MIT Infinite Dimensional Algebra Seminar (Fall 2022)

I will explain the definition of unitarity of a representation of a conformal vertex algebra and describe the classification of unitary representations of minimal quantum affine W-algebras. I will also explain how to compute the characters of these representations.

We define a natural, purely geometrical bijection between the set of monodromy-free sl_2-opers (aka projective structures) on the projective line with the prescribed type of regular singularities at prescribed real marked points and the set of arc diagrams of Frenkel-Kirillov-Varchenko. The former set indexes the solutions of Bethe ansatz equations for the Gaudin model (according to Feigin and Frenkel), while the latter one indexes the canonical base in a tensor product of U_q(sl_2)-modules (via the Schechtman-Varchenko isomorphism). Both sets carry a natural action of the cactus group, i.e., the fundamental group of the real Deligne-Mumford space of stable rational curves with marked points. We prove that our bijection is compatible with this cactus group action. Moreover, this cactus group action coincides with that on an appropriate tensor product of sl_2-crystals coming from the coboundary category formalism (joint work with Nikita Markarian).

The talk is based on the joint work with Roman Bezrukavnikov and Victor Kac (arXiv:2209.08865). Let g be a simple Lie algebra and let $\hat{g}$ be the corresponding affine Lie algebra. It is known that characters of irreducible (highest weight) representations of $\hat{g}$ can be computed in terms of values at q=1 of affine (inverse) Kazhdan-Lusztig polynomials. These values can be computed recursively but there are no explicit formulas for them in general. The goal of this talk is to describe certain cases when we can compute the values above explicitly resulting in explicit formulas for characters of certain irreducible $\hat{g}$-modules (partly generalizing results of Kac and Wakimoto). The calculation relies on the description of the corresponding module over the affine Hecke algebra in terms of the equivariant $K$-theory of the Springer resolution. Time permitting we will discuss possible generalizations.

This talk in based on a joint work with V. Serganova and V. Hinich, arXiv:2209.06253 [1].

We introduce a notion of a root groupoid as a replacement of the notion of Weyl group for (Kac-Moody) Lie superalgebras. The objects of the root groupoid classify certain root data, the arrows are defined by generators and relations. As an abstract groupoid the root groupoid has many connected components and we show that to some of them one can associate an interesting family of Lie superalgebras which we call root superalgebras. Classical Kac-Moody Lie superalgebras appear as minimal root superalgebras in fully reflectable components. We classify all root superalgebras in these components.

To each connected component we associate a graph (called skeleton) generalizing the Cayley graph of the Weyl group. The skeleton satisfies a version of Coxeter property generalizing the fact that the Weyl group of a Kac-Moody Lie algebra is Coxeter.

Links:
[1] https://arxiv.org/abs/2209.06253.

To certain 2-Calabi-Yau categories A one may associate a BPS Lie algebra, which in certain cases can be shown to generate the Borel-Moore homology of the stack of objects in A under a PBW-type theorem. An example of such a case is the category of representations of a preprojective algebra associated to a quiver Q, where this Lie algebra recovers the Kac polynomials of Q as the cohomologically graded dimensions of its (dimension vector)-graded pieces. In this talk I will concentrate on "totally negative" 2CY categories. Examples are provided by categories of representations of the preprojective algebra of Q in the case in which Q has enough arrows, or the category of semistable Higgs bundles on a curve C of genus of at least 2, or the category of representations of the fundamental group of the same C. For totally negative 2CY categories, in joint work with Hennecart and Schlegel-Mejia, we prove that the BPS Lie algebra is a free Lie algebra, with generators identified with the intersection cohomology of coarse moduli spaces. As applications, we prove

• The following extension of nonabelian Hodge theory to moduli stacks: for a curve C of genus g, there is a canonical isomorphism between the stack of rank r degree d Higgs bundles, and the stack of r-dimensional d-twisted representations of the fundamental group of C.
• The following strengthening of the Kac positivity conjecture: the Bozec-Schiffmann polynomial counting absolutely cuspidal representations of the preprojective algebra of a totally negative quiver has positive coefficients.

I will present new constructions of several of the exceptional simple Lie super- algebras with integer Cartan matrix in characteristic p = 3 and p = 5, which were classified in [1]. These include the Elduque and Cunha Lie superalgebras. Specifi- cally, let αp denote the kernel of the Frobenius endomorphism on the additive group scheme Ga over an algebraically closed field of characteristic p. The Verlinde cate- gory Verp is the semisimplification of the representation category Rep αp, and Verp contains the category of super vector spaces as a full subcategory. Each exceptional Lie superalgebra we construct is realized as the image of an exceptional Lie algebra equipped with a nilpotent derivation of order at most p under the semisimplification functor from Rep αp to Verp. The content of this talk can primarily be found in [2] and [3].

Keywords: modular Lie superalgebras, symmetric tensor categories Mathematics Subject Classification 2020: 17B, 18M20

References [1] S. Bouarroudj, P. Grozman, and D. Leites, Classification of simple finite- dimensional modular Lie superalgebras with Cartan matrix, Symmetry, Inte- grability and Geometry: Methods and Applications (SIGMA) v. 5 (2009), no. 060, 63 pages. [2] A.S. Kannan, New Constructions of Exceptional Simple Lie Superalgebras with Integer Cartan Matrix in Characteristics 3 and 5 via Tensor Categories, Trans- formation Groups (2022). [3] P. Etingof, and A.S. Kannan, Lectures On Symmetric Tensor Categories, arXiv:2103.04878 (2021).

Lecture Notes

Nakajima and Grojnowski initiated the relationship between infinite-dimensional Lie algebras and the Hilbert scheme of points on a complex surface. In physics, a similar correspondence between a certain conformal field theory and the cohomology of the moduli space of instantons was laid out by Alday, Gaiotto, and Tachikawa. One way to contextualize these relationships is through the topological twist of a mysterious six-dimensional superconformal field theory. We will use a new mathematical understanding of this theory to propose, and provide evidence for, enhancements of our understanding of the cohomology of instanton moduli spaces which will involve appearances of certain exceptional super Lie algebras.

In recent work, the study of partial flag varieties and the Schubert bases of their equivariant cohomology has been extended to cotangent bundles and Segre-Schwartz-MacPherson classes. I will discuss the behavior of these bases in the restriction in cohomology from type A to type C Grassmannians. When considering their cotangent bundles, this behavior has a further geometric interpretation in terms of Maulik-Okounkov stable envelopes and Lagrangian correspondences. Both settings can be considered from the combinatorial perspective of puzzle rules, which in turn are interpreted as quantum integrable systems via R-matrices for the sl(3) Yangian. This is joint work with Allen Knutson and Paul Zinn-Justin.

Feigin-Loktev's fusion product of cyclic graded modules over the current algebra was introduced in 1998. Although its definition is elementary, very little is known about its properties in general. We introduce a way to study it geometrically. This approach also gives Borel-Weil-type theorems for the Beilinson-Drinfeld Grassmannian and the global convolution diagram. We will also discuss a relation to quantum loop group representations and the category of coherent perverse sheaves on the affine Grassmannian.​