Commun. Math. Phys. 308, 281–301 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1345-3
Communications in
Mathematical Physics
Schur Polynomials and The Yang-Baxter Equation Ben Brubaker1 , Daniel Bump2 , Solomon Friedberg3 1 Department of Mathematics, MIT, Cambridge, MA 02139-4307, USA. E-mail:
[email protected] 2 Department of Mathematics, Stanford University, Stanford, CA 94305-2125, USA.
E-mail:
[email protected]
3 Department of Mathematics, Boston College, Chestnut Hill, MA 02467-3806, USA.
E-mail:
[email protected] Received: 29 January 2010 / Accepted: 25 May 2011 Published online: 1 October 2011 – © Springer-Verlag 2011
Abstract: We describe a parametrized Yang-Baxter equation with nonabelian parameter group. That is, we show that there is an injective map g → R(g) from GL(2, C) × GL(1, C) to End(V ⊗ V ), where V is a two-dimensional vector space such that if g, h ∈ G then R12 (g)R13 (gh) R23 (h) = R23 (h) R13 (gh)R12 (g). Here Ri j denotes R applied to the i, j components of V ⊗ V ⊗ V . The image of this map consists of matrices whose nonzero coefficients a1 , a2 , b1 , b2 , c1 , c2 are the Boltzmann weights for the non-field-free six-vertex model, constrained to satisfy a1 a2 + b1 b2 − c1 c2 = 0. This is the exact center of the disordered regime, and is contained within the free fermionic eight-vertex models of Fan and Wu. As an application, we show that with boundary conditions corresponding to integer partitions λ, the six-vertex model is exactly solvable and equal to a Schur polynomial sλ times a deformation of the Weyl denominator. This generalizes and gives a new proof of results of Tokuyama and Hamel and King. Baxter’s method of solving lattice models in statistical mechanics is based on the startriangle relation, which is the identity R12 S13 T23 = T23 S13 R12 ,
(1)
where R, S, T are endomorphisms of V ⊗ V for some vector space V . Here Ri j is the endomorphism of V ⊗ V ⊗ V in which R is applied to the i th and j th copies of V and the identity map to the k th component, where i, j, k are 1, 2, 3 in some order. If the endomorphisms R, S, T are all equal, this is the Yang-Baxter equation (cf. [15,25]). A related construction is the parametrized Yang-Baxter equation R12 (g)R13 (g · h)R23 (h) = R23 (h)R13 (g · h) R12 (g),
(2)
where the endomorphism R now depends on a parameter g in a group G and g, h ∈ G in (2). There are many such examples in the literature in which the group G is an abelian
282
B. Brubaker, D. Bump, S. Friedberg
group such as R or R× . In this paper we present an example of (2) having a non-abelian parameter group. The example arises from a two-dimensional lattice model—the six-vertex model. We now briefly review the connection between lattice models and instances of (1) and (2). In statistical mechanics, one attempts to understand global behavior of a system from local interactions. To this end, one defines the partition function of a model to be the sum of certain locally determined Boltzmann weights over all admissible states of the system. Baxter (see [1] and [2], Chap. 9) recognized that instances of the star-triangle relation allowed one to explicitly determine the partition function of a lattice model. The six-vertex, or ‘ice-type,’ model is one such example that is much studied in the literature, and we revisit it in detail in the next section. For the moment, we offer a few general remarks needed to describe our results. In our presentation of the six-vertex model, each state is represented by a labeling of the edges of a finite rectangular lattice by ± signs, called spins. If the Boltzmann weights are invariant under sign reversal the system is called field-free, corresponding to the physical assumption of the absence of an external field. For field-free weights, the six-vertex model was solved by Lieb [23] and Sutherland [32], meaning that the partition function can be exactly computed. The papers of Lieb, Sutherland and Baxter assume periodic boundary conditions, but nonperiodic boundary conditions were treated by Korepin [18] and Izergin [14]. Much of the literature assumes that the model is field-free. In this case, Baxter shows there is one such parametrized Yang-Baxter equation with parameter group C× for each value of a certain real invariant , defined below in (7) in terms of the Boltzmann weights. One may ask whether the parameter subgroup C× may be enlarged by including endomorphisms whose associated Boltzmann weights lie outside the field-free case. If = 0 the group may not be so enlarged. However we will show in Theorem 4 that if = 0, then the group C× may be enlarged to GL(2, C) × GL(1, C) by expanding the set of endomorphisms to include non-field-free ones. In this expanded = 0 regime, R(g) is not field-free for general g. It is contained within the set of exactly solvable eight-vertex models called the free fermionic model by Fan and Wu [8,9]. Our calculations suggest that it is not possible to enlarge the group G to the entire free fermionic domain in the eight vertex model. As an application of these results, we study the partition function for ice-type models having boundary conditions determined by an integer partition λ. More precisely, we give an explicit evaluation of the partition function for any set of Boltzmann weights chosen so that = 0. This leads to an alternate proof of a deformation of the Weyl character formula for GLn found by Hamel and King [12,13]. That result was a substantial generalization of an earlier generating function identity found by Tokuyama [33], expressed in the language of Gelfand-Tsetlin patterns. Our boundary conditions depend on the choice of a partition λ. Once this choice is made, the states of the model are in bijection with strict Gelfand-Tsetlin patterns having a fixed top row. These are triangular arrays of integers with strictly decreasing rows that interleave (Sect. 3). In its original form, Tokuyama’s formula expresses the partition function of certain ice models as a sum over strict Gelfand-Tsetlin patterns. This connection between states of the ice model and strict Gelfand-Tsetlin patterns has one historical origin in the literature for alternating sign matrices. (An independent historical origin is in the Bethe Ansatz. See Baxter [2] Chap. 8 and Kirillov and Reshetikhin [17].) The bijection between the set of alternating sign matrices and strict Gelfand-Tsetlin patterns having the smallest possible top row is in Mills, Robbins and Rumsey [27], while the connection with what are recognizably states of the six-vertex
Schur Polynomials and The Yang-Baxter Equation
283
model is in Robbins and Rumsey [29]. This connection was used by Kuperberg [19] who gave a second proof (after Zeilberger) of the alternating sign matrix conjecture of Mills, Robbins and Rumsey. It was observed by Okada [28] and Stroganov [31] that the number of n × n alternating sign matrices, that is, the value of Kuperberg’s ice (with particular Boltzmann weights involving cube roots of unity) is a special value of the particular Schur function in 2n variables with λ = (n, n, n − 1, n − 1, . . . , 1, 1) divided by a power of 3. Moreover Stroganov gave a proof using the Yang-Baxter equation. This occurrence of Schur polynomials in the six-vertex model is different from the one we discuss, since Baxter’s parameter is nonzero for those investigations. There are other works relating symmetric function theory to vertex models or spin chains. Lascoux [20,21] gave six-vertex model representations of Schubert and Grothendieck polynomials of Lascoux and Schützenberger [22] and related these to the Yang-Baxter equation. Fomin and Kirillov [10,11] also gave theories of the Schubert and Grothendieck polynomials based on the Yang-Baxter equation. Tsilevich [34] gives an interpretation of Schur polynomials and Hall-Littlewood polynomials in terms of a quantum mechanical system. Jimbo and Miwa [16] give an interpretation of Schur polynomials in terms of two-dimensional fermionic systems. (See also Zinn-Justin [35].) McNamara [26] has clarified that the Lascoux papers are potentially related to ours at least in that the Boltzmann weights [21] belong to the expanded = 0 regime. Moreover, he is able to show based on Lascoux’ work how to construct models of factorial Schur functions. Ice models for factorial Schur functions were further investigated by Bump, McNamara and Nakasuji, who found more general constructions. 1. The Six-Vertex Model We review the six-vertex model from statistical mechanics. Let us consider a lattice (or sometimes more general graph) in which the edges are labeled with “spins” ±. Each vertex will be assigned a Boltzmann weight, which depends on the spins on its adjacent edges. Let us denote the Boltzmann weights as follows:
All remaining Boltzmann weights are taken to be zero; in particular the Boltzmann weight will be zero unless the number of adjacent edges labeled ‘−’ is even. We will consider the vertices in two possible orientations, as shown above, and arrange these Boltzmann weights into a matrix as follows: ⎛ ⎜ R=⎝
⎞
a1
⎛
⎟ ⎜ ⎠=⎝
b1 c1 c2 b2 a2
a1 (R)
⎞ b1 (R) c1 (R) c2 (R) b2 (R)
⎟ ⎠. a2 (R)
(3)
284
B. Brubaker, D. Bump, S. Friedberg
If the edge spins are labeled ν, β, γ , θ ∈ {+, −} as follows:
θγ
++ = a (R), then we will denote by Rνβ the corresponding Boltzmann weight. Thus R++ 1 etc. Because we will sometimes use several different systems of Boltzmann weights within a single lattice, we label each vertex with the corresponding matrix from which the weights are taken. Alternately, R may be thought of as an endomorphism of V ⊗ V , where V is a two-dimensional vector space with basis v+ and v− . Write θγ Rνβ vθ ⊗ vγ . (4) R(vν ⊗ vβ ) = θ,γ
Then the ordering of basis vectors: v+ ⊗ v+ , v+ ⊗ v− , v− ⊗ v+ , v− ⊗ v− gives (4) as the matrix (3). If φ is an endomorphism of V ⊗ V we will denote by φ12 , φ13 and φ23 the endomorphisms of V ⊗ V ⊗ V defined as follows. If φ = φ ⊗ φ , where φ , φ ∈ End(V ) then φ12 = φ ⊗ φ ⊗ 1, φ13 = φ ⊗ 1 ⊗ φ and φ23 = 1 ⊗ φ ⊗ φ . We extend this definition to all φ by linearity. Now if φ, ψ, χ are three endomorphisms of V ⊗ V we define the Yang-Baxter commutator φ, ψ, χ = φ12 ψ13 χ23 − χ23 ψ13 φ12 . Lemma 1. The vanishing of R, S, T is equivalent to the star-triangle identity
(5)
for every fixed combination of spins σ, τ, α, β, ρ, θ . The term star-triangle identity was used by Baxter. The meaning of equation (5) is as follows. For fixed σ, τ, α, β, ρ, θ, μ, ν, γ , the value or Boltzmann weight of the left-hand side is by definition the product of the Boltzmann weights at the three vertices, νμ θγ ρα that is, Rσ τ Sνβ Tμγ , and similarly for the right-hand side. Hence the meaning of (5) is that for fixed σ, τ, α, β, ρ, θ, ψδ φα θρ θγ ρα Rσνμτ Sνβ Tμγ = Tτβ Sσ δ Rφψ . (6) γ ,μ,ν
δ,φ,ψ
It is not hard to see that this is equivalent to the vanishing of R, S, T . In [2], Chap. 9, Baxter considered conditions for which, given S and T , there exists a matrix R such that R, S, T = 0. We will slightly generalize his analysis. He considered mainly the field-free case where a1 (R) = a2 (R) = a(R), b1 (R) = b2 (R) = b(R) and
Schur Polynomials and The Yang-Baxter Equation
285
c1 (R) = c2 (R) = c(R). The condition c1 (R) = c2 (R) = c(R) is easily removed, but with no gain in generality. The other two conditions a1 (R) = a2 (R) = a(R), b1 (R) = b2 (R) = b(R) are more serious restrictions. In the field-free case, let a(R)2 + b(R)2 − c(R)2 , 2a(R) b(R)
(R) =
a1 (R) = a2 (R) = a(R), etc .
(7)
Then Baxter showed that given any S and T with (S) = (T ), there exists an R such that R, S, T = 0. Generalizing this result to the non-field-free case, we find that there are not one but two parameters a1 (R)a2 (R) + b1 (R)b2 (R) − c1 (R)c2 (R) , 2a1 (R)b1 (R) a1 (R)a2 (R) + b1 (R)b2 (R) − c1 (R)c2 (R) 2 (R) = 2a2 (R)b2 (R) 1 (R) =
to be considered. Theorem 2. Assume that a1 (S), a2 (S), b1 (S), b2 (S), c1 (S), c2 (S), a1 (T ), a2 (T ), b1 (T ), b2 (T ), c1 (T ) and c2 (T ) are nonzero. Then a necessary and sufficient condition for there to exist parameters a1 (R), a2 (R), b1 (R), b2 (R), c1 (R), c2 (R) such that R, S, T = 0 with c1 (R), c2 (R) nonzero is that 1 (S) = 1 (T ) and 2 (S) = 2 (T ). Proof. Suppose that 1 (S) = 1 (T ) and 2 (S) = 2 (T ). Then we may take b2 (S)a1 (T )b1 (T ) − a1 (S)b1 (T )b2 (T ) + a1 (S)c1 (T )c2 (T ) a1 (T ) a1 (S)b1 (S)a2 (T ) − a1 (S)a2 (S)b1 (T ) + c1 (S)c2 (S)b1 (T ) , = b1 (S) b1 (S)a2 (T )b2 (T ) − a2 (S)b1 (T )b2 (T ) + a2 (S)c1 (T )c2 (T ) a2 (R) = a2 (T ) a2 (S)b2 (S)a1 (T ) − a1 (S)a2 (S)b2 (T ) + c1 (S)c2 (S)b2 (T ) , = b2 (S) a1 (R) =
b1 (R) = b1 (S)a2 (T ) − a2 (S)b1 (T ), c1 (R) = c1 (S)c2 (T ),
b2 (R) = b2 (S)a1 (T ) − a1 (S)b2 (T ), c2 (R) = c2 (S)c1 (T ).
(8)
(9)
(10) (11)
Using 1 (S) = 1 (T ) and 2 (S) = 2 (T ) it is easy to see that the two expressions for a1 (R) agree, and similarly for a2 (R). One may check that R, S, T = 0. On the other hand, it may be checked that the relations required by R, S, T = 0 are contradictory unless 1 (S) = 1 (T ) and 2 (S) = 2 (T ).
In the field-free case, these two relations reduce to a single one, (S) = (T ), and then (R) has the same value: (R) = (S) = (T ). The equality (5) has important implications for the study of row-transfer matrices, one of Baxter’s original motivations for introducing the star-triangle relation. Given Boltzmann weights a1 (R), a2 (R), . . . , we associate a 2n × 2n matrix V (R). The entries
286
B. Brubaker, D. Bump, S. Friedberg
in this matrix are indexed by pairs α = (α1 , . . . , αn ), β = (β1 , . . . , βn ), where αi , βi ∈ {±}. The coefficient V (R)α,β is computed by first calculating the products of the Boltzmann weights of the vertices of the configuration
for each choice of spins ε1 , . . . , εn ∈ {±}, and then summing these products over all possible states (that is, all assignments of the εi ). Note that in this configuration the righthand spin is denoted ε1 rather than εn+1 ; that is, the boundary conditions are periodic. It follows from Baxter’s argument that if R can be found such that R, S, T = 0 then V (S) and V (T ) commute, and can be simultaneously diagonalized. We will not review Baxter’s argument here, but variants of it with non-periodic boundary conditions will appear later in this paper. In the field-free case when R, S, T = 0, V (R) belongs to the same commuting family as V (S) and V (T ). This gives a great simplification of the analysis in Chapter 9 of Baxter [2] over the analysis in Chapter 8 using different methods based on the Bethe Ansatz. In the non-field-free case, however, the situation is different. If 1 (S) = 1 (T ) and 2 (S) = 2 (T ) then by Theorem 2 there exists R such that R, S, T = 0, and so one may use Baxter’s method to prove the commutativity of V (S) and V (T ). However 1 (R) and 2 (R) are not necessarily the same as 1 (S) = 1 (T ) and 2 (S) = 2 (T ), respectively, and so V (R) may not commute with V (S) and V (T ). In addition to the field-free case, however, there is another case where V (R) necessarily does commute with V (S) and V (T ), and it is that case which we turn to next. This is when a1 a2 + b1 b2 − c1 c2 = 0. The next theorem will show that if the weights of S and T satisfy this condition, then R exists such that R, S, T = 0, and moreover the weights of R also satisfy the same condition. Thus not only V (S) and V (T ) but also V (R) lie in the same space of commuting transfer matrices. In this case, with a1 = a1 (R), etc., we define ⎞ ⎛ ⎞ ⎛ c1 a1 b1 c1 a1 b2 ⎟ ⎜ ⎟ ⎜ (12) π(R) = π ⎝ ⎠=⎝ ⎠. c2 b2 −b1 a2 a2 c2 Theorem 3. Suppose that c1 (S) c2 (S) and c1 (T ) c2 (T ) are nonzero and a1 (S)a2 (S) + b1 (S)b2 (S) − c1 (S)c2 (S) = a1 (T )a2 (T ) + b1 (T )b2 (T ) − c1 (T )c2 (T ) = 0. (13) Then the R ∈ End(V ⊗ V ) defined by π(R) = π(S) π(T )−1 satisfies R, S, T = 0. Moreover, a1 (R) a2 (R) + b1 (R) b2 (R) − c1 (R) c2 (R) = 0.
(14)
Proof. We will use Theorem 2, where it was assumed that a1 (S), a2 (S), b1 (S), b2 (S), c1 (S), c2 (S), a1 (T ), a2 (T ), b1 (T ), b2 (T ), c1 (T ) are all nonzero. Now we are only assuming that the ci are nonzero. It is enough to prove Theorem 3 assuming also that
Schur Polynomials and The Yang-Baxter Equation
287
the ai and bi are nonzero, so that Theorem 2 applies, since the case where the ai and bi are possibly zero will then follow by continuity. The matrix R will not be the matrix in Theorem 2, but will rather be a constant multiple of it. We have ⎛ ⎞ c2 (T ) 1 ⎜ a2 (T ) −b2 (T ) ⎟ π(T )−1 = ⎝ ⎠, b1 (T ) a1 (T ) D c1 (T ) where D = a1 (T )a2 (T ) + b1 (T )b2 (T ) = c1 (T )c2 (T ). With notation as in Theorem 2, Eqs. (8) and (9) may be rewritten, using (13), as the equations a1 (R) = a1 (S)a2 (T ) + b2 (S)b1 (T ), a2 (R) = a2 (S)a1 (T ) + b1 (S)b2 (T ). Combined with (10) and (11) these imply that π(R) = π(S) D π(T )−1 . We are free to multiply R by a constant without changing the validity of R, S, T = 0, so we divide it by D.
We started with S and T and produced R such that R, S, T = 0 because this is the construction motivated by Baxter’s method of proving that transfer matrices commute. However it is perhaps more elegant to start with R and T and produce S as a function of these. Thus let R be the set of endomorphisms R of V ⊗ V of the form (3), where a1 a2 + b1 b2 = c1 c2 . Let R∗ be the subset consisting of such R such that c1 c2 = 0. Theorem 4. There exists a composition law on R∗ such that if R, T ∈ R∗ , and if S = R ◦ T is the composition then R, S, T = 0. This composition law is determined by the condition that π(S) = π(R)π(T ), where π : R∗ −→ GL(4, C) is the map (12). Then R∗ is a group, isomorphic to GL(2, C) × GL(1, C). Proof. This is a formal consequence of Theorem 3.
It is interesting that, in the non-field-free case, the group law occurs when 1 = 2 = 0. In the application to statistical physics for field-free weights, phase transitions occur when = ±1. If || > 1 the system is “frozen” in the sense that there are correlations between distant vertices. By contrast −1 < < 1 is the disordered range where no such correlations occur, so our group law occurs in the analog of the middle of the disordered range. 2. Boundary Conditions and Partition Functions In this section, we describe the global model to be studied using the local Yang-Baxter relation from the previous section. Let λ = (λ1 , . . . , λr +1 ) be a fixed integer partition with λr +1 = 0 and let ρ = (r, r − 1, . . . , 0). Consider a rectangular lattice with r + 1 rows and λ1 + r + 1 columns. Number the columns of the lattice in descending order from left to right, λ1 + r to 0. We attach boundary conditions to this lattice according to the choice of λ + ρ. This amounts to a choice of spin ± to every edge along the boundary prescribed as follows. Boundary Conditions Determined by λ. On the left and bottom boundary edges, assign spin +; on the right edges assign spin −. On the top, assign spin − at every column labeled λi + n − i (1 i n), that is, for the columns labeled with values in λ + ρ; assign spin + at every column not labeled by λi + n − i for any i.
288
B. Brubaker, D. Bump, S. Friedberg
Table 1. Square ice and their associated Boltzmann weights
For example, suppose that n = 3 and λ = (3, 1, 0), so that λ + ρ = (5, 2, 0). Then the spins on the boundary are as in the following figure:
(15)
The column labels have been written along the top. (The use of the row labelling will be explained shortly.) The choice of λ = (0, . . . , 0) would give all − signs across the top row, and is referred to in the literature as “domain wall boundary conditions.” A state of the model will consist of an assignment of spin ± to each internal edge, pictured above as open circles. The interior spins are not entirely arbitrary, since we require that every vertex “•” in the configuration has adjacent edges whose spins match one of the six admissible configurations in Table 1 under “Square ice” in the table below. The set of all such states with boundary conditions corresponding to λ as above will be called Sλ . Each of the six types of vertex is assigned a Boltzmann weight, which is allowed to depend on the row i in which it occurs. We have emphasized this dependence in the notation of Table 1. To each state x ∈ Sλ , the Boltzmann weight w(x) of the state x is then the product of the Boltzmann weights of all vertices in the state. The partition function Z (Sλ ) is defined to be the sum of the Boltzmann weights over all states: Z (Sλ ) = w(x). x∈Sλ
Note: The word “partition” occurs in two different senses in this paper. The partition function in statistical physics is different from partitions in the combinatorial sense. So for us a reference to a “partition” without “function” refers to an integer partition. As an example, suppose that r = 1 and λ = (0, 0) so λ + ρ = (1, 0). In this case Sλ has cardinality two. The states and their associated Boltzmann weights are:
Schur Polynomials and The Yang-Baxter Equation
289
Hence (1) (1)
Z (Sλ ) = c1 c2
(2) (1) (1) (2) a1 a2 + b1 b2 .
The partition function for general λ of arbitrary rank r will be evaluated in Theorem 9, assuming the free-fermionic condition (19). 3. Tokuyama’s Deformation of The Weyl Character Formula Let us momentarily consider a piece of square ice with just one layer of vertices. Let α1 , . . . , αm be the column numbers (from left to right) of −’s along the top boundary and let β1 , . . . , βm be the column numbers of −’s along the bottom boundary. For example, in the ice
we have m = 3, m = 2, (α1 , α2 , α3 ) = (5, 2, 0) and (β1 , β2 ) = (3, 0). Since the columns are labeled in decreasing order, we have α1 > α2 > · · · and β1 > β2 > · · ·. Lemma 5. Suppose that the spin at the left edge is +. Then m = m or m + 1 and α1 β1 α2 · · ·. If m = m then the spin at the right edge is +, while if m = m + 1 it is −. We express the condition that α1 β1 α2 · · · by saying that the sequences α1 , . . . , αm , and β1 , . . . , βm , interleave. This lemma is essentially the line-conservation principle in Baxter [2], Sect. 8.3. Proof. The spins along horizontal edges are determined by a choice of spins along the top and bottom boundary and a choice of spin at the left boundary edge (which is assumed to be +). This is clear since, according to the six vertices appearing in Table 1, the edges at each vertex have an even number of + spins. If the rows do not interleave then one of the illegal configurations (i.e., not one of the six in Table 1)
will occur. It follows that α1 β1 since if not the vertex in the β1 column would be surrounded by spins in the first illegal configuration. Similarly β1 α2 , since otherwise the vertex in the α2 column would be surrounded by spins in the second above illegal configuration, and so forth. The last statement is a consequence of the observation that the total number of spins must be even.
A Gelfand-Tsetlin pattern is a triangular array of dominant weights (or equivalently integer vectors whose entries are weakly decreasing), in which each row has length one less than the one above it, and the rows interleave. The pattern is called strict if the rows are strictly dominant (i.e., the integer components are strictly decreasing).
290
B. Brubaker, D. Bump, S. Friedberg
It follows from Lemma 5 that taking the column indices of − spins along vertical edges gives a sequence of strictly dominant weights forming a strict Gelfand-Tsetlin pattern. For example, given the state
the corresponding pattern is T=
⎧ ⎨5 ⎩
⎫ 0⎬
2 3
0
⎭
3
.
(16)
It is not hard to see that this gives a bijection between strict Gelfand-Tsetlin patterns having fixed top row λ + ρ and states with boundary conditions determined by λ. We recall some further definitions from Tokuyama [33]. An entry of a Gelfand-Tsetlin pattern (not in the top row) is classified as left-leaning if it equals the entry above it and to the left. It is right-leaning if it equals the entry above it and to the right. We will call a pattern leaning if all entries below the top row are right- or left-leaning. A pattern is special if it is neither left- nor right-leaning. Thus in (16), the 3 in the bottom row is left-leaning, the 0 in the second row is right-leaning and the 3 in the middle row is special. If T is a Gelfand-Tsetlin pattern, let l(T) be the number of left-leaning entries. Let dk (T) be the sum of the k th row of T, and dr +2 (T) = 0. Theorem 6 (Tokuyama). We have r +1 dk (T)−dk+1 (T) zk (z i + t z j )sλ (z 1 , . . . , zr +1 ), t l(T) (t + 1)s(T) = T
k=1
(17)
i< j
where the sum is over all strict Gelfand-Tsetlin patterns with top row λ + ρ. As Tokuyama [33] explains, if t = −1, this reduces to the Weyl character formula, while if t = 0 it reduces to the combinatorial description of Schur polynomials. See also [3], Chap. 5 for further discussion of Tokuyama’s formula. Later in this paper we will give a new proof of this theorem and of a generalization of it by Hamel-King, and we will generalize yet further by evaluating the partition function Z (Sλ ) for Boltzmann weights in the free-fermionic regime. 4. Evaluation of The Partition Function Z(Sλ ) We begin by recording a version of the parametrized Yang-Baxter equation using the notation from Sect. 2.
Schur Polynomials and The Yang-Baxter Equation (i)
291 ( j)
(i)
( j)
Lemma 7. Let S(i) = (a1 , . . . , c2 ) and T ( j) = (a1 , . . . , c2 ) be sets of Boltzmann weights corresponding to rows i and j, respectively, satisfying (13). If we choose Boltzmann weights for R(i, j) as follows: ⎛ (i) ( j) ⎞ ( j) (i) a1 a2 + b1 b2 ⎜ ⎟ ( j) (i) (i) ( j) (i) ( j) a2 b1 − a2 b1 c1 c2 ⎜ ⎟ R(i, j) =⎜ ⎟, ( j) (i) ( j) (i) (i) ( j) ⎝ ⎠ c1 c2 a1 b2 − a1 b2 ( j) (i) (i) ( j) a1 a2 + b1 b2 then the star-triangle identity holds: R(i, j), S(i), T ( j) = 0. Proof. This is just a restatement of Theorem 3 using notation for the Boltzmann weights to reflect the dependence on rows.
Let us record this in tabular form for later reference.
(18)
Lemma 8. Let Sλ be an ensemble with boundary conditions corresponding to λ and Boltzmann weights satisfying (13). Then for i < j, the expression (i) ( j)
( j) (i)
(a1 a2 + b1 b2 )Z (Sλ ) is invariant under the interchange of spectral parameters i and j. Proof. We modify the boundary conditions by introducing a single R vertex at the left edge of the ice connecting rows i and j. For simplicity, we illustrate with an ensemble Sλ with λ = (3, 1, 0) and i = 2, j = 3:
Comparing with the admissible configurations for the R vertex given in the table in (18), the only possible values for a and b are +. Thus every state of this new boundary value
292
B. Brubaker, D. Bump, S. Friedberg
problem determines a unique state of the original problem, and the partition function for each such state is the original partition function multiplied by the Boltzmann weight ( j) (i) (i) ( j) of the R-vertex, which is (a1 a2 + b1 b2 ). Now we apply the star-triangle identity, and obtain equality with the the following configuration (again pictured for our special case):
Thus if S denotes this ensemble then its partition function is ( j)
( j)
Z (S ) = (a1(i) a2 + b1 b2(i) )Z (Sλ ). Repeatedly applying the star-triangle identity, we eventually obtain the configuration in which the R-vertex is moved entirely to the right:
Now there is only one admissible configuration for the R-vertex on the right-hand side, ( j) (i) (i) ( j) namely c = d = −. The Boltzmann weight for this R-vertex is a1 a2 + b1 b2 . Note that upon moving through the ice, the roles of i and j have been interchanged. Com( j) ( j) paring partition functions, this proves that (a1(i) a2 + b1 b2(i) )Z (Sλ ) is unchanged by switching the spectral parameters i and j.
Theorem 9. Let λ be a partition with r + 1 parts, largest part λ1 and smallest part 0. Let Sλ be the corresponding ensemble, with r + 1 rows. Suppose that a1(i) a2(i) + b1(i) b2(i) = c1(i) c2(i) . Then
⎡
Z (Sλ ) = ⎣
r +1 k=1
(k)
(k) (a1 )λ1 c2
⎤ ( j) (i) (a1 a2
(i) ( j) + b1 b2 )⎦ sλ
i< j
(19)
b2(1) b2(2) b2(r +1) , , . . . , (1) (2) (r +1) a1 a1 a1
, (20)
where sλ is the Schur polynomial corresponding to λ.
Schur Polynomials and The Yang-Baxter Equation
293
Remark 4.1. Equation (19) describes the free-fermionic regime in the six-vertex model. More generally, see Fan and Wu [8,9] for the free-fermionic eight-vertex model. The identity (20) is an equality of homogeneous polynomials of degree λ1 + r + 1 (i) (i) in a1 , . . . , c2 for each i. The Schur polynomial is expressed in terms of the vari(i) (i) (i) ables b2 /a1 , but if a1 = 0 for some i then, due to this homogeneity, one may clear denominators before evaluating and Eq. (20) still makes sense. Turning to the proof of Theorem 9, also due to the homogeneity of (20), we may assume c2(i) = 1 for all i. Lemma 10. Given any partition λ, the expression ⎡ ⎤−1 ( j) (i) de f ( j) (i) sSλ = ⎣ (a1 a2 + b1 b2 )⎦ Z (Sλ ) i< j
is symmetric with respect to the spectral parameters and expressible as a polynomial in (i) (i) (i) (i) the variables a1 , a2 , b1 , b2 with integer coefficients. Proof. It suffices to show this function is invariant under transpositions (i.e. interchang(k) (k+1) (k+1) (k) + b1 b2 )Z (Sλ ) is invariant ing k and k + 1). By Lemma 8, the function (a1 a2 under the interchange k ↔ k + 1. It follows that ⎡ ⎤ ⎡ ⎤ (i) ( j) ⎣ (a a + b( j) b(i) )⎦ Z (Sλ ) = ⎣ (a (i) a ( j) + b( j) b(i) )⎦ sS (21) λ 1 1 1 2 2 1 2 2 i= j
i< j
(k) (k+1)
+ is invariant under k ↔ k + 1 since the left-hand side of (21) is a product of (a1 a2 ( j) (i) (k+1) (k) (i) ( j) b1 b2 )Z (Sλ ) and factors (a1 a2 + b1 b2 ) that are permuted under k ↔ k + 1. Thus sS,λ must also be symmetric. (i) (i) (i) (i) (i) (i) The identity (19) with c2 = 1 becomes a1 a2 + b1 b2 = c1 . This allows one to (i) eliminate c1 (i = 1, . . . , r + 1) from Z (Sλ ), regarding it as a polynomial in the ring (1)
(1)
(r +1)
R = Z[a1 , . . . , b2 , . . . , a1
( j)
(r +1)
, . . . , b2
].
( j)
The left-hand side of (21) is divisible by (a1(i) a2 + b1 b2(i) ) with i < j and may be regarded as an element of the unique factorization domain R. As the left-hand side ( j) (i) (i) ( j) of (21) is symmetric, we conclude that it is also divisible by (a1 a2 + b1 b2 ) with (i) (i) (i) (i)
i < j. This shows sSλ is a polynomial in the a1 , a2 , b1 , b2 . (i)
Proof of Theorem 9. Since sSλ defined in Lemma 10 is independent of c1 for all i, we (i) may take c1 = 0 for all i in computing the partition function Z (Sλ ). Upon doing this, the remaining states of ice with non-zero Boltzmann weights (i.e., those without any (i) c1 ) are in bijection with leaning Gelfand-Tsetlin patterns. We will show that in this case, the function sSλ is, up to a constant multiple, a Schur (i) (i) (i) (i) polynomial in the variables b2 /a1 = −a2 /b1 by comparing our expression with the Weyl character formula. First, we demonstrate that the product ⎡ ⎤ ( j) (i) ( j) (i) ⎣ (a a + b b )⎦ 1 2 1 2 i< j
294
B. Brubaker, D. Bump, S. Friedberg (i)
(i)
can be expressed in terms of the Weyl denominator in the b2 /a1 . Indeed, for any pair i, j with i < j, write (i) (i) ( j) (i) b2 ( j) (i) ( j) −b1 b2 ( j) (i) b2 (i) ( j) (i) ( j) − (i) , a1 a2 + b1 b2 = a1 + b1 b2 = a1 b1 (i) ( j) a1 a1 a1 (i) (i)
(i) (i)
(i)
where we have used the identity (19) in the form a1 a2 + b1 b2 = 0 since c1 = 0. Performing this for all such pairs i, j, we have (i) ( j) (i) ( j) (i) b( j) b (i) ( j) 2 (a1 a2 + b1 b2 ) = a1 b1 − 2(i) ( j) a1 a i< j i< j 1 r +1 (i) b( j) b2 (k) k−1 (k) r +1−k 2 (a1 ) (b1 ) − (i) . = (22) ( j) a1 k=1 i< j a1 As we argued in Lemma 10, after setting c1(i) = 0, the function Z (Sλ ) is a polynomial (i) (i) (i) (i) (i) (i) in the variables a1 , a2 , b1 , b2 . We make the substitution a2 = −b1 z i for all i where the z i are (for the moment) just formal parameters. Call the resulting function NSλ = NSλ (z 1 , . . . , zr +1 ), which is a polynomial in the z i whose coefficients are poly(i) (i) (i) (i) nomial expressions in the a1 , b1 , b2 . We claim that the power of b1 appearing in (i) each coefficient of NSλ is equal to r + 1 − i. Indeed, this is the total number of a2 (i) and b1 appearing in the Boltzmann weight of each state of Z (Sλ ). These weights are contributed by the two vertices having north and south spins both equal to −. In leaning patterns, the number of such vertices in row i is always r + 1 − i. Thus de f
NS = NSλ λ
r +1
(k)
(b1 )−(r +1−i)
k=1 (k) is independent of b1
N
for all k. Hence Sλ is a polynomial in the z i with coefficients that are polynomials in the a1(i) , b2(i) . These two Boltzmann weights are the unique pair with north and south spins both equal to +. In states of ice corresponding to leaning GelfandTsetlin patterns, the total number of such vertices in row i is equal to 1 +i −(r +1), where 1 = λ1 +· · ·+λr +r is the index of the left-most column in ice in Sλ . Thus we may write NS = λ
r +1
(a1(i) )1 +i−(r +1) NS , λ
i=1 (i)
(i)
is a polynomial in the z and b /a where NS i 2 1 with integer coefficients. λ (i) (i) (i) (i) (i) (i) Initially we set z i = −a2 /b1 . However, in light of the relation a1 a2 + b1 b2 = is a polynomial in the z with integer c1(i) = 0, we also have z i = b2(i) /a1(i) . Thus NS i λ coefficients. Combining with (22), we have ⎡ ⎤−1 r +1 λ1 r +1 (i) NSλ (z 1 , . . . , zr +1 ) ( j) ( j) (i) (i) ⎣ ⎦ , sSλ = (a1 a2 + b1 b2 ) Z (Sλ ) = a1 i< j z j − z i i=1 i< j i=1
(23) where λ1 is the largest part of the partition.
Schur Polynomials and The Yang-Baxter Equation
295
The weight μ of a state, i.e., its degree of as a monomial in z i , is given by counting the number of vertices having Boltzmann weight a2 or b2 in row i. These are the unique pair of vertices having west spin equal to − (since c1(i) = 0). It is easy to see that the weight μ of any non-zero state of ice is a permutation σ of the top row of T, that is, of λ + ρ. These weights are all distinct since λ + ρ is strongly dominant, i.e. without repeated entries, are all ±1. Since N is skew-symmetric (because so in fact the coefficients of NS Sλ λ is equal to the sum the denominator is skew-symmetric while sSλ is symmetric), ±NS λ σ ( j) over permutations σ of terms of the form sgn(σ ) z j , where i = λi + ρi for all i. To determine the sign, we may take the state whose corresponding Gelfand-Tsetlin pattern consists entirely of right-leaning entries: ⎫ ⎧ 1 2 · · · r 0⎪ ⎪ ⎪ ⎪ ⎨ 2 · · · r 0 ⎬ . T= .. . ⎪ ⎪ . .. ⎪ ⎪ ⎭ ⎩ 0 This has Boltzmann weight
z j j and so
(z 1 , . . . , zr +1 ) = NS λ
sgn(σ )
z jσ ( j) .
σ ∈Sr +1
The theorem then follows by combining the above with (23) and invoking the Weyl character formula.
5. Another Proof of Tokuyama-Hamel-King This section gives a new proof of results of Tokuyama and Hamel-King. Proof of Theorem 6. Using the specialization of the weights in Theorem 9 as follows (for all rows 1 ≤ i ≤ r + 1): a1(i) = 1, a2(i) = z i , b1(i) = ti , b2(i) = z i , c1(i) = z i (ti + 1), c2(i) = 1.
(24)
(These weights are called S (i) in Table 2 below.) If all ti = t then the resulting partition function simplifies to the right-hand side of Tokuyama’s theorem (17), and in general ti z j + z i sλ (z 1 , . . . , zr +1 ). (25) Z (Sλ ) = i< j
It remains to use the bijection between states in Sλ and strict Gelfand-Tsetlin patterns with top row λ + ρ to show that the summands on the left-hand side of (17) are equal to the Boltzmann weights of the corresponding states. Given Boltzmann weights as in (24), we say that z-weight of a state is (μ1 , . . . , μn ) the μ if the Boltzmann weight is the monomial z μ = z i i times a polynomial in the variables ti . Recall that if T is a Gelfand-Tsetlin pattern, we set dk (T) to be the sum of the k th row and dr +2 (T) = 0. The following lemma shows that the powers of z i appearing in the Boltzmann weight of a state agree with the corresponding summand in Tokuyama’s theorem.
296
B. Brubaker, D. Bump, S. Friedberg
Lemma 11. If T is the Gelfand-Tsetlin pattern corresponding to a state of z-weight μ, then μk = dk (T) − dk+1 (T). Proof. From Table 1, μk is the number of vertices in the k th row that have an edge configuration of one of the three forms:
Let αi ’s (respectively βi ’s) be the column numbers for which the top edge spin (respectively, the bottom edge spin) of vertices in the k th row is − (with columns numbered in descending order, as always). By Lemma 5 we have α1 β1 α2 · · · αr +2−k . It is easy to see that the vertex in the j-column has one of the above configurations if and only if its column number j satisfies αi > j βi for some i. Therefore the number of such j is αi − βi = dk (T) − dk+1 (T).
Finally, it is easy to see that if an entry in the k th row of T is left leaning (respectively special), and that entry is j, then the configuration in the j-column and the k th row of the ice is
so from our specialization of weights above, it follows that the powers of ti in the Boltzmann weight of a state match the powers of ti in the corresponding summands on the left-hand side of (17). Setting all ti = t gives the result of Tokuyama.
It is clear from the proof that we’ve shown something a bit stronger, in that the ti ’s are allowed to be independent variables. Indeed, the case of ti independent is easily seen to be equivalent to Hamel and King’s generalization of Tokuyama’s theorem (cf. Proposition 1.1 of [13].) 6. A Relation Between Partition Functions It is natural to ask whether other models and choices of Boltzmann weights exist for which the resulting partition function matches the right-hand side of (17). Let us denote by Sλ the ice model with boundary conditions as in Sect. 2 and assignment of Boltzmann weights as in (24). So we may write Z (Sλ ) = (z i + ti z j )sλ (z 1 , . . . , zr +1 ). i< j (i)
(i)
If we simply replace the choices of a2 and b1 appearing in (24) with (i)
(i)
a2 = ti z i , b1 = 1, then by Theorem 9, the resulting partition function will be Z (Sλ ) = (ti z i + z j )sλ (z 1 , . . . , zr +1 ). i< j
Schur Polynomials and The Yang-Baxter Equation
297
Table 2. Boltzmann weights for (26) and its proof
However, if we also change the model slightly by renumbering the rows in descending order from top to bottom and call the resulting model S λ then Theorem 9 implies that Z (S λ)
i< j
(ti z j + z i ) = Z (Sλ )
(t j z j + z i ),
(26)
i< j
and setting all ti = t, we may cancel the products on either side giving a second expression for Tokuyama’s result. The equality of partition functions in (26) is deceptively subtle as there is probably no bijective proof which matches Boltzmann weights of states in Sλ and S λ. Four proofs of this identity are known. The first has already been given—we may exactly solve both models using (9) and then conclude they are equal. Second, we may prove (20) by realizing each side as the same Whittaker coefficient of a minimal parabolic Eisenstein series—a certain integral over a unipotent group inductively calculated in two different ways. This method will not be described in detail but see [4] for one of the two inductive calculations. (The other is not written down but similar.) The identity corresponds to the special case n = 1 where n is the degree of the cover that occurs in [4]. Third, this is the special case of Statement B in [3] in which the degree n of the Gauss sums that appear in that statement equals 1. There, sums over lattice points in two different polytopes are compared by a combinatorial procedure related to the Schützenberger involution of a crystal graph. We conclude this section with a fourth proof of (26) using another instance of the parametrized Yang-Baxter equation. This allows us to compare two partition functions without having to explicitly evaluate either. As mentioned above, a generalization of (26) is known using n th order Gauss sums which specialize, when n = 1, to the Boltzmann weights above. For general n, ice models exist for both sides of the identity in Statement B in [3] but no corresponding Yang-Baxter equation is known. See [5] and Chap. 19 of [3] for further discussion of this. We turn to the proof of (26). Recall that the Boltzmann weights S (i) and S (i) used in the systems Sλ and S λ are as in Table 2. This table also defines a third type of Boltzmann weight R (i, j) that we will require for the proof of (26).
298
B. Brubaker, D. Bump, S. Friedberg
As in Table 2, vertices in a given state having Boltzmann weights corresponding to will be labeled with a black dot (•) and those corresponding to Boltzmann weights will be labeled with an open dot (◦). Lemma 12. Consider the Boltzmann weights in Table 2. Then the following star-triangle identity holds: R (i, j), S (i), T ( j) = 0. Proof. This is just a special case of Lemma 7. We have ⎛ t j z j + zi z i − ti t j z j z i (ti + 1) ⎜ R (i, j) = ⎝ z j (t j + 1) zi − z j
⎞ ⎟ ⎠. z i + ti z j
Proposition 13. Let Boltzmann weights for two ice models S and S be chosen as in Table 2, both having boundary conditions corresponding to a partition λ as in Sect. 2, but with rows in states of S labeled in ascending order from top to bottom, and rows in states of S in descending order. Then Z (S (ti z j + z i ) = Z (Sλ ) (t j z j + z i ). λ) i< j
i< j
Proof. Begin with an state x of Sλ , say (for example with λ = (3, 1, 0)):
(We’re focusing on the bottom row for the moment, so the unlabeled edges can be filled in arbitrarily.) We wish to transform this into a state having a row of Delta ice so that we may use the star-triangle relation in Lemma 12. The vertices in the bottom row all have south spin equal to + and the Boltzmann weights for and given in Table 2 differ only for a2 and b1 vertices, which both have south spin equal to −. Hence we may simply consider the bottom vertices to be Boltzmann weights without affecting the Boltzmann weight of the entire state:
Schur Polynomials and The Yang-Baxter Equation
299
Note that this would not work in any row but the last because it is essential that there be no − on the bottom edge spins. Now we add a Gamma-Delta R-vertex.
If we call this model S , then we claim that Z (S ) = (t3 z 3 + z 2 )Z (Sλ ). Indeed, from Lemma 12, the values of spins a and b indicated in the figure must both be + and so the value of the R-vertex is t3 z 3 + z 2 for every state in the model. Now repeatedly using the star-triangle relation, Z (S ) = Z (S ), where S is the model with boundary:
Here we must have spins c, d in the figure above both equal to − which implies that (t3 z 3 + z 2 )Z (Sλ ) = Z (S ) = (t2 z 3 + z 2 )Z (S ), where S is the model with boundary of form
We repeat the process, first moving the row of Delta ice up to the top, then introducing another row of Delta ice by simple replacement at the bottom, etc., until we arrive at the model S
λ and obtain (26). Acknowledgements. We are grateful to Gautam Chinta and Tony Licata for stimulating discussions and to the referee for insightful comments. This work was supported by NSF grants DMS-0652609, DMS-0652817, DMS-0652529, DMS-0702438, DMS-1001079, DMS-1001326 and NSA grant H98230-10-1-0183. SAGE [30] was very useful in the preparation of this paper.
References 1. Baxter, R.J.: The inversion relation method for some two-dimensional exactly solved models in lattice statistics. J. Stat. Phys. 28(1), 1–41 (1982)
300
B. Brubaker, D. Bump, S. Friedberg
2. Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], 1982 3. Brubaker, B., Bump, D., Friedberg, S.: Weyl Group Multiple Dirichlet Series: Type A Combinatorial Theory, Annals of Mathematics Studies, Vol. 175, Princeton, NJ: Princeton University Press, 2011 4. Brubaker, B., Bump, D., Friedberg, S.: Weyl group multiple Dirichlet series, Eisenstein series and crystal bases. Ann. of Math. 173, 1081–1120 (2011) 5. Brubaker, B., Bump, D., Chinta, G., Friedberg, S., Gunnells, P.: Metaplectic ice. http://arxiv.org/abs/ 1009.1741v1 [math.RT], 2010 6. Drinfeld, V.G.: Quantum groups. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), Providence, RI: Amer. Math. Soc., 1987, pp. 798–820 7. Faddeev, L.D., Reshetikhin, N.Yu., Takhtajan, L.A.: Quantization of Lie groups and Lie algebras. In: Algebraic Analysis, Vol. I, Boston, MA: Academic Press, 1988, pp. 129–139 8. Fan, C., Wu, F.Y.: Ising model with next-neighbor interactions. I. Some exact results and an approximate solution Phys. Rev. 179, 560–570 (1969) 9. Fan, C., Wu, F. Y.: General lattice model of phase transitions. Phys. Revi. B 2(3), 723–733 (1970) 10. Fomin, S., Kirillov, A.N.: The Yang-Baxter equation, symmetric functions, and Schubert polynomials. In: Proc. of the 5th Conference on Formal Power Series and Algebraic Combinatorics (Florence, 1993), Disc. Math. 153, 123–143 (1996) 11. Fomin, S., Kirillov, A.N.: Grothendieck polynomials and the Yang-Baxter equation. In: Formal power series and algebraic combinatorics/Séries formelles et combinatoire algébrique, Piscataway, NJ: DIMACS, 1994, pp. 183–189 12. Hamel, A.M., King, R.C.: U-turn alternating sign matrices, symplectic shifted tableaux and their weighted enumeration. J. Alg. Comb. 21(4), 395–421 (2005) 13. Hamel, A.M., King, R.C.: Bijective proofs of shifted tableau and alternating sign matrix identities. J. Alg. Comb. 25(4), 417–458 (2007) 14. Izergin, A.G.: Partition function of a six-vertex model in a finite volume. Dokl. Akad. Nauk SSSR 297(2), 331–333 (1987) 15. Jimbo, M.: Introduction to the Yang-Baxter equation. Internat. J. Modern Phys. A 4(15), 3759–3777 (1989) 16. Jimbo, M., Miwa, T.: Solitons and infinite-dimensional Lie algebras. Publ. Res. Inst. Math. Sci. 19(3), 943–1001 (1983) 17. Kirillov, A.N., Reshetikhin, N.Yu.: The Bethe ansatz and the combinatorics of Young tableaux. J. Soviet Math. 41(2), 925–955 (1988) 18. Korepin, V.E.: Calculation of norms of Bethe wave functions. Commun. Math. Phys. 86(3), 391–418 (1982) 19. Kuperberg, G.: Another proof of the alternating-sign matrix conjecture. Int. Math. Res. Notices 1996(3), 139–150 (1996) 20. Lascoux, A.: Chern and Yang through ice. Preprint, 2002 21. Lascoux, A.: The 6 vertex model and Schubert polynomials. SIGMA Symmetry Integrability Geom. Methods Appl. 3, Paper 029, 12 pp. (electronic) (2007) 22. Lascoux, A., Schützenberger, M.-P.: Symmetry and flag manifolds. In: Invariant theory (Montecatini, 1982), Volume 996 of Lecture Notes in Math., Berlin: Springer, 1983, pp. 118–144 23. Lieb, E.: Exact solution of the problem of entropy in two-dimensional ice. Phys. Rev. Lett. 18, 692–694 (1967) 24. Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs. New York: The Clarendon Press/Oxford University Press, Second edition, 1995, (with contributions by A. Zelevinsky) 25. Majid, S.: Quasitriangular Hopf algebras and Yang-Baxter equations. Int. J. Mod. Phys. A 5(1), 1–91 (1990) 26. McNamara, P.J.: Factorial Schur functions via the six-vertex model. http://arxiv.org/abs/0910.5288v2 [math.co], 2009 27. Mills, W.H., Robbins, D.P., Rumsey, H. Jr.: Alternating sign matrices and descending plane partitions. J. Comb. Th. Ser. A 34(3), 340–359 (1983) 28. Okada, S.: Alternating sign matrices and some deformations of Weyl’s denominator formulas. J. Alg. Comb. 2(2), 155–176 (1993) 29. Robbins, D.P., Rumsey, H. Jr.: Determinants and alternating sign matrices. Adv. in Math. 62(2), 169–184 (1986) 30. Stein, W., et al.: SAGE Mathematical Software, Version 4.1. http://www.sagemath.org, 2009 31. Stroganov, Yu.G.: The Izergin-Korepin determinant at a cube root of unity. Teoret. Mat. Fiz. 146(1), 65–76 (2006) 32. Sutherland, B.: Exact solution for a model for hydrogen-bonded crystals. Phys. Rev. Lett. 19(3), 103–104 (1967)
Schur Polynomials and The Yang-Baxter Equation
301
33. Tokuyama, T.: A generating function of strict Gelfand patterns and some formulas on characters of general linear groups. J. Math. Soc. Japan 40(4), 671–685 (1988) 34. Tsilevich, N.V.: The quantum inverse scattering problem method for the q-boson model, and symmetric functions. Funkt. Anal. i Priloz. 40(3), 53–65, 96 (2006) 35. Zinn-Justin, P.: Six-vertex loop and tiling models: Integrability and combinatorics. Habilitation thesis http://arxiv.org/abs/0901.0665v2 [math-ph], 2009 Communicated by P. Forrester
Commun. Math. Phys. 308, 303–323 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1347-1
Communications in
Mathematical Physics
Cyclic Monopoles, Affine Toda and Spectral Curves H. W. Braden School of Mathematics, Edinburgh University, Edinburgh, UK. E-mail:
[email protected] Received: 2 March 2010 / Accepted: 22 June 2011 Published online: 1 October 2011 – © Springer-Verlag 2011
Abstract: We show that any cyclically symmetric monopole is gauge equivalent to Nahm data given by Sutcliffe’s ansatz, and so obtained from the affine Toda equations. Further the direction (the Ercolani-Sinha vector) and base point of the linearising flow in the Jacobian of the spectral curve associated to the Nahm equations arise as pull-backs of Toda data. A theorem of Accola and Fay then means that the theta-functions arising in the solution of the monopole problem reduce to the theta-functions of Toda.
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 2. Monopoles . . . . . . . . . . . . . . . . . . . . . . . . 3. The Sutcliffe Ansatz . . . . . . . . . . . . . . . . . . . 4. Flows and Solutions . . . . . . . . . . . . . . . . . . . 5. The Base Point . . . . . . . . . . . . . . . . . . . . . . 6. Fay-Accola Factorization . . . . . . . . . . . . . . . . 7. Discussion . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . Appendix A. Proof of Theorem 6.2 via Poincaré Reducibility References . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
303 305 308 310 314 316 319 320 320 322
1. Introduction Magnetic monopoles, the topological soliton solutions of Yang-Mills-Higgs gauge theories in three space dimensions with particle-like properties, have been the subject of considerable interest over the years. BPS monopoles, arising from a limit in which the the Higgs potential is removed but a remnant of this remains in the boundary conditions,
304
H. W. Braden
satisfy the first order Bogomolny equation Bi =
3 1 i jk F jk = Di 2 j,k=1
and have merited particular attention (see [MS04] for a recent review). This focus is in part due to the ubiquity of the Bogomolny equation. Here Fi j is the field strength associated to a gauge field A, and is the Higgs field. We shall focus on the case when the gauge group is SU (2). The Bogomolny equation may be viewed as a dimensional reduction of the four dimensional self-dual equations upon setting all functions independent of x4 and identifying = A4 ; they are also encountered in supersymmetric theories when requiring certain field configurations to preserve some fraction of supersymmetry. The study of BPS monopoles is intimately connected with integrable systems. Nahm gave a transform of the ADHM instanton construction to produce BPS monopoles [Nah82] and ˆ This the resulting Nahm’s equations have Lax form with corresponding spectral curve C. curve, investigated by Corrigan and Goddard [CG81], was given a twistorial description by Hitchin [Hit82] where the same curve lies in mini-twistor space, Cˆ ⊂ TP1 . Just as Ward’s twistor transform relates instanton solutions on R4 to certain holomorphic vector bundles over the twistor space CP3 , Hitchin showed that the dimensional reduction leading to BPS monopoles could be made at the twistor level as well and was able to prove that all monopoles could be obtained by this approach [Hit83] provided the curve Cˆ was subject to certain nonsingularity conditions. Bringing methods from integrable systems to bear upon the construction of solutions to Nahm’s equations for the gauge group SU (2) Ercolani and Sinha [ES89] later showed how one could solve (a gauge ˆ transform of) the Nahm equations in terms of a Baker-Akhiezer function for the curve C. Although many general results have now been obtained few explicit solutions are known. This is for two reasons, each coming from a transcendental constraint on the ˆ The first is that the curve Cˆ is subject to a set of constraints whereby the periods curve C. of a meromorphic differential on the curve are specified. This type of constraint arises in many other settings as well, for example when specifying the filling fractions of a curve in the AdS/CFT correspondence. Such constraints are transcendental in nature and until quite recently these had only been solved in the case of elliptic curves (which correspond to charge 2 monopoles). In [BE06,BE07] they were solved for a class of charge 3 monopoles using number theoretic results of Ramanujan. The second type of constraint is that the linear flow on the Jacobian of Cˆ corresponding to the integrable motion only intersects the theta divisor in a prescribed manner. In the monopole setting this means the Nahm data will yield regular monopole solutions but a similar constraint also appears in other applications of integrable systems. In Hitchin’s approach (reviewed below) this may be expressed as the vanishing of a real one parameter family of cohoˆ L λ (n − 2)) = 0 for λ ∈ (0, 2). Viewing the mologies of certain line bundles, H 0 (C, line bundles as points on the Jacobian this is equivalent to a real line segment not intersecting the theta divisor of the curve. Indeed there are sections for λ = 0, 2 and the flow is periodic (mod 2) in λ and so we are interested in the number of times a real line intersects . While techniques exist that count the number of intersections of a complex line with the theta divisor we are unaware of anything comparable in the real setting and again solutions have only been found for particular curves [BE09]. Thus the application of integrable systems techniques to the construction of monopoles and (indeed more generally) encounters two types of problems that each merit further study.
Cyclic Monopoles
305
The present paper will use symmetry to reduce these problems to ones more manageable. Long ago monopoles of charge n with cyclic symmetry Cn were shown to exist [OR82] and more recently such monopoles were reconsidered [HMM95] from a variety of perspectives. The latter work indeed considered the case of monopoles with more general Platonic symmetries and for the case of tetrahedral, octahedral and icosahedral symmetry (where such monopoles exist) the curves were reduced to elliptic curves. (See [HS96a,HS96b,HS97] for development of this work.) Our first result is to strengthen work of Sutcliffe [Sut96]. Motivated by Seiberg-Witten theory Sutcliffe gave an ansatz for Cn symmetric monopoles in terms of su(n) affine Toda theory. The spectral curve Cˆ of a Cn symmetric monopole yields an n-fold unbranched cover of the hyperelliptic spectral curve C of the affine Toda theory, a spectral curve that arises in Seiberg-Witten theory describing the pure gauge N = 2 supersymmetric su(n) gauge theory. (We shall recall some properties of the Nahm construction and this relation between curves in Sect. 2.) Sutcliffe’s ansatz (Sect. 3) shows how solutions to the affine Toda equations yield cyclically symmetric monopoles. Our first result proves that any cyclically symmetric monopole is gauge equivalent to Nahm data given by Sutcliffe’s ansatz, and so obtained from the affine Toda equations. We mention that Hitchin in an unpublished note had, prior to Sutcliffe, observed that cyclic charge 3 monopoles were equivalent to solutions of the affine Toda equations. The remainder of this paper shows that the relation between the Nahm data and the affine Toda system is much closer than simply that they yield the same equations of motion. The solution of an integrable system is typically expressed in terms of the straight line motion on the Jacobian of the system’s spectral curve. Such a line is determined both by its direction and a point on the Jacobian. We shall show that both the direction (given by the Ercolani-Sinha vector, Sect. 4) and point relevant for monopole solutions (Sect. 5) are obtained as pull-backs of Toda data. This connection is remarkable and ties the geometry together in a very tight manner. Section 6 recalls a theorem of Accola and Fay that holds in precisely this setting, showing how the thetafunctional solutions of the monopole reduce to precisely the theta-functional solutions of Toda. At this stage we have reduced the problem of constructing cyclically symmetric monopoles to one of determining hyperelliptic curves that satisfy the transcendental constraints described above. Though more manageable the problems are still formidable and a construction in the charge 3 setting will be described elsewhere [BDE]. We conclude with a discussion. 2. Monopoles We shall briefly recall the salient features for constructing su(2) monopoles of charge n. We begin with Nahm’s construction [Nah82]. In generalizing the ADHM construction of instantons Nahm established an equivalence between nonsingular monopoles and what is now referred to as Nahm data: three n × n matrices Ti (s) with s ∈ [0, 2] satisfying N1 Nahm’s equation 3 1 dTi = i jk [T j , Tk ], ds 2
(2.1)
j,k=1
N2 Ti (s) is regular for s ∈ (0, 2) and has simple poles at s = 0 and s = 2, the residues of which form an irreducible n-dimensional representation of su(2), Ti (s) = Tit (2 − s). N3 Ti (s) = −Ti† (s),
306
H. W. Braden
Upon defining A(ζ ) = T1 + i T2 − 2i T3 ζ + (T1 − i T2 )ζ 2 , M(ζ ) = −i T3 + (T1 − i T2 )ζ, we find that Nahm’s equation is equivalent to the Lax equation 3 dTi 1 d = i jk [T j , Tk ] ⇐⇒ [ + M, A] = 0. ds 2 ds j,k=1
Here ζ is a spectral parameter. Following from the Lax equation we have the invariance of the spectral curve Cˆ : 0 = P(η, ζ ) := det(η1n + A(ζ )),
(2.2)
where P(η, ζ ) = ηn + a1 (ζ )ηn−1 + · · · + an (ζ ),
deg ar (ζ ) ≤ 2r.
(2.3)
As with any spectral curve presented in the form (2.2) one should always ask where Cˆ lies. Typically the spectral curve lies in a surface, Cˆ ⊂ S, and properties of the surface are closely allied with the integrable system encoded by the Lax equation. For the case at hand Cˆ ⊂ T P1 := S,
(η, ζ ) → η
d ∈ T P1 , dζ
and monopoles admit a minitwistor description: the curve Cˆ corresponds to those lines in R3 which admit normalizable solutions of an appropriate scattering problem in both directions [Hit82,Hit83]. This latter description makes clear that Cˆ comes equipped with an antiholomorphic involution or real structure coming from the reversal of orientation of lines (η, ζ ) → (−η/ ¯ ζ¯ 2 , −1/ζ¯ ). This means the coefficients of (2.3) are such that ar (ζ ) = (−1)r ζ 2r ar (−1/ζ )
(2.4)
and so each may be expressed in terms of 2r + 1 (real) parameters r r αr,l 1/2 1 ar (ζ ) = χr (ζ − αr,k )(ζ + ), αr,k ∈ C, χr ∈ R. αr,l αr,k l=1
k=1
We remark that a real structure constrains the form of the period matrix of a curve and that while in general there may be between 0 and gˆ + 1 ovals of fixed points of an antiˆ for the case at hand there are no fixed holomorphic involution (gˆ being the genus of C) points. For the monopole spectral curve (2.3) we have (generically) gˆ = (n − 1)2 . Although in many situations the solution of the integrable system encoded by a Lax pair (with spectral parameter) only depends on intrinsic properties of the spectral curve the monopole physical setting means that extrinsic properties of our curve in T P1 are
Cyclic Monopoles
307
relevant here. Spatial symmetries act on the monopole spectral curve via fractional linear transformations. Although a general Möbius transformation does not change the period matrix of a curve Cˆ only the subgroup P SU (2) < P S L(2, C) preserves the reality properties necessary for a monopole spectral curve. These reality conditions are an extrinsic feature of the curve (encoding the space-time aspect of the problem) whereas the intrinsic properties of the curve are invariant under birational transformations or the full Möbius group. Such extrinsic aspects are not a part of the usual integrable system story. Thus S O(3) spatial rotations induce an action on T P1 via P SU (2): if p q ∈ P SU (2), (| p|2 + |q|2 = 1) then q¯ p¯ ζ → ζ˜ :=
p¯ ζ − q¯ , qζ + p
η → η˜ :=
η (q ζ + p)2
(2.5)
corresponds to a rotation by θ around n ∈ S 2 where n 1 sin (θ/2) = Im q, n 2 sin (θ/2) = −Re q, n 3 sin (θ/2) = Im p, cos (θ/2) = −Re p. This S O(3) action commutes with the standard real structure on T P1 . The action on the spectral curve may be expressed as P(η, ˜ ζ˜ ) =
˜ P(η, ζ) , (q ζ + p)2n
˜ P(η, ζ ) = ηn +
n
ηn−r a˜ r (ζ ),
(2.6)
r =1
where in terms of the parameterization above ⎡
1/2 ⎤ r r α˜ r,l a˜ r (ζ ) χ˜r 1 ⎣ ⎦ ar (ζ ) → ≡ (ζ − α˜ r,k )(ζ + ) (q ζ + p)2r (q ζ + p)2r α˜ r,l α ˜ r,k l=1 k=1 with pαr,k + q¯ , p¯ − αr,k q r ( p¯ − αr,k q)( p − α¯ r,k q)( ¯ α¯ r,k p¯ + q)(αr,k p + q) ¯ 1/2 χr → χ˜r ≡ χr . αr,k α¯ r,k
αk → α˜ r,k ≡
k=1
In particular the form of the curve does not change under a rotation: that is, if ar = 0 then so also a˜ r = 0. Hitchin, Manton and Murray [HMM95] showed how curves invariant under finite subgroups of S O(3) or their binary covers yield symmetric monopoles. Suppose we have a symmetry; the spectral curve 0 = P(η, ζ ) is transformed to the same curve, ˜ ˜ 0 = P(η, ˜ ζ˜ ) = P(η, ζ )/(q ζ + p)2n . Then P(η, ζ ) = P(η, ζ ), or equivalently ar (ζ ) = a˜ r (ζ ). Relevant for us is the example of cyclically symmetric monopoles. Let ω = exp(2πi/n). A rotation of order n is then given by p¯ = ω1/2 , q = 0 which yields φ : (η, ζ ) → (ωη, ωζ ). Correspondingly ηi ζ j is invariant for i + j ≡ 0 mod n and the spectral curve ηn + a1 ηn−1 ζ + a2 ηn−2 ζ 2 + · · · + an ζ n + βζ 2n + γ = 0
308
H. W. Braden
is invariant under the cyclic group Cn generated by this rotation. Imposing the reality conditions (2.4) and centering the monopole (setting a1 = 0) then gives us the spectral curve in the form ηn + a2 ηn−2 ζ 2 + · · · + an ζ n + βζ 2n + (−1)n β¯ = 0,
ai ∈ R,
(2.7)
and by an overall rotation we may choose β real. Now the Cn -invariant curve Cˆ (2.7) of genus gˆ = (n − 1)2 is an n-fold unbranched cover of a genus g = n − 1 curve C. The Riemann-Hurwitz theorem yields the relation gˆ = n(g − 1) + 1. Introduce the rational invariants x = η/ζ, ν = ζ n β, then x n + a2 x n−2 + · · · + an + ν +
(−1)n |β|2 = 0, ν
and upon setting y = ν − (−1)n |β|2 /ν we obtain the curve y 2 = (x n + a2 x n−2 + · · · + an )2 − 4(−1)n |β|2 .
(2.8)
This curve is the spectral curve of su(n) affine Toda theory in standard hyperelliptic form. ˆ j above the point ζ = ∞ project For future reference we note that the n-points ∞ to one of the infinite points, ∞+ , of the curve (2.8), while the n-points above the point ˆ j we have η/ζ ∼ ρ j ζ as ζ ∼ ∞ ˆ j , with ζ = 0 project to the other infinite point. At ∞ ρ j = β 1/n exp(2πi[ j + 1/2]/n). The n = 2 example. The reality conditions for n = 2 and a2 (ζ ) = βζ 4 + γ ζ 2 + δ means that δ = β¯ and γ = γ¯ and (2.7) becomes η2 + βζ 4 + γ ζ 2 + β¯ = 0. This is an elliptic curve. If β = |β|e2iθ let U = ζ eiθ and V = iηeiθ /|β|1/2 and this may be rewritten as V 2 = U 4 + t U 2 + 1,
t = γ /|β|.
(2.9) For irreducibility t = 2. Now the curve (2.8) becomes (with Y = y/ γ 2 − 4|β|2 ), Y 2 = x 4 + t x 2 + 1,
2t t = √ . 2 t −4
(2.10)
These two curves (2.9, 2.10) are 2-isogenous: if we quotient the former curve under the involution (U, V ) → (−U, −V ) we obtain the latter. 3. The Sutcliffe Ansatz Some years ago Sutcliffe [Sut96] introduced the following ansatz for cyclically symmetric monopoles. Let ⎞ ⎛ 0 e(q1 −q2 )/2 0 ... 0 ⎟ ⎜ 0 0 0 e(q2 −q3 )/2 . . . ⎟ ⎜ ⎟ ⎜ .. .. .. (3.1) T1 + i T2 = ⎜ ⎟, . . . ⎟ ⎜ (q −q )/2 ⎠ ⎝ n n−1 0 0 0 ... e e(qn −q1 )/2 0 0 ... 0
Cyclic Monopoles
309
⎛
0
... ... ... .. .
0 0
⎜e(q1 −q2 )/2 ⎜ ⎜ 0 e(q2 −q3 )/2 T1 − i T2 = − ⎜ ⎜ .. ⎝ . 0 0 ... ⎞ ⎛ p1 0 . . . 0 i ⎜ 0 p2 . . . 0 ⎟ T3 = − ⎜ , . .. ⎟ .. 2 ⎝ .. . . ⎠ 0
0
...
0 0 0 e(qn−1 −qn )/2
⎞ e(qn −q1 )/2 ⎟ 0 ⎟ ⎟ 0 ⎟ , (3.2) ⎟ .. ⎠ . 0
(3.3)
pn
where pi , qi are real. Then Ti (s) = −Ti† (s) and Nahm’s equations yield ⎧ ⎪ ⎨ p1 − p2 = q˙1 − q˙2 , d .. ⇒ (T1 + i T2 ) = i[T3 , T1 + i T2 ] . ⎪ ds ⎩ pn − p1 = q˙n − q˙1 , ⎧ q1 −q2 + eqn −q1 , ⎪ ⎨ p˙ 1 = −e d i . .. T3 = [T1 , T2 ] = [T1 + i T2 , T1 − i T2 ] ⇒ ⎪ ds 2 ⎩ p˙ n = −eqn −q1 + eqn−1 −qn . These equations then follow from the equations of motion of the affine Toda Hamiltonian H=
1 2 p1 + · · · + pn2 − eq1 −q2 + eq2 −q3 + · · · + eqn −q1 . 2
(3.4)
Sutcliffe’s observation is that particular solutions of these equations will then yield cyclically invariant monopoles. In fact the monopole Lax operator A(ζ ) here is essentially the usual Toda Lax operator and 1 Tr A(ζ )2 = ζ 2 H. 2 The spectral curve of the affine Toda system is then (2.8) upon restricting the center of mass motion i pi = 0 = i qi . The constant β may be related to the coefficient of the scaling element when the Toda equations are expressed in terms of the affine algebra sln . In fact we may strengthen Sutcliffe’s ansatz substantially. At this stage we only have that solutions of the Toda equations will yield some solutions of the Nahm equations with cyclic symmetry. First we will show that any Cn invariant solution of Nahm’s equations (for charge n su(2) monopoles) are given by solutions of the affine Toda equations. Then we will very concretely relate the solutions. We have that G ⊂ S O(3) acts on triples t = (T1 , T2 , T3 ) ∈ R3 ⊗ S L(n, C) via the natural action on R3 and conjugation on S L(n, C). This natural action may be identified with the SU (2) action on O(2) given above. If g ∈ S O(3) and g = ρ(g ) is its image in S L(n, C) then we have " ! g ◦ η + (T1 + i T2 ) − 2i T3 ζ + (T1 − i T2 )ζ 2 ! " = ω η + ω−1 g(T1 + i T2 )g −1 − 2igT3 g −1 ζ + ωg(T1 − i T2 )g −1 ζ 2 .
310
H. W. Braden
Thus invariance of the spectral curve gives g(T1 + i T2 )g −1 = ω(T1 + i T2 ), gT3 g −1 = T3 , g(T1 − i T2 )g −1 = ω−1 (T1 − i T2 ). Now Hitchin, Manton and Murray [HMM95] have described how the S O(3) action on S L(n, C) decomposes as the direct sum 2n − 1 ⊕ 2n − 3 ⊕ . . . ⊕ 5 ⊕ 3, where 2k − 1 denotes the S O(3) irreducible representation of dimension 2k − 1. We may identify S O(3) and its image in S L(n, C) and because this decomposition has rank S L(n, C) = n − 1 summands then, by a theorem of Kostant [K], the Lie algebra of this S O(3) is a principal three-dimensional By ⎞ conjugation we may express our gener⎡ subalgebra. ⎛ ⎤ 0 1 0 2π ⎝ −1 0 0⎠⎦ and then g = ρ(g ) = exp 2π ator g of Cn as g = exp ⎣ H , n n 0 0 0 where H is semi-simple and the generator of the principal three-dimensional algebra’s Cartan subalgebra. Kostant described the action of such elements on arbitrary semi-simple Lie algebras and their roots. For the case at hand we have that g is equivalent to Diag(ωn−1 , . . . , ω, 1) and that g E i j g −1 = ω j−i E i j . Therefore at this stage we know that for a cyclically invariant monopole we may write T1 + i T2 =
ˆ α∈
˜ e(α,q)/2 Eα ,
T3 = −
i p˜ j H j , 2 j
ˆ are the simple roots together with minus the where in principle q˜i , p˜ i ∈ C, and α ∈ highest root. (The sum over Hi may be taken as either the Cartan subalgebra of S L(n, C) or, by reinstating the center of mass, the Cartan subalgebra of G L(n, C).) The Sutcliffe ansatz follows if the q˜i and p˜ i may be chosen real. Now by an SU (n) transformation Diag(eiθ1 , eiθ2 , . . . , eiθn ) (where i θi = 0) together with an overall S O(3) rotation the reality of q˜i may be achieved. The reality of p˜ i follows upon imposing Ti (s) = −Ti† (s) which also fixes T1 − i T2 . At this stage we have established the following. Theorem 3.1. Any cyclically symmetric monopole is gauge equivalent to Nahm data given by Sutcliffe’s ansatz, and so obtained from the affine Toda equations. 4. Flows and Solutions The relation between the Nahm data and the affine Toda system is much closer than simply that they yield the same equations of motion. Let Cˆ denote the genus (n − 1)2 spectral curve of the monopole and C denote the genus n − 1 spectral curve of the Toda ˆ n and the natural projection π : Cˆ → C is theory. We have already noted that C = C/C an n-fold unbranched cover. The solution of an integrable system is typically expressed in terms of the straight line motion on the Jacobian of the system’s spectral curve. Such a line is determined both by its direction and a point on the Jacobian. We shall now show that both the direction and point relevant for monopole solutions are obtained as pull-backs of Toda data.
Cyclic Monopoles
311
First we recall that meromorphic differentials describe flows, and that a meromorphic differential on a Riemann surface is uniquely specified by its singular parts together with $gˆ # ˆ Z), some normalisation conditions. If aˆ i , bˆ i form a canonical basis for H1 (C, i=1
aˆ i ∩ bˆ j = −bˆ j ∩ aˆ i = δi j , then one such normalisation condition is that the aˆ -periods of the meromorphic differential vanish. (Thus the freedom to add to the meromorphic differential a holomorphic differential without changing its singular part is eliminated.) In what follows we denote g by {ai , bi }i=1 a similar canonical basis for H1 (C, Z). ˆ j to For the monopole the Lax operator A(ζ ) has poles at ζ = ∞. If we denote ∞ be the n points on the spectral curve above ζ = ∞ (and these may be assumed distinct) ˆ j . Consequently in terms of a local coordinate t then we find that η/ζ = ρ j ζ as ζ ∼ ∞ ˆ j , ζ = 1/t, then at ∞ ρj η = − 2 + O(1) dt. d ζ t Thus on the monopole spectral curve we may uniquely define a meromorphic differential ˆ j and normalization by the pole behaviour at ∞ % ρ j γ∞ = 2 + O(1) dt, 0= γ∞ . t aˆi ˆ The vector of b-periods,
% = 1 γ∞ , U 2iπ bˆ known as the Ercolani-Sinha vector [ES89], determines the direction of the monopole ˆ This vector is in fact constrained. Let us first recall Hitchin’s conditions flow on Jac(C). on a monopole spectral curve, equivalent to the Nahm data already given. These are H1 Reality conditions ar (ζ ) = (−1)r ζ 2r ar (−1/ζ ). H2 Let L λ denote the holomorphic line bundle on T P1 defined by the transition function g01 = exp(−λη/ζ ), and let L λ (m) ≡ L λ ⊗ π ∗ O(m) be similarly defined in terms of the transition function g01 = ζ m exp (−λη/ζ ). Then L 2 is trivial on Cˆ and L 1 (n − 1) is real. ˆ L λ (n − 2)) = 0 for λ ∈ (0, 2). H3 H 0 (C, We have already seen the reality conditions. Here the triviality of L 2 means that there exists a nowhere-vanishing holomorphic section. The following are equivalent [ES89, HMR00]: ˆ (1) L 2 is trivial on C. & T & ∈ ⇐⇒ U = 1 (2) 2U γ , . . . , γ = 21 n + 21 τˆ m. ∞ ∞ ˆ ˆ 2πı b b 1
gˆ
(3) There exists a 1-cycle es = n · aˆ + m · bˆ such that for every holomorphic differential % β0 ηn−2 + β1 (ζ )ηn−3 + · · · + βn−2 (ζ ) = dζ, = −2β0 . ∂P/∂η es
312
H. W. Braden
Here τˆ is the period matrix of Cˆ and is the associated period lattice of the curve. Thus is constrained to be a half-period. These are known as the Ercolani-Sinha constraints U and they impose gˆ transcendental constraints on the curve yielding n (2 j + 1) − gˆ = (n + 3)(n − 1) − (n − 1)2 = 4(n − 1) j=2
degrees of freedom. We now turn to consider the behaviour of the Ercolani-Sinha vector under a symmetry. Clearly our group acting on the curve leads to an action on divisors and consequently on the Jacobian. We now show that the Ercolani-Sinha vector describing the flow is fixed under the symmetry. This means the vector may be obtained from the pull-back of a vector on the Jacobian of the quotient (Toda) curve. Suppose we have a symmetry 0 = P(η, ζ ) = P(η, ˜ ζ˜ ) =
˜ P(η, ζ) . (q ζ + p)2n
In particular ∂η˜ P(η, ˜ ζ˜ ) = (q ζ + p)2 ∂η P(η, ˜ ζ˜ ) =
˜ ζ) ∂η P(η, ∂η P(η, ζ ) = . 2n−2 (q ζ + p) (q ζ + p)2n−2
(4.1)
Using d ζ˜ =
dζ (q ζ + p)2
we see then that ζ˜ r η˜ s d ζ˜ ( pζ ¯ − q) ¯ r (q ζ + p)2n−4−r −2s ηs dζ . = ∂η P(η, ζ ) ∂η˜ P(η, ˜ ζ˜ ) Bringing these together Lemma 4.1. The differential ωˆ r,s = only if
ζ r ηs dζ ∂η P(η,ζ )
is invariant under the rotation (2.5) if and
ζ r = ( pζ ¯ − q) ¯ r (q ζ + p)2n−4−r −2s . This always has a solution, the holomorphic differential ωˆ =
ηn−2 dζ . ∂η P(η, ζ )
For the particular case of interest here, for rotations given by q = 0, | p|2 = 1, then r s ζ η dζ ζ r ηs dζ = ωr +s+2 , (4.2) φ∗ ∂η P(η, ζ ) ∂η P(η, ζ ) and we also have solutions for each s (0 ≤ s ≤ n − 2) and r = n − 2 − s. These give us g = n − 1 Cn -invariant holomorphic differentials which are pullbacks of the holomorphic differentials on C. We remark also that the symmetry always fixes the subspaces
Cyclic Monopoles
313
g−1 ˆ
μr ωr,s for fixed s. Thus on the space of holomorphic differentials {ωˆ I } I =1 ∪{ωˆ 0,n−2 } (for appropriate I = (r, s) whose order does not matter) we have ⎛ ⎞ ∗ ∗ 0 φ ∗ (ωˆ 1 , . . . , ωˆ g−1 ˆ 0,n−2 ) = (ωˆ 1 , . . . , ωˆ g−1 ˆ 0,n−2 ) ⎝∗ ∗ 0⎠ ˆ ,ω ˆ ,ω 0 0 1 r
:= (ωˆ 1 , . . . , ωˆ g−1 ˆ 0,n−2 )L , ˆ ,ω
(4.3)
where L is a gˆ × gˆ complex matrix. As L n = 1, the matrix is both invertible and diagonalizable. With {ˆai , bˆ i } the canonical homology basis introduced earlier and {uˆ j } a basis of holomorphic differentials for our Riemann surface Cˆ we have the matrix of periods & ˆ &aˆ i uˆ j = A = 1 Aˆ (4.4) τˆ Bˆ ˆb uˆ j i
ˆ Z) with τˆ = Bˆ Aˆ −1 the period matrix. If σ is any automorphism of Cˆ then σ acts on H1 (C, and the holomorphic differentials by aˆ i aˆ i A B , σ ∗ uˆ j = uˆ k L kj , σ∗ ˆ = C D bi bˆ i A B ∈ Sp(2g, ˆ Z) and L ∈ G L(g, ˆ C). Then from where C D % % uˆ = σ ∗ uˆ σ∗ γ
we obtain
A C
B D
γ
Aˆ Aˆ = ˆ L. ˆ B B
(4.5)
With the ordering of holomorphic differentials of (4.3) the second of the equivalent conditions for the Ercolani-Sinha vector says there exist integral vectors n, m such that Aˆ (n, m) ˆ = −2(0, . . . , 0, 1). (4.6) B Now suppose σ corresponds to a symmetry coming from a rotation. Then the form of L in (4.3) gives Aˆ Aˆ (n, m) ˆ = −2(0, . . . , 0, 1) = −2(0, . . . , 0, 1).L = (n, m) ˆ .L B B ˆ A A B , = (n, m) C D Bˆ and so
A (n, m) − (n, m) C
B D
Aˆ = 0. Bˆ
314
H. W. Braden
Aˆ As the rows of the lattice generated by ˆ are independent over Z we therefore have B that A B (n, m) = (n, m) C D A B for all symplectic matrices representing the symmetries coming from spatial C D rotations. In particular (n, m) is invariant under the group of symmetries. Therefore the Ercolani-Sinha vector is invariant and so as an element of the Jacobian, this will reduce to a vector of the Jacobian of the quotient curve. Viewing this vector as a divisor on the curve it projects to a divisor on the quotient curve. Thus we have established Theorem 4.2. The Ercolani-Sinha vector is invariant under the group of symmetries of the spectral curve arising from rotations (2.5), = π ∗ (U), U
U ∈ Jac(C).
(4.7)
For the cyclic symmetry under consideration we have from (−1)n |β|2 dζ dζ dy = n ν + = −n(x n + a2 x n−2 + · · · + an ) , ν ζ ζ ∂η P(η, ζ ) = ζ n−1 ∂x (x n + a2 x n−2 + · · · + an ), that
ζ n−2−s ηs dζ 1 xsdx = π∗ − . ∂η P(η, ζ ) n y
(4.8)
Thus each of the invariant differentials (for 0 ≤ s ≤ n − 2) reduce to hyperelliptic differentials. 5. The Base Point ˆ that Hitchin In the construction of monopoles there is a distinguished point ' K ∈ Jac(C) ˆ For n ≥ 3 this point is a singular uses to identify degree gˆ − 1 line bundles with Jac(C). point of the theta divisor, ' K ∈ singular [BE06]. If we denote the Abel map by ˆ = A Qˆ ( P) then
(
Pˆ Qˆ
uˆ i ,
' K = Kˆ Qˆ + A Qˆ (n − 2)
n
ˆk . ∞
(5.1)
k=1
ˆ If K ˆ is the canonical divisor Here Kˆ Qˆ is the vector of Riemann constants for the curve C. C of the curve then A ˆ (K ˆ ) = −2 Kˆ ˆ . The righthand side of (5.1) is in fact independent Q
C
Q
of the base point Qˆ in its definition.
Cyclic Monopoles
315
The point K is the base point of the linear motion in the Jacobian referred to earlier and we shall now relate this to a point in the Jacobian of the Toda spectral curve C. Let A Q (KC ) = −2K Q be the corresponding quantities for the curve C with basis of ˆ = Q is holomorphic differentials {u a }. We first relate π ∗ K Q and Kˆ Qˆ , where π( Q) n ˆ ˆ some preimage of Q. Let our symmetry be φ : C → C, φ = 1, and observe that (with ˆ = P, π( Q) ˆ = Q) π( P) ( P n−1 ( φ s ( P) n−1 ! ˆ " ˆ − A ˆ φ s ( Q) ˆ A Qˆ φ s ( P) π ∗ (A Q (P)) = π ∗ u = uˆ = . Q s ˆ s=0 φ ( Q)
Q
s=0
(This is actually independent of the base-point chosen for the Abel map, so well-defined.) 2g−2 2g−2 n−1 s ˆ Now if α=1 Pα is a canonical divisor for C then α=1 s=0 φ ( Pα ) is a canonical ˆ Thus divisor for C. ) * π ∗ (−2K Q ) = π ∗ A Q (KC ) ⎞ ⎛ 2g−2 ( Pα u⎠ = π∗ ⎝ α=1
Q
= A Qˆ (Kˆ Cˆ ) − 2(g − 1) = −2 Kˆ Qˆ − 2(g − 1)
n−1
ˆ A Qˆ φ s ( Q)
s=0 n−1
ˆ . A Qˆ φ s ( Q)
s=0
Therefore π ∗ (K Q ) = Kˆ Qˆ + (g − 1)
n−1
ˆ + e, A Qˆ φ s ( Q) ˆ
(5.2)
s=0
where 2eˆ ∈ is a half-period. This expression may be rewritten as π (K Q ) = Kˆ Qˆ + (g − 1) ∗
n−1
ˆ + eˆ A Qˆ φ s ( Q)
s=0 n−1 " s ˆ ˆ − (gˆ − 1)A ˆ ( P) ˆ + (g − 1) φ A ( Q) + eˆ = Kˆ Qˆ + (gˆ − 1)A Qˆ ( P) ˆ Q Q
!
s=0
ˆ + (g − 1) = Kˆ Pˆ − n(g − 1)A Qˆ ( P)
n−1
ˆ + eˆ A Qˆ φ s ( Q)
s=0
= Kˆ Pˆ + (g − 1)
n−1
ˆ + e, A Pˆ φ s ( Q) ˆ
s=0
showing the left-hand side is independent of the choice of base-point for the Abel map. Comparison of (5.1) and (5.2) now shows that ' K = π ∗ (K ∞+ ) − e, ˆ
(5.3)
316
H. W. Braden
ˆ k ) = ∞+ as noted earlier. Now the half-period eˆ can be identified and is of where π(∞ the form eˆ = π ∗ (e). The actual identification depends on an homology choice and will be given in the next section, but for the moment we simply note the form ' K = π ∗ (K ∞+ − e).
(5.4)
6. Fay-Accola Factorization The standard reconstruction of solutions for an integrable system with spectral curve Cˆ proceeds by constructing the Baker-Akhiezer functions for this curve. These may be calculated in terms of theta functions for the curve and for our present purposes we may focus on the theta function θ (λU − ' K | τˆ ). This describes a flow on the Jacobian of Cˆ in the direction of the Ercolani-Sinha vector U with base point ' K . We have observed that we have a cyclic unramified covering π : Cˆ → C of the affine Toda spectral curve by ˆ which the monopole spectral curve. The map π leads to a map π ∗ : Jac(C) → Jac(C) ∗ g g ˆ may be lifted to π : C → C . Further we have established that λU − ' K = π ∗ (λU − K ∞+ + e). We now are in a position to make use of a remarkable factorization theorem due to Accola and Fay [Acc71,Fay73] and also observed by Mumford. When zˆ = π ∗ z the theta functions on Cˆ and C are related by this factorization theorem, Theorem 6.1 (Fay-Accola). With respect to the ordered canonical homology bases {ˆaic , bˆ ic } described below and for arbitrary z =∈ Cg we have θ [e](π ˆ ∗ z; τˆ c ) = c0 (τ c ) +n−1 0 0 ... 0 (z; τ c ) k=0 θ k 0 ... 0 n
(6.1)
which is a non-zero modular constant c0 (τˆ c ) independent of z. Here τˆ c is the a-normalized period matrix for the curve Cˆ in this homology basis and 0 0 ... 0 0 0 ... 0 ∗ ∗ eˆ = n−1 . = π (e) = π n−1 0 ... 0 0 ... 0 2n 2 The significance of this theorem for our setting is that it means we can reduce the construction of solutions to that of quantities purely in terms of the hyperelliptic affine Toda spectral curve. The theorem is expressed in terms of a particular choice of homology basis which is well adapted to the symmetry at hand. In terms of the conformal automorphism φ : Cˆ → Cˆ of Cˆ that generates the group Cn = {φ s | 0 ≤ s ≤ n − 1} of cover transformations of Cˆ and the projection π : Cˆ → C there exists a basis {ˆac0 , bˆ c0 , aˆ c1 , bˆ c1 , . . . , aˆ cg−1 , bˆ cg−1 } of ˆ ˆ c c c c c c homology cycles for Cˆ and {a0 , b0 , a1 , b1 , . . . , ag−1 , bg−1 } for C such that (for 1 ≤ j ≤ g − 1, 0 ≤ s ≤ n) π(ˆac0 ) = ac0 , π(ˆacj+s(g−1) ) = acj , π(bˆ c0 ) = n bc0 , π(bˆ cj+s(g−1) ) = bcj , φ s (ˆac0 ) ∼ aˆ c0 , φ s (ˆacj ) = aˆ cj+s(g−1) , φ s (bˆ 0 ) = bˆ c0 , φ s (bˆ cj ) = bˆ cj+s(g−1) .
Cyclic Monopoles
317
ˆ then Here φ s (ˆa0 ) is homologous to aˆ 0 . If vˆi are the aˆ -normalized differentials for C, ( ( ( ( vˆi = vˆi = (φ s )∗ vˆi = vˆi−s(g−1) , δi, j+s(g−1) = aˆ j+s(g−1)
ˆ j) φ s (a
aˆ j
aˆ j
and we find that (φ s )∗ vˆ0 = vˆ0 ,
(φ s )∗ vˆi = vˆi−s(g−1) .
If vi are the normalized differentials for C, then ( ( ( vi = vi = δi j = aj
ˆ j+s(g−1) ) π(a
a j+s(g−1)
(6.2)
π ∗ (vi )
shows that π ∗ (vi ) = vˆi + (φ)∗ vˆi + . . . + (φ p−1 )∗ vˆi and similarly that π ∗ (v0 ) = vˆ0 . We may use the characters of Cn to construct the remaining linearly independent differˆ entials on C. ˆ which lifts to an automorphism From (6.2) we have an action of Cn on Jac(C) g ˆ of C by φ s (ˆz ) = (ˆz 0 , zˆ 1−s(g−1) , . . . , zˆ g−1−s(g−1) , . . . , zˆ 1+(n−s−1)(g−1) , . . . , zˆ n−1+(n−s−1)(g−1) ). (6.3) Now (6.3) together with the invariance of the Ercolani-Sinha vector mans that in this cyclic homology basis we have (n, m) = (r0 , r, . . . , r, s0 , s, . . . , s),
(6.4)
where the vectors r = (r1 , . . . , r g−1 ) and similarly s are each repeated n times. We also have π∗ (es) = r0 a0 + nr · a + ns0 b0 + ns · b. π∗
(6.5)
ˆ With the choices above (things are different for b-normalization) we may lift the map ∗ g g ˆ ˆ : Jac(C) → Jac(C) to π : C → C , π ∗ (z) = π ∗ (z 0 , z 1 , . . . , z g−1 ) = (n z 0 , z 1 , . . . , z g−1 , . . . , z 1 , . . . , z g−1 ) = zˆ .
With this homology basis the period matrices for the two curves are related by the block form ⎞ ⎛ c n τ00 τ0c j τ0c j . . . τ0c j c (1) (n−1) ⎟ ⎜ τ j0 M M M ⎟ ⎜ τˆ c = ⎜ . ⎟ . ⎠ ⎝ . M τ cj0 M(1)
318
where M(s) =
H. W. Braden
,
The (r, s) block here has entry M(s−r ) and (M(s−r ) )T = (s) M(r −s) by the bilinear identity. Then τicj = n−1 s=0 Mi j . The case n = 3 is instructive, for here the n − 2 block matrices are just numbers and we have ˆ j ) vˆi . φ −s (b
⎛
a ⎜b c τˆ = ⎝ b b
b c d d
b d c d
⎞ b d⎟ , d⎠ c
τc =
1
3a
b
b c + 2d
.
(6.6)
The point to note is that although the period matrix for Cˆ involves integrations of differentials that do not reduce to hyperelliptic integrals, the combination of terms appearing in the reduction can be expressed in terms of hyperelliptic integrals. This is a definite simplification. Further the function defined by τˆ c has the symmetries (ˆz |τˆ c ) = (φ s (ˆz )|τˆ c ) for all zˆ ∈ Cgˆ . In particular, the divisor is fixed under Cn . If we are to reduce the construction of cyclic monopoles to a problem involving only hyperelliptic quantities we must describe the Ercolani-Sinha constraints in the context of the curve C. Theorem 6.2. The Ercolani-Sinha constraint on the curve Cˆ yields the constraint A − 2(0, . . . , 0, 1) = (r0 , nr, ns0 , ns) B
(6.7)
on the curve C with respect to the differentials u s = −x s d x/(ny) (s = 0, . . . , n − 2). Proof. The invariance of the Ercolani-Sinha vector means that φ ∗ (es) = es. Thus ( es
( ωˆ r,s =
( φ ∗ (es)
ωˆ r,s =
∗
es
φ ωˆ r,s = ω
( r +s+2
es
ωˆ r,s ,
where we have used (4.2). Thus the integral of any noninvariant differential around the cycle es must vanish, while from (4.8) and the Ercolani-Sinha condition we have that ( 1 xsdx 1 xsdx = . = π − − n y y es π∗ (es) n (
−2 δs,n−2
∗
The theorem then follows upon using (6.5).
In actual calculations it is convenient to use the unnormalized differentials ωˆ r,s and x s d x/(ny) rather than Fay’s normalized differentials vˆi . An alternate proof of Theorem 6.2 via Poincaré’s reducibility theorem is given in the Appendix, which provides further useful relations amongst the periods of the two curves.
