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Anomalies of NonInvertible Symmetries in (3+1)d
by Clay Cordova, PoShen Hsin, Carolyn Zhang
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Authors (as registered SciPost users):  PoShen Hsin 
Submission information  

Preprint Link:  https://arxiv.org/abs/2308.11706v2 (pdf) 
Date submitted:  20240423 04:39 
Submitted by:  Hsin, PoShen 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
Anomalies of global symmetries are important tools for understanding the dynamics of quantum systems. We investigate anomalies of noninvertible symmetries in 3+1d using 4+1d bulk topological quantum field theories given by Abelian twoform gauge theories, with a 0form permutation symmetry. Gauging the 0form symmetry gives the 4+1d "inflow" symmetry topological field theory for the noninvertible 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 "FrobeniusSchur indicator" of the noninvertible symmetry (generalizing the FrobeniusSchur indicator of 1+1d fusion categories) may be incompatible with trivial boundary dynamics. As a consequence we derive conditions for noninvertible 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 timereversal 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 noninvertible timereversal symmetry that permutes anyons. As a concrete realization of our general discussion, we construct new lattice Hamiltonian models in 3+1d with noninvertible symmetry, and constrain their dynamics.
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Report
This work provides a new way to determine the anomaly of noninvertible symmetry in higher dimensions. The key idea is to consider the abelian symmetry TQFT associated with the abelian part of the symmetry, and examine its topological boundary conditions.
The first level of anomaly comes from the nonexistence of duality invariant topological boundary condition of the abelian symmetry TQFT.
The second level of anomaly comes from the FS indicator, which, by decorated domain wall construction, reduces to the anomaly of time reversal symmetry of the noninvertible defect worldvolume theory. Then (1) the noninvertible symmetry being anomaly free (in the sense of admitting a trivially gapped phase) is argued to be equivalent to (2) the existence of a 3d world volume TQFT that matches the time reversal anomaly from the FS indicator. This equivalence between (1) and (2) are scattered in different parts of the paper, and I would suggest the authors to establish the equivalence better by, e.g. having a dedicated subsection, and explain the proof of (1) > (2) and (2) > (1) directions explicitly, therefore showing their equivalence. I believe this would make the paper, especially Section 3, more readable.
Apart from the above suggestion, there is a small gap for the argument around eq 2.2. One needs to argue that simple surface induces only 1 copy of identity line. Otherwise eq 2.2 can be modified to $n \times n’ = \sum_i n_i$, which does not lead to contradiction.
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Strengths
1  The paper studies anomalies for noninvertible symmetries in 3+1d. The authors identify two levels of anomalies, generalizing analogous results for anomalies of TambaraYamagami noninvertible 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 noninvertible version of time reversal symmetry
3  The authors present and study explicit lattice models for 3+1d theories that enjoy a ZN 1form symmetry and are invariant under its gauging
Weaknesses
1  At times, the discussion is rather technical, especially in section 3
Report
Uncovering dynamical consequences of noninvertible 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 KramersWannierlike 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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