2025 Simons Collaboration on Global Categorical Symmetries Annual Meeting

Our collaboration focuses on topological symmetries in quantum field theory (QFT). These are governed by higher algebraic-categorical and homotopical data, and control and constrain features of genuine, physically inspired QFTs. We have developed the mathematical foundations of these symmetries. This study was made possible in arbitrary dimensions by the cobordism hypothesis, but in low dimensions it interfaced with the theory of fusion and structured categories, and their generalizations, to which the work of our collaboration has contributed significantly. In parallel, we have been pursuing applications to physical theories of interest: condensed matter and via lattice models, the standard model and its variations, QED, strongly interacting theories without Lagrangian formulation, and super-conformal theories of stringy origin. The connection between topological information hiding quantum field theories and homotropy theory is developing rapidly and growing ever closer and greatly facilitated by the work of our affiliates and co-workers, as well as the events we are organizing regularly.

This year’s presentations were given by collaboration members: six PIs and two formal associates. After last year’s meeting, which gave an overview of four years of results, this year we aimed to showcase the new directions in our research; necessarily the results presented had a more preliminary flavor.

One major recurring theme was that of topological continuous symmetries of quantum field theory. Proposals to capture their calculus, in higher dimension, ranged from holography methods (Bah), to BF-BV theory (Garcia Etxebarria and Cattaneo), and derived methods (Freed). Applications included criteria for gap lessness (Ohmori) as well as refinements of the phase diagram of QED in 2+1 dimensions (Dumitrescu). Finally, new developments in 2- and 3-dimensional TQFT were presented, pertaining to elliptic cohomology (Yamashita) and the diagrammatic calculus of Hopf algebras (Johnson-Freyd).

Ibrahima Bah offered a holographic description of symmetries generated by compact Lie groups via the geometry of non-compact branes attached at infinity in space-time.

Iñaki Garcia-Etxebarria presented a SymTFT, or ‘quiche’ picture for continuous symmetries; intriguingly, in relation to gravity, he proposed an extension of this picture to space-time symmetries.

Daniel Freed related continuous group symmetries to derived moduli spaces of flat connections, with examples of effective action of the derived part via point operators.

Mayuko Yamashita announced new results motivated by the long-conjectured relations of the spectra of topological modular forms (TMF, tmf) with moduli of 2D conformal field theories (CFTs). She was able to relate the equivariant versions to CFTs with group symmetry by meshing the latter with the modular tensor categories arising from the underlying vertex algebras.

Alberto Cattaneo exploited the realization of BF theory, an implementation of continuous symmetry, as a limit of Yang-Mills theory. In four dimensions, it led to a surprising application, whereby the electric flux of 4D Maxwell theory, as well as its non-abelian analogue, may be recovered from BF theory.

Theo Johson-Freyd presented a classification of Hopf algebras in infinity-categories from a construction of the Ising model going back to Severa and Freed-Teleman, which starts from a 3D TQFT and a pair of transversal boundary conditions. This was vastly generalized in joint work with Reutter, dispensing with the customary and very restrictive full dualizability constraints.

Kantaro Ohmori presented a novel criterion for gap lessness in 1+1-dimensional models based on a continuous group G of symmetries and its interaction with a fusion category symmetry, mediated by a correspondence diagram controlled by a finite subgroup H of G. Its importance was supported with examples from compact bosons, WZW-related models, and bosonized free fermions in lattice models.

Thomas Dumitrescu closed the meeting describing new developments in 2+1 dimensional QED. With the addition of mass parameters and/or strong background magnetic fields, he was able, by exploiting the symmetries of the theory, to refine the (previously rudimentary) phase diagram of this theory.

As in prior years, the meeting was preceded by a three-day workshop at NYU, co-organized by Del Zotto, Teleman, and Yifan Wang (NYU), with advice and help from the collaboration. The satellite meeting was supported by collaboration funds, and the foundation kindly helped with the logistics of housing for the guests.

The meeting opened with seven short lectures by collaboration postdoctoral fellows and a close associate (Krylov, Grigoletto, Stewart, McNamara, Riva, Yang, Yu) covering recent work on mathematical applications of topological symmetries to geometric representation theory, higher categorical structures and lattice models.

It was followed by seven lectures over two days, given by invited researchers external to the collaboration (Shao, Wang, Sopenko, Snyder, Tantivasadakarn), plus our PIs Reshetikhin and Córdova. As intended and expected, the added days of interaction and discussion multiplied the impact of the annual meeting.

Ibrahima Bah
Johns Hopkins University

Hanging Branes and Continuous Symmetries in AdS/CFT

I will describe holographic duals of topological operators for continuous symmetries. These can be obtained by considering branes that hang on the boundary. I will comment on how the physics of these branes captures fusion rules and measurements of charges.
 