Cyclic Monopoles
319
7. Discussion In this paper we have shown that any cyclically symmetric monopole is gauge equivalent to Nahm data obtained via Sutcliffe’s ansatz from the affine Toda equations. Further, the data needed to reconstruct the monopole, the Ercolani-Sinha vector and base point for linear flow on the Jacobian, may also be obtained from data on the affine Toda equation’s hyperelliptic spectral curve C. A theorem of Fay and Accola then enables us to express the theta functions for the monopole spectral curve in terms of the theta functions for the curve C. Finally the transcendental constraints on the monopole’s spectral curve can be recast as transcendental constraints for the hyperelliptic curve C (Theorem 6.2). At this stage then the construction of cyclically symmetric monopoles has been reduced to one entirely in terms of hyperelliptic curves. Although analogues of both the transcendental constraints still exist this is a significant simplification. We note that the structure of the theta divisor is better understood in the hyperelliptic setting [V95] and the hyperelliptic integrals are somewhat simpler than the general integrals appearing in the Ercolani-Sinha constraint for the full monopole curve. Other approaches to constructing monopoles are known. In particular [HMM95] describe cyclically symmetric monopoles within the rational map approach (see also [MS04, §8.8]). These authors show that the rational map for monopoles with Cn invariance about the x3 -axis takes the form R(z) =
μz l , −ν
zn
where 0 ≤ l ≤ n − 1. The complex quantity ν determines μ when the monopoles are strongly centered. Here ν = (−1)n−1 β¯ of Eq. (2.7). The moduli space Mln is a 4-dimensional totally geodesic submanifold of the full moduli space. It is interesting that both the rational map description and the description we have presented lead to extra discrete parameters (l in the case of rational maps, and k in 6.1). The connection, if any, between these will be pursued elsewhere [BDE]. Clearly the ansatz for monopoles extends to other algebras. If we construct the spectral curve from the Dn Toda system using the 2n dimensional representation we find a spectral curve Cˆ of the form η2n + a1 η2n−2 ζ 2 + a2 η2n−4 ζ 4 + · · · + an ζ 2n + αη2 (
1 + ζ 4n−4 w) = 0. w
Letting x = η/ζ the curve (upon dividing by ζ 2n ) becomes x 2n + a1 x 2n−2 + a2 x 2n−4 + · · · + an + αx 2 (
1 wζ 2n−2
and so we get with ν = αwζ 2n−2 , Pn (x 2 ) + x 2 (ν +
α2 )=0 ν
leading to a hyperelliptic curve C˜ y 2 = Pn (x 2 )2 − 4α 2 x 4 .
+ ζ 2n−2 w) = 0,
320
H. W. Braden
This curve has cyclic symmetry C2n−2 from the appearance of ζ 2n−2 and C2 due to the appearance of x 2 . The genus of Cˆ is (2n − 1)2 − 2n. The genus of C˜ is 2n − 1. Finally C˜ covers a genus n − 1 curve C, y 2 = Pn (u)2 − 4α 2 u 2 . Here we expect the Toda motion to lie in the Prym of this covering, but the general theory warrants further study. Acknowledgements. I have benefited from many discussions with Antonella D’Avanzo, Victor Enolskii and Timothy P. Northover. The results presented here were described at the MISGAM supported meeting “From Integrable Structures to Topological Strings and Back”, Trieste 2008, and the Lorentz Center meeting “Integrable Systems in Quantum Theory”, Leiden 2008. I am grateful to the organisers of these meetings for providing such a stimulating and pleasant environment.
Appendix A. Proof of Theorem 6.2 via Poincaré Reducibility It is instructive to see an alternative proof of Theorem 6.2 in terms of Poincaré’s reducibility condition, which we now recall. Consider Riemann matrices Aˆ A 1 1 ˆ ˆ A, = = A, = ˆ = B τ τˆ B ˆ where Aˆ and Bˆ are the gˆ × gˆ matrices of aˆ -periods and b-periods respectively for ˆ 2g ˆ the curve C with similarly named quantities for the curve C. If {γˆa }a=1 is a basis for ˆ Z), {ωˆ μ }gˆ ˆ and {γi }2g a basis for a basis of holomorphic differentials of C, H1 (C, μ=1 g
i=1
H1 (C, Z), {ωα }α=1 a basis of holomorphic differentials of C, these are related by π∗ (γˆa ) = Mai γi ,
π ∗ (ωμ ) = ωˆ α λαμ .
Here λ is a complex gˆ × g-matrix of maximal rank and M is a 2gˆ × 2g-matrix of integers of maximal rank. Then from % % % % ˆ aμ (M)aμ = Mai ωμ = ωμ = π ∗ ωμ = ωˆ α λαμ = (λ) γi
π∗ γˆa
γˆa
γˆa
we obtain Poincaré’s reducibility condition ˆ = M. λ
(A.1)
For the cyclic homology basis and corresponding aˆ -normalized differentials vˆi of Fay this takes the form 1 1 1 I 0 I = := M c , τc τˆ c τ 0 I where we define the gˆ × g matrices I, I and (to be used shortly) P, ⎛ ⎞ ⎞ ⎛ ⎞ ⎛ n 0 1g 1 0 0 ⎟ 0 1g−1 ⎟ ⎜0 1g−1 ⎟ ⎜ ⎜ ⎟ , I = ⎜ . ⎟, P = ⎜ . ⎟. I=⎜ . . . ⎝ .. ⎝ .. ⎠ ⎝ .. .. ⎠ .. ⎠ 0 1g−1
0 1g−1
0
Cyclic Monopoles
321
For the same cyclic homology basis but an arbitrary basis of holomorphic differentials we obtain (A.1) with λ = Aˆ −1 I A. Now bringing together the Ercolani-Sinha constraint (4.6) with (A.1) we find Aˆ A A . −2(0, . . . , 0, 1)λ = (n, m) ˆ λ = (n, m)M = (r0 , nr, ns0 , ns) B B B where we have used (6.4) and that (n, m)M = (r0 , nr, ns0 , ns). Here Aˆ has been constructed from the differentials ωˆ r,s = ζ r ηs dζ /∂η P(η, ζ ) (which are not Fay’s normalized differentials vˆi ) while the differentials for A are as yet unspecified and we wish to construct λ. Using (4.8) it is convenient to choose u s = −x s d x/(ny) (so that π ∗ (u s ) = ωˆ n−2−s,s ) and to order the differentials with the noninvariant differentials before the invariant differentials, {ωˆ r,s }r +s =n−2 ∪ {ωˆ n−2,0 , . . . , ωˆ 0,n−2 }. Then we find the matrix of periods ⎛ ⎞ 0 ... 0 ∗ ... ∗ ⎜ ⎟ D(0) A ⎜ ⎟ (1) ⎜ ⎟ D A Aˆ = ⎜ ⎟. ⎜ ⎟ .. .. ⎝ ⎠ . . D(n−1)
A
Here the first row has zero entries for the periods of the noninvariant differentials over the invariant cycle aˆ 0 while D(k) is the (g − 1) × (gˆ − g) matrix of periods of the noninvariant differentials over the cycles aˆ i+k(g−1) (i = 1, . . . , g − 1). Thus ( ( ( ( (k) ωˆ r,s = ωˆ r,s = (φ k )∗ ωˆ r,s = ωk(r +s+2) ωˆ r,s Di,(r,s) = ai+k(g−1)
=ω
k(r +s+2)
φ k (ai )
ai
ai
(0) Di,(r,s) .
The matrix of periods A of the invariant differentials over the same cycles is such that ( ( ( ( ωˆ n−2−s,s = π ∗ (u s ) = us = us , aˆi
aˆi
π∗ (aˆi )
ai
and the matrix of periods A for the curve C appearing above is precisely the submatrix ∗ ... ∗ A= . A Next we note that we may write (0, . . . , 0, 1)λA−1 = (0, . . . , 0, 1)Aˆ −1 I = (0, . . . , 0, 1)Aˆ −1 C P = (0, . . . , 0, 1) −1 × C −1 Aˆ P −1 −1 = C −1 Aˆ , . . . , C −1 Aˆ g,1 ˆ
g,g ˆ
322
with
H. W. Braden
⎛
1 ⎜0 ⎜0 C =⎜ ⎜. ⎝ ..
0
0 0
1g−1 1g−1 .. .
1g−1
0
1g−1
0
... ... ... .. . ...
⎛ ⎞ 1 0 0 ⎟ ⎜0 ⎜ 0 ⎟ ⎟ , C −1 = ⎜0 ⎜. ⎟ ⎝ .. ⎠ 1g−1 0
0 1g−1 −1g−1 .. .
0 0 1g−1
... ... ... .. .
0
...
This factorization was motivated by the observation that ⎛ ⎞ 0 ... 0 ∗ ... ∗ ⎜ ⎟ D(0) A ⎜ ⎟ (1) (0) E ⎜ ⎟ D −D 0 C −1 Aˆ = ⎜ ⎟= F ⎜ ⎟ .. .. ⎝ ⎠ . . (n−1) (n−2) −D 0 D
A , 0
−1g−1
⎞ 0 0 ⎟ 0 ⎟ ⎟. ⎟ ⎠ 1g−1
ˆ = |A| |F| we have the cofactor and so upon noting that gˆ − g is even and |C −1 A| expression −1 1 1 C −1 Aˆ = = Cof C −1 Aˆ Cof (A) j,g = A−1 g, j . j,gˆ g, ˆ j |A| |F| |A| Thus (0, . . . , 0, 1)λA−1 = (0, . . . , 0, 1)A−1 , where the right-hand row vector is g-dimensional and the left is g-dimensional. ˆ Bringing these results together establishes the theorem. References [Acc71]
Accola Robert, D.M.: Vanishing Properties of Theta Functions for Abelian Covers of Riemann Surfaces. In: Advances in the Theory of Riemann Surfaces: Proceedings of the 1969 Sony Brook Conference, edited by L.V. Ahlfors, L. Bers, H.M. Farkas, R.C. Gunning, I. Kra, H.E. Rauch, Princeton, NJ: Princeton University Press, 1971, pp. 7–18 [BDE] Braden, H.W., D’Avanzo A., Enolski, V.Z.: On charge-3 cyclic monopoles. Nonlinearity 24, 643–675 (2011) [BE06] Braden, H.W., Enolski, V.Z.: Remarks on the complex geometry of 3-monopole, math-ph/ 0601040. Part I appears as “some remarks on the Ercolani-Sinha construction of monopoles”. Theor. Math. Phys. 165, 1567–1597 (2010) [BE07] Braden, H.W., Enolski, V.Z.: Monopoles, Curves and Ramanujan. Reported at Riemann Surfaces, Analytical and Numerical Methods, Max Planck Instititute (Leipzig), #2007. Matem. Sborniki. 201, 19–74 (2010) [BE09] Braden, H.W., Enolski, V.Z.: On the tetrahedrally symmetric monopole. Commun. Math. Phys. 299, 255–282 (2010) [CG81] Corrigan, E., Goddard, P.: An n monopole solution with 4n − 1 degrees of freedom. Commun. Math. Phys. 80, 575–587 (1981) [ES89] Ercolani, N., Sinha, A.: Monopoles and baker functions. Commun. Math. Phys. 125, 385–416 (1989) [Fay73] Fay, J.D.: Theta functions on Riemann surfaces. Lectures Notes in Mathematics, Vol. 352, Berlin: Springer, 1973 [Hit82] Hitchin, N.J.: Monopoles and geodesics. Commun. Math. Phys. 83, 579–602 (1982) [Hit83] Hitchin, N.J.: On the construction of monopoles. Commun. Math. Phys. 89, 145–190 (1983) [HMM95] Hitchin, N.J., Manton, N.S., Murray, M.K.: Symmetric monopoles. Nonlinearity 8, 661–692 (1995)
Cyclic Monopoles
[HMR00] [HS96a] [HS96b] [HS97] [K] [MS04] [Nah82] [OR82] [Sut96] [V95]
323
Houghton, C.J., Manton, N.S., Romão, N.M.: On the constraints defining bps monopoles. Commun. Math. Phys. 212, 219–243 (2000) Houghton, C.J., Sutcliffe, P.M.: Octahedral and dodecahedral monopoles. Nonlinearity 9, 385– 401 (1996) Houghton, C.J., Sutcliffe, P.M.: Tetrahedral and cubic monopoles. Commun. Math. Phys. 180, 343–361 (1996) Houghton, C.J., Sutcliffe, P.M.: su(n) monopoles and platonic symmetry. J. Math. Phys. 38, 5576–5589 (1997) Kostant, B.: The principal three-dimensional subgroup and the betti numbers of a complex simple lie group. Amer. J. Math. 81, 973–1032 (1959) Manton, N., Sutcliffe, P.: Topological Solitons. Cambridge: Cambridge University Press, 2004 Nahm, W.: The construction of all self-dual multimonopoles by the ADHM method. In: Monopoles in Quantum Field Theory, edited by N.S. Craigie, P. Goddard, W. Nahm, Singapore: World Scientific, 1982 O’Raifeartaigh, L., Rouhani, S.: Rings of monopoles with discrete symmetry: explicit solution for n = 3. Phys. Lett. 112, 143 (1982) Sutcliffe, P.M.: Seiberg-witten theory, monopole spectral curves and affine toda solitons. Phys. Lett. B 381, 129–136 (1996) Vanhaecke, P.: Stratifications of hyperelliptic jacobians and the sato grassmannian. Acta. Appl. Math. 40, 143–172 (1995)
Communicated by N.A. Nekrasov
Commun. Math. Phys. 308, 325–364 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1349-z
Communications in
Mathematical Physics
Upper Bound on the Density of Ruelle Resonances for Anosov Flows Frédéric Faure1 , Johannes Sjöstrand2 1 Institut Fourier, UMR 5582, 100 rue des Maths, BP 74, 38402 St Martin d’Hères, France.
E-mail:
[email protected]
2 IMB, UMR 5584, Université de Bourgogne, 9, Avenue Alain Savary, BP 47870, 21078 Dijon Cedex, France.
E-mail:
[email protected] Received: 9 March 2010 / Accepted: 1 July 2011 Published online: 13 October 2011 – © Springer-Verlag 2011
Abstract: Using a semiclassical approach we show that the spectrum of a smooth Anosov vector field V on a compact manifold is discrete (in suitable anisotropic Sobolev spaces) and then we provide an upper bound for the density of eigenvalues of the operator (−i)V , called Ruelle resonances, close to the real axis and for large real parts. Résumé: Par une approche semiclassique on montre que le spectre d’un champ de vecteur d’Anosov V sur une variété compacte est discret (dans des espaces de Sobolev anisotropes adaptés). On montre ensuite une majoration de la densité de valeurs propres de l’opérateur (−i)V , appelées résonances de Ruelle, près de l’axe réel et pour les grandes parties réelles. Contents 1. 2. 3.
4. 5.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of Lemma 1.2 on the Escape Function . . . . . . . . . . . . . . . . Proof of Theorem 1.4 about the Discrete Spectrum of Resonances . . . . . on 3.1 Conjugation by the escape function and unique closed extension of P L 2 (X ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . has empty spectrum for (z) large enough . . . . . . . . . . . . . 3.2 P is discrete on (z) ≥ − (Cm − C) with some C ≥ 0 3.3 The spectrum of P independent of m . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of Theorem 1.5 that the Eigenvalues are Intrinsic to the Anosov Vector Field V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of Theorem 1.8 for the Upper Bound on the Density of Resonances . 5.1 Semiclassical symbols . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The symbol of the conjugated operator . . . . . . . . . . . . . . . . . 5.3 Main idea of the proof . . . . . . . . . . . . . . . . . . . . . . . . . .
326 327 335 340 340 342 343 345 346 346 346 349
326
5.4 Proof of Theorem 1.8 . . . . . . . . . . . . A. Some Results in Operator Theory . . . . . . . . A.1 On minimal and maximal extensions . . . . A.2 The sharp Gårding inequality . . . . . . . . A.3 Quadratic partition of unity on phase space . A.4 FBI transform and Toeplitz operators . . . .
F. Faure, J. Sjöstrand
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
350 358 358 359 359 362
1. Introduction Chaotic behavior of certain dynamical systems is due to hyperbolicity of the trajectories. This means that the individual trajectories are unstable under small perturbations of the initial point [9,29] and have a complicated and unpredictable behavior. However, the evolution of a cloud of points seems to be simpler: it will spread and equidistribute according to an invariant measure, called an equilibrium measure (or S.R.B. measure). Also from the physical point of view, such a cloud reflects the unavoidable lack of knowledge about the initial point. Following this idea, D. Ruelle [40,41], has shown in the 70’s that instead of considering individual trajectories, it is much more natural to study the evolution of densities under a linear operator called the Ruelle transfer operator or the Perron-Frobenius operator. For dynamical systems with strong chaotic properties, such as uniformly expanding maps or uniformly hyperbolic maps, Ruelle, Bowen, Fried, Rugh and others, using symbolic dynamics techniques (Markov partitions), have shown that the transfer operator has a discrete spectrum of eigenvalues. This spectral description has an important meaning for the dynamics since each eigenvector corresponds to an invariant distribution (up to a time factor). From this spectral characterization of the transfer operator, one can derive other specific properties of the dynamics such as decay of time correlation functions, central limit theorem, mixing, etc. In particular a spectral gap implies exponential decay of correlations. This spectral approach has recently (2002–2005) been improved by M. Blank, S. Gouëzel, G. Keller, C. Liverani [6,10,22,32], V. Baladi and M. Tsujii [3,4] (see [4] for some historical remarks) and in [17], through the construction of functional spaces adapted to the dynamics, independent of every symbolic dynamics. The case of flows i.e. dynamical systems with continuous time is more delicate (see [18] for historical remarks). This is due to the direction of time flow which is neutral (i.e. two nearby points on the same trajectory will not diverge from one another). In 1998 Dolgopyat [13,14] showed the exponential decay of correlation functions for certain Anosov flows, using techniques of oscillatory integrals and symbolic dynamics. In 2004 Liverani [31] adapted Dolgopyat’s ideas to his functional analytic approach, to treat the case of contact Anosov flows. In 2005 M. Tsujii [50] obtained an explicit estimate for the spectral gap for the suspension of an expanding map and in 2008 [51,52] he obtained such an estimate in the case of contact Anosov flows. Microlocal approach to transfer operators. It also appeared recently [15–17] that for hyperbolic dynamics on a manifold X , the transfer operators are Fourier integral operators and using standard tools of microlocal analysis, some of their spectral properties can be obtained from the study of “the associated classical symplectic dynamics”, namely the initial hyperbolic dynamics on X lifted to the cotangent space T ∗ X (the phase space). The simple idea behind this, crudely speaking, is that a transfer operator transports a “wave packet” associated to a point in phase space into another wave packet corresponding to the image point under the symplectic dynamics.
Upper Bound on Density of Ruelle Resonances for Anosov Flows
327
Following this approach, we studied hyperbolic diffeomorphisms in [16,17]. The aim of the present paper is to show that microlocal analysis in the semi-classical limit is also well adapted to hyperbolic systems with a neutral direction and with the inverse of the Fourier component in the neutral direction being a natural semi-classical parameter. In the paper [15] one of us has considered a partially expanding map and showed that a spectral gap develops in the limit of large oscillations in the neutral direction (which is a semiclassical limit). In this paper we consider a hyperbolic flow on a manifold X , generated by a vector field V . In this paper as well as in [17], one aim is to make more precise the connection between the spectral study of Ruelle resonances and the spectral study in quantum chaos [35,55], in particular to emphasize the importance of the symplectic properties of the dynamics in the cotangent space T ∗ X on the spectral properties of the transfer operator, and long time behavior of the dynamics. 1.1. The main results. The results of this paper will concern the following situation. Let X be an n-dimensional smooth compact connected Riemannian manifold, with n ≥ 3. Let φt be the flow on X generated by a smooth vector field V ∈ C ∞ (X ; T X ): V (x) =
d (φt (x))/t=0 ∈ Tx X, dt
x ∈ X.
(1.1)
We assume that the flow φt is Anosov. Let us first review some facts about Anosov flows and the dynamics induced on the cotangent space T ∗ X . 1.1.1. Anosov flows. We recall the definition (see [29] p. 545, or [37] p. 8). See Fig. 1. Definition 1.1. On a smooth Riemannian manifold (X, g), a vector field V generates an Anosov flow (φt )t∈R (or uniformly hyperbolic flow) if: • For each x ∈ X , there exists a decomposition Tx X = E u (x) ⊕ E s (x) ⊕ E 0 (x),
(1.2)
where E 0 (x) is the one dimensional subspace generated by V (x). • The decomposition (1.1) is invariant by φt for every t: ∀x ∈ X, (Dx φt )(E u (x)) = E u (φt (x)) and (Dx φt )(E s (x)) = E s (φt (x)). • There exist constants c > 0, θ > 0 such that for every x ∈ X , |Dx φt (vs )|g ≤ ce−θt |vs |g , |Dx φt (vu )|g ≤ ce
−θ|t|
|vu |g ,
∀vs ∈ E s (x), t ≥ 0 ∀vu ∈ E u (x), t ≤ 0,
(1.3)
meaning that E s is the stable distribution and E u the unstable distribution for positive time. Let us recall some facts: • Standard examples of Anosov flows are suspensions of Anosov diffeomorphisms (see [37] p. 8), or geodesic flows on manifolds M with sectional negative curvature (see [37] p. 9, or [29] p. 549, p. 551). Notice that in this case, the geodesic flow is Anosov on the unit cotangent bundle X = T1∗ M.
328
F. Faure, J. Sjöstrand
Fig. 1. Picture of an Anosov flow in X and instability of trajectories
• The global hyperbolic structure of Anosov flows or Anosov diffeomorphisms is a very strong geometric property, so that manifolds carrying such dynamics satisfy strong topological conditions and the list of known examples is not so long. See [7] for a detailed discussion and references on that question. • Let du = dim E u (x),
ds = dim E s (x),
(they are independent of x ∈ X ). Equation (1.2) implies du + ds + 1 = dim X = n. For every du , ds ≥ 1 one may construct an example of an Anosov flow: one considers a suspension of a hyperbolic diffeomorphism of S L n−1 (Z) on Tn−1 , with n = du + ds + 1, such that there are ds eigenvalues with modulus |λ| < 1, and du eigenvalues with modulus |λ| > 1. Anosov one form α and regularity of the distributions E u (x), E s (x). The distribution E 0 (x) is smooth since V (x) is assumed to be smooth. The distributions E u (x), E s (x) and E u (x) ⊕ E s (x) are only Hölder continuous in general (see [37] p. 15, [21] p. 211). The above hypothesis on the flow implies that there is a particular continuous 1-form on X , denoted α called the Anosov 1-form and defined by ker(α(x)) = E u (x) ⊕ E s (x),
(α(x))(V (x)) = 1,
∀x ∈ X.
(1.4)
Since E u and E s are invariant by the flow, then α is invariant as well and (in the sense of distributions) LV (α) = 0,
(1.5)
where LV denotes the Lie derivative. We discuss now some known results about the smoothness of the distributions E u (x), E s (x) in some special cases. • In the case of a geodesic flow on a smooth negatively curved Riemannian manifold M (with X = T1∗ M), the Anosov one form α is the smooth canonical Liouville one form α = nj=1 ξ j d x j on T ∗ X and restricted to T1∗ M. We have used here local coordinates x j on X and related coordinates ξ j on Tx∗ X . Therefore E u (x) ⊕ E s (x) is C ∞ . The distributions E u (x), E s (x) are C 1 individually (see [20] p. 252).
Upper Bound on Density of Ruelle Resonances for Anosov Flows
329
• The previous example is a special case of a contact flow: the flow φt is a contact flow (or Reeb vector field, see [34] p. 106, [42, p. 55]) if the associated one form α defined in Eq. (1.4) is C ∞ and if dα | (E u ⊕E s ) is non degenerate (i.e. symplectic)
(1.6)
meaning that α is a contact one form. Equivalently, d x := α ∧ (dα)d is a volume form on X with d := dim(E u ) = dim(E s ). Notice that (1.5) implies that the volume form is invariant by the flow: LV (d x) = 0.
(1.7)
In that case, E u (x) ⊕ E s (x) = ker(α) is C ∞ and α determines V by dα(V ) = 0 and α(V ) = 1. Anosov flow lifted on the cotangent space T ∗ X . We first review how one can lift the vector field V and its flow to the cotangent bundle. We may view V as a 1st order difn ferential operator, in local coordinates, V = j=1 V j (x) ∂∂x j , where V j are smooth and := −i V takes the form H = real-valued. The operator H V j (x)Dx j , Dx j = 1 ∂ i ∂x j
∞ ∗ and it has the real-valued principal symbol H0 (x, ξ ) = V (ξ ) ∈ C (T X ). In local coordinates H0 (x, ξ ) = V j (x)ξ j . Let ∂ H0 ∂ ∂ H0 ∂ − (1.8) X= ∂ξ j ∂ x j ∂ x j ∂ξ j
be the Hamilton field of H0 and let φt = exp t V : X → X and Mt = exp tX : T ∗ X → T ∗ X be the flows generated by the vector fields V and X, well defined for t ∈ R. Then Mt is a lift of φt in the sense that π ◦ Mt = φt ◦ π , where π : T ∗ X → X denotes the natural projection. Mt is a symplectic map in the sense that Mt∗ ω = ω, where ω = j dξ j ∧ d x j is the symplectic 2-form. In fact Mt is the map naturally induced on T ∗ X by φt : we have Mt (x, ξ ) = (φt (x), ((Dφt (x))t )−1 (ξ )).
(1.9)
Tx∗ X = E u∗ (x) ⊕ E s∗ (x) ⊕ E 0∗ (x)
(1.10)
Let
be the decomposition dual to (1.2) in the sense that (E 0∗ (x))(E u (x) ⊕ E s (x)) = 0, (E u∗ (x))(E u (x) ⊕ E 0 (x)) = 0, (E s∗ (x))(E s (x) ⊕ E 0 (x)) = 0.
(1.11)
Here E 0∗ (x) is dual to E 0 (x) spanned by α (x), while E u∗ (x) and E s∗ (x) are dual to E s (x) and E u (x) respectively, so that dim E 0∗ (x) = dim E 0 (x) = 1, dim E u∗ (x) = dim E s (x) = ds , dim E s∗ (x) = dim E u (x) = du .
(1.12)
330
F. Faure, J. Sjöstrand
Fig. 2. Picture of Tx∗ X for a given x ∈ X , and a given energy E ∈ R
From (1.9) it follows that Mt (x, ξ ) is linear in ξ and combining (1.9) with (1.3) we see that with the notation ρ = (ρx , ρξ ) to denote a point ρ in the cotangent bundle, |Mt (ρ)ξ | ≤ Ce−θt |ρξ |, ρ ∈ E s∗ , t ≥ 0,
(1.13)
|Mt (ρ)ξ | ≤ Ce−θ|t| |ρξ |, ρ ∈ E u∗ , t ≤ 0. Here | · | denotes the natural dual norm on the fibers of the cotangent bundle. We remark that for every E ∈ R, the energy shell E := H0−1 (E)
(1.14)
is invariant under the flow Mt (since H0 is invariant under its own Hamilton flow). In each fiber T ∗ X, E is of the form1 E (x) := E ∩ Tx∗ X = Eα(x) + E s∗ (x) ⊕ E u∗ (x),
(1.15)
where α is the Anosov 1 form in (1.4). The set K E := E ∩ E 0∗ = {Eα(x); x ∈ X }
(1.16)
is the trapped set for the flow Mt restricted to E in the sense that if ρ ∈ E then {Mt (ρ); t ∈ R} is bounded if and only if ρ ∈ K E . See Fig. 2. = −i V , defined in the sense of distributions as an unbounded operThe operator H t , given ator on L 2 (X ), is the generator of the group of Ruelle transfer operators M t u = u ◦ φ−t , t ∈ R. We remark that M t is also a Fourier integral operator that by M quantizes the canonical transformation Mt [28].
1 Since E E ∗ ⊕ E ∗ = 0, V ∈ E and H (x, ξ ) = V (ξ ), then H E ∗ ⊕ E ∗ = 0. Also H (α (x)) = 1, 0 0 0 0 0 0 u s u s then H0 (Eα (x)) = E.
Upper Bound on Density of Ruelle Resonances for Anosov Flows
331
1.1.2. Anisotropic Sobolev spaces. The escape function in our approach is provided by the following lemma that will be proved in Sect. 2 using very much the contraction and expansion properties (1.13). Roughly speaking, an escape function on phase space T ∗ X should decrease along the flow Mt outside the trapped set (1.16). Escape functions are used in order to establish existence of resonances in specific Sobolev spaces and have been introduced by B. Helffer and J. Sjöstrand [25] and used in many situations [43,45–47]. In [24], the authors consider the geodesic flow associated to Schottky groups and provide an upper bound for the density of Ruelle resonances, see also [8]. Lemma 1.2. Let u, n 0 , s ∈ R with u < n 0 < s and u < 0 < s. Let N0 be an arbitrarily small conic neighborhood in T ∗ X \0 of the neutral direction E 0∗ . There exists a smooth function m(x, ξ ) ∈ C ∞ (T ∗ X ) called an order function, taking values in the interval [u, s], and an escape function on T ∗ X defined by:
G m (x, ξ ) := m(x, ξ ) log 1 + ( f (x, ξ ))2 ,
(1.17)
where f ∈ C ∞ (T ∗ X ) and for |ξ | ≥ 1, f > 0 is positively homogeneous of degree 1 in ξ (i.e. f (x, λξ ) = λ f (x, ξ ) for λ ≥ 1). f (x, ξ ) = |ξ | in a conical neighborhood of E u∗ and E s∗ . f (x, ξ ) = |H0 (x, ξ )| in a conical neighborhood of E 0∗ , such that: 1. For |ξ | ≥ 1, m(x, ξ ) depends only on the direction |ξξ | ∈ Sx∗ X and takes the value u (respect. n 0 , s) in a small neighborhood of E u∗ (respect. E 0∗ , E s∗ ). See Fig. 3(a). 2. G m decreases strictly and uniformly along the trajectories of the flow Mt in the cotangent space, except in a conical vicinity N0 of the neutral direction E 0∗ and for small |ξ |: there exists R > 0 such that ∀ (x, ξ ) ∈ T ∗ X, if |ξ | ≥ R, ξ ∈ / N0
then X(G m )(x, ξ ) < −Cm < 0, (1.18)
with Cm := c min(|u|, s)
(1.19)
and c > 0 independent of u, n 0 , s. 3. More generally ∀ (x, ξ ) ∈ T ∗ X with |ξ | ≥ R,
X(G m )(x, ξ ) ≤ 0.
(1.20)
See Fig. 3(b). The order function m(x, ξ ) belongs to the classical symbol space S 0 = S10 (T ∗ X ) in the sense that for every choice of coordinates x1 , . . . , xn that identifies an open set ⊂ Rn and for the corresponding dual coordinates ξ1 , . . . , ξn , U ⊂ X with an open set U we have for all multi-indices α, β ∈ Nn , ∂ξα ∂xβ m(x, ξ ) = O(1)ξ −|α| , . uniformly on K × Rn for every K U
with ξ :=
1 + |ξ |2
332
F. Faure, J. Sjöstrand
(a)
(b)
t on the cosphere bundle S ∗ X := T ∗ X \ {0} /R+ which is the bundle of Fig. 3. (a) The induced flow M ∗ directions of cotangent vectors ξ/ |ξ |. (Here the picture is restricted to a fiber Sx X ). (b) Picture in the cotangent space Tx∗ X which shows in grey the sets outside of which the escape estimate (1.18) holds
Definition 1.3. Let m(x, ξ ) ∈ S 0 be a real-valued and let and 21 < ρ ≤ 1. A function m(x,ξ ) of variable order if for every choice of coorp ∈ C ∞ (T ∗ X ) belongs to the class Sρ ⊂ Rn and for dinates x1 , . . . , xn that identifies an open set U ⊂ X with an open set U the corresponding dual coordinates ξ1 , . . . , ξn , we have for all multi-indices α, β ∈ Nn , |∂ξα ∂xβ p(x, ξ )| ≤ O(1)ξ m(x,ξ )−ρ|α|+(1−ρ)|β|
(1.21)
. uniformly on K × Rn for every K U We refer to [17, Sect. A.2.2] for a precise description of theorems related to symbols with variable orders. With m as in Lemma 1.2, we see that the symbol Am = exp G m belongs to the m m symbol space S1−0 := 1>ρ>1/2 Sρ . As explained in the Appendix in the paper m : [17, Lem. 6], we can associate to this symbol a pseudodifferential operator A D (X ) → D (X ), C ∞ (X ) → C ∞ (X ) whose distribution kernel is smooth outside ⊂ Rn is a local coordinate chart, then the diagonal and such that if κ : X ⊃U →U ) u → A m (u ◦ κ) ◦ κ −1 ∈ C ∞ (U ) is a pseudodifferential the operator Bˆ : C0∞ (U operator which up to a smoothing operator is given by Weyl quantization2 [49, (14.5) p. 60]: 2 In this paper we choose Weyl quantization because it has some well known interesting properties. First a real symbol pW ∈ S μ ,μ ∈ R, is quantized in a formally self-adjoint operator Pˆ = Op ( pW ). Secondly, a change of coordinate systems preserving the volume form changes the symbol at a subleading order S μ−2
only. In other words, on a manifold with a fixed smooth density d x, the Weyl symbol pW of a given pseudodifferential operator Pˆ is well defined modulo terms in S μ−2 . The Weyl symbol of the operator Hˆ = −i V considered in this paper is i div (V ) . 2 Indeed from [49, (14.7) p. 60], in a given chart where V = V j (x) ∂ j ≡ V (x) ∂x , HW (x, ξ ) = V (ξ ) +
(1.22)
∂x
i i i ∂x ∂ξ (V (x) .ξ ) = V (x) .ξ + ∂x V = V (ξ ) + div (V ) HW (x, ξ ) = exp 2 2 2
and div (V ) depends only on the choice of the volume form, see [48, p. 125]. Notice that this symbol does not depend on the choice of coordinates systems provided the volume form is expressed by d x = d x1 . . . d x˙ n . The first term H0 (x, ξ ) = V (ξ ) in (1.22) belongs to S 1 and is called the principal symbol of Hˆ . The second term 2i div (V ) in (1.22) belongs to S 0 and is called the subprincipal symbol of Hˆ .
Upper Bound on Density of Ruelle Resonances for Anosov Flows
ˆ Bu(x) = Op (b) u (x) :=
1 (2π )n
ei(x−y)·ξ b(
333
x+y , ξ )u(y)dydξ, 2
(1.23)
m (T ∗ X ) (if we identify T ∗ X × Rn in the where the Weyl symbol b ∈ S1−0 |U | U with U m−1+0 m−1+ . natural way) and coincides with Am modulo a symbol of class S1−0 := >0 S1−ε In addition, as we have seen in [17], the quantization Am can be chosen to be essentially −1 self-adjoint and bijective with an inverse A m which is the quantization of the symbol 1/Am in a similar way. We define the anisotropic Sobolev space H m (X ) of variable order m by 2 −1 H m (X ) := A m (L (X )),
(1.24)
and equip it with the natural norm m u, u H m = A where we follow the convention that all norms are in L 2 if nothing else is specified. Some basic properties of the space H m , such as embedding properties, are given in [17, Sect. 3.2]. 1.1.3. The theorems. Let V be a smooth Anosov vector field on a smooth compact manifold X (we do not assume that the Anosov one form α is contact nor that it is smooth). The first two results are very close to the results already obtained by [10, Thm. 1] (with the slight difference that the authors use Banach spaces). The novelty here is to show that this resonance spectrum fits with the general theory of semiclassical resonances developed by B. Helffer and J. Sjöstrand [25] and initiated by Aguilar, Baslev, Combes [1,5]. Theorem 1.4. “Discrete spectrum”. Let m be a function which satisfies the hypothesis = −i V defines a maximal closed unbounded of Lemma 1.2 p. 9. The generator H operator on the anisotropic Sobolev space H m , : Hm → Hm H in the sense of distributions with domain given by ) := {ϕ ∈ H m , H ϕ ∈ H m }. D( H It coincides with the closure of (−i V ) : C ∞ → C ∞ in the graph norm for operators. For z ∈ C such that (z) > −(Cm − C) with Cm defined in (1.19) and some C inde − z) : D( H ) ∩ H m → H m is a Fredholm operator with pendent of m, the operator ( H has a discrete specindex 0 depending analytically on z. Consequently the operator H trum in the domain (z) > −(Cm − C), consisting of eigenvalues λi of finite algebraic multiplicity. Recall that if |u| , s are chosen large then Cm is large. See Fig. 4. Concerning Fredholm operators we refer to [12, p.122] or [25, App. A p. 220]. The proof of Theorem 1.4 is given in Sect. 3. Remarks. of • In Proposition 3.1 we shall see that the space H m is invariant under the operator C and C anti-commute it follows that the set of Ruelle complex conjugation. Since H resonances is invariant under reflection in the imaginary axis.
334
F. Faure, J. Sjöstrand
Fig. 4. Spectrum
of Ruelle resonances of H = −i V . From Theorem 1.8 the number of eigenvalues in the rectangle is o E n−1/2 for E → ∞
• We a simple and well known result (which follows from the property that recall M t = 1), that there is no eigenvalue in the upper half plane (z) > 0 and no ∞ Jordan block on the real axis. See [10, Thm. 1][17]. The next theorem shows that the spectrum is intrinsic. Theorem 1.5. “Thediscrete spectrum is intrinsic to the Anosov vector field”. Let m = m m , f,G log 1 + f 2 be another set of functions as in Lemma 1.2 so that Theorem m has discrete spectrum in the set (z) > −(C m 1.4 applies and H : H → H m − C). m m Then in the set (z) > − min((Cm − C), (Cm − C)) the eigenvalues of H : H → H counted with their multiplicity and their respective eigenspaces coincide with those of → Hm . : Hm H The eigenvalues λi are called the Ruelle Resonances and we denote the set by ). The resolvent (z − H )−1 viewed as an operator C ∞ (X ) → D (X ) has a Res( H meromorphic extension from (z) 1 to C. The poles of this extension are the Ruelle resonances. The proof of Theorem 1.5 is given in Sect. 4. The next theorem describes the wavefront set of the eigenfunctions associated to λi . The wavefront set of a distribution has been introduced by Hörmander. The wavefront set corresponds to the directions in T ∗ X where the distribution is not C ∞ (i.e. the local Fourier transform is not rapidly decreasing). The wavefront set of a PDO is the directions in T ∗ X where the symbol is not rapidly decreasing: Definition 1.6 ([23, p. 77, 49, p. 27]). If (x0 ,ξ0 ) ∈ T ∗ X \0, we say that A ∈ S m is non characteristic (or elliptic) at (x0 , ξ0 ) if A (x, ξ )−1 ≤ C |ξ |−m for (x, ξ ) in a small conic neighborhood of (x0 , ξ0 ) and |ξ | large. If u ∈ D (X ) is a distribution, we say that u is C ∞ at (x0 , ξ0 ) ∈ T ∗ X \0 if there exists A ∈ S m non characteristic (or elliptic) at (x0 , ξ0 ) such that Op (A) u ∈ C ∞ (X ). The wavefront set of the distribution u is WF (u) := {(x0 , ξ0 ) ∈ T ∗ X \0, u is not C ∞ at (x0 , ξ0 )}. The wavefront set of the operator Op (A) is the smallest closed cone ⊂ T ∗ X \0 such that A/ ∈ S −∞ . is Theorem 1.7. The wavefront set of the associated generalized eigenfunctions of H contained in the unstable direction E u∗ . . In Lemma 1.2, if we let Proof. Let us consider an eigen-distribution ϕ ∈ H m of H n 0 , s → +∞, the order function m (x, ξ ) can be made arbitrarily large for |ξ | ≥ 1, in every direction outside a small vicinity of E u∗ . As a result ϕ (which remains unchanged)
Upper Bound on Density of Ruelle Resonances for Anosov Flows
335
is smooth in every direction except E u∗ . From the definition of the wave-front set of a distribution this means that the wave-front set of ϕ is contained in the unstable direction E u∗ . Theorem 1.7 follows. The following theorem is the main result of this paper: Theorem 1.8. “Upper bound for the density of resonances”. For every β > 0, in the limit E → +∞ we have √ ), | (λ) − E| ≤ E, (λ) > −β} ≤ o(E n−1/2 ), {λ ∈ Res( H (1.25) with n = dim X . Remarks. • Since the spectrum is symmetric with respect to the imaginary axis, the estimates (1.25) also hold for E → −∞. • The upper bound given in (1.25) results from our method and choice of escape function A (x, ξ ). In the proof, o E n−1/2 comes from an estimate of a V ⊂ T ∗ X which contains the trapped set E and whose symplectic volume is of order Vol (V) δ E −1/2 , with δ arbitrarily small. Using Weyl inequalities we obtain an upper bound of order E n Vol (V) δ E n−1/2 in (1.25). It is expected that a better choice of the escape function G m could improve this upper bound. This is a work in preparation by the authors. For specific models, e.g. geodesic flows on a surface with constant
n negative curvature, it is observed from zeta functions that the upper bound is O E 2 (see [30]). We reasonably expect this in general. • From the upper bound (1.25), one can deduce upper bounds in larger spectral domains. For example: for every β > 0, in the semiclassical limit E → +∞ we have , (λ) ∈ [−E, E] , (λ) > −β ≤ o E n , λ ∈ Res H with n = dim X . 2. Proof of Lemma 1.2 on the Escape Function In this section we construct a smooth function G m on the cotangent space T ∗ X called the escape function and prove Lemma 1.2. We will denote, |ξξ | the direction of a cotangent vector ξ and S ∗ X := (T ∗ X \ {0}) /R+ the cosphere bundle which is the bundle of directions of cotangent vectors ξ/ |ξ |. S ∗ X is a compact space. The images of E u∗ , E s∗ , E 0∗ , N0 ⊂ T ∗ X by the projection T ∗ X \ {0} → S ∗ X are denoted respectively u∗ , E s∗ , E ∗ , N 0 ⊂ S ∗ X , see Fig. 3(a) in Subsect. 1.1.2. E 0 Remarks on Lemma 1.2. • It is important to notice that we can choose m such that the value of Cm is arbitrarily 0 is arbitrarily small. large (by making s, |u| → ∞) and that the neighborhood N • The value of n 0 could be chosen to be n 0 = 0 for simplicity but it is interesting to observe that letting n 0 , s → +∞, the order function m (x, ξ ) can be made arbitrarily large for |ξ | ≥ 1, outside a small vicinity of E u∗ . We will use this in the proof of Theorem 1.7 in order to show that the wavefront of the eigen-distributions are included in E u∗ .
336
F. Faure, J. Sjöstrand
Fig. 5. Illustration for the proof of Lemma 2.1. The horizontal axis is a schematic picture of M and this shows the construction and properties of the sets Vu , Vs and Wu , Ws
• Inspection of the proof shows that with an adapted norm |ξ | obtained by averaging, c can be chosen arbitrarily close to θ , defined in (1.13). • The constancy of m in the vicinity of the stable/unstable/neutral directions allows us to have a smooth escape function G m despite the fact that the distributions E s∗ (x) , E u∗ (x) , E 0∗ (x) have only Hölder regularity in general. We first define a function m (x, ξ ) called the order function following closely [17] Sect. 3.1 (and [19] p. 196). The function m. The following lemma is useful for the construction of escape functions. Let M be a compact manifold and let v be a smooth vector field on M. We denote exp (tv) : M → M the flow at time t generated by v. Let u , s be compact disjoint subsets of M such that dist (exp (tv) (ρ), s ) → 0, t → +∞ when ρ ∈ M \ u , dist (exp (tv) (ρ), u ) → 0, t → −∞ when ρ ∈ M \ s . Lemma 2.1. Let Vu , Vs ⊂ M be open neighborhoods of u and s respectively and let ε > 0. Then there exist Wu ⊂ Vu , Ws ⊂ Vs , m ∈ C ∞ (M; [0, 1]), η > 0 such that v(m) ≥ 0 on M, v(m) > η > 0 on M \ (Wu ∪ Ws ) , m (ρ) > 1 − ε for ρ ∈ Ws and m (ρ) < ε for ρ ∈ Wu . Proof. After shrinking Vu , Vs we may assume that Vu ∩ Vs = ∅ and t ≥ 0 ⇒ exp (tv) (Vs ) ⊂ Vs , and t ≤ 0 ⇒ exp (tv) (Vu ) ⊂ Vu .
(2.1)
Let T > 0 and let Ws := M\exp (T v) (Vu ) = exp (T v) (M\Vu ) and Wu := M\exp (−T v) (Vs ) = exp (−T v) (M\Vs ). See Fig. 5. If T is large enough one has Wu ⊂ Vu , Ws ⊂ Vs and Ws ∩ Wu = ∅. Let m 0 ∈ C ∞ (M; [0, 1]) be equal to 1 on Vs and equal to 0 on Vu . Put m=
1 2T
T
−T
m 0 ◦ exp (tv) dt.
(2.2)
Then v(m)(ρ) =
1 (m 0 (exp (T v) (ρ)) − m 0 (exp (−T v) (ρ))) . 2T
• Let ρ ∈ M \ (Wu ∪ Ws ). From (2.3) we see that v (m) (ρ) =
1 2T
(1 − 0)) =
(2.3) 1 2T
> 0.
Upper Bound on Density of Ruelle Resonances for Anosov Flows
337
For ρ ∈ M let I (ρ) := {t ∈ R, exp (tv) (ρ) ∈ M\ (Vu ∪ Vs )} . This is a closed connected interval by (2.1) and moreover its length is uniformly bounded: ∃τ > 0, ∀ρ ∈ M, |max (I (ρ)) − min (I (ρ))| ≤ τ. In other words, τ is an upper bound for the travel Time in the domain M\ (Vu ∪ Vs ). To prove the lemma, we have to consider two more cases: • Let ρ ∈ Wu . If t ≤ T − τ then m 0 (exp (tv) (ρ)) = 0 and ⎛ ⎞ T T −τ τ 1 ⎜ ⎟ < ε, m 0 (exp (tv) ρ) dt + m 0 (exp (tv) ρ) dt ⎠ ≤ m (ρ) = ⎝ 2T 2T −T T −τ =0
≤1
where the last inequality holds if one chooses T large enough. One has m 0 (exp (−T v) (ρ)) = 0 therefore (2.3) implies that v(m)(ρ) ≥ 0. • Let ρ ∈ Ws . One shows similarly that ⎛ ⎞ T −T +τ 1 ⎜ ⎟ 2T − τ m (ρ) = m 0 (exp (tv) ρ) dt + m 0 (exp (tv) ρ) dt ⎠ ≥ ⎝ 2T 2T −T −T +τ ≥0
=1
> 1 − ε, for T large enough, and v(m)(ρ) ≥ 0.
We now apply Lemma 2.1 to the case when M = S ∗ X and v is the image X on S ∗ X of our Hamilton field X. See Fig. 6. s∗ and let s = s1 ⊂ M be the set of limit points • We first take u = u1 = E ∗ , E u∗ lim j→+∞ exp t j v(ρ), where ρ ∈ M \ u1 and t j → +∞. s1 is the union of E 0 and all trajectories exp(Rv)(ρ), where ρ has the property that exp tv(ρ) converges to ∗ ∗ ∗ when t → −∞ and to E u∗ when t → +∞. Equivalently, s1 is the image E E u ⊕ E0 0 ∗ ∗ ∗ ∞ in S X of E u ⊕ E 0 . Applying the lemma, we get m 1 = m ∈ C (M; [0, 1]) such s∗ , m 1 > 1 − ε that m 1 < ε outside an arbitrarily small neighborhood Wu1 of u1 = E ∗ ∗ outside an arbitrarily small neighborhood Ws1 of s1 = E u ⊕ E 0 and X(m 1 ) ≥ 0 1 everywhere with strict inequality X(m 1 ) > η > 0 outside Ws ∪ Wu1 . • Similarly, we can find m 2 = m ∈ C ∞ (M; [0, 1]), such that m 2 < ε outside an ∗ arbitrarily small neighborhood Wu2 of u2 = E s ⊕ E 0 , m 2 > 1 − ε outside an arbi2 2 ∗ trarily small neighborhood Ws of s = E u and X(m 2 ) ≥ 0 everywhere with strict inequality X(m 2 ) > η > 0 outside Ws2 ∪ Wu2 . Let u < n 0 < s and put m := s + (n 0 − s) m 1 + (u − n 0 ) m 2 , 0 := Ws1 ∩ Wu2 , N u := Ws1 ∩ Ws2 . Ns := Wu1 ∩ Wu2 , N
338
F. Faure, J. Sjöstrand
Fig. 6. Representation of different sets on S ∗ X used in the proof
Then s ∪ N 0 ∪ N u = S ∗ X \ Wu1 ∪ Ws1 ∪ S ∗ X \ Wu2 ∪ Ws2 we have • on S ∗ X \ N X (m 1 ) > η or X (m 2 ) > η, therefore X (m 1 ) + (u − n 0 ) X (m 2 ) X ( m ) = (n 0 − s) < −η min (|n 0 − s| , |u − n 0 |) . 1 2 • on Ns = Wu ∩ Wu we have m 1 < ε and m 2 < ε therefore m > s + (n 0 − s) ε + (u − n 0 ) ε s = s (1 − ε) + uε > , 2 where the last inequality holds if ε is chosen small enough. u = Ws1 ∩ Ws2 we have m 1 > 1 − ε and m 2 > 1 − ε therefore • on N m < s + (n 0 − s) (1 − ε) + (u − n 0 ) (1 − ε) u = εs + u (1 − ε) < , 2 where the last inequality holds if ε is chosen small enough. • on S ∗ X we have X (m 1 ) + (u − n 0 ) X (m 2 ) ≤ 0. X ( m ) = (n 0 − s)
(2.4)
(2.5)
(2.6)
(2.7)
T∗M
satisfying ξ , if |ξ | ≥ 1, m (x, ξ ) = m |ξ | =0 if |ξ | ≤ 1/2.
We construct a smooth function m on
The symbol G m . Let
G m (x, ξ ) := m (x, ξ ) log 1 + ( f (x, ξ ))2
with f ∈ C ∞ (T ∗ X ) such that for |ξ | ≥ 1, f > 0 is positively homogeneous of degree 1 in ξ , and ξ s ⇒ f (x, ξ ) := |ξ | , u ∪ N ∈N |ξ | ξ 0 ⇒ f (x, ξ ) := |H0 (x, ξ )| . ∈N |ξ |
Upper Bound on Density of Ruelle Resonances for Anosov Flows
339
The consequences of these choices are:
• Since X (H0 ) = 0 then X log 1 + ( f (x, ξ ))2 = 0 for
ξ |ξ |
0 . ∈N
• Since E s∗ is the stable direction and E u∗ the unstable one, ∃C > 0,
ξ s ⇒ X (log ξ ) < −C, ∈N |ξ |
ξ u ⇒ X (log ξ ) > C. ∈N |ξ | (2.8)
• In general X log 1 + f 2 is bounded: ∃C2 > 0,
∀ξ ∈ T ∗ X, X log 1 + ( f (x, ξ ))2 < C2 .
We will show now the uniform escape estimate Eq.(1.18), in Lemma 1.2. One has
X (G m ) = X (m) log 1 + f 2 + mX log 1 + f 2 .
(2.9)
We will first consider each term separately assuming |ξ | ≥ 1. s ∪ N u ∪ N 0 then using (2.4) and the fact that X log 1 + f 2 • If ξ ∈ S∗ X \ N and m are bounded, there exists R > 0 large enough such that for |ξ | ≥ R, X (G m ) (x, ξ ) < −c min (s, |u|) with c > 0 independent of u, n 0 , s. u then from (2.8) and (2.6) there exists c > 0 such that • If ξ∈N m X (log ξ ) < −c |u| < 0. X (G m ) = X (m) log ξ + ≤0
≥0
< u2
>C
s then from (2.8) and (2.5) there exists c > 0 such that • If ξ∈N m X (log ξ ) < −cs < 0. X (G m ) = X (m) log ξ + ≤0
≥0
> 2s
<−C
We have obtained the uniform escape estimate Eq. (1.18), Lemma 1.2. Finally for 0 , we have ξ∈N
X (G m ) = X (m) log 1 + f 2 +m X log 1 + f 2 ≤ 0, ≤0 ≥0
=0
and we deduce (1.20), Lemma 1.2. We have finished the proof of Lemma 1.2.
340
F. Faure, J. Sjöstrand
3. Proof of Theorem 1.4 about the Discrete Spectrum of Resonances Here are the different steps that we will follow in the proof. 2 −1 on the Sobolev space H m = A 1. The operator H m L (X ) is unitarily equivalent to 2 := A m H A −1 the operator P m on L (X ). We will show that P is a pseudo-differential operator. We will compute the symbol P (x, ξ ) of P in Lemma 3.2. The important fact is that the derivative of the escape function appears in the imaginary part of the symbol P (x, ξ ). − z is invertible 2. For (z) 1, using the Gårding inequality, we will show that P has no spectrum in the domain (z) 1. and therefore that P 3. Using the Gårding inequality again for a modified operator and analytic Fredholm − z is invertible for (z) > − (Cm − C) for some theory we will show that P constant C independent of m, except for a discrete set of points z = λi that are eigenvalues of finite multiplicity. We begin by the following proposition which is a very simple observation. ˆ defined by Cϕ Proposition 3.1. “Symmetry”. The conjugation operator C = ϕ ∞ m on C (X ) and extended to D (X ) by duality, leaves the space H invariant. If ψ = λψ, ψ ∈ H m then ψ ∈ H m is also an eigenfunction with eigenvalue := Cψ H λ = −λ. The spectrum of Ruelle resonances is therefore symmetric under reflexion in the imaginary axis. Proof. Observe that if Bˆ is a pseudo-differential operator with symbol b (x, ξ ) then from (1.23) Cˆ Bˆ Cˆ is a pseudo-differential operator with symbol b (x, −ξ ). From its construcA C = A. tion the symbol Am (x, ξ ) is real and Am (x, −ξ ) = Am (x, ξ ). Therefore C −1 2 ˆ ˆ Since C = AC A is an isometry on L (X ) we conclude that C is an isometry on the −1 L 2 (X ) . We have the following relation: space H m (X ) = A C + C H = 0. H
(3.1)
Cϕ = Indeed since the vector field V is real, for every ϕ ∈ C ∞ (X ) one has H H ϕ. Then using (3.1), one has H ψ = −i V (ϕ) = i V (ϕ) = −(−i V (ϕ)) = −C Cψ = −C H ψ = −λψ . H on L 2 (X ). 3.1. Conjugation by the escape function and unique closed extension of P Let us define := A m H A −1 P m .
(3.2)
on L 2 (X ) is unitarily equivalent to H on H m . Recall that a smooth Then the operator P μ ∗ function p(x, ξ ) on T X belongs to the classical symbol space S μ = S1 (T ∗ X ) if for every choice of coordinates x1 , . . . , xn that identifies an open set U ⊂ X with an open ⊂ Rn and for the corresponding dual coordinates ξ1 , . . . , ξn , we have set U ∂ξα ∂xβ p(x, ξ ) = O(1)ξ μ−|α| , with ξ := 1 + |ξ |2 . uniformly on K × Rn for every K U
Upper Bound on Density of Ruelle Resonances for Anosov Flows
341
m(x,ξ )
The symbol classes with variable order Sρ have been given in Definition 1.3. In the following lemma, the notation Om S −1+0 means that the term is a symbol in S −1+0 . We add the index m to emphasize that it depends on the escape function m whereas O S 0 means that the term is a symbol in S 0 which does not depend on m. defined in (3.2) is a PDO in Op S 1 . With respect to every Lemma 3.2. The operator P given system of coordinates its symbol is equal to
P (x, ξ ) = H (x, ξ ) + i (X (G m )) (x, ξ ) + Om S −1+0 , (3.3) : where H (x, ξ ) is the symbol of H
H (x, ξ ) = V (ξ ) + O S 0 , with principal symbol V (ξ ) ∈ S 1 and X (G m ) ∈ S +0 . X is the Hamiltonian vector field of H defined in (1.8). Proof. The proof consists in making the following two lines precise and rigorous: −1
= A H A −1 = Op e G m H Op e G m P (1 − Op (G m ) + . . .) (1 + Op (G m ) + . . .) H = H + Op (G m ) , H +. . . = Op (H − i {G m , H } + . . .) = Op (H + iX (G m ) + . . .) . In order to avoid to work with exponentials of operators, let us define
tm , m,t := Op et G m = Op e G tm = A 0 ≤ t ≤ 1, A and m,t := A m,t H A −1 H m,t = H m,0 and P = H m,1 . We have Op (G m ) ∈ Op S +0 , which interpolates between H −1 m,t ∈ Op S −tm(x,ξ )+0 , H ∈ Op S 1 . We deduce that3 m,t ∈ Op S tm(x,ξ )+0 , A A 1+0 m,t ∈ Op S . Then H
m,t dA = Op G m et G m = Op (G m ) Op et G m + Op Om S tm−1+0 dt !
−1 dA d Am,t −1 m,t m,t Am,t = − A = Op G m + rm,t dt dt 3 From the theorem of composition of pseudodifferential operators (PDO), see [49, Prop. (3.3) p. 11], if m m A ∈ Sρ 1 and B ∈ Sρ 2 then
m +m −(2ρ−1) ) , Op ( A) Op (B) = Op (AB) + O Op(Sρ 1 2 m +m 2
i.e. the symbol of Op (A) Op (B) is the product AB and belongs to Sρ 1
m +m 2 −(2ρ−1)
modulo terms in Sρ 1
.
342
F. Faure, J. Sjöstrand
with rm,t ∈ S −1+0 and4
d d Am,t −1 −1 −1 Hm,t = Am,t Am,t H Am,t + Am,t H Am,t dt dt
m,t ∈ Op S +0 . = Op G m + rm,t , H m,t − H = Therefore H
"
t d 0 ds
−1 dA m,t m,t A dt
!
m,s ds ∈ Op S +0 and H
d + Op G m + rm,t , H m,t − H + Op rm,t , H Hm,t = Op (G m ) , H dt
+ Om Op S −1+0 . = Op (G m ) , H We deduce that = H + P
1 0
Since
we get
d Hm,t dt dt
+ Op (G m ) , H + Om Op S −1+0 . =H
= Op i (X (G m )) (x, ξ ) + Om S −1+0 , Op (G m ) , H
= H + Op i (X (G m )) (x, ξ ) + Om S −1+0 . P
= Op V (ξ ) + O S 0 with a remainder in S 0 which is independent Finally since H of the escape function m, we get (3.3). is a unbounded PDO of order 1 that we may first equip with the We have shown that P has a domain C ∞ (X ) which is dense in L 2 (X ). Lemma A.1, Sect. A.1 shows that P on L 2 (X ). The adjoint P ∗ is also unique closed extension as an unbounded operator P a PDO of order 1 and it is the unique closed extension from C ∞ (X ). has empty spectrum for (z) large enough. Let us write 3.2. P 2 = P 1 + i P P ∗ + P ∗ , P 2 := i P −P 1 := 1 P formally self-adjoint. From (3.3) and (1.20), with P 2 2 2 is the symbol of the operator P
(3.4) P2 (x, ξ ) = X (G m ) (x, ξ ) + O S 0 + Om S −1+0 which belongs to S +0 and satisfies ∃C0 , ∀ (x, ξ ) , (P2 (x, ξ )) ≤ C0 . 4 From [49, Eqs. (3.24) (3.25) p. 13], if A ∈ S m 1 and B ∈ S m 2 then the symbol of Op (A) , Op (B) is ρ ρ m +m −2(2ρ−1) m +m −(2ρ−1) , which belongs to Sρ 1 2 . We also recall the Poisson bracket −i {A, B} modulo Sρ 1 2 [48, (10.8) p. 43] that {A, B} = −X B (A) where X B is the Hamiltonian vector field generated by B.
Upper Bound on Density of Ruelle Resonances for Anosov Flows
343
From the sharp Gårding inequality (A.2), Sect. A.2 applied here with order μ = 1 (since2 2 u|u ≤ (C0 + C) u P2 ∈ S +0 ⊂ S 1 ) we deduce that there exists C > 0 such that P which is written: 2 − (C0 + C) u|u ≤ 0. (3.5) P Notice that C and C0 depend on m a priori. Lemma 3.3. From the inequality (3.5) we deduce that for every z ∈ C, (z) > C + C0 , has − z −1 exists (as a bounded operator on L 2 (X )). Therefore P the resolvent P empty spectrum for (z) > C + C0 . Proof. Let ε = (z) − (C0 + C) > 0. Then for u ∈ C ∞ (X ), − z u|u = P 2 − (C0 + C) u|u − ( (z) − (C0 + C)) u2 ≤ −ε u2 . P By Cauchy-Schwarz inequality, P − z u|u ≥ ε u2 . − z u|u ≥ P − z u u ≥ P Hence for u ∈ C ∞ (X ),
P − z u ≥ ε u .
(3.6) and it follows that P − z is injective with By density this extends to all u ∈ D P − z . closed range R P ∗ = P 1 − i P 2 gives The same argument for the adjoint P ∗ ∗ P , − z u ≥ ε u , ∀u ∈ D P (3.7) ∗ −z is also injective. If u ∈ L 2 (X ) is orthogonal to R P − z then u belongs to the so P ∗ 2 − z : D P → L 2 (X ) kernel of P − z which is 0. Hence R P − z = L (X ) and P is bijective with bounded inverse. is discrete on (z) ≥ − (Cm − C) with some C ≥ 0 indepen3.3. The spectrum of P dent of m. As usual [39, p. 113], in order to obtain a discrete spectrum for the operator to construct a relatively compact perturbation χ of the operator such that P, it suffices − iχ P has no spectrum on (z) ≥ − (Cm − C). Let χ0 : T ∗ X → R+ be a smooth non negative function with χ0 (x, ξ ) = Cm > 0 for (x, ξ ) ∈ N0 and χ0 (x, ξ ) = 0 outside a neighborhood of N0 where N0 is defined in Eq. (1.18), Lemma 1.2. See also Fig. 3(b). We can assume that χ0 ∈ S 0 . Let χ 0 := Op (χ0 ). We can assume that χ 0 is self-adjoint. From Eq. (1.18), for every (x, ξ ) ∈ T ∗ X, |ξ | ≥ R, (X (G m ) (x, ξ ) − χ0 (x, ξ )) ≤ −Cm , hence (3.4) gives for every (x, ξ ) ∈ T ∗ X :
P2 (x, ξ ) − χ0 (x, ξ ) ≤ −Cm + C + Om S −1+0 ,
(3.8)
with some C ∈ R independent of m, coming from the O S 0 term in (3.4). Notice that −1+0 the remainder term Om S can be bounded by a constant but which depends on m.
344
F. Faure, J. Sjöstrand
Since P2 ∈ S μ , ∀μ, 0 < μ < 1, the sharp Gårding inequality (A.2) implies that for every u ∈ C ∞ (X ) there exists Cμ > 0 (Cμ depends on m a priori) such that 2 − χ P 0 + (Cm − C) u|u ≤ Cμ u2 μ−1 . H
2
5 term A ∈ We have also applied the Calderon Vaillancourt theorem2 to the remainder −1+0 Om S in (3.8) to see that |(Op (A) u|u)| ≤ C A u −1+0 ≤ C A u2 μ−1 (since H 2
# $H 2 μ−1 μ−1 −1+0 2 u ξ < ). The right hand side can be written C = C u|u = μ μ μ−1 2 2 H
2
1 = Op (χ1 ) , χ1 = Cμ ξ μ−1 ∈ S μ−1 and can be absorbed on the left χ1 u|u) with χ ( by defining χ := χ0 + χ1 ,
χ := Op (χ ) .
We can assume that χ is self-adjoint. We obtain: 2 − χ P + (Cm − C) u|u ≤ 0. As in the proof of Lemma 3.3, Sect. 3.2, we obtain that the resolvent
− iχ P − z
−1
: L 2 (X ) → L 2 (X ) is bounded for (z) > − (Cm − C) . (3.9)
The following lemma is the central observation in the proof of Theorem 1.4. Lemma 3.4. For every z ∈ C such that (z) > − (Cm − C), the operator −1 − iχ : L 2 (X ) → L 2 (X ) is compact. χ P − z Proof. Let z ∈ C such that (z) > − (Cm − C). On the cone N0 , the operator − iχ P − z is elliptic of order 1. We can therefore invert it micro-locally on N0 (that is its principal symbol V (ξ ) is non vanishing away from zero, see for instance [23, Chaps. 6,7]), namely construct E ∈ S −1 and R1 , R2 ∈ S 0 such that − iχ = 1+ R 1 , P − iχ 2 , P − z E E − z = 1 + R j ∩ N0 = ∅. ∀ j = 1, 2, WF R (3.10) (The wavefront set of a PDO has been defined in Subsect. 1.1.3). In particular j = ∅, therefore χ 2 is a compact operator. Also E is a compact operator WF χ R R −1 (since E ∈ S ). Then from (3.10), we write: −1 − R 2 P − iχ − z −1 = E − iχ , P − z −1 −1 − iχ − iχ χ P − χ χ P − z − z , = χ R E 2 bounded compact
compact
−1 − iχ and deduce that χ P − z is a compact operator.
bounded
The following lemma finishes the proof of Theorem 1.4.
5 If A ∈ S μ then there exists C > 0 such that ∀u ∈ C ∞ (X ) , |(Op (A) u|u)| ≤ C μ A A Op ξ u|u =
C A u
μ
H2
.
Upper Bound on Density of Ruelle Resonances for Anosov Flows
345
has discrete spectrum with locally finite multiplicity on (z) > Lemma 3.5. P − (Cm − C). Proof. For every z ∈ C, with (z) > − (Cm − C), we have obtained in (3.9) that − iχ P − z is invertible. We write:
−1 − iχ − z = 1 + iχ − iχ P − z . P P − z − iχ → L 2 (X ) is bijective with bounded inverse and hence Here P − z : D P
−1 − iχ : L 2 (X ) → L 2 (X ) is FredFredholm of index 0. Similarly 1 + i χ P − z holm of index 0 by Lemma 3.4. Thus − z : D P → L 2 (X ) , (z) > Cm − C, P is a holomorphic family of Fredholm operators (of index 0) invertible for (z) 0. It then suffices to apply the analytic Fredholm theorem ([38, p. 201, case (b)], see also [25, p. 220 App. A]). 4. Proof of Theorem 1.5 that the Eigenvalues are Intrinsic to the Anosov Vector Field V Let m and G m be as in Lemma 1.2. Let m = f (m), where f ∈ C ∞ (R) , f (t) ≥ viewed as max (0, t) , f ≥ 0, f (t) = 0 for t ≤ u/2 and f (t) = t for t ≥ s/2. H m a closed unbounded operator in H has no spectrum in the half plane (z) ≥ C1 for ⊂ L 2 so if : L 2 → L 2 . Since m C1 0. The same holds for H ≥ 0 we have H m m v ∈ H then R L 2 (z) v = R H m (z) v for (z) ≥ C1 , where R L 2 denotes the resolvent : L 2 → L 2 and similarly for R H m . of H ⊂ H m , and hence R Since m ≥ m we also have H m (z) v = R H m (z) v for Hm m (z) ≥ C1 , v ∈ H . Especially when v ∈ C ∞ , we get R L 2 (z) v = R H m (z) v, (z) ≥ C1 . Applying Theorem 1.4, we conclude that R L 2 (z), viewed as an operator C ∞ → D has a meromorphic extension R (z) from the half plane (z) ≥ C1 to the half plane (z) > − (Cm − C) and this extension coincides with R H m restricted to C ∞ . If γ is a simple positively oriented closed curve in the half plane (z) > − (Cm − C) : H m → H m , then the spectral projection, associated which avoids the eigenvalues of H inside γ , is given by to the spectrum of H 1 m πγH = R H m (z) dz. 2πi γ For v ∈ C ∞ , we have m πγH v
1 = πγ v := 2πi
γ
R (z) dz v.
Now C ∞ is dense in H m and πγH is of finite rank, hence its range is equal to the image πγ (C ∞ ) of C ∞ . The latter space is independent of the choice of H m . More precisely if m , f are as in Theorem 1.4 and we choose γ as above, now in the half plane : Hm → Hm (z) > − min ((Cm − C) , (Cm − C)) and avoiding the spectra of H m m → Hm , then the spectral projections π H and π H have the same range. : Hm and H γ γ Theorem 1.5 follows. m
346
F. Faure, J. Sjöstrand
5. Proof of Theorem 1.8 for the Upper Bound on the Density of Resonances The asymptotic regime (z) 1 which is considered in Theorem 1.8 is a semiclassical regime in the sense that it involves large values of H (ξ ) = V (ξ ) 1, hence large values of |ξ | in the cotangent space T ∗ X . For convenience, we will switch to h-semiclassical analysis. Let 0 < h # 1 be a small parameter (we will set E = 1/ h in Theorem 1.8). In h-semiclassical analysis a symbol a (x, ξ ) is quantized according to O ph (a (x, ξ )) = Op (a (x, hξ )) ,
(5.1)
where Op has been defined in (1.23). In this section we first recall the definition of symbols in h-semiclassical analysis. In with respect to this new calculus. In Lemma 5.3 we derive the symbol of the operator P Sect. 5.3 we give the main idea of the proof and the next sections give the details. 5.1. Semiclassical symbols. We first define the symbol classes we will need in h-semiclassical analysis. We will use calligraphic symbols S to distinguish these from the symbols of the homogeneous theory S defined in (1.21). μ Definition 5.1. The symbol class h −k Sρ with 1/2 < ρ ≤ 1, order μ ∈ R and k ∈ R consists of C ∞ functions p (x, ξ ; h) on T ∗ X , indexed by 0 < h # 1 such that in every trivialization (x, ξ ) : T ∗ X |U → R2n , for every compact K ⊂ U , ∀α, β, ∂ξα ∂xβ p ≤ C K ,α,β h −k+(ρ−1)(|α|+|β|) ξ μ−ρ|α|+(1−ρ)|β| . (5.2) μ μ For short we will write Sρ instead of h −k Sρ when k = 0, and write S μ instead of μ Sρ when ρ = 1. For symbols of variable orders we have: Definition 5.2. Let m (x, ξ ) ∈ S 0 , 21 < ρ ≤ 1 and k ∈ R. A family of ∞ ∗ functions
p (x, ξ ; h) ∈ C (T X ) indexed by 0 < h # 1, belongs to the m(x,ξ ) class h −k Sρ of variable order if in every trivialization (x, ξ ) : T ∗ X |U → R2n , for every compact K ⊂ U and all multi-indices α, β ∈ Nn , there is a constant C K ,α,β such that α β (5.3) ∂ξ ∂x p (x, ξ ) ≤ C K ,α,β h −k−(1−ρ)(|α|+|β|) ξ m(x,ξ )−ρ|α|+(1−ρ)|β| , for every (x, ξ ) ∈ T ∗ X |U . 5.2. The symbol operator. In the 1-quantization we have seen in (1.22) of the conjugated that Hˆ = Op V (ξ ) + 2i div (V ) . We rescale the spectral domain z ∈ C by defining: z h := hz, From (5.1) we get
h := h H . H
i Hh = Op V (hξ ) + h div (V ) 2
= Oph V (ξ ) + O h S 0 ∈ Oph S 1 .
(5.4)
Upper Bound on Density of Ruelle Resonances for Anosov Flows
347
From now on we will work with these new variables and we will often drop the indices h for short. We will take again the escape function to be G m (x, ξ ) := m (x, ξ ) log 1 + ( f (x, ξ ))2 as in (1.17) but quantized by the h-quantization giving the h-PDO Oph (G m ) with hsemiclassical symbol G m ∈ S +0 . Also m := Oph (exp (G m )) A m(x,ξ ) m is automatic if h is small is a h-PDO with symbol Am ∈ S1−0 (the invertibility of A enough). Notice that the Sobolev space defined now by
−1 2 −1 L 2 (X ) = Oph (Am ) L (X ) Hhm := A m
is identical6 to (1.24) as a linear space. However the norm in Hhm depends on h. In the following lemma, we will use again the notation Om h 2 S −1+0 which means that the term is a symbol in h 2 S −1+0 . We add the subscript m to emphasize that it depends on the escape function m whereas O hS 0 means that the term is a symbol in hS 0 which does not depend on m. Lemma 5.3. We define := A m H A −1 P m , as in (3.2). Its symbol P ∈ S 1 is
P (x, ξ ) = V (ξ ) + i hX (G m ) (x, ξ ) + O hS 0 + Om h 2 S −1+0 .
(5.5)
Proof. The proof is very similar to the proof of Lemma 3.2. Let us define
tm , m,t := Oph et G m = Oph e G tm = A 0≤t ≤1 A and m,t H A −1 m,t := A H m,t , = H m,0 and P = H m,1 . We have7 A m,t ∈ Oph S tm+0 , which interpolates between H −tm+0 ∈ Oph S 1 , therefore H m,t ∈ Oph S 1+0 . Then −1 ,H A m,t ∈ Oph S
d d −1 −1 = − A Am,t A A m,t m,t m,t = Oph G m + rm,t dt dt
6 Let us show that the space H m := Op (A ) −1 L 2 (X ) using h-quantization does not depend on m h h
h and is therefore identical to (1.24) obtained with h = 1. We have Oph (Am ) ∈ Op Sρm (symbol class
−1 without h). Therefore Bˆ := Oph (Am ) Oph=1 (Am ) ∈ Op S 0 is invertible on L 2 with continuous
−1 2 −1 2 m . Bˆ L (X ) = Oph=1 ( Am ) inverse: Bˆ L 2 = L 2 . Hence Hhm := Oph ( Am ) L (X ) = Hh=1 7 The theorem of composition of h-semiclassical PDO [17] says that if A ∈ S m 1 and B ∈ S m 2 , then the ρ ρ m +m m +m −(2ρ−1) symbol of Oph (A) Oph (B) is the product AB and belongs to Sρ 1 2 modulo hSρ 1 2 .
348
F. Faure, J. Sjöstrand
with rm,t ∈ hS −1+0 and d m,t . Hm,t = Oph G m + rm,t , H dt
" t d m,t − H = We deduce that dt Hm,t ∈ Oph hS +0 , therefore H 0 Oph hS +0 also and
d ds
m,s ds H
∈
d + Oph G m + rm,t , H m,t − H + Oph rm,t , H Hm,t = Oph (G m ) , H dt
+ Om Oph h 2 S −1+0 . = Oph (G m ) , H We deduce that
= H + P 0
Since8
1
d Hm,t dt dt
+ Oph (G m ) , H + Om Oph h 2 S −1+0 . =H
= Oph i h (X (G m )) (x, ξ ) + Om h 2 S −1+0 , Op (G m ) , H
we get
= H + Oph i h (X (G m )) (x, ξ ) + Om h 2 S −1+0 . P = Oph V (ξ ) + O hS 0 with a remainder in hS 0 which is indeFinally, since H pendent of the escape function m, we get (5.5). We recall the main properties of the different terms in (5.5). First V (ξ ) ∈ S 1 is real. In each fiber Tx∗ X, V (ξ ) is linear in ξ and for every E ∈ R the characteristic set E := {(x, ξ ) , V (ξ ) − E = 0} is the energy shell defined in (1.14) and transverse to E 0∗ . The second term i hX (G m ) ∈ hS +0 is purely imaginary and we recall some properties of X (G m ) obtained in Lemma 1.2: ⎧ ⎪ for |ξ | ≥ R ⎨≤ 0 X (G m ) (x, ξ ) is ≤ Om (1) for |ξ | < R (5.6) ⎪ ⎩≤ −C , Cm > 0, for (x, ξ ) ∈ / (D R ∪ N0 ) , m where D R = {ξ, |ξ | ≤ R} and N0 is defined in Lemma 1.2. With a convenient choice of the order function m (x, ξ ) we have: ) N0 arbitrarily small conical vicinity of E 0∗ , (5.7) Cm > 0 arbitrarily large.
8 If A ∈ S m 1 and B ∈ S m 2 , then the symbol of Op ( A) , Op (B) is the Poisson bracket −i h {A, B} = ρ ρ h h m +m −(2ρ−1) m +m −2(2ρ−1) modulo h 2 Sρ 1 2 . Here X B is the Hamiltonian vector i hX B (A) and belongs to hSρ 1 2
field generated by B.
Upper Bound on Density of Ruelle Resonances for Anosov Flows
349
Fig. 7. The objective is to bound from above the number of eigenvalues λi in the domain Zβ . For that purpose, we will bound the number of resonances in the disk of radius 1 + bh and center z E = E + i
5.3. Main idea of the proof. Before giving the details of the proof we give here the main arguments that we will use in order to prove (1.25). Let us consider the following complex valued function p (x, ξ ) ∈ S 1 made from the first two leading terms of the symbol (5.5): p (x, ξ ) := V (ξ ) + i hX (G m ) . Let E > 0, h # 1 and β > 0. We define the spectral domain Z ⊂ C by: * + Z := λ ∈ C, | (λ) − E| ≤ βh, (λ) ≥ −βh .
(5.8)
(5.9)
See Fig. 7. Let VZ := (x, ξ ) ∈ T ∗ X, p (x, ξ ) ∈ Z . We have from (5.8), (5.6) and assuming Cm > β, ) ) (x, ξ ) ∈ [ E−√βh,E+√βh ] |V (ξ ) − E|2 ≤ βh ⇒ (x, ξ ) ∈ VZ ⇔ hX (G m ) (x, ξ ) ≥ −βh (x, ξ ) ∈ (D R ∪ N0 )
(5.10)
, (5.11)
,
√ where [ E−√βh,E+√βh ] := |E −E|≤ βh E is a union of energy shells (1.14). We deduce that the symplectic volume of VZ is √ (5.12) Vol (VZ ) ≤ C Vol (N0 ∩ E ) h,
with some constant C > 0 (independent of E and m). See Fig. 8. (Notice that Vol (N0 ∩ E ) is a “Liouville volume” inherited from the symplectic volume on T ∗ X and the energy function H ). We suppose that E > R and that h is small enough so that [ E−√βh,E+√βh ] D R = ∅. Using a max-min formula and Weyl inequalities we will obtain an upper bound for the number of eigenvalues in terms of this upper bound (and choosing Cm > 4β): {λi ∈ Z} ≤
Cm Vol (VZ ) = Cm C Vol (X ) Vol (N0 ∩ E ) h 1/2−n . hn
350
F. Faure, J. Sjöstrand
Fig. 8. Picture in Tx∗ X with x ∈ X , of the volume V Z which supports micro-locally the eigenvalues λi ∈ Z of Fig. 7
Using that Vol (N0 ∩ E ) can be chosen arbitrarily small, from Eq. (5.7), we deduce that
{λi ∈ Z} ≤ o h 1/2−n , which is precisely (1.25) with α = 1/ h. The proof below follows these ideas but is not so simple because P (x, ξ ) in Eq. (5.5) is a symbol and not simply a function (symbols belong to a non commutative algebra of star product) and because the term hX (G m ) is subprincipal. We will have to decompose the phase space T ∗ X in different parts in order to separate the different contributions as √ in (5.11). Another technical difficulty is that the width of the volume VZ is of order h. We will √ use FBI quantization which is convenient for a control on phase space at the scale h.
5.4. Proof of Theorem 1.8. We present in reverse order the main steps we will follow in the proof. Steps of the proof:
of the operator P in the • Our purpose is to bound the cardinal of the spectrum σ P rectangular domain Zβ given by (5.9). But as suggested by Fig. 7 and confirmed by in the disk Lemma 5.4 below, it suffices to bound the number of eigenvalues of P D (z E , 1 + bh) := {z ∈ C, |z − z E | ≤ (1 + bh)} , with radius (1 + bh) and center: z E := E + i ∈ C. Lemma 5.4. If b ≥ 2β and h small enough then
- σ P Zβ ⊂ D (z E , 1 + bh) .
b > 0,
Upper Bound on Density of Ruelle Resonances for Anosov Flows
351
⇒ (z) ≤ 0. Also, the Pythagoras Theorem in the Proof. We know that z ∈ σ P √ 2 corner of Z gives the condition (1 + bh)2 ≥ (1 + βh)2 + βh which is fulfilled if b ≥ 2β and h small enough. in the disk D (z E , 1 + bh), we will • In order to bound the number of eigenvalues of P use Weyl inequalities in Corollary 5.9 and a bound for the number of small singular − zE ∗ P − z E ) obtained − z E (i.e. eigenvalues of P values of the operator P in Lemma 5.8. • In order to get this bound on singular values, we will bound from below the expres 2 ∗ sions P − z E u = P − z E P − z E u|u . From symbolic calculus (see footnote 7) we can compute the symbol of this operator and get:
− zE ∗ P − z E = Op |V (ξ ) − E|2 + |1 − hX (G m )|2 P (5.13)
+Op O hS 1 + Om h 2 S +0 (5.14)
= Op |V (ξ ) − E|2 + 1 − 2hX (G m ) + O hS 1
+Om h 2 S +0 . However it is not possible to from this symbol because for deduce directly estimates large |ξ | the remainders O hS 1 and Om h 2 S +0 may dominate the important term 2hX (G m ) ∈ hS +0 . Therefore we first have to perform a partition of unity on phase space. Partition of unity on phase space. Let K 0 ⊂ T ∗ X be a compact subset (independent of h) such that VZ ⊂ K 0 with VZ defined in (5.10). See Fig. 8. Lemma A.3 associates a “quadratic partition of unity of PDO” to the compact set K 0 : there exist symbols χ0 ∈ S −∞ and χ1 ∈ S 0 of self-adjoint operators χ 0 , χ 1 such that χ 02 + χ 12 = 1 + Op h ∞ S −∞
(5.15)
supp (χ0 ) is compact and the compact set K 0 , χ0 = 1 + O (h ∞ ) , χ1 = O (h ∞ ). Then from Lemma A.4 called the “IMS localization formula” we have: for every u ∈ L 2 (X ),
2 2 P − zE χ − zE χ − z E u 2 = P 0 u + P 1 u + O h 2 u2 . (5.16) Notice that the remainder is of order h 2 , that is of order one higher than one would expect at first sight. We will now study the different terms of (5.16) separately. Informal remark.. In order to show that Lemma 5.5 below is expected, let us give an informal remark (not necessary for the proof). Using the function p (x, ξ ) := V (ξ ) + i hX (G m ), as in (5.8), which is the dominant term of the symbol P (x, ξ ), we write: | p (x, ξ ) − z E |2 = |V (ξ ) − E|2 + |1 − hX (G m )|2
= |V (ξ ) − E|2 + 1 − 2hX (G m ) + O h 2 S +0 .
(5.17) (5.18)
352
F. Faure, J. Sjöstrand
Fig. 9. Picture in Tx∗ X with x ∈ X , which shows the partition of unity of phase space used in the proof of Lemma 5.5. The h-essential support of χ1 is outside the set K 0
If (x, ξ ) ∈ / K 0 there are two cases, according to (5.6): 1.
X (G m ) (x, ξ ) ≤ −Cm . Then | p (x, ξ ) − z E |2 ≥ 1 + 2hCm .
2.
|V (ξ ) − E|2 ≥ C0 > 0 and X (G m ) ≤ O (1) from (5.6). Then | p (x, ξ ) − z E |2 ≥ 1 + C0 + O (h) . In both cases we have | p (x, ξ ) − z E |2 ≥ 1 + 2hCm .
(5.19)
Since χ1 is negligible on K 0 , the following Lemma 5.5 is not surprising in the light of property (5.19). It gives a lower bound for the second term in the right side of (5.16). Lemma 5.5. For every u ∈ L 2 (X ), 2 P − zE χ χ1 u2 − O h ∞ u2 . 1 u ≥ (1 + 2h (Cm − C))
(5.20)
Proof. In order to prove (5.20) we have to consider a partition of unity in order to take into account two contributions as in the discussion after (5.17). Let 0 ∈ S 0 which has its support inside the region where χ1 = 1 and we set 0 = 1 away from a conical neighborhood of the energy shell E , Eq. (1.14), which is the characteristic set V (ξ ) − E = 0. See Fig. 9. 1 Since (V (ξ ) − E) is the principal symbol of (P (x, ξ ) − E) ∈ S and is non van ∈ Op S −1 such that ishing on the support of 0 , there exists Q ∈ Op h ∞ S −∞ . P − E = 0 + R, R Q is continuous in L 2 (X ), there exists C0 > 0 such that for every v ∈ L 2 (X ) , Since Q 2 2 , hence for every u ∈ L 2 (X ), v ≥ C10 Qv 2 2 2 1 1 Q P − E χ χ P − E χ 0 + R 1 u ≥ 1 u 1 u = C0 C0 2 ∞ 1 0 χ ≥ 1 u − O h u2 . 2C0
(5.21)
Upper Bound on Density of Ruelle Resonances for Anosov Flows
353
= P 1 + i P 2 with P i self-adjoint, we have Writing P 2
P − zE χ − E ∗ + i − E − i χ 1 u = P P 1 u| χ1 u 2 − E χ 2 χ 1 u + χ1 u2 − 2 P 1 u| χ1 u . = P
(5.22)
Using (5.21) in (5.22) we get for every a > 0: 2 2 2 P − E χ − E χ − zE χ 1 u + ah P 1 u + χ1 u2 1 u = (1 − ah) P 2 χ − 2P 1 u| χ1 u 2 − E χ 2 χ 1 u + ≥ (1 − ah) P χ1 u2 − 2 P 1 u| χ1 u 2 ah 0 χ + 1 u − O h ∞ u2 2C0 2 − E χ 1 u + = (1 − ah) P χ1 u2
ah ∗ 1 u| 0 χ + −2 P2 + χ1 u − O h ∞ u2 . 2C0 0 (5.23) Recall from (5.5) that
2 = Op hX (G m ) + O hS 0 ∈ Op hS +0 . P Therefore
2 + ah ∗ 0 −2 P 2C0 0
∈ Op hS +0 .
Assume a ≥ 4C0 (Cm − C). Then from (5.6) and the hypothesis on 0 , for every (x, ξ ) ∈ supp (χ1 ) we have
ah ∗ ah −2P2 + ≥ 2h (Cm − C) . 0 0 (x, ξ ) ≥ min 2h (Cm − C) , 2C0 2C0 0 1 ∈ We add ∞can−∞ a symbol 1 ∈ S positive, which∗vanishes on supp (χ1 ) so that 1 χ Op h S and such that for every (x, ξ ) ∈ T X we have
ah ∗ ah −2P2 + ≥ 2h (Cm − C) . 0 + 1 (x, ξ ) ≥ min 2h (Cm − C) , 2C0 0 2C0 The semiclassical sharp Gårding inequality implies that:
ah ∗ 1 u| 0 0 χ ∀u ∈ L (X ) , χ1 u2 χ1 u ≥ 2h (Cm −C)−O h 2 −2 P2 + 2C0 −O h ∞ u2 , 2
354
F. Faure, J. Sjöstrand
1 χ where the remainder term O (h ∞ ) u2 comes from 1 u| χ1 u . With (5.23) we get: 2 2 P − zE χ − E χ 1 u + 1 u ≥ (1 − ah) P χ1 u2
χ1 u2 − O h ∞ u2 + 2h (Cm − C) − O h 2
χ1 u2 − O h ∞ u2 . ≥ 1 + 2h (Cm − C) − O h 2 The term O h 2 χ1 u2 can be absorbed in the constant C.
Lemma 5.6. We have 2 P − zE χ χ0 u2 − O h ∞ u2 , 0 u ≥ (1 − O (h))
(5.24)
where the term O (h) does not depend on m. There exists a family of trace class operators Bh (depending on h) such that Bh Tr ≤ O (1) Cm Vol (N0 ∩ E ) h 1/2−n , Bh ≥ 0, (5.25) (where the constant O (1) does not depend on the escape function m) and for every u ∈ L 2 (X ), 2 P − zE χ χ0 u2 − O h ∞ u2 . Bh u|u ≥ (1 + 2h (Cm − O (1))) 0 u + h (5.26) Remarks. Lemma 5.6 concerns the first term of the right hand side of (5.16). In order to obtain (5.26), which is similar to (5.20), it has been necessary to add a new term which involves a trace class operator Bh to gain positivity in the domain VZ (5.11). Equation (5.24) shows that without this term the lower bound is smaller. Proof. The construction is based on ideas around Anti-Wick quantization, Berezin quantization, FBI transforms, Bargmann-Segal transforms, Gabor frames and Toeplitz operators, see e.g. [26]. We review some definitions in Appendix A.4. We will use the following properties for an operator obtained by Toeplitz quantization of a function A (x, ξ ; h) (such that the following expression makes sense). Let OpT (A) := A (x, ξ ; h) πx,ξ d xdξ. Then for u ∈ C ∞ A (x, ξ ) ≥ 0 ⇒ OpT (A) u|u ≥ 0, also if A ∈ L 1 , O (1) Tr OpT (A) = hn
(5.27)
A (x, ξ ) d xdξ,
(5.28)
and (∀ (x, ξ ) ,
A (x, ξ ) ≥ 0)
⇒
Op (A) = Tr Op (A) . T T Tr
(5.29)
Upper Bound on Density of Ruelle Resonances for Anosov Flows
355
∈ Oph (S m ) is a PDO with principal symbol a0 (modulo hS m−1 ), then there exists If A a ∈ S m such that = OpT (a) + R, A
∈ Op h ∞ S is negligible and a = a0 mod hS m−1 . where R From (5.13) we have − zE ∗ P − zE χ χ 0 P 0 = χ 0 S χ0 + R,
∈ Op h ∞ S with R
−∞
−∞
(5.30)
and S = OpT (S) ,
with the Toeplitz symbol
S (x, ξ ; h) = |V (ξ ) − E|2 + 1 − 2hX (G m ) (x, ξ ) + O h S −∞ + Om h 2 S −∞ (the remainders are in S −∞ since χ0 has compact support in (5.30). Since X (G m ) ≤ 0 from (5.6), we deduce (5.24) using Gårding’s inequality (5.27). In order to get (5.26), let 0 ≤ Bh ∈ C0∞ (T ∗ X ) be such that (x, ξ ) ∈ VZ ⇒ Bh (x, ξ ) ≥ 2Cm .
(5.31)
In view of (5.12) Bh can be chosen such that √ Bh (x, ξ ) d xdξ ≤ O (1) Cm Vol (VZ ) = O (1) Cm Vol (X ) Vol (N0 ∩ E ) h. T∗X
(5.32) Let Bh := OpT (Bh ). From (5.29), (5.28) and (5.32) we deduce (5.25). Recall that from (5.11) we have / VZ ⇒ |V (ξ ) − E|2 ≥ hCm or − hX (G m ) ≥ hCm . (x, ξ ) ∈ Therefore in view of (5.31) for every (x, ξ ) ∈ T ∗ X we have S (x, ξ ; h) + h Bh (x, ξ ) = |V (ξ ) − E|2 + 1 − 2hX (G m ) + h Bh (x, ξ )
+O h S −∞ + Om h 2 S −∞
≥ 1 + 2hCm + O hS −∞ + Om h 2 S −∞ . After multiplying both sides by χ 0 , using χ 0 0 = Bh + Op h ∞ S −∞ and Gårding’s Bh χ inequality we deduce that ∀u ∈ L 2 (X ) , χ Bh u|u ≥ (( χ0 (1 + 2h (Cm − O (1))) χ 0 ) u|u) 0 S χ0 u|u + h ∞ 2 +O h u . Replacing the first term by (5.30) this gives (5.26).
356
F. Faure, J. Sjöstrand
Equation (5.16) with (5.20), (5.26), (5.15) gives: Corollary 5.7. We have P − z E u 2 + h ∀u ∈ L 2 (X ) , Bh u|u ≥ (1 + 2 (Cm − O (1)) h) u2 , (5.33) where O (1) does not depend on m. Moreover P − z E u 2 ≥ (1 − O (h)) u2 . ∀u ∈ L 2 (X ) ,
(5.34)
Here we have used (5.24) to get (5.34). Let us show that these imply last relations an upper bound for the number of eigen − zE ∗ P − z E smaller than (1 + 2 (Cm − O (1)) h). values of the operator P − z E sorted from below. Lemma 5.8. Let s1 ≤ s2 ≤ · · · be the singular values P More precisely, s12 ≤ s22 ≤ · · · are the eigenvalues of the positive self-adjoint operator − z E below the infimum of the essential spectrum of A, possibly := P − zE ∗ P A completed with an infinite repetition of that infimum if there are only finitely many such eigenvalues. Then the first eigenvalue is s1 ≥ 1 − O (h)
(5.35)
and 1
if j > O (1) Cm Vol (N0 ∩ E ) h 2 −n then s j ≥ 1 + (Cm − O (1)) h,
(5.36)
where O (1) means some constant independent of m. In other words the number of sin − z E below 1 + (Cm − O (1)) h is O (1) Cm Vol (N0 ∩ E ) h 21 −n . gular values of P Proof. Equation (5.35) is a direct consequence of (5.34). We use the “max-min formula” for self-adjoint operators [39, p. 78] and Eq. (5.33). Put λm := Cm − O (1). We have for every j, s 2j = u, Au max min U ⊆L 2 (X ), codim(U )≤ j−1
≥ 1 + 2λm h +
u∈U,u=1
max
U ⊆L 2 (X ), codim(U )≤ j−1
= 1 + 2λm h − h
min
U ⊆L 2 (X ), codim(U )≤ j−1
min
u∈U,u=1
max
− h Bh u|u
u∈U,u=1
Bh u|u
= 1 + 2λm h − hb j , where U varies in the set of closed subspaces of L 2 (X ) and b1 ≥ b2 ≥ . . . denote the eigenvalues of Bh (possibly completed with an infinite repetition of 0 if there are only finitely many such eigenvalues). We have Bh = b1 + b2 + . . . . Bh Tr = Tr Equation (5.25) implies that for every ε0 > 0, if b j ≥ ε0 then 1 jε0 ≤ Tr Bh ≤ O (1) Cm Vol (N0 ∩ E ) h 2 −n . 1
Equivalently if j > ε10 O (1) Cm Vol (N0 ∩ E ) h 2 −n , then b j < ε0 and s 2j ≥ 1+2λm h − hb j ≥ 1 + 2 (λm − ε0 ) h. Taking the square root we get (5.36).
