2022 Simons Collaboration on Global Categorical Symmetries Annual Meeting

Clay Córdova
University of Chicago

Generalized Symmetry in Quantum Field Theory

Clay Córdova will survey recent developments in the notion of symmetries in quantum field theory. Viewing symmetries as topological operators embeds the rich framework of topological quantum field theory and higher category theory in the setting of more general field theories. This provides a powerful framework which organizes dynamics in physical systems, as well as applications of field theory to geometry and representation theory. Córdova will illustrate these themes in several concrete settings and discuss open questions.
 

Michele Del Zotto
Uppsala University

Categorical Symmetries from Higher Dimensions

A central goal in the study of categorical symmetries is constructing explicit examples of non-topological quantum field theories (QFTs) exhibiting such structure. In this talk, Michele Del Zotto will discuss progress on the subject, building on higher dimensional systems. In the first part of the talk, Del Zotto will focus on applications of stringy techniques to geometrically engineer categorical symmetries. In the second part, applications of six-dimensional superconformal systems will be discussed. This talk reports on several joint projects written with subsets of the following list of co-authors during year one of the collaboration: Vladimir Bashmakov, Iñaki García Etxebarria, Azeem Hasan, Jonathan J. Heckman, Justin Kaidi, Shani Nadir Meynet, Robert Moscrop, Sakura Schäfer-Nameki, Ethan Torres and Hao Zhang.
 

Dan Freed
University of Texas at Austin

What are topological symmetries in QFT

Motivated by analogs in representation theory, together with Constantin Teleman and Greg Moore we introduce a framework for topological symmetries in field theory together with a calculus of topological defects based on techniques in topological field theory. Crucial is the treatment of symmetries and anomalies as quantum, rather than classical, as I will illustrate in an example.
 

Theo Johnson-Freyd
Dalhousie University & Perimeter Institute

Homotopy Quantum Groups

Systems of global categorical symmetry can be thought of as quantum higher groups; Theo Johnson-Freyd will define and describe their homotopy quantum groups in which two operators represent the same class if they are related by a quantum (noninvertible) homotopy. When the categorical symmetry is a usual higher group, these homotopy quantum groups recover its usual homotopy groups. For fusion higher categories, two operators are in the same “quantum homotopy class” if and only if they are related by a condensation of operators of higher homotopical degree. Although in general the homotopy quantum groups can reflect the noninvertible nature of categorical symmetries, it happens remarkably often that they are honest groups — that all operators are “invertible up to quantum homotopy,” but that the homotopy itself is noninvertible. This provides an interesting middle ground between fully invertible and fully noninvertible symmetries. This talk is based on joint work with David Reutter.
 

David Jordan
University of Edinburgh

Extended Symmetries in Skein Theory and in M-Theory

This talk will highlight insights between mathematics and physics of higher form symmetries.
In joint work with Gunningham and Safronov, David Jordan will provide physical insight about higher form symmetries that help to confirm a mathematical conjecture relating the dimensions of skein modules for a group G and its Langlands dual group GL, in two broad classes of examples.

Mathematical insights from homotopy theory will clarify and help to analyze higher form symmetries in the context of geometric engineering in M-theory. This is joint with Freed, Etxebarria and Del Zotto.
 

Gregory Moore
Rutgers University

Two Projects Of Possible Interest To This Collaboration

Gregory Moore will review work done with Roman Geiko on time-reversal symmetry in 3d Chern-Simons theory. Moore will prove that in the case of a torus gauge group the theory has a time-reversal symmetry iff all the higher Gauss sums are real. He conjectures that a similar criterion applies to the nonabelian case.

Moore will conclude by addressing some questions that arise when using topologically twisted supersymmetric Yang-Mills theory to give physical derivations of diffeomorphism invariants of smooth compact four-manifolds. Among other things there will be a discussion of the so-called “K-theoretic Donaldson invariants” obtained from 5d SYM theory. (This last part is work in progress with Heeyeon Kim, Jan Manschot, Runkai Tao, and Xinyu Zhang.)
 

Julia Plavnik
Indiana University

Galois Theory Techniques in the Classification of Modular Categories

In this talk, Julia Plavnik will discuss how Galois symmetry plays a fundamental role in determining the structure of a modular category. Plavnik will show general results on how the Galois action interacts with fusion subcategories. They will also show applications of these results to the classification of modular categories.
 

Constantin Teleman
University of California, Berkeley

Towards a Universal Target for TQFTs

Topological quantum field theories (TQFTs) of Reshetikhin-Turaev (RT) type were not constructed as fully local theories, thus not immediately within reach of the powerful calculus of the Lurie’s cobordism hypothesis; yet, they behave as if they were. In joint work with Dan Freed and Claudia Scheimbauer, Constantin Teleman will resolve this puzzle by constructing a dualizable 3-category in which every RT theory is fully local. Related to this, Teleman will modify a natural target for 4-dimensional TQFTs that eliminates the ‘Witt group’ of modular tensor categories. This is consistent with a conjecture of Hopkins that the universal target for TQFTs should have the Pontrjagin dual of the sphere as spectrum of units.