Cluster structure of the quantum Coxeter–Todasystem

Gus Schrader

Columbia University

CAMP, Michigan State University

May 10, 2018

Slides available atwww.math.columbia.edu/∼ schrader

Gus Schrader Cluster structure of quantum Coxeter–Toda system

Motivation

I will talk about some joint work with Alexander Shapiro on anapplication of cluster algebras to quantum integrable systems.

Motivation: a conjecture from quantum higher Teichmullertheory.

Gus Schrader Cluster structure of quantum Coxeter–Toda system

Setup for higher Teichmuller theory

S = a marked surface

G = PGLn(C)

XG ,S := moduli of framed G–local systems on S

Gus Schrader Cluster structure of quantum Coxeter–Toda system

XG ,S is a cluster Poisson variety

Fock–Goncharov: XG ,S is a cluster Poisson variety: it’scovered up to codimension 2 by an atlas of toric charts

TΣ : (C∗)d −→ XG ,S

,

labelled by quivers QΣ. The Poisson brackets are determined bythe adjacency matrix εjk of QΣ:

{Yj ,Yk} = εkjYjYk .

Gus Schrader Cluster structure of quantum Coxeter–Toda system

Canonical quantization of cluster varieties

Letq = eπib

2, b2 ∈ R>0 \Q.

Promote each cluster chart to a quantum torus algebra

T qΣ =

⟨Y1, . . . , Yd

⟩/{Yj Yk = q2εkj Yk Yj}.

Gus Schrader Cluster structure of quantum Coxeter–Toda system

Positivity for quantum cluster varieties

Embed each quantum cluster chart T q into a Heisenberg algebraH generated by y1, . . . , yd with relations

[yj , yk ] =i

2πεjk ,

by the homomorphism

Yj 7→ e2πbyj .

H has irreducible Hilbert space representations in which thegenerators Yj act by positive self-adjoint operators.

Gus Schrader Cluster structure of quantum Coxeter–Toda system

Representations of X qG ,S

The quantum cluster algebra X qG ,S has positive representations

Vλ[S ]

labelled by all possible real eigenvalues λ of monodromies aroundthe punctures of S .

Gus Schrader Cluster structure of quantum Coxeter–Toda system

Quantum mutation

The non–compact quantum dilogarithm function ϕ(z) is theunique solution of the pair of difference equations

ϕ(z − ib±1/2) = (1 + e2πb±1z)ϕ(z + ib±1/2)

Gus Schrader Cluster structure of quantum Coxeter–Toda system

Quantum mutation

Sincez ∈ R =⇒ |ϕ(z)| = 1,

and each yk is self–adjoint we have unitary operators

ϕ(yk)↔ quantum mutation in direction k

Gus Schrader Cluster structure of quantum Coxeter–Toda system

Representation of cluster modular group

The quantum dilogarithm satisfies the pentagon identity:

[p, x ] =1

2πi

=⇒ ϕ(p)ϕ(x) = ϕ(x)ϕ(p + x)ϕ(p).

So we get a unitary representation of the cluster modulargroup(oid).

For XG ,S , this group contains the mapping class group of thesurface (realized by flips of ideal triangulation)

Gus Schrader Cluster structure of quantum Coxeter–Toda system

Modular functor conjecture

So we have an assignment:

marked surface S 7−→{algebra X qG ,S , representation of X q

G ,S on V ,

unitary action of MCG(S) on V}.

Conjecture: (Fock–Goncharov ‘09) This assignment is functorialunder gluing of surfaces.

Gus Schrader Cluster structure of quantum Coxeter–Toda system

Modular functor conjecture

i.e. if S = S1 ∪γ S2, there should be a mapping–class–groupequivariant isomorphism

V [S ] '∫ ⊕ν

Vν [S1]⊗ Vν [S2]

ν = eigenvalues of monodromy around loop γ

Gus Schrader Cluster structure of quantum Coxeter–Toda system

Relation with quantum Toda system

So we need to understand the spectral theory of the operatorsH1, . . . , Hn quantizing the functions on XG ,S sending a localsystem to its eigenvalues around γ.

