Conference

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. "

"A unitary 1d QFT consists of a Hilbert space and a Hamiltonian. A group acting on a 1d QFT is a group acting on the Hilbert space, commuting with the Hamiltonian. Note that the *data* of an action only involves the Hilbert space. The Hamiltonian is only there to provide a constraint. Moreover, all 1d QFT have isomorphic Hilbert spaces (except in special cases, e.g. in the case of a 1d TQFT, when the Hilbert space is finite dimensional). A unitary 2d QFT consists of the 0-dimensional and 1-dimensional part of the QFT, along with the data of the Stress-energy tensor. An action of a fusion category on a 2d QFT is again something where the *data* only involves the 0-dimensional and 1-dimensional part of the QFT, while the Stress-energy tensor is only there to provide a constraint. The upshot is that it makes sense to act on the 0-dimensional and 1-dimensional part of the QFT. Moreover, I conjecture that all 2d QFTs have isomorphic 0-dimensional + 1-dimensional parts (except in special cases, e.g. in the case of a chiral CFT)."

"It is possible to construct interesting field theories by placing string theory on suitable singular geometries, and adding branes. In the fairly special cases where Lagrangians are known for the resulting theories, field theory arguments often show that these theories have generalised symmetry structures. In this talk I will review recent work developing a dictionary, valid even in the absence of a known Lagrangian description, between properties of the string theory geometry and generalised symmetries of the associated field theories."

" I will talk about a monoidal localization theorem for the small quantum group u_q(G), where G is a reductive algebraic group and q is a root of unity. In joint work with Julia Pevtsova, we show that the category of representations for u_q(G) admits a fully faithful tensor embedding into the category of coherent sheaves over a “quantum” flag variety. This quantum flag variety is, essentially, some finitely fibered space over the classical flag variety G/B. I will explain how this embedding theorem codifies certain relationships between the small quantum group and its quantum Borels. "

I will review the analytic component of the geometric Langlands correspondence, developed recently in my joint work with E. Frenkel and D. Kazhdan (based on previous works by other authors, including A. Braverman, R. Langlands, J. Teschner, M. Kontsevich), with a special focus on archimedian local fields, especially R. This is based on our work with E. Frenkel and D. Kazhdan and insights shared by D. Gaiotto and E. Witten.

We identify infinitely many non-invertible generalized global symmetries in QED and QCD for the real world in the massless limit. In QED, while there is no conserved Noether current for the axial symmetry because of the ABJ anomaly, for every rational angle, we construct a conserved and gauge-invariant topological symmetry operator. Intuitively, it is a composition of the axial rotation and a fractional quantum Hall state coupled to the electromagnetic U(1) gauge field. These conserved symmetry operators do not obey a group multiplication law, but a non-invertible fusion algebra over TQFT coefficients. These non-invertible symmetries lead to selection rules, which are consistent with the scattering amplitudes in QED. We further generalize our construction to QCD, and show that the neutral pion decay can be understood from a matching condition of the non-invertible global symmetry.