a ribbon QUE algebra Uq is put into correspondence to each plate: Pk −→ ak ∈ Uq . The variables attached to different plates are distinct elements of Uq , hence their matrix elements are co-commutative. 2. All the characteristic curves are colored with irreducible finite-dimensional representations of Uq . 3. If on a boundary of the k th plate there are nk disjoint cuts of the characteristic curves colored with representations j1 , j2 , . . . , jnk , we construct a gauge field tangle by repeatedly applying the co-multiplication: Fk = ∆nk −1 (ak ) : Vj1 ⊗ . . . ⊗ Vjnk → Vj1 ⊗ . . . ⊗ Vjnk or, graphically,
=⇒ j1
j2
jnk
ak
ak
j1
j2
···
ak
jnk
.
One has to respect the cyclic order and mutual orientations of the cuts. A reversion of an orientation of a cut corresponds to the conjugation? of the corresponding matrix element. 4. One puts into correspondence to each ball Bi ∈ {B} carrying a pattern of the characteristic curves on its boundary a vertex tangle by using the Reshetikhin-Turaev functor ctang → repUq . ?
With respect to some fixed ∗-structure.
Quantum Deformation of Lattice Gauge Theory
311
5. In the end, the pieces are attached together. To do it, one embeds the handlebody into R3 in such a way that the cuts of the characteristic curves on boundaries of the plates project to distinct points on the (x, y) plane and onto disjoint segments on the (x, z) plane. Then one can use the (x, z) projection of the vertex tangles to complete the construction in terms of elements of a ribbon Hopf algebra as was depicted in the previous section. A result is a functional taking values in C. Remarks. 1) Let us notice that the initial data are colorings and directions of the characteristic curves as well as Uq elements attached to the plates. 2) One can sum over all the colorings with arbitrary weights as in Eq. (2.4). As we are restricted to real forms of Uq (sl(n)), the result has to be independent of the directions of the characteristic curves. 3) Modules appearing in different vertex tangles are independent. However, after the assemblage, all pieces are fit together and, permuting matrix elements adjacent via a vertex tangle, one has to deform the tangle, which means some effective non-cocommutativity. We shall dwell at this point later. Now, let us simply illustrate what may happen by the example shown in Fig. 1. 4) Clearly, the construction gives different results for non-isotopic embeddings of H into R3 . This lack of self-consistency will disappear after the integration over gauge fields (see the next section).
a
b
a
b
=
a
=
b
a
b
= c
c
b
a
b
a
c
c
= c
c
Fig. 1. An illustration of the non-cocommutativity of fields adjacent via a vertex.
Let us now discuss the issue of gauge invariance within our framework. In the general settings of gauge theory the basic object is a fiber bundle over a base manifold. Gauge field performs a parallel transport of fibers and thus is interpreted as a G-connection in sections of the bundle. Gauge transformations act by automorphisms of the fibers. To
312
D.V. Boulatov
make it explicit, one has to choose some G-basis at each point of the base. A quantity is gauge invariant if it is independent of a particular choice of the bases. In LGT a base manifold is substituted by a finite cell complex. Therefore, instead of a fiber bundle, one has a tensor product of G-modules, one for each 0-cell in the complex. In our construction of qQCD3 , 0-cells are associated with the vertex tangles. One chooses bases of the Uq -modules for each tangle independently and then sandwiches the matrix elements between them. Thus, we can reformulate gauge invariance as the requirement of independence from particular choices of all the bases. As in the quantum case the notion of the group manifold is absent, one cannot translate a change of a frame into a group rotation. Although these changes can be given a matrix form, their status is quite different from the one of gauge fields. Therefore, we lose a contact with the explicit formula (2.1).
2.5. Partition function. We connect integration with the co-unit. By definition, it is a linear functional projecting onto the trivial representation. For the matrix elements, we have
R
j
da D (a) = δj,0
∼ =
ε( a ) = δj,0
.
(2.51)
j
An integral of an arbitrary product of matrix elements can be reduced to the basic one (2) by subsequently applying the tensor product decomposition with the Clebsch-Gordan coefficients. For example, the orthogonality of matrix elements reads β1
β1 β2 β1
β2
β2
( α
α ) =
j1 α1 j2 α2
P
α
(
) =
δj1,j−2 dj1
j1 α1
(2.52)
j1 α1 j2 α2 ,
where dj is the q-dimension of a module Vj . The selfconsistency of the definition can be easily checked
ε( a ε( b
a ) b )
=
1 d2j
=
1 dj
(2.53)
Quantum Deformation of Lattice Gauge Theory
313
The main property of the Haar integral is the right/left invariance: a
( abc
abc ) =
P
a
( b ) =
c
a
a
1 dj
c
=
c
1 dj
.
(2.54)
c
In the general case, the invariance easily follows from the properties of the ClebschGordan coefficients and the antipode. The integral should be used with some caution. For example, a reader has to be aware that
ε( a )
∼ 6 = ε(
a
)
(2.55)
and the r.h.s. of this formula makes R no sense. Otherwise, one could easily arrive at contradictions. Expressions like dx dy f (x, y) are inadequate in the quantum case. To exclude ambiguity, we shall always connect the integration with the linear operator acting on a tensor product of modules constructed with help of the co-multiplication: Definition 2. Z
ε ∆n := ε ∆n ( ) : (V1 ⊗ . . . ⊗ Vn+1 −→ V1 ⊗ . . . ⊗ Vn+1 ) −→ C.
It can be calculated recursively:
ε( a · · · a a ) =
X
ε( a · · · a )
.
(2.56)
To complete our construction we need to specify a real form of Uq (sl(n)) with respect to some fixed ∗-structure. We are interested in 2 cases: (i) Uq (su(n)) which makes sense for real q and (ii) Uq (sl(n, R)) for |q| = 1. The Hopf ∗-algebra Uq (su(n)) has been well investigated starting from the pioneering works of Woronowicz and Vaksman and Soibelman [21]. We need the following facts: 1. There is the one-to-one correspondence between finite dimensional irreducible representations of Uq (su(n)) and the classical algebra U (su(n)). 2. The representation ring of Uq (su(n)) spanned by matrix elements of finite-dimensional irreducible representations can be regarded as the q-deformation of the algebra of regular functions on SU (N ) (the quantum Peter-Weyl theorem). In particular, there exists a q-analog of the group δ-function.
314
D.V. Boulatov
3. There exists a q-analog of the Haar measure. The matrix elements are orthogonal with respect to it. An explicit representation of them can be given in terms of qspecial functions. In this case the group integration is performed with help of the so-called Jackson integral from the q-special function theory. Now, we are in a position to define the qQCD3 partition function. We shall denote the number of k-cells in a complex as Nk . Definition 3. We take the construction of the qQCD3 functor introduced in the previous section. Then 1. We color the characteristic curves with Uq (su(n)) irreps: γi → ji , i = 1, . . . , N2 . 2. We put into correspondence to every plate Pk (k = 1, . . . , N1 ) the integral Pk −→ ε(∆nk −1 (ak )), ak ∈ Uq (su(n)). 3. By applying Eqs. (2), (2.52) and (2.56) we obtain a collection of closed 3-valent ribbon graphs {τ }, the number of which equals the number of 0-cells in a base cell complex. By using the Reshetikhin-Turaev functor, we calculate the quantum invariant?? , J(τk ), for each connected component, τk . Let us denote their product as Zj1 ...jN2 =
N0 Y
J(τk ).
(2.57)
k=1
4. The partition function equals the sum over all colorings of the characteristic curves Zβ =
X
N2 Y
djk e−βCjk Zj1 ...jN2 ,
(2.58)
{j1 ...jN2 } k=1
where dj is the quantum dimension and Cj is a second Casimir eigenvalue. Remarks. 1) If q = 1, this definition reduces to the one given in Eqs. (2.3) and (2.4). 2) Zβ is a gauge invariant quantity in the sense described in the previous section. Indeed, as any vertex tangle after the integration gives a closed ribbon 3-valent graph, the choice of a basis attached to it is irrelevant. 3) After the integration, all non-isotopic embeddings of a handlebody H into R3 become equivalent and the consideration can be restricted to isotopies of the handlebody itself. 4) If one considers a cell complex dual to a simplicial one, the ribbon graph invariants J(τk ) in Eq. (2.57) coincide with the quantum 6-j symbols in the Racah-Wiegner normalization. 2.6. The root of unity case. For applications most interesting is the case when q equals a primitive root of unity: q ` = 1. Then Uq (sl(n)) possesses the real form Uq (sl(n, R)). This case is rather complicated technically. One has to work with the restricted specialization Uqres (sl(n)) of Uq (sl(n)) and the issue of the duality between the QF and QUE algebras becomes quite subtle. Fortunately, one can go on with the notion of the modular Hopf algebra [15]. ??
Often called the generalized Jones polynomial.
Quantum Deformation of Lattice Gauge Theory
315
ˇ v) equipped with a distinguished Definition 4. Consider a ribbon Hopf algebra (A, R, family {Vj }j∈S of irreducible A-modules indexed by a finite set S including the trivial ˇ v) is called a modular Hopf algebra if the following requirerepresentation V0 . (A, R, ments are fulfilled: 1. qdim Vj 6= 0, ∀j ∈ S. 2. The set {Vj }j∈S is equipped with an involution j → j ∗ such that Vj ∗ = Vj∗ and Vj∗∗ = Vj . 3. For any sequence j1 , . . . , jn ∈ S M Vi⊕mi ⊕ I, mi ∈ N V j 1 ⊗ Vj 2 ⊗ . . . V j n = i∈S
as A-modules and for all A-module endomorphisms, f , of the ideal I qtr(f ) = 0. 4. Let sij be the quantum invariant of the Hopf link, two components of which are colored with irreps i and j ∈ S h i , sij = qtr i j
then the matrix (sij )i,j∈S is invertible. Let us take the row of the inverse matrix s−1 corresponding to the trivial representation V0 , then X (s−1 )0j 0 sj 0 j = δ0,j . (2.59) j 0 ∈S
We can consider Eq. (2.59) as an analog of the basic integral (2) with the obvious action of the co-multiplication:
∆
=
.
(2.60)
Owing to the third condition in the definition of the modular Hopf algebra, we find the following analog of the orthogonality of matrix elements
i
X i∈S
(s−1 )0i qtr
i
i
h
= f j1 j2
X i∈S
(s−1 )0i qtr
i
h
= f j1 j2
δj1 ,j2 dj1
f
(2.61)
316
D.V. Boulatov
for any endomorphism f : V ⊗ V → V ⊗ V . In these formulas, the q-trace is necessary to project out the ideal I. An example of the modular Hopf algebra has been given by Reshetikhin and Turaev [15] in the sl2 case. In Ref. [18] the notion of the quasi-modular Hopf algebra has been introduced by slightly weakening the irreducibility condition on the modules from {Vj }j∈S . It still leads to 3-manifold invariants of the Reshetikhin-Turaev type and therefore sufficient for our purposes as well. Turaev and Wenzl have constructed examples of quasi-modular Hopf algebras associated with Uq (g), q ` = 1, for all g of the A, B and D types. Thus, we define the qQCD3 partition function at a root of unity in the same way as in the previous section, using the given above definition of the integral. One can describe the quantity Zj1 ...jN2 appearing in Eq. (2.57) as follows. We consider a Heegaard splitting S e M = H h H. Let us continue a homomorphism h into a small neighborhood of ∂H. T e e = M 2 × 0; {α} ∈ ∂H In other words, H H = Mg2 × [0, 1], ∂H = Mg2 × 1 and ∂ H g e Then for any standard embedding of H into R3 , the characteristic and {γ} ∈ ∂ H. curves and the α-cycles form a non-trivial link, L. They are colored with two sets of representations j1 , . . . , jN2 ∈ S and i1 , . . . , iN1 ∈ S. By using the Reshetikhin-Turaev i1 ...iN functor, we calculate the quantum invariant of the link, Jj1 ...jN1 (L), and sum over the 2 colors of the α-cycles with the weights (s−1 )0i : Zj1 ...jN2 =
X
N1 Y
i1 ...iN1 ∈S k=1
i1 ...iN
(s−1 )0ik Jj1 ...jN1 (L).
(2.62)
2
2.7. The topological limit. If in the root of unity case one chooses the Boltzmann weight coefficients equal to vj = (s−1 )0j , one finds the partition function
Z0 (C) =
X
N2 Y
j1 ...jN2 ∈S k=1
vjk Zj1 ...jN2 =
X
N2 Y
j1 ...jN2 ∈S k=1
v jk
X
N1 Y
i1 ...iN1 ∈S k=1
i1 ...iN
vik Jj1 ...jN1 (L), 2
(2.63) i1 ...iN
where Nk is the number of k-dimensional cells in a complex C; Jj1 ...jN1 (L) is the 2 quantum invariant of aPlink L given by a Heegaard diagram associated to the complex C. Let us denote ω = i∈S vi di . Theorem 1. I(M) = Z0 (C)/ω N0 +N3 −2 is a topological invariant of a manifold M represented by a complex C. I(M) is multiplicative with respect to the connected sum: I(M) = I(M1 )I(M2 ),
if M = M1 #M2
and I(S 3 ) = 1. Proof. The Heegaard splitting associated to a cell complex C having Nk cells in the k’th dimension gives a handlebody, Hg , of the genus g = N1 − N0 + 1. Let us fix g independent α-cycles of the Heegaard diagram and take the corresponding integrals (i.e., sums over i’s in Eq. (2.63)) first. By applying the CGC decomposition and then using the orthogonality (2.61), we deform the set of the characteristic curves in the link L into
Quantum Deformation of Lattice Gauge Theory
317
some 3-valent ribbon graph G. Every application of Eq. (2.61) destroys a handle of Hg . Therefore, having taken the g integrals, we obtain a spherical ribbon 3-valent graph G plus a collection of N1 − g = N0 − 1 disjoint unlinked loops corresponding to the rest of the α-cycles. The integrals associated to them give ω N0 −1 . Now, we can recover the intial configuration of the characteristic curves??? by restoring the g integrals corresponding to the independent α-cycles. In this way we obtain a cell decomposition of M with only one 0-cell and every 1-cell corresponding to a generator of π1 (M). This procedure is the direct analog of fixing an axial gauge in LGT. The Heegaard splitting is obviously symmetric with respect to the Poincar´e duality, therefore we can repeat the previous procedure with roles of the α-cycles and the characteristic curves interchanged. In this way we fix a set of g independent characteristic curves and pick up the factor ω N3 −1 . Thus, we finish with some balanced presentation of π1 (M). To prove the topological invariance, we have to show that I(M) is not changed [i] by Dehn twists on contractible loops, [ii] by the cycle slide and [iii] by the stabilization. i) Invariance under the Dehn twists on loops contractible inside Hg is obvious. By taking the g integrals, we cut all handles and always get the same 3-valent ribbon graph G. ii) Invariance under the cycle slide follows from the analog of Haar measure invariance as illustrated in Fig. 2.
X
vi
X
j
i
vj qtr
X
=
vi
X
j i
=
vj qtr f
f
1X d
i
X
=
vi qtr
vi
X
j i
vj qtr
f f
Fig. 2. Invariance under the cycle slide.
iii) The stabilization consists in adding a handle to Hg and extending a gluing homomorphism h by the identity on its boundary. It amounts to the addition to L of one α-cycle and one characteristic curve forming the Hopf link. Therefore, the integration associated to the new handle attaches the trivial representation to the new characteristic curve and it is unimportant how it is linked with the other α-cycles. To show the multiplicative nature of the invariant, let us choose such a cell decomposition of M = M1 #M2 that a sphere dividing M1 and M2 consists of only one 0-cell and one 2-cell. Then the corresponding characteristic curve is linked with no α-cycle and the corresponding link L in Eq. (2.63) has two connected components. ???
or another one equivalent to it.
318
D.V. Boulatov
The normalization I(S 3 ) = 1 follows from the observation that the Hopf link corresponds to a genus 1 Heegaard splitting of the sphere. t u Remarks. 1) The meaning of the choice of the Boltzmann weight coefficients made in Eq. (2.63) is clear. They correspond to the δ-function weights. Therefore, Z0 (M) can be regarded as a generalization of Eq. (2.10). In contrast to the finite group partition function, the q-deformed model is obviously self-dual with respect to the Poincar´e duality of 3folds. 2) Let us consider a simplicial complex C (s) . If we take in the expression (2.63) for the partition function Z0 (C (s) ) all the sums associated to triangles in C (s) prior to the others, then the answer is identical to the definition of the Turaev-Viro invariant. Indeed, for each triangle we find the tangle equivalent to the product of two 3j-symbols:
X
vi
=
X
vi
j1 j2 j3
=
1 dj3
.
(2.64)
j1 j2 j3
j1 j2 j3
By closing all the tangles, we obtain a Racah-Wiegner 6j-symbol
j1 j4
j2 j5
j3 j6
j1
j2
1 p = dj6 dj2 dj5
j3
(2.65)
j6 j4 j5
inside each tetrahedron of the simplicial complex. The indices, j1 , . . . , jN1 , are attached to 1-simplexes of C (s) . Taking the sums over them we arrive at the Turaev-Viro state sum invariant [17]: I0 (C (s) ) = ω N1 −N2 +2
N1 X Y {jk ∈S} k=1
v jk
N3 Y j t1 j t4 t=1
j t2 j t5
j t3 j t6
,
(2.66)
where the 6-tuple (t1 , . . . , t6 ) denotes six edges of the t-th tetrahedron. Explicit expressions for (s−1 )ij in the sl2 case are given in Ref. [15]. Thus, Eq. (2.63) can be regarded as a general definition of the Turaev-Viro invariant. 3) The expression for Z0 (M) given in Eq. (2.63) coincides with the Reshetikhin-Turaev construction of 3-fold invariants IRT via the surgery representation [15]. Therefore, Z0 (M) is automatically invariant under the Kirby calculus applied formally to the link L. It means that, given a manifold M, there exists another one N such that I(M) = IRT (N ). As I(M) = |IRT (M)|2 , we conclude that N ∼ = M#M (M is M with the opposite orientation). A simple illustration in the case of lens spaces can be found in Ref. [6].
Quantum Deformation of Lattice Gauge Theory
319
2.8. Bounded manifolds and links. Every set of disjoint simple closed curves {γ} on a handlebody H determines a bounded 3-manifold M constructed by gluing plates to annular neighborhoods of the curves. It can be shown that every orientable bounded 3-manifold can be obtained in this way. The handlebody in this construction is a tubular neighborhood of a 1-skeleton of M. Therefore we can straightforwardly apply the qQCD3 functor in the bounded case. For it, we [i] fix a system of α-cycles on ∂H; [ii] color curves from {γ} with Uq irreps; [iii] repeat steps 3, 4 and 5 from Definition 1 without any modification. It suggests the following interpretation of our construction. A spine is a 2-dimensional polyhedron which can be embedded in some 3-manifold. Any 3-manifold with a boundary collapses to a spine. Let us delete a ball from every 3-cell of a closed complex C. In such a way we obtain a bounded manifold which collapses to a 2-skeleton K2 of C. If C is dual to a simplicial complex, K2 is called a standard spine. Matveev has introduced two moves which relate all standard spines of the same manifold [13]. It can be easily shown that I(M) from the previous section is invariant under the Matveev moves. The definition of the qQCD3 functor uses an immersion of K2 into R3 and depends on it. It seems to be an intrinsic feature of q-deformed LGT rather than a defect of our presentation. Only gauge invariant singlet quantities (the partition function, for example) are independent of a way K2 is immersed into R3 . One of the advantages of our presentation of qQCD3 is a relative simplicity of introducing Wilson loops in it. In classical LGT, a loop average is defined as
A(L1 , . . . , Lm ) =
1 Zβ
Z Y G `
dg`
Y f
Wβ (h∂f )
m Y i=1
tr Vj [hLi ], i
(2.67)
where {L} are m closed curves embedded into a 1-skeleton K 1 of a complex C. We color the ith curve with a representation ji of a gauge group G. The holonomy hL is defined in Eq. (2.2). In the q-deformed case, we have to specify a link formed by the collection of curves in a manifold M ∼ = C. For it we represent the curves {L} by a set of disjoint ribbon loops on a boundary ∂H 0 of a handlebody H 0T⊂ H (as usual,SH 0 and H are tubular neigborhoods of K 1 and H 0 lies inside H: H H 0 = H 0 , H H 0 = H). If it is not possible, then one has to take a finer subdivision of M. In the case of links in R3 , it is a standard technical trick to realize a link as a system of disjoint loops on a handlebody embedded into R3 . And we simply use it as a definition. We can apply the qQCD3 functor to such a composite handlebody without any additional modification. Loops from {L} enters on equal footing with characteristic curves. One can repeat the same argument as in the partition function case to prove that an answer is independent of an embedding of H in R3 . However, it does not mean that the q-deformation of Eq. (2.67) gives no non-trivial knot invariant. Let us consider a link in R3 . There has to exist a trivial embedding such that characteristic curves of a Heegaard diagram lying on ∂H are unlinked and contractible in R3 \ H. Therefore the sums over their colors disjoin α-cycles on ∂H and the link of curves {L} on ∂H 0 (in other words, cut handles of the complementary handlebody S 3 \ H). What remains is exactly the Jones polynomial associated to the link {L}. The comprehensive treatment of quantum invariants of links and 3-valent graphs in 3-manifolds can be found in Ref. [16].
320
D.V. Boulatov
3. qQCD2 We define a qQCD2 functor by applying the qQCD3 one to an embedding of an oriented 2-manifold Mg2 in R3 . In the topological limit, we can consider the simplest cell decomposition of Mg2 consisting of a single 2-cell and 2g 1-cells. A tubular neigborhood of its 1-skeleton is a handlebody H of the genus 2g. The Heegaard diagram has only one characteristic curve. Each integral destroys a handle of H and contributes a factor 1/dj to an answer. The calculation is reduced to a repeated application of Eq. (2.61) and one easily gets I(Mg2 ) = ω 2g−1 Z0 (Mg2 ) = ω 2g−1
X
vj Zj =
j∈S
X
vj
j∈S
dj ω
1−2g .
(3.1) 2πi
Let us consider a concrete example of the quantum group Uq (sl(2)), at q = e k+2 . In this case, the set of modules in the definition of the modular Hopf√algebra is given by sin( 2j+1 π)
the fusion ring Vj (j = 0, 21 , 1, 23 , . . . , k2 ) and dj = sink+2π ; ω = k+2 We find k/2 2 sin2 2j+1 π 1−g X k+2 . I(Mg2 ) = k + 2 1
(k+2)/2 ; π sin k+2
vj = dj /ω.
(3.2)
j=0, 2 ,...
These are known as Verlinde’s numbers. They are all integer and equal to the dimensions of spaces of conformal blocks in the WZW model on a genus g Riemann surface. If one starts with a more complicated cell decomposition of a Riemann surface, then one has simply to apply the orthogonality relation (2.61) till all handles of H are destroyed. In two dimensions, local properties of the qQCD2 functor can be formalized in a pure algebraic way. For it, let us cut from Mg2 a piece which can be projected on a plane R2 . It gives a subdivision (triangulation, say) of some region on the plane. There is a natural cyclic order of edges incident to a vertex. Following Fock and Rosly [11], one introduces a ciliation at every vertex, i.e., breaks this order. Let us say that an edge `1 goes after `2 (`1 > `2 ), if an anti-clockwise angle ϕ(`1 ) between the edge `1 and the x-axis is bigger than an angle ϕ(`2 ) between `2 and the x-axis: ϕ(`1 ) > ϕ(`2 ). We assume that no edge is parallel to the x-axis, and orient edges in the y-direction. Say, put an arrow at an end having a bigger y coordinate. Assuming that any two vertices are connected at most by i one edge, we can numerate edges by ordered pairs of vertices (i, j) ∼ = j% . Alekseev, Grosse and Schomerus have introduced the following algebra of gauge fields U(i,j) [3]: i) If two edges have no common vertices, fields are co-commutative: 1
2
2
1
U (i,j) U (n,m) =U (n,m) U (i,j) ; here i, j, n and m are all distinct. ii) If two edges share a vertex, then 1 2 1 2 U (k,j) U (i,j) U (i,j) U (k,j) = 1 2 U (k,j) U (i,j)
Rˇ 12 Rˇ −1 12
if (i, j) > (k, j) if (i, j) < (k, j)
.
Quantum Deformation of Lattice Gauge Theory
321
We picture these relations as i
i
k
a
b
=
k
if (i, j) > (k, j)
j
j i
k
a
a
b
b
i
k
=
a
b
j
if (i, j) < (k, j) .
j
We have drawn in solid lines Uq -elements figuring in the AGS relations associated to the j th vertex. All attached to other vertices are dashed. iii) Fields attached to the same edge form a quasi-triangular Hopf algebra: 2
1
2
1
−1 1. R12 U (i,j) U (i,j) R12 =U (i,j) U (i,j) . 1
2
2. · U (j,i) U (i,j) = 1. This property is sometimes called the cancellation of a backtracking (c.f. Eq. (2.27)). Remarks. 1) The properties [i] and [iii] are obviously in agreement with our definition of qQCD2 (see, e.g., , the pictorial illustration in Fig. 1, and the discussion preceding it). The relations [ii] follow from transformations of modules associated to vertex tangles. Of course, being made, such a move has to be compensated somewhere by its reciprocal for a whole construction to remain invariant. 2) As a closed surface can be projected onto R2 only locally, one has to use gluing homomorphisms to assemble a Riemann surface out of flat pieces. These homomorphisms match the gauge field algebra relations on different pieces and have to be added in order to complete the construction. 3) The set of the AGS relations is distinguished by an observation that they generate a lattice Kac-Moody algebra in the sense of Ref. [2]. However, they do not constitute all possible symmetries of the qQCD2 functor. 4. Concluding Remarks 1. The first natural question to ask is whether the results of this paper could be generalized to higher dimensions. The answer is certainly “No”! The reason for it is that, in dimensions bigger than 3, there is no natural ordering of faces incident to an edge in a complex. It restricts the class of acceptable Hopf algebras to triangular ones. Then the corresponding construction essentially coincides with the classical Wilsonian LGT.
322
D.V. Boulatov
2. It is tempting to interpret the topological invariant considered in this paper as some suitable generalization of Eq. (2.10) and the construction of topological qQCD3 as a generalization of H 1 (C, G). Unfortunately, we are able to say nothing constructive about it. However, in the 2-dimensional case, a notion of a quantum moduli space could presumably be formulated [11], which leaves some hope for the future. 3. We conjecture that qQCD3 with Uq (su(n)) gauge group possesses a continuum limit equivalent to a gauge theory whose action includes both Yang-Mills and ChernSimons terms. One could introduce a non-zero coupling constant in the root-of-unity case as well, which implies some deformation of Chern-Simons theory. The meaning of this procedure is absolutely unclear to us. Acknowledgement. I thank V.Turaev for the fruitful discussion. This work was supported by the EEC program “Human Capital and Mobility” under the contract ERBCHBICT9941621.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
Alekseev, A.: St.-Peterburg Math. J. 6, 1 (1994) Alekseev, A., Faddeev, L., Semenov-Tian-Shansky, M.: Commun. Math. Phys. 149, 335 (1992) Alekseev, A., Grosse, H. ,Schomerus, V.: Commun. Math. Phys. 172, 317 (1995) Buffenoir, E., Roche, P.: Commun. Math. Phys. 170, 669 (1995) Boulatov, D.V.: Mod. Phys. Lett. A7, 1629 (1992) Boulatov, D.V.: Int. J. Mod. Phys. A8, 3139 (1993) Boulatov, D.V.: 3D Gravity and Gauge Theories. In NATO Advanced Studies Institutes Series B, 328, 39 (1995) Boulatov, D.V.: Mod. Phys. Lett. A8, 3491 (1993) Crane, L.: Commun. Math. Phys. 135, 615 (1991) Drinfeld, V.G.: Quantum Groups. In Proc. ICM (1987) 798; Faddeev, L.D., Reshetikhin, N., Takhtajan, L.: Leningrad Math. J. 1, 193 (1990) Fock, V.V., Rosly, A.A.: Poisson structure on moduli of flat connections on Riemann surfaces and r-matrix. Preprint ITEP 72–92 (1992); Teor. mat. Fiz. 95, 228 (1993) Kohno, T.: Topology 31, 203 (1992) Matveev, S.V.: Math. USSR Izvestiya 31, 423 (1988) Ponzano, G., Regge, T.: In: F.Bloch (ed.) Spectroscopic and group theoretical methods in physics, 1968 Reshetikhin, N.Yu., Turaev, V.G.: Commun. Math. Phys. 124, 307 (1989); Invent. Math. 103, 547 (1991) Turaev, V.: Publ. Math. IHES 77, 121 (1993) Turaev, V.G., Viro, O.Y.: Topology 31, 865 (1992); Turaev, V.G.: C.R. Acad. Sci. Paris 313, 395 (1991); J. Diff. Geom. 36, 35 (1992) Turaev, V., Wenzl, H.: Int. J. Math. 4, 323 (1993) Wilson, K.: Phys. Rev. D10, 2445 (1974) Witten, E.: Commun. Math. Phys. 121, 351 (1989); Nucl. Phys. B311, 46 (1988/89) Woronowicz, S.L.: Commun. Math. Phys. 111, 613 (1987); Vaksman, L.L., Soibelman, Ya.S.: Func. Anal. Appl. 22, 170 (1988)
Communicated by A. Connes
Commun. Math. Phys. 186, 323 – 379 (1997)
Communications in
Mathematical Physics c Springer-Verlag 1997
Distribution of Overlap Profiles in the One-Dimensional Kac–Hopfield Model? Anton Bovier1 , V´eronique Gayrard2 , Pierre Picco2 1
Weierstraß-Institut f¨ur Angewandte Analysis und Stochastik, Mohrenstraße 39, D-10117 Berlin, Germany. E-mail: [email protected] 2 Centre de Physique Th´ eorique – CNRS, Luminy, Case 907, F-13288 Marseille Cedex 9, France. E-mail: [email protected]; [email protected] Received: 14 February 1996 / Accepted: 30 September 1996
Abstract: We study a one-dimensional version of the Hopfield model with long, but finite range interactions below the critical temperature. In the thermodynamic limit we obtain large deviation estimates for the distribution of the “local” overlaps, the range of the interaction, γ −1 , being the large parameter. We show in particular that the local overlaps in a typical Gibbs configuration are constant and equal to one of the meanfield equilibrium values on a scale o(γ −2 ). We also give estimates on the size of typical “jumps”, i.e. the regions where transitions from one equilibrium value to another take place. Contrary to the situation in the ferromagnetic Kac-model, the structure of the profiles is found to be governed by the quenched disorder rather than by entropy. 1. Introduction Models of statistical mechanics where particles (or spins) interact through potentials Jγ (r) ≡ γ d J(γr), r ∈ Rd , with J some function that either has bounded support or is rapidly decreasing were introduced by Kac et al. [KUH] in 1963 as links between shortrange, microscopic models and mean field theories such as the van der Waals theory of the liquid-gas transition. The main success of these models can be seen in that they explain, through the Lebowitz-Penrose theorem, the origin of the Maxwell rule that has to be invoked in an ad hoc way to overcome the problem of the non-convexity of the thermodynamic functions arising in mean-field theories. Recently, there has been renewed interest in this model in the context of attempting to obtain a precise description of equilibrium configurations [COP] and their temporal evolution [DOPT] in magnetic systems at low temperatures. In [COP] large deviation techniques were used to describe precisely the profiles of local magnetization in a one dimensional Ising model with Kac potential in infinite volume in the limit γ ↓ 0. It ? Work partially supported by the Commission of the European Union under contract No. CHRX-CT930411
324
A. Bovier, V. Gayrard, P. Picco
turned out that this apparently simple system exhibits a surprisingly rich structure when considered at appropriate scales and it appears that the Kac-type models can still offer an interesting test ground for the study of low-temperature phenomena. The purpose of the present paper is to extend such an analysis to a class of models with random interactions. Spin systems where spins at sites i and j interact through a random coupling Jij whose mean value is zero (or close to zero) are commonly termed spin glasses. The prototype models are the Sherrington-Kirkpatrick model (SK-model) [SK] where the lattice is the completely connected graph on N vertices and the couplings Jij are i.i.d. centered gaussian variables with variance N −1/2 , and the Edwards-Anderson model [EA], defined on the lattice Zd and with Jij i.i.d. centered random variables with variance 1 if i and j are nearest neighbors in the lattice, whereas Jij ≡ 0 otherwise. These systems are notoriously difficult to analyse and little is known on a firm basis about their low temperature properties. The situation is somewhat better in the case of the mean-field SK-model, for which there is at least a rather elaborate picture based on the so-called replica-method (for a review see [MPV]) which is quite commonly accepted, although almost no results exist that are mathematically rigorous. Exceptions concern the high-temperature phase [ALR, FZ, CN, T1] and some self-averaging properties of the thermodynamic quantities [PS, BGP3]. For short-range models (the Edwards-Anderson model [EA]) the situation is much worse, and there exist conflicting theories on such fundamental questions as the upper and lower critical dimension and the number of low temperature phases, all of which are more or less supported by heuristic arguments (see e.g. [FH, BF, vE, NS]), and the interpretation of numerical simulations on finite systems (for a recent analysis and a critical assessment of the situation see [MPR]). The difficulties with the SK-model have soon prompted the proposal of simplified models for spin-glasses in which the statistics of the random couplings was changed while some of the features are conserved. The Mattis-model [Ma] where Jij ≡ i j with i independent symmetric Bernoulli variables was realized to be trivially equivalent to a ferromagnet and lacking the essential feature of frustration; Luttinger [Lu] amended this by setting Jij ≡ ξi1 ξj1 + ξi2 ξj2 while Figotin and Pastur [FP1, FP2] proposed and analysed a generalization of this interaction with an arbitrary fixed number of summands and more general distribution of the random variables ξiµ . While these models could be solved exactly, they lacked essential features expected for real spin glasses and thus did not become very popular until they were again proposed in a quite different context by Hopfield [Ho] as models for autoassociative memory. Hopfield also considered the number of summands, M , to be a function of the size, N , of the graph (“network”) and observed numerically a drastic change of behaviour of the system as the ratio α ≡ M/N exceeded a certain threshold. This was confirmed by Amit et al. [AGS] through a theoretical analysis using the replica trick. Indeed, the Hopfield model can be seen as a family of models depending on the different growth rate of M (N ) that mediates between simple ferromagnets and the SK spin-glass. The Hopfield model offers the advantage to be more amenable to a mathematically rigorous analysis than the SK-model, at least as long as M (N ) does not grow too fast with N . By now we have a fairly complete understanding of the structure of the low temperature Gibbs states [BGP1, BGP3, BG4] in the case where limN ↑∞ M/N ≤ α0 , for α0 sufficiently small. It is thus interesting to take advantage of this situation in order to get some insight into the relation between finite dimensional spin-glasses and the corresponding mean field models by studying the finite dimensional version of the Hopfield model with a Kac-type interaction. It should be noted that such a model had
Distribution of Overlap Profiles in 1-D Kac–Hopfield Model
325
already been considered by Figotin and Pastur [FP3] in 1982 in the case of bounded M . In a recent paper [BGP2] we have proven the analogue of the classical Lebowitz-Penrose theorem for this model, i.e. we have proven the convergence of the thermodynamic functions to the convex hulls of those of the mean-field model as γ ↓ 0 under the condition that limγ↓0 M (γ)/γ = 0. In the present paper we turn to the more detailed analysis of the Gibbs states of the Kac-Hopfield model and consider, as a first step, the one dimensional case along the lines of [COP]. Let us start by defining our model in a precise way and by fixing our notations. Let (, F , P) be an abstract probability space. Let ξ ≡ {ξiµ }i∈Z,µ∈N be a two-parameter family of independent, identically distributed random variables on this space such that P(ξiµ = 1) = P(ξiµ = −1) = 21 . (The precise form of the distribution of ξiµ is not really essential and far more general distributions can be considered.) We denote by σ a function σ : Z → {−1, 1} and call σi , i ∈ Z the spin at site i. We denote by S the space of all such functions, equipped with the product topology of the discrete topology in {−1, 1}. We choose the function Jγ (i − j) ≡ γJ γ|i − j| , and 1, if |x| ≤ 1/2 J(x) = . (1.1) 0, otherwise (Note that other choices for the function J(x) are possible. They must satisfy the conR ditions J(x) ≥ 0, dxJ(x) = 1, and must decay rapidly to zero on a scale of order unity. For example, the original choice of Kac was J(x) = e−|x| . For us, the choice of the characteristic function is particularly convenient.) The interaction between two spins at sites i and j will be chosen for given ω ∈ , as M (γ) 1 X µ ξi [ω]ξjµ [ω]Jγ (i − j)σi σj . (1.2) − 2 µ=1
and the formal Hamiltonian will be Hγ [ω](σ) = −
1 2
X
M (γ) X
ξiµ [ω]ξjµ [ω]Jγ (i − j)σi σj .
(1.3)
(i,j)∈Z×Z µ=1
As usual, to make mathematically meaningful statements, we have to consider restrictions of this quantity to finite volumes. We will do this in a particular way which requires some prior discussion. Note that the parameter γ introduces a natural length scale γ −1 into our model which is the distance over which spins interact directly. We will be interested later in the behaviour of the system on that and larger scales and will refer to it as the macroscopic scale, whereas the sites i of the underlying lattice Z are referred to as the microscopic scale. In the course of our analysis we will have to introduce two more intermediate, mesoscopic scales, as shall be explained later. We find it convenient to measure distances and to define finite volumes in the macroscopic rather than the microscopic scale, as this allows to deal with volumes that actually do not change with γ. Although this will require some slightly unconventional looking definitions, we are convinced the reader will come to appreciate the advantages of our conventions later on. Let thus 3 = [λ− , λ+ ] ⊂ R be an interval on the real line. Thus for points i ∈ Z referring to sites on the microscopic scale we will write i∈3
iff λ− ≤ γi ≤ λ+ .
(1.4)
326
A. Bovier, V. Gayrard, P. Picco
Note that we will stick very strictly to the convention that the letters i, j, k always refer to microscopic sites. The Hamiltonian corresponding to a volume 3 (with free boundary conditions) can then be written as Hγ,3 [ω](σ) = −
1 2
X
M (γ) X
ξiµ [ω]ξjµ [ω]Jγ (i − j)σi σj .
(1.5)
(i,j)∈3×3 µ=1
We shall also write in the same spirit S3 ≡ ×i∈3 {−1, 1} and denote its elements by σ3 . The interaction between the spins in 3 and those outside 3 will be written as Wγ,3 [ω](σ3 , σ3c ) = −
(γ) XXM X i∈3
j∈3c
ξiµ [ω]ξjµ [ω]Jγ (i − j)σi σj .
(1.6)
µ=1
The finite volume Gibbs measure for such a volume 3 with fixed external configuration σ3c (the ‘local specification’) is then defined by assigning to each σ3 ∈ S3 the mass σ3c Gβ,γ, [ω](σ3 ) ≡ 3
1
e σ3c Zβ,γ, [ω] 3
−β [Hγ,3 [ω](σ3 )+Wγ,3 [ω](σ3 ,σ3c )]
(1.7)
σ3c [ω] is a normalizing factor usually called partition function. We will also where Zβ,γ, 3 denote by 1 e−βHγ,3 [ω](σ3 ) (1.8) Gβ,γ,3 [ω](σ3 ) ≡ Zβ,γ,3 [ω]
the Gibbs measure with free boundary conditions. It is crucial to keep in mind that we are always interested in taking the infinite volume limit 3 ↑ R first for fixed γ and to study the asymptotic of the result as γ ↓ 0 (this is sometimes referred to as the ‘Lebowitz-Penrose limit’). In [BGP2] we have studied the distribution of the global ‘overlaps’ mµ3 (σ) ≡ µ γ P i∈3 ξi σi under the Gibbs measure (1.7). Here we are going into more detail in that |3| we want to analyse the distribution of local overlaps. To do this we will actually have to introduce two intermediate mesoscopic length scales, 1 `(γ) L(γ) γ −1 . Note that both `(γ) and L(γ) will tend to infinity as γ ↓ 0 while `(γ)/L(γ) as well as γL(γ) tend to zero. We will assume that `, L and γ −1 are integer multiples of each other. Further conditions on these scales will be imposed later. To simplify notations, the dependence on γ of ` and L will not be made explicit in the sequel. We now divide the real line into boxes of length γ` and γL, respectively, with the first box, called 0, being centered at the origin. The boxes of length γ` will be called x, y, or z, and labelled by the integers. That is, the box x is the interval of length γ` centered at the point γ`x. No confusion should arise from the fact that we use the symbol x as denoting both the box and its label, since again x, y, z are used exclusively for this type of boxes. In the same way, the letters r, s, t are reserved for the boxes of length γL, centered at the points γLZ, and finally we reserve u, v, w for boxes of length one centered at the integers. With these conventions, it makes sense to write e.g. i ∈ x shorthand for `x − `/2 ≤ i ≤ `x + `/2, etc.1 In this spirit we define the M (γ) dimensional vector m` (x, σ) and mL (r, σ) whose µth components are 1 On a technical level we will in fact have to use even more auxiliary intermediate scales, but as in [COP] we will try to keep this under the carpet as far as possible.
Distribution of Overlap Profiles in 1-D Kac–Hopfield Model
327
mµ` (x, σ) ≡
1X µ ξ σi ` i∈x i
(1.9)
mµL (r, σ) ≡
1X µ ξ σi L i∈r i
(1.10)
and
respectively. Note that we have, for instance, that mµL (r, σ) =
` X µ m (x, σ). L x∈r `
(1.11)
We will also have to be able to indicate the box on some larger scale containing a specified box on the smaller scale. Here we write simply, e.g., r(x) for the unique box of length L that contains the box x of length `. Expressions like x(i), u(y) or s(k) have corresponding meanings. Remark . It easy to connect from our notation to the continuum notation used in [COP]. For instance, (1.9) can be rewritten as m` (x, u) =
1 X µ γ ξ σi , γ` i∈x i
(1.12)
P where γ i∈x can be interpreted as a Riemann sum; the same occurs in all other expressions. The rˆole of the different scales will be the following. We will be interested in the typical profiles of the overlaps on the scale L, i.e. the typical mL (r, σ) as a function of r; we will control these functions within volumes on the macroscopic scale γ −1 . The smaller mesoscopic scale ` enters only in an auxiliary way. Namely, we will use a block-spin approximation of the Hamiltonian with blocks of that size. We will see that it is quite crucial to use a much smaller scale for that approximation than the scale on which we want to control the local overlaps. This was noted already in [COP]. We want to study the probability distribution induced by the Gibbs measure on the functions mL (r) through the map defined by (1.10). The corresponding measure space is for fixed γ simply the discrete space {−1, −1 + 2/L, . . . , 1 − 2/L, 1}M (γ)×Z , which should be equipped with the product topology. Since this topology is quite non-uniform with respect to γ (note that both L and M tend to infinity as γ ↓ 0), this is, however, not well adapted to take the limit γ ↓ 0. Thus we replace the discrete topology on {−1, −1 + 2/L, . . . , 1 − 2/L, 1}M (γ) by the Euclidean `2 -topology (which remains meaningful in the limit) and the product topology P corresponding to Z is replaced by thenweak local L2 topology w.r.t. the measure γL r∈· ; that is to say, P a family of profiles mL (r) converges to the profile mL (r), iff for all finite R ∈ R, γL r∈[−R,R] kmnL (r) − mL (r)k2 ↓ 0 as n ↑ ∞. While for all finite γ this topology is completely equivalent to the product topology of the discrete topology, the point here is that it is meaningful to ask for uniform convergence with respect to the parameter γ. We will denote this space by Tγ , or simply T and call it the space of profiles (on scale L). Before presenting our results, it may be useful to discuss in a somewhat informal way the heuristic expectations based on the the work of [COP] and the results known from [BGP1, BGP3, BG4]. In [COP] it was shown that the typical magnetization profiles are such that almost everywhere, mL (r, σ) is very close to one of the two equilibrium
328
A. Bovier, V. Gayrard, P. Picco
values of the mean field model, ±a(β); moreover, the profile is essentially constant −1 over macroscopic distances of the order eγ . The distances between jumps are actually independent exponentially distributed random variables. Heuristically, this picture is not too difficult to understand. First, one approximates the Hamiltonian by a blockspin version by replacing the interaction potential by a function that is constant over blocks of length L. Ignoring the error term, the resulting model depends on σ only through the variables mL (r, σ). In fact, at each block r there is a little mean-field model and these mean field models interact through a ferromagnetic interaction of the form JγL (r − s)(mL (r) − mL (s))2 . This interaction can only bias a given block to choose between the two possible equilibrium values, but never prevent it from taking on an equilibrium value over a longer interval. Moreover, it tends to align the blocks. To jump from one equilibrium into the other costs in fact an energy of the order of γ −1 , so that the −1 probability that this happens in a given unit interval is of the order e−γ . This explains why the entropy can force this to happen only on distances of the order of the inverse of this value. Finally, the Markovian character of a one-dimensional model leaves only a Poisson-distribution as a candidate for the distribution of the jumps. The main difficulty in turning these arguments into rigorous proofs lies in the control of the error terms. It is crucial for the above picture that there is a complete symmetry between the two equilibrium states of the mean field model. As we have shown in [BGP2], the KacHopfield model can be approximated by a blocked model just the same, and in [BGP1] we have shown that the mean field Hopfield model has its equilibrium states sharply concentrated at the 2M points m(µ,s) ≡ sa(β)eµ ,
µ ∈ {1, . . . , M }, s ∈ {−1, 1},
(1.13)
where eµ is the µth standard unit vector and a(β) is the largest solution of a = tanh(βa). Thus we can again expect the overlap profiles to be over long distances constant close to one of these values. What is different here, however, is that due to the disorder the different equilibrium positions are not entirely equivalent. We have shown in [BGP3] that the fluctuations are only of the order of the square root of the volume, but since they are independent from block to block, they can add up over a long distance and effectively enforce jumps to different equilibrium positions at distances that are much shorter than those between entropic jumps. In fact, within the blocked approximation, it is not hard to estimate that the typical distance over which the profiles remain constant should be of the order γ −1 on the macroscopic scale (i.e. γ −2 on the microscopic scale). Using a concentration of measure estimates in a form developed by M. Talagrand [T2], we extend these estimates to the full model. Our main results on the typical profiles can then be summarized (in a slightly informal way) as follows: Assume that limγ↓0 γM (γ) = 0. Then there is a scale L γ −1 such that with P-probability tending to one (as γ ↓ 0) the following holds: (i) In any given macroscopic finite volume in any configuration that is “typical” with respect to the infinite volume Gibbs measure, for “most” blocks r, mL (r, σ) is very close to one of the values ±a(β)eµ (we will say that mL (u, σ) is “close to equilibrium”). (ii) In any macroscopic volume 1 that is small compared to γ −1 , in a typical configuration, there is at most one connected subset J (called a “jump”) 1 on which mL is not close to equilibrium. Moreover, if such with |J| ∼ γL a jump occurs, then there exist (s1 , µ1 ) and (s2 , µ2 ), such that for all u ∈ 1 to the left of J, mL (u, σ) ∼ s1 a(β)eµ1 and for all u ∈ 1 to the right of J, mL (u, σ) ∼ s2 a(β)eµ2 .
Distribution of Overlap Profiles in 1-D Kac–Hopfield Model
329
The precise statement of these facts will require more notation and is thus postponed to Sect. 6 where it will be stated as Theorem 6.15. That section contains also the large deviation estimates that are behind these results. We should mention that we have no result that would prove the existence of a “jump” in a sufficiently large region. We discuss this problem in Sect. 7 in some more detail. We also remark that the condition limγ↓0 γM (γ) = 0 will be imposed thoughout the paper. It could be replaced with lim supγ↓0 γM (γ) ≤ αc (β) for some strictly positive αc (β) for all β > 1. However, an actual estimate of this constant would be outrageously tedious and does not really appear, in our view, to be worth the trouble. The remainder of the paper is organized in the following way. The next two sections provide some technical tools that will be needed throughout. Sect. 2 introduces the mesoscopic approximation of the Hamilitonian and corresponding error estimates. Sect. 3 contains large deviation estimates for the standard Hopfield model that are needed to analyse the mesoscopic approximation introduced before. Here we make use of some fundamental results from [BGP2] and [BG3] but present them in a somewhat different form. In Sect. 4 we begin the actual analysis of typical profiles. Here we show that for events that are local, we can express their probabilities in terms of a finite volume measure with random boundary conditions (see Corollary 4.2). In Sect. 5 we derive estimates on the random fluctuations of the free energies corresponding to these measures. In Sect. 6 we make use of these estimates to show that local events can be analysed using the mesoscopic approximation introduced in Sect. 2. This section is divided into three parts. Sect. 6.1 contains an analysis of measures with free boundary condition in macroscopic volumes of order o γ −1 . It is shown that they are asymptotically concentrated on constant profiles (see Theorem 6.1). This result is already quite instructive, and technically rather easy. In Sections 6.2 and 6.3 the measures with non-zero boundary conditions are studied. In Sect. 6.2 the case where the boundary conditions are the same on both sides of the box is studied. It is shown that here, too, the profiles are typically constant and take the value favored by the boundary conditions (see Theorem 6.9). In Sect. 6.3 the case with different boundary conditions is treated. Here we show that the typical profile has exactly one “jump” and is constant otherwise (see Theorem 6.14). The results of Sections 4 and 6 are then combined to yield Theorem 6.15 which gives a precise statement the result announced above. In Sect. 7 we discuss some of the open points of our analysis. In particular we argue, that typical profiles are non-constant on a sufficiently large scale and that their precise form is entirely disorder determined (up to the global sign). We also formulate some conjectures for the model in dimensions greater than one. In Appendix A we give a proof of a technical estimate on the minimal energy associated to profiles that contain “jumps” between different equilibrium positions that is needed in Sect. 6.
2. Block-Spin Approximations While mean-field models are characterized by the fact that the Hamiltonian is a function of global averages of the spin variables, in Kac-models the Hamiltonian is “close”, but not identical to a function of “local” averages. In this section we make this statement precise by introducing the block version of the Hamiltonian and deriving the necessary estimates on the error terms. We define X 1 ` Jγ` (x − y)(m(x), m(y)), (2.1) Eγ, 3 (m) ≡ − γ` 2 (x,y)∈3×3
330
A. Bovier, V. Gayrard, P. Picco `,L Eγ, ˜ ≡ −γ`L 3 (m, m)
XX
Jγ (`x − Lr)(m(x), m(r)), ˜
(2.2)
x∈3 r∈3c
and
` −1 ` 1Hγ, Eγ,3 (m` (σ)), 3 (σ3 ) ≡ Hγ,3 (σ3 ) − γ
(2.3)
`,L −1 `,L Eγ,3 (m` (σ), mL (σ)). 1Wγ, 3 (σ3 , σ3c ) ≡ Wγ,3 (σ3 , σ3c ) − γ
(2.4) −
For our purposes, we only need to consider volumes 3 of the form 3 = [λ , λ ] with |3| > 1. For such volumes we set ∂3 ≡ ∂ − 3∪∂ + 3, ∂ − 3 ≡ [λ− − 21 , λ− ), and ∂ + 3 ≡ `,L (λ+ , λ+ + 21 ]. Thus, obviously, Wγ,3 (σ3 , σ3c ) = Wγ,3 (σ3 , σ∂ 3 ) and 1Wγ, 3 (σ3 , σ3c ) = `,L 1Wγ,3 (σ3 , σ∂ 3 ). +
Lemma 2.1. For all δ > 0, i) "
# √ √ |3| γ |1H3 (σ)| ≥ γ`(γ)8 2(log 2 + δ) + 2 2γM (γ) ≤ 16e−δ γ , P sup |3| σ∈S3 (2.5) ii) h `,L P supσ∈S3∪∂ 3 γ|1Wγ, 3 (σ3 , σ∂ 3 )| > (4γL(γ)(log 2 + δ) + γM (γ)) 1 +
` L
21 i
δ
≤ 8e− γ . (2.6) Proof. We will give the proof of (ii) only; the proof of (i) is similar and can be found in [BGP2]. Since |3| > 1, the spins inside ∂ − 3 do not interact with those inside ∂ + 3 `,L and 1Wγ, 3 (σ3 , σ∂ 3 ) can be written as `,L `,L `,L 1Wγ, 3 (σ3 , σ∂ 3 ) = 1Wγ,3 (σ3 , σ∂ − 3 ) + 1Wγ,3 (σ3 , σ∂ + 3 ),
where `,L 1Wγ, 3 (σ3 , σ∂ ± 3 ) = −
X X XX
(2.7)
[Jγ (i−j)−Jγ (`x−Lr)](ξi , ξj )σi σj . (2.8)
x∈3 r∈∂ ± 3 i∈x j∈r `,L Both terms in (2.7) being treated similarly, we will only consider 1Wγ, 3 (σ3 , σ∂ + 3 ). First notice that since
Jγ (i− j) − Jγ (`x − Lr) (2.9) = γ 1I{|i−j|≤(2γ)−1 } 1I{|`x−Lr|>(2γ)−1 } − 1I{|i−j|>(2γ)−1 } 1I{|`x−Lr|≤(2γ)−1 } h i `,L `,L `,L 1 2 we can write 1Wγ, 3 (σ3 , σ∂ + 3 ) = γ 1 Wγ,3 (σ3 , σ∂ + 3 ) − 1 Wγ,3 (σ3 , σ∂ + 3 ) with `,L 11 Wγ, 3 (σ3 , σ∂ + 3 ) = −
X X XX
1I{|i−j|≤(2γ)−1 } 1I{|`x−Lr|>(2γ)−1 } (ξi , ξj )σi σj
x∈3 r∈∂ + 3 i∈x j∈r
(2.10) and
Distribution of Overlap Profiles in 1-D Kac–Hopfield Model `,L 12 Wγ, 3 (σ3 , σ∂ + 3 ) = −
X X XX
331
1I{|i−j|>(2γ)−1 } 1I{|`x−Lr|≤(2γ)−1 } (ξi , ξj )σi σj .
x∈3 r∈∂ + 3 i∈x j∈r
(2.11) `,L `,L 2 + + (σ , σ ) and 1 W (σ , σ ) can be treated in the same Again, both terms 11 Wγ, 3 ∂ 3 ∂ 3 3 γ,3 3 way so that we only present an estimate of the former. Using the identity 1I{|i−j|≤(2γ)−1 } 1I{|`x−Lr|>(2γ)−1 } = 1I{|i−j|≤(2γ)−1 } 1I{(2γ)−1 <|`x−Lr|≤(2γ)−1 +(`+L)/2} (2.12) and setting µ gγ, 3 (r) =
XX
X
1I{|i−j|≤(2γ)−1 } ξiµ ξjµ σi σj
(2.13)
x∈3: i∈x j∈r (2γ)−1 <|`x−Lr|≤(2γ)−1 +(`+L)/2
we have `,L 11 Wγ, 3 (σ3 , σ∂ + 3 ) = −
M X X
µ gγ, 3 (r).
(2.14)
µ=1 r∈∂ + 3
Note that the right-hand side of (2.14) is a sum of independent random variables since for any two distinct r1 , r2 in ∂ + 3, the sets {x ∈ 3 : (2γ)−1 < |`x − Lr1 | ≤ (2γ)−1 +(` + L)/2} and {x ∈ 3 : (2γ)−1 < |`x − Lr2 | ≤ (2γ)−1 + (` + L)/2} are disjoint. Therefore, # M X X µ −1 gγ,3 (r) > γ −2 , ≤ 2(γ +1) P > P sup 4 4 σ∈S3∪∂ + 3 µ=1 r∈∂ + 3 (2.15) where the probability in the right-hand side is independent of the chosen spin configuration σ3∪∂ + 3 . For convenience we will choose the configuration whose spins are all one’s. Using the exponential Markov inequality together with the independence, we get # " `,L 2 1 P sup γ |1 Wγ,3 (σ3 , σ∂ + 3 )| > 4 σ∈S3 ∪∂ + 3 " #M Y 1 (2.16) tgγ, (r) (γ −1 +1) −tγ −2 4 inf e Ee 3 . ≤2 "
`,L γ 2 |11 Wγ, 3 (σ3 , σ∂ + )|
t≥0
r∈∂ + 3
1 + Thus we have to estimate the Laplace-transform of gγ, 3 (r) for any r ∈ ∂ 3. We write
Ee
1 tgγ, 3 (r)
X = E exp t ξj1 j∈r
X
X
x∈3: i∈x (2γ)−1 <|`x−Lr|≤(2γ)−1 +(`+L)/2
1I{|i−j|≤(2γ)−1 } ξi1
.
(2.17) Note that all the ξj1 with j ∈ r are independent of the ξi1 with i ∈ x for x satisfy−1 + (` + L)/2, and that, conditioned on these lating (2γ)−1 < |`x − Lr| ≤ (2γ) P 1 ter variables, the variables ξj x∈3 1I{(2γ)−1 <|`x−Lr|≤(2γ)−1 +(`+L)/2} 1I{|i−j|≤(2γ)−1 } ξi1
332
A. Bovier, V. Gayrard, P. Picco
are independent. If we denote by Ej the expectation w.r.t. ξj1 , this allows us to write Ee
1 tgγ, 3 (r)
=E
Y
Ej exp
j∈r
≤E
Y j∈r
exp
X
tξj1 −1
(2γ)
X
x∈3: i∈x <|`x−Lr|≤(2γ)−1 +(`+L)/2
t2 2
1I{|i−j|≤(2γ)−1 } ξi1
X x∈3: (2γ)−1 <|`x−Lr|≤(2γ)−1 +(`+L)/2
2 (2.18) X 1 1I{|i−j|≤(2γ)−1 } ξi , i∈x
where we have used that ln cosh x ≤ 21 x2 . Using the H¨older-inequality on the last line, we arrive at Y 1 Eetgγ,3 (r) ≤ j∈r
2 E exp Lt 2
X x∈3: (2γ)−1 <|`x−Lr|≤(2γ)−1 +(`+L)/2
2 L1 X (2.19) 1 1I{|i−j|≤(2γ)−1 } ξi . i∈x
Now Lt2 E exp 2
X x∈3: (2γ)−1 <|`x−Lr|≤(2γ)−1 +(`+L)/2
Lt2 ≤ E exp 2
2 X 1 1I{|i−j|≤(2γ)−1 } ξi i∈x
X (2γ)
−1
x∈3: <|`x−Lr|≤(2γ)−1 +(`+L)/2
2 X 1 ξi i∈x
(2.20)
1 , ≤p 2 1 − t L(L + `)/2 where we have used the Khintchine inequality and the fact that, for all r ∈ ∂ + 3, XX
1I{(2γ)−1 <|`x−Lr|≤(2γ)−1 +(`+L)/2} ≤
x∈3 i∈x
√ Since for 0 ≤ x ≤ 1/2, 1/ 1 − x ≤ ex , for t2 ≤ (2.18)–(2.20), 1
Eetgγ,3 (r) ≤ et
2 L(L+`) 2
1 L(L+`) ,
.
L+` . 2
(2.21)
we finally get, collecting (2.22)
Distribution of Overlap Profiles in 1-D Kac–Hopfield Model
333
Therefore, since ]{r ∈ ∂ + 3} ≤ (2γL)−1 , choosing t = √
1 L(L+`)
in (2.22) yields
i h `,L P supσ∈S3∪∂ + 3 γ 2 |11 Wγ, 3 (σ3 , σ∂ + 3 )| > 4 n o 1 M 2/γ+1 √ exp − γ exp 4γL . ≤2 4γ
(2.23)
L(L+`)
√ Choosing in 2.6 as = 4γ L(L + `) log 2 +
M (γ) 4`(γ)
+ δ for some δ > 0, gives (2.6).
3. Some Large Deviation Estimates for the Hopfield Model In the preceding chapter we have introduced the block-approximation for the Hamiltonian of the Kac-Hopfield model. To make use of these, we need some large deviation results for the standard Hopfield model. They are essentially contained in [BGP1] and [BGP2], but we present them here in a slightly different way that focuses on our actual needs. We set M N ≡ α throughout this section. Recall that we have to consider the quantities X βN 2 e 2 kmN [ω](σ)k2 1I{kmN (σ)−mk2 ≤ρ} . (3.1) ZN,β,ρ (m) ≡ 2−N σ∈SN 1 ln ZN,β,ρ (m). In this paper we are mostly interested in the We set fN,β,ρ (m) ≡ − βN localization of the minima of the functions fN,β,ρ (m). Thus we will only need the following estimates:
Lemma 3.1. Define the random function ΦN,β (m) ≡
N 1 1 X kmk22 − ln cosh (β(ξi , m)) . 2 βN
(3.2)
i=1
Then and for ρ ≥
√
fN,β,ρ (m) ≥ ΦN,β (m) − 21 ρ2
(3.3)
∗
2α, if m is a critical point of ΦN,β (m), fN,β,ρ (m∗ ) ≤ ΦN,β (m∗ ) +
ln 2 βN .
(3.4)
eσ on {−1, 1}N through Proof. To prove Lemma 3.1, we define probability measures P e σ , given by their expectation E Eσ eβN (m,mN (σ)) · e Eσ · ≡ . (3.5) Eσ eβN (m,mN (σ)) We have obviously that eσ e ZN,β,ρ (m) = E = e− =e
βN 2
kmk22 e
Eσ e
βN 2
βN 2
1 βN − 21 kmk22 + βN
kmN (σ)k22 −βN (m,mN (σ))
kmN (σ)−mk22
PN i=1
1I{kmN (σ)−mk2 ≤ρ} Eσ eβN (m,mN (σ))
1I{kmN (σ)−mk2 ≤ρ} Eσ eβN (m,mN (σ))
ln cosh β(ξi ,m)
2 e σ e βN 2 kmN (σ)−mk2 1 E I{kmN (σ)−mk2 ≤ρ} .
(3.6)
334
A. Bovier, V. Gayrard, P. Picco
But 1I{kmN (σ)−mk2 ≤ρ} ≤ e
βN 2
kmN (σ)−mk22
1I{kmN (σ)−mk2 ≤ρ} ≤ e
so that we get on the one hand
−βN ΦN,β (m)− 21 ρ2
ZN,βρ (m) ≤ e
βN 2
ρ2
1I{kmN (σ)−mk2 ≤ρ} , (3.7)
,
(3.8)
which yields (3.3), and on the other hand eσ kmN (σ) − mk2 ≤ ρ . ZN,βρ (m) ≥ e−βN ΦN,β (m) P
(3.9)
But, using Chebychev’s inequality, we have that e σ kmN (σ) − mk2 eσ kmN (σ) − mk2 ≤ ρ ≥ 1 − 1 E P 2 ρ2
(3.10)
and e σ kmN (σ) − mk2 = E 2 QN β(m,ξi σi ) P −2 P P ν ν ν ν −1 ν 2 Eσ i=1 e ν N j,k ξj ξk σj σk − 2m N j µj σj + (m ) QN i=1 cosh β(ξi , m) X X X X 1 1 1+ 2 tanh(β(m, ξj )) tanh(β(m, ξk ))ξjν ξkν = 2 N ν j N ν j6=k (3.11) X 2 XX ν − m tanh(β(m, ξj ))ξjν + (mν )2 N j ν ν !2 X 1 X M X 1 X 2 ν ν − = tanh (β(m, ξi )) + ξ tanh(β(m, ξi )) − m . N N i N i i ν ν If m∗ is a critical point of Φ, m∗ =
1 X ξi tanh(β(m∗ , ξi )), N i
and so the last terms in (3.11) vanish and we remain with
(3.12)
e σ kmN (σ) − mk2 ≤ M E 2 N
!
1 X 1− tanh2 (β(ξi , m)) N i
from which (3.4) follows immediately.
≤ α,
(3.13)
Given the upper and lower bounds in terms of Φ, it remains to show that this function takes its absolute minima only near the points m(µ,s) ≡ sa(β)eµ defined in (1.13). This was done in [BGP1] and, with sharper estimates in [BG3] and [BG4]. We recall this result in a form suitable for our purposes. Recall that a(β) is the positive solution of the equation a = tanh(βa). For any ρ > 0, we define the balls n o (3.14) Bρ(µ,s) ≡ m ∈ RM km − m(µ,s) k2 ≤ ρ .
Distribution of Overlap Profiles in 1-D Kac–Hopfield Model
335
We will moreover write Bρ ≡
[
Bρ(µ,s)
(3.15)
(µ,s)∈{1,...,M }×{−1,1}
and denote by Bρc the complement of Bρ in RM . Then Proposition 6.1 of [BG4] specializes to Proposition 3.2. There exists γa > 0 and constants c0 < 1/2, c1 > 0, such √ that for all β > 1 for α/a(β)2 < γa , there exists a subset 4 (N ) ⊂ with P(4 (N )) ≥ 1 − e−cM such that for all ω ∈ 4 , for all m ∈ Bcc0 √α/a(β) , ΦN,β [ω](m) − ΦN,β [ω](m(µ,s) ) ≥ c1 a(β)2
inf
µ∈{1,...,M },s∈{−1,1}
km − m(µ,s) k22 . (3.16)
4. Local Effective Measures In Sect. 2 we have seen that the Kac-Hopfield Hamiltonian can be approximated by a block-spin Hamiltonian up to errors that are essentially proportional to γ` times the volume. This means of course that we cannot use this approximation throughout the entire volume 3 if we are interested in controlling local observables, as the errors would grow without bounds in the thermodynamic limit. A clever way to solve this difficulty was given in [COP] for the ferromagnetic Kac-model. The crucial point is that if one is interested in local observables in a box V , it is possible to show that with large probability (w.r.t. the Gibbs measure) not far away from this volume, there are intervals of macroscopic length 1 where the mesoscopic magnetization profiles are very close to one of the “equilibrium” values of the mean-field model. This knowledge allows to effectively decouple the system inside and outside this region, with the outside acting only as a “boundary condition”. Due to the randomness of the interaction, an additional difficulty presents itself in terms of the randomness of the effective boundary conditions. This makes it necessary to perform this analysis on two separate length scales: in this section we consider a rather large volume (which we will see later can be chosen of order o(γ −1 ) (on the macroscopic scale); in Sect. 6 these measures will be further analyzed by localizing them to much smaller boxes. To begin, we imitate [COP] by defining variables η that serve as a decomposition of the configuration space through seµ if ∀r∈u km(µ,s) − mL (r, σ)k2 ≤ ζ (4.1) η(u, σ) ≡ ηζ,L (u, σ) = 0 if ∀µ,s ∃r∈u : km(µ,s) − mL (r, σ)k2 > ζ √ (This definition is unequivocal if ζ is chosen small enough, i.e. ζ < 2a(β)). For a given configuration σ, η determines whether a unit interval is close to equilibrium on the scale L. For a given volume V ≡ [v− , v+ ] ⊂ 3, with |V | > 1, we set n τ + = inf{u ≥ v+ : η(u, σ) 6= 0} (4.2) ∞ if no such u exists and τ− =
n
sup{u ≤ v− : η(u, σ) 6= 0} −∞ if no such u exists.
(4.3)
336
A. Bovier, V. Gayrard, P. Picco
For a given configuration σ, τ ± indicates the position of the first unit interval to the right, resp. the left, of V where the configurations σ is close to equilibrium. Let us introduce the indices µ+ , µ− , s+ , s− , w+ , w− , where µ± ∈ {1, . . . , M (γ)}, ± s ∈ {−1, 1} and w+ ∈ [v+ , ∞), w− ∈ (−∞, v− ]. In the sequel, if not otherwise specified, all sums and unions over these indices run over the above sets. The expression (µ± , s± ), resp. (µ± , s± , w± ), are abbreviations for (µ+ , s+ , µ− , s− ), resp. (µ+ , s+ , w+ , µ− , s− , w− ). With these notations we define a partition of the configuration space S whose atoms are given by ± ± , s , w± ) ≡ A(µ o n + − σ ∈ S : τ + = w+ , τ − = w− , η(τ + , σ) = s+ eµ , η(τ − , σ) = s− eµ .
and we denote by
[
SR =
(4.4)
A(µ± , s± , w± ).
(4.5)
µ± ,s± ,w
± 0≤±(w± −v± )≤R
Notice that c SR = A+ (R) ∪ A− (R),
(4.6)
where A+ (R) ≡ {σ ∈ S : τ + > v+ + R} = {σ ∈ S : ∀v+ ≤w≤v+ +R η(w, σ) = 0}
(4.7)
and A− (R) ≡ σ ∈ S : τ − < v− − R = σ ∈ S : ∀v− −R≤w≤v− η(w, σ) = 0 . (4.8) Before stating the main results of this chapter we need some further notations. For given indices µ± , s± , w± we write 1 ≡ [w− + 21 , w+ − 21 ] and we set o n b ± , s± , w± ) ≡ σ ∈ S : η(w+ , σ) = s+ eµ+ , η(w− , σ) = s− eµ− . A(µ
(4.9)
We define the Gibbs measure on 1 with mesoscopic boundary conditions m(µ − − w+ and m(µ ,s ) at w− as the measure that assigns, to each σ1 ∈ S1 , the mass, (µ± ,s± ) [ω](σ1 ) Gβ,γ,1
±
=
1 ±
±
(µ ,s ) Zβ,γ,1 [ω]
−β Hγ,1 [ω](σ1 )+Wγ,1 [ω](σ1 ,m(µ e
± ,s± )
+
,s+ )
at
)
,
(4.10)
±
(µ ,s ) [ω] is the corresponding normalization factor and where Zβ,γ,1 ±
±
Wγ,1 [ω](σ1 , m(µ ,s ) ) − P P ≡ − i∈1 s− a(β)ξiµ σi j∈∂ − 1 Jγ (i − j) + P P − i∈1 s+ a(β)ξiµ σi j∈∂ + 1 Jγ (i − j).
