MIT Infinite Dimensional Algebra Seminar (Fall 2023)

Let g be a symmetrisable Kac-Moody algebra. The corresponding quantum Weyl group operators give an action of the braid group of g on integrablemodules over the quantum group U_hg. Answering a question of Pavel Etingof, we prove that the restriction of this action to the pure braid group extends to all (not necessarily integrable) category O modules for Uhg. We also show that this action describes the monodromy of the Casimir connection of g on category O modules. This is joint work with Andrea Appel, and is based on https://arxiv.org/abs/2208.05331 [1]

The moduli space of rational curves with n+1 marked points and a tangent vector at the first of them (called framing) has a natural compactification F_n analogous to the Deligne-Mumford one for the space of non-framed curves, called moduli space of cactus flower curves. Contrary to the usual Deligne-Mumford compactification, it is singular -- but still has many nice properties. I will explain how the real locus of this moduli space controls coboundary monoidal categories that are finite and concrete (i.e. endowed with a faithful monoidal functor to finite sets). The main example of such a category is the category of Kashiwara crystals for a semisimple Lie algebra g. Using the moduli space of cactus flower curves, we can determine the category of Kashiwara g-crystals from some quantum integrable spin chains related to g (namely, XXX Heisenberg chain and its Gaudin degeneration), without referring to canonical bases or to Stembridge axioms. This also describes the monodromy of solutions to Bethe ansatz in the corresponding integrable systems in terms of the Schutzenberger involutions on Kashiwara crystals.

This is a joint work in progress with Aleksei Ilin, Joel Kamnitzer, Yu Li, and Piotr Przytycki.

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It turns out that the most involved step in the recently obtained proof of the geometric Langlands conjecture is concerned with the compatibility of the Langlands functor with Constant Terms. We will explain how to prove it via a representation-theoretic result about representations of affine algebras at the critical level.

Following Gaitsgory's talk on Eisenstein series from the week before, we will explain some tricks for how to conclude that the geometric Langlands functor is an equivalence. This work is all joint with Gaitsgory, and some parts are joint with Arinkin, Beraldo, Chen, Faergeman, and Lin.

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Homological mirror symmetry predicts an equivalence between the derived category of equivariant coherent sheaves on the additive Coulomb branch X and a version of the wrapped Fukaya category on the multiplicative Coulomb Y branch with a superpotential. Under decategorification, we get an interesting identification between the equivariant K-theory of X and homology of Y. In this talk, we will show how special bases in the equivariant K-theory (fixed points, stable envelopes) are mirrored by cycles on Y. This is work in progress with Andrey Smirnov, with insights from a joint project with Mina Aganagic, Yixuan Li, Vivek Shende, and Peng Zhou.

Analytic Langlands correspondence was proposed by Etingof, Frenkel and Kazhdan based on ideas and results of Langlands, Teschner, Braverman-Kazhdan and Kontsevich. Let $X$ be a smooth irreducible projective curve over $\mathbb{C}$, $G$ be a semisimple group. On one side of this conjectural correspondence there are $G^{\vee}$-opers on $X$ satisfying a certain condition ({\it real} opers), where $G^{\vee}$ is Langlands dual group. On the other side there are certain operators on $L^2(Bun_G)$, called Hecke operators, where $Bun_G$ is the variety associated to the moduli stack of stable $G$-bundles on $X$ and $L^2(Bun_G)$ is a Hilbert space of square-integrable half-densities. I will describe the main picture and present new results in this direction. Partially based on joint projects with A. Wang and S. Raman.

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Let $G$ be a split reductive group over a finite field $k$. A well-known result of Prasad says that the space of (complex-valued) functions with finite support on the moduli space of $G$-bundles on $P^1$ over $k$ endowed with Borel structures at $0$ and infinity is naturally isomorphic to the regular bi-module over the Iwahori-Hecke algebra of $G$ over the field $k((t))$. We shall present a generalization of this result in two directions: first, we replace Borel structures at $0$ and infinity with a full trivialization at those points, and also we replace the finite field $k$ with a local non-archimedian field $F$.

We shall also discuss two types of applications of this generalization:

1. We show how the relativistic Calogero-Moser variety of $G$ arises as the spectrum of Hecke operators over $F$ in a certain special case
2. We give a conjectural explicit description of the character of the lifting of a cuspidal representation of $G(F)$ to the group $G(E)$ for any finite extension $E$ of $F$.

This is a joint work in progress with D. Kazhdan.

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The amplituhedron $A_{n,k,m}(Z)$ is the image of the positive Grassmannian under a map induced by a positive linear map. It was originally introduced in physics by Arkani-Hamed and Trnka in order to give a geometric interpretation of scattering amplitudes and in particular the Britto-Cachazo-Feng-Witten recurrence. The amplituhedron has since been linked to cluster algebras and interesting objects from combinatorics. I will give an introduction to the amplituhedron and discuss our recent proof of the cluster adjacency conjecture for BCFW tiles, as well as the BCFW tiling conjecture. Based on joint work with Even-Zohar, Lakrec, Parisi, Sherman-Bennett, and Tessler.

Lecture Slides

This talk is complementary to Lauren Williams' recent talk. We'll discuss several motivating examples from algebra/geometry/physics and related combinatorial objects (plabic graphs, tilings, and positroids). We'll emphasize the connections between these objects and polyhedral geometry.

This combinatorial structure can be described in elementary terms accessible to high school students as a game on a certain class of tilings of polygons by triangles.

This game on tilings is related to Lusztig's theory of total positivity (in type A) and to works by Arkani-Hamed et al on scattering amplitudes. It helps to construct the geometric RSK correspondence, to prove the Littlewood-Richardson rule, to describe Berenstein-Zelevinsky polytopes and string cones of Kashiwara's parametrizations of gln-crystals, to generalize the octahedron recurrence, etc.

We'll mention different directions to extend these combinatorial structures to other Cartan-Killing types and beyond the planar limit.

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Quantum connections of symplectic manifolds X are flat connections defined from Gromov—Witten theory of X, acting on singular cohomology of X. When the target manifold X is a symplectic resolution, the quantum connection of X is known to recover and generalize well-known flat connections from representation theory.

On cohomology of X with coefficients in positive characteristic, Fukaya and Wilkins have defined new operations that deform the action of the classical Steenrod operations using symplectic Gromov—Witten theory. I will discuss results from the study of these quantum Steenrod operations for (conical) symplectic resolutions, in particular their relationship with the mod p quantum connection and its p-curvature.

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