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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.
Gus Schrader Quantum total positivity
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 .
Gus Schrader Quantum total positivity
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.
Gus Schrader Quantum total positivity
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)?
Gus Schrader Quantum total positivity
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.
Gus Schrader Quantum total positivity
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.
Gus Schrader Quantum total positivity
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.
Gus Schrader Quantum total positivity
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.
Gus Schrader 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
Gus Schrader Quantum total positivity
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.
Gus Schrader Quantum total positivity
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
Gus Schrader Quantum total positivity
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.
Gus Schrader Quantum total positivity
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.
Gus Schrader Quantum total positivity
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.
Gus Schrader Quantum total positivity
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.
Gus Schrader Quantum total positivity
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.
Gus Schrader Quantum total positivity
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.
Gus Schrader Quantum total positivity
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).
Gus Schrader Quantum total positivity
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.
Gus Schrader Quantum total positivity
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.
Gus Schrader Quantum total positivity
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.
Gus Schrader Quantum total positivity
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).
Gus Schrader Quantum total positivity
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.
Gus Schrader Quantum total positivity
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).
Gus Schrader Quantum total positivity
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.
Gus Schrader Quantum total positivity
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).
Gus Schrader Quantum total positivity
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.
Gus Schrader Quantum total positivity
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.
Gus Schrader Quantum total positivity
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 .
Gus Schrader Quantum total positivity
Geometric approach to Frenkel and Ip’s conjecture
First observation: central character of Pλ is determined byeigenvalues of holonomy around the puncture.
Gus Schrader Quantum total positivity
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.
Gus Schrader Quantum total positivity
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.
Gus Schrader Quantum total positivity
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).
Gus Schrader Quantum total positivity
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.
Gus Schrader Quantum total positivity
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.
Gus Schrader Quantum total positivity
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.
Gus Schrader Quantum total positivity
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.
Gus Schrader Quantum total positivity
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