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 ${\mathcal{C}}^{⊠n}$ of a modular tensor category $\mathcal{C}$ for any $n\ge 2$. This is accomplished by checking that a particular family of ${\mathcal{C}}^{⊠n}$-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 $\mathcal{C}⊠\mathcal{C}$ discussed by Fuchs and Schweigert in hep-th/1310.1329 , 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.

arXiv comment51 pages with TikZ figures; uses braids.sty by Andrew Stacey. v2: minor corrections, revisions and additions; cross-references added; "Preliminaries" expanded

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