Upper Bound on Density of Ruelle Resonances for Anosov Flows
357
We deduce now an upper bound for the number of eigenvalues of P. Corollary 5.9. We have .
/ 1 Cm σ P ∩ D zE, 1 + h ≤ O (1) Cm Vol (N0 ∩ E ) h 2 −n . 2
(5.37)
sorted such that j → λ j − z E Proof. Let λ1 , λ2 , λ3 . . . denote the eigenvalues of P is increasing. The Weyl inequalities (see [43, (a.8), p. 38] for a proof) give N 0 j=1
sj ≤
N 0 λ j − z E ,
∀N ,
(5.38)
j=1
where s j j are the singular values defined in Lemma 5.8 above. Let / .
/ . Cm Cm h = σ (P) ∩ D z E , 1 + h N := λ j : λ j − z E ≤ 1 + 2 2 and let := O (1) Cm Vol (N0 ∩ E ) h 2 −n M 1
be the term which appears in (5.36). We want to show the bound:
≤ 2+O 1 N M Cm
(5.39)
≤ M then (5.39) is true. Conversely let us suppose that N ≥ M. for Cm 1. If N Using (5.38) we have ⎛ ⎞⎛ ⎞
M N 0 0 Cm N ⎝ h sj⎠ ⎝ sj⎠ ≤ 1 + . 2 j=1
j= M+1
Then using (5.35) and (5.36) we have M
(1 − O (h)) (1 + (Cm − O (1)) h)
− M N
Cm h ≤ 1+ 2
N
We take the logarithm and since h # 1 we get: Cm h (h) + N − M (Cm − O (1)) h ≤ N − MO 2
C m (Cm + O (1)) . − O (1) ≤ M ⇔N 2 Now since Cm 1,
1 + O C1m ≤ M 2+O 1
= M , ⇔N 1 1 Cm + O 2 Cm
so we have obtained (5.39). This implies (5.37).
.
358
F. Faure, J. Sjöstrand
From Lemma 5.4 with b = C2m and β = C4m we deduce that the upper bound (5.37) implies an upper bound: ) 2 1 C C m m , | (λi − E)| ≤ λi ∈ σ P h, (λi ) ≥ − h 4 4 1
= O (1) Cm Vol (N0 ∩ E ) h 2 −n . Here we take E = 1 and h1 1 and return to the original spectral variable z = zh function m such that h after the scaling (5.4). From (5.7) we can choose the escape 1 Cm 1 is arbitrarily large and Vol (N0 ∩ E ) < o Cm is arbitrarily small so that
1 1 O (1) Cm Vol (N0 ∩ E ) h 2 −n = o h 2 −n . Since the spectrum does not depend on the escape function m, we get (1.25) with E = h1 . We have finished the proof of Theorem 1.8. Acknowledgements. This work has been supported by “Agence Nationale de la Recherche” under the grants JC05_52556 and ANR-08-BLAN-0228-01. The authors thank the referee for the very detailed and accurate review.
A. Some Results in Operator Theory A.1. On minimal and maximal extensions. We show here that the pseudodifferential defined in Eq.(3.2), has a unique closed extension on L 2 (X ). This is well operator P known for elliptic PDO [53, Chap.13, p.125]. The fact that P has order 1 (since it is defined from a vector field on X ) is important in the present non elliptic case. min of the operator P with domain The domain of the minimal closed extension P ∞ C (X ) is * + j → v ∈ L 2 (X ) . Dmin := u ∈ L 2 (X ) , u j ∈ C ∞ (X ) → u in L 2 (X ) and Pu (A.1) max has domain The maximal closed extension P * + ∈ L 2 (X ) . Dmax := u ∈ L 2 (X ) , Pu is defined a priori on C ∞ (X ) and D (X )). (Recall that P of order 1 (i.e P ∈ Op S 1 ), the minimal and maximal Lemma A.1. For a PDO P := Dmin = Dmax , i.e. there is a unique closed extension of extensions coincide: D P in L 2 (X ). the operator P Proof. Dmin ⊂ Dmax is clear. Let us check that Dmax ⊂ Dmin . Let u ∈ Dmax , i.e. ∈ L 2 (X ). We will construct a sequence u h ∈ C ∞ (X ) with u ∈ L 2 (X ) , v := Pu h → v in L 2 . h → 0, such that u h → u in L 2 (X ) and show that Pu ∗ + ∞ Let χ : T X → R be a C function such that χ (x, ξ ) = 1 for |ξ | ≤ 1, and χ (x, ξ ) = 0 for |ξ | ≥ 2. For h > 0, let the function χh on T ∗ X be defined by χh (x, ξ ) = χ (x, hξ ). Let the truncation operator be: χ h := Op (χh ) .
Upper Bound on Density of Ruelle Resonances for Anosov Flows
359
Notice that χ h is a smoothing operator which truncates large components in ξ (larger than 1/ h), χ h is similar to a convolution in x coordinates. Let u h := χ h u. It is clear that u h → u in L 2 (X ) as h → 0. We have χh u = χ + P, χ h = P h Pu h u. Pu → v = Pu as h → 0. The principal symbol of the PDO h Pu term converges χ The first P, χ h is 1 1 {P, χh } = ∂ξ P∂x χh − ∂x P∂ξ χh . i i Now we use the fact that P ∈ S 1 has order 1. In the first term, ∂ξ P ∈ S 0 is bounded and ∂x χh is non-zero only on a large ring h1 ≤ |ξ | ≤ h2 . In the second term ∂x P ∈ S 1 has order 1 but ∂ξ χh = h∂ξ χ (x, hξ ) is non-zero on the same large ring and therefore χ of order (−1) (since h |ξ |−1 on the ring). Therefore the PDO P, h converges χ strongly to zero in L 2 (X ) as h → 0. Hence P, h u → 0 as h → 0. We deduce that h → v = Pu, and that u ∈ Dmin . Pu A.2. The sharp Gårding inequality. References: [23, p. 52] or (A.8), [33, p. 99], [54, p. 1157] for a short proof using Toeplitz quantization. is a PDO with symbol P ∈ S μ , μ ∈ R, (P) ≥ 0, then there Proposition A.2. If P exists C > 0 such that ∀u ∈ C ∞ (X ) , Pu|u (A.2) ≥ −C u2 μ−1 , H
where u2H μ
2
# $μ # $μ := ξ u| ξ u L 2 (X ) denotes the norm in the Sobolev space H μ .
:= Oph (A) for a A.3. Quadratic partition of unity on phase space. We denote A symbol A. Lemma A.3. Let K 0 ⊂ T ∗ X compact. There exist symbols χ0 ∈ S −∞ and χ1 ∈ S 0 of self-adjoint operators χ 0 , χ 1 such that χ 02 + χ 12 = 1 + R.
The symbol R ∈ h ∞ S −∞ is negligible, supp (χ0 ) is compact and on K 0 , χ1 (x, ξ ) = ∞ O (h ) , χ0 (x, ξ ) = 1 + O (h ∞ ). Proof. Let K 0 ⊂ T ∗ X be compact. We can find symbols 0 ≤ χ0 ∈ C0∞ (T ∗ X ) (with compact support) and 0 ≤ χ1 ∈ C ∞ (T ∗ X ) such that ) 0 on K 0 χ1 = 1 for |ξ | 1
360
F. Faure, J. Sjöstrand
and
) A := χ02 + χ12 is
>0 =1
everywhere for |ξ | 1.
We replace χ0 , χ1 respectively by χ0 A−1/2 , χ1 A−1/2 . We obtain 1 = χ02 + χ12 . := χ ∈ Oph hS −∞ . We write R = hr0 (x, ξ ) + h 2 . . .. Let R 02 + χ 12 − 1. Then R We replace χ j , j = 0, 1 by
χ j := (1 + h r0 )−1/4 χ j (1 + h r0 )−1/4 , which is also self-adjoint. We obtain
χ 02 + χ 12 = (1 − h r0 ) χ r1 ) χ 02 + (1 − h 12 + O Oph h 2 S −∞
= 1 + O Oph h 2 S −∞ . If we iterate this algorithm, we obtain the lemma.
A.3.1. I.M.S. localization formula. The following lemma is similar to the “I.M.S localization formula” given in [11, p. 27]. It uses the quadratic partition of phase space obtained in Lemma A.3 above. ∈ Oph (S μ ) for some μ ∈ R and that (P − P ∗ ) ∈ Lemma A.4. Suppose that P μ Oph (hS ). Then for every u ∈ L 2 (X ) , z ∈ C,
2 2 P − z χ − z χ − z u 2 = P 0 u + P 1 u + O h 2 u2 . (A.3) Proof. For simplicity, we suppose z = iβ with β ∈ R, i.e. (z) = 0 (this is equivalent − (z) by some operator P ). We use (5.15) and write to replacing P
2 P − iβ u 2 = P − iβ ∗ χ − iβ u|u = − iβ u|u − iβ ∗ P P k P k=0,1
+O h
∞
(A.4) (A.5)
u . 2
The aim is to move the operators χ k outside. One has for k = 0, 1: ∗ 2 − iβ − χ − iβ ∗ P − iβ χ − iβ χ k P k k P P ∗ ∗ − iβ − χ − iβ ∗ P − iβ χ k P k P k =χ k P − iβ χ k P − iβ + P , χ k χ ∗ ∗ ,χ − iβ − χ − iβ χ = P k χ k P k P P, k ⎞ ⎛ ⎞ ⎛ ⎟ ⎟ ⎜ ∗ ⎜ ∗ χ χ − χ ∗ P, k P k ⎠ − iβ ⎝ P k + χ k ⎠ . (A.6) ,χ k χ ,χ k χ k P, =⎝ P k P Ik
IIk
∈ Oph (S μ ) with some μ ∈ R, then First remark that for every PDO A χ A, k ∈ Oph hS −∞ .
Upper Bound on Density of Ruelle Resonances for Anosov Flows
361
This is obvious for k = 0 since χ ∈ Oph S −∞ and for k = 1 this is because −∞ 0 1 = 0. We have assumed that χ1 − 1) ∈ Oph S and A, (
∗ − P ∈ Oph hS μ , P
therefore
∗ − P, χ P k ∈ Oph h 2 S −∞ .
Also
χ P, k ∈ Oph hS −∞ .
The first term of (A.6) is ∗ χ ,χ − χ ∗ P, Ik = P k P k k χ k P
χ − χ P, χ = P, k χ k P k P k + O Oph h 2 S −∞
χ + O Oph h 2 S −∞ = P, k , χ k P
= O Oph h 2 S −∞ . The second term of (A.6) is ∗ χ ,χ IIk = P k + χ k k χ k P,
χ χ = P, k χ k + χ k + O Oph h 2 S −∞ k P, 3 4
χ = P, k2 + O Oph h 2 S −∞ . Therefore using (5.15), 4
3 χ 02 + χ 12 + O Oph h 2 S −∞ II0 + II1 = P,
= O Oph h 2 S −∞ . We have shown that k=0,1
− iβ P
∗
− iβ = − iβ ∗ P − iβ χ k + O Oph h 2 S −∞ . χ k2 P χ k P k=0,1
Coming back to (A.4) we get (A.3).
362
F. Faure, J. Sjöstrand
A.4. FBI transform and Toeplitz operators. References: [33,54]. The manifold X is equipped with a smooth Riemannian metric so that we have a well-defined exponential map expx : Tx X → X which is a diffeomorphism from a neighborhood of 0 ∈ Tx X onto a neighborhood of x ∈ X . Define the coherent state at point (x, ξ ) ∈ T ∗ X to be the function of y ∈ X : 1 i −1 2 ξ dist (x, y) , ξ expx (y) − (y) := χ (x, y) exp h 2h
ex,ξ
ξ := (1 + |ξ |)1/2 ,
where χ ∈ C ∞ (X × X ) is a standard cutoff to a small neighborhood of the diagonal. In the Euclidean case X = Rn , the cutoff is often superfluous and we get the complex Gaussian “wave packet” i 1 2 ξ |y − x| . ξ. (y − αx ) − (y) = exp h 2h
ex,ξ
We have the following known facts [26,44]: 3n
• There exists a0 (x, ξ ; h) ∈ h − 2 S n/2 elliptic and a0 > 0 such that u= πx,ξ u d xdξ + Ru, ∀u ∈ L 2 (X ) T∗X
(A.7)
with πx,ξ := a0 (x, ξ ; h) ex,ξ ex,ξ |. ∈ Oph h ∞ S −∞ negligible. d x is the Riemannian volume on X . Equation and R (A.7) is called resolution of identity. • We can define the FBI-transform of u ∈ C ∞ (X ) by ex,ξ (y)u (y) dy, (T u) (x, ξ ; h) := a0 (x, ξ ; h) ex,ξ |u = a0 (x, ξ ; h) X
which is asymptotically isometric from (A.7):
ˆ = u L 2 (X ) + O h ∞ . T u L 2 (T ∗ X ) = u L 2 (X ) + u| Ru • πx,ξ ≥ 0 and 2 πx,ξ tr = Tr πx,ξ = a0 (x, ξ ; h) ex,ξ = O (1) h −n . • If B ∈ Oph (S m ) is a PDO with principal symbol b0 (modulo hS m−1 ), then B=
T∗X
b (x, ξ ; h) πx,ξ d xdξ + R,
is negligible as above, b ∈ S m and b = b0 mod hS m−1 . where R
Upper Bound on Density of Ruelle Resonances for Anosov Flows
363
• For a function A (x, ξ ; h) (such that the following expression makes sense), we define the Toeplitz quantization of A by OpT (A) := A (x, ξ ; h) πx,ξ d xdξ, then the previous results imply a “Gårding’s inequality”: A (x, ξ ) ≥ 0 ⇒ OpT (A) u|u ≥ 0 and O (1) Tr OpT (A) = hn
(A.8)
A (x, ξ ) d xdξ.
References 1. Aguilar, J., Combes, J.M.: A class of analytic perturbations for one-body Schrödinger Hamiltonians. Commun. Math. Phys. 22, 269–279 (1971) 2. Arnold, V.I., Avez, A.: Méthodes ergodiques de la mécanique classique. Paris: Gauthier Villars, 1967 3. Baladi, V.: Anisotropic Sobolev spaces and dynamical transfer operators: C ∞ foliations. In: Kolyada, S. (ed.) et al., Algebraic and topological dynamics. Proceedings of the conference, Bonn, Germany, May 1-July 31, 2004. Providence, RI: Amer. Math. Soc., Contemporary Mathematics, 385, 2005, pp. 123–135 4. Baladi, V., Tsujii, M.: Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms. Ann. Inst. Fourier 57, 127–154 (2007) 5. Balslev, E., Combes, J.M.: Spectral properties of many-body Schrödinger operators with dilatationanalytic interactions. Commun. Math. Phys. 22, 280–294 (1971) 6. Blank, M., Keller, G., Liverani, C.: Ruelle-Perron-Frobenius spectrum for Anosov maps. Nonlinearity 15, 1905–1973 (2002) 7. Bonatti, C., Guelman, N.: Transitive anosov flows and axiom-a diffeomorphisms. Erg. Th. Dyn. Sys. 29(3), 817–848 (2009) 8. Borthwick, D.: Spectral theory of infinite-area hyperbolic surfaces. Basel-Boston: Birkhauser, 2007 9. Brin, M., Stuck, G.: Introduction to Dynamical Systems. Cambridge: Cambridge University Press, 2002 10. Butterley, O., Liverani, C.: Smooth Anosov flows: correlation spectra and stability. J. Mod. Dyn. 1(2), 301–322 (2007) 11. Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger operators, with application to quantum mechanics and global geometry. (Springer Study ed.). Texts and Monographs in Physics. BerlinHeidelberg-New York: Springer-Verlag, 1987 12. Davies, E.B.: Linear operators and their spectra. Cambridge: Cambridge University Press, 2007 13. Dolgopyat, D.: On decay of correlations in Anosov flows. Ann. of Math. (2) 147( 2), 357–390 (1998) 14. Dolgopyat, D.: On mixing properties of compact group extensions of hyperbolic systems. Israel J. Math. 130, 157–205 (2002) 15. Faure, F.: Semiclassical origin of the spectral gap for transfer operators of a partially expanding map. Nonlinearity 24, 1473–1498 (2011) 16. Faure, F., Roy, N.: Ruelle-Pollicott resonances for real analytic hyperbolic map. Nonlinearity 19, 1233–1252 (2006) 17. Faure, F., Roy, N., Sjöstrand, J.: A semiclassical approach for Anosov diffeomorphisms and Ruelle resonances. Open Math. Journal. 1, 35–81 (2008) 18. Field, M., Melbourne, I., Török, A.: Stability of mixing and rapid mixing for hyperbolic flows. Ann. of Math. (2) 166(1), 269–291 (2007) 19. Gérard, C., Sjöstrand, J.: Resonances en limite semiclassique et exposants de Lyapunov. Commun. Math. Phys. 116(2), 193–213 (1988) 20. Ghys, E.: Flots d’Anosov dont les feuilletages stables sont différentiables. Ann. Sci. École Norm. Sup. (4) 20(2), 251–270 (1987) 21. Ghys, E.: Déformations de flots d’Anosov et de groupes fuchsiens. Ann. Inst. Fourier (Grenoble) 42(1-2), 209–247 (1992) 22. Gouzel, S., Liverani, C.: Banach spaces adapted to Anosov systems. Erg. Th. Dyn. Sys. 26, 189–217 (2005) 23. Grigis, A., Sjöstrand, J.: Microlocal analysis for differential operators. Volume 196 of London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press, 1994 24. Guillope, L., Lin, K., Zworski, M.: The Selberg zeta function for convex co-compact. Schottky groups. Commun. Math. Phys. 245(1), 149–176 (2004)
364
F. Faure, J. Sjöstrand
25. Helffer, B., Sjöstrand, J.: Résonances en limite semi-classique. (resonances in semi-classical limit). Memoires de la S.M.F., 24/25, 1986 26. Hitrik, M., Sjöstrand, J.: Rational invariant tori, phase space tunneling, and spectra for non-selfadjoint operators in dimension 2. Ann. Scient. de l’école normale supérieure. http://arxiv.org/abs/math/ 0703394v1 [math.SP], 2008 27. Hurder, S., Katok, A.: Differentiability, rigidity and Godbillon-Vey classes for Anosov flows. Publ. Math., Inst. Hautes étud. Sci. 72, 5–61 (1990) 28. Hörmander, L.: Fourier integral operators. I. Acta Math. 127(1), 79–183 (1971) 29. Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge: Cambridge University Press, 1995 30. Leboeuf, P.: Periodic orbit spectrum in terms of Ruelle-Pollicott resonances. Phys. Rev. E (3) 69(2), 026204 (2004) 31. Liverani, C.: On contact Anosov flows. Ann. of Math. (2) 159(3), 1275–1312 (2004) 32. Liverani, C.: Fredholm determinants, Anosov maps and Ruelle resonances. Disc. Cont. Dyn. Sys. 13(5), 1203–1215 (2005) 33. Martinez, A.: An Introduction to Semiclassical and Microlocal Analysis. Universitext. New York, NY: Springer 2002 34. McDuff, D., Salamon, D. Introduction to symplectic topology, 2nd edition. Oxford: Clarendon press, 1998 35. Nonnenmacher, S.: Some open questions in ‘wave chaos’. Nonlinearity 21(8), T113–T121 (2008) 36. Nonnenmacher, S., Zworski, M.: Distribution of resonances for open quantum maps. Comm. Math. Phys. 269(2), 311–365 (2007) 37. Pesin, Y.: Lectures on Partial Hyperbolicity and Stable Ergodicity. Zünch: European Mathematical Society, 2004 38. Reed, M., Simon, B.: Mathematical methods in physics, Vol. I: Functional Analysis. New York: Academic Press, 1972 39. Reed, M., Simon, B.: Mathematical methods in physics, Vol. IV: Analysis of operators. New York: Academic Press, 1978 40. Ruelle, D.: Thermodynamic formalism. The mathematical structures of classical equilibrium. Statistical mechanics. With a foreword by Giovanni Gallavotti. Reading, MA: Addison-Wesley Publishing Company, 1978 41. Ruelle, D.: Locating resonances for axiom A dynamical systems. J. Stat. Phys. 44, 281–292 (1986) 42. Cannas Da Salva, A.: Lectures on Symplectic Geometry. Berlin-Heidelberg-New York: Springer, 2001 43. Sjöstrand, J.: Geometric bounds on the density of resonances for semiclassical problems. Duke Math. J. 60(1), 1–57 (1990) 44. Sjöstrand, J.: Density of resonances for strictly convex analytic obstacles. Canad. J. Math. 48(2), 397–447, (1996) (with an appendix by M. Zworski) 45. Sjöstrand, J.: Lecture on resonances. Available on http://www.math.polytechnique.fr/~sjoestrand/ NowListe070411.html, 2002 46. Sjöstrand, J.: Resonances associated to a closed hyperbolic trajectory in dimension 2. Asym. Anal. 36(2), 93–113 (2003) 47. Sjöstrand, J., Zworski, M.: Fractal upper bounds on the density of semiclassical resonances. Duke Math. J. 137, 381–459 (2007) 48. Taylor, M.: Partial differential equations, Vol. I. Berlin-Heidelberg-New York: Springer, 1996 49. Taylor, M.: Partial differential equations, Vol. II. Berlin-Heidelberg-New York: Springer, 1996 50. Tsujii, M.: Decay of correlations in suspension semi-flows of angle-multiplying maps. Erg. Th. Dyn. Sys. 28, 291–317 (2008) 51. Tsujii, M.: Quasi-compactness of transfer operators for contact Anosov flows. Nonlinearity 23, 1495 (2010) 52. Tsujii, M.: Contact Anosov flows and the FBI transform. http://arXiv.org/abs/1010.0396v2 [math.DS], 2010 53. Wong, M.W.: An introduction to pseudo-differential operators. 2nd ed., River Edge, NJ: World Scientific Publishing Co. Inc., 1999 54. Wunsch, J., Zworski, M.: The FBI transform on compact C ∞ manifolds. Trans. Am. Math. Soc. 353(3), 1151–1167 (2001) 55. Zelditch, S.: Quantum ergodicity and mixing of eigenfunctions. Elsevier Encyclopedia of Math. Phys., vol. 1, Oxford: Elsevier, 2006, pp. 183–196 56. Zworski, M.: Resonances in physics and geometry. Notices of the A.M.S. 46(3), 319–328 (1999) Communicated by S. Zelditch
Commun. Math. Phys. 308, 365–413 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1357-z
Communications in
Mathematical Physics
Dimensional Reduction Over the Quantum Sphere and Non-Abelian q -Vortices Giovanni Landi1,2 , Richard J. Szabo3,4 1 Dipartimento di Matematica e Informatica, Università di Trieste, Via A. Valerio 12/1, 34127 Trieste, Italy.
E-mail:
[email protected]
2 INFN, Sezione di Trieste, Trieste, Italy 3 Department of Mathematics, Heriot-Watt University, Colin Maclaurin Building,
Riccarton, Edinburgh EH14 4AS, UK. E-mail:
[email protected]
4 Maxwell Institute for Mathematical Sciences, Edinburgh, UK
Received: 27 April 2010 / Accepted: 18 July 2011 Published online: 21 October 2011 – © Springer-Verlag 2011
Abstract: We extend equivariant dimensional reduction techniques to the case of quantum spaces which are the product of a Kähler manifold M with the quantum twosphere. We work out the reduction of bundles which are equivariant under the natural action of the quantum group SUq (2), and also of invariant gauge connections on these bundles. The reduction of Yang–Mills gauge theory on the product space leads to a q-deformation of the usual quiver gauge theories on M. We formulate generalized instanton equations on the quantum space and show that they correspond to q-deformations of the usual holomorphic quiver chain vortex equations on M. We study some topological stability conditions for the existence of solutions to these equations, and demonstrate that the corresponding vacuum moduli spaces are generally better behaved than their undeformed counterparts, but much more constrained by the q-deformation. We work out several explicit examples, including new examples of non-abelian vortices on Riemann surfaces, and q-deformations of instantons whose moduli spaces admit the standard hyper-Kähler quotient construction. Contents 0. 1.
2.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . SUq (2)-Equivariant Bundles on the Quantum Projective Line . 1.1 Quantum projective line CPq1 . . . . . . . . . . . . . . . . 1.2 Equivariant line bundles on CPq1 . . . . . . . . . . . . . . SUq (2)-Invariant Gauge Fields on the Quantum Projective Line 2.1 Left-covariant forms on SUq (2) . . . . . . . . . . . . . . 2.2 Holomorphic forms on CPq1 . . . . . . . . . . . . . . . . . 2.3 Connections on equivariant line bundles over CPq1 . . . . . 2.4 Holomorphic structures . . . . . . . . . . . . . . . . . . . 2.5 Unitarity and gauge transformations . . . . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
366 368 368 371 374 374 375 377 379 380
366
G. Landi, R. J. Szabo
2.6 SUq (2)-invariant connections and gauge transformations . . . . 2.7 K-theory charges . . . . . . . . . . . . . . . . . . . . . . . . . 3. Dimensional Reduction of Invariant Gauge Fields . . . . . . . . . . 3.1 Dimensional reduction of SUq (2)-equivariant vector bundles . . 3.2 Covariant hermitian structures . . . . . . . . . . . . . . . . . . 3.3 Decomposition of covariant connections . . . . . . . . . . . . . 3.4 SUq (2)-invariant connections and gauge transformations . . . . 3.5 Integrable connections . . . . . . . . . . . . . . . . . . . . . . 4. Quiver Gauge Theory and Non-Abelian Coupled q-Vortex Equations 4.1 Metrics on SUq (2)-equivariant vector bundles . . . . . . . . . . 4.2 Dimensional reduction of the Yang–Mills action functional . . . 4.3 Holomorphic chain q-vortex equations . . . . . . . . . . . . . . 4.4 Vacuum structure . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Stability conditions . . . . . . . . . . . . . . . . . . . . . . . . 5. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Deformations of holomorphic triples and stable pairs . . . . . . 5.2 q-vortices on Riemann surfaces . . . . . . . . . . . . . . . . . . 5.3 q-instantons . . . . . . . . . . . . . . . . . . . . . . . . . . . . Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
382 383 385 386 389 389 392 395 396 397 398 401 404 408 409 409 410 410 411 412
0. Introduction Let M be a smooth manifold. In this paper we define and characterize vector bundles over the quantum space M := CPq1 × M which are equivariant under an action of the quantum group SUq (2). Here CPq1 is the quantum projective line which is defined in §1.1. The vector bundles will be given as (finitely-generated and projective) SUq (2)equivariant modules over the algebra of functions A( M ) = A(CPq1 ) ⊗ A(M). We will describe the dimensional reduction of invariant connections on the SUq (2)-equivariant modules over the algebra A( M ). In particular, we will reduce Yang–Mills gauge theory on A( M ) to a type of Yang–Mills–Higgs theory on the manifold M. The vacuum equations of motion for this model give q-deformations of some known vortex equations, whose solutions possess, as we shall see, some remarkable properties. In the q = 1 case, a general and systematic treatment of SU(2)-equivariant dimensional reduction over the product CP1 × M of the ordinary complex projective line CP1 with a Kähler manifold M was first carried out in [1]. Here SU(2) acts in the standard way by isometries of the homogeneous space CP1 and trivially on M. It was shown in [1] that there is a one-to-one correspondence between SU(2)-equivariant vector bundles over CP1 × M and U(1)-equivariant vector bundles over M, with U(1) acting trivially on M. The reduced vector bundle has the structure of a quiver bundle, in this case a representation of the linear Am+1 quiver chain in the category of complex vector bundles over M. Moreover, certain natural first order gauge theory equations on CP1 × M reduce to generalizations of vortex equations called holomorphic chain vortex equations, which contain a multitude of BPS-type integrable equations as special cases [20]. These include standard abelian and non-abelian vortex equations in two dimensions, and the self-duality and perturbed abelian Seiberg–Witten monopole equations in four dimensions. With suitable notions of stability for holomorphic bundles over CP1 × M and the corresponding quiver bundles over M, a variant of the Hitchin–Kobayashi correspondence identifies the necessary and sufficient conditions for the existence of solutions to
Dimensional Reduction Over the Quantum Sphere
367
these equations [1]. This particular reduction has been further developed in [12,20,31], where some physical applications are also considered. The reduction generalizes to the product of M with any homogeneous space which is a flag manifold; see [2,21] for the general theory. In this paper we will study how equivariant dimensional reduction and the ensuing vortex equations are modified when the ‘internal’ sphere CP1 is replaced with a particular noncommutative deformation. Dimensional reduction over the fuzzy sphere CP1F was considered in [5,4,16], where it was shown that the deformation significantly alters the vacuum structure of the induced Yang–Mills–Higgs theory, which in some instances may not coincide with the standard vortex models in the commutative limit. In particular, solutions of abelian vortex equations are studied in [16] which correspond to instantons in the original Yang–Mills theory on CP1F × M but are nevertheless non-BPS states of the dimensionally reduced field theory. In the following we will demonstrate that a similar vacuum structure emerges when the dimensional reduction is performed over a quantum sphere CPq1 . As discussed in [7], a basic problem with standard vortex equations is that it is not possible to reach the zeroes of the corresponding Yang–Mills–Higgs action functional by means of non-trivial vortex solutions, due to topological obstructions. In [3] it was shown that one can improve this functional by using the formalism of twisted quiver bundles, which yields zeroes of the action for bundles admitting flat connections. In the present paper we show that, in contrast to the usual quiver gauge theories that arise through dimensional reduction, the same is true for the Yang–Mills–Higgs models which are systematically obtained via SUq (2)-equivariant dimensional reduction over CPq1 . In order to rigorously carry out the dimensional reduction in parallel to the commutative case, it is necessary to extend the equivariant decompositions of [1,31] within the algebraic framework of noncommutative geometry and in a Hopf algebraic framework appropriate to the action of the quantum group SUq (2). This is the content of §1–§3 of the present paper. In §1 and §2 we extend the requisite geometry of the projective line CP1 to the q-deformed case CPq1 , using the fact that there are finitely-generated projective modules over the quantum sphere that correspond to the canonical line bundles on the Riemann sphere in the q → 1 limit. In §3 we generalize the decompositions of [1,31] to invariant gauge fields for the action of SUq (2) on M = CPq1 × M. In §4 we study the reduction of Yang–Mills theory on M . In particular, we formulate a suitable notion of generalized instanton on the quantum space M which coincides with solutions of the vortex equations associated to minima of the induced q-deformed Yang–Mills–Higgs action functional on M; we call the (gauge equivalence classes of) solutions to these equations ‘q-vortices’. We also examine in detail the structure of the corresponding vacuum moduli spaces and the topological stability conditions for the existence of solutions to the q-vortex equations, finding in general that these moduli spaces are much more constrained than their classical q → 1 limits. In §5 we study some explicit examples and compare with analogous results in the literature for the case q = 1, showing that the q-deformation generically improves the geometrical structure of the associated moduli spaces. In particular, we analyse moduli spaces of q-vortices on Riemann surfaces giving new examples of non-abelian vortices, and show that our q-deformations of instantons on Kähler surfaces are analogous to those of some previous noncommutative deformations of the self-duality equations. Conventions. In the following we shall use the terminology covariance to mean both covariance for an action and ‘co-covariance’ for a coaction. The q-number
368
G. Landi, R. J. Szabo
[s] = [s]q :=
q s − q −s , q − q −1
(0.1)
is defined for q = 1 and any s ∈ R. For a coproduct we use the conventional Sweedler notation (x) = x(1) ⊗ x(2) (with implicit summation). This convention is iterated to give (id ⊗) ◦ (x) = ( ⊗ id) ◦ (x) = x(1) ⊗ x(2) ⊗ x(3) , and so on. 1. SUq (2)-Equivariant Bundles on the Quantum Projective Line The quantum projective line CPq1 is defined as a quotient of the sphere Sq3 SUq (2) with respect to an action of the group U(1). It is the standard Podle´s sphere Sq2 of [28] with additional structure. The construction we need is the well-known quantum principal U(1)-bundle over Sq2 , whose total space is the manifold of the quantum group SUq (2). 1.1. Quantum projective line CPq1 . We begin with the algebras of Sq3 and CPq1 . The manifold of Sq3 is identified with the manifold of the quantum group SUq (2). The deformation parameter q ∈ R can be restricted to the interval 0 < q < 1 without loss of generality. The coordinate algebra A(SUq (2)) is the ∗-algebra generated by elements a and c with the relations a c = q c a and c∗ a ∗ = q a ∗ c∗ , a c∗ = q c∗ a and c a ∗ = q a ∗ c, c c∗ = c∗ c and a ∗ a + c∗ c = a a ∗ + q 2 c c∗ = 1. (1.1) These relations are equivalent to requiring that the ‘defining’ matrix a −q c∗ U= c a∗ is unitary, U U ∗ = U ∗ U = 1. The Hopf algebra structure for A(SUq (2)) is given by the coproduct a −q c∗ a −q c∗ a −q c∗ = ⊗ , c a∗ c a∗ c a∗ with a ‘tensor product’ of rows by columns, e.g. (a) = a ⊗ a − q c∗ ⊗ c, etc., the antipode ∗ c∗ a a −q c∗ , = S c a∗ −q c a and the counit
a −q c∗ c a∗
=
1 0 . 0 1
The quantum universal enveloping algebra Uq (su(2)) is the Hopf ∗-algebra generated as an algebra by four elements K , K −1 , E, F with K K −1 = 1 = K −1 K and relations K ± 1 E = q ± 1 E K ± 1,
K ± 1 F = q ∓ 1 F K ± 1 and [E, F] =
K 2 − K −2 . (1.2) q − q −1
Dimensional Reduction Over the Quantum Sphere
369
The ∗-structure is simply K∗ = K ,
E∗ = F
and
F ∗ = E,
and the Hopf algebra structure is provided by the coproduct , the antipode S, and the counit defined by (K ± 1 ) = K ± 1 ⊗ K ± 1 , (E) = E ⊗ K + K −1 ⊗ E, (F) = F ⊗ K + K −1 ⊗ F, S(K ) = K −1 , S(E) = −q E, S(F) = −q −1 F, (K ) = 1, (E) = (F) = 0. There is a bilinear pairing between Uq (su(2)) and A(SUq (2)) given on generators by
K , a = q −1/2 , K −1 , a = q 1/2 , K , a ∗ = q 1/2 and K −1 , a ∗ = q −1/2 ,
E, c = 1 and F, c∗ = −q −1 , with all other couples of generators pairing to 0. One regards Uq (su(2)) as a subspace of the linear dual of A(SUq (2)) via this pairing. There are canonical left and right Uq (su(2))-module algebra structures on A(SUq (2)) such that [36]
g , h x := g h , x
and
g , x h := h g , x
for all g, h ∈ Uq (su(2)), x ∈ A(SUq (2)). They are given by h x := (id ⊗h) , (x) and x h := (h ⊗ id) , (x) , or equivalently and x h := h , x(1) x(2) h x := x(1) h , x(2) in the Sweedler notation. These right and left actions mutually commute, (h x) g = x(1) h , x(2) g = g , x(1) x(2) h , x(3) = h g , x(1) x(2) = h (x g), and since the pairing satisfies
S(h)∗ , x = h , x ∗
for all h ∈ Uq (su(2)), x ∈ A(SUq (2)), the ∗-structure is compatible with both actions, ∗ ∗ and x ∗ h = x S(h)∗ h x ∗ = S(h)∗ x for all h ∈ Uq (su(2)), x ∈ A(SUq (2)). The left action for any s ∈ N0 is given explicitly by s
K ± 1 as = q ∓ 2 as
s
and
K ± 1 a∗ s = q ± 2 a∗ s ,
K ± 1 cs = q ∓ 2 cs
and
K ± 1 c∗ s = q ± 2 c∗ s ,
F as = 0
and
F a ∗ s = q (1−s)/2 [s] c a ∗ s−1 ,
F cs = 0
and
F c∗ s = −q −(1+s)/2 [s] a c∗ s−1 ,
E a s = −q (3−s)/2 [s] a s−1 c∗
and
E a ∗ s = 0,
E cs = q (1−s)/2 [s] cs−1 a ∗
and
E c∗ s = 0.
s
s
(1.3)
370
G. Landi, R. J. Szabo
The right action is given explicitly by s
as K ± 1 = q ∓ 2 as
s
and
a∗ s K ± 1 = q ± 2 a∗ s ,
cs K ± 1 = q ± 2 cs
and
c∗ s K ± 1 = q ∓ 2 c∗ s ,
a s F = q (s−1)/2 [s] c a s−1
and
a ∗ s F = 0,
cs F = 0
and
c∗ s F = −q −(s−3)/2 [s] a ∗ c∗ s−1 ,
as E = 0
and
a ∗ s E = −q (3−s)/2 [s] c∗ a ∗ s−1 ,
cs E = q (s−1)/2 [s] cs−1 a
and
c∗ s E = 0.
s
s
(1.4)
Now we describe the U(1)-principal bundle over Sq2 , whose total space is the manifold of the quantum group SUq (2). It is an example of a quantum homogeneous space [10] constructed as follows. If A(U(1)) := C[ζ, ζ ∗ ] ζ ζ ∗ − 1 denotes the (commutative) algebra of coordinate functions on the group U(1), the map ζ 0 a −q c∗ π : A(SUq (2)) −→ A(U(1)), = (1.5) π 0 ζ∗ c a∗ is a surjective Hopf ∗-algebra homomorphism, so that A(U(1)) becomes a quantum subgroup of SUq (2) with a right coaction R := (id ⊗π ) ◦ : A(SUq (2)) −→ A(SUq (2)) ⊗ A(U(1)).
(1.6)
The coinvariant elements for this coaction, i.e. elements {x ∈ A(SUq (2)) | R (x) = x ⊗ 1}, generate a subalgebra of A(SUq (2)) which is the coordinate algebra A(Sq2 ) of the standard Podle´s sphere Sq2 first described in [28]. For the purposes of the present paper, it will be useful to also have an equivalent description of the bundle by taking an action (irrelevantly right or left) of the abelian group U(1) = {z ∈ C | z z ∗ = 1} on the algebra A(SUq (2)), i.e. we consider the map α : U(1) −→ Aut A(SUq (2))
(1.7)
defined on generators by αz (a) = a z αz (c) = c z
and and
αz (a ∗ ) = a ∗ z ∗ , αz (c∗ ) = c∗ z ∗ ,
(1.8)
and extended as an algebra map, αz (x y) = αz (x) αz (y) for x, y ∈ A(SUq (2)) and z ∈ U(1). Here the complex number z is the evaluation of the function ζ ∈ A(U(1)). The coordinate algebra A(Sq2 ) is then regarded as the subalgebra of invariant elements in A(SUq (2)),
A(Sq2 ) := A(SUq (2))U(1) := x ∈ A(SUq (2)) αz (x) = x .
(1.9)
As a set of generators for A(Sq2 ) we may take B− := a c∗ ,
B+ := c a ∗
and
B0 := c c∗ ,
(1.10)
Dimensional Reduction Over the Quantum Sphere
371
for which one finds relations and B− B0 = q 2 B0 B− B− B+ = q 2 B0 1 − q 2 B0
B+ B0 = q −2 B0 B+ , and
B+ B− = B0 1 − B0 ,
and ∗-structure (B0 )∗ = B0 and (B+ )∗ = B− . The algebra inclusion A(Sq2 ) → A(SUq (2)) is a quantum principal bundle and can be endowed with compatible calculi [10], a construction that we shall illustrate later on. In §2.2 we will describe a natural complex structure on the quantum two-sphere Sq2 for the unique two-dimensional covariant calculus on it. This will transform the sphere Sq2 into a quantum riemannian sphere or quantum projective line CPq1 . Having this in mind, with a slight abuse of ‘language’ we will speak of CPq1 rather than Sq2 from now on. The sphere Sq2 (and hence the quantum projective line CPq1 ) is a quantum homogeneous space of SUq (2) and the coproduct of A(SUq (2)) restricts to a left coaction of A(SUq (2)) on A(Sq2 ) (or A(CPq1 )): L : A(CPq1 ) −→ A(SUq (2)) ⊗ A(CPq1 ). In particular, the elements Y− := −a c∗ ,
Y+ := q c a ∗
and
−1 Y0 := q 2 1 + q 2 − q 2 c c∗
transform according to the fundamental ‘vector corepresentation’ of SUq (2) given by L (Y− ) = a 2 ⊗ Y− − 1 + q −2 Y− ⊗ Y0 + c∗ 2 ⊗ Y+ , L (Y0 ) = q a c ⊗ Y− + 1 + q −2 Y0 ⊗ Y0 − c∗ a ∗ ⊗ Y+ , L (Y+ ) = q 2 c2 ⊗ Y− + 1 + q −2 Y+ ⊗ Y0 + a ∗ 2 ⊗ Y+ .
(1.11)
The following result is evident. Proposition 1.12. The element 1 ∈ A(CPq1 ) is the only coinvariant element for this coaction, i.e. the only x ∈ A(CPq1 ) for which L (x) = 1 ⊗ x. 1.2. Equivariant line bundles on CPq1 . Let ρ : U(1) → V be a representation of U(1) on a finite-dimensional complex vector space V . The corresponding space of ρ-equivariant elements is given by
A(SUq (2)) ρ V := ϕ ∈ A(SUq (2)) ⊗ V (α ⊗ id)ϕ = (id ⊗ρ −1 ) ϕ , (1.13) where α is the action (1.7) of U(1) on A(SUq (2)). The space (1.13) is an A(CPq1 )bimodule. We shall think of it as the module of sections of the vector bundle associated with the quantum principal U(1)-bundle on CPq1 via the representation ρ. There is a natural SUq (2)-equivariance, in that the left coaction of A(SUq (2)) on itself extends in a natural way to a left coaction on A(SUq (2)) ρ V given by ρ = ⊗ id : A(SUq (2)) ρ V −→ A(SUq (2)) ⊗ A(SUq (2)) ρ V .
(1.14)
372
G. Landi, R. J. Szabo
The irreducible representations of U(1) are labelled by an integer n ∈ Z. If Cn C is the irreducible one-dimensional left U(1)-module of weight n, they are given by ρn : U(1) −→ Aut(Cn ),
Cn v −→ z n v ∈ Cn .
(1.15)
The corresponding spaces of equivariant elements are well-known and amount to a vector space decomposition [24, Eq. (1.10)] A(SUq (2)) = Ln , (1.16) n∈Z
where
Ln := A(SUq (2)) ρn C x ∈ A(SUq (2)) αz (x) = x (z ∗ )n .
(1.17)
In particular, L0 = A(CPq1 ). One has L∗n = L−n and Ln Lm = Ln+m . Each Ln is clearly a bimodule over A(CPq1 ) and is naturally isomorphic to A(SUq (2)) ρn Cn . It was shown in [33, Prop. 6.4] that each Ln is a finitely-generated projective left (and right) A(CPq1 )-module of rank one. They give the modules of SUq (2)-equivariant elements or of sections of line bundles over the quantum projective line CPq1 with monopole charges −n. One has the following results (cfr. [17, Prop. 3.1]). Lemma 1.18. (1) Each Ln is the bimodule of equivariant elements associated with the irreducible representation of U(1) with weight n. (2) The natural map Ln ⊗ Lm → Ln+m defined by multiplication induces an isomorphism of A(CPq1 )-bimodules Ln ⊗A(CPq1 ) Lm Ln+m , and in particular HomA(CPq1 ) (Lm , Ln ) Ln−m . Proof. These results follow by using the representation theory of U(1) as well as the relations a ⊗A(CPq1 ) c = q c ⊗A(CPq1 ) a, a ⊗A(CPq1 ) c∗ = q c∗ ⊗A(CPq1 ) a, c ⊗A(CPq1 ) c∗ = c∗ ⊗A(CPq1 ) c, and so on, which are easily established. From the transformations in (1.8), it follows that an A(CPq1 )-module generating set for Ln is given by elements
∗ μ ∗ n−μ
(n) for n ≥ 0, μ = 0, 1, . . . , n, c a
= (1.19) μ c|n|−μ a μ for n ≤ 0, μ = 0, 1, . . . , |n| . Then one writes equivariant elements as ⎧ n n ⎪ ⎪ ∗ μ ∗ n−μ ⎪ c a f = for n ≥ 0, f˜μ c∗ μ a ∗ n−μ ⎪ μ ⎪ ⎪ ⎪ μ=0 μ=0 ⎨ ϕf = ⎪ |n| |n| ⎪ ⎪ ⎪ |n|−μ μ ⎪ c a fμ = for n ≤ 0, f˜μ c|n|−μ a μ ⎪ ⎪ ⎩ μ=0
μ=0
(1.20)
Dimensional Reduction Over the Quantum Sphere
373
with f μ and f˜μ generic elements in A(CPq1 ). The elements in (1.19) are not independent over A(CPq1 ) since the bimodules Ln are not free modules. A generic finite-dimensional representation (V, ρ) for U(1) is given by a weight decomposition V =
Cn ⊗ Vn ,
ρ=
n∈W (V )
ρn ⊗ id .
(1.21)
n∈W (V )
Here (Cn , ρn ) is the one-dimensional irreducible representation of U(1) with weight n ∈ Z given in (1.15), the spaces Vn = HomU(1) (Cn , V ) are the multiplicity spaces, and the set W (V ) = {n ∈ Z | Vn = 0} is the set of weights of V . For the corresponding space of ρ-equivariant elements we have a corresponding decomposition
A(SUq (2)) ρ V =
Ln ⊗ Vn ,
(1.22)
n∈W (V )
with Ln the irreducible modules in (1.17) giving sections of line bundles over CPq1 . The left action of the group-like element K on A(SUq (2)) allows one to give a dual presentation of the line bundles Ln as
Ln = x ∈ A(SUq (2)) K x = q n/2 x .
(1.23)
Indeed, if H is the infinitesimal generator of the U(1)-action α, the group-like element K can be written as K = q −H/2 . Then from the relations (1.2) of Uq (su(2)) one finds E Ln ⊂ Ln+2
and
F Ln ⊂ Ln−2 .
(1.24)
On the other hand, commutativity of the left and right actions of Uq (su(2)) yields Ln h ⊂ Ln
(1.25)
for all h ∈ Uq (su(2)). It was shown in [33, Thm. 4.1] that there is also a decomposition Ln =
(n)
VJ ,
(1.26)
|n| |n| |n| J = 2 , 2 +1, 2 +2,... (n)
with V J the spin J representation space (for the right action) of Uq (su(2)). Combined with (1.16), we get a Peter-Weyl decomposition for A(SUq (2)) [36]. A PBW-basis for A(SUq (2)) is given by monomials a m ck c∗ l for k, l = 0, 1, . . . and m ∈ Z, with the convention that a −m is short-hand notation for a ∗ m when m > 0. Furthermore, a similar basis for Ln is given by the monomials a l−k ck c∗ l+n , since from (1.3) it follows that K (a m ck c∗ l ) = q (−m−k+l)/2 a m ck c∗ l and the requirement that −m − k + l = n is met by redefining l → l + n, forcing in turn m = l − k. In particular, the monomials a l−k ck c∗ l are the only K -invariant elements, thus providing a PBW-basis for L0 = A(CPq1 ).
374
G. Landi, R. J. Szabo
2. SUq (2)-Invariant Gauge Fields on the Quantum Projective Line We will now describe connections on the quantum projective line. For this, we will need an explicit description of the calculi on the quantum principal bundle over CPq1 . The principal bundle (A(SUq (2)), A(CPq1 ), A(U(1))) is endowed with compatible nonuniversal calculi [10,11] obtained from the three-dimensional left-covariant calculus on SUq (2) [36], which we present first. We then describe the unique left-covariant twodimensional calculus on the quantum projective line CPq1 [29] obtained by restriction, and also the projected calculus on the structure group U(1). The calculus on CPq1 can be canonically decomposed into a holomorphic and an anti-holomorphic part. All the calculi are compatible in a natural sense. These constructions will produce a connection on the quantum principal bundle over CPq1 with respect to the left-covariant calculus • (CPq1 ), also with a natural holomorphic structure. This connection will determine a covariant derivative on the module of equivariant elements Ln , which can be shown [19] to correspond to the canonical Grassmann connection on the associated projective modules over A(CPq1 ). We also briefly recall how to compute the monopole number n ∈ Z by means of a Fredholm module. On the other hand, to integrate the gauge curvature one needs a ‘twisted integral’ and the result is no longer an integer but rather its q-analogue. 2.1. Left-covariant forms on SUq (2). The first differential calculus we take on the quantum group SUq (2) is the left-covariant calculus developed in [36]. It is three-dimensional, such that its quantum tangent space is generated by the three elements Xz =
1 − K4 , 1 − q −2
X − = q −1/2 F K
and
∗ X + = q 1/2 E K = X − .
Their coproducts and antipodes are easily found to be (X z ) = 1 ⊗ X z + X z ⊗ K 4
and
S(X z ) = −X z K
−4
(X ± ) = 1 ⊗ X ± + X ± ⊗ K 2 , S(X ± ) = −X ± K
and
−2
.
(2.1) (2.2)
The dual space of one-forms 1 (SUq (2)) has a basis βz = a ∗ da + c∗ dc,
β− = c∗ da ∗ − q a ∗ dc∗
and
β+ = a dc − q c da (2.3)
of left-invariant forms. The differential d : A(SUq (2)) → 1 (SUq (2)) is given by d f = (X − f ) β− + (X + f ) β+ + (X z f ) βz
(2.4)
for all f ∈ A(SUq (2)). If (1) is the (left) coaction of A(SUq (2)) on itself extended to forms, the left-coinvariance of the basis forms is the statement that (1) (βs ) = 1 ⊗ βs , while the left-covariance of the calculus is stated as ( ⊗ id) ◦ (1) = (1) ⊗ id ◦ (1) and
(2.5)
( ⊗ id) ◦ (1) = 1.
Dimensional Reduction Over the Quantum Sphere
375
The requirement that it is a ∗-calculus, i.e. d( f ∗ ) = (d f )∗ , yields ∗ β− = −β+
and
βz∗ = −βz .
The bimodule structure is given by βz a = q −2 a βz , βz a ∗ = q 2 a ∗ βz , β± a = q −1 a β± and β± a ∗ = q a ∗ β± ,
βz c = q −2 c βz , βz c∗ = q 2 c∗ βz , β± c = q −1 c β± and β± c∗ = q c∗ β± . (2.6)
Higher degree forms can be defined in a natural way by requiring compatibility with the commutation relations (the bimodule structure (2.6)) and that d2 = 0. One has dβz = −β− ∧ β+ , dβ+ = q 2 1 + q 2 βz ∧ β+ and dβ− = −q −2 1 + q 2 βz ∧ β− (2.7) together with the commutation relations β+ ∧ β+ = β− ∧ β− = βz ∧ βz = 0, β− ∧ β+ + q −2 β+ ∧ β− = 0, βz ∧ β− + q 4 β− ∧ βz = 0,
(2.8)
βz ∧ β+ + q −4 β+ ∧ βz = 0. Finally, there is a unique top form β− ∧ β+ ∧ βz . We may summarize the above results as follows. Proposition 2.9. For the three-dimensional left-covariant differential calculus on SUq (2), the bimodules of forms are all trivial (left) A(SUq (2))-modules given explicitly as 0 (SUq (2)) = A(SUq (2)), 1 (SUq (2)) = A(SUq (2)) β− , β+ , βz , 2 (SUq (2)) = A(SUq (2)) β− ∧ β+ , β− ∧ βz , β+ ∧ βz , 3 (SUq (2)) = A(SUq (2)) β− ∧ β+ ∧ βz . The exterior differential and commutation relations are obtained from (2.7) and (2.8), whereas the bimodule structure is obtained from (2.6). 2.2. Holomorphic forms on CPq1 . The restriction of the three-dimensional calculus of §2.1 to the quantum projective line CPq1 yields the unique left-covariant two-dimensional calculus on CPq1 [22]. Further development of this approach has led to a description of this calculus in terms of a Dirac operator [32]. The ‘cotangent bundle’ 1 (CPq1 ) is shown to be isomorphic to the direct sum L−2 ⊕ L+2 of the line bundles with degree (monopole charge) ± 2. Since the element K acts as the identity on A(CPq1 ), the differential (2.4) restricted to A(CPq1 ) becomes d f = (X − f ) β− + (X + f ) β+ = q −1/2 (F f ) β− + q 1/2 (E f ) β+
376
G. Landi, R. J. Szabo
for f ∈ A(CPq1 ). This leads to a decomposition of the exterior differential into a holomorphic and an anti-holomorphic part, d = ∂ + ∂, with ∂ f = (X − f ) β−
∂ f = (X + f ) β+
and
for f ∈ A(CPq1 ). An explicit computation on the generators (1.10) of CPq1 yields ∂ B− = −q −1 a 2 β− , ∂ B+ = q a
∗2
∂ B0 = −q −1 c a β− β+ ,
∗
∗
∂ B0 = c a β+
and and
∂ B+ = c2 β− ,
∂ B− = −q 2 c∗ 2 β+ .
It follows that 1 (CPq1 ) = 0,1 (CPq1 ) ⊕ 1,0 (CPq1 ), where 0,1 (CPq1 ) L−2 β− ∂(A(CPq1 )) is the A(CPq1 )-bimodule generated by ∂ B− , ∂ B0 , ∂ B+ = a 2 , c a , c2 β− = q 2 β− a 2 , c a , c2 and 1,0 (CPq1 ) L+2 β+ ∂(A(CPq1 )) is the A(CPq1 )-bimodule generated by ∂ B+ , ∂ B0 , ∂ B− = a ∗ 2 , c∗ a ∗ , c∗ 2 β+ = q −2 β+ a ∗ 2 , c∗ a ∗ , c∗ 2 . That these two modules of forms are not free is also expressed by the existence of relations among the differentials given by ∂ B0 − q −2 B− ∂ B+ + q 2 B+ ∂ B− = 0
and
∂ B0 − B+ ∂ B− + q −4 B− ∂ B+ = 0.
The two-dimensional calculus on CPq1 has then quantum tangent space generated by the two elements X + and X − (or, equivalently F and E). It has a unique (up to scale) top invariant form β, which is central, β f = f β for all f ∈ A(CPq1 ), and 2 (CPq1 ) is the free A(CPq1 )-bimodule generated by β, i.e. one has 2 (CPq1 ) = βA(CPq1 ) = A(CPq1 )β. Both β± commute with elements of A(CPq1 ) and so does β− ∧ β+ , which may be taken as the natural generator β = β− ∧ β+ of 2 (CPq1 ) (cfr. [22] or [32, App.]). Writing any one-form as α = x β− + y β+ ∈ L−2 β− ⊕ L+2 β+ , the product of one-forms is given by (x β− + y β+ ) ∧ (t β− + z β+ ) = x z − q 2 y t β− ∧ β+ . By (2.7) it is natural (and consistent) to demand dβ− = dβ+ = 0 when restricted to CPq1 . Then the exterior derivative of any one-form α = x β− + y β+ ∈ L−2 β− ⊕ L+2 β+ is dα = d(x β− + y β+ )
= ∂ x ∧ β− + ∂ y ∧ β+ = X − y − q 2 X + x β− ∧ β+ ,
(2.10)
since K acts as q ± 1 on L± 2 . Notice that in (2.10), both X + x and X − y belong to A(CPq1 ), as they should. We may summarize these results as follows. Proposition 2.11. The two-dimensional differential calculus on the quantum projective line CPq1 is given by • (CPq1 ) = A(CPq1 ) ⊕ (L−2 β− ⊕ L+2 β+ ) ⊕ A(CPq1 )β− ∧ β+ . Moreover, the splitting 1 (CPq1 ) = 1,0 (CPq1 )⊕ 0,1 (CPq1 ), together with the two maps ∂ and ∂ given above, constitute a complex structure for the differential calculus.
Dimensional Reduction Over the Quantum Sphere
377
A Hodge operator at the level of one-forms is constructed in [22] via a left-covariant map ˆ : 1 (CPq1 ) → 1 (CPq1 ) which squares to the identity id. In the description of the calculus as given in Proposition 2.11, it is defined by ˆ (∂ f ) = ∂ f and ˆ ∂ f = −∂ f (2.12) for all f ∈ A(CPq1 ). One then demonstrates its compatibility with the bimodule structure, i.e. the map ˆ is a bimodule map. Thus ˆ has values ± 1 on holomorphic or antiholomorphic one-forms respectively, i.e. one has ˆ = ± id on 1,0 (CPq1 ) or 0,1 (CPq1 ) respectively. In particular, ˆ β± = ± β± . The calculus has one central top two-form and the Hodge operator is naturally extended by requiring ˆ 1 = β− ∧ β+
and
ˆ (β− ∧ β+ ) = 1.
(2.13)
We conclude this section by mentioning the calculus on U(1) which makes all three calculi compatible from the quantum principal bundle point of view. The strategy [10] consists in defining the calculus on the coordinate algebra A(U(1)) via the Hopf projection π in (1.5). One finds that the projected calculus is one-dimensional and bicovariant. Its quantum tangent space is generated by X = Xz =
1 − K4 1 − q −2
(2.14)
with dual one-form given by βz . Explicitly, one finds βz = z ∗ dz,
dz = z βz
and
dz ∗ = −q 2 z ∗ βz
along with the noncommutative commutation relations βz z = q −2 z βz ,
βz z ∗ = q 2 z ∗ βz
and
z dz = q 2 dz z.
The data (A(SUq (2)), A(CPq1 ), A(U(1))) defines a ‘topological’ quantum principal bundle. There are differential calculi both on the total space A(SUq (2)) (the threedimensional left-covariant calculus) and on A(U(1)) (obtained from it via the same projection π in (1.5) giving the bundle structure). Moreover, from the calculus on A(SUq (2)) one also obtains by restriction a calculus on the base space A(CPq1 ). The three calculi are compatible with the bundle structure [10] (see also [19]), thus constructing a quantum principal bundle with non-universal calculi. The vector field X z is vertical for the fibration. 2.3. Connections on equivariant line bundles over CPq1 . The most efficient way to define a connection on a quantum principal bundle (with given calculi) is by decomposing the one-forms on the total space into horizontal and vertical forms [10,11]. Since horizontal one-forms are given in the structure group of the principal bundle, one needs a projection onto forms whose range is the subspace of vertical one-forms. The projection is required to be covariant with respect to the right coaction of the structure Hopf algebra. For the principal bundle over the quantum projective line CPq1 that we are considering, a principal connection is a covariant left module projection : 1 (SUq (2)) → 1ver (SUq (2)), i.e. 2 = and (x α) = x (α) for α ∈ 1 (SUq (2)) and
378
G. Landi, R. J. Szabo
x ∈ A(SUq (2)). Equivalently, it is a covariant splitting 1 (SUq (2)) = 1ver (SUq (2)) ⊕ 1hor (SUq (2)). The covariance of the connection is the requirement that (1) α (1) R ◦ = ◦ αR , (1)
with α R the extension to one-forms of the action α R in (1.7)–(1.8) of the structure Hopf algebra U(1). It is not difficult to see that with the left-covariant three-dimensional calculus on A(SUq (2)), a basis for 1hor (SUq (2)) is given by β− , β+ . Furthermore, one has α (1) R (βz ) = βz ,
∗2 α (1) R (β− ) = β− z
and
2 α (1) R (β+ ) = β+ z ,
and so a natural choice of connection = z is to define βz to be vertical [10,22], whence z (βz ) := βz
and
z (β± ) := 0.
With a connection, one has a covariant derivative acting on right A(CPq1 )-modules E of equivariant elements, ∇ := (id −z ) ◦ d : E −→ E ⊗A(CPq1 ) 1 (CPq1 ), and one readily proves the Leibniz rule ∇(ϕ · f ) = (∇ϕ) · f + ϕ ⊗ d f for all ϕ ∈ E and f ∈ A(CPq1 ). We shall take for E the line bundles Ln of (1.17). Then with the left-covariant two-dimensional calculus on A(CPq1 ) (coming from the left-covariant three-dimensional calculus on A(SUq (2)) as described in §2.2), we have ∇ϕ = (X + ϕ) β+ + (X − ϕ) β−
(2.15)
with X ± ϕ ∈ Ln±2 for ϕ ∈ Ln . Using Lemma 1.18 we conclude that ∇ϕ ∈ Ln−2 β− ⊕ Ln+2 β+ Ln ⊗A(CPq1 ) 1 (CPq1 ) as required. A generic covariant derivative on the module Ln is of the form ∇α = ∇ + α, with α an element in HomA(CPq1 ) (Ln , Ln ⊗A(CPq1 ) 1 (CPq1 )). For later use it is helpful to characterize this space. More generally, from Lemma 1.18 we can infer the following results. Lemma 2.16. For any n ∈ Z one has HomA(CPq1 ) Ln , Ln ⊗A(CPq1 ) 1 (CPq1 ) L−2 β− ⊕ L+2 β+ = 1 (CPq1 ),
(2.17)
while for any two distinct integers n, m ∈ Z one has HomA(CPq1 ) Ln , Lm ⊗A(CPq1 ) 1 (CPq1 ) Lm−n−2 β− ⊕ Lm−n+2 β+ Lm−n ⊗A(CPq1 ) 1 (CPq1 ).
(2.18)
Given the connection, we can work out an explicit expression for its curvature, defined to be the A(CPq1 )-linear (by construction) map ∇ 2 := ∇ ◦ ∇ : Ln −→ Ln ⊗A(CPq1 ) 2 (CPq1 ).
Dimensional Reduction Over the Quantum Sphere
379
n be the connection on the line bundle Ln defined in (1.17), Proposition 2.19. Let ∇ given in (2.15) for the canonical left-covariant two-dimensional calculus on A(CPq1 ). Then, with ϕ ∈ Ln , its curvature is given by n2 ϕ = (X z ϕ) β− ∧ β+ . ∇
(2.20)
As an element in HomA(CPq1 ) (Ln , Ln ⊗A(CPq1 ) 2 (CPq1 )), one has n2 = −q n+1 [n] β− ∧ β+ . ∇
(2.21)
Proof. Using (2.15), (2.8) and the fact that dβ± = 0 on CPq1 , by the Leibniz rule we have n ϕ = (X − X + ϕ) β− ∧ β+ + (X + X − ϕ) β+ ∧ β− n ∇ ∇ = (X − X + − q 2 X + X − ) ϕ β− ∧ β+ , and (2.20) follows from the relation X − X + − q 2 X + X − = X z . Since X z ϕ = −q n+1 [n] ϕ for ϕ ∈ Ln , one has (2.21). Since X z A(CPq1 ) = 0, the curvature is A(CPq1 )-linear. We can also derive an explicit expression for the corresponding gauge potential an defined by ϕ an = ∇ϕ − dϕ for ϕ ∈ Ln . With X z the vertical vector field in (2.14), using (2.4) and (2.15) we find ϕ an = − (X z ϕ) βz = q n+1 [n] ϕ βz , or an = q n+1 [n] βz .
(2.22)
As usual, an is not defined on CPq1 but rather on the total space SUq (2) of the bundle, i.e. an ∈ HomA(CPq1 ) (Ln , Ln ⊗A(CPq1 ) 1 (SUq (2))). In terms of the gauge potential, the curvature is given by n2 = d an fn := ∇
(2.23)
as a direct consequence of the first identity in (2.7). 2.4. Holomorphic structures. The connection given in §2.3 can be naturally decomposed into a holomorphic and an anti-holomorphic part, ∇ = ∇ ∂ + ∇ ∂ . They are given by ∇ ∂ ϕ = (X + ϕ) β+
and
∇ ∂ ϕ = (X − ϕ) β−
with the corresponding Leibniz rules ∇ ∂ (ϕ · f ) = ∇ ∂ ϕ · f + ϕ ⊗ ∂ f
and
∇ ∂ (ϕ · f ) = ∇ ∂ ϕ · f + ϕ ⊗ ∂ f,
(2.24)
for all ϕ ∈ Ln and f ∈ A(CPq1 ). They are both flat, i.e. (∇ ∂ )2 = 0 = (∇ ∂ )2 , and so the connection ∇ is integrable. Holomorphic ‘sections’ are elements ϕ ∈ Ln which satisfy ∇ ∂ ϕ = 0.
380
G. Landi, R. J. Szabo
From the actions given in (1.3) we see that F a s = 0 and F cs = 0 for any s ∈ N0 , while F a ∗ s = 0 and F c∗ s = 0 for any s ∈ N. Then, from the expressions (1.20) for generic equivariant elements, we see that there are no holomorphic elements in Ln for n > 0. On the other hand, for n ≤ 0 the elements c|n|−μ a μ , μ = 0, 1, . . . , |n| are holomorphic, ∇ ∂ c|n|−μ a μ = 0. Since ker ∂ = C (as only the constant functions on CPq1 do not contain the generator a ∗ or c∗ ), so that the only holomorphic functions on CPq1 are the constants, these are the only invariants in degree n. We may conclude that holomorphic equivariant elements are all polynomials in two variables a, c with the commutation relation a c = q c a, which defines the coordinate algebra of the quantum plane. Further aspects of these holomorphic structures are reported in [17]. 2.5. Unitarity and gauge transformations. On each line bundle Ln , n ≥ 0 there is an A(CPq1 )-valued hermitian structure hˆ n : Ln × Ln −→ A(CPq1 ) defined by ⎛ ⎞ n n n c∗ μ a ∗ n−μ f μ , c∗ ν a ∗ n−ν gν ⎠ = f μ∗ a n−μ cμ c∗ μ a ∗ n−μ gμ hˆ n ⎝ μ=0
ν=0
μ=0
(2.25) A(CPq1 )-module
in the basis (1.19)–(1.20). Having taken the right A(CPq1 )-module structure for Ln , the hermitian structure (2.25) is right A(CPq1 )-linear and left A(CPq1 )antilinear. It is covariant under the natural left coaction of A(SUq (2)) on Ln induced by the inclusion Ln ⊂ A(SUq (2)). There is an analogous formula for n ≤ 0. By composing hˆ n with the Haar functional of A(SUq (2)) restricted to A(CPq1 ), one obtains a C-valued inner product on Ln . Since the Haar functional of A(SUq (2)) is invariant under the coaction of A(SUq (2)) on itself [18, §4.2.6], we get an SUq (2)-invariant inner product on each Ln . If we write elements ϕ ∈ Ln as vector-valued functions ϕ = (ϕμ , μ = 0, 1, . . . , |n|), the hermitian structure is simply hˆ n (ϕ, ψ) = μ ϕμ∗ ψμ . n is unitary, i.e. it is compatible with the hermitian Lemma 2.26. The connection ∇ structure hˆ n , n ϕ , ψ + hˆ n ϕ , ∇ n ψ = d hˆ n (ϕ, ψ) hˆ n ∇ for any ϕ, ψ ∈ Ln . Proof. On the one hand, d hˆ n (ϕ, ψ) = X + hˆ n (ϕ, ψ) β+ + X − hˆ n (ϕ, ψ) β− . Using the coproducts (2.1) we have
|n| |n| ∗ ϕμ X ± ψμ + X ± ϕμ∗ K 2 ψμ X ± hˆ n (ϕ, ψ) = X ± ϕμ∗ ψμ = μ=0
μ=0
=
|n| ϕμ∗ X ± ψμ + q n X ± ϕμ∗ ψμ ,
μ=0
Dimensional Reduction Over the Quantum Sphere
381
and in turn |n| ∗ ˆ ϕμ X ± ψμ + q n X ± ϕμ∗ ψμ β± . d h n (ϕ, ψ) =
±
μ=0
∗ = −β we have On the other hand, using the antipodes (2.2) and β± ± |n| |n| ∗ ∗ n ϕ , ψ = hˆ n ∇ β± X ± ϕμ ψμ = q n q ∓ 2 β± X ± ϕμ∗ ψμ ±
±
μ=0
= qn
μ=0
|n| ±
μ=0
X ± ϕμ∗ ψμ β± ,
and in turn |n| n ˆh n ∇ n ϕ , ψ + hˆ n ϕ , ∇ n ψ = q X ± ϕμ∗ ψμ + ϕμ∗ X ± ψμ β± . ±
μ=0
A direct comparison now gives the result.
We already know that any other connection is written as ∇α = ∇ + α with α a generic element in HomA(CPq1 ) (Ln , Ln ⊗A(CPq1 ) 1 (CPq1 )) which, for a unitary connection ∇, is necessarily anti-hermitian, hˆ n (αϕ, ψ) + hˆ n (ϕ, αψ) = 0
for ϕ, ψ ∈ Ln .
Lemma 2.27. Unitary elements α ∈ HomA(CPq1 ) (Ln , Ln ⊗A(CPq1 ) 1 (CPq1 )) are of the form α = x β− + q 2 x ∗ β+ = x β− − (x β− )∗ , with x a generic element in L−2 . Proof. From the identification (2.17), we seek elements in 1 (CPq1 ) = L−2 β− ⊕ L+2 β+ which are unitary. It is straightforward to verify that a generic one-form α = x− β− + x+ β+ with x∓ ∈ L∓ 2 is unitary with respect to the hermitian structure hˆ n if and only if it is written as claimed. The group U(Ln ) of gauge transformations consists of unitary elements in EndA(CPq1 ) (Ln ) (with respect to the hermitian structure hˆ n ). It acts on a connection ∇ by (u, ∇) −→ ∇ u = u ◦ ∇ ◦ u ∗ . An arbitrary connection ∇α = ∇ + α will then transform to (∇α )u = ∇ + α u with α u = u (∇u ∗ ) + u α u ∗ . We know from Lemma 1.18 that EndA(CPq1 ) (Ln ) L0 = A(CPq1 ). Thus U(Ln ) consists of unitary elements in the coordinate algebra A(CPq1 ). Of these there are none which are nontrivial. Indeed, in the coordinate algebra of A(SUq (2)) there are no nontrivial invertible elements [15, App.]. Since A(CPq1 ) is a subalgebra of the latter, it cannot contain any nontrivial invertible (hence unitary) elements either.
382
G. Landi, R. J. Szabo
2.6. SUq (2)-invariant connections and gauge transformations. Recall that there is a coaction (1.14) of A(SUq (2)) on modules of sections. Let us denote by (n) the coaction on Ln , (n) : Ln −→ A(SUq (2)) ⊗ Ln ,
(n) (ϕ) = ϕ(−1) ⊗ ϕ(0) ,
(2.28)
(1)
with implicit summation as usual. By combining it with the coaction L of A(SUq (2)) on the bimodule of one-forms 1 (CPq1 ) we get an analogous coaction (1) (n) : Ln ⊗A(CPq1 ) 1 (CPq1 ) −→ A(SUq (2)) ⊗ Ln ⊗A(CPq1 ) 1 (CPq1 ) , (1) (n) (ω) = ω(−1) ⊗ ω(0) .
(2.29)
Next we give an ‘adjoint’ coaction of A(SUq (2)) on the space C(Ln ) of unitary connections, C : C(Ln ) −→ A(SUq (2)) ⊗ C(Ln ), defined by (1) C (−) = m 12 ◦ id ⊗(n) ◦ id ⊗(−) ◦ S ⊗ id ◦ (n) with m 12 the multiplication in the first two factors of the tensor product and S the antipode. Thinking of C (−) as acting on 1 ⊗ ϕ with ϕ ∈ Ln , and using (2.28), we get the ‘explicit’ expression C (∇α )(ϕ) = S ϕ(−1) ∇α (ϕ(0) ) (−1) ⊗ ∇α (ϕ(0) ) (0) . n in (2.15) is the unique invariant connection Lemma 2.30. The canonical connection ∇ for this coaction, i.e. the unique element ∇ ∈ C(Ln ) for which C (∇) = 1 ⊗ ∇. In particular, there is no non-trivial element in HomA(CPq1 ) (Ln , Ln ⊗A(CPq1 ) 1 (CPq1 )) which is invariant. n is most easily seen from Proof. The left-coinvariance of the canonical connection ∇ the corresponding gauge potential in (2.22). This is clearly left-coinvariant from the properties (2.5) of the basis one-forms, and in particular of βz . Since a unitary element α ∈ HomA(CPq1 ) (Ln , Ln ⊗A(CPq1 ) 1 (CPq1 )) is of the form given in Lemma 2.27, it is evident that α = 0 is the only such left-invariant element. An ‘adjoint’ coaction of A(SUq (2)) on the group U(Ln ) of gauge transformations, U : U(Ln ) −→ A(SUq (2)) ⊗ U(Ln ), WOULD be defined analogously as above by U (−) = m 12 ◦ id ⊗(n) ◦ id ⊗(−) ◦ S ⊗ id ◦ (n) , and thinking of U (−) as acting on 1 ⊗ ϕ, with ϕ ∈ Ln , one has. U (u)(ϕ) = S ϕ(−1) u(ϕ(0) ) (−1) ⊗ u(ϕ(0) ) (0) .
Dimensional Reduction Over the Quantum Sphere
383
In fact, we already know that U(Ln ) consists of unitary elements in A(CPq1 ). Then, the A(SUq (2))-coaction U is just the restriction to U(Ln ) of the canonical A(SUq (2))coaction on A(CPq1 ) given in (1.12). Also, as U(Ln ) is made only of complex numbers of modulus one, the following result is immediate. Lemma 2.31. The element 1 ∈ U(Ln ) is the unique invariant gauge transformation for this coaction, i.e. the unique element u ∈ U(Ln ) for which U (u) = 1 ⊗ u. This also follows from Proposition 1.12 giving 1 as the only SUq (2)-invariant element in the algebra A(CPq1 ). 2.7. K-theory charges. The line bundles on the sphere CPq1 described in §1.2 are classified by their monopole number n ∈ Z. One writes Ln = p(n) (A(CPq1 ))|n|+1 with suitable projections p(n) in Mat |n|+1 (A(CPq1 )). They are given explicitly by p(n) μν =
⎧√ n−μ a μ a ∗ ν c∗ n−ν , ⎪ ⎨ αn,μ αn,ν c
n≥0 (2.32)
⎪ ⎩ β
∗ μ ∗ |n|−μ |n|−ν ν a a c , n,μ βn,ν c
n ≤ 0,
with μ, ν = 0, 1, . . . , |n| and the numerical coefficients αn,μ =
n−μ−1 j=0
1 − q 2(n− j) 1 − q 2( j+1)
and
βn,μ = q
2μ
μ−1 j=0
1 − q −2(|n|− j) . 1 − q −2( j+1)
We use the convention −1 j=0 (−) := 1. The projections in (2.32) are representatives of classes in the K-theory of CPq1 , i.e. [p(n) ] ∈ K0 (CPq1 ). One computes the corresponding monopole number by pairing them with a non-trivial element in the dual K-homology, i.e. with (the class of) a non-trivial Fredholm module [μ] ∈ K0 (CPq1 ). For this, one first calculates the corresponding Chern characters in the cyclic homology ch• (p(n) ) ∈ HC• (CPq1 ) and cyclic cohomology ch• (μ) ∈ HC• (CPq1 ) respectively, and then uses the pairing between cyclic homology and cohomology. The Chern character of the projections p(n) has a non-trivial component in degree zero ch0 (p(n) ) ∈ HC0 (CPq1 ) given simply by a (partial) matrix trace
ch0 (p(n) ) := tr(p(n) ) =
⎧ n μ−1 ⎪ ⎪ ∗ n−μ ⎪ α (c c) 1 − q 2 j c∗ c , ⎪ n,μ ⎪ ⎪ ⎪ j=0 ⎨ μ=0 ⎪ ⎪ |n| ⎪ ⎪ ⎪ ⎪ βn,μ (c∗ c)μ ⎪ ⎩ μ=0
|n|−μ−1 j=0
1 − q −2 j c∗ c ,
n≥0
n ≤ 0,
384
G. Landi, R. J. Szabo
and ch0 (p(n) ) ∈ A(CPq1 ). Dually, one needs a cyclic zero-cocycle, i.e. a trace on A(CPq1 ). This was obtained in [23] and it is a trace on A(CPq1 )/C, i.e. it vanishes on C ⊂ A(CPq1 ). On the other hand, its values on powers of the element c∗ c is given by −1 , k > 0. μ (c∗ c)k = 1 − q 2k The pairing was computed in [14] and results in [μ] , [p(n) ] := μ ch0 (p(n) ) = −n.
(2.33)
This integer is a topological quantity that depends only on the bundle, both over the quantum sphere and over its classical limit which is an ordinary two-sphere. In this limit it could also be computed by integrating the curvature two-form of any connection. However, in order to integrate the gauge curvature on the quantum sphere CPq1 one requires a ‘twisted integral’, and the result is no longer an integer but rather a q-integer. We recall here the main facts, referring to [19] for additional details. It is known [18, Prop. 4.15] that the modular automorphism associated with the Haar state H on the algebra A(SUq (2)) when restricted to the subalgebra A(CPq1 ) yields a faithful, invariant state on A(CPq1 ), i.e. H (a X ) = H (a) (X ) for a ∈ A(CPq1 ) and X ∈ Uq (su(2)), with modular automorphism ϑ(g) = g K 2
for g ∈ A(CPq1 ),
(2.34)
such that H (a b) = H ϑ(b) a
(2.35)
for a, b ∈ A(CPq1 ). With β− ∧ β+ the central generator of 2 (CPq1 ), H the Haar state on A(CPq1 ), and ϑ its modular automorphism in (2.34), it was proven in [32] that the linear functional CPq1
: 2 (CPq1 ) −→ C,
CPq1
a β− ∧ β+ := H (a)
(2.36)
defines a non-trivial ϑ-twisted cyclic two-cocycle τ on A(CPq1 ) given by τ (a0 , a1 , a2 ) :=
1 a0 da1 ∧ da2 . 2 CPq1
(2.37)
This means that bϑ τ = 0 and λϑ τ = τ , where bϑ is the ϑ-twisted coboundary operator (bϑ τ )( f 0 , f 1 , f 2 , f 3 ) := τ ( f 0 f 1 , f 2 , f 3 ) − τ ( f 0 , f 1 f 2 , f 3 ) + τ ( f 0 , f 1 , f 2 f 3 ) − τ ϑ( f 3 ) f 0 , f 1 , f 2 , and λϑ is the ϑ-twisted cyclicity operator
(λϑ τ )( f 0 , f 1 , f 2 ) := τ ϑ( f 2 ) , f 0 , f 1 .
The non-triviality means that there is no twisted cyclic one-cochain α on A(CPq1 ) such that bϑ α = τ and λϑ α = α, where here the operators bϑ and λϑ are defined by formulae
Dimensional Reduction Over the Quantum Sphere
385
like those above (and directly generalize in any degree). Thus τ is a class in HC2ϑ (CPq1 ), the degree two twisted cyclic cohomology of the quantum space CPq1 . In terms of the projections p(n) , the curvature (2.21) of the connection (2.15) is given by (n) dp(n) ∧ dp(n) = −q n+1 [n] p(n) β− ∧ β+ . F∇ n := p
(2.38)
Using the normalization H (1) = 1 for the Haar state on A(CPq1 ), its integral (2.36) is computed to be q −1 (2.39) trq F∇ n = −[n]. CPq1
Here trq stands for the twisted or ‘quantum’ trace defined as follows [35]. Given an element M ∈ Mat |n|+1 (A(CPq1 )), its (partial) quantum trace is the element trq (M) ∈ A(CPq1 ) defined by |n| M jl σ|n|/2 (K 2 ) , trq (M) := tr M σ|n|/2 (K 2 ) = j,l=0
lj
where σ|n|/2 (K 2 ) is the matrix formof the spin J = |n|/2 representation of the modular element K 2 . In particular, trq p(n) = q −n . The q-trace is ‘twisted’ by the automorphism ϑ, trq (M1 M2 ) = trq (M2 K 2 ) M1 = trq ϑ(M2 ) M1 . From the definition (2.37) of the ϑ-twisted cyclic two-cocycle τ and the expression (2.38) of the curvature F∇ n , the integral (2.39) is also found to coincide with the coupling of the cocycle τ to the projection p(n) as −1 (2.40) 2q τ ◦ trq p(n) , p(n) , p(n) = −[n]. The pairing in (2.33) is the index of the Dirac operator on CPq1 . In parallel, the pairing in (2.40) can be obtained [27,35] as the q-index of the same Dirac operator, i.e. the difference between the quantum dimensions of its kernel and cokernel computed using trq . Thus the q-integer (2.39) may be naturally regarded as a quantum Fredholm index computed from the pairing between the ϑ-twisted cyclic cohomology and the (Hopf U (su(2)) algebraic) SUq (2)-equivariant K-theory K0 q (CPq1 ) [27,35]. 3. Dimensional Reduction of Invariant Gauge Fields For a smooth manifold M, let M denote the quantum space CPq1 × M. By this we mean the family of quantum projective lines CPq1 ×{ p} CPq1 parametrized by points p ∈ M. Let A(M) = C ∞ (M) be the commutative algebra of smooth functions on M. Then the algebra of M is given by A( M ) := A(CPq1 ) ⊗ A(M). Using the connections on the quantum principal bundle over CPq1 given in §2, we will now construct invariant connections on SUq (2)-equivariant modules over the algebra A( M ) and describe their dimensional reduction over CPq1 .
386
G. Landi, R. J. Szabo
3.1. Dimensional reduction of SUq (2)-equivariant vector bundles. We start by giving a coaction of the quantum group SUq (2) on A( M ), by coacting trivially on A(M) and with the canonical coaction L on A(CPq1 ) given in (1.12). This gives a map defined by : A( M ) −→ A(SUq (2)) ⊗ A( M ), b ⊗ f −→ m 13 L (b) ⊗ (1 ⊗ f ) = b(−1) ⊗ b(0) ⊗ f
(3.1)
for b ∈ A(CPq1 ), f ∈ A(M), where we use the Sweedler-like notation L (b) = b(−1) ⊗ b(0) (with implicit summation), and m 13 denotes multiplication in the first and third factors of the tensor product. In parallel with the description (1.9) of the twosphere algebra A(CPq1 ) as the subalgebra of invariant elements in A(SUq (2)), there is an analogous description of the algebra A( M ) in terms of invariant elements in A(SUq (2))⊗A(M). For this, we let U(1) act trivially on A(M) with corresponding map ! αz : U(1) −→ Aut A(SUq (2)) ⊗ A(M) ,
! αz (x ⊗ f ) = αz (x) ⊗ f,
(3.2)
with αz the U(1)-action on A(SUq (2)) given in (1.7)–(1.8). It is then evident that
U(1) αz ( f ) = f . := f ∈ A(SUq (2)) ⊗ A(M) ! A( M ) = A(SUq (2)) ⊗ A(M) (3.3) It is also useful to regard the algebra A(M) itself as coming from A( M ) via a projection related to the map π in (1.5) that establishes the ‘quantum group’ A(U(1)) as a quantum subgroup of A(SUq (2)). Indeed, by restricting π to the subalgebra A(CPq1 ) ⊂ A(SUq (2)) one gets a one-dimensional representation π : A(CPq1 ) −→ C,
π(B− ) = π(B+ ) = π(B0 ) = 0
(3.4)
on the generators and π(1) = 1, which is none other than the counit restricted to A(CPq1 ). We then have a surjective algebra homomorphism ! π = π ⊗ id : A( M ) −→ A(M),
x ⊗ f −→ (x) f.
(3.5)
A right A( M )-module E is said to be SUq (2)-equivariant if it carries a left coaction δ : E −→ A(SUq (2)) ⊗ E of the Hopf algebra A(SUq (2)) which is compatible with the coaction of A(SUq (2)) on A( M ), δ(ϕ · f ) = δ(ϕ) · ( f )
for all ϕ ∈ E , f ∈ A( M ).
Similarly, one defines SUq (2)-equivariant left A( M )-modules. The remainder of this section is devoted to relating A(SUq (2))-equivariant bundles E on the quantum space M to U(1)-equivariant bundles E over the manifold M. Let E → M be a smooth, U(1)-equivariant complex vector bundle, with U(1) acting trivially on M. This induces an action ρ of the group U(1) on the (right) A(M)-module E = C ∞ (M, E) of smooth sections of the bundle E, making it U(1)-equivariant. By the classical Serre–Swan theorem, the module E is a finitely-generated (right) projective
Dimensional Reduction Over the Quantum Sphere
387
module over A(M). Consider now the space E of equivariant elements, generalizing those in (1.13), given by
E = A(SUq (2)) ρ E := ϕ ∈ A(SUq (2)) ⊗ E (α ⊗ id)ϕ = (id ⊗ρ −1 ) ϕ . (3.6) There is a natural SUq (2)-equivariance. Again the left coaction of A(SUq (2)) on itself extends naturally to a left coaction on A(SUq (2)) ρ E given by ρ = ⊗ id : A(SUq (2)) ρ E −→ A(SUq (2)) ⊗ A(SUq (2)) ρ E . This coaction is naturally compatible with the corresponding SUq (2)-coaction in (3.1). The space (3.6) is an A( M )-bimodule. Any ϕ ∈ A(SUq (2)) ρ E can be written as ϕ = ϕ (1) ⊗ ϕ (2) with ϕ (1) ∈ A(SUq (2)) and ϕ (2) ∈ E (and an implicit sum understood). Then the bimodule structure is given as (b ⊗ f ) ϕ (1) ⊗ ϕ (2) = b ϕ (1) ⊗ f ϕ (2) and (1) ϕ ⊗ ϕ (2) (b ⊗ f ) = ϕ (1) b ⊗ ϕ (2) f for b ⊗ f ∈ A(CPq1 ) ⊗ A(M). As a right (or left) A( M )-module, it is finitely-generated and projective when it is defined with the tensor product of modules E which are finitely-generated and projective, respectively. Conversely, let E be a finitely-generated SUq (2)-equivariant right (or left) projective A( M )-module. The surjective algebra homomorphism ! π : A( M ) → A(M) in (3.5) (together with the quantum group surjection in (1.5)) induces a map sending A( M )modules to A(M)-modules, with a residual coaction of the ‘quantum group’ A(U(1)) which is trivial on A(M). From E we obtain one such module E, such that the coaction of A(U(1)) is an action of U(1) on E. Again by the Serre–Swan theorem, E is the A(M)-module of smooth sections E = C ∞ (M, E) of a complex vector bundle E → M which is equivariant with respect to the action of U(1) lifting the trivial action on M. An alternative way to understand this correspondence between SUq (2)-equivariant modules over A( M ) and U(1)-equivariant bundles over M is as follows. Given p ∈ M, consider the evaluation map ev p : A(M) → C defined by ev p ( f ) = f ( p) for f ∈ A(M). By U(1)-equivariance, it induces a surjective algebra homomorphism ev p : A( M ) → A(CPq1 ). Let E be a finitely-generated SUq (2)-equivariant projective right (or left) A( M )-module. Then the surjection ev p induces a finitely-generated SUq (2)equivariant projective right (or left) module E p over A(CPq1 ). We may in this way regard E also as a family of finitely-generated SUq (2)-equivariant projective right (or left) A(CPq1 )-modules E p of the type described in §1.2, parametrized by points p ∈ M. The module E p is in correspondence with the representations of U(1) via the construction of §1.2, and admits a decomposition (1.22) into irreducible rank one modules (1.17). We are now ready to formulate the fundamental statement of dimensional reduction, which will enable us to think of E = A(SUq (2)) ρ E as the module of sections of an SUq (2)-equivariant vector bundle on CPq1 × M. We begin with the following preliminary decomposition. Lemma 3.7. Let M be a smooth manifold with trivial U(1)-action. Let Cn , n ∈ Z, denote the irreducible U(1)-module of weight n as given in (1.15). Then every U(1)-equivariant A(M)-bimodule E is isomorphic to a finite direct sum Cn ⊗ En , (3.8) E n∈W (E )
388
G. Landi, R. J. Szabo
where W (E) ⊂ Z is the set of eigenvalues for the U(1)-action on E, and En are A(M)bimodules with trivial U(1)-coaction. If E is finitely-generated (resp. projective) then the modules En are also finitely-generated (resp. projective). Proof. Denote by Cn , with n ∈ Z, the A(M)-bimodule of sections of the trivial bundle over M with typical fibre Cn . It is naturally U(1)-equivariant. Using the decomposition (1.21) of a generic finite-dimensional representation (V, ρ) for U(1), the dual formulation of [1, Prop. 1.1] then gives a finite isotopical decomposition E Cn ⊗A(M) En , n∈W (E )
where W (E) ⊂ Z is the set of eigenvalues for the U(1)-action on E, so that
W (E) = n ∈ Z En = 0 are the weights of E, and En = HomU(1) (Cn , E) are A(M)-bimodules with trivial U(1)action. Since Cn is associated to the trivial bundle, it is of the form Cn Cn ⊗ A(M) and the decomposition (3.8) follows. Proposition 3.9. Every finitely-generated SUq (2)-equivariant projective bimodule E over A( M ) can be equivariantly decomposed, uniquely up to isomorphism, as E =
m i=0
Ei =
m
Lm−2i ⊗ Ei
(3.10)
i=0
for some m ∈ N0 , where Ei are bimodules of sections of smooth vector bundles E i over M with trivial SUq (2) coactions and Ln are bimodules (1.17) of sections of the SUq (2)-equivariant line bundles over CPq1 , together with morphisms i ∈ HomA( M ) ( E i−1 , E i ),
i = 1, . . . , m
of A( M )-bimodules. Proof. Since the U(1)-action on A(M) is trivial, by Lemma 3.7 we have that every U(1)-equivariant A(M)-bimodule F is isomorphic to a finite direct sum F C n ⊗ Fn , n∈W (F )
where W (F) = n ∈ Z Fn = 0 are the weights of F for the U(1)-action, and Fn = HomU(1) (Cn , F) are A(M)-bimodules with trivial U(1)-action. Putting this together with the decomposition (1.22) in terms of the line bundles Ln , we arrive at a decomposition for the corresponding induced bimodule over A( M ) given by F = A(SUq (2)) ρ F = Ln ⊗ Fn .
n∈W (F )
This decomposition describes the U(1)-action on F . The rest of the left SUq (2)-coaction is incorporated by using the dual right Uq (su(2))-action. From (1.25) the latter leaves each line bundle Ln alone but this is not the case for the bimodules Fn . From relations (1.2) the right action of E sends Ln ⊗ Fn to Ln ⊗ Fn−2 with corresponding ϕn : Fn →
Dimensional Reduction Over the Quantum Sphere
389
Fn−2 that are A(M)-bimodule morphisms. In particular, every indecomposable bimodule F has weight set of the form W ( F ) = {m − , m − +2, . . . , m + −2, m + } consisting of consecutive even or odd integers. By defining m = 21 (m + − m − ), E = L−m − −m ⊗ F , Ei = Fm + −2i , and E i = Lm−2i ⊗ Ei , we find that the K -action is given by (3.10), while the E-action is determined by a chain of A(M)-bimodule morphisms, φ1
φ2
φm
E0 −→ E1 −→ · · · −→ Em , with φi := ϕm + −2i . By fixing A(M)-valued hermitian structures h i : Ei × Ei → A(M) on the modules Ei , the action of F = E ∗ is given by the adjoint morphisms φi∗ in HomA(M) (Ei , Ei−1 ). 3.2. Covariant hermitian structures. We will now give a gauge theory formulation of the equivalence between the SUq (2)-equivariant bundles over A( M ) = A(CPq1 ) ⊗ A(M) and the module chains over A(M) described in Proposition 3.9. We first describe the reduction of SUq (2)-covariant hermitian structures on the SUq (2)-equivariant bimodules E of §3.1. On each line bundle Ln , there is the A(CPq1 )-valued hermitian structure defined in (2.25). Since we require an element in A(CPq1 ), any two modules Ln and Lm with m = n are taken to be orthogonal. Let E be a finitely-generated SUq (2)-equivariant projective right module over the algebra A( M ), with corresponding equivariant decomposition (3.10). On each A(M)module Ei in this decomposition we fix an A(M)-valued hermitian structure h i : Ei × Ei −→ A(M). Combined with (2.25) this gives an A( M )-valued hermitian structure on E i defined by h i = hˆ m−2i ⊗ h i : E i × E i −→ A(CPq1 ) ⊗ A(M), and in turn a left SUq (2)-covariant hermitian structure on E by h =
m
h i : E × E −→ A( M ).