If we can find a complete set of eigenfunctions for H1, . . . , Hn

that are simultaneous eigenfunctions for the Dehn twist around γ,we can construct the isomorphism and prove the conjecture.

Gus Schrader Cluster structure of quantum Coxeter–Toda system

Relation with quantum Toda system

Theorem (S.–Shapiro ’17)

There exists a cluster for XG ,S in which the operators H1, . . . , Hn

are identified with the Hamiltonians of the quantumCoxeter–Toda integrable system.

Gus Schrader Cluster structure of quantum Coxeter–Toda system

Classical Coxeter–Toda system

Fix G = PGLn(C), equipped with a pair of opposite Borelsubgroups B±, torus H = B+ ∩ B−, and Weyl group W ' Sn.

We have Bruhat cell decompositions

G =⊔u∈W

B+uB+

=⊔v∈W

B−vB−.

The double Bruhat cell corresponding to a pair (u, v) ∈W ×Wis

Gu,v := (B+uB+) ∩ (B−vB−) .

Gus Schrader Cluster structure of quantum Coxeter–Toda system

Double Bruhat cells

Berenstein–Fomin–Zelevinsky: the double Bruhat cells Gu,v ,and their quotients Gu,v/AdH are a cluster varieties.

e.g. G = PGL4, u = s1s2s3, v = s1s2s3. The quiver for Gu,v/AdHis

Gus Schrader Cluster structure of quantum Coxeter–Toda system

Poisson structure on G

There is a natural Poisson structure on G with the following keyproperties:

Gu,v ⊂ G are all Poisson subvarieties, whose Poissonstructure descends to quotient Gu,v/AdH .

The algebra of conjugation invariant functions C[G ]AdG

(generated by traces of finite dimensional representations ofG ) is Poisson commutative:

f1, f2 ∈ C[G ]AdG =⇒ {f1, f2} = 0

Gus Schrader Cluster structure of quantum Coxeter–Toda system

Cluster Poisson structure on G

Gekhtman–Shapiro–Vainshtein: the Poisson bracket onC[Gu,v ] is compatible with the cluster structure: ifY1, . . . ,Yd are cluster X–coordinates,

{Yj ,Yk} = εjkYjYk ,

Gus Schrader Cluster structure of quantum Coxeter–Toda system

Classical Coxter–Toda system

Now suppose u, v are both Coxeter elements in W (each simplereflection appears exactly once in reduced decomposition)e.g. G = PGL4, u = s1s2s3, v = s1s2s3.

Then

dim(Gu,v/AdH) = l(u) + l(v)− dim(H)

= 2dim(H),

and we have dim(H)–many independent generators of our Poissoncommutative subalgebra C[G ]G

=⇒ we get an integrable system on Gu,v/AdH , called theCoxter–Toda system.

Gus Schrader Cluster structure of quantum Coxeter–Toda system

Quantization of Coxeter–Toda system

Consider the Heisenberg algebra Hn generated by

x1, . . . , xn; p1, . . . , pn,

[pj , xk ] =δjk2πi

acting on L2(Rn), via

pj 7→1

2πi

∂

∂xj

Gus Schrader Cluster structure of quantum Coxeter–Toda system

Quantization of Coxeter–Toda system

A representation of the quantum torus algebra for theCoxeter–Toda quiver:

e.g. Y2 acts by multiplication by e2πb(x2−x1).

Gus Schrader Cluster structure of quantum Coxeter–Toda system

Construction of quantum Hamiltonians

Let’s add an extra node to our quiver, along with a “spectralparameter” u:

Gus Schrader Cluster structure of quantum Coxeter–Toda system

Construction of quantum Hamiltonians

Theorem (S.–Shapiro)

Consider the operator Qn(u) obtained by mutating consecutively at0, 1, 2, . . . , 2n. Then

1 The quiver obtained after these mutations is isomorphic to theoriginal;

2 The unitary operators Qn(u) satisfy

[Qn(u),Qn(v)] = 0,

3 If A(u) = Qn(u − ib/2)Qn(u + ib/2)−1, then one can expand

A(u) =n∑

k=0

HkUk , U := e2πbu

and the commuting operators H1, . . . ,Hn quantize theCoxeter–Toda Hamiltonians.