(4.11)
Proposition 4.1. Let F be a cylinder event with base contained in [v− , v+ ]. Then
Distribution of Overlap Profiles in 1-D Kac–Hopfield Model
i)
337
There exists a positive constant c such that, for all integer R, there exists −1 R with P(R ) ≥ 1 − Re−cγ such that for all µ± , s± , w± , v+ ≤ w+ ≤ v+ + R, v− − R ≤ w ≤ v− and ω ∈ R for all 3 ⊃ [v− − R, v+ + R], Gβ,γ,3 [ω] F ∩ A(µ± , s± , w± ) (4.12) (µ± ,s± ) b ± , s± , w± ) e8βγ −1 (ζ+2γL) [ω] (F ) Gβ,γ,3 [ω] A(µ ≤ Gβ,γ,1 and for any u+ ≥ v+ , u− ≤ v− , b ± , s ± , u± ) Gβ,γ,3 [ω] F ∩ A(µ (µ± ,s± ) b ± , s± , u± ) e−8βγ −1 (ζ+2γL) . (F ) [ω] G [ω] A(µ ≥ Gβ,γ,[u β,γ, 3 ,u ] − +
(4.13)
i h √ α 1 , 2 a(β) set (ζ) ≡ c1 a(β)2 ζ 2 , where c0 and c1 are the conii) For ζ ∈ c0 a(β)
stants appearing in Proposition 3.2. There exist a positive constant c0 such 0 that for all integer R, there exists R with P(R ) ≥ 1 − γ −1 Re−c Mqand
there exist finite positive constants c2 and c3 such that if ζ(ζ)γL > 2c2 then for all ω ∈ R and 3 ⊃ [v− − R, v+ + R], c ) ≤ exp (−βLRc3 ζ(ζ)) . Gβ,γ,3 [ω](F ∩ SR
M ` ,
(4.14)
Corollary 4.2. Let F be a cylinder event with base contained in [v− , v+ ]. Then there exist a positive constant c0 such that for all integer R, there exists R with 0 M P(R ) ≥ 1 − γ −1 Re−c q and there exist finite positive constants c1 and c2 such that if ζ(ζ)γL > 2c1
then for all ω ∈ R and 3 ⊃ [v− − R, v+ + R],
M ` ,
Gβ,γ,3 [ω](F ) ≤
P
±
µ± ,s±
−R<w− ≤v− v+ ≤w+
±
(µ ,s ) Gβ,γ,1 [ω] (F )
b ± , s± , w± ) e8βγ −1 (ζ+2γL) Gβ,γ,3 [ω] A(µ + exp (−βLRc2 ζ(ζ)) ,
(4.15)
and there exist u± with ±(u± − v± ) ≤ R such that for all 3 ⊃ [v− − R, v+ + R] Gβ,γ,3 [ω](F ) P (µ± ,s± ) b ± , s± , u± ) e−8βγ −1 (ζ+2γL) , (F ) ≥ µ± ,s± Gβ,γ,[u [ω] G [ω] A(µ β,γ, 3 ,u ] − + moreover there exists (µ± , s± ) and u± with ±(u± − v± ) ≤ R such that 1 b ± , s ± , u± ) ≥ Gβ,γ,3 [ω] A(µ . 8R2 M 2
(4.16)
(4.17)
Remark . Corollary 4.2 tells us that in order to estimate the probability of some local event in V with respect to the infinite volume Gibbs measure we only need to control finite volume Gibbs measures in volumes |1| with all possible boundary conditions corresponding to one of the mean field equilibrium states. This analysis will be performed in Sect. 6. On the other hand, it appears quite hopeless to get a more precise information b ± , s± , u± ) appearing in both bounds. This is, than (4.17) on the terms Gβ,γ,3 [ω] A(µ
338
A. Bovier, V. Gayrard, P. Picco
after some thought, not surprising, but reflects the fact that the exact shape of typical profiles depends strongly on the disorder and only some of their properties on relatively short scales can be effectively controlled. In particular, it is clear that we cannot hope to get something like a full large deviation principle (in the sense of the results of [COP] in the deterministic case) for the infinite volume Gibbs measures. Proof. The first assertion of Corollary 4.2 is obvious from (4.12) and (4.14). To prove the second, we need to show that 1 b ± , s± , u± ) ≥ sup Gβ,γ,3 [ω] A(µ . (4.18) sup 2M 2 8R ± ± ±(u −v )≤R µ ,s ± ± But from (4.14) we see that c ≤ 1 − exp (−βLRc2 ζ(ζ)) ≤ 1 − Gβ,γ,3 [ω](SR ) ≤ Gβ,γ,3 [ω] τ+ ≤ v+ + R, τ− ≥ v− − R P ≤ ±(u± −v± )≤R Gβ,γ,3 [ω] τ− = u− , τ+ = u+ P ≤ ±(u± −v± )≤R Gβ,γ,3 [ω] η(u− , σ) 6= 0, η(u+ , σ) 6= 0 P P − + ≤ ±(u± −v± )≤R µ± ,s± Gβ,γ,3 [ω] η(u− , s) = s− eµ , η(u+ , s) = s+ eµ ˆ ± , s ± , u± ) ≤ 4R2 M 2 sup±(u± −v± )≤R supµ± ,s± Gβ,γ,3 [ω] A(µ (4.19) which gives (4.18). 1 2
In order to prove Proposition 4.1, we need the following lemmata. Lemma 4.3. For any finite subset 0 ⊂ Z we denote by A(0) the M × M -matrix with elements 1 X µ ν ξ ξ (4.20) Aµ,ν (0) = |0| i∈0 i i and let B be the N × N -matrix with entries Bi,j =
M 1 X µ µ ξi ξj . N
(4.21)
µ=1
Set N = |0| and assume that M ≥ N 1/6 . Then, (i) EkA(0) − 1Ik ≤
q
M N
q 2+
M N
ln N +CN 1/6
(4.22)
and (ii) There exists a universal constant K < ∞ such that for all 0 ≤ δ ≤ 1, δ2 . (4.23) P |kB(0)k − EkB(0)k| > δ ≤ K exp −N 2K In particular,
Distribution of Overlap Profiles in 1-D Kac–Hopfield Model
"
P kA(0)k ≥
q 1+
2 M N
339
#
δ2 . (1 + δ) ≤ K exp −N 2K
(4.24)
Proof. For the proof of this lemma, see [BG3], Sect. 2. Somewhat weaker estimates were previously obtained in [Ge,ST,BG1,BGP1]. Lemma 4.4. Let {Xi (n), i ≥ 1} be independent random variables with Xi (n) ≥ 0, satisfying, for any z ≥ 0, P [Xi (n) ≥ (1 + z)an ] ≤ cn e−zbn ,
(4.25)
where an , bn , cn are strictly positive parameters satisfying bn ↑ ∞ and (ln cn )/bn ↓ 0 as n ↑ ∞. Then, ln cn (4.26) E(Xi (n)) ≤ an 1 + bn and, for all > 0 and n sufficiently large, # " K 1 X Xi (n) ≥ (1 + z + )an ≤ e−zbn (1−η)K , P K
(4.27)
i=1
where η ≡ η(, bn , cn ) ↓ 0 as n ↑ ∞. Proof. Setting Yi (n) ≡ Xi (n)/an , we have, Z ∞ Z E(Yi (n)) = E 1I{y≤Yi (n)} dy = 0
∞
P(Yi (n) ≥ y)dy.
(4.28)
0
Thus, for any x ≥ 0, Z E(Yi (n)) ≤ 1 + x +
∞
P(Yi (n) ≥ y)dy.
(4.29)
1+x
Performing the change of variable y = 1 + z and making use of (4.25) yields Z ∞ cn E(Yi (n)) ≤ 1 + x + cn e−bn z dz = 1 + x + e−xbn . bn x
(4.30)
Now, choosing x = (ln cn )/bn minimizes the r.h.s. of (4.42) and gives (4.26). To prove (4.27) we first use that, by the exponential Markov inequality, for any t > 0, # " K K Y 1 X Yi (n) ≥ 1 + z + ≤ e−Kt(1+z+) EetYi (n) . (4.31) P K i=1
i=1
To estimate the Laplace transform of Yi (n), we write that, using integration by parts, Z ∞ Z ∞ EetYi (n) = E(1 + tety 1I{y≤Yi (n)} dy) = 1 + tety P(Yi (n) ≥ y)dy (4.32) 0
and, for any x ≥ 0,
0
340
A. Bovier, V. Gayrard, P. Picco
EetYi (n) = 1 +
R 1+x 0
≤ et(1+x) +
tety P(Yi (n) ≥ y)dy + R∞ 1+x
≤ et(1+x) + cn te
R∞ 1+x
tety P(Yi (n) ≥ y)dy
tety P(Yi (n) ≥ y)dy R t ∞ x
(4.33)
e−z(bn −t) dz,
where we used (4.25) in the last line after having performed the change of variable y = 1 + z. Choosing t = bn (1 − η) for some 0 < η ≤ 1, we get h i −xbn EetYi (n) ≤ ebn (1−η)(1+x) 1 + cn 1−η e η (4.34) −xbn ≤ exp bn (1 − η)(1 + x) + cn 1−η , η e and finally, inserting (4.46) in (4.43) yields h P i K P K1 Y (n) ≥ 1 + z + i i=1 ≤e
−zbn (1−η)K
h
exp −(1 − η)K bn ( − x) −
cn −xbn η e
i
(4.35) .
For n large enough, choosing x = /2, one can always choose η ≡ η(, bn , cn ) such that the last exponential in (4.27) is less than 1 and η(, bn , cn ) ↓ 0 as n ↑ ∞. Lemma 4.5. There exists a positive constant c such that, for all integer R, there −1 exists R with P(R ) ≥ 1 − Rγ −1 e−cγ such that for all µ± , s± , w± , v+ ≤ w+ ≤ v+ + R, v− − R ≤ w− ≤ v− and ω ∈ R (i) ± ± 1,L supσ:η(w± ,σ)=s± eµ± γ −1 Eγ,1 [ω](σ1 , mL (σ∂1 )) − Wγ,1 [ω](σ1 , m(µ ,s ) ) √ √ ≤ ζγ −1 (1 + 2γM (γ)) 2 (4.36) and (ii)
2 p sup |Wγ,1 [ω](σ1 , σ∂1 )| ≤ γ −1 4 1 + M/` ,
(4.37)
σ
where 1 = [w− + 21 , w+ − 21 ]. Proof. We first prove (i). We set Wγ,1 [ω](σ1 , m(µ
±
,s± )
+ ) = Wγ,1 [ω](σ1 , m(µ
+
,s+ )
− ) + Wγ,1 [ω](σ1 , m(µ
−
,s− )
), (4.38)
where − [ω](σ1 , m(µ Wγ,1
−
,s− )
) ≡ −L
X i∈1
and
−
s− a(β)ξiµ σi
X r∈∂ − 1
Jγ (i − Lr)
(4.39)
Distribution of Overlap Profiles in 1-D Kac–Hopfield Model + Wγ,1 [ω](σ1 , m(µ
+
,s+ )
) ≡ −L
X
341 +
s+ a(β)ξiµ σi
X
Jγ (i − Lr).
(4.40)
r∈∂ + 1
i∈1
We will consider only the terms corresponding to the interaction with the right part of 1, the other ones being similar. We have + + −1 1,L + [ω](σ1 , m(µ ,s ) ) 1I{σ∈A(µ γ Eγ,1 [ω](σ1 , mL (σ∂ + 1 )) − Wγ,1 b ± ,s± ,w± )} P i h P + + ≤ L i∈1 r∈∂ + 1 Jγ (i − Lr)σi ξi , mL (r, σ∂ + 1 ) − m(µ ,s ) 1I{σ∈A(µ b ± ,s± ,w± )}
P
P + + ≤ L r∈∂ + 1 i∈1 Jγ (i − Lr)ξi σi 2 mL (r, σ∂ + 1 ) − m(µ ,s ) 1I{σ∈A(µ b ± ,s± ,w± )} 2
P
P ≤ ζL r∈∂ + 1 i∈1 Jγ (i − Lr)ξi σi 2 ≡ T + (σ). (4.41) T − (σ) is defined in an analogous way. Recalling the definition (4.21) we have P T + (σ) = ζL r∈∂ + 1 P 21 P i∈[w+ −1,w+ − 21 ] j∈[w+ −1,w+ − 21 ] (ξi , ξj )σi σj Jγ (i − Lr)Jγ (j − Lr) 21 P P (4.42) ≤ ζL r∈∂ + 1 γ −1 kBk i∈[w+ −1,w+ − 1 ] (σi Jγ (i − Lr))2 2 P 1 ≤ ζL r∈∂ + 1 kBk 2 ≤ ζ(2γ)−1 kBk 2 , 1
where we have used in the last inequality that #{r ∈ ∂ + 1} = (2γL)−1 . Thus, by Lemma 4.3, for all > 0, p √ 2 + −1 −1 (4.43) P sup T (σ) ≥ ζ(2γ) (1 + 2γM ) 1 + ≤ 2Kγ exp − 2Kγ σ∈S from which (i) follows. We turn to the proof of (ii). Using (2.4) we have, for all > 0, P supσ∈S |Wγ,1 [ω](σ1 , σ∂1 )| ≥ 42 h i `,` (m` (σ1 ), m` (σ∂1 )) ≥ 22 ≤ P supσ∈S γ −1 Eγ,1
(4.44)
i h `,` (σ1 , σ∂1 ) ≥ 22 . +P supσ∈S 1Wγ,1 Let us consider the first probability in the r.h.s. of (4.44). By definition, X X `,` (m` (σ1 ), m` (σ∂1 )) = γ` Jγ` (x − y)(m` (x, σ1 ), m` (y, σ∂1 )). (4.45) Eγ,1 x∈1 y∈∂1
Now
(m` (x, σ1 ), m` (y, σ∂1 )) ≤ km` (x, σ1 )k2 km` (y, σ∂1 )k2 (4.46) 1
1
≤ kB(x)k 2 kB(y)k 2 ,
342
A. Bovier, V. Gayrard, P. Picco
where B(x) is the ` × `-matrix B(x) = {B(x)i,j }i∈x,j∈x with B(x)i,j = Thus
1 `
PM
µ µ µ=1 ξi ξj .
`,` Eγ,1 (m` (σ1 ), m` (σ∂1 )) ≤ (γ`)2
P
P x∈1
1
y∈∂1
1
1I{|`x−`y|≤(2γ)−1 } kB(x)k 2 kB(y)k 2
P P 1 1 γ` y∈[w+ − 1 ,w+ +1] kB(y)k 2 ≤ γ` x∈[w+ −1,w+ − 1 ] kB(x)k 2 2
(4.47)
2
P P 1 1 γ` y∈[w− ,w− + 1 ] kB(y)k 2 + γ` x∈[w− + 1 ,w− +1] kB(x)k 2 2
2
≡ T 1 T2 + T3 T4 and,
X 4 `,` 2 P sup Eγ,1 (m` (σ1 ), m` (σ∂1 )) ≥ 2 ≤ P(Tk ≥ ), σ∈S
(4.48)
k=1
where the last equality in (4.47) defines the quantities Tk . All four probabilities on the right-handnside of (4.48) o will be bounded in the same way. Let us consider P(T1 ≥ ). 1 2 are independent random variables. It follows from Note that kB(x)k 1 x∈[w+ −1,w+ − 2 ]
Lemma 4.3 that, for all ˜ > 0,
2 i h p 1 ˜ ` , P kB(x)k 2 > 1 + M/` (1 + ˜) ≤ 4K` exp − K
(4.49)
and by Lemma 4.4, we get that for large enough `, p 1 ˜ . P T1 ≥ (1 + M/`)(1 + ˜) ≤ K exp − 2 2Kγ
Therefore, choosing ≡ 21 (1 +
p
(4.50)
M/`)(1 + ˜) in (4.44), (4.48) yields
i h p `,` (m` (σ1 ), m` (σ∂1 )) ≥ (2γ)−1 (1 + M/`)2 (1 + ˜)2 P supσ∈S γ −1 Eγ,1 ˜ . ≤ 4K exp − 2Kγ
(4.51)
Choosing ˜ = 1 and using Lemma 2.1 to bound the second term in (4.44) we get (4.37) which concludes the proof of Lemma 4.5. We are now ready to prove Proposition 4.1. Proof of Proposition 4.1, Part i). Setting 1c ≡ 3 \ 1 and denoting by σ˜ and σ¯ independent copies of σ, some simple manipulations allow us to write
Distribution of Overlap Profiles in 1-D Kac–Hopfield Model
343
b ± , s± , w± )) Gβ,γ,3 [ω](F ∩ A(µ± , s± , w± )) ≤ Gβ,γ,3 [ω](F ∩ A(µ ± ± 1 −β Hγ,1 [ω](σ1 )+Wγ,1 [ω](σ1 ,m(µ ,s ) ) Eσ1 e = Zβ,γ,3 [ω] ± ± −β Hγ,1c [ω](σ1c )+ Wγ,1 [ω](σ1 ,σ1c )−Wγ,1 [ω](σ1 ,m(µ ,s ) ) × Eσ1c e ×1I{σ∈F ∩A(µ b ± ,s± ,w )} ±
= Eσ1
1 ±
±
(µ ,s ) Zβ,γ,1 [ω]
× Eσ1c Eσ˜ 1
−β Hγ,1 [ω](σ1 )+Wγ,1 [ω](σ1 ,m(µ e
1 Zβ,γ,3 [ω]
± ,s± )
)
e
(4.52)
−β Hγ,1c [ω](σ1c )+Hγ,1 [ω](σ˜ 1 )+Wγ,1 [ω](σ˜ 1 ,σ1c ) ∗
∗∗
−β(W +W ) ×1I{σ∈F ∩A(µ b ± ,s± ,w± )} e µ± ,s± = Eσ1 Gβ,γ,1 [ω](σ1 )1I{σ∈F } Eσ¯ 3 Gβ,γ,3 [ω](σ¯ 3 )1I{σ∈ b ± ,s± ,w± )} ¯ A(µ † †† e−β(W +W ) ,
h i ± ± W ∗ = Wγ,1 [ω](σ1 , σ1c ) − Wγ,1 [ω](σ1 , m(µ ,s ) ) , h i ± ± W ∗∗ = Wγ,1 [ω](σ˜ 1 , m(µ ,s ) ) − Wγ,1 [ω](σ˜ 1 , σ1c ) , h i ± ± W † = Wγ,1 [ω](σ1 , σ¯ 1c ) − Wγ,1 [ω](σ1 , m(µ ,s ) ) , h i ± ± W †† = Wγ,1 [ω](σ¯ 1 , m(µ ,s ) ) − Wγ,1 [ω](σ¯ 1 , σ¯ 1c ) ,
where
b ± , s± , w± ), Now, if σ¯ ∈ A(µ h i ± ± Wγ,1 [ω](σ1 , σ¯ 1c ) − Wγ,1 [ω](σ1 , m(µ ,s ) ) h i ± ± + Wγ,1 [ω](σ¯ 1 , m(µ ,s ) ) − Wγ,1 [ω](σ¯ 1 , σ¯ 1c ) ± ± ≤2 sup Wγ,1 [ω](σ¯ 1 , σ¯ 1c ) − Wγ,1 [ω](σ¯ 1 , m(µ ,s ) ) b ± ,s± ,w± ) σ∈ ¯ A(µ ± ± (4.53) −1 1,L ≤2 sup γ Eγ,1 [ω](σ¯ 1 , mL (σ¯ ∂1 ))−Wγ,1 [ω](σ¯ 1 , m(µ ,s ) ) b ± ,s± ,w± ) σ∈ ¯ A(µ 1,L + 2 sup 1Wγ,1 [ω](σ¯ 1 , σ¯ ∂1 ) . σ∈S ¯
Finally, by Lemma 4.5 and Lemma 2.1, the supremum over µ± , s± and w± , v+ ≤ w+ ≤ v+ + R v− − R ≤ w− ≤ v− , of the last line of (4.53) is bounded from above by 8γ −1 (ζ + 2γL) with a Pξ -probability, greater than 1 − 4γ −1 R exp(−cγ −1 ) for some positive constant c. Thus from (4.52) and (4.53) follow both (4.12) and (4.13).
344
A. Bovier, V. Gayrard, P. Picco
Proof of Proposition 4.1 part ii). Using (4.6) the l.h.s. of (4.14) is bounded from above by Gβ,γ,3 [ω](A+ (R)) + Gβ,γ,3 [ω](A− (R)). We estimate the first term, the second one being similar. Since the spin configuration is away from the equilibria on a length R, we can decouple the interaction between this part and the rest of the volume 3, by making a rough estimate of those interaction terms. The fact that we are out of equilibrium will give terms proportional to R that will be dominant if R is chosen large enough. More precisely, calling 1R ≡ [v+ , v+ + R], we have, for all fixed R, Gβ,γ,3 A+ (R) =
1 Zβ,γ,3
E σ3
−β Hγ,1 (σ1 )+Wγ,1 (σ1 ,σ3\1 ) R R R R R × e−βHγ,3\1R (σ3\1R ) e 1I{σ∈A+ (R)} ≤e
1
4cγ −1
Zβ,γ,1R
Eσ1
h R
e−βHγ,1R (σ1R ) 1I{σ∈A+ (R)}
i
(4.54)
−1
with a Pξ -probability greater than 1 − 4γ −1 e−cγ for some positive constant c, where we have used Lemma 4.5 to bound the interaction between 1R and 3 \ 1R . To estimate the last term in (4.54), we express it in terms of block spin variables on the scale `. Using (2.5) we get
Gβ,γ,1R A (R) ≤ e +
2cγ −1 |1R |(4γ`+γM )
Eσ1 e
` −βγ −1 Eγ, 1 (m` (σ)) R
R
Eσ1 e
−βγ −1 E `
γ,1R
R
1I{σ∈A+ (R)}
(m` (σ))
(4.55)
−1 with a Pξ -probability greater than 1 − e−cγ |1R | We derive first a lower bound on the denominator which will be given effectively by restricting the configurations to be in the neighborhood of a constant profile near one of the equilibrium positions sa(β)eµ . We will choose without loss of generality s = 1, µ = 1. Recalling the definitions of Bρ(1,1) and Bρ from (3.14) and (3.15) we have that, obviously,
Eσ1 e
` −βγ −1 Eγ, 1 (m` (σ))
≥ Eσ1 e
R
R
` −βγ −1 Eγ, 1 (m` (σ))
Y
R
R
1I{m` (x,σ)∈Bρ(1,1) } . (4.56)
x∈1R
It can easily be shown that, on the set {m` (x, σ) ∈ Bρ , ∀x ∈ 1R }, ` (m` (σ)) ≥ −γ −1 Eγ,1 R
` X (km` (x, σ)k22 − 4ρ2 ), 2 x∈1
(4.57)
R
from which (4.56) yields Eσ1 e
` −βγ −1 Eγ, 1 (m` (σ)) R
R
≥ e−4βγ = e−4βγ
−1
−1
|1R |ρ2
|1R |ρ2
Q x∈1R
Q x∈1R
E σx e
β` 2 2 km` (x,σ)k2
Zx,β,ρ a(β)e1 ,
1I
{m` (x, σ) ∈ Bρ(1,1) }
(4.58)
Distribution of Overlap Profiles in 1-D Kac–Hopfield Model
345
provided that ρ is sufficiently large so that Bρ(1,1) contains the lowest minimum of Φ in q the neighborhood of a(β)e1 , which is the case if ρ ≥ c M ` , for some finite constant c with a Pξ -probability ≥ 1 − e−cM . Next we derive an upper bound for the numerator of the ratio in (4.55). Using the inequality ab ≤ 21 (a2 + b2 ) we get ` −γ −1 Eγ,1 (m` (σ)) ≤ R
` X km` (x, σ)k22 , 2 x∈1
(4.59)
R
and whence Eσ1 e
` −βγ −1 Eγ, 1 (m` (σ))
R
≤ Eσ1 e
β` 2
R
P
x∈1R
R
1I{σ∈A+ (R))} km` (x,σ)k22 1I{σ∈A+ (R)} .
(4.60)
Let us now recall that, by definition, A+ (R) = σ ∈ S ∀u∈1R ∃r∈u : inf km(µ,s) − mL (r, σ)k2 > ζ .
(4.61)
µ,s
Using that mL (r, σ) =
` L
P x∈r
m` (x, σ) we have
km(µ,s) − mL (r, σ)k2 ≤ so that
(
A (R) ⊂ +
` X km(µ,s) − m` (x, σ)k2 L x∈r
` X σ ∈ S ∀u∈1R ∃r∈u : inf km(µ,s) − m` (x, σ)k2 > ζ µ,s L x∈r
(4.62)
) .
(4.63)
We will use the following fact: Lemma 4.6. Let {Xk , k = 1, 2, . . . , K} be a sequence of real numbers satisfying 0 ≤ Xk ≤ c for some c < ∞. Let ζ < c and assume that K 1 X Xk > ζ. K
(4.64)
k=1
Then |{1 ≤ k ≤ K : Xk > δζ}| ≥ K
ζ(1 − δ) . c − δζ
(4.65)
c Proof. For 0 ≤ δ ≤ 1, define the set Vδ,ζ ≡ {k|Xk ≤ δζ} and set Vδ,ζ ≡ {1, . . . , K} \ Vδ,ζ . Then 1 K
=
PK k=1 1 K (c
Xk ≤
1 K
P k∈Vδ,ζ
− δζ)|Vδ,ζ | + δζ,
Xk +
1 K
P
c k∈Vδ,ζ
Xk ≤
which, together with (4.64) implies the bound (4.65).
1 K c|Vδ,ζ |
+
1 c K δζ|Vδ,ζ |
(4.66)
346
A. Bovier, V. Gayrard, P. Picco
Let us denote by Vδ,ζ (r) the set of all subsets S ⊂ {x ∈ r} with cardinality respectively volume ζ(1 − δ) . |S| ≥ γL 2 − δζ
L ζ(1−δ) ` 2−δζ ,
(4.67)
Then, since km(µ,s) − m` (x, σ)k2 < 2, Lemma 4.6 implies that there exists a set S ∈ Vδ,ζ (r) such that for all x ∈ S, km(µ,s) − m` (x, σ)k2 > δζ. That is to say o n c A+ (R) ⊂ σ ∈ S ∀u∈1R ∃r∈u ∃S∈Vδ,ζ (r) : ∀x∈S , m` (x, σ) ∈ Bδζ . (4.68) Therefore Eσ1 e
` −βγ −1 Eγ, 1 (m` (σ)) R
R
≤
Q u∈1R
×1I{∃
E σu e
β` 2
P x∈u
1I{σ∈A+ (R)}
km` (x,σ)k22
c : ∀x ∈ S , m` (x, σ) ∈ Bδζ } P β` P P Q km` (x,σ)k22 x∈u ≤ u∈1R r∈u S∈Vδ,ζ (r) Eσu e 2
Q x∈S
r∈u ∃S∈Vδ,ζ (r)
(4.69)
1I{m (x, σ) ∈ B c } . ` δζ
Inserting this and (4.58) into (4.55) we have
Gβ,γ,1R [ω] A+ (R) −1
≤ eγ |1R |(16γ`+4γM +4βρ ) Y X X Y 2
c Y Zx,β,δζ Zx,β Z (a(β)e1 ) Zx,β,ρ (a(β)e1 ) u∈1R r∈u S∈Vδ,ζ (r) x∈u\S x,β,ρ x∈S Y X X −1 2 TS(1) TS(2) , ≡ eγ |1R |(16γ`+4γM +4βρ )
(4.70)
u∈1R r∈u S∈Vδ,ζ (r)
where we have defined c ≡ Eσx e Zx,β,δζ
β` 2 2 km` (x,σ)k2
1I{m (x, σ) ∈ B c } . ` δζ
It follows from Proposition 2.3 of [BGP1] that q Zx,β ≤ exp −β` φ(a(β)) − c M . `
(4.71)
(4.72)
so that using Lemma 3.1 we get that TS(1)
≤
Y
exp +β`c
q
M `
≤ e+βγ
−1
c
√M `
,
(4.73)
x∈u\S c with a Pξ -probability ≥ 1 − (γ`)−1 e−cM On the other hand, to bound Zx,β,δζ , we 2 proceed as in [BG2] and first note that km` (x, σ)k2 ≤ 2 for all σ. Next, we introduce
Distribution of Overlap Profiles in 1-D Kac–Hopfield Model
347
√ the lattice W`,M with spacing 1/ ` in RM and we denote by W`,M (2) the intersection of this lattice with the ball of radius 2 in RM . We have 2` . (4.74) |W`,M (2)| ≤ exp M ln M p Now, we may cover the ball of radius 2 in RM with balls of radii ρˆ ≡ M/` centered at the points of W`,M (2). Supposing that δζ > ρˆ this yields, P c Zx,β,ρˆ (m)[ω] ≤ m∈W`,M (2) 1I{m ∈ B c Zx,β,δζ δζ−ρˆ } (4.75) P exp −β` Φx,β (m)[ω] − 21 ρˆ2 . ≤ m∈W`,M (2) 1I{m ∈ B c } δζ−ρˆ Let us now assume that δζ − ρˆ satisfies the hypothesis of Proposition 3.2, then c Zx,β,δζ q 1 2 ≤ exp −β` φ(a(β)) + (δζ − ρ) ˆ − 4(δζ − ρ) ˆ M ˆ − ` − 2ρ
M β`
ln
2` M
(4.76)
with a Pξ -probability ≥ 1 −q e−cM , where (·) is the function defined in Proposition 4.1. We will assume that δζ
M ` .
Thus
q c Zx,β,δζ M (δζ ≤ exp −β` − ρ) ˆ − cδζ ` Zx,β,ρ (a(β)e1 )
(4.77)
0
with a Pξ -probability ≥ 1 − e−c M . Thus the product TS(1) TS(2) defined in (4.70) is bounded by q M − (ζ)|S| (4.78) TS(1) TS(2) ≤ exp βγ −1 c ` 0
with a Pξ -probability ≥ 1 − (γ`)−1 |S|e−c M . Hence Y X X TS(1) TS(2) u∈1R r∈u S∈Vδ,ζ (r)
≤
Y X
X
u∈1R r∈u S∈Vδ,ζ (r)
q exp −βγ −1 c |S|(ζ) − M `
ln 2 −c ≤ exp −βγ −1 |1R | γLζc(ζ) − γ| ln(γL)| − γL `
q
(4.79)
M `
0
with a Pξ -probability ≥ 1 − (γ)−1 Re−c M , and finally, inserting (4.70) in (4.59) we arrive at Gβ,γ,1R [ω] (A+ (R)) ≤ h q i M 2 + 8γ` + 2ρ exp −βγ −1 R γLcζ(ζ) − c0 ` 0
(4.80)
with a Pξ -probability ≥ 1 − (γ`)−1 Re−c ` , where we have used the fact that M `.
348
A. Bovier, V. Gayrard, P. Picco
5. Self Averaging Properties of the Free Energy In this chapter we study the self averaging properties of the free energy of the HopfieldKac model with mesoscopic boundary conditions. We denote the partition function on the volume 1 with boundary condition − + s− a(β)eµ on the left of 1 and s+ a(β)eµ on the right of 1 by − − + + −β Hγ,1 (σ)+Wγ,1,∂ − 1 (σ1 |m(µ ,s ) )+Wγ,1,∂ + 1 (σ1 )|m(µ ,s ) ) (µ± ,s± ) ≡ Eσ1 e Z1 , (5.1) and the corresponding free energy f1(µ
±
,s± )
≡ f1 = −
± ± γ ln Z1(µ ,s ) . β|1|
(5.2)
To include the case of free boundary conditions, we set m(0,0) ≡ 0. ± ± We are interested in the behavior of the fluctuations of f1(µ ,s ) around its mean value. We will use Theorem 6.6 of Talagrand [T2] that we state for the convenience of the reader. We denote by MX a median of the random variable X. Recall that a number x is called the median of a random variable X if both P[X ≥ x] ≥ 21 and P[X ≤ x] ≥ 21 . Theorem 5.1. [T2] Consider a real valued function f defined on [−1, +1]N . We assume that, for each real number a the set {f ≤ a} is convex. Consider a convex set B ⊂ [−1, +1]N , and assume that for all x, y ∈ B, |f (x)−f (y)| ≤ kkx−yk2 for some positive k. Let X denote a random vector with i.i.d. components {Xi }1≤i≤N taking values in [−1, +1]. Then for all t > 0, 4 t2 P |f (X) − Mf (X)| ≥ t ≤ 4b + exp − , (5.3) 1 − 2b 16k 2 where b ≡ P [X 6∈ B] and we assume that b < 21 . The main result of this chapter is the following proposition: Proposition 5.2. If γ`, M/` and γM are small enough, then for all t > 0, there exists a universal numerical constant K such that i h ± ± p ± ± −1 P f1(µ ,s ) − Ef1(µ ,s ) ≥ t + K γ −1 |1| (5.4) −1 √ ≤ K exp − γ 8 |1|( 1 + t2 − 1) . Proof. Note first that the set {f1 ≤ a} is convex. This follows from the fact that the Hamiltonian Hγ,1 is a convex function of the variable ξ. The main difficulty that remains is to establish that f1 is a Lipshitz function of the independent random variables ξ with a constant k that is small with large probability. To prove the Lipshitz continuity of f1 it is obviously enough to prove the corresponding bounds for Hγ,1 (σ) and ± ± Wγ,1,∂ ± 1 (σ1 |m(µ ,s ) ). Let us first prove that Hγ,1 (σ) is Lipshitz in the random variable ξ. Let us write ξ ≡ ξ[ω] and ξˆ ≡ ξ[ω 0 ]. Denoting by ξ µ σ the coordinatewise product of the two vectors ξ µ and σ and Jγ (i − j) the symmetric γ −1 |1| × γ −1 |1| matrix with i, j entries, we have
Distribution of Overlap Profiles in 1-D Kac–Hopfield Model
349
X M h h i i ξ µ σ − ξˆµ σ , Jγ ξ µ σ + ξˆµ σ . |Hγ,1 [ω](σ) − Hγ,1 [ω 0 ](σ)| = µ=1 1/2
Since Jγ is a symmetric and positive definite matrix, its square root Jγ using the Schwarz inequality we may write P M µ=1 [ξ µ σ − ξˆµ σ], Jγ [ξ µ σ + ξˆµ σ] ≤
P µ=1
(5.5)
exists. Thus
1/2 1/2 kJγ [ξ µ σ − ξˆµ σ]k2 kJγ [ξ µ σ + ξˆµ σ]k2
(5.6)
≤ J +J − where
1/2 M X J + ≡ ([ξ µ σ + ξˆµ σ], Jγ [ξ µ σ + ξˆµ σ])
(5.7)
µ=1
and J−
1/2 M X ˆ 2. ≡ ([ξ µ σ − ξˆµ σ], Jγ [ξ µ σ − ξˆµ σ]) ≤ kξ − ξk
(5.8)
µ=1
The last inequality in (5.8) follows since kJγ k ≤ 1. On the other hand, by convexity PM PM (J + )2 ≤ 2 µ=1 (ξ µ σJγ ξ µ σ) + 2 µ=1 (ξˆµ σJγ ξˆµ σ)
(5.9)
0
= 2Hγ,1 [ω](σ) + 2Hγ,1 [ω ](σ) Collecting, we get |Hγ,1 [ω](σ) − Hγ,1 [ω 0 ](σ)| ≤
√
1/2
ˆ 2 Hγ,1 [ω](σ) + Hγ,1 [ω 0 ](σ) 2kξ − ξk
. (5.10)
This means that as in [T2], we are in a situation where the upper bound for the Lipshitz norm of Hγ,1 [ω](σ) is not uniformly bounded. However the estimates of Sect. 2, allow us to give reasonable estimates on the probability distribution of this Lipshitz norm. Recalling (2.5) we have # " P
sup |1Hγ,1 (σ)| ≥ γ −1 |1|(16(1 + c))γ` + 4γM ) ≤ 16e−cγ
−1
|1|
.
(5.11)
σ∈S1
Therefore, using (2.1) we get h i P supσ∈S1 |Hγ,1 (σ)| ≥ γ −1 |1|(C + (16(1 + c))γ` + 4γM ) ≤ 16e
h
i
(5.12)
2(m` (x, σ), m` (y, σ)) ≤ km` (x, σ)k22 + km` (y, σ)k22 .
(5.13)
−Cγ −1 |1|
+ P supσ∈S1 |γ
−1
` Eγ,1 (m` (σ))|
≥ Cγ
−1
1 .
To estimate this last probability, we notice that by convexity
350
A. Bovier, V. Gayrard, P. Picco
Therefore
P ` (m` (σ))| = 1/2 x,y∈1 Jγ` (x − y)(m` (x, σ), m` (y, σ)) |γ −1 Eγ,1 ≤ `/2
P
(5.14)
2 x∈1 km` (x, σ)k2
Now we have i h P P supσ∈S1 ` x∈1 km` (x, σ)k22 ≥ 2Cγ −1 |1| ≤ 2γ ≤2
−1
|1|
P P ` x∈1 km` (x, σ)k22 ≥ 2Cγ −1 |1|
γ −1 |1|
inf 0≤t<1/2 e
−2Cγ −1 |1|t
Q
QM x∈1
µ=1
Ee
(5.15) t`
1 `
P i∈x
ξiµ σi
2 .
Using the well known inequality [BG1] !2 X µ 1 1 ξi σ i ≤ √ E exp t` , ` i∈x 1 − 2t
(5.16)
and choosing t = 1/4, the r.h.s of (5.15) is bounded from above by C − (1 + M/2`) ln 2 . exp −γ −1 |1| 2
(5.17)
Collecting, we get # " X −1 2 −1 km` (x, σ)k2 ≥ γ |1|2 c + (1 + M/2`) ln 2 ≤ e−cγ |1| , (5.18) P sup ` σ∈S1
x∈1
which implies, if γ`, γM and M/` are small enough, that # " P
sup |Hγ,1 (σ)| ≥ γ −1 |1|(4c + 1) ≤ 17e−cγ
−1
|1|
,
(5.19)
σ∈S1
which is the estimate we wanted. − + [ω] (respectively Wγ,1 [ω]) the terms To treat the boundary terms let us call Wγ,1 corresponding to interactions with the left (respectively right) part of the boundary ∂1. − + [ω], the one of Wγ,1 [ω] being completely We estimate first the Lipshitz norm of Wγ,1 identical, − [ω](σ1 , m(µ |Wγ,1
−
,s− )
− ) − Wγ,1 [ω 0 ](σ1 , m(µ
−
,s− )
)|
P P − − J (i − j) ≤ a(β) i∈1 σi (ξiµ − ξˆiµ ) − j∈∂ 1 γ ≤ a(β)
P
−
−
µ − ξˆiµ )2 i∈1 (ξi
ˆ 2 ≤ γ 1/2 a(β)kξ − ξk 2 ˆ 2, ≤ γ 1/2 kξ − ξk 2
1/2 P
P i∈1
j∈δ − 1 Jγ (i − j)
2 1/2
(5.20)
Distribution of Overlap Profiles in 1-D Kac–Hopfield Model
351
where we have used the Schwarz inequality and 2 X X Jγ (i − j) ≤ γ −1 . i∈1
(5.21)
j∈∂ − 1
Therefore if we denote by ( B ≡
) γ −1 |1|M
ξ ∈ [−1, +1]
| sup |Hγ,1 (σ)| ≤ γ
−1
|1|(4c + 1) ,
(5.22)
σ∈S1
using (5.3), (5.19), (5.20) and some easy computations, we get i h ± ± ± ± −1 −1 t2 P f1(µ ,s ) − Mf1(µ ,s ) ≥ tγ −1 |1| ≤ 68e−cγ |1| + 68e− 16(4c+2) γ |1| . (5.23) Choosing c such that c =
t2 16(4c+2)
we get −1 h ± ± i √ ± ± γ P f1(µ ,s ) − Mf1(µ ,s ) ≥ tγ −1 |1| ≤ 136 exp − |1|( 1 + t2 − 1) . 8 (5.24) Finally, a simple calculation shows that (5.24) implies that q ± ± ± ± (5.25) |Mf1(µ ,s ) − Ef1(µ ,s ) | ≤ 544 γ −1 |1|, and this implies the claim of Proposition 5.2.
We will mainly use Proposition 5.2 in the following form: Corollary 5.3. If |1| ≤ γ −1 g(γ) for some g(γ) with g(γ) ↓ 0 and γ −1 g(γ) > c, for −1/2 for all γ small enough then there exists a set g with P[g ] ≥ 1 − Ke−c(g(γ)) some positive constants c and K, such that for all ω ∈ g , i h ± ± (µ± ,s± ) − E ln Z1(µ ,s ) ≤ βγ −1 (g(γ))1/4 . (5.26) ln Z1 Proof. The Corollary follows from Proposition 5.2 by choosing t = γ 1/2 |1|−1/2 (g(γ))−1/4 6. Localization of the Gibbs Measures II: The Block-Scale 6.1. Finite volume, free boundary conditions. Instead of dealing with the measures (µ± ,s± ) [ω] immediately, we will first consider the simpler case of Gibbs measures in Gβ,γ, 3 a finite volume 3 ≡ [v− , v+ ] of order |3| = o(γ −1 ) with free (Dirichlet) boundary conditions. This will be considerably simpler and the result will actually be needed as a basic input in order to deal with the full problem. On the other hand, the result may be seen as interesting in its own right and exhibits, to a large extent, the main relevant features of the model. This may indeed satisfy many readers who may not wish to follow the additional technicalities. With this in mind, we give a more detailed exposition of this case. Our basic result here will be that the free boundary conditions measure in volumes small compared to γ −1 are concentrated on “constant profiles” with very large probability. More precisely, we have
352
A. Bovier, V. Gayrard, P. Picco
Theorem 6.1. Assume that γ|3| ↓ 0, β large enough (β > 1) and γM (γ) ↓ 0. Then we can find γ −1 Lˆ 1 and ζˆ ↓ 0, such that on a subset 3 ⊂ −1/2 with P(c3 ) ≤ e−cg (γ) , where g(γ) ↓ 0 and γ −1 g(γ) > c, we have that for all ω ∈ 3 , ˆ ˆ ζ) −Lh( Gβ,γ,3 [ω] ∃u∈3 ηζ, (6.1) ˆL ˆ (u, σ) = 0 ≤ e and
ˆ ˆ ζ) −Lh( , Gβ,γ,3 [ω] ∃u∈3 ηζ, ˆL ˆL ˆ (u, σ) 6= ηζ, ˆ (u + 1, σ) ≤ e
(6.2)
where h(ζ) = cβζ(ζ) and (ζ) is defined in Proposition 4.1. The proof of this theorem relies on a large deviation type estimate for events that take place on a scale much smaller than the size of 3. We will consider events F that are in the cylinder algebra with base I = [u− , u+ ] ⊂ 3, where |I| 1/(γ`) is very small compared to 3 and that in addition are measurable with respect to the sigma-algebra ± ± generated by the variables {m` (σ, x)}x∈I . Let us define the functions U1(µ ,s ) and ± ± (µ ,s ) F1,β,ρ by U1(µ
±
,s± )
(m` ) ≡ γ`
P
+ γ` and
±
±
x,y∈1
P
(µ ,s ) (m` ) ≡ U1(µ F1,β,ρ
Jγ` (x − y)
km` (x)−m` (y)k22 4
x∈1,y∈∂1 Jγ` (x − y)
±
,s± )
(m` ) + γ`
X
km` (x)−m(µ 2
± ,s± ) 2 k2
fx,β,ρ (m` (x)),
(6.3)
(6.4)
x∈1
where fx,β,ρ (m` (x)) ≡ −
β` 2 1 ln Eσ e 2 km` (σ,x)k2 1I{km` (σ,x)−m` (x)k2 ≤ρ} . β`
(6.5)
For any δ > 0 define the δ-covering Fδ of F as Fδ ≡ {σ|∃σ0 ∈F : ∀x∈I km` (σ, x) − m` (σ 0 , x)k2 < δ}.
(6.6)
With these notations we have the following large deviation estimates: Theorem 6.2. Let F and Fδ be as defined above. Assume that |3| ≤ g(γ)γ −1 where g(γ) satisfies the hypothesis of Corollary 5.3. Then there exist `, L, ζ, R −1/2 all depending on γ and a set 3 ⊂ with P[c3 ] ≤ Ke−c(g(γ)) + e−cR/γ such that for all ω ∈ 3 , γ β
ln Gβ,γ,3 [ω](F )
h i (µ± ,s± ) (1,1,1,1) ≤ − inf µ± ,s± ,±(w± −u± )≤R inf m` ∈F F[w (m ) − inf F (m ) ` m ` ` [w− ,w+ ],β,γ − ,w+ ],β,γ +er(`, L, M, ζ, R), (6.7) and for any δ > 0, for γ small enough
Distribution of Overlap Profiles in 1-D Kac–Hopfield Model γ β
353
ln Gβ,γ,3 [ω](Fδ )
≥ − inf µ± ,s± ,±(w± −u± )≤R h i (µ± ,s± ) (1,1,1,1) inf m` ∈F F[w (m` ) − inf m` F[w (m` ) − ,w+ ],β,γ − ,w+ ],β,γ
(6.8)
−er(`, L, M, ζ, R), ˆ R) is a function of α ≡ γM that tends to zero as α ↓ 0. where er(`, L, M, ζ, Proof. Relative to the interval I we introduce again the partition S from Sect. 4. While we will use again the estimate (4.10) we treat the terms corresponding to SR somewhat differently. Let us introduce the constrained partition functions Zβ,γ,3 [ω](F ) ≡ Gβ,γ,3 [ω](F )Zβ,γ,3 [ω].
(6.9)
Just as in Proposition 4.1 we have that −
Zβ,γ,3 (F ∩ A(µ± , s± , w± )) ≤ Zβ,γ,3− ({η(w− , σ) = s− eµ }) ±
±
+
(µ ,s ) (F )Zβ,γ,3+ ({η(w+ , σ) = s+ eµ })e8γ ×Zβ,γ,1
−1
(6.10) (ζ+2γL)
and −
Zβ,γ,3 (F ∩ A(µ± , s± , w± )) ≥ Zβ,γ,3− ({η(w− , σ) = s− eµ }) (µ± ,s± ) (F )Zβ,γ,3+ ({η(w+ , σ) ×Zβ,γ,1
+ µ+
= s e })e
−8γ −1 (ζ+2γL)
(6.11) ,
where 1 = [w− + 21 , w+ − 21 ] and 3± are the two connected components of the complement of 1 in 3. Using the trivial observation that Zβ,γ,3 ≥ Zβ,γ,3 (A(µ± = 1, s± = 1, w± )),
(6.12)
this combines to ±
±
±
Gβ,γ,3 (F ∩ A(µ , s , w± )) ≤
±
(µ ,s ) Zβ,γ,1 (F ) (1,1,1,1) Zβ,γ,1 −
×
Zβ,γ,3− ({η(w− , σ) = s− eµ }) Zβ,γ,3+ ({η(w+ , σ) = s+ eµ }) Zβ,γ,3− ({η(w− , σ) = e1 }) Zβ,γ,3+ ({η(w+ , s) = e1 })
×e16γ
−1
+
(6.13)
(ζ+2γL)
The point is now that the ratios of partition functions on 3± are in fact “close” to one. Indeed we have Lemma 6.3. Let 3 = [w− − 21 , w+ + 21 ] with |3| ≤ γ −1 g(γ), where g(γ) ↓ 0 and g(γ)/γ ≥ c > 0. Then − ln Zβ,γ,3 ({η(w− , σ) = s− eµ }) − ln Zβ,γ,3 ({η(w− , σ) = e1 }) (6.14) −1 1/4 (g(γ)) + 10ζ + 48γL ≤ βγ with probability greater than 1 − e−cγ
−1
−1/2
− Ke−c(g(γ))
.
354
A. Bovier, V. Gayrard, P. Picco −
−
(0,0,µ ,s ) Proof. Let us denote by Zβ,γ, the partition function with free boundary condi3\w− −
−
tion on ∂ + 3 and mesoscopic boundary condition m(µ ,s ) at w− (see the lines following (2.4) and (4.9) for the notation). Introducing a carefully chosen zero and using the triangle inequality, we then see that − ln Zβ,γ,3 ({η(w− , σ) = s− eµ }) − ln Zβ,γ,3 ({η(w− , σ) = e1 }) − ≤ ln Zβ,γ,3 ({η(w− , σ) = s− eµ }) (0,0,µ− ,s− ) (0,0,1,1) 1 + ln Z − ln Z ({η(w , σ) = e }) − ln Zβ,γ, β,γ, 3 − 3\w− β,γ,3\w−
(0,0,µ− ,s− ) (0,0,µ− ,s− ) + ln Zβ,γ, − E ln Z 3\w− β,γ,3\w−
(6.15)
(0,0,µ− ,s− ) (0,0,1,1) − E ln Zβ,γ, + E ln Zβ,γ, 3\w− 3\w−
(0,0,1,1) (0,0,1,1) − ln Z + E ln Zβ,γ, 3\w− β,γ,3\w− .
The third term on the right-hand side of (6.15) is zero by symmetry, while the second and fourth are bounded by Corollary 5.3 by γ −1 (g(γ))−1/4 with probability at least −1 −1/2 . To bound the first term we proceed as in the proof of 1 − e−cγ − Ke−c(g(γ)) Proposition 4.1, Part i, that is we use the same decomposition as in (4.4) and (4.53). This gives that −
−
−
−
(0,0,µ ,s ) = ln Zw− ,β,γ ({η(w− , σ) = s− eµ }) ln Zβ,γ,3 ({η(w− , σ) = s− eµ }) − ln Zβ,γ, 3\w−
+ O 4γ −1 (ζ + 2γL)
(6.16) The constraint partition function on the block w− is easily dealt with. First, we note that by (2.5) with probability greater than 1 − exp(−cγ −1 ) we can replace the Hamiltonian by its blocked version on scale L at the expense of an error of order γ −1 (16γL). Then we can repeat the steps (4.56) to (4.58) and use Lemma 3.1 to get that with the same probability, −
ln Zw− ,β,γ ({η(w− , σ) = s− eµ }) (6.17) ≥ −βγ −1 φ(a(β)) + ζ 2 + lnln L2 − βγ −1 (16γL), q provided ζ ≥ 2 M L . Using (4.59) and the large deviation bound (3.3), we also get − ln Zw− ,β,γ ({η(w− , σ) = s− eµ }) ≤ −βγ −1 φ(a(β)) − 21 ζ 2 + βγ −1 (16γL). (6.18) The same bounds hold of course for the term with (s− , µ− ) replaced by (1, 1), so that we get an upper bound 3 2 −1 48γL + 8ζ + ζ (6.19) βγ 2 for the first term on the right of (61.9). Putting all things together, we arrive at the assertion of the lemma.
Distribution of Overlap Profiles in 1-D Kac–Hopfield Model
355
Lemma 6.3 asserts that to leading order, only the first ratio of partition functions is relevant in (6.13). On the other hand, since by Proposition 4.1, Part (ii), we only need to consider the case |1| ≤ R, we can use the block approximation on scale ` for those, committing an error of order βγ −1 (Rγ`) only. We will make this precise in the next lemma. Lemma 6.4. For any (µ± , s± , w± ) and I ⊂ 1 ⊂ 3 and any F that is measurable with respect to the sigma algebra generated by {m` (σ, x)}x∈I ,
γ β
ln
± ,s± ) (F ) β,γ,1 Z (1,1,1,1) β,γ,1
Z (µ
±
±
(µ ,s ) (1,1,1,1) ≤ − inf m` ∈F F1,β,ρ (m` ) + inf m` F1,β,ρ (m` ) 2` | + |1| M + c0 |1|γ` + |1|γM | ln M `
(6.20)
and ∀δ > 0 for sufficiently small γ
γ β
ln
± ,s± ) (Fδ ) β,γ,1 Z (1,1,1,1) β,γ,1
Z (µ
±
±
(µ ,s ) (1,1,1,1) ≥ − inf m` ∈F F1,β,ρ (m` ) + inf m` F1,β,ρ (m` ) 2` | + |1| M − c0 |1|γ` + |1|γM | ln M `
(6.21)
with probability greater than 1 − e−c|1|/γ . Proof. Using Lemma 2.1, we see that
(µ± ,s± ) (F ) Zβ,γ,1
≤ Eσ 1I{m` (σ)∈F } × eβγ
−1
`,L ` (µ −βγ −1 Eγ, 1 (m` (σ))+Eγ,1 m` (σ1 ),m e
± ,s± )
(6.22)
40|1|γ`
and
(µ± ,s± ) (F ) Zβ,γ,1
≥ Eσ 1I{m` (σ)∈F } × e−βγ
Now
−1
`,L ` (µ −βγ −1 Eγ, 1 (m` (σ))+Eγ,1 m` (σ1 ),m e
40|1|γ`
.
± ,s± )
(6.23)
356
A. Bovier, V. Gayrard, P. Picco
± ± `,L ` (m` (σ1 )) + E1,∂1 E1 m` (σ1 ), m(µ ,s ) ± ± `,L ` (m` (σ1 )) + E1,∂1 = E1 m` (σ1 ), m(µ ,s ) +γ`
X km` (σ, x)k2 X [a(β)]2 2 + γ` 2 2 x∈1 x∈∂1
X km` (σ, x)k2 X [a(β)]2 2 −γ` − γ` 2 2 x∈1 x∈∂1 X 1 Jγ` (x − y) (m` (σ, x), m` (σ, y)) = − γ` 2 (x,y)∈1×1 X ± ± Jγ` (x − y) m` (x, σ), m(µ ,s ) − γ`
(6.24)
x∈1,y∈∂1
X 1 X1 ± ± ± ± (m` (x, σ), m` (x, σ)) + γ` m(µ ,s ) , m(µ ,s ) + γ` 2 2 x∈1 x∈∂1
X [a(β)]2 X km` (σ, x)k2 2 − γ` . − γ` 2 2 x∈1 x∈∂1
On the other hand γ`
X
Jγ` (x − y)
x,y∈1
+ γ`
X
km` (σ, x) − m` (σ, y)k22 4
Jγ` (x − y)
x∈1,y∈∂1
km` (σ, x) − m(µ 2
±
,s± ) 2 k2
X
1 Jγ` (x − y) (m` (σ, x), m` (σ, y)) 2 x,y∈1 X ± ± 1 m` (σ, x), m(µ ,s ) Jγ` (x − y) − γ` 2
= −γ`
x∈1,y∈∂1
X
1 Jγ` (x − y) km` (σ, x)k22 2 x,y∈1 X 1 1 km` (σ, x)k22 + [a(β)]2 + γ` Jγ` (x − y) 2 2
+ γ`
x∈1,y∈∂1
X
1 Jγ` (x − y) (m` (σ, x), m` (σ, y)) 2 x,y∈1 X ± ± 1 m` (σ, x), m(µ ,s ) Jγ` (x − y) − γ` 2
= −γ`
x∈1,y∈∂1
X1 km` (σ, x)k22 + γ` + γ` 2 x∈1
X x∈1,y∈∂1
1 Jγ` (x − y) [a(β)]2 . 2
(6.25)
Distribution of Overlap Profiles in 1-D Kac–Hopfield Model
357
Comparing (6.24) and (6.25) we find that ± ± `,L ` (m` (σ1 )) + E1,∂1 m` (σ1 ), m(µ ,s ) E1 + γ`
X km` (σ, x)k2 X [a(β)]2 2 + γ` 2 2 x∈1 x∈∂1
X
= γ`
Jγ` (x − y)
x,y∈1
X
+ γ`
Jγ` (x − y)
x∈1,y∈∂1
X
− γ`
x∈1,y∈∂1
≡ U1µ
±
,s±
km` (σ, x) − m` (σ, y)k22 4 km` (σ, x) − m(µ 2
±
,s± ) 2 k2
(6.26)
1 Jγ` (x − y) [a(β)]2 2
(m` (σ1 )) − C(|1|, β),
where C(|1|, β) is an irrelevant σ-independent constant that will drop out of all relevant formulas and may henceforth√be ignored. For suitably chosen ρ we introduce a lattice WM,ρ in RM with spacing ρ/ M . Then for any domain D ⊂ RM , the balls of radius ρ centered at the points q of WM,ρ ∩D cover D. For reasons that should be clear from Sect. 3, we choose ρ = 2
M ` .
With probability greater than 1 − exp(−c`), fx,β,ρ (m` (x)) = ∞
kmk22
> 2, while if the number of lattice points within the ball of radius 2 are bounded 2` . But this implies that by exp M ln M ` ± ± −βγ −1 E1 m` (σ1 ),m(µ ,s ) (m` (σ1 )+E `,L 1,∂ 1 ln Eσ1 1I{m` (σ)∈F } e (6.27) h i (µ± ,s± ) 2` (m` ) − C(|1|, β) + |1| M | ln M | + 2M , ≤ −γ −1 β inf m` ∈F F1,β,ρ ` q and also, if δ > 2
M ` ,
` ± ± −βγ −1 E1 m` (σ1 ),m(µ ,s ) (m` (σ1 )+E `,L 1,∂ 1 ln Eσ1 1I{m` (σ)∈Fδ } e h i (µ± ,s± ) (m` ) − C(|1|, β) − |1|2 M ≥ −γ −1 β inf m` ∈F F1,β,ρ ` .
(6.28)
Treating the denominator in the first line of (6.13) in the same way and putting everything together concludes the proof of the lemma. An immediate corollary of Lemma 6.4 is Lemma 6.5. For any (µ± , s± , w± ), |3| ≤ γ −1 g(γ) and any F that is measurable with respect to the sigma algebra generated by {m` (σ, x)}x∈I ,
358
A. Bovier, V. Gayrard, P. Picco γ β
˜ ± , s± , w± )) ln Gβ,γ,3 (F ∩ A(µ ±
±
(µ ,s ) (1,1,1,1) (m` ) + inf m` F1,β,ρ (m` ) ≤ − inf m` ∈F F1,β,ρ
+ c0 γL + (g(γ))1/4 + ζ + |1|γ` + |1|γM | ln −1/2
with probability greater than 1 − Ke−c(g(γ)) numerical constants c, c0 , K.
2` M|
(6.29) + |1| M `
− 2e−c/γ for some finite positive
Proof. This is an immediate consequence of (6.13) and Lemmata 6.3 and 6.4.
We are now set to prove the upper bound in Theorem 6.2. Using the notation of Sect. 4 we have that c ) ln Gβ,γ,3 (F ) ≤ ln Gβ,γ,3 (F ∩ SR ) + Gβ,γ,3 (F ∩ SR c Gβ,γ,3 (F ∩ SR ) = ln Gβ,γ,3 (F ∩ SR ) + ln 1 + Gβ,γ,3 (F ∩ SR ) ≤ 4M 2 2R
sup µ± ,s± ,±(w± −u± )≤R
+ ln 1 +
ln Gβ,γ,3 (F ∩ A(µ± , s± , w± ))
exp (−c2 βLRζ(ζ)) Gβ,γ,3 (F ∩ SR )
(6.30)
,
where we used (4.14). We see that the last term can be made irrelevantly small by choosing R sufficiently large. In fact, since we will consider events F whose probability will be at least of order exp(−γ −1 βC), it will suffice to choose R
1 . γLζ(ζ)
(6.31)
On the other hand, in order for the error terms in (6.20) to go to zero, we must assure that (note that |1| = |I| + 2R is of order R) R(γ` + M ` ) tends to zero. With α ≡ γM , this means α ↓ 0. (6.32) R γ` + γ` √ √ From this we see that ` should be chosen as γ` = α while R must satisfy R α ↓ 0. (6.31) and (6.32) impose conditions on L and ζ, namely that √ α γLζ(ζ)
↓ 0.
(6.33)
Of course we also need that ζ ↓ 0 and γL ↓ 0, but clearly these constraints can be satisfied provided that α ↓ 0 as γ ↓ 0. Thus the upper bound of Theorem 6.2 follows. To prove the lower bound, we will actually need to make use of the upper bound. To do so, we need more explicit control of the functional F , i.e. we have to use the explicit bounds on fx,β,ρ (m` (x)) in terms of the function Φ from Lemma 3.1. Lemma 6.6. The functional F defined in (6.4) satisfies ±
±
(µ ,s ) (m` ) ≥ U1(µ F1,β,ρ
and
±
,s± )
(m` ) + γ`
X
1 Φx,β (m` (x)) − |1|ρ2 2 x∈1
(6.34)
Distribution of Overlap Profiles in 1-D Kac–Hopfield Model
359
(1,1,1,1) inf F1,β,ρ (m` ) ≤ |1|φβ (a(β)) + |1| m`
where φβ (a) ≡
a2 2
ln 2 `β
(6.35)
− β −1 ln cosh(βa).
Proof. Equation (6.34) follows straightforward from (3.3). To get (6.35), just note that U is non-negative and is equal to zero for any constant m` , while from Lemma 3.1 it follows that inf fx,β,ρ (m` (x)) ≤ inf Φx,β (m` (x)) +
m` (x)
m` (x)
ln 2 `β
ln 2 `β ln 2 = φβ (a(β)) + . `β
≤ Φx,β (m(1,1) ) +
(6.36)
This concludes the derivation of the upper bound. We now turn to the corresponding lower bound. What is needed for this is an upper bound on the partition function that would be comparable to the lower bound (6.12). Now X Eσ e−βH3 (σ3 ) 1I{η(w± ,σ)=s± eµ± } Zβ,γ,3 = (µ± ,s± )
Zβ,γ,3
×P =
X
(µ± ,s± )
Eσ e
−βH3 (σ3 ) Eσ e{η(w ,σ)=s± eµ± } ±
−βH3 (σ3 )
(µ± ,s± )
×
=
1I{η(w± ,σ)=s± eµ± }
Zβ,γ,3 1 − 1I{η(w± ,σ)=0}
(6.37)
e−βH3 (σ3 )
Eσ X Eσ e−βH3 (σ3 ) (µ± ,s± )
−1 ×1I{η(w± ,σ)=s± eµ± } 1 − Gβ,γ,3 {η(w± , σ) = 0} . This is almost the same form as the one we want, except for the last factor. The point is now that we want to use our upper bound from Theorem 6.2 to show that Gβ,γ,3 {η(w± , σ) = 0} is small, e.g. smaller than 1/2, so that this entire factor is negligible on our scale. Remembering our estimate (4.10), one may expect an estimate of the order exp(−c2 βLζ(ζ)), up to the usual errors. Unfortunately, these errors are of order exp(±βγ −1 (ζ + γL)) and thus may offset completely the principal term. A way out of this apparent dilemma is given by our remaining freedom of choice in the parameters ζ and L; that is to say, to obtain the lower bound, we will use a ζˆ and a Lˆ such that first ˆ ζ) ˆ γ −1 ζ + L. This is they still satisfy the requirement (61.223) while second c2 Lˆ ζ(( clearly possible. With this in mind we get Lemma 6.7. With the same probability as in Lemma 6.5,
360
A. Bovier, V. Gayrard, P. Picco
γ ˆ ˆ ˆ 1 − δ (δ ζ) ln Gβ,γ,3 {ηζ, ˆL ˆ (w± , σ) = 0} ≤ −γ Lζ β 2 − δ ζˆ 2` . | + RM + c0 γL + (g(γ))1/4 + ζ + Rγ` + RγM | ln M `
(6.38)
Proof. The proof of this lemma is very similar to the proof of (ii) of Proposition 4.1, except that in addition we use the upper bound of Lemma 6.5 to reduce the error terms. We will skip the details of the proof.
Choosing Lˆ and ζˆ appropriately, we can thus achieve that −1 ≤ 2 so that 1 − Gβ,γ,3 {η(w± , σ) = 0} Zβ,γ,3 ≤ 2
X (µ± ,s± )
Eσ e−βH3 (σ3 ) 1I{η(w± ,σ)=s± eµ± } −
≤ 2(2M )2 sup Zβ,γ,3− ({η(w− σ) = s− eµ }) µ± ,s±
(µ± ,s± ) ×Zβ,γ,1 Zβ,γ,3+ ({η(w+ σ)
+
= s+ eµ })e+8γ
(6.39) −1
ˆ ˆ β(ζ+2γ L)
(we will drop henceforth the distinction between Lˆ and L and ζˆ and ζ). The first and third factor in the last line are, by Lemma 6.3, independent of µ± , s± , up to the usual errors. The second partition function is maximal for (µ+ , s+ ) = (µ− , s− ), (this will be −1 −1/2 , shown later). Thus with probability greater than 1 − e−cγ − Ke−c(g(γ)) ±
±
±
Gβ,γ,3 (F ∩ A(µ , s , w± )) ≥
±
(µ ,s ) Zβ,γ,1 (F ) (1,1,1,1) Zβ,γ,1
0
e−c βγ
−1
(ζ+γL+(g(γ))1/4 )
(6.40)
for some numerical constant c, c0 . Using the second assertion of Lemma 6.4 allows us to conclude the proof of Theorem 6.2. We are now ready to prove Theorem 6.1: Proof of Theorem 6.1. Notice first that the first assertion (6.1) follows immediately from Lemma 6.7. Just note that P G [ω] {η (u, s) = 0} Gβ,γ,3 [ω] ∃u∈3 ηζ, ˆL ˆ ˆ (u, σ) = 0 ≤ ˆ β,γ, 3 u∈3 ζ,L (6.41) ˆ ˆ ˆ ≤ |3|e−cβ Lζ((ζ)) ˆ z. for suitably chosen L, ˆ To prove (6.2), note that we need only consider the case where both η(u, σ) and η(u+1, σ) are non-zero. This follows then simply from the upper bound of Theorem 6.2 and the lower bound ±
±
inf µ± ,s± inf m` :η(u,m` )6=η(u+1,m` )6=0 U1(µ ,s ) (m` ) P P ≥ 41 γ` x∈u y∈u+1 Jγ` (x − y)km` (x) − m` (y)k22 Using convexity, we see that
(6.42)
Distribution of Overlap Profiles in 1-D Kac–Hopfield Model
γ`
P
P x∈u
≥ (γ`)2
Jγ` (x − y)km` (x) − m` (y)k22 P P 2 r∈u,s∈u+1 x∈r y∈s km` (x) − m` (y)k2
y∈u+1
P
ˆ −1 −2 |r−s|≤(γ L)
P ˆ 2 ≥ (γ L) ˆ 2 = (γ L)
P
361
r∈u,s∈u+1 ˆ −1 −2 |r−s|≤(γ L)
r∈u,s∈u+1 ˆ −1 −2 |r−s|≤(γ L)
P
`
Lˆ x∈r m` (x) − kmLˆ (r) −
2 mLˆ (s)k2
` ˆ L
P
2
m (y)
` y∈s
(6.43)
2
.
Inserting this inequality into (6.42) gives immediately that ±
±
inf µ± ,s± inf m` :η(u,m` )6=η(u+1,m` )6=0 U1(µ ,s ) (m` ) P (a(β))2 − 2a(β)ζˆ ≥ 41 r∈u,s∈u+1 ˆ −1 −2 |r−s|≤(γ L) ˆ 2 (a(β))2 − 2a(β)ζˆ . ≥ 18 (1 − 2γ L) From here the proof of (6.2) is obvious.
(6.44)
This concludes our analysis of the free boundary condition measure in volumes of order o(γ −1 ). We have seen that these measures are concentrated on constant profiles on some scale Lˆ γ −1 (microscopic scale). In the next subsection we will analyse the measures with fixed equilibrium boundary conditions. 6.2. Finite volume, fixed symmetric boundary conditions. To proceed in order of increasing difficulty, we consider first the case where the boundary conditions are the same on both sides of the box 3. Since these are compatible with one of the preferred constant profiles of the free boundary conditions measures and since the size of the box 3 we consider is so small that by our self-averaging results we know that the random fluctuations do not favour one of the constant values by a factor on the scale exp(βγ −1 ), we expect that the optimal profile will be the constant profile compatible with the boundary conditions. Indeed, we will prove Theorem 6.8. Assume that |3| ≤ g(γ)γ −1 , where g(γ) satisfies the hypothesis of Corollary 5.3. Then there exist `, L, ζ, R all depending on γ and a set 3 ⊂ −1/2 with P[c3 ] ≤ Ke−c(g(γ)) + e−cR/γ such that for all ω ∈ 3 , γ β
(µ,s,µ,s) ln Gβ,γ, [ω](F ) 3
h i (µ,s,µ,s) (1,1,1,1) ≤ − inf ±(w± −u± )≤R inf m` ∈F F[w (m ) − inf F (m ) ` m ` ` [w− ,w+ ],β,γ − ,w+ ],β,γ + er(`, L, M, ζ, R) (6.45) and for any δ > 0, for γ small enough γ β
(µ,s,µ,s) ln Gβ,γ, [ω](Fδ ) 3
i h (µ,s,µ,s) (1,1,1,1) ≥ − inf ±(w± −u± )≤R inf m` ∈F F[w (m ) − inf F (m ) ` m ` ` [w− ,w+ ],β,γ − ,w+ ],β,γ − er(`, L, M, ζ, R), (6.46)
362
A. Bovier, V. Gayrard, P. Picco
where er(`, L, M, ζ, R) is a function of α ≡ γM that tends to zero as α ↓ 0 and where Fδ is defined in (6.6). An immediate corollary of Theorem 6.8 is the analog of Theorem 6.1 for the measures (µ,s,µ,s) Gβ,γ, [ω]: 3 Theorem 6.9. Assume that γ|3| ↓ 0, β large enough (β > 1) and γM (γ) ↓ 0. Then we can find γ −1 Lˆ 1 and ζˆ ↓ 0, such that on a subset 3 ⊂ with −1/2 P(c3 ) ≤ e−cg (γ) , where g(γ) ↓ 0 and γ −1 g(γ) > c, we have that for all ω ∈ 3 ˆ ˆ (µ,s,µ,s) µ ≤ e−Lg(ζ) , Gβ,γ, [ω] ∃u∈3 ηζ, (6.47) ˆL ˆ (u, σ) 6= se 3 where h(ζ) = cβζ(ζ) and (ζ) is defined in Proposition 4.1. Remark . Equation (6.47) implies that with P-probability one (µ,s,µ,s) µ lim Gβ,γ, = 1. [ω] ∀u∈3 ηζ, ˆL ˆ (u, σ) = se 3 γ↓0
(6.48)
Proof of Theorem 6.8. Many of the technical steps in this proof are similar to those of the preceeding subsection, and we will stress only the new features here. Let us fix without restriction of generality (µ, s) = (1, 1). We consider again the upper bound first. Proceeding as in (6.1), the first major difference is that (6.13) is replaced by (1,1,1,1) (F ∩ A(µ± , s± , w± )) Gβ,γ, 3 −
≤
−
±
±
+
+
(1,1,µ ,s ) (µ ,s ,1,1) (µ ,s ) Zβ,γ, (F ) Zβ,γ,3+ \w+ 3− \w− Z1,β,γ (1,1,1,1) Zβ,γ, 3− \w−
(1,1,1,1) Z1,β,γ
(1,1,1,1) Zβ,γ, 3+ \w+
ecγ
−1
(ζ+γL)
,
(6.49)
where we have also used (6.16) through (6.18) to replace partition functions with boundary condition on one side and constraint on the other by partition functions with two-sided boundary conditions. While in the free boundary condition case, by symmetry, the ratios of partition functions on 3± were seen to be negligible, we will show here that they favour (µ± , s± ) = (1, 1). To make this precise, define for any box 3 ≡ [λ− , λ+ ] with |3| = o(γ −1 ), (µ, ˜ s,µ,s) ˜ Zβ,γ, 3 (µ, ˜ s,µ,s) ˜ Pβ,γ, ≡ . (6.50) 3 (1,1,1,1) Zβ,γ,3 In the case of symmetric boundary conditions, Corollary 5.3 provides the following estimates: −1 1/4 −1 1/4 (µ,s,µ,s) ≤ ecβγ (g(γ)) . (6.51) e−cβγ (g(γ)) ≤ Pβ,γ, 3 ˜ s,µ,s) ˜ for (µ, ˜ s) ˜ 6= (µ, s). Without loss All we need are thus estimates on the quantity P3(µ, of generality we may consider the case (µ, ˜ s, ˜ µ, s) = (1, 1, 2, 1) only. As shown in the forthcoming lemma, the quantity
P0 ≡
sup
[w− ,w+ ]⊂3∪∂ 3 |w− −w+ |<2R
(1,1,2,1) Zβ,γ,[w − +1,w+ −1] (1,1,1,1) Zβ,γ,[w − +1,w+ −1]
(6.52)
Distribution of Overlap Profiles in 1-D Kac–Hopfield Model
363
(1,1,2,1) with R chosen as in (6.32), will prove to be of special relevance in estimating Pβ,γ, . 3 It has in a reasonable sense the interpretation of the probability of having a “jump”. Note that the logarithm of P0 is self-averaging so that, up to the usual error terms, by Corollary 5.3, the random quantity can be replaced by the following deterministic one: (1,1,2,1) (1,1,1,1) sup exp E ln Zβ,γ,[w − E ln Z (6.53) P¯0 ≡ β,γ,[w− +1,w+ −1] . − +1,w+ −1] [w− ,w+ ]⊂3∪∂ 3 |w− −w+ |<2R
With this notation we have the Lemma 6.10. Assume that R satisfies (6.31) and that |3| < γ −1 g(γ) where g(γ) ˆ ζ, ζˆ all depending on γ is chosen as in Corollary 5.3. Then, there exists `, L, L, 00 −1/2 −c0 γ −1 R such that, with a probability greater than 1 − e − Ke−c (g(γ)) , where K, c, c0 and c00 are strictly positive numerical constants, (1,1,2,1) Pβ,γ, ≤ P¯0 eγ 3
−1
er 0 (`,L,M,ζ,R)
(6.54)
and, if in addition |3| > R, for γ small enough, −1 0 ˆ ˆ (1,1,2,1) ≥ P¯0 e−γ er (`,L,M,ζ,R) Pβ,γ, 3
(6.55)
ˆ M, ζ, R) is a function of α that tends to zero as α ↓ 0. where er0 (`, L, Remark . Lemma 6.10 states a very crucial result that can be paraphrased as follows: If the boundary conditions over a volume 3 with |3| = o(γ −1 ) require a “jump. then this jump takes place somewhere in the volume over a region smaller than 2R; in particular, and this will become evident in the proof, there will occur one single “jump”. Note that we cannot determine the precise location of this jump. The optimal position will be determined by the randomness. The proof of Lemma 6.10. relies on the important fact that, as stated in the next lemma, the quantity P0 is exponentially small. Lemma 6.11. With the notations of Lemma 6.4 we have: i)
With a probability greater than 1 − e−cM , for some constant c > 0, P0 ≥ e− 2 βγ 1
−1 2
a (β) −cβγ −1 (Rγ`+RγM | ln
e
2` M
ln 2 |+R M ` +2R ` )
.
