Organized by Mina Aganagic, Ivan Danilenko, Andrei Okounkov, and Peng Zhou

Weekly on **Mondays 2:10 PM** (Pacific Time)

Meetings are in-person, at **402** Physics South

We have a lunch on the 4th floor of Physics South before the seminar

If you want to be added to the seminar **mailing list**, use this link https://forms.gle/Qk7Vpz4Uxv7rSyWaA

For those joining us remotely, we have a Zoom link: https://berkeley.zoom.us/j/99373923587?pwd=SUhXamdtbHhJMUJERlJ4NVJHL1Jtdz09

*Special Day/Time/Location.

This is a research seminar, intended for mathematicians and physicists. For the speaker to successfully reach the audience in both fields, it is important to explain, as clearly as possible: the motivations for the work, questions addressed, key ideas. The audience may fail to appreciate the glory of the result, otherwise.

Spring 2022 Fall 2021, Spring 2021, Fall 2020, Summer 2020, Spring 2020, Fall 2019, Spring 2019, Fall 2018, Fall 2017, Spring 2017, Fall 2016

This talk is a report of work in progress with Mina Aganagic, Yixuan Li and Ivan Danilenko. The B-side is the (resolved) additive Coulomb branch \(X\), and A-side is the (deformed) multiplicative Coulomb branch \(Y\) with a superpotential \(W\), and HMS says \( \mathrm{Coh}(X) = \mathrm{Fuk}(Y,W) \) where the RHS is the wrapped Fukaya category. Webster has shown \( \mathrm{Coh}(X) \) is equivalent to the category of \(A_{G,N}\)-mod, where \(A_{G,N}\) is the cKLRW algebra. We will show that \(\mathrm{Fuk}(Y,W)\) is also equivalent to \(A_{G,N}\)-mod. As a first step, we consider the gauge group \(G = \mathrm{GL}(k)\) and representation \(N =(\mathbb{C}^k)^n \), where we use Honda-Tian-Yuan's technique to turn endomorphism algebra of Lagrangian to certain subquotient of skeins on \(S^1 \times S^1 \times [0,1] \).

The spectral decomposition of the Hilbert space of automorphic functions is a very old and central topic in number theory, and mathematics in general. In particular, the Eisenstein series produce automorphic functions on a group G from automorphic functions on its Levi subgroups and one is interested in spectrally decomposing them. I will review some classical as well as recent results in this area, with an emphasis on some potential points of contact with the more traditional topics discussed in this seminar.

I will explain how a recent “universal wall-crossing” framework of Joyce works in equivariant K-theory, which I view as a multiplicative refinement of equivariant cohomology. Enumerative invariants, possibly of strictly semistable objects living on the walls, are controlled by a certain (multiplicative version of) vertex algebra structure on the K-homology groups of the ambient stack. In very special settings like refined Vafa-Witten theory, one can obtain some explicit formulas. For moduli stacks of quiver representations, this geometric vertex algebra should be dual in some sense to the quantum loop algebras that act on the K-theory of stable loci.

Ozsváth-Szabó's Heegaard Floer homology is a holomorphic curve analogue of the Seiberg-Witten Floer homology of closed 3-manifolds. Bordered Heegaard Floer homology is an extension of (one version of) Heegaard Floer homology to 3-manifolds with boundary, developed jointly with Ozsváth and Thurston. This talk is an overview of bordered Heegaard Floer homology. We will start by describing the structure and aspects of the construction of Heegaard Floer homology and bordered Floer homology, and then talk about some reinterpretations and refinements of bordered Floer homology. Some of this is work in progress with Ozsváth and Thurston.

I will discuss a symplectic version of annular Khovanov homology, taking place in Fukaya-Seidel categories of `horizontal' Hilbert schemes of type A Milnor fibres. This talk reports on joint work with Cheuk Yu Mak.

