cluster structure of the quantum coxeter toda systemmath.columbia.edu/~schrader/camp.pdf · cluster...
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Example: G = PGL4
Gus Schrader Cluster structure of quantum Coxeter–Toda system
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Example: G = PGL4
Gus Schrader Cluster structure of quantum Coxeter–Toda system
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Example: G = PGL4
Gus Schrader Cluster structure of quantum Coxeter–Toda system
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Example: G = PGL4
Gus Schrader Cluster structure of quantum Coxeter–Toda system
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Example: G = PGL4
Gus Schrader Cluster structure of quantum Coxeter–Toda system
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Example: G = PGL4
Gus Schrader Cluster structure of quantum Coxeter–Toda system
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Example: G = PGL4
Gus Schrader Cluster structure of quantum Coxeter–Toda system
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Example: G = PGL4
Gus Schrader Cluster structure of quantum Coxeter–Toda system
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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
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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
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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).
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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
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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
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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