Categorical Symmetries in Quantum Field Theory (Conference and School)

27 Aug 2023, 09:00 → 8 Sept 2023, 12:15 Europe/Zurich

SRS

Alberto Cattaneo (University of Zurich), Claudia Scheimbauer (TU Munich), Constantin Teleman (University of Berkeley), Dan Freed (U. of Texas in Austin), Lennart Döppenschmitt (University of Zurich), Lukas Müller (Perimeter Institute), Michele Delzotto (Uppsala Universitet), Thomas Dumitrescu (UCLA)

Program

28 August – 1 September 2023: Research Conference

Speakers:

• David Ayala (Montana State University)
• Maissam Barkeshli (University of Maryland)
• Mathew Bullimore (Durham University)
• Owen Gwilliam (University of Massachusetts Amherst)
• Kenneth Intriligator (University of California San Diego)
• Anton Kapustin (California Institute of Technology)
• Zohar Komargodski* (Simons Center for Geometry and Physics)
• Catherine Meusburger (Universität Erlangen-Nürnberg)
• Kantaro Ohmori ( University of Tokyo)
• Sakura Schafer-Nameki (University of Oxford)
• Christoph Schweigert (Universität Hamburg)
• Peter Teichner (Max Planck Institut für Mathematik)
• Ashvin Vishwanath (Harvard University)
• Kevin Walker (Microsoft Station Q)
• Chelsea Walton (Rice University)
• Mayuko Yamashita (Kyoto University)

 

(*to be confirmed)

The aim of this workshop is to bring together researchers from both physics- in topological quantum field theory, high energy physics, and condensed matter- and mathematics- higher categories, representation theory, and topology- to report on recent progress and open questions in the study of generalised notions of symmetry. There have been significant recent developments in exploring categorical symmetries in each of these fields, so the time is ripe to bring researchers from these communities together to promote a cross-fertilisation of ideas and insights. The primary theme of the workshop is to develop a common framework for defining and working with categorical symmetry in quantum field theory.

3-8 September 2023: School

Lectures:

• Applied Cobordism Hypothesis - David Jordan (University of Edinburgh)
• Non-invertible Symmetries - Shu-Heng Shao (Stony Brook University)
• The Mathematics of TQFTs and Defects - Constantin Teleman (University of California Berkeley)
• Symmetry Categories 101 - Michele Del Zotto (Uppsala Universitet)

 

This will be a hybrid mode event: most speakers will be in-person, with some virtual speakers; about half of the participants will be in-person, and talks will also be live-streamed on Zoom. Videos will be archived on a youtube channel. 

Immediately preceeding GCS2023 there are two distinct events, a Nordita Program (August 14-25) and a conference in Regensburg (August 14-18). Although these events are logically separate, with different organizing teams and registrations, we purposefully scheduled these events close to each other to avoid too many transatlantic flights.

 

2023.simons_dialogue_handout.pdf

Monday 28 August

1 Christoph Schweigert: String-net methods for CFT correlators

Based on a graphical calculus for pivotal bicategories, we develop a string-net construction of a modular functor. We show that a rigid separable Frobenius functor between strictly pivotal bicategories induces a linear map between the corresponding bicategorical string-net spaces that is compatible with the mapping class group actions and with sewing. This result implies that correlators of two-dimensional conformal field theories factorize over string-net spaces constructed from defect data.

10:00 Coffee break

2 Anton Kapustin: Symmetries, anomalies, and the bulk-boundary correspondence

’t Hooft anomalies are obstructions to gauging a global symmetry of a QFT. In one spatial dimension ’t Hooft anomaly of a Lie group symmetry can also be described purely algebraically, without a reference to gauging: it manifests itseld a non-trivial central extension of the current algebra. In higher dimensions, there is no completely satisfactory algebraic reformulation of ’t Hooft anomaly. In this talk, I will argue that such an algebraic reformulation should involve higher-form symmetries. To support this claim I will discuss analogous issues for gapped lattice systems in one dimension higher which are related to QFT via the bulk-boundary correspondence.

