Ivan Ip : Positive representations of split real quantum groups

In this lecture series, I will discuss the historic development of the theory of positive representation of quantum groups in the split real case. I will survey the necessary background on its analytic aspects and cluster algebraic realization, in order to construct (in type An) a new class of braided tensor category (due to Schrader-Shapiro), as well as to prove an analogue of Peter-Weyl theorem (joint work with Schrader-Shapiro). The geometric realization of the positive representations will be explained in the lectures (#3-#4) by Linhui Shen.

The tentative plan of the lectures is as follows

Lecture #1 Motivation and construction of the positive representations of split real quantum groups

Lecture #2 The cluster realization of positive representations and factorization of the universal R operator

Lecture #3-4 Continuous braided tensor category structure and the Peter-Weyl theorem.

References:

I. Frenkel and I. Ip, Positive representations of split real quantum groups and future perspectives, Int. Math. Res. Not. IMRN 2014, no. 8, 2126–2164. (arXiv)

I. Ip, Positive representations of split real simply-laced quantum groups, Publ. Res. Inst. Math. Sci. 56 (2020), no. 3, 603–646. (arXiv)

I. Ip, Cluster realization of Uq(g) and factorizations of the universal R-matrix, Selecta Math. (N.S.) 24 (2018), no. 5, 4461–4553. (arXiv)

B. Ponsot and J. Teschner, Clebsch-Gordan and Racah-Wigner coefficients for a continuous series of representations of Uq(sl(2,R)), Comm. Math. Phys. 224 (2001), no. 3, 613–655. (arXiv)

G. Schrader and A. Shapiro, Continuous tensor categories from quantum groups I: algebraic aspects, arXiv:1708.08107. (arXiv)

G. Schrader and A. Shapiro, On b-Whittaker functions, arXiv:1806.00747. (arXiv)

David Jordan : Decorated character stacks and quantum cluster algebras via factorization homology

In these lectures, I will offer a new construction of decorated character stacks -- moduli spaces of G-local systems on a surface, with B-reductions near punctures and marked points -- in the framework of stratified factorization homology, as recently introduced by Ayala, Francis and Tanaka. I will explain a toolkit called "monadic reconstruction" which allows one to compute concretely, in this framework, and I will detail how quantum cluster algebras studied by Fock--Goncharov and Goncharov--Shen arise naturally as an "open subvariety" of the quantized decorated character stack. Roughly, I will try and discuss:

First lecture: Geometry and the basic definition

1. The formulation of the stack, and how to think about that kind of stack concretely, some examples

2. What is decorated factorization homology, and the theorem that QC(stack) is computed by it.

3. The definition of the quantization involving quantum groups in place of classical groups.

Second lecture: Category theory and representation theory background

1. Stuff about categories: colimits, completeness, functors, adjoints, monads

2. Stuff about tensor categories: module categories, algebra objects, Tannakian and Barr-Beck reconstruction

3. Some easy examples of Barr-Beck reconstruction.

4. Examples in commutative/noncommutative algebraic geometry.

Third and fourth lectures: Construction of cluster charts

1. Explaining how to use the prior machinery to construct cluster charts in the stratified factorization homology categories, with lots of examples

2. Relation to A-varieties, P-varieties, and X-varieties.

References:

D. Jordan, I. Le, G. Schrader, and A. Shapiro, Quantum decorated character stacks, arxiv:2102.12283. (arXiv)

A. Brochier, D. Jordan, and N. Snyder, On dualizability of braided tensor categories, arxiv:1804.07538. (arXiv)

D. Ben-Zvi, A. Brochier, and D. Jordan, Integrating quantum groups over surfaces, J. Topol. 11 (2018), no. 4, 874–917. (arXiv)

A. Amabel, A. Kalmykov, L. Müller, and H.L. Tanaka. Lectures on Factorization Homology, ∞-Categories, and Topological Field Theories, DOI: https://doi.org/10.1007/978-3-030-61163-7. (arXiv)

Hyun Kyu Kim

lecture 1 title : Higher Teichmüller spaces and cluster varieties

abstract : For a surface S and an algebraic group G, Fock and Goncharov defined certain moduli spaces of G-local systems on S. These moduli spaces have special charts, which give structures of cluster varieties, and lead to the notions of higher Teichmüller spaces. I will sketch basic constructions.

lecture 2 title : Quantization problems for cluster varieties

abstract : Given a scheme with a Poisson structure, I will formulate what it means to quantize this scheme. Applying to cluster X-varieties, I will explain the notion of the quantum cluster variety. I will formulate the deformation quantization in terms of algebras and of operators on Hilbert spaces, and introduce relevant problems.

lecture 3 title : Deformation quantization of enhanced Teichmüller spaces

abstract : The enhanced Teichmüller space of a surface is a prominent example of a cluster X-variety. I will recall the regular functions on it, which are classical observables to be quantized. I will explain how these can be quantized using Bonahon-Wong's SL2 quantum trace map. If time allows, I will explain some properties of this quantization, including certain positivity.

lecture 4 title : Deformation quantization of PGL3 higher Teichmüller spaces

abstract : The ring of regular functions on the PGL3 higher Teichmüller space of a surface has a basis parametrized by SL3-laminations, which are SL3 generalization of Fock-Goncharov's laminations and which are based on Kuperberg's webs. I will briefly explain these basic functions, and discuss how to quantize them using SL3 quantum trace map.

