Talks

In this talk, I will talk about the representation theory of the spherical double affine Hecke algebra (DAHA) of C∨ C1, using brane quantization. By showing a one-to-one correspondence between Lagrangian A-branes with compact support and finite-dimensional representations of the spherical DAHA, we provide evidence of derived equivalence between the A-brane category of SL(2,C)-character variety of a four-punctured sphere and the representation category of DAHA of C∨ C1. The D4 root system plays an essential role in understanding both the geometry and representation theory. In particular, this A-model approach reveals the action of an affine braid group of type D4 on the category.

We discuss a sequence of projects (some in progress) constructing "line operator categories for 4d HT theories" from CY3 manifolds and categories. Concretely - in certain settings, the CY3 cohomological Hall algebra B (due to Kontsevich--Soibelman any many others) we show admits a Drinfeld-style vertex coproduct coloured by the cohomology of the CY3. We explain how to apply Majid vertex bosonisation to this structure to get a category B-Mod(H-Mod) factorising over the complex line C. This involves proving that B is a "vertex quantum group". Examples include quivers with potential - where we compare to Drinfeld/Yang-Zhao/Davison coproducts - canonical bundles of CY2s, deformed CY3 completions (and by work in progress - a larger class of examples). Joint with Šarūnas Kaubrys, Shivang Jidnal, (and Pierre Descombes)

In my talk, I will present a version of the de Rham isomorphism for derived manifolds and derived differentiable stacks. I will also show how the naive version of the de Rham theorem fails for general derived manifolds, but holds in analytic contexts, such as for derived analytic manifolds. The talk is based on a recent preprint: arXiv:2505.03978.

There is a family of Fukaya categories, or categories of A-branes, labeled by a choice of a Lie algebra, which categorify Chern-Simons invariants of links. I will describe how categorified quantum groups emerge from a certain natural set of gluing operations which these Fukaya categories have. This solves a long standing open problem, which is to provide the categorified co-product, providing a monoidal structure for the categorified representations of the quantum group. As a byproduct, this illuminates how quantum group symmetry emerges in Chern-Simons theory.

I will describe a new method for constructing conformal blocks for the Virasoro vertex algebra with central charge c=1, by relating them to conformal blocks for the Heisenberg algebra on a branched double cover. The construction is joint work with Qianyu Hao. It fits into the "holomorphic-topological" theme of the conference only inasmuch as vertex algebras are holomorphic objects and nonabelianization involves topological line defects. One concrete output is new formulas for tau functions in the sense of integrable systems -- most concretely, for Painleve equations -- as I will briefly explain; this part is joint work in progress with Dawit Belayneh and Davide Gaiotto.