3 Mayuko Yamashita: Topological modular forms and heretoric string theory

In this talk I will explain my works with Y. Tachikawa to study anomaly in heterotic string theory via homotopy theory, especially the theory of Topological Modular Forms (TMF). TMF is an E-infinity ring spectrum which is conjectured by Stolz-Teichner to classify two-dimensional supersymmetric quantum field theories in physics. In the previous work (https://arxiv.org/abs/2108.13542), we proved the vanishing of anomalies in heterotic string theory mathematically by using TMF.
Furthermore, we have a recent update (https://arxiv.org/abs/2305.06196) on the previous work. Because of the vanishing result, we can consider a secondary transformation of spectra, which is shown to coincide with the Anderson self-duality morphism of TMF. This allows us to detect subtle torsion phenomena in TMF by differential-geometric ways.

16:00 Coffee break

4 Simons Dialogue

Tuesday 29 August

5 Maissam Barkeshli: TBA

10:00 Coffee break

6 Mathew Bullimore: Representation theory for categorical symmetries

My talk is about how categorical symmetries act on and organise the spectrum of non-topological extended operators in QFT. I will address this question from two equivalent perspectives: the representation theory of higher tube algebras and the sandwich construction of symmetries via the Drinfeld center. I'll discuss a variety of examples in three dimensions: invertible symmetries and higher twisted Drinfeld doubles, Ising-like symmetries, and braided fusion symmetries. Based on https://arxiv.org/abs/2305.17165.

7 Chelsea Walton: Reflective centers of module categories and quantum K-matrices

This talk will be on recent joint work with Robert Laugwitz and Milen Yakimov (arXiv:2307.14764) that is motivated by obtaining solutions to the quantum reflection equation (qRE). To start, given a braided monoidal category C and C-module category M, we introduce a version of the Drinfeld center Z(C) of C adapted for M. We refer to this category as the "reflective center" E_C(M) of M. Just like Z(C) is a canonical braided monoidal category attached to C, we show that E_C(M) is a canonical braided module category attached to M. When C is the category of modules over a quasitriangular Hopf algebra H, and M is the category of modules over an H-comodule algebra A, we show that E_C(M) is equivalent to a category of modules over an explicit algebra, which we call the "reflective algebra" R_H(A) of A. Here, R_H(A) is akin to Drinfeld double of H. We show that reflective algebras are quasitriangular H-comodule algebras, and examine their corresponding quantum K-matrices (which are solutions to the qRE).

16:00 Coffee break

8 Gong Show

Wednesday 30 August

9 Gong Show

10:00 Coffee break

10 Catherine Meusburger: Turaev-Viro-Barrett-Westbury state sums with defects

We define a Turaev-Viro-Barrett-Westbury state sum model of triangulated 3-manifolds with surface defects (oriented 2d surfaces), line defects and point defects (graphs on the defect surfaces). Surface defects are labeled with bimodule categories over spherical fusion categories with bimodule traces, line and point defects with bimodule functors and
bimodule natural transformations. The state sum uses generalised 6j symbols that encode the coherence isomorphisms of the defect data. We prove the triangulation independence of the state sum and show that it can be computed with polygon diagrams that satisfy the cutting and gluing identities for polygon presentations of oriented surfaces. State
sums detect the genus of a defect surface and are sensitive to its embedding. Defect lines on defect surfaces with trivial defect data define ribbon invariants for the centre of the underlying spherical fusion category. Reference: C. Meusburger, State sum models with defects based on spherical fusion categories, Adv. Math. 429 (2023), DOI:10.1016/j.aim.2023.109177

11 Sakura Schafer-Nameki: Categorical Symmetries and Generalized Charges

I will give a summary of recent works on global categorical symmetries in QFTs, with a focus on non-invertible symmetries and their description in terms of fusion higher categories. In addition to discussing the symmetries, I will also introduce the notion of a generalized charge, and provide several examples of this for both invertible and non-invertible symmetries.

16:00 Coffee break

Thursday 31 August

12 Owen Gwilliam: Symmetries via factorization algebras

Factorization algebras arose in topology and representation theory but also provide a framework for describing the observables and symmetries of a field theory. This talk will begin by surveying factorization algebras and what they do well -- and do not -- in physics, at least at present. In the latter part we will describe work in progress with Araminta Amabel on selection rules for line operators of gauge theories, transposing Dan Freed's argument into the key of factorization algebras.

