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Highergroup symmetry in finite gauge theory and stabilizer codes
by Maissam Barkeshli, YuAn Chen, PoShen Hsin, Ryohei Kobayashi
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Submission summary
Authors (as registered SciPost users):  YuAn Chen · PoShen Hsin · Ryohei Kobayashi 
Submission information  

Preprint Link:  scipost_202307_00028v1 (pdf) 
Date submitted:  20230721 20:56 
Submitted by:  Kobayashi, Ryohei 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
A large class of gapped phases of matter can be described by topological finite group gauge theories. In this paper we show how such gauge theories possess a highergroup global symmetry, which we study in detail. We derive the $d$group global symmetry and its 't Hooft anomaly for topological finite group gauge theories in $(d+1)$ spacetime dimensions, including nonAbelian gauge groups and DijkgraafWitten twists. We focus on the 1form symmetry generated by invertible (Abelian) magnetic defects and the higherform symmetries generated by invertible topological defects decorated with lower dimensional gauged symmetryprotected topological (SPT) phases. We show that due to a generalization of the Witten effect and chargeflux attachment, the 1form symmetry generated by the magnetic defects mixes with other symmetries into a higher group. We describe such highergroup symmetry in various lattice model examples. We discuss several applications, including the classification of fermionic SPT phases in (3+1)D for general fermionic symmetry groups, where we also derive a simpler formula for the $[O_5] \in H^5(BG, U(1))$ obstruction that has appeared in prior work. We also show how the $d$group symmetry is related to faulttolerant nonPauli logical gates and a refined Clifford hierarchy in stabilizer codes. We discover new logical gates in stabilizer codes using the $d$group symmetry, such as a ControlledZ gate in (3+1)D $\mathbb{Z}_2$ toric code.
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Reports on this Submission
Report #2 by Anonymous (Referee 2) on 202419 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202307_00028v1, delivered 20240109, doi: 10.21468/SciPost.Report.8388
Strengths
1. Clearly written and systematic.
2. Many concrete examples are presented.
Report
The paper studies the invertible symmetries of DijkgraafWitten gauge theories in all dimensions. It is clearly written with numerous examples presented to illustrate the conclusions. I recommend the paper wholeheartedly for publication with minimal changes. My suggestions pertain only to setting the results in a mathematical context.
 My understanding is that DijgraafWitten theory for a finite group $G$ in D = d+1 dimensions should have $(D1)$fusion category symmetry (in the bosonic case and putting aside issues of unitarity) $\Sigma Z((D2)Vect_G^{\omega_D})$ where $Z$ denote the Drinfeld center and \Sigma condensation completion of the delooping. Then from a mathematical perspective, I expect the authors are computing the invertible part of this symmetry category. Is that correct?
 Following from the above, I expect general genuine codimension2 defects (genuine = not attached to a codimension1 topological defect) are labelled by a conjugacy class in $G$ and a projective $(D2)$representation of the centraliser with projective $(D1)$cocyle given by the transgression of $\omega_D$. This seems compatible with the discussion at the beginning of section 2, and includes magnetic defects stacked with gauged SPT phases for the unbroken subgroup. However, it may be nice to relate the discussion in 2,2.1 to this mathematical classification. (I appreciate starting with 2.2 the author's consider instead twisted sector magnetic defects attached to codimension1 topological defects.)
Requested changes
I suggest some discussion of how the results relate to the aforementioned mathematical framework. However, as this is primarily a physics oriented journal, this should be taken as a suggestion only!
Report #1 by Anonymous (Referee 1) on 2023922 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202307_00028v1, delivered 20230922, doi: 10.21468/SciPost.Report.7851
Strengths
detailed study of highergroup symmetry in finite gauge theories and several exciting applications, such as the classification of fermionic SPTs and faulttolerant gates
very well written and clear
Weaknesses
outlook with possible future directions could be mentioned
Report
This paper comprehensively studies the higher group symmetry and its 't Hooft anomaly in finite gauge theories. It describes the highergroup symmetry as arising from a generalization of the Witten effect and the chargeflux attachment. It provides several explicit examples, including field theories and lattice models. The applications discussed in the paper are exciting. For instance, studying the faulttolerant logical gates from this perspective could lead to insights for practical applications.
The paper's presentation and results are excellent. I wholeheartedly recommend publishing it in SciPost Physics.