MIT Infinite Dimensional Algebra Seminar (Fall 2017)

The talk deals with representations of rational Cherednik algebras of type A over fields of characteristic p>>0. There are several reasons to be interested in this kind of representations. An "ideological" reason is that passing from complex numbers to fields of characteristic p>>0 "affinizes" the representation theory, as evidenced by the representation theory of semisimple Lie algebras. The representation theory of rational Cherednik algebras over C is already of "affine type" so passing to characteristic p>>0 should result in a representation theory of "double affine type". Such kinds of representation theory presently and not understood but it seems that they should appear in several other contexts. A "practical" reason to be interested in characteristic p representations of type A rational Cherednik algebras is that structural results in this area should lead to (re)proving some difficult results in the combinatorics of Macdonald polynomials, such as Haiman's n! theorem.

In my talk I will concentrate on analogs of categories O in characteristic p>>0. By definition, they consist of finite dimensional graded modules. It is more or less classical result that this category is highest weight in a suitable sense (roughly speaking, highest weight means certain upper triangularity properties). I will define filtrations on these categories (standardly stratified structures) and relate the associated graded categories to more classical categories O from characteristic 0. Time permitting I will explain a relation of wall-crossing functors to these standardly stratified structures. No preliminary knowledge of the representation theory of rational Cherednik algebras or of highest weight categories is required.

Abstract. To a smooth surface, we associate the W-algebra of type gl_r with two deformation parameters equal to the Chern roots of the cotangent bundle of S. We expect that the resulting algebra acts on the K-theory groups of moduli spaces of semistable rank r sheaves on S, and one can compute commutation relations between the algebra and the Carlsson-Okounkov Ext operator. When the surface is S=A^2, this allows one to present the Ext operator as a vertex operator for deformed W-algebras, thus yielding a mathematical proof of the 5d AGT relations with matter for the gauge group U(r)

Four dimensional N=2 superconformal fields theories in physics produce some interesting mathematical invariants such as Schur indices and Higgs branches. In my talk I will explain some remarkable relations of these invariants with vertex algebras and their consequences.

A vertex operator algebra and a local conformal net (of operator algebras) are two mathematical formulations of chiral conformal field theory. We present a construction of the latter from the former, with unitarity, and going back to the former. We also discuss representation theoretic aspects of this construction.

We will discuss some results about modules over quantized symplectic resolutions from a joint project with A. Okounkov aimed at describing automorphisms of their derived categories.

I will describe some gauge theory constructions of Vertex Operator Algebras. Dualities between gauge theories imply non-trivial relations for the corresponding VOAs, such as Feigin-Frenkel duality or coset constructions of W-algebras. I will discuss the implications of these constructions for the Geometric Langlands program and Symplectic Duality

An important problem in vertex algebra theory is to study modular properties of characters of representations. By now this is well-understand for regular (after Y.Zhu) and C_2-cofinite (after Miyamoto) vertex algebras. But there are vertex algebras that do not belong to either group yet their characters satisfy interesting modular properties. An important family of examples come from integrable irreducible highest weight modules over affine superalgebras (after Kac and Wakimoto). These characters are known to be related to mock theta functions of Ramanujan.

In my talk I'll focus on a family of W-algebras coming from certain extensions of affine W-algebras. Although their representation theory is poorly understood their characters have been proposed and studied. I'll explain two approaches to modular invariance of characters and Verlinde-type formula - both based on iterated integrals of modular forms. In the first approach (joint work with T. Creutzig) modular invariance is formulated with the help of "charge" variables. In the second approach (joint work with K. Bringmann and J. Kaszian), we work in a non-holomorphic setup and replace characters with better behaved "companions". This approach is intimately linked to mock-modularity, quantum modular forms, etc. We shall see how these approaches elucidate and enrich representation theory of vertex algebras.

In this talk we will report on a recent progress made by B.Feigin and the speaker of constructing a certain chiral algebra A that carries a Kac-Moody symmetry for a G and its Langlands dual G^L (at Langlands dual levels \kappa and \kappa^L). The localization of A on Bun_G\times Bun_{G^L} provides a kernel for the global quantum Langlands equivalence D_{\kappa}(Bun_G) -> D_{\kappa^L}(Bun_{G^L}). The category of chiral modules over A is a category acted on by the product of the loop groups G((t)) \times G^L((t)) and provides a kernel for the local quantum Langlands equivalence G((t))-Cat_{\kappa} -> G^L((t))-Cat_{\kappa^L}.

We introduce a new one-parameter centralizer algebra as a Fourier cousin of the Birman-Wenzl-Murakami algebra and prove unitarity at roots of unity. We find a new type of Schur-Weyl duality between this centralizer algebra and two families of quantum subgroups. We construct its irreducible representations and compute their quantum dimensions in a closed form. Its Bratteli diagram gives two families of Dynkin diagrams for Ocneanu's higher representation theory.