Cluster structure of quantum Coxeter Toda system

Transcription

1 Cluster structure of the quantum Coxeter Toda system Columbia University June 5, 2018 Slides available at schrader

2 I will talk about some joint work (arxiv: ) with Alexander Shapiro on an application of cluster algebras to quantum integrable systems.

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 Modular functor conjecture As Sasha explained, Fock and Goncharov constructed an assignment S {algebra X q G,S, Hilbert space representation V of X q G,S, unitary rep of MCG(S) on V}. Conjecture: (Fock Goncharov 09) This assignment is local with respect to gluing of surfaces.

5 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

6 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.

7 Rank 1 case For G = PGL 2, this amounts to simultaneously diagonalizing the geodesic length operator H = e 2πb ˆp + e 2πb ˆp + e 2πbˆx (b R) and the Dehn twist operator D. Kashaev 00: the operators H, D have common eigenfunctions Ψ λ, λ R >0, with eigenvalues H Ψ λ = 2 cosh(λ)ψ λ, D Ψ λ = e 2πiλ2 Ψ λ.

8 Rank 1 case The eigenfuctions Ψ λ are orthogonal and complete: δ(λ µ) Ψ λ (x)ψ µ (x)dx = m(λ) where the spectral measure is and q = e πib2. R R >0 Ψ λ (x)ψ λ (y)m(λ)dλ = δ(x y), m(λ) = sinh(bλ) sinh(b 1 λ),

9 Higher rank: quantum Coxeter Toda system How to generalize Kashaev s result to higher rank gauge groups? First, we need to identify the operators Ĥ 1,..., Ĥ n. Theorem (S. Shapiro 17) When G = PGL n, there exists a cluster for X q G,S in which the operators Ĥ1,..., Ĥn are identified with the Hamiltonians of the quantum Coxeter Toda integrable system.

10 Classical Coxeter Toda system Fix G = PGL n (C), equipped with a pair of opposite Borel subgroups B ±, torus H, 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 ).

11 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

12 Classical Coxter Toda systems Hoffmann Kellendonk-Kutz-Reshetikhin 00: Suppose u, v are both Coxeter elements in W (each simple reflection appears exactly once in their reduced decompositions) e.g. G = PGL 4, u = s 1 s 2 s 3, v = s 1 s 2 s 3. In particular, Coxeter elements have length l(u) = dim(h) = l(v)

13 Classical Coxeter Toda systems In this Coxeter case, dim(g u,v /Ad H ) = l(u) + l(v) = 2dim(H). And we have dim(h) many independent generators of our Poisson commutative subalgebra C[G] G (one for each fundamental weight) = we get an integrable system on G u,v /Ad H, called the Coxter Toda system.

14 Double Bruhat cells Berenstein Fomin Zelevinsky: the double Bruhat cells G u,v, and their quotients G u,v /Ad H are cluster Poisson 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

15 Cluster Poisson structure on G The Poisson bracket on C[G u,v ] is compatible with the cluster structure: if Y 1,..., Y d are cluster coordinates in chart labelled by quiver Q, {Y j, Y k } = ɛ jk Y j Y k, where ɛ jk = #(j k) #(k j) is the signed adjacency matrix of the quiver.

16 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

17 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 positive operator e 2πb(x 2 x 1 ).

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

19 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.

20 Example: G = PGL 4

21 Example: G = PGL 4

22 Example: G = PGL 4

23 Example: G = PGL 4

24 Example: G = PGL 4

25 Example: G = PGL 4

26 Example: G = PGL 4

27 Example: G = PGL 4

28 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

29 Quantum Coxeter Toda eigenfunctions Kharchev Lebedev Semenov-Tian-Shansky 02: recursive construction of common eigenfunctions for gl n Hamiltonians H 1,..., H n via Mellin Barnes integrals: Ψ gl n+1 λ 1,...,λ n+1 (x 1,..., x n+1 ) = R n K(λ, γ x n+1 )Ψ gl n γ 1,...,γ n (x 1,..., x n )dγ. The b Whittaker functions Ψ gl n λ 1,...,λ n satisfy H k Ψ gl n λ 1,...,λ n = e k (λ)ψ gl n λ 1,...,λ n, where e k (λ) is the k th elementary symmetric function in e 2πbλ 1,..., e 2πbλn.

30 Cluster construction of quantum Toda eigenfunctions Using the cluster realization of the Toda Hamiltonians, we give a dual recursive construction of the b Whittaker functions: we expand Ψ gl n+1 λ 1,...,λ n+1 (x 1,..., x n+1 ) = R n L(x, y λ n+1 )Ψ gl n λ 1,...,λ n (y 1,..., y n )dy. Using this construction, we can compute the eigenvalues of the Dehn twist D n ; and establish orthogonality and completeness of the b Whittaker functions.

31 Construction of the eigenfunctions Problem: Construct complete set of joint eigenfunctions 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))

32 Recursive construction of the eigenfunctions Recursive construction: we want to find an operator R n+1 n (λ) such that R n+1 n (λ n+1 ) Ψ gl n λ 1,...,λ n = Ψ gl n+1 λ 1,...,λ n+1 Set where λ = i(b+b 1 ) 2 λ. Pengaton identity = 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).

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

34 Dehn twist eigenvalue Similarly, the pentagon identity implies D n+1 R n+1 n (λ) = e πiλ2 R n+1 n (λ)d n, so by the recursion we derive the Dehn twist spectrum D n Ψ λ1,...,λ n = e πi(λ λ2 n ) Ψ λ1,...,λ n.

35 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 ),

36 Orthogonality and completeness Using the cluster recursive construction of Ψ gl n, we can prove the orthogonality and completeness relations R n Ψ gl n λ (x)ψgl n µ (x)dx = δ(λ µ), m(λ) R n Ψ gl n λ (x)ψgl n µ (y)m(λ)dλ = δ(x y), with spectral measure m(λ) = j<k sinh(πb(λ j λ k )) sinh(πb 1 (λ j λ k )).

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