Constantin Teleman, UC Berkeley
Symmetry has long played a prominent role in constraining and governing physical laws. The projective symmetries of quantum mechanics and field theory ushered in the notion of higher symmetries effected by higher groups. Inspired by the TQFT of the last decades, a theory of “non-invertible” higher symmetries has been developed, involving notions of higher structured algebras and categories. The Simons Collaboration on Global Categorical Symmetries develops the general calculus of such higher symmetries, their action on, and implications for, quantum field theories.
The Simons Collaboration on Global Categorical Symmetries inaugural annual meeting will discuss progress on the following goals:
- describing the action of a TQFT of symmetries on a general QFT, with the concomitant calculus of topological defects and operators of varying codimension
- ongoing work on the higher categorical notions of semi-simplicity allowing for an algebraic treatment of high-dimensional TQFTs
- examples of TQFT actions on QFTs, with consequences for the dynamics and renormalization flow
- the constructions of such actions, starting from features present in the dynamics of the QFT.
Collaboration PIs wish to set the stage for a systematic exploration of further questions, including continuous higher symmetries, the study of non-perturbative features of conventional and unconventional (S)QFTs, implications of higher symmetry for the RG flow, and a methodical understanding of condensation of topological defects. They hope all participants will want to share their insights, ideas and contributions in informal interaction around the structured presentations.
9:30 AM Clay Córdova | Generalized Symmetry in Quantum Field Theory 11:00 AM David Jordan | Extended Symmetries in Skein Theory and in M-Theory 1:00 PM Dan Freed | What are topological symmetries in QFT? 2:30 PM Julia Plavnik | Galois Theory Techniques in the Classification of Modular Categories 4:00 PM Constantin Teleman | Towards a Universal Target for TQFTs
9:30 AM Theo Johnson-Freyd | Homotopy Quantum Groups 11:00 AM Michele Del Zotto | Categorical Symmetries from Higher Dimensions 1:00 PM Gregory Moore | Two Projects Of Possible Interest To This Collaboration
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
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.
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.
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.
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.
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.)
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.
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.