Mathematical Physics

Recently, Bozec–Calaque–Scherotzke have articulated a noncommutative version of the AKSZ construction, which associates to a smooth Calabi–Yau category, a fully extended TQFT valued in a category of iterated Calabi–Yau cospans. In this talk, I will study a class of examples of such theories that arise in the context of conjectures of Costello and Li, which describe Type II strings in certain Ramond–Ramond backgrounds as topological strings. These TQFTs capture structural features of the BPS physics of D-branes that are universal in Chan–Paton factors. As an illustration, I will sketch a construction in which the outputs of such TQFTs give rise to Hall-type algebras. Examples of this paradigm include CoHAs associated to complex 3-folds, CoHAs attached to local systems on 3-manifolds, and the categorified Hall algebras of Porta–Sala.

We formulate the holomorphic twists of the 6d N=(0,1) and (0,2) abelian tensor multiplets as moduli spaces in derived geometry, using Deligne cohomology as a key tool. This description allows one to mimic the Beilinson-Drinfeld construction of lattice chiral algebras to quantize these 6d theories; their factorization homology on closed 3-folds relates to Witten's construction of line bundles on intermediate Jacobians. This is work in progress with Chris Elliott, Ingmar Saberi, and Brian Williams.

The moduli space of quasimaps gives a partial compactification of maps from an algebraic curve to a variety. In physics, the cohomology of this moduli space can be viewed as the state space of a (twisted) supersymmetric gauge theory. It is shown by Bullimore-Dimofte-Gaiotto-Hilburn-Kim that when the target space is a flag variety, this cohomology admits an action of the Lie algebra gl_n and can be identified with the Verma module for generic parameters. I will describe a further compactification of these moduli spaces called relative quasimaps, first introduced by Ciocan-Fontanine-Kim-Maulik, and explain how their cohomology is related to the tilting module of gl_n.

In QFT, there exists a well-known set of relations between the scattering amplitudes of gluons in gauge theory and those of gravitons in gravity, known collectively as the double copy. There are many beautiful ways in which this is manifested, famous examples being the KLT double copy, the double copy in the CHY formalism, and the BCJ double copy. In this talk I will show how the double copy relation can be seen arising from amplitude expressions derived using twistor sigma models and the RSVW and Cachazo-Skinner formulae. This talk is based on 2406.04539, with Tim Adamo.

We show that specific exponential integrals serve as generating functions of labeled edge-colored graphs. Based on this, we derive asymptotics for the number of edge-colored graphs with arbitrary weights assigned to different vertex structures. The asymptotic behavior is governed by the critical points of a polynomial. As an application, we discuss the Ising model on a random graph and show how its phase transitions arise from our formula.

K-theoretical Coulomb branches are expected to have cluster structure. Cautis and Williams categorified this expectation. In particular, they conjecture (and prove in type A) that the category of perverse coherent sheaves on the affine Grassmannian is a cluster monoidal categorification. We discuss recent progress on this conjecture. In particular, we construct cluster short exact sequences of certain perverse coherent sheaves. We do that by constructing a bridge, relating this (geometric) category to the (algebraic) category of finite dimensional modules over the quantum affine group. This is done by relating both categories to the notion of Feigin--Loktev fusion product.