(6.56) q
ii) There exists ζ˜0 > 0 depending on β such that for all ζ˜0 ≥ ζ˜ ≥ 2a(β)
M ` ,
0
with a probability greater than 1 − e−c M , for some constant c0 > 0, √ √ √ −1 ˜ ˜ 2` M −βγ −1 (ζ) 12((a(β))2 −4ζ˜ 2 )−3 (ζ) P0 ≤ e ecβγ (Rγ`+RγM | ln M |+R ` ) . (6.57) We will assume in the sequel that the parameters `, L, M and R satisfy the set of conditions (6.31) to (6.33) from Sect. 6.1. It is then clear that the parameter ζ˜ in part ii) of Lemma 6.11 can always be chosen in such a way that the exponential decrease of the first term in the r.h.s. of (6.57) compensates the increase of the second one. We will postpone the proof of Lemma 6.11 to the end of this subsection.
364
A. Bovier, V. Gayrard, P. Picco
Proof of Lemma 6.10. Without loss of generality we will, for convenience, consider only sets 3 of the form 3 ≡ [λ− − 21 , λ+ + 21 ], where λ± are assumed to be integers. We start with the proof of the upper bound (6.54). Let us define the set (6.58) B ≡ σ : ∀u∈3 η(u, σ) ∈ {0, e1 , e2 } . We further define 1 1 1 , if such u exists , u1 (σ) ≡ sup u ∈ [λ− − 2 , λ+ + 2 ] | η(u, σ) = e , otherwise λ− − 1 1 2 , if such u exists , u2 (σ) ≡ inf u ∈ (u1 (σ), λ+ + 2 ] | η(u, σ) = e , otherwise λ+ + 1 and we set
B(u1 , u2 ) ≡ {σ ∈ B | u1 (σ) = u1 , u2 (σ) = u2 } .
(6.59) (6.60)
(6.61)
A piece of profile between locations u1 (σ) and u2 (σ) will be called a “jump” between equilibrium (1,1) and (2, 1). For R chosen as in (6.32), we will set moreover [ B(u1 , u2 ) (6.62) C≡ λ− −1≤u1
and
[
D≡
B(u1 , u2 ).
(6.63)
λ− −1≤u1
Note that B = C ∪ D. Thus, obviously,
and
(1,1,2,1) (1,1,2,1) (1,1,2,1) = Zβ,γ, (B) + Zβ,γ, (B c ) Zβ,γ, 3 3 3
(6.64)
(1,1,2,1) (1,1,2,1) (1,1,2,1) (B) = Zβ,γ, (C) + Zβ,γ, (D). Zβ,γ, 3 3 3
(6.65)
Now, on the one hand, we have (1,1,2,1) (D) Zβ,γ, 3 (1,1,1,1) Zβ,γ, 3
≤
X
≤
(1,1,2,1) Zβ,γ, ∀u1
λ− −1≤u1
X λ− −1≤u1
(1,1,1,1) Zβ,γ, 3
Zβ,γ,[u1 ,u2 ] ∀u1
e
+cβγ −1 γL+ζ+(g(γ))1/4
,
(6.66)
where we have proceeded by complete analogy with the proof of the upper bound of Theorem 6.2 (see (6.9)–(6.13) and Lemma 6.3) to chop out the partition functions in [λ− − 21 , u1 ] and [u2 , λ+ + 21 ], and where we have dropped the boundary conditions of Zβ,γ,[u1 ,u2 ] in the numerator of the last line. This holds with a probability greater than −1/2 . Up to some minor modifications, it then follows from the proof of 1 − Ke−c(g(γ)) 0 −1/2 , Proposition 4.1, Part ii), that, with a probability greater than 1 − e−c M − Ke−c(g(γ))
Distribution of Overlap Profiles in 1-D Kac–Hopfield Model (1,1,2,1) Zβ,γ, (D) 3 (1,1,1,1) Zβ,γ, 3
≤ |3|2 e−βLRζ(ζ) e
365
cβγ −1 γL+ζ+(g(γ))1/4
.
(6.67) −1/2
On the other hand, we have also that, with a probability greater than 1 − Ke−c(g(γ)) (1,1,2,1) (1,1,2,1) X (C) Zβ,γ,[u ∀u1
,
|u2 −u1 |<2R
×ecβγ
−1
(γL+ζ+(g(γ))1/4 )
≤ 2R|3|P0 e
cβγ −1 γL+ζ+(g(γ))1/4
(6.68)
From now on we will abstain from specifying the probability with which our various estimates hold; this will straightforwardly follow from the different results called into play. Now, using the lower bound (6.56) of Lemma 6.11 and recalling that R is chosen large enough to satisfy the constraint (6.31), we see that the r.h.s of (6.67) is negligibly small compared with that of (6.68). Combining this with (6.64) and using Corollary 5.3 we then arrive at (1,1,2,1) (B) Zβ,γ, 0 −1 1/4 3 ≤ P¯0 ec βγ (γL+ζ+(g(γ)) ) . (6.69) (1,1,1,1) Zβ,γ,3 (1,1,2,1) We are therefore left to consider the constrained partition function Zβ,γ, (B c ). 3 c µ By definition, for any σ ∈ B , there must exist u ∈ 3 such that η(u, σ) = se , with (s, µ) 6∈ {(1, 1), (1, 2)}. This means that we can define the four random locations (6.70) u+1 (σ) = sup u ∈ [λ− − 21 , λ+ + 21 ] | η(u, σ) 6∈ {0, e2 } , 1 + 2 , if such u exists and , (6.71) u+2 (σ) ≡ inf u ∈ [u1 (σ), λ+ + 2 ] | η(u, σ) = e , otherwise λ+ + 1
and
1 1 1 u− (6.72) 1 (σ) ≡ inf u ∈ [λ− − 2 , λ+ + 2 ] | η(u, σ) 6∈ {0, e }, . 1 , if such u exists and , sup u ∈ [λ− − 21 , u− 2 (σ)] | η(u, σ) = e u− 2 (σ) ≡ , otherwise λ− − 1 (6.73) + (σ) ≤ u (σ). In other words, any configuration and can be sure that, for all σ ∈ B c , u− 1 1 − + in B c contains two “jumps”. The following two events, B − (u− 2 ) and B (u2 ), describe, − respectively the leftmost and rightmost of these jumps. For u2 > λ− − 1 we set B − (u− 2 ) n o 1 − − η(u, σ) = 0 ≡ σ ∀λ− −1
366
A. Bovier, V. Gayrard, P. Picco
n o B + (u+2 ) ≡ σ ∀u+2
×
(1,1,µ Zβ,γ,[λ
≤
Z (µ−− ,s1− ,µ+ ,s1 + ) sup
[u1 + 2 ,u+1 − 2 ],β,γ (1,1,1,1) Zβ,γ,[u − 1 + 2 ,u+1 − 21 ] 1
sup
λ− −1≤u−
,s− )
(µ ,s ,2,1) (B − (u− 2 )) Zβ,γ,[u+ + 1 ,λ +
− 1 1 − − 2 ,u1 − 2 ]
1
(1,1,1,1) Zβ,γ,[λ − 1 ,u− − 1 ] −
2
+
1 ++ 2 ]
(B + (u+2 ))
(1,1,1,1) Zβ,γ,[u + + 1 ,λ 2
2
1
2
2
(6.78)
.
+]
2 cβγ −1 (γL+ζ+(g(γ))1/4 )
×|3| M e 4
Clearly, each of the two terms in the second line of (6.78) is bounded as in (6.69), so that, up to the error term, we get the relation2 (1,1,2,1) Zβ,γ, (B c ) 3 (1,1,1,1) Zβ,γ, 3
≤ (P¯0 )2
sup
sup
λ− −1≤u−
−
−
+
+
1
2
1
(µ ,s ,µ ,s ) Pβ,γ,[u , − 1 + ,u+ + 1 ]
(6.79)
2
and combining this with (6.64) and (6.69) we arrive, still up to the error term, at P3(1,1,2,1) −
−
+
+
1
2
1
(µ ,s ,µ ,s ) ≤ P¯0 + (P¯0 )2 supλ− −1≤u−
1
1
2
(µ+ ,s+ )6=(1,2)
(6.80)
2
We immediately see from this recursion that the supremum over the µ± , s± will be realized for (µ− , s− ) = (µ+ , s+ ). But putting the estimates (6.51) together with the upper bound (6.57) of Lemma 6.11 and Corollary 5.3 we get that, for small enough γ, (µ,s,µ,s) P¯0 Pβ,γ,[u 1. From this the upper bound (6.54) is readily obtained. − 1 + ,u+ + 1 ] 1
3,
2
1
2
We now turn to the proof of the lower bound (6.55). First note that for any [w− , w+ ] ⊂ (1,1,2,1) (1,1,2,1) ≥ Zβ,γ, (A(1, 1, 2, 1, w± )) Zβ,γ, 3 3 (1,1,0,0) (1,1,2,1) (0,0,2,1) ({η(w− , σ) = e1 })Z1,β,γ Zβ,γ,3+ ({η(w+ , σ) = e2 })e−8γ ≥ Zβ,γ, 3−
−1
(ζ+2γL)
, (6.81) where we obtained the second inequality by proceeding just as in (6.10). The difficulty (1,1,1,1) . thus lies in establishing a corresponding upper bound for the partition function Zβ,γ, 3 2 Observe that this inequality shows in particular that the probability of having more than one jump is bounded by the square of the probability of having one jump.
Distribution of Overlap Profiles in 1-D Kac–Hopfield Model
367
But this can be done by, basically, repeating the proof of the lower bound of Theorem 6.2. I.e., we first use the decomposition (6.37) to write P
(1,1,1,1) = Zβ,γ, 3
Eσ eHγ,3 (σ3 )+Wγ,3 (σ3 ,m
µ± ,s±
(µ± ,s± )
)
1I{η(w± ,σ± )=s± eµ± } (6.82)
i−1
h
(1,1,1,1) ({η(w± , σ ± ) = 0}) 1 − Gβ,γ, 3
(1,1,1,1) anticipating that Gβ,γ, ({η(w± , σ ± ) = 0}) can be shown to be very small. We will 3 prove that Lemma 6.7 still holds when the Gibbs measure with free boundary conditions (1,1,1,1) . Assuming for the moment that this is true we get, as in (6.38) is replaced by Gβ,γ, 3 ˆ in (6.39) and with the same choices of the parameters ζˆ and L, (1,1,1,1) (1,1,0,0) Zβ,γ, ≤ 2(2M )2 sup Zβ,γ, ({η(w− , σ) 3 3− µ± ,s±
=
(µ± ,s± ) (0,0,1,1) s− eµ })Z1,β,γ Zβ,γ,3+ ({η(w+ , σ) −
= s+ eµ })e−8γ +
−1
ˆ ˆ (ζ+2γ L)
(6.83)
Next, proceeding as in (6.49) to replace constrained partition functions with free boundary condition on one side, by partition functions with two-sided boundary conditions, we have ±
−
(1,1,0,0) (µ ,s ({η(w− , σ) = s− eµ }) Z1,β,γ Zβ,γ, 3− (1,1,0,0) Zβ,γ, ({η(w− , σ) = e1 }) 3− −
−
±
±
+
≤ ec γ
−1
1/4
(ζ+2γL+(g(γ)
)
(1,1,1,1) Z1,β,γ +
(1,1,µ ,s ) (µ ,s ) (µ ,s ,1,1) cγ ≤ Pβ,γ, P Pβ,γ,3+ \w+ e 3− \w− 1,β,γ 0
±
−1
+
(0,0,1,1) Z,β,γ, ({η(w+ , σ) = s+ eµ }) 3+ (0,0,1,1) Zβ,γ, ({η(w+ , σ) = e1 }) 3+
(ζ+2γL)
(6.84)
))
where, to obtain the last line, we used (6.51) to treat terms with symmetric boundary conditions while we used, in the case of asymmetric boundary conditions, the upper bound (6.54) of Lemma 6.10 together with Lemma 6.11 and Corollary 5.3. From this and (62.30) it follows that, up to the error term, (1,1,1,1) (1,1,0,0) ≤ 2(2M )2 supµ± ,s± Zβ,γ, ({η(w− , σ) = e1 }) Zβ,γ, 3 3−
(6.85)
(1,1,1,1) (0,0,1,1) Z1,β,γ Zβ,γ,,3+ ({η(w+ , σ) = e1 }).
Following a procedure with which the reader is now well acquainted, (6.81) together with (6.85) easily yields (6.55) by choosing w− and w+ as those which satisfy the constrained supremum problem in (6.52). Of course we must ask that |1| > R to ensure that this choice is always possible. To complete the proof of (6.55) it remains to show that Lemma 6.7 holds when (1,1,1,1) . To do so, all we need is to prove the analogoe of Lemma replacing Gβ,γ,3 by Gβ,γ, 3 (1,1,1,1) . But this is an immediate consequence of (6.49). Indeed, we 6.5 for the measure Gβ,γ, 3 can for the present purpose be content with the bounds from Corollary 5.3. to estimate (although roughly) the first and third factor in the r.h.s. of (6.49), while using Lemma 6.4 to treat the middle term. This concludes the proof of Lemma 6.10. We are now ready to continue the proof of the upper bound of Theorem 6.8. Remember that we were left in (6.49) to estimate the ratios of partition functions in 3± .
368
A. Bovier, V. Gayrard, P. Picco
In the case of asymmetric boundary conditions, i.e., (µ± , s± ) 6= (1, 1), Lemma 6.10 enables us to replace these quantities by the corresponding ratios in boxes of length at least R. More precisely, consider two boxes 30 and 3 such that 30 ⊂ 3 and R < |30 | < |3| < γ −1 g(γ), where g(γ) is chosen as in Corollary 5.3. Then, Lemma 6.10 implies, for any (µ, ˜ s) ˜ 6= (µ, s), that (µ, ˜ s,µ,s) ˜ (µ, ˜ s,µ,s) ˜ Pβ,γ, ≤ Pβ,γ, e+γ 3 30
−1
ˆ (er 0 (`,L,M,ζ,R)+er 0 (`,L,M,ζ,R))
.
(6.86)
e − ≡ [w− + 1 −R, w− + 1 ] and 3 e + ≡ [w+ − 1 , w+ − 1 +R], Therefore, defining the boxes 3 2 2 2 2 adjacent to 1 on its left, respectively right-hand side, we have, up the the error terms, (1,1,1,1) Gβ,γ, (F 3
±
±
∩ A(µ , s , w± )) ≤
(µ± ,s± ) (1,1,µ− ,s− ) Z1,β,γ (F ) (µ+ ,s+ ,1,1) P − P + . (1,1,1,1) e 3 \w+ ,β,γ e 3 \w− ,β,γ Z1,β,γ
(6.87)
By (6.51) a relation of the form (6.85), and hence (6.87), trivially holds in the case of symmetric boundary conditions. From here we can easily reconstruct the ratio of e + with (1, 1, 1, 1)-boundary conditions. I.e., e ≡ 3 e− ∪ 1 ∪ 3 partition functions in 1 proceeding much along the line of the proof of the upper bound of Lemma 6.10 (using in particular (6.85)) we obtain, up to the usual error term, ±
Z (1,1,1,1) (F ∩ {η(w± , σ) = s± eµ }) e,β,γ 1 (1,1,1,1) ± ± Gβ,γ,3 (F ∩ A(µ , s , w± )) ≤ Z (1,1,1,1) e,β,γ 1 (6.88)
Z (1,1,1,1) (F ) e,β,γ 1 ≤ . Z (1,1,1,1) e,β,γ 1
The upper bound of Theorem 6.8 then follows from (6.88) and Lemma 6.4 just as the upper bound of Theorem 6.2 follows from Lemma 6.5. At this point, the proof of the lower bound (6.55) is a simple matter. In full generality, for arbitrary 1 ≡ [w− + 21 , w+ − 21 ] and any (µ, s), (µ,s,µ,s) Gβ,γ, (F ) ≥ 3
(µ,s,µ,s) Zβ,γ, (F ∩ A(µ± = µ, s± = s, w± )) 3 (µ,s,µ,s) Zβ,γ, 3
.
(6.89)
Proceeding as in (6.10) to bound the numerator in (6.88) and using (6.85) to treat the denominator we get (µ,s,µ,s) Zβ,γ, (F ∩ A(µ± = µ, s± = s, w± )) 3 (µ,s,µ,s) Zβ,γ, 3 (µ,s,0,0) (µ,s,µ,s) (0,0,µ,s) Zβ,γ, ({η(w− , σ) = seµ }) Z1,β,γ (F ) Zβ,γ, ({η(w+ , σ) = seµ }) (6.90) 3− 3+ ≈ (µ,s,0,0) . (µ,s,µ,s) (0,0,µ,s) Zβ,γ,3− ({η(w− , σ) = e1 }) Z1,β,γ Zβ,γ, ({η(w+ , σ) = e1 }) 3+ (µ,s,µ,s) for which Again, we recognise in the first and last factor above the quantities Pβ,γ, 3 we have the estimates (6.51). Thus, up to the usual error term,
Distribution of Overlap Profiles in 1-D Kac–Hopfield Model
(µ,s,µ,s) Gβ,γ, (F ) ≥ 3
369
(µ,s,µ,s) Z1,β,γ (F )
(6.91)
(µ,s,µ,s) Z1,β,γ
Now, (6.91) and Lemma 6.4 yield (6.46) by choosing w± as the solutions of the variational problem in (6.45). This completes the proof of Theorem 6.8. We finally are left to give the proof of Lemma 6.11 Proof of Lemma 6.11. To prove the lower bound (6.56), just note that for any event F satisfying the assumptions of Lemma 6.4, and, making use of the lower bound (6.21), we have h i (1,1,2,1) (1,1,1,1) (m` ) − inf m` F[λ (m` ) P0 ≥ exp −βγ −1 inf m` ∈F F[λ − ,λ+ ] − ,λ+ ] (6.92) −1 2` M ×e−cβγ (Rγ`+RγM | ln M |+R ` ) . Now, choosing the event F as F ≡ {{m` (x)}x∈I |m` (x) = a(β)e1
∀x ≤ 0, m` (x) = a(β)e2
∀x > 0},
(6.93)
it easily follows from the definition (6.4) of F together with the estimates of Lemma 3.1 and Proposition 3.2 that, under their respective assumptions, (1,1,2,1) (1,1,1,1) (m` ) − inf m` F[λ (m` ) inf m` ∈F F[λ − ,λ+ ] − ,λ+ ] (1,1,2,1) ≤ inf m` ∈F U[λ (m` ) + 2R lnβ`2 + R M ` − ,λ+ ]
≤
a2 (β) 2
(6.94)
+ 2R lnβ`2 + R M `
from which (6.56) follows. To prove the lower bound (6.57) we make use of the bound (6.20) of Lemma 4 to write P0 ≤ sup λ− −1≤u1
(6.95)
−1 2` M ×ecβγ (Rγ`+RγM | ln M |+R ` ) .
Equation (6.57) is then an immediate consequence of (6.95) together with the following proposition Proposition 6.12. There exists ζ˜0 > 0 depending on β such that for all ζ˜0 ≥ ζ˜ ≥ q M 2a(β) ` and for all boxes 1, inf m` F1(1,1,2,1) (m` ) − inf m` F1(1,1,1,1) (m` ) p p p ˜ ˜ 12((a(β))2 − 4ζ˜2 ) − 3 (ζ) ≥ (ζ) with probability greater than 1 − e−cM .
(6.96)
370
A. Bovier, V. Gayrard, P. Picco
The proof of Proposition 6.12, which is somewhat technical, will be the object of Appendix A. With this, the proof of Lemma 6.11 is concluded. 6.3. Finite volume, fixed asymmetric boundary conditions. In this last subsection we consider the case where the boundary conditions to the right and to the left of the box 3 are distinct. We would expect here that the optimal profile will be the “jump” profile compatible with these conditions. We will prove Theorem 6.13. Assume that |3| ≤ g(γ)γ −1 , where g(γ) satisfies the hypothesis of Corollary 5.3. Then there exist `, L, ζ, R all depending on γ and a set 3 ⊂ −1/2 with P[c3 ] ≤ Ke−c(g(γ)) +e−c/γ such that for all ω ∈ 3 , for any (µ, ˜ s) ˜ 6= (µ, s), γ β
(µ, ˜ s,µ,s) ˜ ln Gβ,γ, [ω](F ) 3
i h (µ, ˜ s,µ,s) ˜ (1,1,2,1) ≤ − inf ±(w± −u± )≤R inf m` ∈F F[w (m ) − inf F (m ) ` m` [w− ,w+ ] ` − ,w+ ]
(6.97)
+er(`, L, M, ζ, R), and for any δ > 0, for γ small enough, γ β
(µ, ˜ s,µ,s) ˜ ln Gβ,γ, [ω](Fδ ) 3
h i (µ, ˜ s,µ,s) ˜ (1,1,2,1) ≥ − inf ±(w± −u± )≤R inf m` ∈F F[w (m ) − inf F (m ) ` m ` ` [w− ,w+ ] − ,w+ ]
(6.98)
−er(`, L, M, ζ, R), ˆ R) is a function of α ≡ γM that tends to zero as α ↓ 0. where er(`, L, M, ζ, Proof. The proof of Theorem 6.13 presents no additional technical difficulties compared to that of Theorem 6.8. We shall thus be very brief and restrict ourselves to detail the only subtle step. This one enters in the proof of the upper bound for the quantity (µ, ˜ s,µ,s) ˜ (F ∩A(µ± , s± , w± )). From now on we will place ourselves on the subset of the Gβ,γ, 3 probability space on which our various estimates from Sect. 6.1 and 6.2 hold. Without loss of generality we may only consider the case (µ, ˜ s, ˜ µ, s) = (1, 1, 2, 1). It is a simple matter to establish that (1,1,2,1) (F ∩ A(µ± , s± , w± )) Gβ,γ, 3 −
≤
−
±
±
+
+
(1,1,µ ,s ) (µ ,s ,2,1) (µ ,s ) Zβ,γ, (F ) Z3+ \w+ ,β,γ 3− \w− Z1,β,γ (1,1,1,1) Zβ,γ, 3− \w−
(1,1,2,1) Z1,β,γ
Z3(2,1,2,1) + \w β,γ +
× ecγ
−1
(ζ+γL)
.
(6.99)
Just as in (6.87) we replace the ratios of partition functions in boxes 3± above by the e ± of length R. Thus, up to negligible errors, corresponding ratios in boxes 3 (1,1,2,1) (F Gβ,γ, 3
±
±
∩ A(µ , s , w± )) ≤
(µ± ,s± ) (1,1,µ− ,s− ) Z1,β,γ (F ) (µ+ ,s+ ,2,1) P − P + . (1,1,2,1) e 3 \w+ ,β,γ e 3 \w− ,β,γ Z1,β,γ −
(6.100) +
e with e ≡ 3 e ∪1∪3 From this we want to reconstruct the Gibbs measure in 1 (1, 1, 2, 1)-boundary conditions. Treating the numerator just as in (6.88), all we need is to show that, still up to negligible errors,
Distribution of Overlap Profiles in 1-D Kac–Hopfield Model
371
(1,1,2,1) (0,0,2,1) Z (1,1,0,0) ({η(w− , σ) = e1 })Z1,β,γ Z + ({η(w− , σ) = e2 }) ≥ Z (1,1,2,1) . (6.101) − e,β,γ e 3 ,β,γ 1 e 3 ,β,γ
To prove this we start by piecing together the first and second term in the l.h.s. of (6.101). Using in turn the lower bound (6.55) and the upper bound (6.54), (1,1,2,1) ({η(w− , σ) = e1 })Z1,β,γ Z (1,1,0,0) − e 3 ,β,γ (1,1,2,1) (1,1,1,1) = Z (1,1,0,0) ({η(w− , σ) = e1 })P1,β,γ Z1,β,γ − e 3 ,β,γ −1 0 ˆ P¯0 e−γ er (`,L,M,ζ,R) ≥ Z (1,1,1,1) − e 3 ∪1,β,γ
≥ Z (1,1,2,1) e−γ − e 3 ∪1,β,γ
−1
(6.102)
0 ˆ (er 0 (`,L,M,ζ,R)+er (`,L,M,ζ,R))
,
where we have in addition used (6.85) in the first inequality. In the same manner Z (1,1,2,1) Z (0,0,2,1) ({η(w− , σ) = e2 }) + − 3 ,β,γ e 3 ∪1,β,γ e ≥ Z (1,1,2,1) e−γ e,β,γ 1
−1
0 ˆ (er 0 (`,L,M,ζ,R)+er (`,L,M,ζ,R))
(6.103) .
Combining (6,102) and (6.103) thus gives (6.101) and making use of the later with (6.100) yields, up to the usual error term, (1,1,2,1) (F Gβ,γ, 3
Z (1,1,2,1) (F ) e,β,γ 1 ∩ A(µ , s , w± )) ≤ . Z (1,1,2,1) e 1,β,γ ±
±
(6.104)
The upper bound (6.97) of Theorem 6.13 now simply follows from (6.104) just as the upper bound of Theorem 6.8 follows from (6.88). The proof of the lower bound is a mere repetition of that of the lower of Theorem 6.8: simply substitute the boundary conditions (1, 1, 1, 1) by (1, 1, 2, 1) and use (6.10) instead of (6.84). This concludes the proof of Theorem 6.13. Finally, we want to give a characterization of the typical profile in the case of asymmetric boundary conditions. The relevant estimates and notations for this have been introduced already in the proof of Lemma 6.10. Let us define the events 0
0
(µ,s,µ ,s ) E1, 3 µ 0 µ0 . ≡ σ ∃ u0 ≤u1 ∈3 ∀λ− ≤u
(6.105)
Theorem 6.14. Assume that γ|3| ↓ 0, β large enough (β > 1) and γM (γ) ↓ 0. Then we can find γ −1 Lˆ 1 and ζˆ ↓ 0, such that on a subset 3 ⊂ with −1 0 P(c3 ) ≤ e−γ f (ζ ) we have that for all ω ∈ 3 , ˆ ˆ (µ,s,µ0 ,s0 ) (µ,s,µ0 ,s0 ) Gβ,γ, ≥ 1 − 2Re−Lc(ζ) . [ω] E1, 3 3
(6.106)
372
A. Bovier, V. Gayrard, P. Picco
Remark . This theorem implies that for any volume 3 such that γ|3| ↓ 0, we have P-almost surely, (µ,s,µ0 ,s0 ) (µ,s,µ0 ,s0 ) [ω] E = 1. (6.107) lim Gβ,γ, 1,3 3 γ↓0
(Here one may, to avoid complications with the “almost sure” statement due to the uncountability of the number of possible sequences γn , assume for simplicity that limγ↓0 is understood to be taken along some fixed discrete sequence, e.g. γn = 1/n. To show that the convergence holds also with probability one for all sequences tending to zero, one can use a continuity result as given in Lemma 2.3 of [BGP2]). Proof. The proof of this theorem follows from Lemma 6.10 and its proof. We leave the details to the reader. We are now ready to state a precise version of the main result announced in the introduction. We define the events o n (µ,s) µ . (6.108) σ ∀u∈3 ηζ, E0, ˆL ˆ (u, σ) = se 3 ≡ and set
(µ,s) E0,3 ≡ ∪(µ,s) E0, 3 ,
(6.109) 0
0
(µ,s,µ ,s ) E1,3 ≡ ∪(µ,s)6=(µ0 ,s0 ) E1, . 3
(6.110)
With this notation we have Theorem 6.15. For any macroscopic box V such that limγ↓0 γ|V k = 0, P-almost surely, (6.111) lim lim Gβ,γ,3 [ω] E0,V ∪ E1,V = 1. γ↓0 3↑Z
Proof. This theorem follows immediately Corollary 4.2 and Theorems 6.9 and 6.14. The remark after (6.107) also applies here. 7. Conclusions and Conjectures In the preceding sections we have labored hard to prove that typical profiles in the one dimensional Kac–Hopfield model are constant on a scale of the order o(γ −1 ). The careful reader will have noticed the conspicuous absence of any argument that would prove that they are non-constant on any larger scale. The reason for the absence of such an argument (µ± ,s± ) lies in Sect. 5. There, we prove upper bounds on the p fluctuations of the quantities f1 that imply that they are typically not larger than γ −1 |1|. What is not shown, and what would be needed, is that these fluctuations are actually of that size, and in particular that 0 0 for µ 6= µ0 , f1(µ,s,µ,s) − f1(µ ,s,µ ,s) typically differ by a random amount of that order. We certainly believe that this is true, but rigorous proofs of such statements are notoriously difficult to obtain and many problems in the theory of disordered systems are unsolved for very similar reasons. To our knowledge the only known method in this direction is the work of Aizenman and Wehr [AW] that yields, however, no good quantitative results for finite volume objects. In fact, it appears that even the uniqueness of the Gibbs state in two dimensions (which one should expect to be provable with this method) cannot be shown using their approach (just as, and for similar reasons, is the case in the two
Distribution of Overlap Profiles in 1-D Kac–Hopfield Model
373
dimensional spin glass). A general method that would allow to get lower bounds on fluctuations corresponding to Theorem 5.1 is thus still a great desideratum. A natural question that poses itself is of course, “What about dimensions greater than one?” Here, again, conjectures come easy, but some of them may be provable. First, as mentioned, we would expect that in dimension d = 2 we still have a unique Gibbs state. This is motivated by the fact that at least the block-approximation looks very much like a multi-state random field model, for which this result would follow from Aizenman–Wehr. But as for a proof, see above..... The same argument suggests, on the basis of the results of Bricmont–Kupiainen [BrKu] and Bovier–K¨ulske [BK] that in dimension d ≥ 3 we will have many Gibbs states, at least one for every pattern and its mirror image. We would expect that this can actually be proven, although technically this would be quite hard. To our surprise, it turns out that such a result is not even known in the ferromagnetic Kac model (see Cassandro, Marra and Presutti [CMP] for a conjecture), and techniques to take into account the weak but long range interaction in proofs of phase-stability have still to be worked out. However, this problem appears to be solvable. This entire line of research is very interesting and will be pursued in forthcoming publications.
Appendix A: Proof of Proposition 6.12 In this section we prove a lower bound on the infimum of the free energy functional over all the profiles that form an interface between a “phase” where the local overlaps are − + close to a(β)s− eµ and another one where they are close a(β)s+ eµ . In the case of the ferromagnetic Kac model, the shape of the interface was described in [COP] Chap. 6. In the case of the Kac-Hopfield model due to our restricted knowledge of the Hopfield model free energy with fixed overlaps, we cannot perform such a detailed analysis. Instead of working with the full free energy functional F defined in (61.15) we will replace it by a lower bound (that is also suitably normalized to have its minimal value e that could be chosen equal to zero) defined as follows: Given a macroscopic volume 1 without loss of generality to be [1, u3 ] we denote by Feµ e 1
±
,s±
≡Vµ e 1
±
,s±
+ γ`
X
ΦT (m` (x)),
(8.1)
e x∈int 1
where Vµ e 1
±
,s±
X
≡ γ`
Jγ` (x − y)
e x,y∈int 1 X
+ γ`
e,y∈∂ 1 e x∈int 1
km` (x) − m` (y)k22 4
Jγ` (x − y)
km` (x) − m(µ 2
±
,s± ) 2 k2
,
(8.2)
1/2 and for any ζ ≥ 2a(β)( M (cf. Proposition 3.1), ` )
Φ (m` (x)) ≡ T
0, (ζ),
if ∃µ,s km` (x) − a(β)seµ k2 ≤ ζ; otherwise.
(8.3)
374
A. Bovier, V. Gayrard, P. Picco
where (ζ) is defined in Proposition 4.1. The set of profiles that form an interface between ˜ is denoted by the (s− , µ) and the (s+ , µ+ ) within the volume 1 e ≡ T ± (1) o n − + e | m` (x) = a(β)s− eµ , ∀x≤0 , m` (x) = a(β)s+ eµ , ∀x≥y3 , m` (x), x ∈ 1
(8.4)
where y3 ≡ sup{y|y ∈ u3 } Proposition 6.12 then follows immediately from Proposition 7.1. There exists a ζ0 ≡ ξ(β, M, `) such that for all ζ, ζ0 ≥ ζ ≥ 1/2 , we have 2a(β)( M ` ) p p p ± ± 12((a(β))2 − 4ζ 2 ) − 3 (ζ) . (8.5) Feµ ,s ≥ (ζ) inf e e) 1 m` ∈T ± (1 e we denote by Proof. For any given profile in T ± (1), −
y1 ≡ sup{x ≥ 0|km` (x − 1) − a(β)s− eµ k2 ≤ ζ}.
(8.6)
the last exit of the ζ neighborhood of the (s− , µ− ) phase and +
y2 = inf{x ≥ y1 , km` (x) − a(β)s+ eµ k2 ≤ ζ}
(8.7)
the first entrance in the ζ neighborhood of the (s+ , µ+ ) phase after y1 . Notice that by e y1 and y2 exist and satisfy 0 ≤ y1 ≤ y2 ≤ y3 . We defined also the definition of T ± (1), overlap increments: (8.8) D(x) ≡ m` (x) − m` (x − 1). We write for 1 = y0 ≤ x1 ≤ y1 , −
m` (x1 ) − a(β)s− eµ =
x1 X
D(x),
(8.9)
x=1
and for i = 1, 2 and all yi ≤ xi+1 ≤ yi+1 m` (yi ) − m` (xi ) = m` (xi+1 ) − m` (yi ) =
P yi
and at last + µ+
a(β)s e
x=xi +1
Pxi+1
− m` (x3 ) =
x=yi +1 y3 X
D(x), (8.10) D(x),
D(x).
(8.11)
x=x3 +1
We define now the quantity L≡
3 X i=1
2
2
X
X
yi
xi
D(x) D(x) .
+
xi =yi−1 +1 x=yi−1 +1 x=xi +1 2 yi X
(8.12)
2
±
±
We first show that L can be bounded from above in term of V µ ,s . Then we will bound e 1 from below L by solving elementary variational problems. Putting those two bounds ± ± together will give a lower bound for V µ ,s . e 1
Distribution of Overlap Profiles in 1-D Kac–Hopfield Model
375
Let us start with the upper bound. We first perform for each value of b ∈ 1, . . . , [1/γ`] a block summation with blocks of length b, the location of the leftmost part of the first block being a point z ∈ 1, . . . , b. Explicitly, calling ub+z X
D(u, b, z) ≡
D(x),
(8.13)
x=(u−1)b+z+1
we write x1 X
D(x) =
x=1
z X
D(x) +
x −z 1 b X
x=1
D(u, b, z) + D+ (x1 , b, z),
(8.14)
u=1
where
x1 X
D+ (x1 , b, z) =
x
x=
1 −z b
D(x).
(8.15)
b+z+1
The second sum in the r.h.s of (8.14) will be called the bulk term, while the first sum and D+ will be called boundary terms. We have also yi −z b yi X X D(x) = D− (xi , b, z) + D(u, b, z) + D+ (yi , b, z) (8.16) xi +1−z x=x +1 i
u=
+2
b
with the “boundary” terms
D− (xi , b, z) =
xi +1−z b+b+z+1 b X
D(x)
(8.17)
D(x).
(8.18)
x=xi +1
and D+ (yi , b, z) =
yi X
yi −z
x=
We have also xi X
b+z+1
xi −z b X
D(x) = D− (yi−1 , b, z) +
x=yi−1 +1
b
u=
yi−1 +1−z b
D(u, b, z) + D+ (xi , b, z).
(8.19)
+2
Here the “boundary” terms are: D− (yi−1 , b, z) ≡
yi−1 +1−z b+b+z+1 b X
D(x)
(8.20)
x=yi−1 +1
and D+ (xi , b, z) ≡
xi X
x
x=
i −z b
D(x). b+z+1
(8.21)
376
A. Bovier, V. Gayrard, P. Picco
For a given b and z, the Schwarz Inequality implies
P
2
xi
x −y +1
x=yi−1 +1 D(x) ≤ i bi−1 + 2 2
×
kD− (yi−1 , b, z)k22
+
P
xi −z (8.22) b 2 2 yi−1 +1−z kD(u, b, z)k2 + kD+ (xi , b, z)k2 .
u=
+2
b
We want to take the mean of the two sides of (8.22) over all the possible choices of block lengths b in 1, . . . , [(γ`)−1 ] and z in 1, . . . , b. To do this we use a weighted mean for the block lengths and an uniform mean for the z ∈ 1, . . . b. We use P
(γ`)−1
b=1
b2 =
(γ`)−1
(γ`)−1 +1)(2 (γ`)−1 +1 6
= ( (γ`)−1 )3 31 (1 + O(γ`))
(8.23)
to define a weighted mean on 1, . . . , [(γ`)−1 ]. Performing explicitly these weighted means gives
2
P
xi
x=yi−1 +1 D(x) ≤ 2
× (γ`)−1 33(1+O(γ`)) [ ] ×
Pb
xi − yi−1 + 1 +
2 γ`
P(γ`)−1 b=1
P xib−z yi−1 +1−z kD(u, b, z)k22 + kD+ (xi , b, z)k22 , kD− (yi−1 , b, z)k22 +
z=1
u=
b
+2
(8.24) and by the very same argument
P yi
x=xi +1
×
2 D(x) 2 ≤
3 [(γ`)−1 ]3 (1+O(γ`))
yi − xi + 1 +
2 γ`
P(γ`)−1 b=1
(8.25)
Pb P yib−z 2 xi −z kD(u, b, z)k22 + D+ (yi , b, z)k22 . × z=1 kD− (xi , b, z)k2 + u=
b
+2
Collecting the “bulk” terms in (8.24) and (8.25) to bound L, it is not difficult to check that
Distribution of Overlap Profiles in 1-D Kac–Hopfield Model
3 [(γ`)−1 ]3 (1+O(γ`))
P yi
P
xi =yi−1
(γ`)−1
Pb z=1
b=1
377
xi − yi−1 + 1 +
P xib−z yi−1 +1−z kD(u, b, z)k22 + yi − xi + 1 + u=
+2
b
≤ 3 [γ`]3 (1 + O(γ`)) yi − yi−1 ×
P
(γ`)−1
b=1
Pb
P
z=1
P yi−1 ≤x,y≤yi
yi − yi−1 +
2 γ`
P yi −z bxi −z kD(u, b, z)k22 u=
b
+2
yi −z byi−1 +1−z kD(u, b, z)k22
u=
b
≤ 3 (1 + O(γ`)) γ`(yi − yi−1 ) ×
2 γ`
2 γ`
γ`(yi − yi−1 ) + 2
2
Jγ` (x − y) km` (x) − m` (y)k2 .
(8.26) It remains to consider the “boundary” terms, putting together the terms D+ (xi , b, z) and D− (xi , b, z), it is not too difficult to check that P yi P (γ`)−1 Pb 3 2 x − y + 1 + 3 i i−1 z=1 γ` [(γ`)−1 ] (1+O(γ`)) xi =yi−1 b=1
2
2
P xi +1−z
P xi b+b+z+1 b 2
+ y i − xi + 1 +
xi −z D(x) D(x) × x=xi +1 γ`
x= b b+z+1 2 2 P 2 2 ≤ 3 (yi − yi−1 + γ` )γ` yi−1 ≤x,y≤yi Jγ` (x − y) km` (x) − m` (y)k2 (8.27) Therefore we get P3 L ≤ 4 i=1 γ`(yi − yi−1 )(γ`(yi − yi−1 ) + 3) (8.28) P 2 × yi−1 ≤x,y≤yi Jγ` (x − y) km` (x) − m` (y)k2 , which is the upper bound we wanted. Now we want to bound from below L. Notice first that by solving explicitly the variational problem we have that for all m1 , m2 ∈ RM , inf m` (x) {km` (x) − m1 k22 + km` (x) − m2 k22 } (8.29) ≥ 21 km1 − m2 k22 . Using (8.6) and (8.7) and convexity, we get 2
2
L ≥ (y1 − y0 ) ζ2 + 21 (y2 − y1 )km` (y1 ) − m` (y2 )k22 + (y3 − y2 ) ζ2 ≥ 21 (y2 − y1 ) (a(β))2 − 4ζ 2 . On the other hand, cf. (8.3), we have X ΦT (m` (x)) ≥ γ`(y2 − y1 )(ζ). γ` e x∈int 1
(8.30)
(8.31)
378
A. Bovier, V. Gayrard, P. Picco
Therefore, introducing the macroscopic variables ui = γ`yi we get Feµ e 1
±
,s±
12 ≥ (u2 − u1 )(ζ) + u2 −u (a(β))2 − 4ζ 2 1 +3 q √ √ ≥ (ζ) 12 (a(β))2 − 4ζ 2 − 3 (ζ) ,
(8.32)
where the last step follows from the explicit computation of the infimum over all possible values of u2 − u1 . References [AGS] [ALR] [AW] [BG1] [BG2] [BG3] [BG4]
[BGP1] [BGP2] [BGP3] [BF] [BK] [BrKu] [CMP] [CN] [COP] [DOPT]
[EA] [vE] [FH] [FP1] [FP2]
Amit, D.J., Gutfreund, H., Sompolinsky, H.: Statistical mechanics of neural networks near saturation. Ann. Phys. 173, 30–67 (1987) Aizenman, M., Lebowitz, J.L., and Ruelle, D.: Some rigorous results on the Sherrington-Kirkpatrick spin glass model. Commun. Math. Phys. 112, 3–20 (1987) Aizenman, M., Wehr, J.: Rounding effects on quenched randomness on first-order phase transitions. Commun. Math. Phys. 130, 489 (1990) Bovier, A., Gayrard, V.: Rigorous results on the thermodynamics of the dilute Hopfield model. J. Stat. Phys. 69, 597–627 (1993) Bovier, A., Gayrard, V.: An almost sure large deviation principle for the Hopfield model. Ann. Probab 24, 1444–1475 (1996) Bovier, A. Gayrard, V.: The retrieval phase of the Hopfield model, A rigorous analysis of the overlap distribution. Probab. Theor. Rel. Fields 107, 61–98 (1996) Bovier, A., Gayrard, V.: Hopfield models as generalized random mean field models. WIAS preprint No 253, to appear in Mathematics of spin glasses and neural networks. A. Bovier and P. Picco eds, Progress in Probability. Basel–Boston: Birkh¨auser 1997 Bovier, A., Gayrard, V., Picco, P.: Gibbs states of the Hopfield model in the regime of perfect memory. Prob. Theor. Rel. Fields 100, 329–363 (1994) Bovier, A., Gayrard, V., Picco, P.: Large deviation principles for the Hopfield model and the KacHopfield model. Prob. Theor. Rel. Fields 101, 511–546 (1995) Bovier, A., Gayrard, V., Picco, P.: Gibbs states of the Hopfield model with extensively many patterns. J. Stat. Phys. 79, 395–414 (1995) Bovier, A., Fr¨ohlich, J.: A heuristic theory of the spin glass phase. J. Stat.Phys. 44, 347–391 (1986) Bovier, A., K¨ulske, Ch.: A rigorous renormalization group method for interfaces in random media. Rev. Math. Phys. 6, 413–496 (1994) Bricmont, J., Kupiainen, A.: Phase transition in the 3d random field Ising model. Commun. Math. Phys. 116, 539–572 (1988) Cassandro, M., Marra, R., Presutti, E.: Corrections to the critical temperature in 2d Ising systems with Kac potentials. J. Stat. Phys. 78, 1131–1138 (1995) Comets, F., Neveu, J.: The Sherrington-Kirkpatrick model of spin glasses and stochastic calculus, the high temperature case. Commun. Math. Phys. 166, 549–564 (1995) Cassandro, M., Orlandi, E., Presutti, E.: Interfaces and typical Gibbs configurations for onedimensional Kac potentials. Prob. Theor. Rel. Fields 96, 57–96 (1993) De Masi, A., Orlandi, E., Presutti, E., Triolo, L.: Glauber evolution with Kac potentials, I. Mesoscopic and macroscopic limits, interface dynamics. Nonlinearity 7, 633–696 (1994); II. Spinodal decomposition. to appear Edwards, P., Anderson, W.: Theory of spin glasses. J. Phys. F 5, 965–974 (1975) van Enter, A.C.D.: Stiffness exponent, number of pure states, and Almeida-Thouless line in spin glasses. J. Stat. Phys. 60, 275–279 (1990) Fisher, D.S., Huse, D.A.: Pure phases in spin glasses. J. Phys. A 20, L997-L1003 (1987); Absence of many states in magnetic spin glasses. J. Phys. A 20, L1005-L1010 (1987) Pastur, L.A., Figotin, A.L.: Exactly soluble model of a spin glass. Sov. J. Low Temp. Phys. 3(6), 378–383 (1977) Pastur, L.A., Figotin, A.L.: On the theory of disordered spin systems. Theor. Math. Phys. 35, 403–414 (1978).
Distribution of Overlap Profiles in 1-D Kac–Hopfield Model
[FP2]
379
Pastur, L.A., Figotin, A.L.: Infinite range limit for a class of disordered spin systems. Theor. Math. Phys. 51, 564–569 (1982) [FZ] Fr¨ohlich, J., Zegarlinski, B.: Some comments on the Sherrington-Kirkpatrick model of spin glasses. Commun. Math. Phys. 112, 553–566 (1987) [Ge] Geman, S.: A limit theorem for the norms of random matrices. Ann. Probab. 8, 252–261 (1980) [Ho] Hopfield, J.J.: Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci. USA 79, 2554–2558 (1982) [K] Koch, H.: A free energy bound for the Hopfield model. J. Phys. A A 26, L353–L355 (1993) [KUH] Kac, M., Uhlenbeck, G., Hemmer, P.C.: On the van der Waals theory of vapour-liquid equilibrium. I. Discussion of a one-dimensional model, J. Math. Phys. 4, 216–228 (1963); II. Discussion of the distribution functions, J. Math. Phys. 4, 229–247 (1963); III. Discussion of the critical region, J. Math. Phys. 5, 60–74 (1964). [LP] Lebowitz, J., Penrose, O.: Rigorous treatment of the Van der Waals Maxwell theory of the liquidvapour transition. J. Math. Phys. 7, 98–113 (1966) [Lu] Luttinger, J.M.: Exactly Soluble Spin-Glass Model. Phys.Rev. Lett. 37, 778–782 (1976) [Ma] Mattis, D.C.: Solvable spin system with random interactions. Phys. Lett. 56A, 421–422 (1976) [MPR] Marinari, E., Parisi, G., Ritort, F.: On the 3D Ising spin glass. J. Phys. A 27, 2687–2708 (1994) [MPV] M´ezard, M., Parisi, G., Virasoro, M.A.: Spin-glass theory and beyond. Singapore: World Scientific, 1988 [NS] Newman, Ch.M., Stein, D.L.: Non-mean-field behaviour in realistic spin glasses. Phys. Rev. Lett. 76, 515–518 (1996) [PS] Pastur, L., Shcherbina, M.: Absence of self-averaging of the order parameter in the SherringtonKirkpatrick model. J. Stat. Phys. 62 , 1–19 (1991) [SK] Sherrington, D., Kirkpatrick, S.: Solvable model of a spin glass. Phys. Rev. Lett. 35, 1792–1796 (1972) [ST] Shcherbina, M., Tirozzi, B.: The free energy for a class of Hopfield models. J. Stat. Phys. 72, 113–125 (1992) [T1] Talagrand, M.: Concentration of measure and isoperimetric inequalities in product space. Publ. Math. I.H.E.S. 81, 73–205 (1995) [T2] Talagrand, M.: A new look at independence. Ann. Probab. 23, 1–37 (1996) Communicated by D. Brydges
Commun. Math. Phys. 186, 381 – 392 (1997)
Communications in
Mathematical Physics c Springer-Verlag 1997
A Hierarchy of Defects in Biaxial Nematics Paolo Biscari, Giovanna Guidone Peroli Dipartimento di Matematica, Universit`a di Pisa, Via Buonarroti, 2 – 56127 Pisa, Italy. E-mail: [email protected], E-mail: [email protected] Received: 28 November 1994 / Accepted: 30 September 1996
Abstract: We study the topological properties of the manifolds which describe uniaxial and biaxial nematics in order to compare and classify all types of bulk and surface defects that can arise in a nematic liquid crystal. We explain how topological charges can be assigned both to isotropic defects in uniaxial nematics and to uniaxial defects in biaxial nematics; we further see how it is possible to distinguish between isolated surface defects and surface defects which extend in the bulk, and between surface defects which can relax in the bulk and defects forced to stay on the surface.
1. Introduction The topological theory of defects in structured media and its applications have been widely studied over the past years [1–3]. A body with microstructure is a continuum system endowed with some kind of microscopic order; its local properties are described by an order parameter that takes values in a manifold R. The configuration of an ordered continuum is then determined by an application from the region in space Ω occupied by the body to the manifold R. Defects in the usual sense are all singularities of such an application. The classical topological theory of defects shows that both the existence and the stability of defects depend on the topological structure of the manifold R. Nonetheless, interpreting defects as singularities is meaningful only when the degree of order is the same everywhere in Ω, as is the case within the classical theory of nematic liquid crystals which employs the real projective plane RP2 as R. When the degree of order may vary, a manifold richer in states is needed to describe transitions between differently ordered phases which take different places in space. Accordingly, defects are to be seen in a new light [4–7]. We decompose R into the union of disjoint submanifolds with different degree of order: they define a hierarchy of degeneracy in R. For nematic liquid crystals, the submanifolds of R comprise the biaxial, uniaxial, and isotropic states,
382
P. Biscari, G. G. Peroli
with a decreasing degree of order. We interpret defects as degeneracies in the order, which occur when the application of Ω into R, though smooth, traverses different submanifolds of R. The classical problem of the topological theory of defects now regards points in Ω which happen to lie among others bearing a higher degree of order. A complete study of defects for uniaxial nematics with fixed degree of orientation, both in the bulk and on a surface, has been performed in [4, 8–10]. On the other hand, bulk defects in biaxial nematics, again with fixed eigenvalues, were studied in [1–3, 11, 12]. We summarize these results, we extend them to the case of variable eigenvalues of the order tensor, and we give a complete and unifying description of surface defects that may arise both in uniaxial and in biaixal nematics. We frame our study within de Gennes’ theory [13]: this allows us to treat in the same setting both transitions: biaxial uniaxial and uniaxial isotropic. We shall describe the manifold where de Gennes’ order tensor takes values: it is a five-dimensional manifold, but we shall see that the study of its topological properties can be quite simplified by homotopy considerations (see Theorems 3.1 and 3.2). We shall therefore be able to describe the whole hierarchy of defects topologically induced in the bulk of nematic liquid crystals. Furthermore, assuming that surfaces induce in the order tensor the additional symmetry described in [14], we will give an exhaustive classification of line and point defects that can occur on a surface bounding a nematic liquid crystal. In addition, by studying the relative homotopy groups, we can decide whether a defect can leave the surface, and whether it is the trace on the surface of a bulk defect. This paper is organized as follows. In the next section we describe the manifold R where the order tensor in the bulk takes values; we see how a surface affects it, and recall the main results of the topological theory of defects. Section 3 is devoted to the study of R and of its biaxial and uniaxial submanifolds. Section 4 deals with the relative homotopy groups and gives their complete description also for the case of pure biaxial states (see Theorem 4.2), a case which to our knowledge has not yet been considered. Finally, in Sect. 5, we use the theoretical results of the preceding sections to give a complete classification of the defects that can arise in the bulk and on the surface of a nematic. 2. A Hierarchy of Nematic Defects The microscopic state of a nematic liquid crystal can be described by means of a symmetric traceless order tensor Q, that takes values in the set ) ( 3 3 X X 1 , (2.1) λi e i ⊗ e i : λi = 0, λi ≥ − Q := Q = 3 i=1
i=1
so that we can represent a configuration of a nematic liquid crystal as a continuous application from the region in space occupied by the body to the manifold Q. The local physical properties of the nematic depend on the degree of symmetry of the tensor Q. In particular, there are three disjoint subsets of Q which give information on quite different behaviours of the nematic liquid crystal: the class B := {Q ∈ Q : λi 6= λj ∀i 6= j}, which corresponds to biaxial distributions; (ii) the class U ∗ := {Q ∈ Q : λi = λj 6= λk for i 6= j 6= k} of uniaxial distributions; (i)
Hierarchy of Defects in Biaxial Nematics
383
(iii) the singleton {0} ⊂ Q, which describes the isotropic distribution. Note that every Q ∈ U ∗ can be written [15] as 1 Q=s n⊗n− I (2.2) 3 with s ∈ − 21 , 1 \ {0} and n · n = 1. It is useful to introduce also the manifold Q∗ := B ∪ U ∗ = Q \ {0}, which contains all anisotropic distributions. Let us now describe the submanifold of Q fit to represent the states attained on a surface. Following [14], we suppose that on a surface the probability distribution fx obeys the additional symmetry: fx (Rπ `) = fx (`),
∀` ∈ S2 ,
(2.3)
where Rπ is the rotation of angle π about the normal ν to the surface. Condition (2.3) is verified if and only if ν is an eigenvector of Q so that the surface order parameter manifold is (2.4) C := {Q ∈ Q : Qν = λν ν}. The class C (see [14] for further details) can be represented by a cone and contains both uniaxial and biaxial distributions, as well as the isotropic state (see Fig. 1). The uniaxial subclass CU ∗ := C ∩ U ∗ consists of the axis of the cone, which contains the uniaxial distributions with director along ν, and of a two-fold inner cone, where the uniaxial distributions with director orthogonal to ν are represented. The vertex of the two-fold cone, which lies on the axis of C, represents the isotropic distribution. The biaxial subclass CB := C ∩ B has three connected components, separated by the uniaxial two-fold cone. As above, we also define C ∗ := CB ∪ CU ∗ = C \ {0}.
Uniax. distr. with director // normal, and s > 0
Uniax. distr. with dir. orth. normal, and s < 0
Isotropic distribution
Uniax. distr. with dir. orth. normal, and s > 0
Uniax. distr. with director // normal, and s < 0
Fig. 1.
384
P. Biscari, G. G. Peroli
We call nematic defect any point in the body where the degree of symmetry of Q increases, and so the degree of order decreases. There are then three different types of nematic defects: uniaxial defects in biaxial states, isotropic defects in uniaxial states, and isotropic defects in both. We call topologically stable the defects that cannot be removed by local surgery, that is by a continuous change of the field Q(x) in a neighbourhood of the defect: they are forced to appear by topological reasons. The topological structure of the manifold where the order parameter takes values determines the type of topologically stable defects that can arise in a body [1]: topologically stable wall, line, and point defects in an ordered system are described by elements of the homotopy groups π0 (R), π1 (R), and π2 (R), respectively, where R is the manifold spanned by the order parameter. Taking R as B, U ∗ , or Q∗ , we shall be able to explore the three types of defects described above. The situation becomes more complex when we wish to study surface defects, namely defects that arise on a surface bounding the body. In this case we must consider, in e ⊂ R, addition to the manifold R of the states accessible in the bulk, a submanifold R determined by the boundary conditions, that contains the states Q accessible on the surface. There are three different kinds of surface defects. (D1) Point (line) defects that are the restriction to the surface of line (wall) defects in the volume. (D2) Isolated point (line) defects that cannot be removed from the surface; they have a neighbourhood that does not contain any further defect point. (D3) Isolated point (line) defects that can relax in the bulk. They can be removed from the surface by local surgery, although they are topologically stable. Defects of types (D2) and (D3) are characterized by elements of the relative homotopy e π1 (R, R) e , as described in [8]; furthermore, by studying the algebraic groups π2 (R, R) structure of these groups, we shall be able to distinguish between them. To construct the groups that characterize defects of type (D1), we introduce the homomorphism e −→ πk (R), j] : πk (R)
k = 0, 1,
(2.5)
e → R; then, π1 (R)/ e ker j] and π0 (R)/ e ker j] deinduced by the inclusion map j : R scribe, respectively, surface point defects which have a “tail” in the bulk and surface line defects that determine a wall defect inside the material. In words, point defects have tails when the loops that surround them on the surface cannot be shrunk to a point even in R, and line defects determine walls when they border upon domains that belong to different connected components of R.
3. Homotopy We now examine the topological structure of the manifolds Q∗ , B, and U ∗ . The following results will allow us to determine the bulk defects that come into being by mere topological reasons. The set Q is a five-dimensional manifold. It is path-connected and convex, so that it is homotopic to the five-dimensional unit ball B 5 . Since 0 ∈ Q is an interior point, from elementary topology we obtain the following
Hierarchy of Defects in Biaxial Nematics
385
Theorem 3.1. Q∗ is homotopic to S 4 , the four dimensional unit sphere; its first three homotopy groups are: (1) π0 (Q∗ ) = 0; (2) π1 (Q∗ ) = 0; (3) π2 (Q∗ ) = 0. We now take into account the manifold B. A preliminary result about the local structure of B is given in [3,11]: P Lemma 3.1. Let γ = (γ1 , γ2 , γ3 ) ∈ R3 be such that γ1 < γ2 < γ3 , i γi = 0, and γi ≥ − 13 for i = 1, 2, 3. The subset Bγ ⊂ B of the tensors that have these eigenvalues is homeomorphic to SO(3)/D4 , where D4 is the abelian dihedral group containing four elements, the identity and the rotations of angle π about the three axes of the trihedron. In the next theorem we show that the whole structure of the five dimensional manifold B is indeed equivalent, from a topological point of view, to that of Bγ . Thus, we can apply to B the results known for Bγ [16, Chap. 10]. Theorem 3.2. The first three homotopy groups of B are: (1) π0 (B) = 0; (2) π1 (B) = H, the group of quaternion unities; (3) π2 (B) = 0. P P Proof. Let Q1 = i λi ei ⊗ ei and Q2 = i µi fi ⊗ fi be two arbitrary points in B. < µ3 . The path Without loss P of generality wecan assume λ1 < λ2 < λ3 and µ1 < µ2 P Q(t) := i tµi + (1 − t)λi ei ⊗ ei joins Q(0) = Q1 and Q(1) = i µi ei ⊗ ei ; furthermore, it lies in B because of the assumption on the order of the {λi } and {µi }. As Q(1) can be joined to Q2 by the preceding lemma, Q1 can be joined to Q2 by a path in B, and so π0 (B) = 0. Now, let γ1 < γ2 < γ3 be three real numbers chosen as in Lemma 3.1, and let B γ be the subset of B containing all tensors of the form Q = γ1 e1 ⊗e1 +γ2 e2 ⊗e2 +γ3 e3 ⊗e3 . The application F : B × [0, 1] → B γ defined by 3 X F Q, t := tγi + (1 − t)λi ei ⊗ ei ,
(3.1)
i=1
P if Q = i λi ei ⊗ ei with λ1 < λ2 < λ3 , endows B γ of the structure of strong deformation retract of B, that means that both spaces share the same topological structure. Note that the continuity of F follows from the fact that the application which assigns to every tensor Q the triad of its eigenvectors is continuous in B (while it would not be so in the whole Q, or even in Q∗ ). Thus, we have πk (B) = πk (B γ ) = πk (SO(3)/D4 ) for k = 1, 2; statements (2) and (3) of the theorem are then obtained by considering the universal covering SU (2) of SO(3) and applying to it well-known results of algebraic topology [16, Chap. 10]. Using the same techniques, we now explore the structure of U ∗ . First, we study the properties of the submanifold composed of all tensors which have a fixed s = s. By associating to every Q = s(n ⊗ n − 13 I) the direction of n, we have the following
386
P. Biscari, G. G. Peroli
Lemma 3.2. Let s ∈ − 21 , 1 \ {0}. The subset of U ∗ consisting in all tensors of the form Q = s(n ⊗ n − 13 I), with n · n = 1, is homeomorphic to RP2 , the real projective plane. The following theorem describes the main topological properties of U ∗ . Theorem 3.3. The first three homotopy groups of U ∗ are: (1) π0 (U ∗ ) = Z/2Z; (2) π1 (U ∗ ) = Z/2Z; (3) π2 (U ∗ ) = Z. Proof. U ∗ has at least two path-connected components, because within it elements with different sign of s cannot be connected. On the other hand, reasoning as in the first part of the proof of Theorem 3.2, it is easy to see that elements whose s have the same sign can be connected by a path in U ∗ , so that π0 (U ∗ ) = Z/2Z. Furthermore, let s ∈ ]0, 1], and let U be the subset of U ∗ composed by all uniaxial tensors with s = s. The application F : U ∗+ × [0, 1] → U defined by 1 (3.2) n⊗n− I , F Q, t := ts + (1 − t)s 3 if Q = s n ⊗ n − 13 I , gives a continuous deformation of the connected component U ∗+ with positive s onto one subspace where s is fixed. Again, the continuity of F follows from the fact that the application which assigns S the director n to a uniaxial tensor is continuous in U ∗+ (while it would not be so in U ∗+ {0}). Since, by the preceding lemma, U is topologically equivalent to RP2 , we can call upon known results on the homotopy groups of RP2 [16, cap. 10] to conclude the proof. Clearly, the same arguments allow us to prove the topological equivalence between U ∗− , the other connected component of U, and RP2 . 4. Relative Homotopy We assume now that the interior states of the nematic liquid crystal are described by Q∗ , B, or U ∗ , while the surface states are described by elements of C ∗ , C B , or CU ∗ , respectively. As remarked in Sect. 2, information about the nature of isolated surface stable defects come from the structure of the relative homotopy groups. Here we determine these groups. To study the structure of the relative groups, we write the exact homotopy sequence: e . . . π2 (R) e → π1 (R) e → π0 (R)
e → → π2 (R) → π2 (R, R) e → → π1 (R) → π1 (R, R) → π0 (R).
(4.1)
e has not a natural group structure: it is a pointed set, that is a set with Note that π1 (R, R) a distinguished element (see also [17, Chap. 7]). e turns out to be quite simple. In general, we The structure of the groups π2 (R, R) only know that the groups e → π2 (R) (4.2) π2 (R) / im π2 (R)
Hierarchy of Defects in Biaxial Nematics
387
e but when the sequence splits1 in π2 (R, R), e this latter is are normal divisors of π2 (R, R), e merely the direct product between (4.2) and the image group ker π1 (R) → π1 (R) . In e because the π1 (R) e are projective moduli the cases studied here, (4.1) splits in π2 (R, R) over Z (i.e. free abelian groups). This fact helps us distinguish between isolated point defects of types (D2) and (D3): a point defect can relax in the bulk if, and only if, it is e → π1 (R) ; in that case, the associated to the trivial element of the group ker π1 (R) e → π2 (R) to which it is associated characterizes it as element of π2 (R)/ im π2 (R) a point defect in the bulk. In words, a point defect can leave the surface when a loop e A similar result stands also for line defects: they that encloses it can be shrunk in R. can relax in the bulk if, and only if, the domains they border upon belong to the same e connected component of R. Before studying the relative homotopy groups, we recall here the homotopy groups of the topological spaces we are interested in.
π0 π1 π2
Q∗
C∗
B
0 0 0
0 0
0 H 0
CB
U∗
CU ∗
Z/3Z Z/2Z n Z/4Z Z Z Z/2Z 0 Z
(4.3)
The topological properties of C ∗ , CB , and CU ∗ come straight from the representation of C as a cone: C ∗ is homotopic to the sphere S2 , CB to the disjoint union of three circles S1 , and CU ∗ to the disjoint union of two circles and two points. Note that π1 (CU ∗ ) is either Z or 0, depending on whether we fix the base point on a component homotopic to S1 or not. Theorem 4.1. The homotopy groups of Q∗ , relative to its submanifold C ∗ , are: (1) π2 (Q∗ , C ∗ ) = 0; (2) π1 (Q∗ , C ∗ ) = 0. e = C ∗ we have π2 (Q∗ , C ∗ ) = P2 ⊕ Q2 , where Proof. Considering (4.1) for R = Q∗ and R P2 := ker π1 (C ∗ ) → π1 (Q∗ ) = 0, and (4.4) Q2 := π2 (Q∗ ) / im π2 (C ∗ ) → π2 (Q∗ ) = 0. Moreover, π1 (Q∗ , C ∗ ) = P1 ⊕ Q1 , where
P1 := ker π0 (C ∗ ) → π0 (Q∗ ) = 0, and Q1 := π1 (Q∗ ) / im π1 (C ∗ ) → π1 (Q∗ ) = 0.
(4.5)
In the following theorem we describe the relative homotopy groups πk (B, CB ), k = 1, 2, which will lead in Sect. 5.4 to a new classification of uniaxial defects on the surface of a biaxial nematic that enjoys the symmetry (2.3). Theorem 4.2. The homotopy groups of the manifold B, relative to its submanifold CB , are: (1) π2 (B, CB ) = Z; 1
in the sense of [18, §IV.1]
388
P. Biscari, G. G. Peroli
(2) π1 (B, CB ) = Z/3Z ⊕ Z/2Z. Proof. We write the sequence (4.1) fit to this case: 0 −→ π2 (B, CB ) −→ α −→ π1 (CB ) −→ π1 (B) −→ π1 (B, CB ) −→ −→ π0 (CB ) −→ 0.
(4.5)
The map α : π1 (CB ) → π1 (B) sends the generator 1 ∈ π1 (CB ) = Z into one element of order four in π1 (B) = H, because the lift on SO(3) of any of the S1 in CB corresponds to a rotation of angle π about the normal ν, which is an order-four element in π1 (B). Then, ker α = 4Z ' Z, and so π2 (B, CB ) = 0 ⊕ Z = Z.
(4.7)
Furthermore, if we call e1 := α(1), we have that im α = {e1 , −1, −e1 , 1} ⊂ H. The group H/ im α is then Z/2Z, and π1 (B, CB ) = π0 (CB ) ⊕ H/ im α = Z/3Z ⊕ Z/2Z. Finally, we consider a uniaxial material. The first part of the following theorem has been known for a while [9]; here we add the computation of π1 (U ∗ , CU ∗ ), which will lead us in Sect. 5.2 to some new results about line defects on a surface. Theorem 4.3. The homotopy groups of U ∗ , relative to its subset CU ∗ , are: Z ⊕ Z for surface point defects surrounded by uniaxial states with n ⊥ ν, ∗ (1) π2 (U , CU ∗ ) = Z for surface point defects surrounded by uniaxial states with n k ν; Z/2Z if we fix the base point in a component where n ⊥ ν, (2) π1 (U ∗ , CU ∗ ) = Z/3Z if we fix the base point in a component where n k ν. Proof. Let ϑ be the angle between n and ν. Then, π2 (U ∗ , CU ∗ ) = P2 ⊕ Q2 where, in this case, π n P2 = ker π1 (CU ∗ ) → π1 (U ∗ ) = Z if ϑ = 2 , and (4.8) 0 if ϑ = 0. Q2 = π2 (U ∗ ) / im π2 (CU ∗ ) → π2 (U ∗ ) = Z. The group π1 (U ∗ , CU ∗ ) requires care, because the different connected parts of CU ∗ have different topological properties. We recall that an element of π1 (U ∗ , CU ∗ ), with base point in p, is a path γ : [0, 1] → U ∗ such that γ(0) = p ∈ CU ∗ , γ(1) ∈ CU ∗ , and γ(t) ∈ U ∗ ∀t ∈ [0, 1]. First suppose that p is taken in one of the two components of CU ∗ in which n ⊥ ν (for example, in that with s > 0); the end point γ(1) must then belong to one of the two components of CU ∗ where s > 0, since points where s < 0 cannot be reached by a path in U ∗ . Furthermore, in this case π1 (CU ∗ ) = Z, so that π1 (U ∗ )/ im π1 (CU ∗ ) → π1 (U ∗ ) = 0, and π1 (U ∗ , CU ∗ ) has only two elements: the identity (corresponding to an unstable element), and a path associated to a line defect that separates a domain where n ⊥ ν from a domain where n k ν. On the other hand, if p belongs to a component where the director is parallel to the normal, π1 (CU ∗ ) = 0, π1 (U ∗ )/ im π1 (CU ∗ ) → π1 (U ∗ ) = Z/2Z, and π1 (U ∗ , CU ∗ ) possesses three elements: a trivial path, a path associated to a line defect that separates a domain where n k ν and a domain where n ⊥ ν (this path belongs to π1 (U ∗ , CU ∗ ) for any choice of the basepoint p), and a non-trivial path between two domains where n k ν. This latter path will be described at length in Sect. 5.4.
