In this talk I will review some recent progress in the study of non-invertible symmetries in dimensions d>2. After introducing known constructions and describing how they lead to constraints on RG flows, I will discuss how non-invertible symmetries can also be used to obtain new RG flows. This involves the notion of ``non-invertible twisted compactification,” which can be used to construct...
This will be an introductory discussion of our joint work with Roman Bezrukavnikov. Given a symplectic resolution X, one may study its Gromov-Witten theory and the monodromy group of the curve-counting functions in the K\"ahler variables. There is also a large group of derived autoequivalences of X coming from its quantization in large prime characteristic, as studied by Bezrukavnikov and...
"Motivated by applications to soft supersymmetry breaking, we revisit the Seiberg-Witten solution for N=2 super Yang-Mills theory in four dimensions with gauge group SU(N). We present a simple exact Taylor series expansion for the periods obtained at the origin of moduli space, thereby generalizing earlier results for SU(2) and SU(3). With the help of these analytic results and others, we...
I'll discuss a vertex algebra whose correlators are scattering amplitudes (and form factors) of self-dual Yang-Mills theory, for certain gauge groups and matter. The vertex algebra is a kind of vertex quantum group, and is a cousin of the affine Yangian. This is joint work with Natalie Paquette.
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...
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...
A frequent theme in mathematical approaches to quantum field theory is being able to draw intuitive but rigorous pictures of particle interactions. For example, in theories like QED and QCD, Feynman diagrams can be organized into a Hopf algebra structure in such a way that models renormalization of Feynman integrals. One can draw an analogy with the role of Hopf algebras in (2+1)D topological...
" 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,...
"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...
"We will review the role of Quantum Field Theory (QFT) in modern physics. We will highlight how QFT uses a reductionist perspective as a powerful quantitative tool relating phenomena at different length and energy scales. We will then discuss various examples motivated by string theory and lattice models that challenge this separation of scales and seem outside the standard framework of QFT....
"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,...
"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...
"Every braided fusion category has a framed S-matrix pairing' which records the braiding between simple objects. Non-degeneracy/Morita invertibility of the category (akamodularity' 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...
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...
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...
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...
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.
