Mathematical Physics

I will discuss some of the (higher) structure of TQFT's that can be deformed by flat connections for continuous global symmetries, focusing on examples coming from twists of 3d supersymmetric theories, and the manifestation of this structure in boundary VOA's.

I will consider four-dimensional gauge theories whose global symmetries admit certain discrete ’t Hooft anomalies that are intimately related to the (fractionalized) global-symmetry quantum numbers of Wilson-’t Hooft line defects in the theory. Determining these quantum numbers is typically straightforward for Wilson lines, but requires a careful analysis of fermion zero modes for ’t Hooft lines, which I will describe for several classes of examples. This in turn leads to a calculation of the anomaly. Along the way I will comment on how this understanding relates to some classic and recent examples in the literature.

"I will discuss a proposal for generating non-invertible symmetries in QFTs in d>2, by gauging outer automorphisms. First this will be illustrated in 3d, where the framework is relatively well established, and then extended to higher dimensions. For 4d gauge theories, a comparison to other approaches to non-invertible symmetries is provided, in particular the map to gauging theories with mixed anomalies. This talk is based on work that appeared in 2204.06564 and in progress, with Lakshya Bharwaj (Oxford), Lea Bottini (Oxford) and Apoov Tiwari (Stockholm)."

String theory constructions allow one to realize vast classes of non-trivial quantum field theories (QFTs), including many strongly coupled models that elude a conventional Lagrangian description. ’t Hooft anomalies for global symmetries are robust observables that are particularly well-suited to explore QFTs realized in string theory. In this talk, I will discuss systematic methods to compute anomalies of theories engineered with branes, using as input the geometry and flux configuration transverse to the non-compact directions of the branes worldvolume. Examples from M-theory and Type IIB string theory illustrate the versatility of this approach, which can capture both ordinary and generalized symmetries, continuous or discrete.

"Every braided fusion category has a `framed S-matrix pairing' which records the braiding between simple objects. Non-degeneracy/Morita invertibility of the category (aka `modularity' in the oriented case) is equivalent to non-degeneracy of this pairing. I will define higher-dimensional versions of S-matrices which pair morphisms of complementary dimension in higher semisimple categories and sketch a proof that these pairings are non-degenerate if and only if the higher category is. Along the way, I will introduce higher semisimple categories and higher fusion categories and interpret these results in terms of the associated anomalous topological quantum field theories. This is based on joint work in progress with Theo Johnson-Freyd."

"Khovanov showed in ‘99 that the Jones polynomial arises as the Euler characteristic of a homology theory. The knot categorification problem is to find a general construction of knot homology groups and to explain their meaning: what are they homologies of? Homological mirror symmetry, formulated by Kontsevich in ’94, naturally produces hosts of homological invariants. Sometimes, it can be made manifest, and then its striking mathematical power comes to fore. Typically though, it leads to invariants which have no particular interest outside of the problem at hand. I will explain that there is a vast new family of mirror pairs of manifolds for which homological mirror symmetry does lead to interesting invariants, and solves the knot categorification problem. "