Cluster structure of quantum Coxeter Toda system

Transcription

1 Cluster structure of the quantum Coxeter Toda system Columbia University CAMP, Michigan State University May 10, 2018 Slides available at schrader

2 Motivation I will talk about some joint work with Alexander Shapiro on an application of cluster algebras to quantum integrable systems. Motivation: theory. a conjecture from quantum higher Teichmüller

3 Setup for higher Teichmüller theory S = a marked surface G = PGL n (C) X G,S := moduli of framed G local systems on S

4 X G,S is a cluster Poisson variety Fock Goncharov: X G,S is a cluster Poisson variety: it s covered up to codimension 2 by an atlas of toric charts T Σ : (C ) d X G, Ŝ, labelled by quivers Q Σ. The Poisson brackets are determined by the adjacency matrix ɛ jk of Q Σ : {Y j, Y k } = ɛ kj Y j Y k.

5 Canonical quantization of cluster varieties Let q = e πib2, b 2 R >0 \ Q. Promote each cluster chart to a quantum torus algebra T q Σ Ŷ1 =,..., Ŷ d /{Ŷ j Ŷ k = q 2ɛ kj Ŷ k Ŷ j }.

6 Positivity for quantum cluster varieties Embed each quantum cluster chart T q into a Heisenberg algebra H generated by ŷ 1,..., ŷ d with relations by the homomorphism [ŷ j, ŷ k ] = i 2π ɛ jk, Ŷ j e 2πbŷ j. H has irreducible Hilbert space representations in which the generators Ŷj act by positive self-adjoint operators.

7 Representations of X q G,S The quantum cluster algebra X q G,S has positive representations V λ [S] labelled by all possible real eigenvalues λ of monodromies around the punctures of S.

8 Quantum mutation The non compact quantum dilogarithm function ϕ(z) is the unique solution of the pair of difference equations ϕ(z ib ±1 /2) = (1 + e 2πb±1z )ϕ(z + ib ±1 /2)

9 Quantum mutation Since z R = ϕ(z) = 1, and each ŷ k is self adjoint we have unitary operators ϕ(ŷ k ) quantum mutation in direction k

10 Representation of cluster modular group The quantum dilogarithm satisfies the pentagon identity: [ˆp, ˆx] = 1 2πi = ϕ( ˆp)ϕ(ˆx) = ϕ(ˆx)ϕ( ˆp + ˆx)ϕ( ˆp). So we get a unitary representation of the cluster modular group(oid). For X G,S, this group contains the mapping class group of the surface (realized by flips of ideal triangulation)

11 Modular functor conjecture So we have an assignment: marked surface S {algebra X q G,S, representation of X q G,S on V, unitary action of MCG(S) on V}. Conjecture: (Fock Goncharov 09) This assignment is functorial under gluing of surfaces.

12 Modular functor conjecture i.e. if S = S 1 γ S 2, there should be a mapping class group equivariant isomorphism V [S] ν V ν [S 1 ] V ν [S 2 ] ν = eigenvalues of monodromy around loop γ

13 Relation with quantum Toda system So we need to understand the spectral theory of the operators Ĥ 1,..., Ĥ n quantizing the functions on X G,S sending a local system to its eigenvalues around γ. If we can find a complete set of eigenfunctions for Ĥ 1,..., Ĥ n that are simultaneous eigenfunctions for the Dehn twist around γ, we can construct the isomorphism and prove the conjecture.

14 Relation with quantum Toda system Theorem (S. Shapiro 17) There exists a cluster for X G,S in which the operators Ĥ 1,..., Ĥ n are identified with the Hamiltonians of the quantum Coxeter Toda integrable system.

15 Classical Coxeter Toda system Fix G = PGL n (C), equipped with a pair of opposite Borel subgroups B ±, torus H = B + B, and Weyl group W S n. We have Bruhat cell decompositions G = B + ub + u W = v W B vb. The double Bruhat cell corresponding to a pair (u, v) W W is G u,v := (B + ub + ) (B vb ).

16 Double Bruhat cells Berenstein Fomin Zelevinsky: the double Bruhat cells G u,v, and their quotients G u,v /Ad H are a cluster varieties. e.g. is G = PGL 4, u = s 1 s 2 s 3, v = s 1 s 2 s 3. The quiver for G u,v /Ad H

17 Poisson structure on G There is a natural Poisson structure on G with the following key properties: G u,v G are all Poisson subvarieties, whose Poisson structure descends to quotient G u,v /Ad H. The algebra of conjugation invariant functions C[G] Ad G (generated by traces of finite dimensional representations of G) is Poisson commutative: f 1, f 2 C[G] Ad G = {f 1, f 2 } = 0

18 Cluster Poisson structure on G Gekhtman Shapiro Vainshtein: the Poisson bracket on C[G u,v ] is compatible with the cluster structure: if Y 1,..., Y d are cluster X coordinates, {Y j, Y k } = ɛ jk Y j Y k,

19 Classical Coxter Toda system Now suppose u, v are both Coxeter elements in W (each simple reflection appears exactly once in reduced decomposition) e.g. G = PGL 4, u = s 1 s 2 s 3, v = s 1 s 2 s 3. Then dim(g u,v /Ad H ) = l(u) + l(v) dim(h) = 2dim(H), and we have dim(h) many independent generators of our Poisson commutative subalgebra C[G] G = we get an integrable system on G u,v /Ad H, called the Coxter Toda system.

