Schedule and Abstracts
Mina Aganagic (UC Berkeley)
Homological Knot Invariants from Mirror Symmetry
In 1999, Khovanov showed that a link invariant known as the Jones polynomial is the Euler characteristic of a homology theory. The knot categorification problem is to find a general construction of knot homology groups, and to explain their meaning -- what are they homologies of?
Homological mirror symmetry, formulated by Kontsevich in 1994, naturally produces hosts of homological invariants. Typically though, it leads to invariants which have no particular interest outside of the problem at hand.
I showed recently that there is a new family of mirror pairs of manifolds, for which homological mirror symmetry does lead to interesting invariants and solves the knot categorification problem. The resulting invariants are computable explicitly for any simple Lie algebra, and certain Lie superalgebras.
Mykola Dedushenko (Stony Brook)
Vertex algebra of the interval-reduced theory and its applications
Boundary conditions of a 3d N=2 QFT preserving (0,2) supersymmetry are known to support boundary vertex algebras in the Q-cohomology. I will discuss a 3d N=2 theory reduced on an interval with (0,2) boundary conditions on both ends. This configuration flows to some two-dimensional N=(0,2) theory, and the vertex operator algebra in its Q-cohomology is interesting, subtle, and much richer than in the one-boundary setup. I will also explain one fun application of this construction: It turns out that interval reductions of certain simple 3d N=2 gauge theories compute VOA(M_4) (which will be briefly reviewed) for some four-manifolds M_4, such as products of Riemann surface and a two-sphere, CP^2, Hirzebruch surface, four-sphere, and presumably more.
Gurbir Dhillon (Yale)
On the log Kazhdan--Lusztig correspondence
An influential conjecture of Feigin--Gainutdinov--Semikhatov--Tipunin from the mid 2000s relates representations of small quantum groups and triplet vertex algebras at positive integer values of the Kac--Moody level. We will formulate an extension of the conjecture to all Kac--Moody levels and will describe a way to prove it conditional on some (in-reach) conjectures in the quantum geometric Langlands program.
Hiraku Nakajima (Kavli-IPMU)
Equivariant intersection cohomology of instanton moduli spaces
We consider instanton moduli spaces of the Taub-NUT space / finite cyclic group, which are conjecturally Coulomb branches of quiver gauge theories of affine type. Geometric Satake for affine Lie algebras predicts that sum of hyperbolic stalks of their intersection cohomology sheaves has a structure of a representation of an affine Lie algebra. We give a conjectural description of costalks of equivariant intersection cohomology sheaves, whose sum is a representation of a coset VOA.
Cris Negron (USC)
Quantum SL(2) and non-rational VOAs at (p,1)-central charge
I will talk about recent work with Terry Gannon, where we relate modules for non-rational vertex operator algebras at central charge c=1-6(p-1)^2/p to representations for quantum SL(2) at a 2p-th root of 1. For example, we show that the vertex tensor category of integrable modules for the Virasoro VOA, at the given central charge, is equivalent to the ribbon tensor category of (big) quantum SL(2)-representations. Our results resolve the so-called "logarithmic Kazhdan-Lusztig conjecture" in type A_1, as formulated by Gainutdinov-Semikhatov-Tipunin-Feigin in '06..
Sunghyuk Park (UT Austin)
Knot lattice homology and q-series
I will tell a story combining two invariants that can be defined for generalized algebraic knots. One is knot lattice homology, introduced by Ozsvath, Stipsicz, and Szabo, which is believed to agree with knot Floer homology. The other one is the two-variable series of Gukov and Manolescu, from which various sl(2) quantum invariants such as colored Jones polynomials and Akutsu-Deguchi-Ohtsuki (ADO) invariants can be recovered. This is a work in progress with Ross Akhmechet and Peter K. Johnson.
Du Pei (CQM Denmark)
Conformal field theories and topological modular forms
We will discuss how the theory of topological modular forms can be used to learn about VOAs and conformal field theories. In particular, we will focus on a type of constraints that can be used to rule out the existence of an infinite set of "extremal CFTs", including those with central charges c = 48, 72, 96 and 120.
Pavel Putrov (ICTP)
Non-semisimple TQFTs and BPS q-series.
In my talk I will describe a conjectural relation between the topological invariant of 3-manifolds of Costantino--Geer--Patureau-Mirand (CGP) constructed using a non-semisimple category of representations of a quantum group, and counting of BPS states in a certain M-theory setup.
Fei Yan (Rutgers)
Line defects, link invariants, and VOA
In this talk we will discuss aspects of line defects in 4d N=2 theories of class S, focusing on the connections to skein theory and the construction of link invariants. We will also comment on the connections to VOAs.