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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