Permutation actions on modular tensor categories of topological multilayer phases

TL;DR

This paper constructs a non-trivial symmetric group action on the n-fold Deligne product of a modular tensor category, linking algebraic structures to permutation defects in topological phases relevant for quantum computation.

Contribution

It demonstrates a new symmetric group representation on tensor categories and characterizes permutation actions as bimodule categories, with implications for topological quantum phases.

Findings

Established an $S_n$-action on $oxtimes n$ of a modular tensor category.

Proved the uniqueness of the permutation bimodule categories based on $S_2$ action.

Connected algebraic structures to permutation twist defects in topological phases.

Abstract

We find a non-trivial representation of the symmetric group $S_n$ on the $n$-fold Deligne product $\mathcal{C}^{\boxtimes n}$ of a modular tensor category $\mathcal{C}$ for any $n \geq 2$. This is accomplished by checking that a particular family of $\mathcal{C}^{\boxtimes n}$-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 $\mathcal{C}\boxtimes\mathcal{C}$ 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.