Asymptotic properties of the quantum representations of the modular group

TL;DR

This paper investigates the large level asymptotics of quantum representations of the modular group, showing they act as Fourier integral operators and connecting classical and quantum Chern-Simons theories, with implications for Witten-Reshetikhin-Turaev invariants.

Contribution

It demonstrates that modular group elements act as Fourier integral operators in the large level limit, linking classical and quantum topological theories.

Findings

Modular group actions are Fourier integral operators at large levels.

Established connection between classical and quantum Chern-Simons theories.

Derived asymptotic expansion of Witten-Reshetikhin-Turaev invariants for hyperbolic torus bundles.

Abstract

We study the asymptotic behaviour of the quantum representations of the modular group in the large level limit. We prove that each element of the modular group acts as a Fourier integral operator. This provides a link between the classical and quantum Chern-Simons theories for the torus. From this result we deduce the known asymptotic expansion of the Witten-Reshetikhin-Turaev invariants of the torus bundles with hyperbolic monodromy.