2023 Simons Collaboration on Global Categorical Symmetries Annual Meeting

Our collaboration investigates topological symmetries, governed by higher categorical and homotopical data, as they control and constrain features of genuine, physically inspired models of quantum field theories (QFTs), with applications to physical systems of various kinds, from condensed matter to high energy physics. We are developing both the mathematical foundations of these symmetries — a study made possible in arbitrary dimension by the cobordism hypothesis, but interfacing in low dimensions with the theory of fusion and structured categories and generalizations, and the applications to physical theories of interest, such as lattice models, the standard model, QED, as well as strongly interacting theories without Lagrangian formulation and super-conformal theories of stringy origin.

This year’s presentations offered three distinct strands: one foundational, on the mathematics and calculus of topological theories and defects (Scheimbauer and Teleman), and two ‘applied’ directions, one on the construction of symmetries in QFTs (by holography and string theory methods, García Etxebarria and Bah) and one on the use of symmetries to derive physical consequences (Seiberg, Dumitrescu, Córdova).

Nathan Seiberg opened the morning session with the discussion of “emanant symmetries,” an exact kind of symmetry emerging in the low-energy sector of a QFT. This was illustrated with fermions in the Ising system, specifically the twisted and untwisted Majorana chains. This triggered a lively post-lecture debate on the need (or lack of) spin-statistics in low dimension.

Ashvin Vishwanath updated us on recent advances in quantum computing. In collaboration with the Quantinuum group, efficient methods were found to prepare entangled states of ‘non-abelian topological order’ based on the fusion categories for certain solvable groups. The construction exploits the equivalent description of the relevant symmetry as a Dijkgraaf-Witten theory for an abelian group.

Thomas Dumitrescu reviewed examples of confining and Higgs phases in gauge theories with matter, where the relevant phase diagram is controlled by invertible TQFTs (symmetry-protected topological phases). In particular, the analysis of Dumitrescu revealed a gapless SPT point in the phase diagram of QCD, with potential new application to the physics of neutron stars.

Claudia Scheimbauer reported on recent work with Walde, constructing TQFTs from finite homotopy types as fully local theories. This was accomplished using the notion of higher categorical semi-additivity, developed by Hopkins and Lurie in the context of chromatic homotopy, and used there to construct higher cyclotomic extensions of Morava theories. These TQFTs are the backbone of all known TQFTs in dimensions higher than
3. One consequence is the functoriality of the older construction of Fred-Hopkins-Lurie-Teleman.

Constantin Teleman reported on ongoing work with Freed and Hopkins, developing the mathematical frame for the conjectural operation of condensation of topological defects, long used (with a partly heuristic basis) in condensed matter and low-dimensional QFT calculation of symmetries and gauging. This opened a (continuing) discussion of reinterpreting older work of Moore and Seiberg in dimensions 2 and 3.

Iñaki García Etxebarria gave an overview construction of symmetry operators in QFT in the SymTFT model, using a working example of the generators of SU(2) symmetry in a 4-dimensional N=4 SUSY theory. This has applications to the study of gapped phases. He followed that with the more sophisticated construction of SymTFTs from string theory by the ‘geometric engineering’ method of reduction along singular manifolds with special (Calabi-Yau) geometry.

Ibrahima Bah presented methods of constructing non-invertible symmetry operators in QFT from string theory. The method involves reduction of string theory branes to symmetry operators and, in many cases, permits some calculus of these symmetries (such as the fusion rules) to be executed from the calculus of branes in the bulk, via the AdS/CFT correspondence.

Clay Córdova closed the meeting with explicit studies of duality defects in low-dimensional lattice models, leading to a proposed definition of anomalous symmetry, specific to categorical symmetries dimensions 3 and 2 but offering natural generalizations, related to the existence of fiber functors and the possibility of gauging categorical symmetries.

This year’s meeting was preceded by a two-day workshop at NYU, co-organized by Bah, Del Zotto, Teleman and Wang, with advice and help from the GCS collaboration. The satellite meeting was supported by GCS collaboration sources and the Simons Foundation kindly helped with the logistics of housing for the guests. We hosted 6 structured presentations: Shu-Heng Shao, Michele Del Zotto, Victor Ostrik, Jonathan Heckman, Yifan Wang and Tomer Schlank, many selected to connect with the formal presentations at the annual meeting: Shao with Seiberg, Heckman with Etxebarria and Bah, Schlank with Scheimbauer, and Del Zotto and Wang with Córdova. New ideas were also introduced: asymptotic representation theory (Ostrik) and a ‘branching’ of topological symmetries on different energy regimes of a TQFT (Heckman), and a new technique to obtain selection rules on curved spacetimes from higher symmetries (Del Zotto). Two afternoon discussions featured a presentation by Zohar Komargodski, on the appearance of anomalies and obstructions in the calculus of topological operators, and a (very animated) session on the general topic of anomalies led by Teleman. The added days of interaction multiplied the impact and exchange of ideas, especially allowing fruitful exchanges between collaboration members and external participants.

Motivated by the advances in topological quantum field theory over the past decade, its close relationship with higher algebraic structures, and the growing role of extended operators’ in Quantum Field Theory, the Simons Collaboration on Global Categorical Symmetries centers on a systematic study of the interconnection between these notions. The collaboration is summarized under the heading of ‘categorical’ and ‘non-invertible’ symmetries, as they generalize the symmetries defined from group actions.

Recent progress includes the calculus of (fully local) topological operators in TQFT, the analysis of duality defects in QFT and lattice system, the role of higher groups in charge fractionalization, and the construction of topological defects by dimensional reduction along non-compact manifolds. Areas of current and further study inspired by physical problems include the role of extended operators in constraining the RG flow, topological phases of lattice theories, the nature and role of (higher) categorical calculus beyond the purely topological QFT context, as well as their application to stringy and holographic settings. On the mathematical side, the non-semi simple theories extend the reach of the methods to physically non-standard problems.

