Non-invertible Symmetries and Higher Representation Theory II

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

• Fibre functors: The second bullet point on page six is intended to give an alternative interpretation of the category Rep(G) for a finite group G, which may be interpreted as the category of functors F: BG -> Vec from the delooping of G into Vec. We refrain from calling these “fibre functors” as the latter term is usually reserved for cases where the pre-image of the functor is a (higher) fusion category corresponding to the symmetry category of a given theory (and not the delooping of a finite group, which is not a fusion category).

• Condensation defects: Condensation defects are (partly) labelled by subgroups: While simple objects in 2Rep(G) 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, a symmetry defect labelled by (H,c) corresponds to a surface where the bulk gauge symmetry G is broken down to H and supplemented by an SPT phase c. In particular, 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 on the defect. This will be clarified in an updated version of Part I and is stated in the first paragraph of page 25 of Part II.

• Bimodule classification: The definition of \Phi_g as in eq. (2.11) is the natural one given the left- and right-morphisms l and r of a bimodule, and is also the standard definition appearing in the mathematical literature on the classification of such bimodules [24]. We implement the redefinition (2.13) to achieve a more canonical form of the 2-cocycle (2.14), but retain its original form for comparison.

• Cohomology of 2-groups: We are not aware of a group-cohomology classification of the cohomology of the classifying space of a 2-group. Our notation H^n(\mathcal{G}, U(1)) is explained in footnote 5.

• Gauging extensions in 2d: We included section 2.4 with the presented level of detail to make completely apparent the analogy with the gauging of sub-2-groups discussed in section 3.4 (in particular the analogy between figure 13 and figures 30/31).

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

• TQFT coefficients: Given a surface defect X in three dimensions, one may in principle obtain new surface defects by stacking X with decoupled 2d TQFTs. However, since 2d fully extended stable TQFTs are (up to equivalence) completely classified by positive integers n (corresponding to their number of vacua), 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 (3.1). In particular, the mathematical fusion rules are not associated to particular topologies of the surface defects but internal to the fusion 2-category under consideration. We clarified our treatment of Euler terms in a footnote on page 23.

• Typos:
- Added (−1) to \alpha|_H = (d\psi)^(−1) above eq. (1.1) in the introduction. We choose this convention to interpret \psi(h_1, h_2) as the phase assigned to the junction of two H-lines, as illustrated in figure 2.
- Further explained the notation used in eq. (1.2).
- Fixed typo (”now” repeated) in first sentence of section 4.5.
- Clarified our usage of twisted group cohomology by a footnote on page 35.
- Added “orbifold branch” to the first sentence of the second paragraph of section 2.6.