10:00 Coffee break

13 Kenneth Intriligator: Anomalies of 4d Spin_G theories

We consider ’t Hooft anomalies for a variety of 4d gauge theories whose fermion matter content admits Spin_G(4) generalized spin structure, with G gauged or global.
We discuss and compute the w2 w3 type ’t Hooft anomalies that arise in this context, and aspects of anomaly matching in the possible IR phases. Based on work with T. Daniel Brennan.

14 Kevin Walker: Categorified idempotent completion, topological symmetries of QFTs, and generalized Kramers-Wannier duality

Any (n-pivotal) n-category C can be embedded in a Morita-equivalent completion C^#. Because of the Morita equivalence, any module/action of C automatically leads to one of the larger category C^#. In particular, discrete k-form symmetries of d-dimensional QFTs correspond to actions of C(G, d+1, k+1) = \pi_{\le d+1}(B^{k+1}(G)), and therefore give rise to actions of the completed (d+1)-category C(G, d+1, k+1)^#. While C(G, d+1, k+1) is built out of invertible morphisms, C(G, d+1, k+1)^# typically contains many non-invertible morphisms leading to non-invertible symmetries of the original QFT. I’ll also discuss how completed n-categories can be used to construct many new examples of Kramers-Wannier-type dualities. This is joint work with Fiona Burnell.

16:00 Coffee break

15 Zohar Komargodski: Applications of Symmetries for Conformal Defects

Friday 1 September

16 Kantaro Ohmori: Non-supersymmetric heterotic branes, bordisms, 2d SCFTs

The no-bordism conjecture by McNamara and Vafa states that the bordism group with tangential structure and branes (singularity types) for a consistent quantum gravity should vanish.
This predicted previously unknown non-supersymmetric branes in string theory which should cancel the apparently nontrivial bordism classes.
In this talk I will propose the worldsheet theories of a string in the throat region of some of the predicted new branes in heterotic string theory.
I will also describe the relation to Stolz-Teichner conjecture as a connection to Yamashita’s talk.

The no-bordism conjecture by McNamara and Vafa states that the bordism group with tangential structure and branes (singularity types), for consistent quantum gravity, should vanish. As a corollary of the conjecture they predicted previously unknown non-supersymmetric branes in string theory, which are required to cancel the apparently nontrivial bordism classes. In this presentation, I will propose the worldsheet theories of a string in the throat region of some of the predicted branes in heterotic string theory.
Additionally, I will mention the relationship to the Stolz-Teichner conjecture and connect it to Yamashita’s talk.

10:00 Coffee break

17 David Ayala: Derived Skein modules via factorization homology

In this talk, I will motivate derived Skein modules, which recover (classical) Skein modules while possessing a host of expected continuous symmetries, functorialities, and local-to-global formulae. Then, using factorization homology, I will outline how to construct derived Skein modules.

Monday 4 September

18 Shu-Heng Shao: Non-invertible Symmetries

10:00 Coffee break

19 Shu-Heng Shao: Non-invertible Symmetries

20 David Jordan: Applied Cobordism Hypothesis

21 David Jordan: Applied Cobordism Hypothesis

16:30 Coffee break

22 Discussion

Tuesday 5 September

23 Shu-Heng Shao: Non-invertible Symmetries

10:00 Coffee break

24 Shu-Heng Shao: Non-invertible Symmetries

25 Constantin Teleman: The Mathematics of TQFTs and Defects

26 Constantin Teleman: The Mathematics of TQFTs and Defects

16:30 Coffee break

27 Shu-Heng Shao: Non-invertible Symmetries

28 Discussion

29 Gong Show

Wednesday 6 September

30 Michele Del Zotto: Symmetry Categories 101

10:00 Coffee break

31 Michele Del Zotto: Symmetry Categories 101

32 David Jordan: Applied Cobordism Hypothesis

16:30 Coffee break

33 David Jordan: Applied Cobordism Hypothesis

34 Discussion

Thursday 7 September

35 Michele Del Zotto: Symmetry Categories 101

10:00 Coffee break

36 Michele Del Zotto: Symmetry Categories 101

37 Constantin Teleman: The Mathematics of TQFTs and Defects

16:30 Coffee break

38 Constantin Teleman: The Mathematics of TQFTs and Defects

39 Discussion

Friday 8 September

40 David Jordan: Applied Cobordism Hypothesis

10:00 Coffee break

41 Constantin Teleman: The Mathematics of TQFTs and Defects

42 Michele Del Zotto: Symmetry Categories 101