Hierarchy of Defects in Biaxial Nematics
389
This theorem completes the description of isolated surface defects. We conclude this section by analyzing the algebraic objects which give us information about surface defects which are the trace of line or wall defects in the bulk. As we said in Sect. 2, we e ker j] and π0 (R)/ e ker j] . Only few cases give non-trivial results should look at π1 (R)/ regarding these groups. We summarize them in the following theorem, whose proof comes straightforwardly from the study of (4.1). Theorem 4.4. The only non-trivial groups which describe the surface defects that invade the volume are: (1) π1 (CB ) / ker π1 (CB ) → π1 (B) = Z/4Z; Z for surface point defects surrounded by uniaxial states with n ⊥ ν, ∗ (2) π1 (CU ∗ ) / ker π1 (CU ∗ ) → π1 (U ) = 0 for surface point defects surrounded by uniaxial states with n k ν; ∗ (3) π0 (CU ∗ ) / ker π0 (CU ∗ ) → π0 (U ) = Z/2Z. 5. Classification of Nematic Defects In Sects. 3 and 4, we have determined all the homotopy and relative homotopy groups we are interested in. We are now in a position to classify the corresponding types of defects that can arise in a nematic. 5.1. Isotropic defects in the bulk of uniaxial nematics. From Theorem 3.3 we have that π2 (U ∗ ) = Z; this allows us to classify point defects in the bulk by an integer topological charge NB that can be computed as in [9]. Line defects in the bulk can be associated with elements of π1 (U ∗ ) = Z/2Z, so that only one topologically stable line defect exists, which corresponds to the non-trivial element of π1 (U ∗ ). To see whether a given loop encloses such a line, we assign a director n0 to the initial point of the loop, we follow the continuous variation of the director n along the path and we check whether at the end the director is equal to n0 or to −n0 (topologically, we consider the lift to S2 of the loop in RP2 and we check whether it is a closed path or not). Only in the latter case we have a topologically stable line defect. Finally, since U ∗ has two path-connected components, wall defects in the bulk of a uniaxial nematic arise whenever there are points with different sign for s. 5.2. Isotropic defects on the surface of uniaxial nematics. The set CU ∗ , where Q is supposed to take values, has four path-connected components: two with director along ν and opposite sign for s (the two parts of the axis of the cone in Fig. 1, separated by the isotropic distribution), and the other two with director orthogonal to ν and opposite sign for s (the two components of the two-fold cone in Fig. 1). To study line defects on a surface, we must first consider the types of domains that the line can border upon. A line defect can separate two regions with different sign of s; this type of line is the trace of a wall defect in the bulk (see Theorem 4.4(3)). If the sign of s is the same on both sides of the line, but the director is parallel to ν on one side and perpendicular to it on the other, we have an isolated isotropic line that cannot leave the surface by Theorem 4.3(2). Finally, if a line defect separates two uniaxial domains with the same sign of s and the same tilt angle, Theorem 4.3(2) tells us that every line defect can be removed if the tilt angle is equal to π2 , and that a stable line defect exists if the tilt
390
P. Biscari, G. G. Peroli
angle is equal to zero. In the latter case, we can decide whether a line defect is stable or not using the same technique as in Sect. 5.1: we assign the director n0 = ν to a surface point on one side of the line defect, we follow the continuous variation of the director n along a path in the bulk that ends on the other side and we check whether the director so obtained is equal to ν (topologically unstable line) or −ν (topologically stable line). To classify point defects on the surface, we must again consider separately regions in which the director is parallel and orthogonal to ν. Inside a domain where the director is parallel to ν, point defects can be classified by an integer topological charge, since π2 (U ∗ , CU ∗ ) = Z (see Theorem 4.3(1)); however, they are not genuine surface defects, for they can be removed from the surface just by local surgery: they can leave the surface to create a bulk point defect of the same topological charge. To compute this charge, we take a hemisphere σ + seeping in the volume that surrounds the defect, and we perform the integral Z 1 ∂n ∂n ∧ da, (5.1) n· NB = 2π σ+ ∂ϕ ∂ϑ where φ and ϑ are arbitrary coordinates on σ + and n(ϕ, ϑ) is the director of Q(ϕ, ϑ). By Theorem 4.4(2), no point defect of this type has a tail in the volume. The situation becomes more complex in a domain where the director is perpendicular to ν. In this case, π2 (U ∗ , CU ∗ ) = Z ⊕ Z by Theorem 4.3(1): a point defect must now be assigned two topological charges. To do so, we take a hemisphere σ + in the volume, whose boundary is a circle γ on the surface. First, we define the surface topological charge of the defect: Z dn 1 du, (5.2) NS = ± du π γ
where u is the natural parameter on the circle, and the sign has to be chosen as to whether dn the pair {n, dn du / | du |} has the same orientation as a fixed reference frame on the surface or not (n is the director of Q and lies on the surface, as well as dn du ). If NS 6= 0, the defect cannot be removed from the surface; furthermore, if NS is odd, the point defect has a tail in the bulk, since γ cannot be shrunk to a point even in U ∗ (that is, γ is the non-trivial element of π1 (U ∗ )/ ker j] in Theorem 4.4(2)). Then we compute the integral (5.1), and now we arrive at the integer number NB − NS , where NB is the bulk topological charge, because NS 6= 0. Note that, although the defect can disappear from the surface only if NS = 0, a point defect with NS 6= 0 can generate two defects with topological charges (NS , NB0 ) and (0, NB00 ), with NB0 + NB00 = NB , the second of which can relax in the volume. 5.3. Uniaxial defects in the bulk of biaxial nematics. Theorem 3.2 states that π2 (B) = 0, so that no point defect is topologically stable in a biaxial nematic. In other words, a biaxial nematic can always avoid uniaxial melting in the interior, whichever boundary condition is imposed. On the other hand, since π1 (B) = H, line defects can arise. To decide which element of H corresponds to a given loop that encloses a uniaxial line defect in a biaxial nematic, we assign three unit vectors to the triad of Q at the initial point of the loop and we follow the continuous variation of the triad along the path. Upon return to the initial point, one of the following cases arises: (i) one or more of the unit vectors have opposite sign; in this case, the line defect is associated to an element of order four in H;
Hierarchy of Defects in Biaxial Nematics
391
(ii) the three unit vectors take their initial value; in this case, the path is homotopic to a rotation of 2kπ about some axis; if k is odd, the line defect is associated to the element of order two in H, while, if k is even, the line defect is associated to the trivial element of H and it can be removed by local surgery. Since there are six elements of order four in H, six different line defects of type (i) can arise in a biaxial nematic: the combination and crossing of these lines are described in [3,12]. Finally, B is path-connected, so that no wall defect is topologically stable in biaxial nematics. 5.4. Uniaxial defects on the surface of biaxial nematics. Theorem 4.2(2) tells us that every surface line defect in a biaxial nematic has two topological charges associated with it. The first of them indicates whether the domains adjacent to the line belong to the same connected component of CB or not; lines that divide two different components are always composed of uniaxial distributions with director orthogonal to the normal, and cannot leave the surface. To recognize one of these lines it suffices to read the eigenvalues of Q at both sides of the line, since in one component of CB the eigenvalue associated to the normal is the largest one, in another it is intermediate, and in the third it is the smallest one. The second topological charge shows that there are two different types of lines which divide every given pair of connected components. In fact, following the continuous variation of the eigenvector ν around one of those lines, we find that around the nontrivial line ν reverses its sign, while around the trivial line it does not do so. Regarding point defects, Theorem 4.2(1) tells us that π2 (B, CB ) = Z, so that topologically stable point defects can be classified by an integer; moreover, they cannot relax in the bulk, as π2 (B) = 0. To compute the topological charge of a point defect, we take a circle γ around it on the surface and we perform the integral Z de 1 du, (5.3) NS = ± π γ du where u is again the natural parameter on the circle, the sign is chosen as in (5.2), and e(u) follows continuously the variation of any of the two eigenvectors of Q different from ν. Theorem 4.4(1) tells us that surface point defects in biaxial nematics can be of four different types, three of which have non-trivial tails in the bulk. In fact, NS counts the number of semicycles that e performs along γ, so that if we write NS = 4m + n with m ∈ Z and n ∈ {0, 1, 2, 3}, one of the following possibilities arises: n = 1 or n = 3: e reverses its sign along γ, which is then a non-trivial loop in B; the point defect has a tail of order four in H (see point (i) in Sect. 5.3); (ii) n = 2: e performs an odd number of cycles along γ, which is still a non-trivial loop in B; the point defect has a tail of order two in H (see point (ii) in Sect. 5.3); (iii) n = 0: e performs an even number of cycles along γ, which is now a trivial loop in B; the point defect has no tail (see point (ii) in Sect. 5.3). (i)
5.5. Isotropic defects. Finally, we consider here the case of a nematic liquid crystal that is free to choose between uniaxial and biaxial states. Theorems 3.1 and 4.1 state that all homotopy and relative homotopy groups are trivial in this case, so that there are no
392
P. Biscari, G. G. Peroli
isotropic point, line, or wall defects that cannot be removed by local surgery when both uniaxial and biaxial states are accessible. A consequence of this result is that a uniaxial nematic, forced to have an isotropic defect, can always avoid the isotropic melting by becoming biaxial in a neighbourhood of the putative singular point. Acknowledgement. The authors thank the referee of Communications in Mathematical Physics for his useful suggestions on this paper. One of us (P.B.) thanks the Istituto Nazionale di Alta Matematica for financial support.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
Mermin, N.D.: Rev. Mod. Phys. 51, 591 (1979) Toulouse, G., Kleman, M.: J. Physique Lettres 37, L149 (1976) Poenaru, V., Toulouse, G.:J. Physique 38, 887 (1977) Trebin, H.-R.: Adv. Phys. 31, 195 (1982) R Kutka, H-R Trebin, & M Kiemes, J. Physique 50, 861 (1989) Bais, F.: Phys. Lett. 98B, 437 (1981) Kibble, T.W.B., Lazarides, G., Shafi, Q.: Phys. Lett. 113B, 237 (1982) Volovik, G.E.: JETP Lett. 28, 59 (1978) Volovik, G.E., Lavrentovich, O.D.: Sov. Phys. JETP 58, 1159 (1983) Misirpashaev, T.Sh.: Sov. Phys. JETP 72, 973 (1991) Toulouse, G.: J. Physique Lettres 38, L67 (1977) De’Neve, T., Kleman, M., Navard, P.: J. Physique II 2, 187 (1992) de Gennes, P.G.: The Physics of Liquid Crystals. Oxford: Clarendon Press, 1974 Biscari, P., Capriz, G., Virga, E.G.: Liquid Crystals. 16, 479 (1994) Ericksen, J.L.: Arch. Rat. Mech. Anal. 113, 97 (1991) Kleman, M.: Points, Lines and Walls, New York: John Wiley & Sons, 1983 Spanier, E.H.: Algebraic Topology, New York: Mc Graw-Hill, 1966 Hungerford, T.W.: Algebra, New York: Springer, 1974
Communicated by A. Jaffe
Commun. Math. Phys. 186, 393 – 411 (1997)
Communications in
Mathematical Physics c Springer-Verlag 1997
W -Realization of Lie Algebras: Application to so(4, 2) and Poincar´e Algebras F. Barbarin1 , E. Ragoucy1 , P. Sorba1,2 1 Laboratoire de Physique Th´ eorique ENSLAPP? , Groupe d’Annecy: LAPP, Chemin de Bellevue BP 110, F-74941 Annecy-le-Vieux Cedex, France
Laboratoire de Physique Th´eorique ENSLAPP, Groupe de Lyon: ENS Lyon, 46 all´ee d’Italie, F-69364 Lyon Cedex 07, France
2
Received: 10 June 1996 / Accepted: 8 October 1996
Abstract: The property of some finite W-algebras to appear as the commutant of a particular subalgebra in a simple Lie algebra G is exploited for the obtention of new G-realizations from a “canonical” differential one. The method is applied to the conformal algebra so(4, 2) and therefore yields also results for its Poincar´e subalgebra. Unitary irreducible representations of these algebras are recognized in this approach, which is naturally compared -or associated- to the induced representation technique. 1. Introduction As shown in [1, 2], the construction of finite W-algebras achieved in the framework of Hamiltonian reduction [3] also leads to the determination of the commutants, in the enveloping algebra1 U(G), of particular subalgebras of a simple Lie algebra G. Such a property can be used to build, from a special realization of G, a large class of G-representations. Let us be more specific, and consider a graded decomposition of the simple Lie algebra G of the type G = G −1 ⊕ G 0 ⊕ G +1
(1.1)
with the corresponding commutation relations: [G 0 , G ±1 ] ⊂ G ±1 ; [G ±1 , G ±1 ] = {0} ; [G +1 , G −1 ] ⊂ G 0 .
(1.2)
As recalled in Sect. 2, it is possible to recognize in the commutant of a G-subalgebra Ge of the form Ge = G −1 ⊕ Ge0 , with Ge0 ⊂ G 0 , a particular finite W-algebra. The determination ?
URA 14-36 du CNRS, associ´ee a` l’Ecole Normale Sup´erieure de Lyon et a` l’Universit´e de Savoie Strictly speaking, we consider a generalization (localization) of U (G), which contains, apart from U (G) itself, quotients u−1 v, vu−1 , and also ur , r ∈ Q, where u, v ∈ U (G), u 6= 0. 1
394
F. Barbarin, E. Ragoucy, P. Sorba
of the realization of the W-algebra needed for our purpose can be obtained in a systematic way after some modification [2] of the usual Hamiltonian reduction technique [4]. Moreover, one knows how to construct a realization of G with differential operators on the space of smooth functions ϕ(y1 , . . . , ys ) with s = dimG − . In this picture, the abelianity of the G −1 -part allows each G −1 generator to act by direct multiplication: ϕ(y1 , . . . , ys ) → yi ϕ(y1 , . . . , ys ) with i = 1, . . . , s,
(1.3)
cf. action of the translation group, while the generators of the G 0 ⊕ G + part will be represented by polynomials in the yi and ∂yi (see Sect. 3). It is from this particular – canonical – differential realization of G that new representations will be constructed with the use of the finite W-algebra mentioned above. Realization of the Ge generators will not be affected in this approach. On the contrary, to the differential form of each generator in the G \ Ge part will be added a sum of Wgenerators, the coefficients of which are functions f (yi , ∂yi ). By associating a matrix differential realization to each irreducible finite dimensional W-representation, one will get an action of G on vector functions ϕ ~ = (ϕ1 , . . . , ϕd ), with ϕi = ϕi (y1 , . . . , ys ), where d is the dimension of the considered W-representation. Such an approach has already been considered with some success [1] for the study of the Heisenberg quantization for a system of two particles in 1 and 2 dimensions [5]. The corresponding algebras are respectively sp(2) and sp(4), and in each case, it has been possible to relate the anyonic parameter to the eigenvalues of a W-generator. It has seemed to us of some interest to apply our technique on a rather well-known algebra, the so(4, 2) one, also called the 4-dimensional conformal algebra. The decomposition of interest is the one in which the G −1 part is constituted by the four translations, G 0 by the Lorentz algebra so(3, 1) plus the dilatation, and G + by the four special conformal generators. The representation space is then the set of smooth functions F(M 4 , C), where M 4 is the usual Minkowski space, and the finite W-algebra of interest appears as a non-linear deformation of the G 0 subalgebra. This algebra is the commutant in U (G) of the part G − ⊕ Ge0 , where Ge0 is a four dimensional subalgebra of G 0 . Note also that the commutant of G − can be seen as the direct sum of this W-algebra and of G − , i.e. W ⊕ G − . As presented in Sect. 4, a suitable basis for this non-linear algebra is the one made of three generators, components of the J~ vector constituting an so(3) part, and ~ under the J~ part, the commutation three other generators forming a second vector S relation among the Si ’s producing a seventh generator belonging to the center of W. It is interesting to remark that, restricting to the Poincar´e subalgebra the general realization so obtained, only the J~ part contributes and appears to be the spin algebra directly connected to the (Wµ ) Pauli-Lubanski-Wigner four vector. The usual Poincar´e representations can therefore easily be recognized in this approach, the W-part being related to the usual Wigner rotations. Then, realizations of the special conformal generators involve naturally the other W-generators. In a rather elegant way, a (Σµ ) four ~ vector, conformal analogous to the (Wµ ) one, can be exhibited and connected to the S part of the W-algebra. Apart from this geometrical picture, one can recognize through the finite dimensional representation of the W-algebra, the characterization of the unitary irreducible representations of the conformal algebra, as computed in [6] in case of positive energy. It is clear, from the considered example, that the proposed method provides the same results as those given by the usual induced representation techniques. This is discussed at the end of the paper.
W-Realization of Lie Algebras: Application to so(4, 2) and Poincar´e Algebras
395
2. W-Algebras as Commutants 2.1. General results. We recall here the framework which leads to the main result obtained in [2], namely Theorem. Any finite W(G,S) algebra, with S = µs`(2) direct sum of µ regular s`(2) subalgebras of G, can be seen as the commutant in the enveloping algebra U (G) of some e G-subalgebra G. Moreover, let H be the Cartan generator of the diagonal s`(2) in S. If we call G − , G 0 and G + the eigenspaces of respectively negative, null, and positive eigenvalues under H, then Ge decomposes as Ge = G − ⊕ Ge0 , where Ge0 is a subalgebra of G 0 which can be uniquely determined (see below). We start with a simple Lie algebra G, and consider an s`(2)-embedding defined through the regular subalgebra S in G. Together with this s`(2)-embedding, we get a gradation of G, using the action of the s`(2) Cartan generator H. We will call G − , G 0 and G + the eigenspaces of respectively negative, null, and positive eigenvalues under H. Below, we will consider the case where S is a (sum of) regular subalgebra(s) s`(2), which leads to Abelian G − subalgebras. In [2], we have shown that, for integral gradations, the commutant of G − in the enveloping algebra U (G) is isomorphic to the W(G, µ s`(2)) algebra plus a center. Moreover, we have also shown that one can get rid of the center when looking at the commutant of a wider subalgebra Ge = G − ⊕ Ge0 , where Ge0 is a subalgebra of G 0 and G − = G −1 . In the case of half-integral gradations, we should use the halving procedure. However, when S = µs`(2), it can be shown that we still have Ge = G − ⊕ Ge0 with G − = G −1 ⊕ G − 1 . 2 Note that more general finite W(G,S) algebras may be obtained as a commutant, but one has to extend Ge0 to the G + part of G (see the example in [2]). The case of the “affine” W(G, S) algebra can also be treated with this framework [2]. As an illustration, let us consider S = µ s`(2) in G = s`(n). The simple roots of G will be noted αi with 1 ≤ i ≤ n − 1. Then, one can take Ge = ⊕µi=1 0i , where 0i is the subalgebra Pj βj = k=i αk Pj 0i = {Eβj , i ≤ j ≤ n − i ; Eγj , i ≤ j ≤ n − i − 1 ; Hβn−i } with γj = k=i αn−k (2.1) 2.2. Explicit calculations. We start with a general element J = J a ta in G. J a are the coordinate-functions (J a ∈ G ∗ ), and it is known that one can provide them with a Poisson Bracket structure that mimics the commutation relations of G: [ta , tb ] = fab c tc while {J a , J b } = f ab c J c , indices being lowered via the Killing metric ηab =< ta , tb >, and raised by its inverse η ab . Thanks to this metric (and these PB) we identify G ∗ with G so that the decompositions of G (like G = G − ⊕ G 0 ⊕ G + ) will also be valid for G ∗ , that we will loosely write G. We will also use the Cartan involution Θ defined by Θ(Eαi ) = −E−αi for any simple root αi . It allows us to define another scalar product (X, Y ) =< X, Θ(Y ) >. We consider the decomposition G = G − ⊕ G 0 ⊕ G + obtained by the Cartan generator of the diagonal s`(2)diag in µ regular s`(2). As a notation, we use latin indices a, b, c, . . . to label the generators of G, and greek (resp. overlined-greek) to label generators in Ge (resp. in G, its orthogonal complementary w.r.t. the scalar product (., .) ). When needed,
396
F. Barbarin, E. Ragoucy, P. Sorba
we will also use tA , ti and tA to denote the generators of G − , G 0 , and G + respectively. Thus, we note ta ∈ G . J = J a ta = J α tα + J α tα = J A tA + J i ti + J A tA with tα ∈ Ge ; tα ∈ G t A ∈ G − ; ti ∈ G 0 ; tA ∈ G + (2.2) To compute the commutant of Ge in U (G), we consider the group transformations e where LieΘ(G) e = Θ(G), e G e being the connected J → J g = gJg −1 with g ∈ Θ(G), x b tb e e component of the Lie group whosealgebra is G. In fact, if g = e , with tb ∈ Θ(G), −1 b one can rewrite gJg as exp xb J , . (J), where now the PB J b , . acts on J a ∗ (instead of ta ), J b belongs to Ge , and xb = ηbc xc . Then, one can show that, starting b e from any generic element J, there exists a unique group generator g = ex tb ∈ Θ(G) g α such that J = e− + W tα , where e− is the negative root generator of the s`(2)diag under consideration and tα ∈ G. Let us remark that the parameters xb depend only on e the components J α ∈ G. e 0 ) (in the neighborhood of the identity), e as Θ(G− ) Θ(G In fact, decomposing Θ(G) e it is easy to show that Θ(G0 ) is defined as the subgroup of G0 which transforms J into J 0 with e 0 ) : J → J h = J 0 with J 0 | (2.3) h ∈ Θ(G G − = e− , while Θ(G− ) transforms J 0 into J g = e− + W α tα : e − ) : J 0 → J g = e − + W α tα . g ∈ Θ(G
(2.4)
This property is still true when G − is not Abelian, but in that case one has to include e to satisfy (2.3). elements of Θ(G+ ) in order for Θ(G) α Altogether, the generators W are the “gauge-fixed” components of J g . Hence, they e and generate the commutant of G. e They form have vanishing PB with any element in G, α a classical (PB) version of the W-algebra. Note that the W are linear in J β : we will come back on this property in Sect. 3. A quantum version of this algebra will be obtained by using the map i : G ∗ → G defined by i(J a ) = η ab tb and extended to U(G) by symmetrization of the products (see [2] for details). Explicitly, one uses a matrix representation for the ta to get a simple expression for e which act on J. We determine g such that J g takes the the group generators g ∈ Θ(G) form (2.4). The coefficients W α of tα are then expressed in terms of all the J a ’s, contrary to the usual Hamiltonian reduction approach, since we have not imposed any constraint on the current components. Finally, from the classical expression of the W-generators, we use the above mentioned quantization procedure to obtain the commutant of Ge in U (G). 2.3. Example: s`(2). We make explicit here the results just presented in the case of G = s`(2). We start with a generic element 1 0 J J+ a 2 . (2.5) J = J ta = J − − 21 J 0 As normalization, we take
W-Realization of Lie Algebras: Application to so(4, 2) and Poincar´e Algebras
[t0 , t± ] = ±t± ; [t+ , t− ] = 2t0 , η00 = 21 ; η+− = 1 so that η 00 = 2 ; η +− = 1, {J 0 , J ± } = ±2J ± ; {J + , J − } = J 0 .
397
(2.6)
Here the eigenspaces G ±,0 are one-dimensional. Thus, Ge0 must be G 0 itself, and indeed we have: ! √ 00 0+ 00 1 0 J− 0 J J J = 2J −1 0 = J = . is such that hJh with h= 00 0 √1 − J 0+ = J + J − 1 −J J Now, acting with G+ leads to 1 0 W 1 −J 00 gJ 0 g −1 = . with W = J 0+ + (J 00 )2 = J + J − + (J 0 )2 and g = 1 0 0 1 4 After quantization, we recover the well-known result that the Casimir operator 1 C2 = t20 + (t+ t− + t− t+ ) = t20 + t0 + t− t+ 2
(2.7)
commutes with t− and t0 . Note however that we get also the non-trivial information that the only elements in U(s`(2)) that commute with both t− and t0 (not necessarily t+ ) are polynomials in W . 3. W-Realization of Lie Algebras The above presentation allows us to provide a new technique for the construction of representations of a Lie algebra G starting from a given (canonical) realization of G. 3.1. Regular representation of a Lie algebra G. We recall here the regular representation of a Lie algebra G. We start with G, the Lie group of G, and consider F = F (G, C) the space of smooth functions on G. One can construct two isomorphic representations of G on F . They are associated to the right and left actions of G onto itself. The left representation UL is defined through the map g ∈ G → UL (g) with (UL (g)ϕ) (g 0 ) = ϕ(g −1 g 0 ) , ∀ϕ ∈ F , ∀g 0 ∈ G,
(3.1)
while the right representation UR uses the map g → UR (g) with (UR (g)ϕ) (g 0 ) = ϕ(g 0 g) , ∀ϕ ∈ F , ∀g 0 ∈ G.
(3.2)
These representations induce naturally two representations of G on F : d ϕ(e−t g 0 ) , t ∈ G → TL (t) with TL (t)ϕ(g 0 ) = d =0 d t ∈ G → TR (t) with TR (t)ϕ(g 0 ) = ϕ(g 0 et ) . d =0 From now on, we will focus on UR and TR , and the way the following construction can be done for UL and TL is analogous.
398
F. Barbarin, E. Ragoucy, P. Sorba
3.2. Differential realization of G. We now use the regular representation to build a differential realization (as used in [7]) adapted to the decomposition2 G = G − ⊕ G 0 ⊕ G + that we have used in the previous section. We first remark that P = G 0 ⊕G + is a parabolic subalgebra of G, and that any element g ∈ G can be decomposed in a unique way as g = pg− with p ∈ P and g− ∈ G− . Now, starting from the regular representation TR on F , we can restrict it to the subspace F of functions ϕ ∈ F such that ϕ(pg) = ϕ(g) ∀g ∈ G and ∀p ∈ P . We will call this realization T R . Note that F is not empty because F (G− , C) can be canonically embedded in F . Thus T R is not the trivial representation. Any g ∈ G can be decomposed into g = pg− with p ∈ P and g− ∈ G− , in a unique way. Moreover, the set {tA } being a basis of G − , we can label any g− ∈ G− A with dimG − = s variables xA such that g− = ex tA . Thus we will have ϕ(g) = ϕ(pg− ) = ϕ(g− ) = ϕ(ex
A
tA
) ≡ ϕ(x1 , x2 , . . . , xs ).
(3.3)
Let us look more carefully at T R . For tB ∈ G − , we have:
d T R (tB )ϕ(g) = ϕ(getB ) d
=0
d = ϕ(pg− etB ) d
=0
d = ϕ(g− etB ) d
, (3.4) =0
where we have used the decomposition g = pg− . Now, as G − is Abelian, we have ϕ(getB ) = ϕ(ex
A
tA +tB
) = ϕ(x1 , x2 , . . . , xB + , . . . , xs ),
so that T R (tB )ϕ(x1 , . . . , xs ) = ⇒ T R (tB ) = ∂B .
d
d ϕ(x
1
, x2 , . . . , xB + , . . . , xs )
=0
=
∂ ϕ(x1 , . . . , xs ) ∂xB
(3.5) To be complete, let us add that, using e−B eA eB = ee
−B
AeB
(3.6)
the same calculation shows that ∀tj ∈ G 0 , T R (tj ) = −fjA B xA ∂B where [tj , tA ] = fjA B tB .
(3.7)
Then, it is easy to check that T R (tA ) =
1 f j fjB C xA xB ∂C 2 AA
(3.8)
is solution of [tj , tA ] = fjA B tB and [tA , tA ] = fAA i ti . We have seen that one can identify F with a space of functions of dimG − variables. We would like G − elements to act multiplicatively instead of acting through differential operators. Thus, we perform a Fourier transformation: 2
In the case of half-integral gradation G − = G −1 ⊕ G − 1 we will perform a halving to get an integral one. 2
W-Realization of Lie Algebras: Application to so(4, 2) and Poincar´e Algebras
e ϕ∈F → ϕ e ∈ F, ϕ(x1 , . . . , xs ) → ϕ(y e 1 , . . . , ys ) =
Z
399
ds x ei~y·~x ϕ(x1 , . . . , xs ),
e , the G − part acts multiplicatively, with ~y · ~x = yA xA . On the new space of functions F and the other generators of G will be differential polynomials in the variables: tA ∈ G − → TeR (tA ) = −iyA , d , tj ∈ G 0 → TeR (tj ) = fjA B yB ∂ A + fjA A with ∂ B = dyB i i tA ∈ G + → TeR (tA ) = fAA j fjB C yC ∂ A ∂ B + (fAA j fjB B + fAB j fjA B )∂ A . 2 2 It is this representation we will use in the following. 3.3. W-realization. We use the techniques of the Sect. 2 to get a wide class of representations of G, starting from the differential representation based on G − . We know that the generators of W(G, µs`(2)) ≡ W form the commutant of Ge = G − ⊕ Ge0 . They can be expressed in terms of all the generators of G. Moreover, we have seen that, in the differential realization, the elements of G − are represented by yA , while the generators in Ge0 are represented by yB ∂ A (up to some constants). Thus, the W-generators commute both with yA and ∂ A (∀ A), i.e. they are constant on the whole differential representation. Let w0α be these constants, where α labels the generators of W. From the general results of Sect. 2, we know that there exists a unique element e such that g ∈ Θ(G) J g = g(J α tα + J α tα )g −1 = e− + W α tα with g = eu
a
ta
e and ua = ua (J α ). , ta ∈ Θ(G) (3.9)
Let us rewrite this formula as J = J α tα + J α tα = g −1 (e− + W α tα )g.
(3.10)
This shows that we can write any element J a in terms of the J α ’s and the W α ’s. We also remark that, in the same way the realization of W α is linear in J α because of (3.9), the J α ’s also depend linearly in the W α ’s because of (3.10). Then starting with a given realization of Ge we construct a realization of the whole algebra G leaving Ge unchanged and expressing the other generators of G in terms of those of Ge and the W-algebra, by applying formula (3.10). Performing the same calculation at the quantum level, we deduce that starting with a given representation of G, and so of Ge , one can obtain other representations of G using representations of the W-algebra. It is natural to start with the differential representation of G. Then, one generalizes it into a “matrix differential representation” using a matrix representation of W. More e , where d is the dimension of precisely, we take as representation space d copies of F the W-representation one chooses. Then, G will be represented by
400
F. Barbarin, E. Ragoucy, P. Sorba
tA ∈ G − → T~R (tA ) = −iyA . 1I, T~R (tα ) = fαA B yB ∂ A + fαA A . 1I, → ~ tα ∈ Ge0 , B A A tj ∈ G 0 . 1I (t ) = f y ∂ + f T R B iA α αA tα ∈ (G 0 \ Ge0 ) → β B β +Rαβ (yA , ∂ ).(M − w0 1I) i tA ∈ G + → T~R (tA ) = f j fjB C yC ∂ A ∂ B + (fAA j fjB B 2 AA + fAB j fjA B )∂ A . 1I + + RAβ (yA , ∂ B ).(M β − w0β 1I), where M I is the d × d matrix representing W I , 1I is the d × d identity matrix, and RαI (yA , ∂ B ) are polynomials in the differentials. The above operators will act on vectors: ϕ1 (y1 , . . . , ys ) ϕ2 (y1 , . . . , ys ) . ϕ ~ (y1 , . . . , ys ) = .. . ϕd (y1 , . . . , ys ) 3.4. Example: s`(2). We continue the example of Sect. 2. We denote an element of the group SL(2) by its coordinates g(x− , x0 , x+ ) = ex
−
t− x 0 t 0 x + t +
e
e
(3.11)
so that the functions read ϕ(x− , x0 , x+ ) on the regular representation. In the differential e and we have representation, we have functions ϕ(y e − ) ≡ ϕ(y) T˜0 (t− ) = −iy ; T˜0 (t0 ) = −y∂ − 1 and T˜0 (t+ ) = −iy∂ 2 − 2i∂
(3.12)
We use the formula computed in Sect. 2 to construct the W-realization for s`(2). We have seen that, at the quantum level, w = t20 + t0 + t− t+ is the generator of the commutant of {t0 , t− } (see Eq. (2.7)). We start with a representation T˜0 corresponding to a value w0 of the Casimir operator. Then, we construct a new representation T keeping for t0 and t− the form T˜0 , T (t0 ) = T˜0 (t0 ) and T (t− ) = T˜0 (t− ), and inverting the formula (2.7) for t+ : w − w0 = T (t0 )2 + T (t0 ) + T (t− )T (t+ ) − T˜0 (t0 )2 + T˜0 (t0 ) + T˜0 (t− )T˜0 (t+ ) = T˜0 (t− ) T (t+ ) − T˜0 (t+ ) ⇒ T (t+ ) = T˜0 (t+ ) +
w − w0 . T˜0 (t− )
Using the differential representation, we get w0 = 0 and w T (t+ ) = −iy∂ 2 − 2i∂ + i with w ∈ R. y
(3.13)
Thus (3.13), together with T (t− ) = −iy and T (t0 ) = −y∂ − 1 is a representation of s`(2) with a value w for the Casimir operator. Note that, for s`(2), a similar (up to a change of variable y x2 ) realization has been given in [5].
W-Realization of Lie Algebras: Application to so(4, 2) and Poincar´e Algebras
401
4. Application to Poincar´e and so(4, 2) Algebras We now apply the results of the previous sections to get realizations of the conformal so(4, 2) algebra (and of its Poincar´e subalgebra). We will show that the W-realization of this latter provides the Pauli-Lubanski-Wigner quadrivector W µ in a very natural way. A generalization to the whole so(4, 2) algebra will exhibit two new quadrivectors which, together with P µ and W µ , form a basis of the Minkowski space. The scalar products of these vectors give all the physical quantities such as the mass or the spin. 4.1. Description. The so(4, 2) algebra is known as the conformal algebra in four dimensions, in the Minkowski space, i.e. with the metric gµν = diag(1, −1, −1, −1). It is a non-compact form of so(6), with maximal compact subalgebra so(4) ⊕ so(2). Note that s`(4) is also a non-compact version of so(6), but with maximal compact subalgebra so(4). Thus, one can connect s`(4) to so(4, 2) through a compactification process, a property which is useful for calculations. The algebra so(4, 2) has fifteen generators: Four translations: Pµ with µ = 0, 1, 2, 3 which form G− Three rotations and three boosts: Mij and M0i respectively (i, j = 1, 2, 3) G0 One dilatation: D Four special conformal transformations: Kµ with µ = 0, 1, 2, 3 which form G + . (4.1) Rotations and boosts constitute the Lorentz algebra G 0 = so(3, 1), and together with the translations those generators form the Poincar´e algebra. In the following we will denote by P = G 0 ⊕ G + the corresponding parabolic algebra. This latter decomposition is actually induced by a natural gradation of the algebra with respect to the element −iD, which assigns a grade −1 to Pµ , 0 to Mµν , and +1 to Kµ . The commutation relations read: [Pµ , Pν ] = 0 [Mµν , Pρ ] = i(gνρ Pµ − gµρ Pν ) [D, Pµ ] = −iPµ , [Mµν , Mρσ ] = i(gνρ Mµσ − gµρ Mνσ + gµσ Mνρ − gνσ Mµρ ) [D, Mµν ] = 0, [Kµ , Kν ] = 0 [Mµν , Kρ ] = i(gνρ Kµ − gµρ Kν ) [D, Kµ ] = iKµ , [Pµ , Kν ] = 2i(−Mµν + gµν D).
(4.2)
so(4, 2) possesses three Casimir operators of degree 2, 3 and 4. We will call them C2 , C3 and C4 respectively. We will need a frame (e0 , e1 , e2 , e3 ) for the Minkowski space: e~1 = (1, 0, 0) e0 = (e0 )µ = (1, ~0) with e~2 = (0, 1, 0) , (4.3) ej = (ej )µ = (0, e~j ) j = 1, 2, 3 e~ = (0, 0, 1) 3
which satisfies eµ · eν = (eµ )ρ (eν )σ gρσ = gµν and (eµ )ρ (eν )σ g µν = g ρσ .
(4.4)
4.2. Momentum representation. We first construct the differential representation associated to G − (see Sect. 3). We start with a coordinate representation, where functions depend on x = (xµ ) and perform a Fourier transformation: Z ϕ(p) e = ds x eip·x ϕ(x).
402
F. Barbarin, E. Ragoucy, P. Sorba
The coordinate differential representation on the space of smooth functions on G/P : ∂ ∂xµ ∂ D = ix · ∂x
Pµ = i
∂ ∂ − xν µ ), ν ∂x ∂x ∂ ∂ Kµ = i(2xµ x · − x2 µ ) ∂x ∂x
Mµν = i(xµ
(4.5)
is thus converted into the momentum representation: P µ = pµ
Mµν = i(pµ ∂ν − pν ∂µ ),
D = −i(p · ∂ + 4) Kµ = (pµ −2p · ∂∂µ − 8∂µ ),
(4.6)
where ∂µ stands for ∂p∂µ , p is the quadrivector (pµ ), ∂ the quadrivector (∂µ ), and = ∂ · ∂ = g µν ∂µ ∂ν = ∂ µ ∂µ . 4.3. The W-algebra . We wish to construct the commutant of the Pµ in order to build a W-representation for G = so(4, 2). We have seen that D is the right grading operator, since G − = {Pµ }. In the case we consider s`(4) the maximally non-compact form of G, this gradation corresponds to the model W(s`(4), 2s`(2)). Then referring to the formula (2.1) we can take Ge = G − ⊕ Ge0 , where Ge0 = {M13 − M01 , M23 − M02 , M03 , D}.
(4.7)
The commutant of Ge can be seen as a compactified form3 of the algebra W(s`(4), 2s`(2)). Indeed, calculations can be done in an easier way when using the s`(4) algebra (instead of so(4, 2)) and then compactifying the resulting algebra. This algebra is a polynomial deformation of G 0 = so(3, 1) ⊕ so(1, 1). It contains seven generators: three generators Jk , k = 1, 2, 3 of degree one in the so(4, 2) elements, which form so(3), a Lie subalgebra of the W-algebra, and also four generators C2 and Sk , k = 1, 2, 3 of degree two. C2 is the second order Casimir of so(4, 2): C2 =
1 µρ νσ (g g Mµν Mρσ + g µν (Pµ Kν + Kµ Pν )) − D2 . 2
(4.8)
For the clarity of the presentation, we postpone the expressions of the other generators in terms of the so(4, 2) generators (formulae (4.24), (4.35) and (4.37)). The commutation relations of the W-algebra read: [Jj , Jk ] = ijkl Jl , [Jj , Sk ] = ijkl Sl ,
[Sj , Sk ] = −ijkl 2(J12 + J22 + J32 ) − C2 − 4 Jl .
(4.9)
4.4. The W-realization. Now using techniques summarized in Sect. 3 as well as the results of 4, one can determine the general expressions of the conformal algebra generators. The generators corresponding to Ge are unchanged with respect to the differential realization (4.6). One remarks that the so(4, 2) generators are indeed linear in the Ji (i = 1, 2, 3) and Zk (k = 0, 1, 2, 3) see (4.22) which also form a basis of the W-algebra. 3
Let us mention the recent work [8] on the real forms of W-algebras.
W-Realization of Lie Algebras: Application to so(4, 2) and Poincar´e Algebras
403
~ One notes also that only the J-part of the W-algebra shows up in the Poincar´e part: this will be discussed in the next paragraph. One can check that every W-generator reduces to zero in the differential realization (4.6). Finally let us emphasize that the realization given just below corresponds to values of p2 > 0; therefore the quantities p0 + p3 and p2 appearing in the denominators never vanish. Let us now give the realization that one gets from our “W-approach” (1I denotes the identity matrix): Pµ = pµ 1I µ = 0, 1, 2, 3, M12 = i(p1 ∂2 − p2 ∂1 ) 1I + J3 , p p2 p2 J2 − J3 , M13 = i(p1 ∂3 − p3 ∂1 ) 1I − p0 + p 3 p0 + p 3 p p1 p2 J1 + J3 , M23 = i(p2 ∂3 − p3 ∂2 ) 1I + p0 + p 3 p0 + p 3 p p2 p2 J2 − J3 , M01 = i(p0 ∂1 − p1 ∂0 ) 1I − p0 + p 3 p0 + p 3 p p1 p2 J1 + J3 , M02 = i(p0 ∂2 − p2 ∂0 ) 1I + p0 + p 3 p0 + p 3 M03 = i(p0 ∂3 − p3 ∂0 ) 1I, D = −i(p · ∂ + 4) 1I, 2 K0 = (p0 −2p · ∂∂0 − 8∂0 ) 1I − Z3 p0 + p 3 p0 1 p (p1 Z1 + p2 Z2 ) + 2 Z0 + p (p0 + p3 ) p2 p 2i p2 5 p1 5 p2 − −( 2 + ∂2 )J1 + ( 2 + ∂1 )J2 p0 + p 3 2p 2p 2i (p2 ∂1 − p1 ∂2 )J3 , − p0 + p 3 1 p1 K1 = (p1 −2p · ∂∂1 − 8∂1 ) 1I + p Z1 + 2 Z0 + 2 p p p 2i p2 5 p0 + p3 − + ∂ 0 + ∂ 3 J2 p 0 + p 3 2 p2 p2 −2i (∂0 + ∂3 ) − ∂2 J3 , p0 + p 3 1 p2 K2 = (p2 −2p · ∂∂2 − 8∂2 ) 1I + p Z2 + 2 Z0 + 2 p p p 2i p2 5 p0 + p3 + + ∂ 0 + ∂ 3 J1 p0 + p 3 2 p2 p1 +2i (∂0 + ∂3 ) − ∂1 J3 , p0 + p 3
(4.10) (4.11) (4.12) (4.13) (4.14) (4.15) (4.16) (4.17)
(4.18)
(4.19)
(4.20)
404
F. Barbarin, E. Ragoucy, P. Sorba
K3 = (p3 −2p · ∂∂3 − 8∂3 ) 1I +
2 p3 Z3 + 2 Z0 p0 + p 3 p
1
p (p1 Z1 + p2 Z2 ) + (p0 + p3 ) p2 p 2i p2 5 p2 5 p1 + −( 2 + ∂2 )J1 + ( 2 + ∂1 )J2 p0 + p 3 2p 2p 2i (p2 ∂1 − p1 ∂2 )J3 , + p0 + p 3 −
where
Z1 = 2S1 + J3 J1 + J1 J3 Z2 = 2S2 + J3 J2 + J2 J3 , Z0 = 2S3 + C2 − J32 − 2(J12 + J22 ), Z3 = S3 − (J12 + J22 )
(4.21)
(4.22)
4.5. The Poincar´e algebra. Let us focus on the expressions of the Poincar´e generators (4.10)- (4.16). We recall the expressions of the Pauli-Lubanski-Wigner quadri-vector W µ = 21 µνρσ Pν Mρσ which satisfies: [Wµ , Pν ] = 0 [Wµ , Wν ] = iµνρσ W ρ P σ W · P = 0.
(4.23)
It is well-known that the irreducible representations of the Poincar´e algebra are labelled by the eigenvalues of P 2 = p2 and W 2 = −s(s + 1)p2 , where s is the spin of the particle. Because of the relation W · P = 0, the quadrivector W µ possesses only 3 independent components. These 3 independent components generate the spin algebra so(3) when p2 is positive [9]. This is recovered in a very natural way in our W-algebra framework. Indeed, the generators Jk really play the role of the spin generators, since they can be rewritten as: X 1 1 nk · W = − (nk )µ Wµ and W µ = −m (nk )µ Jk (since P · W = 0) , m m k=1 (4.24) p with m = p2 and W · W = −P 2 J~2 , (4.25) k=3
Jk = −
and where we have introduced the frame of the “particle” of momentum p: p0 p1 p2 p3 1 p = ( , , , ), m m m m m 1 (p1 , p0 + p3 , 0, −p1 ), n1 = (n1 )µ = p0 + p 3 1 n2 = (n2 )µ = (p2 , 0, p0 + p3 , −p2 ), p0 + p 3 −m 1 n3 = (n3 )µ = p, (1, 0, 0, −1) + p0 + p 3 m
n0 = (n0 )µ =
which obeys to
(4.26) (4.27) (4.28) (4.29)
nµ · nν = (nµ )ρ (nν )σ gρσ = gµν ,
(4.30)
(nµ )ρ (nν )σ g µν = g ρσ .
(4.31)
and also to The Lorentz transformation, denoted by [p] in [9] and L(p) in [6], which moves the rigid referential frame (e0 , e1 , e2 , e3 ) to the p-frame (n0 , n1 , n2 , n3 ):
W-Realization of Lie Algebras: Application to so(4, 2) and Poincar´e Algebras L(p)
(e0 , e1 , e2 , e3 ) −→ (n0 , n1 , n2 , n3 )
405
(4.32)
can explicitly be written as a rotation followed by a boost ~ ~ p~ · B p~⊥ · R ) and R(p) = exp(−iθ ), k~ pk k~ p⊥ k (4.33) k~ p⊥ k2 k~ pk , cos(θ) = 1 − and p~⊥ = (p2 , −p1 , 0), with sh(λ) = m (m + p0 )(p0 + p3 )
L(p) = B(p)R(p) with B(p) = exp(−iλ
~ = (M23 , M31 , M12 ). This transfor~ = (M01 , M02 , M03 ) and R where we have defined B mation also relates the three vector (Ji ) with the four vector (Wµ ): L(p) ~ −→ W = (Wµ ). mJ = (0, mJ)
(4.34)
Actually, L(p) belongs to the set of Lorentz transformations which send the four vector (m, 0, 0, 0) into p = (p0 , p1 , p2 , p3 ). It is through L(p) that the representations of the Poincar´e group can be constructed from representations of the rotation subgroup. Indeed, following [9] and [6], the Lorentz transformation 3 acting on the function ϕ˜ of the p-variable in the U representation: 3 ϕ(p) ˜ = U (3)ϕ(3 ˜ −1 p) is written more conveniently on the Wigner functions ψ defined by ˜ ψ(p) = U (L(p)−1 )ϕ(p) as
3 ψ (p) = U L(p)−1 3L(3−1 p) ψ(3−1 p).
We recognize in the product L(p)−1 3L(3−1 p) a Wigner rotation, element of the (m, 0, 0, 0)-vector stabilizer, itself isomorphic to the SO(3) group when p2 > 0. Let us emphasize that the transformation L(p) can be chosen up to a R-rotation leaving (m, 0, 0, 0) invariant. In particular, the L(p) used in the computations of [6] is the pure boost B(p). This of course produces a new but equivalent frame (n00 , n01 , n02 , n03 ) and different expressions for the Lie algebra realization. 4.6. Generalization to so(4, 2). We now look at the other generators of the conformal algebra. We proceed as for the Poincar´e algebra. In the same way we have introduced the Pauli-Lubanski-Wigner vector W µ , let us define: 1 1 Σµ = −W 2 Pµ + P 2 P α Mαµ (D + i) − (Pµ P ·K − P 2 Kµ ) − µνρσ W ν M ρσ . 2 2 (4.35) It satisfies Σ · P = 0, [Σµ , Pν ] = 0 , [Σµ , Wν ] = iµνρσ Σ ρ P σ [Σµ , Σν ] = iµνρσ P 2 W ρ P σ (P 2 (4 + C2 ) + 2W 2 ) . The generators Si are connected to the quadrivector (Σ µ ) through
(4.36)
406
F. Barbarin, E. Ragoucy, P. Sorba
k=3 X 1 1 µ µ 3 Sk = − 3 nk · Σ = − 3 (nk ) Σµ and Σ = −m (nk )µ Sk (since P · Σ = 0) m m k=1 (4.37) and also: L(p) ~ −→ Σ = (Σµ ). (4.38) m3 S = (0, m3 S)
Let us remark that the third and fourth Casimir operators of so(4, 2) can be expressed as W-polynomials: ~ = − W · Σ = − 1 C3 , J~ · S (P 2 )2 2 1 1 2 2 2 2 ~ + J~ (C2 − J~ + 2) = Σ + P 2 W 2 W 2 + P 2 (C2 + 2) = − C4 + S 2 3 (P ) 4
(4.39) 2 C2 3
with the following expression for the Casimir operators: 1 fµν + 1 (K · W + P · W ), (4.40) C3 = − DMµν M 4 2 1 1 fµν )2 + C4 = 2(W · V + V · W ) + (P 2 K 2 + K 2 P 2 ) − (P · K)2 + (Mµν M 2 16 4 −2(D2 + 2)Mµν M µν + 4DP µ K ν Mµν−8iD(C2 −D2 ) + C2 −20D2 , (4.41) 3 where we have introduced Vµ =
1 µνρσ fµν = µνρσ Mρσ . Kν Mρσ and M 2
(4.42)
Note that (e0 , J, S) on the one hand, and (P, W, Σ) on the other hand form three ~ independent quadrivectors: it is natural to look for a fourth quadrivector X = (0, X) such that (e0 , J, S, X) is a frame. Then, we will get a quadrivector χ such that L(p)
m4 X −→ χ and (P, W, Σ, χ) frame.
(4.43)
Calculations show that X1 = (J2 S3 + S3 J2 ) − (J3 S2 + S2 J3 ) − 3iS1 , ~ = (X1 , X2 , X3 ), X2 = (J3 S1 + S1 J3 ) − (J1 S3 + S3 J1 ) − 3iS2 with X X3 = (J1 S2 + S2 J1 ) − (J2 S1 + S1 J2 ) − 3iS3
(4.44)
~ is a non-commutative version of the is such a vector. Note that the leading term of X ~ ~ exterior product J ∧ S. From Eqs. (4.25) and (4.39), one sees that all the “physical informations” of the Poincar´e algebra as well as of the conformal algebra so(4, 2) are contained in these frames. 4.7. Case m = 0. Up to now, the results we have presented are valid for p2 > 0. One can wonder whether there exists some limiting process that could lead to the case of zero mass representations. At that point, we have to make a distinction between the so(4, 2) algebra and its Poincar´e subalgebra. Indeed, in the Poincar´e algebra, the mass is an invariant (p2 = m2 ), so that it is very natural to look at the limit point m = 0. In the so(4, 2) algebra, this is no more the case, but the sign (p2 ) is still an invariant. By sign of p2 , we mean (p2 ) = +, 0, − when p2 is respectively positive, zero
W-Realization of Lie Algebras: Application to so(4, 2) and Poincar´e Algebras
407
or negative. One easily guesses that a limiting process on the three points = +1, 0, −1 is hard to find. In other words, in the Poincar´e algebra, the (positive mass) irreducible representations are hyperboloids in the Minkowski space, while in the so(4, 2) algebra, the irreducible representation = +1 is the whole interior of the light cone. Then, if, in the Poincar´e algebra, it is natural to look at the light cone as the limit of the family of hyperboloids when m → 0, we have to face a problem of dimensional reduction of the space of representations in the so(4, 2) (see [10] for a direct treatment of the case m = 0 in so(4, 2)). Thus, we will focus below on the Poincar´e algebra to study the limit m → 0. The ~ and C2 W-algebra then reduces to a three-dimensional algebra since the generators S do not enter in the realization (4.10-4.16). Moreover, in the case m = 0, one cannot introduce the vector J~ and the frame attached to the particle, which is now of zero mass. However, we can define: Qj = nj · W for j = 1, 2 while Q3 =
1 (e0 − e3 ) · W, p0 + p 3
(4.45)
which form an E(2) subalgebra [Q3 , Q1 ] = iQ2 [Q3 , Q2 ] = −iQ1 [Q1 , Q2 ] = 0
(4.46)
in accordance with the general results on zero mass particles. Moreover, (4.45) shows that for W µ proportional to P µ , and using the definitions (4.27-4.28), we have Q1 = Q2 = 0. Let us recall that this latter condition is the one we need to select the physical cases when m = 0, i.e. the cases of finite spin (more exactly helicity). By definition, these latter cases are the ones that correspond to finite dimensional representations of the E(2) algebra. Thus, they are also natural in the W-realizations, since we consider a priori matricial representations of the W-algebra. In that case, the realization reads: Pµ = pµ 1I µ = 0, 1, 2, 3, M12 = i(p1 ∂2 − p2 ∂1 ) 1I + Q3 , p2 Q3 , M13 = i(p1 ∂3 − p3 ∂1 ) 1I − p0 + p 3 p1 Q3 , M23 = i(p2 ∂3 − p3 ∂2 ) 1I + p0 + p 3 p2 Q3 , M01 = i(p0 ∂1 − p1 ∂0 ) 1I − p0 + p 3 p1 Q3 , M02 = i(p0 ∂2 − p2 ∂0 ) 1I + p0 + p 3 M03 = i(p0 ∂3 − p3 ∂0 ) 1I,
(4.47) (4.48) (4.49) (4.50) (4.51) (4.52) (4.53)
where Q3 corresponds to the helicity. Let us note that although the denominator p0 + p3 may vanish when m = 0, the above expressions can still be well-defined using limits p0 + p3 → 0 and pj = o(p0 + p3 ) (j = 1, 2). 4.8. Comparison with induced representations. It is time to compare the above obtained W-realization of the so(4, 2) algebra with the usual ones constructed via the induced representation method. As could be expected, the results are equivalent. More is given in [6] where the classification of all the unitary ray representation of the SU (2, 2) group with positive energy is achieved. It will then be possible for us, using these results,
408
F. Barbarin, E. Ragoucy, P. Sorba
to select the finite dimensional representations of the W-algebra which lead to such unitary representations of the conformal one, and therefore to unitary representations of the conformal group. Let us start by introducing a realization of the W-algebra under consideration known as the Miura realization, and particularly useful in what follows. Indeed, it is the Miura realization of the W-algebra which can be unearthed in [6]. Its knowledge also provides a simple algorithm for the construction of representations of the W-algebra. 4.8.1. The Miura realization The Miura transformation provides a realization of the W-algebra relative to the gradation G = G −1 ⊕ G 0 ⊕ G +1 in terms of elements of the enveloping algebra U (G 0 ) [3]. There, the G 0 part is the Lorentz algebra to which have been added the dilatation: so(3, 1) ⊕ so(1, 1), and the Miura realization reads simply: ~ J~ = R
~=R ~ ×B ~ − i(D − 1)B ~ S
~2 − B ~ 2 + D(D − 4), C2 = R
(4.54)
using again the notations of Sect. 4 for the Lorentz generators. One immediately remarks that this realization is much simpler than the one above determined and which involves all the so(4, 2) generators: (4.24) and (4.37). Thus, to each finite dimensional G 0 representation can be associated, via Eq. (4.54), a finite dimensional W representation. Labelling in a rather natural way (j1 , j2 ; d) an so(3, 1) ⊕ so(1, 1) irreducible representation, the center element C2 in W will get the scalar value: c2 = 2j1 (j1 + 1) + 2j2 (j2 + 1) + d(d − 4). 4.8.2. Unitary representations The unitary representations with positive energy of the universal covering group Gˆ of G = SU (2, 2) are presented in [6], see also [11]. We note the chain of isomorphisms: conformal group of ˆ (4.55) ' SO(4, 2)/ Z2 ' SU (2, 2)/ Z4 ' G/(0 1 × 02 ), Minkowski space where 01 ' Z2 and 02 ' Z. Such unitary representations possess a lowest weight, and are interpolation of the holomorphic discrete series. These latter are realized by considering functions on the bounded symmetric domain SU (2, 2)/K, where K = SU (2) × SU (2) × U (1) is the maximal compact subgroup of SU (2, 2). Actually they are induced by the finite dimensional (unitary) representations of K. However, Mack [6] described them as induced representations by the parabolic subgroup P = 02 M AN , where following his notations M , A, N are the subgroups of Lorentz, dilatation and special conformal transformations respectively. The Lie algebra of P is exactly G 0 ⊕ G + in our language. We note that P is actually the little group ˆ . of the point x = 0 in the compactified Minkowski space, itself isomorphic to G/P The induced representations on this space are then constructed in the usual way. We take finite dimensional representations Dλ of the parabolic subgroup P , where N acts trivially. They are labelled by two non-negative (half-)integers (j1 , j2 ), associated to spinor representations of the Lorentz group Dj2 j1 , and by d a real number associated to the dilatation:
W-Realization of Lie Algebras: Application to so(4, 2) and Poincar´e Algebras
409
Dλ (man) = |a|d−2 Dj2 j1 (m). It acts on a (2j1 +1)(2j2 +1) dimensional vector space E λ . From now on we will disregard the action of the discrete central part 0, which of course will not affect the Lie algebra realization. We then consider the space of functions ϕ on the group G with values in E λ which have the covariance property ϕ(gman) = Dλ (man)−1 ϕ(g)
∀g ∈ SU (2, 2).
The conformal group acts on these functions by the left regular representation: (T (g)ϕ)(g 0 ) = ϕ(g −1 g 0 ). Since the decomposition g = g− man is unique, functions ϕ can be viewed as functions ϕ(x) defined on G/P , the Minkowski space. The action of SU (2, 2) then takes the form: (T (g)ϕ)(x) = |a|2 Dλ (man)−1 ϕ(x0 ), where x0 (∈ G− ), m, a, n are defined through the unique decomposition g −1 x = x0 man. Note that this way to describe unitary representations of SU (2, 2) is unusual since we induced by non-unitary representations of a parabolic subgroup which is not cuspidal [12]. By cuspidal, we mean that the Cartan subalgebra of the semi-simple part M of P is made only with compact generators, which is not the case here. To get the action of the Lie algebra so(4, 2), one has simply to take the infinitesimal form: ∂ ∂xµ
∂ ∂ − xν µ − iΣµν ), ∂xν ∂x ∂ ∂ ∂ ) Kµ = i((8 − 2d)xµ + 2xµ x · − x2 µ − 2ixν Σµν ). D = i(4 − d + x · ∂x ∂x ∂x (4.56) Here Σµν is a (2j1 +1)(2j2 +1) representation of the Lorentz algebra with Ri = 21 ijk Σ jk and Bi = Σ0i . The comparison with the W-realization proceeds in several steps. First of all, we have to perform a Fourier transformation in order to get momentum representations with functions ϕ(p): ˜ Pµ = i
P µ = pµ D = −i(p · ∂ + d)
Mµν = i(xµ
Mµν = i(pµ ∂ν − pν ∂µ − iΣµν ), Kµ = pµ −2p · ∂∂µ − 2d∂µ − 2iΣµν ∂ ν .
(4.57)
Then we study the action of the conformal algebra on the functions ψ(p) defined by ˜ ψ(p) = Dj2 j1 L(p)−1 md−4 ϕ(p) where L(p) is a Lorentz transformation that transports the frame (e0 , e1 , e2 , e3 ) to the frame (n0 , n1 , n2 , n3 ). If X is the form of a generator of so(4, 2) acting on functions ϕ(p) ˜ then, when acting on functions ψ(p) it becomes Xˆ = D j2 j1 L(p)−1 md−4 X m−(d−4) Dj2 j1 L(p) . Note that when restricted to the Poincar´e algebra, since the mass is constant on an irreducible representation, this whole trick amounts to working on the Wigner wave function (cf. Sect. 4). We then recover the well-known representations of the Poincar´e
410
F. Barbarin, E. Ragoucy, P. Sorba
group induced by the group of rotations. Explicitly, Lorentz transformations are given by (T (3)ψ)(p) = Dj2 j1 L(p)−1 3L(3−1 p))ψ(3−1 p , where D j2 j1 (L(p)−1 3L(3−1 p)) reduces simply to representations of spin s = |j1 − j2 |, . . . , j1 + j2 of so(3). This realization of the Poincar´e algebra corresponds exactly to the form we find for the W-representation (4.11)-(4.16), by identifying Ji with Ri . ˆ we first compute md−4 Xm−(d−4) . For the other so(4, 2) generators, to calculate X, Then, we perform a second order Taylor expansion of L(p) around (p0 , p~) = (m, ~0). ˆ using the fact that Kµ transforms as a fourFinally, we get the general form of X, vector under a Lorentz transformation, and D as a scalar. We obtain for the dilatation the realization (4.17) computed in the W-representation. We also get the W-realization (4.18)-(4.21) for Kˆµ , provided we identify ~ J~ = R
~=R ~ ×B ~ − i(d − 1)B ~ S
~2 − B ~ 2 + d(d − 4), C2 = R
(4.58)
which is precisely the Miura realization given in (4.54). Thus, we have made the link between the W-realization and the induced representation techniques. As a further consequence, the unitarity conditions on j1 ,j2 and d given in [6] can be seen as conditions on the W-algebra representations. In particular the class of representations of positive masses j1 j2 6= 0,
d ≥ j1 + j2 + 2
with
d > j1 + j2 + 1
with j1 j2 = 0
(4.59)
gives rise respectively to the C2 conditions: c2 ≥ 2j1 (j1 + 1) + 2j2 (j2 + 1) + (j1 + j2 )2 − 4, c2 > 3(j1 + j2 + 1)(j1 + j2 − 1).
(4.60)
For j1 j2 6= 0 and d > j1 +j2 +2, each representation contains spins s = |j1 −j2 |, · · · , j1 +j2 , while for d = j1 + j2 + 2 or j1 j2 = 0 each representation contains only the spin s = j1 + j2 .
Acknowledgement. We would like to thanks Philippe Caldero for valuable discussions about commutants in Lie algebras. We are also indebted to Raymond Stora for his interest and numerous comments on induced representations of the Poincar´e and conformal algebras and Xo Luc Frappat for a careful reading of the manuscript.
References 1. 2. 3. 4. 5. 6. 7. 8.
Barbarin, F., Ragoucy, E., Sorba, P.: Nucl. Phys. B442, 425 (1995) Barbarin, F., Ragoucy, E., Sorba, P.: Int. Journ. Mod. Phys. A11, 2835 (1996) de Boer, J., Harmsze, F., Tjin, T.: Phys. Rep. 272, 139 (1996) Feher, L., O’Raifeartaigh, L., Ruelle, P., Tsutsui, I., Wipf, A.: Phys. Rep. 222, 1 (1992) and references therein Leinaas, J.M. and Myrheim, J.: Int. Journ. Mod. Phys. A8, 3649 (1993) Mack, G.: Commun. Math. Phys. 55, 1 (1977) Kostant, B.: Lecture Notes in Mathematics. 466, Berlin, Heidelberg, New-York: Springer 1974, p 101 Evans, J.M., Madsen, J.O.: Phys. Let. 384, 131 (1996)
W-Realization of Lie Algebras: Application to so(4, 2) and Poincar´e Algebras
411
9. Moussa, P., Stora, R.: In “Methods in Subnuclear Physics”. London: Gordon and Breach, Herceg-Novi Summer School, 1966 10. Mack, G., Todorov, I.: Journ. Math. Phys. 10, 2078 (1969) 11. R¨uhl, W.: Commun. Math. Phys. 30, 287 (1973) 12. Knapp, A.W., Speh, B.: J. Funct. Anal. 45, 41 (1982) Communicated by R.H. Dijkgraaf
Commun. Math. Phys. 186, 413 – 449 (1997)
Communications in
Mathematical Physics c Springer-Verlag 1997
On the Stability of Realistic Three-Body Problems? Alessandra Celletti1 , Luigi Chierchia2 1 Dipartimento di Matematica, Universit` a dell’Aquila, 67100–Coppito, L’Aquila, Italy. E-mail: [email protected] 2 Dipartimento di Matematica, Universit` a di Roma Tre, Largo San Murialdo 1, 00146 Roma, Italy. E-mail: [email protected]
Received: 1 April 1996 / Accepted: 25 October 1996
Abstract: We consider the system Sun–Jupiter–Ceres as an example of a planar, circular, restricted three-body problem and, after substituting the mass ratio of Jupiter/Sun (which is approximately 10−3 ) with a parameter ε, we prove the existence of stable quasiperiodic motions with frequencies close to the observed (average) frequencies reported in “The Astronomical Almanac” for |ε| ≤ 10−6 . The proof is “computer-assisted”. Table of Contents 1 Introduction and Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 2 Restricted, Circular, Planar Three-Body Problem . . . . . . . . . . . . . . . . . . 418 3 A Model from the Solar System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 4 A KAM Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 4.1 Algebraic Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 4.2 Analytic Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 4.3 KAM Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 4.4 KAM Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 5 Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 5.1 Step 1: Formal ε-expansion and initial approximate solution . . . . . . . . . 438 5.2 Step 2: Norm bounds relative to the initial approximate solution . . . . . . 441 5.3 Step 3: Application of the KAM algorithm and of the KAM Theorem . . 445 A Some Computer-Assisted Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 ? This work is part of the research program of the European Network on: “Stability and Universality in Classical Mechanics”, # ER–BCHRXCT940460.
414
A. Celletti, L. Chierchia
1. Introduction and Theorem 1.1 1) Since human kind developed a definite mathematical taste (whatever this means) the “stability” of planetary motions might be considered as one of the central questions in mathematics. Nowadays consciousness about new phenomena (pollution, just to name one) has drawn the attention of scientists and non-scientists to other types of stabilities (in other words, extinction of living species will depend more and more upon the chaotic effects of pollution rather than on “the sky falling on our heads”). Nevertheless the stability problem for many-body systems interacting only through gravitation still stems out as one of the more intriguing and rich problems in mathematics. In modern times outstanding contributions came, above all, from H. Poincar´e [13], V. I. Arnold [2] and J. Moser [12]. In particular the so-called KAM (Kolmogorov, Arnold, Moser) theory (see [1] and references therein) gave a “positive” answer to the above stability problem in the sense that it proved ([2]) the possibility of the existence of many-body systems (“planetary systems”) whose time evolution may be described by a linear flow on a torus (“quasi-periodic motion,” here, synonymous of “stable motion”). The drawback of this beautiful result is that, taking the estimates contained in it seriously, it turns out that the mass ratio of the planets of such a hypothetical planetary system with their star should be more or less comparable to the mass ratio of a proton with the Sun.1 One then may pose the question of stability of realistic many-body systems. Of course, to give a mathematical content to the word “realistic” is clearly impossible and the best we could do was to get inspiration from our own planetary system. The “simplest non-trivial” three-body problem is the so-called planar, circular, restricted three-body problem (see Sect. 2 for definitions): we considered one of the most popular three-body problems of the Solar system, namely Sun, Jupiter and Ceres (one of the major bodies of the so-called asteroid belt). Following Delaunay [6], we then derived a Hamiltonian model. Clearly, in deriving a model one simplifies Nature2 quite a bit. We then took our model seriously (from the mathematical point of view) and replacing the Jupiter/Sun mass ratio, which is approximately 10−3 , with a “perturbative parameter” ε, we asked for how large values of ε one could find quasi-periodic motions with “frequencies” close to the observed frequencies of the Sun–Jupiter–Ceres system (which may be found in the ephemeris [16]). For such a model we proved stability for |ε| ≤ 10−6 , being therefore away from “reality” by three orders of magnitude. We leave it to the reader to judge if this is realistic or not. We believe, however, that with some more efforts one should indeed be able to prove stability up to ε = 10−3 and we regard our result as a first step towards a proof of the mathematical stability of realistic many-body problems. Our result relies basically on two techniques: (i) a (new) KAM scheme presented in Sect. 4 (for the experts: a KAM result in Hamiltonian setting in the style of Moser, Salamon and Zehnder [15] with emphasis on analytical dependence upon parameters); (ii) computer-assisted (rigorous) estimates, which are needed in order to apply “effectively” the KAM scheme to our three-body problem. It is well known that computers may be used to prove theorems (see, e.g., [7, 11, 10] or think of the famous “four-colour theorem”). We were not really enthusiastic to rely 1 Mass of proton/mass of Sun= 1.6724 · 10−21 /8.4078 · 1030 ' 1.9891 · 10−52 ; compare [8]. For some recent applications of Arnold’s result to three-body problems see [14]. 2 For example, to consider Sun–Jupiter–Ceres as a planar, circular, restricted three-body problem means, in particular, that one is assuming the Jupiter orbit circular: this is a rather crude approximation and more “realistic” models would include the Jupiter eccentricity, the Saturn “secular” effects on Jupiter, etc. See Sect. 3 for a (partial) justification of our model.
Stability of Three-Body Problems
415
on machines to prove our result but couldn’t get away without it. In fact we are pretty sure that with more refined techniques and/or with new ideas one might get better results without the (essentially trivial but) lengthy computations which are the only reason to call in machines. 2) We give now a precise formulation of the main result. Let (`, g) ∈ T2 ≡ R2 /(2πZ2 ); let3 L0 ≡ 0.729305 ,
G0 ≡ 0.727162 ,
r0 ≡ 0.001 ,
(1.1)
and let B ≡ {(L, G) ∈ C2 : |L − L0 | ≤ r0 , |G − G0 | ≤ r0 },
B0 ≡ B ∩ R2 .
(1.2)
On T2 × B0 , endowed with the standard symplectic form d` ∧ dL + dg ∧ dG, consider the one-parameter family of Hamiltonian functions given by H(`, g, L, G; ε) ≡ (
1 1 − G)2 + 2ε ( 2 − G) R(`, g, L, G) 2 2L 2L
≡ h(L, G) + ε f (`, g, L, G) , with the “perturbing function” R defined as X Rn (L, G) cos(n1 ` + n2 g), R(`, g, L, G) ≡
(1.3)
(1.4)
n∈Z2 0≤|n1 |+|n2 |≤10
where Rn ≡ Rn1 n2 vanishes unless it belongs to the following list: R00 ≡ R11 ≡ R22 ≡ R33 ≡ R55 ≡
9 L4 3 (1 + L4 + e2 ) , 4 16 2 3 6 5 L (1 + L4 ) , 8 8 5 4 L4 (3 + L ) , 4 4 5 6 7 L (1 + L4 ) , 8 16 63 10 L , 128
9 L4 e (1 + L4 ) , 2 8 L4 e (9 + 5L4 ) , R12 ≡ − 4 3 R32 ≡ L4 e , 4 35 8 L , R44 ≡ 64
R10 ≡ −
(1.5) (1.6)
(1.7)
and the “eccentricity” e, which is a function of the “action variables” (L, G), is defined as4 r G2 e ≡ e(L, G) ≡ 1 − 2 . L Let α be the golden mean (α ≡ − ≡ 3
√
5−1 2 )
and let
1 5 + = 2.573432... , 2 13 + α
+ ≡
1 5 + = 2.579251... . 2 12 + α
All the following numbers, quantities and functions will be physically motivated in Sects. 2 and 3. In this paper the letter e will always refer to “eccentricities” and never to the Neper number; the exponential function will be denoted exp(x). 4
416
A. Celletti, L. Chierchia
The “observed average frequency” of Ceres is approximately −c ≡ −2.577107 (see Sect. 3 below) so that − < c < + . Let q −1/3 L± ≡ ± , G± ≡ L± 1 − e20 , where e0 = 0.0766 is the “observed” eccentricity of Ceres as found in [16]. Finally, we define the h-frequencies ω (±) ≡ (E± ± , E± ) ,
E± ≡ −2(
1 − G± ) . 2L2±
(1.8)
For later use we point out that ω (±) is a “Diophantine vector” satisfying |ω (±) · n| ≥ (γ± |n|)−1 , with γ± given by5
∀ n ∈ Z2 \{0} ,
√ γ± = 2|E± |( 5 + 24 ± 1) .
(1.9)
(1.10)
The result discussed in 1) above can now be formulated as follows. Theorem 1.1. Let H be as in (1.3)–(1.5); let B0 be as in (1.1), (1.2); let ω (±) be as in (1.8). Then, for all 0 ≤ |ε| ≤ 10−6 there exist (unique) two-dimensional analytic tori Sε (ω (±) ) ⊂ B0 × T2 , depending analytically also on the parameter ε (for |ε| ≤ 10−6 ), on which the H-flow is (analytically) conjugated to the linear flow θ ∈ T2 → θ − ω (±) t. Remark 1.1. In Sect. 3 we give a physical motivation for having chosen as a perturbating function a trigonometric polynomial. We believe, however, that considering perturbations with an infinite number of non-vanishing Fourier coefficients would lead to essentially the same results making only technically more involved (and more expensive) the proof. 3) (On the proof) The proof of Theorem 1.1 is given in Sect. 5 and, as already mentioned, is a “computer assisted” application of the KAM scheme6 of Sect. 4. As it is well known KAM schemes are “Newton algorithms”: they are procedures to iteratively construct solutions for certain nonlinear equations (with “loss of regularity”), starting from some initial “approximate solution,” with a quadratic rate of convergence. Our initial approximate solution is a suitable truncation (actually a “fifth order truncation”) of the so-called “Lindstedt series” (see [1] for generalities), i.e., of the formal ε-power series solution for the invariant torus equation associated to the looked for quasiperiodic solution. To this initial datum we apply the “KAM algorithm” presented in Sect. 4. The “KAM algorithm” is based on an algebraic scheme which, starting from a given “approximate solution,” produces a new function solving the invariant torus equation up to an error which is “quadratically smaller” than the one produced by the starting approximate solution (Sect. 4.1). This algebraic scheme (which, as already mentioned, is new) is equipped with a set of “accurate” estimates (Sect. 4.3). The algebraic scheme plus the set of estimates is what we call “the KAM algorithm.” We then work out a criterion (the KAM Theorem of Sect. 4.4) which guarantees the applicability of the KAM 5
In general, numbers of the form a =
p q
±
1 k+α
with p, q, k non negative integers, q > 0 and k ≥ 2,
satisfy |an + m| ≥ (γ|n|)−1 for any n, m ∈ Z, n 6= 0 with γ = q 2 (k + α) (see, e.g., [3]). 6 See [4] and [5] for general information, references and a different “KAM computer-assisted algorithm”.
