Reducibility of quantum representations of mapping class groups

TL;DR

This paper establishes a general criterion for when quantum representations of mapping class groups are reducible, focusing on modular tensor categories with specific algebraic properties, and provides explicit decompositions for certain cases.

Contribution

It introduces a broad condition linking algebraic structures to reducibility of quantum representations, including explicit decompositions for quantum SU(2) at all even levels.

Findings

Quantum representations are reducible under certain algebraic conditions.

Explicit decomposition constructed for quantum SU(2) at even levels.

Conjecture on complete irreducible decomposition at high levels.

Abstract

In this paper we provide a general condition for the reducibility of the Reshetikhin-Turaev quantum representations of the mapping class groups. Namely, for any modular tensor category with a special symmetric Frobenius algebra with a non-trivial genus one partition function, we prove that the quantum representations of all the mapping class groups built from the modular tensor category are reducible. In particular for SU(N) we get reducibility for certain levels and ranks. For the quantum SU(2) Reshetikhin-Turaev theory we construct a decomposition for all even levels. We conjecture this decomposition is a complete decomposition into irreducible representations for high enough levels.