Quantum total positivity and continuous tensor categoriesmath.columbia.edu/~schrader/zmp.pdf · Continuous tensor categories in higher rank? Conjecture (Frenkel and Ip, ’11) : The

Quantum total positivity and continuous tensorcategories

Gus Schrader

Columbia University

ZMP Colloquium, DESY Hamburg

November 16, 2017

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

Gus Schrader Quantum total positivity

The plan

I am going to talk about some recent joint work with AlexanderShapiro (University of Toronto)

Our work involves the application of theory of quantum clusteralgebras and quantum total positivity to some problems inrepresentation theory and topological field theory.

Let’s begin by describing the kind of problems we want to solveusing these techniques.


Quantum group Uq(g)

Let g be any Lie algebra.

Its universal enveloping algebra U(g) is a Hopf algebra; i.e. anassociative algebra with a co-associative algebra map

∆ : U(g)→ U(g)⊗U(g).

called the coproduct.

The Hopf algebra U(g) is co-commutative;

∆op := Flip ∆

= ∆.

So modules for U(g) form a symmetric tensor category: we have

Flip : V ⊗W 'W ⊗ V .


Quantum group Uq(g)

Now suppose g is a simple Lie algebra over C.

The quantum group Uq(g) is a Hopf algebra deformation of theenveloping algebra U(g).

Coproduct in Uq(g) is no longer co-commutative; instead Uq(g) isquasi-triangular : it has an R-matrix R ∈ Uq(g)⊗Uq(g) satisfying

∆op = AdR ∆,

R12R13R23 = R23R13R12 ∈ Uq, (g)⊗3

where ∆op is the opposite coproduct.

This means Uq(g) can be used to construct interestingbraided tensor categories.


Jones-Reshetikhin-Turaev-Witten construction

One interesting class of such categories comes from finitedimensional representations of Uq(g) when q is a root of unity.

Reshetikin and Turaev used this category to define a 3d-TQFT,which yields invariants of 3-manifolds and framed links. Wheng = sl2, these recover the Jones polynomial.

Witten: construct these invariants from geometric quantization ofChern-Simons theory with compact gauge group K . If K = SU2,again obtain Jones polynomials.

Question: Is there an analog of these relations in the case of anon-compact split real gauge group, e.g. G = SLn(R)?


Principal series for Uq(sl2)

Let ~ ∈ R>0 \Q, and set

q = eπi~2.

Ponsot and Teschner, ’99: The split real quantum group Uq(sl2)has a principal series of irreducible ∗-representations

Ps ' L2(R), s ∈ R≥0,

with the following properties:

the Chevalley generators E ,F ,K of the quantum group act onPs by positive essentially self-adjoint operators;

Ps is a bimodule for the quantum group Uq(sl2) and itsmodular dual Uq∨(sl2), where q, q∨ are related by themodular S-duality:

q∨ = eπi/~2.


A continuous tensor category

Most importantly, Ponsot and Teschner showed that the principalseries of Uq(sl2) form a “continuous tensor category”: one has

Ps1 ⊗ Ps2 =

∫ ⊕R≥0

Psdµ(s).

The measure on the Weyl chamber R≥0 is

dµ(s) = 4 sinh (2π~s) sinh(2π~−1s

)ds.


Higher rank principal series

Frenkel and Ip, ’11 : For g of any finite Dyknin type, the splitreal quantum group Uq(g,R) has a family of irreducible principalseries representations Pλ labelled by points of a Weyl chamberλ ∈ C+.

As in the rank 1 case:

the Chevalley generators of the quantum group act on Pλ bypositive essentially self-adjoint operators;

Pλ is a bimodule for the quantum group Uq(g) and itsLanglands dual Uq∨(Lg), where q, q∨ are related by themodular S-duality:

q∨ = eπi/~2.


Continuous tensor categories in higher rank?

