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Non-invertible Symmetries and Higher Representation Theory I

by Thomas Bartsch, Mathew Bullimore, Andrea E. V. Ferrari, Jamie Pearson

Submission summary

Authors (as registered SciPost users): Thomas Bartsch · Andrea Ferrari
Submission information
Preprint Link: scipost_202403_00037v1  (pdf)
Date submitted: 2024-03-26 01:25
Submitted by: Bartsch, Thomas
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
  • Mathematical Physics
Approach: Theoretical


The purpose of this paper is to investigate the global categorical symmetries that arise when gauging finite higher groups in three or more dimensions. The motivation is to provide a common perspective on constructions of non-invertible global symmetries in higher dimensions and a precise description of the associated symmetry categories. This paper focusses on gauging finite groups and split 2-groups in three dimensions. In addition to topological Wilson lines, we show that this generates a rich spectrum of topological surface defects labelled by 2-representations and explain their connection to condensation defects for Wilson lines. We derive various properties of the topological defects and show that the associated symmetry category is the fusion 2-category of 2-representations. This allows us to determine the full symmetry categories of certain gauge theories with disconnected gauge groups. A subsequent paper will examine gauging more general higher groups in higher dimensions.

Author comments upon resubmission

We thank both referees for their detailed reports and for pointing out instances with a need for clarification in our paper. Below, we address some of the questions raised in the reports:

• Action of condensation defects: While it is true that condensation defects in three dimensions act trivially on local operators (i.e. when placed on a 2-sphere), they act non-trivially on extended operators (e.g. when placed on a cyinder wrapping a line operator). Condensation defects (in particular SPT phases stacked on a submanifold when gauging) should thus not be identified with the identity operator.

• Fusion structure on Repc(G): The category Rep^c(G) of projective representations of a finite group G with fixed 2-cocycle c does not have a fusion structure, since 2-cocylces add when taking tensor products of representations. The representation category Rep(G_c) of the central extension G_c of G by U(1) determined by c is not a fusion category in general, since U(1) is a continuous group. Moreover, in order to construct an irrep of G_c one may specify a character n ∈ Z of U(1) together with a projective irrep of G with 2-cocycle c^n. In this sense, projective irreps of G with 2-cocycle c correspond to irreps of G_c with n=1. Taking tensor products of two such irreps then produces an irrep with n=2, so there is no canonical way of associating a projective irrep of G with 2-cocycle c to the latter.

• Non-abelian example: As the gauging of (non-normal) subgroups of a non-abelian group is already part of an extensive case study in our subsequent paper arxiv:2212.07393 (see section 3.6), we refrain from adding an additional non-abelian example to the current instalment. We hope that the added distinction between simple objects and their isomorphism classes in 2Rep(G) as described below clarifies possible confusions w.r.t. subgroups related by conjugation.

List of changes

Below, we summarise the relevant changes that we made in order to address the main points raised in the reports:

• Simple objects in 2Rep(G): Clarified the distinction between simple objects and their isomorphism classes in 2Rep(G) (see pages 26, 32/33, 45, 48): While simple objects are labelled by pairs (H,c) consisting of a subgroup H ⊂ G and a class c ∈ H^2 (H,U(1)) , two such simple objects (H,c) and (H’,c’) are considered equivalent if there exists a group element g ∈ G such that H’ = gHg-1 and c’ = c^g. From a physical point of view, this means that there is no physical distinction between surfaces obtained by gauging conjugate subgroups H and H’, since they only differ by a residual symmetry transformation g.

• TQFT coefficients: Given a surface defect X, one may in principle obtain new surface defects by stacking X with decoupled 2d TQFTs. However, since 2d fully extended stable TQFTs are completely classified by positive integers n (up to equivalence), stacking such a TQFT T_n on top of X simply corresponds to taking the direct sum T_n ⊗ X = X ⊕ … ⊕ X = n ⋅ X. This is the nature of integer fusion coefficients in equations such as (4.52) and (4.77). We clarified this by a footnote on page 33. The above issue becomes more pertinent in 3+1 dimensions, where 3d TQFTs no longer enjoy a simple integer-classification, as was addressed in section 4 of our subsequent paper arxiv:2212.07393.

• Added further details on the induction and restriction of 2-representations in section 4.3.2.

• Added clarification on the product structure in the gauged theory as originating from the tensor product of bimodules for the Frobenius algebra object used for the gauging at the beginning of section 3.2.2.

• Added further comments on the physical interpretation of simple objects in 2Rep(G) to sections 4.3.2 and 5.3.2.

• Notation: Clarified the notation U(1)^O on page 2 and explained the meaning of a “G-orbit” as a set O equipped with a transitive G-action.

• Fixed typos:
- \hat{T} --> T in figure 10 on page 10.
- Included the Frobenius-Schur indicator and bicharacter in the specification of the Tambara-Yamagami symmetry category on page 18.
- H^2(G,U(1)^O) --> H^2(H,U(1)^O) in the second bullet point in the middle of page 3
- “lass” --> “class” at the bottom of page 3
- V_{h,e} --> V_{\chi,e} in equation (3.14)
- Fixed our usage of “e” for the identity group element in section 4.2.1
- Clarified our notation for identity 2-morphisms in equations such as (4.18) by a footnote on page 24
- i --> j below eq. (4.22)
- (n-j) -->( j-n) in eq. (4.25)
- \Phi_{g,j} --> \Phi_{g, \sigma_h(j)} on top of page 28
- Fixed \varphi: H \to Aut(H) on page 39 above equation (5.1)
- Fixed missing reference below eq. (7.5) on page 55
- “gauge \A” --> “gauge H” below eq. (7.7)

• Figures: Added figure 5 to illustrate the consistency conditions displayed in equations (2.10) and (2.11).

• References: added the following references:
- arxiv:1905.09566 on pages 2 & 26 to highlight the mathematical notion of condensations in comparison to the construction presented in the paper.
- arxiv:2206.05646 on page 33, which discusses a similar classification of simple objects after gauging a finite group G in 3d in terms of subgroups H ⊂ G but neglects SPT phases

Current status:
In voting

Reports on this Submission

Anonymous Report 2 on 2024-5-14 (Invited Report)


The authors addressed part of my comments, promising that the missing points are clarified in the part II paper. I recommend the paper for publication.


Publish (meets expectations and criteria for this Journal)

  • validity: good
  • significance: good
  • originality: good
  • clarity: good
  • formatting: excellent
  • grammar: perfect

Anonymous Report 1 on 2024-3-28 (Invited Report)


The authors have addressed all the comments in my previous report appropriately. I recommend the paper to be published.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

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