MIT Infinite Dimensional Algebra Seminar (Fall 2021)

We will suggest a conjectural picture, which follows from the quantum geometric Langlands theory, according to which the minimal model CFT factors into the tensor product of two copies of WZW CFT (for a pair of Langlands dual groups).

I will explain a cluster description of moduli spaces of R-vector bundles with flat connections over topological surfaces, where R is a non-commutative field. Examples include moduli spaces of Stokes data, which appear in the study ​​of differential equations with irregular singularities on Riemann surfaces. This is a joint work with Maxim Kontsevich.

This is a joint work with A.Braverman and H.Nakajima. We give yet another proof (a geometric one) of the famous Kazhdan-Lusztig conjecture on the characters of irreducible modules in the category O over a complex semisimple Lie algebra (in the Koszul-dual formulation). The proof proceeds by analysis of fixed points in the zastava spaces.

We define a new family of commuting operators F_k in Khovanov-Rozansky link homology, similar to the action of tautological classes in cohomology of character varieties. We prove that F_2 satisfies ``hard Lefshetz property" and hence exhibits the symmetry in Khovanov-Rozansky homology conjectured by Dunfield, Gukov and Rasmussen in 2005. This is a joint work with Matt Hogancamp and Anton Mellit.

We will review conditions on a vertex algebra V so that its space of conformal blocks on the torus is finite dimensional. This leads to conditions of V related to C_2 cofiniteness: the zero-th Poisson homology of Zhu's C_2 algebra R_V is finite dimensional. We analyze analogous conditions so that the higher chiral homology of V on the torus is finite dimensional, this leads to the obvious condition on the Poisson homology of Zhu's C_2 algebra, as well as some extra conditions on the full classical limit of V. This is joint work with J. V. Ekeren.

In this talk, I will give a simple description of the closure of the nilpotent orbits appearing as associated varieties of admissible affine vertex algebras in terms of primitive ideals. I will also connect these varieties with the cohomology of the small quantum groups associated with an l-th root of unity.

I will talk about the positive part of a certain affine Springer fiber studied by Goresky, Kottwitz, and MacPherson, and a certain interesting open subvariety. The Hilbert series of their Borel-Moore homology turns out to be related to reproducing kernels of the Bergeron-Garsia nabla operator. This operator is easy to define in the basis of modified Macdonald polynomials, but producing explicit combinatorial evaluations of this operator is usually difficult and (conjecturally) relates to interesting Hilbert series associated to various moduli spaces. Our work is motivated by the nabla positivity conjecture of Bergeron, Garsia, Haiman, and Tesler that predicts that nabla evaluated on a Schur function is sometimes positive, sometimes negative. We categorify this conjecture and reduce it to a vanishing conjecture for the interesting open variety. It turns out, each irreducible S_n representation mysteriously prefers to live in certain degrees and weights in the cohomology. This is a joint work with Erik Carlsson.

In 1988, based on the fundamental conjectures on operator product expansion and modular invariance, Moore and Seiberg observed that there should be tensor categories with additional structures associated to rational conformal field theories. Since then, tensor category structures from conformal field theories have been constructed, studied and applied to solve mathematical problems. Mathematically, conformal field theories can be constructed and studied using the representation theory of vertex operator algebras. In this talk, I will give a survey on the constructions and studies of various tensor category structures on module categories for vertex operator algebras.

I will review an analytic approach to the geometric Langlands correspondence, following my work with E. Frenkel and D. Kazhdan, arXiv:1908.09677, arXiv:2103.01509, arXiv:2106.05243. This approach was developed by us in the last couple of years and involves ideas from previous and ongoing works of a number of mathematicians and mathematical physicists -- Kontsevich, Langlands, Nekrasov, Teschner, Gaiotto-Witten and others. One of the goals of this approach is to understand single-valued real analytic eigenfunctions of the quantum Hitchin integrable system. The main method of studying these functions is realizing them as the eigenbasis for certain compact normal commuting integral operators the Hilbert space of L2 half-densities on the (complex points of) the moduli space Bun(G,X) of principal G-bundles on a smooth projective curve X, possibly with parabolic points. These operators actually make sense over any local field, and over non-archimedian fields are a replacement for the quantum Hitchin system. We conjecture them to be compact and prove this conjecture in the genus zero case (with parabolic points) for G=PGL(2).

I will first discuss the simplest non-trivial example of Hecke operators over local fields, namely G=PGL(2) and genus 0 curve with 4 parabolic points. In this case the moduli space of semistable bundles Bun(G,X)^{ss} is P^1, and the situation is relatively well understood; over C it is the theory of single-valued eigenfunctions of the Lame operator with coupling parameter -1/2 (previously studied by Beukers and later in a more functional-analytic sense in our work with Frenkel and Kazhdan). I will consider the corresponding spectral theory and then explain its generalization to N>4 points and conjecturally to higher genus curves.

The tools of support varieties (initiated by Carlson in 1983) and tensor triangular geometry (initiated by Balmer in 2005) have played an important role in the study of monoidal triangulated categories, with stable categories of finite tensor categories forming one of the principal classes of examples. The relationship between support varieties and tensor triangular geometry has been used in many cases to classify the thick ideals of the category in question, a fundamental problem. We will discuss work of Buan-Krause-Solberg and Nakano-V.-Yakimov, which defined and developed noncommutative versions of Balmer's theory, and will proceed to describe new methods for determining the Balmer spectrum and thick ideals of a monoidal triangulated category. This is joint work with Daniel Nakano and Milen Yakimov.

I will explain an isomorphism between the positive half o a quantum toroidal group and the K-theoretic Hall algebra of a preprojective algebra of affine type. There is an analogue result in the finite type case. For type A_1 this allows to propose a coherent categorification of the quantum affine sl(2). Surprisingly it may be computed using KLR algebras. The talk is based on two joint works, one with E. Vasserot, the other with P. Shan and E. Vasserot.