Gus Schrader Cluster structure of quantum Coxeter–Toda system

Example: G = PGL4

Gus Schrader Cluster structure of quantum Coxeter–Toda system

Example: G = PGL4

Gus Schrader Cluster structure of quantum Coxeter–Toda system

Example: G = PGL4

Gus Schrader Cluster structure of quantum Coxeter–Toda system

Example: G = PGL4

Gus Schrader Cluster structure of quantum Coxeter–Toda system

Example: G = PGL4

Gus Schrader Cluster structure of quantum Coxeter–Toda system

Example: G = PGL4

Gus Schrader Cluster structure of quantum Coxeter–Toda system

Example: G = PGL4

Gus Schrader Cluster structure of quantum Coxeter–Toda system

Example: G = PGL4

Gus Schrader Cluster structure of quantum Coxeter–Toda system

The Dehn twist

The Dehn twist operator Dn from quantum Teichmuller theorycorresponds to mutation at all even vertices 2, 4, . . . , 2n:

We have[Dn,Qn(u)] = 0

Gus Schrader Cluster structure of quantum Coxeter–Toda system

Construction of the eigenfunctions

Problem: Construct complete set of joint eigenfunctions (i.e.b–Whittaker functions) for operators Qn(u),Dn.

e.g. n = 1.Q1(u) = ϕ(p1 + u)

If λ ∈ R,Ψλ(x1) = e2πiλx1

satisfiesQ1(u)Ψλ(x1) = ϕ(λ+ u)Ψλ(x1).

(equivalent to formula for Fourier transform of ϕ(z))

Gus Schrader Cluster structure of quantum Coxeter–Toda system

Recursive construction of the eigenfunctions

Recursive construction: let

Rn+1n (λ) = Qn (λ∗)

e2πiλxn+1

ϕ(xn+1 − xn),

where λ∗ = i(b+b−1)2 − λ.

Pengaton identity =⇒

Qn+1(u) Rn+1n (λ) = ϕ(u + λ) Rn+1

n (λ) Qn(u).

Gus Schrader Cluster structure of quantum Coxeter–Toda system

Recursive construction of the eigenfunctions

So given a gln eigenvector Ψλ1,...,λn(x1, . . . , xn) satisfying

Qn(u)Ψλ1,...,λn =n∏

k=1

ϕ(u + λk) Ψλ1,...,λn ,

we can build a gln+1 eigenvector

Ψλ1,...,λn+1(x1, . . . , xn+1) := Rn+1n (λn+1) ·Ψλ1,...,λn

satisfying

Qn+1(u)Ψλ1,...,λn+1 =n+1∏k=1

ϕ(u + λk) Ψλ1,...,λn+1 .

Gus Schrader Cluster structure of quantum Coxeter–Toda system

A modular b–analog of Givental’s integral formula

Writing all the Rn+1n (λ) as integral operators, we get an explicit

Givental–type integral formula for the eigenfunctions:

Ψ(n)λ (x) = e2πiλnx

∫ n−1∏j=1

(e2πi t j (λj−λj+1)

j∏k=2

ϕ(tj ,k − tj ,k−1)

j∏k=1

dtj ,kϕ(tj ,k − tj+1,k − cb)ϕ(tj+1,k+1 − tj ,k)

),

where tn,1 = x1, . . . , tn,n = xn.e.g. n = 4 we integrate over all but the last row of the array

t11

t21 t22

t31 t32 t33

x1 x2 x3 x4

Gus Schrader Cluster structure of quantum Coxeter–Toda system

Unitarity of the b–Whittaker transform

Using the cluster construction of the b–Whittaker functions, wecan prove

Theorem (S.–Shapiro)

The b–Whittaker transform

(W[f ])(λ) =

∫Rn

Ψ(n)λ (x)f (x)dx

is a unitary equivalence.

This completes the proof of the Fock–Goncharov conjecture forG = PGLn.

Gus Schrader Cluster structure of quantum Coxeter–Toda system