Asymptotic properties of the quantum representations of the mapping class group

TL;DR

This paper investigates the asymptotic behavior of quantum representations of surface mapping class groups, showing they lift to asymptotic representations and analyzing their properties as Fourier integral operators.

Contribution

It provides a detailed analysis of the large level limit of quantum representations, including their lifting, Fourier integral operator structure, and asymptotic character expansion.

Findings

Operators are Fourier integral operators with explicit canonical relations and symbols.

Quantum representations asymptotically unitarize and satisfy the Egorov property.

Characters of the representations have an asymptotic expansion matching Witten's heuristic formula.

Abstract

We establish various results on the large level limit of projective quantum representations of surface mapping class groups obtained by quantizing moduli spaces of flat SU(n)-bundle. Working with the metaplectic correction, we proved that these projective representations lift to asymptotic representations. We show that the operators in these representations are Fourier integral operators and determine explicitly their canonical relations and symbols. We deduce from these facts the Egorov property and the asymptotic unitarity, two results already proved by J.E. Andersen. Furthermore we show under a transversality assumption that the characters of these representations have an asymptotic expansion. The leading order term of this expansion agrees with the formula derived heuristically by E. Witten in "Quantum field theory and the Jones polynomial".