Annual meeting lectures will offer updates on the collaboration’s current research with the expectation of generating lively discussions and contributions from the participants outside of the formal schedule.

Ibrahima Bah
Johns Hopkins University

Aspects of Symmetry and Branes in Holography
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String theory and holography provide important points of view on the topological nature of symmetries acting on quantum systems. In this talk, Ibrahima Bah will discuss how symmetry structure can be described from the bulk in AdS/CFT, both from top down, in string theory, and bottom-up perspectives. Bah will discuss how branes can realize non-invertible symmetry operators and the various properties associated to them, such as fusion rules.
 

Clay Córdova
University of Chicago

Anomalies of Non-Invertible Symmetries
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Non-invertible symmetries, i.e., topological operators with general fusion rules, have recently emerged as a new tool to study dynamics of quantum field theories. Clay Córdova will formulate the notion of anomalies of non-invertible symmetries, which are obstructions to realizing the non-invertible symmetry in a trivially gapped/invertible phase. As Córdova describes, such anomalies are intimately connected to finding topological boundary conditions of an associated topological field theory that encodes the symmetry. Córdova will compute these anomalies both for 2d and 4d quantum field theories which have a duality symmetry captured by a (higher) fusion category. Along the way, Córdova will show an interesting, decorated domain wall construction which is useful for understanding the contribution to the anomaly of Frobenius-Schur indicators.
 

Thomas Dumitrescu
UCLA

Invertible TQFTs and Gauge Theory Phases
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Gauge theory phases are characterized by the breaking pattern of global symmetries. Typically, these are higher symmetries acting on extended operators. Gauge theories with fundamental matter, e.g., QCD, typically do not have such higher symmetries, obstructing sharp definitions of confining and Higgs phases.

In this talk, Thomas Dumitrescu will discuss examples in which confining and Higgs phases are naturally characterized by certain invertible TQFTs, or equivalently via symmetry protected topological (SPT) phases, which involve only ordinary symmetries. Dumitrescu will conclude with applications to the QCD phase diagram.
 

Iñaki García Etxebarria
Durham University

Deriving SymTFTs from Geometry
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Symmetry TFTs (aka, SymTFTs), and the related “sandwich construction,” provide a principled way of discussing various subtle aspects of many systems with categorical symmetries. These objects also appear very naturally when studying quantum field theory using geometric methods in string theory. In this talk, Iñaki García Etxebarria will review some elementary aspects of this description of symmetries and explain how it arises in string theory and holography.
 

Claudia Scheimbauer
TU Munich

Finite Gauge Theory as an Extended Topological Field Theory

Following Freed-Hopkins-Lurie-Teleman’s proposal, we construct finite gauge theory as an extended topological field theory by following two steps: first, from a theory defined depending on choices of principle bundles, construct a theory valued in “spans” or “correspondences,” and then compose with a variant of a finite path integral summing over the choices. Claudia Scheimbauer will explain how to construct the second part in a higher categorical setting using the crucial property “m-semiadditivity” and a universal property of the higher category of spans. This generalizes Harpaz’s result and is joint work in progress with Tashi Walde.
 

Nathan Seiberg
Institute for Advanced Study

Emanant Symmetries
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Based on joint work with Meng Cheng (arXiv:2211.12543), with Shu-Heng Shao (arXiv: 2307.02534) and with Shu-Heng Shao and Sahand Seifnashri (to appear), Nathan Seiberg will discuss some aspects of global symmetries and their ‘t Hooft anomalies.

Seiberg will define a notion of an emanant global symmetry. It is not a symmetry of the UV theory, but unlike emergent (accidental) symmetries, it is not violated by any relevant or irrelevant operators in the IR theory. It is an exact symmetry of the low-energy theory.

Seiberg will demonstrate this notion in several well-known examples and will discuss in detail the Majorana chain, the transverse field Ising model, a continuum system with a chemical potential, and the Heisenberg chain. In all these models emanant symmetries exist, and, in one case, it is a non-invertible emanant symmetry.
 

Constantin Teleman
UC Berkeley

Condensed Thoughts
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One intriguing operation on defects in QFTs is “condensation”, referring to the coalescence of lower-dimensional defects into higher-dimensional ones. Constantine Teleman will review condensation of topological defects in the context of topological symmetries, with emphasis on finite homotopy theories. He will identify two key ingredients in the process: the notion of “Dirichlet boundary condition” and the “Ostrik correspondence” between algebra objects and module categories and will show how condensation reflects the notion of connectivity in topology.

In addition to the multiple physics sources (including but not limited to collaboration colleagues), Teleman’s talk is greatly inspired by ideas of Gaiotto and Johnson-Freyd and is based on a joint project with Dan Freed and Mike Hopkins.
 

Ashvin Vishwanath
Harvard University

Using Measurements to Gauge Symmetries & to Create the First Non-Abelian Topological State on Quantum Hardware
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In quantum systems measurement can actively modify and steer quantum states acting as a quantum chisel to sculpt new patterns of entanglement. Ashvin Vishwanath will describe how our improved understanding of symmetries allows us to leverage the power of measurements to efficiently create the long sought after non-Abelian topological phases. Surprisingly, the particular non-Abelian states which can be created in this way are closely related to Galois’ characterization of solvable polynomial equations.

Finally, Vishwanath will describe our recent collaboration with Quantinuum that implements the entanglement sculpting approach to state creation and demonstrates the non-Abelian statistics of excitations. In particular, Vishwanath will discuss the Borromean braiding of excitations, a signature unique to non Abelian topological order, and its measurement on the Quantinuum platform.