MIT Infinite-Dimensional Algebra Seminar (Spring 2024)

In their study of GL(N)-GL(m) Howe duality, Cautis-Kamnitzer-Morrison observed that the GL(N) Reshetikhin-Turaev link invariant can be computed in terms of quantum gl(m). This idea inspired Cautis and Lauda-Queffelec-Rose to give a construction of GL(N) link homology in terms of Khovanov-Lauda's categorified quantum gl(m). There is a Spin(2n+1)-Spin(m) Howe duality, and a quantum analogue which was first studied by Wenzl. In the first half of the talk I will explain how to use this duality to compute the Spin(2n+1) link polynomial, and present calculations which suggest that the Spin(2n+1) link invariant is obtained from the GL(2n) link invariant by folding. In the second part of the talk, I will introduce the parallel categorified constructions and explain how to use them to define Spin(2n+1) link homology.

This is based on joint work in progress with Ben Elias and David Rose.

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I will explain certain 4-tensors appearing in studies of two-dimensional topological order from a viewpoint of subfactor theory of Jones and alpha-induction there, which is a tensor functor arising from a modular tensor category and a Frobenius algebra in it. They are understood with quantum 6j-symbols and braiding.

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(joint with Vasily Krylov) I will explain the computation of special values of parabolic affine inverse Kazhdan-Lusztig polynomials, which give explicit formulas for certain irreducible representations of affine Lie algebras that generalize Kac and Wakimoto's results. By Bezrukavnikov's equivalence, the canonical basis in the subregular part of the anti-spherical module can be identified with irreducible objects in the exotic t-structure on the equivariant derived category of the subregular Springer fiber. We describe the irreducible objects explicitly using an equivariant derived McKay correspondence. In doing so, we identify the module with a module Lusztig defines, which compatibly extends to the regular cell.

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Let X -->S be a smooth family of algebraic varieties, P an invertible function on X. Consider the relative de Rham cohomology of the asymptotic D-module P^{1/h}: = (O_X, hd + dP/P). In the limit, when h goes to 0, the cohomology depends only on the neighborhood of the critical locus of function P. I will explain how, when working with algebraic varieties over Z/nZ, the above "asymptotic" formula for the cohomology becomes exact. I will also explain some applications to the study of the KZ equations over Z/nZ. The talk is based on a joint work with Alexander Varchenko.

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In an influential 1980 paper of Victor Kac, he proved (among many other things) that the number of absolutely indecomposable representations of a quiver over a finite field behaves like point-counting on a variety over F_q. This variety has been made precise by Crawley-Boevey and Van den Bergh using deformed preprojective algebras.

A parallel problem is to count absolutely indecomposable vector bundles on a curve over a finite field. About 10 years ago, Schiffmann proved that the number of such (with degree coprime to the rank) is equal to the number of stable Higgs bundles of the same rank and degree (up to a power of q). Dobrovolska, Ginzburg and Travkin gave a slightly different formulation of this result and a very different proof. Neither argument obviously generalizes to G-bundles for other reductive groups G.

In joint work with Konstantin Jakob, we generalize the above-mentioned results to G-bundles. Namely, we show that the number of absolutely indecomposable G-bundles (suitably defined) on a curve over a finite field can be expressed using the number of stable (parabolic) G-Higgs bundles on the same curve.

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The focus will be on a recent construction of the motivic superpolynomials for arbitrary singularities (multi-branch and non-square-free). They will be introduced from scratch, which includes the definition of varieties of torsion-free sheaves of any rank over curve singularities. Our motivic superpolynomials are q,t,a-generalizations of orbital integrals associated with Affine Springer Fibers of type A in the case of the most general characteristic polynomials. I will not use the theory of ASF. The key conjecture is their coincidence with the DAHA superpolynomials of the corresponding (colored) algebraic links. The latter (due to Cherednik-Danilenko) will be defined. This coincidence can be seen as a high-level Shuffle Conjecture. As an application, the DAHA vertex will be considered and its relation to the q-theory of Riemann’s zeta. Also, q,t,a-deformations of the modified rho-invariants of algebraic knots will be discussed; classically, rho is defined via the Atiyah-Patodi-Singer eta invariant, but I will need only some formulas in this talk. See https://arxiv.org/abs/2304.02200.

Consider a simple algebraic group $G$ of classical type and its Lie algebra $\mathfrak{g}$. Let $(e,h,f) \subset \mathfrak{g}$ be an $\mathfrak{sl}_2$-triple and $Q_e= C_G(e,h,f)$. The torus $T_e$ that comes from the $\mathfrak{sl}_2$-triple acts on the Springer fiber $\mathcal{B}_e$. Let $\mathcal{B}_e^{gr}$ denote the fixed point loci of $\mathcal{B}_e$ under this torus action. Our main geometric result is that when the partition of $e$ has up to $4$ rows, the derived category $D^b(\mathcal{B}_e^{gr})$ admits a complete exceptional collection that is compatible with the $Q_e$-action. The objects in this collection give us a finite set $Y_e$ that is naturally equipped with a $Q_e$-centrally extended structure. We prove that the set $Y_e$ constructed in this way coincides with a finite set that has appeared in various contexts in representation theory. For example, a direct summand $J_c$ of the asymptotic Hecke algebra is isomorphic to $K_0(Sh^{Q_e}(Y_e\times Y_e)$. The left cells in the two-sided cell $c$ corresponding to the orbit of $e$ are in bijection with the $Q_e$-orbits in $Y_e$. Our main numerical result is an algorithm to compute the multiplicities of the $Q_e$-centrally extended orbits that appear in $Y_e$.