Conjecture (Frenkel and Ip, ’11) : The principal principalseries representations Pλ of Uq(g) are closed under tensor product,and thus form a continuous tensor category.

In recent joint work with Alexander Shapiro, we have proved thisconjecture for g = sln+1.

Our proof is based on a geometric reformulation of the conjecturethat allows us to study the principal series using the tools ofquantum cluster algebras and quantum total positivity.


Moduli spaces of framed local systems

A marked surface S is a compact oriented surface S with a finiteset x1, . . . , xk ⊂ ∂S of marked points on the boundary.

Its punctured boundary is ∂S := ∂S \ x1, . . . , xk.

Let G be a complex semisimple Lie group of adjoint type, andB ⊂ G a Borel subgroup. A framed G -local system on S is:

1 a G -local system L on S , together with

2 a flat section of the restriction of the associated flag bundle(L ×G G/B) |

∂S.

Definition

XG ,S

:= moduli of framed G -local systems on S


Concrete description of framing data

More concretely: The punctured boundary ∂S is a union ofintervals and circles. The framing data consists of

For each interval component of ∂S , a choice of flag F ∈ G/B;

For each S1 component of ∂S , a choice of flag F ∈ G/B suchthat F is preserved by the monodromy around S1.


Example

Let S be a punctured disk with 2 marked points. Then

XG ,S

= (g ,F1,F2,F ) | gF = F /G

whereg ∈ G , F ,F1,F2 ∈ G/B.

F gF1 F2


XG ,S is a cluster variety

Fock-Goncharov: For G = PGLn(C), XG ,S

is a cluster Poissonvariety: it is covered 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:

Xj ,Xk = εkjXjXk .

Different charts are related by subtraction-free birationaltransformations called cluster mutations.


Clusters from ideal triangulations

Shrink closed components of ∂S to punctures. A triangulation ofS is ideal if its vertices are at punctures or marked points.

Each ideal triangulation provides a cluster chart. Fill each trianglewith the following quiver (here G = PGL4):

2 5 9

1 4 8 12

3 7 11

6 10

When two triangles share an edge, we “amalgamate” them, i.e.identify the quiver vertices lying on that edge, and replace thecorresponding pairs of cluster variables by their products.


Cluster mutations

Mutation at a vertex Xi proceeds in 3 steps:

1 reverse all incident edges;

2 for each pair of edges j → i and i → k create an edge k → j ;

3 delete pairs of opposite edges;

For example, for G = PGL2:

1 2

3

4 5

µ3

1 2

3

4 5

Mutations for G = PGL2 correspond to flips of triangulation. ForG = PGLn, one flip is realized by a sequence of

(n+13

)mutations.


Total positivity

Let X be any cluster variety.

Since the gluing maps between cluster charts are subtraction–free ,there is a well–defined notion of the totally positive pointsX+ ⊂ X .

The totally positive locus X+ consists of points in X whose toriccoordinates in every cluster chart are positive.


Higher Teichmuller spaces

So we have a well–defined totally positive subset X+

G ,S⊂ X

G ,S.

If G = PGL2(C), the variety X+

G ,Scan be identified with a

component in the moduli space Mflat(S ,PSL2(R)), isomorphic toa decorated Teichmuller space.

So the positive loci X+

G ,Sare higher rank analogs of Teichmuller

spaces.


Canonical quantization of cluster varieties

Promote each cluster chart to a quantum torus algebra

T qΣ = 〈X1, . . . ,Xd〉 /XjXk = q2εkjXkXj.

They are “glued” by quantum mutations, which are the algebraautomorphisms of conjugation by the quantum dilogarithm Γq(Xk ),where

Γq(X ) =∞∏

n=1

1

1 + q2n+1X.


Quantum mutation

Example: If X2X1 = q2X1X2, then

µ2(X1) = Γq(X2) · X1 · Γq(X2)−1

= X1 · Γq(q2X2) · Γq(X2)−1

= X1(1 + qX2)(1 + q3X2) . . .

