Quantum Topology and Field Theory Seminar

Time: 4:30-5:30pm

Location: KT205

Abstract:  In 2010, Cooper and Krushkal provided a categorification of the Jones-Wenzl projectors. In recent work, we provided a new categorification which succeeds in being compatible with odd Khovanov homology—a variant of Khovanov’s original theory defined initially by Ozsváth, Rasmussen, and Szabó. Among the consequences of this result is the construction of a new ("odd") categorification of the colored Jones polynomial. In this talk, I aim to introduce our categorification of the Temperley-Lieb algebras, highlighting the peculiarities of the odd theory.

Time: 4:30-5:30pm

Location: Zoom (https://yale.zoom.us/j/92973039463)

Abstract:  In this talk, I will explain my joint work with R. Abedin, in which we construct, for each Lie algebra g, a Hopf algebra and a spectral R-matrix satisfying quantum Yang-Baxter equation. This Hopf algebra is a quantization of the Lie bi-algebra structure on T^*g[t] defined by Yang’s r-matrix, and therefore we call it the Yangian of T^*g. The construction is based on the category of coherent sheaves on the equivariant affine grassmannian associated to the formal group of g, and is motivated by the study of the category of line defects in a 4 dimensional holomorphic-topological field theory. 

Time: 4:30-5:30pm

Location: KT801

Abstract:  Multiple polylogarithms appear to be central for many seemingly unrelated areas of mathematics: volumes of hyperbolic polytopes, scissors congruence, algebraic K-theory, special values of zeta functions, etc. Despite the existence of this wide network of connections, the most fundamental properties of these functions, predicted by the Goncharov program, remain conjectural. I will talk about the recent progress in the Goncharov program, which is based on the connection between multiple polylogarithms and the Steinberg module of Q. The talk is based on the joint work with Steven Charlton and Danylo Radchenko. 

Time: 4:30-5:30pm

Location: KT801

Abstract:  We associate to each ciliated bipartite ribbon graph in R^3 an isotopy invariant Laurent polynomial in a single variable q^(1/N), called the SL_N quantum trace, which can be expressed as a quantum deformation of the partition function for the N-dimer model.  The construction is based on Sikora's SL_N quantum traces for N-webs in R^3.  For planar graphs, the quantum trace is moreover a symmetric Laurent polynomial in q, which can be expressed as a quantum deformation of the Kasteleyn determinant of the graph equipped with the trivial connection.  We also provide a similar expression for planar graphs equipped with a general quantum matrix connection (subject to a relatively strong commutativity constraint).  This is joint work with Richard Kenyon, Nicholas Ovenhouse, Sam Panitch, and Sri Tata. 

Time: 4:30-5:30pm

Location: KT801

Abstract:  Many interesting algebraic varieties appearing in low-dimensional topology and representation theory (for example various kinds of surface character varieties, or subvarieties of simple Lie groups or their flag manifolds) are known to admit cluster Poisson structures. Given some geometrically defined morphism between two such varieties, it is natural to ask whether it respects the corresponding cluster structures in a suitable sense. I will explain a kind of 'gluing procedure' for certain special kinds of cluster structures, which leads to a positive answer to the question above for morphisms of character varieties associated to cutting a surface along a simple closed curve, as well for morphisms between BFN Coulomb branches of quiver gauge theories obtained by restricting a factor of the gauge group to its maximal torus. Based on joint work with Alexander Shapiro.

Time: 4:30-5:30pm

Location: KT801

Abstract:  For a Lie group G, the G-skein module of a 3-dimensional manifold M is a fundamental object in Witten’s interpretation of quantum knot invariants in the framework of a topological quantum field theory. It depends on a parameter q and, when this parameter q is a root of unity, the G-skein module contains elements with a surprising “transparency” property, in the sense that they can be traversed by any other skein without changing the resulting total skein. I will describe some (and conjecturally all) of these transparent elements in the case of the special linear group SL_n. The construction is based on the very classical theory of symmetric polynomials in n variables.  

