2024 Simons Collaboration on Global Categorical Symmetries Annual Meeting

Clay Còrdova
University of Chicago

Representation Theory of Solitons

Solitons in two-dimensional quantum field theory exhibit patterns of degeneracies and associated selection rules on scattering amplitudes. We develop a representation theory that captures these intriguing features of solitons. This representation theory is based on an algebra we refer to as the “strip algebra,” which is defined in terms of the non-invertible symmetry, i.e., a fusion category, and its action on boundary conditions encoded by a module category. The strip algebra is a weak Hopf algebra, a fact which can be elegantly deduced by quantizing the three-dimensional Drinfeld center TQFT, on a spatial manifold with corners. These structures imply that the representation category of the strip algebra is also a unitary fusion category which we identify with a dual category. We present a straightforward method for analyzing these representations in terms of quiver diagrams where nodes are vacua and arrows are solitons and provide examples demonstrating how the representation theory reproduces known degeneracies and selection rules of soliton scattering. Our analysis provides the general framework for analyzing non-invertible symmetry on manifolds with boundary and applies both to the case of boundaries at infinity, relevant to particle physics, and boundaries at finite distance, relevant in conformal field theory or condensed matter systems.
 

Michele Del Zotto
Uppsala University

Exploring the higher structure of symmetries

I will review recent progress in understanding the higher structure of topological defects in higher dimensional field theories (as well as some of its applications). In particular, I will discuss and contrast aspects of the worldvolume approach towards the calculus of defects and the topological symmetry theory, as well as some consequences of higher structures on RG flows, allowing to establish dictionaries between UV and IR physics. Along the way, I will also briefly touch upon some aspects of recent proposals to generalize the topological symmetry theory framework to include continuous global symmetries.
 

Michael Hopkins
Harvard University

The mutual influence of homotopy theory and quantum field theory

I will describe some projects in the collaboration that involve the application of homotopy theoretic methods to questions arising from quantum field theory. My focus will be on the ways in which this relationship has generated new research directions in both fields.
 

Kenneth Intriligator
University of California at San Diego

Categorical Symmetries in QFT: Progress and Outlook II

We will continue to review some of the major motivations and themes of our collaboration’s work, highlighting applications of generalized symmetries towards a deeper understanding of quantum field theory, renormalization group flows, and IR phases.
 

David Jordan
University of Edinburgh

The quantum A-polynomial from parabolic defects

I will describe a joint project with Jennifer Brown to construct the so-called quantum A-polynomial invariants of knots functorially using the theory of parabolic induction and restriction for quantum groups, encoded via certain non-invertible defects. We obtain an algorithm computing our invariant from an ideal triangulation of the knot complement; the algorithm is sufficiently explicit that it can be executed in Sage.
 

Julia Plavnik
Indian University

The Homotopy Coherent Classification of Fusion 2-Categories

Fusion 2-categories are a higher categorical analog of fusion categories that have gained a lot of attention in the last years because of their importance in many fields of math and physics, such as TQFTs, condensed matter and high energy physics. The classification of fusion (1-) categories is a very active research area and has provided new examples and led to the development of new invariants and tools to understand these categories.

In this talk, we will present a parametrization of multifusion 2-categories in terms of lower categorical data, involving braided fusion categories, group theory and cohomological data. If time allows, we will also show some applications of this result. This is a joint work in progress with Décoppet, Johnson-Freyd, Huston, Nikshych, Penneys, Reutter and Yu.
 

Tomer Schlank
University of Chicago

Ambidexterity and Quantum Field Theories

Ambidexterity is an ∞-categorical phenomenon that leads to a highly structured form of integration. For certain types of quantum field theories, such as Dijkgraaf-Witten theories, this form of integration enables a mathematically rigorous definition of path integrals. In this talk, we will examine how results from the abstract theory of ambidexterity can be used to compute universal target (∞,n)-categories for quantum field theories with various structures and properties motivated by physics. These include universal targets with well-behaved partition functions or those exhibiting electro-magnetic duality. Additionally, we will discuss ongoing efforts to extend this approach to more general forms of quantum field theories. This talk is based on collaborative projects with Scheimbauer, Del Zotto, Ohmori, and Yanovski.
 

Constantin Teleman
University of California Berkeley

Categorical Symmetries in QFT: Progress and Outlook I

We will review some major themes of our collaboration work, listing some of the significant results and methods: the Quiche/SymTFT calculus of symmetries and defects, the role of homotopy theory, boundary theories, dualities and anomalies; classification of structured categories; and gauge theory dualities and applications to geometry. Many of the ideas introduced will be developed comprehensively in our program lectures. This talk will emphasize the mathematical side story and conclude with open questions building on recent developments.