I will explain how to define a t-structure on the wrapped Fukaya category of a complex conic symplectic manifold, whose heart is the global sections of a perverse sheaf of categories on the core of the symplectic manifold. Here, a perverse sheaf of a category is just a sheaf of categories whose hom sheaves are (shifted) perverse. I will also show that the subcategory of objects with bounded microstalks (morally, the part of the wrapped Fukaya category coming from compact Lagrangians) admits Bridgeland stability conditions compatible with this t-structure.

Many gauge theories in four dimensions are based on PDEs that involve a gauge connection coupled to other fields. The latter are usually a source of a major headache since they lead to non-compactness of the moduli spaces. Today we will discuss two aspects of this major problem and two ways of dealing with it. One will help us understand how to perform cutting and gluing in Vafa-Witten theory, i.e. to what extent this theory is a (decorated) TQFT or, rather, how far it actually is from a decorated TQFT. This question is important for topological applications, such as behavior of the Vafa-Witten invariants under Gluck twist, log-transforms, and other surgery operations. The second part of the talk and another look at non-compactness of moduli spaces will lead us to a new "version" of the Vafa-Witten theory that is closer to vertex algebras VOA[M_4] and related developments.

Using brane quantization, we study the representation theory of the spherical double affine Hecke algebra of type \(A_1\) in terms of the topological A-model on the moduli space of flat \(\mathop{\mathrm{SL}}(2,\mathbb{C})\)-connections on a once-punctured torus. In particular, we provide an explicit match between finite-dimensional representations and A-branes with compact support; one consequence is the discovery of new finite-dimensional indecomposable representations. We proceed to embed the A-model story in an M-theory brane construction, closely related to the one used in the 3d/3d correspondence; as a result, we identify modular tensor categories behind particular finite-dimensional representations with \(\mathop{\mathrm{PSL}}(2,\mathbb{Z})\) action. Using a further connection to the fivebrane system for the class S construction, we go on to study the relationship of Coulomb branch geometry and algebras of line operators in 4d \(\mathcal{N}=2^*\) theories to the double affine Hecke algebra.

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 multiplicative Coulomb branch Y with superpotential W. If one decategorifies both sides by taking K-theory, the construction still gives an interesting identification between well-known objects in the equivariant K-theory of X and cycles with coefficients in local systems on Y. The talk will show how it works for the fixed point basis and the stable envelopes. Work in progress with Andrey Smirnov, with many insights from the joint project with Mina Aganagic, Peng Zhou and Yixuan Li.

I will discuss algebraic structures associated to moduli of sheaves on elliptic surfaces, and describe their relation with other parts of mathematical physics. These algebraic structures control the enumerative geometry of these moduli spaces analogous to how quantum groups control enumerative invariants of quiver varieties. The main results discussed will include a description of the quantum differential equation in these geometries and work in progress describing the relevant algebras as Hopf algebras with generalized versions of R matrices.

In her recent work, Mina Aganagic proposed novel perspectives on computing knot homologies associated with any simple Lie algebra. One of her proposals relies on counting intersection points between Lagrangians in Landau-Ginsburg models on symmetric powers of Riemann surfaces. In my talk, I am going to present a concrete algebraic algorithm for finding such intersection points, turning the proposal into an actual calculational tool. I am going to illustrate the construction on the example of the sl_2 invariant for the Hopf link. I am also going to comment on the extension of the story to homological invariants associated to gl(m|n) super Lie algebras, solving this long-standing problem. The talk is based on our work in progress with Mina Aganagic and Elise LePage.

Topological twists of 3d N=4 gauge theories admit boundary conditions that support vertex operator algebras, initially constructed by Costello and Gaiotto. I will review this construction, then discuss current work (with A. Ballin, T. Creutzig, and W. Niu) proving equivalences of these vertex algebras under abelian 3d mirror symmetry, and relating their module categories (a.k.a. bulk Wilson lines and vortex lines) to representations of certain quantum supergroups. We also directly relate these categories to those that have appeared in recent work of Oblomkov-Rozansky, Hilburn-Raskin, and Gammage-Hilburn on line operators.