Permutation actions on modular tensor categories of topological multilayer phases

Abstract

We find a non-trivial representation of the symmetric group S n on the n-fold Deligne product C ⊠n of a modular tensor category C for any n ≥ 2. This is accomplished by checking that a particular family of C ⊠n -bimodule categories representing adjacent transpositions satisfies the symmetric group relations with respect to the relative Deligne product. The bimodule categories are based on a permutation action of S 2 on C ⊠ C discussed by Fuchs and Schweigert in [FS14], for which we show that it is, in a certain sense, unique. In the context of condensed matter physics, the S n -representation corresponds to the specification of permutation twist surface defects in a (2+1)-dimensional topological multilayer phase, which are relevant to topological quantum computation and could promote the explicit construction of the data of an S n -gauged phase.

Computing fusion rules for spherical G-extensions of fusion categories

“…From an algebraic viewpoint, the o 4 obstruction for permutation actions was shown to vanish in [GJ19], hence these extensions always exist. They have been studied in the /2 case ([BS11, BFRS10], [EMJP18,Pas18]). Very recently, Delaney has given an algorithm for computing the fusion rules of general permutation extensions using the concept of bare defects [Del19].…”