20 Quantization of Coxeter Toda system Consider the Heisenberg algebra H n generated by x 1,..., x n ; p 1,..., p n, acting on L 2 (R n ), via [p j, x k ] = δ jk 2πi p j 1 2πi x j

21 Quantization of Coxeter Toda system A representation of the quantum torus algebra for the Coxeter Toda quiver: e.g. Ŷ 2 acts by multiplication by e 2πb(x 2 x 1 ).

22 Construction of quantum Hamiltonians Let s add an extra node to our quiver, along with a spectral parameter u:

23 Construction of quantum Hamiltonians Theorem (S. Shapiro) Consider the operator Q n (u) obtained by mutating consecutively at 0, 1, 2,..., 2n. Then 1 The quiver obtained after these mutations is isomorphic to the original; 2 The unitary operators Q n (u) satisfy [Q n (u), Q n (v)] = 0, 3 If A(u) = Q n (u ib/2)q n (u + ib/2) 1, then one can expand n A(u) = H k U k, k=0 U := e 2πbu and the commuting operators H 1,..., H n quantize the Coxeter Toda Hamiltonians.

24 Example: G = PGL 4

25 Example: G = PGL 4

26 Example: G = PGL 4

27 Example: G = PGL 4

28 Example: G = PGL 4

29 Example: G = PGL 4

30 Example: G = PGL 4

31 Example: G = PGL 4

32 The Dehn twist The Dehn twist operator D n from quantum Teichmüller theory corresponds to mutation at all even vertices 2, 4,..., 2n: We have [D n, Q n (u)] = 0

33 Construction of the eigenfunctions Problem: Construct complete set of joint eigenfunctions (i.e. b Whittaker functions) for operators Q n (u), D n. e.g. n = 1. Q 1 (u) = ϕ(p 1 + u) If λ R, Ψ λ (x 1 ) = e 2πiλx 1 satisfies Q 1 (u)ψ λ (x 1 ) = ϕ(λ + u)ψ λ (x 1 ). (equivalent to formula for Fourier transform of ϕ(z))

34 Recursive construction of the eigenfunctions Recursive construction: where λ = i(b+b 1 ) 2 λ. Pengaton identity = let R n+1 n (λ) = Q n (λ e 2πiλx n+1 ) ϕ(x n+1 x n ), Q n+1 (u) Rn n+1 (λ) = ϕ(u + λ) R n+1 n (λ) Q n (u).

35 Recursive construction of the eigenfunctions So given a gl n eigenvector Ψ λ1,...,λ n (x 1,..., x n ) satisfying Q n (u)ψ λ1,...,λ n = n ϕ(u + λ k ) Ψ λ1,...,λ n, k=1 we can build a gl n+1 eigenvector satisfying Ψ λ1,...,λ n+1 (x 1,..., x n+1 ) := R n+1 n (λ n+1 ) Ψ λ1,...,λ n Q n+1 (u)ψ λ1,...,λ n+1 = n+1 k=1 ϕ(u + λ k ) Ψ λ1,...,λ n+1.

36 A modular b analog of Givental s integral formula Writing all the Rn n+1 (λ) as integral operators, we get an explicit Givental type integral formula for the eigenfunctions: Ψ (n) λ n 1 ( (x) = e2πiλnx j=1 j k=1 e 2πit j (λ j λ j+1 ) j ϕ(t j,k t j,k 1 ) k=2 dt j,k ϕ(t j,k t j+1,k c b )ϕ(t j+1,k+1 t j,k ) where t n,1 = x 1,..., t n,n = x n. e.g. n = 4 we integrate over all but the last row of the array t 11 t 21 t 22 t 31 t 32 t 33 x 1 x 2 x 3 x 4 ),

37 Unitarity of the b Whittaker transform Using the cluster construction of the b Whittaker functions, we can prove Theorem (S. Shapiro) The b Whittaker transform (W[f ])(λ) = is a unitary equivalence. Ψ (n) λ R n (x)f (x)dx This completes the proof of the Fock Goncharov conjecture for G = PGL n.