Stability of Three-Body Problems
417
algorithm an infinite number of times yielding a solution of the invariant torus equation. Such criterion is obtained simplifying the estimates and getting a unique stronger condition (the “KAM condition”) ensuring the indefinite applicability of the scheme. As we already pointed out in previous papers (see [5] and references therein), in concrete applications, it is convenient to iterate a few times the KAM algorithm before trying to apply the KAM theorem. Both the computation of the initial approximate solution and the application of the KAM algorithm are computer-assisted. We remark that it is quite different to explicitly compute the initial approximate solution from the “computation” of the new approximate solutions based on the KAM algorithm: the calculation of the truncated Lindstedt series is completely explicit (we compute numbers!) while the construction of the sequence of quadratically better and better approximate solutions is only implicitly described by the KAM algorithm and what we actually compute are bounds on norms relative to such approximate solutions. 4) (On the use of computers) Our proof is “computer-assisted” in the sense that certain formulae (derived below) have been implemented on a computer (a VAX) keeping rigorous control, by means of the so-called “interval arithmetic” (see below), of the numerical errors introduced by the machine. We report in Sects. 5.2 and 5.3 all the computer-aided calculations needed to prove Theorem 1.1. Instead we do not include the computer program which anybody can write by her/himself.7 We are obviously aware of the (phylosofical?) problem of proceeding in such a way: It is clear that writing a program in a slightly different way or using different machines might (better: will) produce slightly different outputs, which in our case are intervals of rational numbers (see below). However we regard the computer-implementation, once it is clearly settled the type of rigorous method used to control the propagation of numerical errors (here “interval arithmetic”) as a detail at the same level, say, of the details needed to work out explicitly the estimates of the KAM algorithm of Sects. 4.3 and 4.4. Of course, we shall be happy to send to interested readers the computer programs contained in this paper. Let us now briefly discuss interval arithmetic which is the technical tool we used to control the numerical errors introduced by the machines. Real numbers are represented by computers as sign-exponent-fraction quantities, with the length of the exponent and of the fraction depending on the machine. Any result among elementary operations (sum, subtraction, multiplication and division) is rounded by the computer up to a certain decimal digit. To rigorously implement on a computer a certain sequence of formulae, one first reduces such formulae to a sequence of elementary operations.8 The idea of the “interval arithmetic” is then to construct an interval (exactly representable on the computer) containing the exact result of an elementary operation and to replace (in the obvious way) algebra on numbers with algebra on intervals. In our FORTRAN 77 programs we define quadruple precision (H-floating) variables, which are allowed to vary in a range between 0.84 · 10−4392 and 0.59 · 104392 . The binary structure of a quadruple precision datum is composed by 128 bits, with 1 sign bit, 15 bits for the exponent and the remaining bits for the fraction. Two extra hidden guard bits are used to guarantee the result of an elementary operation “up to 1/2 of the last significant bit” ([17]). The interval containing the result of an elementary operation is therefore obtained increasing or decreasing by one bit the last bit of the mantissa, eventually taking care of 7 The (11189–line) computer program is “just” a translation in computer language (FORTRAN 77) of the formulae of Sects. 5.1 and 4.3 after the standard arithmetic (basic operations) is replaced by “interval arithmetic” (the “arithmetic routines” may be found, e.g., at pages 153–158 of [4]). 8 Elementary functions (such as roots, exponentials, trigonometric functions, etc.), will be approximated by a finite sequence of elementary operations using Taylor polynomials keeping track of errors; see [5] §8.3.
418
A. Celletti, L. Chierchia
the propagation of the carry. For further information and for the necessary routines we refer to [4 and 5]. We finally mention that in the Appendix we report a few computer-assisted data with the following doublefold aim. From one side the reader reproducing our estimates might check her/his results with ours; from the other side the reader who is not going to waste time performing the computations will have an idea of the type of outputs one needs in this paper. 2. Restricted, Circular, Planar Three-Body Problem Here we recall the Hamiltonian formulation of the “restricted, circular, planar three-body problem” (for general information see [6 or 1]). Consider first a Keplerian two-body problem made up of two material points (“bodies”) P1 and P2 with masses m1 and m2 and let P2 revolve on a circular orbit around P1 . Consider now a third body A moving on the orbital plane of P1 and P2 and subject to the gravitational attraction of P1 and P2 . Let the mass mA of A be much smaller than m1 , m2 and assume that the motion of P1 and P2 is not affected by A. The study of the dynamics associated to such a model is known in the literature as the circular, planar, restricted three-body problem. In particular, we shall be interested in phase space regions for which the resulting motion of A is a nearly circular orbit “around” P1 . A convenient Hamiltonian formulation of such a three-body problem is based upon the classical “planar Delaunay variables” [6]. Let T ≡ R/(2πZ) and consider the phase space9 P = {(λ, γ, ψ) ∈ T3 } × {(3, 0, E) ∈ R3 : 3 6= 0, |0| < |3|} endowed with the standard symplectic form dλ ∧ d3 + dγ ∧ d0 + dψ ∧ dE. Then, the dynamics associated with the circular, planar restricted three-body problem is given by the Hamiltonian flow generated by the Hamiltonian10 H0 (λ, γ, ψ, 3, 0, E) ≡
1 + E + ε R0 (λ, γ − ψ, 3, 0) , 232
(2.1)
where ε ≡ m2 /m1 and the “perturbation function” R0 is given as follows. Let ν ∈ T (the “eccentric anomaly”) be implicitly defined for |e| < 1 (e ∈ R) by the relation (“Kepler’s equation”) λ = ν − e sin ν ; let ϕ ∈ T (the “true anomaly”) be implicitly defined (again for |e| < 1) by the relation tg
1 + e 1/2 ν ϕ−γ = tg , 2 1−e 2
and define the “orbital radius” r as r≡
a(1 − e2 ) , 1 + e cos(ϕ − γ)
where
a ≡ 32 .
The function R0 in (2.1) is then given by 9 The Delaunay coordinates λ, γ, ψ are often called, respectively, the “mean anomaly,” the “argument of the perihelion” and the “longitude” (of the “planet” P2 ). 10 We have chosen the units of measure in such a way that m + m = 1 and that the period of P is 2π. 1 2 A
Stability of Three-Body Problems
419
1 , R0 (λ, γ − ψ, 3, 0) ≡ − r cos(ϕ − ψ) − p 1 + r2 − 2r cos(ϕ − ψ) s
where e is defined as e≡
1−
02 . 32
We recall that a convenient representation of R0 is obtained by means of Legendre polynomials11 : if r < 1 (which will be the case for our specific model) one finds R0 = 1 +
∞ X
rj Pj (cos(ϕ − ψ)) .
j=2
A trivial reduction shows that the dynamics generated by (2.1) may be described by a two–degree–of–freedom Hamiltonian: under the canonical (or “symplectic”) transformation (`, g, τ ) ≡ (λ, γ − ψ, ψ) , (L, G, T ) ≡ (3, 0, 0 + E) , H0 takes the form H1 (`, g, L, G) =
1 − G + ε R0 (`, g, L, G) , 2L2
(2.2)
having omitted the dummy variable T ; the phase space is now T2 × {(L, G) ∈ R2 : L 6= 0, |G| < |L|}. 3. A Model from the Solar System Let us now focus on the case in which P1 is the Sun, P2 is Jupiter and A is Ceres (one of the largest bodies in the asteroid belt). Notice (again) that regarding Sun–Jupiter–Ceres as a planar, circular, restricted three-body problem contains a lot of physical approximations, which we shall not discuss here. But even accepting these basic approximations, the reader will have certainly noticed that the Hamiltonian in (1.3) is different from the Hamiltonian in (2.2): besides the factor ( 2L1 2 − G) (and 2ε in place of ε), the main difference is that R is a trigonometric polynomial of degree 10 while R0 contains infinite non vanishing Fourier harmonics. The “selection rule” which led us to the choice of the “physically relevant” Fourier modes is based on the following trivial observation. Among other things, the gravitational effects on Ceres of asteroids and planets and most notably the attraction exerted by Saturn (which, after Jupiter, is the largest planet in the Solar system12 ) have been neglected. Therefore, after having defined a (rough) measure, GSa , of the Ceres–Saturn attraction, we disregard in the Fourier expansion of R0 the terms exceeding GSa in absolute value. In order to define GSa we first look up a few astronomical data in the ephemeris (see The Astronomical Almanac [16]). In particular we want to define the “reference values” of L0 and G0 for the Sun–Jupiter–Ceres system. Observations of the true motion of Ceres, as found in [16], indicate that Ceres moves on a nearly elliptical orbit of “average eccentricity” (2k+1)P (x)x−kP
(x)
k k−1 P0 (x) = 1; P1 (x) = x; Pk+1 (x) = , (for k ≥ 1). k+1 Even though the orbit of Ceres is closer to the orbit of Mars than to the orbit of Saturn, the difference in mass makes the gravitational attraction of Saturn the largest one after that of Jupiter.
11
12
420
A. Celletti, L. Chierchia
e0 ≡ 0.0766
(3.1)
and whose average semimajor axis is approximately 0.532. Hence, the corresponding “average frequency” of Ceres, computed by Kepler’s third law, yields a value of −c ≡ −2.577107 . Since −c ' ∂L H1 |ε=0 = −L−3 we take as “reference L-value” the quantity 1
L0 ≡ 0.729305 ' c3
and, since G0 = L0
q 1 − e20 , we take, as “reference G-value” the quantity G0 ≡ 0.727162 .
Such reference values have been taken as center of the analyticity domain for the “action variables”; see (1.1) and (1.2). Notice that with our choice of the analyticity radius r0 one finds that the function e(L, G) satisfies 0.019799 < |e(L, G)| < 0.106364 ,
∀ (L, G) ∈ B .
Let us turn now to the definition of GSa . In general, for planets whose orbits have a larger semimajor axis than that of Ceres, the “secular term” of H1 is given by13 ε (≡ mass of the planet/mass of the Sun) times the term R00 ≡ R00 (L; e) in (1.5). Keeping in mind that, in the integrable limit, L is the ratio of the semimajor axis of Ceres with that of the planet we define (for planets whose orbits have a larger semimajor axis than that of Ceres) (3.2) GP ≡ ε(P ) × R00 (L(P ); e0 ), where e0 is the observed “average eccentricity” of Ceres (3.1), ε(P ) is the mass ratio of the planet P and of the Sun and L(P ) is the ratio of the semimajor axis of Ceres with that of the planet P . Looking up the “true” values in [16] one finds GSa ≡ GSaturn = 6.3778 · 10−6 . For comparison purposes we report also the value for Jupiter, which is GJupiter = 7.8850 · 10−5 . Neglecting in the expansion of εR0 those terms whose size is smaller than14 GSa , one is led to consider a “three-body problem” governed by the Hamiltonian H2 (`, g, L, G; ε) ≡
1 − G + ε R(`, g, L, G) 2L2
(3.3)
with R given in (1.4) and (1.5) of Sect. 1. The final modification of H2 which gives the Hamiltonian in (1.3), (1.4) is due to merely technical reasons. As mentioned above, our results are based on computer-assisted KAM theory, and one of the standard hypotheses of KAM theorems is that the unperturbed Hamiltonian 13 The “secular term” of H is the average over the angular variables ` and γ of the “perturbation” εR ; 1 0 the computation is immediately checked using, e.g., the above mentioned expansion in terms of Legendre Polynomials. 14 More precisely we omit all the terms such that ε|R (L , G )| < G . n 0 0 Sa
Stability of Three-Body Problems
421
(ε = 0) is non-degenerate, i.e. has an invertible Hessian matrix on its domain of analyticity. In the case of (3.3), the unperturbed Hamiltonian is given by h0 (L, G) =
1 − G, 2L2
whose Hessian matrix is not invertible. There are a few well known methods to overcome this minor problem15 and it turns out that for our purposes the most convenient one is to follow Poincar´e’s trick [13], which consists in replacing the Hamiltonian H2 by its square.16 Therefore we let H3 ≡ (H2 )2 : 1 1 − G)2 + 2ε ( 2 − G) R(`, g, L, G) 2L2 2L + ε2 [R(`, g, L, G)]2 .
H3 (`, g, L, G; ε) = (
The Hessian of the unperturbed Hamiltonian (H3 |ε=0 ) is equal to 5 6G 2 4 6 − L3 , A ≡ A(L, G) ≡ L 2 L 2 L3
(3.4)
(3.5)
and, if (L, G) ∈ B, one has 12 1 | det A| = 4 ( 2 − G) L 2L 12 1 − (G + r ) ≥ 8.830153 . ≥ 0 0 (L0 + r0 )4 2(L0 + r0 )2 To be consistent with the criterion that led us to the Hamiltonian (3.3), we have to omit the term of order ε2 in (3.4) and this leads us to the Hamiltonian (1.3) introduced in Sects.(1, 2).
4. A KAM Theorem Here we prove a KAM result, in the style of [5], which will be the basis of the proof of Theorem 1.1. First we provide a “KAM algorithm” (in the Hamiltonian context), which yields a sequence of quadratically better and better approximations to the conjugacy function of a maximal invariant (Diophantine) torus, and then we formulate a criterion ensuring the applicability of the algorithm an infinite number of times and hence the existence of an invariant torus. Technically, the algorithm, which does not use symplectic transformations (used, instead, in the original works of the masters), may be viewed as a Hamiltonian version of the Lagrangian approach developed in the eighties by Moser, Salamon and Zehnder (see [5] and references therein). 4.1. Algebraic Scheme. Let us consider a smooth (later real-analytic) Hamiltonian h(x, y), where x varies on the standard N -torus TN ≡ RN /(2πZN ) and y varies in 15 For example, one can replace the non-degeneracy hypothesis with a “iso-energetic non-degeneracy” (see, e.g., [1]), which is satisfied by h0 . 16 Note that the dynamics generated by a Hamiltonian function h = h(q, p) and by h2 coincide up to a time scale: if z(t) = (q(t), p(t)) is an h-motion then z(2Et), with E = h(z(0)) is the corresponding h2 -motion.
422
A. Celletti, L. Chierchia
some open ball B N ⊂ RN ; (x, y) are standard symplectic coordinates.17 The problem is to construct an invariant N -torus S on which the flow is conjugated to the linear flow θ ∈ TN → θ + ωt for some “rationally independent” vector18 ω ∈ RN . The Sembedding function θ ∈ TN → (θ + u(θ), v(θ)) ∈ TN × B N is immediately seen to satisfy the following quasi-linear, degenerate PDE on TN : ω + Du − hy (θ + u, v) = 0 , Dv + hx (θ + u, v) = 0 ,
(4.1)
where D denotes the derivatives in the ω direction: D ≡ ω · ∂θ ≡
N X
ωi
i=1
∂ , ∂θi
(4.2)
and hx , hy denote the gradient of h with respect to x, y. As usual, we assume that ω is a Diophantine vector, i.e. there exist γ > 0 and a positive integer τ such that N X ωi ni ≥ (γ|n|τ )−1 , |ω · n| ≡
∀ n ∈ ZN \{0} .
(4.3)
i=1
The starting point of a KAM algorithm is an approximate solution (u, v), which solves (4.1) up to some “error.” In order to formulate a precise result we need some notations and some assumptions. Given a function u : TN → RN we denote by uθ or by ∂θ u its Jacobian matrix (uθ )ij ≡ hxy denotes the matrix with entries hxy ≡ ij
∂ui ; ∂θj ∂2h ; ∂xi ∂yj
thus hyx is the transpose of hxy , i.e. hyx = hTxy ; if A is a square matrix, we denote by A# the antisymmetric part of A times two: A# ≡ A − AT ; finally, if θ ∈ TN → f (θ) ∈ Rs (s ≥ 1) is a smooth function with vanishing mean value, i.e. Z 1 f (θ) dθ = 0 , hf i ≡ (2π)N TN (and if D is as in (4.2), (4.3)), we denote by D −1 f the unique solution with vanishing mean value of the equation Dg = f ; such a solution in Fourier expansion has the form X fn exp(in · θ) , D−1 f = iω ·n N n∈Z \{0}
where fn denote Fourier coefficients and i = 17 18
√ −1.
That is, the symplectic structure is given by the standard 2-form I.e. if ω · n = 0 for some n ∈ ZN , then n must be 0.
PN i=1
dxi ∧ dyi .
Stability of Three-Body Problems
423
Assumption 4.1. Let θ ∈ TN → (u, v) ∈ RN × RN be a smooth function and let M and h0yy be the matrices19 M ≡ I + uθ , h0yy (θ) ≡ hyy θ + u(θ), v(θ) . (4.4) Denoting20
T ≡ M−1 h0yy M−T ,
(4.5)
we assume that, for any θ ∈ TN , the matrices M, h0yy and hT i are invertible. Proposition 4.1. Let h, u, v and ω satisfy, respectively, Assumption 4.1 and (4.3) and define f and g by ω + Du − hy (θ + u, v) = f , Dv + hx (θ + u, v) = g .
(4.6)
Then, if we define the vector/matrix-valued functions b(θ) and B(θ) by # B ≡ MT gθ − vθT fθ , b ≡ vθT f − MT g , we have hbi = 0 ,
hBi = 0 .
(4.7)
Furthermore, the following equation holds21 : ω + Du0 − hy (θ + u0 , v 0 ) = f 0 , Dv 0 + hx (θ + u0 , v 0 ) = g 0 ,
(4.8)
where u0 , v 0 , f 0 , g 0 are defined at the end of the following list of definitions22 : −1 hM−1 f i − hT D−1 bi , c1 ≡ hT i b0 ≡ T D−1 b + c1 − M−1 f , z ≡ M D−1 b0 + c2 , c2 ≡ −hMD−1 b0 i , −1 Dz − h0yx z + f , w ≡ h0yy q1 ≡ hx (θ + u + z, v + w) − h0x − h0xx z − h0xy w , q2 ≡ hy (θ + u + z, v + w) − h0y − h0yx z − h0yy w , q3 ≡ fθT w − gθT z − MT q1 , f 0 ≡ −q2 , h i g 0 ≡ M−T D (D−1 B)M−1 z + MT (h0yy )−1 fθ M−1 z − q3 , u0 ≡ u + z ,
v0 ≡ v + w .
I is the identity matrix. To be precise we should replace, in (4.4), u with p ◦ u, p being the projection of onto TN ; however we shall omit, here and in other circumstances, such projection. 20 The superscript −T denotes the transpose of the inverse: A−T = (A−1 )T . 21 Here and in what follows, the prime attached to a function will never denote derivates but just new functions. 22 As above if h = h(x, y), h0 (θ) denotes the function h(θ + u(θ), v(θ)). 19
RN
424
A. Celletti, L. Chierchia
Remark 4.1. (i) It is immediate to check that if we replace f and g by εf and εg then z, w = O(ε) and f 0 , g 0 = O(ε2 ), i.e. the errors associated to u0 and v 0 are quadratically smaller than the errors associated to u and v. We shall call a couple (u, v) as in (4.6) an approximate solution for (4.1) and the relative couple (f, g) the error function. (ii) Note that the constants c1 and c2 are defined so that the functions b0 and z have vanishing mean value. Proof. To check the first of (4.7), observe that ∂θ h0 = MT h0x + vθT h0y ; now, multiply the first of (4.6) by vθT , the second by −MT , add them together and use integration by parts to get rid of the terms containing θ-derivatives. To check the second of (4.7), take the θ-gradient of (4.6) to obtain DM = h0yx M + h0yy vθ + fθ , Dvθ = −h0xx M − h0xy vθ + gθ . Let
# A ≡ M T vθ ,
(4.9)
(4.10)
and notice (integration by parts) that hAi = 0 .
(4.11)
From (4.9), it follows that the matrix A satisfies the equation DA = B ,
(4.12)
from which the second of (4.7) follows at once. The first of (4.8) follows immediately from the definitions: ω + Du0 − hy (θ + u0 , v 0 ) = ω + Du + Dz − hy (θ + u + z, v + w) = hy (θ + u, v) − hy (θ + u + z, v + w) + Dz + f = −q2 − h0yx z − h0yy w + Dz + f = f0 .
(4.13)
The check of the second of (4.8) is more tricky. First observe that last identity in (4.13) can be rewritten as Dz = h0yx z + h0yy w − f . (4.14) Next, from the definition of z, it follows that i h D MT (h0yy )−1 MD(M−1 z) + f = b .
(4.15)
Solving for h0yy in the first of (4.9) and inserting the obtained expression in the definition of w, we get h i w = (h0yy )−1 Dz − (DM)M−1 − h0yy vθ M−1 − fθ M−1 z + f = vθ M−1 z + (h0yy )−1 fθ M−1 z + MD(M−1 f ) + f . (4.16)
Stability of Three-Body Problems
425
From the definition of g 0 , (4.15), (4.16), (4.10)÷(4.12), it follows that MT g 0 + q3 + b = D(MT w − vθT z) , i.e. g 0 = M−T D(MT w − vθT z) − b − q3 . From this identity, recalling the definitions of b and q3 and using (4.9) to eliminate fθ and gθ , one obtains i h h i g 0 = M−T vθT h0yx z + h0yy w − f + g + h0xx z + h0xy w + q1 + Dw − M−T vθT Dz , which, in view of (4.14), the definition of q1 and the second of (4.6), yields the second of (4.8). 4.2. Analytic Tools. From now on we shall work in the real-analytic category; in this section we review some basic technical facts. We shall consider the Banach space of periodic functions f real-analytic on the torus TN , admitting (for some prefixed ξ > 0) analytic extension on the closed strip 1ξ ≡ {θ ∈ CN : | Im θi | ≤ ξ , ∀ i = 1, ..., N } , equipped with the “Fourier norm” X
kf kξ ≡
|fn | exp(|n|ξ) ;
(4.17)
n∈ZN
in CN (and its subsets RN and ZN ) we shall use the 1-norm, |y| ≡ |y|1 ≡
N X
|yi | .
i=1
If f : TN → Cs is aPvector valued, real-analytic function, analytic on 1ξ , its norm is defined as kf kξ ≡ i kfi kξ , which coincides with (4.17) if fn denotes the s-vector whose components are given by the Fourier coefficients of the components of f . These definitions are immediately extended to matrix/tensor-valued functions by making use of the standard “operator norm”: e.g. if A(θ) is a matrix-valued periodic functions with analytic extension on 1ξ , we set kAkξ ≡
sup c∈CN :|c|=1
kAckξ ;
or, if ∂x3 f is the tensor of order three of the derivatives of a periodic, real-analytic function f : 1ξ → C, its Fourier norm is given by k∂x3 f kξ
N N X X ≡ sup k |b|=|c|=1 i=1
j,k=1
∂3f bk cj kξ . ∂xi ∂xj ∂xk
Finally, we shall also consider functions (possibly vector/matrix/tensor-valued) h = br (y0 ), where B br (y0 ) h(x, y) periodic in x and real-analytic on the closed domain 1ξ × B
426
A. Celletti, L. Chierchia
b N (y0 ) denotes the closed complex ball of radius r around y0 ∈ CN . For any such ≡B r br (y0 )) function, which admits the expansion23 (convergent on 1ξ × B X h(x, y) = hn,k exp(in · x) (y − y0 )k , (4.18) n∈ZN k∈NN
we set
X
khkξ,r ≡
|hn,k | exp(|n|ξ) r|k| .
(4.19)
n∈ZN k∈NN
The elementary properties of interest in the present context are collected in the following Lemma 4.1. (i) Let f : TN → R have an analytic extension on 1ξ (for some ξ > 0) and let ω ∈ RN be a rationally independent vector. Then, for all 0 < δ ≤ ξ, for any p ∈ Z and for any k ∈ N or any k ∈ NN one has24 kD−p ∂xk f kξ−δ ≤ kf kξ σpk (δ), where, if k = 0, f is assumed to have vanishing mean value, and sup πnk |ω · n|−p exp(−δ|n|) , σpk (δ) ≡ {n∈ZN \{0}:fn 6=0}
πnk ≡
|n|k , if k ∈ N , |nk | , if k ∈ NN .
(ii) Let f, g : TN → R have an analytic extension to 1ξ , then kf gkξ ≤ kf kξ kgkξ . ¯ let h : TN × {y ∈ Rs : |y − y0 | ≤ r} → R have an analytic (iii) Let 0 < ξ < ξ; b extension on 1ξ¯ × Brs (y0 ), f : TN → RN and g : TN → Rs have analytic extension on 1ξ . Assume that kf kξ ≤ ξ¯ − ξ, kg − y0 kξ ≤ r. Then, denoting φ(θ) ≡ (θ + f (θ), g(θ)), one has kh ◦ φkξ ≤ khkξ,r ¯ . Proof. (i) The claim follows immediately by expanding f in Fourier series. (ii) In the following sums the indices n, m run over ZN , X X X kf gkξ ≡ |(f g)n | exp(|n|ξ) = fm gn−m exp(|n|ξ) n
≤
X
n
m
exp |m| exp |n − m||fm | |gn−m |
n,m
= kf kξ kgkξ . (iii) In the following sums the indices n, m run over ZN , the index k runs over Ns and j over N. Using (ii), one gets Qk 23 k k We use the standard notation: (y − y0 ) =
24
If k ∈ N,
∂xk f
i=1
(yi − y0i ) i .
denotes the k-tensor of the derivatives of f ; if k ∈ NN , ∂xk f =
∂ |k| f
k
k
∂x1 1 ···∂xNN
.
Stability of Three-Body Problems
427
X X hm,k ij (m · f )j (g − y0 )k exp(|n|ξ) j! n−m n m,k,j X |hm,k | exp(|m|ξ) ≤ (m · f )j (g − y0 )k exp(|n − m|ξ) j! n−m
|h ◦ φ|ξ =
n,m,k,j
=
X |hm,k | k(m · f )j (g − y0 )k kξ exp(|m|ξ) j!
m,k,j
≤
X |hm,k | |k| km · f kjξ kg − y0 kξ exp(|m|ξ) j!
m,k,j
≤
X |hm,k | |m|j (ξ¯ − ξ)j r|k| exp(|m|ξ) j!
m,k,j
= khkξ,r ¯ .
Remark 4.2. (i) Note that the result might be empty if ω is “too well approximable by rationals vectors.” If ω satisfies (4.3) then one checks easily that σpk (δ) ≤
γ p δ −(|k|+pτ ) (|k| + pτ )! , if p ≥ 0, |p| δ −(|k|+|p|) (|k| + |p|)! , if p < 0 ,
≡ max |ωi | .
(4.20)
(ii) It is easy to check that (i) and (iii) of Lemma 4.1 holds also if f , respectively, h are vector valued.
4.3. KAM Algorithm. Here we describe the “KAM algorithm” associated to the scheme of Sect. 4.1, i.e. we equip the algebraic scheme described in Proposition 4.1 with “accurate” and detailed estimates so as to end up with a map K, which to given bounds on norms of the relevant objects relative to a certain approximate solution (u, v), associates corresponding bounds on the new approximation (u0 , v 0 ). More precisely, let us start by making quantitative the hypotheses formulated in Sect. 4.1. ¯ r, E, Ep,q (p, q ∈ N) be such that (x, y) → h(x, y) is Assumption 4.2. Let 0 < ξ < ξ, N
real analytic on 1ξ¯ × B r (y0 ) and k(hyy )−1 kξ,r ¯ ≤E ,
k∂xp ∂yq hkξ,r ¯ ≤ Ep,q ;
(4.21)
assume that the approximate solution (u, v) is real analytic on 1ξ and let U , V , M , M , Ve , F , G, Te be positive numbers bounding the following norms: kukξ ≤ U , kvkξ ≤ V , kMkξ ≤ M , kM−1 kξ ≤ M , −1 kvθ kξ ≤ Ve , kf kξ ≤ F , kgkξ ≤ G , |hT i | ≤ Te ,
(4.22)
where M, T , f and g are, respectively, as in (4.4), (4.5) and (4.6). Finally assume that U ≤ ξ¯ − ξ ,
ρ ≡ kv − y0 kξ < r .
(4.23)
428
A. Celletti, L. Chierchia
Now let 0 < δ ≤ ξ and define ξ0 ≡ ξ − δ .
(4.24)
In the rest of this section we shall define the KAM map, i.e., an explicit map 0
K : (U, V, M, M , Ve , F, G, Te) → (U 0 , V 0 , M 0 , M , Ve 0 , F 0 , G0 , Te0 ),
(4.25)
where (U 0 , ..., G0 , Te0 ) are bounds on the norms ku0 kξ0 ,..., kg 0 kξ0 and on the number −1 |hT 0 i |, with u0 , v 0 , f 0 , g 0 defined in Proposition 4.1, while M0 and T 0 are (obviously) defined as M0 ≡ I + u0θ ≡ M + zθ ,
T 0 ≡ M0
−1
hyy (θ + u0 , v 0 ) M0
−T
.
We start now to work out the necessary estimates. By Lemma 4.1 and (4.22), we get 2
kT kξ ≤ T ≡ M E0,2 . Remark 4.3. In principle, the bound on kT kξ could be improved replacing E0,2 by k(h0yy )−1 kξ (without invoking point (iii) of Lemma 4.1); in practice, however, such a norm is difficult to evaluate accurately and one would not get significantly better estimates. All other estimates are immediately obtained from Lemma 4.1 (and from the definitions given in Proposition 4.1). Here is the complete list: kT −1 kξ ≤ T ≡ M 2 E ; kbkξ ≤ F∗ ≡ Ve F + M G ;
δ kBkξ− δ ≤ s1 F∗ , ; s1 ≡ 2σ01 2 2 |c1 | ≤ Te M F + T σ10 (ξ)F∗ ; (note that in the last estimate we have used the fact that the supremum norm supTN | · | is dominated by the 0-Fourier norm k · k0 ); δ kAkξ0 ≡ kD−1 Bkξ0 ≤ s2 F∗ , s1 ; s2 ≡ σ10 2 δ kb0 kξ− δ ≤ B∗ ≡ T σ10 F∗ + T Te M F + T F∗ σ10 (ξ) + M F ; 2 2 δ |c2 | ≤ M kD−1 b0 k0 ≤ M σ10 ξ − B∗ ; 2 δ δ kzkξ0 ≤ B∗ s3 , + M 2 σ10 ξ − ; s3 ≡ M σ10 2 2 ku0 kξ0 ≡ ku + zkξ0 ≤ U + kzkξ0 ≤ U 0 ≡ U + B∗ s 3 ; kDzkξ0 = k(DM)(D−1 b0 + c2 ) + Mb0 kξ0 ≤ B∗ s 4 , s4 ≡ s3 σ−10 (δ) + M ; −1 kzθ kξ0 = kMθ (D b0 + c2 ) + MD−1 ∂θ b0 kξ0
Stability of Three-Body Problems
≤ B∗ s 5 , kDzθ kξ0
429
s5 ≡ σ01 (δ)s3 + M σ11
δ
; 2 = k(DMθ )(D−1 b0 + c2 ) + DMD−1 ∂θ b0 + Mθ b0 + M∂θ b0 kξ0 ≤ B∗ s 6 , δ δ i h + σ01 (δ) + σ01 ; s6 ≡ σ−11 (δ)s3 + M σ−10 (δ)σ11 2 2
kwkξ0 ≤ E kDzkξ0 + E1,1 kzkξ0 + F 0
kv kξ0 kwθ kξ0
s7 ≡ s4 + E1,1 s3 ; ≤ E(s7 B∗ + F ) , ≡ kv + wkξ0 ≤ V + kwkξ0 ≤ V 0 ≡ V + E(s7 B∗ + F ) ; = k∂θ (h0yy )−1 (Dz − h0yx z + f ) + (h0yy )−1 (Dzθ − ∂θ h0yx z − h0yx zθ + fθ )kξ0 ≤ σ01 (δ)E(kDzkξ0 + E1,1 kzkξ0 + F ) h i +E kDzθ kξ0 + σ01 (δ)E1,1 kzkξ0 + E1,1 kzθ kξ0 + σ01 (δ)F
kq1 kξ0
≤ EB∗ s8 + EF s9 , s8 ≡ σ01 (δ)s7 + s6 + σ01 (δ)E1,1 s3 + E1,1 s5 , s9 ≡ 2σ01 (δ) ; 1 1 2 2 ≤ khxxx kξ,r ¯ kzkξ 0 + khxxy kξ,r ¯ kzkξ 0 kwkξ 0 + khxyy kξ,r ¯ kwkξ 0 2 2 ≤ s10 B∗2 + s11 B∗ F∗ + s12 F 2 , 1 1 2 s10 ≡ E3,0 s23 + EE2,1 s3 s7 + E1,2 E s27 , 2 2 2
kf 0 kξ0
s11 ≡ s3 E2,1 E + s7 E1,2 E , 1 2 s12 ≡ E E1,2 ; 2 1 2 ≡ kq2 kξ0 ≤ khyxx kξ,r ¯ kzkξ 0 + khyxy kξ,r ¯ kzkξ 0 kwkξ 0 2 1 2 + khyyy kξ,r ¯ kwkξ 0 2 ≤ F 0 ≡ s010 B∗2 + s011 B∗ F + s012 F 2 , 1 1 2 s010 ≡ E2,1 s23 + EE1,2 s3 s7 + E0,3 E s27 , 2 2 2
kq3 kξ0
s011 ≡ s3 E1,2 E + s7 E0,3 E , 1 2 s012 ≡ E E0,3 ; 2 ≤ σ01 (δ)F kwkξ0 + σ01 (δ)Gkzkξ0 + M kq1 kξ0 ≤ s13 B∗ F + s14 B∗ G + s15 F 2 + s16 B∗2 , s13 ≡ σ01 (δ)Es7 + M s11 , s15 ≡ σ01 (δ)E + M s12 ,
s14 ≡ σ01 (δ)s3 , s16 ≡ M s10 ;
430
A. Celletti, L. Chierchia
h kg 0 kξ0 = kM−T BM−1 z + A(DM−1 )z + AM−1 Dz i +(DMT )(h0yy )−1 fθ M−1 z +D(h0yy )−1 fθ M−1 z + (h0yy )−1 Dfθ M−1 +(h0yy )−1 fθ (DM−1 ) z + (h0yy )−1 fθ M−1 Dz − M−T q3 kξ0 h ≤ M s1 F∗ M B∗ s3 + s2 F∗ σ−10 (2δ)M B∗ s3 + s2 F∗ M B∗ s4 i +σ−10 (2δ)Eσ01 (2δ)F M B∗ s3 +σ−10 (δ)Eσ01 (2δ)F M B∗ s3 + Eσ−11 (2δ)F M B∗ s3 +Eσ01 (2δ)F σ−10 M B∗ s3 + Eσ01 (2δ)F M B∗ s4 +M s13 B∗ F + s14 B∗ G + s15 F 2 + s16 B∗2 ≤ G0 ≡ s17 F∗ B∗ + s18 EF B∗ + s14 M B∗ G + s15 M F 2 + s16 M B∗2 , 2
s17 ≡ M (s1 s3 + σ−10 (2δ)s2 s3 + s2 s4 ) , h s18 ≡ M s3 M σ−10 (2δ)σ01 (2δ) + σ−10 (δ)σ01 (δ) + σ−11 (2δ) i −1 +σ01 (2δ)σ−10 (2δ) + M σ01 (2δ)s4 + M E s13 ; kM0 kξ0 ≡ kM + zθ kξ0 ≤ M + kzθ kξ0 ≤ M 0 ≡ M + B∗ s 5 . 0
To define M we have to distinguish two cases according to whether M B∗ s5 is greater 0 or smaller than 1. In the first case we define M to be infinite: M B∗ s5 ≥ 1
0
=⇒ M ≡ ∞ ,
while if M B∗ s5 < 1,
(4.26)
we proceed as follows: kM0
−1
kξ0 = k(I + M−1 zθ )−1 M−1 kξ0 ≤ M (1 − M kzθ kξ0 )−1 0
≤ M ≡ M (1 − M B∗ s5 )−1 . Next, kvθ0 kξ0 ≡ kvθ + wθ kξ0 ≤ Ve + kwθ kξ0 ≤ Ve 0 ≡ Ve + EB∗ s8 + EF s9 . 0
Of course, as it is clear from the definition of M , the KAM algorithm will be of some use only if (4.26) is satisfied. To complete the computation of K, it remains to bound 0 −1 |hT 0 i |, i.e. to compute Te0 . If M = ∞, we define also Te0 ≡ ∞, otherwise we proceed as follows. A bit of algebra shows that if we define
Stability of Three-Body Problems
431
C ≡ (I + M−1 zθ )−1 − I =
∞ X
(−M−1 zθ )k ,
k=1 0
−1
C ≡ CM , C 00 ≡ hyy (θ + u0 , v 0 ) − h0yy , then we can write T 0 as
T 0 = T + C∗ ,
where T
T
C∗ ≡ M−1 h0yy C 0 + M−1 C 00 M−1 + M−1 C 00 C 0 + C 0 h0yy M−1 T
T
+C 0 h0yy C 0 + C 0 C 00 M−1 + C 0 C 00 C 0 . Thus, since supTN |zθ | ≤ kzθ k0 (and a bound on kzθ k0 is obtained exactly as above replacing δ with ξ), we obtain sup |C| ≤ M kzθ k0 (1 − M kzθ k0 )−1 TN
≤ C ≡ M B∗0 s05 (1 − M B∗0 s05 )−1 ;
sup |C 0 | ≤ C 0 ≡ CM ; TN
sup |C 00 | ≤ E1,2 kzk0 + E0,3 kwk0 TN
≤ C 00 ≡ E1,2 B∗0 s03 + E0,3 E(s07 B∗0 + F ) ; 2
sup |C∗ | ≤ C∗ ≡ 2M E0,2 C 0 + M C 00 + 2M C 00 C 0 TN
+C 0 E0,2 + C 0 C 00 , 2
2
(4.27)
where B∗0 , s05 , s03 , s07 are defined as above but with δ replaced by ξ. Now, if TeC∗ ≥ 1 otherwise, if
=⇒ Te0 = ∞ , TeC∗ < 1 ,
(4.28)
we have |hT 0 i
−1
| = |(I + hT i
−1
hC∗ i)−1 hT i
−1
| ≤ Te0 ≡ Te(1 − TeC∗ )−1 .
(4.29)
The computation of the map K is completed. Remark 4.4. (“KAM algorithm”). If (4.23) is satisfied when U, v, ξ are replaced by U 0 , v 0 , ξ 0 then the map K can be re-applied and, iterating when possible, one obtains a sequence (u(j) , v (j) ) of approximate solutions [with relative error functions (f (j) , g (j) )] and corresponding norm-bounds Uj , Vj ,...,Tej . More precisely, fix numbers δj & 0 such P that δj < ξ (where δ0 ≡ δ as above); let, for j ≥ 1, ξj ≡ ξj−1 − δj−1 ; call the above “first approximate solution” (u(0) , v (0) ), (f (0) , g (0) ) the relative error function and attach to M, T and to their norm bound an index 0. Then, given, for j ≥ 0, (u(j) , v (j) )
432
A. Celletti, L. Chierchia
and the relative norm bounds (Uj , Vj , ..., Tej ) we let (u(j+1) , v (j+1) ) be the approximate solution constructed in25 Proposition 4.1 and, if conditions (4.23), (4.26) and (4.28) are satisfied (see, again, footnote 25), then (Uj+1 , ..., Tej+1 ) = K(Uj , ..., Tej ) are the norm bounds controlling the new approximate solution (u(j+1) , v (j+1) ). 4.4. KAM Theorem. described above.26
Here we prove a KAM theorem based on the KAM algorithm
Theorem 4.1. Let ω satisfy (4.3), let Assumption 4.2 hold, let 0 < ξˆ < ξ ,
1 ξ − ξˆ , δˆ ≡ 2 ξ
let η be the following norm on (f, g) η ≡ max{E E0,2 , E0,2 Te} max{γF , γ 2 E0,2 Ve F , γ 2 E0,2 G} , define the following parameters related to ω and to the quantities introduced in Assumption 4.2: ≡ max |ωi | , 1≤i≤N
1 ≡ max{ , E1,1 } , 2
∗ ≡ max{E2,1 , EE1,2 1 , E0,3 E 21 } , 2
H∗ ≡ max{E3,0 , EE2,1 1 , E1,2 E 21 } , H∗0 ≡ max{H∗ , E1 }, H∗00 ≡ max{H∗0 , Ve ∗ } , α∗ ≡ max{E E0,2 , E0,2 Te} · max{γ 2 E0,2 H∗00 , γ∗ } , ˆ 2τ +2 } , ˆ 2τ , (ξ − ξ) θ ≡ max{(ξ − ξ) E1 E1 e , , T E0,2 , TeE1,2 , TeE0,3 E1 } , r−ρ Ve α ≡ max{α∗ , α∗∗ } .
α∗∗ ≡ θ max{1 ,
ˆ satisfying27 There exists a polynomial ν in (ξ, δ) 5 ˆ ≤ 21 + 88 max{ξ , ξ 6 } , ≤ ν(ξ, δ) 4
∀ξ > 0 , ∀ 0 < δˆ <
1 , 2
such that, if 25 Of course, the unprimed quantities in Proposition 4.1 (besides E and E p,q which remain unchanged) correspond here to the index j while the primed ones correspond here to the index j + 1. Analogously, one has to attach in the obvious way an index j or j + 1 in the formulae defined in this section; for example ej Fj + Mj Gj , s1 = 2σ01 (δj /2), etc. T j = Mj2 E, F∗j = V 26 We recall that in order to apply such a theorem in an “effective way,” one should apply it only after a few iterations of the map K: see [5] for more information. 27 The polynomial ν (of degree 12 and with positive coefficients) is explicitly given in the proof below. For later use we report also the following values:
1 ν(1, ) = 17.4281... , 4
1 ν(0.0025, ) = 6.5190... 4
(4.30)
Stability of Three-Body Problems
433
9 ˆ −2(2τ +1) 28τ +13 τ !4 ν(ξ, δ) ˆ ≤1, ηα M 7 M (ξ − ξ)
(4.31)
then there exists a solution (u, ˜ v) ˜ of (4.1), which is real-analytic on 1ξˆ and is η-close to (u, v). Furthermore hui ˜ = hui and the solution (u, ˜ v) ˜ is the unique solution in an η neighbourhood of (u, v) such that hui ˜ = hui. For us it will be particularly important to investigate dependence upon parameters: assume that the Hamiltonian h = h(x, y; µ) depends analytically also upon a set of parameters varying in some complex domain µ ∈ D ⊂ Cp . Then the coefficients hn,k in (4.18) will be analytic functions of µ ∈ D and we shall redefine the norm in (4.19) by X (4.32) sup |hn,k | exp(|n|ξ) r|k| . khkξ,r ≡ µ∈D
n∈ZN k∈NN
Analogously, for a function f (x; µ) analytic in 1ξ × D we shall redefine the Fourier norm (4.17) by X sup |fn | exp(|n|ξ) . kf kξ ≡ n∈ZN
µ∈D
Note that we are not changing the symbols of the norms since the domain D will remain unchanged in the proofs. Finally, Assumptions 4.1 and 4.2 will be modified as follows: Assumption 4.3. Let (θ, µ) ∈ TN × D → (u, v) ∈ CN × CN be a regular function of θ and let M and h0yy be the matrices h0yy ≡ hyy θ + u(θ; µ), v(θ; µ); µ . (4.33) M ≡ I + uθ , We assume that, if we denote T ≡ M−1 h0yy M−T ,
(4.34)
then for any (θ, µ) ∈ TN × D, the matrices M, h0yy and hT i are invertible. ¯ r, E, Ep,q (p, q ∈ N) be such that (x, y, µ) → Assumption 4.4. Let 0 < ξ < ξ, N
h(x, y; µ) is real analytic on 1ξ¯ × B r (y0 ) × D and such that (4.21) holds; assume that (u, v) is real analytic on 1ξ × D and let U, V, M, M , Ve , F, G, Te be positive numbers for which (4.22) holds28 (with M, T , f and g as in (4.33), (4.34) and (4.6)). Finally, assume that (4.23) holds. With these modifications one obtains ˆ δ, ˆ ν be as in Theorem 4.1; assume that h, (f, g) satisfy Theorem 4.2. Let ω, ξ, ξ, Assumption 4.4. Let α and η be defined as in Theorem 4.1 and let (4.31) hold. Then there exists a solution (u, ˜ v) ˜ of (4.1), which is real-analytic on 1ξˆ × D and is η-close to (u, v). Furthermore hui ˜ = hui and the solution (u, ˜ v) ˜ is the unique solution in an η neighbourhood of (u, v) such that hui ˜ = hui. 28
Last inequality must be rewritten as supµ∈D |hT i−1 | ≤ Te.
434
A. Celletti, L. Chierchia
Remark 4.5. (i) We shall not prove here the local uniqueness (under the constraint hui ˜ = hui) of the solution (u, ˜ v) ˜ and we refer the reader to [5] (where a similar statement is proven in full detail). (ii) Precise estimates concerning the η closeness (which is of course in the sense of the k · kξˆ norm) can be easily deduced from the detailed estimates given in the proof. Proof. (of Theorem 4.1.) The argument is by induction. We define, for j ≥ 0, ξj ≡ ξˆ +
ξ − ξˆ , 2j
δj ≡ ξj − ξj+1 =
ξ − ξˆ . 2j+1
We assume (by induction) to have constructed (by iterating j ≥ 0 times the scheme of Proposition 4.1 starting with (u(0) , v (0) ) ≡ (u, v)) the approximate solution (u(j) , v (j) ); denote by (Uj , ..., Tej ) the relative norm-bounds29 and assume that, for 0 ≤ i ≤ j, ku(i) kξi ≤ ξ¯ − ξi , Vei ≤ 2Ve0 ,
Mi ≤ 2M0 ,
kv (i) − y0 kξi ≤ r , M i ≤ 2M 0 , Tei ≤ 2Te0 .
(4.35)
We want to show that (4.35) is satisfied also for i = j + 1 and that the norms of z (i) , w , f (i) and g (i) decay exponentially fast as i tends to infinity so that the claim of the theorem follows. We now list estimates on various quantities entering the definition of the map K (i.e. on the definitions of the parameters, introduced in the previous section, which bound the norms of the relevant objects associated to30 u(i) , v (i) ); the estimates are completely elementary and we shall give details only for some of the first estimates (all the other estimates being obtained in a complete analogous way). In the following estimates we shall make systematic use of (4.35), (4.20), of the fact that τ ≥ 1 and of the following simple observations31 : (i)
γ ≥ 1 , where
M0 ≥ 1 ,
M0 ≥ 1 ,
EE0,2 ≥ 1 ,
δj ξj+1 =1− ≤ δˆ , ξj ξj
δ0 1 ξ − ξˆ . = δˆ ≡ ξ0 2 ξ
(4.36)
(4.37)
Here is the list: δ i ≤ tδi−1 , (∀ t > 0) ; σ10 (ξi ) ≤ γξi−τ τ ! ; σ01 t δ δi i σ10 ≤ 2τ γδi−τ τ ! , σ10 ξi − ≤ σ10 (ξi ) ≤ γξi−τ τ ! ; 2 2 29 See the previous section. Note that, with these definitions, ξ = ξ, U = U ,...,M = M , etc. The “norm0 0 0 parameters” are defined in the previous section where the primed quantities correspond to u(i+1) , v (i+1) while the unprimed ones correspond to u(i) , v (i) . 30 Recall that the primed quantities of Sect. 4.3 correspond here to the index i + 1 while the unprimed ones to the index i, that the index i = 0 corresponds to the “initial” approximate solution, namely to the quantities defined in Assumption 4.2 and, finally, that Ep,q and E are independent of i. 31 The first relation in (4.36) follows from (4.3) letting n = e(i) , where {e(i) } is the standard orthonormal basis of RN ; the second and third relations follow by observing that e(1) is an eigenvector with eigenvalue 1 of MT (θ0 ) and of M−T (θ0 ), if θ0 is a critical point of u1 (θ); the last two relations are obvious.
Stability of Three-Body Problems
σ−10
δ i
435
≤ t δi−1 , (∀ t > 0);
t δ i σ−11 ≤ 2t2 δi−2 , (∀ t > 0) ; t 2
Ti ≤ 4M 0 E0,2 ; s1 ≤ 4δi−1 ;
σ11
T i ≤ 4M02 E ,
δ i
2
≤ γ2τ +1 δi−(τ +1) (τ + 1)! ;
F∗i ≤ 2Ve0 Fi + 2M0 Gi ;
s2 ≤ 22+τ γδi−(τ +1) τ ! ;
h 1 δ τ i i s3 ≤ M0 2τ +1 γδi−τ τ ! + 4M02 γξi−τ τ ! = M02 2τ +1 γδi−τ τ ! +2 M0 2ξi ≤ M02 2τ +1 γδi−τ τ ! 1 + δˆ ≡ M02 2τ +1 γδi−τ τ ! ν1 , ν1 ≡ 1 + δˆ ; δ02 ; 2 ν 1 ; ≤ γM02 2τ +2 δi−(τ +1) (τ + 1)! ν3 , ν3 ≡ 1 + 4 3 ν1 + δ02 ; ≤ (γ)M02 2τ +2 δi−(τ +2) (τ + 1)! ν4 , ν4 ≡ 1 + 2 16 ≤ (γ1 )M02 2τ +1 δi−(τ +1) τ ! ν5 , ν5 ≡ ν2 + δ0 ν1 ; ν 5 ν 1 δ0 + + ν 3 δ0 ; ≤ (γ1 )M02 2τ +2 δi−(τ +2) (τ + 1)! ν6 , ν6 ≡ ν4 + 4 4 ≤ 2δi−1 ;
s4 ≤ (γ)M02 2τ +1 δi−(τ +1) τ ! ν2 , ν2 ≡ ν1 + s5 s6 s7 s8 s9
s10 ≤ γ 2 H∗ M04 22τ +1 δi−2(τ +1) τ !2 ν7 , ν7 ≡ (ν5 + ν1 δ0 )2 ; s11 ≤ γ 2 H∗ M02 2τ +1 δi−(τ +1) τ ! ν8 , ν8 ≡ ν5 + ν1 δ0 ; γ 2 H∗ ; 2 ≤ γ 2 ∗ M04 22τ +1 δi−2(τ +1) τ !2 ν7 ;
s12 ≤ s010
s011 ≤ γ 2 ∗ M02 2τ +1 δi−(τ +1) τ ! ν8 ; γ 2 ∗ ; 2 ≤ γ 2 H∗0 M03 2τ +1 δi−(τ +2) τ ! ν9 , ν9 ≡ ν5 + 2δ0 ν8 ;
s012 ≤ s13
s14 ≤ γM02 2τ +1 δi−(τ +1) τ ! ν1 ;
s15 ≤ γ 2 H∗0 M0 δi−1 ν10 , ν10 ≡ 1 + δ0 ;
s16 ≤ γ 2 H∗ M05 22τ +2 δi−2(τ +1) τ !2 ν7 ;
ν1 ν1 δ02 + ; 2 2 9 ν 9 ν2 + . ≡ ν1 + 8 2 4
s17 ≤ γ(γ)(M0 M 0 )2 22τ +5 δi−2(τ +1) τ !2 ν11 , ν11 ≡ ν2 + s18 ≤ γ 2 E
−1
H∗0 M03 M 0 M0 2τ +3 δi−(τ +2) τ ! ν12 , ν12 2
Let now ηi be as η (defined in the text of Theorem 4.1) but with Fi , Gi in place of F , G. Then, 4
γB∗i ≤ ηi M0 M 0 2τ +4 δi−τ τ ! ν13 , ν13 ≡ 1 + 4δˆ + 4
9 δ0 ; 16
kz (i) kξi+1 ≤ ηi M03 M 0 22τ +5 δi−2τ τ !2 ν14 , ν14 ≡ ν1 ν13 ;
436
A. Celletti, L. Chierchia
kzθ(i) kξi+1 ≤ ηi M03 M 0 22τ +6 δi−(2τ +1) τ !2 (τ + 1) ν15 , ν15 ≡ ν3 ν13 ; 4
kw(i) kξi+1 ≤ ηi (E1 )M03 M 0 22τ +5 δi−(2τ +1) τ !2 ν16 , ν16 ≡ ν5 ν13 + 4
δ03 ; 27
kwθ(i) kξi+1 ≤ ηi (E1 )M03 M 0 22τ +6 δi−2(τ +1) τ !2 (τ + 1) ν17 4
ν17 ≡ ν6 ν13 +
δ03 ; 28
kf (i+1) kξi+1 ≤ ∗ ηi2 M06 M 0 24τ +9 δi−(4τ +2) τ !4 ν18 8
2 ν18 ≡ ν7 ν13 +
δ03 δ06 ν8 ν13 + 14 ; 6 2 2
kg (i+1) kξi+1 ≤ H∗0 ηi2 M07 M 0 24τ +12 δi−(4τ +2) τ !4 ν19 , 9
δ3 δ05 δ0 δ2 ν11 ν13 + 06 ν12 ν13 + 07 ν1 ν13 + 14 ν10 ; (4.38) 2 2 2 2 (in the last inequality we used also the fact that from EE0,2 ≥ 1 and γ1 ≥ 1 it follows that γ max{Ve F, G} ≤ H∗0 ηi ). We let now 2 ν19 ≡ ν7 ν13 +
1 (4.39) 4 and observe that all the ν 0 s are greater than or equal to 1. Putting together the above ˆ one finds definitions of the various νi ’s and recalling that δ0 = ξ δ, 101 ˆ2 47 ˆ 1161 ˆ3 5 δξ + δ ξ+ δ ξ ν = + 10 δˆ + 33 δˆ2 + 40 δˆ3 + 16 δˆ4 + 4 8 2 8 5257 ˆ2 2 39527 ˆ3 2 50415 ˆ4 2 329 ˆ4 δ ξ + 64 δˆ5 ξ + δ ξ + δ ξ + δ ξ + 2 512 512 256 77319 ˆ3 3 455303 ˆ4 3 6105 ˆ5 3 δ ξ + δ ξ + δ ξ +194 δˆ5 ξ 2 + 64 δˆ6 ξ 2 + 8192 8192 64 1131 ˆ4 4 8415 ˆ5 4 921 ˆ6 4 15447 ˆ5 5 δ ξ + δ ξ + δ ξ + δ ξ +50 δˆ6 ξ 3 + 256 512 64 16384 925 ˆ6 5 1369 ˆ6 6 δ ξ + δ ξ . + 512 16384 We note, for later use, that32 ν18 , ν14 , ν15 , ν16 + δ0 ν14 } . ν ≥ max{ (4.40) 4 Thus, (using the inequalities α ≥ α∗ and ν ≥ max{ ν418 , ν19 }), we find ˆ ≡ ν19 + ν ≡ ν(ξ0 , δ)
ηi+1 ≤ κλi ηi2 , where33
9 ˆ −(4τ +2) τ !4 ν , κ ≡ αM07 M 0 28τ +13 (ξ − ξ) 34 Iterating, for all 0 ≤ i ≤ j + 1, we get
λ ≡ 24τ +2 .
32 As it is immediate to check after having written down explicitly the definition of ν (which, recalling that ˆ turns out to be a polynomial of degree 12 in ξ0 and δˆ with positive (rational) coefficients) and of the δ0 = ξ0 δ, other νk ’s. 33 The reason for having in this formula α in place of α will be plain when we shall check the inductive ∗ assumptions (4.35) for j + 1. P P i−1 k i−1 34 Recall that 2 = 2i − 1, (i − k)2k−1 = 2i − i − 1, (and that η0 = η). k=0 k=1
Stability of Three-Body Problems
437 i
ηi ≤
(ηκλ)2 , κλi+1
whence, in view of the “KAM condition” (4.31) (which is now recognized as equivalent to require ηκλ ≤ 1), 1 . (4.41) ηi ≤ κλi+1 This bound allows to get simple estimates on the norms of z (i) and w(i) for 0 ≤ i ≤ j. Using (4.41) and the facts that ν ≥ max{ν14 , ν15 }
and
ˆ 2τ +2 , (ξ − ξ) ˆ 2τ } , α ≥ max{(ξ − ξ)
from (4.38) we get max{kz (i) kξi+1 , kzθ(i) kξi+1 } ≤ min{δi ,
1 1 1 }. M 0 215 8i
(4.42)
Analogously, using ˆ 2τ +1 max{ 1 , 1 } , α ≥ E1 (ξ − ξ) r − ρ Ve
ν ≥ ν16 , from (4.38) there follows
1 1 kw(i) kξi+1 ≤ min{r − ρ , Ve } 18 i . 2 8
(4.43)
To check the inductive hypotheses (4.35) we shall also need simple bounds on the constants Ci , Ci0 , Ci00 , C∗i [recall (4.27) and (4.29)]. Using ν ≥ max{ν15 , ν16 + δ0 ν14 } and the fact that ˆ 2τ +1 max{1 , TeE0,2 , TeE1,2 , TeE0,3 E1 } , α ≥ (ξ − ξ) and that M i B∗i s5 ≤
1 2 M0
1 1 1 1 min{1 , } ≤ 14 , 14 i e 2 8 2 T E0,2
we obtain Ci ≤ min{1 ,
1
}
1
1 1 , 213 8i
TeE0,2 1 1 1 1 } , Ci0 ≤ min{1 , 12 8i e M T E0,2 0 2 2 M0
4
Ci00 ≤ max{E1,2 , E0,3 E1 } ηi M03 M 0 δi−2τ +1 22τ +5 τ !2 (ν16 + δ0 ν14 ) , 1 1 C∗i ≤ Te0−1 8 i . (4.44) 2 8 We are ready to check (4.35) for i + 1: From (4.35) and (4.42) we find ku(i+1) kξi+1 ≤ ku(i) kξi + kz (i) kξi+1 ≤ ξ¯ − ξi + δi = ξ¯ − ξi+1 .
438
A. Celletti, L. Chierchia
From (4.23) and (4.43), we get kv (i+1) − y0 kξi+1 ≤ kv (0) − y0 kξ +
i X
kw(j) kξi+1
j=0 ∞ r−ρX 1 ≤ ρ + 18 ≤r. 2 8j j=0
Finally, using (4.43), (4.42), (4.44) one easily obtains the remaining inductive hypotheses. Observe that from (4.41), (4.42) and (4.43) it follows that the error functions f (i) and g (i) go to zero exponentially fast, while u(i) and v (i) converge (exponentially fast) to real-analytic functions u˜ and v. ˜ This concludes the proof of Theorem 4.1. Proof. (of Theorem 4.2) The reader will have no difficulty in checking that the previous proof goes through word by word so that the claim follows from uniformity in µ ∈ D. 5. Proof of Theorem 1.1 The proof of Theorem 1.1 will be divided in three steps: (1) construction of the starting approximate solution (Remark 4.1) using ε-expansions; (2) bounds (4.22) on the norms relative to starting approximate solution with D ≡ {ε ∈ C : |ε| ≤ ε0 ≡ 10−6 }, see Assumption 4.4; (3) iteration of the map K (4.25) and application of Theorem 4.2. 5.1. Step 1: Formal ε-expansion and initial approximate solution. The Hamiltonian H in (1.3) contains ε as a parameter: ε corresponds to the parameter µ of Theorem 4.2 and D corresponds to the complex ball {ε ∈ C : |ε| ≤ ε0 } for some ε0 > 0 to be determined below.35 As was well known to Poincar´e, Lindstedt & Co., one may compute formally the ε-expansion of quasi-periodic formal solutions (“Lindstedt series”; see [1]). Our starting approximate solution will be a suitable truncation of such a formal expansion. Here we deduce a few elementary formulae which allow us to explicitly compute recursively the formal solution. Let N = 2, x ≡ (`, g), y ≡ (L, G) and let us rewrite explicitly Eq. (4.1) for the Hamiltonian H: 1 1 1 1 − v2 ) 3 + 2ε( 2 − v2 )RL (θ + u, v) − 2ε 3 R(θ + u, v), 2v12 2v1 v1 v1 1 1 ω2 + Du2 (θ) = −2( 2 − v2 ) + 2ε( 2 − v2 )RG (θ + u, v) − 2εR(θ + u, v), 2v1 2v1 1 Dv1 (θ) = −2ε( 2 − v2 )R` (θ + u, v), 2v1 1 Dv2 (θ) = −2ε( 2 − v2 )Rg (θ + u, v) ; (5.1) 2v1 ω1 + Du1 (θ) = −2(
35 Clearly H is an entire function of ε and the restriction on D is needed in order to meet the basic condition (4.31); the choice of the “best value for ε0 ” (i.e., the largest one) has been done simply by “trial and error”.
Stability of Three-Body Problems
439
here ω and D are short for ω (±) and for ω (±) · ∂θ , θ ∈ T2 and recall (1.8), (1.9) and (1.10). It is well known that (5.1) admits a formal solution u(θ) ˜ ∼
∞ X
u˜ (j) (θ) εj ,
v(θ) ˜ ∼
j=0
∞ X
v˜ (j) (θ)εj ,
(5.2)
j=0
with u˜ (j) and v˜ (j) being vector-valued real-analytic on T2 functions: u˜ (j) ≡ (u˜ (j) ˜ (j) 1 ,u 2 ),
v˜ (j) ≡ (v˜ 1(j) , v˜ 2(j) ) .
Furthermore, such a formal solution is uniquely determined by requiring that the averages of the u˜ (j) ’s vanish: hu˜ (j) i = 0 , (∀ j ≥ 0) . In particular this implies that u˜ (0) ≡ 0 . Instead (as one checks immediately by inserting (5.2) into (5.1) and looking at the order zero in ε) v˜ (0) has to be chosen so that ∂(L,G) H (v˜ (0) ) = ω , i.e. , v˜ (0) ≡ (L± , G± ) . ε=0
Remark 5.1. The formal solvability of (5.1) implies that the right-hand sides of the last two equations of (5.1) have vanishing mean value36 over T2 . The averages of the v˜ (j) ’s have then to be chosen so that the first two equations in (5.1) are solvable. This leaves free the averages of the u˜ (j) ’s, which, as already said, will be taken to be zero. As initial approximate solution we take u (θ; ε) ≡ (0)
5 X
(j)
j
u˜ (θ) ε ,
v (θ; ε) ≡ (0)
j=0
5 X
v˜ (j) (θ) εj
(5.3)
j=0
(recall that, with the above conventions, v˜ (0) ≡ (L± , G± ), u˜ (0) ≡ 0 and hu˜ (j) i = 0). We proceed by writing down the explicit formulae which, implemented on a machine, allow to compute37 the functions u˜ (j) , v˜ (j) or, more precisely, allow to compute intervals of real numbers containing the Fourier coefficients of the (u˜ (j) , v˜ (j) )’s. Recalling the explicit form of the function R [see (1.4), (1.5)], one sees that the right-hand sides of (5.1) have the form M X
ri εsi (v1(0) )pi (v2(0) )qi eσi (v1(0) , v2(0) ) cni (θ + u(0) ) ,
(5.4)
i=1
where: M < ∞; ri are rational numbers; si , pi , qi , σi are integers obeying the constraints 0 ≤ si ≤ 1 ,
−5 ≤ pi ≤ 10 ,
0 ≤ qi ≤ 1 ,
−1 ≤ σi ≤ 1 ;
For any periodic function f (θ), θ ∈ TN , the integral over TN of Df vanishes. Notice that since the Hamiltonian H is a trigonometric polynomial in x, the functions (u˜ (j) , v˜ (j) ) are also trigonometric polynomials. 36 37
440
A. Celletti, L. Chierchia
ni ∈ Z2 with |ni | ≤ 10; finally cn (x) is either cos n · xP or sin n · x. We shall denote by [·]j the operator that acts on a (formal) ε-power series, εk a(k) , by associating to it the j th coefficient a(j) : hX i εk a(k) ≡ a(j) . j
k≥0
Let p, q, σ ∈ Z with q > 0 and |σ| ≤ 1 and let us compute the j th ε-coefficient of, respectively, ap bq eσ (a, b) and of cn (θ + ϕ) where a, b and ϕ are formal ε-power series (with periodic real-analytic coefficients) given by X X ak (θ)εk , b ∼ bk (θ)εk , a∼ k≥0
ϕ∼
X
k≥0
k
ϕ (θ)ε ≡ (k)
X
k≥0
k ϕ(k) 1 (θ)ε ,
k≥0
X
k ϕ(k) . 2 (θ)ε
k≥0
As above, we denote y0 ≡ v˜ (0) and write the expansions of e(y) and of e−1 (y) ≡ 1/e(y) as38 X X e(y) ≡ eh (y − y0 )h , e−1 (y) ≡ e˜h (y − y0 )h . h∈N2
h∈N2
Then, for p ≥ 0, one finds [a−p bq eσ (a, b)]j X p q k −p q−k 1] 2 +k2 ] y 1 y02 2 eσ,h a[h = ˜ k[k41 ] b[h , k3 a k5 k1 k2 01
(5.5)
(k,h)∈I−p,q
where I−p,q ≡ {(k, h) ∈ N5 × N2 : 0 ≤ k1 ≤ p , 0 ≤ k2 ≤ q , k3 ≥ h1 , k4 ≥ k1 , k 5 ≥ k2 + h 2 , k 3 + k 4 + k 5 = j } ; if σ = −1 e˜h if σ = 1 . eσ,h ≡ eh δ 0|h| if σ = 0 P th c[k] (ε) denotes the k th power of a formal power series c ∼ j≥0 cj εj and c[k] j its j ε-coefficient; finally a˜ is the power series defined by a˜ ∼
1 1 − . a a0
Analogously, for p > 0, one gets [ap bq eσ (a, b)]j X p q p−k q−k 1 +k1 ] [h2 +k2 ] = y 1 y02 2 eσ,h a[h bk4 , k3 k1 k2 01
(5.6)
(k,h)∈Ip,q
38
We use standard multiindex notation: if n ∈
n! = n1 ! · · · nN !;
zn
=
n z1 1
n · · · zNN .
NN , |n|
=
PN i=1
ni ; if z ∈
RN , ∂zn
=
∂ |n| n n ∂z1 1 ···∂zn N
;
Stability of Three-Body Problems
441
where: Ip,q ≡ {(k, h) ∈ N4 × N2 : 0 ≤ k1 ≤ p , 0 ≤ k2 ≤ q , k 3 ≥ k 1 + h 1 , k 4 ≥ k2 + h 2 , k 3 + k 4 = j } .
(5.7)
Computing [cn (θ + ϕ)]j is clearly equivalent to evaluate [exp(in · ϕ)]j and letting En(0) ≡ 1 , En(k) (θ)
∀ n ∈ Z2 ,
k 1X ≡ ` En(k−`) (θ) n · ϕ` (θ) , k `=1
one gets
h
i
exp(in · ϕ)
j
= En(j) (θ) .
(5.8)
Inserting (5.2) in (5.1) one obtains recursive relations of the type Du˜ (k) = Av˜ (k) + 8(k) ,
Dv˜ (k) = 9(k) ,
(5.9)
where A is the (constant) matrix given in (3.5) evaluated at (L, G) ≡ (L± , G± ) and the vectors 8(k) and 9(k) depend on u˜ (0) ,...,u˜ (k−1) , v˜ (0) ,...,v˜ (k−1) (and on θ) and can be explicitly written down by using the remark leading to (5.4) and the expansions (5.5), (5.6) and (5.8). Assume now that (u˜ (j) , v˜ (j) ), for j = 0, ..., k − 1 are known and let us determine (k) (k) (u˜ , v˜ ). Inverting the operator D in the second of (5.9) (recalling Remark 5.1) we let v˜ (k) ≡ v¯ (k) + D−1 9(k) , where v¯ (k) denotes the average of v˜ (k) and it has to be determined so that the equations for Du˜ (k) have a right-hand side with vanishing mean value, i.e., u˜ (k) ≡ D−1 Av˜ (k) + 8(k) . v¯ (k) ≡ −A−1 h8(k) i , The formulae for the recursive computation of the functions (u˜ (j) , v˜ (j) ) (and hence of our choice of the initial approximate solution) are complete. In Appendix A we report the number of Fourier coefficients of the functions (u˜ (j) , v˜ (j) ) for j ≤ 5 and, as an example, the list of intervals trapping the Fourier coefficients of the first component of u˜ (1) . 5.2. Step 2: Norm bounds relative to the initial approximate solution. Having defined the initial approximate solution as the fifth order truncation (5.3) of the ε-expansion of the formal solution of (5.1), we want now to estimate the relative norm parameters as defined in (4.22). We attach an index 0 to the quantities related to our starting approximate solution (i.e. the fifth order truncation (5.3) of the ε-expansion of the formal solution of (5.1)); thus the symbols ξ, U, V ,...,Te of Sect. 4.3 correspond here to ξ0 , U0 , V0 ,...,Te0 (see Remark 4.4). Let ε0 ≡ 10−6 , ξ0 ≡ 0.2 , and recall that the norm in (4.32) contains also a supremum taken over the complex parameter region D, D ≡ {ε ∈ C : |ε| ≤ ε0 } .
442
A. Celletti, L. Chierchia
The evaluation of U0 , V0 , M0 and Ve0 are easily obtained having computed the “explicit” (in the sense of interval arithmetic) form of the approximate solution (u, v). Having (u, v) given as u(0) =
5 X
εj
j=1
X
u˜ (j) n exp(in · θ) ,
v (0) =
5 X
εj
j=0
0<|n|≤νj
X
v˜ n(j) exp(in · θ)
|n|≤νj0
(where the νj , νj0 are the trigonometric degrees listed in Appendix A), we let U0 and V0 be upper bounds on 5 X j=1
X
εj0
|u˜ (j) n | exp(|n|ξ0 ) ,
5 X j=0
0<|n|≤νj
εj0
X
|v˜ n(j) | exp(|n|ξ0 ) .
|n|≤νj0
The computer-assisted evaluations of such sums yields the values39 U0(+) ≡ 1.319112913121820055842417988726441 · 10−5 , U0(−) ≡ 1.309817593029817987914398304540215 · 10−5 , V0(+) ≡ 1.45622035987166272215426104777053 , V0(−) ≡ 1.45731724381047957877869538647097 . It is easy to check that these bounds imply sup |θ + u(0) | , θ∈1ξ 0 ε∈D
sup |v (0) | ∈ B . θ∈1ξ 0 ε∈D
Analogously, since40 M≡I+
5 X
εj
j=1
vθ(0) =
5 X j=1
X
i u˜ (j) n ⊗ n exp(in · θ) ,
0<|n|≤νj
εj
X i v˜ n(j) ⊗ n exp(in · θ) , |n|≤νj0
one gets the computer-assisted evaluations 39 Recall that the plus sign corresponds to the case v ˜ (0) = (L+ , G+ ) while the minus sign corresponds to v˜ (0) = (L− , G− ). Note also that interval arithmetic yields (as the name says) intervals with rational endpoints trapping the actual quantity one is computing; but since we need (usually) upper bounds we shall report only the right endpoints of the computed intervals which shall define our norm-bounds. 40 If a, b ∈ CN , we denote by a ⊗ b the N × N matrix with entries (a ⊗ b) = a b . We also recall that ij i j a b by default we use the 1-norm on vectors so that the (“operator”) norm on a 2 × 2 matrix A = may c d be bounded as follows
kAk = sup |Ax| ≤ min |x|1 =1
max{|a|, |b|} + max{|c|, |d|} , max{|a|, |c|} + max{|b|, |d|} .
Stability of Three-Body Problems
443
M0(+) ≡ 1.000021266465664701238677170875501 , M0(−) ≡ 1.000021439075256126271432431716151 , Ve (+) ≡ 7.453571858517957815592730443874432 · 10−7 , 0
Ve0(−) ≡ 7.531270519848394217447212176311194 · 10−7 .
(5.10)
The evaluation of M 0 is immediately obtained using (5.10): kM−1 kξ0 ≡ k(I + uθ )−1 kξ0 ≤ (±)
≤ M0
1 1 ≤ 1 − kuθ kξ0 1 − V˜0(±)
,
which leads to (+)
M 0 ≡ 1.00002126691793688133738303697581 , (−)
M0
≡ 1.000021439534899928447344044122831 .