(3.11)
i=0
By construction, the modules E i , i = 0, 1, . . . , m are SUq (2)-covariantly mutually orthogonal, i.e. h ( E i , E j ) = 0 for i = j. 3.3. Decomposition of covariant connections. Denote the left-covariant calculus on ˆ Let ( 1 (M), d) be the standard ∗-calculus A(CPq1 ) constructed in §2.2 by ( 1 (CPq1 ), d). on A(M), with 1 (M) the vector space of (complex) differential one-forms and d the usual de Rham exterior derivative on the smooth manifold M. Then we define a calculus ( 1 ( M ), d ) on A( M ) = A(CPq1 ) ⊗ A(M) by and d = dˆ ⊗ id + id ⊗d. 1 ( M ) = 1 (CPq1 ) ⊗ A(M) ⊕ A(CPq1 ) ⊗ 1 (M) Let E be a finitely-generated SUq (2)-equivariant projective right A( M )-module. Then we define 1 ( E ) = E ⊗A( M ) 1 ( M ),
390
G. Landi, R. J. Szabo
and from the equivariant decomposition (3.10), E = get a corresponding decomposition 1 ( E ) =
m
" i
Ei =
" i
Lm−2i ⊗ Ei , we
1 ( E i )
i=0
with 1 ( E i ) = E i ⊗A( M ) 1 ( M ) 1 (Lm−2i ) ⊗ Ei ⊕ Lm−2i ⊗ 1 (Ei ) , and obvious notations 1 (Lm−2i ) = Lm−2i ⊗A(CPq1 ) 1 (CPq1 ) and 1 (Ei ) = Ei ⊗A(M) 1 (M). A connection on the right A( M )-module E is given via a covariant derivative ∇ : E −→ 1 ( E ) obeying the Leibniz rule ∇ ϕ · (b ⊗ f ) = ( ∇ ϕ) · (b ⊗ f ) + ϕ ⊗A( M ) d (b ⊗ f ), for ϕ ∈ E and b ⊗ f ∈ A(CPq1 ) ⊗ A(M). The connection is unitary if in addition it is compatible with the hermitian structure h of §3.2, so that h ( ∇ ϕ, ψ) + h (ϕ, ∇ ψ) = d h (ϕ, ψ) (3.12) for ϕ, ψ ∈ E . Here the metric h is naturally extended to a map 1 ( E ) × 1 ( E ) → 2 ( M ) by the formulae h (ϕ ⊗A( M ) η, ψ) = η∗ h (ϕ, ψ)
h (ϕ, ψ ⊗A( M ) ξ ) = h (ϕ, ψ) ξ,
and
for ϕ, ψ ∈ E and η, ξ ∈ 1 ( M ), which respectively define metrics 1 ( E ) × E → 1 ( M ) and E × 1 ( E ) → 1 ( M ). For any p ≥ 0, the connection ∇ is extended to a C-linear map ∇ : p ( E ) → p+1 ( E ) by the graded Leibniz rule, where p( E ) =
m
p( E i )
i=0
with p ( E i ) = Lm−2i ⊗ p (Ei ) ⊕ 1 (Lm−2i ) ⊗ p−1 (Ei ) ⊕ 2 (Lm−2i ) ⊗ p−2 (Ei ) and 0 (Ei ) := Ei . As usual, for any two connections ∇ , ∇ their difference is an element ∇ − ∇ = A ∈ HomA( M ) E , 1 ( E ) , and if the connections are unitary then the ‘matrix of one-forms’ A is in addition antihermitian, h ( A ϕ, ψ) + h (ϕ, A ψ) = 0
for ϕ , ψ ∈ E .
Dimensional Reduction Over the Quantum Sphere
391
The collection of anti-hermitian elements in HomA( M ) ( E , 1 ( E )) will be denoted by HomaA( M ) ( E , 1 ( E )). The group U( E ) of gauge transformations consists of unitary elements in EndA( M ) ( E ), with respect to the hermitian structure h . It acts on a connection ∇ by (u, ∇ ) −→ ∇ u = u ◦ ∇ ◦ u ∗ , where here u acts implicitly as u ⊗A( M ) id 1 ( M ) . A connection ∇ then transform to ( ∇ A )u = ∇ + A u with
A
= ∇ + A will
A u = u ( ∇ u∗) + u A u∗. That each U(Ln ) S 1 , the complex numbers of modulus one, means that the part of a gauge transformation in U( E ) acting on the bundles Lm−2i is trivial. This fact will be used in §4.2 for the gauge invariance of the Yang–Mills action functional. Lemma 3.13. Any unitary connection ∇ on ( E , h ) decomposes as ∇ =
m
∇i +
i=0
β
ji
− β ∗ji
,
j
where: (1) Each ∇ i is a unitary connection on ( E i , h i ), i.e. h i ( ∇ i ϕ, ψ) + h i (ϕ, ∇ i ψ) = d h i (ϕ, ψ) (2) For j = i, β
ji
for ϕ, ψ ∈ E i .
∈ HomA( M ) ( E i , 1 ( E j )) is the adjoint of − β i j , i.e.
h ( β ji ϕ, ψ) + h (ϕ, β i j ψ) = 0
for ϕ ∈ E i , ψ ∈ E j .
Proof. Decompose the connection as ∇ =
m
∇ E
i=0
i
with ∇ E : E i −→ 1 ( E ) i
and
∇
Ei
=
m
β
ji
with β
ji
: E i −→ 1 ( E j ).
j=0
Then, since h ( E i , E j ) = 0 when i = j, the unitarity condition (3.12) for ∇ breaks into pieces giving the claimed decomposition with ∇ i = β ii . In a completely analogous way, one can decompose any given element A of the space HomaA( M ) ( E , 1 ( E )) as A =
m i=0
Ai +
j
A
ji
− A ∗ji
,
392
G. Landi, R. J. Szabo
with A i ∈ HomaA( M ) ( E i , 1 ( E i )) and A a decomposition
ji
∈ HomA( M ) ( E i , 1 ( E j )), leading to
m HomaA( M ) E , 1 ( E ) HomaA( M ) E i , 1 ( E i ) i=0
⊕
HomA( M ) E i , 1 ( E j ) .
j
" 3.4. i Ei = " SUq (2)-invariant connections and gauge transformations. On E = i Lm−2i ⊗ Ei we denote by E the coaction of the Hopf algebra A(SUq (2)) which combines the natural coaction of A(SUq (2)) on the modules Lm−2i given in (2.28) and the trivial coaction on the modules Ei , E : E −→ A(SUq (2)) ⊗ E , (1)
(1)
and by E its lift to 1 ( E ), E : 1 ( E ) → A(SUq (2)) ⊗ 1 ( E ). In complete parallel with §2.6 there are ‘adjoint’ coactions of A(SUq (2)) on the space C( E ) of unitary connections on E , on the group U( E ) of gauge transformations, as well as on the spaces HomA( M ) ( E i , E j ) and HomA( M ) ( E i , 1 ( E j )). We shall denote by C( E )SUq (2) , etc. the corresponding space of coinvariant elements, i.e.
C( E )SUq (2) = ∇ ∈ C( E ) C ( ∇ ) = 1 ⊗ ∇ , and similarly for the other spaces and coactions. The spaces C( E )SUq (2) and U( E )SUq (2) of invariant connections and gauge transformations are described in terms of objects defined on M and of canonical (and unique) objects defined on CPq1 . We begin with the space C( E )SUq (2) . Lemma 3.14. One has
HomaA( M ) ( E i , 1 ( E i ))
SUq (2)
= C ⊗ HomaA(M) (Ei , 1 (Ei )),
while for i = j one has ⎧ if j = i ± 1, ⎨0 SUq (2) ⎪ = Cβ− ⊗ HomA(M) (Ei , Ei−1 ) if j = i − 1, HomA( M ) ( E i , 1 ( E j )) ⎪ ⎩ Cβ ⊗ Hom + A(M) (Ei , Ei+1 ) if j = i + 1.
Proof. For any i, j one has HomA( M ) ( E i , 1 ( E j )) HomA(CPq1 ) (Lm−2i , Lm−2 j ) ⊗ HomA(M) (Ei , 1 (E j )) ⊕ HomA(CPq1 ) (Lm−2i , 1 (Lm−2 j )) ⊗ HomA(M) (Ei , E j )
Dimensional Reduction Over the Quantum Sphere
393
and, since SUq (2) coacts trivially on the bundles Ei , for the coinvariant elements one finds SUq (2) HomA( M ) ( E i , 1 ( E j )) SU (2) HomA(CPq1 ) (Lm−2i , Lm−2 j ) q ⊗ HomA(M) (Ei , 1 (E j )) SU (2) ⊕ HomA(CPq1 ) (Lm−2i , 1 (Lm−2 j )) q ⊗ HomA(M) (Ei , E j ) . In order to proceed we need only the fact that there are no non-trivial SUq (2)-invariant elements in the modules Ln for n = 0, while 1 is the only invariant element in the algebra L0 = A(CPq1 ). Then by Lemma 1.18 one has # SUq (2) SUq (2) C if 2i = 2 j, L2i−2 j = (3.15) HomA(CPq1 ) (Lm−2i , Lm−2 j ) 0 if 2i = 2 j. On the other hand, using (2.18) one finds SU (2) SU (2) HomA(CPq1 ) (Lm−2i , 1 (Lm−2 j )) q L2i−2 j−2 β− ⊕ L2i−2 j+2 β+ q ⎧ ⎪ if 2i − 2 j = ± 2, ⎨0 Cβ− if 2i − 2 j = 2, ⎪ ⎩ Cβ if 2i − 2 j = −2, + and the results now follow.
SUq (2) ∗ = −β , an element A ∈ Hom a 1 ( E )) Using Lemma 3.14 and β− ( E , + A( M ) can be written as A =
m ∗ 1 ⊗ Ai + β+ ⊗ φi+1 + β− ⊗ φi+1 ,
(3.16)
i=0
where the ‘gauge potentials’ Ai ∈ HomaA(M) (Ei , 1 (Ei )) and ‘Higgs fields’ φi+1 in the space HomA(M) (Ei , Ei+1 ) for i = 0, 1, . . . , m with Em+1 := 0. Now let ( E , h ) be an SUq (2)-equivariant hermitian A( M )-module decomposed as in (3.10) with the metric h decomposed as in (3.11). Let (Ei , h i ) for i = 0, 1, . . . , m be the hermitian A(M)-modules composing ( E , h ), and let C(Ei ) be the corresponding spaces of unitary connections. Proposition 3.17. There is a bijection between the spaces C( E )SUq (2) and C ( E ) :=
m C(Ei ) × HomA(M) (Ei , Ei+1 ) i=0
which associates to any element (∇, φ) of C ( E ), given by connections ∇i ∈ C(Ei ) and Higgs fields φi+1 ∈ HomA(M) (Ei , Ei+1 ) for i = 0, 1, . . . , m, the SUq (2)-invariant unitary connection ∇ ∈ C( E )SUq (2) given by ∇ =
m i=0
∗ ∇ i + β+ ⊗ φi+1 + β− ⊗ φi+1 .
(3.18)
394
G. Landi, R. J. Szabo
Here ∇ i is the unitary connection on ( E i , h i ) given by m−2i ⊗ id + id ⊗∇i , ∇i =∇ m−2i is the unique (by Lemma 2.30) SUq (2)-invariant unitary connection on where ∇ the hermitian line bundle (Lm−2i , hˆ m−2i ) given in (2.15) and (2.25). Proof. Fix a unitary connection ∇i0 ∈ C(Ei ) for each i = 0, 1, . . . , m. Define unitary m−2i ⊗id + id ⊗∇ 0 . They are clearly SUq (2)connections on each ( E i , h i ) by ∇ i0 = ∇ i invariant and give rise to an SUq (2)-invariant unitary connection ∇ 0 on ( E , h ) defined by the sum ∇ 0 = i ∇ i0 . All SUq (2)-invariant unitary connections ∇ on ( E , h ) SUq (2) . The general take the form ∇ = ∇ 0 + A with A ∈ HomaA( M ) ( E , 1 ( E )) form of such an element is given in (3.16), from which the expression (3.18) follows. Next we give a similar characterization of the space U( E )SUq (2) . Lemma 3.19. One has
HomA( M ) ( E i , E j )
SUq (2)
# =
C ⊗ EndA(M) (Ei ) if i = j, 0 if i = j.
Proof. For any i, j one has HomA( M ) ( E i , E j ) HomA(CPq1 ) (Lm−2i , Lm−2 j ) ⊗ HomA(M) (Ei , E j ) and, since SUq (2) coacts trivially on the bundles Ei , for the invariant elements one finds SUq (2) HomA( M ) ( E i , E j ) SU (2) HomA(CPq1 ) (Lm−2i , Lm−2 j ) q ⊗ HomA(M) (Ei , E j ). The result now follows from (3.15). In the same setting as in Proposition 3.17, let U(Ei ) be the group of gauge transformations corresponding to the hermitian A(M)-module (Ei , h i ). Proposition 3.20. There is a bijection between the groups U( E )SUq (2) and U ( E ) :=
m
U(Ei ),
i=0
which associates to any element u = (u 0 , u 1 , . . . , u m ) ∈ U ( E ) the SUq (2)-invariant gauge transformation of ( E , h ) given by u =
m
ui
i=0
with u i = 1 ⊗ u i ∈ U( E i )SUq (2) C ⊗ U(Ei ).
Dimensional Reduction Over the Quantum Sphere
395
Proof. This follows from Lemma 2.31 and Lemma 3.19.
The group U(Ei ) acts on both spaces HomA(M) (Ei , Ei+1 ) and HomA(M) (Ei+1 , Ei ) of Higgs fields by u i (φi+1 ) = φi+1 ◦ u i−1
and
∗ ∗ = u i ◦ φi+1 u i φi+1 ,
∗ ∈ Hom for u i ∈ U(Ei ) and φi+1 ∈ HomA(M) (Ei , Ei+1 ), φi+1 A(M) (Ei+1 , Ei ). There is also an induced action of the group U ( E ) on the space C ( E ) of connections. The following result is then immediate.
Proposition 3.21. The bijections between invariant connections and between invariant gauge transformations of Proposition 3.17 and Proposition 3.20, respectively, are compatible with the actions of the groups of Proposition 3.20 on the connections of Proposition 3.17, and there is a bijection between gauge orbits C( E )SUq (2)
U( E )SUq (2) ≡ C ( E ) U ( E ).
3.5. Integrable connections. In the sequel we will need to work with integrable connections as well. Let M be a complex manifold, with the standard complex structure for the complexified de Rham differential calculus. Combined with the complex structure for the differential calculus on A(CPq1 ) described in Proposition 2.11, we get a natural complex structure for the calculus on the algebra A( M ) = A(CPq1 ) ⊗ A(M). If ∇ is a connection on the A( M )-bimodule E with equivariant decomposition (3.10), then 0,2 is an element of HomA( M ) ( E , 0,2 ( E )), the (0, 2)-component of its curvature F ∇ where 0,2 ( E ) m 0,2 (Lm−2i ) ⊗ Ei ⊕ 0,1 (Lm−2i ) ⊗ 0,1 (Ei ) ⊕ Lm−2i ⊗ 0,2 (Ei ) . = i=0 0,2 = 0. In this case the pair ( E , ∇ ) is a The connection ∇ is then integrable if F ∇ holomorphic vector bundle [17, §2]. By (3.18) an SUq (2)-invariant unitary holomorphic connection on ( E , h ) is of the form
∇∂ =
m
∗ ∇ i∂ + β− ⊗ φi+1 ,
(3.22)
i=0
where ∇ i∂ is the holomorphic connection on ( E i , h i ) given by ∂m−2i ⊗ id + id ⊗∇i∂ ∇ i∂ = ∇ ∂ with ∇ m−2i the unique SUq (2)-invariant unitary holomorphic connection on the hermitian line bundle (Lm−2i , hˆ m−2i ) given in (2.24), and ∇ ∂ is a holomorphic unitary i
396
G. Landi, R. J. Szabo
∗ ∈ Hom connection on (Ei , h i ). As before the Higgs fields φi+1 A(M) (Ei+1 , Ei ). Its curvature is readily found to be m 2 2 ∗ 0,2 ∂ ∗ id ⊗ ∇i∂ + β− ⊗ φi+1 F∇ , := ∇ ∂ = ◦ ∇i+1 − ∇i∂ ◦ φi+1
(3.23)
i=0
∂ 2 where we have used ∇ = 0 and β− ∧ β− = 0. m−2i There is a natural coaction of the quantum group SUq (2) on the collection C( E )1,1 of integrable unitary connections on ( E , h ), obtained by restricting the ‘adjoint’ coacSU (2) tion C of A(SUq (2)). Let C( E )1,1 q be the SUq (2)-invariant subspace. This is the space of holomorphic structures on E [17] for which the coaction of the Hopf algebra A(SUq (2)) is holomorphic. Let C(Ei )1,1 be the collection of integrable unitary connections on (Ei , h i ). Proposition 3.24. Let C ( E )1,1 be the subspace of C ( E ) consisting of integrable con∗ ∈ Hom nections ∇i∂ ∈ C(Ei )1,1 and Higgs fields φi+1 A(M) (Ei+1 , Ei ) for i = 0, 1, . . . , m ∂ ∂ on which the holomorphic connection ∇i+1,i on HomA(M) (Ei+1 , Ei ) induced by ∇i+1 and ∇i∂ vanishes, ∂ ∗ ∗ ∂ ∗ ∇i+1,i (φi+1 ) := φi+1 ◦ ∇i+1 − ∇i∂ ◦ φi+1 = 0.
Then the bijection of Proposition 3.17 defines a bijection between the spaces SU (2) C( E )1,1 q and C ( E )1,1 . Proof. This is a direct consequence of the expression (3.23) for the curvature.
4. Quiver Gauge Theory and Non-Abelian Coupled q-Vortex Equations In the remainder of this paper we will assume that M is a connected Kähler manifold of complex dimension d, with fixed Kähler form ω ∈ 1,1 (M). We then proceed to work out the equivariant dimensional reduction of Yang–Mills theory defined on the quantum space M = CPq1 × M. This will produce a q-deformation of the usual quiver gauge theories on M associated to the linear Am+1 quiver [1,2,12,20,21,31]. The vacuum states of the resulting quiver gauge theory are described by q-deformations of chain vortex equations, which arise by dimensional reduction of BPS-type gauge theory equations on M and whose solutions we call ‘q-vortices’. The q-vortices on the manifold M are in a bijective correspondence with generalized instantons on the quantum space M . The data in the space C ( E ) of Proposition 3.17, defining a q-vortex will be referred to as a ‘stable q-quiver bundle’ over M. In contrast to the q = 1 case, the degree of a q-quiver bundle is generically non-zero, and there are q-vortices which are realized as zeroes of the quiver gauge theory action functional. In fact, we will find that the q-deformation of quiver bundles over the manifold M is analogous in some ways to the twistings of quiver bundles considered in [3]. In particular, we will find analogous constraints on the characteristic classes of stable q-quiver bundles over M, which can be used to naturally construct flat connections on M. Henceforth we fix a deformation parameter 0 < q < 1 and an integer m ≥ 0 parametrizing an SUq (2)-equivariant decomposition as in (3.10).
Dimensional Reduction Over the Quantum Sphere
397
4.1. Metrics on SUq (2)-equivariant vector bundles. In the following we will make use of various SUq (2)-invariant metrics defined on the equivariant modules over A( M ) considered in §3. We start by defining a natural Hodge duality operator on the forms p ( M ) = A(CPq1 ) ⊗ p (M) ⊕ 1 (CPq1 ) ⊗ p−1 (M) ⊕ 2 (CPq1 ) ⊗ p−2 (M) with 0 (M) = A(M) and <0 (M) := 0 =: >2d (M). Let : p (M) → 2d− p (M) d be the Hodge operator corresponding to the Kähler metric of M, with 1 = ωd! and 2 = id. Using the left-covariant Hodge operator ˆ on • (CPq1 ) defined in §2.2, we then define the bimodule map := ˆ ⊗ : p ( M ) −→ 2(d+1)− p ( M ) with 2 = id. Using the integration defined in (2.36), we define an integral := M
CPq1
⊗ M
: 2 (CPq1 ) ⊗ 2d (M) −→ C
$ when the integral over M exists. We set M α := 0 whenever α ∈ / 2 (CPq1 ) ⊗ 2d (M). One then introduces a complex inner product on p ( M ) for each p ≥ 0 by (α, α ) p ( M ) :=
α∗ ∧ α
(4.1)
M
for α, α ∈ p ( M ). Forms of different degrees are defined to be orthogonal. This is a natural generalization of analogous inner products (−, −) p (M) for each p ≥ 0 defined via the Hodge operator on M. Let E be a finitely-generated SUq (2)-equivariant projective bimodule over the algebra A( M ), with equivariant decomposition (3.10). Given hermitian structures on its components h i : Ei × Ei → A(M), we define L 2 -metrics and L 2 -norms on the modules of sections Ei for each i = 0, 1, . . . , m by (ϕi , ψi )h i =
h i (ϕi , ψi ) M
ωd d!
and
1/2
ϕi h i = (ϕi , ϕi )h i
for ϕi , ψi ∈ Ei . The hermitian structures h i also induce a metric h i,i+1 on each of the spaces of Higgs fields Hom A(M) (Ei , Ei+1 ) as follows. Since each bimodule Ei is finitely-generated and projective, it is of the form Ei = pi (A(M))n i for some n i ∈ N and a projection pi ∈ Mat n i (A(M)); then EndA(M) (Ei ) pi Mat n i (A(M))pi . We denote by tr the partial matrix trace over ‘internal indices’ of the endomorphism algebra EndA(M) (Ei ) and define ∗ h i,i+1 (φi+1 , φi+1 ) = tr φi+1 ◦ φi+1 ∗ :E for φi+1 , φi+1 ∈ HomA(M) (Ei , Ei+1 ), where φi+1 i+1 → Ei is the adjoint morphism of the Higgs field φi+1 : Ei → Ei+1 with respect to the hermitian metrics h i on Ei and h i+1 on Ei+1 . The associated L 2 -inner products and L 2 -norms are obtained by integrating the hermitian metrics over M to get )h i,i+1 = (φi+1 , φi+1
M
h i,i+1 (φi+1 , φi+1 )
ωd d!
and
1/2
φi+1 h i,i+1 = (φi+1 , φi+1 )h i,i+1 .
398
G. Landi, R. J. Szabo
Using the hermitian structure h in (3.11), we can further define a complex inner product on the bimodules E over A( M ). Let ϕ, ψ ∈ E with decompositions ϕ = i ϕˆi ⊗ϕi and ψ = i ψˆ i ⊗ ψi , where ϕˆi , ψˆ i ∈ Lm−2i and ϕi , ψi ∈ Ei . Using the orthogonality " of the direct sum decomposition E = i E i with respect to h , we define an L 2 -metric and L 2 -norm on E by (ϕ, ψ) h =
m
H hˆ m−2i (ϕˆi , ψˆ i ) (ϕi , ψi )h i
and
1/2
ϕ h = (ϕ, ϕ) h ,
i=0
where H is the Haar functional on A(CPq1 ). Finally, we define SUq (2)-invariant metrics on the spaces HomA( M ) ( E , p ( E )) for each p ≥ 0. Since E is finitely-generated and projective, any A( M )-linear map E → p ( E ) can be regarded as an element in EndA( M ) ( E ) ⊗A( M ) p ( M ), i.e. as a matrix with entries in p ( M ). By composing the hermitian structure h on E with an ordinary (partial) matrix trace tr over ‘internal indices’, one constructs an inner product on EndA( M ) ( E ). By combining this product with the inner product on p ( M ) given in (4.1), one then obtains a natural L 2 -inner product (−, −) h on the space HomA( M ) ( E , p ( E )) with associated L 2 -norm − h . In an analogous way, one defines L 2 -inner products (−, −)h i on the orthogonal components HomA(M) (Ei , p (Ei )) with associated L 2 -norms − h i , and L 2 -inner products and L 2 -norms (−, −)h i,i+1 and − h i,i+1 on the spaces HomA(M) (Ei , p (Ei+1 )). 4.2. Dimensional reduction of the Yang–Mills action functional. Let C( E ) be the space of unitary connections on an SUq (2)-equivariant hermitian A( M )-module ( E , h ). The Yang–Mills action functional YM : C( E ) → [0, ∞) is defined by % %2 YM( ∇ ) = % F ∇ % h ,
(4.2)
where F ∇ = ∇ 2 is the curvature of the connection ∇ , regarded as an element of HomA( M ) ( E , 2 ( E )). Under a gauge transformation, i.e. the action of a unitary endomorphism u ∈ U( E ) of the module E on C( E ), one has F ∇ u = u F ∇ u∗, and by construction the functional (4.2) is gauge invariant, i.e. YM( ∇ u ) = YM( ∇ ) for all ∇ ∈ C( E ) and u ∈ U( E ). Consequently, the Yang–Mills action functional descends to a map on gauge orbits YM : C( E ) U( E ) → [0, ∞). We have already remarked that the part of a gauge transformation that acts on the bundles Lm−2i is trivial. This entails that in proving the invariance of the Yang–Mills functional under gauge transformation, there is no problem coming from the integral over CPq1 not being a trace but rather a twisted trace. In this section we compute the restriction of the functional (4.2) to the corresponding SUq (2)-invariant subspaces C( E )SUq (2) and C( E )SUq (2) U( E )SUq (2) . Proposition 4.3. Under the bijection of Proposition 3.17, the restriction of the Yang– Mills action functional YM |C ( E )SUq (2) on the quantum space M to SUq (2)-invariant
Dimensional Reduction Over the Quantum Sphere
399
unitary connections is equal to the Yang–Mills–Higgs functional YMHq,m : C ( E ) → [0, ∞) on the manifold M defined by YMHq,m (∇, φ) =
m % %2 2 % % % F∇ % + q + 1 %∇i−1,i (φi )%2 i h h i
i=0
i−1,i
% ∗ + %φi+1 ◦ φi+1
%2 −q 2 φi ◦ φi∗ − q m−2i+1 [m − 2i] idEi %h , i
(4.4)
∗ . Here F 2 with φ0 := 0 =: φ0∗ and φm+1 := 0 =: φm+1 ∇i = ∇i is the curvature of the connection ∇i ∈ C(Ei ) on M, regarded as an element of HomA(M) (Ei , 2 (Ei )), while ∇i−1,i is the connection on HomA(M) (Ei−1 , Ei ) induced by ∇i−1 on Ei−1 and ∇i on Ei with
∇i−1,i (φi ) = φi ◦ ∇i−1 − ∇i ◦ φi .
(4.5)
Under the bijection of Proposition 3.21, the functional (4.4) restricts to a map on gauge orbits YMHq,m : C ( E ) U ( E ) → [0, ∞). Proof. From (3.18), any SUq (2)-invariant unitary connection ∇ ∈ C( E )SUq (2) is of the ∗ ) with ∇ = ∇ m−2i ⊗ id + id ⊗∇i . Thus form ∇ = i ( ∇ i + β+ ⊗ φi+1 + β− ⊗ φi+1 i its curvature F ∇ = ∇ ◦ ∇ is given by F∇ =
m
∗ ∇ i2 + β+ ⊗ ∇i−1,i (φi ) − β− ⊗ ∇i+1,i (φi+1 )
i=0
∗ + (β+ ∧ β− ) ⊗ (φi ◦ φi∗ ) + (β− ∧ β+ ) ⊗ (φi+1 ◦ φi+1 )
∗ + (β+ ∧ β+ ) ⊗ (φi+1 ◦ φi ) − (β− ∧ β− ) ⊗ (φi∗ ◦ φi+1 ) .
(4.6)
A straightforward computation gives ∇ i2 = fm−2i ⊗ id + id ⊗F∇i , 2 where fm−2i = ∇ m−2i is the curvature (2.23) of the canonical connection on the SUq (2)equivariant line bundle Lm−2i over CPq1 . By (2.21) one has fm−2i = −q m−2i+1 [m − 2i] β− ∧ β+ as an element of HomA(CPq1 ) (Lm−2i , 2 (Lm−2i )). By (2.8) one has β− ∧ β− = 0 = β+ ∧ β+ and β+ ∧ β− = −q 2 β− ∧ β+ . Substituting everything into (4.6), we arrive at F∇ =
m i=0
∗ id ⊗F∇i + β+ ⊗ ∇i−1,i (φi ) − β− ⊗ ∇i+1,i (φi+1 ) + (β− ∧ β+ )
∗ ◦ φi+1 − q 2 φi ◦ φi∗ − q m−2i+1 [m − 2i] id . ⊗ φi+1
(4.7)
400
G. Landi, R. J. Szabo
We now use the definition of the Hodge operator and integral on M from §4.1, ∗ = −β , to compute the together with ˆ β± = ± β± and (2.13), and the ∗-structure β± ∓ corresponding Yang–Mills action functional (4.2). One finds YM( ∇ ) = M
=
∗ tr F ∇ ∧ F∇
m i, j=0
CPq1
&
⊗
tr M
∗ ∗ id ⊗F∇∗i −β− ⊗ ∇i−1,i (φi )∗ + β+ ⊗ ∇i+1,i (φi+1 )
∗ ∗ + (β− ∧ β+ ) ⊗ φi+1 ◦ φi+1 − q 2 φi ◦ φi∗ − q m−2i+1 [m − 2i] id ∧ (β− ∧ β+ ) ⊗ (F∇ j )−β+ ⊗ ∇ j−1, j (φ j ) −β− ⊗ ∇ j+1, j (φ ∗j+1 ) ' ωd . (4.8) + id ⊗ φ ∗j+1 ◦ φ j+1 − q 2 φ j ◦ φ ∗j − q m−2 j+1 [m − 2 j] id d! " Now use orthogonality of the splitting E = i E i , together with the fact that only the top two-form β− ∧ β+ ∈ 2 (CPq1 ) has a non-zero integral over CPq1 in (4.8). Using the identities (2.8) once again and the normalization H (1) = 1 for the Haar state on A(CPq1 ), one finds that (4.8) coincides with the Yang–Mills–Higgs action functional (4.4). By construction, the functional YMHq,m is invariant under the action of the gauge group U ( E ) of Proposition 3.20, and hence descends to a well-defined functional on the orbit space C ( E ) U ( E ) → [0, ∞). We can rewrite the Yang–Mills–Higgs functional (4.4) in a more suggestive way. For this, consider the direct sum of A(M)-modules with induced connection (E, ∇), E=
m
Ei ,
∇=
i=0
m
∇i .
i=0
" The induced hermitian structure h = i h i on E defines an inner product given by 1/2 (ϕ, ψ)h = i (ϕi , ψi )h i for ϕ, ψ ∈ E, with corresponding norm ϕ h = (ϕ, ϕ)h p and extension to HomA(M) (E, (E)) as described in §4.1. The Higgs fields φi ∈ HomA(M) (Ei−1 , Ei ), with i = 1, . . . , m, induce an element φ=
m
φi
i=1
of the quiver representation module R( E ) ⊂ EndA(M) (E) given by R( E ) =
m i=1
HomA(M) (Ei−1 , Ei ),
" structure h = with i ∇i,i−1 . The induced " induced connection ∇ = hermitian ) ) = h defines an inner product given by (φ, φ (φ , φ h i i i−1,i i i h i−1,i for φ, φ ∈ 1/2 R( E ), with associated L 2 -norm φ " R( E ), we introh = (φ, φ)h . Given φ, φ ∈ " ∗ ∗ ∗ ∗ duce the endomorphisms φ ◦ φ = (φ ◦ φ ) and φ ◦ φ = i i i (φ ◦ φ )i in " End (E ) ⊂ End (E) by A(M) i A(M) i ∗ ∗ ∗ φ ◦ φ i = φi ◦ φi ∗ and φ ◦ φ i = φi+1 ◦ φi+1 .
Dimensional Reduction Over the Quantum Sphere
401
Using these maps we define the q-commutator [φ ∗ , φ ]q ∈ EndA(M) (E) by ) ( ∗ φ , φ q = φ∗ ◦ φ − q 2 φ ◦ φ∗. Finally, define an endomorphism q,m of E by q,m =
m
q m−2i+1 [m − 2i] idEi .
i=0
Then the Yang–Mills–Higgs action functional (4.4) can be succinctly rewritten in compact form as a functional on YMHq,m : C(E) × R( E ) → [0, ∞) given by %2 % %2 % YMHq,m (∇, φ) = YM(∇) + q 2 + 1 %∇(φ)%h + %[φ ∗ , φ]q − q,m %h . 4.3. Holomorphic chain q-vortex equations. Given a unitary connection ∇ ∈ C( E ), • there is a well-defined " map [p∇ , −] from the space HomA( M ) ( E , ( E )) to itself, where • ( E ) = ( E ). On homogeneous morphisms T ∈ HomA( M ) p≥0 ( E , p ( E )) it is defined by [ ∇ , T ] := ∇ ◦ T − (−1) p T ◦ ∇ . For the curvature F ∇ = ∇ 2 ∈ HomA( M ) ( E , 2 ( E )), one then has the Bianchi identity [ ∇ , F ∇ ] = 0. As the space of connections C( E ) is an affine space modeled on HomA( M ) ( E , 1 ( E )), as usual, for the critical points of the Yang–Mills action functional (4.2) one needs to compute it on a one-parameter family ∇ + t η, and equate to zero the linear term in t for the corresponding expansion. By using properties of the complex inner product (−, −) h on HomA( M ) ( E , • ( E )), this variational problem for the Yang–Mills action functional (4.2) shows that its critical points ∇ obey the Euler–Lagrange equation [ ∇ ∗ , F ∇ ] = 0,
(4.9)
where the adjoint operator of [ ∇ , −] is defined with respect to the inner product as ∗ [∇ ,T], T h = T , [∇ ,T ] h (4.10) for T, T ∈ HomA( M ) ( E , • ( E )). Using the definition of the inner product given in §4.1, one easily shows that [ ∇ ∗ , T ] = − [ ∇ , T ]. The purpose of this section is to characterize stable critical points of the Yang–Mills action functional (4.2) on M , and to study their dimensional reduction to configurations on M. For this, following [34, §1.2] we introduce the notion of generalized instanton on the quantum space M . Lemma 4.11. Let ∇ ∈ C( E ) be a unitary connection and ∈ 2d−2 ( M ) a closed form of degree 2d − 2, regarded as the element id E ⊗A( M ) , such that F ∇ = −F ∇ ∧ .
(4.12)
Then ∇ solves the Yang–Mills equation (4.9) and YM( ∇ ) = Top2 ( E , ), where Top2 ( E , ) = − F ∇ , (F ∇ ∧ ) h . (4.13)
402
G. Landi, R. J. Szabo
Proof. Using (4.12) and the graded right Leibniz rule we compute [ ∇ ∗, F ∇ ] = − [ ∇ , F ∇ ] = [ ∇ , F ∇ ∧ ] = [ ∇ , F ∇ ] ∧ + F ∇ ∧ d = 0, where in the last line we used the Bianchi identity and d = 0. The second statement (4.13) follows from direct substitution of Eq. (4.12) into the Yang–Mills action functional YM( ∇ ) = (F ∇ , F ∇ ) h . Using d = 0, the definition of the inner product (−, −) h , and the Bianchi identity [ ∇ , F ∇ ] = 0, one shows in the standard way that the functional (4.13) does not depend on the choice of connection ∇ on E . It thus defines a ‘topological action’ which depends only on the A( M )-module E and the closed form , and hence provides an a priori lower bound on the Yang–Mills action functional. We refer to the gauge invariant equation (4.12) as the -anti-selfduality equation. Gauge equivalence classes in C( E ) U( E ) of solutions to this first order equation are called generalized instantons or -instantons. In the sequel we will use the natural closed (1, 1)-form ω = (β− ∧ β+ ) ⊗ 1 + 1 ⊗ ω on A( M ), and set =
ωd−1 ωd−2 ω d−1 =1⊗ + (β− ∧ β+ ) ⊗ , (d − 1)! (d − 1)! (d − 2)!
(4.14)
where the second term is absent when d = 1. We write Top2 ( E , ω ) for the corresponding topological action functional (4.13). For simplicity, we also assume that tr(F ∇ ) = 0 for any connection ∇ ∈ C( E ) without loss of generality, for otherwise one can consider F˜ ∇ = F ∇ − r1 tr(F ∇ ) p where r is the rank of the projection p defining the bimodule E . We recall that on the algebra A( M ) = A(CPq1 ) ⊗ A(M) there is a natural complex structure which combines the complex structure of M with the complex structure for the differential calculus on A(CPq1 ) given in Proposition 2.11. Using it we write p( E ) = i, j ( E ) i+ j= p
for the corresponding splitting of the space of E -valued p-forms into (i, j)-forms. The ∗-involution maps i, j ( E ) to j,i ( E ), while the Hodge operator maps i, j ( E ) to d+1− j,d+1−i ( E ). Consider then the linear operator L ω : i, j ( E ) −→ i+1, j+1 ( E ),
L ω (α) := α ∧ ω .
Let L ∗ω : i, j ( E ) → i−1, j−1 ( E ) be its adjoint with respect to the L 2 -inner product defined in §4.1. If α ∈ 1,1 ( E ), then α ω := L ∗ω (α) = (α ∗ ∧ ω ) is the component of α (in 0,0 ( E ) = E ) along the closed (1, 1)-form ω . As in §4.1, this definition is extended to the modules EndA( M ) ( E ) over A( M ) and the corresponding modules over A(M) in the obvious ways.
Dimensional Reduction Over the Quantum Sphere
403
For a connection ∇ ∈ C( E ), we decompose the curvature 2,0 1,1 0,2 F∇ = F∇ + F∇ + F∇
(4.15)
2,0 into its (2, 0), (1, 1), and (0, 2) components. Since ∇ is unitary, one has F ∇ = 0,2 ∗ ∗ 1,1 1,1 and F ∇ = − F ∇ . We recall that the connection ∇ is called integra− F∇
0,2 = 0. ble if F ∇
Lemma 4.16. A connection ∇ solves the generalized instanton equation (4.12) if and only if it is an integrable unitary connection in C( E )1,1 which satisfies the hermitian Yang–Mills equation ω
F ∇ = 0.
(4.17)
Proof. The Hodge operator acts with a grading (−1) j on i, j ( E ). Substituting the decomposition (4.15) into (4.12) with the choice (4.14) we thus find 0,2 F∇ = 0,
whence ∇ ∈ C( E )1,1 . For the remaining (1, 1)-component, we note first the identity F∇ ∧
d+1 ωd ω ω = F ∇ ∧ ω = −F ∇ . d! (d + 1)!
But by the -anti-selfduality equation (4.12), the left-hand side is also equal to F∇ ∧
d+1 ωd ω ω = − F ∇ ∧ ω = +F ∇ , d! (d + 1)!
and comparing the two expressions gives (4.17).
We next describe the SUq (2)-equivariant dimensional reduction of the above generalized instanton equations. Proposition 4.18. Under the bijection of Proposition 3.24, the subspace of SUq (2) SU (2) invariant connections ∇ ∂ ∈ C( E )1,1 q solving the generalized instanton equation on the quantum space M corresponds bijectively to the subspace of C ( E )1,1 consisting of elements (∇ ∂ , φ ∗ ) which satisfy the holomorphic chain q-vortex equations on the manifold M given by ∗ F∇ωi = q 2 φi ◦ φi∗ − φi+1 ◦ φi+1 + q m−2i+1 [m − 2i] idEi
(4.19)
for each i = 0, 1, . . . , m. Here F∇ωi = (F∇1,1 )∗ ∧ ω is the component (in i EndA(M) (Ei )) of the curvature of ∇i along the Kähler form of M.
404
G. Landi, R. J. Szabo
Proof. By Proposition 3.24, the SUq (2)-invariant subspace of generalized instanton connections on M described by Lemma 4.16 consists of connections and Higgs fields on M satisfying F∇0,2 =0 i
and
∂ ∗ ∇i+1,i (φi+1 )=0
for i = 0, 1, . . . , m. Whence the (1, 1)-component of the curvature (4.7) for a connection SU (2) ∇ ∂ ∈ C( E )1,1 q is given by 1,1 F∇ =
m
∗ 2 ∗ m−2i+1 id ⊗F∇1,1 +(β ∧ β )⊗ φ ◦ φ −q φ ◦ φ −q [m − 2i] id . − + i+1 i i+1 i i
i=0
We now use (2.13) and multiply this form with ω =1⊗
ωd + (β− ∧ β+ ) ⊗ (ω) d!
to get ω
F∇ =
m
∗ id ⊗ F∇ωi + φi+1 ◦ φi+1 − q 2 φi ◦ φi∗ − q m−2i+1 [m − 2i] id .
i=0
" Using orthogonality of the direct sum E = i E i , the hermitian Yang–Mills equation (4.17) then corresponds to the set of Eqs. (4.19). Equations (4.19) are naturally invariant under the action of the group U ( E ). Under the bijection of Proposition 3.21, an SUq (2)-invariant generalized instanton reduces to a gauge equivalence class of solutions to (4.19) in C ( E ) U ( E ). We call such a class a q-vortex on the manifold M. We can rewrite the system of q-vortex equations collectively in a more suggestive form. Using the definitions given at the end of §4.2, they can be succinctly rewritten in a compact form as the single equation in EndA(M) (E) given by ( ) F∇ω + φ ∗ , φ q = q,m . This equation illustrates the crucial feature of our deformation of standard quiver vortex equations, in that the commutator of Higgs fields is replaced with the q-commutator. We shall find various interesting consequences of this deformation below. 4.4. Vacuum structure. In order to establish a correspondence between the generalized instanton equations on M and the non-abelian q-vortex equations on M, we will now show directly how the latter equations describe stable critical points of the Yang–Mills– Higgs functional (4.4). For this, we will assume in the following that, for each i = 1, . . . , m, the hermitian endomorphism F∇ωi of Ei is non-negative, i.e. it defines a non-negative sesquilinear form on HomA(M) (Ei−1 , Ei ) given by (F∇ωi ◦ φi , φi )h i−1,i for φi , φi ∈ HomA(M) (Ei−1 , Ei ). Summing these forms we then get a non-negative sesquilinear form on R( E ) defined by (φ, φ )R ( E ) :=
m i=1
F∇ωi ◦ φi , φi
h i−1,i
,
Dimensional Reduction Over the Quantum Sphere
405
for φ, φ ∈ R( E ), with corresponding norm φ 2R ( E ) := (φ, φ)R ( E ) ≥ 0 for each φ ∈ R( E ). Theorem 4.20. The restriction of the Yang–Mills–Higgs functional YMHq,m to elements (∇ ∂ , φ ∗ ) ∈ C ( E )1,1 , for which F∇ωi ∈ EndA(M) (Ei ) is non-negative for i = 1, . . . , m, is given by YMHq,m ∇ ∂ , φ ∗ =
m m−2i+1 2q [m − 2i] Top1 (Ei , ω) − Top2 (Ei , ω) − 2 1 − q 2 φ 2R ( E ) i=0 m
+
% ω % % F + φ ∗ ◦ φi+1 − q 2 φi ◦ φ ∗ − q m−2i+1 [m − 2i] idE %2 , i+1 i ∇i i hi
(4.21)
i=0
where Top1 (Ei , ω) := F∇ωi , idEi h
ωd−2 and Top2 (Ei , ω) := − F∇i , F∇i ∧ . (d − 2)! h i
i
Proof. We will apply a Bogomol’nyi-type transformation to the action functional (4.4). Firstly, we have % %2 2 % % % F∇ % + q + 1 %∇i−1,i (φi )%2 i hi h i−1,i % 0,2 %2 % ω %2 %2 2 % ∂ = 4% F∇i %h + % F∇i %h + 2 q + 1 %∇i−1,i (φi )%h − Top2 (Ei , ω) (4.22) i
i
i−1,i
for each i = 0, 1, . . . , m (see e.g. [3, §4]). For (∇ ∂ , φ ∗ ) ∈ C ( E )1,1 , this is equal to % ω %2 % F % − Top (Ei , ω). 2 ∇i h i
We now combine F∇ωi 2h i
with the last set of norms in (4.4). For this, we expand out the last set of inner products in (4.21) to get % % ω % F + φ ∗ ◦ φi+1 − q 2 φi ◦ φ ∗ − q m−2i+1 [m − 2i] id %2 i+1 i ∇i hi % ω %2 % ∗ %2 2 ∗ m−2i+1 % % % = F∇i h + φi+1 ◦ φi+1 − q φi ◦ φi − q [m − 2i] id %h i i ω ∗ 2 ∗ m−2i+1 + 2 F∇i , φi+1 ◦ φi+1 − q φi ◦ φi − q [m − 2i] id h . i
The last term in the inner product here gives −2q m−2i+1 [m − 2i]
Top1 (Ei , ω). The first two terms in this inner product can be evaluated by using (4.5) and the graded commutator to deduce that the curvature of the induced connection ∇i−1,i on HomA(M) (Ei−1 , Ei ) is F∇i−1,i (φi ) := ∇i−1,i ◦ ∇i−1,i (φi ) = φi ◦ F∇i−1 − F∇i ◦ φi . = 0 for each i = 0, 1, . . . , m, we can use standard Kähler identiSince F∇0,2 i ties [3, Eq. (4.10)] for the induced holomorphic structures on the A(M)-bimodules HomA(M) (Ei−1 , Ei ) to get % ∂ %2 % ∂ %2 ω F∇i−1,i (φi ) , φi h = %∇i−1,i (φi )%h − %∇i−1,i (φi )%h , i−1,i
i−1,i
i−1,i
406
G. Landi, R. J. Szabo
which vanishes for (∇ ∂ , φ ∗ ) ∈ C ( E )1,1 by Proposition 3.24. It follows that m
∗ F∇ωi , φi+1 ◦ φi+1 − q 2 φi ◦ φi∗
hi
i=0
=
m
φi ◦ F∇ωi−1 − q 2 F∇ωi ◦ φi , φi
h i−1,i
i=1
=
m *
F∇ωi−1,i (φi ) , φi
h i−1,i
+ 1 − q 2 F∇ωi ◦ φi , φi h
+ i−1,i
i=1
= 1 − q 2 φ 2R ( E ) . Putting everything together yields (4.21).
Corollary 4.23. The minima of the Yang–Mills–Higgs action functional YMHq,m on C ( E ), having values in [0, ∞), are given by elements (∇ ∂ , φ ∗ ) ∈ C ( E )1,1 which satisfy the holomorphic chain q-vortex equations (4.19), and for which the curvature projections F∇ωi in EndA(M) (Ei ) for each i = 1, . . . , m are non-negative with m m−2i+1 1 2q [m − 2i] Top (E , ω) − Top (E , ω) . φ 2R ( E ) ≤ i i 1 2 2 1 − q 2 i=0
(4.24) When equality holds in (4.24), the minima achieve the infemum YMHq,m (∇ ∂ , φ ∗ ) = 0. Proof. From (4.22) it follows that the action functional (4.4) is minimized by taking (∇ ∂ , φ ∗ ) ∈ C ( E )1,1 . From (4.21) one then gets that there is a Bogomol’nyi-type inequality YMHq,m (∇, φ) m m−2i+1 ≥ [m − 2i] Top1 (Ei , ω) − Top2 (Ei , ω) − 2 1 − q 2 φ 2R ( E ) , 2q i=0
with Top1 (Ei , ω) and Top2 (Ei , ω) not dependent on the choice of connection ∇i on the A(M)-module Ei . This inequality is saturated by solutions to the holomorphic q-vortex equations, with the bound (4.24) since YMHq,m (∇, φ) ≥ 0 and 0 < q < 1. Corollary 4.25. A stable q-quiver bundle on M has characteristic classes constrained by the Bogomolov–Gieseker-type inequality 2
m
q m−2i [m − 2i] Top1 (Ei , ω) ≥
i=0
m
Top2 (Ei , ω).
i=0
If equality holds, then the connections ∇i are flat and ∗ ◦ φi+1 − q 2 φi ◦ φi∗ = q m−2i+1 [m − 2i] idEi φi+1
for each i = 0, 1, . . . , m.
(4.26)
Dimensional Reduction Over the Quantum Sphere
407
Proof. The first statement follows from the inequality (4.24) together with φ 2R ( E ) ≥ 0 and 0 < q < 1. For the second statement, we use (4.21) to get YMHq,m (∇, φ) = −2 1 − q 2 φ 2R ( E ) ≤ 0. But from its definition (4.4) the Yang–Mills–Higgs functional is a sum of non-negative terms, whence YMHq,m (∇, φ) = 0 and thus F∇i = 0 for each i = 0, 1, . . . , m. Let us demonstrate explicitly that the vacuum moduli space of the Yang–Mills–Higgs functional is generically non-empty. Proposition 4.27. Suppose that the finitely-generated projective A(M)-bimodules Ei have the same rank for all i = 0, 1, . . . , m. Then a class of explicit solutions to the vacuum equations (4.26) is given by φi = φi 0 := λi u i , i = 1, . . . , m, (4.28) where u i ∈ HomA(M) (Ei−1 , Ei ) are arbitrary holomorphic morphisms unitary with respect to the hermitian structures h i−1 on Ei−1 and h i on Ei , and the induced connec∂ tion ∇i−1,i , and λi are the q-numbers q m−2i+3 [m − 2i + 2] − q 4 [m − 2i] − q 4i [m + 2] − q 4 [m] . λi = 1 − q2 1 − q6 (4.29) Proof. Substituting (4.28) into (4.26) gives the linear recursion relation λi+1 − q 2 λi = q m−2i+1 [m − 2i].
(4.30)
With λ0 = 0 = λm+1 , the solution of (4.30) is given by λi =
i−1
( ) q m−2(i−2k)+3 m − 2(i − k − 1) .
k=0
We now use the definition of the q-integers [m − 2(i − k − 1)] given in (0.1) and perform the sums over k using i−1 k=0
xk =
1 − xi , 1−x
with x = q 6 and x = q 2 . This gives q m−2i+3 1 − q 6i q m−2i+2 − q m−2i+4 λi = −1 6 2 q −q 1−q 1−q − 1 − q 2i q −m+2i−2 − q −m+2i+4 , which is easily manipulated into the form (4.29).
408
G. Landi, R. J. Szabo
4.5. Stability conditions. We can also derive topological obstructions to the existence of solutions to the q-vortex equations (4.19). For this, we suppose that M is compact, and define the degree of a hermitian finitely-generated projective A(M)-module (E, h) by deg(E) =
Top1 (E, ω) , volω (M)
$ d where volω (M) = M ωd! is the Kähler volume of M. The degree depends on the cohomology class of ω. The slope of E is the number μ(E) =
deg(E) , rank(E)
with the rank, rank(E), defined as the trace of the identity endomorphism acting on E. Analogously to [1, §3], we define the (q, m)-degree of a finitely-generated SUq (2)equivariant projective bimodule E over the algebra A( M ), with equivariant decomposition as in (3.10), by degq,m ( E ) =
m
deg(Ei ) − q m−2i+1 [m − 2i] rank(Ei )
(4.31)
i=0
and its (q, m)-slope by μq,m ( E ) =
degq,m ( E ) rank( E )
,
where rank( E ) = i rank(Ei ). The definition (4.31) has a natural topological meaning, when we recall from §2.7 that the q-integers [m − 2i] label classes in the SUq (2)U (su(2))
equivariant K-theory K0 q (CPq1 ). Since the quantum group SUq (2) acts trivially on the manifold M, the (q, m)-degree labels classes in the SUq (2)-equivariant K-theory of M = CPq1 × M. Thus the standard assignment [20,31] of D-brane charges in equivariant K-theory to quiver vortices extends to our q-vortices as well. The parameters q, m and the topology of the bundles Ei over M are then constrained by the following (semi-)stability criteria. Proposition 4.32. A stable q-quiver bundle on M has slopes constrained by the inequalities: (a) μ(E0 ) ≤ q m+1 [m], with equality if and only if E0 admits a holomorphic connection ∇0 solving the hermitian Yang–Mills equation F∇ω0 = q m+1 [m] idE0 . (b) μ(Em ) ≥ −q −m+1 [m], with equality if and only if Em admits a holomorphic connection ∇m solving the hermitian Yang–Mills equation F∇ωm = −q −m+1 [m] idEm . (c) μq,m ( E ) ≤ 0, with equality if and only if Ei admits a holomorphic connection ∇i solving the hermitian Yang–Mills equation F∇ωi = q m−2i+1 [m − 2i] idEi for each i = 0, 1, . . . , m. Proof. Point (a) follows from the q-vortex equation (4.19) for i = 0 after taking inner products on both sides with idE0 , and using (idE0 , idE0 )h 0 = rank(E0 ) volω (M) together with φ0 := 0 and φ1 2h 0,1 ≥ 0. Point (b) follows similarly from (4.19) with i = m and
Dimensional Reduction Over the Quantum Sphere
409
φm+1 := 0. For point (c), we take inner products on both sides of (4.19) with idEi and sum over i = 0, 1, . . . , m to get the constraint m
deg(Ei ) =
i=0
m
q m−2i+1 [m − 2i] rank(Ei ) +
i=0
m q2 − 1 φi 2h i−1,i , volω (M) i=1
and the result follows since 0 < q < 1 and only if φi = 0).
φi 2h i−1,i
≥ 0 (with φi 2h i−1,i = 0 if and
5. Examples In this final section we will briefly consider some explicit examples of the q-vortex equations (4.19). In particular, we will describe how the q-deformations affect stability conditions for the existence of solutions to these equations and the structure of the corresponding moduli spaces. These considerations provide the first step to formulating a general algebro-geometric notion of stability for SUq (2)-equivariant modules over A( M ) and the corresponding q-quiver bundles over M. 5.1. Deformations of holomorphic triples and stable pairs. A holomorphic triple (E0 , E1 , φ) on a compact Kähler manifold (M, ω) consists of a pair of holomorphic vector bundles E0 , E1 over M and a holomorphic morphism φ
→ E1 E0 − which together obey coupled vortex equations [13,8]. For m = 1, our q-vortex equations (4.19) provide a q-deformation of such triples. In this case Eqs. (4.19) read F∇ω0 = q 2 idE0 −q −2 φ ◦ φ ∗ , F∇ω1 = − idE1 −q 2 φ ∗ ◦ φ , (5.1) where φ := φ1 . The topological stability conditions on these triples are governed by Proposition 4.32 for m = 1. Additionally, by taking inner products on both sides of Eqs. (5.1) with idE0 and idE1 respectively, and summing shows that the degrees of the bundles are related by deg(E0 ) + q −2 deg(E1 ) = q 2 rank(E0 ) − q −2 rank(E1 ). Substituting this into the formula for the (q, 1)-degree of the quiver bundle then yields degq,1 ( E ) = 1 − q −2 deg(E1 ) + rank(E1 ) . These criteria are much more stringent than the stability condition of [8]. Let us now denote E := E0 , ∇ := ∇0 , and take E1 to be the A(M)-module of sections of the trivial holomorphic line bundle over M, i.e. E1 = C ⊗ A(M) A(M). Then F∇ω1 = 0, and HomA(M) (E0 , E1 ) E0 so that the Higgs field φ can be regarded as an element of E, i.e. as a holomorphic section. We set φ := q −1 ϕ with ϕ ∈ E, so that ϕ ∗ ⊗A(M) ϕ = idA(M) by the second equation of (5.1). The first equation of (5.1) can then written as F∇ω + q −2 ϕ ⊗A(M) ϕ ∗ = q 2 idE .
(5.2)
410
G. Landi, R. J. Szabo
Thus in this case the triple describes a q-deformation of the stable pairs (E, ϕ) considered in [7]. The stability condition reads deg(E) = q 2 rank(E) − q −2 which is much more restrictive than the undeformed one of [7,13]. 5.2. q-vortices on Riemann surfaces. Let M be a compact oriented Riemann surface of genus g. Then Eqs. (5.1) describe q-deformations of non-abelian vortices on M. For g = 1, the area of M is volω (M) =
8π (1 − g) κ
by the Gauss–Bonnet theorem, where κ is the scalar curvature of M with respect to the Kähler metric corresponding to ω. Let us again consider a particular case. If E := E0 , ∇ := ∇0 and E1 Cr ⊗ A(M) with r = rank(E), then the Higgs field φ = q −1 ϕ 1 can be regarded as an element of Cr ⊗ E. The characteristic class 2π Top1 (M, ω) is the first Chern class c1 (E) of the bundle E. A non-empty moduli space of solutions to the q-vortex equations, formally the same as in (5.2), is ensured in this case by the stability condition 4r 2 q − q −2 (1 − g) c1 (E) = κ 2
for g = 1. Since 0 < q < 1, this degree satisfies the bound c1 (E) < 4qκ (1 − g). Hence the pair (E, ϕ) is τ -stable in the sense of [9], and by the Hitchin–Kobayashi correspondence it is gauge equivalent to a solution of the non-abelian q-vortex equations. The corresponding moduli space of solutions is described explicitly in [6]. For abelian vortices, r = 1, this moduli space coincides with the |n|th symmetric product orbifold of M, i.e. the space of effective divisors on M of degree n = c1 (E). By taking E1 to be a (generically non-trivial) holomorphic line bundle, one also obtains from (5.1) a q-deformation of the non-abelian vortex equations studied in [30]. However, contrary to the q = 1 case, wherein the reduction r = 1, E0 E1 and ∇0 = −∇1 would lead to the standard abelian BPS vortex equations on M, such an abelian reduction in (5.1) is not consistent for q = 1. Indeed, as we first witnessed in point (c) of Proposition 4.32, the q-deformation generically imposes very stringent constraints on the allowed stable quiver bundles. Moreover, the q-vortices do not exist on the complex plane M = C, wherein the formal limit volω (M) = ∞ would necessitate an infinite vortex number and action. The features spelled out in this section are generic properties of the q-vortex equations (5.1). 5.3. q-instantons. Let (M, ω) be a Kähler surface. Set E0 E1 =: E, with r = rank(E), ∂ (φ) = 0, from (4.5) we have and φ = idE . Then since φ is a holomorphic section, ∇0,1 ∇0 = ∇1 =: ∇ and both equations in (5.1) simplify to F∇ω = q 2 − 1 idE . (5.3) For bundles E of vanishing degree and q 2 = 1, this equation gives a deformation of the hermitian Yang–Mills equation on M, and hence of the standard anti-selfduality
Dimensional Reduction Over the Quantum Sphere
411
equations F∇ = −F∇ . Gauge equivalence classes of solutions to (5.3) are thus called q-instantons, and their moduli spaces can be described explicitly in the following way. The fixed points on the space C(E) of unitary connections on E under the action of the group of gauge transformations U(E) are given by integrable connections ∇ ∈ C(E)1,1 . The natural U(E)-invariant symplectic form ωC on C(E) is thus given by ωC (α, α ) = α, α
1 2
ω tr α ∗ ∧ α ω2 M
for ∈ We implicitly use the inclusion of the Lie algebra of A(M) U(E) in its dual space by means of the hermitian structure h on E. Then the corresponding moment map μC : C(E) → (Lie U(E))∗ is given by Homa
(E, 1 (E)).
μC (∇) = F∇ω . The moduli space of q-instantons on M is thus realized as the symplectic quotient 2 U(E), μ−1 C (q − 1) idE 2 and hence the q-vortices in this case correspond to points of μ−1 C (q − 1) idE which lie inside the Kähler submanifold C(E)1,1 (outside the singularities). When M = C2 , the constant shift in the moment map condition from μC = 0 to μC = (q 2 − 1) idE induces a shift in the corresponding real ADHM equation. The effect of this shift is to augment [25] the moduli space of holomorphic instanton bundles to the moduli space of torsion free sheaves on the projective plane CP2 with a trivialization on a fixed projective line CP1 ⊂ CP2 . This resolves the small instanton singularities and turns the instanton moduli space into a hyper-Kähler manifold of complex dimension 4r k, where k = 8π1 2 Top2 (M, ω) = c2 (E). It is well-known [26] that this modification arises explicitly in the equations which determine instantons on a certain noncommutative deformation of R4 . Here we have shown that the same sort of resolution of instanton moduli space is achieved via our q-deformed dimensional reduction procedure over the quantum projective line CPq1 . The essential feature behind such resolutions, provided here by the deformation in (5.3), lies in the content of point (c) of Proposition 4.32. Final Remarks In this paper we have shown that the formalism of SU(2)-equivariant dimensional reduction over the sphere has a natural SUq (2) Hopf algebraic generalization to reductions over the quantum sphere. This was achieved by recasting the standard dimensional reduction procedure into a purely algebraic framework and using the fact that much of the geometry of the projective line survives q-deformation to the quantum projective line (the quantum sphere with additional structure). We obtained a q-deformed Yang–Mills–Higgs theory from the reduction of Yang–Mills theory, and also q-deformations of quiver chain vortex equations from the reduction of natural first order gauge theory equations. We demonstrated that the moduli spaces of solutions to these q-vortex equations are more constrained but generically better behaved than their q → 1 limits. In some instances, the vacuum moduli space can be described as a symplectic or even hyper-Kähler quotient. It would be interesting to explore whether the generic q-vortex equations admit such a moment map interpretation, as they do in the q = 1 case from the action of a unitary group on a representation space of quiver modules (see
412
G. Landi, R. J. Szabo
[3, §2.2]). This presumably involves interpreting the q-commutator terms as moment map equations for a sort of quantum group action on the space of quiver gauge connections. This may also help fill an important gap in our construction, namely the proper formulation of stability conditions and the ensuing Hitchin–Kobayashi-type correspondence which relates the existence of solutions to the gauge equations with a stability criterion. This problem appears to lie in the general realm of extending noncommutative geometry into the algebro-geometric setting, which is not yet fully developed. It would be interesting to see if the q-deformations of the Yang–Mills–Higgs models derived in this paper improve the phenomenological viability of the models constructed in [12]. The somewhat intricate q-dependence of the vacuum Higgs field configurations described by Proposition 4.27 may drastically alter the dynamical mass generation in these models. It would also be interesting to extend our constructions to Hopf algebraic equivariant dimensional reduction over other quantum homogeneous spaces. Acknowledgements. GL was partially supported by the Italian Project ‘Cofin08 – Noncommutative Geometry, Quantum Groups and Applications’. RJS was partially supported by the Grant ST/G000514/1 ‘String Theory Scotland’ from the UK Science and Technology Facilities Council. We thank T. Brzezinski for an email exchange.
References 1. Álvarez-Cónsul, L., García-Prada, O.: Dimensional reduction, SL(2, C)-equivariant bundles and stable holomorphic chains. Int. J. Math. 12, 159–201 (2001) 2. Álvarez-Cónsul, L., García-Prada, O.: Dimensional reduction and quiver bundles. J. Reine Angew. Math. 556, 1–46 (2003) 3. Álvarez-Cónsul, L., García-Prada, O.: Hitchin–Kobayashi correspondence, quivers and vortices. Commun. Math. Phys. 238, 1–33 (2003) 4. Aschieri, P., Grammatikopoulos, T., Steinacker, H., Zoupanos, G.: Dynamical generation of fuzzy extra dimensions, dimensional reduction and symmetry breaking. J. High Energy Phys. 0609, 026 (2006) 5. Aschieri, P., Madore, J., Manousselis, P., Zoupanos, G.: Dimensional reduction over fuzzy coset spaces. J. High Energy Phys. 0404, 034 (2004) 6. Baptista, J.M.: Non-abelian vortices on compact Riemann surfaces. Commun. Math. Phys. 291, 799–812 (2009) 7. Bradlow, S.B.: Special metrics and stability for holomorphic bundles with global sections. J. Diff. Geom. 33, 169–213 (1991) 8. Bradlow, S.B., García-Prada, O.: Stable triples, equivariant bundles and dimensional reduction. Math. Ann. 304, 225–252 (1996) 9. Bradlow, S.B., Daskalopoulos, G., García-Prada, O., Wentworth, R.: Stable augmented bundles over Riemann surfaces. London Math. Soc. Lect. Notes Ser. 208, 15–67 (1995) 10. Brzezi´nski, T., Majid, S.: Quantum group gauge theory on quantum spaces. Commun. Math. Phys. 157, 591–638 (1993); Erratum ibid. 167, 235 (1995) 11. Brzezi´nski, T., Majid, S.: Quantum differential and the q-monopole revisited. Acta Appl. Math. 54, 185–233 (1998) 12. Dolan, B.P., Szabo, R.J.: Dimensional reduction, monopoles and dynamical symmetry breaking. J. High Energy Phys. 0903, 059 (2009) 13. García-Prada, O.: Dimensional reduction of stable bundles, vortices and stable pairs. Int. J. Math. 5, 1–52 (1994) 14. Hajac, P.M.: Bundles over quantum sphere and noncommutative index theorem. K-Theory 21, 141–150 (2000) 15. Hajac, P.M., Majid, S.: Projective module description of the q-monopole. Commun. Math. Phys. 206, 247–264 (1999) 16. Harland, D., Kürkçüoˇglu, S.: Equivariant reduction of Yang–Mills theory over the fuzzy sphere and the emergent vortices. Nucl. Phys. B 821, 380–398 (2009) 17. Khalkhali, M., Landi, G., van Suijlekom, W.D.: Holomorphic structures on the quantum projective line. Int. Math. Res. Not. 4, 851–884 (2010) 18. Klimyk, A., Schmüdgen, K.: Quantum Groups and their Representations. Berlin-Heidleberg-New York: Springer, 1997
Dimensional Reduction Over the Quantum Sphere
413
19. Landi, G., Reina, C., Zampini, A.: Gauged laplacians on quantum Hopf bundles. Commun. Math. Phys. 287, 179–209 (2009) 20. Lechtenfeld, O., Popov, A.D., Szabo, R.J.: Rank two quiver gauge theory, graded connections and noncommutative vortices. J. High Energy Phys. 0609, 054 (2006) 21. Lechtenfeld, O., Popov, A.D., Szabo, R.J.: Quiver gauge theory and noncommutative vortices. Prog. Theor. Phys. Suppl. 171, 258–268 (2007) 22. Majid, S.: Noncommutative riemannian and spin geometry of the standard q-sphere. Commun. Math. Phys. 256, 255–285 (2005) 23. Masuda, T., Nakagami, Y., Watanabe, J.: Noncommutative differential geometry on the quantum twosphere of P. Podle´s. I: An algebraic viewpoint. K-Theory 5, 151–175 (1991) 24. Masuda, T., Mimachi, K., Nakagami, Y., Noumi, M., Ueno, K.: Representations of the quantum group SUq (2) and the little q-Jacobi polynomials. J. Funct. Anal. 99, 357–387 (1991) 25. Nakajima, H.: Heisenberg algebra and Hilbert schemes of points on projective surfaces. Ann. Math. 145, 379–388 (1997) 26. Nekrasov, N.A., Schwarz, A.S.: Instantons on noncommutative R4 and (2, 0) superconformal six-dimensional theory. Commun. Math. Phys. 198, 689–703 (1998) 27. Neshveyev, S., Tuset, L.: A local index formula for the quantum sphere. Commun. Math. Phys. 254, 323–341 (2005) 28. Podle´s, P.: Quantum spheres. Lett. Math. Phys. 14, 193–202 (1987) 29. Podle´s, P.: Differential calculus on quantum spheres. Lett. Math. Phys. 18, 107–119 (1989) 30. Popov, A.D.: Non-abelian vortices on Riemann surfaces: An integrable case. Lett. Math. Phys. 84, 139– 148 (2008) 31. Popov, A.D., Szabo, R.J.: Quiver gauge theory of non-abelian vortices and noncommutative instantons in higher dimensions. J. Math. Phys. 47, 012306 (2006) 32. Schmüdgen, K., Wagner, E.: Dirac operator and a twisted cyclic cocycle on the standard Podle´s quantum sphere. J. Reine Angew. Math. 574, 219–235 (2004) 33. Schmüdgen, K., Wagner, E.: Representations of cross product algebras of Podle´s quantum spheres. J. Lie Theory 17, 751–790 (2007) 34. Tian, G.: Gauge theory and calibrated geometry. I. Ann. Math. 151, 193–268 (2000) 35. Wagner, E.: On the noncommutative spin geometry of the standard Podle´s sphere and index computations. J. Geom. Phys. 59, 998–1016 (2009) 36. Woronowicz, S.L.: Twisted SUq (2) group. An example of a noncommutative differential calculus. Publ. Rest. Inst. Math. Sci., Kyoto Univ. 23, 117–181 (1987) Communicated by A. Connes
Commun. Math. Phys. 308, 415–438 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1359-x
Communications in
Mathematical Physics
Liouville Integrability of a Class of Integrable Spin Calogero-Moser Systems and Exponents of Simple Lie Algebras Luen-Chau Li1 , Zhaohu Nie2,3 1 Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA.
E-mail:
[email protected]
2 Department of Mathematics, Pennsylvania State University, Altoona Campus, 3000 Ivyside Park,
Altoona, PA 16601, USA. E-mail:
[email protected]
3 Current address: Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA
Received: 4 August 2010 / Accepted: 8 June 2011 Published online: 21 October 2011 – © Springer-Verlag 2011
Abstract: In previous work, we introduced a class of integrable spin Calogero-Moser systems associated with the classical dynamical r-matrices with spectral parameter, as classified by Etingof and Varchenko for simple Lie algebras. Here the main purpose is to establish the Liouville integrability of these systems by a uniform method based on evaluating the primitive invariants of Chevalley on the Lax operators with spectral parameter. As part of our analysis, we will develop several results concerning the algebra of invariant polynomials on simple Lie algebras and their expansions.
1. Introduction Systems of spin Calogero-Moser (CM) type are Hamiltonian systems with very rich structures. After the initial example of Gibbons and Hermsen [GH], a variety of such systems have appeared in the literature over the years. (See, for example, [BAB1,BAB2, FP,HH,L1,L3,LX1,LX2,MP,Pech,P,Wo,Y] and the references therein.) This is a testimonial to the relevance of such systems in various areas of mathematics and physics. In [LX1,LX2], as a by-product of an effort to understand conceptually the calculations in [BAB1,BAB2], we introduced a class of spin CM systems associated with the classical dynamical r-matrices with spectral parameter, as defined and classified by Etingof and Varchenko for complex simple Lie algebras [EV]. The classical dynamical r-matrices with spectral parameter in [EV] are solutions of the classical dynamical Yang-Baxter equation (CDYBE) with spectral parameter, which was first introduced and studied by Felder [F]. While Felder studied CDYBE in the context of conformal field theory, we showed how to make use of the solutions of this equation to construct and to study our spin systems. Indeed, in [L2], we showed how to obtain the explicit solutions of the integrable spin CM systems in [LX2] by using the factorization method developed in [L1]. That this is possible is due to some remarkable geometric structures underlying the so-called modified dynamical Yang-Baxter equation (mDYBE) [L1]. This work is a
416
L.-C. Li, Z. Nie
sequel to [LX2] and [L2]. Our main purpose here is to establish the Liouville integrability of the integrable spin CM systems in [LX2] on generic symplectic leaves. The spin CM systems constructed in [LX1,LX2] are of three types: rational, trigonometric, and elliptic, as in the case of their spinless counterparts in [OP]. In the rational case, recall that we have a family of rational spin CM systems parametrized by subsets ⊂ which are closed with respect to addition and multiplication by −1. Here is the root system associated with a complex simple Lie algebra g and a fixed Cartan subalgebra h of g. In the trigonometric case, there is also a family but now the systems are parametrized by subsets π of a fixed simple system π ⊂ . Finally we have an elliptic spin CM system for each complex simple Lie algebra. In [LX1,LX2], generalized Lax operators taking values in the dual bundles A of the corresponding coboundary dynamical Lie algebroids A∗ were constructed for these systems. If we let H denote a connected Lie group corresponding to h, then recall that in each case, the phase space P of the spin CM system is a Hamiltonian H -space (with equivariant momentum map J ) which admits an H -equivariant realization in the corresponding A and the Hamiltonian is the pullback of a natural invariant function on A under the realization map. It is a characteristic of these systems that the pullback of natural invariant functions on A to P do not Poisson commute everywhere, but they do so on J −1 (0) in all cases. Hence we can obtain the integrable spin systems on J −1 (0)/H by Poisson reduction. It should be pointed out that in our setup, the Lax operator L is only part of the generalized Lax operator. Indeed, for the rational (resp. trigonometric) case, if = (resp. the span < π > = ), the equation of motion for L only carries partial information about the dynamics and it is necessary to obtain the missing piece from the other parts of the generalized Lax operator [L2]. Nevertheless, as the reader will see, the Lax operator suffices when we consider Liouville integrability. The method we use to establish the Liouville integrability of the integrable spin CM systems can be explained as follows. Let us begin with a familiar situation. In the usual classical r -matrix theory, it is well-known that if the Lax operator L of a finite-dimensional system takes values in a loop algebra Lg, (i.e., L depends on a spectral parameter), then one can obtain integrals in involution by pulling back the ad-invariant functions on Lg using L . However, due to the finite-dimensionality of the system, we cannot expect the integrals obtained in this way to be functionally independent or nonzero even though the ring of ad-invariant functions of Lg has an infinite number of generators. In the case when g ⊂ gl(n, C) for some n, say, then of course there is a standard way to construct a finite collection of Poisson commuting integrals. Namely, one simply writes down the characteristic polynomial of L(z) and in this case, the completeness of the integrals can be addressed by algebro-geometric means provided certain additional conditions are satisfied [RSTS]. (For an example where there exists a family of additional integrals besides the ones given by the characteristic polynomial of L(z), see [DLT].) In our case, we can of course take concrete representations of the simple Lie algebras; however, an intrinsic way to construct the necessary integrals which works for all simple Lie algebras is clearly preferred. What we essentially do in this work is to substitute the elementary symmetric functions in the matrix case by the primitive invariant polynomials of Chevalley, which are homogeneous with degrees related to the exponents of the simple Lie algebras. (See, for example, [C,K2,V].) To obtain the quantities of interest, we simply evaluate the primitive invariants on the Lax operator and these give Poisson commuting integrals on J −1 (0) by the general theory in [LX2,L1]. To count the number of nontrivial integrals obtained in this manner, our basic realization is that we can appeal to a theorem of Shephard and Todd [ST] relating the sum of exponents of a complex
Liouville Integrability of Integrable Spin CM Systems
417
simple Lie algebra to the number of roots of (g, h). Of course, there remains the task of showing that the nontrivial integrals are functionally independent on an open, dense set of the phase space. As the reader will see, we can also accomplish this in a uniform way due to some common structure which exists among the three types of spin CM systems. To conclude, we remark that the method which we develop here to construct and count the number of integrals is a general method. In principle, it should work for other systems associated with simple Lie algebras and with spectral parameter dependent Lax operators. Thus what we show in this work is just an illustration of this general method. Furthermore, some of our analysis involving invariant polynomials (see Lemma 3.1 and Theorem 6.4) may also be of independent interest in Lie theory. The paper is organized as follows. In Sect. 2, we present for the most part some background material for the reader’s convenience; we also take the opportunity to set up the notations. In the first subsection, we begin by summarizing some basic facts about the invariant polynomials of Chevalley and the exponents of simple Lie algebras which are of relevance here. We also recall some of the tools which we find useful in dealing with these invariant polynomials. (Further tools will be developed in subsequent sections.) In the second subsection, we recall the construction of the class of spin Calogero-Moser systems associated with the classical dynamical r-matrices with spectral parameter. We then explain how Poisson reduction gives rise to the associated integrable models. At the end of the subsection, we conclude with our first result, namely, the connection between the dimension of the maximal dimensional phase spaces of our systems and the exponents of the complex simple Lie algebras. In Sect. 3, we construct the integrals for the rational case by evaluating the primitive invariants on the Lax operators and we count the number of nontrivial integrals. As it turns out, for each primitive invariant Ik , exactly one of the quantities which arise in the expansion of Ik (L(z)) in z is identically zero in this case while another one is a Casimir function. In Sects. 4 and 5, we do the same for the trigonometric case and the elliptic case. Finally, in Sect. 6, we show that the integrals constructed in Sects. 3-5 are functionally independent, thus proving the Liouville integrability of the systems on generic symplectic leaves. 2. Preliminaries In [LX2], we introduced a class of integrable spin Calogero-Moser systems associated with the classical dynamical r-matrices with spectral parameter, as classified by Etingof and Varchenko [EV] for complex simple Lie algebras. Our goal in this section is to establish Proposition 2.2.6 which gives the dimension of the maximal dimensional phase spaces of such systems in terms of the exponents of the complex simple Lie algebras. For the reader’s convenience, we will provide some background material, we will also take the opportunity to set up the notations. In the first subsection, we will begin by summarizing a number of basic facts about the invariant polynomials of Chevalley and the exponents of simple Lie algebras. We will also collect here some of the tools which we find useful in dealing with these polynomials. In the second subsection, we will recall the class of integrable spin Calogero-Moser systems in [LX2]. Then we will present our first result which we alluded to above, thus tying together the two subsections. 2.1. The invariant polynomials of Chevalley and the exponents. Let g be a complex semisimple Lie algebra of rank N , and let G be a connected Lie group with Lie(G) = g. We recall that the group G acts on the algebra P(g) of polynomial functions on g by g · P = P g , g ∈ G, P ∈ P(g), where
418
L.-C. Li, Z. Nie
P g (x) = P(Adg−1 x), x ∈ g.
(2.1.1)
Let I (g) denote the ring of polynomial functions on g invariant under the above action of G. Then the well-known theorem of Chevalley [C] asserts that I (g) is generated by N algebraically independent homogeneous polynomials I1 , . . . , I N . In other words, if C[Y1 , . . . , Y N ] denotes the polynomial ring in the N variables Y1 , . . . , Y N , then I (g) = C[I1 , . . . , I N ].
(2.1.2)
Let us denote by h a fixed Cartan subalgebra of g, and let W be the Weyl group of the pair (g, h) generated by reflections in the hyperplanes in h. Then indeed the restriction of I (g) to h is an algebra isomorphism of I (g) onto the algebra P(h)W of polynomials on h which are invariant under W. Write deg Ik = dk , k = 1, . . . , N .
(2.1.3)
We will assume that the Ik ’s are ordered in the sense that d1 ≤ d2 ≤ · · · ≤ d N .
(2.1.4)
Following Kostant [K2], we will refer to the Ik ’s as the primitive invariants. The numbers m k = dk − 1, k = 1, . . . , N , are called the exponents of g and are basic invariants of g [B]. (See also [CM] and the references therein.) For the purposes of this work, we will need the following results due to Shephard and Todd [ST]. Theorem 2.1.1 [ST]. Let m k be the exponents of g, k = 1, . . . , N . Then N
m k = # of reflections in W
k=1
1 (# of roots of (g, h)) 2 1 = (dim g − N ). 2
=
(2.1.5)
As the reader will see, (2.1.5) is crucial in counting the total number of integrals which we construct by evaluating the primitive invariants on the Lax operators of the integrable spin Calogero-Moser systems. We will explain this in the next subsection below. While the problem of computing the exponents was originally motivated by the problem of computing the Betti numbers of complex simple Lie groups, it turns out that there is a different way to describe these numbers which is relevant for us. For this purpose, let us introduce some notation. First of all, we will assume from now on that g is simple with Cartan sublagebra h and Killing form (·, ·), and let g = h ⊕ α∈ gα be the root space decomposition of g with respect to h. We fix a simple system of roots π = {α1 , . . . , α N }, and denote by ± the corresponding positive/negative system. If α ∈ + , recall that we N n i αi , where n i are non-negative can express α uniquely as a sum of simple roots i=1 integers. The height of α is defined to be the number ht (α) =
N i=1
ni .
(2.1.6)
Liouville Integrability of Integrable Spin CM Systems
419
Theorem 2.1.2 [K1]. If b j is the number of α ∈ + such that ht (α) = j, then (a) b j − b j+1 is the number of times j appears as an exponent of g, (b) N = b1 ≥ b2 ≥ · · · ≥ bh−1 = 1, where h is the Coxeter number. Moreover, the partition of r = |+ | as defined by the above sequence of numbers is conjugate to the partition h − 1 = m N ≥ m N −1 ≥ · · · ≥ m 1 = 1. In the rest of the subsection, we will summarize a number of basic facts from [K2] that we will use in Sect. 3 and more significantly in Sect. 6 below. To begin with, we recall that the symmetric algebra S = S(g∗ ) can be identified with P(g). On the other hand, we can associate to each x ∈ g a differential operator ∂x on g, defined by d f (y + t x), f ∈ C ∞ (g). (2.1.7) ∂x f (y) = dt t=0 In this way we have a linear map x → ∂x which can be extended to an isomorphism from the symmetric algebra S∗ = S(g) to the algebra of differential operators ∂ with constant coefficients on g. From now on we will identify the two spaces and with this identification, we have a nondegenerate pairing between S∗ and S given by
∂, f = ∂ f (0),
(2.1.8)
where ∂ ∈ S∗ , f ∈ S, and ∂ f (0) denote the value of the function ∂ f at 0 ∈ g. It is obvious that both S∗ and S are graded: S∗ = ⊕ j≥0 S j , S = ⊕ j≥0 S j . If f ∈ S m and x ∈ g, it follows from the Taylor expansion that (∂ )m x , f = f (x). (2.1.9) m! Now S is a G-module by (2.1.1). On the other hand, it is clear that the adjoint action of G on g can be naturally extended to an action of G on S∗ . Therefore in view of (2.1.1), we have
g · ∂, g · f = ∂, f ,
(2.1.10)
for all g ∈ G, ∂ ∈ S∗ and f ∈ S. By differentiation, S and S ∗ become g-modules and the actions of g on both spaces are by derivations. Therefore we have the “product rule” and the “power rule”: x · (∂δ) = (x · ∂)δ + ∂(x · δ), x · ∂ n = n∂ n−1 (x · ∂),
(2.1.11) (2.1.12)
for all x ∈ g, ∂, δ ∈ S∗ , and n ∈ N. For y ∈ g, we also have x · ∂ y = ∂[x,y] .
(2.1.13)
Since the pairing between S∗ and S obeys (2.1.10), it follows that
x · ∂, f + ∂, x · f = 0
(2.1.14)
for all x ∈ g, f ∈ S. In particular, this implies that
x · ∂, f = 0, for all f ∈ I (g) since x · f = 0 for f ∈ I (g).
(2.1.15)
420
L.-C. Li, Z. Nie
Now let x0 be the unique element in h such that αi (x0 ) = 1, i = 1, . . . , N . Then α(x0 ) = ht(α), and [x0 , eα ] = ht(α)eα for all α ∈ . For each j ∈ Z, let ( j)
S∗ = {∂ ∈ S∗ |x0 · ∂ = j∂}. Then S∗ =
( j) j∈Z S∗ ,
(2.1.16)
( j)
and if ∂ ∈ S∗ , we will say ∂ has weight j. Clearly, we have (ht (α)) ∂eα ∈ S∗ , ∂ p ∈ S∗(0)
(2.1.17)
for α ∈ , p ∈ h. Also, ( j)
(i+ j)
S∗(i) S∗ ⊆ S∗
.
(2.1.18)
The following consequence of (2.1.15) is very important to us in Sect. 6 below, namely, ( j) it follows from the definition of S∗ in (2.1.16) that ( j)
∂ ∈ S∗ for j = 0 ⇒ ∂, f = 0 for all f ∈ I (g).
(2.1.19)
Analogously, we let g( j) be the eigenspace of ad x0 for the eigenvalue j. Then g = ⊕ j∈Z g( j) and [g(i) , g( j) ] ⊂ g(i+ j) .
(2.1.20)
2.2. A class of integrable spin Calogero-Moser systems and their phase spaces. We recall that g is a complex simple Lie algebra with Cartan subalgebra h, and ± are the positive/negative system relative to a fixed simple system π of roots. For each positive root α ∈ + , let eα ∈ gα and e−α ∈ g−α be root vectors which are dual with respect to (·, ·) so that [eα , e−α ] = Hα , where the latter is the unique element in h which corresponds to α under the isomorphism induced by the Killing form (·, ·). Wealso fix an orthonormal basis (xi )1≤i≤N of h, and write p = i pi xi , ξ = i ξi xi + α∈ ξα eα for p ∈ h and ξ ∈ g. Lastly, we let H be a connected Lie subgroup of G with Lie(H ) = h. Let r be a classical dynamical r-matrix with spectral parameter in the sense of [EV], with coupling constant equal to 1. We will fix a simply connected set U ⊂ h on which r (·, z) is holomorphic. By Proposition 4.5 of [LX2], we can construct the associated H -equivariant classical dynamical r-matrix R : U −→ L(Lg, Lg), where Lg is the loop algebra of g and L(Lg, Lg) is the space of linear maps on Lg. Indeed, it was established in [LX2] that R is a solution of the modified dynamical Yang-Baxter equation (mDYBE). Hence we can equip A∗ = T ∗ U × Lg∗ T U × Lg (Lg∗ is the restricted dual of Lg) with a Lie algebroid structure, the so-called coboundary dynamical Lie algebroid associated to R. Our construction of the class of spin Calogero-Moser system and its realization is based on the following result. Theorem 2.2.1 [LX2]. The map ρ = (m, τ, L) : A∗ T U × g −→ T U × Lg A given by # ρ(q, p, ξ ) = (q, − hξ, p + r− (q)ξ )
(2.2.1)
is an H-equivariant Poisson map, when the domain is equipped with the Lie-Poisson structure corresponding to the trivial Lie algebroid A T U × g, and the target is equipped with the Lie-Poisson structure corresponding to A∗ . Here, H acts on T U × g
Liouville Integrability of Integrable Spin CM Systems
421
and T U × Lg by acting on the second factors with the adjoint action and the map # (q) : g −→ Lg is defined by r− # (q)ξ )(z), η) = (r (q, z), η ⊗ ξ ) ((r−
(2.2.2)
for ξ, η ∈ g. Remark 2.2.2. (a) The Lie-Poisson structure on the dual of the trivial Lie algebroid A T U × g is given by {φ, ψ} A∗ (q, p, ξ ) = (δ2 φ, δ1 ψ) − (δ1 φ, δ2 ψ) + (ξ, [δφ, δψ]) [L1]. Thus the Poisson structure is a product structure, where T ∗ U T U is equipped with the canonical structure, and g∗ g is equipped with the LiePoisson structure. Moreover, the H -action on T U × g above is a canonical action with equivariant momentum map J : T U × g −→ h, (q, p, ξ ) → − hξ, where h is the projection map to h relative to the splitting g = h ⊕ h⊥ . (b) As a special case of Proposition 3.1 in [L1], the dual bundle A equipped with the Lie-Poisson structure and H -action as defined in the above theorem is also a Hamiltonian H -space. The corresponding equivariant momentum map γ : A −→ h is given by the simple formula γ (q, p, X ) = p. (c) The map ρ = (m, τ, L) in the above theorem is to be regarded as the generalized Lax operator of the corresponding spin CM system whose construction we recall below. The map L, on the other hand, is the Lax operator. Let Q be the quadratic function 1 Q(ξ ) = 2
(ξ(z), ξ(z)) c
dz , 2πi z
(2.2.3)
where c is a small circle around the origin. Clearly, Q is an ad-invariant function on Lg. Definition 2.2.3 [LX2]. Let r be a classical dynamical r-matrix with spectral parameter with coupling constant equal to 1. Then the Hamiltonian system on A∗ T U × g (equipped with the Lie-Poisson structure as in Theorem 2.2.1) generated by the H invariant Hamiltonian 1 dz H(q, p, ξ ) = (L ∗ Q)(q, p, ξ ) = (2.2.4) (L(q, p, ξ )(z), L(q, p, ξ )(z)) 2 c 2πi z is called the spin Calogero-Moser system associated to r . Note that the pullback of ad-invariant functions on Lg by the Lax operator L does not Poisson commute everywhere. In order to construct the integrable spin CM systems, we have to invoke Poisson reduction [MR,OR]. For this purpose, let Pri be the projection map onto the i th factor of U × h × Lg A, i = 1, 2, 3, and let π0 : J −1 (0) −→ J −1 (0)/H, π H : γ −1 (0) −→ γ −1 (0)/H be the canonical projections. If f is an ad-invariant function on Lg, the unique function on γ −1 (0)/H determined by Pr3∗ f |γ −1 (0) will be denoted by f¯, while the unique function on J −1 (0)/H determined by L ∗ f |J −1 (0) will be denoted by F0 . Because the map ρ is an H -equivariant Poisson map, it induces a unique Poisson map ρ : J −1 (0)/H −→ γ −1 (0)/H characterized by π H ◦ ρ|J −1 (0) = ρ ◦ π0 . From the various definitions, we have ∗ f¯, F0 ◦ π0 = L ∗ f |J −1 (0). F0 = ρ
(2.2.5)
In particular, the Hamiltonian H of the spin CM system in (2.2.4) drops down to H0 = ρ ∗ Q¯ on J −1 (0)/H.
422
L.-C. Li, Z. Nie
Theorem 2.2.4 [LX2,L1]. (a) The pullback of ad-invariant functions on Lg by L Poisson commute on J −1 (0). (b) Functions F0 = ρ ∗ f¯ corresponding to ad-invariant functions f on Lg Poisson commute on the reduced space J −1 (0)/H. Remark 2.2.5. (a) The reduced spaces J −1 (0)/H and γ −1 (0)/H are Poisson varieties in the sense of [OR]. (b) The existence of the Poisson map ρ and the formulation of Theorem 2.2.4 (b) follow from the general result in [L1]. In [LX2], we made an additional assumption, we also did not have γ −1 (0)/H available at the time. Thus the reduction picture obtained there was not an intrinsic one. (c) In [L1], we obtain an intrinsic expression for the Lie-Poisson structure on the dual bundle of a coboundary dynamical Lie algebroid, from which it is clear that functions which are obtained as pullback of ad-invariant functions under the map Pr3 Poisson commute on γ −1 (0). Thus in hindsight, the result in Theorem 2.2.4 (a) is just a consequence of this fact, Theorem 2.2.1, and the property that ρ(J −1 (0)) ⊂ γ −1 (0). We now restrict to a smooth component of J −1 (0)/H and for this purpose, we consider the following open submanifold of g: U = { ξ ∈ g | ξ−αi = (ξ, eαi ) = 0, i = 1, . . . , N }.
(2.2.6)
(Note the convention in (2.2.6) is opposite to that in [LX2].) Then the H -action above induces a Hamiltonian H -action on T U × U and we denote the corresponding momentum map also by J so that J −1 (0) = T U × (h⊥ ∩ U). Now recall from [LX2] that there exists an H -equivariant map g : U −→ H. Using this map, we can identify the reduced space J −1 (0)/H = T U ×(h⊥ ∩U/H ) with T U ×gr ed , where gr ed = + α∈−π Ceα , and = Nj=1 e−α j . Indeed, the identification map is given by (q, p, [ξ ]) → (q, p, Adg(ξ )−1 ξ ).
(2.2.7)
Thus the natural projection π0 : J −1 (0) −→ T U × gr ed is the map (q, p, ξ ) → (q, p, s = Adg(ξ )−1 ξ ).
(2.2.8)
Consequently, by Poisson reduction [MR], the Poisson structure on T U × gr ed is a product structure, where the second factor gr ed is equipped with the reduction (at 0) of the Lie-Poisson structure on U. Thus the symplectic leaves of T U × gr ed are of the form T U × Or ed , where Or ed = (O ∩ U ∩ h⊥ )/H and O is an orbit in g. Proposition 2.2.6. The generic symplectic leaves in T U × gr ed have dimension equal N mk . to dim g − N = 2 k=1 Proof. Clearly, the generic symplectic leaves in T U × gr ed correspond to generic orbits in g. So let O be a generic orbit in g and let Or ed be the corresponding reduction in gr ed . It is well-known that dim O = dim g − N . (See, for example, [K2].) Therefore, dim Or ed = dim O − 2N = dim g − 3N . Consequently, dimension of T U × Or ed = 2N + dim Or ed = dim g − N . To complete the proof, it remains to establish the equality dim g − N = 2 But this is just the assertion in (2.1.5).
(2.2.9) N
k=1 m k .
Liouville Integrability of Integrable Spin CM Systems
423
According to the above proposition, in order to establish the Liouville integrability of the integrable models associated with our spin Calogero-Moser systems, we have to N exhibit k=1 m k nontrivial integrals in involution which are functionally independent on open dense sets of the generic symplectic leaves of T U × gr ed . But as the reader will see, each of the primitive invariants Ik when evaluated on the Lax operators will give rise to m k = dk − 1 such integrals. Hence the total number of nontrivial conserved N quantities with the required properties is exactly k=1 m k . This explains the importance of (2.1.5).
3. The Rational Spin Calogero-Moser Systems The rational spin Calogero-Moser systems are associated with the rational dynamical r-matrices with spectral parameter r (q, z) =
1 + eα ⊗ e−α , z α(q)
(3.1)
α∈
where ⊂ is any set of roots which is closed with respect to addition and multiplication by −1, and ∈ (S 2 g)g is the Casimir element corresponding to the Killing form (·, ·). Therefore, the Hamiltonians are given explicitly by H(q, p, ξ ) =
1 2 1 ξα ξ−α pi − 2 2 α(q)2
(3.2)
α∈
i
and the corresponding Lax operators are of the form L(q, p, ξ )(z) = p +
ξα ξ eα + . α(q) z
(3.3)
α∈
From the homogeneity of Ik and the form of L(q, p, ξ )(z) above, Ik (L(q, p, ξ )(z)) can be expanded as Ik (L(q, p, ξ )(z)) =
dk
Ik j (q, p, ξ ) z − j .
(3.4)
j=0
As the reader will see, the Ik1 ’s are actually identically zero on J −1 (0). In order to demonstrate this for all cases, we need to establish the following lemma which is a refinement of (2.1.19) using just weights. Lemma 3.1. Let X ⊂ . Then for all f ∈ I (g), p ∈ h,
n mα ∂p ∂e α , f = 0 α∈X
unless
α∈X
m α α = 0.
