Notes on generalized global symmetries in QFT

Sharpe, Eric

doi:10.1002/prop.201500048

Cited by 130 publications

(156 citation statements)

References 167 publications

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“…When M 5 has a boundary, the standard Wu formula may no longer hold, and they differ at most by a co-boundary term. 44 We denote the co-boundary term as δ(...) in the second line. Since (...) is a well-defined term of background gauge fields, it is a 4d invertible TQFT, which does not contribute to the 4d dynamics.…”

Section: (812)mentioning

confidence: 99%

Quantum 4d Yang-Mills theory and time-reversal symmetric 5d higher-gauge topological field theory

2019

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We explore various 4d Yang-Mills gauge theories (YM) living as boundary conditions of 5d gapped short/long-range entangled (SRE/LRE) topological states. Specifically, we explore 4d time-reversal symmetric pure YM of an SU(2) gauge group with a second-Chern-class topological term at θ = π (SU(2) θ=π YM), by turning on the background fields for both the time-reversal (i.e., on unorientable manifolds) and the 1-form center global symmetry. We find Four Siblings of SU(2) θ=π YM with distinct couplings to background fields, labeled by (K 1 , K 2 ): K 1 = 0, 1 specifies Kramers singlet/doublet Wilson line and new mixed higher 't Hooft anomalies; K 2 = 0, 1 specifies boson/fermionic Wilson line and a new Wess-Zumino-Witten-like counterterm. Higher anomalies indicate that to realize all higher nglobal symmetries locally on n-simplices, the 4d theory becomes a boundary of a 5d higher-symmetryprotected topological state (SPTs, as an invertible topological quantum field theory in math, or as a 5d higher-symmetric interacting topological superconductor in condensed matter). Via Weyl's gauge principle, by dynamically gauging the 1-form symmetry, we transform a 5d bulk SRE SPTs into an LRE symmetry-enriched topologically ordered state (SETs); thus we obtain the 4d SO(3) θ=π YM-5d LREhigher-SETs coupled system with dynamical higher-form gauge fields. We further derive new exotic anyonic statistics of extended objects such as 2-worldsheet of strings and 3-worldvolume of branes, physically characterizing the 5d SETs. We discover triple and quadruple link invariants potentially associated with the underlying 5d higher-gauge TQFTs, hinting a new intrinsic relation between nonsupersymmetric 4d pure YM and topological links in 5d. We provide 4d-5d lattice simplicial complex regularizations and bridge to 4d SU(2) and SO(3)-gauged quantum spin liquids as 3+1 dimensional realizations.

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Section: (812)mentioning

confidence: 99%

Quantum 4d Yang-Mills theory and time-reversal symmetric 5d higher-gauge topological field theory

2019

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“…6 • When D = 3, the resulting mixed symmetry is a 2-group, and the corresponding 4d TQFT was studied in [15]. An appearance of higher groups in physics was also noted in [3].…”

Section: Miscellaneous Remarksmentioning

confidence: 99%

On gauging finite subgroups

2020

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We study in general spacetime dimension the symmetry of the theory obtained by gauging a non-anomalous finite normal Abelian subgroup A of a Γ-symmetric theory. Depending on how anomalous Γ is, we find that the symmetry of the gauged theory can be i) a direct product of G = Γ/A and a higher-form symmetryÂ with a mixed anomaly, whereÂ is the Pontryagin dual of A; ii) an extension of the ordinary symmetry group G by the higher-form symmetryÂ; iii) or even more esoteric types of symmetries which are no longer groups.We also discuss the relations to the effect called the H 3 (G,Â) symmetry localization obstruction in the condensed-matter theory and to some of the constructions in the works of Kapustin-Thorngren and Wang-Wen-Witten.

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“…We do not claim that this is the ultimate concept for the 0-form finite symmetry in two dimensions; there still might be a generalization in the future. For example, in other spacetime dimensions, a proposed generalization was to use the concept of p-groups, see e.g [4],. and one might want to unify the two approaches.…”

mentioning

confidence: 99%

On finite symmetries and their gauging in two dimensions

Bhardwaj¹,

Tachikawa²

2018

J. High Energ. Phys.

210

301

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It is well-known that if we gauge a Z n symmetry in two dimensions, a dual Z n symmetry appears, such that re-gauging this dual Z n symmetry leads back to the original theory. We describe how this can be generalized to non-Abelian groups, by enlarging the concept of symmetries from those defined by groups to those defined by unitary fusion categories. We will see that this generalization is also useful when studying what happens when a non-anomalous subgroup of an anomalous finite group is gauged: for example, the gauged theory can have non-Abelian group symmetry even when the original symmetry is an Abelian group. We then discuss the axiomatization of two-dimensional topological quantum field theories whose symmetry is given by a category. We see explicitly that the gauged version is a topological quantum field theory with a new symmetry given by a dual category. 2 Recently in [5], Gaiotto, Kapustin, Komargodski and Seiberg performed an impressive study of the phase structure of thermal 4d su(2) Yang-Mills theory. One important step in the analysis is the symmetry structure of the thermal system, which is essentially three-dimensional. As a dimensional reduction from 4d, the system has a Z 2 × Z 2 0-form symmetry and a Z 2 1-form symmetry, with a mixed anomaly. Then the authors gauged the Z 2 1-form symmetry, and found that the total 0-form symmetry is now D 8 . This D 8 was then used very effectively to study the phase diagram, but that part of their paper does not directly concern us here. Their analysis of turning an anomalous Abelian symmetry by gauging a non-anomalous subgroup into a non-Abelian symmetry is a 3d analogue of what we explain in 2d. See their Sec. 4.2, Appendix B and Appendix C. Clearly an important direction to pursue is to generalize their and our constructions to arbitrary combinations of possibly-higher-form symmetries in arbitrary spacetime dimensions, but that is outside of the scope of this paper.

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Notes on generalized global symmetries in QFT

Cited by 130 publications

References 167 publications

Quantum 4d Yang-Mills theory and time-reversal symmetric 5d higher-gauge topological field theory

Quantum 4d Yang-Mills theory and time-reversal symmetric 5d higher-gauge topological field theory

On gauging finite subgroups

On finite symmetries and their gauging in two dimensions

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