A quantum ergodic theorem for mapping class groups action on character variety

TL;DR

This paper establishes a quantum ergodic theorem linking the ergodic action of mapping class groups on character varieties to the asymptotic behavior of invariant subspaces, with applications in TQFT spin decompositions.

Contribution

It introduces a quantum ergodic theorem connecting mapping class group actions with invariant subspace asymptotics in Witten-Reshetikhin-Turaev representations.

Findings

Proves ergodicity criteria for subgroup actions on character varieties.

Derives asymptotic behavior of invariant subspaces in TQFT representations.

Provides new insights into spin decomposition in quantum topology.

Abstract

We state a theorem relating the ergodicity of the action of a given subgroup of the mapping class group of a surface on the character variety, to the asymptotic of its invariant subspaces through the Witten-Reshetikhin-Turaev representations. As application we give an asymptotic result on the spin decomposition arising in TQFT.