(3.5)
424
L.-C. Li, Z. Nie
Proof. For any h ∈ h, it follows from (2.1.15) that
∂emαa , f = 0. h · ∂ np
(3.6)
α∈X
Since g acts on S∗ by derivation, we can use (2.1.11) and (2.1.12) and (2.1.13) to expand the left-hand side of (3.6). This gives
h · ∂ np ∂emαa , f α∈X
∂ n−1 p (h
=n
· ∂p)
α∈X
=n =
α∈X
m α α(h)
∂emαα ,
∂ np
f +
α∈X
α∈X
f +
∂emαα ,
α∈X
Therefore if
∂ n−1 p ∂[h, p]
mα
α∈X
mα
α∈X
∂ np
∂ np
β=α
β=α
m ∂eββ ∂emαα −1 (h
· ∂eα ), f
m ∂eββ ∂emαα −1 ∂[h,eα ] ,
f
∂emαα , f .
m α α = 0, we must have ∂ np α∈X ∂emαα , f = 0.
Proposition 3.2. For each 1 ≤ k ≤ N , Ik,dk (q, p, ξ ) = Ik (ξ ). Moreover, for (q, p, ξ ) ∈ J −1 (0), we have Ik1 (q, p, ξ ) = 0. Hence the number of nontrivial inteN grals Ik j (q, p, s) which Poisson commute on T U × Or ed is equal to k=1 m k , where Or ed is the reduction of a generic orbit O in U. Proof. From the homogeneity of Ik and the relation L(q, p, ξ )(z) = L(q, p, ξ )(∞) + ξz (see (3.3) above), we have Ik (L(q, p, ξ )(z)) = Ik (L(q, p, ξ )(∞)) + · · · +
1 Ik (ξ ) z dk
from which it is immediate that Ik,dk (q, p, ξ ) = Ik (ξ ). From the same expansion above and the definition of Ik1 , we also have Ik1 (q, p, ξ ) = lim z[Ik (L(q, p, ξ )(z)) − Ik (L(q, p, ξ )(∞))] z→∞ d = Ik (L(q, p, ξ )(∞) + tξ ) dt t=0 = (δ Ik (L(q, p, ξ )(∞)), ξ ).
(3.7)
Let us first consider the case where = . For (q, p, ξ ) ∈ J −1 (0), it is clear that we have ξα ξ = q, eα . α(q) α∈
Liouville Integrability of Integrable Spin CM Systems
425
Therefore, upon substituting into the above expression for Ik1 (q, p, ξ ), we find Ik1 (q, p, ξ ) = (δ Ik (L(q, p, ξ )(∞)), [q, L(q, p, ξ )(∞)]) =0 ¯ = \ be the complement as Ik is invariant. In the other case where = , let of , then ⎛ ⎞ Ik1 (q, p, ξ ) = ⎝δ Ik (L(q, p, ξ )(∞)), ξβ eβ ⎠ (3.8) ¯ β∈
because δ Ik (L(q, p, ξ )(∞)), β∈ ξβ eβ = 0 by the same reasoning as in the pre vious case. Now it is clear that the right-hand side of (3.8) is linear in β∈¯ ξβ eβ . In view of this, it suffices to show that ¯ δ Ik (L(q, p, ξ )(∞)), eβ = 0 for all β ∈ . (3.9) To this end, observe that (δ Ik (x), y) =
1
∂ dk −1 ∂ y , Ik (dk − 1)! x
(3.10)
for all x, y ∈ g. If we put x = L(q, p, ξ )(∞) and y = eβ in the above expression and dk −1 invoke the multinomial expansion to calculate ∂ L(q, p,ξ )(∞) , the result is δ Ik (L(q, p, ξ )(∞)), eβ m α ξα
α∈ α(q) m m ∂eαα ∂eβ , Ik . = ∂p m! α∈ m α ! m+
α∈
(3.11)
α∈
m α =dk −1
¯ we have But for β ∈ , β+
jα α = 0
(3.12)
α∈
for any choice of jα ∈ N, α ∈ because is closed under addition and multiplication by −1. Hence it follows from Lemma 3.1 that each individual term of the sum in (3.11) is equal to zero. This completes the proof that Ik1 (q, p, ξ ) = 0 for (q, p, ξ ) ∈ J −1 (0). On the other hand, it is a consequence of Proposition 6.8 in Sect. 6 that all the other Ik j ’s are not identically zero. Finally, since Ik (ξ ) are Casimir functions for 1 ≤ k ≤ N , the number of nontrivial integrals for each k is dk − 1 = m k . Remark 3.3. (a) Because the height function ht : −→ Z is not one-to-one, for this ¯ reason, we cannot conclude from (3.12) that ht (β)+ α∈ jα ht (α) = 0 for β ∈ , jα ∈ N, α ∈ . This is why we cannot infer the sum in (3.11) is zero from (2.1.19).
426
L.-C. Li, Z. Nie
(b) For the rational spin CM systems considered in this section, it was pointed out in [L2] that there exists a second realization in the dual bundle of a coboundary dynamical Lie algebroid. More precisely, define R : U −→ L(g, g) by ξα eα , R(q)ξ = − (3.13) α(q) α∈
then R is a solution of the CDYBE. Let A∗ T U × g be the coboundary dynamical Lie algebroid associated with R and let A T U × g be the trivial Lie algebroid. Then according to [L1], R : A∗ −→ A , (q, p, ξ ) → (q, hξ, − p + R(q)ξ )
(3.14)
is a morphism of Lie algebroids. Consequently, the dual map R∗ is an H -equivariant Poisson map, when the domain and target are equipped with the corresponding Lie-Poisson structures. Explicitly, R∗ (q, p, ξ ) = (q, − hξ, p − R(q)ξ ) = (q, − hξ, L(q, p, ξ )(∞)).
(3.15)
We would like to point out that the (spectral parameter independent) Lax operator L ∞ (q, p, ξ ) = L(q, p, ξ )(∞) coming out from this picture is of no use in proving Liouville integrability. This is because the number of integrals it gives is far from sufficient. The same remark also applies to the Lax operators of the hyperbolic spin CM systems in [L1] and the Lax operators of the spin CM systems associated with the Alekseev-Meinrenken dynamical r-matrices [AM] in [FP]. 4. The Trigonometric Spin Calogero-Moser Systems The trigonometric spin Calogero-Moser systems are the Hamiltonian systems in Definition 2.2.3 associated to the following trigonometric dynamical r-matrices with spectral parameter: r (q, z) = c(z) xi ⊗ xi − φα (q, z)eα ⊗ e−α , (4.1) α∈
i
where
and
c(z) = cot z
(4.2)
⎧ sin(α(q)+z) − , α ∈ < π > ⎪ ⎪ ⎨ sin α(q) sin z e−i z φα (q, z) = α ∈ π + − sin z, ⎪ ⎪ ⎩ ei z − sin α ∈ π −. z,
(4.3)
In (4.3) above, π is an arbitrary subset of the simple system π ⊂ , < π > is the root span of π and π ± = ± \ < π >± . Accordingly, the Lax operators are given by L(q, p, ξ )(z) = p + c(z) ξi xi − φα (q, z)ξα eα i
= p + c(z)ξ +
α∈
α∈
ψα (q)ξα eα ,
(4.4)
Liouville Integrability of Integrable Spin CM Systems
where
427
⎧ ⎨ c(α(q)), α ∈< π > −i, α ∈ π + ψα (q) = ⎩ +i, α ∈ π −.
(4.5)
Hence we have a family of dynamical systems parametrized by subsets π of π with Hamiltonians of the form: 1 1 2 1 1 ξα ξ−α pi − H(q, p, ξ ) = − 2 2 sin2 α(q) 3 i
5 − 6
α∈<π >
ξα ξ−α −
α∈\<π >
1 2 ξi . 3
(4.6)
i
Now, from the homogeneity of Ik and (4.4), we have the expansion Ik (L(q, p, ξ )(z)) =
dk
Ik j (q, p, ξ )(c(z)) j .
(4.7)
j=0
Proposition 4.1. For each 1 ≤ k ≤ N , Ik,dk (q, p, ξ ) = Ik (ξ ). If in addition, (q, p, ξ ) ∈ J −1 (0), then the following relation holds: Ik j (q, p, ξ ) i j = 0. (4.8) j odd
Therefore, the number of nontrivial integrals Ik j (q, p, s), j = 1, which Poisson commute on T U × Or ed is equal to k=1 m k , where Or ed is the reduction of a generic orbit O in U. Proof. As in the proof of Proposition 3.2, it is easy to show that Ik,dk (q, p, ξ ) = Ik (ξ ) and this is a Casimir function for each k. To establish the relation (4.8) for (q, p, ξ ) ∈ J −1 (0), we divide into two cases. First, consider π = π. In this case, we have L(q, p, ξ )(±i∞) = p + c(α(q))ξα eα ∓ iξ. α∈
Therefore, on using the relation (c(α(q)) − i)e2iα(q) = c(α(q)) + i, we find that Ade2iq L(q, p, ξ )(i∞) = L(q, p, ξ )(−i∞). As a consequence, we obtain Ik (L(q, p, ξ )(i∞)) = Ik (L(q, p, ξ )(−i∞)) from which (4.8) follows upon using (4.7). Now, consider the case π = π. We will establish (4.8) in this case through a limiting procedure. For this purpose, we define c(α(q − q0 ))ξα eα + c(z)ξ, q0 ∈ h. L q0 (q, p, ξ )(z) = p + α∈
428
L.-C. Li, Z. Nie
Then as above, if Ik (L q0 (q, p, ξ )(z)) =
dk
q
Ik j0 (q, p, ξ )(c(z)) j ,
j=0
we have
q
Ik j0 (q, p, ξ ) i j = 0.
j odd
. . . , ω N be the fundamental weights (with respect to π ). We set q0 = Now, let ω1 , q0 (t) = −it α j ∈π / Hω j (cf. [EV]). By using the relation αi (Hω j ) = (αi , ω j ) =
(αi , α j ) δi j , 2
we find that lim c(α(q − q0 (t))) = ψα (q).
t→∞
Therefore, lim L q0 (t) (q, p, ξ )(z) = L(q, p, ξ )(z),
t→∞
and so Ik j (q, p, ξ ) = limt→∞ I q0 (t) (q, p, ξ ). Hence we obtain (4.8) upon passing to q (t) the limit as t → ∞ in the relation j odd Ik j0 (q, p, ξ ) i j = 0. By the same reason as in Proposition 3.2, all the Ik j ’s are not identically zero in this case. Finally, since we can express Ik1 in terms of Ik3 , · · · through (4.8), the count follows. 5. The Elliptic Spin Calogero-Moser Systems Let ℘ (z) be the Weierstrass ℘-function with periods 2ω1 ,2ω2 ∈ C, and let σ (z), ζ (z) be the related Weierstrass sigma-function and zeta-function, respectively. The elliptic spin Calogero-Moser system is the spin Calogero-Moser system associated with the elliptic dynamical r-matrix with spectral parameter r (q, z) = ζ (z) xi ⊗ xi − l(α(q), z)eα ⊗ e−α , (5.1) α∈
i
where l(w, z) = −
σ (w + z) . σ (w)σ (z)
(5.2)
Explicitly, the Hamiltonian is given by 1 2 1 pi − ℘ (α(q))ξα ξ−α H(q, p, ξ ) = 2 2 and its Lax operator is of the form L(q, p, ξ )(z) = p + ζ (z)
(5.3)
α∈
i
i
ξi xi −
l(α(q), z)ξα eα .
α∈
From now on, we will restrict our attention to (q, p, ξ ) ∈ J −1 (0).
(5.4)
Liouville Integrability of Integrable Spin CM Systems
429
Proposition 5.1. For each 1 ≤ k ≤ N , Ik (L(q, p, ξ )(z)) is an elliptic function of z with poles of order dk at the points of the rank 2 lattice = 2ω1 Z + 2ω2 Z.
(5.5)
Hence Ik (L(q, p, ξ )(z)) can be expanded in the form dk (−1) j Ik j (q, p, ξ )℘ ( j−2) (z). Ik (L(q, p, ξ )(z)) = Ik0 (q, p, ξ ) + ( j − 1)!
(5.6)
j=2
Proof. Let ηi = ζ (ωi ), i = 1, 2. Then from l(α(q), z + 2ωi ) = e2ηi α(q)l(α(q), z) and e2ηi α(q) eα = Ade2ηi q eα , we have L(q, p, ξ )(z + 2ωi ) = Ade2ηi q L(q, p, ξ )(z), i = 1, 2. Therefore, Ik (L(q, p, ξ )(z)) is a doubly-periodic function of z. As L(q, p, ξ )(z) is meromorphic with simple poles at the points of the lattice = 2ω1 Z + 2ω2 Z, it follows from the homogeneity of Ik that Ik (L(q, p, ξ )(z)) is an elliptic function of z with poles of order dk at the points of . The expansion of Ik (L(q, p, ξ )(z)) then follows from standard argument in the theory of elliptic functions. Proposition 5.2. For each 1 ≤ k ≤ N , Ik,dk (q, p, ξ ) = Ik (ξ ). Hence the number of nontrivial integrals Ik j (q, p, s) which Poisson commute on T U × Or ed is equal to N k=1 m k , where Or ed is the reduction of a generic orbit O in U. Proof. In a deleted neighborhood of z = 0, we have 1 l(α(q), z) = − + ζ (α(q)) + higher order terms z from which it follows that L(q, p, ξ )(z) = p +
ξ + ζ (α(q))ξα eα + higher order terms. z
(5.7)
α∈
Therefore, on invoking the homogeneity of Ik , we obtain the following expansion in a deleted neighborhood of z = 0: Ik (L(q, p, ξ )(z)) =
1 Ik (ξ ) + O(1). z dk
But on the other hand, we have ℘ ( j−2) (z) = (−1) j
( j − 1)! + O(1) zj
for j = 2, . . . , dk . Consequently, it follows from (5.6) that we also have Ik (L(q, p, ξ )(z)) = z −dk Ik,dk (q, p, ξ ) + O(1) in a deleted neighborhood of z = 0. Comparing the two expansions of Ik (L(q, p, ξ )(z)), the first assertion follows. The second assertion is now obvious as none of the coefficients in the expansion (5.6) is identically zero by Proposition 6.8.
430
L.-C. Li, Z. Nie
6. Functional Independence of the Integrals and Liouville Integrability As the reader will see, we can establish the functional independence of the integrals for all three cases in a uniform way. For (q, p, ξ ) ∈ J −1 (0), we begin with the observation (see (3.3), (4.4) and (5.7)) that the Lax operator can be expressed in the following form: L(q, p, ξ )(z) = p + h(z)ξ + k0 (q, ξ ) + k1 (q, ξ, z) in a deleted neighborhood of 0, where 1 , in the rational/elliptic case h(z) = z c(z), in the trigonometric case,
(6.1)
(6.2)
and k0 (q, ξ ) =
⎧ ⎪ ⎨ α∈ α∈ ⎪ ⎩
ξα α(q) eα ,
in the rational case
ψα (q)ξα eα ,
α∈ ζ (a(q))ξα eα ,
in the trigonometric case
(6.3)
in the elliptic case,
and lastly, k1 (q, ξ, z) =
0, ∞
in the rational/trigonometric case
i=1 k1i (q, ξ )z
i,
in the elliptic case.
(6.4)
By using (2.1.9) and the above, it follows from the multinomial expansion that Ik (L(q, p, ξ )(z)) 1 a j = ∂ p ∂ξ (∂k0 (q,ξ ) + ∂k1 (q,ξ,z) )b , Ik h(z) j . j!a!b!
(6.5)
a+b+ j=dk
We will split the second line of the above expression into a sum of two terms Ik (L(q, p, ξ )(z)) = Fk ( p, ξ, z) + Rk (q, p, ξ, z),
(6.6)
where Fk ( p, ξ, z) =
a+ j=dk
1 a j ∂ ∂ , Ik h(z) j j!a! p ξ
(6.7)
and Rk (q, p, ξ, z) =
a+b+ j=dk
1 a j ∂ p ∂ξ (∂k0 (q,ξ ) + ∂k1 (q,ξ,z) )b , Ik h(z) j . j!a!b!
(6.8)
b≥1
Clearly, we have Fk ( p, ξ, z) =
dk j=0
Fk j ( p, ξ )h(z) j ,
(6.9)
Liouville Integrability of Integrable Spin CM Systems
431
where Fk j ( p, ξ ) =
1 d −j j ∂ pk ∂ξ , Ik j!(dk − j)!
(6.10)
for each j. Therefore these functions are the same in all three cases and the degree of Fk j ( p, ξ ) in the variable p is equal to dk − j. On the other hand, Rk (q, p, ξ, z) =
dk
Rk j (q, p, ξ )h(z) j + Rk (q, p, ξ, z),
(6.11)
j=0
where Rk (q, p, ξ, z) is identically zero in the rational/trigonometric case and is given by a power series in z which vanishes at 0 in the elliptic case. From the formulas in (6.3), (6.4), it is clear that Rk j is given by a different formula for each of the three cases. However, these play no role in our analysis. For us, the only piece of information which is needed is the degree of Rk j in the variable p and according to (6.8) and (6.4), this is at most equal to dk − j − 1 (and hence is less than that of Fk j ). We next turn to the definitions of the Ik j ’s in (3.4), (4.7) and (5.6) for the three cases. By comparing these expressions with (6.6), (6.9)–(6.11), we find that Ik j (q, p, ξ ) = Fk j ( p, ξ ) + Rk j (q, p, ξ ),
j = 1, . . . , dk
(6.12)
in all three cases. The relation also holds for j = 0 for the rational/trigonometric case but for the elliptic case, we have Ik0 (q, p, ξ ) ≡ Fk0 ( p, ξ ) + Rk0 (q, p, ξ ),
(6.13)
where ≡ means the two sides differ by a linear combination of Ik j (q, p, ξ ) for j ≥ 4 and even. That this is so is due to contributions from the constant terms in the Laurent series expansions of ℘ ( j−2) (z) on the right-hand side of (5.6) for j ≥ 4 and even. Proposition 6.1. The functional independence of Fk j ( p, ξ ), j = 0, 1, . . . , dk , k = 1, . . . , N on an open dense set of h × (U ∩ h⊥ ) implies the functional independence of Ik j (q, p, ξ ), j = 0, 1, . . . , dk , k = 1, . . . , N on an open dense set of T U × (U ∩ h⊥ ). Proof. Suppose the Ik j ’s are functionally dependent. Then there exists an analytic function f (u 1 , . . . , u d ) depending on d = 21 (dim g+N ) variables such that f (Ik j (q, p, ξ )) = 0. Fix a point q = q0 ∈ U , then f (Ik j (q0 , p, ξ )) = 0 is a functional dependence relation q among the polynomials Ik j0 ( p, ξ ) := Ik j (q0 , p, ξ ) in p and ξ. Since analytic dependence implies algebraic dependence for polynomials (see, for example, [W] and the references therein), we can assume that f is a polynomial in the variables u 1 , . . . , u d . Now the highest order term in p in the expression f (Ik j (q0 , p, ξ )) is of the form g(Fk j ( p, ξ )) n1 · · · I Nn dd N , the highest order term for a summand g of f, since for each monomial I10 n1 in p is given by F10 · · · FNn dd N . Furthermore, since f is not identically zero, neither is g. But f (Ik j (q, p, ξ )) = 0 implies g(Fk j ( p, ξ )) = 0, hence the Fk j ’s are functionally dependent. In what follows, we will establish the functional independence of Fk j ( p, ξ ), j = 0, 1, . . . , dk , k = 1, . . . , N on an open dense set of h × (U ∩ h⊥ ). The following is a lemma which is very useful in some of our calculations.
432
L.-C. Li, Z. Nie
Lemma 6.2. Let f ∈ I (g), then for all x, y, z ∈ g, and all m, n ≥ 0, we have
∂xm ∂[x,y] ∂zn , f =
n
∂ m+1 ∂[y,z] ∂zn−1 , f , m+1 x
(6.14)
where by convention the right-hand side of the formula is zero when n = 0. Proof. By using (2.1.13), (2.1.12) back and forth and (2.1.11), we find for n ≥ 1 that
∂xm ∂[x,y] ∂zn , f = − ∂xm (y · ∂x )∂ n , f 1
(y · ∂xm+1 )∂zn , f =− m+1 1 1
y · (∂xm+1 ∂zn ), f +
∂ m+1 (y · ∂zn ), f =− m+1 m+1 x n
∂ m+1 (y · ∂z )∂zn−1 , f = m+1 x n
∂ m+1 ∂[y,z] ∂zn−1 , f , = m+1 x where we have used (2.1.15) in addition to the “power rule” in going from the third line to the fourth line. When n = 0, the calculation stops in the second line for we can invoke (2.1.15) to conclude that the resulting expression is equal to zero. Proposition 6.3. For all 1 ≤ k ≤ N , ( p, ξ ) ∈ h × (U ∩ h⊥ ), (a) Fk0 ( p, ξ ) = Ik ( p), (b) Fk1 ( p, ξ ) = 0, (c) Fk,dk ( p, ξ ) = Ik (ξ ). Proof. The assertions in (a) and (c) are obvious. For (b), we use the representation in (6.10) together with the fact that ∂ξ has no weight zero part for ξ ∈ h⊥ . The assertion therefore is a consequence of (2.1.19). In order to set up our calculation, we will arrange the variables and the functions Fk j in some definite order. Note that for each 1 ≤ k ≤ N , the number of Fk j ( p, ξ )’s with j = 1 is equal to dk . Therefore we have a partition given by the sequence h = d N ≥ d N −1 ≥ · · · ≥ d1 = 2.
(6.15)
Since dk = m k + 1, it is easy to show from Theorem 2.1.2 that the above sequence is conjugate to the partition N = b0 = b1 ≥ b2 ≥ · · · ≥ bh−1 = 1.
(6.16)
The ordering of the Fk j ’s which we will use is the following: F10 , . . . , FN 0 ; F12 , . . . , FN 2 ; FN −b2 +1,3 , . . . , FN 3 ; · · · ; FN ,d N .
(6.17)
Clearly, for each value of j ≥ 2, the number of functions in each group {Fk j } is precisely b j−1 from our discussion above. Now for each 1 ≤ j ≤ h − 1, let us denote the roots with height equal to j by α j,i , i = 1, . . . , b j . We will order the variables as depicted in the following: p1 , . . . , p N ; ξα1 , . . . , ξα N ; ξα2,1 , . . . , ξα2,b2 ; · · · ; ξαh−1,1 .
(6.18)
Liouville Integrability of Integrable Spin CM Systems
433
Theorem 6.4. The functions Fk j ( p, ξ ), j = 0, 1, . . . , dk , k = 1, . . . , N are functionally independent on an open dense set of h × (U ∩ h⊥ ). To prove this assertion, we will compute the coefficient of dp1 ∧ · · · ∧ dp N ∧ dξα1 ∧ · · · ∧ dξα N ∧ · · · ∧ dξαh−1,1 in the expression for d F10 ∧ · · · ∧ d FN 0 ∧ d F1,2 ∧ · · · ∧ d FN 2 ∧ · · · ∧ d FN d N at the points of h × ( + n), where is as in Sect. 2.2 and n is the nilpotent subalgebra α∈+ gα . Note that the choice of + n follows Kostant in [K2]. Indeed, if e+ = α∈π cα eα , cα = 0 for all α ∈ π, then Kostant showed that the N -dimensional plane o = + ge+ ⊂ + n is a global cross-section of the generic orbits in g in the sense that each such orbit intersects o at precisely one point and no two distinct points in o are conjugate. This is the reason why it suffices to consider h × ( + n). Remark 6.5. For g = sl(N +1, C), the generic orbits can be characterized as those orbits through matrices whose characteristic polynomial and minimal polynomial coincide. In this case, we can take o to be the set of companion matrices and the result of Kostant which we quoted above is well-known in matrix theory. (See, for example, [HJ].) The computation which we referred to above will be achieved in a sequence of propositions. First of all, the coefficient which we want to compute is the determinant of a square (block) matrix D of partial derivatives whose diagonal blocks are given by D0 =
∂ Fl0 ∂ pi
N , Dj =
∂ FN −b j +l, j+1 ∂ξα j,i
l,i=1
b j ,
j = 1, . . . , h − 1,
(6.19)
l,i=1
in that order. Proposition 6.6. At the points ( p, ξ ) ∈ h × ( + n), (a) Fk0 does not depend on ξα for all α ∈ + , (b) for j ≥ 2, Fk j does not depend on ξα for those α with ht (α) ≥ j, and it depends linearly on ξα for those α with ht (α) = j − 1, (c) D is block lower-triangular, i.e., ⎛ ⎜ ⎜ D=⎜ ⎝
⎞
D0 D1 ∗
⎟ ⎟ ⎟, ⎠
0 D2
···
(6.20)
Dh−1 and the square blocks D j defined in (6.19) depend only on p. Proof. (a) This is just a consequence of Proposition 6.3 (a). (b) Thispart follows from weight consideration. Apply (6.10) with ξ = + ξ + = + α∈+ ξα eα and apply the binomial expansion to calculate (∂ + ∂ξ + ) j ; we have
434
L.-C. Li, Z. Nie d −j
Fk j ( p, ξ ) ≡ ∂ pk
∂j , Ik + j
d −j
ξα ∂ pk
∂j−1 ∂eα , Ik + O(ξ 2 ).
(6.21)
α∈+
Here the notation a ≡ b is a shorthand for a = λb for some λ = 0 and we will henceforth use this shorthand. On the other hand, the reminder term O(ξ 2 ) involves terms which are at least quadratic in the components of ξ + . From (2.1.17) d −j j and (2.1.18), ∂ pk has weight 0 while ∂ has weight − j. Therefore the first term j−1 in (6.21) is zero by (2.1.19). If ht (α) ≥ j, then ∂ ∂eα has weight strictly bigger dk − j j−1 than 0 and therefore the corresponding term ∂ p ∂ ∂eα , Ik in (6.21) is zero j−1 by (2.1.19). On the other hand, if ht (α) = j − 1, the operator ∂ ∂eα has weight 0 and therefore the corresponding ξα appears linearly in Fk j . Finally, it is clear that the term O(ξ 2 ) does not depend on ξα for α with height greater or equal to j. This completes the argument. (c) This immediately follows from the assertions in (a), (b) and (6.19). We next compute the values of the determinants |D j |, j = 0, . . . , h − 1. For this purpose, we have to study the diagonal blocks of D more closely. Proposition 6.7. At the points ( p, ξ ) ∈ h × ( + n), the following properties hold. (a) For 1 ≤ l, i ≤ b j , the element D j (l, i) of D j in the (l, i) position has degree d N −b j +l − j − 1 in p. (b) The first b j − b j+1 rows of D j are constants. (When b j+1 = b j , this just means that there are no constant rows.) Indeed, when b j − b j+1 > 0, we have the formula D j (l, i) ≡ ∂j ∂eα j,i , I N −b j +l , for 1 ≤ l ≤ b j − b j+1 .
(6.22)
Proof. (a) For j = 0, the assertion is clear because Fl0 is homogeneous of degree dl in p by Proposition 6.3 (a). For j ≥ 2, it follows from (6.10), (6.19) and (6.21) that D j (l, i) = ≡
∂ FN −b j +l, j+1 ∂ξα j,i d N −b +l − j−1 j
∂ p j ∂ ∂eα j,i , I N −b j +l .
(6.23)
Hence the degree of D j (l, i) in p is d N −b j +l − j − 1. (b) If b j − b j+1 > 0, we have m k = j for N − b j + 1 ≤ k ≤ N − b j+1 from Theorem 2.1.2 (a) which implies d N −b j +l = j + 1 for l = 1, . . . , b j − b j+1 . Thus D j (l, i) is of degree 0 in p for l = 1, . . . , b j − b j+1 from (6.23), i.e., they are constants. Proposition 6.8. Let +j denote the set of positive roots of height j. Then on h, we have the recursion relations: N αi ≡ |D0 |, (a) |D1 | i=1 (b) |D j | α∈+ α ≡ |D j−1 | for j ≥ 2, j
where the proportionality constants in (a) and (b) are independent of p ∈ h.
Liouville Integrability of Integrable Spin CM Systems
Therefore |D j |( p) ≡
435
α( p), j = 0, 1, . . . , h − 1
(6.24)
ht (α)> j with the convention that |Dh−1 |( p) ≡ 1. Hence |D j |( p) = 0 for p ∈ h , j = 0, 1, . . . , h − 1, where h is the open, dense set of regular points of h. Proof. It is a classical result that |D0 |( p) ≡ α∈+ α( p) and the regular points of h are precisely those points where |D0 |( p) = 0. (See [S] and [K2].) Therefore, if we can establish the recursion relations, it will follow from this result that |D j |( p) = 0 for p ∈ h , j = 0, 1, . . . , h − 1. (a) Let Hi = Hαi , i = 1, . . . , N . Since the Hi ’s form a basis of h, the determinant |D0 | in (6.19) can be computed in this basis up to a nonzero scale. That is, |D0 |( p) = ∂ dpl −1 ∂xi , Il l,i ≡ ∂ dpl −1 ∂ Hi , Il l,i . But from (6.23), (6.14) and the relation [eαi , ] = Hi , we have αi ( p)D1 (l, i)( p) ≡ αi ( p) ∂ dpl −2 ∂ ∂eαi , Il = ∂ dpl −2 ∂ ∂[ p,eαi ] , Il ≡ ∂ dpl −1 ∂[eαi ,] , Il = ∂ dpl −1 ∂ Hi , Il . Hence the formula follows from the property of determinants. (b) Consider the root vector eα j,i . Clearly we have eα j,i ∈ g( j) and ∈ g(−1) . (See the definition at the end of Sect. 2.1.) Therefore [ eα j,i , ] ∈ g( j−1) by (2.1.20). Hence we can write b j−1
[ eα j,i , ] =
a j,n,i eα j−1,n ,
(6.25)
n=1
where the coefficients on the right-hand side are not all zero. Indeed, it follows from (4.4.3) and the proof of Proposition 19 in [K1] that ker (ad ) ∩ n = 0 and therefore the b j−1 × b j matrix A j = (a j,n,i )n,i is of full rank. Now, by making use of the formula for D j (l, i) in (6.23), it follows by applying (6.14) and (6.25) that d N −b j +l − j−1 j ∂ ∂eα j,i , I N −b j +l
α j,i ( p)D j (l, i)( p) ≡ α j,i ( p) ∂ p
d N −b j +l − j−1 j ∂ ∂[ p,eαi,i ] , I N −b j +l
= ∂ p
d N −b j +l − j
≡ ∂ p
b j−1
=
∂j−1 ∂[eα j,i ,] , I N −b j +l d N −b j +l − j
a j,n,i ∂ p
∂j−1 ∂eα j−1,n , I N −b j +l
n=1 b j−1
=
a j,n,i D j−1 (l + b j−1 − b j , n)( p).
n=1
We now divide the proof into two cases.
(6.26)
436
L.-C. Li, Z. Nie
Case 1. b j−1 = b j . In this case, we have D j ( p) diag (α j,1 ( p), . . . , α j,b j ( p)) ≡ D j−1 ( p)A j
(6.27)
from (6.26) above and the matrix A j is invertible. Therefore, when we take the determinant of both sides of (6.27), we obtain the desired formula. Case 2. b j−1 > b j . In this case, (6.26) can be rewritten as D j ( p) diag (α j,1 ( p), . . . , α j,b j ( p)) ≡ D j−1 ( p)A j ,
(6.28)
where D j−1 ( p) is the b j × b j−1 submatrix of D j−1 ( p) obtained by deleting its first b j−1 − b j rows. Now, recall that the first b j−1 − b j rows of D j are constants in this case by Proposition 6.7 (b). Consequently, for 1 ≤ l ≤ b j−1 − b j , it follows by using (6.22) and by reversing the steps in the kind of calculation in (6.26) that b j−1
b j−1
a j,n,i D j−1 (l, n)( p) ≡
n=1
=
a j,n,i ∂j−1 ∂eα j−1,n , I N −b j−1 +l
n=1
∂j−1 ∂[eα j,i ,] , I N −b j−1 +l
= 0,
(6.29)
where we have used (6.14) in the n = 0 case and (6.25) in going from the first line to the second line. By combining (6.28) and (6.29), we conclude that 0 = D j−1 ( p)A j . (6.30) D j ( p) diag (α j,1 ( p), . . . , α j,b j ( p)) But since the b j−1 × b j matrix A j has full rank, we can extend it to an invertible b j−1 × b j−1 matrix " A j by adjoining b j−1 − b j column vectors from the canonical basis of Cb j−1 on the right-hand side of A j . In this way, we obtain from (6.30) that 0 # = D j−1 ( p) " Aj. (6.31) D j ( p) diag (α j,1 ( p), . . . , α j,b j ( p)) ∗ Therefore, on taking the determinants of both sides of (6.31), we again obtain the desired formula. This proves Theorem 6.4 as |D|( p) ≡
h−2
⎛ ⎝
j=0
=
⎞ α( p)⎠
ht (α)> j α( p)ht (α) = 0
(6.32)
α∈+
for p ∈ h . As a consequence of Theorem 6.4 and Proposition 6.1, we obtain the following corollary.
Liouville Integrability of Integrable Spin CM Systems
437
Corollary 6.9. The Poisson commuting integrals Ik j (q, p, ξ ), j = 0, 1, . . . , dk , k = 1, . . . , N on T U × (U ∩ h⊥ ) are functionally independent on an open dense set of T U × (U ∩ h⊥ ). Finally we are ready to state the main theorem of this work. Theorem 6.10. The reduction of the rational, trigonometric and elliptic spin Calogero-Moser systems to J −1 (0)/H T U × gr ed are Liouville integrable on the generic symplectic leaves of T U × gr ed . Proof. With the identification J −1 (0)/H T U × gr ed , the conserved quantities in involution are given by Ik j (q, p, s), where s ∈ gr ed . Therefore the number of nontrivial integrals required for Liouville integrability is exactly one-half the dimension of the generic symplectic leaves of T U × gr ed for each of the three cases. (See Proposition 3.2, 4.1 and 5.2) Finally, the functional independence of the integrals follows from Corollary 6.9 above. References [AM]
Alekseev, A., Meinrenken, E.: Clifford algebras and the classical dynamical Yang-Baxter equation. Math. Res. Lett. 10, 253–268 (2003) [B] Bott, R.: An application of Morse theory to the topology of Lie groups. Bull. Soc. Math. Fr. 84, 251–258 (1956) [BAB1] Billey, E., Avan, J., Babelon, O.: The r-matrix structure of the Euler-Calogero-Moser model. Phys. Lett. A 186, 263–271 (1994) [BAB2] Billey, E., Avan, J., Babelon, O.: Exact Yangian symmetry in the classical Euler-Calogero-Moser model. Phys. Lett. A 188, 263–271 (1994) [C] Chevalley, C.: Invariants of finite groups generated by reflections. Amer. J. Math. 77, 778–782 (1955) [CM] Collingwood, D., McGovern, W.: Nilpotent orbits in semisimple Lie algebras. New York: Van Nostrand Rheinhold (1993) [DLT] Deift, P., Li, L.-C., Tomei, C.: Matrix factorization and integrable systems. Comm. Pure Appl. Math. 42, 443–521 (1989) [EV] Etingof, P., Varchenko, A.: Geometry and classification of solutions of the classical dynamical Yang-Baxter equation. Commun. Math. Phys. 192, 77–120 (1998) [GH] Gibbons, J., Hermsen, T.: A generalization of the Calogero-Moser systems. Physica D 11D, 337– 348 (1984) [F] Felder, G.: Conformal field theory and integrable systems associated to elliptic curves. Proc. ICM Zurich, Basel: Birkhäuser, 1994, pp. 1247–1255 [FP] Feher, L., Pusztai, G.: Spin Calogero-Moser models obtained from dynamical r-matrices and geodesic motion. Nucl. Phys. B 734, 304–325 (2006) [HH] Ha, Z.N.C., Haldane, F.D.M.: On models with inverse-square exchange. Phys. Rev. B 46, 9359– 9368 (1992) [HJ] Horn, R., Johnson, C.: Matrix analysis. Cambridge: Cambridge University Press, 1985 [K1] Kostant, B.: The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group. Amer. J. Math. 81, 973–1032 (1959) [K2] Kostant, B.: Lie group representations on polynomial rings. Amer. J. Math. 85, 327–404 (1963) [L1] Li, L.-C.: A family of hyperbolic spin Calogero-Moser systems and the spin Toda lattices. Comm. Pure Appl. Math. 57, 791–832 (2004) [L2] Li, L.-C.: A class of integrable spin Calogero-Moser systems II:exact solvability. IMRP Int. Math. Res. Pap. 2006, Art. ID 62058, 53 pp [L3] Li, L.-C.: Poisson involutions, spin Calogero-Moser systems associated with symmetric Lie subalgebras and the symmetric space spin Ruijsenaars-Schneider models. Commun. Math. Phys. 265, 333–372 (2006) [LX1] Li, L.-C., Xu, P.: Spin Calogero-Moser systems associated with simple Lie algebras. C. R. Acad. Sci. Paris, Série I 331, 55–60 (2000) [LX2] Li, L.-C., Xu, P.: A class of integrable spin Calogero-Moser systems. Commun. Math. Phys. 231, 257–286 (2002)
438
[MR] [MP] [OP] [OR] [P] [Pech] [RSTS]
[ST] [S] [V] [W] [Wo] [Y]
L.-C. Li, Z. Nie
Marsden, J., Ratiu, T.: Reduction of Poisson manifolds. Lett. Math. Phys. 11, 161–169 (1986) Minahan, J.A., Polychronakos, A.: Interacting Fermion systems from two-dimensional QCD. Phys. Lett. B 326, 288–294 (1994) Olshanetsky, M., Perelomov, A.M.: Completely integrable Hamiltonian systems connected with semisimple Lie algebras. Invent. Math. 37, 93–108 (1976) Ortega, J.-P., Ratiu, T.: Singular reduction of Poisson manifolds. Lett. Math. Phys. 46, 359–372 (1998) Polychronakos, A.: Calogero-Moser systems with noncommutative spin interactions. Phys. Rev. Lett. 89, 126403 (2002) Pechukas, P.: Distribution of energy eigenvalues in the irregular spectrum. Phys. Rev. Lett. 51, 943–946 (1983) Reyman, A., Semenov-Tian-Shansky, M.: Group-theoretical methods in the theory of finite-dimensional integrable systems. Dynamical Systems VII, Encyclopaedia of Mathematical Sciences, V.I. Arnold, S.P. Novikov eds., Vol. 16, Berlin-Heidelberg-New York: Springer-Verlag, 1994, pp. 116– 225 Shephard, G.C., Todd, J.A.: Finite unitary reflexion groups. Can. J. Math. 6, 274–304 (1954) Steinberg, R.: Invariants of finite reflection groups. Can. J. Math. 12, 616–618 (1960) Varadarajan, V.S.: On the ring of invariant polynomials on a simple Lie algebra. Amer. J. Math. 90, 308–317 (1968) Whitney, H.: Complex analytic varieties. Reading, MA.-London-Don Mills, Ont: Addison-Wesley, 1972 Wojciechowski, S.: An integrable marriage of the Euler equations with the Calogero-Moser systems. Phys. Lett. A 111, 101–103 (1985) Yukawa, T.: New approach to the statistical properties of energy levels. Phys. Rev. Lett. 54, 1883– 1886 (1985)
Communicated by P. Forrester
Commun. Math. Phys. 308, 439–456 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1319-5
Communications in
Mathematical Physics
Morrey Potentials and Harmonic Maps David R. Adams1 , Jie Xiao2, 1 Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027, USA.
E-mail:
[email protected]
2 Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s,
NL A1C 5S7, Canada. E-mail:
[email protected] Received: 19 October 2010 / Accepted: 1 May 2011 Published online: 7 September 2011 – © Springer-Verlag 2011
Abstract: This paper discusses trace estimates for Morrey potentials (i.e., Riesz potential integrals of Morrey functions), leading to a consideration of the C ∞ smoothness of a class of generalized harmonic maps. 1. Introduction Inspirited by a well-developed theory of the Newton potentials and their energy integrals in mathematical physics (cf. [6]) but also a deep influence of the Morrey regularity theory on harmonic map equations (cf. [21]), this paper is devoted to a study of the integrability or regularity properties of the (0, N ) α th order Riesz potential integrals of Morrey functions f ∈ L p,λ : f (y) dy Iα f (x) := N |x − y| N −α R existing as weak solutions u to
α th 2
order Laplace equation,
α
(−) 2 u = κ N ,α f with κ N ,α
N π 2 2α α2 . = N −α 2
In the above and below, (·) represents the Gamma function. And, the function f being in the Morrey space L p,λ (where 1 ≤ p < ∞, 0 < λ ≤ N ) on the N -dimensional Euclidean space R N means: 1 f L p,λ =
sup x∈R N & r >0
r λ−N
p
B(x,r )
| f (y)| p dy
< ∞,
where B(x, r ) represents the Euclidean ball with center x and radius r . Jie Xiao was in part supported by NSERC of Canada.
440
D. R. Adams, J. Xiao
More precisely, our principal concern is to settle the following embedding problem: For what non-negative Borel measure μ and exponents: p ≥ 1; q > 1; α > 0, does it hold that sup
f L p,λ ≤1
1 q |Iα f | dμ < ∞ ? q
B(0,1)
(1.1)
Based on the special and essential structure of L p,λ , a consideration of (1.1) makes sense only when 1 ≤ p < αλ . But, in the case 1 ≤ p = αλ , one is required to determine: When does there exist a constant c such that the following inequality
(1.2) exp c|Iα f |q dμ < ∞ is valid for q > 0? sup f L p,λ ≤1 B(0,1)
Because one is restricting the potential Iα f to the support of μ which, for instance, may be a lower dimensional Hausdorff measure on a given smooth manifold in R N , such embeddings are referred to as trace estimates for the Morrey potentials (or the Morrey-Sobolev spaces) Iα L p,λ . Obviously, we should first ask ourselves if there is anything like this in the literature? To the best of our knowledge, except the following weak-type trace estimate due to Adams (cf. [3, Theorem 5.1]): p(N −λ−d) μ {y ∈ B(x, r ) : |Iα f (y)| > t} p,λ f ∈L ⇒ sup t λ−αp ∀ t > 0 N −λ r x∈R N & r >0 under λ 0 < λ, d ≤ N , 1 ≤ p < , αp > N − d, & α
μ B(x, r ) sup < ∞, rd x∈R N & r >0
little has been published beyond the case λ = N for which the inequality (1.1) goes to Adams’ analogue of the trace result which first appeared in [1] for L p potentials with 1 < p < N /α; and while under 1 < p = N /α and p = p/( p − 1), the estimate (1.2) is replaced by
(1.3) exp c|Iα f | p dμ < ∞. sup f L p ≤1 B(0,1)
See also Adams [2] and Maz’ya [17, Corollary 8.6.2] as well as Adams-Hedberg [6, p. 210, 7.6.4]. Due to the preceding observation and the following Morrey type embedding (cf. [8, Remark 3.4]) Iα : L p,λ → C
α− λp
under
N λ <α≤ p p
which extends the famous Morrey Lemma: ∇u ∈ L p,λ & λ < p ⇒ u is H older ¨ continuous o f ex ponent α = 1 −
λ , p
Morrey Potentials and Harmonic Maps
441
thanks to (cf. [23, Sect. V.2.3]) u(x) =
N
∇u(y) · (x − y) dy, |x − y| N RN
2
2π
N 2
we will study this trace problem in accordance with two cases: αp < λ and αp = λ, whose details (including the sharp estimates) are presented in Section 2. In Section 3, as an interesting compensation for the case αp > λ, we will apply the above Morrey type embedding to prove that any bounded weak solution u (sending B(0, 1) to a bounded subset of Rn ) to the generalized harmonic map equation −u = |∇u| p u is of C ∞ on B(0, 1) provided ∇u ∈ L p, p(1−α) with 1 ≤ p ≤ 2 ≤ N & 0 < α < 1. Notation: From now on, X Y, X Y, and X ≈ Y represent that there exists a constant c > 0 such that X ≤ cY, X ≥ cY, and c−1 Y ≤ X ≤ cY, respectively. Additionally,
· · · is taken into action while the considered integral domain is evident. 2. Traces of Morrey Potentials In this section, we handle trace estimates for the Morrey potentials. 2.1. Special cases. According to Adams [3, Thm. 3.1] or [4, Thm. 4.1 & Remark 4.1], we have the following Morrey-Sobolev type embedding. Theorem 2.1. (Adams). Let α ∈ (0, N ) and λ ∈ (0, N ]. (i) If 1 < p < αλ , then Iα f
λp
L λ−αp
,λ
f L p,λ .
(2.1)
(ii) If 1 < p = αλ , then Iα f BMO f L p,λ .
(2.2)
Note that the previous critical case (ii) can be improved to Iα L 2,2α = Q α ⊆ BMO when 0 < α < 1 (cf. [26]), where BMO is the John-Nirenberg space with bounded mean oscillation [16], namely, f ∈ BMO if and only if f − < ∞, f BMO = sup f
x∈R N & r >0
B(x,r )
B(x,r )
where B(x,r ) g stands for the average of g over B(x, r ) with respect to the N -dimensional Lebesgue measure. So, the improved embedding yields that each element of Iα L p,λ is exponentially integrable with respect to the Lebesgue N -measure. This integrability can be made more explicit via the following Morrey-Moser-Trudinger type inequality (2.3). Theorem 2.2. Let α, λ ∈ (0, N ), 1 < p = αλ , and f ∈ L p,λ be supported in B(0, 1). If ω N −1 stands for the surface area of B(0, 1) and cα,λ =
λ ω N −1 1p −1 , N −α N
442
D. R. Adams, J. Xiao
then
c|Iα f (x)| dx < ∞ exp f L p,λ B(0,1)
(2.3)
is valid for any constant c < cα,λ . Furthermore, if (2.3) is valid for some c > 0 then c<
N ω N 1p . ω N −1 N − λ
(2.4)
Proof. For ∈ (0, 2), r ∈ [ , 2] and x ∈ B(0, 1) let φ(r, ) = | f (y)| dy. B(x,r )\B(x, )
Note that B(x, r ) \ B(x, ) is an annulus. Using the polar coordinate system we find r N −1 t | f (t, ς )| dς dt, φ(r, ) = ∂ B(0,1)
where dς represents the N − 1 dimensional Hausdorff measure on the unit sphere ∂ B(0, 1). Furthermore, if α (r, ) = | f (y)||x − y|α−N dy, B(x,r )\B(x, )
i.e., a cut-off of the Riesz potential of | f | on B(x, r ) \ B(x, ), then d d α (r, ) = r α−N φ(r, ), dr dr and hence an integration by part gives α (2, ) = 2α−N φ(2) − (α − N )
2
t α−N −1 φ(t, ) dt,
(2.5)
which can be found in [22]. Note that the Hölder inequality implies φ(r, ) ≤ where p =
p p−1 .
ω
N −1
N
1 p
r
N − λp
f L p,λ ,
So, (2.5) is employed to derive
α (2, ) ≤
ω
N −1
N
1 p
α− λ f L p,λ 2 p + (N − α)
2
t
α− λp −1
dt .
(2.6)
Now, using the hypothesis p = λ/α we achieve α (2, ) ≤
ω
N −1
N
1 p
2 f L p,λ 1 + (N − α) ln ,
(2.7)
thereby finding Iα (| f |)(x) ≤
1 ω 2 ω N −1 α N −1 p r M( f )(x) + f L p,λ 1 + (N − α) ln α N r
(2.8)
Morrey Potentials and Harmonic Maps
443
when r < 2. Here M( f ) is the Hardy-Littlewood maximal function of f . Without loss of generality, we may assume M( f ) > 0. Then, given δ > 0, let r = min{2, η} where η =
α
ω N −1
δ M( f )
1
α
.
Using (2.8) we get Iα (| f |)(x) ≤ δ +
ω
N −1
1 p
N
N − α 2 λ ln 1 + . f L p,λ 1 + λ η
(2.9)
Suppose c < cα,λ . Then there is a τ > 0 such that c ≤ cα,λ τ (1 + τ )−1 . The following two cases are considered: Case 1. Iα (| f |)(x) ≥ (1 + τ )δ. Clearly, this and (2.9) imply cIα (| f |)(x) ≤ c
1 + τ
Iα (| f |)(x) − δ
τ 1 2 λ ω N − α N −1 p ln 1 + ≤ cα,λ f L p,λ 1 + N λ η 2 λ N − α λ f L p,λ 1 + ln 1 + = . N −α λ η
Case 2. Iα (| f |)(x) < (1 + τ )δ. This yields exp cIα (| f |)(x) ≤ exp cα,λ (1 + τ )δ . These two cases, along with the boundedness of M on L p (where p = λ/α > 1) yield (2.3) via I (| f |)(x) α dx exp c f L p,λ B(0,1) = (· · · ) x∈B(0,1) & Iα (| f |)(x)≥(1+τ )δ + (· · · ) x∈B(0,1) & Iα (| f |)(x)<(1+τ )δ M f (x) p λ ω N −1 2ω N −1 λ + ≤ exp dx N −α N α δ B(0,1) cα,λ (1 + τ )δ ω N −1 . exp + N f L p,λ For the remaining part of Theorem 2.2, we just note that if f α (x) = |x|−α χ B(0,1) (x)
444
D. R. Adams, J. Xiao
then
Iα f α (x) =
B(0,1)
|x − y|α−N |y|−α dy = ω N −1 ln
1 + ψ(x), |x|
(2.10)
where ψ is a continuous and bounded function on R N ; see also [11]. With (2.10) and the norm estimate 1 ω N −1 p f α L p,λ ≤ , N −λ we easily get that
1
t
N −cω N −1
N −λ ω N −1
1p
0
dt t
|Iα f α (x)| dx < ∞ exp cα,N f α L p,λ B(0,1)
implies (2.4). Remark 2.3. Since λ < N and cα,N <
N ω N 1p , ω N −1 N − λ
which amounts to
N 1 N α1 λ < ∀α < λ, N −α N −λ
we may view (2.4) as an asymptotically sharp inequality. Perhaps it is worth mentioning that if p → ∞ then α → 0, p → 1, and hence cα,N → N /ω N −1 . This last constant is the sharp one in Adams’ inequality (cf. [5, Thm. 2]): 1 N Iα f (x) p exp d x < ∞. (2.11) ω N −1 f L p B(0,1) 2.2. General cases. Theorems 2.1-2.2 suggest an investigation of traces of the Morrey potentials. To do so, we recall 1 p p H p,λ = f ∈ L loc : f H p,λ = inf | f (y)| p w(y)1− p dy <∞ , w
RN
p where L loc means the class of all p-locally integrable functions on R N and the infimum (N −λ) comprising all non-negative weights w that belong to the weight class is over A1 A1 (whose definition and related A p -weights can be found in e.g. [12,24 or 25]) and
satisfy
RN
(∞)
w d N −λ =
∞ 0
(∞)
N −λ ({x ∈ R N : w(x) > t}) dt ≤ 1. (∞)
Here and henceforth, for 0 < λ < N the symbol N −λ stands for the Hausdorff capacity with order N − λ, as a set function on R N . The following estimates (cf. [8, Thm. 6.2]) for Wolff type potentials are useful for our treatment.
Morrey Potentials and Harmonic Maps
445
Theorem 2.4 (Adams-Xiao). Let (α, λ, p) ∈ (0, N ) × (0, N ) × (1, ∞), αp < N , p p = p−1 , and μ a non-negative Borel measure on R N . Then (i) Iα μ
p L p ,λ
≈
μ
Wα, p,λ (y) dμ(y);
(ii) Iα μ
p H p ,λ
≈
inf
(N −λ)
w∈A1
μ,w
Wα, p,λ (y) dμ(y).
Here Iα μ(x) =
μ Wα, p,λ (y)
|x − y|α−N dμ(y),
RN
∞
=
0
p −1 μ B(y, r ) dr , r r λ+ p(N −λ−α)
and μ,w Wα, p,λ (y)
∞
=
0
p −1 r αp μ B(y, r ) dr
. r w(z) dz B(y,r )
For the case p < λ/α we prove the following assertion: Theorem 2.5. Let μ be a non-negative Borel measure on R N such that μ(B(x, r )) < ∞ for α p < λ < N & N ≥ d > λ − α p. rd (r,x)∈(0,∞)×R N sup
(2.12)
Then sup
f L p,λ ≤1
B(0,1)
1/q |Iα f |q dμ <∞
holds for ∀ q < p˜ =
dp . λ − αp
Moreover, (2.13) fails when q = p, ˜ λ < N , and μ is the Lebesgue N -measure.
(2.13)
446
D. R. Adams, J. Xiao
Proof. As a matter of fact, it is enough to prove (2.13) under the assumption that μ B(0, 1) is sufficiently small. This is because B(0, 1) can be covered by a finite number of balls {B j } Jj=1 , i.e., B(0, 1) ⊆ ∪ Jj=1 B j for which μ(B j ) is small enough, and consequently (2.13) will follow for some bigger but finite constant. To do so, we may, without loss of generality, assume 0 ≤ f ∈ L p,λ . We need to estimate the Wolff potential ∞ αp t μ(B(x, t)) 1/ p−1 dt μ,w , Wα, p,λ (x) = ω(B(x, t)) t 0 where
ω(B(x, t)) =
w(y) dy, B(x,t)
and we expect – for μ supported in a set E ⊆ R N , there is a constant c > 0 such that μ,w Wα, p,λ (x)
αp − λ −
d( p − 1) ≤ c μ(E)
+
1 p−1
(2.14)
when > 0 and μ(E) are sufficiently small. −λ) Now to get the estimate (2.14) we must choose a weight w ∈ A(N . Our choice is 1 |x|−σ1 , |x| < 1, w(x) = |x|−σ2 , |x| ≥ 1, where
σ1 < N − λ < σ2 so that
(∞)
RN
w d N −λ < ∞.
Indeed, if w1 (x) = |x|−σ1 & w2 (x) = |x|−σ2 , then our weight is w1 ∧ w2 = min{w1 , w2 }, and a calculation shows that this weight is still an A1 weight. The Wolff potential is estimated by
δ
0
t α p+d t N −σ1
1 p−1
+μ(E)
dt t
+μ(E)
1
1/ p−1 δ
1/ p−1 1
δ
α p+d−N +σ1 p−1
tα p
1/ p−1
t N −σ1
∞
tα p
t N −σ2 1
+ μ(E) p−1 δ
for δ sufficiently small. Here σ1 < N − λ < N − α p,
1 p−1
dt t dt t
α p−N +σ1 p−1
Morrey Potentials and Harmonic Maps
447
so α p − N + σ1 < 0 & d > N − σ1 − α p
⇒
d > λ − α p,
which implies α p + d − N + σ1 > 0. Note that we may have to choose N − λ < σ2 < N − α p. So finally, setting δ = μ(E)1/d we get the estimate (2.14) as promised. Using Theorem 2.4 (ii) and the argument for (2.14), we obtain Iα μ
p H p ,λ
1+ d+αp−N +σ1 d( p−1) , μ(E)
thereby deriving that p
d+αp−N +σ1 Iα μ(x) f (x) d x f p p,λ μ(E) 1+ d( p−1) L holds for f ∈ L p,λ . So, if Iα f ≥ 1 on E, then the definition of the capacity Cα (E; L p,λ ) (cf. [7]) yields
N −αp−σ1 d Cα (E; L p,λ ). μ(E) Now, for t > 0 let
E = E t = B(0, 1) ∩ {x ∈ R N : Iα f (x) > t} & μ B(0, 1) be small.
Then p¯
μ(E t ) f L p,λ t − p¯ , where p¯ =
dp < p˜ & σ1 = N − λ −
N − αp − σ1
for arbitrarily small > 0. And then we easily get (cf. [3, Lemma]) that for
σ1 +αp−N dp , η = f L p,λ μ(B(0, 1)) the following estimate: (Iα f )q dμ = B(0,1)
+ 0
∞
η
η
μ(E t ) dt q
αp N −αp−σ μ B(0, 1) ηq + f L p,λ 1
1− q p f q μ B(0, 1) L p,λ
∞ η
t
dp q− N −αp−σ
is valid whenever q < p¯ but p¯ can be chosen arbitrarily close to p. ˜
1
dt t
448
D. R. Adams, J. Xiao
For the failure of (2.13) with q = p, ˜ λ < N and μ being the Lebesgue N -measure, we only need to note [7, Ex. 5.5]: f λ, p (x) = |x|
− λp
χ B(0,1) (x) ⇒ Iα f λ, p (x) ≈ |x|
α− λp
as |x| → 0.
Remark 2.6. Two more facts below are worth pointing out. (i) (2.13) always implies that for any B(x, r ) ⊆ B(0, 1), q(λ−αp) q μ B(x, r ) Cα B(x, r ); L p,λ p r p ,
which is an endpoint case of (2.12). (ii) In a similar manner, we can establish the “dual form” of (2.13), namely, sup |Iα f |q dμ < ∞ f H p,λ ≤1 B(0,1)
with sup (r,x)∈(0,∞)×R N
(2.15)
r −d μ B(x, r ) < ∞
holding for d > λ + p(N − λ − α) > 0 &
1 λ + p(N − λ − α) − > 0. q dp
Although Theorem 2.5 is relatively optimal, it can be improved to the following form. Theorem 2.7. Let μ be a non-negative Borel measure on R N with (2.12). Then 1/ p˜ |Iα f | p˜ [ln(1 + |Iα f |)]−γ dμ <∞ (2.16) sup f L p,λ ≤1
B(0,1)
holds for p˜ =
dp & γ > 2. λ − αp
Proof. The argument is very similar to that for Theorem 2.5. So, only the key steps are listed below. First of all, taking ⎧ λ−N (− ln |x|)−β , |x| ≤ 1/2, ⎪ ⎨|x| w(x) = |x|λ−N , 1/2 < |x| < 1, ⎪ ⎩|x|λ−N − , |x| ≥ 1, where β > 1 is given and > 0 is small enough, we get 1 ∞ 1/2 dr dr (∞) −β dr + + < ∞. w d N −λ (− ln r ) r −
r r 0 1/2 r 1
Morrey Potentials and Harmonic Maps
449
Next, we estimate the Wolff potential as in Theorem 2.5: δ 1 dt t αp+d p−1 S:= t λ (− ln t)−β t 0 1 αp 1
1 t p−1 dt + μ(E) p−1 λ t t δ ∞ αp 1
1 t p−1 dt + μ(E) p−1 λ−
t t 1 αp+d−λ β
1 αp−λ δ p−1 (− ln δ) p−1 + μ(E) p−1 δ p−1 for α < λ & d > λ − αp & μ(E) → 0.
1 Now, if δ = μ(E) d ln
−β/d 1 , μ(E)
then
β(λ−αp
αp−λ+d S μ(E) d( p−1) − ln μ(E) d( p−1) , and hence via the usual normalization,
β(λ−αp)
λ−α p μ(E t ) d − ln μ(E) d( p−1) t − p f L p,λ . This, in turn, yields μ(E t ) t − p˜ (− ln t)β as t → ∞ and f L p,λ ≤ 1. Finally, if φ(t) = t p˜ (ln t)−γ ∀ t ≥ 1, then
∞
and thus the result follows.
μ(E t ) dφ(t)
∞
(ln t)β−γ
dt , t
Remark 2.8. It is conjectured that γ > 2 can be reduced to γ > 1 since the function f λ, p (x) = |x|
− λp
χ B(0,1) (x) is an element of L p,λ and
B(0,1)
−γ |Iα f λ, p | p˜ ln(1 + |Iα f λ, p |) dμ < ∞ ∀γ > 1.
Because p = λ/α is the endpoint of p < λ/α, we have the following result. Theorem 2.9. Let μ be a non-negative Borel measure on R N with sup (r,x)∈(0,∞)×R N
r −d μ(B(x, r )) < ∞ holding for 0 < d ≤ N .
(2.17)
450
D. R. Adams, J. Xiao
(i) If α p = λ < N , then for any q ∈ (0, 1] there exists a constant c > 0 such that (2.18) exp c|Iα f |q dμ < ∞ sup f L p,λ ≤1 B(0,1)
holds. Moreover, (2.18) fails when c > 0, q > 1, λ < N , and μ is the Lebesgue N -measure. (ii) If α p = λ = N , then for any q ∈ (0, NN−1 ] there exists a constant c > 0 such that (2.18) holds. Moreover, (2.18) fails when c > 0, q > NN−1 , λ = N and μ obeys (2.17). Proof. (i) The argument for Theorem 2.5 can be partially used for this case p = λ/α. Here the above type estimates – now with λ−N |x| (ln 1/|x|)−κ , |x| < 21 , w(x) = |x| ≥ 21 , c1 |x|−σ2 , for a constant κ > 1. On the other hand, c1 can be chosen so that w is continuous if necessary, and σ2 is still greater than N − λ. Now we see that when estimating the Wolff potentials we get
δ
0
t λ+d t λ (ln 1/t)−κ
1 p−1
d κ dt δ p−1 (ln 1/δ) p−1 t
and δ
1
tλ t λ (ln 1/t)−κ
1 p−1
κ dt +1 (ln 1/δ) p−1 . t
So again if δ = μ(E)1/d , we then get an estimate like μ,w
κ
1
1
κ
+1
Wα, p,λ (x) μ(E) p−1 (ln 1/μ(E)) p−1 + μ(E) p−1 (ln 1/μ(E)) p−1 .
(2.19)
Using (2.19) we eventually obtain (ln 1/μ(E))κ+ p−1 t − p , assuming f p,λ ≤ 1. So, there is a constant c > 0 such that p μ(E t ) exp −c t κ+ p−1 . Notice that κ > 1 ensures
p κ+ p−1
< 1. This is significant because we are getting
sup
f L p,λ ≤1 B(0,1)
exp(c |Iα f |q ) dμ < ∞
for any q < 1, namely, to exponentially integrate Iα f against any lower d-dimensional measure μ, we are apparently required to lower the exponent on Iα f .
Morrey Potentials and Harmonic Maps
451
Next, we handle q = 1. In fact, if we write σ1 = N − λ − & σ2 = N − λ +
for sufficiently small > 0, as above we can similarly derive
d−
μ,w Wα, p,λ (x) −1 μ(E) d( p−1) .
(2.20)
But, this (2.20) yields a constant ι > 0 such that d
∀ t ≥ 1. μ(E t ) ≤ ι 1− p t − p Choosing = ( p − 1)d j −1 , where j is natural number, we find ∞ j p − 1 j . μ(E t )t j−1 dt ≤ ι j/( p−1) ( p − 1)d j 1 By Sterling’s formula for the Gamma function (·), we obtain j j j ( p − 1)(cι) p−1 d( p−1) (cι) j ≈ 3 . j j + 1 j2
But then, as long as cι < 1 we get sup
f L p,λ ≤1 B(0,1)
exp(c |Iα f |) dμ < ∞.
For the failure of (2.18) with c > 0, q > 1, λ < N , and μ being the Lebesgue N -measure, we just notice [7, Ex. 5.5] again but with αp = λ or the example in [18, Remark 3.2]: f α (x) = |x|−α χ B(0,1) (x) ⇒ − ln |x| Iα f α (x) as |x| → 0. (ii) The finiteness of (2.18) under this circumstance follows from [6, p. 210, 7.6.4]. For its failure, let us consider the following example: 1 1 −γ N & γ > . χ B(0,1) (x), α = f α,γ (x) = |x|−α ln |x| p N Note that 1 1−γ f α,γ ∈ L p & Iα f α,γ (x) ln as |x| → 0. |x| On the other hand, for σ ≤ 1 and c > 0 (which will be chosen to be small enough when σ = 1) one has ∞ ∞ c −1 1 σ σ > t dt dt < ∞. μ |x| < 1 : exp c ln exp − d |x| ln t 1 1 Thus Iα f α,γ is not exp(cL q )-integrable when q >
N N −1 .
452
D. R. Adams, J. Xiao
Remark 2.10. Two comments are in order: (i) With this test function, we see readily that if B(0, r ) ⊆ B(0, 1) then (2.18) ensures μ B(0, r ) r d for some d > 0. (ii) If ∀r > 0 & ∀x ∈ R N ,
μ(B(x, r )) ≈ r d then
sup
f L p,λ ≤1 B(0,1)
exp(c|Iα f |) dμ < ∞
must imply d
c<
1
1− 1
.
(N − λ) p ω N −1p 3. Regularity of Generalized Harmonic Maps Given p ∈ [1, ∞), consider the following harmonic map equation of type p (cf. [13]): − u = |∇u| p u
(3.1)
for mappings u : B(0, 1) → Rn . Note that (3.1) is different from the usual p-harmonic map equation −div(|∇u| p−2 ∇u) = |∇u| p u except p = 2 under which the weak solution to (3.1) is called a harmonic map existing as the critical point of the Dirichlet energy 1 E(u) = |∇u|2 . 2 B(0,1) In the forgoing equations and forthcoming ones, ∇u stands for the gradient of u = k (u 1 , · · · , u n ), i.e., the N × n matrix with entries ∂ j u k = ∂u ∂ x j , and ⎛
N
|∇u| = ⎝
⎞1 2
n
(∂ j u k )
2⎠
.
j=1 k=1
Furthermore, u is the Laplacian, i.e., u = (u 1 , · · · , u n ) =
N j=1
∂ 2j u 1 ∂ x 2j
N
,··· , j=1
∂ 2j u n . ∂ x 2j
It is well-known that when N = 2, the harmonic maps with the unit sphere Sn−1 (n ≥ 2) of Rn as their target space are Hölder continuous (cf. [14]), but this is no longer true for any stationary harmonic map on B(0, 1) with N ≥ 3 under which the singular set has vanishing N − 2 dimensional Hausdorff measure (cf. [10], [19] and its related references). The following result extends the well-known regularity result for p = 2 (cf. [9,15,20]).
Morrey Potentials and Harmonic Maps
453
Theorem 3.1. Let 1 ≤ p ≤ 2 ≤ N and 0 < α < 1. If u is a bounded weak solution to (3.1) with ∇u ∈ L p, p(1−α) , then u ∈ C ∞ B(0, 1) , i.e., u is of C ∞ on B(0, 1). Proof. Note that u is extendable to a map on R N via setting u = 0 on R N \ B(0, 1). With this extension ∇u ∈ L p, p(1−α) is meaningful and yields u ∈ C α by Morrey’s lemma mentioned in Sect. 1. To prove the assertion, it suffices to show that u is of C ∞ on any small open ball B(x0 , r0 ) ⊆ B(0, 1). Define ψ p (r ) = sup |∇u − (∇u) B(x,r ) | p , x∈B(x0 ,r0 /2) B(x,r )
where
(∇u) B(x,r ) =
B(x,r )
∇u 1 , · · · ,
B(x,r )
∇u n .
By hypothesis we have ψ p (r ) r N − p(1−α) ,
(3.2)
and then iteratively increase the power of r in (3.2). To do so, we notice that a slight modification of Giaquinta’s iteration lemma (cf. [13, p. 86, Lemma 2.1]) and its argument derives the following Lemma 3.2. If 0 ≤ φ is a quasi-increasing function on [0, ∞), i.e., 0 ≤ x1 < x2 < ∞ implies φ(x1 ) φ(x2 ), and is such that t β φ(T ) + bT γ ∀t ≤ T ≤ T0 & γ < β (3.3) φ(t) ≤ a T holds for some constants a, b, T0 , then there exists a constant c such that φ(t) ≤ ct γ ∀t ∈ [0, ∞).
(3.4)
In what follows, we may assume u(x1 ) = 0 for a given point x1 ∈ B(x0 , r0 /2), and write u = u1 + u2 , where u1 = 0 on the boundary of B(0, 1), as well as u1 = u & u2 = 0 on B(x1 , r1 ) with r1 < r0 /2. Note that u2 is harmonic. So an application of the maximum principle yields u2 L ∞ (B(x1 ,r1 )) ≤ u2 L ∞ (∂ B(x1 ,r1 )) uC α (B(x1 ,r1 )) r1α ,
(3.5)
and hence u1 L ∞ (B(x1 ,r1 )) = u − u2 L ∞ (B(x1 ,r1 )) r1α . Note that Campanato’s estimates for harmonic functions (cf. [13, p. 78, Theorem 2.1 and p. 80, Prop. 2.2]) ensure |∇u2 − (∇u2 ) B(x1 ,r ) | p B(x1 ,r ) r N + p |∇u2 − (∇u2 ) B(x1 ,s) | p ∀r < s. (3.6) s B(x1 ,s)
454
D. R. Adams, J. Xiao
This last fact derives that ψ p is quasi-increasing on [0, ∞). Meanwhile 2 |∇u1 | d x = − u1 · u B(x1 ,r1 ) B(x1 ,r1 ) = u1 · (|∇u| p u) B(x1 ,r1 )
# # #u1 # L ∞ (B(x
1 ,r1 ))
# #p #∇u# p, p(1−α) r N − p(1−α) 1
L
N − p(1−α)+α r1 .
Applying Hölder’s inequality to the last estimate with p ≤ 2, we then get |∇u1 | p B(x1 ,r1 )
p
N (1− 2p ) r1
N + p [α− p(1−α)] r1 2 .
B(x1 ,r1 )
|∇u1 |
2
2
(3.7)
By (3.6) and (3.7) we get that for r ∈ (0, r1 ), |∇u − (∇u) B(x1 ,r ) | p B(x1 ,r ) |∇u1 − (∇u1 ) B(x1 ,r ) | p + |∇u2 − (∇u2 ) B(x1 ,r ) | p B(x1 ,r ) B(x1 ,r ) r N + p |∇u2 − (∇u2 ) B(x1 ,r1 ) | p r1 B(x1 ,r1 ) + |∇u1 − (∇u1 ) B(x1 ,r1 ) | p , B(x1 ,r1 )
thereby finding ψ p (r )
r N + p r1
N + 2p [α− p(1−α)]
ψ p (r1 ) + r1
.
By Lemma 3.2 we get p
ψ p (r ) r N + 2 [α− p(1−α)] , and so ∇u ∈ L p, p(1−α1 ) (Campanato space), where α1 = 1 −
p(1 − α) − α . 2
If α1 < 1, then L p, p(1−α1 ) = L p, p(1−α1 ) , and hence the original hypothesis holds for α1 . Via iterating the above procedure for finitely many steps, since α1 > α we can get some αk > 1 and then ∇u ∈ L p,− pβ for some β ∈ (0, 1), and hence ∇u is of C β . Since the right-hand side of (3.1) is now of C β , we get that u is of C 2,β . But then the right-hand 3,β side of (3.1) is of C 1,β . This in turn yields that u is of C . Continuing this process, we eventually reach that u ∈ C ∞ B(0, 1) .
Morrey Potentials and Harmonic Maps
455
Remark 3.3. It remains open to settle the case p > 2 of Theorem 3.1. But, after this paper was completed, C. Wang from the University of Kentucky wrote us that the above proof p actually gives that if α ∈ ( p+1 , 1), ∇u ∈ L p, p(1−α) , then the bounded weak solution to (3.1) enjoys r N +2 N − p(1−α)+α ψ2 (r ) ψ2 (r1 ) + r1 . r1 This, plus Lemma 3.2, derives ψ2 (r ) r N − p(1−α)+α , namely, r −N and thus ∇u ∈ C
( p+1)α− p 2
B(x1 ,r )
|∇u − (∇u) B(x1 ,r ) |2 r ( p+1)α− p ,
on B(0, 1), implying u ∈ C ∞ B(0, 1) .
References 1. Adams, D.R.: Traces of potentials arising from translation invariant operators. Ann. Scuola Norm. Sup. Pisa 25(3), 203–217 (1971) 2. Adams, D.R.: Traces of potentials. II. Indiana Univ. Math. J. 22, 907–918 (1973) 3. Adams, D.R.: A note on Riesz potentials. Duke Math. J. 42, 765–778 (1975) 4. Adams, D.R.: Lectures on L p -Potential Theory, Volume 2, Department of Mathematics, University of Umeå, 1981 5. Adams, D.R.: A sharp inequality of J. Moser for higher order derivatives. Ann. Math. 128, 385–398 (1988) 6. Adams, D.R., Hedberg, L.I.: Function Spaces and Potential Theory. Berlin-Heidelberg: Springer-Verlag, 1996 7. Adams, D.R., Xiao, J.: Nonlinear analysis on Morrey spaces and their capacities. Indiana Univ. Math. J. 53, 1629–1663 (2004) 8. Adams, D.R., Xiao, J.: Morrey spaces in harmonic analysis. Ark. Mat. to appear, doi:10.1007/s11512010-0134-0, 30 pp, March 2011 9. Angelsberg, G., Pumberger, D.: A regularity result for polyharmonic maps with higher integrability. Ann. Glob. Anal. Geom. 35, 63–81 (2009) 10. Bethuel, F.: On the singular set of stationary harmonic maps. Manus. Math. 78, 417–443 (1993) 11. Fuglede, B.: The logarithmic potential in higher dimensions. Mat. Fys. Medd. Dan. Cid. Selsk. 33, 1– 14 (1960) 12. Garcia-Cuerva, J., Rubio de Francia, J.L.: Weighted Norm Inequalities and Related Topics. North-Holland Mathematics Studies, 116. Notas de Matemática [Mathematical Notes], 104. Amsterdam: North-Holland Publishing Co., 1985 13. Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Annals of Math. Studies 105, Princeton, NJ: Princeton University Press, 1983 14. Hélein, F.: Regularity of weakly harmonic maps between a surface and a Riemannian manifold. C. R. Acad. Sci. Paris Sr. I Math. 312, 591–596 (1991) 15. Hélein, F.: Harmonic maps, conservation laws and moving frames. Cambridge Tracts in Mathematics 150, Cambridge: Cambridge University Press, 2002 16. John, F., Nirenberg, L.: On functions of bounded mean oscillation. Comm. Pure Appl. Math. 14, 415– 426 (1961) 17. Mazýa, V.G.: Sobolev Spaces. Berlin-New York: Springer, 1985 18. Mizuta, Y., Shimomura, T., Sobukawa, T.: Sobolev’s inequality for Riesz potentials of functions in nondoubling Morrey spaces. Osaka J. Math. 46, 255–271 (2009) 19. Rivi`re, T., Struwe, M.: Partial regularity for harmonic maps and related problems. Comm. Pure Appl. Math. 61, 451–463 (2008) 20. Rupflin, M.: What is a ... Morrey space? What is a ... Campanato space? Preprint, April 22, 2008 21. Schoen, R., Yau, S.T.: Lectures on Harmonic Maps. Conference Proc. & Lecture Notes in Geometry & Topology, Vol. II, Somerville, MA: International Press, Inc., 1997
456
D. R. Adams, J. Xiao
22. Serrin, J.: A remark on the Morrey potential. Contemp. Math. 426, 307–315 (2007) 23. Stein, E.M.: Singular Integrals and Differentiability of Functions. Princeton, NJ: Princeton Univ. Press, 1970 24. Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton, NJ: Princeton Univ. Press, 1993 25. Torchinsky, A.: Real-variable Methods in Harmonic Analysis. New York: Dover Publications, Inc., 2004 26. Xiao, J.: Homothetic variant of fractional Sobolev space with application to Navier-Stokes system. Dyn. Part. Diff. Eq. 4, 227–245 (2007) Communicated by P. Constantin
Commun. Math. Phys. 308, 457–478 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1300-3
Communications in
Mathematical Physics
A Finite Analog of the AGT Relation I: Finite W -Algebras and Quasimaps’ Spaces Alexander Braverman1 , Boris Feigin2,3 , Michael Finkelberg3,4,5 , Leonid Rybnikov3,5 1 Department of Mathematics, Brown University, 151 Thayer St., Providence, RI 02912, USA.
E-mail:
[email protected]
2 L.D. Landau Institute for Theor. Phys., Kosygina St., 2, Moscow 119339, Russia 3 Department of Mathematics, National Research University Higher School of Economics,
20 Myasnitskaya St., Moscow 101000, Russia. E-mail:
[email protected];
[email protected];
[email protected]
4 Independent Moscow University, Bol’shoj Vlas’evskii Pereulok, 11, Moscow 119002, Russia 5 Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoy Karetny per 19,
Moscow 127994, Russia Received: 30 October 2010 / Accepted: 18 March 2011 Published online: 7 September 2011 – © Springer-Verlag 2011
Abstract: Recently Alday, Gaiotto and Tachikawa [2] proposed a conjecture relating 4-dimensional super-symmetric gauge theory for a gauge group G with certain 2-dimensional conformal field theory. This conjecture implies the existence of certain structures on the (equivariant) intersection cohomology of the Uhlenbeck partial compactification of the moduli space of framed G-bundles on P2 . More precisely, it predicts the existence of an action of the corresponding W -algebra on the above cohomology, satisfying certain properties. We propose a “finite analog” of the (above corollary of the) AGT conjecture. Namely, we replace the Uhlenbeck space with the space of based quasi-maps from P1 to any partial flag variety G/P of G and conjecture that its equivariant intersection cohomology carries an action of the finite W -algebra U (g, e) associated with the principal nilpotent element in the Lie algebra of the Levi subgroup of P; this action is expected to satisfy some list of natural properties. This conjecture generalizes the main result of [5] when P is the Borel subgroup. We prove our conjecture for G = G L(N ), using the works of Brundan and Kleshchev interpreting the algebra U (g, e) in terms of certain shifted Yangians. 1. Introduction 1.1. The setup. Let G be a semi-simple simply connected complex algebraic group (or, more generally, a connected reductive group whose derived group [G, G] is simply connected) and let P be a parabolic subgroup of G. We shall denote by L the corresponding Levi factor. Let B be a Borel subgroup of G contained in P and containing a maximal torus T of G. We shall also denote by the coweight lattice of G (which is the same as the lattice of cocharacters of T ); it has a quotient lattice G,P = Hom(C∗ , L/[L , L]), which ˇ of the Langlands dual can also be regarded as the lattice of characters of the center Z ( L) ˇ Note that G,B = . The lattice G,P contains the canonical sub-semi-group group L. +G,P spanned by the images of positive coroots of G.
458
A. Braverman, B. Feigin, M. Finkelberg, L. Rybnikov
Set now GG,P = G/P. There is a natural isomorphism H2 (GG,P , Z) G,P . Let C be a smooth connected projective curve over C. Then the degree of a map f : C → GG,P can be considered as an element of G,P ; it is easy to see that actually deg f must lie in +G,P . For θ, θ ∈ G,P we shall write θ ≥ θ if θ − θ ∈ +G,P . Let eG,P ∈ GG,P denote the image of e ∈ G. Clearly eG,P is stable under the action of P on GG,P . Let MG,P denote the moduli space of based maps from (P1 , ∞) to (GG,P , eG,P ), i.e. the moduli space of maps P1 → GG,P which send ∞ to eG,P . This space is acted on by the group P × C∗ , where P acts on GG,P preserving the point eG,P and C∗ acts on P1 preserving ∞; in particular, the reductive group L ×C∗ acts on MG,P . Also for any θ ∈ +G,P let MθG,P denote the space of maps as above of degree θ . For each θ as above one can also consider the space of based quasi-maps (or Zastava space in the terminology of [9,15 and 16]; cf. also [6] for a review of quasi-maps’ θ . This is an affine algebraic variety containing Mθ spaces) which we denote by QMG,P G,P as a dense open subset. Moreover, it possesses a stratification of the form θ QMG,P = MθG,P × Symθ−θ A1 , 0≤θ ≤θ
where for any γ ∈ +G,P we denote by Symγ A1 the variety of formal linear combina tions λi xi , where xi ∈ A1 and λi ∈ +G,P such that λi = γ . In most cases the θ variety QMG,P is singular. 1.2. Equivariant integration. For a connected reductive group G with a maximal torus T let AG = HG∗ ( pt, C). This is a graded algebra which is known to be canonically isomorphic to the algebra of G-invariant polynomial functions on the Lie algebra g of G. We shall denote by KG its field of fractions. Let now Y be a variety endowed with an action of G such that Y T is proper. Then as was remarked e.g. in [5], we have a well-defined integration map : IH∗G (Y ) → KG , Y
which is a map of AG -modules. In particular, it makes sense to consider the integral Y 1 ∈ KG of the unit cohomology class. Let us also set IH∗G (Y )loc = IH∗G (Y ) ⊗ KG . Then IH∗G (Y )loc is a finite-dimensional AG
vector space over KG endowed with a non-degenerate KG -valued (Poincaré) pairing ·, · Y,G . θ In particular, all of the above is applicable to Y = QMG,P and G = L × C∗ . In θ ∗ particular, if we let 1G,P denote the unit class in the L × C -equivariant cohomology of θ , then we can define QMG,P ZG,P =
θ∈θG,P
qθ
θ QMG,P
1θG,P .
(1.1)
Finite Analog of AGT Relation I: Finite W -Algebras and Quasimaps’ Spaces
459
ˇ with values in the field K L×C∗ of L-invariant rational This is a formal series in q ∈ Z ( L) functions on l × C. In fact the function ZG,P is a familiar object in Gromov-Witten theory: it is explained in [5] that up to a simple factor ZG,P is the so-called equivariant J -function of GG,P (cf. Sect. 6 of [5]). In particular, this function was studied from many different points of view (cf. [5,19,20] for the case when P is the Borel subgroup). It was conjectured in [5,6] that the function ZG,P should have an interpretation in terms of representation theory related to the Langlands dual Lie algebra gˇ . More generally, let us set θ IHθG,P = IH L×C∗ (QMG,P )loc , IHG,P = IHθG,P , θ∈+G,P
·, · θG,P = ·, · QM θ ,L×C∗ , ·, · G,P = G,P
ˇ (−1) θ,ρ ·, · θG,P ,
θ∈+G,P
where ρˇ denotes the half-sum of the positive roots of g, and we view θ as the positive integral linear combination of (the images of) the simple coroots out of l. Then one would like to interpret the +G,P -graded vector space IHG,P together with the intersection pairing ·, · and the unit cohomology vectors 1θG,P ∈ IHθG,P in such terms. A complete answer to this problem in the case when P is a Borel subgroup was given in [5]. The purpose of this paper is to suggest a conjectural answer in the general case and to prove this conjecture for G = G L(N ). The relevant representation theory turns out to be the representation theory of finite W -algebras which we recall in Sect. 2. We should also note that our conjecture is motivated by the so-called Alday-Gaiotto-Tachikawa (or AGT) conjecture [2] which relates 4-dimensional supersymmetric gauge theory to a certain 2-dimensional conformal field theory (more precisely, one may view some part of the AGT conjecture as an affine version of our conjecture; this point of view is explained in Sect. 6). In fact, the current work grew out of an attempt to create an approach to the AGT conjecture in terms of geometric representation theory. We hope to pursue this point of view in future publications. 1.3. The main conjecture. In the remainder of this introduction we are going to give a more precise formulation of our main conjecture and indicate the idea of the proof for G = G L(N ). To do this, let us first recall the corresponding result from [5] dealing with the case when P is a Borel subgroup. In what follows we shall denote it by B instead of P. First, it is shown in [5] that the Lie algebra gˇ acts naturally on IHG,B . Moreover, this action has the following properties. First of all, let us denote by ·, · G,B the direct sum ˇ ·, · θ . of the pairings (−1) θ,ρ G,B Recall that the Lie algebra gˇ has its triangular decomposition gˇ = nˇ + ⊕ hˇ ⊕ nˇ − . Let κ : gˇ → gˇ denote the Cartan anti-involution which interchanges nˇ + and nˇ − and acts as ˇ For each λ ∈ h = (h) ˇ ∗ we denote by M(λ) the corresponding Verma an identity on h. module with highest weight λ; this is a module generated by a vector vλ with (the only) relations t (vλ ) = λ(t)vλ for t ∈ hˇ and n(vλ ) = 0 for n ∈ nˇ + . Theorem 1.4. (1) IHG,B (with the above action) becomes isomorphic to M(λ), where λ = − a − ρ.
460
A. Braverman, B. Feigin, M. Finkelberg, L. Rybnikov
(2) IHθG,B ⊂ IHG,B is the − a − ρ − θ -weight space of IHG,B . (3) For each g ∈ gˇ and v, w ∈ IHG,B we have g(v), w G,B = v, κ(g)w G,B . (4) The vector θ 1θG,B (lying is some completion of IHG,B ) is a Whittaker vector (i.e. a n+ -eigen-vector) for the above action. a
As a corollary we get that the function q ZG,B is an eigen-function of the quantum Toda hamiltonians associated with gˇ with eigen-values determined (in the natural way) by a (we refer the reader to [13] for the definition of (affine) Toda integrable system and its relation with Whittaker functions). In fact, in [5] a similar statement is proved also when G is replaced by the corresponding affine Kac-Moody group — cf. Sect. 6 for more detail. Our main conjecture gives a generalization of the statements 1)–3) above to arbitrary P. Namely, to any nilpotent element e ∈ gˇ one can associate the so-called finite W -algebra U (ˇg, e). We recall the definition in Sect. 2.1 (this definition is such that when e = 0 we have U (ˇg, e) = U gˇ and when e is regular, then U (ˇg, e) is the center of U gˇ ). Roughly speaking, we conjecture that analogs of 1)–3) hold when U gˇ is replaced by U (ˇg, e Lˇ ) (we refer the reader to Sect. 2 for the definition of Verma module and Whittaker vectors for finite W -algebras). The main purpose of this paper is to formulate this conjecture more precisely and to prove it for G of type A. 1.5. Organization of the paper. The paper is organized as follows: in Sect. 2 we recall basic definitions about finite W -algebras for general G; we also recall the basic results about parabolic quasi-maps’ spaces and formulate our main conjecture. In Sect. 3 we recall the results of Brundan and Kleshchev who interpret finite W -algebras in type A using certain shifted Yangians and in Sect. 5 we discuss the notion of Whittaker vectors for finite W -algebras from this point of view. In Sect. 4 we use it in order to prove our main conjecture for G = S L(N ) (and any parabolic). One important ingredient in the proof is this: we replace the intersection cohomology of parabolic quasi-maps’ spaces by the ordinary cohomology of a small resolution of those spaces (which we call parabolic Laumon spaces). Finally in Sect. 6 we discuss the relation between the above results and the AGT conjecture. 2. Finite W -Algebras 2.1. W -algebra. Let e be a principal nilpotent element of the Levi subalgebra l ⊂ g. Let U (g, e) denote the finite W -algebra associated to e, see e.g. [11]. We recall its definition for the readers’ convenience. Choose an sl2 -triple (e, h, f ) in g. We introduce a grading on g by eigenvalues of adh : g= g(i), g(i) := {ξ ∈ g : [h, ξ ] = iξ }. i∈Z
The Killing form (·, ·) on g allows to identify g with the dual space g∗ . Let χ = (e, ?) be an element of g∗ corresponding to e. Note that χ defines a symplectic form ωχ on g(−1) as follows: ωχ (ξ, η) := χ , [ξ, η] . We fix a Lagrangian subspace l ⊂ g(−1) with respect to ωχ , and set m := l ⊕ i≤−2 g(i). We define the affine subspace mχ ⊂ U g
Finite Analog of AGT Relation I: Finite W -Algebras and Quasimaps’ Spaces
461
as follows: mχ := {ξ − χ , ξ , ξ ∈ m}. Finally, we define the W -algebra U (g, e) := (U g/U g · mχ )ad m := {a + U g · mχ : [m, a] ⊂ U g · mχ }. It is easy to see that U (g, e) = End U g(U g ⊗ Cχ ), Um
where Cχ denotes the natural 1-dimensional module over m corresponding to the character χ . In this description the algebra structure on U (g, e) becomes manifest. It is equipped with the Kazhdan filtration F0 U (g, e) ⊂ F1 U (g, e) ⊂ . . ., see e.g. Sect. 3.2 of [11]. We recall its definition for the readers’ convenience. We denote the standard PBW filtration on U g (by the order by Fist U g. The Kazhdan of a monomial) filtration on U g is defined by Fi U g := 2k+ j≤i Fst k U g ∩ U g( j), where U g( j) is the eigenspace of adh on U g with eigenvalue j. Being a subquotient of U g, the W -algebra U (g, e) inherits the Kazhdan filtration Fi U (g, e). We also consider the shifted Kazhdan filtration Fi U (g, e) := Fi+1 U (g, e), and we define the C[]-algebra U (g, e) as the Rees algebra of the filtered algebra (U (g, e), F• ). Abusing notation, we will sometimes call U (g, e) just a W -algebra. Let us extend the scalars to the field K := K L× ˇ C∗ , and let ∈ K stand for the generator of HC2 ∗ ( pt, Z). Thus K = C(t∗ /W L × A1 ), where W L stands for the Weyl group of l, and A1 is the affine line with coordinate . Given a point ∈ t∗ /W L we consider the Verma module M(−−1 , e) over U (g, e) ⊗ K U (g, e) ⊗ K introduced in Sects. 4.2 and 5.1 of [11]. As ∈ t∗ /W L varies, these modules form a family over C[t∗ /W L × (A1 − 0)]. Localizing to K we obtain the universal Verma module M(g, e) over U (g, e) ⊗ K. Note that a certain ρ-shift is incorporated into the definition of M(g, e), cf. the text right after Lemma 5.1 of [11].1 In what follows we shall often abbreviate M := M(g, e). 2.2. Whittaker vector. Let te stand for the centralizer of e in t. Recall from [11] that the e e collection of nonzero weights of t on g is called a restricted root system , and the weights on p := i≥0 g(i) form a positive root system e+ ⊂ e . Let I e be the set of simple roots, i.e. positive roots which are not positive linear combinations of other positive roots. According to Theorem 6 of [10], the simple roots form a base of (te )∨ . Let us choose a linear embedding : ge → U (g, e) as in Theorem 3.6 of [11]. For a simple root α ∈ I e , we consider the corresponding weight space (geα ). The Kazhdan filtration induces the increasing filtration on the root space (geα ). We define a positive integer m α so that Fm α (geα ) = (geα ), but Fm α −1 (geα ) = (geα ). The following conjecture holds for g of type A by the work of J. Brundan and A. Kleshchev (cf. Sect. 3.7 below), and for all exceptional types according to computer calculations by J. Brundan (private communication): Conjecture 2.3. dim Fm α (geα )/ Fm α −1 (geα ) = 1. Definition 2.4. A linear functional ψ on α∈I e (geα ) is called regular if for any α ∈ I e we have ψ(Fm α −1 (geα )) = 0, but ψ(Fm α (geα )) = 0. If Conjecture 2.3 is true, then T e acts simply transitively on the set of regular functionals. 1 The definition of the module M(g, e) from [11] is (unfortunately) quite involved and we are not going to recall it here. On the other hand for g = gl(N ) there is another (in some sense, more explicit) definition of this module which we are going to recall in Sect. 3.
462
A. Braverman, B. Feigin, M. Finkelberg, L. Rybnikov
e Definition 2.5. Given a regular functional ψ on α∈I e (gα ), a ψ-eigenvector w in a completion θ∈ ˇ ˇ Mθ of the universal Verma module M is called a G, P ψ−Whittaker vector. 2.6. Shapovalov form. Let σ stand for the Cartan antiinvolution of g identical on t. Let w0l stand for the adjoint action of a representative of the longest element of the Weyl group of the Levi subalgebra l. Then the composition w0l σ preserves e and everything else entering the definition of the finite W -algebra and gives rise to an antiisomorphism ∼ U (g, e)−→U (g, e), where U (g, e) (see Sect. 2.2 of [11]) is defined just as U (g, e), only with left ideals replaced by right ideals. Composing this antiisomorphism with the ∼ isomorphism U (g, e)−→U (g, e) of Corollary 2.9 of [11] we obtain an antiinvolution ς of U (g, e). Definition 2.7. The Shapovalov bilinear form (·, ·) on the universal Verma module M with values in K is the unique bilinear form such that (x, yu) = (ς (y)x, u) for any x, u ∈ M, y ∈ U (g, e) with value 1 on the highest vector.
