Mathematical Physics

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.