(1 + q3X2)(1 + q5X2) . . .

= X1(1 + qX2).


Positivity for quantum cluster varieties?

Now let’s suppose that q = eπi~2, with ~ ∈ R>0 \Q.

We can embed a quantum cluster chart T q into a Heisenbergalgebra H generated by x1, . . . , xd with relations

[xj , xk ] =i

2πεjk ,

by the homomorphism

Xj 7→ e2π~xj .

The algebra H has a family of irreducible Hilbert spacerepresentations Vχ parameterized by central charactersχ ∈ Hom(ker ε,R), in which the generators Xj act by positiveessentially self-adjoint operators.


Modular duality

Now consider q∨ = eπi/~2, obtained from q = eπi~2

by thetransformation ~ 7→ 1/~.

We also have an embedding of T q∨ into the Heisenberg algebra Hgiven by

Xj = e2π~−1xj ,

soXk Xj = (q∨)2εkj Xj Xk .

Note that the generators Xj commute with the original ones

Xj = e2π~xk :Xj Xk = e2πiεjk XkXj = XkXj ,

since εjk ∈ Z.


Quantum total positivity

Now we’ve defined a positive representation of a single quantumcluster chart, we need the quantum analog of ‘subtraction-freegluing maps’.

i.e. We want to promote quantum mutations to unitaryequivalences between the Hilbert spaces for mutation-equivalentquantum torus algebras.

Problem: The formal power series Γq(X ) does not make sensewhen |q| = 1.


Non-compact quantum dilogarithm

Faddeev’s solution: We can replace Γq(X ) by the non-compactquantum dilogarithm

Φ~(z) = exp

(−∫R+i0

etz

(e~t − 1)(e~−1t − 1)

dt

t

).

The function Φ~(z) is well-defined for ~ ∈ R, and satisfies thedifference equations

Φ~(z + i~) = (1 + qe2π~z )Φ~(z),

Φ~(z + i~−1) = (1 + q∨e2π~−1z )Φ~(z).


Non-compact quantum dilogarithm

Therefore cluster mutation in direction k can be realized byconjugation by Φ~(xk ).

Moreover, we have

Φ~(z) =1

Φ~(z).

So since xk is self-adjoint, the operator Φ~(xk ) is a unitaryoperator on Vχ.

Thus the mutated operators µk (Xj ) are also positive self-adjoint,and we get a unitary representation of the groupoid of clustertransformations.


Quantum groups and quantum total positivity

Next: I want to present a theorem that lets us apply quantumtotal positivity to study the principal series for Uq(sln).


Cluster realization of Uq(sln)

Theorem (S.-Shapiro ’16)

Let S be a punctured disk with two marked points, and let X q

PGLn,S

be the corresponding quantum cluster algebra. Then there is anembedding of algebras

Uq(sln) → X q

PGLn,S,

with the property that for each Chevalley generator of thequantum group, there is a cluster in which that generator is acluster monomial.


Example: Uq(sl2)

4

1

2

3

E 7→ X1(1 + qX2), K 7→ q2X1X2X3,

F 7→ X3(1 + qX4), K ′ 7→ q2X3X4X1.

KK ′ is central.

Note thatE = µ2(X1) and F = µ4(X3).


Example: Uq(sl3)

1

2

3

4

5

6

7

8

9

10

E1 7→ X1(1 + qX2), K1 7→ q2X1X2X3,

E2 7→ X4(1 + qX5(1 + qX6(1 + qX7))), K2 7→ q4X4X5X6X7X8,

F1 7→ X3(1 + qX7(1 + qX10(1 + qX5))), K ′1 7→ q4X3X7X10X5X1,

F2 7→ X8(1 + qX9), K ′2 7→ q2X8X9X4.

K1K′1 and K2K

′2 are central.