2.8. Parabolic Zastava spaces and finite W -algebras: the main conjecture. We are now ready to formulate our main conjectures. We are going to change slightly the notations. Namely, the symbols G, P, L , g, p, l will denote the same things as in the Introduction. However, we are now going to denote by e the principal nilpotent element in the Langlands dual Lie algebra ˇl ⊂ g; similarly, M will now denote the universal Verma module over U (ˇg, e). Note that in this case M becomes naturally graded by +G,P . Also, the universal coefficient field K is now nothing else but the field K L×C∗ which appeared in the Introduction. With these conventions we formulate the following Conjecture 2.9. (1) There is an isomorphism of G,P -graded K-vector spaces U (ˇg, e)-action on IHG,P . IHG,P and M; in particular, there is a natural θ (2) The isomorphism takes the vector θ∈G,P 1θ ∈ θ∈G,P IH G,P to a ψ Whittaker vector w ∈ θ∈G,P Mθ for a certain regular functional ψ. (3) For x, u ∈ Vθ we have ((x), (u)) = (−1)|θ| x, u θG,P . In what follows we shall give a more precise formulation of this conjecture when g is of type A (in that case we shall also prove the conjecture). In particular, we shall specify the regular functional ψ in that case. 3. Shifted Yangians and Finite W -Algebras In this section we recall an explicit realization of finite W -algebras in type A using shifted Yangians (due to Brundan and Kleshchev). 3.1. Shifted Yangian. Let π = ( p1 , . . . , pn ), where p1 ≤ p2 ≤ · · · ≤ pn , and p1 + · · · + pn = N . Recall the shifted Yangian Yπ(gln ) introduced by J. Brundan and (r ) A. Kleshchev (see [12] and [18]). It is an associative C[]-algebra with generators di ,
Finite Analog of AGT Relation I: Finite W -Algebras and Quasimaps’ Spaces (r )
463
(r )
i = 1, . . . , n, r ≥ 1; fi , i = 1, . . . , n − 1, r ≥ 1; ei , i = 1, . . . , n − 1, r ≥ pi+1 − pi + 1, subject to the following relations: (r )
(s)
[di , d j ] = 0, +s−1 (t) (r +s−t−1) (r ) (s) [ei , f j ] = −δi j rt=0 di di+1 ,
(3.1) (3.2)
where di(0) := 1, and the elements di (r ) are found from the relations δr 0 , r = 0, 1, . . .; (r )
(s)
r −1
(s)
t=0 r −1
[di , e j ] = (δi j − δi, j+1 ) (r )
[di , f j ] = (δi, j+1 − δi j )
(t) (r +s−t−1)
di e j
r
(t) (r −t) t=0 di di
,
(r +s−t−1) (t) di ,
fj
=
(3.3)
(3.4)
t=0
[ei(r ) , ei(s+1) ] − [ei(r +1) , ei(s) ] = (ei(r ) ei(s) + ei(s) ei(r ) ), [fi(r +1) , fi(s) ] − [fi(r ) , fi(s+1) ] = (fi(r ) fi(s) + fi(s) fi(r ) ), (r ) (s+1) (r +1) (s) (r ) (s) , ei+1 ] = −ei ei+1 , [ei , ei+1 ] − [ei (r +1) (s) (r ) (s+1) (s) (r ) , fi+1 ] − [fi , fi+1 ] = −fi+1 fi , [fi (r ) (s) [ei , e j ] = 0 if |i − j| > 1, (r ) (s) [fi , f j ] = 0 if |i − j| > 1, (r ) (s) (t) (s) (r ) (t) [ei , [ei , e j ]] + [ei , [ei , e j ]] = 0 if |i − j| = 1, (r ) (s) (t) (s) (r ) (t) [fi , [fi , f j ]] + [fi , [fi , f j ]] = 0 if |i − j| = 1.
(3.5) (3.6) (3.7) (3.8) (3.9) (3.10) (3.11) (3.12)
We introduce the generating series dk (u) = 1 +
∞
(s) −s+1 −s u , s=1 dk
fk (u) :=
ek (u) =
∞
∞
(s) −s+1 −s u , s= pk+1 − pk +1 ek
(s)
fk −s+1 u −s .
s=1
Finally, we define ak (u) := d1 (u)d2 (u − 1) . . . dk (u − k + 1), Ak (u) := u p1 (u − 1) p2 . . . (u − k + 1) pk ak (u), and also Bk (u) := (u − k + 1) pk+1 − pk Ak (u)ek (u − k + 1), Ck (u) := fk (u − k + 1)Ak (u).
(3.13) (3.14)
3.2. W -algebra U (g, e) and its universal Verma module. Let p ⊂ g := gl N = End (W ) be the parabolic subalgebra preserving the flag 0 ⊂ W1 ⊂ · · · ⊂ Wn−1 ⊂ W . Here W = w1 , . . . , w N , and Wi = w1 , . . . , w p1 +···+ pi . Let e be a principal nilpotent element of a Levi factor l of p. Let U (g, e) denote the finite W -algebra associated to e, see e.g. [12]. It is equipped with the Kazhdan filtration F0 U (g, e) ⊂ F1 U (g, e) ⊂ . . ., see e.g. Sect. 3.2 of [12]. We also consider the shifted Kazhdan filtration Fi U (g, e) :=
464
A. Braverman, B. Feigin, M. Finkelberg, L. Rybnikov
Fi+1 U (g, e), and we define the C[]-algebra U (g, e) as the Rees algebra of the filtered algebra (U (g, e), F• ). Abusing notation, we will sometimes call U (g, e) just a W -algebra. According to Sect. 3.4 of [12], U (g, e) is the quotient of Yπ(gln ) by the relations (r ) d1 = 0, r > p1 . Let us denote the standard coordinates in the diagonal Cartan subalgebra t of gl N by x1 , . . . , x N . Then the universal Verma module M = M(gl N , e) over U (g, e) ⊗ K is a vector space over the field K of rational functions in , x1 , . . . , x N symmetric in the groups (x1 , . . . , x p1 ), (x p1 +1 , . . . , x p1 + p2 ), . . . , (x p1 +···+ pn−1 +1 , . . . , x N ). Let us consider the field extension K ⊂ K := C(t∗ × A1 ) = C(, x1 , . . . , x N ). We denote by M = M (gl N , e) := M ⊗K K the universal Verma module with the extended scalars. 3.3. The Gelfand-Tsetlin module. According to [18], the module M admits a rather explicit description. More precisely, the authors of [18] construct a Gelfand-Tsetlin module V over U (g, e) ⊗ K equipped with a Gelfand-Tsetlin base numbered by the Gelfand-Tsetlin patterns, and write down explicitly the matrix coefficients of the generators of U (g, e) in this base. We recall these results here. To a collection d = (di(a) j ), n − := 1 ≥ i ≥ j, p j ≥ a ≥ 1, we associate a Gelfand-Tsetlin pattern = (d) (a)
(a)
(λi j ), n ≥ i ≥ j, p j ≥ a ≥ 1, as follows: λn j := −1 x p1 +···+ p j−1 +a + j − 1, n ≥
−1 (a) j ≥ 1; λi(a) j := − pi j + j − 1, n − 1 ≥ i ≥ j ≥ 1 (see (3.15)). The corresponding base element ξ = ξ(d) will be denoted by ξd for short. Thus, the set {ξd } (over all forms a basis of V . collections d) We have used the following notation: given d, (l)
(l)
pik := dik − x p1 +...+ pk−1 +l , 1 ≤ l ≤ pi .
(3.15)
Also, for n ≥ i ≥ j we introduce the monic polynomials λi j (u) := (u + λi(1) j )... (p )
(u + λi j j ). Finally, we define the action of the generators of U (gl N , e) on V by their matrix elements in the Gelfand-Tsetlin base: e(s)
i[d,d ]
s−1− pi+1 + pi = −−1− pi (pi(a) j − i)
(a) (a) (b) (b) × (pi j − pik )−1 (pi j − pi+1,k ), k≤i, b≤ pk (k,b)=( j,a)
(a)
if di j
(a)
= di j − 1 for certain j ≤ i; (s) d ] i[d,
f
(a)
= −1+ pi (pi j + (1 − i))s−1
(a)
(a) (b) (b) × (pi j − pik )−1 (pi j − pi−1,k ), k≤i, b≤ pk (k,b)=( j,a)
(a)
(3.16)
k≤i+1 b≤ pk
(a)
(3.17)
k≤i−1 b≤ pk (s)
(s)
if di j = di j + 1 for certain j ≤ i. All the other matrix coefficients of ei , fi vanish. The following proposition is taken from [18].
Finite Analog of AGT Relation I: Finite W -Algebras and Quasimaps’ Spaces
465
Proposition 3.4. The formulas (3.16), (3.17) give rise to the action of U (gl N , e) ⊗ K on V . Moreover, Ai (u), Ci (u), Bi (u) are polynomials in u of degrees p1 + · · · + pi (resp. p1 + · · · + pi − 1, p1 + · · · + pi − 1) such that
Ai (u)ξd = λi1 (u) . . . λii (u − i + 1)ξd ,
(3.18)
for i = 1, . . . , n, and
Bi (−1 pi(a) j )ξd −1 (a) −1 (a) = −λi+1,1 (−1 pi(a) (a) , j )λi+1,2 ( pi j − 1) . . . λi+1,i+1 ( pi j − i)ξd+δ
(3.19)
ij
Ci (−1 pi(a) j )ξd (a)
(a)
(a)
= λi−1,1 (−1 pi j )λi−1,2 (−1 pi j − 1) . . . λi−1,i−1 (−1 pi j − i + 2)ξd−δ (a) , (3.20) ij
(a) (a) (a) for i = 1, . . . , n − 1, where d ± δi j is obtained from d by replacing di j by di j ± 1.
Proof. The formulas (3.19) and (3.16) (resp. (3.20) and (3.17)) are equivalent by the Lagrange interpolation. So it suffices to prove that the formulas (3.18), (3.19), (3.20) give rise to the action of U (gl N , e) ⊗ K on V . Now V admits a certain integral form over C[±1 , x1 , . . . , x N ] which can be specialized to the values of parameters xi satisfying certain integrality and positivity conditions. These specializations admit finite-dimensional subspaces spanned by certain finite subsets of the Gelfand-Tsetlin base. Theorem 4.1 of [18] describes the action of U (gl N , e) in these finite-dimensional subspaces by the formulas (3.18), (3.19), (3.20). It follows that the action of generators given by (3.18), (3.19), (3.20) in these subspaces satisfies the relations of U (gl N , e). both d and d ± δ (a) enter the above finite subsets for quite a few For each given d, ij specializations: more precisely, the set of special values of x1 , . . . , x N such that both d (a) and d ± δ enter the corresponding finite subsets is Zariski dense in t∗ × A1 . It follows ij
that the relations of U (gl N , e) are satisfied for all values of x1 , . . . , x N .
Proposition 3.5. The Gelfand-Tsetlin module V is isomorphic to the universal Verma module with extended scalars M . Proof. First, V is an irreducible module over U (gl N , e) ⊗ K . In effect, by Propo(r ) sition 3.4 the Gelfand-Tsetlin subalgebra of U (gl N , e) ⊗ K generated by di acts diagonally in the Gelfand-Tsetlin base with pairwise distinct eigenvalues. Therefore, it suffices to check the following two things:
466
(1) (2)
A. Braverman, B. Feigin, M. Finkelberg, L. Rybnikov (s)
for each d there are indices i, s such that fi ξd = 0; (s) for each d = 0 there are indices i, s such that ei ξd = 0.
Both follow directly from the formulas (3.16), (3.17). Second, M is an irreducible module over U (gl N , e) ⊗ K . In effect, for a general highest weight ∈ t∗ /W L the Verma module M(−−1 , e) is irreducible according to [11]. Hence the universal Verma module is irreducible as well. ∼ Now to construct the desired isomorphism M −→V it suffices to produce a nonzero homomorphism M → V . By the universal property of Verma modules, it suffices to identify the highest weights of M and V . By the argument in the proof of Theorem 5.5 (r ) of [11], di acts on the highest vector of M by multiplication by the r th elementary symmetric polynomial in the variables i −1−−1 x j , p1 + · · · + pi−1 + 1 ≤ j ≤ p1 + · · · + pi . On the other hand, it follows from the formula (3.18) that di (u) acts on the highest vector p1 +···+ pi −1 ξ0 by multiplication by u − pi j= p1 +···+ pi−1 +1 (u + i − 1 − x j ). The coincidence of highest weights completes the proof of the proposition. Recall that the Galois group of K over K is W L . By the irreducibility of U (gl N , e)⊗ K -module V , there is a unique semilinear action of W L on V intertwining the action of U (gl N , e) ⊗ K , and trivial on the highest vector ξ0 . Corollary 3.6. The universal Verma module M(gl N , e) is isomorphic to V W L . 3.7. The characters of positive subalgebra. Recall the notations of [11]. So t stands for the diagonal Cartan subalgebra of g, and te stands for the centralizer of e in t, and ge stands for the centralizer of e in g. The collection e of non-zero weights of te on g is a restricted root system, see e.g. Sect. 3.1 of [11]. The roots appearing in p ⊂ g form a system e+ ⊂ e of positive roots. Let us denote by ge+ ⊂ ge the subspace spanned by the positive root vectors. Recall a linear space embedding : ge → U (g, e) of Theorem 3.6 of [11]. We define U+(g, e) as the subalgebra of U (g, e) generated by (ge+ ). In terms of the shifted Yangian, U+(g, e) is generated by {ei(s) }, 1 ≤ i ≤ n −1, s ≥ pi+1 − pi +1. We are interested in the (additive) characters of U+(g, e), that is maximal ideals of U+ (g, e)ad := U+(g, e)/[U+(g, e), U+(g, e)]. We have U+(g, e)ad Sym( (ge+ )ad ), where (ge+ )ad := (ge+ )/ (ge+ ) ∩ [U+(g, e), U+(g, e)]. In terms of roots, (ge+ )ad is spanned by the simple positive root spaces in (ge+ ). In terms of the shifted Yangian, (s)
(ge+ )ad is spanned by {ei }, 1 ≤ i ≤ n − 1, pi+1 − pi + 1 ≤ s ≤ pi+1 . The Kazhdan filtration induces the increasing filtration on the root space (geα ) for a simple positive root α ∈ e+ . In terms of the shifted Yangian, Fr (geαi ) is spanned by {ei(s) }, 1 ≤ i ≤ n − 1, r ≥ s ≥ pi+1 − pi + 1. For a simple root α we define m α so that Fm α (geα ) = (geα ), but Fm α −1 (geα ) = (geα ). Clearly, for α = αi we have m αi = pi+1 . We say that an additive character χ : U+(g, e) → C(), that is a linear function
(ge+ )ad → C(), is regular if χ (Fm α −1 (geα )) = 0, but χ (Fm α (geα )) = 0 for any simple root α. Let T e stand for the centralizer of e in the diagonal torus of G. Then the adjoint action of T e on the set of regular characters is transitive. We specify one particular regular character in terms of the shifted Yangian: χ(ei(s) ) = 0 for (p ) pi+1 − pi + 1 ≤ s < pi+1 , and χ(ei i+1 ) = −1 .
Finite Analog of AGT Relation I: Finite W -Algebras and Quasimaps’ Spaces
467
in a completion of the universal Verma Definition 3.8. The Whittaker vector w ∈ M module (the product of the weight spaces) is the unique eigenvector for U+(g, e) with the eigenvalue χ whose highest weight component coincides with the highest vector. For a weight d we denote by wd the weight d component of w. 3.9. The Shapovalov form in terms of shifted Yangians. We consider the antiinvolution (s+ p − p ) (s) (s) (s) ς : Yπ(gln ) → Yπ(gln ) taking di to di , and fi to ei i+1 i . This is nothing else than the composition of isomorphism (2.35) of [12] and anti-isomorphism (2.39) of [12]. It descends to the same named antiinvolution ς : U (g, e) → U (g, e). According to Sect. 3.5 of [12], this antiinvolution can be alternatively described as follows. Let σ stand for the Cartan antiinvolution of g (transposition). Let w0l stand for the adjoint action of a representative of the longest element of the Weyl group of the Levi subalgebra l. Then the composition w0l σ preserves e and everything else entering the definition of the finite ∼ W -algebra and gives rise to an antiisomorphism U (g, e)−→U (g, e), where U (g, e) (see Sect. 2.2 of [11]) is defined just as U (g, e), only with left ideals replaced by right ∼ ideals. Composing this antiisomorphism with the isomorphism U (g, e)−→U (g, e) of Corollary 2.9 of [11] we obtain an antiinvolution of U (g, e). This antiinvolution coincides with ς . Definition 3.10. The Shapovalov bilinear form (·, ·) on the universal Verma module M with values in C(, x1 , . . . , xn ) is the unique bilinear form such that (x, yu) = (ς (y)x, u) for any x, u ∈ M, y ∈ U (g, e), with value 1 on the highest vector. 4. Parabolic Laumon Spaces and Correspondences: Proof of the Main Conjecture for G = G L(n) In this section we prove Conjecture 2.9 for G = G L N . Note that in this case G ˇ P P, ˇ L L, ˇ T Tˇ . G, 4.1. We recall the setup of [17]. Let C be a smooth projective curve of genus zero. We fix a coordinate z on C, and consider the action of C∗ on C such that v(z) = v −1 z. We ∗ have CC = {0, ∞}. We consider an N -dimensional vector space W with a basis w1 , . . . , w N . This defines a Cartan torus T ⊂ G = G L N = Aut (W ) acting on W as follows: for T t = (t1 , . . . , t N ) we have t(wi ) = ti wi . 4.2. We fix an n-tuple of positive integers p1 ≤ p2 ≤ · · · ≤ pn such that p1 + · · · + pn = N . Let P ⊂ G be a parabolic subgroup preserving the flag 0 ⊂ W1 := w1 , . . . , w p1 ⊂ W2 := w1 , . . . , w p1 + p2 ⊂ · · · ⊂ Wn−1 := w1 , . . . w p1 +···+ pn−1 ⊂ Wn := W . Let G/P be the corresponding partial flag variety. Given an (n − 1)-tuple of nonnegative integers d = (d1 , . . . , dn−1 ), we consider Laumon’s parabolic quasiflags’ space Qd , see [21], 4.2. It is the moduli space of flags of locally free subsheaves 0 ⊂ W1 ⊂ · · · ⊂ Wn−1 ⊂ W = W ⊗ OC such that rank(Wk ) = p1 + · · · + pk , and deg(Wk ) = −dk .
468
A. Braverman, B. Feigin, M. Finkelberg, L. Rybnikov
It is known to be a smooth connected projective variety of dimension dim(G/P) + n−1 i=1 di ( pi + pi+1 ), see [21], 2.10. 4.3. We consider the following locally closed subvariety Qd ⊂ Qd (parabolic quasiflags based at ∞ ∈ C) formed by the flags 0 ⊂ W1 ⊂ · · · ⊂ Wn−1 ⊂ W = W ⊗ OC such that Wi ⊂ W is a vector subbundle in a neighbourhood of ∞ ∈ C, and the fiber of Wi at ∞ equals the span w1 , . . . , w p1 +···+ pi ⊂ W . It is known to be a smooth connected quasiprojective variety of dimension n−1 d i=1 di ( pi + pi+1 ). Moreover, there is a natural proper morphism Qd → QMG,P and according to A. Kuznetsov, this morphism is a small resolution of singularities d (cf. Remark after Theorem 7.3 of [9]), so that HT• ×C∗ (Qd ) = IH•T ×C∗ (QMG,P ), and • • HL× C∗ (Qd ) = IH L×C∗ (QMG,P ). d
4.4. Fixed points The group G × C∗ acts naturally on Qd , and the group T × C∗ acts naturally on Qd . The set of fixed points of T × C∗ on Qd is finite; its description is absolutely similar to [17], 2.2, which we presently recall. (p ) (1) Let d be a collection of nonnegative integral vectors di j = (di j , . . . , di j j ), n −1 ≥ p j (l) di j , and for i ≥ k ≥ j we have i ≥ j ≥ 1, such that di = ij=1 |di j | = ij=1 l=1 (l) (l) dk j ≥ di j , i.e. for any 1 ≤ l ≤ p j we have d ≥ d . Abusing notation we denote by kj
d the corresponding T × C∗ -fixed point in Qd :
ij
(p )
(1) · 0)w1 ⊕ · · · ⊕ OC (−d11 1 · 0)w p1 , W1 = OC (−d11 (p )
(1)
(1)
W2 = OC (−d21 · 0)w1 ⊕ · · · ⊕ OC (−d21 1 · 0)w p1 ⊕ OC (−d22 · 0)w p1 +1 (p )
... ... ..., Wn−1 =
(1) OC (−dn−1,1
⊕ . . . ⊕ OC (−d22 2 · 0)w p1 + p2 ,
(p )
1 · 0)w1 ⊕ · · · ⊕ OC (−dn−1,1 · 0)w p1 ⊕ . . .
(1)
(p
)
n−1 · · · ⊕OC (−dn−1,n−1 · 0)w p1+···+pn−2 +1 ⊕· · ·⊕OC (−dn−1,n−1 · 0)w p1 +···+ pn−1 .
4.5. Correspondences For i ∈ {1, . . . , n − 1}, and d = (d1 , . . . , dn−1 ), we set d + i := (d1 , . . . , di + 1, . . . , dn−1 ). We have a correspondence Ed,i ⊂ Qd × Qd+i formed by the pairs (W• , W• ) such that for j = i we have W j = Wj , and Wi ⊂ Wi , cf. [17], 2.3. In other words, Ed,i is the moduli space of flags of locally free sheaves 0 ⊂ W1 ⊂ · · · Wi−1 ⊂ Wi ⊂ Wi ⊂ Wi+1 · · · ⊂ Wn−1 ⊂ W such that rank(Wk ) = p1 +· · ·+ pk , and deg(Wk ) = −dk , while rank(Wi ) = p1 +· · ·+ pi , and deg(Wi ) = −di − 1. According to [21], 2.10, Ed,i is a smooth projective algebraic variety of dimension n−1 dim(G/P) + i=1 di ( pi + pi+1 ) + pi .
Finite Analog of AGT Relation I: Finite W -Algebras and Quasimaps’ Spaces
469
We denote by p (resp. q) the natural projection Ed,i → Qd (resp. Ed,i → Qd+i ). We also have a map r : Ed,i → C, (0 ⊂ W1 ⊂ · · · Wi−1 ⊂ Wi ⊂ Wi ⊂ Wi+1 · · · ⊂ Wn−1 ⊂ W) → supp(Wi /Wi ). The correspondence Ed,i comes equipped with a natural line bundle Li whose fiber at a point (0 ⊂ W1 ⊂ · · · Wi−1 ⊂ Wi ⊂ Wi ⊂ Wi+1 · · · ⊂ Wn−1 ⊂ W) equals (C, Wi /Wi ). Let q stand for the character of T × C∗ : (t, v) → v. We define the line bundle Li := q 1−i Li on the correspondence Ed,i , that is Li and Li are isomorphic as line bundles but the equivariant structure of Li is obtained from the equivariant structure of Li by the twist by the character q 1−i . Finally, we have a transposed correspondence T Ed,i ⊂ Qd+i × Qd . 4.6. Restricting to Qd ⊂ Qd we obtain the correspondence Ed,i ⊂ Qd × Qd+i together with line bundle Li and the natural maps p : Ed,i → Qd , q : Ed,i → Qd+i , r : Ed,i → C − ∞. We also have a transposed correspondence T Ed,i ⊂ Qd+i × Qd . It is a n−1 smooth quasiprojective variety of dimension i=1 di ( pi + pi+1 ) + pi . 4.7. We denote by IHG,P,T the direct sum of equivariant (complexified) cohomology: IHG,P,T = ⊕d HT• ×C∗ (Qd ). It is a module over HT• ×C∗ ( pt) = C[t ⊕ C] = C[x1 , . . . , x N , ]. Here t⊕C is the Lie algebra of T ×C∗ . We define as the positive generator of HC2 ∗ ( pt, Z). Similarly, we define xi ∈ HT2 ( pt, Z) in terms of the corresponding one-parametric subgroup. We define IH G,P,T = IHG,P,T ⊗ HT• ×C∗ ( pt) Frac(HT• ×C∗ ( pt)). We have an evident grading IHG,P,T =
d
IHG,P,T , where
d d IHG,P,T
= HT• ×C∗ (Qd ) ⊗ HT• ×C∗ ( pt) Frac(HT• ×C∗ ( pt)).
According to the Thomason localization theorem, restriction to the T × C∗ -fixed point set induces an isomorphism HT• ×C∗ (Qd ) ⊗ HT• ×C∗ ( pt) Frac(HT• ×C∗ ( pt)) ∗
→ HT• ×C∗ (QdT ×C ) ⊗ HT• ×C∗ ( pt) Frac(HT• ×C∗ ( pt)). of the T × C∗ -fixed points d (see Sect. 4.4) form a basis The fundamental cycles [d] ∗ T × C ) ⊗ HT• ×C∗ ( pt) Frac(HT• ×C∗ ( pt)). The embedding of a point d into in ⊕d HT• ×C∗ (Qd Qd is a proper morphism, so the direct image in the equivariant cohomology is well ∈ IHd defined, and we will denote by [d] G,P,T the direct image of the fundamental cycle forms a basis of IHG,P,T . The set {[d]} of the point d.
470
A. Braverman, B. Feigin, M. Finkelberg, L. Rybnikov
4.8. For any 0 ≤ i ≤ n we will denote by Wi the tautological ( p1 + · · · + pi )dimensional vector bundle on Qd × C. By the Künneth formula we have HT• ×C∗ (Qd × C) = HT• ×C∗ (Qd ) ⊗ 1 ⊕ HT• ×C∗ (Qd ) ⊗ τ , where τ ∈ HC2 ∗ (C) is the first Chern class of O(1). Under this decomposition, for the Chern class c j (Wi ) we have c j (Wi ) =: ( j) ( j−1) ( j) 2j ( j−1) c j (Wi ) ⊗ 1 + c j (Wi ) ⊗ τ , where c j (Wi ) ∈ HT ×C∗ (Qd ), and c j (Wi ) ∈ 2 j−2
HT ×C∗ (Qd ). For 0 ≤ m ≤ n we introduce the generating series Am (u) with coefficients in the equivariant cohomology ring of Qd as follows: Am (u) := u
p1 +···+ pm
+
p1 +···+ pm
(−)−r cr(r ) (Wm ) − cr(r −1) (Wm ) u p1 +···+ pm −r .
r =1
In particular, A0 (u) := 1. We also define the operators (r +1+ pk+1 − pk )
ek
(r +1)
fk
:= p∗ (c1 (Lk )r · q∗ ) : IHG,P,T → IHG,P,T , r ≥ 0,
(4.1)
:= −q∗ (c1 (Lk )r · p∗ ) : IHG,P,T → IHG,P,T , r ≥ 0.
(4.2)
d
d−k
d
d+k
We consider the following generating series of operators on IHG,P,T : dk (u) = 1 + →
∞
(s)
dk −s+1 u −s := ak (u + k − 1)ak−1 (u + k − 1)−1 : IHG,P,T d
s=1 d IHG,P,T [[u −1 ]],
(4.3)
where 1 ≤ k ≤ n and ak (u) := u − p1 (u − 1)− p2 . . . (u − k + 1)− pk Ak (u); (4.4) ∞ d d−k (s) ek (u) = ek −s+1 u −s : IHG,P,T → IHG,P,T [[u −1 ]], 1 ≤ k ≤ n − 1; s=1+ pk+1 − pk
(4.5) fk (u) =
∞
(s)
fk −s+1 u −s : IHG,P,T → IHG,P,T [[u −1 ]], 1 ≤ k ≤ n − 1. d
d+k
(4.6)
s=1
We also introduce the auxiliary series Bk (u), Ck (u) by the formulas (3.13), (3.14). The following theorem is a straightforward generalization of Theorem 2.9 and the proof of Theorem 2.12 of [17], which are in turn its particular case for p1 = · · · = pn = 1. (s)
(s)
Theorem 4.9. The matrix coefficients of the operators ei , fi in the fixed point base of IHG,P,T are as follows: {[d]}
(a) (s) (a) (a) (b) (b) (pi j − pik )−1 (pi j − pi+1,k ), e = −1 (pi j − i)s−1− pi+1 + pi i[d,d ]
(a)
if di j
k≤i, b≤ pk (k,b)=( j,a) (a)
= di j − 1 for certain j ≤ i;
k≤i+1 b≤ pk
Finite Analog of AGT Relation I: Finite W -Algebras and Quasimaps’ Spaces (s) d ] i[d,
f
(a)
= −−1 (pi j + (1 − i))s−1
(a)
(b)
(pi j − pik )−1
k≤i, b≤ pk (k,b)=( j,a)
471
(a)
(b)
(pi j −pi−1,k ),
k≤i−1 b≤ pk
(a) (s) (s) if di(a) j = di j + 1 for certain j ≤ i. All the other matrix coefficients of ei , fi vanish. equals Furthermore, the eigenvalue of Ai (u) on [d]
(a) (u − −1 pi j ). j≤i a≤ p j n−1
∼ → (−1)|d| i=1 di pi ξ Proposition 4.10. The isomorphism : IHG,P,T −→V, [d] d intertwines the same named operators di , ei , fi , etc. In particular, the operators di , ei , fi defined in (4.3), (4.1), (4.2), turn IHG,P,T into the Gelfand-Tsetlin module over U (g, e) ⊗ K .
Proof. A straightforward comparison of Theorem 4.9 and Proposition 3.4.
4.11. Now we return to the localized L × C∗ -equivariant cohomology IHG,P = WL . Note that the action of W L on IHG,P,T is semilinear with respect to the structure IHG,P,T of K -module, and also it commutes with the action of correspondences since both the correspondences and the line bundles Li are equipped with the action of L. Hence under ∼ the identification : IHG,P,T −→V the W L -action on IHG,P,T goes to the W L -action on V introduced just before Corollary 3.6. Combining Corollary 3.6 with Proposition 4.10 we arrive at the following theorem proving Conjecture 2.9(1) in the case g = gl N . ∼
Theorem 4.12. The isomorphism : IHG,P,T −→V restricted to W L -invariants gives ∼ the same named isomorphism of U (gl N , e) ⊗ K-modules : IHG,P −→M(gl N , e). 5. Whittaker Vector and Shapovalov Form Proposition 5.1. If 1d stands for the unit cohomology class of Qd , then (1d ) = wd . (s)
Proof. For pi+1 − pi + 1 ≤ s < pi+1 , we have ei 1d = 0 for degree reasons (it would (p ) have had a negative degree). Similarly, ei i+1 1d+i , having degree 0, must be a constant multiple of 1d . More precisely, we decompose the projection p : Ed,i → Qd into composition of the proper p × r : Ed,i → Qd × (C − ∞), and further projection pr : Qd ×(C−∞) → Qd with fibers C−∞ = A1 . We have p∗ (c1 (Li ) pi −1 ·q∗ 1d+i ) = pr ∗ (p × r)∗ (c1 (Li ) pi −1 · q∗ 1d+i ). Now (p × r)∗ (c1 (Li ) pi −1 · q∗ 1d+i ) is well defined in nonlocalized equivariant cohomology, and for the degree reasons must take 1d+i to a constant multiple c of the unit class in the equivariant cohomology of Qd × (C − ∞). Furthermore, pr ∗ c = −1 c1d . So it remains to calculate the constant c. This can be done over the open subset U ⊂ Qd , where Wi /Wi−1 has no torsion, and hence p × r is a fibration with a fiber P pi −1 . More precisely, the correspondence Ed,i over U × (C − ∞) is just the projectivized vector bundle P(Wi /Wi−1 ), and Li is nothing else than O(1). We conclude that c = 1. The proposition is proved. d Proposition 5.2. For x, u ∈ IHG,P we have ((x), (u)) = (−1)|d| Qd (xu).
472
A. Braverman, B. Feigin, M. Finkelberg, L. Rybnikov (s)
Proof. Evidently, the operators fi pairing (?·?).
(s+ pi+1 − pi )
and −ei
are adjoint with respect to the
Thus we have fully proved Conjecture 2.9 for g = gl(N ). 6. Relation to the AGT Conjecture 6.1. The Uhlenbeck spaces of A2 . Let G be an almost simple simply connected complex algebraic group with maximal torus T and let g, t be the corresponding Lie algebras. For an integer a ≥ 0 let BundG (A2 ) denote the moduli space of principal G-bundles on P2 of second Chern class a with a chosen trivialization at infinity (i.e. a trivialization on the “infinite” line P1∞ ). It is shown in [8] that this space has the following properties: a) BundG (A2 ) is non-empty if and only if a ≥ 0; b) for a ≥ 0 the space BundG (A2 ) is an irreducible smooth quasi-affine variety of dimenˇ where hˇ denotes the dual Coxeter number of G. sion 2a h, In [8] we construct an affine scheme UdG (A2 ) containing BundG (A2 ) as a dense open subset which we are going to call the Uhlenbeck space of bundles on A2 . The scheme UdG (A2 ) is still irreducible but in general it is highly singular. The main property of UdG (A2 ) is that it possesses the following stratification: BundG (A2 ) = BunbG (A2 ) × Syma−b (A2 ). (6.1) 0≤b≤a
Here each BunbG (A2 )×Syma−b (A2 ) is a locally closed subset of UdG (A2 ) and its closure is equal to the union of similar subsets corresponding to all b ≤ b. We shall denote by BunG (A2 ) (resp. UG (A2 )) the disjoint union of all BundG (A2 ) (resp. of UdG (A2 )). Let us note that the group G × GL(2) acts naturally on BundG (A2 ): here the first factor acts by changing the trivialization at ∞ and the second factor acts on A2 . It is easy to deduce from the construction of [8] that this action extends to an action of the same group on the Uhlenbeck space UdG (A2 ). The group G × GL(2) acts naturally on UdG , where G acts by changing the trivialization at P1∞ and GL(2) acts on P2 preserving P1∞ . 6.2. Instanton counting. We may now consider the equivariant integral 1d UdG
of the unit G × GL(2)-equivariant cohomology class (which we denote by 1d ) over UdG ; the integral takes values in the field K which is the field of fractions of the algebra ∗ A = HG×GL(2) ( pt). Note that A is canonically isomorphic to the algebra of polynomial functions on the Lie algebra g × gl(2) which are invariant with respect to the adjoint action. Thus each Ud 1d may naturally be regarded as a rational function of a ∈ t and G
(ε1 , ε2 ) ∈ C2 ; this function must be invariant with respect to the natural action of W on t and with respect to interchanging ε1 and ε2 .
Finite Analog of AGT Relation I: Finite W -Algebras and Quasimaps’ Spaces
473
Consider now the generating function Z=
∞
Q
d
d=0
UdG
1d .
It can (and should) be thought of as a function of the variables q and a, ε1 , ε2 as before. The function Z (Q, a, ε1 , ε2 ) is called the Nekrasov partition function of pure N = 2 supersymmetric gauge theory.
6.3. The AGT conjecture. In [2] Alday, Gaiotto and Tachikawa suggested a relation between 4-dimensional supersymmetric gauge theory for G = SL(2) and the so-called Liouville 2-dimensional conformal field theory; some generalizations to other groups were suggested in [4,23]. Here we are going to formulate a few mathematical statements suggested by the AGT conjecture (it is not clear to us whether from the physics point of view they should be perceived as direct corollaries of it). Consider the above Uhlenbeck space UdG and let IHG× G L(2) (UdG ) denote its equivariant intersection cohomology. This is a module over the algebra AG×GL(2) := ∗ HG×GL(2) ( pt); this algebra is just the algebra of polynomial functions on g × gl(2) which are invariant under the adjoint action. We denote by KG×GL(2) its field of fractions and we let IHd,aff = IH∗G×GL(2) (UdG ) G
⊗
AG×GL(2)
KG×GL(2) .
d,aff We also set IHaff G to be the direct sum of all the IH G . This is a vector space over KG×GL(2) which informally we may think of as a family of vector spaces parametrized by a ∈ t/W and (ε1 , ε2 ) ∈ C2 /Z2 . Also, each IHdG is endowed with a perfect symmetric ˇ
pairing ·, · , which is equal to the Poincaré pairing multiplied by (−1)hd . Consider now the case G = SL(2). Then we can identify the Cartan subalgebra h of sl(2) with C. Thus a ∈ h can be thought of as a complex number. Warning. It is important to note that if we think about a as a weight of the Langlands dual sl(2)-algebra with standard generators e, h, f , then by definition the value of this weight on h is equal to α(a) = 2a (where α denotes the simple root of sl(2). This observation will be used below. Let Vir denote the Virasoro Lie algebra; it has the standard generators {L n }n∈Z and c, where c is central and L n ’s satisfy the standard relations. Given a field K of characteristic 0, for every ∈ K , c ∈ K we may consider the Verma module M,c over Vir on which c acts by c and which is generated by a vector m ,c such that L 0 (m ,c ) = m ,c ;
L n (m ,c ) = 0 for n > 0.
In addition, the Verma module M,c is equal to the direct sum of its L 0 -eigen-spaces M,c,d , where n ∈ Z≥0 and L 0 acts on M,c,d by + d. It is easy to see that there exists a unique collection of vectors w,c,d ∈ M,c,d (for all d ∈ Z≥0 ) such that 1) w,c,0 = m ,c ; 2) we have L i · w,c,d = 0 for i > 1 and L 1 · w,c,d = w,c,d−1 .
474
A. Braverman, B. Feigin, M. Finkelberg, L. Rybnikov
We let w,c denote the sum of all the w,c,d ; this is an element of the completed ˆ ,c = Verma module M d≥0 M,c,d . In addition the module M,c possesses unique symmetric bilinear form ·, · such that m ,c , m ,c = 1 and L n is adjoint to L −n . Then the AGT conjecture implies the following (the statement below is often referred to as “non-conformal limit” of the AGT conjecture, cf. [24]): Conjecture 6.4. Let =− Then
a2 (ε1 + ε2 )2 6(ε1 + ε2 )2 + ; c =1+ . ε1 ε2 4ε1 ε2 ε1 ε2
(6.2)
UdG
1 = w,c,d , w,c,d .
In other words, ˇ
Z (a, ε1 , ε2 , (−1)h Q) = Q − w,c , Q L 0 w,c . In fact, it is quite natural to expect that the following stronger result holds: Conjecture 6.5. (1) There exists an action of the Virasoro algebra Vir on IHG such that with this action IHaff G becomes isomorphic to M,c , where (2) the intersection pairing is Vir-invariant, i.e. the adjoint operator to L n is L −n , (3) L n · 1d = 0 for any n > 1 and d ≥ 0, (4) L 1 · 1d = 1d−1 for any d > 0. One can generalize Conjecture 6.4 and Conjecture 6.5 to arbitrary G. We are not going to give details here, but let us stress one thing: when G is simply laced the Virasoro algebra Vir has to be replaced by the W -algebra corresponding to the affine Lie algebra gaff ; in fact for general G (not necessarily simply laced) we believe that the W -algebra associated with the Langlands dual affine Lie algebra g∨ aff should appear (cf. the next subsection for some motivation). In other words, we expect that for general G the space IHG carries a natural action of the W -algebra W (g∨ aff ), which makes it into a Verma module over this algebra; analogs of Properties 2,3,4 above are also expected.2 It is easy to deduce from the results of [8] that the character of IHG is equal to the character of the Verma module for the W -algebra. Before we explain the connection of Conjecture 6.5 with the rest of the paper, let us recall some (known) modification of it. 6.6. The flag case. Choose a parabolic subgroup P ⊂ G and let C denote the “horizontal” line in P2 (i.e. we choose a straight line in P2 different from the one at infinity and call). Let Bun G,P denote the moduli space of the following objects: 1) a principal G-bundle FG on P2 ; 2) a trivialization of FG on P1∞ ; 3) a reduction of FG to P on C compatible with the trivialization of FG on C. 2 A modification of this conjecture exists also for g = gl(N ); this conjecture will be proved in [25].
Finite Analog of AGT Relation I: Finite W -Algebras and Quasimaps’ Spaces
475
Let us describe the connected components of Bun G,P . We are going to use the notations of Sect. 1.1. Let also aff G,P = G,P × Z be the lattice of characters of aff ∗ ˇ Z ( L) × C . Note that G,G = Z. aff,+ The lattice aff G,P contains a canonical semi-group G,P of positive elements (cf. [8]). It is not difficult to see that the connected components of BunG,P are parameterized by the elements of aff,+ G,P : aff BunθG,P . Bun G,P = θaff ∈aff,+ G,P
Typically, for θaff ∈ aff G,P we shall write θaff = (θ, d), where θ ∈ G,P and d ∈ Z.
aff is naturally acted upon by P × (C∗ )2 ; embedding L into P we get Each BunθG,P
aff . In [8] we define for each θaff ∈ aff,+ an action of L × (C∗ )2 on BunθG,P G,P a certain aff aff which contains BunθG,P as a dense open subset. The scheme Uhlenbeck scheme UθG,P aff still admits an action of L × (C∗ )2 . UθG,P Following [5] define θ aff aff ZG,P = qaff
θ∈aff G,P
aff 1θG,P .
(6.3)
θaff UG,P
Remark. In addition to [5 and 7] various examples of functions Zaff G,P were studied recently in the physical literature (cf. for example [1,3]) as (the instanton part of) the Nekrasov partition function in the presence of surface operators. ∗ ˇ One should think of Zaff G,P as a formal power series in qaff ∈ Z ( L) × C with values in the space of ad-invariant rational functions on l × C2 . Typically, we shall write ˇ and Q ∈ C∗ . Also we shall denote an element of l × C2 qaff = (q, Q), where q ∈ Z ( L) by (a, ε1 , ε2 ) or (sometimes it will be more convenient) by (a, , ε) (note that for general P (unlike in the case P = G) the function Zaff G,P is not symmetric with respect to switching ε1 and ε2 ). aff the localized L × (C∗ )2 -equivariant interAs before, let us now denote by IHθG,P θaff aff ; we also set IHaff section cohomology of UθG,P G,P to be the direct sum of all the IH G,P ; aff,+ aff aff note that IHaff G,G = IH G . Then IH G,P is G,P -graded K L×(C∗ )2 vector space. We can endow with a non-degenerate pairing ·, · . The following result is proved in [5]:
Theorem 6.7. Let P = B be a Borel subgroup. Then IHaff G,B possesses a natural action of the Lie algebra gˇ aff such that (a,ε1 ) (1) As a gˇ aff -module IHaff G,B is isomorphic to M(λaff ), where λaff = − ε2 − ρaff . aff 3 aff 1) (2) IHθG,B is the − (a,ε ε2 − ρaff − θaff -weight space of IH G,B .
aff (3) The isomorphism of (1) takes the pairing ·, · on IHθG,B to (−1)|θaff | times the 1) Shapovalov pairing (·, ·) on the − (a,ε ε2 − ρaff − θaff -weight space of M(λaff ).
3 Here by (a, ε ) we mean the weight of g∨ whose “finite” component is a and whose central charge is 1 aff ε1 . Also ρaff is a weight of g∨ aff which takes value 1 on every simple coroot.
476
A. Braverman, B. Feigin, M. Finkelberg, L. Rybnikov
6.8. Interpretation via maps and the “finite-dimensional” analog. Choose now another smooth projective curve X of genus 0 and with two marked points 0X , ∞X . Choose also a coordinate x on X such that x(0X ) = 0 and x(∞X ) = 0. Let us denote by GG,P,X the scheme classifying triples (FG , β, γ ), where 1) FG is a principal G-bundle on X; 2) β is a trivialization of FG on the formal neighborhood of ∞X ; 3) γ is a reduction to P of the fiber of FG at 0X . aff We shall usually omit X from the notations. We shall also write Gaff G for GG,G . aff aff Let eG,P ∈ GG,P denote the point corresponding to the trivial FG with the natural β and γ . It is explained in [8] that the variety Bun G,P is canonically isomorphic to the aff 1 scheme classifying based maps from (P1 , ∞) to (Gaff G,P , eG,P ) (i.e. maps from P to aff ). GG,P sending ∞ to eG,P aff The scheme GG,P may (and should) be thought of as a partial flag variety for gaff . Thus the scheme BunG,P should be thought of as an affine analog of MG,P . Also the flag Uhlenbeck scheme UG,P should be thought of as an affine analog of the scheme QMG,P (this analogy is explained in more detail in [8]). Thus Theorem 6.7 can be considered as an affine version of Theorem 1.4 and Conjecture 6.5 (together with its generalization to arbitrary g mentioned above) is an affine version of Conjecture 2.9 (in fact, there should be a more general version of this conaff jecture, dealing not only with IHaff G but with arbitrary IH G,P ). To conclude the paper we are going to explain how to use this analogy with Conjecture 2.9 in order to derive the formulas (6.2) from Theorem 6.7. For χ , k ∈ C let M(χ , k) denote the Verma module over sl(2)aff with central charge k and highest weight χ (i.e. the standard generator h of sl(2) acts on the highest weight vector as multiplication by χ ). According to [14] the algebra Vir is obtained by certain BRST reduction from U (sl(2)aff ); the corresponding BRST reduction of M(χ , k) is equal to M,c , where =
(χ + 1)2 − (k + 1)2 ; 4(k + 2)
c =1−
6(k + 1)2 . k+2
(6.4)
According to Theorem 6.7 we should take4 χ =−
2a − 1; ε2
k=−
ε1 − 2. ε2
Thus k + 2 = −ε1 /ε2 and (ε1 + ε2 )2 (k + 1)2 =− . k+2 ε1 ε2 Hence c =1+
6(ε1 + ε2 )2 , ε1 ε2
and =
a2 (χ + 1)2 − (k + 1)2 (ε1 + ε2 )2 =− + , 4(k + 2) ε1 ε2 4ε1 ε2
which coincides with (6.2). 4 The appearance of the factor 2 is explained in the Warning in Section 6.3.
(6.5)
Finite Analog of AGT Relation I: Finite W -Algebras and Quasimaps’ Spaces
477
Acknowledgements We are grateful to A. Molev for the explanation of the results of [18]. We are also grateful to J. Brundan, A. Kleschev and I. Losev for their explanations about W -algebras. Thanks are due to A. Tsymbaliuk for the careful reading of the first draft of this note and spotting several mistakes, and to S. Gukov, D. Maulik, A. Nietzke, A. Okounkov, V. Pestun, Y. Tachikawa for very helpful discussions on the subject. A. B. was partially supported by the NSF grants DMS-0854760 and DMS-0901274. B. F, M. F., and L. R. were partially supported by the RFBR grants 09-01-00239-a, 09-01-00242, 10-01-92104-JSPS-a, and RFBR-CNRSL-a 10-01-93111, the Ministry of Education and Science of Russian Federation grant No. 2010-1.3.1-111-017-029, and the AG Laboratory HSE, RF government grant, ag. 11.G34.31.0023, and HSE science foundation grants 10-01-0078 and 11-09-0033.
References 1. Alday, L.F., Gaiotto, D., Gukov, S., Tachikawa, Y., Verlinde, H.: Loop and surface operators in N=2 gauge theory and Liouville modular geometry. JHEP 1001, 113 (2010) 2. Alday, L.F., Gaiotto, D., Tachikawa, Y.: Liouville correlation functions from four-dimensional gauge theories. Lett. Math. Phys. 91(2), 167–197 (2010) 3. Alday, L.F., Tachikawa, Y.: Affine SL(2) conformal blocks from 4d gauge theories. Lett. Math. Phys. 94(1), 87–114 (2010) 4. Awata, H., Yamada, Y.: Five-dimensional AGT Relation and the deformed β-ensemble. Prog. Theor. Phys. 124, 227–262 (2010) 5. Braverman, A.: Instanton counting via affine Lie algebras. I. Equivariant J -functions of (affine) flag manifolds and Whittaker vectors. In: Algebraic structures and moduli spaces, CRM Proc. Lecture Notes 38, Providence, RI: Amer. Math. Soc., 2004, pp. 113–132 6. Braverman, A.: Spaces of quasi-maps and their applications. In: International Congress of Mathematicians. Vol. II, Zürich: Eur. Math. Soc., 2006, pp. 1145–1170 7. Braverman, A., Etingof, P.: Instanton counting via affine Lie algebras II: from Whittaker vectors to the Seiberg-Witten prepotential. In: Studies in Lie theory, Progr. Math., 243, Boston, MA: Birkhäuser Boston, 2006, pp. 61–78 8. Braverman, A., Finkelberg, M., Gaitsgory, D.: Uhlenbeck spaces via affine Lie algebras. In: The unity of mathematics (volume dedicated to I. M. Gelfand’s 90th birthday), Progr. Math. 244, Boston, MA: Birkhäuser Boston, 2006, pp. 17–135 9. Braverman, A., Finkelberg, M., Gaitsgory, D., Mirkovi´c, I.: Intersection cohomology of Drinfeld’s compactifications. Selecta Math. (N.S.) 8(3), 381–418 (2002) 10. Brundan, J., Goodwin, S.M.: Good grading polytopes. Proc. London Math. Soc. 94, 155–180 (2007) 11. Brundan, J., Goodwin, S., Kleshchev, A.: Highest weight theory for finite W -algebras. Int. Math. Res. Not. 2008, Art. ID rnn051, 53 pp. (2008) 12. Brundan, J., Kleshchev, A.: Representations of shifted Yangians and finite W -algebras. Mem. Amer. Math. Soc. 196, no. 918, Providence, RI: Amer. Math. Soc., 2008 13. Etingof, P.: Whittaker functions on quantum groups and q-deformed Toda operators. In: Differential topology, infinite-dimensional Lie algebras, and applications, Amer. Math. Soc. Transl. Ser. 2, 194, Providence, RI: Amer. Math. Soc., 1999, pp. 9–25 14. Feigin, B., Frenkel, E.: Representations of affine Kac-Moody algebras, bosonization and resolutions. Lett. Math. Phys. 19, 307–317 (1990) 15. Finkelberg, M., Mirkovi´c, I.: Semi-infinite flags. I. Case of global curve P1 . In: Differential topology, infinite-dimensional Lie algebras, and applications, Amer. Math. Soc. Transl. Ser. 2, 194, Providence, RI: Amer. Math. Soc., 1999, pp. 81–112 16. Feigin, B., Finkelberg, M., Kuznetsov, A., Mirkovi´c, I.: Semi-infinite flags. II. Local and global intersection cohomology of quasimaps’ spaces. In: Differential topology, infinite-dimensional Lie algebras, and applications, Amer. Math. Soc. Transl. Ser. 2, 194, Providence, RI: Amer. Math. Soc., 1999, pp. 113–148 17. Feigin, B., Finkelberg, M., Negut, A., Rybnikov, L.: Yangians and cohomology rings of Laumon spaces. Selecta Math. http://arxiv.org/abs/0812.4656v4 [math.AG], (2011, to appear) 18. Futorny, V., Molev, A., Ovsienko, S.: Gelfand-Tsetlin bases for representations of finite W -algebras and shifted Yangians. In: “Lie theory and its applications in physics VII”, H. D. Doebner, V. K. Dobrev, eds., Proceedings of the VII International Workshop, Varna, Bulgaria, June 2007, Sofia: Heron Press, 2008, pp. 352–363 19. Givental, A., Kim, B.: Quantum cohomology of flag manifolds and Toda lattices. Commun. Math. Phys. 168(3), 609–641 (1995) 20. Kim, B.: Quantum cohomology of flag manifolds G/B and quantum Toda lattices. Ann. of Math. 149 (2), 129–148 (1999) 21. Laumon, G.: Un Analogue Global du Cône Nilpotent. Duke Math. J 57, 647–671 (1988)
478
A. Braverman, B. Feigin, M. Finkelberg, L. Rybnikov
22. Laumon, G.: Faisceaux Automorphes Liés aux Séries d’Eisenstein. Perspect. Math. 10, 227–281 (1990) 23. Mironov, A., Morozov, A.: On AGT relation in the case of U(3). Nucl. Phys. B 825, 1–37 (2010) 24. Marshakov, A., Mironov, A., Morozov, A.: On non-conformal limit of the AGT relations. Phys. Lett. B 682(1), 125–129 (2009) 25. Maulik, D., Okounkov, A.: In preparation 26. Taki, M.: On AGT Conjecture for Pure Super Yang-Mills and W-algebra. JHEP 1105, 038 (2011) Communicated by N.A. Nekrasov
Commun. Math. Phys. 308, 479–510 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1342-6
Communications in
Mathematical Physics
Limit Theorems for Dispersing Billiards with Cusps P. Bálint1 , N. Chernov2 , D. Dolgopyat3 1 Institute of Mathematics, Budapest University of Technology and Economics, H-1111, Egry Jozsef u. 1,
Budapest, Hungary. E-mail:
[email protected]
2 Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294, USA.
E-mail:
[email protected]
3 Department of Mathematics, University of Maryland, College Park, MD 20742, USA.
E-mail:
[email protected] Received: 15 November 2010 / Accepted: 6 April 2011 Published online: 23 September 2011 – © Springer-Verlag 2011
Abstract: Dispersing billiards with cusps are deterministic dynamical systems with a mild degree of chaos, exhibiting “intermittent” behavior that alternates between regular and chaotic patterns. Their statistical properties are therefore weak and delicate. They are characterized by a slow (power-law) decay of correlations, and as a result the classical central limit theorem fails. √ We prove that a non-classical √ central limit theorem holds, with a scaling factor of n log n replacing the standard n. We also derive the respective Weak Invariance Principle, and we identify the class of observables for which the classical CLT still holds. 1. Introduction We study billiards, i.e., dynamical systems where a point particle moves in a planar domain D (the billiard table) and bounces off its boundary ∂D according to the classical rule “the angle of incidence is equal to the angle of reflection”. The boundary ∂D is assumed to be a finite union of C 3 smooth compact curves that may have common endpoints. Between collisions at ∂D, the particle moves with a unit speed and its velocity vector remains constant. At every collision, the velocity vector changes by v+ = v− − 2v− , nn,
(1.1)
where v− and v+ denote the velocities before and after collision, respectively, n stands for the inward unit normal vector to ∂D, and ·, · designates the scalar product. If the boundary ∂D is entirely smooth and concave, and the curvature of ∂D does not vanish, the billiard is said to be dispersing. Such billiards were studied by Sinai [22], and now they are known as Sinai billiards. A classical example is a unit torus T2 with finitely many fixed disjoint convex obstacles Bi , i = 1, . . . , k, i.e., the table is D = T2 \ ∪i (int Bi ).
480
P. Bálint, N. Chernov, D. Dolgopyat
Fig. 1. Billiard table with three cusps
Sinai proved that the resulting billiard dynamics in D is uniformly hyperbolic, ergodic, and K-mixing. By uniform hyperbolicity we mean that the expansion rates of unstable vectors are uniform, i.e., they expand exponentially fast. Gallavotti and Ornstein [15] proved that Sinai billiards are Bernoulli. Young [24] proved that correlations decay exponentially fast. The central limit theorem and other limit laws were derived in [4,6]. All these results have been extended to dispersing billiards with piecewise smooth boundaries, i.e., to tables with corners, provided the boundary components intersect each other transversally, i.e., the angles made by the walls at corner points are positive. A very different picture arises if some boundary components converge tangentially at a corner, i.e., make a cusp. Dispersing billiards with cusps were first studied by Machta [19] who investigated a billiard table made by three identical circular arcs tangent to each other at their points of contact (Fig. 1). He found (based on heuristic arguments) that correlations for the collision map decay slowly (only as 1/n, where n denotes the collision counter). The hyperbolicity is non-uniform meaning that there are no uniform bounds on the expansion of unstable vectors: when the trajectory falls into a cusp (Fig. 1), it may be trapped there for quite a while, and during long series of collisions in the cusp unstable vectors expand very slowly. Rigorous bounds on the decay of correlations were derived recently in [11,13]. It was shown that if A is a Hölder continuous function (“observable”) on the collision space M, then for all n ∈ Z, 2 ζn (A) := μ A · (A ◦ F n ) − μ(A) = O(1/|n|), (1.2) where F : M → M denotes the collision map (billiard map) and μ its invariant measure; we use standard notation μ(A) = M A dμ. We refer the reader to [10] for a comprehensive coverage of the modern theory of dispersing billiards and to [11] for a detailed description of dispersing billiards with cusps. Billiards with cusps are among the very few physically realistic chaotic models where correlations decay polynomially as in (1.2) leading to a non-classical Central Limit Theorem. However, the proofs of limit theorems (Theorems 1, 2 and 3 presented here) require much tighter control over the underlying dynamics than the proof of (1.2) does. As for the general strategy, our arguments follow the scheme developed in [8]. Nonetheless, concerning the specifics of billiards with cusps, we implement new ideas in the following sense.
Limit Theorems for Dispersing Billiards with Cusps
481
Machta’s original argument [19] consists of approximating the dynamics in the cusps by differential equations. The proofs in [11] involve direct, though technically complicated, estimates of the deviations of the actual billiard trajectories from the solutions of Machta’s equations. We employ a novel approach: we integrate Machta’s differential equation and find a conserved quantity, then show that the corresponding dynamical quantity is within O(1) of that ideal quantity. This gives us the necessary tight control over the dynamics. Here we summarize some issues that provide the main motivation for proving Theorems 1, 2 and 3. Billiards with cusps can be obtained by a continuous transformation of Sinai billiards. Suppose we enlarge the obstacles Bi on the torus T2 until they touch each other. At that moment cusps are formed on the boundary ∂D and the billiard ceases to be a Sinai billiard. Thus billiards with cusps appear on a natural boundary ∂S of the space S of all Sinai billiards. Strong statistical properties of Sinai billiards deteriorate near that boundary and one gets slow nonuniform hyperbolicity with ‘intermittent chaos’. Billiards within the class S but near its boundary ∂S are also interesting, because the obstacles nearly touch one another leaving narrow tunnels (of width ε > 0) in between. A periodic Lorentz gas with narrow tunnels was first examined by Machta and Zwanzig [20] who analyzed (heuristically) the diffusion process as ε → 0. We plan to investigate billiards with tunnels rigorously, and the current work is a first step in that direction. Our interest in billiards with cusps also comes from the studies [9] of a Brownian motion of a heavy hard disk in a container subject to a bombardment of fast light particles. When the slowly moving disk collides with a wall of the container, the area available to the light particles turns into a billiard table with (two) cusps, see [9, p. 193], and some light particles may be caught in one of them, and they would be hitting the disk at an unusually high rate. An important task is then to estimate the overall effect produced by those rapid collisions in the cusp. At each collision the light particle transfers some momentum to the heavy disk, and the total momentum transferred to the disk can be represented by a Birkhoff sum of a certain function. We study the limit behavior of Birkhoff sums Sn A = A + A ◦ F + · · · + A ◦ F n−1
(1.3)
for Hölder continuous functions on M. As usual, we consider centered sums, i.e., Sn A − nμ(A) = Sn (A−μ(A)), so we will always assume that μ(A) = 0; otherwise we replace A with A − μ(A). Because correlations decay as 1/n, the central limit theorem (CLT) fails. Indeed, due to (1.2), n−1 μ [Sn A]2 = (n − |k|)ζk (A) = O(n log n),
(1.4)
k=−n+1
√ so the proper normalization factor for Sn A must be n log n, rather than the classical √ n. Our main goal is to establish a non-classical central limit theorem: Theorem 1 (CLT). Let D be a planar dispersing billiard table with a cusp. Let A be a Hölder continuous1 function on the collision space M. Then we have a (nonclassical) 1 The function A may be piecewise Hölder continuous, provided its discontinuity lines coincide with discontinuities of F ±k for some k ≥ 1.
482
P. Bálint, N. Chernov, D. Dolgopyat
Fig. 2. Orientation of r and ϕ on ∂D
Central Limit Theorem Sn A ⇒ N (0, σ 2 ) √ n log n
(1.5)
for some σ 2 = σ A2 ≥ 0, which is given by explicit formula (1.7). The convergence (1.5) means precisely that for every z ∈ R,
z 2 Sn A 1 − s μ √ e 2σ 2 ds
(1.6)
as n → ∞. In the degenerate case σ = 0, the left-hand side of (1.5) converges to zero in probability; see also Theorem 3. Remark. Since the map F is ergodic, the limit law (1.5) is mixing in the sense of probability theory, i.e., the limit in (1.6) holds true if we replace μ with any measure that is absolutely continuous with respect to μ; see [14, Sect. 4.2]. The limit law (1.5) can be interpreted, in physical terms, as superdiffusion, and σ 2 = σ A2 as superdiffusion coefficient; see similar studies in [1,8,23]. In [1], an analogue of Theorem 1 was proved for another billiard model where correlations decay as O(1/n) – the Bunimovich stadium. In [8,23] a similar superdiffusion law was proved for the Lorentz gas with infinite horizon, which is also characterized (after a proper reduction [12,13]) by O(1/n) correlations. Our proofs follow the lines of [8]. There are natural coordinates r and ϕ in the collision space M, where r denotes the arc length parameter on ∂D and ϕ the angle of reflection, i.e., the angle between v+ and n in the notation of (1.1). Note that −π/2 ≤ ϕ ≤ π/2; the orientation of r and ϕ is shown in Fig. 2. The billiard map F preserves measure μ on M given by dμ = cμ cos ϕ dr, dϕ, where cμ = [2 length(∂D)]−1 is the normalizing factor. In these coordinates, M is a union of rectangles [ri , ri
] × [−π/2, π/2], where the intervals [ri , ri
] correspond to smooth components (arcs) of ∂D. The Hölder continuity of a function A : M → R means that |A(r, ϕ) − A(r , ϕ )| ≤ K A |r − r |α A + |ϕ − ϕ |α A for some α A > 0 (the Hölder exponent) and K A > 0 (the Hölder norm) provided r and r belong to one interval [ri , ri
]. The function A need not change continuously from one interval to another, even if the corresponding arcs have a common endpoint.
Limit Theorems for Dispersing Billiards with Cusps
483
The cusp is a common terminal point of two arcs, i i and i 2 , of ∂D; thus the coordinate r takes two values at the cusp, r = ri 1 and r
= ri
2 . Now the coefficient σ A2 is given by σ A2 =
cμ 8a¯
π/2 −π/2
[A(r , ϕ) + A(r
, ϕ)]
√
2 cos ϕ dϕ
,
(1.7)
where a¯ = (a1 + a2 )/2 and a1 , a2 denote the curvatures of the two arcs making the cusp measured at the vertex of the cusp. Remark. Our results easily extend to dispersing billiards with more than one cusp. To account for the total effect of all the cusps, σ A2 must be the sum of expressions (1.7), each corresponding to one cusp. The non-classical limit theorem (1.5) leads to the following lower bound on the correlations ζn (A) defined by (1.2): Corollary 1.1. If σ A2 = 0, then the sequence |n|ζn (A) cannot converge to zero as n → ±∞. Proof. This is an analogue of Corollary 1.3 in [1]. then If we had ζn (A) = o(1/|n|), A the first identity in (1.4) would imply μ [Sn A]2 = o(n log n), hence √nSnlog would n converge to zero in probability. This would contradict (1.5). Our next result reinforces the central limit theorem (1.5): Theorem 2 (WIP). Let A satisfy the assumptions of Theorem 1 and σ A2 = 0. Then the following Weak Invariance Principle holds: the process Ss N A , W N (s) = σ A2 N log N
0 < s < 1,
(1.8)
converges, as N → ∞, to the standard Brownian motion. As usual, Ss N A here is defined by (1.3) for integral values of s N and by linear interpolation in between. The same remark as we made after Theorem 1 applies here: the limit distribution of the left hand side of (1.8) is the same with respect to any measure that is absolutely continuous with respect to μ. Lastly we investigate the degenerate case σ A2 = 0 (which occurs when the integral in (1.7) vanishes). Theorem 3 (Degenerate case). Let A satisfy the assumptions of Theorem 1 and σ A2 = 0. Then we have the classical Central Limit Theorem Sn A √ ⇒ N (0, σˆ 2 ) n for some σˆ 2 = σˆ A2 ≥ 0 (see Theorem 6 for a precise formula).
(1.9)
484
P. Bálint, N. Chernov, D. Dolgopyat
The remark made after Theorem 1 applies here, too, i.e., the limit distribution of the left hand side of (1.9) is the same with respect to any measure that is absolutely continuous with respect to μ. Also, if a billiard table has several cusps, then σ A2 = 0 if and only if the expression (1.7) vanishes for every cusp. Lastly, it is standard that σˆ A2 = 0 if and only if A is a coboundary, i.e., there exists a function g ∈ L 2μ (M) such that A = g − g ◦ F almost everywhere (this follows from general results; see, e.g., [18] and [17, Theorem 18.2.2]). Remark. The function Ap is a coboundary if and only if for every periodic point x ∈ M, F p x = x, we have i=1 A(F i x) = 0; see [7]. And in dispersing billiards periodic points are dense [5]. Thus coboundaries make a subspace of infinite codimension in the space of Hölder continuous functions, i.e., the situation σˆ A2 = 0 is extremely rare. On the other hand, σ A2 = 0 occurs whenever the integral in (1.7) vanishes, hence such functions make a subspace of codimension one (for billiards with k cusps it would be a subspace of codimension k), so such functions are not so exceptional. Remark. Correlations decay slowly in discrete time, when each collision counts as a unit of time. The picture is different in physical, continuous time: the collisions inside a cusp occur in rapid succession, thus their effect is much less pronounced. In fact, the corresponding billiard flow is rapid mixing in the sense that correlations for smooth observables decay faster than any polynomial rate, and a classical Central Limit Theorem holds [2]. It is worth noting that any observable A˜ on the phase space of the flow, D × S1 , can be reduced to an observable A : M → R for the billiard map by integrating A˜ between collisions. If A˜ is bounded, then clearly A → 0 near cusps, hence σ A2 = 0, and therefore ˜ satisfies a classical CLT. A, too (just like A), On the other hand, some physically important observables for the flow are not bounded near cusps, and for them the classical CLT may fail; for example the number of collisions during the time interval (0, T ), as T → ∞, does not satisfy a classical CLT. 2. Induced Map It is standard in the studies of nonuniformly hyperbolic maps to reduce the dynamics onto a subset M ⊂ M so that the induced map F : M → M will be strongly hyperbolic and have exponential decay of correlations. In the present case the hyperbolicity is slow only because of the cusp. So we cut out a small vicinity of the cusp; i.e., we remove from M two rectangles, R1 = [ri 1 , ri 1 + ε0 ] × [−π/2, π/2] and R2 = [ri
2 − ε0 , ri
2 ] × [−π/2, π/2], with some small ε0 > 0 and consider the induced map F on the remaining collision space M = M\(R1 ∪ R2 ). It preserves the conditional measure ν on M, where ν(B) = μ(B)/μ(M) for any B ⊂ M. The map F : M → M is strongly hyperbolic and has exponential decay of correlations [11]. On the other hand, the induced map F is rather complex and has infinitely many discontinuity lines. Now let R(x) = min{m ≥ 1 : F m x ∈ M} denote the return time function on M. The domains Mm = {x ∈ M : R(x) = m}
(2.1)
Limit Theorems for Dispersing Billiards with Cusps
485
for m ≥ 1 are called cells; note that M = ∪m≥1 Mm . Cells are separated by the discontinuity lines of F. Given a function A on M we can construct the “induced” function on M as follows: A(x) =
R (x)−1
A(F m x).
(2.2)
m=0
We also denote by Sn A its Birkhoff sums: Sn A = A + A ◦ F + · · · + A ◦ F n−1 .
(2.3)
It is standard that ν(R) = 1/μ(M) (Kac’s formula) and ν(A) = μ(A)/μ(M). Since we always assume μ(A) = 0, we also have ν(A) = 0. If the original function A is continuous, then the induced function A will be continuous on each cell Mm , but it may have countably many discontinuity lines that separate cells Mm ’s from each other. So the discontinuity lines of A will coincide with those of the map F. Theorem 4 (CLT for the induced map). Let A : M → M satisfy the assumptions of Theorem 1 and A be the induced function on M constructed by (2.2). Then Sn A 2 ), ⇒ N (0, σA √ n log n
(2.4)
2 = ν(R)σ 2 . where σA A
Remark. The function R itself (more precisely, its “centered” version R0 = R − ν(R)) satisfies the above limit theorem, i.e., Sn R − nν(R) 2 ), ⇒ N (0, σR √ n log n
(2.5)
where 2 σR =
cμ 2a¯
π/2
−π/2
√
2 cos ϕ dϕ
.
(2.6)
Indeed, define a function A by 1 − ν(R) for x ∈ M . A= 1 for x ∈ M\M Then by (2.2) we have A = R − ν(R). Note that A is piecewise constant, with a single discontinuity line that separates M from M\M (hence its discontinuity line coincides with that of F), and μ(A) = 0. Thus Theorem 4 applies and gives (2.5). The formula (2.6) follows from (1.7), because A ≡ 1 in the vicinity of the cusp. The remark made after Theorem 1 applies here, too. Indeed, the ergodicity of F implies that of F, hence the limit law (2.4) is mixing, i.e., the limit distribution of the left-hand side of (2.4) is the same with respect to any measure that is absolutely continuous with respect to ν.
486
P. Bálint, N. Chernov, D. Dolgopyat
Proof of Theorem 1 from Theorem 4. Our argument is similar to [8, Sect. 3.1]; see also [10, Thm. 7.68] and [16, Thm. A.1]. First, according to the previous remark, the limit law (2.4) holds with respect to the measure ν˜ defined by d ν˜ /dν = R/ν(R). Our next step is to prove the limit law (1.5) with respect to ν˜ . Given n ≥ 1 we fix n1 = [n/ν(R)]. For every x ∈ M let n2 = n2 (x) be the number of returns to M of the trajectory of x within the first n iterations, i.e., n2 satisfies Sn2 R(x) ≤ n < Sn2 +1 R(x). Then we have Sn A = Sn1 A + (Sn2 A − Sn1 A) + (Sn A − Sn2 A).
(2.7)
Due to Theorem 4, we have Sn A ⇒ N (0, σ A2 ), √ 1 n log n
(2.8)
thus it is enough to show that the other two terms in (2.7) are negligible, i.e. χ1 =
Sn2 A − Sn1 A , √ n log n
χ2 =
Sn A − Sn2 A √ n log n
(2.9)
both converge to zero in probability. It is enough to prove the convergence to zero with respect to ν, because ν˜ is an absolutely continuous measure. To deal with χ1 we use (2.5), which implies that for any ε > 0 there is a C = Cε > 0 such that ν |n2 − n1 | ≤ C n log n ≥ 1 − ε. (2.10) Now the desired result χ1 → 0 would follow if both expressions max √ 1≤ j≤C n log n
1 √ n log n
n 1+ j i A(F x)
(2.11)
i=n1
and max √ 1≤ j≤C n log n
n1 1 i A(F x) √ n log n
(2.12)
i=n1 − j
converged to zero in probability. Because F preserves the measure ν, we can replace n1 with 0 in the above expressions. Then their convergence to zero easily follows from the Birkhoff Ergodic Theorem. To deal with χ2 we note that A is bounded, hence |Sn A −Sn2 A| ≤ A∞ (n −Sn2 R). Note that n − (Sn2 R)(x) = k if and only if F n (x) ∈ F k (Mm ) for some m > k, hence
ν x : n − (Sn2 R)(x) = k =
∞
ν(Mm ).
m=k
Thus the distribution of n − Sn2 R does not depend on n, which implies χ2 → 0 in probability. We just proved (1.5) with respect to the measure ν˜ . The latter is defined on M, but it corresponds to a representation of the space (M, μ) as a tower over M, whose levels are made by the images F i (Mm ), 0 ≤ i < m. Thus the space (M, ν) ˜ is naturally
Limit Theorems for Dispersing Billiards with Cusps
487
isomorphic to (M, μ), and the limit law (1.5) on the space (M, ν) ˜ can be restated as follows: with respect to the measure μ on M we have Sn A ◦ (2.13) ⇒ N (0, σ 2 ), √ n log n where (x) = F ζ (x) (x) and ζ (x) = max{m ≤ 0 : F m x ∈ M}; so plays the role of the “projection” on the base of the tower. Lastly, the effect of is negligible and can be handled in the same way as the difference Sn A − Sn2 A above. Theorem 5 (WIP for the induced map). Let A satisfy the assumptions of Theorem 1 and σ A2 = 0. Let A be the induced function on M constructed by (2.2). Then the following Weak Invariance Principle holds: the process Ss N A W N (s) = , 0 < s < 1, (2.14) 2 N log N σA converges, as N → ∞, to the standard Brownian motion. We derive Theorem 2 from Theorem 5 in Sect. 8. Theorem 6 (Degenerate CLT for the induced map). Let A satisfy the assumptions of Theorem 1 and σ A2 = 0. Let A be the induced function on M constructed by (2.2). Then we have a classical Central Limit Theorem Sn A 2 ), (2.15) √ ⇒ N (0, σˆ A n 2 = ν(R)σ where σˆ A ˆ A2 . The latter satisfies the standard Green-Kubo formula 2 = σˆ A
∞
ν(A · (A ◦ F n )).
(2.16)
n=−∞
This series converges exponentially fast. We derive Theorem 3 from Theorem 6 in Sect. 7. Remark. Even in the non-degenerate case, σ A2 = 0, all the terms in the series (2.16), except the one with n = 0, are finite and their sum converges (see our Lemma 3.2 2 given by the series (2.16) is finite if and only if its central term (with below). Hence σˆ A n = 0) is finite. The latter occurs if and only if σ A2 = 0 (see our Lemma 4.3 and Sect. 7 below), so we have 2 σˆ A < ∞ ⇔ σ A2 = 0.
In the bulk of the paper we prove Theorem 4. The degenerate case is treated in Sect. 7. The more specialized limit law (Theorem 5) is proved in the last Sect. 8. The underlying map F : M → M is strongly hyperbolic and has exponential decay of correlations for bounded Hölder continuous functions [11]. But we have to deal with a function A that has infinitely many discontinuity lines and is unbounded; in fact its second moment is usually infinite; see below. There are two strategies for proving limit theorems for such functions. One is based on Young’s tower and spectral properties of the corresponding transfer operator on functional spaces [1]. The other is more direct – it truncates the unbounded function A and then uses probabilistic moment estimates [8]. We follow the latter approach. In Sects. 3–4 we describe the general steps of the proof, which can be applied to many similar models. In Sects. 5–6 we provide model-specific details.
488
P. Bálint, N. Chernov, D. Dolgopyat
3. Truncation of A The constructions and arguments in Sects. 3–4 are rather general, they are based on a minimal collection of properties of the underlying dynamical system. Thus our arguments can be easily applied to other models. The necessary model-specific facts are stated as lemmas here; they will be all proved in Sects. 5–6. Lemma 3.1. We have ν(Mm ) m −3 . The notation P Q means that C1 < P/Q < C2 for some positive constants C2 > C1 > 0. This lemma is in fact proved in [11]. The power 3 here is the minimal integral power for this estimate. Indeed, the sets F i (Mm ), 1 ≤ i ≤ m, are disjoint, hence the series ∞ m=1 mν(Mm ) converges; its sum is ν(R) = 1/μ(M). Thus, if ν(Mm ) m −a for some a > 0, then a > 2. In most interesting systems with weak hyperbolicity (such as stadia, semi-dispersing billiards, etc.; see [12,13]), we have either ν(Mm ) = O(m −3 ) or ν(Mm ) = O(m −4 ); in the latter case one expects the classical central limit theorem to hold. Next we note that A|Mm = O(m), because the original function A is bounded. This implies that ν(|A|) < ∞, but usually the second moment of A is infinite, i.e., ν(A2 ) = ∞. To cope with this difficulty we will truncate the function A (in two different ways). To fix our notation, for each 1 ≤ p ≤ q we denote M p,q = ∪ p≤m
q
m −3 p −2 .
(3.1)
m= p
We also put A p,q = A · 1M p,q , where 1 B denotes the indicator of the set B, and 1 ˆ A p,q = A − A dν 1M p,q , ν(M p,q ) M p,q the “centered” version of A p,q . Note that both A p,q and Aˆ p,q vanish outside M p,q . Also, ν(Aˆ p,q ) = 0. Now we choose a large constant ω (say, ω > 10) and fix two levels at which we will truncate our function: √ √ n and q = n log log n, (3.2) p= ω (log n) so that A = A1, p + A p,q + Aq,∞ . Due to (3.1) we have ν ∃i ≤ n : F i (x) ∈ Mq,∞ ) = O (log log n)−2 → 0,
(3.3)
so the values of Aq,∞ ◦ F i can be disregarded because their probabilities are negligibly small. Thus we can replace A with A1,q . Again, due to Lemma 3.1, 1 . ν(A1,q ) = − A dν = O √ n log log n Mq,∞
Limit Theorems for Dispersing Billiards with Cusps
489
Next, we show that A1,q can be further replaced with Aˆ := A1,q − Aˆ p,q .
(3.4)
To this end we need to prove that the overall contribution of the values Aˆ p,q ◦ F i is negligible because they tend to cancel each other. Our proof involves a bound on correlations: Lemma 3.2. For each k ≥ 1 and any 1 ≤ p ≤ q ≤ ∞ and 1 ≤ p ≤ q ≤ ∞ we have ν Aˆ p,q · (Aˆ p ,q ◦ F k ) ≤ Cθ k
(3.5)
for some C > 0 and θ ∈ (0, 1) that are determined by the function Aˆ but do not depend on p, q, p , q or k. k ≥ 1 here is essential, because for k = 0 the resulting integral The condition ν Aˆ p,q Aˆ p ,q is not uniformly bounded (and it actually turns infinite for q = q = ∞). n−1 Aˆ p,q ◦ F i can be We now return to (3.4). The second moment of Sn Aˆ p,q = i=0 estimated by ν [Sn Aˆ p,q ]2 = O(n log log n),
(3.6)
where the main contribution comes from the “diagonal” terms 2 ν Aˆ p,q ◦ F i = O log(q/ p) = O(log log n), as all the other terms sum up to O(n) due to (3.5). Now by Chebyshev’s inequality for any ε > 0, const · n log log n → 0. ν |Sn Aˆ p,q | ≥ ε n log n ≤ ε2 n log n
(3.7)
Hence we can replace A1,q with Aˆ given by (3.4), i.e., Theorem 4 would follow if we prove that Sn Aˆ 2 ) ⇒ N (0, σA √ n log n
(3.8)
with respect to the measure ν. We prove (3.8) in Sect. 4. We record several useful facts. The function Aˆ is constant on the set M p,q , and its value on this set is ˆM = A| p,q
1 A dν = O( p). ν(M p,q ) M p,q
490
P. Bálint, N. Chernov, D. Dolgopyat
The first few moments of Aˆ can be roughly estimated as ˆ = ν(A1,q ) = −ν(Aq,∞ ) = O(1/q), ν(A) 2 p p m2 2 ˆ + O 2 = O(log p), ν(A ) = O 3 m p m=1 p p3 m3 3 ˆ + O 2 = O( p), ν(|A| ) = O m3 p m=1 4 p p m4 ν(Aˆ 4 ) = O + O = O( p 2 ). m3 p2
(3.9) (3.10)
(3.11)
(3.12)
m=1
ˆ hence due to (3.9) and the correlation bound Also note that Aˆ = Aˆ 1,q − Aˆ p,q + ν(A), (3.5) we have ν Aˆ · (Aˆ ◦ F k ) ≤ 4Cθ k + C /q 2 (3.13) for some constant C > 0 and all k ≥ 1. 4. Moment Estimates Here we begin the proof of the CLT for the truncated function, i.e., (3.8). Our truncations have removed all excessively large values of the function A, so now we can apply probabilistic arguments. We shall use Bernstein’s classical method based on the “big small block” technique. That is, we partition the time interval [0, n − 1] into a sequence of alternating big intervals (blocks) of length P = [n a ] and small blocks of length Q = [n b ] for some 0 < b < a < 1. The number of big blocks is K = [n/(P + Q)] ∼ n 1−a . There may be a leftover block in the end, of length L = n − K P − (K − 1)Q < P + Q. We denote by k , 1 ≤ k ≤ K , our big blocks and set S P(k) =
Aˆ ◦ F i ,
Sn =
i∈k
and Sn
= Sn Aˆ − Sn =
K
S P(k)
k=1
Aˆ ◦ F i .
i∈[0,n−1]\∪k
The second sum Sn
contains no more than n
= K Q + P ≤ 2n h terms, where h = max{a, 1 − a + b} < 1. Just as in the proof of (3.6), we estimate ν [Sn
]2 = O(n
log n) = O(n h log n), where the main contribution comes from the “diagonal” terms 2 ν Aˆ ◦ F i = O(log p) = O(log n),
Limit Theorems for Dispersing Billiards with Cusps
491
as all the other terms sum up to O(n
) due to (3.13). Now by Chebyshev’s inequality for any ε > 0, const · n h log n ν |Sn
| ≥ ε n log n ≤ → 0. ε2 n log n Hence we can neglect the contribution from the small blocks, as well as from the last leftover block. Thus Theorem 4 is equivalent to √
Sn 2 ). ⇒ N (0, σA n log n
(4.1)
By the Lévy continuity theorem, it suffices to show that the characteristic function K itS (k) itS =ν φn (t) = ν exp √ n exp √ P n log n n log n k=1
2 ), i.e., to exp(− 1 σ 2 t 2 ). converges, pointwise, to that of the normal distribution N (0, σA 2 A First we need to decorrelate the contributions from different big blocks, i.e., we will prove that
φn (t) =
K itS (k) + o(1). ν exp √ P n log n
(4.2)
k=1
This requires bounds on multiple correlations defined below. Let f be a Hölder continuous function on M which may have discontinuity lines coinciding with those of F. Consider the products f − = ( f ◦ F − p1 ) · ( f ◦ F − p2 ) · · · ( f ◦ F − pk ) for some 0 ≤ p1 < · · · < pk and f + = ( f ◦ F q1 ) · ( f ◦ F q2 ) · · · ( f ◦ F qr ) for some 0 ≤ q1 < · · · < qr . Note that f − depends on the values of f taken in the past, and f + on the values of g taken in the future. The time interval between the future and the past is p1 + q1 . Lemma 4.1. Suppose f is Hölder continuous on each cell Mm with Hölder exponent α f and Hölder norm K f . Then ν( f − f + ) − ν( f − )ν( f + ) ≤ Bθ | p1 +q1 | , (4.3) where θ = θ (α f ) ∈ (0, 1), and B = C0 K f f k+r ∞ , where C0 = C0 (D) > 0 is a constant.
(4.4)
492
P. Bálint, N. Chernov, D. Dolgopyat
√ ˆ n log n). We To prove (4.2) we apply Lemma 4.1 to the function f = exp(it A/ obviously have f ∞ = 1; for the Hölder exponent we have α f = αAˆ and for the √ Hölder norm K f = tKAˆ / n log n. Now the proof of (4.2) goes on by splitting off block at a time, and the √ one big accumulated error in the end will be O K KAˆ θ Q / n log n . This is small enough due to the following lemma: Lemma 4.2. The induced function A is Hölder continuous on each cell Mm . Its restriction to Mm has Hölder exponent αA > 0 determined by α A alone (i.e., independent of m) and Hölder norm KA,m = O(m d ) for some d > 0. √ Since we truncated our function A at the level q = n log log n, we have KAˆ = O(n d ), and since we chose Q = n b for some b > 0, the factor θ Q will suppress KAˆ and K . This completes the proof of (4.2). Due to the invariance of ν we can rewrite (4.2) as K itS P + o(1), φn (t) = ν exp √ n log n
(4.5)
where S P = S P(1) corresponds to the very first big block. Next we use the Taylor expansion t 2 S P2 itS P |S P |3 itS P =1+ √ (4.6) +O exp √ − 2n log n (n log n)3/2 n log n n log n and then integrate it. For the linear term, we use (3.9) and get ν(S P ) = O(P/q).
(4.7)
ν(S P2 ) = Pν(Aˆ 2 ) + O(P) = O(P log p).
(4.8)
For the quadratic term, we have
Indeed, the main contribution comes from the “diagonal” terms, ν(Aˆ 2 ) = O(log p), as all the other terms sum up to O(P) due to (3.13). Moreover, ν(Aˆ 2 ) = ν(A21, p ) + O(1), hence (4.8) can be rewritten as ν(S P2 ) = Pν(A21, p ) + O(P).
(4.9)
The value ν(A21, p ) must be computed precisely, to the leading order: 2 log p + O(1). Lemma 4.3. We have ν(A21, p ) = 2σA
Therefore, (4.9) takes the form 2 ν(S P2 ) = 2PσA log p + O(P).
(4.10)
For the cubic term, we apply the Cauchy-Schwartz inequality: 1/2 ν(|S P |3 ) ≤ ν(S P2 )ν(S P4 ) .
(4.11)
Limit Theorems for Dispersing Billiards with Cusps
493
For the fourth moment we use the expansion ν(S P4 ) = ν Aˆ j1 Aˆ j2 Aˆ j3 Aˆ j4 ,
(4.12)
j , for brevity. We will consider ordered sets of indices, i.e., ˆ where we denote Aˆ j = A◦F 0 ≤ j1 ≤ j2 ≤ j3 ≤ j4 < P. We fix a large constant C1 1 and divide the products Aˆ j1 Aˆ j2 Aˆ j3 Aˆ j4 into several types depending on the gaps between indices
D1 = j2 − j1 ,
D2 = j3 − j2 ,
D3 = j4 − j3 .
Case 1 (most significant). |Di | ≤ C1 log p for all i = 1, 2, 3. Then by the Hölder inequality and (3.12), ν Aˆ j Aˆ j Aˆ j Aˆ j ≤ ν(Aˆ 4 ) = O( p 2 ), 1 2 3 4 thus the total contribution of such terms is O P p 2 log3 p . Case 2 (of moderate significance). |D2 | > C1 log p and |Di | ≤ C1 log p for i = 1, 3. ˆ then use Lemma 4.2 and the Hölder We again apply Lemma 4.1 to the function f = A, inequality: 2 ˆ 4∞ n d θ C1 log p . ν Aˆ j1 Aˆ j2 Aˆ j3 Aˆ j4 = ν Aˆ 2 + O A It follows from (3.10) that the first term is O(log2 p), and if C1 is large enough, the second term will be, say, o( p −10 ). Hence the total contribution of all the above terms is O(P 2 log4 p). Other cases (least significant): If |D1 | > C1 log p and |Di | ≤ C1 log p for i = 2, 3, then the same argument gives, due to (3.9) and (3.11), ˆ ˆ 3 ) = O(q −1 p), ν Aˆ j1 Aˆ j2 Aˆ j3 Aˆ j4 = O |ν(A)|ν(| A| (here and below we suppress correlations as they are just o( p −10 )), so the total contribution of all these terms is O(P 2 q −1 p log2 p). If |Di | > C1 log p for i = 1, 2 and |D3 | ≤ C1 log p, then we get ˆ 2 ν(Aˆ 2 ) = O(q −2 log p), ν Aˆ j1 Aˆ j2 Aˆ j3 Aˆ j4 = O |ν(A)| so the total contribution of all these terms is O(P 3 q −2 log2 p). Lastly, if |Di | > C1 log p for i = 1, 2, 3, then we get ˆ 4 = O(q −4 ), ν Aˆ j1 Aˆ j2 Aˆ j3 Aˆ j4 = O |ν(A)| so the total contribution of all these terms is O(P 4 q −4 ). Summarizing all the above cases gives an overall bound: ν(S P4 ) = O P p 2 log3 p .
(4.13)
Then (4.11) becomes, due to (3.10) and (4.13), ν(|S P |3 ) = O(P p log2 p).
(4.14)
494
P. Bálint, N. Chernov, D. Dolgopyat
Now integrating (4.6) gives 2 P t 2 σA itS P P =1− ν exp √ +O √ 2n n log n n log n 2 2 t σA P P . +O √ = exp − 2n n log n Finally, (4.5) can be rewritten as 2 2 t σ PK PK φn (t) = exp − A + o(1), +O √ 2n n log n
(4.15)
(4.16)
2 t 2 ), as desired. which converges to exp(− 21 σA
Remark. Now we can justify the need for the second truncation at level p. If we did not use it, our estimates on the first and second order terms in (4.6) would be still adequate, but the estimate on the third order term would not be satisfactory. Indeed, if we just replace p with q in (4.14), then the first error term in (4.16) would diverge. 5. Basic Facts, Hölder Norms, and Correlations In this section we begin our proofs of the model-specific facts stated as lemmas in the previous sections. Dispersing billiards with cusps have been studied in [19], then with mathematical rigor in [11]; see also [2] and [13]. Here we briefly summarize the basic facts; the reader is advised to check [11] for more details. The map F : M → M is uniformly hyperbolic, i.e., it expands unstable curves and contracts stable curves at an exponential rate. More precisely, if u is an unstable tangent vector at any point x ∈ M, then Dx F n (u) ≥ cn u for some constants c > 0 and > 1 and all n ≥ 1. Similarly, if v is a stable tangent vector, then Dx F −n (v) ≥ cn v for all n ≥ 1. There is no uniform upper bounds on the expansion and contraction rates, because those approach infinity near grazing (tangential) collisions. The singularities of the original map F : M → M are made by trajectories hitting corner points (other than cusps) or experiencing grazing (tangential) collisions with ∂D. The singularities of F lie on finitely many smooth compact curves. Those curves are stable in the sense that their tangent vectors belong to stable cones. Likewise, the singularities of F −1 are unstable curves. The singularities of the induced map F are those of F plus the boundaries of the cells Mm , m ≥ 1. Those boundaries form a countable union of smooth compact stable curves that accumulate near the (unique) phase point whose trajectory runs directly into the cusp. The structure of cells Mm and their boundaries are described in [11]. Each cell has length m −7/3 in the unstable direction and length m −2/3 in the stable direction. Its measure is μ(Mm ) m −7/3 × m −2/3 = m −3 . Incidentally, this is our Lemma 3.1. The map F = F m expands the cell Mm in the unstable direction by a factor m 5/3 and contracts it in the stable direction by a factor m 5/3 , too. So the image F(Mm ) has ‘unstable size’ m −2/3 and ‘stable size’ m −7/3 . The images accumulate near the (unique) phase point whose trajectory emerges directly from the cusp.
Limit Theorems for Dispersing Billiards with Cusps
495
A characteristic feature of hyperbolic dynamics with singularities is the competition between hyperbolicity and the cutting by singularities. The former causes expansion of unstable curves, it makes them longer. The latter breaks unstable curves into pieces and thus produces shorter curves. One of the main results of [11] is a so called one-step expansion estimate [11, Eq. (5.1)] for the induced map F, which guarantees that the expansion is stronger than the cutting by singularities, i.e., “on average” the unstable curves grow fast, at an exponential rate. The one-step expansion estimate is a main tool in the subsequent analysis of statistical properties for the map F. It basically implies the entire spectrum of standard facts: the growth lemmas, the coupling lemma for standard pairs and standard families, equidistribution estimates, exponential decay of correlations (including multiple correlations) for bounded Hölder continuous functions, limit theorems for the same type of functions, etc. All these facts with detailed proofs are presented in [10, Chap. 7] for general dispersing billiards (without cusps), but those proofs work for our map F almost verbatim (see [11, p. 749]). In particular, our Lemma 4.1 follows by a standard argument (see [10, Thm. 7.41]), so we will not repeat its proof here. Proof of Lemma 4.2. Given x, y ∈ Mm , we obviously have |A(x) − A(y)| ≤ ≤
m−1 i=0 m−1
|A(F i x) − A(F i y)| K A [dist(F i x, F i y)]α A .
(5.1)
i=0
The images F i (Mm ), i = 1, . . . , m − 1, keep stretching in the unstable direction and shrinking in the stable direction, as i increases (see [11, pp. 750–751]), thus we can assume that x, y lie on one unstable curve. It was shown in [11, Eq. (4.5)] that unstable vectors u at points x ∈ Mm are expanded under F = F m by a factor Dx F m (u)/u mλ1 λm−1 ,
(5.2)
where λ1 = Dx F(u)/u,
λm−1 = Dx F m−1 (u)/Dx F m−2 (u)
are the one-step expansion factors at two “special” iterations at which the corresponding points F(x) and F m−1 (x) may come arbitrarily close to ∂ M, i.e., experience almost grazing collisions. For this reason λ1 and λm−1 do not admit upper bounds, they may be arbitrarily large (see [11, p. 741]). For those two iterations with unbounded expansion factors we can use the Hölder continuity (with exponent 1/2) of the original billiard map F, i.e., dist(F x, F y) ≤ C1 [dist(x, y)]1/2 for some C1 > 0 (see, e.g., [10, Ex. 4.50]). Then due to (5.2) for all i = 2, . . . , m − 2 we have dist(F i x, F i y) ≤ C2 m dist(F x, F y) ≤ C1 C2 m[dist(x, y)]1/2
496
P. Bálint, N. Chernov, D. Dolgopyat
for some C2 > 0. Lastly, again by the Hölder continuity of F, dist(F m−1 x, F m−1 y) ≤ C1 [dist(F m−2 x, F m−2 y)]1/2 3/2
1/2
≤ C1 C2 m 1/2 [dist(x, y)]1/4 . Adding it all up according to (5.1) gives |A(x) − A(y)| ≤ KA,m [dist(x, y)]α A /4 with KA,m = O(m 2 ). Lemma 4.2 is proved.