To estimate Te0 write the matrix T as T ≡ A0 + B , where A0 ≡ A(L± , G± ) [A(L, G) being the invertible matrix defined in (3.5)]. Hence −1 −1 khT i−1 k = k(I + A−1 0 hBi) A0 k ≤
kA−1 0 k . 1 − kA0 k−1 kBk0
To estimate kBk0 we write explicitly the definition of B and use the following computerassisted bounds kRk0 kRL k0 kRG k0 kRLL k0 kRLG k0 kRGG k0
≤ ≤ ≤ ≤ ≤ ≤
1.43780490153865715166812155290155 , 68.8071599209183467048665730038651 , 64.3060545612114280147776693441263 , 56614.1072116429904165317710256315 , 56755.0234606639063731168541598635 , 28205.7872421132384032285133772315 .
In this way one obtains Te0(+) ≡ 8.567100579602744072852019085722171 , Te(−) ≡ 8.559621778472845438209667021368791 . 0
A bit more delicate is the evaluation of F0 and G0 , i.e. of the norms of the error functions f and g. We shall follow the classical Cauchy’s majorant method (see [9], chapter 5 for generalities). We recall that if X X αk (y − y0 )k and α(y) ¯ ≡ α¯ k (y − y0 )k α(y) ≡ k∈Np
k∈Np
444
A. Celletti, L. Chierchia
are smooth functions of y in a neighbourhood of y0 ∈ Rp , α¯ is said to majorize α and is denoted α ≺ α¯ , if |αk | ≤ α¯ k ,
∀k.
For example, for any n ∈ Zp , cos n·x ≺ cosh(|n1 |x1 +· · ·+|np |xp ) ,
sin n·x ≺ sinh(|n1 |x1 +· · ·+|np |xp ) . (5.11)
We need a little technical lemma. Lemma 5.1. Let ν, p ∈ Z+ and let X
α(x, y) ≡
cn (x)αn (y) ,
n∈Zp :|n|≤ν
where cn (x), for each n, is either cos n · x or sin n · x (x ∈ Tp ) and the αn ’s are analytic functions on a complex ball around y0 ∈ Rp . Fix M ∈ N. Let b0 ≡ y0 and, for 1 ≤ j ≤ M , let a(j) (θ) and b(j) (θ) be given Rp -valued functions of θ ∈ Tp analytic on 1ξ for some ξ > 0. Let α¯ n (y) be analytic functions such that αn ≺ α¯ n ,
(∀ |n| ≤ ν) .
Let a¯ 0 ≡ (ξ, ..., ξ), b¯ 0 ≡ (|y01 |, ..., |y0p |) and, for 1 ≤ j ≤ M let a¯ (j) and b¯ (j) be p-vectors with nonnegative components such that ¯ (j) ka(j) i kξ ≤ a i ,
¯ (j) kb(j) i kξ ≤ b i ,
(∀ 1 ≤ j ≤ M ,
∀ 1 ≤ i ≤ p) .
Finally, denote c¯n (x) ≡
cosh(|n1 |x1 + · · · + |np |xp ) if cn = cos n · x sinh(|n1 |x1 + · · · + |np |xp ) if cn = sin n · x
and let M M X X εj a(j) (θ), εj b(j) (θ) , α(ε, ˜ θ) ≡ α θ +
β(ε, θ) ≡
j=1
h
X
˜ θ) εj α(·,
j
j≥M +1
ϕ(ε) ≡
X |n|≤ν
c¯n
M X
j=0
i
εj a¯ (j)
,
j=0
α¯ n
M X
εj b¯ (j) ,
j=0
where, of course, the operator [·]j refers to ε-expansions. Then, for all ε¯ > 0, k[α(·, ˜ θ)]j kξ ≤ [ϕ]j , ¯ − sup kβkξ ≤ ϕ(ε)
|ε|≤ε¯
M X j=0
∀j ≥0; ε¯j [ϕ]j .
(5.12)
Stability of Three-Body Problems
445
Proof. Observe that, by (5.11), for any fixed θ ∈ 1ξ , one has cn
M X j=0
and also that41 αn
M X j=0
M X εj a(j) (θ) ≺ c¯n εj a¯ (j) , j=0
M X εj b(j) (θ) ≺ α¯ n εj b¯ (j) . j=0
These relations imply that for all θ ∈ 1ξ , α(·, ˜ θ) ≺ ϕ, which implies immediately (5.12). In order to apply the lemma to our situations we need the following obvious majorizations. Let z ∈ C with |z| < 1; let42 y10 , y20 be positive numbers and let |y1 − y10 | < 1, |y2 − y20 | < 1; finally let s be a positive integer. Then √ √ 1 1 1−z ≺2− 1−z , (1 − z)− 2 ≺ (1 − z)− 2 ; y 2 2 y2 y2 y2 2 ≺ , ≺ ; y1 2y10 − y1 y1 2y10 − y1 s 2 y2 e(y1 , y2 ) ≺ e(y ¯ 1 , y2 ) ≡ 2 − 1 − ; 2y10 − y1 2 −1/2 y2 ; e−1 (y1 , y2 ) ≺ 1 − 2y10 − y1 1 s 2 1 1 , −2 − y + ≺ 2y . y1−s ≺ 2 2 2y10 − y1 2y10 − y1 2y12 Using the above observations, one gets the following evaluations of the norm bounds F0 and G0 : F0(+) ≡ 1.875530182753837192126197355672179 · 10−23 , F0(−) ≡ 1.881443228644026073096600123652624 · 10−23 , −20 , G(+) 0 ≡ 1.524340973886308626744645639896068 · 10
G(−) ≡ 1.573241367452682546591639160273634 · 10−20 . 0 5.3. Step 3: Application of the KAM algorithm and of the KAM Theorem. We proceed to apply the KAM algorithm and the KAM Theorem worked out in Sect. 4.3: recall 3) of Sect. 1 and Remark 4.4. Also this step is computer-assisted; however double precision (rather than quadruple precision) will usually be good enough. We discuss in detail the case with initial data (L+ , G+ ); the case with initial data (L− , G− ) is completely analogous43 and we spare the reader more data which would not shed much more light. 41 Recall the basic facts of majorization theory (see, e.g., [9]): sum, multiplication and composition are preserved by majorizations. 42 To avoid confusion with the norm parameter G we denote here the action-variables by (y , y ). 0 1 2 43 Actually, the (L , G )-case “converges” after three iterations of K while the (L , G ) case after four. + + − −
446
A. Celletti, L. Chierchia
Remark 5.2. The quantities with an index 0 refer (as in Step 2 above ) to our choice of the initial approximate solution. To such initial approximate solution we apply a few times [respectively three times for the case with initial data (L+ , G+ ) and four times for the case with initial data (L− , G− )] the KAM algorithm obtaining new approximate solutions (u(1) , v (1) ),...,(u(3) , v (3) ) [for the case with initial data (L+ , G+ ), while for the case with initial data (L− , G− ) we consider also (u(4) , v (4) )]. The input of the KAM Theorem will be respectively (u, v) ≡ (u(3) , v (3) ) (with the relative norm bounds given by the KAM algorithm) for the case with initial data (L+ , G+ ) and (u, v) ≡ (u(4) , v (4) ) for the case with initial data (L− , G− ). Quantities referring to the input of the KAM Theorem will carry no index (and beware of the difference between u˜ (j) and u(j) ). Since we shall consider only the case with initial data (L+ , G+ ) we shall drop from the notation the suffix (+) . To apply the KAM algorithm we have to fix the values of the “angle” analyticity widths, namely the values of ξ¯ and ξ0 [see (4.23)] and of ξ1 ≡ ξ 0 [see (4.24)] and44 ξ2 ≡ ξ10 , ξ3 ≡ ξ20 . We choose45 ξ¯ ≡ 0.201 , ξ0 ≡ 0.2 , ξ1 = 0.1 , ξ2 = 0.05 , ξ3 ≡ 0.0025 . We also fix the value r appearing in (4.23) as r ≡ 0.001. After three steps of the KAM algorithm we shall apply the KAM Theorem, we therefore let ξ ξˆ ≡ ≡ 0.00125 . ξ ≡ ξ3 ≡ 0.0025 , 2 Next, we need estimates on the derivatives of the Hamiltonian h(x, y) appearing in the definitions of ,...η in Theorem 1.1 [recall (4.21)]. These estimates (which do not change in the iteration) are straightforward and one obtains: E ≤ 0.834569178062801416845925922143676 , E0,2 ≤ 56.5486438633162084677890669788006 , E1,1 E0,3 E1,2 E2,1
≤ 1.186694583405233167666977446730261 · 10−3 , ≤ 24606.1421916127359268759358546612 , ≤ 3.96265819827614919370110457943010 , ≤ 2581.05325817214297337992701624725 ,
E3,0 ≤ 1.747949724005794044987232216957351 · 10−4 . Now we iterate the map K three times: conditions (4.23), (4.26) and (4.28) are satisfied for j = 0, 1, 2, 3 (as it is easy to check using the values (Uj , ..., Tej ) reported here). The (double precision) iterated values of the map K are the following. After the first iteration:
44 45
U1 ≤ 1.309817593322689 · 10−5 , V1 ≤ 1.45622036006177 , M1 ≤ 1.00002126646582 , M 1 ≤ 1.00002126691809 , V˜1 ≤ 7.453583421789768 · 10−7 , F1 ≤ 4.006354647833255 · 10−24 , G1 ≤ 1.368656716949379 · 10−25 , T˜1 ≤ 0.834604675233853 . P
Recall that ξj+1 = ξj − δj , where the δj ’s are such that δ0 > δ1 > δ2 ... and δj < ξ. These values have been chosen by “optimizing” (trial and error) the KAM algorithm.
Stability of Three-Body Problems
447
After the second iteration: U2 ≤ 1.309817593322718 · 10−5 , V2 ≤ 1.45622036006177 , M2 ≤ 1.00002126646582 , M 2 ≤ 1.00002126691809 , V˜2 ≤ 7.453583426051443 · 10−7 , F2 ≤ 1.209224397282491 · 10−31 , G2 ≤ 2.301233775774239 · 10−33 , T˜2 ≤ 0.834604675233862 . After the third iteration: U3 ≤ 1.309817593322718 · 10−5 , V3 ≤ 1.45622036006177 , M3 ≤ 1.00002126646582 , M 3 ≤ 1.00002126691809 , V˜3 ≤ 7.453583426051448 · 10−7 , F3 ≤ 8.855523608162042 · 10−44 , G3 ≤ 3.174732732716713 · 10−46 , T˜3 ≤ 0.834604675233862 . The quantities 1 ,...,η defined in Theorem 4.2 (recalling Theorem 4.1) are immediately computed using the above values and one obtains α ≤ 88530999255.1887 ,
η ≤ 1.185668436207269 · 10−38 .
With such values condition (4.31) is satisfied, in fact we obtained46 9
ηα M 7 M ξ −2(2τ +1) 216τ +23 τ !4 ≤ 3.584973875295102 · 10−8 < 1 , so that Theorem 1.1 holds.
A. Some Computer-Assisted Data We first report the trigonometric degrees νj , νj0 appearing in the Fourier-Taylor expansion of the approximate solution (u(0) , v (0) ) (obtained as the fifth ε-order truncation of the formal solution) u(0) =
5 X j=1
εj
X
u˜ (j) n exp in · θ ,
0<|n|≤νj
v (0) =
5 X j=0
εj
X
v˜ n(j) exp in · θ .
0<|n|≤νj0
We give the result by components: for u˜ (j) we found: Order j 1 2 3 4 5
Fourier deg. of u˜ (j) 1 5 15 25 35 45
Fourier deg. of u˜ (j) 2 5 15 25 35 45
For v˜ (j) : 46
According to the above convention M and M are, in fact, the values M3 and M 3 .
448
A. Celletti, L. Chierchia
Order j 0 1 2 3 4 5
Fourier deg. of v˜ 1(j) 0 5 14 24 34 44
Fourier deg. of v˜ 2(j) 0 5 11 21 31 41
The number of Fourier components of u(0) and of v (0) is given for each order of the Taylor series expansion in powers of the perturbing parameter ε. We report only those components with Fourier index (n, m) with n > 0 or n = 0 and m ≥ 0. For u(0) we found: Order j 0 1 2 3 4 5
Fourier coeff. of u˜ (j) 1 0 5 35 93 179 293
Fourier coeff. of u˜ (j) 2 0 5 35 93 179 293
For v (0) we found: Order j 0 1 2 3 4 5
Fourier coeff. of v˜ 1(j) 6 34 92 178 292 436
Fourier coeff. of v˜ 2(j) 5 26 76 156 264 400
Finally, we report now the intervals containing the five components of the function 47 . u˜ (1) 1 For the initial data (L+ , G+ ) we obtained : 0.6730562643923955199965449172780 + [58, 87] · 10−33 , 0.2979062447302117191526526718571 + [77, 85] · 10−33 , 1.482946598095398011285563691781 + [19, 25] · 10−32 , −0.4589344174885537366679779722512 + [53, 41] · 10−33 , −0.2010745230469404094053836635164 + [22, 14] · 10−33 . For the initial data (L− , G− ) we obtained: 47
The notation x + δ[a, b], with δ > 0 and a < b, means [x + δa, x + δb]: for example, 0.6730562643923955199965449172780 + [58, 87] · 10−33 = [0.673056264392395519996544917278058, 0.673056264392395519996544917278087] .
Stability of Three-Body Problems
449
0.6766043577880158539656402750183 + [17, 48] · 10−33 , 0.3061845611995878973674729226746 + [45, 54] · 10−33 , 1.488798647829940896171271103877 + [83, 90] · 10−32 , −0.4659443843426364961646113772143 + [22, 11] · 10−33 , −0.2018899370701671721933153371788 + [70, 62] · 10−33 . References 1. Arnold, V. I. (editor): Encyclopaedia of Mathematical Sciences. Dynamical Systems III, BerlinHeidelberg-New York: Springer-Verlag 3, 1988 2. Arnold, V. I.: Small divisor problems in classical and Celestial Mechanics. Russ. Math. Surv. 18, 85–191 (1963) 3. Celletti, A.: Analysis of resonances in the spin-orbit problem in Celestial Mechanics: The synchronous resonance (Part I). J. of App. Math. and Phys. (ZAMP) 41, 174 (1990) 4. Celletti, A., Chierchia, L.: Construction of Analytic KAM Surfaces and Effective Stability Bounds. Commun. Math. Phys. 118, 119–161 (1988) 5. Celletti, A., Chierchia, L.: A Constructive Theory of Lagrangian Tori and Computer-assisted Applications. In: Dynamics Reported, C.K.R.T. Jones, U. Kirchgraber, H.O. Walther, Managing Editors, Vol. 4 (New Series), Berlin-Heidelberg-New York: Springer–Verlag, 1995 6. Delaunay, C.: Th´eorie du Mouvement de la Lune. M´emoires de l’Acad´emie des Sciences 1, Tome XXVIII, Paris (1860) 7. Eckmann, J.-P., Wittwer, P.: Computer methods and Borel sommability applied to Feigenbaum’s equation. Berlin-Heidelberg-New York: Springer Lecture Notes in Physics 227, 1985 8. H´enon, M.: Explorationes num´erique du probl`eme restreint IV: Masses egales, orbites non periodique. Bullettin Astronomique 3, no. 1, fasc. 2, 49–66 (1966) 9. John, F.: Partial Differential Equations. Berlin-Heidelberg-New York: Springer-Verlag, 1982 10. Koch, H., Schenker, A., Wittwer, P.: Computer-Assisted Proofs in Analysis and Programming in Logic: A Case Study. Preprint Universit´e de Gen`eve (1994) 11. Meyer, K.R., Schmidt, D. (editors): Computer Aided Proofs in Analysis. Berlin-Heidelberg-New York: Springer-Verlag, 1991 12. Moser, J.: Stable and random motions in dynamical systems, with special emphasis on Celestial Mechanics. Ann. Math. Studies; 77, Princeton, NJ: Princeton University Press, 1973 13. Poincar`e, H.: Les Methodes Nouvelles de la Mechanique Celeste. Paris: Gauthier Villars, 1892 14. Robutel, P.: Stability of the planetary three-body problem. II. KAM Theory and existence of quasi-periodic motions. Celestial Mechanics 62, 219–261 (1995) 15. Salamon, D., Zehnder, E.: KAM theory in configuration space. Comment. Math. Helveticae 64, 84 (1989) 16. (no author listed) The Astronomical Almanac. Washington, D.C.: U.S. Government Printing Office, 1990 17. (no author listed) Vax Architecture handbook. Digital Equipment Corporation, 1981 Communicated by Ya.G. Sinai
Commun. Math. Phys. 186, 451 – 479 (1997)
Communications in
Mathematical Physics c Springer-Verlag 1997
An Analogue of the Kac–Wakimoto Formula and Black Hole Conditional Entropy Roberto Longo? Dipartimento di Matematica, Universit`a di Roma “Tor Vergata”, via della Ricerca Scientifica, I–00133 Roma, Italy, and Centro Linceo Interdisciplinare, Accademia Nazionale dei Lincei via della Lungara 10, I–00165 Roma, Italy. E-mail: [email protected] Received: 31 May 1996 / Accepted: 8 November 1996
Abstract: A local formula for the dimension of a superselection sector in Quantum Field Theory is obtained as vacuum expectation value of the exponential of the proper Hamiltonian. In the particular case of a chiral conformal theory, this provides a local analogue of a global formula obtained by Kac and Wakimoto within the context of representations of certain affine Lie algebras. Our formula is model independent and its version in general Quantum Field Theory applies to black hole thermodynamics. The relative free energy between two thermal equilibrium states associated with a black hole turns out to be proportional to the variation of the conditional entropy in different superselection sectors, where the conditional entropy is defined as the Connes–Størmer entropy associated with the DHR localized endomorphism representing the sector. The constant of proportionality is half of the Hawking temperature. As a consequence the relative free energy is quantized proportionally to the logarithm of a rational number, in particular it is equal to a linear function of the logarithm of an integer once the initial state or the final state is taken fixed. Introduction In the first part of this paper we shall derive a formula for the dimension of a superselection sector in Conformal Quantum Field Theory. However, as we shall see, the role played by conformal invariance is not essential, and indeed we will subsequently deal with general Quantum Field Theory and apply our formula to the computation of the relative free energy between two thermal equilibrium states of the background system for a black hole. The reader mainly interested in the latter topic may at first read the second part of this introduction and then go directly to Sect. 3. A local analogue of the Kac–Wakimoto formula. There is a general phenomenon relating the distribution of the Hamiltonian density levels to a dimension, a classical example be?
Supported in part by MURST and CNR-GNAFA.
452
R. Longo
ing given by Weyl’s theorem on the asymptotic distribution of the Laplacian eigenvalues on a compact Riemann manifold. A similar occurrence appears in the context of lowest weight representations of certain affine Kac–Moody algebras. If L0 and Lρ are the conformal Hamiltonians in the vacuum representation and in the representation ρ of such an infinite dimensional Lie algebra, then there exists the limit lim+
β→0
Tr(e−βLρ ) = d(ρ), Tr(e−βL0 )
(0.1)
and the thus defined d(ρ) has the formal properties of a dimension [30]. One expects such a formula to hold in more generality in conformal Quantum Field Theory on S 1 with ρ a superselection sector and d(ρ) the statistical dimension of ρ (one has to assume at least that Tr(e−βL0 ) < ∞ or any structural property to guarantee this), but no result in this direction has been so far obtained (cf. [43]). However, if we restrict the vacuum state ω to the local von Neumann algebra A(I) associated with an interval I, then ω is faithful by the Reeh-Schlieder theorem and hence, by the Tomita–Takesaki theory, ω is a Gibbs state with respect to its modular group1 . Such a modular group has a geometric meaning and we may interpret it as a local dynamics, in other words the logarithm of the modular operator can be regarded as a local Hamiltonian. One can then argue from [9] that in this local situation the distribution of the energy density levels should be tested in mean, at a specific value of the inverse temperature parameter β, and hence also a local version of formula (0.1) should not be asymptotic as β → 0+ , but evaluated at a specific β. Let 3I (t) be the one-parameter group of special conformal transformations of S 1 associated with the interval I of S 1 (formula (2.1) below) and Kρ the generator of the corresponding one-parameter unitary group in the representation ρ. We shall obtain the formula (0.2) (e−2πKρ ξ, ξ) = d(ρ), where ξ is any cyclic vector for ρ(A(I 0 )) such that (ρ(·) ξ, ξ) is the vacuum state on A(I 0 ) and I 0 = S 1 \I. In comparison with the formula (0.1), we first note that the right-hand side d(ρ) in (0.2) has not only the properties of a dimension, but it is actually identified with the Doplicher–Haag–Roberts [17] statistical dimension of ρ. Moreover the local formula (0.2) holds in full generality, independently of any requirement on the growth of Hamiltonian spectral density. Hopefully it might lead to a model independent proof of the formula (0.1), but it has its own interest. As we shall see, its proof makes use of the knowledge of the modular structure of the local von Neumann algebras [7, 21], the description of the conjugate sector in terms of the modular involutions [22, 23], and a result on actions of groups on tensor categories that determines Kρ as a linear function of the logarithm of the relative modular operator (as in formula (3.8) below). Finally the validity of formula (0.2) goes beyond the context of Conformal Quantum Field Theory. Indeed the same structure is present in the general setting Quantum Field Theory on Minkowski space [47], provided we consider the local von Neumann algebra associated with a wedge region, because in this case the modular structure has the geometric interpretation described by the Bisognano–Wichmann theorem. We then treat this case, where a further physical interpretation can be given. Mutatis mutandis, formulae 1
The reader should be aware of the different meaning of the term modular in refs. [48] and [30].
Kac–Wakimoto Formula and Black Hole Conditional Entropy
453
in Sect. 2 are also valid in Sect. 3 and vice versa, with the exception of Corollary 3.6. We avoid repetitions and state in each of them the results closer to the spirit of the section. Quantum numbers for the relative free energy associated with a black hole. As was indicated in [3, 4], a black hole appears from the outside to have some features of a thermodynamical system in equilibrium. In particular Bekenstein suggested the entropy of a black hole to be equal to λA, where A is the surface area of the black hole and λ a constant, an hypothesis related to the Generalised Second Law of Thermodynamics, dS + λdA ≥ 0,
(0.3)
where, in any process, dA is the increment of A and dS is the increment the entropy of the outside region. Taking into account Quantum effects and General Relativity, Hawking [27] was led to the conclusion that the black hole has a surface temperature T =
~ a , kc 2π
where a is the acceleration of a freely falling object at the surface of the black hole2 . This computation was made in the context of a free Quantum Field Theory on a curved space-time. Sewell [44] noticed that, at least in the case of a spherically symmetric eternal black hole, a model independent derivation of the Hawking temperature is possible, also in analogy with the Unruh effect [51], see also [16], by means of the Bisognano–Wichmann theorem [5] on the Minkowski space-time. One considers the Rindler space-time W as an approximation of the Schwarzschild space-time and realizes W as a wedge region in the Minkowski space-time, say W = {x, x1 > |x0 |}. The evolution x0 (t) = a−1 sht x1 (t) = a−1 cht corresponds to an observer moving within W with uniform acceleration a, and his proper time is equal to t/a. W is a natural horizon for this observer, since he cannot send a signal out of W and receive it back. The von Neumann algebra A(W ) of the observables on the Minkowski space localized in W is therefore the proper (global) observable algebra for such a mover. The proper time translations for him are thus given by the one-parameter automorphism group αat of A(W ) corresponding to the rescaled pure Lorentz transformations leaving W invariant. By the Bisognano–Wichmann theorem αat satisfies the KMS condition at inverse temperature β=
2π a
with respect to the (restriction of the Minkowski) vacuum state ω to A(W ), i.e. the latter is a thermal equilibrium state [25]. By the Einstein equivalence principle one can identify W with the outside region of the black hole and interpret the thermal outcome as a gravitational (black hole) effect. We refer to [24, 31, 52] for a more complete account of these arguments and further references. We note here that this description has certain restrictions. One is due to the 3
c ~ 2 T = in terms of the mass M of the star and the gravitational constant G. In the following we 8πkM G shall always use natural units so that the Plank costant ~, the speed of light c and the Boltzmann costant k are all set equal to 1, thus T = a/2π.
454
R. Longo
appearance of the Minkowskian vacuum tied up with the Poincar´e symmetries that do not exist globally on a general curved space-time. Another point concerns the choice of the Rindler space-time, that only near the horizon is a good approximation of the more appropriate Schwarzschild space-time. We shall briefly discuss these aspects in the final comments. Despite its limitations, this viewpoint is strikingly simple and powerful. Let us now consider a thermodynamical system Σ placed in the asympotically flat outside region of a black hole B. The Hawking radiation creates a heat bath for Σ and therefore Σ is an open system. Taking into account only observable quantities, Sewell [45] inferred that the right themodynamical potential for Σ is the Gibbs free energy, rather than the entropy, and rederived the generalised second law (0.3), where the area term represents now a contribution to the mechanical work done by Σ on B. Due to the Hawking effect, we have spontaneous creation of particles, so that the system undergoes a change of quantum numbers. From the Quantum Field Theory point of view, the system goes in a different superselection sector [53]. Following [26, 17] we thus consider a representation ρ of the quasi-local C∗ -algebra A = ∪A(O)− that is localizable in any space-like cone and has finite statistics. Under general conditions ρ is (and we assume to be) Poincar´e covariant with positive energy-momentum [22]. We may assume that ρ is localized within W and therefore the restriction of ρ to A(W ) ∩ A has a normal extension to an endomorphism of its weak closure A(W ), that will still be denoted by ρ|A(W ) . The index-statistics theorem [36, 39] shows that the map ρ −→ ρ|A(W ) is a faithful full functor of tensor categories, namely all the information on the superselection structure (charge transfers, statistics,. . . ) is visible within A(W ), in particular 1
d(ρ) = Ind(ρ) 2 , where d(ρ) is the DHR statistical dimension, i.e. the order of the parastatistics, and Ind(ρ) is the Jones index of ρ|A(W ) (more precisely the minimal index, see [34, 39]). In the sector ρ the proper time evolution is given by a one-parameter automorphism ρ of A(W ) corresponding to the pure Lorentz transformations leaving W ingroup αat variant, αtρ · ρ = ρ · αt . ρ admits a unique thermal equilibrium normal state ϕρ at the same As we shall see, αat inverse Hawking temperature β = 2π/a. If Kρ is the generator of the unitary implementation of the pure Lorentz in the x1 -direction, then
Hρ = aKρ is the proper Hamiltonian for our system Σ in the sector ρ. A similar structure appears in the analysis of the chemical potential [2]. Adding one particle is not a drastic change as our thermodynamical system has essentially infinitely many particles, so we do not obtain an inequivalent representation, namely ρ is normal on A(W ). There is however an important difference. The chemical potential labels different equilibrium states at the same temperature for the same dynamics, while we ρ ,a look at the ϕρ ’s as equilibrium states with respect to their own different dynamics αat fact compatible with the General Relativity context where the dynamics should depend on the state as the matter has influence on the metric tensor.
Kac–Wakimoto Formula and Black Hole Conditional Entropy
455
Motivated by the above discussion, we will consider the relative free energy between two thermal equilibrium states for the system Σ associated with the external region of a black hole. Guided by the thermodynamical expression dF = dE − T dS, we express the relative free energy of the states ω and ϕρ by F (ω|ϕρ ) = ϕρ (Hρ ) − β −1 S(ω|ϕρ ), where S is the Araki relative entropy [1] of the two states and ϕρ (Hρ ) is the relative mean (internal) energy. We shall find the relation 1 F (ω|ϕρ ) = − β −1 Sc (ρ), 2
(0.4)
where Sc (ρ) is the Connes-Størmer [15] conditional entropy of ρ(A(W )) ⊂ A(W ). Here we use the fact that, by the Pimsner–Popa theorem [41] and the index-statistics theorem [36], 21 Sc (ρ) equals the logarithm of the statistical dimension d(ρ) of ρ. As the latter takes only integral values by the DHR theorem [17], we conclude that the possible values for the relative free energy are F (ω|ϕρ ) = −β −1 log(n),
n = 1, 2, 3, . . . ,
namely the integer n = d(ρ) here acquaints the different meaning of a quantum number labeling the relative free energy levels. The vacuum state ω should play no specific physical role in (0.4) and it is only a convenient reference state in our setting. Indeed we will extend the formula to the case of two arbitrary thermal equilibrium states ϕσ and ϕρ , F (ϕσ |ϕρ ) =
1 −1 β (Sc (σ) − Sc (ρ)), 2
(0.5)
n and therefore F (ϕσ |ϕρ ) = β −1 log( m ) is β −1 -times the logarithm of a rational number n m , where n depends only on σ and m only on ρ. Finally we observe that formula (0.5) is consistent with the above recalled interpretations of the increment dA of surface area of the black hole [4, 45] and, in a sense, it unifies different points of view: the increments of the conditional entropy of Σ, an information theoretical concept, is indeed proportional to the increment of its free energy, a statistical mechanics concept. We summarize the conceptual scheme in the proof of our result in the following diagram:
Connes–Størmer Entropy Th. 3.4
1 −1 2β
?
Relative Free Energy
Pimsner–Popa Th.
exp −1 log β
Cor. 2.2
-
Jones Index √ · Index-statistics Th. ? DHR dimension
456
R. Longo
1. Connes Cocycles and Endomorphisms Let M be a von Neumann algebra and ϕ, ω faithful normal positive linear functionals on M and denote by σ ω and σ ϕ their modular group given by the Tomita–Takesaki theory [48]. The Connes Radon–Nikodym cocycle [12] u(t) = (Dϕ : Dω)t is the map (t ∈ R → u(t) unitary of M ) such that u(t + s) = u(t)σtω (u(s))
(1.1)
characterized by: for any given x, y ∈ M there exists a complex function F bounded and continuous in {0 ≤ Imz ≤ 1} and analytic in its interior such that F (t) = ϕ(σtϕ (y)u(t)x),
F (t + i) = ω(xu(t)σtω (y)).
The relevant property is that u(t) intertwines σtω and σtϕ
=
σtϕ ,
(1.2)
namely
u(t)σtω (·)u(t)∗ .
(1.3)
ω
Conversely a continuous unitary σ -cocycle u (i.e. (1.1) is valid for u) is the Connes cocycle with respect to a unique faithful normal positive linear functional or semifinite weight ϕ of M . If M is a factor, a continuous unitary σ ω -cocycle u satisfying (1.3) is uniquely determined up to a one-dimensional character of R, hence, in order to check whether u(t) = (Dϕ : Dω)t , one may test Eq. (1.2) in the special case x = y = 1: there must exist a function F bounded and continuous in {0 ≤ Imz ≤ 1} and analytic in its interior such that F (t) = ϕ(u(t)), F (t + i) = ω(u(t)). (1.4) In particular, given the σ ω -cocycle u, the positive functional ϕ such that u(t) = (Dϕ : Dω)t may be computed by (1.2) as ϕ(x) = anal.cont. ω(xu(t)). t→−1
(1.5)
Let now M be an infinite factor and denote by End(M ) the set finite-index (or finitedimensional) endomorphisms of M . Namely ρ ∈ End(M ) if ρ is an endomorphism of M whose intrinsic dimension d(ρ) is finite ([40], see Appendix A). Equivalently ρ(M ) is a subfactor of M with finite index in the sense of [34, 32], indeed, as shown in [37], 1
d(ρ) = Ind(ρ) 2 , where Ind(ρ) denotes the minimal index of ρ(M ) ⊂ M . We refer to [39] and references therein for the notions of index theory of our use. We fix a normal faithful state ω of M . Given ρ ∈ End(M ) we denote by 8ρ the minimal left inverse of ρ and set 9ρ = d(ρ)8ρ ,
ψρ = ω · 9ρ
so that ψρ is a normal faithful positive functional of M . We define the cocycle of an endomorphism ρ by u(ρ, t) = (Dψρ : Dω)t
(1.6)
(see also [33]). As will be apparent, several of the results in this section are valid (essentially with the same proofs) for endomorphisms of factors with a normal faithful conditional expectation onto the range, but for simplicity we just treat the finite-index case.
Kac–Wakimoto Formula and Black Hole Conditional Entropy
457
ω Proposition 1.1. Let ρ ∈ End(M ) and set ρt = σtω ρσ−t . Then u(ρ, t) satisfies
Adu(ρ, t) · ρt = ρ.
(1.7)
In particular, if ρ is irreducible, then u(ρ, t) is characterized by (1.7) up to the multiplication by a one-dimensional character of R. Proof. The minimal expectation Eρ = ρ8ρ onto ρ(M ) leaves ψρ invariant since ψρ ρ8ρ = ω8ρ ρ8ρ = ω8ρ = ψρ ; thus
σ ψρ |ρ(M ) = σ ψρ |ρ(M ) = σ ωρ
−1
= ρσ ω ρ−1 ,
namely ψ
σt ρ ρ(x) = ρσtω (x), and since
ψ σt ρ
x ∈ M,
= Adu(ρ, t)σtω we have ω Adu(ρ, t) · σtω · ρ · σ−t = ρ.
(1.8)
Now Eq. (1.8) detemines the restriction of Adu(ρ, t) to ρ(M ), hence it determines u(ρ, t) up to muliplication by a unitary in ρ(M )0 ∩ M and therefore, if ρ is irreducible, it determines u(ρ, t) up to a phase. As recalled in Appendix A, the elements of End(M ) are the objects of a tensor C∗ category where the arrows are given by (A.0). The following in this section has a natural interpretation in the setting of tensor C∗ -categories, although here below we use the explicit formulas of our setting (see [37]). As we shall use the greek letter σ to denote an element of End(M ), the modular group will be always denoted with explicit reference to the functional (e.g. σtω ). Proposition 1.2. Let ρ ∈ End(M ) be irreducible and contained in σ ∈ End(M ), namely there exists an isometry w in (ρ, σ) (i.e. w ∈ M and wρ(x) = σ(x)w, ∀x ∈ M ), then 9ρ = 9σ (w · w∗ ).
(1.9)
(i) If σ = ⊕N i=1 ni ρi is an irreducible decomposition of σ and for each i {wk , k = 1, . . . ni } is an orthonormal basis of isometries in (ρi , σ), then
9σ =
ni N X X
9ρi (wk(i)∗ · wk(i) ).
(1.10)
i=1 k=1
Proof. Let 90σ be defined by the right-hand side of (1.10). Then 90σ (σ(x)) =
ni N X X i=1 k=1
9ρi (wk(i)∗ σ(x)wk(i) ) =
ni N X X
9ρi (ρi (x)) =
i=1 k=1
N X
ni d(ρi )x = d(σ)x,
i=1
thus d(σ)−1 90σ is a left inverse of σ. As 90σ (wk(i) wk(i)∗ ) = 9ρi (1) = d(ρi ), we have that 90σ = 9σ , showing the validity of Eq. (1.10). Formula (1.9) is obtained similarly. Proposition 1.3. If T is an arrow in (ρ, σ), then T u(ρ, t) = u(σ, t)σtω (T ) .
(1.11)
458
R. Longo
Proof. Let first ρ be an irreducible component of σ and w an isometry in (ρ, σ). Then by (1.9) we have u(ρ, t) = (Dψρ : Dω)t = (Dψσ (w · w∗ ) : Dω)t , therefore by the Radon–Nikodym chain rule and using the relation (Dψσ (w∗ · w) : Dψσ )t = w∗ σtψσ (w) we have u(ρ, t) = (Dψσ (w∗ · w) : Dψσ )t (Dψσ : Dω)t = w∗ σtψσ (w)u(σ, t) = w∗ u(σ, t)σtω (w), and after multiplying on the left by w all members in the above expression, and using the σtψσ -invariance of ww∗ , we obtain the special case of (1.11), wu(ρ, t) = u(σ, t)σtω (w).
(1.12)
(i) If now σ = ⊕N i=1 ni ρi is an irreducible decomposition of σ and the wk are as in Proposition 1.2, we have from (1.12) u(σ, t)wk(i) wk(i)∗ = wk(i) u(ρ, t)σ ω (wk(i)∗ ) and summing up over i and k we have X (i) wk u(ρ, t)σ ω (wk(i)∗ ). (1.13) u(σ, t) = i,k
If now T is an arrow in (ρ, σ), we may decompose both σ and ρ into irreducibles so that the ranges of the wk(i) ’s of ρ (resp. of σ) are either orthogonal or contained in the range of T (resp. of T ∗ ). Then multiplying (1.13) on the left by T (resp. on the right by σtω (T )) will kill the indices corresponding to the orthogonal part and the result is obtained by Eq. (1.12). Proposition 1.4. u(ρσ, t) = ρ(u(σ, t))u(ρ, t). Proof. We first note that ρ(u(ρ, t)) = (Dψρσ : Dψρ )t .
(1.14)
Indeed as both functionals ψρσ and ψρ leave invariant the conditional expectation ρ8ρ onto ρ(M ), their Radon–Nikodym cocycle coincides with the cocycle of their restriction to ρ(M ), thus (Dψρσ : Dψρ )t = (Dψρσ |ρ(M ) : Dψρ |ρ(M ) )
t
= (Dψσ · ρ−1 |ρ(M ) : Dω · ρ−1 |ρ(M ) )t = ρ((Dψσ : Dω)t ) = ρ(u(ρ, t)). The proposition then follows by the Radon–Nikodym chain rule for the Connes cocycles. Propositions 1.3 and 1.4 state that u(ρ, t) is a two-variable-cocycle (Appendix A), with respect to the action of R on End(M ) given by ω t → ρt = σtω · ρ · σ−t ,
t → σtω (T ),
(1.15)
where ρ is an object and T is an arrow in End(M ). Recall now that each ρ ∈ End(M ) has a conjugate object ρ, ¯ namely there exist Rρ ∈ (ι, ρρ) ¯ and R¯ ρ ∈ (ι, ρρ) ¯ standard solutions of Eq. (A.1), i.e.
Kac–Wakimoto Formula and Black Hole Conditional Entropy
R¯ ρ∗ ρ(Rρ ) = 1,
459
Rρ∗ ρ( ¯ R¯ ρ ) = 1
√ and kRρ k = kR¯ ρ k (= d(ρ)) is minimal. As explained in Appendix A, given an arrow T ∈ (ρ1 , ρ2 ), the conjugate arrow T • ∈ (ρ¯1 , ρ¯2 ) is defined by T • = ρ¯2 (R¯ ρ∗1 T ∗ )Rρ2 , where Rρi and R¯ ρi give a standard solution for the conjugate equation defining the conjugate ρ¯i . Note now that given ρ ∈ End(M ), once we choose ρ¯ defined by the Rρ and Rρ¯ , then (ρ) ¯ t is a conjugate of ρt defined by Rρt = σtω (Rρ ), R¯ ρt = σtω (R¯ ρ ), that we may simply denote by ρ¯t . In the following u(ρ, t)• is defined by this choice of the R-operators. Proposition 1.5. u(ρ, ¯ t) = u(ρ, t)• . Proof. By definition ¯ R¯ ρ∗t u(ρ, t)∗ )Rρ u(ρ, t)• = ρ( = ρ(σ ¯ tω (R¯ ρ∗ )u(ρ, t)∗ )Rρ = ρ(σ ¯ tω (R¯ ρ∗ )σtω (u(ρ, −t)))Rρ = ρ(σ ¯ tω (R¯ ρ∗ u(ρ, −t)))Rρ = u(ρ, ¯ t)σtω (ρ( ¯ R¯ ρ∗ u(ρ, −t)))u(ρ, ¯ t)∗ Rρ = u(ρ, ¯ t)σtω (ρ( ¯ R¯ ρ∗ u(ρ, −t)))σtω (u(ρ, ¯ −t))Rρ = u(ρ, ¯ t)σtω (ρ( ¯ R¯ ρ∗ u(ρ, −t))u(ρ, ¯ −t))Rρ . ω ¯ R¯ ρ∗ u(ρ, −t))u(ρ, ¯ −t))Rρ = 1 or, by applying σ−t , that Thus we have to show that σtω (ρ( ∗ ¯ ¯ −t)Rρ−t = 1. Indeed we have ρ( ¯ Rρ u(ρ, −t))u(ρ,
¯ −t)Rρ−t = ρ( ¯ R¯ ρ∗ )ρ(u(ρ, ¯ −t))u(ρ, ¯ −t)Rρ−t ρ( ¯ R¯ ρ∗ u(ρ, −t))u(ρ, ∗ = ρ( ¯ R¯ ρ )u(ρρ, ¯ −t)Rρ−t = ρ( ¯ R¯ ρ∗ )Rρ = 1 , where u(ρρ, ¯ −t)Rρ−t = Rρ by Proposition 1.3.
Lemma 1.6. If j is an ω-preserving anti-automorphism of M , then u(ρ, t) = j(u(j · ρ · j −1 , −t)). ω , thus Proof. Since j preserves ω, by the KMS condition jσtω j −1 = σ−t −1 ω j(u(j · ρ · j , −t)) is a σt -cocycle and it coincides with u(ρ, t) because it satifies Eq. (1.5).
Proposition 1.7. Let T be a C ∗ -tensor subcategory with conjugates of End(M ) and z a two-variable cocycle for the action R → AutT given by the modular group σ ω (Eq. (1.15)). Suppose there is an anti-automorphism j of M such that j −1 ρj is a conjugate of ρ and z(j −1 ρj, t) = j(z(ρ, −t)) for a given ρ ∈ T . Then z(ρ, t) coincides with the Connes cocycle u(ρ, t) defined by (1.5). As a consequence, d(ρ) = anal.cont. ω(z(ρ, t)). t→−i
(1.16)
460
R. Longo
Proof. We have z(ρ, t) = µ(ρ, t)u(ρ, t) for some character µ(ρ, t) of R. Since ρ¯ = j −1 ρj is a conjugate of ρ we have by Lemma 1.6, z(ρ, ¯ t) = j(z(ρ, −t)) = j(µ(ρ, −t)u(ρ, −t)) = µ(ρ, t)j(u(ρ, −t)) = µ(ρ, t)u(ρ, ¯ t), On the other hand by Lemma A.3 of Appendix A, z(ρ, ¯ t) = z(ρ, t)• = µ(ρ, t)• u(ρ, t)• = µ(ρ, t)u(ρ, ¯ t), thus µ(ρ, t) = 1. Formula (1.16) is thus a consequence of (1.5) in the case x = 1.
Before concluding this section we recall the notions of entropy for later use. If N ⊂ M is an inclusion of II1 -factors, the Connes-Størmer (conditional) entropy H(M |N ) is defined in [15]. By the Pimsner–Popa Theorem [41], H(M |N ) = log[M : N ],
(1.17)
where [M : N ] is the Jones index of N ⊂ M (if N ⊂ M is irreducible or extremal). H(M |N ) is generalized to the type III setting in [14]. If ϕ is a normal faithful state of M one defines X
Hϕ (M |N ) = sup{ (ϕi )
S(ϕ|ϕi ) − S(ϕ| N |ϕi| N )},
i
sets of finitely many normal positive linear functionals where (ϕi ) varies among theP n ϕ1 , ϕ2 , . . . ϕn of M such that i=1 ϕi = ϕ, and S(·|·) denotes the Araki relative entropy between states, see [6] and Eq. (3.7) below. If E is a normal conditional expectation of M onto N one sets HE (M |N ) = sup{Hϕ (M |N ), ϕ · E = ϕ}. If moreover N is a III1 -factor, then HE (M |N ) = Hϕ (M |N ) for any normal faithful state ϕ such that ϕ · E = ϕ [28]. HE (M |N ) depends on the choice of a normal conditional expectation E, but we simply write H(M |N ) = HEmin (M |N ), where Emin is the minimal conditional expectation. The Pimsner–Popa equality (1.17) holds true without restrictions, provided [M : N ] denotes the minimum index, see [28]. Finally, if ρ is an endomorphism of M , we consider the conditional entropy of ρ, Sc (ρ) = H(M |ρ(M )).
(1.18)
Kac–Wakimoto Formula and Black Hole Conditional Entropy
461
2. The Formula in Conformal Field Theory 2.1. Finite index case. We now consider a precosheaf (net) A of von Neumann algebras associated with a chiral conformal quantum field theory. Namely A is a map I −→ A(I) from the (proper) intervals of S 1 to von Neumann algebras A(I) on a fixed Hilbert space H that satisfies: ˜ Isotony: if I ⊂ I˜ then A(I) ⊂ A(I), Locality: A(I) and A(I 0 ) commute elementwise, where I 0 = S 1 \I, M¨obius covariance with positive energy: there exists a unitary reperesentation U of M¨obius group SL(2, R)/{1, −1}, that for convenience we regard as a representation of its universal covering group G, such that U (g)A(I)U (g)∗ = A(gI) and the generator of the one-parameter rotation subgroup is positive. We set αg (X) = U (g)XU (g)∗ where X is a local operator, namely X belongs to some A(I), Existence and uniqueness of the vacuum: there exists a unique (up to a phase) U invariant unit vector ∈ H, and it is cyclic for the algebra generated by local operators. We denote by ω = ( ·, ) the vacuum state. For a discussion of these properties and their consequences, see [23]. Let ρ be a covariant positive energy representation of A on a separable Hilbert space Hρ , namely for every proper interval I we have a representation ρI of A(I) on Hρ so that = ρI if I ⊂ I˜ ρI˜ |A(I)
and a positive energy representation Uρ of G on Hρ such that ρgI (αg (X)) = Uρ (g)ρI (X)Uρ (g)∗ , X ∈ A(I) (if A is strongly additive the covariance property is automatic [22]). We shall always assume the representations to be covariant with positive energy. Let 3I (t) be the special conformal one-parameter group associated with I. If I is the upper semi-circle then 3I (t)z =
(z + 1) + e−t (z − 1) , (z + 1) − e−t (z − 1)
(2.1)
while if I0 is any other interval 3I0 is well defined by conjugation of the transformation (2.1) with a conformal transformation g ∈ SL(2, R) such that gI = I0 . We denote by KρI = −i
d Uρ (3I (t))| t=0 dt
the infinitesimal generator of Uρ (3I (t)).
462
R. Longo
Theorem 2.1. Let ρ be a representation of A, I ∈ S 1 an interval and ξ ∈ Hρ a cyclic vector for ρI 0 (A(I 0 )) such that ω(X) = (ρI 0 (X)ξ, ξ) for all X ∈ A(I 0 ). Then3 I
(e−2πKρ ξ, ξ) = d(ρ). A vector ξ as above always exists. As we are interested in the representation ρ up to unitary equivalence, we may identify Hρ with H so that, due to Haag duality (see below), ρ becomes an endomorphism of ˜ ⊂ A(I) ˜ if A localized in a given interval I, namely ρI 0 acts identically, and ρI˜ (A(I)) ˜ see [23]. Because of the Reeh-Schlieder theorem (see [20]), the vacuum vector I ⊂ I, is cyclic for any local von Neumann algebra, in particular for A(I 0 ), therefore satisfies in this representation the properties required for the vector ξ in the statement of Theorem 2.1, showing its existence. As ρ is covariant, there is a unitary α-cocycle z(ρ, g) such that Adz(ρ, g) · αg · ρ · αg−1 = ρ,
(2.2)
where αg = AdU (g). More precisely the equation z(ρ, g) = Uρ (g)U (g)∗ defines z(ρ, g) if g belongs to a neighbourhood of the identity of G, and z(ρ, g) is localized in the sense ˜ if I˜ is any interval containing both I and gI [22], (see also [54]). that it belongs to A(I) Then z(ρ, g) is defined for arbitrary g ∈ G as an element of the universal algebra C ∗ (A) by the cocycle identity z(ρ, gh) = z(ρ, g)ρ(z(ρ, h)), but we do not need this fact. It is important to note that if ρ is irreducible or a finite direct sum of irreducibles, as is the case of a finite-index ρ, then z is uniquely determined by the formula (2.2) as a localized cocycle, because G has no non-trivial unitary finite-dimensional representation, see [23]. We consider the tensor C ∗ -category EI whose objects are the (covariant, positive energy) endomorphisms of A with finite index, namely d(ρ) = d(ρI ) < ∞, localized in an interval whose closure is contained in the interior of a given interval I and the arrows (ρ, σ) are the local operators T such that T ρI1 (X) = σI1 (X)T for all intervals I1 and local operators X ∈ A(I1 ). By [36] (see also [23]) the restriction map ρ ∈ EI → ρI ∈ EndA(I) is a faithful full functor, therefore we may identify EI with a tensor C ∗ -subcategory of EndA(I), so that d(ρ) is identified with the DHR statistical dimension of ρ. Then z is in a natural sense a local two-variable cocycle for the local action of G on EI given by ρ → αg ραg−1 (ρ ∈ EI ) and T → αg (T ) (T arrow), namely the properties a) and b) defining a two-variable cocycle after Lemma A.3 of Appendix A hold for z(ρ, g), but only if g belongs to a neighbourhood of the identity of G (depending on ρ). For example if ρ, σ ∈ EI then ρ(z(ρ, g))z(σ, g) exists if g lies in a neighbourhood of the identity of G and satisfies (2.2) for ρσ, hence it agrees with z(ρσ, g) as the α-cocycle property and formula (2.2) determines it. We thus have: Lemma 2.2. z(ρ, g) is a local two-variable cocycle for the action of G on the tensor C ∗ -category EI . We shall use the following convention: if A is a positive selfadjoint operator and η a vector, then (Aη, η) = 1 1 kA 2 ηk2 if η belongs to the domain of A 2 , and (Aη, η) = +∞ otherwise. 3
Kac–Wakimoto Formula and Black Hole Conditional Entropy
463
Proof. This follows by the above discussion and an elementary direct verification of the two-variable cocycle property. Recall now that each localized endomorphism ρ has a conjugate localized endomorphism given by the formula [22] ρ¯ = j · ρ · j, (2.3) where j = AdJ is the anti-automorphism of A implemented by the modular conjugation J of (A(I1 ), ) for any choice of the interval I1 . To be definite let I be the upper halfcircle and I1 the right half-circle. Due to the geometrical meaning of J, j implements an anti-automorphism of A(I). If g is in the M¨obius group, we denote by g j its conjugate by the anti-automorphism given by the reflection on S 1 corresponding to j. Proposition 2.3. Let ρ be a finite-index endomorphism of A localized in the interval I. With the above notations, we have z(ρ, ¯ g) = z(ρ, g)• = j(z(ρ, g j )) (see Sect. 1 and Appendix A for the definition of the • -mapping). Proof. The first equality follows from Lemma 2.1 and Lemma A.3. However one may see directly the validity of both equalities by the uniqueness of z(ρ, ¯ g) by checking that also z(ρ, g)• and j(z(ρ, g j )) are local α-cocycles and both satisfy formula (2.2) for ρ¯ = jρj. Now the modular structure of A is computed in [29, 7], in particular we have 1it I = U (3I (−2πt)), where 1I is the modular operator of (A(I), ). An important consequence is Haag duality A(I 0 ) = A(I)0 , moreover the A(I)’s are type III1 factors. The next theorem computes the modular structure of A in the representation ρ, see also Proposition 3.5. We set uI (ρ, t) = u(ρI , t) = (Dω9ρI : Dω|A(I) )t as in (1.6), with ω the vacuum state. Theorem 2.4. Let ρ be finite-index endomorphism of A localized in the interval I. Then uI (ρ, t) = z(ρ, 3I (−2πt)), t ∈ R. Proof. By the formula (2.3) for the conjugate sector and Lemma 2.2, the theorem follows immediatly by Proposition 1.7. Corollary 2.5. Let ρ be an irreducible finite-index endomorphism of A localized in the interval I of S 1 . Then I
I
d(ρ) = (e−2πKρ , ) = ke−πKρ k2 .
(2.4)
∗
Proof. Since zI (ρ, 3I (t)) = Uρ (3I (t))U (3I (t)) , Proposition 1.7 and Theorem 2.3 show that the function t → ω(z(ρ, 3I (−2πt))) extends to a function bounded and continuous in the strip {−1 ≤ Imz ≤ 0} and analytic in its interior so that d(ρ) = anal.cont. ω (z(ρ, 3I (−2πt))) = anal.cont. Uρ (3I (−2πt)), . t→−i
t→−i
Standard functional analysis arguments then show that the right hand side of the above I expression is equal to (e−2πKρ , ).
464
R. Longo
Let V : Hρ → H be the unitary given
Proof of Theorem 2.1 in the finite index case. by V ρ(X)ξ = X,
X ∈ A(I 0 )
so that V ρ(X)V ∗ = X if X ∈ A(I 0 ), thus ρ0 = V ρ(·)V ∗ is an endomorphism of A localized in I. By Corollary 2.5 we then have I
I
I
d(ρ) = (e−2πKρ0 , ) = (V e−2πKρ V ∗ , ) = (e−2πKρ ξ, ξ), in case ρ has finite index. We have already commented on the existence of ξ. The infinite index case is discussed here below. 2.2. General case: A criterium for finite index. We now show that formula (2.4) gives a criterium for the finiteness of the index of a sector, namely that Theorem 2.1 holds without restrictions. We start with a general fact. Proposition 2.6. Let N ⊂ M be an inclusion of factors and assume that there exist a normal faithful conditional expectation E of M onto N and a normal faithful conditional expectation E 0 of N 0 onto M 0 . Then N 0 ∩ M is a discrete type I von Neumann algebra, i.e. a (possibly infinite) direct sum of type I factors. Moreover for each minimal projection p of N 0 ∩ M the inclusion N p ⊂ pM p has finite index. Proof. Setting R = N 0 ∩M , E restricts to a faithful expectation of N ∨R onto N , hence N ∨ R is canonically isomorphic to the von Neumann tensor product N ⊗ R and we can assume this isomorphism to be spatial (by tensoring N and M by a type III factor, if necessary). On the other hand E 0 factors through a faithful normal expectation of N 0 onto (N ∨ R)0 = N 0 ∩ R0 by Takesaki’s theorem [49], hence, with the above spatial identification, we have a normal faithful expectation of N 0 ⊗ B(H) onto N 0 ⊗ R0 , that restricts to a normal expectation of C ⊗ B(H) onto C ⊗ R0 , that implies R to be be a type I von Neumann algebra. As R is a direct sum of homogeneous type In von Neumann algebras, by considering the reduced inclusion corresponding to an abelian projection of R (fixed by the modular group of the expectation) we are left to prove our statement in the case R is an abelian von Neumann algebra, namely we have to prove that R is totally atomic, for in this case the finiteness of the index of the reduced inclusions corresponding to minimal projections of R would follow by [37, Proposition 4.4]. By decomposing R into its diffuse and atomic part, we may then assume R to be diffuse abelian and find an absurd. To this end, for notational convenience, we may identify N with M (see [38]), i.e. we set N = σ(M ) for some endomorphism σ of M . R⊕ σλ dµ(λ) into irreducibles and as R is abelian σλ is disjoint We may decompose σ = to σλ0 for almost all pairs (λ, λ0 ), hence σ¯ λ σλ0 does not contain the identity, except for (λ, λ0 ) in a set of product measure 0 and we conclude that σσ ¯ does not contain the identity too. By [36] this shows that there exists no normal faithful expectation onto N contradicting our hypothesis. Lemma 2.7. Let N ⊂ M be an inclusion of von Neumann algebras on a Hilbert space H, a cyclic separating vector for M and 1M , 1N the corresponding modular 1
1
1
2 2 2 operators on H and on N . Then k1M ξk = k1N ξk for all ξ in the domain of 1N .
Kac–Wakimoto Formula and Black Hole Conditional Entropy
465
Proof. If x ∈ N we have with usual notations 1
1
1
1
2 2 2 2 xk = kJN 1N xk = kx∗ k = kJM 1M xk = k1M xk, k1N 1
1
2 2 the equality holds for all the vectors in the domain of 1N . and as N is a core for 1N
Corollary 2.8. Let T (t) and V (s) be two unitary one-parameter groups on a Hilbert space H such that (2.5) V (s)T (t)V (−s) = T (e−2πs t), t, s ∈ R and assume −i d T (t)|t=0 to be positive. Then dt ke−πD ξk = ke−πD T (t)ξk
(2.6)
for all ξ in the domain of e−πD and all t ≥ 0, where D is the generator of V . Proof. The projection P onto the T -fixed vectors commutes both with T and V and on such vectors Eq. (2.6) trivially holds, hence we may assume that P = 0. With this assumption all non-zero representations of the commutation relation (2.5) are quasi-equivalent by the von Neumann uniqueness theorem (D and the logarithm of the generator of T (t) satisfy the Heisenberg commutation relations), hence we may verify Eq. (2.6) in any given representation. Let B be the conformal net on R = S 1 \{−1} given by the current algebra (see e.g. [19]), T and V the translation and dilation unitary groups. Then e−2πD = 1M , where M = B(0, +∞) and T (t)e−2πD T (−t) = 1N , where N = B(t, +∞). Lemma 2.7 then applies to the modular operators 1M and 1N with respect to the vacuum and gives Eq. (2.6). We shall now denote by T I (t) the one-parameter unitary group of translations associated with I, namely cutting S 1 and identifying it with R so that I is identified with R+ , then T I (t) correspond to the translations on R. If ρ is an endomorphism of A localized in I, we shall then denote by TρI , the corresponding one-parameter unitary group in the representation ρ. Corollary 2.9. Let ρ be an endomorphism of A localized in the interval I. Then I
I
ke−πKρ ξk = ke−πKρ TρI (t)ξk I
for all ξ in the domain of e−πKρ and t ≥ 0 . Proof. Immediate by Corollary 2.8.
Proposition 2.10. Let ρ be an endomorphism of A localized in the interval I. If I (e−2πKρ , ) < ∞, then the formula I
I
ψρ (XY ∗ ) = (e−πKρ X, e−πKρ Y ) determines a positive normal linear functional ψρ (Dψρ : Dω|A(I) )t = z(ρ, 3I (−2πt)).
on A(I) such that
466
R. Longo
Proof. By Connes’ theorem [12] there exists a unique normal faithful semifinite weight ψρ on A(I) such that (Dψρ : DωI )t = z(ρ, 3I (−2πt)), where we shorten notations like ω|A(I) = ωI . Moreover 1(ωI 0 |ψρ )it = z(ρ, 3I (2πt))1(ωI 0 |ωI )it , where 1(·|·) denotes the Connes spatial derivative [13] (between a weight on a von Neumann algebra and a weight on its commutant), and since by (2.2) I
−2πitK , 1it (ωI 0 |ωI ) = 1it I =e
we have
I
e−2πKρ = 1(ωI 0 |ψρ ).
(2.7) 1 2
By assumptions thus belongs to the domain of 1(ωI 0 |ψρ ) and this implies that 1 ψρ (1) = k1(ωI 0 |ψρ ) 2 k2 < +∞, namely ψρ is everywhere defined. The proposition now readly follows by formula (1.4): ψρ (XY ∗ ) = anal.cont. ω(Y ∗ u(ρ, t)σtω (X)) t→−i
= anal.cont. ω Y ∗ z(ρ, 3I (−2πt))σtω (X) t→−i = anal.cont. e−i2πtKρ X, Y . t→−i
Alternatively one could use directly the expression given by Proposition 3.5.
Proposition 2.11. Let ρ and ψρ be as in Proposition 2.10 and identify I with R+ as above. Then (2.8) ψρ (Uρ (g)XUρ (g)∗ ) = ψρ (X), X ∈ A(I), provided g ∈ G is a dilation or a positive translation associated with I. As a consequence there exists a positive linear functional ψ˜ ρ on Aζ = ∪` A(`, +∞)− , normal on any A(`, +∞), translation and dilation invariant in the representation ρ, defined by ψ˜ ρ (X) = ψρ (TρI (t)XTρI (t)∗ ), where X ∈ A(`, +∞) and t + ` > 0. Proof. The second assertion clearly follows from the first one. Formula (2.8) holds if g is a dilation, as the dilations correspond to the modular automorphisms of A(I) with respect to ψρ , due to the construction of ψρ . By Proposition 2.10 we thus have to show that I I (2.9) ke−πKρ TρI (t)XTρI (−t)k = ke−πKρ Xk, where X ∈ A(I) and t ≥ 0. Indeed by Corollary 2.9 for all X ∈ A(I) and t ≥ 0 we have I
I
ke−πKρ TρI (t)XTρI (−t)k = ke−πKρ XTρI (−t)k I
I
= ke−πKρ XTρI (−t)k = ke−πKρ Xξk,
(2.10)
where ξ = TρI (−t) = TρI (−t)T I (t). On the other hand z(ρ, t) = TρI (t)T I (−t) belongs to A(I) if t ≥ 0 as ρ(T I (t)·T I (−t)) is also localized in I. Therefore (X 0 ξ, ξ) = (X 0 , ) if X 0 ∈ A(I 0 ).
Kac–Wakimoto Formula and Black Hole Conditional Entropy
467
Now ξ is cyclic for A(I 0 ) if t ≥ 0, because ξ = u with u = z(ρ, t) a unitary in A(I). By Proposition 2.10, Eq. (2.10) gives I
I
I
ke−πKρ TρI (t)XTρI (−t)k = ke−πKρ Xξk = ke−πKρ Xuk = ψρ (Xuu∗ X ∗ ) = ψρ (XX ∗ ) showing (2.9) as desired.
Proposition 2.12. Let ρ = ⊕k ρk be a direct sum of endomorphisms of A all localized in a given interval I. Then X −2πK I I ρk (e , ). (e−2πKρ , ) = k
Proof. Let vk be a family of isometries of A(I) such that ρ = X I I vk e−2πKρk vk∗ , e−2πKρ =
P k
vk ρk (·)vk∗ . Then
k
hence by Proposition 2.10 we have X −2πK I X I ρk ∗ (e vk , vk∗ ) = ψρk (vk∗ vk ) (e−2πKρ , ) = k
=
X k
ψρk (1) =
X
k
d(ρk ).
k
Proposition 2.13. Let ρ be as in Proposition 2.10. Then ρI (A(I))0 ∩ A(I) is equal to the commutant {∪I0 ρI0 (A(I0 ))}0 of the representation ρ. Proof. The proof is based on the arguments given in the proof of [23, Theorem 2.3] that concerned the case ρ had a priori finite index. In that context the proof relied on the construction in [23, Corollary 2.5] of a locally normal faithful positive linear functional invariant under dilations and translations in the representation ρ. Proposion 2.11 provides us with such a functional ψ˜ ρ in our setting, therefore, with obvious modifications, the rest of the proof of [23, Theorem 2.3] is valid here. Lemma 2.14. Let ρ be an endomorphism of A localized in the interval I. If there exists a normal faithful conditional E expectation of A(I) onto ρ(A(I)), then ρI is a (possibly infinite) direct sum of irreducible endomorphisms of A(I) with finite index. Proof. By conformal invariance we may assume that I is the upper semi-circle. We now use the formula ρ¯ = j · ρ · j for the conjugate sector ρ, ¯ where j = AdJ with J the modular conjugation associated with the right semi-circle. Due to its geometrical meaning, j is an anti-automorphism of A(I), so that j · E · j is a normal faithful expectation onto ρ(A(I)). ¯ Now the inclusion ρ(A(I)) ¯ ⊂ A(I) is dual to the inclusion ρ(A(I)) ⊂ A(I), hence the lemma follows by Proposition 2.6. Theorem 2.15. Let ρ be an endomorphism of A localized in the interval I. Then I (e−2πKρ , ) < +∞ if and only if ρ has finite index. Therefore the equality d(ρ) = I (e−2πKρ , ) holds regardless if d(ρ) is finite or infinite.
468
R. Longo I
Proof. We only have to show that if (e−2πKρ , ) < +∞ then ρ has finite index. Now in this case Proposition 2.10 gives us a faithful positive normal linear functional on A(I) whose modular group leaves ρ(A(I)) globally invariant by Proposition 1.1. By the Takesaki theorem [49] we have a normal faithful expectation onto ρ(A(I)), whence by Proposition 2.12 and Lemma 2.14 ρ is a direct sum of irreducible finite index sectors I ρk . As for each ρk the formula (e−2πKρk , ) = d(ρk ) holds true by Corollary 2.5, the results follows by the additivity expressed in Proposition 2.12. Before concluding the section, we mention further applications of the above methods to the analysis of superselection sectors with infinite index [22, Sect. 11], in particular regarding the positivity of the energy in these representations. This matter will be discussed in [55].
3. Hawking Temperature in a Charged State and Conditional Entropy 3.1. General setting and a first expression. Following the discussion made in the introduction, we consider a Quantum Field Theory on the Minkowski space R4 , identify the Rindler space-time W with a wedge region in R4 , and look at W in analogy with the Schwarzschild space-time. For convenience we fix the Lorentz frame so that W = {x ∈ R4 , x1 > |x0 |}, and denote by 3W (t) the corresponding one-parameter group of pure Lorentz transformations in the x1 -direction: ch(t) sh(t) 0 0 sh(t) ch(t) 0 0 3W (t) = 0 0 1 0 0 0 01 Let A(O) be the von Neumann algebra on the Hilbert space H of the observables localized within the region O of the Minkowski space. Let U denote the unitary covariant, positive energy, representation of the Poincar´e group P˜ + on H and the vacuum vector. We assume the local algebras to be generated by a Wightman field [47], in order to have the Bisognano–Wichmann theorem that identifies the Tomita–Takesaki modular operator 1 and the modular conjugation J associated with (A(W ), ): 1it W = U (3W (−2πt)), t ∈ R, and J is the PCT anti-unitary composed with the unitary implementation of the change of sign of the x2 , x3 -coordinates. Therefore U (3W (t)) implements a one-parameter automorphism group αt of A(W ) that satisfies the Kubo–Martin–Schwinger equilibrium condition at inverse temperature β = 2π with respect to the restriction of the vacuum state ω = (., ) to A(W )4 ; in other words, by restriction to A(W ), the pure ground state ω becomes faithful (by the Reeh–Schlieder theorem) and thermal for the geodesic evolution on the Rindler space provided boost transformations. 4 One may start with a modular covariance condition and encode the space-time symmetries intrinsically into the net structure [8].
Kac–Wakimoto Formula and Black Hole Conditional Entropy
469
As already explained, there is a relation of this setting with the Hawking and the Unruh effects, first noted by Sewell [44]. The space-time W can be identified with the outside region of a black hole. Then the observable algebra for the background system of the black hole is A(W ), the corresponding proper Hamiltonian is H = aK = −i
d U (3W (at))|t=0 , dt
where a is the surface gravity of the black hole, and the dynamics in the Heisenberg picture is given by αat (X) = e−iHt XeiHt , X ∈ A(W ). Accordingly ω|A(W ) is a KMS state (i.e. Gibbs state at infinite volume [25]) at inverse a . We refer to [24, 46] for more details and further literature. Hawking temperature β = 2π We shall consider the black hole as a heat reservoir for its background system and treat the latter as an open system. Because of the particle production due to the Hawking effect, the background system undergoes a change in its quantum numbers, namely the system goes in different superselection sectors, and we shall consider the thermal equilibrium charged state corresponding to a given sector. We thus consider a superselection sector, namely the unitary equivalence class of a representation ρ of the quasi-local C∗ -algebra A, the norm closure ∪A(O)− of the union of all local algebras associated to bounded regions O. The representation ρ is assumed to be localizable in each space-like cone S 5 , namely ρ and the identity (vacuum) representation have unitarily equivalent restrictions to the C ∗ -algebra ∪{A(O), O ⊂ S 0 , O bounded}− generated by the local observables in S 0 . We may then assume S ⊂ W and, by identifying the representation Hilbert spaces, ρ to act as the identity on A(O) if O ⊂ S 0 , namely ρ is a DHR localized endomorphism [17]. By wedge duality (consequence of the Bisognano–Wichmann theorem) A(W 0 ) = A(W )0 , therefore ρ restricts to a normal endomorphism of A(W ), still denoted by ρ (more precisely ρ restricts to A(W ) ∩ A and has a normal extension to A(W )). We assume that ρ is irreducible and Poincar´e covariant with positive energymomentum, namely there exists a unitary representation Uρ of universal covering group e+ of the the Poincar´e group such that P Uρ (g)ρ(X)Uρ (g)∗ = ρ(U (g)XU (g)∗ ),
e+ . X ∈ A, g ∈ P
(3.1)
The covariance is automatic under general conditions [22]. As shown in [17], the notion statistics is intrinsically associated with ρ, in particular the statistical dimension d(ρ) is defined and turns out to be a positive integer or +∞. We shall assume d(ρ) < ∞. By the index-statistics theorem [36] Ind(ρ) = d(ρ)2 , where Ind(ρ) is the minimal index of ρ|A(W ) , so we may equivalently assume that the restriction of ρ to A(W ) has finite index. 5 This class exahusts all the translation covariant, positive energy representations with an isolated mass shell [11], but possibly not charges with long range interactions.
470
R. Longo
The representation Uρ giving the covariance is uniquely defined by formula (3.1): since ρ is irreducible any other representation would differ from Uρ by a one-dimensional character of P˜ + , that has to be trivial because P˜ + has no non-trivial finite-dimensional unitary representation. Now z(ρ, g) = Uρ (g)U (g)∗ is a AdU (g)-cocycle and is localized, in particular z(ρ, g) belongs to A(W ) if also ρ · AdU (g) is localized in W . In particular z(ρ, 3W (t)) is a αt -cocycle localized in W . As a consequence we have: Lemma 3.1. αtρ (X) = Uρ (3W (t))XUρ (3W (−t)), X ∈ A(W ) defines a one-parameter automorphism group αtρ of A(W ). Proof. We have αtρ (A(W )) = z(ρ, 3W (t))αt (A(W ))z(ρ, 3W (t))∗ = z(ρ, 3W (t))A(W )z(ρ, 3W (t))∗ = A(W ) because z(ρ, 3W (t)) belongs to A(W ).
ρ The one-parameter automorphism group αat is the dynamics of our system in the sector ρ and the corresponding proper Hamiltonian is given by
Hρ = aKρ = −i
d Uρ (3W (at))| t=0 . dt
Theorem 2.1, or equivalently formula (2.4), has its version here, by an analogous proof, d(ρ) = (e−βHρ , )|β= 2π = (e−2πKρ , ), a
(3.2)
where is the vacuum vector or any other cyclic vector for A(W ) such that ( ·, ) coincides with the vacuum state ω on A(W 0 ). As in Proposition 2.8 we have a normal faithful state of A(W ) given by ϕρ (XX ∗ ) = d(ρ)−1 ke−πKρ Xk2 , X ∈ A(W ).
(3.3)
Lemma 3.2. If ρ is irreducible then the one-parameter automorphism group αtρ is ergodic on A(W ), namely its fixed points are the scalars. Proof. The proof is similar to the one given in [23] in the case of a conformal theory. Details will be given somewhere else. Next we show that the system in the sector ρ admits a thermal equilibrium state at the same Hawking inverse temperature β = 2π/a. ρ admits a unique normal KMS state ϕρ at inverse temperature β = Theorem 3.3. αat The state ϕρ is given by Eq. (3.3) or equivalently by
ϕρ = ω8ρ ,
2π a .
(3.4)
where 8ρ is the minimal left inverse of ρ on A(W ) and, if ρ is irreducible, ϕρ is the ρ unique normal αρ -invariant state of A(W ). If β 6= 2π a , no αat -KMS normal state exists.
Kac–Wakimoto Formula and Black Hole Conditional Entropy
471
Proof. The situation is similar to the one discussed in the previous section. Again, relying on formula (3.2) and Proposition 1.7, we see that the Connes cocycle (Dϕρ : Dω|A(W ) )t , where ϕρ is defined by Eq. (3.4), is equal to d(ρ)−it z(ρ, 3W (−2πt)), thus the modular group of ϕρ is given by ϕ ρ σt ρ = α−2πt , ρ i.e. ϕρ is αat -KMS at β = 2π/a. The non-existence of normal states at different temperatures is an immediate consequence of the outerness of σ ϕρ (cf. [12]), since A(W ) is a type III1 factor (see [35]). The uniqueness of ϕρ as a normal αρ -invariant state is equivalent to the ergodicity of αρ , hence a consequence of Lemma 3.2.
