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.

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.

Recent work by Davide Gaiotto and collaborators introduced a new type of parametric Feynman integrals to compute BRST anomalies in topological and holomorphic quantum field theories. The integrand of these integrals is a certain differential form in Schwinger parameters. In a new article together with Simone Hu, we showed that this "topological" differential form coincides with a "Pfaffian" differential form that had been used by Brown, Panzer, and Hu, to compute cohomology of the odd graph complex and of the linear group. In my talk, I will review some aspects of the graph complex and the role played by the Pfaffian form there, sketch the proof of equivalence, and comment on various observations on either side of the equivalence and their natural counterparts on the other side.

In this talk, I will discuss a birational realization of King's conjecture which is indeed true, and its connections with noncommutative algebraic geometry and mirror symmetry. In particular, I will also establish the A-side analog of this result using constructible sheaves and promote the celebrated Coherent-Constructible Correspondence to a global family. The talk is based on recent joint work https://arxiv.org/abs/2501.00130 and work in preparation with David Favero.

In 2010, Kontsevich and Soibelman defined Cohomological Hall Algebras for quivers and potential as a mathematical construction of the algebra of BPS states. These algebras are modeled on the cohomology of vanishing cycles, which makes these algebras particularly hard to study but often result in interesting algebraic structures. A deformation of a particular case of them gives rise to a positive half of Maulik-Okounkov Yangians. The goal of my talk is to give an introduction to these ideas and explain how for the case of tripled cyclic quiver with canonical cubic potential, this algebra turns out to be one-half of the universal enveloping algebra of the Lie algebra of matrix differential operators on the torus, while its deformation turn out be one half of an explicit integral form of the Affine Yangian of gl(n).

Vertex operator algebras (VOAs) arise in many corners of supersymmetric quantum field theory. One particularly influential instance is in 4d N=2 superconformal field theories, whereby the VOA is realized as the cohomology of a suitable supercharge. Unitarity of the underlying SCFT imposes strong constraints on the structure of the resulting VOA. In this talk, I will describe one aspect of how the unitary of the underlying SCFT constrains this VOA: in the context of superconformal gauge theories, the resulting BRST complex shares a striking resemblance to the de Rham complex of a compact Kähler manifolds. I will finish with several consequences of this observation, e.g. the formality of these BRST complexes as in the work of Deligne-Griffiths-Morgan-Sullivan on compact Kähler manifold. This is based on work in progress with C. Beem.