References:

V.V. Fock and A.B. Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. No. 103 (2006), 1–211. (arXiv)

V.V. Fock and A.B. Goncharov, The quantum dilogarithm and representations of quantum cluster varieties, Invent. Math. 175 (2009), no. 2, 223–286. (arXiv)

D.G.L. Allegretti and H. Kim, A duality map for quantum cluster varieties from surfaces, Adv. Math. 306 (2017), 1164–1208. (arXiv)

H. Kim, SL3-laminations as bases for PGL3 cluster varieties for surfaces, arXiv:2011.14765. (arXiv)

Alexander Shapiro : K-theoretic Coulomb branches and cluster algebras

Braverman, Finkelberg, and Nakajima has recently proposed a mathematical definition of the Coulomb branch of a 3d N=4 SUSY gauge theory of cotangent type. In their work, the (quantized K-theoretic) Coulomb branch is defined as the equivariant K-theory of a certain moduli space specified by a complex reductive group G and its complex representation N. It was subsequently conjectured by Gaiotto, that such a Coulomb branch carries the structure of a quantum cluster variety. I will outline a proof of this conjecture in the case of quiver gauge theories, i.e. when N is a representation of a quiver, and G is its gauge group. My talks will be based on joint work with Gus Schrader.

The very rough plan for the lectures is as follows.

Lecture 1. Introduction to the Braverman–Finkelberg–Nakajima construction: basic definitions and properties

Lecture 2. Quiver gauge theories: explicit formulas and examples in the quiver case

Lecture 3. Quantum Toda system, its cluster structure, and the outline of the proof of Gaiotto's conjecture for quivers without loops

Lecture 4. DT-transformations, spherical DAHA, and a proof for quivers with loops

References:

A. Braverman, M. Finkelberg, and H. Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional N=4 gauge theories, II, Adv. Theor. Math. Phys. 22 (2018), no. 5, 1071–1147. (arXiv)

A. Braverman, M. Finkelberg, and H. Nakajima, Coulomb branches of 3d N=4 quiver gauge theories and slices in the affine Grassmannian, with two appendices by A. Braverman, M. Finkelberg, J. Kamnitzer, R. Kodera, H. Nakajima, B. Webster, and A. Weekes, Adv. Theor. Math. Phys. 23 (2019), no. 1, 75–166. (arXiv)

M. Finkelberg and A. Tsymbaliuk, Multiplicative slices, relativistic Toda and shifted quantum affine algebras, Representations and nilpotent orbits of Lie algebraic systems, 133–304, Progr. Math., 330, Birkhäuser/Springer, Cham, 2019. (arXiv)

G. Schrader and A. Shapiro, K-theoretic Coulomb branches of quiver gauge theories and cluster varieties, arXiv:1910.03186. (arXiv)

Linhui Shen : Moduli spaces and cluster theory

Lecture 1: Double Bott-Samelson Cells

Double Bott-Samelson (BS) Cells are moduli spaces of chains of flags with prescribed relative positions. We show that double BS cells carry intrinsic cluster structures invariant under twist automorphisms. As an application, we obtain a geometric proof of Zamolodchikov periodicity conjecture of Y-systems. This lecture is based on joint work with Daping Weng.

Lecture 2: Augmentation varieties of Legendrian knots

Classifications of Legendrian knots and their exact Lagrangian fillings are central questions in low-dimensional contact and symplectic topology. In this lecture, we focus on Legendrian links obtained as the rainbow closure of positive braids. We show that their augmentation varieties are cluster varieties. As an application, we prove that any positive braid Legendrian link not isotopic to a standard finite type link admits infinitely many exact Lagrangian fillings. This lecture is based on joint work with Honghao Gao and Daping Weng.

Lecture 3: Moduli spaces of G-local systems I: cluster structures

In 2003, Fock and Goncharov introduced a pair of moduli spaces of G-local systems over decorated surfaces. These spaces have found significant applications in higher Teichmüller theory, representation theory, quantum field theory, etc. In this lecture, we present a systematic construction of cluster structures on these moduli spaces for all semisimple groups. This result generalizes the work of Fock-Goncharov for type A cases and the work of I.Le for classical groups. This lecture is based on joint work with Alexander Goncharov.

