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Discrete gauge anomalies revisited

Chang-Tse Hsieh

TL;DR

This work analyzes discrete gauge anomalies in 3+1d chiral fermion theories with Spin(4)×Z_n or Spin^{Z_{2m}}(4) symmetry, using the Dai-Freed framework to map anomalies to 5d spin cobordism groups and eta invariants. It derives explicit anomaly-free conditions for both untwisted and twisted cyclic symmetries, reveals how anomaly cancellation depends on symmetry extensions, and connects these results to Ibáñez–Ross criteria. The authors also explore gapped boundary states for anomalous Z_n global symmetries via symmetry extensions and weak-coupling constructions, linking 4d anomalies to 5d fermionic SPT phases. Overall, the paper provides a cohesive, cobordism-based foundation for discrete gauge anomalies and their possible resolution by symmetry extensions and topological boundary physics.

Abstract

We revisit discrete gauge anomalies in chiral fermion theories in $3+1$ dimensions. We focus on the case that the full symmetry group of fermions is $\mathrm{Spin}(4)\times\mathbb{Z}_n$ or $(\mathrm{Spin}(4)\times\mathbb{Z}_{2m})/\mathbb{Z}_{2}$ with $\mathbb{Z}_2$ being the diagonal $\mathbb{Z}_2$ subgroup. The anomalies are determined by the consistency condition --- based on the Dai-Freed theorem --- of formulating a chiral fermion theory on a generic spacetime manifold with a structure associated with either one of the above symmetry groups and are represented by elements of some finite abelian groups. Accordingly, we give a reformulation of the anomaly cancellation conditions, and compare them with the previous result by Ibáñez and Ross. The role of symmetry extensions in discrete symmetry anomalies is clarified in a formal fashion. We also study gapped states of fermion with an anomalous global $\mathbb{Z}_n$ symmetry, and present a model for constructing these states in the framework of weak coupling.

Discrete gauge anomalies revisited

TL;DR

This work analyzes discrete gauge anomalies in 3+1d chiral fermion theories with Spin(4)×Z_n or Spin^{Z_{2m}}(4) symmetry, using the Dai-Freed framework to map anomalies to 5d spin cobordism groups and eta invariants. It derives explicit anomaly-free conditions for both untwisted and twisted cyclic symmetries, reveals how anomaly cancellation depends on symmetry extensions, and connects these results to Ibáñez–Ross criteria. The authors also explore gapped boundary states for anomalous Z_n global symmetries via symmetry extensions and weak-coupling constructions, linking 4d anomalies to 5d fermionic SPT phases. Overall, the paper provides a cohesive, cobordism-based foundation for discrete gauge anomalies and their possible resolution by symmetry extensions and topological boundary physics.

Abstract

We revisit discrete gauge anomalies in chiral fermion theories in dimensions. We focus on the case that the full symmetry group of fermions is or with being the diagonal subgroup. The anomalies are determined by the consistency condition --- based on the Dai-Freed theorem --- of formulating a chiral fermion theory on a generic spacetime manifold with a structure associated with either one of the above symmetry groups and are represented by elements of some finite abelian groups. Accordingly, we give a reformulation of the anomaly cancellation conditions, and compare them with the previous result by Ibáñez and Ross. The role of symmetry extensions in discrete symmetry anomalies is clarified in a formal fashion. We also study gapped states of fermion with an anomalous global symmetry, and present a model for constructing these states in the framework of weak coupling.

Paper Structure

This paper contains 17 sections, 77 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Discrete symmetry anomalies in chiral fermion theories in 4d
  3. Review of the modern perspective on anomalies of fermions
  4. 't Hooft anomalies of $\mathrm{Spin}(4)\times\mathbb{Z}_n$
  5. 't Hooft anomalies of $\mathrm{Spin}^{\mathbb{Z}_{2m}}(4)$
  6. Further discussion of anomalies of 4d chiral gauge theory associated with discrete symmetries
  7. Review of the Ibáñez-Ross conditions for discrete symmetry anomalies
  8. Connection to the Ibáñez-Ross conditions
  9. Trivialization of anomalies by symmetry extensions
  10. Massive fermions with anomalous global symmetries in 4d
  11. Some information from $\Omega^{5}_{\mathscr{G}}$
  12. A model via weak coupling
  13. Examples
  14. Summary
  15. Details of some derivations
  16. ...and 2 more sections

Figures (2)

  • Figure 1: Two manifolds $X$ and $X'$ with the same boundary $M$ are glued together along $M$ to make a closed manifold $X^*$.
  • Figure 2: Constructing different boundary theories of the same SPT phase by changing the degrees of freedom on the boundary.