Non-invertible Symmetries and Higher Representation Theory I

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.

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