MIT Infinite Dimensional Algebra Seminar (Fall 2015)

Deligne categories \(\text{Rep}(GL_t)\) (for a complex parameter \(t\)) have been constructed by Deligne and Milne in 1982 as a polynomial extrapolation of the categories of algebraic representations of the general linear groups \(GL_n(\mathbb{C})\). In this talk, we will show how to construct a "free abelian tensor category generated by one object of dimension \(t\)", which will be, in a sense, the smallest abelian tensor category which contains the respective Deligne's category \(\text{Rep}(GL_t)\). The construction is based on an interesting stabilization phenomenon occurring in categories of representations of supergroups \(GL(m\mid n)\) when \(t\) is an integer and \(m-n=t\). This is a joint project with V. Seganova and V. Hinich.

Let \(QH(X)\) be a quantum cohomology ring of some variety \(X\). The operation of quantum multiplication defines a flat connection on \(H^2(X)\) also known as quantum differential equation. In my talk I will discuss the generalization of this picture to the quantum K-theory of X given by a quiver variety. The corresponding differential equation is now substituted by a difference equation, which can be considered as a "flat difference connection" on a lattice (Picard group of \(X\)). I will discuss the relation of this difference connection with enumerative geometry, qKZ-equations, monodromy problem (for quantum connection), and quantum dynamical Weyl group.

Homological algebra plays a major role in noncommutative ring theory. One important homological construct related to a noncommutative ring \(A\) is the dualizing complex, which is a special kind of complex of \(A\)-bimodules. When \(A\) is a ring containing a central field \(K\), this concept is well-understood now. However, little is known about dualizing complexes when the ring \(A\) does not contain a central field (I shall refer to this as the noncommutative arithmetic setting). The main technical issue is finding the correct derived category of \(A\)-bimodules.

In this talk I will propose a promising definition of the derived category of \(A\)-bimodules in the noncommutative arithmetic setting. Here \(A\) is a (possibly) noncommutative ring, central over a commutative base ring \(K\) (e.g. \(K = \mathbb{Z}\)). The idea is to resolve \(A\): we choose a DG (differential graded) ring \(A'\), central and flat over \(K\), with a DG ring quasi-isomorphism \(A' \to A\). Such resolutions exist. The enveloping DG ring \(A'^{\text{en}}\) is the tensor product over \(K\) of \(A'\) and its opposite. Our candidate for the "derived category of \(A\)-bimodules" is the category \(D(A'^{\text{en}})\), the derived category of DG \(A'^{\text{en}}\)-modules. A recent theorem says that the category \(D(A'^{\text{en}})\) is independent of the resolution \(A'\), up to a canonical equivalence. This justifies our definition.

Working within \(D(A'^{\text{en}})\), it is not hard to define dualizing complexes over \(A\), and to prove all their expected properties (like when \(K\) is a field). We can also talk about rigid dualizing complexes in the noncommutative arithmetic setting.

What is noticeably missing is a result about existence of rigid dualizing complexes. When the \(K\) is a field, Van den Bergh had discovered a powerful existence result for rigid dualizing complexes. We are now trying to extend Van den Bergh's method to the noncommutative arithmetic setting. This is work in progress, joint with Rishi Vyas.

In this talk I will explain, in broad strokes, what are DG rings, DG modules, and the associated derived categories and derived functors. Also, I will try to go into the details of a few results and examples, to give the flavor of this material.

We'll discuss the algebraic geometry of the moduli space of vector bundles with connection (possibly with irregular singularities) on the formal punctured disc.

Previous studies have seen that (in higher rank) this moduli space exhibits quite pathological behavior, as we will recall in the talk. We will then explain that it appears much kinder through the lens of homological algebra (or more poetically, noncommutative geometry), exhibiting much better features than all its close relatives.

Finally, we will discuss how these results lend credence to the existence of a de Rham Langlands program incorporating arbitrary singularities (the usual story is unramified, or the worst has Iwahori ramification). Moreover, we will be able to use these results to formulate a precise conjecture that is a first approximation to a local geometric Langlands conjecture.

I will discuss an ongoing project that seeks to understand the Carlsson-Okounkov Ext operators in a variety of settings. These are geometric correspondences in the cohomology (or K-theory) rings of various quiver varieties, and my main focus is to show how they can be realized as intertwiners of Kac-Moody and Virasoro algebras (or quantum groups). As an application, we will obtain formulas for Nekrasov partition functions in N=2 gauge theory or for conformal blocks in CFT, as predicted by the AGT conjectures.

I will introduce quantizations of symplectic resolutions and their categories O that generalize classical BGG categories O in the representation theory of semisimple Lie algebras. Once a quantization is fixed, categories O are paramerized by chambers in a real vector space. The main result of this talk is that categories O for different chambers are derived equivalent (via so called cross-walling functors). Time permitting I will also discuss properties of these functors and their connection to Maulik-Okounkov geometric R-matrices. The talk is based on http://arxiv.org/abs/1502.00595.

In 1995 Alexander Kirillov-Jr. described a representation of the mapping class group of a torus, whose matrix elements are given by special values of Macdonald polynomials. This representation is a one-parameter deformation of the Reshetikhin-Turaev TQFT representation and gives rise to a one-parameter deformation of quantum invariants of torus knots. We will describe a construction of a representation of a genus two mapping class group that directly generalizes the Kirillov representation and gives rise to a one-parameter deformation of quantum invariants of genus two knots. The construction is built upon a one-parameter deformation of the square of q-6j symbol of U_q(sl_2), which we define using the somewhat mysterious Macdonald version of Fourier duality. The talk is based upon http://arxiv.org/abs/1504.02620.

Motivated by logarithmic conformal field theory and Gromov-Witten theory, I will introduce a notion of a twisted module of a vertex algebra relative to any (not necessarily semisimple) automorphism. Two features of such twisted modules are that they involve the logarithm of the formal variable and the action of the Virasoro operator \(L_0\) on them is not semisimple. I will derive a Borcherds identity and commutator formula for twisted modules. Examples for affine vertex algebras, free bosons, and symplectic fermions will be presented.

Recently, Anton Mellit and I gave a proof of the famous shuffle conjecture of Haglund, Haiman, Loehr, Ulyanov, and Remmel, which predicts a combinatorial formula for the character of the diagonal coinvariant algebra, and other quantities in algebraic geometry. I'll explain what this conjecture is about, and explain the algebraic structures that go into this recent proof. Hopefully if there's time, I'll explain some of remarkable unsolved generalizations, and their role in algebraic geometry.

Kontsevich and Soibelman defined Donaldson-Thomas invariants of a 3d Calabi-Yau category with a stability condition. Any cluster variety gives rise to a family of such categories. Their DT invariants are encapsulated in single formal automorphism of the cluster variety, called the DT-transformation.

An oriented surface S with punctures, and a finite number of special points on the boundary give rise to a moduli space, closely related to the moduli space of PGL(m)-local systems on S, which carries a canonical cluster Poisson variety structure. We determine the DT-transformation of this space.

This is a joint work with Linhui Shen.

Given an interesting sequence of objects X_0, X_1, X_2, ... , it's fun to ponder: is that subscript just a natural number, or is it hiding deeper structure? Maybe these objects ought to be indexed by finite sets, or finite dimensional vector spaces, or finite total orders, or whatever makes sense. The point is, a map relating two of the indexing gadgets should induce a map on the objects themselves. We use this method of argument to give new results on the homotopy groups of configuration spaces. Next, we give a characterization of indexing categories wherein finitely generated representations are finite length. Finally, we show how computations with these representations can be made effective. Part of this talk is joint work with Jordan Ellenberg.