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Anomalies of Non-Invertible Symmetries in (3+1)d

by Clay Cordova, Po-Shen Hsin, Carolyn Zhang

Submission summary

Authors (as registered SciPost users): Po-Shen Hsin
Submission information
Preprint Link:  (pdf)
Date submitted: 2024-04-23 04:39
Submitted by: Hsin, Po-Shen
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
  • Mathematical Physics
Approach: Theoretical


Anomalies of global symmetries are important tools for understanding the dynamics of quantum systems. We investigate anomalies of non-invertible symmetries in 3+1d using 4+1d bulk topological quantum field theories given by Abelian two-form gauge theories, with a 0-form permutation symmetry. Gauging the 0-form symmetry gives the 4+1d "inflow" symmetry topological field theory for the non-invertible symmetry. We find a two levels of anomalies: (1) the bulk may fail to have an appropriate set of loop excitations which can condense to trivialize the boundary dynamics, and (2) the "Frobenius-Schur indicator" of the non-invertible symmetry (generalizing the Frobenius-Schur indicator of 1+1d fusion categories) may be incompatible with trivial boundary dynamics. As a consequence we derive conditions for non-invertible symmetries in 3+1d to be compatible with symmetric gapped phases, and invertible gapped phases. Along the way, we see that the defects characterizing $\mathbb{Z}_{4}$ ordinary symmetry host worldvolume theories with time-reversal symmetry $\mathsf{T}$ obeying the algebra $\mathsf{T}^{2}=C$ or $\mathsf{T}^{2}=(-1)^{F}C,$ with $C$ a unitary charge conjugation symmetry. We classify the anomalies of this symmetry algebra in 2+1d and further use these ideas to construct 2+1d topological orders with non-invertible time-reversal symmetry that permutes anyons. As a concrete realization of our general discussion, we construct new lattice Hamiltonian models in 3+1d with non-invertible symmetry, and constrain their dynamics.

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Current status:
In refereeing

Reports on this Submission

Anonymous Report 1 on 2024-5-14 (Invited Report)


1 - The paper studies anomalies for non-invertible symmetries in 3+1d. The authors identify two levels of anomalies, generalizing analogous results for anomalies of Tambara-Yamagami non-invertible symmetries in 1+1d
2 - Along the way, the authors study anomalies in the time reversal symmetry of an interesting class of 2+1d TQFTs, and construct an infinite family of 2+1d TQFTs enjoying a non-invertible version of time reversal symmetry
3 - The authors present and study explicit lattice models for 3+1d theories that enjoy a ZN 1-form symmetry and are invariant under its gauging


1 - At times, the discussion is rather technical, especially in section 3


Uncovering dynamical consequences of non-invertible symmetries in 3+1d dimensions is an important endeavor. This paper makes substantial progress in this direction by performing a systematic analysis of 't Hooft anomalies for Kramers-Wannier-like symmetries.

This paper meets this Journals' acceptance criteria and is recommended for publication, after addressing some minor points listed below.

Requested changes

1 - The notation GL(n,Z) is often used to indicate n x n matrices with integer entries that are invertible and whose inverses are also matrices with integer entries. Thus, these matrices have determinant equal to +1, -1. The authors use the notation GL(2r,Z) around (2.8), but remark that U needs not have determinant +1 or -1. The authors could comment briefly on this for clarification purposes.
2 - Some minor typos: "that the the Lagrangian" beginning of sec 2.1.2; "an quadratic" above (2.12); "an symmetric" above (2.24); "affect out discussion" below (3.19); "odd N cause" 4th line page 29; "symmetry symmetry" between (3.42) and (3.43); "can can produce" above Thm 2; "J0 and and J1" bottom page 36; "the anomaly field theory that [...] form symmetries with is given by" 2nd bullet page 40


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