Proof of Lemma 3.2. Our argument is analogous to the proof of a similar correlation bound for the Lorentz gas with infinite horizon [8, Prop. 9.1]. The domain F(Mm ) can be foliated by unstable curves of length m −2/3 . Thus the conditional measure ν on F(Mm ) can be represented by a standard family Gm such that Z (Gm ) = O(m 2/3 ); see [10, Sect. 7.4] for the definition and properties of standard families and the respective Z -function. We just remind the reader that given a standard family G = {(W, νW )} of unstable curves {W } with smooth probability measures {νW } on them, and a factor measure λG that defines a probability measure μG on ∪W , the Z -function is defined by Z (G) := sup ε>0
μG (rG < ε) , ε
where rG (x) denotes the distance from a point x ∈ W ∈ G to the nearer endpoint of W , i.e., rG (x) = dist(x, ∂ W ). If the curves W ∈ G have lengths L, then Z (G) 1/L (see [10, p. 171]). The images Gn = F n (G) are also standard families, and their Z -function satisfies Z (Gn ) ≤ c1 ϑ n Z (G) + c2 ,
(5.3)
where ϑ ∈ (0, 1) and c1 , c2 > 0 are constants. The further images F n (M), n ≥ 1, have the same property: the conditional measure ν on F n (Mm ) can be represented by a standard family (for example, by F n−1 (Gm )) whose Z -function is O(m 2/3 ) (in fact, the Z -function decreases exponentially under F due to (5.3)). Now since the size of Mk in the instable direction is k −7/3 , we have (5.4) ν Mk ∩ F n (Mm ) = ν(Mm ) · O(m 2/3 k −7/3 ) = O(m −7/3 k −7/3 ) for all n ≥ 1 (this estimate was first derived in [13, p. 320]). Next we turn to the estima n ˆ ˆ tion of correlations ν A p,q · (A p ,q ◦ F ) that are involved in Lemma 3.2. For brevity, we denote A(1) = Aˆ p,q and A(2) = Aˆ p ,q . Recall that A(i) |Mm ≤ cm for i = 1, 2 and some c > 0. We truncate the functions A(i) at two levels, p < q, which will be chosen later, i.e., we consider (i)
(i) A(i) = A1,p + A(i) p,q + Aq,∞ . (i)
(i)
The functions A1,q are bounded (their ∞-norm is Ci := A1,q ∞ = O(q)) and have Hölder norm K(i) = O(qd ) by Lemma 4.2. Thus the standard correlation estimate
Limit Theorems for Dispersing Billiards with Cusps
497
[10, Thm. 7.37] (which is our Lemma 4.1 with k = r = 1, applied to two different functions) gives (1) (2) (1) (2) ν A1,q · (A1,q ◦ F n ) = O (K(1) + K(2) )C1 C2 θ n + ν A1,q ν A1,q = O(qd+2 θ n ) + O(q−2 ).
(5.5)
Next, p ∞ (1) ν A · (A(2) ◦ F n ) ≤ c2 mkν(Mk ∩ F n (Mm )) q,∞ 1,p m=1 k=q p ∞
≤ c2 p ≤ c2 p
kν(Mk ∩ F n (Mm ))
m=1 k=q ∞
kν(Mk ) = O(p/q),
(5.6)
k=q
(2) n and a similar estimate holds for ν A(1) q,∞ · (A1,p ◦ F ) . Lastly, by (5.4) ∞ ∞ (1) (2) n 2 ν A ≤ c · (A ◦ F ) mkν(Mk ∩ F n (Mm )) p,∞ p,∞ m=p k=p
≤ c2
∞ ∞
m −4/3 k −4/3
m=p k=p
= O(p−2/3 ).
(5.7)
Combining our estimates (5.5)–(5.7) gives ν A(1) · (A(2) ◦ F n ) = O qd+2 θ n + q−2 + p/q + p−2/3 (1) (2) (the shrewd reader shall notice that ν Ap,q · (Ap,q ◦ F n ) is accounted for twice – once in (5.5) and once in (5.7) – but since (5.7) estimates absolute values, such a duplication cannot hurt). Now choosing q = θ −n/(d+3) and p = q1/2 gives the desired exponential bound on correlations. 6. Second Moment Calculation Here we prove Lemma 4.3. This is the only place where we need a precise asymptotic formula, rather than just an estimate of the order of magnitude. This entails a detailed analysis of the ‘high’ cells Mm (where m is large) which are made by trajectories that go deep into the cusp and after exactly m − 1 bounces off its walls exit it. We use the results and notation of [11]. Let a cusp be made by two boundary components 1 , 2 ⊂ ∂D. Choose the coordinate system as shown in Fig. 3, then the equations of 1 and 2 are, respectively, y = f 1 (x) and y = − f 2 (x), where f i are convex C 3
498
P. Bálint, N. Chernov, D. Dolgopyat
Fig. 3. A cusp made by two curves, 1 and 2
functions, f i (x) > 0 for x > 0, and f i (0) = f i (0) = 0 for i = 1, 2. We will use the Taylor expansion for the functions f i and their derivatives: f i (x) =
1 ai x 2 + O(x 3 ), 2
f i (x) = ai x + O(x 2 ),
f i
(x) = ai + O(x),
where ai = f i
(0). Since the curvature of the boundary of dispersing billiards must not vanish, we have ai > 0 for i = 1, 2. Consider a billiard trajectory entering the cusp and making a long series of N reflections there (so it belongs to M N +1 ). We denote reflection points by (xn , yn ), where yn = f 1 (xn ) or yn = − f 2 (xn ) depending on which side of the cusp the trajectory hits. We also denote by γn = π/2 − |ϕn | the angle made by the outgoing velocity vector with the line tangent to ∂D at the reflection point (xn , yn ). When the trajectory goes down the cusp, xn decreases but γn grows. Then γn reaches π/2 and the trajectory turns back and starts climbing out of the cusp. During that period xn grows back, but γn decreases. Denote by N2 the deepest collision (closest to the vertex of the cusp), then x1 > x2 > · · · > x N2 ≤ x N2 +1 < x N2 +2 < · · · < x N . It was shown in [11] that N2 = N /2 + O(1). The following asymptotic formulas were also proven in [11]: xn n −1/3 N −2/3
∀n = 1, . . . , N2 .
(6.1)
Also, γ1 = O(N −2/3 ) and γn nxn n 2/3 N −2/3
∀n = 2, . . . , N2 .
(6.2)
During the exiting period (N2 ≤ n ≤ N ), we have, due to time reversal symmetry, xn (N − n)−1/3 N −2/3 and γn (N − n)2/3 N −2/3 , with the exception of γ N = O(N −2/3 ). We also note that γ N2 = π/2 + O(1/N ). The sequence (xn , γn ) satisfies certain recurrence equations (that follow from elementary geometry). If we assume that yn = f 1 (xn ), and hence yn+1 = − f 2 (xn+1 ), then γn+1 = γn + tan−1 f 1 (xn ) + tan−1 f 2 (xn+1 )
(6.3)
Limit Theorems for Dispersing Billiards with Cusps
499
and xn+1 = xn −
f 1 (xn ) + f 2 (xn+1 ) . tan γn + tan−1 f 1 (xn )
(6.4)
If, on the other hand, yn = − f 2 (xn ) (and hence yn+1 = f 1 (xn+1 )), then the above equations hold, but f 1 and f 2 must be interchanged. This is all proven in [11]. To motivate our further analysis we note that Eqs. (6.3)–(6.4) can be approximated, to the leading order, by γn+1 − γn ≈ 2ax ¯ n
xn+1 − xn ≈ −
ax ¯ n2 , tan γn
(6.5)
where a¯ = (a1 +a2 )/2. Now (6.5) can be regarded as discrete versions of two differential equations γ˙ = 2ax, ¯
x˙ = −
ax ¯ 2 . tan γ
These equations were first derived (and solved) by Machta [19]. They have an integral I = x 2 sin γ (i.e., I˙ = 0). This suggests that the quantity In = N 2 xn2 sin γn should remain almost constant (the factor N 2 is included so that to make In 1). Indeed, we have for all n = 2, . . . , N − 2, In+1 − In = O(N 2 xn4 /γn ),
(6.6)
which follows by Taylor expansion of the functions involved in (6.3)–(6.4) and using the asymptotic formulas (6.1)–(6.2). (The largest error terms comes from the approximation of tan γn + tan−1 f 1 (xn ) by tan γn .) As a result, we have In+1 − In = O max{n −2 , (N − n)−2 } , hence |In − I N2 | = O(n −1 ). Next we use an elliptic integral to introduce a new variable γ √ s = (γ ) := sin z dz
(6.7)
(6.8)
0
and accordingly we put sn = (γn ) for n ≤ N2 (i.e., while γn keeps increasing). Then γn+1 √ sn+1 − sn = sin z dz = sin γn∗ (γn+1 − γn ) γn
for some γn∗ ∈ (γn , γn+1 ). Again using the Taylor expansion and (6.1)–(6.4) we obtain sn+1 − sn = 2a¯ N −1 In + O(1/n) (6.9) = 2a¯ N −1 I N2 1 + O(1/n) .
500
P. Bálint, N. Chernov, D. Dolgopyat
Summing up from 1 to n gives sn =
log n 2an ¯ . I N2 + O N N
(6.10)
In particular, for n = N2 = N /2 + O(1) we get
π/2 √
0
sin z dz = a¯ I N2 + O(N −1 log N ),
thus I N2 =
π/2
2 √ 1 log N sin z dz + O a¯ 0 N
and (6.10) becomes, with notation κ =
π √ 0
(6.11)
sin z dz,
sn = κn/N + O N −1 log N .
(6.12)
N2 We now estimate the sum S N2 = n=1 A(rn , ϕn ), where (rn , ϕn ) are the standard coordinates of the reflection points (rather than (xn , γn )). If the n th collision occurs at the curve in , i n = 1, 2, then rn = r¯in + rin (xn ), where r¯1 = ri 1 and r¯2 = ri
2 are the r -coordinates of the vertex of the cusp, on the curves 1 and 2 , respectively (see Sect. 1), and
x
ri (x) := (−1)
i n +1 0
1 + [ f i (x)]2 d x.
Also, ϕn = (−1)in (π/2 − γn ), which can be verified by direct inspection. Now
S N2
N2 = A r¯in + rin (xn ), (−1)in (π/2 − −1 (sn )) . n=1
First, recall that the function A is Hölder continuous with exponent α A in the variables r and ϕ. It has the same Hölder continuity with respect to x, but in terms of s we have A r, π/2 − −1 (s ) − A r, π/2 − −1 (s
) = O |s − s
|2α A /3 because −1 (s) ∼ s 2/3 for small s, so our Hölder exponent reduces to 2α A /3. Also note that the collisions at the curves 1 and 2 alternate, and the angle ϕ is negative when colliding at 1 and positive when colliding at 2 (see Fig. 3). Thus it is convenient to introduce 1 ¯ A(ϕ) := [A(ri 1 , −ϕ) + A(ri
2 , ϕ)]. 2
Limit Theorems for Dispersing Billiards with Cusps
501
Approximating an integral by Riemann sums gives S N2 =
N2 A¯ π/2 − −1 (κn/N ) n=1
+ O N (N −1 log N )2α A /3 + xnα A −1 (κ/2) N A¯ π/2 − −1 (s) ds + O N 1−α A /2 = κ 0 π/2 N ¯ = A(π/2 − γ ) sin γ dγ + O N 1−α A /2 κ 0 π/2 N ¯ √cos ϕ dϕ + O N 1−α A /2 . = A(ϕ) κ 0
(6.13)
By the time reversibility, the trajectory going out of the cusp during the period N2 ≤ n ≤ N has similar properties, but now the angle ϕ is positive when colliding at 1 and negative when colliding at 2 . Thus N n=N2
N A(rn , ϕn ) = κ
0
−π/2
¯ √cos ϕ dϕ + O N 1−α A /2 . A(ϕ)
(6.14)
¯ √cos ϕ dϕ + O N 1−α A /2 , A(ϕ)
(6.15)
Combining (6.13) and (6.14) gives N n=1
where again κ =
N A(rn , ϕn ) = κ
π/2 −π/2
π/2 √ −π/2 cos ϕ dϕ. This can be written as
A|M N +1 = J A N + O N 1−α A /2 ,
where JA =
1 2κ
π/2 −π/2
√ A(ri 1 , ϕ) + A(ri
2 , ϕ) cos ϕ dϕ.
(6.16)
(6.17)
For example, if A is a constant function (A ≡ A0 ), then the left hand side of (6.16) is (N + 1)A0 , and on the other hand J A = A0 . Next we turn to the proof of Lemma 4.3 per se. By (6.16) ν(A21, p ) = J A2 =
p
ν(Mm )m 2 + O(1)
m=1 p 2J A2 ν(Hm )m m=1
+ O(1),
where Hm = ∪∞ k=m Mk is the union of ‘high’ cells. Note that
), μ(Hm ) = μ(Hm
[k/2] Hm = ∪∞ (Mk ), k=m F
(6.18)
502
P. Bálint, N. Chernov, D. Dolgopyat
because the domains F [k/2] (Mk ) do not overlap. They consist of phase points deep in the cusp that are nearly half way in their excursions into the cusp. More precisely, if for
we denote by i + the number of forward collisions in the cusp before exiting it x ∈ Hm and by i − the number of backward (past) collisions in the cusp before exiting it, then either i + = i − or i + = i − + 1.
consist of two parts, one on (where the r -coordinates are Also, the domain Hm 1
near and above ri1 ) and the other on 2 (where the r -coordinates are near and below ri
2 ). On both parts the ϕ-coordinates are near zero. More precisely, in our previous notation we have I N2 = N 2 x N2 2 sin γ N2 = N 2 x N2 2 1 + O(1/N 2 ) (because γ N2 = π/2+O(1/N )), hence (6.11) implies x N2 = κ/(2a¯ N )+O(N −2 log N ). Thus our domains have the range of x-coordinates 0 < x < κ/(2am) ¯ + O(m −2 log m). The same bounds obviously hold for |r − ri 1 | and |r − ri
2 |. Now for each fixed x, the range of the ϕ-coordinate corresponds to the change of that coordinate during one iteration of F (indeed, at every iteration ϕ changes by x, while x only changes by O(x 2 ), cf. (6.3)–(6.4); besides, x is near its “stationary point” at iteration N /2, because it stops decreasing and starts increasing). So the range of ϕ can be estimated from (6.8)–(6.9): ϕ ∈ [ϕ1 , ϕ2 ] with ¯ N2 + O(N −2 log N ). ϕ2 − ϕ1 = κ/N + O(N −2 log N ) = 2ax
consists of two parts) Thus (remembering that Hm
μ(Hm ) = 2cμ
=
log m κ 2am ¯ +O ( m 2 )
2ar ¯ + O(r 2 | log r |) dr
0
log m cμ . +O 2 2am ¯ m3 κ2
We disregard the density sin γ of the measure μ because γ N2 = π/2 + O(1/N ). Now recall that ν(B) = ν(R)μ(B) for any set B. Therefore (6.18) becomes ν(A21, p ) = ν(R)cμ κ 2 a¯ −1 J A2 log p + O(1) which completes the proof of Lemma 4.3.
7. Degenerate Case Here we prove the classical Central Limit Theorem (Theorem 3) for degenerate functions A, which are characterized by σ A2 = 0. We begin with Theorem 6 that deals with the induced map.
Limit Theorems for Dispersing Billiards with Cusps
503
Proof of Theorem 6. As it follows from (6.16), we now have A|Mm = O(m 1−α A /2 ), because J A = 0. Hence ν(A2 ) < ∞. Moreover ν(|A|2+δ ) < ∞ for some small δ > 0. So the proof of the CLT will be much easier than it was in Sects. 3–4 for the generic case J A = 0. Still, we will have to truncate A at least once, for two reasons: (i) the third and fourth moments of A may be infinite, and (ii) there is no uniform upper bound on the Hölder norm of A|Mm , thus the multiple correlations √ estimate (Lemma 4.1) does not apply to A. We truncate A at a single level q = n log log n, i.e., we replace A with A1,q . Due to (3.3), this truncation does not affect any limit laws. The formulas (3.9)–(3.12) are now replaced with ν(A1,q ) = −ν(Aq,∞ ) = O
∞ m=q
m 1−α/2 m3
= O q −1−α/2 ,
2 ν(A21,q ) = ν(A2 ) − ν(Aq,∞ ) ∞ m 2−α = ν(A2 ) + O = ν(A2 ) + O(q −α ), 3 m m=q q m 3−3α/2 ν(|A1,q |3 ) = O = O q 1−3α/2 , 3 m m=1 q m 4−2α = O q 2−2α , ν(A41,q ) = O 3 m
(7.1)
(7.2)
(7.3)
(7.4)
m=1
where we denote α = α A for brevity. Next, because A1,q = Aˆ 1,q + ν(A1,q ) we have by Lemma 3.2 and (7.1), ν A1,q · (A1,q ◦ F k ) = O θ k + q −2−α . (7.5) The estimate (7.5) remains valid if we replace either one of the A1,q ’s (or both) with Aq,∞ . We also have A = A1,q + Aq,∞ , thus (7.6) ν A · (A ◦ F k ) = ν A1,q · (A1,q ◦ F k ) + χn,k , where the remainder term can be bounded as follows: |χn,k | = O min{θ k + q −2−α , q −α/2 } .
(7.7)
The first bound follows from the above modification of (7.5), and the second bound, 2 ) = O(q −α ); q −α , comes just from the Cauchy-Schwartz inequality, because ν(Aq,∞ see (7.2). We will use the first bound for k > log n and the second one for k ≤ log n. Now the √analysis of Sect. 4 carries through with√a few changes described below. First, naturally, n log n is replaced everywhere with n. The Taylor expansion (4.6) now reads t 2 S P2 itS P |S P |3 itS P =1+ √ − +O . (7.8) exp √ 2n n 3/2 n n For the linear term we have ν(S P ) = O Pq −1−α/2 due to (7.1).
504
P. Bálint, N. Chernov, D. Dolgopyat
For the main, quadratic term we have (according to (1.4)) ν(S P2 ) = Pν(A21,q ) + 2
P−1
(P − k)ν A1,q · (A1,q ◦ F k ) .
k=1
By using (7.2) and (7.5)–(7.7) we can replace A1,q with A and get ν(S P2 ) = Pν(A2 ) + 2
P−1
(P − k)ν A · (A ◦ F k ) + χn,P
k=1
= O(Pq −α with χn,P
log n+P 2 q −2−α ). Since
Lastly, by Lemma 3.2,
= O(Pq −α log n). P q 2 we have χn,P
2
+ O(1) + χn,P , ν(S P2 ) = P σˆ A 2 is given by (2.16). where σˆ A For the cubic term we apply (4.11) and then analyze the fourth order term as in Sect. 4 (except log p is now replaced with log q). The most significant case gives a contribution of O(Pq 2−2α log3 q), and the moderate significance case gives O(P 2 log2 q) which can be neglected if we choose P = [n a ] with a < 1 − α. Thus we get ν |S P3 | = O(Pq 1−α log2 q).
Now integrating (7.8) gives 2 P t 2 σˆ A itS P P =1− ν exp √ + O 1+α/4 2n n n 2 2 t σˆ P P . = exp − A + O 1+α/4 2n n
(7.9)
Finally, raising to the power K gives the desired result, just like in the end of Sect. 4.
Proof of Theorem 3. Our argument is very similar to the derivation of Theorem 1 from Theorem 4 in Sect. 2, so we only describe the differences. The decomposition (2.7) remains valid, but (2.8) changes to Sn1 A √ ⇒ N (0, σˆ A2 ), n which follows from Theorem 6 that we just proved. Now (2.9) become χ1 =
Sn2 A − Sn1 A , √ n
χ2 =
Sn A − Sn2 A , √ n
which must both converge to zero in probability. The argument for χ2 is the same as it was in Sect. 2, but our analysis of χ1 requires more work. Indeed, since (2.10) remains valid, we now have to show that j 1 i (7.10) max A(F x) √ √ n 1≤ j≤C n log n i=0
converges to zero in probability (note that (7.10) replaces (2.11), while (2.12) should be replaced similarly and we omit it).
Limit Theorems for Dispersing Billiards with Cusps
505
√ √ Since the number of iterations C n log n in (7.10) exceeds the normalization factor n, we cannot use the Birkhoff Ergodic Theorem anymore. Instead, we use specific features of our function A to prove an even stronger fact: Proposition 7.1. The function max1≤k≤n |Sk A|/n 5/6 converges to zero in probability, as n → ∞. Proof. First, we truncate the function A replacing it with A1,q as before. Note that A1,q = O(q 1−α/2 ) = O(n 1/2−α/4 ) (we drop less important logarithmic factors). Hence Sk A1,q − Sk A1,q = o(n 5/6 ) whenever |k − k | ≤ := [n 1/3+α/5 ]. Thus it is enough to show that max |S j A1,q |/n 5/6
1≤ j≤n/
(7.11)
converges to zero in probability. By Chebyshev’s inequality const · j ν |S j A1,q | ≥ εn 5/6 ≤ ε2 n 5/3
because ν [Sk A1,q ]2 = O(k). Summing up over j = 1, . . . , n/ gives ν
const · n 2 / → 0. max |S j A1,q | ≥ εn 5/6 ≤ 1≤ j≤n/ ε2 n 5/3
This proves the proposition, which guarantees that (7.10) converges to zero in probability. The rest of the proof of Theorem 3 is the same as the proof of Theorem 1 in Sect. 2. 8. Weak Invariance Principle Here we turn to the more specialized limit law, the WIP (Theorems 2 and 5). First we derive Theorem 2 from Theorem 5. Our argument is similar to [8, Sect. 3.1]. Proof of Theorem 2 from Theorem 5. To prove that a family of stochastic processes W N (s) weakly converges to a limit process W (s), 0 < s < 1, one needs to verify two conditions: (i) finite-dimensional distributions of W N (s) converge to those of W (s), and (ii) the family {W N (s)} is tight. The first condition means that the random vector {W N (s1 ), . . . , W N (sk )} converges in distribution to {W (s1 ), . . . , W (sk )} for every k ≥ 1 and 0 ≤ s1 < · · · < sk ≤ 1. For k = 1 this is just Theorem 1 derived in Sect. 2, and our argument extends to k > 1 easily. The tightness means that the family of probability measures {PN } on the space C[0, 1] of continuous functions on [0, 1] induced by the processes W N have the following property: for any ε > 0 there exists a compact subset K ε ⊂ C[0, 1] such that PN (K ε ) > 1−ε for all N . The compactness of K ε means that the functions {F ∈ K ε } are uniformly bounded at s = 0 and equicontinuous on [0, 1].
506
P. Bálint, N. Chernov, D. Dolgopyat
All our functions vanish at s = 0, hence we only need to worry about the equicontinuity. That is, we need to verify that for any ε > 0 there exists δ > 0 such that ν sup |W N (s ) − W N (s)| > ε < ε 0<s<s <1 |s −s|<δ
for all N ≥ N0 (ε). Since the function W N (s) is continuous and piecewise linear, we only need to compare its values at “breaking points”, where ns and ns are integers. We denote n = s N and use notation n1 and n2 introduced in the previous derivation of Theorem 1 from Theorem 4. Similarly we denote n = s N and use the respective values n1 , n2 . We have the decomposition (2.7) for both n and n . Now Sn1 A Sn1 A −√ ν sup √ >ε <ε N log N N log N due to the tightness of the family {W N (s)} (which follows from Theorem 5). Next, due to the Birkhoff Ergodic Theorem, Sm R = mν(R) + o(m), hence sup |n2 − n1 |/N converges to zero in probability, i.e., for any ε > 0 and δ > 0 there exists N0 such that for all N > N0 , ν sup |n2 − n1 | > δ N < ε. 0<s<1
Thus, again due to the tightness of the family {W N (s)}, we get Sn1 Sn2 ν sup √ −√ > ε < ε. N log N N log N A similar estimate obviously holds for n2 and n1 . Lastly, due to Lemma 3.1, we have ∞ ν |Sn A − Sn2 A| > ε N log N = O √
hence ν
N log N
1 m3
1 =O , N log N
sup |Sn A − Sn2 A| > ε N log N = O(1/ log N ) → 0,
0
and a similar estimate holds for n and n2 . This completes the proof of Theorem 2 from Theorem 5. Note that at the last point of the proof we had to use Lemma 3.1, i.e., a specific power law for the measures of the cells Mm . The derivation of Theorem 1 from Theorem 4 did not require such specifics. Proof of Theorem 5. First we note that N now plays the role of n in the proof of the CLT (Theorem 4). In particular, the truncation levels p and q must be defined by √ √ N and q = N log log N . (8.1) p= ω (log N )
Limit Theorems for Dispersing Billiards with Cusps
507
Recall that the function A was replaced by its truncated version Aˆ defined by (3.4) based on the estimates (3.3) and (3.7). Now, since the WIP requires us to control the entire path {Sn A}, 0 ≤ n ≤ N , not just its final state S N A, the estimate (3.7) must be upgraded to ν max |Sn Aˆ p,q | ≥ ε N log N → 0. (8.2) 1≤n≤N
The proof of (8.2) resembles the reflection principle in the theory of random walks. We consider “bad” sets Bn,ε = |Sn Aˆ p,q | ≥ ε N log N n−1 Bi,ε . Due to (3.7), ν(B N ,ε ) → 0, thus it is and “new additions” Nn,ε = Bn,ε \ ∪i=1 enough to show that
ν
N ∪n=1 Bn,ε \B N ,ε/2
=
N
ν(Nn,ε \B N ,ε/2 ) → 0.
(8.3)
n=1
Each point x ∈ Nn,ε \B N ,ε/2 satisfies (simultaneously) three conditions: |Sn Aˆ p,q | ≥ ε N log N , Aˆ p,q ◦ F n−1 = 0, 1 |S N Aˆ p,q − Sn Aˆ p,q | ≥ ε N log N . 2
(8.4) (8.5) (8.6)
Since Aˆ p,q ◦ F n−1 = 0, we have F n−1 (x) ∈ Mm for some p ≤ m < q, i.e., the point √ F n−1 (x) lies in a ‘high’ cell with index m ∼ n (modulo less significant logarithmic factors; see (8.1)). √ The image F(Mm ) intersects cells Mm with c1 m < m < c2 m 2 for some c1 , c2 > 0, see [11,13], but typical points y ∈ Mm are mapped into cells Mm with m ∼ m 1/2 . Their further images are typically mapped into cells Mm
with m
∼ m 1/4 , etc. Since m 1/2 ∼ n 1/4 p, we typically have Aˆ p,q ◦ F n+i = 0 for several small i’s. Precisely, there exist a, b > 0 such that for any large C > 0 there is a subset M∗m ⊂ Mm of measure ν(Mm \M∗m ) < K m −a ν(Mm ), where K = K (C) > 0 and such that for every y ∈ M∗m the images F i (y) for i = 1, . . . , C log m never appear in cells Mk with k > m 1−b . This was proved in [13, p. 320]. We will apply this fact to cells Mm with p ≤ m < q, hence we can replace C log m with C log N . The points falling into Mm \M∗m for p ≤ m < q make a set of a negligibly small measure: N q ν ∪n=1 F −n ∪m= p (Mm \M∗m ) = O(N p −2−a ) → 0. Hence we can assume that whenever Aˆ p,q ◦ F n−1 = 0, we have Aˆ p,q ◦ F n+i = 0 for all i = 0, 1, . . . , C log N .
508
P. Bálint, N. Chernov, D. Dolgopyat
For i > C log N , the correlations between Aˆ p,q ◦ F j , j < n, and Aˆ p,q ◦ F n+i are small due to Lemmas 4.1 and 4.2. One can easily check that they are < N −50 if C > 0 is large enough. In the following, we use shorthand notation Fn = |Sn Aˆ p,q |1Nn,ε ,
G n = |S N Aˆ p,q − Sn Aˆ p,q |,
√ where 1 B stands for the indicator of the set B, and we also denote D N = ε N log N . Now we have 1 ν(Nn,ε \B N ,ε/2 ) ≤ ν Fn ≥ D N & G n ≥ D N 2 1 2 ≤ ν Fn G n ≥ D N 2 ≤ 4ν(Fn2 G 2n )/D 4N ≤ 4 ν(Fn2 )ν(G 2n ) + O(N −50 ) /D 4N . The term O(N −50 ) accounts for correlations and is negligibly small. The second moment estimate (3.6) can be easily adapted to ν(G 2n ) = O(n log log N ) ≤ cN log log N for some c > 0. Also note that Fn = |Sn Aˆ p,q |1Nn,ε ≤ (D N + Aˆ p,q ∞ )1Nn,ε ≤ 2D N 1Nn,ε , because Aˆ p,q ∞ ≤ A∞ q D N for large N ’s. Thus N
ν(Nn,ε \B N ,ε/2 ) ≤ 16cN log log N /D 2N + o(1) → 0,
n=1
which completes our proof of (8.3) and that of (8.2). ˆ Thus we again can replace the unbounded function A with its truncated version A. That is, Theorem 5 would follow if we prove that ˆ ˆ N (s) = Ss N A W , 2 N log N σA
0 < s < 1,
(8.7)
converges, as N → ∞, to the standard Brownian motion. Our proof of (8.7) is analogous to that of a similar property of the Lorentz gas with infinite horizon [8, Sect. 11]. The proof consists of two parts: (i) the weak convergence ˆ N (s) to those of the Brownian Motion, and (ii) of finite-dimensional distributions of W the tightness, see below. To derive (i), by the Lévy continuity theorem it is enough to show that for any 0 < s1 < · · · < sk ≤ 1, any sequences n1 n2 nk → s1 , → s2 , . . . , → sk , N N N and any fixed t1 , t2 , . . . , tk we have k k 2 (s − s 2 2 σA i j=1 t j Sn j Aˆ j j−1 ) T j ν exp exp − , → √ 2 N log N j=1
(8.8)
Limit Theorems for Dispersing Billiards with Cusps
509
where s0 = 0 and T j = rk= j tr . This convergence can be proved by the same big small block technique as in Sect. 4: small blocks allow us to decorrelate the contributions from big blocks, and in particular the contributions from the intervals s j − s j−1 , which imply (8.8). It remains to show that the family of functions ˆ N log N }, ˆ N (s)} = {Ss N A/ {W
0<s<1
is tight. By the standard argument (see, e.g., [3, Chap. 2]) it is enough to show that there ˜ K ,n ) → 0 as K → ∞ uniformly exists a sequence {δk } with k δk < ∞ such that ν(M in N , where ˆN ˜ K ,N = ∃ j, k : j < 2k and W M
j +1 2k
ˆN −W
j > K δk . k 2
(8.9)
Let δk = 1/k 2 . First we estimate the ν-measure of ˆ ≥ 1 K N log N , ˜ K ,N ,k, j = |Sn 1 Aˆ − Sn 2 A| M k2
(8.10)
where n 1 = [ j N /2k ] and n 2 = [( j + 1)N /2k ]. Recall that Aˆ = O( p), hence Sn 2 Aˆ − Sn 1 Aˆ = O p(n 2 − n 1 ) = O pN /2k . Thus the set (8.10) is empty if 2k /k 2 > N , in particular if k > 100 log N . For k < 100 log N , we use the fourth moment estimate (4.13) and the Markov inequality to get ˜ K ,N ,k, j ) ≤ ν(M
ˆ4 k 8 ν [Sn 2 Aˆ − Sn 1 A]
=O
K 4 N 2 log2 N k 8 (n − n ) p 2 log3 N 2
1
K 4 N 2 log2 N k8 . =O 99 K 4 2k log N ˜ K ,N ) = Summing over j = 0, . . . , 2k − 1 and then over k ≤ 100 log N gives ν(M O(1/K 4 ) → 0 as K → ∞, uniformly in N , which implies the tightness. This completes the proof of Theorem 5. Acknowledgements. P. Bálint and N. Chernov acknowledge the hospitality of the University of Maryland which they visited and where most of this work was done. P. Bálint was partially supported by the Bolyai scholarship of the Hungarian Academy of Sciences and Hungarian National Fund for Scientific Research (OTKA) grants F60206 and K71693. N. Chernov was partially supported by NSF grant DMS-0969187. D. Dolgopyat was partially supported by NSF grant DMS-0555743.
510
P. Bálint, N. Chernov, D. Dolgopyat
References 1. Bálint, P. Gouëzel, S.: Limit theorems in the stadium billiard. Commun. Math. Phys. 263, 461–512 (2006) 2. Bálint, P., Melbourne, I.: Decay of correlations and invariance principles for dispersing billiards with cusps, and related planar billiard flows. J. Stat. Phys. 133, 435–447 (2008) 3. Billingsley, P. Convergence of probability measures. New York-London-Sydney: John Wiley & Sons, Inc., 1968 4. Bunimovich, L.A., Sinai, Ya.G.: Statistical properties of Lorentz gas with periodic configuration of scatterers. Commun. Math. Phys. 78, 479–497 (1980/81) 5. Bunimovich, L.A., Sinai, Ya.G., Chernov, N.I.: Markov partitions for two-dimensional billiards. Russ. Math. Surv. 45, 105–152 (1990) 6. Bunimovich, L.A., Sinai, Ya.G., Chernov, N.I.: Statistical properties of two-dimensional hyperbolic billiards. Russ. Math. Surv. 46, 47–106 (1991) 7. Bunimovich, L.A. Spohn, H., Viscosity for a periodic two disk fluid: an existence proof. Commun. Math. Phys. 176, 661–680 (1996) 8. Chernov, N., Dolgopyat, D.: Anomalous current in periodic Lorentz gases with infinite horizon. Russ. Math. Surv., 64: 73–124 (2009) 9. Chernov, N., Dolgopyat, D.: Brownian Brownian Motion-1. Memoirs AMS 198, no. 927, 2009 (193 pp) 10. Chernov, N., Markarian, R.: Chaotic Billiards. Mathematical Surveys and Monographs, 127, Providence, RI: Amer. Math. Soc., 2006 (316 pp) 11. Chernov, N., Markarian, R.: Dispersing billiards with cusps: slow decay of correlations. Commun. Math. Phys., 270, 727–758 (2007) 12. Chernov, N., Zhang, H.-K.: Billiards with polynomial mixing rates. Nonlinearity, 18, 1527–1553 (2005) 13. Chernov, N., Zhang, H.-K.: Improved estimates for correlations in billiards. Commun. Math. Phys. 277, 305–321 (2008) 14. Eagleson, G.K.: Some simple conditions for limit theorems to be mixing. Teor. Verojatnost. i Primenen. 21, 653–660 (1976) 15. Gallavotti, G., Ornstein, D.: Billiards and Bernoulli schemes. Commun. Math. Phys. 38, 83–101 (1974) 16. Gouëzel, S.: Statistical properties of a skew product with a curve of neutral points. Ergod. Th. Dynam. Syst. 27, 123–151 (2007) 17. Ibragimov, I.A., Linnik, Yu.V.: Independent and stationary sequences of random variables. Gröningen: Wolters-Noordhoff, 1971 18. Leonov, V.P.: On the dispersion of time-dependent means of a stationary stochastic process. Th. Probab. Appl., 6, 87–93 (1961) 19. Machta, J.: Power law decay of correlations in a billiard problem. J. Stat. Phys. 32, 555–564 (1983) 20. Machta, J., Zwanzig, R.: Diffusion in a periodic Lorentz gas. Phys. Rev. Lett. 50, 1959–1962 (1983) 21. Melbourne, I., Török, A.: Statistical limit theorems for suspension flows. Israel J. Math. 144, 191–209 (2004) 22. Sinai, Ya.G.: Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Russ. Math. Surv. 25, 137–189 (1970) 23. Szasz, D., Varju, T.: Limit laws and recurrence for the planar lorentz process with infinite horizon. J. Stat. Phys. 129, 59–80 (2007) 24. Young, L.-S.: Statistical properties of dynamical systems with some hyperbolicity. Ann. Math. 147, 585–650 (1998) Communicated by G. Gallavotti
Commun. Math. Phys. 308, 511–542 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1320-z
Communications in
Mathematical Physics
Surface Gap Soliton Ground States for the Nonlinear Schrödinger Equation Tomáš Dohnal1 , Michael Plum2 , Wolfgang Reichel2 1 Institut für Wissenschaftliches Rechnen und Mathematische Modellbildung,
Karlsruhe Institute of Technology (KIT), 76128 Karlsruhe, Germany. E-mail:
[email protected]
2 Institut für Analysis, Karlsruhe Institute of Technology (KIT), 76128 Karlsruhe, Germany.
E-mail:
[email protected];
[email protected] Received: 17 November 2010 / Accepted: 16 April 2011 Published online: 10 September 2011 – © Springer-Verlag 2011
Abstract: We consider the nonlinear Schrödinger equation (− + V (x))u = (x)|u| p−1 u, x ∈ Rn with V (x) = V1 (x), (x) = 1 (x) for x1 > 0 and V (x) = V2 (x), (x) = 2 (x) for x1 < 0, where V1 , V2 , 1 , 2 are periodic in each coordinate direction. This problem describes the interface of two periodic media, e.g. photonic crystals. We study the existence of ground state H 1 solutions (surface gap soliton ground states) for 0 < min σ (− + V ). Using a concentration compactness argument, we provide an abstract criterion for the existence based on ground state energies of each periodic problem (with V ≡ V1 , ≡ 1 and V ≡ V2 , ≡ 2 ) as well as a more practical criterion based on ground states themselves. Examples of interfaces satisfying these criteria are provided. In 1D it is shown that, surprisingly, the criteria can be reduced to conditions on the linear 2 2 Bloch waves of the operators − ddx 2 + V1 (x) and − ddx 2 + V2 (x). 1. Introduction The existence of localized solutions of the stationary nonlinear Schrödinger equation (NLS) (− + V (x))u = (x)|u| p−1 u, x ∈ Rn
(1.1)
with a linear potential V and/or a nonlinear potential is a classical problem of continued interest. The present paper deals with the existence of ground states in the case of two periodic media in Rn joined along a single interface, e.g. along the hyperplane {x1 = 0} ⊂ Rn . Both the coefficients V and are then periodic on either side of the interface but not in Rn . Exponentially localized solutions of this problem are commonly
512
T. Dohnal, M. Plum, W. Reichel
called surface gap solitons (SGSs) since they are generated by a surface/interface, and since zero necessarily lies in a gap (including the semi-infinite gap) of the essential spectrum of L := − + V . Ground states in the case of purely periodic coefficients, where the solutions are referred to as (spatial) gap solitons (GSs), were shown to exist in [17] in all spectral gaps of L. The proof of [17] does not directly apply to the interface problem due to the lack of periodicity in Rn . Also, in contrast to the purely periodic case the operator L in the interface problem can have eigenvalues [5,13]. The corresponding eigenfunctions are localized near the interface so that it acts as a waveguide. In this paper we restrict our attention to ground states in the semiinfinite gap to the left of all possible eigenvalues, i.e. 0 < min σ (L). Using a concentration-compactness argument, we prove an abstract criterion ensuring ground state existence based on the energies of ground states of the two purely periodic problems on either side of the interface. We further provide a number of interface examples that satisfy this criterion. Moreover, in the case n = 1 we give a sufficient condition for the criterion using solely linear Bloch waves of the two periodic problems. The physical interest in wave propagation along material interfaces stems mainly from the possibilities of waveguiding and localization at the interface. The problem with two periodic media is directly relevant in optics for an interface of two photonic crystals. Gap solitons in nonlinear photonic crystals are of interest as fundamental ‘modes’ of the nonlinear dynamics but also in applications due to the vision of GSs being used in optical signal processing and computing [15]. The NLS (1.1) is a reduction of Maxwell’s equations for monochromatic waves in photonic crystals with a p th order nonlinear susceptibility χ ( p) when higher harmonics are neglected. In the following let c be the speed of light in vacuum and εr the relative permittivity of the material. In 1D crystals, i.e., εr = εr (x1 ), χ ( p) = χ ( p) (x1 ), Eq. (1.1) arises for the electric field ansatz E(x, t) = (0, u(x1 ), 0)T ei(kx3 −ωt) +c.c. and 2 2 the potentials become V (x1 ) = k 2 − ωc2 εr (x1 ), (x1 ) = ωc2 χ ( p) (x1 ). In 2D crystals, i.e., εr = εr (x1 , x2 ), χ ( p) = χ ( p) (x1 , x2 ), the ansatz E(x, t) = (0, 0, u(x1 , x2 ))T e−iωt +c.c. 2 2 leads to V (x1 , x2 ) = − ωc2 εr (x1 , x2 ) and (x1 , x x ) = ωc2 χ ( p) (x1 , x2 ). The physical condition εr > 1, valid for dielectrics, however, clearly limits the range of allowed potentials V . On the other hand, the NLS is also widely used by physicists as an asymptotic model for slowly varying envelopes of wavepackets in 1D and 2D photonic crystals. In cubically nonlinear crystals the governing model is i∂x3 ϕ + ⊥ ϕ + V˜ (x⊥ )ϕ + (x⊥ )|ϕ|2 ϕ = 0,
(1.2)
where x⊥ = x1 in 1D and x⊥ = (x1 , x2 ) in 2D, see e.g. [7,10]. The ansatz ϕ(x) = eikx3 u(x⊥ ) then leads to (1.1) with V (x) = k − V˜ (x). GSs are also widely studied in Bose-Einstein condensates, where (1.2) is the governing equation for the condensate wave function without any approximation (with x3 playing the role of time) [14]. It is referred to as the Gross-Pitaevskii equation and the periodic potential is typically generated by an external optical lattice. SGSs of the 1D and 2D NLS have been studied numerically in a variety of geometries and nonlinearities including case where only V has an interface and is periodic in Rn [10,16], or vice versa [4,6] (1D), or where both V and have an interface [11]. Optical SGSs have been also observed experimentally in a number of studies, as examples we list: SGSs at the edge of a 1D [19] and a 2D [25,27] photonic crystal as well as at the interface of two dissimilar crystals [24,26].
Surface Gap Soliton Ground States for the NLS
513
2. Mathematical Setup and Main Results Let V1 , V2 , 1 , 2 : Rn → R be bounded functions, p > 1, and consider the differential operators L i := − + Vi , i = 1, 2 on D(L i ) = H 2 (Rn ) ⊂ L 2 (Rn ) and denote their spectra with σ (L i ). Our basic assumptions are: (H1) V1 , V2 , 1 , 2 are Tk -periodic in the xk -direction for k = 1, . . . , n with T1 = 1, (H2) ess sup i > 0, i = 1, 2, (H3) 1 < p < 2∗ − 1, 2n where, as usual, 2∗ = n−2 if n ≥ 3 and 2∗ = ∞ if n = 1, 2. Let us also mention a stronger form of (H2), namely
(H2’) ess inf i > 0 , i = 1, 2. Conditions (H2), (H2’) will be commented below. (H3) is commonly used in the variational description of ground states. In order to have an energy-functional J , which is well-defined on H 1 (Rn ), one needs 1 ≤ p ≤ 2∗ − 1. The assumption p > 1 makes the problem superlinear and the assumption p < 2∗ − 1 provides some compactness via the Sobolev embedding theorem. Although we restrict our attention to 1 < p < 2∗ − 1, problem (1.1) for 0 < p < 1 or for p ≥ 2∗ − 1, n ≥ 3 is also of interest. Consider the two purely periodic (stationary) nonlinear Schrödinger equations L i u = i (x)|u| p−1 u in Rn .
(2.1)
Their solutions arise as critical points of the functionals 1 1 |∇u|2 + Vi (x)u 2 − i (x)|u| p+1 d x, u ∈ H 1 (Rn ). Ji [u] := p+1 Rn 2 If (H1), (H2’), (H3) hold and if 0 ∈ σ (L i ) then it is well known, cf. Pankov [17], that the purely periodic problem (2.1) has a ground state wi , i.e. a function wi ∈ H 1 (Rn ) which is a weak solution of (2.1) such that its energy ci := Ji [wi ] is minimal among all nontrivial H 1 solutions. However, under the additional assumption 0 < min σ (L i ) a ground state of (2.1) exists under the weaker hypotheses (H1), (H2), (H3). This can be seen from an inspection of the proof in [17], which we leave to the reader. In the following, we use (H1), (H2), (H3). Of course our results are also valid under the stronger hypotheses (H1), (H2’), (H3), which do not require any extension of [17]. In the present paper we are interested in ground states for a nonlinear Schrödinger equation modeling an interface between two different materials. For this purpose we define the composite functions V1 (x), x ∈ Rn+ , 1 (x), x ∈ Rn+ , V (x) = (x) = n V2 (x), x ∈ R− , 2 (x), x ∈ Rn− , where Rn± = {x ∈ Rn : ±x1 > 0} and the differential operator L := − + V on D(L) = H 2 (Rn ) ⊂ L 2 (Rn ).
514
T. Dohnal, M. Plum, W. Reichel
We will prove existence of ground states for the nonlinear Schrödinger equation of interface-type Lu = (x)|u| p−1 u in Rn .
(2.2)
Solutions of (2.2) are critical points of the energy functional 1 1 J [u] := |∇u|2 + V (x)u 2 − (x)|u| p+1 d x, u ∈ H 1 (Rn ). p+1 Rn 2 Since J is unbounded from above and below, minimization/maximization of J on all of H 1 (Rn ) is impossible. Therefore we seek solutions of the following constrained minimization problem: find w ∈ N such that J [w] = c := inf J [u], u∈N
(2.3)
where N is the Nehari manifold given by N = {u ∈ H 1 (Rn ) : u = 0, G[u] = 0}, G[u] =
Rn
|∇u|2 +V (x)u 2 −(x)|u| p+1 d x.
Note that N contains all non-trivial H 1 (Rn )-solutions of (2.2) and (H2) ensures that N = ∅.1 The stronger condition (H2’) makes N a topological sphere, i.e, ∀u ∈ H 1 (Rn )\{0} ∃t > 0 such that tu ∈ N . Moreover, one of the advantages of N is that the Lagrange multiplier introduced by the constraint turns out to be zero as checked by a direct calculation. In our case of a pure power nonlinearity one could alternatively use the constraint Rn (x)|u| p+1 d x = 1. Here the Lagrange parameter can be scaled out a posteriori. A third possibility would be the constraint Rn u 2 d x = μ > 0, which generates a Lagrange parameter λ(μ). The additional term λ(μ)u in (2.1) cannot be scaled out. This approach is, moreover, restricted to 1 < p < 1 + n4 , cf. [23]. Definition 1. The following terminology will be used throughout the paper. (a) A bound state is a weak solution of (2.1) in H 1 (Rn ). (b) A ground state is a bound state such that its energy is minimal among all nontrivial bound states. (c) A strong ground state is a solution to (2.3). Note that a strong ground state is a also a ground state because N contains all non-trivial bound states. For the success in solving (2.3) we need to assume additionally to (H1)–(H3) that 0 < min σ (L), which is, e.g., satisfied if there exists a constant v0 > 0 such that V1 , V2 ≥ v0 . Note that σ (L) ⊃ σ (L 1 )∪σ (L 2 ), and hence the assumption 0 < min σ (L) implies in particular 0 < min σ (L i ) for i = 1, 2. The additional spectrum of L may be further essential spectrum or, as described in [5], so-called gap-eigenvalues. As we show in Lemma 14, one always has c ≤ min{c1 , c2 }. Our main result shows that if the strict inequality holds, then strong ground states exist. 1 For a proof, construct a sequence (u ) 1 n p+1 (Rn ) to the characteristic k k∈N in H (R ) which converges in L n : (x) > 0, |x| ≤ R} for some large R. For sufficiently large k ∈ N one finds function of {x ∈ R p+1 d x > 0 and thus ∃t > 0 such that tu ∈ N . k Rn (x)|u k |
Surface Gap Soliton Ground States for the NLS
515
Theorem 2. Assume (H1)–(H3) and 0 < min σ (L). If c < min{c1 , c2 }, where c is defined in (2.3) and c1 , c2 are the ground state energies of the purely periodic problems (2.1), then c is attained, i.e., there exists a strong ground state for the interface problem (2.2). Remark 1. We state the following two basic properties of every strong ground state u 0 of (2.2). (i) u 0 is exponentially decaying. The proof given in [17] can be applied. (ii) Up to multiplication with −1, u 0 is strictly positive. For the reader’s convenience a proof is sketched in Lemma A2 of the Appendix. Let us also note a result which excludes the existence of strong ground states. Theorem 3. Assume (H1)–(H3) and 0 < min σ (L). If V1 ≤ V2 and 1 ≥ 2 and if one of the inequalities is strict on a set of positive measure, then there exists no strong ground state of (2.2). Remark 2. The non-existence result in Theorem 3 can be extended to more general interfaces (not necessarily manifolds) as follows. Let separate Rn into two disjoint sets 1 and 2 with 1 unbounded such that Rn = 1 ∪ 2 and = ∂ 1 = ∂ 2 and suppose 1 (x), x ∈ 1 , V1 (x), x ∈ 1 , (x) = V (x) = V2 (x), x ∈ 2 , 2 (x), x ∈ 2 . Then the previous non-existence result holds if there exists a sequence (q j ) j∈N in 1 ∩ Z (n) such that dist(q j , 2 ) → ∞ as j → ∞, where Z (n) = T1 Z × T2 Z × · · · × Tn Z. This is, for instance, satisfied if there exists a cone C1 in Rn such that outside a sufficiently large ball B R the cone lies completely within the region 1 , i.e., there is a sufficiently large radius R > 0 such that C1 ∩ B Rc ⊂ 1 . In the case n = 2, where the cone becomes a sector, an example of such an interface is plotted in Fig. 1. Note that (to the best of our knowledge) the fact that no strong ground state exists does not preclude the existence of a ground state. Moreover, under the assumptions of Theorem 3 there may still exist bound states of (2.2), i.e. critical points of J in H 1 (Rn ), cf. [4,6]. To verify the existence condition c < min{c1 , c2 } of Theorem 2, it suffices to determine a function u ∈ N such that J [u] < min{c1 , c2 }. The following theorem shows that a suitable candidate for such a function is a shifted and rescaled ground state corresponding to the purely periodic problem with the smaller of the two energies. Using an idea of Arcoya, Cingolani and Gámez [3], we shift the ground state far into the half-space representing the smaller ground state energy. The rescaling is needed to force the candidate to lie in N . The following theorem thus offers a criterion for verifying existence of strong ground states. Theorem 4. Assume (H1)–(H3) and 0 < min σ (L). Let wi be a ground state of (2.1) for i = 1, 2 and let e1 be the unit vector (1, 0, . . . , 0) ∈ Rn .
516
T. Dohnal, M. Plum, W. Reichel
Fig. 1. An example of a curved interface in 2D
(a) If c1 ≤ c2 and
( p + 1) <2
Rn−
Rn−
(V2 (x) − V1 (x))w1 (x − te1 )2 d x
(2 (x) − 1 (x))|w1 (x − te1 )| p+1 d x
(2.4)
for all t ∈ N large enough, then c < min{c1 , c2 } and therefore (2.2) has a strong ground state. Condition (2.4) holds, e.g., if ess sup(V2 − V1 ) < 0. (b) If c2 ≤ c1 , then in the above criterion one needs to replace w1 (x −te1 ) by w2 (x +te1 ), integrate over Rn+ and switch the roles of V1 , V2 and 1 , 2 . Using the criterion in Theorem 4, we have found several classes of interfaces leading to the existence of strong ground states. These are listed in the following theorems. The first example considers potentials related by a particular scaling. Theorem 5. Assume (H1)–(H3), 0 < min σ (L), and V1 (x) = k 2 V2 (kx), 1 (x) = γ 2 2 (kx) for some k ∈ N as well as sup V2 < k 2 inf V2
and k
n+2− p(n−2) p−1
≤γ
4 p−1
.
(2.5)
Then (2.2) has a strong ground state. The next theorem guarantees existence for interfaces with a large jump in . Theorem 6. Assume (H1)–(H3), 0 < min σ (L). (a) Let ess sup(V2 − V1 ) < 0. Then there exists a value β0 > 0 depending on c2 such that if 1 (x) ≥ β0 almost everywhere, then (2.2) has a strong ground state. (b) A similar result holds for ess sup(V2 − V1 ) > 0 and 2 (x) ≥ β0 with β0 = β0 (c1 ). Finally, for the case n = 1 we provide sufficient conditions for criterion (2.4). Instead of ground states w1 , w2 themselves the new sufficient conditions use solely the linear Bloch modes of the operators L 1 , L 2 on a single period. The coefficient in front of the nonlinear term has no influence in this criterion besides allowing the correct ordering between c1 and c2 . Moreover, we show that if 0 is sufficiently far from σ (L), these conditions can be easily checked from the behavior of V1 and V2 near x = 0.
Surface Gap Soliton Ground States for the NLS
517 2
Theorem 7. Let n = 1 and consider the operator L = − ddx 2 + V (x) − λ on D(L) = H 2 (R) ⊂ L 2 (R) with λ ∈ R. Assume (H1)–(H3) and 0 < min σ (L). For i = 1, 2 define 2 by ci the ground state energy of (− ddx 2 + Vi (x) − λ)u = i (x)|u| p−1 u on R and assume c1 ≤ c2 . (a) A sufficient condition for the existence of a strong ground state for Lu = (x)|u| p−1 u in R is
0 −1
V2 (x) − V1 (x) p− (x)2 e2κ x d x < 0,
(2.6)
2
where p− (x)eκ x is the Bloch mode decaying at −∞ of − ddx 2 + V1 (x) − λ. (b) If for some ε > 0 the potentials V1 , V2 are continuous in [−ε, 0) and satisfy V2 (x) < V1 (x) for all x ∈ [−ε, 0), then condition (2.6) holds for λ sufficiently negative. In particular, if V1 , V2 are C 1 -functions near x = 0, then V2 (0) < V1 (0) or
V2 (0) = V1 (0) and V2 (0) > V1 (0)
(2.7)
implies (2.6) for λ sufficiently negative. Remark 3. In the case c2 ≤ c1 the condition corresponding to (2.6) is
1
V1 (x) − V2 (x) p+ (x)2 e−2κ x d x < 0,
0 2
where p+ (x)e−κ x is the Bloch mode decaying at +∞ of − ddx 2 + V2 (x) − λ. It holds if λ is sufficiently negative and V1 , V2 satisfy the conditions of continuity and V1 (x) < V2 (x) in (0, ε] for some ε > 0. The condition corresponding to (2.7) is V1 (0) < V2 (0) or V1 (0) = V2 (0) and V2 (0) > V1 (0). Note that the condition on the derivatives is the same as in (2.7). In Sect. 6 we apply Theorem 7 to a so-called ‘dislocation’ interface where V1 (x) = V0 (x + τ ), V2 (x) = V0 (x − τ ) with τ ∈ R. The rest of the paper is structured as follows. In Sect. 3, using a concentration compactness argument, we prove the general criteria for existence/nonexistence of strong ground states in Rn , i.e. Theorems 2, 3, and 4. Our two n−dimensional examples (Theorems 5 and 6), which satisfy these criteria, are proved in Sect. 4. In Sect. 5 we prove Theorem 7, i.e. a refinement of Theorem 4 for the case n = 1. Section 6 firstly presents a 1D example (n = 1), namely a dislocation interface, satisfying the conditions of Theorem 7, and secondly provides a heuristic explanation of the 1D existence results for λ sufficiently negative. Finally, Sect. 7 discusses some open problems and the application of our results to several numerical and experimental works on surface gap solitons.
518
T. Dohnal, M. Plum, W. Reichel
3. n Dimensions: General Existence Results 3.1. Proof of Theorem 2. According to Pankov [17] ground states wi ∈ H 1 (Rn ) for the purely periodic problem (2.1) are strong ground states, so that they are characterized as wi ∈ Ni
such that
Ji [wi ] = ci := inf Ji [u] u∈Ni
with Ni being the Nehari-manifold Ni = {u ∈ H 1 (Rn ) : u = 0, G i [u] = 0}, G i [u] = |∇u|2 + Vi (x)u 2 − i (x)|u| p+1 d x Rn
for i = 1, 2. The proof of Theorem 2 consists in showing that a strategy similar to that of [17] for purely periodic problems is successful in finding strong ground states of (2.2) provided the basic inequality c < min{c1 , c2 } for the corresponding minimal energy levels holds. In the following we use the scalar product ϕ, ψ = Rn ∇ϕ · ∇ψ + ϕψ d x for ϕ, ψ ∈ H 1 (Rn ). For u ∈ H 1 (Rn ) the bounded linear functional J [u] can be represented by its gradient denoted by ∇ J [u], i.e. J [u]ϕ = ∇ J [u], ϕ for all ϕ ∈ H 1 (Rn ). Lemma 8. There exists a sequence (u k )k∈N on the Nehari-manifold N such that J [u k ] → c and J [u k ] → 0 as k → ∞. Moreover, (u k )k∈N is bounded and bounded away from zero in H 1 (Rn ), i.e., there exists , K > 0 such that ≤ u k H 1 (Rn ) ≤ K for all k ∈ N. Proof. Let |||u|||2 = Rn |∇u|2 + V (x)u 2 d x. Clearly ||| · ||| is equivalent to the standard norm on H 1 (Rn ) since min σ (L) > 0. Note that for u ∈ N one finds J [u] = η|||u|||2 1 with η = 21 − p+1 . This explains why every minimizing sequence of J on N has to be bounded. Moreover every element u ∈ N satisfies |||u|||2 = Rn (x)|u| p+1 d x ≤ p+1 p+1 , and since u = 0, the lower bound on u follows. ¯ Cu H 1 (Rn ) ≤ C|||u||| Now consider a sequence 0 < k → 0 and a minimizing sequence (vk )k∈N of J on N such that J [vk ] ≤ c + k2 . By using Ekeland’s variational principle, cf. Struwe [22], there exists a second minimizing sequence (u k )k∈N in N such that J [u k ] ≤ J [vk ] and J [u k ] < J [u] + k u − u k H 1 (Rn ) for all u ∈ N , u = u k . Consider the splitting ∇ J [u k ] = sk + tk with sk ∈ (Ker G [u k ])⊥ and tk ∈ Ker G [u k ]. Due to the following Lemma 9 we know that tk H 1 (Rn ) ≤ k . Note that the range Rg G [u k ] = R because G [u k ]u k = (1 − p)|||u k |||2 = 0. Furthermore, span(∇G[u k ]) = (Ker G [u k ])⊥ . Hence there exist real numbers σk ∈ R such that sk = −σk ∇G[u k ], i.e., ∇ J [u k ], u k +σk ∇G[u k ], u k = tk , u k .
=0
Thus, |σk |( p − 1)|||u k |||2 ≤ k u k H 1 (Rn ) , which shows that σk → 0 as k → ∞ since |||u k ||| is bounded away from zero. Hence we have proved that ∇ J [u k ] = sk + tk → 0 as k → ∞.
Surface Gap Soliton Ground States for the NLS
519
Lemma 9. Suppose > 0 and u 0 ∈ N are such that J [u 0 ] − J [u] ≤ u 0 − u H 1 (Rn ) for all u ∈ N . If ∇ J [u 0 ] = s + t with s ∈ (Ker G [u 0 ])⊥ and t ∈ Ker G [u 0 ], then t H 1 (Rn ) ≤ . Proof. We split u ∈ H 1 (Rn ) such that u = τ u 0 + v with v ∈ span(u 0 )⊥ . The Fréchet derivative G [u 0 ] may be split into the partial Fréchet derivatives ∂1 G[u 0 ] := G [u 0 ]|span(u 0 ) , ∂2 G[u 0 ] := G [u 0 ]|span(u 0 )⊥ , so that G [u 0 ](τ u 0 + v) = τ ∂1 G[u 0 ]u 0 + ∂2 G[u 0 ]v.
(3.1)
Since G [u 0 ]u 0 = (1 − p)|||u 0 |||2 = 0, we have that ∂1 G[u 0 ] is bijective, and hence by the implicit function theorem there exists a ball B(0) ⊂ span(u 0 )⊥ and a C 1 -function τ : B(0) → R such that G(τ (v)u 0 + v) = 0,
τ (0) = 1
and (τ (0)v)u 0 = −(∂1 G[u 0 ])−1 ∂2 G[u 0 ]v for all v ∈ span(u 0 )⊥ .
(3.2)
Define the linear map ϕ : span(u 0 )⊥ → H 1 (Rn ) by ϕ(v) := (τ (0)v)u 0 + v, v ∈ span(u 0 )⊥ . We claim that ϕ is a bijection between span(u 0 )⊥ and Ker(G [u 0 ]). First note that indeed ϕ maps into Ker(G [u 0 ]), which can be seen from (3.2). Let us prove that ϕ is injective: if ϕ(v) = (τ (0)v)u 0 + v = 0, then clearly v = 0. To see that ϕ is surjective, take u ∈ Ker(G [u 0 ]) and write u = θ u 0 + v for some θ ∈ R and some v ∈ span(u 0 )⊥ . Then, by (3.1) and (3.2), θ u 0 = −(∂1 G[u 0 ])−1 ∂2 G[u 0 ]v = (τ (0)v)u 0 . Hence u = ϕ(v), and we have proved the bijectivity of the map ϕ. Next, we compute for u ∈ N near u 0 , where u = τ (v)u 0 + v with v ∈ B(0) ⊂ span(u 0 )⊥ , that u − u 0 = (τ (v) − 1)u 0 + v = (τ (0)v)u 0 + v + o(v) = ϕ(v) + o(v) as v → 0. Therefore J [u 0 ] − J [u] = J [u 0 ](u 0 − u) + o(u − u 0 ) = −J [u 0 ]ϕ(v) + o(v)
(3.3)
J [u 0 ] − J [u] ≤ u 0 − u H 1 (Rn ) = ϕ(v) H 1 (Rn ) + o(v)
(3.4)
and
by assumption. Setting v = λv¯ with v¯ ∈ span(u 0 )⊥ and letting λ → 0+, we obtain from (3.3), (3.4), −J [u 0 ]ϕ(v) ¯ ≤ ϕ(v) ¯ H 1 (Rn ) for all v¯ ∈ span(u 0 )⊥ . Due to the bijectivity of ϕ and by considering both v¯ and −v¯ we get |J [u 0 ]w| ≤ w H 1 (Rn ) for all w ∈ Ker(G [u 0 ]), which implies the claim.
520
T. Dohnal, M. Plum, W. Reichel
Theorem 2 will follow almost immediately from the next result. Proposition 10. For δ > 0 let Sδ = (−δ, δ) × Rn−1 ⊂ Rn denote a strip of width 2δ. If there exists δ > 0 such that for the sequence of Lemma 8 one has lim inf k∈N u k H 1 (Sδ ) = 0, then c ≥ min{c1 , c2 }. Proposition 10 will be proved via some intermediate results. We define a standard one-dimensional C ∞ cut-off function such that χδ (t) = 1 for t ≥ δ, χδ (t) = 0 for t ≤ 0 and 0 ≤ χδ ≤ 1, χδ ≥ 0. From χδ we obtain further cut-off functions χδ+ (x) := χδ (x1 ), χδ− (x) := χδ (−x1 ), x = (x1 , . . . , xn ) ∈ Rn . Note that χδ± is supported in the half-space Rn± and ∇χδ± is supported in the strip Sδ . Lemma 11. Let (u k )k∈N be a bounded sequence in H 1 (Rn ) such that u k H 1 (Sδ ) → 0 as k → ∞ and define vk (x) := u k (x)χδ+ (x), wk (x) := u k (x)χδ− (x). Then (i) u k 2 = vk 2 + wk 2 + o(1) where · can be the L 2 -norm or the H 1 -norm on Rn , (ii) J [u k ] = J1 [vk ] + J2 [wk ] + o(1), (iii) J [u k ] = J1 [vk ] + J2 [wk ] + o(1), where o(1) denotes terms converging to 0 as k → ∞ and convergence in (iii) is understood in the sense of H −1 (Rn ) := (H 1 (Rn ))∗ . Proof.
(i) First note that u 2k = u 2k (χδ+ + χδ− )2 + u 2k (1 − χδ+ − χδ− )2 + 2u 2k (χδ+ + χδ− )(1 − χδ+ − χδ− ) = vk2 + wk2 + u 2k (1 − χδ+ − χδ− )2 + 2u 2k (χδ+ + χδ− )(1 − χδ+ − χδ− ).
Furthermore, since ∇u k = ∇vk + ∇wk + (1 − χδ+ − χδ− )∇u k − u k ∇(χδ+ + χδ− ),
(3.5)
we find |∇u k |2 = |∇vk |2 + |∇wk |2 + (1 − χδ+ − χδ− )2 |∇u k |2 + u 2k (|∇χδ+ |2 + |∇χδ− |2 )
+ 2(1−χδ+ −χδ− )(∇vk +∇wk ) · ∇u k −2u k (∇vk +∇wk ) · ∇(χδ+ +χδ− )
− 2u k (1 − χδ+ − χδ− )∇u k · ∇(χδ+ + χδ− ).
Integrating these expressions over Rn and observing that terms involving (1 − χδ+ − χδ− ) or ∇(χδ+ + χδ− ) are supported in Sδ , where u k H 1 (Sδ ) tends to zero, and that the sequences (vk )k∈N , (wk )k∈N are bounded in H 1 (Rn ), we obtain the claim (i).
Surface Gap Soliton Ground States for the NLS
(ii) Let us compute (x)|u k | p+1 d x = Rn
521
Rn−
2 (x)|wk | p+1 d x +
Rn+
1 (x)|vk | p+1 d x + I,
where
(x) |u k | p+1 − |wk | p+1 − |vk | p+1 d x.
I = Sδ
By the assumption that u k H 1 (Sδ ) tends to zero and by the Sobolev-embedding theorem I converges to 0 as k → ∞. A similar computation shows V (x)u 2k d x = V2 (x)wk2 d x + V1 (x)vk2 d x + o(1) as k → ∞. Rn
Rn−
Rn+
Together with (i) we get the claim in (ii). (iii) Using (3.5), we obtain J [u k ]ϕ = J [vk ]ϕ + J [wk ]ϕ (∇u k · ∇ϕ + V (x)u k ϕ)(1 − χδ+ (x) − χδ− (x)) + Sδ −u k ∇ϕ · (∇χδ+ (x) + ∇χδ− (x)) d x + I˜k (ϕ), where I˜k (ϕ) =
(x) |u k | p−1 u k − |vk | p−1 vk − |wk | p−1 wk ϕ d x, Sδ
and hence I˜k tends to 0 in H −1 (Rn ) as k → ∞. Thus
˜ (a1 (x)∇u k + a2 (x)u k · ∇ϕ +a3 (x)u k ϕ d x, J [u k ]ϕ = J1 [vk ]ϕ + J2 [wk ]ϕ + Ik (ϕ)+ Sδ
where the functions a1 , . . . , a3 are bounded on Sδ . Using I˜k → 0 in H −1 (Rn ) and once more that u k → 0 in H 1 (Sδ ) as k → ∞ we obtain the claim of (iii). In order to proceed with the proof of Proposition 10, we quote the following famous concentration-compactness result, cf. Lions [9]. With a minor adaptation of the proof given in Willem [28] one can state the following version. Lemma 12 (P.L.Lions, 1984). For 0 < a ≤ ∞ let Sa = (−a, a)×Rn−1 . Let 0 < r < a, s0 ∈ [2, 2∗ ) and assume that (u k )k∈N is a bounded sequence in H 1 (S2a ) such that lim sup |u k |s0 d x = 0. k→∞ ξ ∈Sa
Br (ξ )
Then u k → 0 as k → ∞ in L s (Sa ) for all s ∈ (2, 2∗ ).
522
T. Dohnal, M. Plum, W. Reichel
Proof of Proposition 10. By assumption we may select a subsequence (again denoted by (u k )k∈N ) from the sequence of Lemma 8 such that limk→∞ u k H 1 (Sδ ) = 0. Recall that u k ∈ N satisfies 2 |||u k ||| = (x)|u k | p+1 d x Rn
and that |||u k ||| is bounded and bounded away from 0 by Lemma 8. Hence no subsequence of u k L p+1 (Rn ) converges to 0 as k → ∞. By the concentration-compactness result of Lemma 12 with a = ∞ we have that for any r > 0 there exists > 0 such that lim inf sup u 2k d x ≥ 2, k∈N
ξ ∈Rn
Br (ξ )
and hence that there exists a subsequence of u k (again denoted by u k ) and points ξk ∈ Rn such that u 2k d x ≥ for all k ∈ N. (3.6) Br (ξk )
Next we choose vectors z k ∈ Z (n) = T1 Z × T2 Z × · · · × Tn Z such that (z k − ξk )k∈N is bounded (recall that T1 = 1, T2 , . . . , Tn > 0 denote the periodicities of the functions V1 , V2 , 1 , 2 in the coordinate directions x1 , . . . , xn ). Then there exists a radius ρ ≥ r + supk∈N |z k − ξk | such that u 2k d x ≥ for all k ∈ N. (3.7) Bρ (z k )
We show that (z k )k∈N is unbounded in the x1 -direction. Assume the contrary and define for x ∈ Rn , u¯ k (x) := u k (x + z k ), z k = (0, (z k )2 , . . . , (z k )n ). By the boundedness of (z k )1 there exists a radius R ≥ ρ such that u¯ 2k d x ≥ for all k ∈ N. B R (0)
(3.8)
By taking a weakly convergent subsequence u¯ k u¯ 0 in H 1 (Rn ) and using the compactness of the embedding H 1 (B R (0)) → L 2 (B R (0)), we have u¯ 0 = 0. Moreover, if ϕ ∈ C0∞ (Rn ) and if we set ϕk (x) := ϕ(x − z k ), then we can use the periodicity of V, in the directions x2 , . . . , xn to see that o(1) = J [u k ]ϕk = J [u¯ k ]ϕ as k → ∞, where the first (u k )k∈N as stated in Lemma 8. On equality is a property of the sequence one hand, Rn ∇ u¯ k · ∇ϕ + V (x)u¯ k ϕ d x → Rn ∇ u¯ 0 · ∇ϕ + V (x)u¯ 0 ϕ d x by the weak convergence of the sequence u¯ k u¯ 0 . On the other hand, by the compact Sobolev embedding H 1 (K ) → L p+1 (K ) with K = suppϕ and the continuity of the Nemytskii p+1
operator u → |u| p−1 u as a map from Lp+1 (K ) to L p (K ), cf. Renardy-Rogers [18], we find that Rn (x)|u¯ k | p−1 u¯ k ϕ d x → Rn (x)|u¯ 0 | p−1 u¯ 0 ϕ d x. Hence we have verified that the limit function u¯ 0 is a weak solution of L u¯ 0 = (x)|u¯ 0 | p−1 u¯ 0 in Rn . Standard
Surface Gap Soliton Ground States for the NLS
523 2,q
elliptic regularity implies that u¯ 0 is a strong Wloc (Rn )-solution for any q ≥ 1. Since we also know that u¯ 0 ≡ 0 on Sδ , we can apply the unique continuation theorem, cf. Schechter, Simon [20] or Amrein et al. [2], to find the contradiction u¯ 0 ≡ 0 in Rn . Thus, (z k )k∈N is indeed unbounded in the x1 -direction. Let vk , wk be defined as in Lemma 11 and define v¯k (x) := vk (x + z k ), w¯ k (x) := wk (x + z k ), x ∈ Rn , and observe that both v¯k and w¯ k are bounded sequences in H 1 (Rn ). Moreover, for almost all k we have √ √ or w¯ k L 2 (B R (0)) ≥ v¯k L 2 (B R (0)) ≥ 3 3 by Lemma 11(i) and (3.8). Taking weakly convergent subsequences, we get that v¯k v¯0 and w¯ k w¯ 0 , where v¯0 = 0 or w¯ 0 = 0. Since z k is unbounded in the x1 -direction, we may assume that either (z k )1 → +∞ or (z k )1 → −∞ as k → ∞. In the first case w¯ k 0, while in the second case v¯k 0 as k → ∞. In the following, let us consider only the case (z k )1 → +∞. Then, from Lemma 11 and the periodicity of V1 , V2 , 1 , 2 we have for any bounded sequence ϕk ∈ H 1 (Rn ) that o(1) = J [u k ]ϕk = J1 [vk ]ϕk + J2 [wk ]ϕk + o(1) = J1 [v¯k ]ϕk (· + z k ) + J2 [w¯ k ]ϕk (· + z k ) + o(1) as k → ∞.
(3.9)
If we apply (3.9) to ϕk (x) := ϕ(x − z k ), where ϕ ∈ C0∞ (Rn ), then o(1) = J1 [v¯k ]ϕ + J2 [w¯ k ]ϕ + o(1) = J1 [v¯k ]ϕ + o(1), since w¯ k 0 as k → ∞ (where we have again used the compact Sobolev embedding and the continuity of the Nemytskii operator). From this we can deduce that v¯0 is a nontrivial solution of L 1 v¯0 = 1 (x)|v¯0 | p−1 v¯0 in Rn .
(3.10)
Applying (3.9) with ϕk = u k , one obtains o(1) = J [u k ]u k = J1 [vk ]u k + J2 [wk ]u k + o(1) = J1 [vk ]vk + J2 [wk ]wk + o(1) = J1 [v¯k ]v¯k + J2 [w¯ k ]w¯ k + o(1), which together with (3.10) implies lim inf (J1 [v¯k ] + J2 [w¯ k ]) k∈N
= lim inf J1 [v¯k ] + J2 [w¯ k ] − k∈N
1 (J1 [v¯k ]v¯k + J2 [w¯ k ]w¯ k )
p+1 =o(1) as k→∞
1 1 |∇ v¯k |2 + V1 (x)v¯k2 + |∇ w¯ k |2 + V2 (x)w¯ k2 d x = lim inf − 2 p + 1 Rn k∈N 1 1 − ≥ |∇ v¯0 |2 + V1 (x)v¯02 d x 2 p + 1 Rn = J1 [v¯0 ]. (3.11)
524
T. Dohnal, M. Plum, W. Reichel
Lemma 11(ii) also implies J [u k ] = J1 [vk ] + J2 [wk ] + o(1) = J1 [v¯k ] + J2 [w¯ k ] + o(1) as k → ∞, which together with (3.11) yields the result c = lim J [u k ] ≥ J1 [v¯0 ] ≥ c1 . k→∞
In the case where (z k )1 → −∞ we would have obtained c = limk→∞ J [u k ] ≥ c2 . Hence, in any case we find c ≥ min{c1 , c2 }, which finishes the proof of Proposition 10. Proof of Theorem 2. As in Lemma 8 let (u k )k∈N be a minimizing sequence of J on the Nehari-manifold N such that J [u k ] → 0 as k → ∞. From Proposition 10 we know that lim inf k∈N u k H 1 (Sδ ) > 0 for any δ > 0. Let us fix δ > 0. By the following Lemma 13 we know that for 0 < R < 2δ, lim inf sup |u k |2 d x > 0. (3.12) k∈N
ξ ∈S2δ
B R (ξ )
Thus, there exist centers ξk ∈ S2δ and > 0 such that u 2k d x ≥ for all k ∈ N, B R (ξk )
and by choosing suitable vectors z k ∈ Z (n) with (z k )1 = 0 and a radius ρ ≥ R, we may achieve that u 2k d x ≥ for all k ∈ N. Bρ (z k )
Setting u¯ k (x) := u k (x + z k ), x ∈ Rn , we find
Bρ (0)
u¯ 2k d x ≥ for all k ∈ N.
Taking a weakly convergent subsequence u¯ k u¯ 0 in H 1 (Rn ), we obtain by the argument given in the proof of Proposition 10 that u¯ 0 is a non-trivial weak solution of L u¯ 0 = (x)|u¯ 0 | p−1 u¯ 0 in Rn . Finally, as seen before in the proof of Proposition 10, one obtains 1 c = lim J [u k ] = lim J [u k ] − J [u k ]u k k→∞ k→∞ p+1 1 J [u¯ k ]u¯ k = lim J [u¯ k ] − k→∞ p+1 1 1 |∇ u¯ k |2 + V (x)u¯ 2k d x = lim − n k→∞ 2 p+1 R 1 1 − |∇ u¯ 0 |2 + V (x)u¯ 20 d x ≥ 2 p + 1 Rn = J [u¯ 0 ].
Surface Gap Soliton Ground States for the NLS
525
Since u¯ 0 is non-trivial, it belongs to the Nehari manifold N . Thus, equality holds in the last inequality and u¯ 0 is a strong ground state. Lemma 13. With the notation of the proof of Theorem 2, 2 |u k | d x > 0. lim inf sup k∈N
ξ ∈S2δ
(3.12)
B R (ξ )
Proof. Otherwise, by concentration-compactness Lemma 12 with a = 2δ we find a subsequence such that u k L s (S2δ ) → 0 as k → ∞ for all s ∈ (2, 2∗ ). This is impossible / σ (L), there as the following argument shows. Since J [u k ] → 0 in (H 1 (Rn ))∗ and 0 ∈ exists a sequence (ζk )k∈N in H 1 (Rn ) such that Lζk = J [u k ] in Rn and ζk H 1 (Rn ) → 0 as k → ∞. In particular θk := u k − ζk is a weak solution of Lθk = (x)|u k | p−1 u k in Rn
(3.13)
u k L s (S2δ ) , θk L s (S2δ ) → 0 as k → ∞ for all s ∈ (2, 2∗ ).
(3.14)
and L −1
L q (Rn )
: → is a bounded linear As 0 ∈ σ (L), we may use the fact operator for all q ∈ (1, ∞). This fact may be well known; for the reader’s convenience we have given the details in the Appendix, cf. Lemma A1. Thus, due to the boundedness of (u k )k∈N in H 1 (Rn ), 2 2∗ p , . (3.15) θk W 2,q (Rn ) ≤ const. u k L q (Rn ) = O(1) if q ∈ max 1, p p Because 1 < p < 2∗ − 1, we can choose 2 < t < 2∗ , such that t lies in the range given in (3.15). Therefore θk2 d x ≤ θk L t (S2δ ) θk L t (S2δ )
W 2,q (Rn )
2n n+2
< t < 2 with
1 t
+
1 t
=1
S2δ
≤ θk L t (S2δ ) θk W 2,t (Rn ) = o(1) as k → ∞
(3.16)
because of (3.14) and (3.15). Define a C0∞ -function ξ : R → R with support in [−2δ, 2δ] with 0 ≤ ξ ≤ 1 and ξ |[−δ,δ] = 1 and use the test-function ξ(x1 )θk in (3.13). This leads to
|∇θk | d x ≤
|∇θk |2 ξ d x
2
Sδ
S2δ
= S2δ
∂θk ξ dx ∂ x1 2 + V ∞ θk L 2 (S ) )
(x)|u k | p−1 u k θk ξ − V (x)θk2 ξ − θk
p ≤ ∞ u k L s (S ) θk L s (S 2δ 2δ ∂θk r θk r +ξ L ∞ , L (S2δ ) ∂ x1 L (S2δ )
2δ
(3.17)
where 1s + s1 = 1, r1 + r1 = 1. Since 1 < p < 2∗ −1, we may arrange that ps, s ∈ (2, 2∗ ). ∗ Furthermore we can choose r in the range given r ∈ (2, 2 ).
in (3.15) and additionally Estimating θk L s (S2δ ) ≤ Cθk H 1 (Rn ) ≤ C u k H 1 (Rn ) + ζk H 1 (Rn ) = O(1) and
∂∂θxk1 L r (S2δ ) ≤ θk W 2,r (Rn ) = O(1) by (3.15) and using (3.14), (3.16), we deduce from (3.17) that θk H 1 (Sδ ) → 0 as k → ∞, which together with u k = θk + ζk and ζk H 1 (Rn ) → 0 yields u k H 1 (Sδ ) → 0 as k → ∞ in contradiction to Proposition 10. Hence we now know that (3.12) holds.
526
T. Dohnal, M. Plum, W. Reichel
3.2. Proof of Theorem 3. First we prove c = c1 by showing the two inequalities c ≤ c1 , c ≥ c1 . The first inequality follows from the next lemma and holds always (independently of the ordering of V1 , V2 and 1 , 2 assumed in Theorem 3). Lemma 14. Assume (H1)–(H3) and 0 < min σ (L). Then c ≤ min{c1 , c2 }. Proof. Let w1 be a ground state for the purely periodic problem with coefficients V1 , 1 1 and define u t (x) := w1 (x − te1 ), t ∈ N. Then (setting η = 21 − p+1 ) we compute |∇u t |2 +V1 (x)u 2t d x + |∇u t |2 +V (x)|u t | p+1 d x = (V2 (x)−V1 (x)) u 2t d x Rn
Rn
Rn−
c1 = + o(1) η and Rn
(x)|u t |
p+1
dx =
Rn
1 (x)|u t |
p+1
dx +
Rn−
(2 (x)−1 (x)) |u t | p+1 d x =
c1 +o(1), η
since the integrals over Rn− converge to 0 as t → ∞. Note that c1 = η Rn (|∇u t |2 + V1 (x)u 2t ) d x > 0 because 0 < min σ (L 1 ). Thus for large t we find Rn (x)|u t | p+1 d x > 0,and hence we can determine s ∈ R such that su t ∈ N , i.e., 2 2 n |∇u t | + V (x)u t d x s p−1 = R . p+1 d x Rn (x)|u t | Thus
J [su t ] = ηs
2
|∇u t | + 2
Rn
V (x)u 2t d x
=η
p+1 |∇u t |2 + V (x)u 2t d x p−1
2 p+1 d x p−1 Rn (x)|u t |
Rn
p+1
=η
(c1 /η + o(1)) p−1 2
(c1 /η + o(1)) p−1
→ c1 as t → ∞.
This shows that c ≤ c1 . Similarly, if w2 is a ground state of the purely periodic problem with coefficients V2 , 2 , we can define u t (x) := w2 (x + te1 ) with t ∈ N. Letting t tend to ∞, we can see as above that c ≤ c2 . Next we prove that under the assumptions of Theorem 3 one has c1 ≤ c. Let u ∈ N . Then p+1 p+1 1 (x)|u| dx ≥ (x)|u| dx = |∇u|2 + V (x)u 2 d x > 0, Rn
Rn
Rn
and therefore we can determine τ ∈ R such that τ u ∈ N1 , i.e., 2 2 n |∇u| + V1 (x)u d x p−1 τ = R p+1 d x n 1 (x)|u| R
p+1 + V (x) − V (x) u 2 d x 1 Rn (x)|u| = p+1 d x Rn 1 (x)|u| ≤ 1,
Surface Gap Soliton Ground States for the NLS
527
Fig. 2. An example of the sequence of points q j
since ≤ 1 and V1 ≤ V in Rn . Therefore c1 ≤ J1 [τ u] = ητ 2 |∇u|2 + V1 (x)u 2 d x Rn ≤η |∇u|2 + V (x)u 2 d x Rn
= J [u] since u ∈ N . Since u ∈ N is arbitrary, we see that c1 ≤ c. Now suppose for contradiction that a minimizer u¯ 0 ∈ N of the functional J exists. Then the value τ s.t. τ u¯ 0 ∈ N1 in the above calculation is strictly less than 1 since we may assume u¯ 0 > 0 almost everywhere on Rn (cf. Remark 1 and Lemma A2) and also < 1 and/or V1 < V on a set of nonzero measure. However, τ < 1 implies c1 ≤ J1 [τ u¯ 0 ] < J [u¯ 0 ] = c, which contradicts Lemma 14. This shows that no strong ground state of (2.2) can exist and the proof of Theorem 3 is thus finished. We explain next the statement of Remark 2. The distance of q j to 2 diverges to ∞ as j → ∞ and we can thus use the same argument as in the proof of Theorem 3 with u t (x) := w1 (x − q j ), j 1 and with Rn− replaced by 2 , cf. Fig. 2. 3.3. Proof of Theorem 4. Let us first treat the case c1 ≤ c2 . Similarly to the approach of Arcoya, Cingolani and Gámez [3] we consider u t (x) := w1 (x − te1 ) for large t ∈ N, i.e., we shift the ground state w1 far to the right. Recall from [17] that |w1 (x)| ≤ Ce−λ|x| for appropriate C, λ > 0. (3.18) As in the proof of Lemma 14 we have Rn (x)|u t | p+1 d x = c1 /η + o(1) > 0 for large t ∈ N. Therefore we can determine s ∈ R such that su t ∈ N , i.e., 2 2 n |∇u t | + V (x)u t d x p−1 s . (3.19) = R p+1 d x Rn (x)|u t |
528
T. Dohnal, M. Plum, W. Reichel
Next we compute (using again η =
1 2
−
1 p+1 )
p+1 2 + V (x)u 2 d x p−1 |∇u | t t J [su t ] = ηs 2 |∇u t |2 + V (x)u 2t d x = η 2 p+1 d x p−1 Rn Rn (x)|u t | p+1 2 + V (x)u 2 d x + 2 d x p−1 |∇u | (V (x) − V (x))u t 1 1 t t Rn Rn− 2 = η 2 p+1 d x + p+1 d x p−1 (x)|u | ( (x) − (x))|u | n n 1 t 2 1 t R R− −1 p+1 2 2 1 + p−1 Rn |∇u t | + V1 (x)u 2t d x Rn− (V2 (x)−V1 (x))u t d x(1 + o(1)) = c1 , 2 p+1 d x −1 p+1 d x(1 + o(1)) 1 + p−1 Rn 1 (x)|u t | Rn (2 (x)−1 (x))|u t |
Rn
−
where o(1) → 0 as t → ∞. The last equality holds since both integrals over Rn− converge to 0 as t → ∞ and (1 + ε)α = 1 + αε(1 + o(1)) as ε → 0. Hence, we obtain J [su t ] < c1 if (V2 (x) − V1 (x))w1 (x − te1 )2 d x ( p + 1) Rn−
<2
Rn−
(2 (x) − 1 (x))|w1 (x − te1 )| p+1 d x
(3.20)
for sufficiently large t∈N. This establishes condition (2.4). Moreover, rewriting (3.20) as w1 (x − te1 )2 ( p + 1)δV (x) − 2δ(x)|w1 (x − te1 )| p−1 d x < 0 Rn−
with δV = V2 − V1 and δ = 2 − 1 and using the decay (3.18) of w1 , we see that the above condition is always satisfied for large t ∈ N provided ess sup δV < 0. The case c2 ≤ c1 is symmetric to that above. One simply needs to shift a ground state w2 to the left. Hence, the proof is the same but with w1 , V1 , 1 , c1 , t and Rn− replaced by w2 , V2 , 2 , c2 , −t, and Rn+ . 4. n Dimensions: Examples Let us state the assumptions on the coefficients once for all the examples below. Namely, we take for V1 , V2 , 1 and 2 bounded functions such that (H1)–(H2) are satisfied. The exponent p is assumed to satisfy (H3).
4.1. Proof of Theorem 5 (Left and Right States Related by Scaling). Due to the specific scaling of V1 , V2 and 1 , 2 the ground states w1 , w2 of the purely periodic problems (2.1) are related as follows: given a ground state w2 the transformation 2 k p−1 w2 (kx) w1 (x) = γ
Surface Gap Soliton Ground States for the NLS
529
produces a corresponding ground state w1 . Hence, with η = c1 = η |∇w1 |2 + V1 (x)w12 d x
1 2
−
1 p+1
we find
Rn
4 k p−1 2 =η k |∇w2 |2 (kx) + V2 (kx)w22 (kx) d x γ Rn 4 k p−1 2−n k c2 . = γ n+2− p(n−2)
4
+2−n
4
By assumption k p−1 = k p−1 ≤ γ p−1 . Then min{c1 , c2 } = c1 . Now we can achieve (2.4) in Theorem 4 for large t ∈ N if we assume ess sup(V2 − V1 ) < 0. In our case, where V1 (x) = k 2 V2 (kx), this is ensured by assumption (2.5). 4.2. Proof of Theorem 6 (Large Enough Jump in ). We start this section by a lemma, which explains that the ground-state energy of the periodic problem L 1 u = 1 |u| p−1 u depends monotonically on the coefficient 1 if we keep V1 fixed. To denote the dependence on the coefficient 1 , let us write c1 (1 ) for the ground-state energy, w1 (x; 1 ) for a ground-state, N1 (1 ) for the Nehari-manifold, and J1 [u; 1 ] for the energy-functional. Lemma 15. Assume 1 ≥ 1∗ with ess sup 1∗ > 0. Then c1 (1 ) ≤ c1 (1∗ ). Proof. Let us select s such that sw1 (·; 1∗ ) ∈ N1 (1 ), i.e., ∗ ∗ p+1 d x n (x)|w1 (x; 1 )| . s p−1 = R 1 ∗ p+1 d x Rn 1 (x)|w1 (x; 1 )| Clearly, s ≤ 1 and thus we get c1 (1 ) ≤ J1 [sw1 (·; 1∗ ); 1 ] = s2η |∇w1 (x; 1∗ )|2 + V1 (x)(w1 (x; 1∗ )2 d x Rn
= s J1 [w1 (·; 1∗ ); 1∗ ] ≤ J1 [w1 (·; 1∗ ); 1∗ ] = c1 (1∗ ). 2
Now we can give the proof of Theorem 6. By assumption we have V1 > V2 almost everywhere. Once we have checked c1 (1 ) ≤ c2 (2 ), then we can directly apply Theorem 4 to deduce the existence of a strong ground state. Using ess inf 1 ≥ β0 and applying Lemma 15 with 1∗ = β0 , we get −2
c1 (1 ) ≤ c1 (β0 ) = β0p−1 c1 (1). Hence, by choosing β0 =
c2 (2 ) c1 (1)
1− p 2
,
we obtain c1 (1 ) ≤ c2 (2 ). This finishes the proof of Theorem 6.
530
T. Dohnal, M. Plum, W. Reichel
5. One Dimension: General Existence Result (Proof of Theorem 7) In the case of one dimension we introduce a spectral parameter λ ∈ R into the problem, i.e., we consider the differential operator L λ := −
d2 + V − λ on D(L λ ) = H 2 (R) ⊂ L 2 (R) dx2
(5.1)
and look for strong ground states of L λ u = (x)|u| p−1 u in R.
(5.2)
The functions V and are defined via the bounded periodic functions V1 , V2 , 1 , 2 as before. The statement of Theorem 7 uses Bloch modes of the linear equation − u + V˜ (x)u = 0 in R
(5.3)
with a 1-periodic, bounded function V˜ . We define them next. If we assume that 0 < 2 min σ (− ddx 2 + V˜ (x)), then (5.3) has two linearly independent solutions (Bloch modes) of the form u ± (x) = p± (x)e∓κ x for a suitable value κ > 0 and 1-periodic, positive functions p± . We use the normalization p± ∞ = 1. We summarize next the structure of the proof of Theorem 7. According to Theorem 4, in the case c1 ≤ c2 , we have to check the condition 0 0 ( p + 1) δV (x)w1 (x − t)2 d x < 2 δ(x)|w1 (x − t)| p+1 d x for t 1, (5.4) −∞
−∞
where δV (x) = V2 (x)−V1 (x) and δ(x) = 2 (x)−1 (x). We first show in Lemma 18, via a comparison principle, that for ±x → ∞ a ground state w1 behaves like the Bloch mode u ± of (5.1). Then in Lemma 19 we compute the asymptotic behavior of the two sides of the inequality (5.4) as t → ∞. Since the left hand side behaves like e−2κt whereas the right-hand side behaves like e−( p+1)κt , the verification of (5.4) may be reduced to 0 δV (x)w1 (x − t)2 d x < 0 for t 1. −∞
In fact, Lemma 19 provides an asymptotic formula for the left hand side of (5.4) where w1 (x − t) is replaced by the Bloch mode u − (x − t) = p− (x − t)eκ(x−t) and, using a geometric series, the integral over the interval (−∞, 0) is replaced by a single period (−1, 0). As a result, (5.4) is equivalent to (2.6). This finishes the proof of part (a) of Theorem 7. In√order to prove √ part (b) of Theorem 7 for λ −1, we show in Lemma 20-22 that κ − |λ| = O(1/ |λ|) and that the periodic part p− of the Bloch mode u − converges uniformly to 1 as λ → −∞. As a result, for λ −1 the sign of the integral in (2.6) is dominated by the local behavior of V2 (x) − V1 (x) near x = 0 as detailed in Lemma 23. We begin our analysis with the following version of the comparison principle.
Surface Gap Soliton Ground States for the NLS
531
Lemma 16 (Comparison principle). Assume that V˜ : R → R is bounded, 1-periodic 2 such that 0 < min σ (− ddx 2 + V˜ (x)). Let p± e∓κ x be the Bloch modes for the operator 2 sup p − d 2 + V˜ (x) and set P± := [0,1] ± . inf [0,1] p±
dx
(a) Let u > 0, u ∈ H 1 (R) solve −u + V˜ (x)u ≤ 0 for |x| ≥ |x0 | > 0 for some fixed x0 ∈ R. If we set P := max{P+ , P− }, then 0 < u(x) ≤ Pe−κ(|x|−|x0 |) max u
for all x ∈ R,
I
where I = [−|x0 |, |x0 |]. (b) Let u > 0, u ∈ H 1 (R) solve −u + V˜ (x)u ≥ 0 for |x| ≥ |x0 | > 0 for some fixed x0 ∈ R. If we set Q := min{ P1+ , u(x) ≥ Qe−κ(|x|−|x0 |) min u I
1 P− },
then
for all x ∈ R.
Proof. The proof is elementary and may be well known. We give the details for the reader’s convenience. Let us concentrate on the case (a) and the estimate on the interval [x0 , ∞) and suppose that x0 ≥ 0. The estimate for the interval (−∞, −x0 ] is similar. 2 Due to the assumption 0 < min σ (− ddx 2 + V˜ (x)) we have the positivity of the quadratic form, i.e., 2 ϕ + V˜ (x)ϕ 2 d x ≥ 0 for all ϕ ∈ H 1 (R). (5.5) R
Let ψ := u − su + with u + (x) = p+ (x)e−κ x being a Bloch mode satisfying d2 − 2 + V˜ (x) u + = 0 on R, dx and choose eκ x0 max I u s := inf [0,1] p+ so that ψ(x0 ) ≤ 0. Since ψ satisfies −ψ + V˜ (x)ψ ≤ 0 on (x0 , ∞), testing with ψ + := max{ψ, 0} yields ∞ (ψ + )2 + V˜ (x)(ψ + )2 d x = x0
∞
ψ ψ + + V˜ (x)ψψ + d x ≤ 0
x0
which, after extending ψ + by 0 to all of R, together with the positivity of the quadratic form in (5.5) yields ψ + ≡ 0. This implies the claim in case (a). In case (b) on the interval [x0 , ∞) one considers the function ψ = u − su + with s :=
eκ x0 min I u sup[0,1] p+
and shows that ψ − = max{−ψ, 0} ≡ 0 similarly as above.