Now a finite volume consideration (see Appendix B) suggests to regard (e−βHρ , ) as the ratio of the (here undefined) partition functions Z0 (β) of the state ω and Zρ (β) of the state ϕρ , namely log(e−βHρ , ) = log Z0 (β) − log Zρ (β),
(3.5)
whence we expect the quantity F (ω|ϕρ ) = −β −1 log(e−βHρ , ) to represent the increment of the free energy between ω and ϕρ . We shall see the above formula to hold true in a precise sense. We define the relative free energy F (ω|ϕρ ) between the states ω and ϕρ by F (ω|ϕρ ) = ϕρ (Hρ ) − β −1 S(ω|ϕρ ),
(3.6)
where S(ω|ϕρ ) = S(ω|A(W ) |ϕρ ) is the Araki relative entropy of the two states on A(W ), S(ω|ϕρ ) = −(log 1,ξρ ξρ , ξρ );
(3.7)
here 1,ξρ is Araki’s relative modular operator of A(W ) associated with the two cyclic separating vectors and ξρ , where ξρ is any cyclic vector such that ϕρ = (· ξρ , ξρ ) on A(W ). In particular we may assume ξρ to belong to the natural cone associated with (A(W ), ). The quantity ϕρ (Hρ ) = (Hρ ξρ , ξρ ) in (3.6) represents the relative mean energy in between ω and ϕρ , indeed according to formula (3.5) this has to be given, anticipating Proposition 3.5, by 1/2
1/2
(e−βHρ Hρ , ) d(ρ)(Hρ 1,ξρ , 1,ξρ ) d log(e−βHρ , ) = = − dβ (e−βHρ , ) (e−βHρ , ) 1/2
1/2
= (Hρ J1,ξρ , J1,ξρ ) = (Hρ ξρ , ξρ ) = ϕρ (Hρ ), where J is the modular conjugation of both and ξρ and we have set β = 2π/a and applied formula (3.2). More directly one may define the relative mean energy by the formal expression ϕ˜ ρ (Hρ ) = ϕ˜ ρ (Hρ − H), where Hρ − H is the relative Hamiltonian, ϕ˜ ρ = (·ξρ , ξρ ) and one sets ϕ˜ ρ (H) = 0 motivated by the fact that (e−iHt ξρ , ξρ is a real even function (because 1it preserves the natural cone) picking a maximum at t = 0.
472
R. Longo
The relative entropy S(ω|ϕρ ) is always non-negative, but it may be equal to +∞, as no volume renormalization has been made (cf. Appendix B); also the relative mean energy ϕρ (Hρ ) may be infinite, but we shall show that the relative free energy between ϕρ and ω is finite, so in particular ϕρ (Hρ ) shall be bounded below. Formula (3.6) will have the obvious rigorous meaning as F (ω|ϕρ ) = (Hρ + β −1 log 1,ξρ ξρ , ξρ ). Theorem 3.4. The relative free energy between the thermal equilibrium states ω and ϕρ is proportional to the Connes-Størmer entropy of the sector ρ: 1 F (ω|ϕρ ) = − β −1 Sc (ρ). 2 Here Sc (ρ) denotes the conditional entropy H(A(W )|ρ(A(W )) (see (1.17), (1.18)) n X S(ϕρ |ϕi ) − S(ϕρ · ρ|ϕi · ρ)}, Sc (ρ) = sup{ (ϕi )
i=1
of finitely many normal positive linear functionals where (ϕi ) varies among the setsP n ϕ1 , ϕ2 , . . . ϕn of A(W ) such that i=1 ϕi = ϕρ . We note the extensive property of Sc : Sc (ρ1 ρ2 ) = Sc (ρ1 ) + Sc (ρ2 ) [38]. Proposition 3.5. We have 1 βHρ = 2πKρ = − log 1,ξρ − Sc (ρ). 2
(3.8)
Proof. By Araki’s formula −it (Dϕρ : Dω|A(W ) )t = 1it ,ξρ 1 ,
therefore
−it uW (ρ, t) = (Dd(ρ)ϕρ : Dω|A(W ) )t = d(ρ)it 1it ,ξρ 1 .
On the other hand by our formula uW (ρ, t) = z(ρ, 3W (−2πt)) = e−i2πKρ t ei2πKt , therefore
−it e−i2πKρ t ei2πKt = d(ρ)it 1it ,ξρ 1 ,
−iKt and as 1it we see that = U (3W (−2πt)) = e
e−i2πtKρ = d(ρ)it 1it ,ξρ so the proposition is obtained by differentiating this expression at t = 0. An alternative argument will appear in Lemma 3.10. Proof of Theorem 3.4.
By evaluating on ϕ˜ ρ both sides of formula (3.8) we have
β(Hρ ξρ , ξρ ) = −(log 1,ξρ ξρ , ξρ ) − log d(ρ). On the other hand d(ρ) is the square root of the minimal index of ρ|A(W ) [36], thus by the Pimsner–Popa equality (1.18) it follows that log d(ρ) = 21 Sc (ρ), hence proving the theorem.
Kac–Wakimoto Formula and Black Hole Conditional Entropy
473
Corollary 3.6. The possible values of the relative free energy with initial state ω are 1 F (ω|ϕρ ) = − β −1 log(n), n = 1, 2, 3, . . . . 2 Proof. Immediate by the DHR theorem [17] to the effect that the statistical dimension is a positive integer or +∞. Therefore the integer n, expressing the order of the parastatistics in [17], here appears as a quantum number labeling the relative free energy levels. In low space-time dimensions the quantization of the conditional entropy is less restrictive. By Jones theorem [34] and the results in [38,42] we have however restrictions for the possible values of the relative free energy associated with a planar black hole: Corollary 3.7. In low dimensions the possible values of e−βF (ω|ϕρ ) are restricted to 4 cos2 ( nπ ) in the interval (0, 4). No value in (4, 5) is possible. In the interval (5, 6) only 3 values are possible. Proof. The first assertion follows from [34], because e−βF (ω|ϕρ ) is an index. The rest of the statement is a consequence of the further restrictions on the index values due to the occurrence of the braid group symmetry [38,42]. 3.2. The increment of the free energy between arbitrary thermal equilibrium states. Let σ be another endomorphism localized in W and ξσ the cyclic separating vector for A(W ) such that (Xξσ , ξσ ) = ϕσ (X), X ∈ A(W ) lying in the natural cone associated with (A(W ), ). To extend the definition (3.6) for the free relative energy to the case the initial state is an arbitrary thermal state ϕσ , we note first that the formal relative Hamiltonian between ϕσ and ϕρ is Hρ − Hσ and hence the relative mean internal energy should be formally given by ϕ˜ ρ (Hρ − Hσ ) = ϕ˜ ρ (Hρ − Hσ − H) = ϕ˜ ρ (Hρ + Hσ¯ − H). Here the conjugate charge given by σ¯ = j · σ · j (see [22]) is localized in W 0 and Hσ¯ = JHσ J = −Hσ . These premises and the following lemma will motivate definition (3.9) below. Lemma 3.8. We have
eitHρσ¯ = eitHρ e−itH eitHσ¯ .
Proof. By the cocycle property ¯ 3W (t)))z(ρ, 3W (t)) = z(σ, ¯ 3W (t))z(ρ, 3W (t)) z(ρσ, ¯ 3W (t)) = ρ(z(σ, because z(σ, ¯ 3W (t)) is localized in W 0 and ρ acts identically on A(W 0 ) so the lemma is obtained by multiplying on the right by e−itK the above expression. We thus define the relative free energy between ϕσ and ϕρ by F (ϕσ |ϕρ ) = ϕ˜ ρ (Hρσ¯ ) − β −1 S(ϕσ |ϕρ ),
(3.9)
and we give to this expression a rigorous meaning as is done for expression (3.6). Alternatively, extending the considerations in the previous subsection, we could interpret directly log(e−βHρσ¯ ξσ , ξσ ) as the increment of the logarithm of the partition function between the states ϕσ and ϕρ , leading to the expression F (ϕσ |ϕρ ) = −β −1 log(e−βHρσ¯ ξσ , ξσ ).
474
R. Longo
Theorem 3.9. The relative free energy is given by F (ϕσ |ϕρ ) =
1 −1 β (Sc (σ) − Sc (ρ)). 2
As a consequence e−βF (ϕσ |ϕρ ) is equal to the rational number
d(σ) d(ρ) .
Lemma 3.10. Let ρ be as above, ρ0 an endomorphism localized in W 0 and ϕρ0 = ω · 8ρ0 |A(W 0 ) , where 8ρ0 is minimal left inverse of ρ0 . Then 1(ϕρ0 |ϕρ ) =
d(ρ) −2πKρρ0 e , d(ρ0 )
where 1(·|·) denotes the Connes spatial derivative. Proof. Setting ω 0 = ω|A(W 0 ) and using [13] one has 1(ϕρ0 |ϕρ )it = (Dϕρ0 : Dω 0 )t 1(ω 0 |ϕρ )it = d(ρ)it (Dϕρ0 : Dω 0 )t e−i2πtKρ = d(ρ)it d(ρ0 )−it e−i2πtKρ0 ei2πtK e−i2πtKρ = d(ρ)it d(ρ0 )−it z(ρ0 , 3W (−2πt))z(ρ, 3W (−2πt))e−i2πtK = d(ρ)it d(ρ0 )−it ρ(z(ρ0 , 3W (−2πt)))z(ρ, 3W (−2πt))e−i2πtK = d(ρ)it d(ρ0 )−it z(ρρ0 , 3W (−2πt))e−i2πtK = d(ρ)it d(ρ0 )−it e−i2πtKρρ0 . where we have used Proposition 2.5 in our context.
Corollary 3.11. We have 1ξσ ,ξρ =
d(ρ) −2πKρσ¯ e . d(σ)
Proof. Immediate by the above discussion and the relation 1ξσ ,ξρ = 1(ϕσ · AdJ|ϕρ ), where ϕσ · AdJ is the vector state (·ξσ , ξσ ) on A(W 0 ). Proof of Theorem 3.9.
By the above Corollary we have
1 βHρσ¯ = βaKρσ¯ = − log 1ξσ ,ξρ + (Sc (σ) − Sc (ρ)), 2 and this clearly implies the desired relation.
Appendix A. Tensor Categories and Cocycles The purpose of this appendix is to shed light on part of the mathematical structure underlying our results. Indeed a good part of our results depends only on the tensor categorical structure provided by the superselection sectors and are therefore visible without a more detailed description of the theory. Let T be a strict C ∗ -tensor category. We assume (ι, ι) = C, where ι is the identity object and (·, ·) denotes the intertwiner space. We refer to [40] for the basic notions used here.
Kac–Wakimoto Formula and Black Hole Conditional Entropy
475
A basic and originating example for T , appearing in [17, 18], is obtained by taking End(M ), M a unital C ∗ -algebra with trivial centre, to be the set of objects, and arrows between objects ρ and ρ0 given by (ρ, ρ0 ) = {T ∈ M, T ρ(x) = ρ0 (x)T, ∀x ∈ M },
(A.0)
while the tensor product is given by the composition of maps ρ ⊗ ρ0 = ρ · ρ0 , T ⊗ S = ρ02 (T )S = Sρ01 (T ), T ∈ (ρ1 , ρ2 ), S ∈ (ρ01 , ρ02 ). The reader unfamiliar with abstract tensor categories might at first focus on this particular case. Given an object ρ of T , an object ρ¯ of T is said to be a conjugate of ρ if there exist ¯ and R¯ ρ ∈ (ι, ρρ) ¯ such that Rρ ∈ (ι, ρρ) ∗ R¯ ρ ⊗ 1ρ ◦ 1ρ ⊗ Rρ = 1ρ ; Rρ∗ ⊗ 1ρ¯ ◦ 1ρ¯ ⊗ R¯ ρ = 1ρ¯ . (A.1) We shall assume that each object ρ has a conjugate ρ¯ (this is automatic in End(M ) [37] if M is an infinite factor and ρ has finite index) and shall refer to (A.1) as the conjugate equation for ρ and ρ. ¯ Equation (A.1) has then a standard solution, namely √ one can choose multiples of isometries Rρ and R¯ ρ in (A.1) so that kRρ k = kR¯ ρ k = d(ρ) is minimal. This formula defines d(ρ), the dimension of ρ [40]. ¯ ρ¯0 ) is Now recall that given an arrow T ∈ (ρ, ρ0 ), the conjugate arrow T • ∈ (ρ, defined by ¯ ρ¯0 ), T • = 1ρ¯ 0 ⊗ R¯ ρ∗ ◦ 1ρ¯ 0 ⊗ T ∗ ⊗ 1ρ¯ ◦ Rρ0 ⊗ 1ρ¯ ∈ (ρ, where R¯ ρ and Rρ0 are multiples of isometries in the standard solution for the conjugate equations defining the conjugates ρ¯ and ρ¯0 . The mapping T 7→ T • is antilinear and enjoys in particular the following properties: a) 1•ρ = 1ρ¯ , b) S • ◦ T • = (S ◦ T )• , c) T •∗ = T ∗• . We shall say that α is an (anti-)automorphism of T if α is an invertible functor of T with itself, (anti-)linear on the arrows, commuting with the ∗ -operation and preserving tensor products. The action of α on the object ρ and on the arrow T will be denoted by ρα and T α . Given an automorphism α of T a cocycle u with respect to α is a map ρ ∈ Obj(T ) → u(ρ) unitaries in (ρ, ρα ) such that a) u(ρ ⊗ ρ0 ) = u(ρ) ⊗ u(ρ0 ) . b) If T ∈ (ρ, ρ0 ), then the following diagram commutes: ρ u(ρ)
T
?
ρα
-
Tα
ρ0 u(ρ0 ) ?
ρ0α
(A.2)
476
R. Longo
Proposition A.1. Let u be a cocycle with respect to α as above. Then u(ρ) ¯ = u(ρ)• .
Proof. The proof is obtained similarly as in Proposition 1.5.
Lemma A.2. Let u be a cocycle with respect to α as above and j be an antiautomorphism of T . Then ρ → u(ρj )j is a cocycle with respect to jαj −1 . Proof. The statement is checked by a direct verification.
We now give two uniqueness results that are at the basis of the identifications of the covariance cocycle and the Connes cocycle in this paper. Lemma A.3. With the notations in Lemma A.2, assume that jαj −1 = α−1 and ρj is a conjugate of ρ for all objects ρ. Then the cocycle u with respect to α is unique. Proof. If u0 is a cocycle with respect to α and ρ is irreducible, then u0 (ρ) = µ(ρ)u(ρ) for ¯ = µ(ρ), some phase µ(ρ). By Proposition A.1 µ(ρ) ¯ = µ(ρ), while by Lemma A.2 µ(ρ) hence µ = 1 on the irreducibles, thus always because a cocycle is determined by its value on the irreducible objects. Let now G be a group and α an action of G on T , namely a homomorphism g → αg of G into the automorphism group of T (for simplicity we omit topological assumptions). For any ρ ∈ T and g ∈ G, let u(ρ, g) be a unitary in (ρg , ρ) (where ρg ≡ ραg ). We shall say that u is a two-variable cocycle if: a) For any fixed g ∈ G, u(·, g)∗ is a cocycle with respect to the automorphism αg . b) For any fixed ρ ∈ T , u(ρ, ·) is a α-cocycle, namely u(ρ, gh) = u(ρ, g)u(ρ, h)αg . Proposition A.4. Let u be a two-variable cocycle as above. If G is perfect (i.e. G has no non-trivial one-dimensional unitary representation), then u is unique. Proof. As in the proof of Lemma A.3, if ρ is an irreducible object, a second two-variable cocycle would give rise to a phase µ(ρ, g) that, for a fixed ρ, would be a one-dimensional character of G, and thus had to be trivial. Appendix B. The Relative Free Energy at Finite Volume A finite volume computation with canonical distribution may clarify the notion of relative free energy F in (3.6). Let the Hamiltonians of the evolutions α(0) and α(1) be given by positive selfadjoint operators H0 and H1 , so that α(k) is implemented by eitHk . The Gibbs state ωβ(k) for α(k) is given by ωβ(k) = Tr(ρk ·) with density matrix
ρk = Zk (β)−1 e−βHk ,
where Zk (β) = Tr(e−βHk ) is the partition function. Then Fk = ωk (Hk ) − β −1 S(ρk ) = −β −1 log Zk (β) is the Helmholtz free energy in the state ω (k) , where the entropy in state ω (k) is given by
Kac–Wakimoto Formula and Black Hole Conditional Entropy
477
S(ρk ) = −Tr(ρk log ρk ). Then the relative entropy is given by (see [50]) S(ωβ(0) |ωβ(1) ) = −Tr(ρ0 log ρ0 − ρ0 log ρ1 ) = −ωβ(0) (log ρ0 − log ρ1 ) = βωβ(0) (Hrel ) + log Z0 (β) − log Z1 (β), where Hrel = H0 − H1 is the relative Hamiltonian. The relative free energy is thus given by F (ωβ(0) |ωβ(1) ) = ωβ(0) (Hrel ) − β −1 S(ωβ(0) |ωβ(1) ) = β −1 log Z0 (β) − β −1 log Z1 (β) = F1 − F0 . Had we considered a grand canonical distribution on the Fock space, the Hamiltonian for αt(k) would have been implemented by eit(Hk −µk Nk ) , with µk the chemical potential and Nk the number operator, and the above expression for Fk would have been accordingly modified. We note explicitly that (0)
(1)
e−βF (ωβ |ωβ ) =
Tr(e−βH1 ) , Tr(e−βH0 )
providing evidence to the analogy between formulae (0.1) and (0.2).
Final Comments As mentioned, the Rindler space-time is a good approximation of the Schwarzschild space-time only near the horizon. However a version of our results within the context of the Kruskal extension of the Schwarzschild space-time should be possible, as a version of the Bisognano–Wichmann theorem and a model independent derivation of the Hawking temperature has been given in this setting [44]. Another point to comment on is related to the use of the Minkowski vacuum associated with Poincar´e symmetries. As is known there exists no vacuum state for a quantum field theory on a general curved space-time. In such a general theory the relative free energy could however be defined by consistency with the fusion rules of the superselection structure and it seems that our results may be achieved in wider contexts. Concerning the expression (3.6) for the relative free energy, it would be physically meaningful to derive it by a finite volume approximation, where its expression is given in Appendix B. To this end one should use the split property and the Noether currents, see [10], and this approach might also be useful for the above-discussed extension to more general curved spacetimes. Moreover the development of such techniques could bring up a model independent derivation of the formula (0.1). Acknowledgement. It is a pleasure to thank Claudio D’Antoni, Bernard Kay, John E. Roberts and Rainer Verch for various conversations.
478
R. Longo
References 1. Araki, H.: Relative Hamiltonians for faithful normal states of a von Neumann algebra. Pub. R.I.M.S., Kyoto Univ. 9, 165–209 (1973) 2. Araki, H., Haag, R., Kastler, D., Takesaki, M.: Extensions of KMS states and chemical potential. Commun. Math. Phys. 53, 97–134 (1977) 3. Bardeeen, J.M., Carter, B., Hawking, S.W.: The four laws of black holes mechanics. Commun. Math. Phys. 31, 161 (1973) 4. Bekenstein, J.D.: Black holes and entropy. Phys. Rev. D7, 2333 (1973) 5. Bisognano, J., Wichmann, E.: On the duality condition for a Hermitian scalar field. J. Math. Phys. 16, 985–1007 (1975) and J. Math. Phys. 17, 303–321 (1976) 6. Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics, II. BerlinHeidelberg-New York: Springer-Verlag, 1981 7. Brunetti, R., Guido, D., Longo, R.: Modular structure and duality in conformal quantum field theory. Commun. Math. Phys. 156, 201–219 (1993) 8. Brunetti, R., Guido, D., Longo, R.: Group cohomology, modular theory and spacetime symmetries. Rev. Math. Phys. 7, 57–71 (1995) 9. Buchholz, D., D’Antoni, C., Longo, R.: Nuclear maps and modular structures. I. J. Funct. Anal. 88, 223–250 (1990), II Commun. Math. Phys. 129, 115–13 (1990) 10. Buchholz, D., Doplicher, S., Longo, R.: On Noether’s theorem in quantum field theory. Ann. Phys. 170, 1–17 (1986) 11. Buchholz, D., Fredenhagen, K.: Locality and structure of particle states. Commun. Math. Phys. 84, 1–54 (1982) 12. Connes, A.: Une classification des facteurs de type III. Ann. Sci. Ec. Norm. Sup. 6, 133–252 (1973) 13. Connes, A.: On a spatial theory of von Neumann algebras. J. Funct. An. 35, 153–164 (1980) 14. Connes, A.: Entropie de Kolmogoroff-Sinai et m´ecanique statistique quantique. C. R. Acad. Sci. Paris S´er. I 301, 1–6 (1985) 15. Connes, A.: Størmer, E. Entropy for automorphisms of II1 von Neumann algebras. Acta Math. 134, 288–306 (1975) 16. Davies, P.C.W.: Scalar particle production in Schwarzschild and Rindler metrics. J. Phys. A8, 608 (1975) 17. Doplicher, S., Haag, R., Roberts, J.E.: Local observables and particle statistics I. Commun. Math. Phys. 23, 199–230 (1971), II. Commun. Math. Phys. 35, 49–85 (1974) 18. Doplicher, S., Roberts, J.E.: Endomorphisms of C∗ -algebras, crossed products and duality for compact groups. Ann. Math. 170, 75 (1989) 19. Evans, D., Kawahigashi, K.: Quantum Symmetries on Operator Algebras. In press 20. Fredenhagen, K., J¨orß, M.: Conformal Haag-Kastler nets, pointlike localized fields and the existence of operator product expansion. Commun. Math Phys. 176, 541 (1996) 21. Fr¨ohlich, J., Gabbiani, F.: Operator algebras and conformal field theory. Commun. Math. Phys. 155, 569–640 (1993) 22. Guido, D., Longo, R.: Relativistic invariance and charge conjugation in quantum field theory. Commun. Math. Phys. 148, 521–551 (1992) 23. Guido, D., Longo, R.: The conformal spin and statistics theorem. Commun. Math. 181, 11–35 (1996) 24. Haag, R.: Local Quantum Physics. Heidelberg-Berlin: Springer Verlag, 1992 25. Haag, R., Hugenoltz, N.M., Winnink, M.: On the equilibrium states in quantum statistical mechanics. Commun. Math. Phys. 5, 215 (1967) 26. Haag, R., Kastler, D.: An algebraic approach to Quantum Field Theory. J. Math. Phys. 5, 848–861 (1964) 27. Hawking, S.W.: Particle creation by black holes. Commun. Math. Phys. 43, 199 (1975) 28. Hiai, F.: Minimum index for subfactors and entropy. II. J. Math. Soc. Japan 43, 673–678 (1991) 29. Hislop, P.D., Longo, R.: Modular structure of the von Neumann algebras associated with the free scalar massless field theory. Commun. Math. Phys. 84, 71–85 (1982) 30. Kac, V.G., Wakimoto, M.: Modular and conformal invariance constraints in representation theory of affine algebras. Adv. in Math. 70, 156–236 (1988) 31. Kay, B., Wald, R.: Some recent developments related to the Hawking effect. In: Proc. Int. Conf. Diff. Geom. Meth. in Theor. Phys., Doebner and Hennin (eds.) Singapore: World Scientific, 1987 32. Kosaki, H.: Extension of Jones’ theory on index to arbitrary subfactors. J. Funct. Anal. 66, 123–140 (1986), 123–140 33. Izumi, M.: Canonical extension of endomorphisms of factors. In: “Subfactors”, Proc. Taniguchi Symposium, Araki, Kawahigashi, Kosaki (eds.), Singapore: World Scientific, 1994
Kac–Wakimoto Formula and Black Hole Conditional Entropy
479
34. Jones, V.R.F.: Index for subfactors. Invent. Math. 72, 1–25 (1983) 35. Longo, R.: Algebraic and modular structure of von Neumann algebras of Physics. Proc. Symp. Pure Math. 38, Part 2, 551 (1982) 36. Longo, R.: Index of subfactors and statistics of quantum fields. I . Commun. Math. Phys. 126, 217–247 (1989) 37. Longo, R.: Index of subfactors and statistics of quantum fields. II. Correspondences, braid group statistics and Jones polynomial. Commun. Math. Phys. 130, 285–309 (1990) 38. Longo, R.: Minimal index and braided subfactors. J. Funct. Anal. 109, 98–112 (1992) 39. Longo, R.: Von Neumann algebras and quantum field theory. Proceedings of the ICM, Z¨urich 1994, Basel, Switzerland: Birkh¨auser Verlag, 1995 40. Longo, R., Roberts, J.E.: A theory of dimension. K-Theory 11, 103–159 (1997) 41. Pimsner, M., Popa, M.: Entropy and index for subfactors. Ann. Sci. Ec. Norm. Sup. 19, 57–106 (1986) 42. Rehren, K.H.: On the range of the index of subfactors. J. Funct. Anal. 134, 183–193 (1995) 43. Schroer, B.: Some useful properties of rotational Gibbs states in chiral conformal QFT. Manuscript, 1995 44. Sewell, G.L.: Quantum fields on manifolds: PCT and gravitationally induced thermal states. Ann. Phys. 141, 201 (1982) 45. Sewell, G.L.: On the generalised second low of thermodynamics. Phys. Lett. 122A, 309–311 (1987) 46. Summers, S., Verch, R.: Modular inclusion, the Hawking temperature and Quantum Field Theory in curved space-time. Lett. Math. Phys. 37, 145–158 (1996) 47. Streater, R.F., Wightman, A.S.: PCT, Spin and Statistics, and all that. Reading, MA: Benjamin, 1964 48. Takesaki, M.: Tomita theory of modular Hilbert algebras. Lect. Notes in Math. 128, New York-HeidelbergBerlin: Springer Verlag, 1970 49. Takesaki, M.: Conditional expectation in von Neumann algebras. J. Funct. Anal. 9, 306–321 (1972) 50. Thirring, W.: A Course in Mathematical Physics. Vol. 4, New York-Heidelberg-Berlin: Springer Verlag, 1981 51. Unruh, W.G.: Notes on black hole evaporation. Phys. Rev. D14, 870–892 (1976) 52. Wald, R.M.: Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. Chicago: Univ. Chicago Press, 1994 53. Wick, G.C., Wightman, A.S., Wigner, E.P.: The intrinsic parity of elementary particles. Phys. Rev. 88, 101–105 (1952), 101-105 54. Wiesbrock, H.W.: Superselection structure and localized cocycles. Rev. Math. Phys. 7, 127 (1995) 55. Bertozzini, P., Conti, R., Longo, R.: Covariant sectors with infinite dimension and positivity of the energy. - preprint. Communicated by A. Connes
Commun. Math. Phys. 186, 481 – 493 (1997)
Communications in
Mathematical Physics c Springer-Verlag 1997
Some Schr¨odinger Operators with Power-Decaying Potentials and Pure Point Spectrum Christian Remling? Universit¨at Osnabr¨uck, Fachbereich Mathematik/Informatik, D-49069 Osnabr¨uck, Germany. E-mail: [email protected] Received: 8 November 1996 / Accepted: 8 January 1997
Abstract: We construct (deterministic) potentials V (x) = O(x−c ) such that the Schr¨odinger equation −y 00 + V y = Ey on x ∈ [0, ∞) has dense pure point spectrum in (0, ∞) for almost all boundary conditions at x = 0. As a by-product, we also obtain power-decaying potentials for which the spectrum is purely singular continuous on (0, ∞) for all boundary conditions. 1. Introduction In this paper, we will construct power-decaying potentials V (x) = O(x−c ) (c > 0) such that the Schr¨odinger equation for positive energies E = k 2 , − y 00 (x) + V (x)y(x) = k 2 y(x)
(1)
on the half-axis x ∈ [0, ∞) has square integrable solutions for almost all k > 0 with respect to Lebesgue measure. By well-known arguments (using spectral averaging, see d2 e.g. [16]), this implies that the corresponding Hamiltonian Hα = − dx 2 + V (x) with boundary condition y(0) cos α + y 0 (0) sin α = 0 has dense pure point spectrum in (0, ∞) (i.e. σess = [0, ∞), σc (Hα ) = ∅) for almost all α ∈ [0, π). As far as I am aware, this is even the first explicit example of a Schr¨odinger operator with this spectral type and V → 0. The existence of such potentials, however, is not a new result: The work on decaying random potentials [8, 14] has shown that, roughly speaking, random potentials decaying as Vω (x) ∼ x−c with c < 1/2 almost surely lead to dense pure point spectrum if the values of the potential at different sites are independent with zero expectation. (In fact, c = 1/2 is generally believed to be the borderline between absolutely continuous and singular spectrum in the sense that V (x) = O(x−c ) with ? Current address (until May 31, 1997): 253-37 Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA
482
C. Remling
c > 1/2 should imply σac = [0, ∞). Kiselev has recently proved this for c > 2/3 [6], but in the region 1/2 < c ≤ 2/3, this question is open.) There is, of course, still much point in looking for deterministic examples, since the proofs in the random case strongly depend on averaging procedures and independence assumptions and therefore do not give much insight into the mechanisms with which a fixed potential generates point spectrum. The first deterministic potentials with pure point spectrum were constructed in [3, 5]. However, these examples rely on tunneling through high barriers and therefore lim sup V (x) = ∞. Later, these methods were extended to handle the case of wide barriers (see [19] and also [5] where results of this type are stated without proof), thereby obtaining potentials with lim sup V < ∞ and pure point spectrum in (lim inf V, lim sup V ) (a typical example is V (x) = cos xγ (0 < γ < 1); see also [17]). Although not directly related to this work, I should also mention the construction of potentials with dense point spectrum and |V (x)| ≤ f (x)/x (f → ∞ arbitrarily slowly) in [10, 15]. However, here one has σac = [0, ∞). As an almost immediate consequence of our construction, we will also obtain powerdecaying potentials with purely singular continuous spectrum in (0, ∞) for all boundary conditions. Apparently, only one other class of decaying potentials with this spectral behavior is known at present, namely the type of sparse potentials introduced by Pearson in his classic [11], and although the conditions on the barrier separations could be considerably weakened recently [7], the decay of these potentials is still slower than (say) (ln x)−1/2 . The main tools we will use in this paper are the (modified) Pr¨ufer transformation and transfer matrices. In some respects, our general strategy will be similar to the proof of [3, Theorem 2]. For other papers discussing some of the techniques used here, see, for instance, [7, 8, 10, 12]. This paper is organized as follows. In Sect. 2, we describe the model and state the result we will prove. We state some preliminary observations and explain the general strategy in Sect. 3. The technical details are presented in the following three sections. We conclude with the modification which yields singular continuous spectrum. I would like to thank A. Gordon, B. Simon and T. Wolff for useful discussions, the Deutsche Forschungsgemeinschaft for financial support, and I am grateful for the hospitality of Caltech where this work was done.
2. Description of the Model Let Ln ∈ N, gn > 0, ln ≥ 0, and let W (x) be a bounded, measurable and periodic function with period p. Define a1 = 0, bn = an + p
Ln , gn
an+1 = bn + ln
(n ≥ 1).
The potential we will study is given by V (x) =
0 Wgn (x − an )
if x ∈ (bn , an+1 ) , if x ∈ (an , bn )
where Wg is the rescaled function Wg (x) = g 2 W (gx).
Schr¨odinger Operators with Power-Decaying Potentials
483
Theorem 2.1. Ln , gn , ln and W (x) can be chosen such that: 1) V (x) = O(x−c ) for some c > 0, 2) For almost all k > 0, (1) has an L2 -solution. It follows from V → 0 that σess = [0, ∞). Moreover, as pointed out above, 2) implies that for almost all boundary conditions α, the corresponding half-line operator Hα has no continuous spectrum, hence it must have dense pure point spectrum in (0, ∞). Note also that by recently discovered general principles on the instability of point spectrum [1], there is a dense Gδ set of boundary conditions for which the spectrum is purely singular continuous in (0, ∞)! 3. Preliminary Observations
Write Y (x, k) :=
y(x, k) y 0 (x, k)/k
,
where y is a solution of (1) satisfying an initial condition of the form y(0, k) = − sin α, y 0 (0, k) = cos α. We will use the modified Pr¨ufer variables R, ϕ defined by the equation sin ϕ(x, k) Y (x, k) = R(x, k) , cos ϕ(x, k) and by requiring that ϕ be continuous and R > 0. The transfer matrix T (t, s; k) is defined by the property that it takes Y (s, k) to Y (t, k), i.e. Y (t, k) = T (t, s; k)Y (s, k). Hence it is given by u(t, k) kv(t, k) , (2) T (t, s; k) = u0 (t, k)/k v 0 (t, k) where u, v are the solutions of (1) with u(s) = v 0 (s) = 1, u0 (s) = v(s) = 0. If the potential is periodic, then obviously T (t + p, s + p; k) = T (t, s; k), (this observation is roughly one half of what is usually called Floquet theory). Abbreviating T (k) := TW (p, 0; k) (the transfer matrix over one period for the potential W ), we get the transfer matrix for the potential V described in the preceding section Lemma 3.1. T (an + m2 pgn−1 , an + m1 pgn−1 ; k) = T (kgn−1 )m2 −m1 ,
mi ∈ {0, 1, . . . , Ln }
Proof. By “Floquet theory”, it suffices to prove the assertion for m1 = 0, m2 = 1. Moreover, T (an + pgn−1 , an ; k) = Tˆ (pgn−1 , 0; k), where Tˆ is the transfer matrix of −y 00 + Wgn (x)y = k 2 y. d2 y For a function y(x) and g > 0, let w(t) := y(t/g). Then we have: y solves − dx 2 + d2 w 2 Wg (x)y = k y and satisfies (y, dy/dx)(0) = (a, b) if and only if w solves − dt2 + W (t)w = (k/g)2 w and satisfies (w, dw/dt)(0) = (a, g −1 b). Hence, if ug , vg denote the solutions of −y 00 + Wg y = k 2 y with ug (0) = vg0 (0) = 1, u0g (0) = vg (0) = 0, then vg (x, k) = g −1 v1 (gx, k/g), vg0 (x, k) = v10 (gx, k/g).
ug (x, k) = u1 (gx, k/g), u0g (x, k) = gu01 (gx, k/g), Now the assertion follows from (2).
484
C. Remling
Next, we recall some basic facts about periodic potentials; for further background information on this topic, please consult [2]. Since det T (k) = 1 by constancy of the Wronskian, the eigenvalues of T are determined by its trace D(k) := tr T (k) = u(p, k) + v 0 (p, k), (u, v solve −y 00 + W y = k 2 y and u(0) = v 0 (0) = 1, u0 (0) = v(0) = 0). More explicitly, the eigenvalues are µ, µ−1 with r D(k) D(k)2 ± − 1. (3) µ(k) = 2 4 We choose the sign so that |µ| ≥ 1. µ is real and larger than 1 in magnitude if and only if |D| > 2; if |D| < 2, then |µ| = 1, µ 6= ±1. In either case, the eigenvalues are distinct and hence T is diagonalizable. Let U (k) be a diagonalizing matrix, i.e. µ(k) 0 −1 . U (k)T (k)U (k) = 0 µ−1 (k) This representation leads to a simple estimate on the norm of the transfer matrix. In the following lemma, we use the l2 -norm k(v1 , v2 )t k2 = |v1 |2 + |v2 |2 . Lemma 3.2. a) If |D(k)| < 2, then kT (k)m vk ≥
kvk . kU (k)k kU −1 (k)k
b) If |D(k)| > 2, then there is an angle β(k) ∈ [0, π) such that |ϕ − β(k) − nπ| ≥ ∀n ∈ Z implies 2 kT (k)m eϕ k ≥ |µ(k)|m π (where eϕ = (sin ϕ, cos ϕ)t ). Proof. Obviously,
T (k)m = U (k)
µ(k)m 0
0 µ(k)−m
U −1 (k).
(4)
Hence kT −m wk ≤ |µ|m kU k kU −1 k kwk which implies a) since |µ| = 1 in this case. In order to prove b), notice that because T (k) is real and has real, distinct eigenvalues if |D(k)| > 2, we may take U in the form sin α sin β U= cos α cos β with α, β ∈ [0, π), α 6= β. Now an elementary calculation using (4) leads to (writing γ = α − β) kT m eϕ k2
=
sin−2 γ µ2m sin2 (ϕ − β)
+ µ−2m sin2 (ϕ − α) − 2 cos γ sin(ϕ − β) sin(ϕ − α) .
Viewing the right-hand side as a (quadratic) function of sin(ϕ − α) and determining the minimum shows kT m eϕ k2 ≥ µ2m sin2 (ϕ − β). This yields the desired estimate, since | sin x| ≥ 2|x|/π if |x| ≤ π/2.
Schr¨odinger Operators with Power-Decaying Potentials
485
Now we are in a position to explain the general idea of our construction. By Lemma 3.1, the transfer matrix for x ∈ (an , bn ) is given by powers of T (k/gn ). We will take a function W with infinitely many “gaps” (i.e. intervals with |D| > 2), and we will choose the gn such that for every k, k/gn is in some gap for sufficiently many n. Then Lemma 3.2b) shows that for those n the solution grows by a factor |µ|Ln unless the phase ϕ(an ) is close to β. If k/gn is not in a gap, then the solution can be controlled with the aid of Lemma 3.2a). The point of this argument is that the estimate of Lemma 3.2a) is independent of m. Therefore, after everything else has been chosen, we can take the Ln sufficiently large to obtain increasing solutions. Of course, one then needs an additional argument to deduce the existence of L2 -solutions. To this end, we will use a Ruelle type Theorem (see [13]) recently proved in [9]. We will make no attempt to optimize the exponent c from Theorem 2.1 because it does not seem possible to reach the critical value 1/2. In particular, we will sometimes use rather crude estimates when this does not destroy the power decay. The following lemma allows us to adjust the phases ϕ(an , k) if necessary. For k > 0 with |D(k)| > 2, define S (k) = (β(k) − , β(k) + ) with β(k) from Lemma 3.2b). Lemma 3.3. Suppose V (x) has been chosen for x ≤ bn . Let M be a measurable subset of {k ∈ [k1 , k2 ] : |D(k/gn+1 )| > 2}. Then there is an ln ∈ [0, π/k2 ] such that |{k ∈ M : ϕ(an+1 , k) ∈ S (k/gn+1 )}| ≤
2k2 |M |. πk1
Of course, the phase ϕ has to be evaluated modulo π here. Proof. First note that since V = 0 on (bn , an+1 ), we have ϕ(an+1 , k) = ϕ(bn , k)+(an+1 − bn )k = ϕ(k) + ln k (using the abbreviation ϕ(k) = ϕ(bn , k)). Denote the set defined in the lemma by J(ln ) and integrate |J(ln )| with respect to dln . Fubini-Tonelli shows Z
π/k2 0
Z
Z
|J(ln )| dln =
π/k2
dk M
0
dln χS (k/gn+1 ) (ϕ(k) + kln ).
(ϕ, β are continuous in k, so there is certainly no problem with measurability.) χS denotes the characteristic function of the set S. The second integral on the right-hand side can be estimated by k1−1 2, because ϕ(k) + lk (0 ≤ l < π/k2 ) takes on every value R π/k in [0, π) at most once. Hence 0 2 |J(ln )| dln ≤ k1−1 |M |2, and therefore the claimed inequality must hold for some ln . 4. Asymptotics According to the above remarks, we need to study the asymptotics of D(k) as k → ∞. Since the higher order terms in this asymptotic expansion depend in Ra complicated way on p the Fourier coefficients and also on higher moments of W such as 0 W (x)xeinπx/p dx for a general W (cf. [2]), we now specialize to 0 (0 < x < 1) W (x) = , 1 (1 < x < 2) p = 2. Then an elementary calculation shows
486
C. Remling
T (k) =
cos ω cos k − kω −1 sin ω sin k − cos ω sin k − ωk −1 sin ω cos k
cos ω sin k + kω −1 sin ω cos k cos ω cos k − ωk −1 sin ω sin k
,
where ω 2 = k 2 − 1. In particular, D(k) = 2 cos(ω + k) −
(k − ω)2 sin ω sin k. ωk
The computations in [2, Sect. 4.5] imply that |D(k)| > 2 can only hold if k = nπ/2 + (2nπ)−1 + δ with n ∈ N and δ = O(n−2 ). For these k, a Taylor expansion leads to (we omit this lengthy, but completely elementary computation) D(k) = −2 + 4δ 2 −
4 + O(n−6 ) n4 π 4
(5)
if n is odd and
8δ 2 1 + + O(n−7 ) n2 π 2 n6 π 6 if n is even. With these formulae, we can now show D(k) = 2 − 4δ 2 −
(6)
Lemma 4.1. Let kn = nπ/2 + (2nπ)−1 (n ∈ N) and (n odd) (nπ)−2 δn = . (n even) 2−1 (nπ)−3 Then, for sufficiently large n: a) If kn−1 + δn−1 + (n − 1)−7/2 ≤ k ≤ kn − δn − n−7/2 , then |D(k)| ≤ 2 − π −2 n−13/2 . b) If |k − kn | ≤ δn − n−7/2 , then |D(k)| ≥ 2 + π −2 n−13/2 . Remark. Subsequently, we will only need that |D| − 2 can be estimated by some power of n if the distance of k to the band edges is ≥ Cn−3− ( > 0). Thus, for simplicity, we did not distinguish between odd and even n in the statement; in fact, for odd n, the exponent in b) is 11/2 rather than 13/2. Note also that both estimates are very crude if k is not close to one of the band edges. Proof. The lemma is an easy consequence of (5), (6). The details are left to the reader. In order to establish a), recall that D(k) is monotone in intervals where |D| < 2 [2, Theorem 2.3.1]. In order to be able to use Lemma 3.2, we also need to know the asymptotics of the diagonalizing transformations U (k): Lemma 4.2. If k satisfies either of the assumptions of Lemma 4.1, then U (k) can be chosen such that kU (k)k kU −1 (k)k ≤ Cn13/2 . t1 t2 Proof. Write T (k) = . If t2 6= 0, then t3 t4 −1/2 t2 t2 U = t2 (µ−1 − µ) µ − t1 µ−1 − t1 is a diagonalizing transformation. Notice that |µ − µ−1 |2 = |4 − D2 | ≥ cn−13/2 because of Lemma 4.1. Since |ti |, |µ| are bounded from above, we get
Schr¨odinger Operators with Power-Decaying Potentials
kU k2 ≤ C1
487
n13/4 . |t2 |
Moreover, det U = 1 implies kU −1 k = kU k. Similarly, if t3 6= 0, then µ − t4 µ−1 − t4 −1 −1/2 U = t3 (µ − µ ) t3 t3 is also a diagonalizing transformation with det U = 1 and kU k2 ≤ C2 n13/4 |t3 |−1 . So it suffices to show that max{|t2 |, |t3 |} ≥ C3 n−13/4 . In the situation of Lemma 4.1a), the conditions t1 t4 − t2 t3 = 1, |t1 + t4 | ≤ 2 − π −2 n−13/2 imply that |t2 t3 | ≥ Cn−13/2 , hence max{|t2 |, |t3 |} ≥ C 0 n−13/4 , as desired. If the hypothesis of Lemma 4.1b) holds, consider ω k − sin ω cos k. t2 + t3 = ω k A Taylor expansion similar to that leading to (5), (6) shows t2 +t3 = −2(nπ)−3 +O(n−4 ), so in particular max{|t2 |, |t3 |} ≥ Cn−3 . 5. Choosing the Parameters Up to now, we have fixed the basic periodic potential W . We proceed by choosing the gn . As discussed above, the choice has to be made so that for every k, k/gn is in some gap for sufficiently many n. Lemma 5.1. Let [k1 , k2 ] be a subinterval of (0, ∞). Then there are constants G0 , C > 0 such that for any G ∈ (0, G0 ] we can find numbers g1 , . . . , gN with the following properties: 1) N ≤ CG−3 2) G/2 ≤ gi ≤ G for all i = 1, . . . , N . 3) For all k ∈ [k1 , k2 ], there is an i ∈ {1, . . . , N } so that |k/gi − kr | ≤ δr − r−7/2 for some r ∈ N. Proof. In this proof, we write ci
=
k2i+1 − δ2i+1 + (2i + 1)−7/2 ,
di
=
k2i+1 + δ2i+1 − (2i + 1)−7/2 .
(We work with the gaps corresponding to odd n only, because the gaps with even n are by a factor ∼ n−1 smaller, see Lemma 4.1.) For G > 0, let r1 r2
= =
max{i ∈ N : Gci ≤ k1 }, min{i ∈ N : Gdi ≥ k2 }.
Obviously, ri → ∞ as G tends to zero. From the definition of ri (and that of ki , δi ) one deduces easily that ri = ki (πG)−1 + O(1), where O(1) denotes an expression that remains bounded as G → 0. We take g1 = G and gn = gn−1 cr2 /dr2 for n = 2, . . . , m, where m is the smallest integer with gm cr2 ≤ Gdr2 −1 . By construction, this covers I := [Gcr2 −1 , Gdr2 ] in the sense that for all k ∈ I there is an i ≤ m such that k/gi ∈ [cr2 , dr2 ]. Moreover,
488
C. Remling
gi = G 1 −
i−1
1 + O(r2−4 ) 2π 3 r23
,
(7)
provided that r2 is large enough, i.e. G has to be sufficiently small. In this case, one infers easily from (7) that m can be estimated by m ≤ 4π 3 r22 . Using this and again (7) shows gi ≥ G(1 − r2−1 ) ≥ G/2 if, again, G is small enough. The intervals [Gci , Gdi+1 ] with r1 ≤ i ≤ r2 − 2 can now be covered in the same way by appropriately scaled copies of [ci+1 , di+1 ]. Note that the total number of gi ’s is N ≤ 4π 3 (r1 + 1)2 + . . . + r22 ≤ 4π 3 r23 . Hence this set of gi ’s has the properties stated in the lemma.
In order to treat the whole half-axis k ∈ (0, ∞), let G0 (m), C(m) be the constants of Lemma 5.1 for [k1 , k2 ] = [m−1 , m] (m ∈ N). Given any functions F, G > 0 with F (s) → ∞, G(s) → 0, we can find integers m(s) → ∞ such that m(s) ≤ F (s) and C(m(s)) ≤ F (s), G0 (m(s)) ≥ G(s) for all s ∈ N with m(s) > 1 (take m(1) = . . . = m(s0 − 1) = 1, then m(s0 ) = 2, where s0 is sufficiently large so that the inequalities hold, etc.). We can now choose the whole sequence (gn )∞ n=1 : Let F (s), G(s) > 0 be prescribed sequences with F → ∞, G → 0. Determine m(s) as explained in the preceding paraPs−1 graph, and for s = 1, 2, . . ., pick gj(s)+1 , . . . , gj(s)+N (s) (where j(s) = t=1 N (t)) so that these g’s and N = N (s) satisfy 1)-3) of Lemma 5.1 for G = G(s) and [k1 , k2 ] = [m(s)−1 , m(s)]. These definitions of j(s), N (s), gn as well as the meaning of F (s), G(s), m(s) should be kept in mind throughout the rest of this paper. We will also write s(n) for the unique s determined by j(s) + 1 ≤ n ≤ j(s) + N (s). Next, we investigate the set of k for which k/gn is close to one of the band edges infinitely often (abbreviated as i.o. in the sequel), i.e. for infinitely many n. More precisely, consider An = {k > 0 : |k/gn − ki − δi | < i−7/2 or |k/gn − ki + δi | < i−7/2 for some i}. Fix [k1 , k2 ] ⊂ (0, ∞). If n is sufficiently large, then by construction of the gn and by Lemma 5.1 2), the condition defining An can only hold for i ≥ C0 (k1 )/G(s) (writing s = s(n) for brevity). Hence |An ∩ [k1 , k2 ]| ≤ 4G(s)
X
i−7/2 ≤ CG(s)7/2 .
(8)
i≥C0 /G(s)
Lemma 5.1 1) says that there are at most F (s)G(s)−3 indices n with s(n) = s. Thus summing (8) over n yields ∞ X n=1
|An ∩ [k1 , k2 ]| ≤ C
∞ X
F (s)G(s)1/2 .
s=1
Now the Borel-Cantelli Lemma implies P∞ Lemma 5.2. Assume s=1 F (s)G(s)1/2 < ∞. Then |{k > 0 : k ∈ An i.o.}| = 0.
Schr¨odinger Operators with Power-Decaying Potentials
489
In the same way, we can control the measure of the set where |D| > 2 but at the same time the phase is close to β (recall the definitions from Sect. 3). For a sequence (s) > 0, let (again, with s = s(n)) Bn
=
{k ∈ [m(s)−1 , m(s)] : |k/gn − ki | ≤ δi − i−7/2 for some i, ϕ(an , k) ∈ S(s) (k/gn )}.
Lemma 3.3 says that |Bn | ≤
2 2 π F (s) |M |(s)
for appropriate ln−1 ∈ [0, π/m(s)]. Here,
M = {k ∈ [m(s)−1 , m(s)] : |k/gn − ki | ≤ δi − i−7/2 for some i}. In order to estimate |M |, proceed as above. This leads to |Bn | ≤ CF (s)3 G(s)2 (s), and summing over n proves P∞ Lemma 5.3. Assume s=1 F (s)4 G(s)−1 (s) < ∞. Then there are ln → 0 so that |{k > 0 : k ∈ Bn i.o.}| = 0. Note the somewhat complicated logic behind this statement: If the summability assumption holds, then for any choice of the Ln (see Sect. 2), we can find ln with the stated properties. In particular, recall that ln depends on the values of the potential V (x) for x ≤ bn . However, this does not cause any difficulties because we may as well think of the parameters Ln , gn as being fixed from the beginning (with the values given below, of course). The representation given here tries to avoid unmotivated choices. There will be no further conditions on F, G, , therefore we now fix these parameters. We take F (s) = ln s, G(s) = s−3 , (s) = s−5 (we remark that powers are by no means the only possible choice), hence N (s) ≤ s9 ln s by Lemma 5.1 1). Ln will also depend on s only, thus we can write L(s).
6. Conclusion of the Proof We are now in a position to estimate the solutions of (1). More specifically, we consider Rni := R(xni , k) at the points xni = an + 2i/gn (n ∈ N, i = 0, 1, . . . , Ln ). Fix a k which is not in the exceptional sets of Lemmas 5.2, 5.3, and let n0 be sufficiently large so that k ∈ [m(s)−1 , m(s)] if s ≥ s(n0 ). Furthermore, take n0 also greater than the finitely many n for which k ∈ An ∪ Bn , and let i0 ∈ {0, 1, . . . , Ln0 }. For some of the deductions below, it may be necessary to restrict n0 to even larger values; we will not mention this explicitly in the sequel. Finally, write s0 = s(n0 ), R0 = Rn0 i0 . By combining our previous results, we will obtain the following estimate: If xni ≥ xn0 i0 , then ln Rni − ln R0 ≥ K1
s(n)−1 X t=s0 +1
L(t)t−39/4 − K2
s(n) X
t9 ln2 t.
(9)
t=s0
Here, K1/2 are positive constants which are independent of n0 , i0 , n, i. If xni < aj(s0 +2)+1 , then the first sum is zero by convention. In order to establish (9), we argue as follows: Let m be an index with n0 ≤ m ≤ n. By Lemma 3.1, we have T (xm,i+j , xmi ; k) = T (k/gm )j . Furthermore, R is constant on (bm , am+1 ) = (xmLm , xm+1,0 ) since V = 0 on these intervals. Because of the definition of the sets Am , one of the two parts of Lemma 4.1 applies to k/gm (instead of k).
490
C. Remling
In the situation of Lemma 4.1a), we deduce from Lemma 3.2a) and Lemma 4.2 that kT (k/gm )j vk ≥ Cr−13/2 kvk, where r is the unique integer for which kr−1 + δr−1 + (r − 1)−7/2 ≤ k/gm ≤ kr − δr − r−7/2 holds. Since gm ≥ G(s(m))/2 (see Lemma 5.1 2)), r ≤ C(k)s(m)3 . In terms of R, we thus get (10) ln Rm,i+j − ln Rmi ≥ −C ln s(m), where C is independent of m, i, j. In particular, taking i = 0, j = Lm we see that ln R(am+1 ) − ln R(am ) ≥ −C ln s(m).
(11)
Summing up the contributions from (10), (11) gives the second term of (9). / Assume now that Lemma 4.1b) applies to k/gm . Observe that in this case ϕ(am , k) ∈ S(s(m)) (k/gm ) because of the definition of Bm (compare Lemma 5.3). Moreover, (3) together with Lemma 4.1b) implies that |µ(k/gm )| ≥ 1+π −1 r−13/4 , where r is determined by |k/gm −kr | ≤ δr −r−7/2 , hence, estimating r as above, |µ(k/gm )| ≥ 1+Cs(m)−39/4 . Now Lemma 3.2b) shows ln R(am+1 ) − ln R(am ) ≥ C1 L(s(m))s(m)−39/4 − C2 ln s(m), and since for all t, there is at least one m ∈ {j(t) + 1, . . . , j(t) + N (t)} for which k/gm is in a gap (by Lemma 5.1 3)), we also get the first sum of (9). It remains to treat the contribution to (9) that comes from the beginning of the interval under consideration, namely ln Rn0 ,i0 +j − ln R0 , for the case when k/gn0 is in a gap (i.e. Lemma 4.1b) applies to k/gn0 ). This term is not included in the above treatment because Lemma 5.3 gives no information on the phase at xn0 i0 . However, Lemma 3.2b) does show that 2 Rn0 ,i0 +j ≥ (s0 )|µ(k/gn0 )|i0 +j R(an0 ). π On the other hand, R0 ≤ kT (k/gn0 )i0 kR(an0 ) and therefore, taking kT m k ≤ |µ|m kU k kU −1 k into account (see the proof of Lemma 3.2), Rn0 ,i0 +j ≥
2(s0 )|µ(k/gn0 )|j R0 . πkU (k/gn0 )k kU (k/gn0 )−1 k
kU k kU −1 k can be controlled with the aid of Lemma 4.2. In conclusion, we get again an estimate of the form of (10). Note that k/gn might also happen to be in a gap (n is from (9)), but in this case it is obvious from Lemma 3.2b) that (10) holds. Thus the proof of (9) is complete. If we now take, say, L(s) = [s79/4 ] ([x] is the largest integer ≤ x), then the first sum in (9) dominates the second one, i.e. we get increasing solutions. Simple estimates allow us to rewrite (9) in the following more useful form: s(n) = s0 , s0 + 1 −C1 s90 ln2 s0 . (12) ln Rni − ln R0 ≥ s(n) ≥ s0 + 2 C2 s(n)10 In order to establish the existence of L2 -solutions, we use the following version of [9, Theorem 8.1]. This (slight) reformulation takes into account that we do not have accurate upper bounds for the norm of the transfer matrices.
Schr¨odinger Operators with Power-Decaying Potentials
491
Lemma 6.1 ([9]). Assume that there is an increasing sequence xn → ∞ such that: 1) kT (xn+1 , xn ; k)k ≤ C(k) for all Pn 2) There is a vector v ∈ R2 with kT (xn , x0 ; k)vk−2 < ∞. Then either a) There exists another vector u 6= 0 such that (writing ρn = kT (xn , x0 ; k)uk, Rn = kT (xn , x0 ; k)vk) !2 ∞ X (Rn /Ri )2 + Rn−2 . ρ2n ≤ C1 (k) Rn−2 i=n
or b) ρn /Rn → ∞ if u is not a constant multiple of v. Proof. Follow the proof of [9, Theorem 8.1]. If the vector u∞ constructed there is not a constant multiple of v, then a) follows as in [9] (with u = u∞ ). If u∞ = cv, then b) follows directly from Eq. (8.2) of [9] and the fact that the vector u from that statement is exactly u∞ . The hypotheses of this lemma are satisfied for almost every R xn,i+1k if we let xn be the |V (x)| dx = gn , the sequence of the points xni ≥ xn0 i0 defined above: Since xni standard Gronwall estimate [4, Sect. III.1] proves 1) (in fact, it proves much more, namely that T (xn+1 , xn ; k) → 1 uniformly on compact k sets). Furthermore, (12) obviously implies 2). Now let M be the set of those k 2 for which conclusion b) of the lemma holds. Clearly, b) in particular implies that no solution of (1) is polynomially bounded (as a function of x), hence M is a set of spectral measure zero for all boundary conditions α by Schnol’s Theorem (see [18, Proposition 9]). Therefore, using spectral averaging, Rπ |M | = 0 ρα (M ) dα = 0. P Using (12), we can estimate Rn−2 as well as i≥n (Rn /Ri )2 . Fixing n0 , i0 , we immediately get (with positive constants Ci , as usual) 10
2 ≥ C1 eC2 s(n) . Rni
P In order to bound xmj ≥xni (Rni /Rmj )2 (of course, the sum is over m, j, and n, i are fixed), we also use (12), but now with Rni taking the role of R0 . Here it is important that the constants of (12) are independent of n0 , i0 . As in the preceding section, we translate the sum over m, j into a sum over s and estimate the number of terms with the aid of Lemma 5.1 1). The result of these by now familiar arguments is X 9 2 (Rni /Rmj )2 ≤ C3 eC4 s(n) ln s(n) . xmj ≥xni
Thus Lemma 6.1a) implies that for almost all k > 0, there is a solution y satisfying (y 2 + (y 0 /k)2 )(xni ) = ρ2ni ≤ Ce−C
0 10
s
.
(13)
Because of Gronwall’s Lemma (compare the argument above), this estimate also holds for x ∈ [xni , xn,i+1 ] (with a possibly larger C). The number of indices n, i corresponding to the same s and the length of the intervals [xni , xn,i+1 ] can both be bounded by a power of s, hence (13) guarantees that y ∈ L2 .
492
C. Remling
Finally, for x ≥ 0, let s be the index corresponding to x, i.e. aj(s)+1 ≤ x < aj(s+1)+1 . Then by construction of V (see Sect. 2) s X 4L(t) + π ≤ Cs33 . N (t) x < aj(s+1)+1 ≤ G(t) t=1
The second term in the sum estimates the contribution from the ln . On the other hand, 0 ≤ V (x) ≤ s−6 , so indeed V (x) = O(x−2/11 ). This completes the proof of Theorem 2.1. 7. Singular Continuous Spectrum We modify V as follows. For a growing sequence sn → ∞, let V (x) (aj(sn )+1 ≤ x ≤ bj(sn )+N (sn ) ) Vaux (x) = . 0 else Now the modified potential V˜ is obtained from Vaux by readjusting the ln (more precisely, the subsequence of ln corresponding to the sequence sn ) according to Lemma 5.3. The remark following Lemma 5.3 shows that this may be necessary. Since the transfer matrix for zero potential is unitary, the above arguments also apply to V˜ . In particular, an estimate similar to (9) still holds, the sums now being over the subsequence sn . Consequently, one can show along the lines of the preceding section that for almost all k, ! n X 10 si . (14) R(bj(sn )+N (sn ) , k) ≥ C1 (k) exp C2 (k) i=1
A version of Schnol’s Theorem (namely [18, Proposition 9]) states that if h ∈ L2 ∩ L∞ , then h(·)v(·, k 2 ) ∈ L2 for spectrally almost all E = k 2 . Here v is a solution of the Schr¨odinger equation satisfying the boundary condition at x = 0. Let Dn = aj(sn+1 )+1 − bj(sn )+N (sn ) and h(x) = (n2 Dn )−1/2 if bj(sn )+N (sn ) ≤ x ≤ aj(sn+1 )+1 , and h(x) = 0 otherwise. Then (14) shows that hv ∈ / L2 for almost all k, hence the spectrum is supported by a set of Lebesgue measure zero and therefore σac = ∅. On the other hand, if Dn grows sufficiently fast (which can be achieved by taking a sparse sequence sn ), then we can also exclude the existence of L2 -solutions. More specifically, choose the parameters by the following inductive procedure: Take s1 = 1 and pick the corresponding ln ’s. Next, determine D1 large enough so that for all k ∈ [1−1 , 1] the following holds: If y is any solution with R(0, k) = 1, then D1 R2 (bj(s1 )+N (s1 ) , k) ≥ 1. Pick any s2 so that aj(s2 )+1 ≥ bj(s1 )+N (s1 ) + D1 (for instance, take the smallest number with this property) and adjust the ln ’s corresponding to this s2 . Again, choose D2 so that D2 R2 (bj(s2 )+N (s2 ) , k) ≥ 1, but this time for all k ∈ [2−1 , 2], etc. We can also be more explicit; for instance, it is not hard to show (using Gronwall’s Lemma again) that the sequence sn defined by s1 = 1, sn sn+1 = 2(2 ) has the required properties. Finally, note that since the original V satisfies V (x) = O(x−c ), V˜ is also of order O(x−c ). We summarize our result as
Schr¨odinger Operators with Power-Decaying Potentials
493
Theorem 7.1. The potential V˜ constructed above satisfies: 1) V˜ (x) = O(x−c ) (c > 0). 2) For all boundary conditions at x = 0, the spectrum is purely singular continuous in (0, ∞). Note added in proof The open question mentioned in the introduction has been answered by myself (paper in preparation): If V (x) = O(x−c ) with c > 1/2, then oac = [0, ∞). References 1. del Rio, R., Makarov, N., Simon, B.: Operators with Singular Continuous Spectrum, II. Rank One Operators. Commun. Math. Phys. 165, 59–67 (1994) 2. Eastham, M.S.P.: The Spectral Theory of Periodic Differential Equations. London: Scottish Academic Press, 1973 3. Gordon, A.Y., Molchanov, S.A., Tsagani, B.: Spectral Theory of One-Dimensional Schr¨odinger Operators with Strongly Fluctuating Potentials. Funct. Anal. Appl. 25, 236–238 (1992) and Gordon, A.Y.: Private communication 4. Hartman, P.: Ordinary Differential Equations. Boston: Birkh¨auser, 1982 5. Kirsch, W., Molchanov, S.A., Pastur, L.: One-dimensional Schr¨odinger operators with high potential barriers. Operator Theory: Advances and Applications, Vol.57, Basel: Birkh¨auser Verlag, 1992, pp. 163–170 6. Kiselev, A.: Preservation of the absolutely continuous spectrum of Schr¨odinger equation under perturbations by slowly decreasing potentials and a.e. convergence of integral operators. Preprint (1996) 7. Kiselev, A., Last, Y., Simon, B.: Modified Pr¨ufer and EFGP transforms and the spectral analysis of one-dimensional Schr¨odinger operators. Commun. Math. Phys. (submitted) 8. Kotani, S., Ushiroya, N.: One-dimensional Schr¨odinger operators with random decaying potentials. Commun. Math. Phys. 115, 247–266 (1988) 9. Last, Y., Simon, B.: Eigenfunctions, Transfer Matrices, and Absolutely Continuous Spectrum of OneDimensional Schr¨odinger Operators. Ann. Math. (submitted) 10. Naboko, S.N.: Dense Point Spectra of Schr¨odinger and Dirac Operators. Theor. and Math. Phys. 68, 646–653 (1986) 11. Pearson, D.B.: Singular Continuous Measures in Scattering Theory. Commun. Math. Phys. 60, 13–36 (1978) 12. Remling, C.: A probabilistic approach to one-dimensional Schr¨odinger operators with sparse potentials. Commun. Math. Phys. 13. Ruelle, D.: Ergodic theory of differentiable dynamical systems. Publ. IHES 50, 275–306 (1979) 14. Simon, B.: Some Jacobi matrices with decaying potentials and dense point spectrum. Commun. Math. Phys. 87, 253–258 (1982) 15. Simon, B.: Some Schr¨odinger operators with dense point spectrum. Proc. Am. Math. Soc. bf 125, 203–208 (1997) 16. Simon, B., Wolff, T.: Singular Continuous Spectrum under Rank One Perturbations and Localization for Random Hamiltonians. Commun. Pure Appl. Math. 39, 75–90 (1986) 17. Simon, B., Zhu, Y.: The Lyapunov Exponents for Schr¨odinger Operators with Slowly Oscillating Potentials. J. Funct. Anal. 140, 541–556 (1996) 18. Stolz, G.: Localization for random Schr¨odinger operators with Poisson potential. Ann. Inst. H. Poincar´e 63, 297–314 (1995) 19. Stolz, G.: Localization for Schr¨odinger operators with effective barriers. J. Funct. Anal. (to appear) Communicated by B. Simon
Commun. Math. Phys. 186, 495 – 529 (1997)
Existence and Uniqueness Theorems for Massless Fields on a Class of Spacetimes with Closed Timelike Curves J.L. Friedman, M.S. Morris 1 Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA and Department of Physics, University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA. E-mail: [email protected] 2 Department of Physics & Astronomy, Butler University, 4600 Sunset Ave., Indianapolis, IN 46208, USA. E-mail: [email protected]
Received: 28 November 1994/Accepted: 20 May 1996
Abstract: We study the massless scalar eld on asymptotically at spacetimes with closed timelike curves (CTC’s), in which all future-directed CTC’s traverse one end of a handle (wormhole) and emerge from the other end at an earlier time. For a class of static geometries of this type, and for smooth initial data with all derivatives in L2 on I− , we prove existence of smooth solutions which are regular at null and spatial in nity (have nite energy and nite L2 -norm) and have the given initial data on I− . A restricted uniqueness theorem is obtained, applying to solutions that fall o in time at any xed spatial position. For a complementary class of spacetimes in which CTC’s are con ned to a compact region, we show that when solutions exist they are unique in regions exterior to the CTC’s. (We believe that more stringent uniqueness theorems hold, and that the present limitations are our own.) An extension of these results to Maxwell elds and massless spinor elds is sketched. Finally, we discuss a conjecture whose meaning is essentially that the Cauchy problem for free elds is well de ned in the presence of CTC’s whenever the problem is well-posed in a geometric-optics limit. We provide some evidence in support of this conjecture, and we present counterexamples that show that neither existence nor uniqueness is guaranteed under weaker conditions. In particular, both existence and uniqueness can fail in smooth, asymptotically at spacetimes with a compact nonchronal region. 1. Introduction Although spacetimes with closed timelike curves occur as solutions to the vacuum Einstein equations, they have been regarded as unphysical, in part because the most familiar examples have no well-de ned initial value problem. Morris and Thorne [1], however, introduced a class of wormhole geometries in which, although there are many closed timelike curves, the set of closed timelike and null geodesics has measure zero. On these spacetimes, Morris et al. [2] noted that the evolution of free elds is well de ned in the limit of geometrical optics; and this in turn makes
496
J.L. Friedman, M.S. Morris
Fig. 1. An orientable 3-manifold M is constructed by identifying points of I and points of II that are labeled by the same letter, with subscripts I and II: PII = T(PI )
it seem likely that a multiple scattering series converges to a solution for arbitrary initial data [2–4]. The simplest of the spacetimes they considered are static, with CTC’s present at all times, and we reported an existence and a restricted uniqueness theorem for the massless scalar eld on such static time-tunnel spacetimes [4]. The present paper provides details of these latter results and outlines their extension to spinor and vector elds. A nal section discusses the Cauchy problem on spacetimes in which CTCs are con ned to a compact region. We prove a uniqueness theorem and present a conjecture on the existence of free elds whose data are speci ed on a spacelike hypersurface (partial Cauchy surface) to the past of any CTCs. Until the nal section, the geometry N; g we consider is static in the sense that there is a timelike Killing vector t that is everywhere locally hypersurface orthogonal.1 The manifold has topology N = M × R, where M is a plane with a handle (wormhole) attached: M = R3 #(S 2 × S 1 ). The metric g on N is smooth (C ∞ ), and, for simplicity in treating the asymptotic behavior of the elds, we will assume that outside a compact region R the geometry is at, with metric . 2. Preliminaries A. Foliation of a related spacetime with boundary. One can construct the 3manifold M from R3 by removing two balls and identifying their spherical boundaries, I and T(I ), by a map T, as shown in Fig. 2. In Fig. 2, the map T involves an improper rotation of I , and it yields an orientable handle. Maps T which involve a proper rotation of I yield non-orientable handles. For the rest of this paper, we will in fact assume that T is improper and that the handle is consequently orientable. This matters, however, only to our treatment of a two-component Weyl spinor. Were we to allow non-orientable handles as well, everything would be the same, except that we should have to consider 4-component Dirac spinors. The sphere obtained by the identi cation of I and T(I ) will be called the “throat” of the handle. Its location is arbitrary: after removing any sphere , from the handle 1 Spacetime indices will be lower case Greek, spatial indices lower case Latin, spinor indices upper case Latin. For those familiar with the abstract index notation [5], letters near a (or ) in the alphabet will be abstract while those after i () will be concrete, so that is the th component of the vector . Our signature is − + ++, and Riemann tensor conventions follow [5]. We will use “manifold” as a shorthand for smooth manifold, with or without boundary.
Existence Uniqueness Theorems for Massless Fields
497
Fig. 2. The spacelike hypersurfaces Mt foliate the spacetime whose boundary is the union of two cylinders, CI ∪ CII . In the spacetime N, the cylindrical boundaries are identi ed, and Mt+ is a smooth continuation of Mt across the identi ed spheres I and T(I )
of M one is left with a manifold homeomorphic to R3 \(B3 #B3 ), whose boundary is the disjoint union of two spheres. Let C ∼ = S 2 × R be the history of the throat in the spacetime N; its orbit under the group of time-translations generated by t . Then, after removing the cylinder C from the spacetime N; one is similarly left with a manifold homeomorphic to R4 \[(B3 #B3 ) × R], whose boundary, @(N\C), is the disjoint union, CI t CII , of two timelike cylinders. We will again use the symbol T to denote the map from CI to CII that relates identi ed points; its restriction to a single sphere I ⊂ CI is the map denoted above by T. For the spacetimes we will consider, identi ed points of CI and CII = T(CI ) will be timelike separated, and copies of M in the spacetime cannot be everywhere spacelike. Thus, although t is locally hypersurface-orthogonal, no complete hypersurface of N with a single asymptotically at region is orthogonal to every trajectory of t . One can, however, foliate N\C by spacelike hypersurfaces Mt orthogonal to t . Each Mt can be chosen to agree asymptotically with a t = constant surface of the at metric and to intersect CI and CII in isometric spheres I and II . The sphere T(I ) identi ed with I is a time-translation of II , obtained by moving II a parameter distance along the trajectories of t (see Fig. 2). The manifold Mt near I is a smooth extension of Mt+ near T(I ). That is, if UI ⊂ Mt and Ut+ ⊂ Mt+ are spacelike neighborhoods of I and T(I ), their union UI ∪ Ut+ is a smooth, spacelike submanifold of N. Because the time-
498
J.L. Friedman, M.S. Morris
translation of Ut+ to Mt is a neighborhood UII ⊂ Mt of II isometric to Ut+ , by arti cially identifying the spheres I and II of Mt , one obtains a copy of M (a plane with a handle) with a metric that is everywhere smooth and spacelike. Of course this spacelike copy of M is not a submanifold of the spacetime N. A static metric on N is given by g = −e−2 t t + h ;
(2.1)
where h t = 0. If the Minkowski coordinate t is extended to N\C by making Mt a t = constant surface, then t ∇ t = 1, ∇ t = −e−2 t , and the metric (2.1) can be written on N\C in the form g = −e2 @ t@ t + h :
(2.2)
It will be convenient to single out a representative hypersurface, M := M0 :
(2.3)
We will denote by hab the corresponding spatial metric on M; that is, hab is the pullback of h (or g ) to M. B. An initial value problem 1. Free massless elds on N. We consider the wave equation ≡ ∇ ∇ = 0 ;
(2.4)
for a massless scalar eld , and the corresponding wave equations for massless vector and spinor elds. A Maxwell eld F can be written in terms of a vector potential A satisfying the Lorentz gauge condition, F = ∇ A − ∇ A ;
∇ A = 0 :
(2.5)
The equation ∇ F = 0, governing a free Maxwell eld, is then equivalent to ∇ ∇ A − R A = 0 :
(2.6)
We will adopt the 2-component spinor notation given in Penrose and Rindler [6], with ∇AA0 = AA √0 ∇ , where AA0 has components equal to entries of A(the usual Pauli spin matrices)= 2. The free- eld equation for a massless spinor is given by ∇AA0 A = 0 :
(2.7)
Although N has no spacelike hypersurface which could play the role of a Cauchy surface, one can pose initial data to the massless wave equations at past null in nity, I− . We will show for the scalar wave equation (and outline a proof for the other elds) that for all data in H∞ (I− ), there is a solution to Eq. (2.4). Because the geometry is at outside a region of xed spatial size, I is a copy of the Minkowski space I. Our treatment of the initial value problem will be closely related to a way of expressing a solution of the wave equation in Minkowski space in terms of initial data on I− .