Lecture 4: Moduli spaces of G-local systems II: applications

We will investigate several applications of the moduli spaces of G-local systems studied in Lecture 3. Possible topics include Donaldson Thomas Transformations, Quantum groups, Canonical bases, etc. This lecture is based on joint work with Alexander Goncharov.

References:

[Lecture 1]
V.V. Fock and A.B. Goncharov, Cluster X-varieties, amalgamation and Poisson-Lie groups, Algebraic geometry and number theory, 27–68, Progr. Math., 253, Birkhäuser Boston, Boston, MA, 2006. (arXiv)

L. Shen and D. Weng, Cluster structures on double Bott-Samelson cells, arXiv:1904.07992. (arXiv)

[Lecture 2]

H. Gao, L. Shen, and D. Weng, Augmentations, fillings, and clusters, arXiv:2008.10793. (arXiv)

H. Gao, L. Shen, and D. Weng, Positive braid links with infinitely many fillings, arXiv:2009.00499. (arXiv)

[Lectures 3 & 4]

V.V. Fock and A.B. Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. No. 103 (2006), 1–211. (arXiv)

A.B. Goncharov and L. Shen, Donaldson-Thomas transformations of moduli spaces of G-local systems, Adv. Math. 327 (2018), 225–348. (arXiv)

A.B. Goncharov and L. Shen, Quantum geometry of moduli spaces of local systems and representation theory, arXiv: 1904.10491. (arXiv)

L. Shen, Duals of semisimple Poisson-Lie groups and cluster theory of moduli spaces of G-local systems, arXiv: 2003.07901. (arXiv)

Tsukasa Ishibashi : Wilson lines and their Laurent positivity

On the moduli space of framed G-local systems with pinnings on a marked surface, we introduce a new class of G-valued functions associated with the homotopy classes of arcs connecting boundary intervals which we call the Wilson line functions. These functions are regular functions, and their matrix coefficients in a finite-dimensional representation of G give rise to Laurent polynomials in each cluster Poisson chart. We show that, for a suitable choice of matrix coefficients, these Laurent polynomials have positive integral coefficients in the Goncharov--Shen coordinate system associated with any decorated triangulation. This talk is based on a joint work with Hironori Oya.

References:

T. Ishibashi and H. Oya, Wilson lines and their Laurent positivity, arXiv: 2011.14260. (arXiv)

A.B. Goncharov and L. Shen, Quantum geometry of moduli spaces of local systems and representation theory, arXiv: 1904.10491. (arXiv)

L. Shen, Duals of semisimple Poisson-Lie groups and cluster theory of moduli spaces of G-local systems, arXiv: 2003.07901. (arXiv)

Shunsuke Kano : Categorical dynamical systems arising from sign-stable mutation loops

The sign stability, which was introduced in the joint work with Tsukasa Ishibashi, is a generalization of pseudo-Anosovness of mapping classes to mutation loops.In fact, pseudo-Anosov mapping classes are sign-stable.In this talk, we will consider the autoequivalence F_\phi of a derived category D of the Ginzburg dg algebra associated to a sign-stable mutation loop \phi and compute the categorical entropy of it. If the time permits, we will also discuss the sufficient condition of that F_\phi is pseudo-Anosov in the sense of [Fan-Filip-Fabian-Haiden-Katzarkov-Liu].

Wataru Yuasa : Skein and cluster algebras of marked surfaces without punctures for $\mathfrak{sl}_3$

A skein algebra is a space of webs on a surface with skein relations associated with some Lie algebra. In particular, a knot diagram on the surface is considered as a web. In this talk, we introduce a skein algebra $\mathscr{S}_{\mathfrak{sl}_{3},\Sigma}^q$ with the boundary skein relations for a surface $\Sigma$ with marked points in the boundary. We realize a subalgebra $\mathrm{CA}_{\mathfrak{sl}_3,\Sigma}^q$ of the quantum cluster algebra associated with a moduli space $\mathcal{A}_{SL_3,\Sigma}$ of the decorated $SL_3$-local systems on $\Sigma$ inside the skein algebra $\mathscr{S}_{\mathfrak{sl}_{3},\Sigma}$. We also prove that any $\mathfrak{sl}_3$-web in the skein algebra is expressed as a Laurent polynomial of a cluster. Moreover, we show that elevation-preserving webs in $\Sigma$ have positive coefficients in a cluster associated with an ideal triangulation of $\Sigma$. This talk is based on joint work with Tsukasa Ishibashi.

References:

T. Ishibashi and W. Yuasa, Skein and cluster algebras of marked surfaces without punctures for sl_3, arXiv:2101.00643. (arXiv)

G. Muller, Skein and cluster algebras of marked surfaces, Quantum Topol. 7 (2016), no. 3, 435–503. (arXiv)