Example: positive representations of Uq(sl2)

Consider the Hilbert space H = L2(R), and the self-adjoint,unbounded operators

p =1

2πi

∂

∂x, x = x .

Then for all s ∈ R, we have positive self-adjoint operators

X1 = e2π~(p− s2

), X2 = e2π~(x+s)

X3 = e2π~(−p− s2

), X4 = e2π~(−x+s)

satisfying the cyclic quiver relations

q2XkXk+1 = Xk+1Xk , k ∈ Z/4Z,

where q = eπi~2.


Example: positive representations of Uq(sl2)

X4

X1

X2

X3

Chevalley generators E ,F of Uq(sl2) act by positive, self-adjointoperators

E 7→ µ2(X1), F 7→ µ4(X3).

The Uq(sl2) Casimir element Ω acts by

Ω 7→ e2π~s + e−2π~s .


Geometric approach to Frenkel and Ip’s conjecture

First observation: central character of Pλ is determined byeigenvalues of holonomy around the puncture.


Geometric realization of Pλ ⊗ Pµ

Next, we realize Pλ ⊗ Pµ as a positive representation as follows:

Action of the center of ∆Uq(sln) is determined by eigenvalues ofholonomy around red loop.


Cutting and gluing isomorphisms

Teichmuller theory interpretation of the decomposition of thetensor product of positive representations Pλ ⊗ Pµ into positiverepresentations Pν :

Pλ ⊗ Pµ =

∫νPν ⊗Mν

λ,µdν.


Cutting and gluing isomorphisms

On the right hand side: we have a “fiber product” of twoquantum Teichmuller spaces:

the quantum Teichmuller space for a for the punctured diskwith two marked points on its boundary, corresponding to thediagonally acting copy of Uq(sln); and

the quantum Teichmuller space for a thrice-punctured sphere,where the eigenvalues of two monodromies are specified to λand µ respectively.

The fiber product is taken over the spectrum of the unspecifiedmonodromy ν around the loop associated to the remaining thirdpuncture of the sphere, which parameterizes the central charactersof Uq(sln).


Multiplicities and the thrice-punctured sphere

The appearance of the ’multiplicity space’ corresponding to thethrice-punctured sphere is a new feature in higher rank.

Indeed, when g = sl2, the positive representation of the quantumTeichmuller space for the thrice-punctured sphere degenerates to aone-dimensional representation.


Holonomies and quantum integrability

Algebraically: the picture let us read off a natural sequence ofcluster transformations, which identifies the traces of holonomiesthat determine the action of the center of Uq(sln) with theHamiltonians of the q-difference open Toda lattice.

In particular, these operators form a quantum integrable system.


q–Whittaker functions

Next step: The eigenfunctions of the quantum TodaHamiltonians have been determined byKharchev-Lebedev-Semenov-Tian-Shansky: they are theq-Whittaker functions.

So passing to the basis of q–Whittaker functions diagonalizes theaction of the center of Uq(sln) on Pλ ⊗ Pµ.

The orthogonality and completeness relations for the Whittakerfunctions yields the desired the direct integral decomposition

Pλ ⊗ Pµ =

∫νPν ⊗Mν

λ,µdν.


Towards an infinite dimensional modular functor?

This cutting and gluing result is the first step towards constructingan infinite dimensional analog of a modular functor from thequantization of higher Teichmuller spaces.

When g = sl2, the construction of such a modular functorinvestigated by Teschner. On the ‘loop group’ side, closely relatedto modular functor coming from Virasoro conformal blocks.

Suggests connections with 4D gauge theory and higher rank AGT(IR line operators ∼ Fock-Goncharov coordinates)

Will be interesting to complete the construction in higher ranksand see if it can shed any light on the story for W-algebraconformal blocks.


Documents

Quantum total positivity and continuous tensor categoriesmath.columbia.edu/~schrader/zmp.pdf · Continuous tensor categories in higher rank? Conjecture (Frenkel and Ip, ’11) : The