MIT Infinite Dimensional Algebra Seminar (Fall 2016)

There is a natural functor form the category of vertex algebras to the category of affine Poisson varieties that sends a vertex algebra V to its associated variety X_V . Although X_V is merely a Poisson variety in general, the recent work of Beem, Lemos, Liendo, Peelaers, Rastelli and van Rees predicts the existence of a large number of vertex algebras whose associated variety is symplectic (in fact, hyperK ̈ahler). This conjecture was in part reformulated by Y. Tachikawa as the existence of 2d TQFT associated with vertex algebras. In my talk I will explain that the vertex algebras corresponding to the cap are equivariant affine W-algebras that are natural affinization of Losev’s equivariant W-algebras, and confirm the conjecture of Tachikawa for types A_1 and A_2 in the principal cases.

I'll report on a joint work with S.Lysenko, where we establish an equivalence between the category of metaplectically twisted Whittaker sheaf on the affine Grassmannian of the group G and the category of representations of the small quantum group. The link between the two categories is provided by the category of factorizable sheaves, introduced and studied by Bezrukavnikov, Finkelberg and Schechtman in the 90's

Symplectic singularities were introduced by Beauville in 2000. These are especially nice singular Poisson algebraic varieties that include symplectic quotient singularities and the normalizations of orbit closures in semisimple Lie algebras. Poisson deformations of conical symplectic singularities were studied by Namikawa who proved that they are classified by points of a vector space. Recently I have proved that quantizations of conical symplectic singularities are still classified by the points of the same vector spaces. I will explain these results and then apply them to establish a version of Kirillov's orbit method for semisimple Lie algebras. The talk is based on http://arxiv.org/abs/1605.00592

It is well-known that Hecke algebras H_q(W) of Coxeter groups W do not have interesting Hopf algebra structures because, first, the only available one would emerge via an extremely complicated isomorphism with the group algebra of W and, second, this would make H_q(W) into yet another cocommutative Hopf algebra. The goal of my talk (based on joint work with D. Kazhdan) is to extend each Hecke algebra H_q(W) to a non-cocommutative Hopf algebra (we call it Hecke-Hopf algebra of W) that contains H_q(W) as a coideal subalgebra. Hecke-Hopf algebras have a number of remarkable properties: they generalize Bernstein presentation of Hecke algebras, provide new solutions to the quantum Yang-Baxter equation and a large class of endo-functors of the category H_q(W)-Mod, and new construction of affine and double affine Hecke algebras and their representations.

Given a factorizable finite ribbon category $D$, one can associate to any punctured surface $M$ a functor $Bl_M$ from a tensor power of $D$ to the category of finite-dimensional vector spaces. For a fixed object $F\in D$, we obtain representations of mapping class groups Map(M) that are compatible with sewing. I will present a natural construction which, given any object $F$ of $D$, selects vectors in all space $Bl_M(F,...,F)$ (i.e. when all punctures on M are labeled by F). If and only if the object $F$ carries a structure of a 'modular' commutative symmetric Frobenius algebra in $D$, the vectors obtained by this construction are invariant under the mapping class group actions and are mapped to each other upon sewing. Thereby, they are natural candidates for the bulk correlators of a conformal field theory with bulk state space given by $F$. (Joint with J. Fuchs, arxiv 1604.01143)

I will describe the results of a joint project with D. Kazhdan and Y. Varshavsky. I will start with an elementary result which generalizes to p-adic groups a well known very simple formula for the restriction of the character of the Steinberg representation of a finite Chevalley group to the set of unipotent elements. The proof of this generalization does not use l-adic sheaves, but the idea was inspired by a sheaf-theoretic construction. I will then discuss that construction and (time permitting) possible relation to recent conjectures of G. Lusztig.

It is known from Beem et al. that there is a construction of a vertex operator algebra (VOA) associated to each four-dimensional conformal field theory with N=2 supersymmetry. We propose a formula for the vacuum character of this VOA from the associated Kontsevich-Soibelman wall-crossing invariant of the four-dimensional field theory. We further generalize this proposal to include extended supersymmetric objects, known as line defects, into the four-dimensional field theory. The resulting wall-crossing invariants with line defects turn out to be interesting linear combinations of different characters of the associated VOA. We observe that these linear combinations are respected by the fusion rules in the VOA. (This talk will be based on the joint work 1506.00265 and 1606.08429 with Clay Cordova and Davide Gaiotto.)

I will discuss invariance of the order of the antipode, and traces of the powers of the antipode, under gauge equivalence. In particular, we will see that these values are in fact gauge invariants for Hopf algebras with the Chevalley property (e.g. Taft algebras and duals of pointed Hopf algebras). I will also discuss how our study relates to recent efforts of Shimizu to produce a categorial approach to the indicators of a non-semisimple tensor category. This is joint work with Richard Ng.

A huge class of Vertex Operator Algebras (VOAs) can be constructed starting from four-dimensional superconformal field theories (SCFTs) with N=2 supersymmetry. The simplest examples in this class are the affine vertex algebras associated with the Deligne exceptional series A_1, A_2, D_4, E_6, E_7, E_8, F_4, G_2 at level h^\vee/6 -1. Physical expectations about the four-dimensional SCFTs translate into precise mathematical conjectures for the corresponding VOAs. I will briefly motivate and state some of these conjectures, notably: 1) The existence of novel Topological Quantum Field Theories valued in VOAs, and (time permitting) 2 ) the modular behavior of characters of this class of VOAs, which follows from their conjectured quasi-lisse property.

I define a bilinear form B on the space of K-finite smooth compactly supported functions on G(A)/G(F) when F is a function field and G is a reductive split group. For G = SL(2), the definition of B generalizes to the case where F is a number field (and this is expected to be true for any G). This form is natural from the viewpoint of the geometric Langlands program via the functions-sheaves dictionary. I will discuss how B is related to Eisenstein series and local asymptotics of boundary degenerations of G.

Rozansky [Ro] and Overbay [Ov] described a spectacular knot polynomial that failed to attract the attention it deserved as the first poly-time-computable knot polynomial since Alexander's (1928) and (in my opinion) as the second most likely knot polynomial (after Alexander's) to carry topological information. With Roland van der Veen, I will explain how to compute the Rozansky polynomial using some new commutator-calculus techniques and a Lie algebra 𝔤_1 which is at the same time solvable and an approximation of the simple Lie algebra sl_2. More information here