Time: 4:30-5:30pm

Location: KT801

Abstract:  This talk is about a new family of geometric quantum link invariants that depend on both a link in S³ and a flat 𝔰𝔩₂ connection on its complement. When the connection is trivial they recover the Kashaev invariant (a certain evaluation of the colored Jones polynomial). More generally they can be understood as a quantization of the complex Chern-SImons invariant of the flat connection (aka complex volume), whose real and imaginary parts are the volume and Chern-Simons invariant of the hyperbolic structure determined by the connection. In this talk I will discuss the construction of these invariants using the representation theory of quantum 𝔰𝔩₂ and how the classical complex Chern-Simons invariant arises naturally in this context, then sketch some proposed connections with quantum SL₂(ℂ) Chern-Simons theory. Given time I will also discuss connections with the Volume Conjecture. This talk is based on joint work with Nicolai Reshetikhin.

Time: 5:15-6:15pm

Location: KT801

Abstract: Skein modules were introduced by Józef H. Przytycki as generalisations of the Jones and HOMFLYPT polynomial link invariants in the 3-sphere to arbitrary 3-manifolds. The Kauffman bracket skein module (KBSM) is the most extensively studied of all. However, computing the KBSM of a 3-manifold is known to be notoriously hard, especially over the ring of Laurent polynomials. With the goal of finding a definite structure of the KBSM over this ring, several conjectures and theorems were stated over the years for KBSMs. We show that some of these conjectures, and even theorems, are not true. In this talk I will briefly discuss a counterexample to Marche’s generalisation of Witten’s conjecture. I will show that a theorem stated by Przytycki in 1999 about the KBSM of the connected sum of two handlebodies does not hold. I will also give the exact structure of the KBSM of the connected sum of two solid tori and show that it is isomorphic to the KBSM of a genus two handlebody modulo some specific handle sliding relations. Moreover, these handle sliding relations can be written in terms of Chebyshev polynomials.

Time: 5:15-6:15pm

Location: KT221

Abstract:  The Fukaya-Seidel category is classically defined in terms of a Kähler manifold X equipped with a holomorphic function W, with the relevant data encoded in the exact one-form dW. However, a natural generalization suggests itself: to replace dW with an arbitrary closed (but not necessarily exact) holomorphic one-form α on X. This more general version arises in many constructions in low-dimensional topology. In this talk, I will explore how the framework of the algebra of the infrared provides a means to understand this extension and illuminates the structure of the resulting category. 

Time: 4:30-5:30pm

Location: KT801

Abstract:  The small quantum connection of a Fano variety is among the most accessible and fundamental objects in enumerative geometry. In this talk, I’ll survey recent results on the structure of the quantum connection, with an emphasis on bounds for the sizes of Jordan blocks of the "regularized monodromy". These bounds can be viewed as mirror analogues of classical results by Borel, Katz, and Varchenko concerning the Jordan blocks of monodromy for Gauss–Manin connections associated to families of varieties. This is joint work—partially in progress—with P. Seidel.

Time: 4:30-5:30pm

Location: KT801

Abstract:  The $sl_n$-skein algebra of a surface provides a quantization of the $SL_n(\mathbb{C})$ character variety. For surfaces with boundary, this framework extends naturally to the stated skein algebra. We demonstrate how various aspects of quantum groups admit simple and transparent geometric interpretations through the lens of stated skein algebras. In particular, we show how the Schapiro–Shrader embedding of the quantized enveloping algebra into a quantum torus algebra arises from the quantum trace map. Time permitting, we will also present a geometric realization of the dual canonical basis of $\mathcal{O}_q(\mathfrak{sl}_3)$ using skeins.

Time: 4:30-5:30pm

Location: KT801

Abstract:  Inverse Hamiltonian reduction refers to a series of conjectural relations between W-algebras corresponding to distinct nilpotent orbits in a Lie algebra. I will outline a proof of this conjecture in type A that relies on novel geometric methods. Along the way, we shall encounter a technique for localising vertex algebras and, time permitting, speak briefly on the deformation theory thereof. To build intuition, I shall focus on the finite type analogue of this story, where such techniques are more commonplace. This talk is based on joint work with Dylan Butson.