499
Existence Uniqueness Theorems for Massless Fields
2. Initial value problem on I− for Minkowski space. In the null chart (v = t + r; ˜r ) for Minkowski space, I− has coordinates (v; rˆ ). A smooth solution 0 with nite energy to the wave equation (2.4) has as initial data on I− the single function [9] f0 (v; rˆ) = lim r0 (v; r r) ˆ : r→∞
(2.8)
It is helpful to write the solution 0 in terms of its positive frequency part, 0 : R 0 = 2Re 0 = (2)−3=2 d3k[a(k)ei(k · x−!t) + a∗ (k)e−i(k · x−!t) ] ; (2.9) where
R 0 = (2)−3=2 d3ka(k)ei(k · x−!t) :
Then the function 0 has initial data on I lim r 0 (v; r rˆ) =
r→∞
i (2)
1 2
−
(2.10)
given by [9]
R∞ d!!a(−! rˆ)e−i!v :
(2.11)
0
The corresponding initial data f0 (v; rˆ) for 0 , has Fourier transform, f˜0 (!; rˆ) =
R∞
1 (2)
1 2
−∞
dvf0 (v; rˆ)ei!v ;
(2.12)
related to a(k) by f˜0 (!; rˆ) = i!a(−! rˆ);
!=0;
f˜0 (−!; rˆ) = f˜0∗ (!; r) ˆ :
(2.13) (2.14)
−
If we de ne L2 (I ) by the norm, R kfk2L2 (I− ) = dvd |f|2 ;
(2.15)
then the L2 norm of 0 on a spacelike hyperplane is equal to the L2 norm of its initial data on I− : R R R lim dvd |r 0 |2 = dk|a|2 = dV | 0 |2 : (2.16) r→∞
The L2 norm of 0 depends on hypersurface, but it is bounded by the constant L2 norm of 0 . The ux of energy at I− is given in terms of the stress tensor, 1 T = ∇ ∇ − g ∇ ∇ ; 2
(2.17)
by R I−
dS T t = lim
r→∞
= =
R R
R
dvd [@v (r0 )]2
(2.18)
d!d !2 |f˜0 |2
(2.19)
dk!2 |a(k)|2 :
(2.20)
500
J.L. Friedman, M.S. Morris
Initial data for vector and spinor elds have a similar character. For a Maxwell eld with nite energy on Minkowski space, initial data on I− has the form, i R (2.21) d!!a (−! rˆ)e−i!v ; lim rA (v; r rˆ) = 2Re 1 r→∞ (2) 2 with a (k)k = 0 = a t . Similarly, data on I− for a massless spinor eld has the form, i R (2.22) d!!aA (−! rˆ) e−i!v ; lim r A (v; r rˆ) = 2Re 1 r→∞ (2) 2 where aA (k)kAA0 = 0: As in (2.16), the L2 norm of the initial data for vector and spinor elds is equal in at space to the L2 norm on a spacelike hyperplane: R R R lim dvd |rA0 |2 = dk|a|2 = dV |A0 |2 ; (2.23) r→∞
lim
r→∞
R
R R dvd |r 0 |2 = dk|a|2 = dV | 0 |2 :
(2.24)
From the form of the energy-momentum tensor for vector and spinor elds, 1 1 F F − g F F ; (2.25) T = 4 4 0
0
T = i AA BB ( (A ∇B)A0 B0 − (A0 ∇B0 )A B ) ;
(2.26)
the energy ux of the elds F and A at I− is 1 R 1 R dk!2 aj (k)∗ a j (k) ; dS F F t = 4 I− 4 and
R I−
R 0 0 dS aAA BB ( (A ∇B)A0 B0 − (A0 ∇B0 )A B ) t = dk!2 |a(k)|2 :
(2.27)
(2.28)
3. Initial value problem on I− for N. In the spacetime N; g that we are considering, we seek a solution to the scalar wave equation, ∇ ∇ = 0 ;
(2.29)
with initial data f de ned as in Eq. (2.8) by f(v; rˆ) = lim r(v; r rˆ) ; r→∞
(2.30)
and we shall assume that f and its derivatives are in L2 (I− ) . Because a neighborhood of I− in N is isometric to a neighborhood of I− in Minkowski space, the null Minkowski chart (v = t + r; ˜r ), can be regarded as a chart on N near I− . It will again be convenient to relate f to a function a(k) as in Eqs. (2.12)–(2.14), in this case de ning a(k) by ˜ f(!; rˆ) = i!a(−! rˆ);
!=0;
(2.31)
501
Existence Uniqueness Theorems for Massless Fields
˜ f(−!; rˆ) = f˜∗ (!; rˆ) :
(2.32)
Initial data on I− for massless vector and spinor elds on N can similarly be de ned as vector and spinor elds f and fA on I− , related to solutions A and A of Eqs. (2.6) and (2.7) by f = lim rA (v; r rˆ)
(2.33)
fA = lim r A (v; r rˆ) :
(2.34)
r→∞
and r→∞
Here the at metric near I− is implicitly used to identify the tangent spaces at each point with a single vector space in order to make the limit meaningful; and the spinor space at each point is similarly regarded as a single 2-dimensional complex space. Equivalently, one can simply replace the abstract indices and A of Eqs. (2.33) and (2.34) by concrete indices along frames that are constant with respect to the at metric. Using the same single vector space, one can again de ne Fourier components f˜ by 1 R∞ dvf (v; rˆ)ei!v ; (2.35) f˜ (!; rˆ) = 1 (2) 2 −∞ and
f˜ (−!; rˆ) = f˜ ∗ (!; rˆ) :
(2.36)
Corresponding vectors a (k) with k a (k) = 0 are related to f˜ by f˜ (!; rˆ) = i!a (−! rˆ);
!=0:
(2.37)
0
Spinors A (k) with k AA aA = 0 are similarly related to f˜A if one replaces each subscript by A in Eqs. (2.36) and (2.37). Because the spacetime N is not simply connected, one must specify a choice of spinor structure in order to make sense of Eq. (2.7). On the simply connected spacetime N\C, the two spinor structures on N correspond to a choice of sign in identifying a spinor at a point PI ∈ CI with a spinor at the corresponding point PII ∈ CII . The choice of spinor structure thus becomes a choice of boundary condition (see Eq. (2.41), below). The two choices give two inequivalent spinor elds on N. With a(k); a (k), and aA de ned in this way, Eqs. (2.18), (2.27), and (2.28) remain valid as expressions for the energy ux through I− of N. C. Boundary conditions.A scalar eld on N satis es at the cylindrical boundaries of N\C conditions expressing the continuity of and its normal derivative along a path that traverses the wormhole. Let PI and PII = T(PI ) be identi ed points on the cylinders CI and CII . The tangent vector to a path that traverses the wormhole, entering at PI and leaving at PII , points inward at PI and outward at PII . When the cylinders CI and CII are identi ed, a unit inward normal to CI is thus identi ed with a unit outward normal to CII . If we denote by nˆI and nˆII the unit outward normals to CI and CII , the boundary conditions can be written (PII ) = (PI ) ;
(2.38a)
nˆII ·∇(PII ) = −ˆ nI ·∇(PI ) :
(2.38b)
502
J.L. Friedman, M.S. Morris
The analogous conditions for vector and spinor elds can be stated in terms of the di erential map T∗ induced by T. If {ˆe (PI )} = {ˆe0 ;eˆ1 ;eˆ2 ; nˆI } is a right-handed orthonormal frame at PI , then the corresponding right-handed frame at PII (if T were a proper rotation, then the corresponding frame would be left-handed) is nII } : {ˆe (PII )} = {T∗eˆ0 ; T∗eˆ1 ; T∗eˆ2 ; −ˆ
(2.39)
The boundary conditions for a vector eld can be expressed in terms of its components along the frame {ˆe }: A (PII ) = A (PI ) ;
(2.40a)
nI ·∇A (PI ) : nˆII ·∇A (PII ) = −ˆ
(2.40b)
A spinor eld has components along a spinor frame, an element of the double covering (' SL(2; C)) of the space of right-handed orthonormal frames (' SO(3,1)) at a point. Two spinor frames correspond to the same orthonormal frame, and the choice of spinor structure is the choice of which spinor frame at PII to identify with a spinor frame at PI . Let (o A ; A ) be a eld of spinor frames covering a eld of frames eˆ that satis es Eq. (2.39) on N\C. We can choose as boundary conditions for the corresponding components J of a spinor eld J (PII ) = J (PI ) ;
(2.41a)
nI ·∇ J (PI ) : nˆII ·∇ J (PII ) = −ˆ
(2.41b)
The opposite spinor structure would be selected by changing the sign of the righthand side of these equations or, equivalently, by keeping the same sign but choosing a homotopically di erent frame eld {˜e }. D. Eigenfunction expansions.Because the geometry is static, we can express a solution as a superposition of functions with harmonic time dependence, R (t; x) = d! (!; x)e−i!t ; (2.42a) R A (t; x) = d!A (!; x)e−i!t ; (2.42b) R (2.42c) A (t; x) = d! A (!; x)e−i!t : Here x is naturally a point of the manifold of trajectories of t , but we can identify it with a point of the simply connected spacelike hypersurface M = M0 , with spherical boundaries I and II . Let (t; xI ) and (t + ; xII ) be points of N\C that are identi ed in N. The harmonic components of elds on N can be regarded as elds on M satisfying the boundary conditions,
with phase = ! .
(!; xII ) = ei (!; xI ) ; nˆII ·∇ (!; xII ) = −einˆI ·∇ (!; xI ) ; A (!; xII ) = ei A (!; xI ) ;
(2.43a) (2.43b) (2.44a)
nˆII ·∇A (!; xII ) = −einˆI ·∇A (!; xI ) ; J (!; xII ) = ei J (!; xI ) ; nˆII ·∇ J (!; xII ) = −einˆI ·∇ J (!; xI ) ;
(2.44b) (2.45a) (2.45b)
503
Existence Uniqueness Theorems for Massless Fields
The harmonic components, , of the scalar eld satisfy on M elliptic equations of the form (!2 + L) = 0 ; (2.46) with boundary conditions (2.43), where L can be de ned by the action of ∇ ∇ on time independent elds f on N; that is Lf := e2 ∇ ∇ f|M ;
(2.47)
for elds satisfying $t f = 0, where $t is the Lie derivative along t . Then L = e Da e Da ;
(2.48)
where Da is the covariant derivative of the 3-metric hab on M. We will denote by L the operator L with boundary conditions (2.43). The analogous operators for spinor and scalar elds are discussed in Sect. III C. We show in Lemma 4 below that one can construct solutions F(; k; x), to the wave equation (2.29), having, on the at geometry outside R, the form of a plane wave plus a purely outgoing wave, corresponding to the scattering of the plane wave by the interior geometry. The existence of solutions for initial data on I− then follows if one can show that a spectral decomposition of the form R (2.49) (x; t) = E(k; x)[e−i!t a(k) + ei!t a∗ (k)]d3k ; converges, where E(k; x) = F( = ! ; k; x). The major diculty lies in the fact that, because the boundary conditions (2.38) involve a time-translation, the corresponding boundary conditions (2.43) depend on the frequency !. If were independent of frequency, the result would follow from the usual spectral theorem for self-adjoint operators. Here, however, E(k; x) and E(k 0 ; x) are, for |k|-|k 0 |, eigenfunctions of di erent operators; they are not orthogonal, and their completeness is not guaranteed by the spectral theorem. The main job of Sect. III is to show that the solution to the scalar wave equation for arbitrary initial data on I− can nevertheless be constructed as a spectral integral of the form (2.49). We adopt throughout the common usage in which the term “eigenfunction” refers not to an element of the domain of L , but to a function in a weighted L2 space, whose L2 norm diverges (an example is the function eikx in R3 ). E. Sobolev spaces.We shall need some standard properties of Sobolev spaces, including the Sobolev embedding and trace theorems. These can be found in Reed and Simon [7]. Denote by Hk (N ) the Sobolev space on a manifold N with volume form dv, so that for k a positive integer, Hk (N ) is the space of functions on N for which the function and its rst k derivatives are square integrable. A Hilbert space norm kkk on the set Hk (N ) can be speci ed by choosing a connection ∇ on N and writing " #1=2 R P k dv : (2.50) |∇ 1 · · · ∇ m f|2 kfkk = m 05m5k N Spaces H−k (N ) can then be de ned as the dual spaces to Hk : De nition. For each integer k = 0, the Hilbert space Hk (N ) is the completion of C0∞ (N ) in the norm (2.50); H−k (N ) = Hk (N )∗ .
504
J.L. Friedman, M.S. Morris
Note that for N = Rn with natural volume element and natural at connection, Hk for both positive and negative values of k can be de ned as the completion of C0∞ in the norm 1=2 R n 2 2 k ˆ d |f()| (1 + || ) : (2.51) kfkk = Rn
(Equation (2.51) de nes Hk (Rn ) for any real k; on N one can extend the de nition to all real k by quadratic interpolation [8].) For the spacelike three-manifold M ≡ M0 , we will use the volume element dv = e− dV , where dV is the proper volume associated with the 3-metric hab . The lapse function, e− , appearing in the measure is needed to make the operator L symmetric. Weighted L2 spaces on M will also be needed. De nition. L2; r (M) is the completion of C0∞ (M) in the norm kfk22; r =
R M
dV e− |f|2 (1 + |x|2 )r :
(2.52)
The space L2; r (R3 ) is similarly de ned, with dv the natural volume on R3 . One can de ne on M a smooth global eld of frames that agrees on the at region outside R with a cartesian frame of R3 . A tensor eld will be said to be in any of the spaces de ned above if its components with respect to the frame are in the space. Corresponding to a global frame eld and a choice of spinor structure is a eld of spinor frames on M. A spinor eld will be said to be in any of the spaces above if its components with respect to the frame are in the space. Finally, it is helpful to de ne spaces that incorporate the boundary conditions on @M = I t II . De nition. H1 is the intersection of H1 (M) with functions satisfying the boundary condition (2.38a). H2 is the intersection of H2 (M) with functions satisfying the boundary conditions (2.38a–2.38b). The Sobolev trace theorem implies that elements of H1 (M) have well-de ned values on @M, so H1 and H2 are completions in the H1 and H2 norms of C ∞ functions satisfying the boundary conditions speci ed in the de nition.
III. Existence and Uniqueness Theorems A. Existence theorem for a massless scalar eld. Let R be large enough that for r ¿ R the geometry is at. De ne an exterior region E by E = {p ∈ N|r(p) ¿ R} and an interior region R = N\E. It will be helpful in what follows to introduce a smoothed step function, , that vanishes on R: 0; if p ∈ R (p) = (3.1) 1; if r(p) = R + ; for some ¿ 0.
505
Existence Uniqueness Theorems for Massless Fields
De nition. A scalar eld on N is asymptotically regular at spatial in nity if ◦ ∈ H1 (M); it is asymptotically regular at null in nity if the limits, f(v; r) ˆ = lim r(v; r r) ˆ ; r→∞
ˆ ; g(u; r) ˆ = lim r(u; r r)
(3.2)
r→∞
exist, with f ∈ H1 (I− ); g ∈ H1 (I+ ), where u is the null coordinate t − r. That is, is asymptotically regular if it is an L2 -function with nite energy on N and has well-de ned data with nite energy on I− and I+ . Our result on existence of solutions to the scalar wave equation has the following form: Proposition 1. For almost all spacetimes N, g of the kind described in Sect. II A ( for almost all parameters ), the following existence theorem holds. Let f be initial data on I− for which f and all its derivatives are in L2 (I− ). Then there exists a solution to the scalar wave equation which is smooth and asymptotically regular at null and spatial in nity and which has f as initial data. The proof is given as a sequence of lemmas. For boundary conditions (2.43) speci ed by a xed phase, , we show that the operators L are self-adjoint on a dense subspace of L2 . We follow a method given by Wilcox [9] to obtain explicit eigenfunctions F(; k; x) in a weighted L2 space. For xed boundary phase the eigenfunctions are complete and orthonormal, but, as mentioned earlier, a solution of Eq. (2.29) is a superposition of eigenfunctions, E(k; x) = F( = ! ; k; x) ;
(3.3)
that are not orthonormal. For each !, the function E(k; x) is an eigenfunction of a di erent operator L because the boundary condition depends on ! through the relation = ! . Lemma 1. The operator L with boundary conditions (2.43) is self-adjoint on the space L2 (M) with domain H2 . Proof. Recall that for an operator A with domain D(A) ⊂ L2 (M), a function f ∈ L2 (M) is in D(A† ) if and only if ∃h ∈ L2 (M) such that hf|Agi = hh|gi∀g ∈ D(A). To prove that H2 ⊂ D(L† ) is easy: for f ∈ H2 , L f ∈ L2 (M). Take h = L f. Then, writing := I t II , we have R ↔a dSa e− (fei )D gei : (3.4) hf|L gi = hL f|gi + I tII
Because f and g satisfy the boundary conditions (2.43), we have R R R ↔a ↔a ↔a dSa e−fD g = − dSa e− (fei )D gei = − dSa e−fD g : II
I
(3.5)
I
Thus L is symmetric, hf|L gi = hL f|gi = hh|gi ; and H2 ⊂ D(L† ).
(3.6)
506
J.L. Friedman, M.S. Morris
To prove that D(L† ) ⊂ H2 , we proceed as follows. For f ∈ D(L† ), there is an h ∈ L2 (M) with (3.7) hf|L gi = hh|gi ∀g ∈ D(L ) : That is, the equation L† f = h is satis ed weakly on M. Then, by elliptic regularity, we have f ∈ H2loc (M) and L f = h is satis ed strongly on M. Asymptotic behavior of f follows from the fact that the exterior region E can be regarded as a subspace of Euclidean 3-space, E 3 . Let be the smooth step ˜ function of Eq. (3.1), and letf˜ = f . Thenf˜ is a function on E 3 satisfying ∇2f˜ = h, with h ∈ L2 (E 3 ). Because ∇2 is self-adjoint on E 3 with domain H2 (E 3 ), we have f˜ ∈ H2 (E 3 ), implying f|E ∈ H2 (E). To check that f satis es the boundary conditions, we use the fact, noted in Sect. I (before Eq. (2.1)), that by identifying I and II , one obtains a spacelike copy Mˆ of M with a smooth spatial metric, gˆab . (Mˆ is an arti cially constructed spacelike surface with a handle; it is not a submanifold of our spacetime, N.) The operatorLˆ constructed from gˆab is smooth and elliptic. Because the projection, p : Mt → M , is an isometry there, it agrees with L on the interior of Mt . Let U be a neighborhood of for which p−1 (U ) = UI t UII , with UI and UII disjoint neighborhoods of I and II . If, as expected, the function f ◦ p−1 has a phase that changes by ei across , then we should get a smooth function fˆ on U by requiring f ◦ p−1 ; P ∈ p(UI ) ˆ : (3.8) f(P) = e−i f ◦ p−1 ; P ∈ p(UII ) To see that this is true, rst smoothly truncate f so that it vanishes outside of U1 and UII , and then de ne fˆ as above. Let g of (3.7) similarly have support on U1 ∪ UII , and let h be the function in L2 (M ) given by (3.8). De ne gˆ and hˆ on U1 ∪ UII according to (3.8), with f replaced by g and h, respectively. Then g ∈ H2 (M) ⇒ gˆ ∈ H2 (Mˆ ), and we have ˆ gi ˆ gi hf|L ˆ = hh| ˆ ;
(3.9)
∀gˆ ∈ H2 (Mˆ ) with support on U1 ∪ UII , implying (again by elliptic regularity) that fˆ is smooth across . Thus any f ∈ D(L† ) is a function in H2 satisfying the boundary conditions (2.43), meaning D(L† ) ⊂ H2 . We now nd a set of eigenfunctions that are complete for data on I− . We consider solutions F(; k; x) to Eq. (2.46) which, for r ¿ R, have the form F = (2)−3=2 eik · x + outgoing waves :
(3.10)
To prove existence of the eigenfunctions F, one rst rewrites Eq. (2.46) in the inhomogeneous form (3.11) (!2 + L )’ = ; where ’ is purely outgoing and has compact support. One can do this by using the steplike function (r) of Eq. (3.1), writing F = (2)−3=2 eik · x + Fout :
(3.12)
Existence Uniqueness Theorems for Massless Fields
Then with
507
(!2 + L )F = (!2 + L )Fout − ;
(3.13)
= −(2)−3=2 eik · x (∇ ∇ − 2ik ∇ ) :
(3.14)
The homogeneous equation, (!2 + L )F = 0, is equivalent to the inhomogeneous equation (3.11) with ’ = Fout . Lemma 2. Let k be a sequence of complex numbers with positive imaginary part, such that k → !2 . Consider families {’k ; k } of smooth elds on M, where for each k, the eld k has support on the region r 5 R + , and ’k is the unique asymptotically regular solution to Eq. (3.11) with !2 replaced by k . If k → in L2 (M), then a subsequence {’m } converges in an H1 norm to a smooth outgoing solution ’ to Eq. (3.11). Proof. Let D be a real number greater than R + , MD = {x ∈ M; r 5 D}. Denote by kks; D the norm of Hs (MD ). Suppose rst that k’k k2; D has a bound C independent of k. Since H2 (MD ) ,→ H1 (MD ) is a compact embedding (by the Sobolev embedding theorem) the set {’k } belongs to a compact set of H1 (MD ). Thus there is a subsequence {’m } that converges in H1 (MD ): (3.15) ’m → ’ : We must show that ’ satis es (i) (!2 + L)’ = on MD , (ii) the boundary conditions (2.43), and (iii) we must extend ’ outside MD . (i): To see that ’ weakly satis es (3.11), de ne ∀ ∈ C0∞ (MD ), Q ≡ − hDa (e )|Da (e ’)i + !2 h |’i − h |i :
(3.16)
Add this Q to the quantity hDa (e )|Da (e ’m )i − m h |’m i + h |m i = 0. Then |Q| = | − hDa (e )|Da (e ’)i + !2 h |’m i − h |i + hDa (e )|Da (e ’m )i − m h |’m i + h |m i| 5 |hDa (e )|Da [e (’ − ’m )]i| + !2 |h |’ − ’m i| + (!2 − m ) h |’m i + |h | − m i| :
(3.17)
The limit of the right-hand side of (3.17) is zero as m → ∞: lim k’ − ’m k1; R = 0 ;
(3.18a)
lim k − m k0 = 0 ;
(3.18b)
lim |!2 − m | = 0 :
(3.18c)
m→∞
m→∞
m→∞
Then Q independent of m implies Q = 0. Hence ’ is a weak solution to (!2 + L)’ = in MD , and elliptic regularity implies that it is a strong solution. (ii): Next de ne a neighbourhood U on Mˆ , as in the proof of Lemma 1. And de ne ’ˆ k on U from ’k on MD in exactly the same way that fˆ was de ned in Lemma 1 from f. Then, because ’k is smooth and satis es the boundary conditions
508
J.L. Friedman, M.S. Morris
(2.43) on MD , we have that ’ˆ k is smooth on U . Also, because k has support only ˆ ’ˆ k = 0 on U . Now, outside of U (away from the boundaries), ’ˆ k satis es (k + L) ˆ which is continuous. But, a subsequence {’ˆ m } converges in H1 (U ) to a solution ’, by elliptic regularity ’ˆ is smooth in U and therefore ’ satis es both boundary conditions (2.43) ∀ ∈ C ∞ (MD ). (iii): For r ¿ R + , we have m = 0. Because the space is at outside r = R, and ’m is asymptotically regular, we can use the explicit Green function for ∇2 at + !2 to write √ ↔ ei m |x−y| R ; (3.19) dSy ’m (y)@y ’m (x) = |x − y| |y|=R √ where Im m ¿ 0. Then for R + ¡ r ¡ D, ’(x) = lim ’m (x) = m→∞
↔ ei!|x−y| ; dSy ’(y)@y |x − y| |y|=R R
(3.20)
an outgoing wave. De ning ’ by Eq. (3.20) for r ¿ D we obtain an outgoing C ∞ solution to (3.11) as claimed. The construction has so far relied on the assumption that k’k k2; D , was bounded. If not, the sequence ’˜ k = ’k =k’k k2; D has unit norm and a source ˜k = k =k’k k2; D whose norm converges to zero: lim k˜k k = 0 :
(3.21)
k→∞
This leads to a contradiction. From the previous paragraph, there is a subsequence ˜ m converging to an outgoing solution ’˜ of L ’˜ = 0. But Rellich’s uniqueness theorem (Lemma 3 below) implies ’˜ = 0, whence limm→∞ k’m k0; D = 0. From L’˜ m = ˜m , we have k’˜ m k2; D 5 Ck’˜ m k0; D + km k0 . Then limm→∞ k’˜ m k2; D = 0, contradicting k’˜ m k2; D = 1. Lemma 3 (Rellich’s uniqueness theorem). Let ’ ∈ H2 be outgoing at spatial in nity and satisfy (L + !2 )’ = 0. Then ’ = 0. Proof. In the exterior region E, ’ is a smooth solution to the at space equation (!2 + ∇2 )’ = 0 and can therefore be written as a sum P (2) (3.22) lm h(1) l (!r)Ylm ( ) + lm hl (wr)Ylm ( ) ; (1) (1)∗ converging in Lloc and h(2) are spherical Hankel functions, satis2 . Here hl l = hl fying the Wronskian relation, (2) (2) (1) h(1) l (y)@y hl (y) − hl (y)@y hl (y) =
2 : iy2
(3.23)
Using this relation and orthonormality of the spherical harmonics Ylm , we have, ↔ R R R dS’∗ @r ’ 0 = dV ’∗ (!2 + L)’ − dV (!2 + L)’∗ ’ = R
R
r=R
↔ (1) ↔ (2) 2 2i P 2 (1) [| lm |2 − | lm |2 ] (3.24) [| lm |2 h(2) = l @r hl + | lm | hl @r hl ]R = ! lm P P ⇒ | lm |2 = | lm |2 : (3.25) P
509
Existence Uniqueness Theorems for Massless Fields
In other words, ingoing and outgoing uxes are equal. Then ’ outgoing implies lm = 0 ⇒ lm = 0 ⇒ ’ = 0 outside R. Aronszjan’s elliptic continuation theorem [10] then implies ’ = 0 everywhere on M. Lemma 4. There is a unique solution, F(; k; x), to the equation (L + !2 )F = 0, for which F = (2)−3=2 eik · x + outgoingwaves : (3.26) The map given by
L2 (M) → L2 (R3 ) ;
(3.27)
R ˆ f(x) 7→ f(k) = dkF(; k; x)f(x) ;
(3.28)
is unitary. Proof. An immediate consequence of Lemmas 2 and 3 is that there exists a unique outgoing smooth solution ’ to (3.11). Then Eq. (3.12), relating F to Fout , gives us existence and uniqueness of a smooth solution F of the claimed form. Unitarity is implied by the self-adjointness of L for xed and the fact that R dk is a spectral measure. A detailed proof of unitarity, applicable with essentially trivial changes to our case is given R in Chap. 6 of Wilcox [9]. (For example, Neumann condition,” d3 x[f∇2 g + ∇f · ∇g] = 0, is replaced by Rthe “generalized dV [fe Da (e Da g) + Da (e f)Da (e g)] = 0.) 3 RLemma 5. For almost all the following holds: Let a(k) ∈ L2; n (R ), and let (x) = dkF( = ! ; k; x)a(k). Then (x) ∈ Hn−3=2− (MD ), all ¿ 0.
Proof. Regard M as a subset of R3 and x ∈ M a vector in R3 . Fourier transform F(; k; x), truncating smoothly at x = R: With the smoothed step function of Eq. (3.1), let gy ∈ L2 (M) be given by gy (x) := (2)3=2 eix · y [1 − (x)] ; with norm
3
kgy kL2 (M) = CR 2 ;
(3.29) (3.30)
some C independent of y. From the fact that F, regarded as a map from L2 (M) to L2 (R3 ) is norm-preserving, the function R ˆ k; y) := dV e− F(; k; x)gy (x) (3.31) F(; has norm in k-space ˆ kF(; · ; y)kL2 (R3 ) = CR3=2 ;
∀; y :
(3.32)
ˆ In order to bound the integral of the Lemma, we will bound a norm of F(! ; k; y) in -k space, for any nite interval I = [ 0 ; 1 ] of . It will be convenient to take 1 = m 0 , for some integer m. Writing Q(; k; y) =
ˆ k; y)|2 |!n F(; ; (1 + !2 )n
(3.33)
510
J.L. Friedman, M.S. Morris
we have R∞ R R∞ ! R1 d! d Q(! ; k; y) = d! d!−1 Q(; k; y) 0
I
! 0
0
5
2 ∞R P
d
j=0 0
5
2( j+1)= R 0
d!( j + 1)
2j= 0
R2
R∞
0
0
d d!
! 1 1 + 0 2
1 −1 ! Q 0
!−1 Q :
(3.34)
In the rst inequality, we have used the fact that Q is periodic in . From this relation, we obtain R
"
R
R
dk d Q(! ; k; y) = I
dk +
!¡2= 0 0
5C +
dk
!¿2= 0
R
R2 dk d
!¿2= 0
0
5 C 0 + 1 =
R
!¿2= 0 0
#
R
R
d Q
I
! 1 1 + 0 2
!−1 Q(; k; y)
R2 dk dQ(; k; y) 0
00 3
5 C +C R ;
(3.35)
where the last inequality follows from Eq. (3.32). Integrating over y, we have, R
d dkdy
ˆ k; y)|2 !2n |F(! ; ¡∞ 2 n (1 + ! ) (1 + y2 )3=2+
(3.36)
ˆ ⇒ !n F(! ; k; y) ∈ L2 (I ) ⊗ L2; −n (R3 ) ⊗ L−3=2− (R3 ) ⇒ ∇n F(! ; k; x) ∈ L2 (I ) ⊗ L2; −n (R3 ) ⊗ H−3=2− (MD ) ⇒ F(! ; k; x) ∈ L2 (I ) ⊗ L2; −n (R3 ) ⊗ Hn−3=2− (MD ) :
(3.37)
Thus, for almost all , F(! ; k; x) ∈ L2; −n (R3 ) ⊗ Hn−3=2− (MD ) R ⇒ dka(k)F(! ; k; x) ∈ Hn−3=2− (MD )
(3.38)
for a(k) ∈ L2; n (R3 ). Lemma 6. Let out be the outgoing eld in Minkowski space of a smooth, spatially bounded source, . Then data for out on I+ is well-de ned and smooth.
Existence Uniqueness Theorems for Massless Fields
511
Proof. The lemma follows from the explicit form of the at space retarded solution, R (t = u + r − |˜x − y ˜ |; y ˜) out (t = u + r;˜x) = d3y ; |˜x − y ˜|
(3.39)
where r = |˜x|. Writing |˜x − y ˜ | = r − xˆ · y ˜ + O(|˜ y=˜x|); we have |(t = u + r − |˜x − y ˜ |; y ˜ ) − (t = u + xˆ · y ˜; y ˜ )| ¡ K|˜ y=˜x|max|| ˙ :
(3.40)
This bound holds for all values of r along the null ray at constant u; ; , with max|| ˙ the maximum value of || ˙ on the (compact) intersection of the support of with the past light cones from points of the null ray. Then data on I+ takes the explicit form, R ˜; y ˜) : (3.41) lim r out (t = u + r;˜x) = d3y(t = u + xˆ · y r→∞
Lemma 7. Let be such that the conclusion to Lemma 5 holds. Let a(k) ∈ L2; n (R3 ), all n ∈ Z, and let R (3.42) (t; x) = dka(k)F(! ; k; x)e−i!t : Then is smooth on N, and has data on I− given by Eq. (2.8), ˆ = lim r (v; r r)
r→∞
i R∞ d!!a(−!r)e ˆ i!v : (2)1=2 0
(3.43)
That is, if has for each harmonic the same ingoing part as does a solution 0 in Minkowski space, then has the same data on I− that 0 has. Proof. The smoothness of a(k) implies, by Lemma 5, that the integral of Eq. (3.42) de nes (0; · ) ∈ Hn (MD )∀n. Since a(k)e−i!t ∈ L2 (R3 ), we have (t; · ) ∈ Hn (Mt; D ) (∀n; t), and (t; · ) is smooth on each surface Mt , because it is smooth on Mt; D for all D. To see that is smooth on N, note rst that it is smooth on the throat that joins Mt to its extension Mt+ . This follows from the fact that the location of the cylinder C removed from N is arbitrary: if, instead of removing C from N, one removes a di erent, cylinder C 0 , disjoint from C, one obtains the same , because the eigenfunctions on M from which is constructed are unique by Lemma 3. Thus (t; · ) is smooth on each Mt0 and, in particular, on the throat . Because !a(k) is similarly in L2; n (R3 )∀n, @t is smooth on each Mt and on the throat. Finally, is smooth on N, because N is covered by globally hyperbolic subspacetimes (U; g|U ) which have as Cauchy surfaces U ∩ (Mt ∪ ∪ Mt+ ), for some t; and on (U; g|U ), satis es the smooth hyperbolic equation ∇ ∇ = 0 with smooth initial data. We will rst relate data on I+ to a(k) and then reverse the argument, deducing the data on I− from that on I+ . As in Eq. (3.12), we can write F(! ; k; x) = (x)F0 (k; x) + Fout (! ; k; x) ;
(3.44)
where F0 (k; x) = (2)−3=2 eik · x . Then = (x) 0− + out ;
(3.45)
512
J.L. Friedman, M.S. Morris
where R out = dka(k)Fout (! ; k; x)e−i!t ;
R 0− = dka(k)F0 (k; x)e−i!t :
(3.46)
The integral de ning out converges in L2 (MD ) to a smooth function, because the integrals de ning and the Minkowski-space solution 0 so converge. and out are smooth on M because they are smooth on MD for all D. We can use the spherical harmonic basis {Ylm } for L2 (S 2 ) to write, for the exterior region E, 1=2 Pl 2 ∗ ˆ i jl (!r)Ylm (k)Ylm (ˆ x) ; F0 (k; x) = lm P x) ; Fout (; k; x) = lm (; k)h(1) l (!r)Ylm (ˆ
(3.47) (3.48)
lm
with the sums converging in L2 (S 2 ). Then P 1 R d!!2 clm (!)h(1) x)e−i!t ; out = √ l (!r)Ylm (ˆ 2 lm where clm (!) =
√ R 2 d k a(k) lm (! ; k) :
(3.49)
(3.50)
We can similarly rewrite the convergent integrals for 0 and in E: P 1 R d!!2 2alm (!)jl (!r)Ylm (ˆ x)e−i!t ; 0− = √ 2 lm P 1 R (2) d!!2 [blm (!)h(1) x)e−i!t ; = √ l (!r) + alm (!)hl (!r)]Ylm (ˆ 2 lm where
(3.51)
(3.52)
R ∗ ˆ (k) alm (!) = il d k a(k)Ylm
(3.53)
blm = alm + clm :
(3.54)
and The construction of out from outgoing waves Fout of Eq. (3.12), satisfying the inhomogeneous equation (3.11) expresses out on E as the retarded solution on Minkowski space, (3.41), to out = , with := 0 . If one writes of Eq. (3.41) as a sum of spherical harmonics and uses Eq. (3.39) to relate clm to lm , data (3.41) on I+ for out becomes P 1 R d!! i−(l+1) clm (!)Ylm (ˆ lim r out (t = u + r;x) ˆ =√ x)e−i!u : r→∞ 2 lm
(3.55)
A nite-energy solution on Minkowski space of the form (3.51) has data on I+ given by P 1 R d!! i−(l+1) alm (!)Ylm (ˆ ˆ =√ lim r 0− (u + r;x) x)e−i!u : 2 lm
r→∞
(3.56)
513
Existence Uniqueness Theorems for Massless Fields
The rst part of the proof of Lemma 3 implies for the solution to L = 0 on E, eigenfunctions of L satisfy, for each !, P P |alm (!)|2 = |blm (!)|2 : (3.57) lm
lm
Thus the data induced by on I satis es the constraints imposed on a(k): b(k) ∈ L2; n (R3 ), all n ∈ Z. As a result, the argument just given, with I− and I+ reversed, implies (3.58) = 0+ + in ; +
with 0+ and in smooth, and given on E by expressions P 1 R d!!2 2blm (!)jl (!r)Ylm (ˆ x)e−i!t ; 0+ = √ 2 lm −1 R d!!2 clm (!)h(2) x)e−i!t ; in = √ l (!r)Ylm (ˆ 2
(3.59) (3.60) (3.61)
converging in L2 (MD ) for each D; and data on I− is well-de ned, with P 1 R lim r (v − r; rˆ d!! i(l+1) (blm + clm )Ylm (ˆ x) = √ x)e−i!v ; 2 lm
r→∞
(3.62)
whence, using Eq. (3.54), we recover (3.43). Lemma 8. Let be a smooth retarded solution on Minkowski space to the massless scalar wave equation whose source has support within a compact spatial region r ¡ R, and suppose that has zero data on I− and data f on I+ with nite energy norm, kfkH1 (I+ ) . Then has nite energy norm k kH1 (H) on a spacelike hyperplane H of Minkowski space. Proof. The proof will follow from conservation of energy, but, because the energy density of a massless scalar eld does not include a term proportional to 2 , we will need to consider both the energy of and of a time integral of to bound k k1 on a spacelike hyperplane. Let H be the surface t = 0. We need only consider the retarded eld of a source that vanishes for t ¿ 0, because the eld on t = 0 depends only on values of for t ¡ 0. Let t be the timelike Killing vector @t , and let J = −T t , with T the energy-momentum tensor of . Let E(u0 ) be the energy of on the part H(u0 ) of H with r ¡ |u0 |: R dS J ; (3.63) E(u0 ) = H(u0 )
with dS along the normal ∇ t. Let Iu0 ; v0 be the part of the past null cone v = v0 lying to the future of t = (u0 + v0 )=2; and let Ju0 ; v0 be the part of the future null cone u = u0 between t = 0 and t = (u0 + v0 )=2. Then H(u0 ), Iu0 ; v0 , and Ju0 ; v0 form the boundary of a source-free region of Minkowski space. Denote by FI (u0 ; v0 ) and FJ (u0 ; v0 ) the ux through Iu0 ; v0 and Ju0 ; v0 , respectively, R R dS J ; FJ = dS J ; (3.64) FI = Iu0 ; v0
Ju0 ; v0
514
J.L. Friedman, M.S. Morris
choosing dS along the normals ∇ u and ∇ v. Then ∇ J = 0 implies E(u0 ) + FJ (u0 ; v0 ) = FI (u0 ; v0 ) :
(3.65)
A massless scalar eld, satis es the dominant energy condition: the vector J is future-directed non-spacelike. Consequently E(u0 ); FI (u0 ; v0 ), and FJ (u0 ; v0 ) are all positive, and we have (3.66) E(u0 ) 5 FI (u0 ; v0 ) : The retarded eld of a smooth, spatially compact source has asymptotic behavior (u; x) = f(u;x)=r ˆ + O(r −2 ) ; ∇ v∇ h @
= @u f(u;x)=r ˆ + O(r −2 ) ; = O(r −2 ) ;
(3.67)
where h = + t t is the projection orthogonal to t . Consequently, lim FI (u0 ; v0 ) = FI+ (u0 ) ;
(3.68)
v0 →∞
where FI+ (u0 ) is the ux through the part of I+ (u0 ) lying to the future of u = u0 : FI+ (U ) =
R0 −u0
R du d |@u f(u;x)| ˆ 2:
(3.69)
Since E(u0 ) is independent of v0 , we have for all u0 , E(u0 ) ¡ FI+ (u0 ) . Hence the scalar eld has nite energy on H: E= In order to bound
lim E(u0 ) 5
lim FI+ (u0 ) ¡ kfkH1 (I+ ) :
u0 →−∞
R
H
dV
2
u0 →−∞
(3.70)
, we essentially repeat the argument for Ru ˜ = du0 (u0 ; x) :
(3.71)
0
Equation (3.67) implies that ˜ has data f˜ on I+ , where Ru f˜ = du0 f(u;x) ˆ :
(3.72)
0
Bounding the energy E˜ of E˜ =
R H
=
R
H
dS J˜ = dV
1 2
2
:
on H bounds the L2 norm of
, because
R
1 dV [(@u ˜ )2 + (@u ˜ − @r ˜ )2 + r −2 (@ ˜ )2 + (r sin )−2 (@’ ˜ )2 ] 2 H (3.73)
Because vanishes for u = 0, ˜ satis es the scalar wave equation with source : ˜ Using 1 1 2 2 2 @’ ; (3.74) = −2@u @r + @r + 2 @ + r sin2
515
Existence Uniqueness Theorems for Massless Fields
and Ru @u ˜ (u; x) = (u; x) = du0 @u0 (u0 ; x) ;
(3.75)
0
we have Ru ˜ = du0
(u0 ; x) = (u; ˜ x) :
(3.76)
0
Then Eqs. (3.64, 3.65, 3.66) hold for the energy uxes F˜I (u0 ; v0 ), and F˜J (u0 ; v0 ), and, in particular, ˜ 0 ) 5 F˜I (u0 ; v0 ) : E(u
(3.77)
However, ˜ is not the retarded solution to the scalar wave equation with source , ˜ so we must check that the asymptotic conditions (3.67) on hold for ˜ . This is easy: A function O(r −n ) has, for r greater than some r0 , the form g(u; x)=r n where, for xed u; r, ˆ g is a bounded function of r. In our case, is smooth, and the corresponding functions g are R u smooth and bounded in a compact domain [0; u] for u (and hence for u; r). ˆ Thus 0 r −n g(u0 )du0 = O(r −n ), and it follows that ˜ satis es (3.67). The analogues of Eqs. (3.69) and (3.70), R0 R R0 R ˜ x)| ˆ 2 = du d |f(u;x)| ˆ 2 ¡∞ ; F˜I+ (u0 ) = du d |@uf(u; u0
(3.78)
u0
and E˜ =
˜ 0) 5 lim E(u
u0 →−∞
lim F˜I+ (u0 ) ¡ kfkL2 (I+ ) ;
u0 →−∞
(3.79)
together with Eq. (3.73), imply k kL2 (H) ¡ ∞. Finally, Eq. (3.70) and the bound on k kL2 (H) implies k kH1 (H) ¡ ∞. Corollary. Under the assumptions of Lemma 7; out of Eq. (3.46) is asymptotically regular at spatial in nity. Proof. As in Lemma 7, out is on E the retarded solution of Eq. (3.39) with smooth spatially bounded source . Then Lemma 8 implies out is regular at spatial in nity. Proof of Proposition 1. The proof is essentially immediate from the lemmas. By Lemmas 6 and 7, with = 2 Re , for almost all ; is a smooth solution on N to the scalar wave equation, with data f on I− . Finally, on E, = 0 + out ; where 0 is asymptotically regular at spatial in nity because it is a solution to the
at-space wave equation with data having nite energy-norm on I− and out is regular by the Corollary to Lemma 8.
516
J.L. Friedman, M.S. Morris
B. Restricted uniqueness theorem for a massless scalar eld.Because the system is linear, uniqueness means that the only solution to Eq. (2.29) with zero data at I− is = 0. This is not true in the geometrical optics limit, because closed null geodesics c() can loop through the wormhole and never reach I; and one might worry that there are analogous smooth solutions that vanish in past and future and have zero data on I− . The following restricted uniqueness theorem rules them out: smoothed versions of a looping zero-rest mass particle spread and reach I. Denote by EKt an energy norm of the eld on a compact submanifold Kt ⊂ Mt , where manifolds in the family {Kt } are related by time-translation along the Killing trajectories: R (3.80) EKt = dV e− [|∇|2 + ||2 ] = kkH1 (Kt ) : Kt
Proposition 2. If (t; x) is a smooth solution to ∇ ∇ = 0; having nite energy and zero initial data on I− ; and if limt→±∞ EKt = 0 for any family of compact (time-translation related) Kt ; then = 0. Proof. The result is a corollary of Rellich’s uniqueness theorem for each mode. The requirement that the energy norm on any compact K vanishes as t → ∞ allows one to transform the solution as follows. Denote by HT a smoothed step function, 1; |t| 5 T (3.81) HT (t) = 0; |t| = T + ; and write the Fourier transform of in the form R ˆ (!; x) = lim (2)−1=2 dtHT (t)e−i!t (t; x) : T →∞
We have 0= =
R R
(3.82)
dtHT (t)e−i!t ∇ ∇ (t; x) e− dte−i!t [HT (!2 + L) + 2i!@t HT − @2t HT ](t) ;
whence R R dte−i!t [HT (!2 + L)(t) 2 5 dt(|2!@t HT (t)| + |@2t HT (t)|) 2 :
(3.83) (3.84)
@2t HT
are bounded functions of t with compact support, we can Because @t HT and write R 2 R R dV dtHT e−i!t (!2 + L)(t; x) 5 C max dV |(t; x)|2 ; (3.85) |t|∈[T; T +] K
K
or
R
dtHT e−i!t (!2 + L)(t; x) 2
L2 (Kt )
5 C max kk2L2 (Kt ) ; t∈[T; T +]
(3.86)
where C is a constant independent of T . From our assumption that k|k2L2 (Kt ) → 0 as t → ∞, we have (3.87) lim max kk2L2 (Kt ) = 0 T →∞ t∈[T; T +]
implying
2
R lim dtHT e−i!t (!2 + L)(t; x) L (K ) = 0 : (3.88) 2 t T →∞ R ˆ It follows that for all !, dte−i!t (!2 + L)(t; x) = (!2 + L)(!; x) vanishes for almost all x. Finally, since nite energy and zero initial data imply purely outgoing at spatial in nity, by Rellich’s theorem must vanish as well.
517
Existence Uniqueness Theorems for Massless Fields
C. Other massless elds.Extending these results to Weyl and Maxwell elds appears straightforward. The statement of the existence theorem is identical, with the scalar eld replaced by a Weyl spinor A and (say) a vector potential for a free Maxwell eld F in a Lorentz gauge with A t = 0. The statement of uniqueness for a Weyl eld is identical to that for a scalar eld; for a Maxwell eld it must be modi ed to exclude the time independent solutions that have nonzero ux threading the handle: If F is a smooth solution to
with
∇ F = 0;
∇[ F ] = 0
R
R∗
F dS = 0 =
S
(3.89)
F dS ;
S
having nite energy and zero initial data on I− , and if limt→±∞ EKt = 0 for any family of compact (time-translation related) Kt , then F = 0. For a static spacetime, the proofs appear to require only minor changes. Recall rst that the wave equations for Weyl and Maxwell elds are conformally invariant [6]: If A and A satisfy the Maxwell and Weyl Equations (2.6) and (2.7), respectively, for the metric g , the elds A and e A satisfy the same equations for the conformally related metric e−2 g . As a result, it suces to look at solutions to the hyperstatic metric of the form, g = −@ t@ t + h :
(3.90)
That is, a static metric of the form, (2.2), is related to a hyperstatic metric of the form (3:90) by a smooth conformal factor e2 ; because e2 has the value 1 in a neighborhood of I, the conformally related solutions have the same data on I− .2 As in the scalar wave case, we can decompose the wave operator for vector and spinor elds in the manner (3.91) ∇ ∇ = −$2t + L ; with L = Da D a , as in Eq. 2.48. Then the harmonic components ; A satisfy (!2 + L) = 0;
(!2 + L) A = 0 :
(3.92)
The operator L is symmetric on L2 (M), de ned for vector and spinor elds on R 0 M by the norm h | i = M dV e− | |2 , with | |2 ≡ a and | |2 ≡ tAA0 A A for vectors and spinors, respectively. The de nition of the Sobolev spaces of Sect. II E are similarly extended automatically to vectors and spinors by contracting vector indices with hab and spinor indices with tAA0 . (In a spinor frame associated with an orthonormal frame for which t is the timelike frame vector, tAA0 has components tII 0 = II 0 . Sobolev spaces of Sect. II E are similarly extended to vector and spinor elds. Finally, the space H2 , is de ned for vectors and spinors as the set of elds in H2 (M) satisfying the boundary conditions (2.40,2.41).) Here is a sketch of the proofs. The existence theorem involves the same set of lemmas. Lemma 1, self-adjointness of L on the spaces H2 of elds in H2 (Mt ) again follows from the symmetry of L , re ecting the fact that boundary conditions imply smoothness of the elds ˆ A and ˆ on Mˆ . 2 We thank the referee for pointing out this way to extend our argument to static from hyperstatic metrics.
518
J.L. Friedman, M.S. Morris
Lemma 2 and its proof can be repeated as written with and regarded as vectors or spinors. In the proof of Lemma 3, each of the components of ( A ) with respect to covariantly constant frames (spinor frames) on the exterior region E are scalars satisfying Eqs. (3.22), (3.24), and (3.25). They therefore vanish on E and, by elliptic continuation, vanish on all of M. In Lemma 4, eigenfunctions for vector and spinor elds are de ned by F A = (2)3=2 A eik · x + ’ A ;
(3.93a)
F = (2)3=2 eik · x + ’ ;
(3.93b)
where and A are covariantly constant on E and satisfy k = 0 = kAA0 A . The rst part of Lemma 4, existence and uniqueness of eigenfunctions F ; F A , is again immediate from Lemmas 2 and 3. The second part, unitarity, again appears to be a straightforward extension of Chap. 6 in Wilcox [9], but here there are details we have not checked. The proof of Lemma 5 goes through as written if a in Lemma 5 is interpreted as a vector (spinor) with a covariant index and their product is read as a dot product: R e.g., f(x) = dkF ( = ! ; k; x)a (k). In Lemma 6, the equations are correct as written for the components of and A , but one must use, in addition, the fact that the elds satisfy the peeling theorem for at space [6] to complete the characterization of their behavior. In Lemma 7, the proof of smoothness can be read as written. The proof of regularity at I− and the recovery of initial data relies on a spherical harmonic decomposition that can be modi ed in a standard way for spinors and vectors. Finally, Lemma 8 and the proof of Propositions 1 and 2, can be read as written, with the change in the statement of Proposition 2 given above.
IV. The Cauchy Problem for More General Spacetimes The work reported above shows the existence of an unexpected class of spacetimes for which an existence theorem and at least a partial uniqueness theorem can be proved. How broad is the class of spacetimes for which a generalized Cauchy problem is well-de ned? Examples of spacetimes with CTCs for which one can prove existence and uniqueness for linear wave equations are not dicult to nd, if one allows singularities and does not require that solutions have nite energy [13], and we will display some examples below. Finding examples of nonsingular geometries with CTCs and a well-de ned Cauchy problem is more dicult, but the earlier work by Morris et. al. [2] is persuasive: Their time-tunnel examples are asymptotically
at spacetimes in which CTCs are con ned to a compact region and for which there appears to be a well-de ned initial value problem for data on a spacelike hypersurface to the past of the nonchronal region, the set of points through which CTCs pass. In the present section we present a uniqueness result complementary to that of Sect. III B, a conjecture on existence and uniqueness, and examples of spacetimes that do or do not have a well-posed initial value problem.
Existence Uniqueness Theorems for Massless Fields
519
A. A result on uniqueness for spacetimes with a compact nonchronal region.In Sect. III B we ruled out, for the static spacetimes considered, a lack of uniqueness corresponding to a eld forever trapped inside the nonchronal region, a smooth analog of a closed null geodesic. Here we show, for spacetimes with a compact nonchronal region, A, that when solutions to the Cauchy problem exist, they are unique outside A. Because data is now given on a spacelike surface (instead of I− ), we need no longer restrict consideration to massless wave equations. Initial data on a spacelike hypersurface M for a solution to the scalar wave equation, (−∇ ∇ + m2 ) = 0 ;
(4.1)
will mean the pair of functions = |M ;
= n ∇a |M ;
(4.2)
where n is a unit normal to M. Proposition 3. Let N; g be a smooth; asymptotically at spacetime with regions F and P to the future and past of a compact 4-dimensional submanifold A de ned by F = N\J − (A) and P = N\J + (A); where both F and P are globally hyperbolic and foliated by complete spacelike 3-manifolds. Suppose for arbitrary smooth data with nite energy and a Cauchy surface MP of P that the scalar wave equation has a solution on N with nite energy on F; and suppose that for arbitrary smooth data with nite energy and a Cauchy surface MF of F that the scalar wave equation has a solution on N with nite energy on P. Then; solutions on N with nite energy in N\A are unique in N\A. Proof. The proof relies on the nondegeneracy of the symplectic form, R !M (f; g) := dS (f ∇ g − g∇ f) ;
(4.3)
M
and the fact that ! is independent of hypersurface. That is, let B ⊂ N be a slab, a 4-dimensional submanifold of N bounded by two submanifolds M and M0 in the foliations of P and F that coincide outside of a compact region. Then, for any two solutions with nite energy, R 0 = d4 Vf(∇ ∇ − m2 )g B R = dS (f ∇ g − g∇ f) @B
= !M (f; g) − !M0 (f; g) :
(4.4)
Suppose the theorem is false. Then there is a solution to Eq. (4.1) with zero initial data on MP (say) and with nonzero and with nite energy somewhere on N\A. Thus a hypersurface M in the foliation F has nonzero data for and we can deform M outside the support of (in the intersection of F and P) to ˙ on M such coincide with MP . Because ! is non-degenerate, there is data ( ; ) that (4.5) !M (; )-0 : But, by hypothesis, a solution exists on N, corresponding to the initial data on M; and the fact that ! is independent of hypersurface implies !MP (; )-0, contradicting the assumption that has vanishing initial data on MP .
520
J.L. Friedman, M.S. Morris
Fig. 3. Misner space is the region between the two null rays ‘ and B‘, with points of the null boundaries identi ed by the boost B. The curve C = CC 0 is a chronology horizon, a closed null geodesic that separates the nonchronal region above it from the globally hyperbolic spacetime to its past. Equivalently, Misner space is the quotient of the half of Minkowski space lying to the left of the null line L by the group of boosts generated by B
Corollary. Proposition 3 holds for the Maxwell; Dirac; and Weyl elds. Proof. Each of the three elds has a conserved symplectic product !. The proof goes through as stated, with the symplectic product and initial data of a scalar eld replaced by that of the Maxwell and Dirac elds and with the energy norm for a scalar eld replaced by vector and spinor energy norms. Because the billiard-ball examples considered by Echeverria et al. [3, 11] have a multiplicity of solutions for the same initial data, uniqueness in spacetimes with CTCs is likely to hold only for free or weakly interacting elds. Because solutions seem always to exist for the billiard ball examples in the spacetimes they considered, it may be that classical interacting elds have solutions on spacetimes for which solutions to the free eld equations exist. B. A conjecture.The solution to the problem posed at the beginning of this section – to delineate the class of spacetimes for which a generalized Cauchy problem is well-de ned – is, of course, not known. We present a conjecture here, motivated by examples of geometries which appear to have a well-de ned Cauchy problem and by examples of others for which either existence or uniqueness fails. We will motivate the conjecture with a brief reminder of some examples that are by now well known; a more detailed discussion of additional spacetimes will be given in Sect. IV C. A helpful 2-dimensional example of a spacetime where the Cauchy problem is not well de ned is Misner space. This is the quotient of the half of Minkowski
Existence Uniqueness Theorems for Massless Fields
521
space on one side of a null line L by the subgroup {1; B±1 ; B±2 ; : : :} generated by a boost B about a point of the null line. Equivalently, if ‘ is a null line parallel to L, and B‘ is its boosted image, then Misner space is the strip between ‘ and B‘, where boundary points related by B are identi ed (see Fig. 3). Misner space has a single closed null geodesic, C = CC 0 in the gure, and the past P of C is globally hyperbolic. The future of C is nonchronal, so C is a chronology horizon, a Cauchy horizon that bounds the nonchronal region. Initial data for the scalar-wave equation can be posed on a Cauchy surface M of P, but solutions have divergent energy on the chronology horizon. The reason solutions diverge is clear in the geometrical optics limit. A light ray , starting from M, loops about the space and is boosted each time it loops. Because loops an in nite number of times before reaching C, its frequency and energy diverge as it approaches the horizon. The ray is an incomplete geodesic: it reaches the horizon in nite ane parameter length, because each boost decreases the ane parameter by the blueshift factor ≡ [(1 + V )=(1 − V )]1=2 , with V the velocity of the boost.3 This behavior is not unique to Misner space: A theorem due to Tipler [12] shows that geodesic incompleteness is generic in spacetimes like Misner space in which CTCs are “created” – spacetimes with a nonchronal region to the future of a spacelike hypersurface. In more than two dimensions, however, the existence of incomplete null geodesics like does not always imply that the chronology horizon is unstable. This is because there may be only a set of measure zero of such geodesics so that the energy may remain nite on the chronology horizon. For the time-tunnel spacetimes considered in refs [2, 3], a congruence of null rays initially parallel to spreads as the rays approach the chronology horizon. When the spreading of the rays overcomes the successive boosts (when the fractional decrease in ux is greater than the fractional increase in squared frequency), the horizon is stable in the geometrical optics approximation. We suggest that this is the generic situation – that, roughly speaking, when a spacetime is stable in the geometric optics limit, its Cauchy problem is well de ned. To obtain a more precise statement of the conjecture, we proceed as follows. Hawking [17] has a related de nition of the optical stability of a Cauchy horizon, involving the divergence and boost of a closed null geodesic generator, but it is not directly applicable to the more general situation we consider. Let N; g be an asymptotically at spacetime, let U; g|U be a globally hyperbolic subspacetime, and let V ⊂ U be compact. Then U; g|U can be smoothly foliated by a family of spacelike hypersurfaces. Let St be the restriction to V of such a foliation of U . Let c(x; ) be a smooth null geodesic congruence, transverse at = 0 to a spacelike surface ⊂ N, with an ane parameter and c(x; 0) = x ∈ . Call c regular if it is a subcongruence of similarly de ned smooth null congruence c(x; ˆ ) transverse to at = 0, where the boundary of the support of cˆ lies entirely outside the support of c on . Let k a (x; ) be the vector eld tangent to c. For xed x0 , the curve c(x0 ; ) intersects each surface St in V at most countably many times, at values {n (x0 )} of the ane parameter . Let Wn be the nth 3 That is, trajectories of a (locally-de ned) timelike Killing vector cross the null geodesic at a sequence of points. The Killing vector can be used to compare the ane parameter at succesive crossing points by time-translating a segment of the geodesic to successively later segments. Compared in this way, the ane parameter of a given segment will be less than that of the next segment by the blueshift factor [(1 + V )=(1 − V )]1=2 .
522
J.L. Friedman, M.S. Morris
intersection of the congruence c with St : S Wn = c(x; n (x)) :
(4.6)
x
To de ne an energy carried by the null congruence, we introduce a density (x; ) by setting (x; 0) = 0 and extending to other values of by the conservation equation, (4.7) ∇a (k a ) = 0 : The energy carried across St by c in the nth crossing is then PR a b k k na nb dS : E=
(4.8)
n Wn
Call a neighborhood V optically stable if, for each St and each regular congruence c, E is nite. Call the spacetime N; g optically stable if for each globally hyperbolic subspacetime U; g|U , every open V with V ⊂ U is optically stable. The existence part of our conjecture can be stated as follows. Existence Conjecture. Let N; g be a smooth, asymptotically at spacetime for which past and future regions P = N\J + (A) and F = N\J − (A) of a compact 4dimensional submanifold A are globally hyperbolic. If N; g is optically stable, the Cauchy problem for massless wave equations (for scalar, Maxwell, and Weyl elds) is well-de ned. A conjecture relating uniqueness for massless elds to uniqueness in a geometicoptics sense is easier to formulate. Uniqueness Conjecture. Again let N; g be a smooth, asymptotically at spacetime for which past and future regions P = N\J + (A) and F = N\J − (A) of a compact 4-dimensional submanifold A are globally hyperbolic. Let S± be Cauchy surfaces for NJ ± (A). If all but a set of measure zero of null geodesics intersect S+ and S− , then solutions to massless wave equations on N are unique for initial data on S− (and for initial data on S+ ). Because the instability of massive elds also corresponds to an instability of particles moving along trajectories that become null as one approaches the chronology horizon, it may be that massive wave equations also have a well-de ned Cauchy problem for the same spacetimes. C. Examples and counterexamples. For no asymptotically at spacetime in 4-dimensions, in which CTCs are con ned to a compact region, are we aware of a rigorous demonstration that nite-energy solutions to the scalar wave equation do exist for arbitrary initial data, or that solutions are unique. There may also be no published counterexamples, 4-dimensional asymptotically at spacetimes with a compact nonchronal region (more precisely, spacetimes satisfying condition (i) of the conjecture) for which the nonexistence or nonuniqueness of solutions to free eld equations is proven; but counterexamples like this are not dicult to nd. We present below two examples of asymptotically at spacetimes which are globally hyperbolic to the past and future of a compact region; one can prove for one of them that no solution exists for generic data and for the other that solutions are not unique. The examples show that, without some requirement akin to the optical stability of our conjecture, there can be too many closed geodesics to allow
Existence Uniqueness Theorems for Massless Fields
523
Fig. 4. A piece of Misner space is used in the construction of a 4-dimensional spacetime N with a compact nonchronal region and an unstable chronology horizon. The lines pq and p0 q0 lie in the covering space, the half of Minkowski space lying to the left of v = 0, and they are identi ed by the boost that de nes Misner space. In N, these lines can be regarded as lying along the trajectory of the wormhole mouth
a well-de ned initial value problem. It might also be straightforward to show, for the time-tunnel spacetimes of Refs. [1–3], that whenever the chronology horizon is optically unstable, it is genuinely unstable for elds with smooth initial data. The rst example is a time-tunnel spacetime like those of Morris et al. but with a metric that is everywhere smooth and is chosen to induce a at 2-metric on the part of the identi ed spheres that face each other. The spacetime is also at between the at pieces of the spheres, so the geometry includes a region isometric to a piece of (Misner space)×E 2 , where E 2 is at Euclidean 2-space. For de niteness, we shall identify it with the following piece of Minkowski space. Regard Misner space as the strip of 2-dimensional Minkowski space between the parallel null lines, v = −A and v = −A (V ), with boundary identi ed by a boost with velocity V as above; and take a nite section S of that strip bounded by the bottom half of the hyperbola uv = A2 and by the left half of the hyperbola uv = −A2 , as in Fig. 4 (here u = t − z; v = t + z are the usual null coordinates). Finally, take as the piece of 4-dimensional Minkowski space S × D, where D is the disk x2 + y2 ¡ B2 , and B ¿ A. Spacelike sections of S × D prior to the horizon, C × D, are cylinders with circular cross section in E 2 . In the full spacetime, N, the surface C × D is again a part of the chronology horizon. The key to showing that the chronology horizon of N is unstable is to note that the past of points on C × D with x = y = 0 lies entirely in C × D. This is most easily seen using the universal covering space of S × D. Misner space has as its universal covering space the half of 2-dimensional Minkowski space to the left of v = 0. The corresponding cover S × D of S × D is the part of 4-dimensional Minkowski space bounded by the 3-surfaces v = −A; v = −A ; uv = A, lower branch; uv = −A,
524
J.L. Friedman, M.S. Morris
left branch; and x2 + y2 ¡ B2 . The past light cone in S × D of a point on C × (0; 0) has maximum value of x2 + y2 , where it meets the boundary uv = A, and calculation shows that the intersection is a surface with x2 + y2 ¡ A2 . Since, by construction, A2 ¡ B2 , the past light cone of a point of C × {(0; 0)} never intersects the boundary x2 + y2 = B2 . Thus every point P ∈ S × D to the past of C × {(0; 0)} is in the domain of dependence of the spacelike boundary uv = A of S × D; and every point P ∈ S × D to the past of C × (0; 0) is then in the domain of dependence of the boundary uv = A of S × D. Data on uv = A that is independent of x and y yields a solution that diverges in S × D because the solution is identical in the domain of dependence of uv = A to the divergent solution in Misner space. Finally, by picking a spacelike hypersurface of the full spacetime that agrees with the uv = A surface in S × D, we obtain data on a spacelike hypersurface to the past of the chronology horizon for which no nite-energy solution exists to the scalar wave equation for generic smooth initial data with nite energy. The example of a geometry with compact nonchronal region for which uniqueness fails depends on a construction suggested by Geroch [18]. Although 2-dimensional geometries obtained by removing slits and identifying sides are singular [19], it is possible in 4-dimensions to build smooth geometries in a similar way. The construction relies on the following observation. Lemma 9. Any smooth compact 4-dimensional spacetime with boundary can be embedded (i) in a smooth compact spacetime without boundary; and (ii) in a smooth spacetime that is isometric to Minkowski space outside a compact region. Proof. The technique is borrowed from references [16, 20] (see also [21]). (i): Any manifold M with boundary can be embedded in a compact manifold M˜ by attaching a second copy M 0 of M to the outward side of . Let U be a collar of M , a neighborhood U ∼ = × I with boundary 0 t . A Lorentzian metric g on M can be extended to a Lorentzian metric g˜ on M˜ precisely when a direction eld tˆ of timelike directions on U can be extended to a timelike direction eld on M˜ . Now ˆt can always be extended to a Morse direction eld on M˜ , a direction eld that has isolated zeroes, at each of which the line-element eld is tangent to a vector eld with index ±1. By cutting out a ball B4 containing each zero and gluing in a copy || P 2 \B4 for each zero of index of RP 4 \B4 for each zero of index 1 and a copy of C −1, we can extend the line element to a nonvanishing eld on the interior. (ii): The proof here is nearly identical. First embed M in M˜ as in (i). If one removes a ball B4 from M 0 then one can put a at metric on a neighborhood V of the spherical boundary @B4 that makes V into a copy of Minkowski space outside a ball. One is again asking to extend a direction eld on a new boundary, U t V to the 4-manifold that it bounds, and the construction proceeds as in (i). Finally any Lorentzian metric g on the compact manifold-with-boundary, M 0 (M 0 \B4 ) that is smooth on U (U t V ) can be deformed to a smooth metric that agrees with g on a neighborhood of the boundary of U and V . Using this construction, we exhibit a smooth, asymptotically at spacetime with compact nonchronal region, for which solutions with nite energy do not exist for generic initial data. Begin with a 4-torus T 4 with a at metric chosen to make two of the generators null and the other two spacelike (see Fig. 5). Explicitly, identify
Existence Uniqueness Theorems for Massless Fields
525
Fig. 5. An example of an asymptotically at spacetime with compact nonchronal region is depicted here with two dimensions suppressed. Balls are removed from N and the torus, and their boundary 3-spheres I and II are identi ed. Arrows at P point along null generators of the torus. The shaded region is the support of a solution to the massless scalar wave equation
by translation opposite faces of the rectangular 4-cell in Minkowski space, 0 5 u 5 A; 0 5 v 5 A; 0 5 x 5 A; 0 5 y 5 A: The geometry T 4 ; is chosen because it has solutions to the wave equation whose support is not all of T 4 ; examples are smooth plane waves, functions = (u); with (u) = 0; u ¡ A=2. Cut a ball out of T 4 and embed it in a spacetime that is isometric to Minkowski space outside a compact region U , using Lemma 9.4 The resulting spacetime satis es condition (i) of the Conjecture, but solutions to the wave equation for data on a Cauchy surface M for the past of the Cauchy horizon are not unique: Zero data on M is consistent with arbitrary solutions (u) whose support on T 4 is disjoint from the removed ball. Although our primary interest is in smooth geometries with CTCs con ned to a compact region, it is worth pointing out that if one allows singularities, there are simple examples of spacetimes with CTCs for which one can easily prove the existence of solutions to free- eld equations for arbitrary initial data. For these geometries, however, solutions for smooth data with nite energy are not smooth and do not in general have nite energy; generic solutions are in Lloc 2 . Consider, for example, two-dimensional Minkowski space with two parallel timelike or spacelike segments of equal length removed, as in Fig. 6, and each side of each segment identi ed with a side of the other segment after translation by a timelike vector V , which will be taken to point up and to the right.5 4 An
example of suitable manifold is T 4 #R4 #CP 2 #CP 2 . b be the manifold obtained from Minkowski precisely, to construct the rst spacetime, let N space by removing the two timelike segments L1 and L2 = T(LI ), where T is translation by a timelike vector V . Call the segments with their endpoints removed L1 and L2 . Formally reattach L1 and L2 by b L1 t L2 . The topology of the rst spacetime, N, is generated by the open sets of Nb writing N =Nt together with neighborhoods of points of L1 and L2 de ned as follows: Let O be any open set in an atlas for Minkowski space intersecting the line through L1 in an open interval ‘ ⊂ L1 . Let OR be the part of O lying to the left (right) of the line through L1 . Let OR0 be the part of the translated open set 5 More
526
J.L. Friedman, M.S. Morris
Fig. 6. Two simple spacetimes with CTCs and a well de ned Cauchy problem are shown these two gures. Two parallel slits are removed from Minkowski space and points labelled by the same letter are identi ed
The resulting geometries are at with two conical singularities, corresponding to the removed endpoints of the segments. Similar spacetimes have been discussed by Geroch and Horowitz [15] and Politzer [13]. Some identi ed points are timelike related, and timelike curves in Minkowski space joining points PI ∈ LI to T(PI ) ∈ LII become CTCs in N. For each spacetime N there is a spacelike hypersurface M to the past of the nonchronal region on which initial data can be set, and we will see it is easy to nd a solution for arbitrary initial data on M. Proposition 4. For any initial data ; ˙ in L2 (M) × L2 (M) there is a unique soto the massless scalar wave equation on spacetimes of the form lution in Lloc 2 described above. In a neighborhood of spatial in nity the solution agrees with the Minkowski space solution for the same initial data. Proof. We can divide initial data on M into a sum of data for right-moving and left-moving waves, f(x − t) and g(x + t), by writing " " # # Rx Rx 1 1 0 ˙ 0 0 ˙ 0 (x) − (x) + dx (x ) ; g(x) = dx (x ) : (4.9) f(x) = 2 2 −∞ −∞ We separately construct solutions for right-moving and left-moving data. On Minkowski space, right-moving data, (f; f˙ = −f0 ) gives the solution f(x − t); equivalently, f(P) = f(p), where p ∈ M is the past endpoint of the right moving O0 = T(O) that lies to the right of the line through L2 ; and let OL0 be the part of the translated open set O0 = T(O) that lies to the left of the line through L2 and to the right of the line through L1 . Then b. With the obvious maps to subsets of R4 , the open OL ∪ ‘ ∪ OR and OR0 ∪ ‘0 ∪ OL0 are open sets of N sets just enumerated form an atlas. Because of the deleted endpoints, N is n