Apr. 20: Thomas Creutzig, University of Alberta

Triality of W-algebras

A vertex algebra is the chiral algebra of a two-dimensional conformal field theory and appears in many exciting problems. W-algebras are the broadest class of these and they are associated to certain data of Lie superalgebras. These W-algebras and their representation categories appear as invariants of 4-dimensional gauge theories and dualities of physics translate into isomorphisms of vertex algebras and equivalences of representation categories of these algebras.

I aim to give an overview of the current understanding of the topic, including a new general triality Theorem by Andrew Linshaw and myself.

May 4: Yan Soibelman, Kansas State University

Cohomological Hall algebras, instantons and vertex algebras

Originally the notion of Cohomological Hall algebra (COHA for short) was introduced in my joint paper with Maxim Kontsevich (arXiv: 1006.2706) for the purposes of motivic Donaldson-Thomas theory of 3-dimensional Calabi-Yau categories. A physicist can think informally of COHA as of quantized enveloping algebra of the Lie algebra of single-particle closed BPS states. From this point of view there should be a class of representations of COHA associated with spaces of open BPS states. Mathematically we are talking about representations of COHAs in the cohomology of the moduli spaces of stable framed objects of the category (e.g. stable framed coherent sheaves on a Calabi-Yau 3-fold).

I am going to discuss recent results which relate this class of representations to the (generalization of the ) AGT conjecture. In particular, a version of COHA of the category of torsion sheaves on the standard complex 3-dimensional vector space acts on the cohomology of the moduli space of Nekrasov spiked instantons. This action can be upgraded to the action on this space of the ``vertex algebra at the corner" of Gaiotto and Rapcak. The underlying geometry is the one of non-reduced toric divisors in toric Calabi-Yau 3-folds. The above-mentioned result is yet another incarnation of the idea that quantum algebras which initially might look as ``2-dimensional objects" (e.g. quiver Yangians) should be upgraded to those in dimension 3.

May 25: Anrew Neitzke, Yale

Abelianization of flat connections, and its q-deformation

Abelianization of flat connections is a construction motivated by supersymmetric quantum field theory, which has turned out to be connected to various bits of geometry -- in particular, to Donaldson-Thomas theory, cluster algebra, the exact WKB method for analysis of ODEs, and hyperkahler geometry. In some of these subjects it is known that there exists a natural q-deformation which takes us from the commutative to the noncommutative world. This suggests that there ought to exist a q-deformation of abelianization as well. I will explain joint work in progress with Fei Yan on constructing this q-deformation in a geometric way using spectral networks. This construction is inspired by related work by various authors, especially Bonahon-Wong, Gabella, Gaiotto-Witten. One byproduct is a new scheme for computing known polynomial invariants of links in R^3, which generalizes the usual "vertex models".

June 8: Anton Zeitlin, Louisiana State University

Geometry of Bethe Equations via q-Opers

Integrable models are known to keep reemerging over time in various mathematical incarnations. Recently, such models based on quantum groups naturally appeared in the framework of enumerative geometry. In this context the so-called Bethe ansatz equations, instrumental for finding the spectrum of the XXZ model Hamiltonian, naturally show up as constraints for the quantum K-theory ring of quiver varieties.

In this talk I will describe another geometric interpretation of Bethe ansatz equations, which is indirectly related to the above. I will introduce the notion of (G,q)-opers, the difference analogue of oper connections for simply connected group G. I will explain the one-to-one correspondence between (G,q)-opers of specific kind and Bethe equations for XXZ models. The key element in this identification is the so-called QQ-system, which has previously appeared in the study of ODE/IM correspondence and the Grothendieck ring of the category O of the relevant quantum algebras. I will speculate on how that fits into recently proposed quantum q-Langlands correspondence by M. Aganagic, E. Frenkel and A. Okounkov.

The talk is based on joint work with E. Frenkel, P. Koroteev and D. Sage ( arXiv:1811.09937, arXiv:2002.07344)

June 22: Marcos Marino, University of Geneva

Spectral theory of quantum curves and topological strings

Many problems in mathematical physics, from the WKB method to knot theory, involve quantum versions of algebraic curves. In this talk I will review an approach to the quantization of local mirror curves which makes it possible to reconstruct topological string theory on toric Calabi-Yau manifolds. In this approach, the quantization of the mirror curve leads to a trace class operator on the real line. The Fredholm determinant and spectral traces of this operator turn out to encode the topological string partition functions. Conversely, one can use enumerative invariants to solve the spectral theory of these operators, which leads to new exact results in quantum mechanics.