532
T. Dohnal, M. Plum, W. Reichel
For the next result note that the Wronskian u+ u− =: ω, det u + u − constructed from the linearly independent Bloch modes u ± of (5.3), is a constant. Lemma 17. Assume that V˜ : R → R is bounded, 1-periodic such that 0 < 2 2 min σ (− ddx 2 + V˜ (x)) and let p± e∓κ x be the Bloch modes for the operator − ddx 2 + V˜ (x). If f (x) = O(e−α|x| ) for |x| → ∞ with α > κ and if u is a solution of −u + V˜ (x)u = f (x) on R, then 1 u(x) = x→±∞ u ± (x) ω lim
∞ −∞
lim u(x) = 0,
|x|→∞
u ∓ (s) f (s) ds.
Proof. By the variation of constants formula for inhomogeneous problems we obtain x ∞ 1 1 u(x) = u − (s) f (s) ds u + (x) + u + (s) f (s) ds u − (x), ω ω −∞ x where the boundary condition lim|x|→∞ u(x) = 0 is satisfied because u ± (x) → 0 as x → ±∞ and the integrals are bounded as functions of x ∈ R due to the assumption f (x) = O(e−α|x| ) with α > κ. The claim of the lemma follows since again the assumption α > κ implies ∞ u − (x) → 0 as x → ∞ and u + (s) f (s) ds u + (x) x x u + (x) → 0 as x → −∞. u − (s) f (s) ds u − (x) −∞ Now we can describe the behavior of ground states w1 of (2.1) for i = 1 with V1 replaced by V1 − λ. 2
Lemma 18. Assume that V1 : R → R is bounded, 1-periodic and let λ < min σ (− ddx 2 + V1 (x)). If 1 : R → R is bounded and if z > 0 is a solution (not necessarily a ground state) of −z + (V1 (x) − λ)z = 1 (x)z p in R, then z(x) 1 = x→±∞ u ± (x) ω
d± (λ; z) := lim
where u ± are the Bloch modes of −
d2 dx2
∞ −∞
lim z(x) = 0,
|x|→∞
u ∓ (s)1 (s)z(s) p ds,
+ V1 (x) − λ.
Surface Gap Soliton Ground States for the NLS
533
Proof. Let > 0. Then there exists x0 = x0 () such that −z + (V1 (x) − λ − )z ≤ 0 for |x| ≥ |x0 |. By the comparison principle of Lemma 16 we get the estimate z(x) ≤ const. z∞ e−κλ+ (|x|−|x0 |) for all x ∈ R. Since the map λ → κλ is continuous, cf. Allair, Orive [1], we can choose > 0 so small that pκλ+ > κλ . Hence the assumptions of Lemma 17 with f (x) = 1 (x)z(x) p are fulfilled and the claim follows. Lemma 19. The integrals in (5.4) have the asymptotic form 0 2 0 2 −2κt d− (λ; w1 ) 2 2κ x δV (x)w1 (x −t) d x = e δV (x) p− (x) e d x + o(1) , 1 − e−2κ −1 −∞
0 −∞
δ(x)|w1 (x − t)| p+1 d x
= e−( p+1)κt
d− (λ; w1 ) p+1 1 − e−( p+1)κ
0 −1
δ(x) p− (x) p+1 e( p+1)κ x d x + o(1) ,
where o(1) → 0 as t → ∞, t ∈ N. Remark. Note that the resulting integrals on the right hand side are independent of the nonlinear ground state. Proof. For exponents r ≥ 2 and a 1-periodic bounded function q let us write 0 I (t) = q(x)w1 (x − t)r d x −∞
= d− (λ; w1 )r
0 −∞
q(x)u − (x − t)r d x + E(t).
(5.6)
For an estimation of the error term E(t) we use Lemma 18. Given > 0, there exists K = K () > 0 such that |w1 (x)r − d− (λ; w1 )r u − (x)r | ≤ u − (x)r for all x ≤ −K , i.e., |w1 (x − t)r − d− (λ; w1 )r u − (x − t)r | ≤ u − (x − t)r for all x ≤ 0, t ≥ K . Hence, for t ≥ K , E(t) ≤ q∞
0
−∞
p− (x − t)r er κ(x−t) d x
e−r κt , ≤ q∞ p− r∞ r κ =1
534
T. Dohnal, M. Plum, W. Reichel
i.e., E(t) = e−r κt o(1) as t → ∞. Next we compute the integral for t ∈ N,
0 −∞
q(x)u − (x − t)r d x = e−r κt =
∞
−k
k=0 −k−1 0 e−r κt
1 − e−r κ
−1
q(x) p− (x)r er κ x d x
q(x) p− (x)r er κ x d x.
This result shows that for large values of t the dominating part in (5.6) is played by the integral w.r.t. the Bloch mode, since in comparison the error term can be made arbitrarily small. This is the claim of the lemma. The above computation explains why it is possible to replace the existence condition (5.4) by (2.6). The reason is that the quadratic term on the left side of (5.4) decays like e−2κt whereas the term on the right side decays like e−( p+1)κt as t → ∞. At this stage note that by Lemma 19 we have proved part (a) of Theorem 7. It remains to consider part (b), i.e., to decide on the sign of
0 −1
δV (x) p− (x)2 e2κ x d x
(5.7)
as λ → −∞. First we investigate the behavior of the linear Bloch modes for λ → −∞. We begin by stating a relation between the spectral parameter and the coefficient of exponential decay of the Bloch modes. Lemma 20. Let κ = κ(λ) be the coefficient of exponential decay of the Bloch modes for 2 − ddx 2 + V1 (x) − λ. Then for λ sufficiently negative we have V1 ∞ −V1 ∞ ≤ κ − |λ| ≤ √ √ . κ + |λ| κ + |λ| Proof. We prove the estimate of the difference between κ and principle. The Bloch modes u ± (x) = p± (x)e∓κ x satisfy −u ± + (V1 ∞ − λ)u ± ≥ 0 and
√ |λ| via the comparison
− u ± + (−V1 ∞ − λ)u ± ≤ 0.
Hence, using the comparison principle of Lemma 16, we get for λ < −V1 ∞ , √
C 1 e−
V1 ∞ −λx
√
≤ u + (x) ≤ C2 e−
−V1 ∞ −λx
for x ∈ R.
This implies
V1 ∞ − λ ≥ κ ≥
−V1 ∞ − λ,
from which the statement easily follows. Next we give a representation of the periodic part of the Bloch mode u − (x) = p− (x)eκ x .
Surface Gap Soliton Ground States for the NLS
535 2
Lemma 21. The periodic part p− of the Bloch mode u − of the operator − ddx 2 +V1 (x)−λ satisfies the differential equation p− + 2κ p− + (κ 2 + λ) p− = V1 (x) p− on [−1, 0]
with periodic boundary conditions, and therefore 0 p− (x) =
√ (κ− |λ|)s p
− (s)V1 (s) ds √ √ 2 |λ|(eκ− |λ| − 1)
−1 e
1 + √ 2 |λ|
x
−1
√
e(κ−
|λ|)s
√
p− (s)V1 (s) ds e(−κ+
|λ|)x
√ 0 x √ √ − −1 e(κ+ |λ|)s p− (s)V1 (s) ds 1 (κ+ |λ|)s √ − √ + e p− (s)V1 (s) ds e(−κ− |λ|)x . √ κ+ |λ| 2 |λ| −1 2 |λ|(e − 1)
√
√
Proof. Starting from the solutions e(−κ+ |λ|)x , e(−κ− |λ|)x of the homogeneous equation p + 2κ p + (κ 2 + λ) p = 0, we get via the variation of constants x √ √ 1 (κ− |λ|)s e p− (s)V1 (s) ds e(−κ+ |λ|)x p− (x) = α + √ 2 |λ| −1 x √ √ 1 (κ+ |λ|)s + β− √ e p− (s)V1 (s) ds e(−κ− |λ|)x . 2 |λ| −1 (0) = p (−1), we obtain Inserting the periodicity conditions p− (0) = p− (−1) and p− − the claim.
Now we can state the asymptotic behavior of p− as λ → −∞. Lemma 22. As λ → −∞, we have that p− → 1 uniformly on [−1, 0]. More precisely as λ → −∞ p− − 1∞ = O √1|λ| and, in addition, 1 0 as λ → −∞. |λ|(κ − |λ|) − V1 (s) ds = O √1|λ| 2 −1 Proof. First one checks by a direct computation using Lemma 21 that p− ∞ = √ O(1/ |λ|) as λ → −∞. Hence, by the normalization p− ∞ = 1 and by continuity of p− we have p− (ξλ ) = 1 for some ξλ ∈ [−1, 0]. By the mean value theorem | p− (x) − 1| ≤ p− ∞ |x − ξλ | = O(1/ |λ|)
as λ → −∞ uniformly for x ∈ [−1, 0]. This proves the first part of the lemma. For the second part one first finds again by direct estimates from Lemma 20 and Lemma 21, 0 (κ−√|λ|)s e p− (s)V1 (s) ds (−κ+√|λ|)x 1 √ p− (x) = −1 √ +O √ e |λ| 2 |λ|(eκ− |λ| − 1) 0 p− (s)V1 (s) ds 1 +O √ , (5.8) = −1 √ √ 2 |λ|(κ − |λ|) |λ| √ √ |λ|) uniformly in s, x ∈ [−1, 0]. where we have used that e(κ− |λ|)(s−x) √ = 1 + O(1/ √ Next we observe by Lemma 20 that |λ|(κ − |λ|) is bounded in absolute value by
536
T. Dohnal, M. Plum, W. Reichel
V1 ∞ , and hence has accumulation points as λ → −∞. Every accumulation point d satisfies 0 1 1= V1 (s) ds. 2d −1 If we set γλ :=
0
√ , then γλ − 1 = O(1/ |λ|) by (5.8) and thus 1 0 |λ|(κ − |λ|) = V1 (s)ds + O(1/ |λ|), 2 −1
p− (s)V1 (s) ds −1 √ √ 2 |λ|(κ− |λ|)
which is the second claim of the lemma. Finally, we can determine the behavior of the integral in (5.7). Lemma 23. Let V1 , V2 be bounded and 1-periodic and δV := V2 − V1 . If there exists ε > 0 such that δV is continuous and negative in [−ε, 0), then for sufficiently negative λ, 0 δV (x) p− (x)2 e2κ x d x < 0. −1
Proof. Since δV is continuous and negative on [−ε, 0), there exist √ α > 0 such that δV (x) < −α for x ∈ [−ε, −ε/2]. Using that p− (x)2 − 1 = O(1/ |λ)|), we estimate 0 δV (x) p− (x)2 e2κ x d x −1
≤
−ε/2 −1 −ε
δV (x)e2κ x d x +
−ε/2
δV (x) p− (x)2 − 1 e2κ x d x
−1 −ε/2
1 −ε/2 δV (x)e d x − α e d x + δV ∞ O √ e2κ x d x ≤ |λ| −1 −1 −ε 1 e−κε δV ∞ −2κε α −κε e (e ≤ − − e−2κε ) + δV ∞ O √ 2κ 2κ |λ| 2κ 1 e−2κε δV ∞ + α − αeκε + δV ∞ O √ eκε , = 2κ |λ|
2κ x
2κ x
which is negative for λ −1 because κ → ∞ as λ → −∞. This proves the claim. With this the proof of Theorem 7 is complete. 6. One Dimension: Examples and Heuristics 6.1. A dislocation interface example. As a particular example of a one-dimensional interface we consider so-called dislocated potentials, i.e., if V0 , 0 are bounded 1-periodic functions, then we set 0 (x + τ ), x > 0, V0 (x + τ ), x > 0, (x) = V (x) = V0 (x − τ ), x < 0, 0 (x − τ ), x < 0, where τ ∈ R is the dislocation parameter. We consider problem (5.2) and analogously to 2 2 the notation in (2.1) we define L λ,1 := − ddx 2 +V0 (x+τ )−λ, L λ,2 := − ddx 2 +V0 (x−τ )−λ, 1 (x) := 0 (x + τ ), and 2 (x) := 0 (x − τ ). Note that c1 = c2 in this case. The following is then a direct corollary of Theorem 7 and Remark 3.
Surface Gap Soliton Ground States for the NLS
537
Corollary 24. Assume that V0 , 0 are bounded 1-periodic functions on the real line with ess sup 0 > 0. Assume moreover that 0 < min σ (L λ ) and 1 < p < ∞. (a) Problem (5.2) in the dislocation case with τ ∈ R has a strong ground state provided 0 V0 (x − τ ) − V0 (x + τ ) p− (x + τ )2 e2κ x d x < 0 (6.1) −1
or
1
V0 (x + τ ) − V0 (x − τ ) p+ (x − τ )2 e−2κ x d x < 0,
(6.2)
0 2
where p± (x)e∓κ x with κ > 0 are the Bloch modes of the operator − ddx 2 + V0 (x)−λ. (b) For λ < 0 sufficiently negative at least one of the conditions (6.1), (6.2) is fulfilled provided V0 is a C 1 -potential and V0 (−τ ) = V0 (τ ) or
V0 (−τ ) = V0 (τ ) and V0 (−τ ) > V0 (τ ).
(6.3)
For |τ | sufficiently small the above condition (6.3) on V0 is fulfilled if V0 (0) = 0 or V0 (0) = 0 and sign τ V0 (0) < 0,
(6.4)
where for the second part of the condition one needs to assume that V0 is twice differentiable at 0. Remark. (1) The case where V0 (0) = 0 and sign τ V0 (0) > 0 is not covered by the above theorem. We believe that for this case strong ground states do not exist. (2) One can also consider the dislocation problem with two parameters, i.e, V0 (x + τ ), x > 0, 0 (x + σ ), x > 0, V (x) = (x) = V0 (x − τ ), x < 0, 0 (x − σ ), x < 0, where τ, σ ∈ R are the dislocation parameters. If V0 , 0 are even, bounded 1-periodic functions and if w1 is a ground state for L λ,1 w1 = 1 (x)|w1 | p−1 w1 in R, then w2 (x) := w1 (−x) is a ground state for the problem L λ,2 w2 = 2 (x)|w2 | p−1 w2 in R. One then easily sees that again we have c1 = c2 .2 The result of Corollary 24 (a) immediately applies. For the result of Corollary 24 (b) one only needs to take the second parts of (6.3), (6.4) into account. Proof of Corollary 24. In the dislocation case the unperturbed energy levels of ground states satisfy c1 = c2 and thus both versions of Theorem 7 are available. If u − (x) = 2 p− (x)eκ x is the Bloch mode decaying at −∞ of − ddx 2 + V0 (x)−λ, then p− (x +τ )eκ(x+τ ) is the corresponding Bloch mode of the operator L λ,1 . Therefore, condition (2.6) of Theorem 7 amounts to 0 V0 (x − τ ) − V0 (x + τ ) p− (x + τ )2 e2κ x d x < 0, e2κτ −1
which is equivalent to (6.1) of Corollary 24. Likewise (2.7) amounts to V0 (−τ ) < V0 (τ ) or V0 (−τ ) = V0 (τ ) and V0 (−τ ) > V0 (τ ). If we apply the version of Theorem 7 given in 2 Note that the λ-dependence of c = c has been dropped. 1 2
538
T. Dohnal, M. Plum, W. Reichel
Remark 3, then we get (6.2) instead of (6.1) and V0 (τ ) < V0 (−τ ) or V0 (−τ ) = V0 (τ ) and V0 (−τ ) > V0 (τ ). This explains (6.1), (6.2) as well as (6.3). The final condition (6.4) follows via Taylor-expansion V0 (τ ) − V0 (−τ ) = 2τ V0 (0) + o(τ ), V0 (τ ) − V0 (−τ ) = 2τ V0 (0) + o(τ ) from (6.3). 6.2. A heuristic explanation of Theorem 7(b) and Corollary 24(b) for λ sufficiently negative. We provide next a heuristic explanation of the existence results in the 1D interface problem for λ −1 in Theorem 7(b) and thus Corollary 24 (b). In the following we show how to quickly find a function in N with energy smaller than c1 (≤ c2 ) so that the criterion of Theorem 2 for the existence of ground states is satisfied. We restrict the discussion to the case 1 ≡ 2 . The heuristic part of the analysis is the use of the ‘common wisdom’3 that as λ → −∞ each ground state of the purely periodic problem −u + V1 (x)u = 1 (x)|u| p−1 u in R is highly localized and concentrates near a point x0 (λ). We assume x0 (λ) → x0∗ ∈ (0, 1] as λ → −∞ and that x0∗ is a point, where V1 assumes its minimum. Moreover, we assume below that even at a small distance (e.g. one half period of V1 ) from the concentration point the ground state decays exponentially fast with increasing distance from x0∗ . Consider a ground state w1 (x; λ). The function sw1 lies in N if ∞ 2 w + (V (x) − λ)w12 d x p−1 s = −∞ ∞1 . p+1 d x −∞ (x)|w1 | ∞ Because 1 ≡ 2 and w1 ∈ N1 , the denominator equals −∞ 1 (x)|w1 | p+1 d x = ∞ 2 2 −∞ w1 + (V1 (x) − λ)w1 d x. Therefore ∞ 2 0 2 2 2 0 w1 + (V1 (x) − λ)w1 d x + −∞ w1 + (V2 (x) − λ)w1 d x p−1 = s . ∞ 2 2 −∞ w1 + (V1 (x) − λ)w1 d x Due to the concentration of w1 near x0∗ > 0 as λ → −∞ and its fast decay as |x − x0∗ | grows, we see that s < 1 if V2 (x) < V1 (x) in a left neighborhood of 0. Finally, ∞ 2 s p+1 s 2 w1 + (V (x) − λ)w12 − 1 (x)|w1 | p+1 d x J [sw1 ] = 2 p + 1 −∞ ∞ 1 2 2 1 − =s w + (V (x) − λ)w12 d x 2 p + 1 −∞ 1 0 1 1 2 2 − (V2 (x) − V1 (x))w1 d x , = s J1 [w1 ] + 2 p + 1 −∞ where the second equality follows from sw1 ∈ N . We get J [sw1 ] < J1 [w1 ] = c1 if V2 (x) < V1 (x) in a left neighborhood of 0, see Fig. 3 for a sketch. Note that this calculation does not, however, imply that the function sw1 is a ground state of the interface problem (5.2) (with 1 ≡ 2 ). 3 In [12] this common wisdom has been proved under similar, but not identical assumptions.
Surface Gap Soliton Ground States for the NLS
539
Fig. 3. An example of a function in N attaining smaller energy than c1 (≤ c2 ) for the 1D interface problem (5.2). Heuristically, the ground state existence conditions of Theorem 2 are thus satisfied
In the dislocation case, i.e., Corollary 24 (b), where c1 = c2 , the discussion applies analogously if we restrict attention to the case ≡ 0 ≡ const. We denote again x0∗ = limλ→−∞ x0 (λ) as the point of concentration of a ground state of the purely periodic problem, which now may be any point (left or right of zero) where V0 attains its global minimum. Thus J [sw1 ] < c1 = c2 is satisfied if V0 (x − τ ) < V0 (x + τ ) for x in a left neighborhood of 0 and if we take x0∗ > 0. Likewise, in the case where V0 (x − τ ) > V0 (x + τ ) for x in a right neighborhood of 0 we may take x0∗ < 0. The three possible scenarios, namely V0 (0) = 0, V0 (0) = 0 and V0 having a local minimum at x = 0, and finally V0 (0) = 0 and V0 having a local maximum at x = 0 are depicted schematically in Fig. 4. The full green lines in the columns τ > 0 and τ < 0 depict functions sw1 ∈ N with energy smaller than c1 = c2 . As the above calculations show, the candidate positioned the closest to x = 0 produces the smallest energy and has the smallest s. It is therefore plotted with the smallest amplitude. Finally, we mention that in the dislocation case with two dislocation parameters τ, σ (cf. Remark 2 after Corollary 24) the above considerations apply if V0 , 0 are even potentials and if we set σ = 0 so that 1 (x) = 0 (x) = 2 (x). In this setting only cases (b) and (c) of Fig. 4 apply. 7. Discussion, Open Problems The above analysis describes the existence of ground state surface gap solitons of (1.1) in the case of two materials meeting at the interface described by the hyperplane x1 = 0. It would be of interest to generalize this analysis to curved interfaces as well as to several intersecting interfaces with more than two materials. Besides looking for strong ground states, i.e. global minimizers of the energy J within the Nehari manifold N , one can also pose the question of existence of bound states, i.e. general critical points of J in H 1 (Rn ), including possibly ground states, i.e. minimizers of J within the set of nontrivial H 1 solutions. The existence of such more general solutions is not covered by our results. It is also unclear whether the ground states found in the current paper are unique (up to multiplication with −1 and translation by Tk ek for k = 2, . . . , n) and what their qualitative properties are. In particular, it would be interesting to determine the location where the above ground states are ‘concentrated’. Although their existence is shown using candidate functions shifted along the x1 -axis to +∞ or −∞, we conjecture that the ground states are concentrated near the interface at x1 = 0. The conditions of our non-existence result in Theorem 3 and Remark 2 agree with the set-up of several optics experiments as well as numerical computations in the literature if one neglects the fact that the periodic structures used in these are finite. In [25] the authors consider an interface of a homogeneous dielectric medium with εr = α > 0 and a photonic crystal with εr = α + Q(x), Q > 0 and provide numerics and experiments for surface gap solitons (SGSs) in the semi-infinite gap of the spectrum. The nonlinearity is
540
T. Dohnal, M. Plum, W. Reichel
(a)
(b)
(c)
Fig. 4. Cases (a), (b) and (c) correspond to V0 ∈ C 1 with V0 (0) = 0, V0 having a local minimum and V0 having a local maximum at x = 0, respectively. In the column τ = 0 the ground states are plotted in black. In the columns τ = 0 the green full and red dashed curves are functions sw1 ∈ N with energy smaller and larger than c1 = c2 respectively
cubic ( p = 3) and ≡ const. The observed SGSs cannot be modelled as strong ground states of (2.2) due to the ordering of V1 , V2 and 1 , 2 , which enables the application of Theorem 3. They could, possibly, be strong ground states if the structure was modelled as a finite block of a photonic crystal with εr = α + Q(x), Q > 0 embedded in an infinite dielectric with εr = α but this situation is outside the reach of our model. A similar situation arises in [24], which studies the interface of two cubically nonlinear photonic crystals with 1 ≡ 2 and either V1 ≥ V2 or V1 ≤ V2 with a strict inequality on a set of nonzero measure. Likewise, in the computations of [4,6], where (2.2) is considered in 1D with V1 ≡ V2 and 1 , 2 ≡ const., 1 = 2 , the SGSs computed in the semi-infinite spectral gap cannot correspond to strong ground states of (2.2). The findings of [4,6,24,25] show that despite the absence of strong ground states bound states may still exist. Acknowledgement. We thank Andrzej Szulkin (Stockholm University) for discussions leading to the improved hypothesis (H2) instead of (H2’).
Appendix Lemma A1. Let V ∈ L ∞ and L = − + V (x) such that 0 ∈ σ (L). If 1 < q < ∞ then L −1 : L q (Rn ) → W 2,q (Rn ) is a bounded linear operator.
Surface Gap Soliton Ground States for the NLS
541
Proof. Since the spectrum of L is stable in L q (Rn ) with respect to q ∈ [1, ∞], cf. Hempel, Voigt [8], we have for all u ∈ D(L) ⊂ L q (Rn ) that u L q (Rn ) ≤ CLu L q (Rn ) . We need to check that D(L) = W 2,q (Rn ). Here we restrict to 1 < q < ∞. Note that since V ∈ L ∞ , for u ∈ D(L) we have that Lu, V u, and hence −u all belong to L q (Rn ). Therefore −u + u = (1 − V )u + Lu and with the help of the Green function G −+1 we find
u = G −+1 ∗ (1 − V )u + Lu . By the mapping properties of the Green function (cf. mapping properties of Bessel potentials, Stein [21]), it follows that u ∈ W 2,q (Rn ) and
¯ uW 2,q (Rn ) ≤ C (1 − V )u L q (Rn ) + Lu L q (Rn ) ≤ CLu L q (R n ) . Since trivially W 2,q (Rn ) ⊂ D(L), the proof is done. Lemma A2. Let V, ∈ L ∞ and let L = − + V (x) be such that 0 < min σ (L). If 1 < p < 2∗ − 1, then every strong ground state u 0 of (1.1) is either positive in Rn or negative in Rn . Sketch of proof. Let u 0 be a strong ground state. Then u¯ 0 := |u 0 | is also a strong ground state and u¯ 0 ≡ 0. Due to the subcritical growth of the nonlinearity we have that locally u¯ 0 is a C 1,α -function. If we define Z = {x ∈ Rn : u¯ 0 (x) = 0}, then Z c = Rn \Z is open and ∇ u¯ 0 ≡ 0 on Z . If we assume that Z is nonempty, then there exists an open ball B ⊂ Z c such that ∂ B ∩ Z = ∅. This contradicts Hopf’s maximum principle. Thus u¯ 0 > 0 in Rn and therefore either u 0 > 0 in Rn or u 0 < 0 in Rn . References 1. Allaire, G., Orive, R.: On the band gap structure of Hill’s equation. J. Math. Anal. Appl. 306, 462–480 (2005) 2. Amrein, W.O., Berthier, A.-M., Georgescu, V.: L p -inequalities for the Laplacian and unique continuation. Ann. Inst. Fourier 31, 153–168 (1981) 3. Arcoya, D., Cingolani, S., Gámez, J.: Asymmetric modes in symmetric nonlinear optical waveguides. SIAM J. Math. Anal. 30, 1391–1400 (1999) 4. Blank, E., Dohnal, T.: Families of Surface Gap Solitons and their Stability via the Numerical Evans Function Method. SIAM J. Appl. Dyn. Syst. 10, 667–706 (2011) 5. Dohnal, T., Plum, M., Reichel, W.: Localized modes of the linear periodic Schrödinger operator with a nonlocal perturbation. SIAM J. Math. Anal. 41, 1967–1993 (2009) 6. Dohnal, T., Pelinovsky, D.: Surface gap solitons at a nonlinearity interface. SIAM J. Appl. Dyn. Syst. 7, 249–264 (2008) 7. Efremidis, N.K., Hudock, J., Christodoulides, D.N., Fleischer, J.W., Cohen, O., Segev, M.: Twodimensional optical lattice solitons. Phys Rev Lett. 91, 213906 (2003) 8. Hempel, R., Voigt, J.: The spectrum of a Schrödinger operator in L p (R ν ) is p-independent. Commun. Math. Phys. 104, 243–250 (1986) 9. Lions, P.L.: The concentration compactness principle in the calculus of variations. The locally compact case, II. Ann. Inst. H. Poincaré. Anal. Non Lin. 1, 223–283 (1984) 10. Kartashov, Y.V., Vysloukh, V.A., Torner, L.: Surface gap solitons. Phys. Rev. Lett. 96, 073901 (2006) 11. Kartashov, Y., Vysloukh, V., Szameit, A., Dreisow, F., Heinrich, M., Nolte, S., Tünnermann, A., Pertsch, T., Torner, L.: Surface solitons at interfaces of arrays with spatially modulated nonlinearity. Opt. Lett. 33, 1120–1122 (2008) 12. Kirr, E., Kevrekidis, P.G., Pelinovsky D.E.: Symmetry-breaking bifurcation in the nonlinear Schrödinger equation with symmetric potentials. http://arXiv.org/abs/1012.3921v1 [math-ph], 2010
542
T. Dohnal, M. Plum, W. Reichel
13. Korotyaev, E.: Schrödinger operator with a junction of two 1-dimensional periodic potentials. Asymptot. Anal. 45, 73–97 (2005) 14. Louis, P.J.Y., Ostrovskaya, E.A., Savage, C.M., Kivshar, Y.S.: Bose-Einstein condensates in optical lattices: Band-gap structure and solitons. Phys. Rev. A 67, 013602 (2003) 15. Mingaleev, S., Kivshar, Y.: Nonlinear Photonic Crystals Toward All-Optical Technologies. Opt. Photon. News 13, 48–51 (2002) 16. Makris, K., Hudock, J., Christodoulides, D., Stegeman, G., Manela, O., Segev, M.: Surface lattice solitons. Opt. Lett. 31, 2774–2776 (2006) 17. Pankov, A.: Periodic nonlinear Schrödinger equation with application to photonic crystals. Milan J. Math. 73, 259–287 (2005) 18. Renardy, M., Rogers, R.C.: An introduction to partial differential equations. Second edition. Texts in Applied Mathematics, 13. New York: Springer-Verlag, 2004 19. Rosberg, Ch.R., Neshev, D.N., Krolikowski, W., Mitchell, A., Vicencio, R.A., Molina, M.I., Kivshar, Y.S.: Observation of surface gap solitons in semi-infinite waveguide arrays. Phys. Rev. Lett. 97, 083901 (2006) 20. Schechter, M., Simon, B.: Unique continuation for Schrödinger operators with unbounded potentials. J. Math. Anal. Appl. 77, 482–492 (1980) 21. Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton Mathematical Series. No. 30, Princeton, N.J.: Princeton University Press, 1970 22. Struwe, M.: Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems. Second edition. Results in Mathematics and Related Areas (3), 34. Berlin: Springer-Verlag, 1996 23. Stuart, C.A.: Bifurcation for Dirichlet problems without eigenvalues. Proc. London Math. Soc. 45, 169–192 (1982) 24. Suntsov, S., Makris, K., Christodoulides, D., Stegeman, G., Morandotti, R., Volatier, M., Aimez, V., Arés, R., Yang, E., Salamo, G.: Optical spatial solitons at the interface between two dissimilar periodic media: theory and experiment. Opt. Express 16, 10480–10492 (2008) 25. Szameit, A., Kartashov, Y., Dreisow, F., Pertsch, T., Nolte, S., Tünnermann, A., Torner, L.: Observation of Two-Dimensional Surface Solitons in Asymmetric Waveguide Arrays. Phys. Rev. Lett. 98, 173903 (2007) 26. Szameit, A., Kartashov, Y.V., Dreisow, F., Heinrich, M., Vysloukh, V.A., Pertsch, T., Nolte, S., Tünnermann, A., Lederer, F., Torner, L.: Observation of two-dimensional lattice interface solitons. Opt. Lett. 33, 663–665 (2008) 27. Wang, X., Bezryadina, A., Chen, Z., Makris, K.G., Christodoulides, D.N., Stegeman, G.I.: Observation of two-dimensional surface solitons. Phys. Rev. Lett. 98, 123903 (2007) 28. Willem, M.: Minimax theorems. Progress in Nonlinear Differential Equations and their Applications, 24. Boston, MA: Birkhäuser Boston, Inc., 1996 Communicated by P. Constantin
Commun. Math. Phys. 308, 543–566 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1289-7
Communications in
Mathematical Physics
Infrared Problem for the Nelson Model on Static Space-Times Christian Gérard1 , Fumio Hiroshima2 , Annalisa Panati3 , Akito Suzuki4 1 Département de Mathématiques, Université de Paris XI, 91405 Orsay Cedex, France.
E-mail:
[email protected]
2 Department of Mathematics, University of Kyushu, 6-10-1, Hakozaki, Fukuoka 812-8581, Japan.
E-mail:
[email protected]
3 UMR 6207, Université Toulon-Var, 83957 La Garde Cedex, France.
E-mail:
[email protected]
4 Department of Mathematics, Faculty of Engineering, Shinshu University, 4-17-1 Wakasato,
Nagano 380-8553, Japan. E-mail:
[email protected] Received: 25 November 2010 / Accepted: 31 December 2010 Published online: 4 September 2011 – © Springer-Verlag 2011
Abstract: We consider the Nelson model on some static space-times and investigate the problem of existence of a ground state. Nelson models with variable coefficients arise when one replaces in the usual Nelson model the flat Minkowski metric by a static metric, allowing also the boson mass to depend on position. We investigate the existence of a ground state of the Hamiltonian in the presence of the infrared problem, i.e. assuming that the boson mass m(x) tends to 0 at spatial infinity. We show that if m(x) ≥ C|x|−1 at infinity for some C > 0 then the Nelson Hamiltonian has a ground state. 1. Introduction The study of Quantum Field Theory on curved space-times has seen important developments since the seventies. Probably the most spectacular prediction in this domain is the Hawking effect [Ha,FH,Ba], predicting that a star collapsing to a black hole asymptotically emits a thermal radiation. A related effect is the Unruh effect [Un,Un-W,dB-M], where an accelerating observer in Minkowski space-time sees the vacuum state as a thermal state. Another important development is the use of microlocal analysis to study free or quasi-free states on globally hyperbolic space-times, which started with the seminal work by Radzikowski [Ra1,Ra2], who proved that Hadamard states (the natural substitutes for vacuum states on curved space-times) can be characterized in terms of microlocal properties of their two-point functions. The use of microlocal analysis in this domain was further developed for example in [BFK,Sa]. Most of these works deal with free or quasi-free states, because of the well-known difficulty to construct an interacting, relativistic quantum field theory, even on Minkowski space-time. However in recent years a lot of effort was devoted to the rigorous study of interacting non-relativistic models on Minkowski space-time, typically obtained by coupling a relativistic quantum field to non-relativistic particles. The two main examples are
544
C. Gérard, F. Hiroshima, A. Panati, A. Suzuki
non-relativistic QED, where the quantized Maxwell field is minimally coupled to a nonrelativistic particle and the Nelson model, where a scalar bosonic field is linearly coupled to a non-relativistic particle. For both models it is necessary to add an ultraviolet cutoff in the interaction term to rigorously construct the associated Hamiltonian. In both cases the models can be constructed on a Fock space with relatively little efforts, and several properties of the quantum Hamiltonian H can be rigorously studied. One of them, which will also be our main interest in this paper, is the question of the existence of a ground state. Obviously the fact that H has a ground state is an important physical property of the Nelson model. For example a consequence of the existence of a ground state is that scattering states can quite easily be constructed. These states describe the ground state of H with a finite number of additional asymptotically free bosons. When H has no ground state one usually speaks of the infrared problem or infrared divergence. The infrared problem arises when the emission probability of bosons becomes infinite with increasing wave length. If the infrared problem occurs, the scattering theory has to be modified: all scattering states contain an infinite number of low energy (soft) bosons (see eg [DG3]). Among many papers devoted to this question, let us mention [AHH,BFS,BHLMS,G,H,LMS,Sp] for the Nelson model, and [GLL] for non-relativistic QED. Our goal in this paper is to study the existence of a ground state for the Nelson model on a static space-time, allowing also for a position-dependent mass. This model is obtained by linearly coupling the Lagrangians of a Klein-Gordon field and of a nonrelativistic particle on a static space-time (see Subsect. 2.2). We believe that this model, although non-relativistic, is an interesting testing ground for the generalization of results for free or quasi-free models on curved space-times to some interacting situations. Let us also mention that for the Nelson model on Minkowski space-time the removal of the ultraviolet cutoff can be done by relatively easy arguments. After removal of the ultraviolet cutoff, the Nelson model becomes a local (although non-relativistic) QFT model. In a subsequent paper [GHPS3], we will show that the ultraviolet cutoff can be removed for the Nelson model on a static space-time. Most of our discussion will be focused on the role of the variable mass term on the ground state existence. Note that when one considers a massive Klein-Gordon field in the Schwarzschild metric, the effective mass tends to 0 at the black hole horizon (see eg [Ba]). We believe that the study of the Nelson model with a variable mass vanishing at spatial infinity will be a first step towards the extension of the rigorous justification of the Hawking effect in [Ba] to some interacting models.
1.1. The Nelson model on Minkowski space-time. In this subsection we quickly describe the usual Nelson model on Minkowski space-time. The Nelson model describes a scalar bosonic field linearly coupled to a quantum mechanical particle. It is formally defined by the Hamiltonian 1 2 1 2 2 2 2 H = p + W (q) + π (x) + (∇ϕ(x)) + m ϕ (x)dx + ϕ(x)ρ(x − q)dx, 2 2 R3 R3 where ρ denotes a cutoff function, p, q denote the position and momentum of the particle, W (q) is an external potential and ϕ(x), π(x) are the canonical field position and momentum.
Infrared Problem for the Nelson Model on Static Space-Times
545
The Nelson model arises from the quantization of the following coupled KleinGordon and Newton system: ( + m 2 )ϕ(t, x) = −ρ(x − qt ), (1.1) q¨t = −∇q W (qt ) − ϕ(t, x)∇x ρ(x − qt )dx, where denotes the d’Alembertian on the Minkowski space-time R1+3 . The cutoff function ρ plays the role of an ultraviolet cutoff and amounts to replacing the point particle by a charge density. To distinguish the Nelson model on Minkowski space-time from its generalizations that will be described later in the Introduction, we will call it the usual (or constant coefficients) Nelson model. For the usual Nelson model the situation is as follows: one assumes a stability condition (see Subsect. 4.5), implying that states with energy close to the bottom of the spectrum are localized in the particle position. Then if the bosons are massive, i.e. if m > 0 H has a ground state (see e.g. [G]). On the contrary if m = 0 and ρ(x)dx = 0 then H has no ground state (see [DG3]). 1.2. The Nelson model with variable coefficients. We now describe a generalization of the usual Nelson model, obtained by replacing the free Laplacian −x by a general second order differential operator and the constant mass term m by a function m(x). We set: h := − c(x)−1 ∂ j a jk (x)∂k c(x)−1 + m 2 (x), 1≤ j,k≤d
for a Riemannian metric a jk dx j dxk and two functions c(x), m(x) > 0, and consider the generalization of (1.1): 2 ∂t φ(t, x) + hφ(t, x) + ρ(x − qt ) = 0, (1.2) 1 q¨t = −∇x W (qt ) − R3 φ(t, x)∇x ρ(x − qt )|g| 2 d3 x. Quantizing the field equations (1.2), we obtain a Hamiltonian H acting on the Hilbert space L 2 (R3 ) ⊗ s (L 2 (R3 )) (see Sect. 3), which we call a Nelson Hamiltonian with variable coefficients. Formally H is defined by the following expression: H =
1 2 p + W (q) 2 1 + π 2 (x) + ∂ j c(x)−1 ϕ(x) a jk (x) ∂k c(x)−1 ϕ(x) + m 2 (x)ϕ 2 (x)dx 2 R3 jk + ϕ(x)ρ(x − q)dx. (1.3) R3
The main example of a variable coefficients Nelson model is obtained by replacing in the usual Nelson model the flat Minkowski metric on R1+3 by a static Lorentzian metric, and by allowing also the mass m to be position dependent. Recall that a static metric on R1+3 is of the form gμν (x)dx μ dx ν = −λ(x)dtdt + λ(x)−1 h αβ (x)dxα dxβ ,
546
C. Gérard, F. Hiroshima, A. Panati, A. Suzuki
where x = (t, x) ∈ R1+3 , λ(x) > 0 is a smooth function, and h α,β (x) is a Riemannian metric on R3 . We show in Subsect. 2.3 that the natural Lagrangian for a point particle coupled to a scalar field on (R1+3 , g) leads (after a change of field variables) to the system (1.2). 1.3. The infrared problem. Assuming reasonable hypotheses on the matrix [a jk ](x) and the functions c(x), m(x) it is easy to see that the formal expression (1.3) can be rigorously defined as a bounded below selfadjoint operator H . The question we address in this paper is the problem of existence of a ground state for H . Variable coefficients Nelson models are examples of an abstract class of QFT Hamiltonians called abstract Pauli-Fierz Hamiltonians (see e.g. [G,BD] and Subsect. 4.1). If ω is the one-particle energy, the constant m := inf σ (ω) can be called the (rest) mass of the bosonic field, and abstract Pauli-Fierz Hamiltonians fall naturally into two classes: massive models if m > 0 and massless if m = 0. For massive models, H typically has a ground state, if we assume either that the quantum particle is confined or a stability condition (see Subsect. 4.5). In this paper we concentrate on the massless case and hence our typical assumption will be that lim m(x) = 0.
x→∞
It follows that bosons of arbitrarily small energy may be present. The main result of this paper is that the existence or non-existence of a ground state for H depends on the rate of decay of the function m(x). In fact we show in Thm. 4.1 that if m(x) ≥ a x −1 , for some a > 0, and if the quantum particle is confined, then H has a ground state. In a subsequent paper [GHPS2], we will show that if 0 ≤ m(x) ≤ C x −1− , for some > 0, then H has no ground state. Therefore Thm. 4.1 is sharp with respect to the decay rate of the mass at infinity. (If h = − + λm 2 (x) for m(x) ∈ O( x −3/2 ) and the coupling constant λ is sufficiently small the same result is shown in [GHPS1].) 1.4. Notation. We collect here some notation for the reader’s convenience. If x ∈ Rd , 1 we set x = (1 + x 2 ) 2 . The domain of a linear operator A on some Hilbert space H will be denoted by Dom A, and its spectrum by σ (A). If h is a Hilbert space, the bosonic Fock space over h denoted by s (h) is s (h) :=
∞
⊗ns h.
n=0
We denote by a ∗ (h), a(h) for h ∈ h the creation/annihilation operators acting on s (h). The (Segal) field operators φ(h) are defined as φ(h) := √1 (a ∗ (h) + a(h)). 2 If K is another Hilbert space and v ∈ B(K, K ⊗ h), then one defines the operators a ∗ (v), a(v) as unbounded operators on K ⊗ s (h) by:
Infrared Problem for the Nelson Model on Static Space-Times
547
√ a ∗ (v) K⊗ ns h := n + 1 (1K ⊗ Sn+1 ) v ⊗ 1 ns h , ∗ a(v) := a ∗ (v) , 1 φ(v) := √ (a(v) + a ∗ (v). 2 They satisfy the estimates 1
a (v)(N + 1)− 2 ≤ v,
(1.4)
where v is the norm of v in B(K, K ⊗ h). If b is a selfadjoint operator on h its second quantization d(b) is defined as: n n 1 · · ⊗ 1 ⊗b ⊗ 1 · · ⊗ 1 . d(b) s h : =
⊗ ·
⊗ · j=1
j−1
n− j
2. The Nelson Model on Static Space-Times In this section we discuss the Nelson model on static space-times, which is the main example of Hamiltonians that will be studied in the rest of the paper. It is convenient to start with the Lagrangian framework. 2.1. Klein-Gordon equation on static space-times. Let gμν (x) be a Lorentzian metric of signature (−, +, +, +) on R1+3 . Set |g| = det[gμν ], [g μν ] = [gμν ]−1 . Consider the Lagrangian L free (φ)(x) =
1 1 ∂μ φ(x)g μν (x)∂ν φ(x) + m 2 (x)φ 2 (x), 2 2
for a function m : R4 → R+ and the associated action: 1 L free (φ)(x)|g| 2 (x)d4 x, Sfield (φ) = R4
where φ : R4 → R. The Euler-Lagrange equations yield the Klein-Gordon equation: g φ + m 2 (x)φ = 0, for 1
1
g = −|g|− 2 ∂μ |g| 2 g μν ∂ν . Usually one has 1 2 1 m (x) = (m 2 + θ R(x)), 2 2 where m ≥ 0 is the mass and R(x) is the scalar curvature of the metric gμν , (assuming of course that the function on the right is positive). In particular if m = 0 and θ = 16 , one obtains the so-called conformal wave equation.
548
C. Gérard, F. Hiroshima, A. Panati, A. Suzuki
We set x = (t, x) ∈ R1+3 . The metric gμν is static if: gμν (x)dx μ dx ν = −λ(x)dtdt + λ(x)−1 h αβ (x)dxα dxβ , where λ(x) > 0 is a smooth function and h αβ is a Riemannian metric on R3 . We assume also that m 2 (x) = m 2 (x) is independent on t. ˜ x), we obtain that φ(t, ˜ x) satisfies the equation: Setting φ(t, x) = λ|h|−1/4 φ(t, ∂t2 φ˜ − λ|h|−1/4 ∂α |h| 2 h αβ ∂β |h|−1/4 λφ˜ + m 2 λφ˜ = 0. 1
1
We note that |h|−1/4 ∂α |h| 2 h αβ ∂β |h|−1/4 is (formally) self-adjoint on L 2 (R3 , dx) and is the Laplace-Beltrami operator h associated to the Riemannian metric h αβ (after the usual density change u → |h|1/4 u to work on the Hilbert space L 2 (R3 , dx)). 2.2. Klein-Gordon field coupled to a non-relativistic particle. We now couple the KleinGordon field to a non-relativistic particle. We fix a mass M > 0, a charge density ρ : R3 → R+ with q = R3 ρ(y)d3 y = 0 and a real potential W : R3 → R. The action for the coupled system is S = Spart + Sfield + Sint , for
Spart = Sint =
R
M |˙x(t)|2 − W (x(t))dt, 2 1
R4
φ(t, x)ρ(x − x(t))|g| 2 (x)d4 x.
The Euler-Lagrange equations are: g φ(t, x) + m 2 (t, x)φ(t, x) + ρ(x − x(t)) = 0, 1 M x¨ (t) = −∇x W (x(t)) − R3 φ(t, x)∇x ρ(x − x(t))|g| 2 d3 x. Doing the same change of field variables as in Subsect. 2.1 and deleting the tildes, we obtain the system: 2 ∂t φ − λh λφ + m 2 λφ + ρ(x − x(t)) = 0, (2.1) M x¨ (t) = −∇W (x(t)) − R3 φ(t, x)∇ρ(x − x(t))d 3 x. 2.3. The Nelson model on a static space-time. If the metric is static, Eqs. (2.1) are clearly Hamiltonian equations for the classical Hamiltonian H = Hpart + Hfield + Hint , where: 1 2 ξ + W (x), 2M 1 Hfield (ϕ, π ) = π 2 (x) − ϕ(x)λ(x)h λ(x)ϕ(x) + m 2 (x)λ(x)ϕ 2 (x)dx, 2 R3 Hint (x, ξ, ϕ, π ) = ρ(y − x)ϕ(y)dy. Hpart (x, ξ ) =
R3
Infrared Problem for the Nelson Model on Static Space-Times
549
The classical phase space is as usual R3 ×R3 ×L 2R (R3 )×L 2R (R3 ), with the symplectic form (x, ξ, ϕ, π )ω(x , ξ , ϕ , π ) = x · ξ − x · ξ + ϕ(x)π (x) − π(x)ϕ (x)dx. R3
The usual quantization scheme leads to the Hilbert space: L 2 (R3 , dy) ⊗ s (L 2 (R3 , dx)), where s (h) is the bosonic Fock space over the one-particle space h, and to the quantum Hamiltonian: 1 1 1 1 H = (− y + W (y)) ⊗ 1 + 1 ⊗ d(ω) + √ a ∗ (ω− 2 ρ(· − y)) + a(ω− 2 ρ(· − y)) , 2 2 where 1
ω = (−λh λ + m 2 λ) 2 , d(ω) is the usual second quantization of ω and a ∗ ( f ), a( f ) are the creation/annihilation operators on s (L 2 (R3 , dx)). 3. The Nelson Hamiltonian with Variable Coefficients In this section we define the Nelson model with variable coefficients that will be studied in the rest of the paper. We will deviate slightly from the notation in Sect. 2 by denoting by x ∈ R3 (resp. X ∈ R3 ) the boson (resp. electron) position. As usual we set Dx = ı−1 ∇x , D X = ı−1 ∇ X . 3.1. Electron Hamiltonian. We define the electron Hamiltonian as: K := K 0 + W (X ), where K0 =
D X j A jk (X )D X k ,
1≤ j,k≤3
acting on K := L 2 (R3 , dX ), where: (E1) C0 1 ≤ [A jk (X )] ≤ C1 1, C0 > 0. We assume that W (X ) is a real potential such that K 0 + W is essentially selfadjoint and bounded below. We denote by K the closure of K 0 + W . Later we will assume the following confinement condition : (E2) W (X ) ≥ C0 X 2δ − C1 , for some δ > 0. Physically this condition means that the electron is confined. As is well known (see e.g. [GLL]) for the question of existence of a ground state , this condition can be replaced by a stability condition, meaning that states near the bottom of the spectrum of the Hamiltonian are confined in the electronic variables by energy conservation. We will discuss the extension of our results when one assumes the stability condition in Subsect. 4.5.
550
C. Gérard, F. Hiroshima, A. Panati, A. Suzuki
3.2. Field Hamiltonian. Let:
h 0 := −
c(x)−1 ∂ j a jk (x)∂k c(x)−1 ,
1≤ j,k≤d
h := h 0 + m 2 (x), with a jk , c, m real functions and: C0 1 ≤ [a jk (x)] ≤ C1 1, C0 ≤ c(x) ≤ C1 , C0 > 0, (B1) ∂xα a jk (x) ∈ O( x −1 ), |α| ≤ 1, ∂xα c(x) ∈ O(1), |α| ≤ 2, ∂xα m(x) ∈ O(1), |α| ≤ 1. Clearly h is selfadjoint on H 2 (R3 ) and h ≥ 0. The one-particle space and one-particle energy are: 1
h := L 2 (R3 , dx), ω := h 2 . The constant: inf σ (ω) =: m ≥ 0, can be viewed as the mass of the scalar bosons. The following lemma is easy; Lemma 1. (1) One has Kerω = {0}. (2) Assume in addition to (B1) that lim x→∞ m(x) = 0. Then inf σ (ω) = 0. Proof. It follows from (B1) that (u|hu) ≤ C1 (c−1 u| − c−1 u) + (c−1 u|c−1 m 2 u), u ∈ H 2 (R3 ). Therefore if hu = 0 u is constant. It follows also from (B1) that c(x)−1 preserves H 2 (R3 ). Therefore by the variational principle m 2 = inf σ (h) ≤ C1 inf σ (− + c−2 (x)m 2 (x)) = 0. This proves (2). The Nelson Hamiltonian defined below will be called massive (resp. massless) if m > 0 (resp. m = 0.) The field Hamiltonian is d(ω), acting on the bosonic Fock space s (h).
Infrared Problem for the Nelson Model on Static Space-Times
551
3.3. Nelson Hamiltonian. Let ρ ∈ S(R3 ), with ρ ≥ 0, q = R3 ρ(y)dy = 0. We set: ρ X (x) = ρ(x − X ) and define the UV cutoff fields as: 1
ϕρ (X ) := φ(ω− 2 ρ X ),
(3.1)
where for f ∈ h, φ( f ) is the Segal field operator: 1 φ( f ) := √ a ∗ ( f ) + a( f ) . 2 Note that setting 1
one has ϕρ (X ) =
ϕ(X ) := φ(ω− 2 δ X ), ϕ(X − Y )ρ(Y )dY .
Remark 1. One can think of another definition of UV cutoff fields, namely: 1
ϕ˜χ (X ) := φ(ω− 2 χ (ω)δ X ), for χ ∈ S(R), χ (0) = 1. In the constant coefficients case where h = − both definitions are equivalent. In the variable coefficients case the natural definition (3.1) is much more convenient. The Nelson Hamiltonian is: H := K ⊗ 1 + 1 ⊗ d(ω) + ϕρ (X ),
(3.2)
acting on H = K ⊗ s (h). Set also: H0 := K ⊗ 1 + 1 ⊗ d(ω), which is selfadjoint on its natural domain. The following lemma is standard. Lemma 2. Assume Hypotheses (E1), (B1). Then H is selfadjoint and bounded below on D(H0 ). Proof. It suffices to apply results on abstract Pauli-Fierz Hamiltonians (see e.g. [GGM, Sect. 4]). H is an abstract Pauli-Fierz Hamiltonian with coupling operator v ∈ B(K, K⊗ h) equal to: 1
L 2 (R3 , dX ) u → ω− 2 ρ(x − X )u(X ) ∈ L 2 (R3 , dX ) ⊗ L 2 (R3 , dx). 1
Applying [GGM, Cor. 4.4], it suffices to check that ω− 2 v ∈ B(K, K ⊗ h). Now 1
1
ω− 2 v B(K,K⊗h) = ( sup ω−1 ρ X 2 ) 2 . X ∈R3
and the Kato-Heinz inequality, we obtain that ω−2 ≤ C|Dx |−2 , Using that h ≥ hence it suffices to check that the map C Dx2
L 2 (R3 , dX ) u → |Dx |−1 ρ(x − X )u(X ) ∈ L 2 (R3 , dX ) ⊗ L 2 (R3 , dx) is bounded, which is well known.
552
C. Gérard, F. Hiroshima, A. Panati, A. Suzuki
4. Existence of a Ground State In this section we will prove our main result about the existence of a ground state for variable coefficients Nelson Hamiltonians. This result will be deduced from an abstract existence result extending the one in [BD], whose proof is outlined in Subsects. 4.1, 4.2 and 4.3. Theorem 4.1. Assume Hypotheses (E1), (B1). Assume in addition that: m(x) ≥ a x −1 , for some a > 0, and (E2) for some δ > 23 . Then inf σ (H ) is an eigenvalue. Remark 2. The condition δ > 23 in Thm. 4.1 comes from the operator bound ω−3 ≤ C x 3+ , ∀ > 0 proved in Thm. A3. Remark 3. From Lemma 1 we know that inf σ (ω) = 0 if lim x→∞ m(x) = 0. Therefore the Nelson Hamiltonian can be massless using the terminology of Subsect. 3.2. Remark 4. In a subsequent paper [GHPS2] we will show that if 0 ≤ m(x) ≤ C x −1− , for some > 0, then H has no ground state. Therefore the result of Thm. 4.1 is sharp with respect to the decay rate of the mass at infinity. 4.1. Abstract Pauli-Fierz Hamiltonians. In [BD], Bruneau and Derezi´nski study the spectral theory of abstract Pauli-Fierz Hamiltonians of the form H = K ⊗ 1 + 1 ⊗ d(ω) + φ(v), acting on the Hilbert space H = K ⊗ s (h), where K is the Hilbert space for the small system and h the one-particle space for the bosonic field. The Hamiltonian H is called massive (resp. massless) if inf σ (ω) > 0 (resp. inf σ (ω) = 0). Among other results they prove the existence of a ground state for H if v is infrared regular. Although most of their hypotheses are natural and essentially optimal, we cannot directly apply their abstract results to our situation. In fact they assume (see [BD, Assumption E]) that the one-particle space h equals L 2 (Rd , dk) and the one-particle energy ω is the multiplication operator by a function ω(k) which is positive, with ∇ω bounded, and limk→∞ ω(k) = +∞. This assumption on the one-particle energy is only needed to prove an HVZ theorem for massive (or massless with an infrared cutoff) Pauli-Fierz Hamiltonians. In our case this assumption could be deduced (modulo unitary equivalence) from the spectral theory of h. For example it would suffice to know that h is unitarily equivalent to −. This last property would follow from the absence of eigenvalues for h and from the scattering theory for the pair (h, −) and require additional decay properties of the [a i j ](x), m(x) and of some of their derivatives. We will replace it by more geometric assumptions on ω (see hypothesis (4.4) below), similar to those introduced in [GP], where abstract bosonic QFT Hamiltonians were considered. Since we do not aim for generality, our hypotheses on the coupling operator v are stronger than necessary, but lead to simpler proofs. Also most of the proofs will be only sketched.
Infrared Problem for the Nelson Model on Static Space-Times
553
Let h, K be two Hilbert spaces and set H = K ⊗ s (h). We fix selfadjoint operators K ≥ 0 on K and ω ≥ 0 on h. We set inf σ (ω) =: m ≥ 0. If m = 0 one has to assume additionally that Kerω = {0} (see Remark 5 for some explanation of this fact). Remark 5. If X is a real Hilbert space and ω is a selfadjoint operator on X , the condition Kerω = {0} is well known to be necessary to have a stable quantization of the abstract Klein-Gordon equation ∂t2 φ(t) + ω2 φ(t) = 0, where φ(t) : R → X . If Kerω = {0} the phase space Y = X ⊕ X for the Klein-Gordon equation splits into the symplectic direct sum Yreg ⊕ Ysing , for Yreg = Kerω⊥ ⊕ Kerω⊥ , Ysing = Kerω ⊕ Kerω, both symplectic spaces being invariant under the symplectic evolution associated to the Klein-Gordon equation. On Yreg one can perform the stable quantization. On Ysing , if for example Kerω is d−dimensional, the quantization leads to the Hamiltonian − on L 2 (Rd ). Clearly any perturbation of the form φ( f ) for 1{0} (ω) f = 0 will make the Hamiltonian unbounded from below. So we will always assume that ω ≥ 0, Kerω = {0}.
(4.1)
Let H0 = K ⊗ 1 + 1 ⊗ d(ω). We fix also a coupling operator v such that: v ∈ B(K, K ⊗ h).
(4.2) 1
The quadratic form φ(v) = a(v)+a ∗ (v) is well defined for example on K⊗DomN 2 . We will also assume that: 1
1
ω− 2 v(K + 1)− 2 is compact.
(4.3)
Proposition 1 ([BD] Thm. 2.2). Assume (4.1), (4.3). Then H = H0 + φ(v) is well defined as a form sum and yields a bounded below selfadjoint operator with 1 1 Dom|H | 2 = Dom|H0 | 2 . The operator H defined as above is called an abstract Pauli-Fierz Hamiltonian. 4.2. Existence of a ground state for cutoff Hamiltonians. We introduce as in [BD] the infrared-cutoff objects vσ = F(ω ≥ σ )v, Hσ = K ⊗ 1 + 1 ⊗ d(ω) + φ(vσ ), σ > 0, where F(λ ≥ σ ) denotes as usual a function of the form χ (σ −1 λ), where χ ∈ C ∞ (R), χ (λ) ≡ 0 for λ ≤ 1, χ (λ) ≡ 1 for λ ≥ 2. An important step to prove that H has a ground state is to prove that Hσ has a ground state. The usual trick is to consider H˜ σ = K ⊗ 1 + 1 ⊗ d(ωσ ) + φ(vσ ), where: ωσ := F(ω ≤ σ )σ + (1 − F(ω ≤ σ ))ω = ω + (σ − ω)F(ω ≤ σ ).
554
C. Gérard, F. Hiroshima, A. Panati, A. Suzuki
Note that since ωσ ≥ σ > 0, H˜ σ is a massive Pauli-Fierz Hamiltonian. Moreover it is well known (see e.g. [G,BD]) Hσ has a ground state iff H˜ σ does. The fact that H˜ σ has a ground state follows from an estimate on its essential spectrum (HVZ Theorem). In [BD] this is shown using the condition that h = L 2 (Rd , dk) and ω = ω(k). Here we will replace this condition by the following more abstract condition, formulated using an additional selfadjoint operator r on h. Similar abstract conditions were introduced in [GP]. We will assume that there exists a selfadjoint operator r ≥ 1 on h such that the following conditions hold for all σ > 0: (i) (z − r)−1 : Domωσ → Domωσ , ∀z ∈ C\R, (ii) [r, ωσ ] defined as a quadratic form on Domr ∩ Domω is bounded, (4.4) 1 (iii) r− (ωσ + 1)− is compact on h for some 0 < < . 2 The operator r, called a gauge, is used to localize particles in h. We assume also as in [BD]: 1
(K + 1)− 2 is compact.
(4.5)
This assumption means that the small system is confined. Proposition 2. Assume (4.1), (4.2), (4.3), (4.4), (4.5). Then σess ( H˜ σ ) ⊂ [inf σ ( H˜ σ ) + σ, +∞[. It follows that H˜ σ (and hence Hσ ) has a ground state for all σ > 0. Proof. By (4.3), φ(vσ ) is form bounded with respect to H0 (and to K ⊗ 1 + 1 ⊗ d(ωσ )) with the infinitesimal bound, hence Hσ , H˜ σ are well defined as bounded below selfadjoint Hamiltonians. We can follow the proof of [DG2, Thm. 4.1] or [GP, Thm. 7.1] for its abstract version. For ease of notation we denote simply H˜ σ by H, ωσ by ω and vσ by v. The key estimate is the fact that for χ ∈ C0∞ (R) one has χ (H ext )I ∗ ( j R ) − I ∗ ( j R )χ (H ) ∈ o(1), when R → ∞.
(4.6)
(The extended operator H ext and identification operator I ( j R ) are defined for example in [GP, Sect. 2.4]). The two main ingredients of the proof of (4.6) are the estimates: r [F( ), ωσ ] ∈ O(R −1 ), F ∈ C0∞ (R), R
(4.7)
1 r ≥ 1)vσ (K + 1)− 2 ∈ o(R 0 ). R
(4.8)
and −1
ωσ 2 F(
1
−1
Now (4.8) follows from the fact that vσ (K + 1)− 2 is compact (note that ωσ 2 is bounded since ωσ ≥ σ ), and (4.7) follows from Lemma 3. The estimate (4.6) can then be proved exactly as in [GP, Lemma 6.3]. Note that here we prove only the ⊂ part of the HVZ Theorem, which is sufficient for our purposes. The details are left to the reader.
Infrared Problem for the Nelson Model on Static Space-Times
555
Lemma 3. Assume conditions (i), (ii) of (4.4). Then for all F ∈ C0∞ (R) one has: F(r) : Domωσ → Domωσ , r [F( ), ωσ ] ∈ O(R −1 ). R Proof. The proof of the lemma is easy, using almost analytic extensions, as for example in [GP]. The details are left to the interested reader.
4.3. Existence of a ground state for massless models. Let us introduce the following hypothesis on the coupling operator ([BD, Hyp. F]): 1
ω−1 v(K + 1)− 2 is compact.
(4.9)
Theorem 4.2. Assume (4.1), (4.2), (4.3), (4.4), (4.5) and (4.9). Then H has a ground state. Proof. We can follow the proof in [BD, Sect. 4]. The existence of ground state for Hσ ([BD, Prop. 4.5]) is shown in Prop. 2. The arguments in [BD, Sects. 4.2, 4.3] based on the pull-through and double pull-through formulas are abstract and valid for any one particle operator ω. The only place where the fact that h = L 2 (Rd , dk) and ω = ω(k) appears is in [BD, Prop. 4.7] where the operator |x| = |i∇k | enters. In our situation it suffices to replace it by our gauge operator r. The rest of the proof is unchanged.
4.4. Proof of Thm. 4.1. We now complete the proof of Thm. 4.1, by verifying the 1 hypotheses of Thm. 4.2. We recall that h = L 2 (Rd dx), ω = h 2 and we will take 1 r = x = (1 + x 2 ) 2 . 1
Proof of Thm. 4.1. We saw in the proof of Lemma 2 that v, ω− 2 v are bounded, hence 1 in particular (4.2) is satisfied. By Hypothesis (E2), (K + 1)− 2 is compact, which implies that conditions (4.3) and (4.5) are satisfied. We now check condition (4.4). Note that ωσ = f (h), where f ∈ C ∞ (R) with 1 f (λ) = λ 2 for λ ≥ 2. Clearly Domωσ = H 1 (Rd ) which is preserved by (z − x )−1 , so (i) of (4.4) is satisfied. Condition (iii) is also obviously satisfied. It remains to check condition (ii). To this end we write ωσ = f (h) = (h + 1)g(h), where g ∈ C ∞ (R) satisfies 1
g (n) (λ) ∈ O( λ − 2 −n ), n ∈ N, and hence [ x , ωσ ] = [ x , h]g(h) + (h + 1)[ x , g(h)].
(4.10)
Since ∇a jk (x), ∇c(x), ∇m(x) are bounded and Domh = H 2 (Rd ) we see that 1
[ x , h](h + 1)− 2 , [[ x , h], h](h + 1)−1 are bounded.
(4.11)
556
C. Gérard, F. Hiroshima, A. Panati, A. Suzuki
In particular the first term in the r.h.s. of (4.10) is bounded. To estimate the second term, we use an almost analytic extension of g satisfying: ∂ g˜ (z)| ≤ C N z −3/2−N |Imz| N , N ∈ N, ∂z suppg˜ ⊂ {z ∈ C||Imz| ≤ c(1 + |Rez|)}, g˜ |R = g, |
(4.12)
(see eg [DG1, Prop. C.2.2]), and write ı ∂ g˜ (z)(z − h)−1 dz ∧ d¯z . g(h) = 2π C ∂ z¯ We perform a commutator expansion to obtain that: [ x , g(h)] = g (h)[ x , h] + R2 , for ∂ g˜ ı (z)(z − h)−2 [[ x , h]h](z − h)−1 dz ∧ d¯z . R2 = 2π C ∂ z¯ Since |g (λ)| ≤ C λ −3/2 , (h+1)g (h)[ x , h] is bounded. To estimate the term (h+1)R2 , we use again (4.11) and the bound (h + 1)α (z − h)−1 ≤ C z α |Imz|−1 , α =
1 , 1. 2
We obtain that (h + 1)R2 ≤ C[[ x , h]h](h + 1)−1
∂ g˜ (z)| z 2 |Imz|−3 dzd¯z . ∂ z ¯ C |
This integral is convergent using the estimate (4.12). This completes the proof of (4.4). It remains to check condition (4.9), i.e. the fact that the interaction is infrared regular. This is the only place where the lower bound on m(x) enters. By Thm. A3 we obtain that ω−3/2 x −3/2− is bounded for all > 0. By condition (E2), we obtain that 1
X 3/2+ (K + 1)− 2 is bounded for all > 0 small enough. Therefore to check (4.9) it suffices to prove that the map L 2 (R3 , dX ) u → x 3/2+ ρ(x − X ) X −3/2− u(X ) ∈ L 2 (R3 , dX ) ⊗ L 2 (R3 , dx) is bounded, which is immediate since ρ ∈ S(R3 ). This completes the proof of Thm. 4.1.
Infrared Problem for the Nelson Model on Static Space-Times
557
4.5. Existence of a ground state for non confined Hamiltonians. In this subsection we state the results on existence of a ground state if the electronic potential is not confining. As explained in the beginning of this section, one has to assume a stability condition, meaning that states near the bottom of the spectrum of H are confined in electronic variables from energy conservation arguments. Definition 1. Let H be a Nelson Hamiltonian satisfying (E1), (B1). We assume for simplicity that the electronic potential W (X ) is bounded. Set for R ≥ 1: D R = {u ∈ Dom H |1{|X |≤R} u = 0}. The ionization threshold of H is (H ) := lim
inf
R→+∞ u∈D R , u=1
(u|H u).
The following theorem can easily be obtained by adapting the arguments in this section. Theorem 4.3. Assume (E1), (B1), W ∈ L ∞ (R3 ) and m(x) ≥ a x −1 for some a > 0. Then if the following stability condition is satisfied: (H ) > inf σ (H ), H has a ground state. Sketch of proof. Assuming the stability condition one can prove using Agmon-type estimates as in [Gr] (see [P] for the case of the Nelson model) that if χ ∈ C0∞ (]−∞, (H )[ then eβ|X | χ (Hσ ) is bounded uniformly in 0 < σ ≤ σ0 for σ0 small enough. From this fact one deduces by the usual argument that Hσ has a ground state ψσ and that sup X N ψσ < ∞.
σ >0
(4.13)
One can then follow the proof in [P, Thm. 1.2]. The key infrared regularity property replacing (4.9) is now sup ω−1 vψσ H⊗h < ∞.
σ >0
This estimate follows as in the proof of (4.9) from Thm. A3 and the bound (4.13). The details are left to the reader.
A. Lower Bounds for Second Order Differential Operators In this section we prove various lower bounds for second order differential operators. These bounds are the key ingredient in the proof of the existence of a ground state for the Nelson model.
558
C. Gérard, F. Hiroshima, A. Panati, A. Suzuki
A.1. Second order differential operators. Let us introduce the class of second order differential operators that will be studied in this section. Let: h0 = c(x)−1 D j a jk (x)Dk c(x)−1 , 1≤ j,k≤d
h = h 0 + v(x), with
a jk , c, v
real functions and: C0 1 ≤ [a jk (x)] ≤ C1 1, C0 ≤ c(x) ≤ C1 , C0 > 0,
∂xα a jk (x) ∈ O( x −1 ), |α| ≤ 1, ∂xα c(x) ∈ O(1), |α| ≤ 2,
(A.1)
v ∈ L ∞ (Rd ), v ≥ 0.
(A.2)
Clearly h 0 and h are selfadjoint and positive with domain H 2 (Rd ). We will always assume that d ≥ 3. A.2. Upper bounds on heat kernels. If K is a bounded operator on L 2 (Rd , c2 dx) we will denote by K (x, y) ∈ D (R2d ) its distribution kernel. In this subsection we will prove the following theorem. We set: α
x 2 ψα (t, x) := , α > 0.
x 2 + t Theorem A1. Assume in addition to (A.2), (A.2) that: v(x) ≥ a x −2 , a > 0, then there exists C, c, α > 0 such that: e−th (x, y) ≤ Cψα (t, x)ψα (t, y)ect (x, y), ∀t > 0, x, y ∈ Rd .
(A.3)
If c(x) ≡ 1 or if h 0 is the Laplace-Beltrami operator for a Riemannian metric on Rd , then Thm. A1 is due to Zhang [Zh]. Remark 6. Conjugating by the unitary U:
L 2 (Rd , dx) → L 2 (Rd , c2 (x)dx), u → c(x)−1 u,
we obtain
h˜ 0 := U h 0 U −1 = c(x)−2
D j a jk (x)Dk ,
1≤ j,k≤d
h˜ := U hU
−1
= h˜ 0 + v(x), ˜
which are selfadjoint with domain H 2 (Rd ). Let e−t h (x, y) for t > 0 the integral kernel ˜ of e−t h i.e. such that ˜ e−t h u(x) = e−th (x, y)u(y)c2 (y)dy, t > 0. Rd ˜
˜
Then since e−th (x, y) = c(x)e−t h (x, y)c(y), it suffices to prove Thm. A1 for e−t h .
Infrared Problem for the Nelson Model on Static Space-Times
559
˜ and denote it again by By the above remark, we will consider the operator h˜ 0 (resp. h) h 0 (resp. h). We note that they are associated with the closed quadratic forms: Q0( f ) = ∂ j f a jk ∂k f dx, Rd j,k
Q( f ) = Q 0 ( f ) +
Rd
| f |2 c2 v dx,
with domain H 1 (Rd ). Let us consider the semi-group {e−th }t≥0 generated by h. Since DomQ 0 = H 1 (Rd ), we can apply [D, Thms. 1.3.2, 1.3.3] to obtain that e−th is positivity preserving and extends as a semi-group of contractions on L p (Rd , c2 dx) for 1 ≤ d ≤ ∞, strongly continuous on L p (Rd , c2 dx) if p < ∞. In other words {e−th }t≥0 is a Markov symmetric semigroup. We first recall two results, taken from [PE] and [D]. Lemma 4. Assume (A.2), (A.2). Then there exist c, C > 0 such that: 0 ≤ e−th (x, y) ≤ Cect (x, y), ∀ 0 < t, x, y ∈ Rd . Proof. Since v(x) ≥ 0 it follows from the Trotter-Kato formula that 0 ≤ e−th (x, y) ≤ e−th 0 (x, y), a.e. x, y. The stated upper bound on e−th 0 (x, y) is shown in [PE, Thm. 3.4]. The following lemma is an extension of [D, Lem. 2.1.2] where the case c(x) ≡ 1 is considered. Lemma 5. Assume (A.2), (A.2). Then: (1) e−th is ultracontractive, i.e. e−th is bounded from L 2 to L ∞ for all t > 0, and ct := e−th L 2 →L ∞ = sup
f ∈L 2
e−th f ∞ ≤ ct −d/4 f 2
with some constant c > 0. (2) e−th is bounded from L 1 to L ∞ for all t > 0 and 2 . e−th L 1 →L ∞ ≤ ct/2
(3) The kernel e−th (x, y) satisfies: 2 . 0 ≤ e−th (x, y) ≤ ct/2
Proof. From Lemma 4 we obtain that e−th f ∞ ≤ Cect | f |∞ ≤ C t −d/4 f 2 , using the explicit form of the heat kernel of the Laplacian. This proves (1). Taking adjoints we see that e−th is also bounded from L 1 to L 2 with e−th L 1 →L 2 ≤ ct . It follows that 2 , e−th L 1 →L ∞ ≤ e−th/2 L 2 →L ∞ e−th/2 L 1 →L 2 ≤ ct/2
which proves (2). Statement (3) is shown in [D, Lem. 2.1.2].
560
C. Gérard, F. Hiroshima, A. Panati, A. Suzuki
We will deduce Thm. A1 from the following result. Theorem A2. Assume the hypotheses of Thm. A1. Then there exists C, α > 0 such that: e−th (x, y) ≤ Ct −d/2 ψα (t, x)ψα (t, y). Proof of Theorem A1. Combining Lemma 4 with Thm. A2 we get: 1− e−th (x, y) = e−th (x, y) e−th (x, y) ≤ Ct −d/2 e−(x−y) ≤ C t −d/2 e−c(x−y)
2 /2t
2 /2t
t −(1−)d/2 ψα (t, x)1− ψα (t, y)1−
ψβ (t, x)ψβ (t, y),
for β = (1 − )α. This completes the proof of Thm. A1. It remains to prove Theorem A2. To this end, we employ the following abstract result. Lemma 6 ([MS, Thm. B]). Let (M, dμ) be a locally compact measurable space with σ -finite measure μ and let A be a non-negative self-adjoint operator on L 2 (M, dμ) such that (i) e−t A1 := (e−t A | L 1 ∩L 2 )clos , t ≥ 0 is a C0 -semi-group of bounded operators, L 1 →L 1 i.e., e−t A1 L 1 →L 1 ≤ c1 , t ≥ 0. (ii) e−t A is bounded from L 1 to L ∞ with: e−t A1 L 1 →L ∞ ≤ c2 t − j , t > 0, for some j > 1. Assume moreover that there exists a family of weights ψ(s, x) (s > 0) such that: (B1) ψ(s, x), ψ(s, x)−1 ∈ L 2loc (M \ N , dμ) for all s > 0, where N is a closed null set. (B2) There is a constant c˜ independent of s such that, for all t ≤ s, ψ(s, ·)e−t A ψ(s, ·)−1 f 1 ≤ c ˜ f 1 ,
f ∈ Ds ,
where Ds := ψ(s, ·)L ∞ c (M \ N , dμ) (B3) There exists 0 < < 1 and constants cˆi > 0, i = 1, 2 such that for any s > 0 there exists a measurable set s ⊂ M with (a) |ψ(s, x)|− ≤ cˆ1 for all x ∈ M \ s , (b) |ψ(s, x)|− ∈ L q (s ) and |ψ(s, ·)|− L q (s ) ≤ cˆ2 s j/q with q = 2/(1 − ) and j > 1 is the exponent in condition (ii). Then there is a constant C such that |e−t A (x, y)| ≤ Ct − j |ψ(t, x)ψ(t, y)|, ∀ t > 0, a.e. x, y ∈ M. To verify Condition (B2) of Lemma 6, we will use the following lemma.
Infrared Problem for the Nelson Model on Static Space-Times
561
Lemma 7 ([MS, Criterion 2]). Let e−t A be a C0 -semi-group on L 2 (M, dμ). Denote by
·, · the scalar product on L 2 (M, dμ). Then: e−t A f L ∞ ≤ f L ∞ ,
f ∈ L 2 ∩ L ∞ , t > 0,
if and only if: Re f − f ∧ , A f ≥ 0,
f ∈ D(A),
(A.4)
where f ∧ = (| f | ∧ 1)sgn f with sgn f (x) := f (x)/| f |(x) if | f |(x) = 0 and sgn f (x) = 0 if f (x) = 0. Proof of Thm. A2. We will prove that there exists α > 0 such that the hypotheses of Lemma 6 are satisfied for (M, dμ) = (Rd , c2 (x) d x), A = h and ψ(s, x) = ψα (s, x). For ease of notation we will often denote ψα simply by ψ. From the discussion before Lemma 5, we see that e−th extends as a C0 −semi-group of contractions of L 1 (Rd , c2 dx), which implies that hypothesis (i) holds with c1 = 1. Hypothesis (ii) with j = d/2 follows from (2) of Lemma 5. Note that d/2 > 1 since d ≥ 3. We now check that (Recall that the constant a is defined in Thm. A1) Conditions (B) 1 are satisfied by ψα provided we choose α = α0 a 2 for some constant α0 . Since ψ, ψ −1 are bounded, Condition (B1) is satisfied for all α > 0. Set s := {x ∈ Rd | x 2 ≤ s}. Then α 2
x + s ψ(x)− = ≤ 2α , ∀ x ∈ s ,
x 2 d so that which proves the bound (a) of (B3) for all α > 0. Take now 0 < < d+4α d − 2αq > 0 for q = 2/(1 − ). If 0 ≤ s < 1, s = ∅ and (b) of (B3) is satisfied. If s ≥ 1 we have: αq 2
x + s q ψ − L q (s ) = c2 (x)dx 2 s
x −2αq ≤ C12 (2s)αq dx √ |x|
= Cs αq
√ s
{|x|≤ s}
r d−2αq−1 dr = C s d/2 .
0
Hence (b) is satisfied for j = d/2. It remains to check (B2). To avoid confusion, we denote by g, f the scalar product in L 2 (Rd , c2 (x)dx) and by (g| f ) the usual scalar product in L 2 (Rd , dx). Since ψ, ψ −1 are C ∞ and bounded with all derivatives, we see that {ψe−th ψ −1 }t≥0 is a C0 −semi-group on L 2 (Rd , c2 dx), with generator h ψ := ψhψ −1 , Domh ψ = H 2 (Rd ). We claim that there exists α > 0 such that e−th ψ L 1 →L 1 ≤ C, uniformly for 0 ≤ t ≤ s.
(A.5)
562
C. Gérard, F. Hiroshima, A. Panati, A. Suzuki
By duality, (A.5) will follow from (A.6): ∗
e−th ψ L ∞ →L ∞ ≤ C, uniformly for 0 ≤ t ≤ s.
(A.6)
To prove (A.6), we will apply Lemma 7. To avoid confusion, ∂ j f (x) will denote a partial derivative of the function f , while ∇ j f (x) denote the product of the operator ∇ j and the operator of multiplication by the function f . Setting bi = ψ −1 ∂i ψ, we have: h ∗ψ = ψ −1 hψ = −c(x)−2
∇ j a jk (x)∇k −
j,k
−c
−2
c−2 (x)b j (x)a jk (x)∇k
j,k
(x)∇ j a (x)bk (x) + v(x) − c−2 (x)
= −c(x)
jk
−2
∇ j a (x)∇k − 2c(x) jk
−2
j,k
w(x) = v(x) − c(x)−2 −c(x)
b j (x)a jk (x)bk (x)
j,k
b j (x)a jk (x)∇k + w(x),
j,k
where:
−2
b j (x)a jk (x)bk (x)
j,k
a (x)∂ j bk (x) − c(x)−2 jk
j,k
(∂ j a jk )(x)bk (x).
j,k
Clearly Domh ∗ψ = H 2 (Rd ). To simplify notation, we set A(x) = [a jk (x)], F(x) = (b1 (x), . . . , bd (x)). The identity above becomes: h ∗ψ = −c−2 ∇x A∇x − c−2 F A∇x − c−2 ∇x AF + v − c−2 F AF, = −c−2 ∇x A∇x − 2c−2 F A∇x + w.
(A.7)
We note that b j (x) = αsx j x −2 ( x 2 + s)−1 , which implies that: |b j (x)| ≤ Cα x −1 , |∇x b j (x)| ≤ Cα x −2 , for some C > 0. Since v(x) ≥ a x −2 , this implies using also (A.2) that: v(x) − c(x)−2 F AF(x) ≥ 0, w(x) ≥ 0,
(A.8)
for α > 0 small enough. This implies that Re f, h ∗ψ f = −(∇x f |A∇x f )+( f |(c2 v− F AF) f ) ≥ 0, for f ∈ H 1 (Rd ). ∗
(A.9)
It follows that h ∗ψ is maximal accretive, hence e−th ψ is a C0 −semi-group of contractions by the Hille-Yosida theorem. To check Condition (A.4) in Lemma 7 we follow [MS], with some easy modifications. We write f − f = sgn f χ , χ := 1{| f |≥1} (| f | − 1),
Infrared Problem for the Nelson Model on Static Space-Times
563
and note that if f ∈ Domh ∗ψ ⊂ H 1 (Rd ) then | f |, sgn f, χ ∈ H 1 (Rd ) with ∇sgn f =
∇f 1 ∇f − f ( f ∇ f + f ∇ f ). (A.10) , ∇χ = 1{| f |≥1} ∇| f |, ∇| f | = |f| | f |2 2| f |
We have:
f − f , h ∗ψ f = (∇( f − f )|A∇ f ) − 2(F( f − f )|A∇ f ) + (( f − f )|c2 w f ) =: C1 ( f ) + C2 ( f ) + C3 ( f ). Using (A.10), we have: C1 ( f ) = (∇( f − f )|A∇ f ) χ f χ A∇ f ) − (∇| f || f A∇ f ) A∇ f ) + (∇χ | |f| | f |2 |f| =: B1 ( f ) + B2 ( f ) + B3 ( f ). = (∇ f |
Clearly B1 ( f ) is real valued. Next: χ 1 χ A∇| f |), (A.11) ReB2 ( f ) = − (∇| f || 2 A( f ∇ f + f ∇ f )) = −(∇| f || 2 |f| |f| using (A.10). Similarly: 1 1 A( f ∇ f + f ∇ f )) = (∇χ |A∇χ ), ReB3 ( f ) = (∇χ | 2 |f|
(A.12)
using again (A.10). We estimate now ReC2 ( f ). We have: ReC2 ( f ) = −2Re(F( f − f )|A∇ f ) =
F 1 (χ | A( f ∇ f + f ∇ f )) = −2(Fχ |A∇χ ). 2 |f| (A.13)
We estimate now ReC3 ( f ). We have: ReC3 ( f ) = Re( f − f |c2 w f ) = Re(χ |c2 w| f |) = (χ |c2 w| f |) = (χ |c2 wχ ) + (χ |c2 w). (A.14) Collecting (A.11) to (A.13), we obtain that: χ χ Re f − f , h ∗ψ f = (∇ f | A∇ f ) − (∇| f || A∇| f |) |f| |f| +(∇χ |A∇χ ) − 2(Fχ |A∇χ ) + (χ |c2 wχ ) +(χ |c2 w). We use now the point-wise identity: ∇ f A∇ f − ∇| f |A∇| f | 1 = ∇ f A∇ f − ( f ∇ f + f ∇| f |)A( f ∇ f + f ∇| f |) 4| f |2 1 2 = (2| f |2 ∇ f A∇ f − f 2 ∇ f A∇ f − f ∇ f A∇ f ) 4| f |2 1 = (Re f ∇Im f − Im f ∇Re f )A(Re f ∇Im f − Im f ∇Re f ) ≥ 0. | f |2
(A.15)
564
C. Gérard, F. Hiroshima, A. Panati, A. Suzuki
Hence the first line in the rhs of (A.15) is positive. Concerning the third line, we recall that (A.8) implies that w ≥ 0 if α = α0 a. Since χ ≥ 0 the third line is also positive. Therefore: Re f − f , h ∗ψ f ≥ (∇χ |A∇χ ) − 2(Fχ |A∇χ ) + (χ |c2 wχ ) = χ , h ∗ψ χ = Re χ , h ∗ψ χ ,
using (A.7) and the fact that χ is real. Using (A.9) we obtain Condition (A.4). This completes the proof of Thm. A2. A.3. Lower bounds for differential operators. We now deduce lower bounds for powers of h from the heat kernel bounds in Subsect. A.2. Theorem A3. Assume Hypotheses (A.2), (A.2) and v(x) ≥ a x −2 , a > 0. Then h −β ≤ C x 2β+ , ∀ 0 ≤ β ≤ d/2, > 0. We start by an easy consequence of Sobolev inequality. Lemma 8. On L 2 (Rd ) the following inequality holds: (−)−γ ≤ C x 2δ , ∀ 0 ≤ γ < d/2, δ > γ . Proof. We have −γ
( f |(−)
f) = C
f (x) f (y) dxdy, ∀ 0 < γ < n/2. |x − y|d−2γ
By the Sobolev inequality ([RS2, Eq. IX.19]): f (x) f (y) dxdy ≤ C f r2 , |x − y|d−2γ for r = 2d/(d + 2γ ). We write then f = x −α x α f and use Hölder inequality to get: f r ≤ x −α p x α f q ,
p −1 + q −1 = r −1 .
We choose q = 2, p = d/γ . The function x −α belongs to L d/γ if α > γ . This implies the lemma. Proof of Thm. A3. We first recall the formula: +∞ 1 λ−1−ν = e−tλ t ν dt, ν > −1. (ν + 1) 0
(A.16)
In the estimates below, various quantities like ( f |h −δ f ) appear. To avoid domain questions, it suffices to replace h by h + m, m > 0, obtaining estimates uniform in m and letting m → 0 at the end of the proof. We will hence prove the bounds ( f |(h + m)−β f ) ≤ C( f | x 2β+ f ), ∀ f ∈ C0∞ (Rd ),
(A.17)
Infrared Problem for the Nelson Model on Static Space-Times
565
uniformly in m > 0. Moreover we note that it suffices to prove (A.17) for f ≥ 0. In fact it follows from (A.16) that (h + m)−β has a positive kernel. Therefore ( f |(h + m)−β f ) ≤ (| f ||(h + m)β | f |) ≤ C(| f || x 2β+ | f |) = C( f | x 2β+ f ), and (A.17) extends to all f ∈ C0∞ (Rd ). We will use the bound (A.3) in Thm. A1, noting that if (A.3) holds for some α0 > 0 it holds also for all 0 < α ≤ α0 . We use the inequality
y 2
y 2
x 2 ≤ ,
x 2 + t
y 2 + t t and get for f ∈ C0∞ (Rd ), f ≥ 0: h −β f (x) = c
+∞
0 ≤C
t β−1 e−th f (x)dt
+∞
t β−α−1 (ect x 2α ) f (x)dt
0
= C (−)β−α x 2α f (x), as long as β > α, using again (A.16). Integrating this point-wise inequality, we get that ( f |h −2β f ) ≤ C( f | x 2α (−)−2(β−α) x 2α f ). We can apply Lemma 8 as long as 2(β − α) < d/2, and obtain ( f |h −2β f ) ≤ C( f | x 4β+ f ), ∀ > 0, if α < β < α + d/4. Since α can be taken arbitrarily close to 0, this completes the proof of the theorem. References [A]
Ammari, Z.: Asymptotic completeness for a renormalized non-relativistic hamiltonian in quantum field theory: the Nelson model. Math. Phys. Anal. Geom. 3, 217–285 (2000) [AHH] Arai, A., Hirokawa, M., Hiroshima, F.: On the absence of eigenvectors of hamiltonians in a class of massless quantum field models without infrared cutoff. J. Funct. Anal. 168, 470–497 (1999) [BFS] Bach, V., Fröhlich, J., Sigal, I.M.: Quantum electrodynamics of confined non-relativistic particles. Adv. Math. 137, 299–395 (1998) [Ba] Bachelot, A.: The Hawking effect. Ann. Inst. H. Poincaré Phys. Théor. 70, 41–99 (1999) [BHLMS] Betz, V., Hiroshima, F., Lörinczi, J., Minlos, R.A., Spohn, H.: Ground state properties of the Nelson Hamiltonian – a Gibbs measure-based approach. Rev. Math. Phys. 14, 173–198 (2002) [BD] Bruneau, L., Derezi´nski, J.: Pauli-Fierz Hamiltonians defined as quadratic forms. Rep. Math. Phys. 54, 169–199 (2004) [BFK] Brunetti, R., Fredenhagen, K., Köhler, M.: The microlocal spectrum condition and wick polynomials of free fields on curved space-times. Commun. Math. Phys. 180, 633–652 (1996) [D] Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge Tracts in Mathematics 92. Cambridge: Cambridge University Press, 1989 [dB-M] de Bièvre, S., Merkli, M.: The Unruh effect revisited. Class. Quant. Grav. 23, 6525–6542 (2006) [DG1] Derezinski, J., Gérard, C.: Scattering Theory of Classical and Quantum N. Particle Systems. Texts and Monographs in Physics, Berlin-Heidelberg-New York: Springer-Verlag, 1997 [DG2] Derezinski, J., Gérard, C.: Asymptotic completeness in quantum field theory. Massive Pauli-Fierz Hamiltonians. Rev. Math. Phys. 11, 383–450 (1999) [DG3] Derezinski, J., Gérard, C.: Scattering theory of infrared divergent Pauli-Fierz Hamiltonians. Annales Henri Poincaré 5, 523–578 (2004)
566
[FH] [GGM] [G] [GHPS1] [GHPS2] [GHPS3] [GP] [Gr] [GLL] [Ha] [H] [LMS] [MS] [Ne] [P] [PE] [Ra1] [Ra2] [RS1] [RS2] [Sa] [Se] [Si] [Sp] [Un] [Un-W] [Zh]
C. Gérard, F. Hiroshima, A. Panati, A. Suzuki
Fredenhagen, K., Haag, R.: On the derivation of Hawking radiation associated with the formation of a black hole. Commun. Math. Phys. 127, 273–284 (1990) Georgescu, V., Gérard, C., Moeller, J.: Spectral theory of massless Nelson models. Commun. Math. Phys. 249, 29–78 (2004) Gérard, C.: On the existence of ground states for massless Pauli-Fierz Hamiltonians. Ann. Henri Poincaré 1, 443–455 (2000) Gérard, C., Hiroshima, F., Panati, A., Suzuki, A.: Infrared divergence of a scalar quantum field model on a pseudo riemannian manifold. Interdisciplinary Information Sciences 15, 399–421 (2009) Gérard, C., Hiroshima, F., Panati, A., Suzuki, A.: Absence of ground state for the Nelson model on static space-times. http://arxiv.org/abs/1012.2655vI [math-ph], 2010 Gérard, C., Hiroshima, F., Panati, A., Suzuki, A.: Removal of UV cutoff for the Nelson model on static space-times. In preparation Gérard, C., Panati, A.: Spectral and scattering theory for some abstract qft hamiltonians. Rev. Math. Phys. 21, 373–437 (2009) Griesemer, M.: Exponential decay and ionization thresholds in non-relativistic quantum electrodynamics. J. Funct. Anal. 210, 321–340 (2004) Griesemer, M., Lieb, E., Loss, M.: Ground states in non-relativistic quantum electrodynamics. Invent. Math. 145, 557–595 (2001) Hawking, S.W.: Particle creation by black holes. Commun. Math. Phys. 43, 199–220 (1975) Hirokawa, M.: Infrared catastrophe for Nelson’s model, non-existence of ground state and softboson divergence. Publ. RIMS, Kyoto Univ. 42, 897–922 (2006) Lörinczi, J., Minlos, R.A., Spohn, H.: The infrared behavior in Nelson’s model of a quantum particle coupled to a massless scalar field. Ann. Henri Poincaré 3, 1–28 (2002) Milman, P.D., Semenov, Y.A.: Global heat kernel bounds via desingularizing weights. J. Funct. Anal. 212, 373–398 (2004) Nelson, E.: Interaction of non-relativistic particles with a quantized scalar field. J. Math. Phys. 5, 1190–1997 (1964) Panati, A.: Existence and nonexistence of a ground state for the massless Nelson model under binding condition. Rep. Math. Phys. 63, 305–330 (2009) Porper, F.O., Eidel’man, S.D.: Two sided estimates of fundamental solutions of second order parabolic equations and some applications. Russ. Math. Surv. 39, 119–178 (1984) Radzikowski, M.: Micro-local approach to the Hadamard condition in quantum field theory on curved space-time. Commun. Math. Phys. 179, 529–553 (1996) Radzikowski, M.: A local-to-global singularity theorem for quantum field theory on curved space-time. Commun. Math. Phys. 180, 1–22 (1996) Reed, M., Simon, B.: Methods of Modern Mathematical Physics, I: Functional Analysis. New York: Academic Press, 1975 Reed, M., Simon, B.: Methods of Modern Mathematical Physics, II: Fourier Analysis, Selfadjointness. New York: Academic Press, 1975 Sanders, K.: Equivalence of the (generalized) Hadamard and microlocal spectrum condition for (generalized) free fields in curved space-time. Commun. Math. Phys. 295, 485–501 (2010) Semenov, Y.A.: Stability of l p −spectrum of generalized Schrödinger operators and equivalence of green’s functions. IMRN 12, 573–593 (1997) Simon, B.: Functional Integration and Quantum Physics. New York: Academic Press, 1979 Spohn, H.: Ground state of a quantum particle coupled to a scalar boson field. Lett. Math. Phys. 44, 9–16 (1998) Unruh, W.G.: Notes on black hole evaporation. Phys. Rev. D 14, 870–892 (1976) Unruh, W.G, Wald, R.: What happens when an accelerating observer detects a Rindler particle. Phys. Rav. D 29, 1047–1056 (1984) Zhang, Q.S.: Large time behavior of Schroedinger heat kernels and applications. Commun. Math. Phys. 210, 371–398 (2000)
Communicated by I.M. Sigal