Alberto Cattaneo
University of Zurich

Surface Observables in 4D BF and Yang–Mills Theories
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Maxwell theory possesses an interesting gauge-invariant observable defined as (the exponential of) the Hodge dual of the curvature integrated on a surface Sigma (this can be interpreted as the electric flux through Sigma). As emphasized by ’t Hooft, a nonabelian version would be of significant interest. In this talk, I will first show how to obtain a surface observable for BF theory with cosmological constant. This is a topological field theory, and an AKSZ model, whose fields are a connection and a 2-form B, with equations of motions simply stating that B is proportional, by the “cosmological constant,” to the curvature. This is a nontrivial task which can be achieved through the BV formalism defining a second field theory on Sigma coupled to the ambient fields of BF theory. (As the previously known version for zero cosmological constant, the expectation value of this observable should yield invariants of 2-knots in 4 dimensions.) Subsequently, thanks to a result with F. Bonechi and M. Zabzine, we can recover Yang–Mills theory (plus quantum corrections) from this BF theory via averaging on certain fields (BV pushforward in the terminology developed with P. Mnev and N. Reshetikhin). This procedure also produces a surface observable for Yang–Mills theory which, in the classical limit, corresponds to the nonabelian electric flux.
 

Thomas Dumitrescu
University of California, Los Angeles

Symmetries and Dynamics in Three-Dimensional Quantum Electrodynamics
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QED in 2+1 dimensions is a strongly-coupled quantum field theory, whose dynamics is relevant to different areas of study. In this talk, I will summarize recent progress on understanding the phase diagram of this theory, e.g., as a function of mass parameters or a magnetic field. Symmetries, anomalies, and non-renormalization theorems play a crucial role.
 

Daniel Freed
Harvard

Lie Group Quiche
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In ongoing work with Greg Moore and Constantin Teleman, we explore quiche with relaxed finiteness conditions, in particular those that encode compact Lie group symmetries of quantum field theories. The modern arsenal of results and techniques in topological field theory, some developed in the Collaboration, are deployed.
 

Iñaki Garcia Etxebarria
Durham University

Symmetry TFTs for Continuous Spacetime Symmetries
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I will review joint work with F. Apruzzi, N. Dondi, H. T. Lam and S. Schäfer-Nameki, in which we propose a SymTFT description for the part of the spacetime symmetry group connected to the identity and discuss its connection to gravity.
 

Theo Johnson-Freyd
Dalhousie University

How to Build a Hopf Algebra
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As observed by Severa, Freed–Teleman, and Dimofte–Niu in various levels of generality, if you take a 3D TQFT with a transverse pair of boundary conditions, then the vector space assigned to a square with alternating boundary conditions carries the structure of a Hopf algebra: the multiplication and comultiplication are intervals times open parts, and the antipode is a 180-degree rotation. I will explain a version of this construction which is fully-\infty-coherent, fully-framed, and essentially sharp: I do not require full dualizability, but rather say exactly the dualizability needed; every Hopf algebra, in every presentable \infty-category, arises from this construction. At this level of generality, the existence of the antipode is highly nontrivial: this version can produce (infinite-dimensional) Hopf algebras with nonbijective antipode. The technology going into the construction and proof is not bordism calculus, but rather the “Gray,” aka “lax,” tensor product of (\infty,\infty)-categories. From this perspective, the construction turns out to arise as a “tensor square” of the construction of a monad from an adjunction. This talk is based on arXiv:2508.16787, joint with David Reutter.
 

Kantaro Ohmori
RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences

Symmetry Spans and Enforced Gaplessness
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When do symmetries enforce gaplessness? In 1+1 dimensions, we give a span criterion that certifies gaplessness from partial symmetry data. For a span C ← Vect_H → Vect_G, where C is a fusion-category symmetry and Vect_G/Vect_H denote the categories of G- and H-graded vector spaces encoding a non-anomalous continuous group G and a finite group H, we provide a simple, checkable condition that rules out any symmetric gapped infrared (IR) phase described by a 2D TQFT. In examples (compact bosons, WZW models, bosonized free fermions on lattices), the span generates a larger symmetry that is known to forbid gapping: either a perturbatively anomalous continuous symmetry (WZW) or an infinite-dimensional Onsager algebra (fermions). Clarifying what symmetries can arise from such spans in general is an interesting direction. This talk is based on ongoing work with Takamasa Ando (Yukawa Institute for Theoretical Physics).
 

Mayuko Yamashita
Perimeter Institute

A Merger of Elliptic Cohomology and Global Categorical Symmetries

In this talk, I will illustrate that the Segal–Stolz–Teichner paradigm and global categorical symmetries are starting to merge, by explaining the ongoing work with Theo Jonhson-Freyd and Daniel Berwick-Evans, titled “Extending equivariant TMF to the cusp using vertex algebras.” The spectrum “TMF,” topological modular forms, is related to 2-dimensional supersymmetric quantum field theories (SQFTs). There is an equivariant refinement of TMF which reflects the group-symmetry in SQFTs. The interpretation of its variant “Tmf” has been a mystery, and its equivariant refinement has not been constructed. We provide an answer to both questions using the theory of vertex operator algebras and modular tensor categories.

Watch a playlist of all presentations from this meeting here.