Geometric quantization and semi-classical limits of pairings of TQFT vectors

TL;DR

This paper uses geometric quantization to analyze the semi-classical behavior of TQFT vectors, showing eigenvector concentration near trace function level sets and deriving asymptotics for quantum mapping class representations.

Contribution

It introduces a geometric quantization framework for TQFT curve operators as Toeplitz operators, linking quantum eigenvectors to classical trace functions and providing asymptotic estimates.

Findings

Eigenvectors concentrate near trace function level sets.

Asymptotic estimates for pairings of eigenvectors.

Asymptotics for matrix coefficients of quantum mapping class images.

Abstract

Using geometric quantization, we represent curve operators in the TQFT of Witten-Reshetikhin-Turaev with jauge group SU_2 as Toeplitz operators with symbols corresponding to trace functions. As an application, we show that eigenvectors of these operators are concentrated near the level sets of these trace functions, and obtain asymptotic estimates of pairings of such eigenvectors. This yields an asymptotic for some matrix coefficients of the image of mapping classes by quantum representations.