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Fusion categories satisfy stringent finiteness conditions: they are finite semisimple, abelian, rigid, etc. General finiteness conditions occur in higher category theory in the form of dualizability in symmetric monoidal categories. Together with Constantin Teleman, we apply topological quantum field theory--in particular, boundary theories therein--to characterize fusion categories among all tensor categories: fusion categories are tensor categories that are (1) dualizable and (2) the regular module category is also dualizable. The talk will include an exposition of relevant parts of topological field theory.

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Algebraic structures behind BPS counting on toric Calabi-Yau threefolds have recently been realized mathematically in terms of the quantum loop group associated to a certain quiver drawn on a torus. In this talk, we give a generators-and-relations presentation of the reduced version of this quantum loop group, assuming the quiver satisfies a technical condition we call "shrubbiness".

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Let S be a topological surface with holes. Let M(S,L) be the moduli space parametrising hyperbolic structures on S with geodesic boundary, and a given set L of lengths of the boundary circles. It carries the Weil-Peterson volume form. The volumes of spaces M(S,L) are finite. M.Mirzakhani proved remarkable recursion formulas for them, related to several areas of Mathematics.

However if S is a surface P with polygonal boundary, e.g. just a polygon, similar volumes are infinite. We consider a variant of these moduli spaces, and show that they carry a canonical exponential volume form. We prove that exponential volumes are finite, and satisfies unfolding formulas generalizing Mirzalkhani's recursions.

This part of the talk is based on the joint work with Zhe Sun.

There is a generalization of these moduli spaces for any split simple real Lie group G, with canonical exponential volume forms. When the modular group of the surface P is finite, our exponential volumes are finite for any G. When P are polygons, they provide a commutative algebra of positive Whittaker functions for the group G. The tropical limits of the positive Whittaker are the (zonal) spherical functions for the group G.

Braided monoidal categories have applications in various situations, in particular their universal R-matrices give solutions of the quantum Yang-Baxter equation and representations of braid groups of type A. There are powerful methods for constructing them: Drinfeld doubles of Hopf algebras and Drinfeld centres of monoidal categories. On the other hand, universal K-matrices, leading to solutions of the reflection equation and representations of braid groups of type B are much less well understood. We will describe a construction of reflective centers of module categories. It gives rise to braided module categories and a quantum double construction (a la Drinfeld) for universal K-matrices. From this perspective, quantum R-matrices come from categorical versions of centers of algebras and quantum K-matrices come from categorical versions of centers of bimondules. This is a joint work with Robert Laugwitz and Chelsea Walton.

L-functions play a central role in the theory of automorphic forms and Langlands conjectures. In particular, the conjectures predict existence of certain functions of one variable, the so called local L-factors, assigned to an irreducible representation of a p-adic group G (with a bit of extra data) and a representation of the dual group. When G=GL(n) equipped with the standard n-dimensional representation of the dual group, such a construction was proposed by Godement and Jacquet in 1972.

In a joint work with Braverman, Finkelberg and Kazhdan we propose a generalization of that construction in the functional field case based on geometry of the global Grassmannian. The main result is that the construction produces the expected answer for representations of G generated by an Iwahori invariant vector, its proof is based on my earlier work on the categorification of the affine Hecke algebra and perverse coherent sheaves on the nilpotent cone

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This talk is based on a joint work in progress with Gurbir Dhillon. A remarkable theorem of Feigin and E. Frenkel from the early 90's describes the center of the universal enveloping algebra of an (untwisted) affine Kac-Moody Lie algebra at the so called critical level proving a conjecture of Drinfeld. The center in question is the algebra of polynomial functions on an infinite dimensional affine space known as the space of opers. In our work we study a part of the center in positive characteristic p at an arbitrary non-critical level. Namely, we prove that the loop group invariants in the completed universal enveloping algebra is still the algebra of polynomials on an infinite dimensional affine space that is ``p times smaller than the Feigin-Frenkel center''. In my talk I will introduce all necessary notions, state the result, explain examples, motivations and some ideas of the proof.

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The calculation of the intersection cohomology of the moduli spaces of semistable bundles on Riemann surfaces goes back to the works of Frances Kirwan in the 1980’s. I will present a simple description of these cohomology groups in any rank, obtained as a component of the Decomposition Theorem applied to the parabolic projection map. This result was obtained in joint work with Camilla Felisetti and Olga Trapeznikova. It was in part motivated by the works of Mozgovoy and Reineke.

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