Geometric quantization of symplectic maps and Witten's asymptotic conjecture

TL;DR

This paper explores the semi-classical quantization of symplectic maps on Kähler manifolds using Berezin-Toeplitz operators, providing a trace formula and applying it to Witten's conjecture on quantum representations.

Contribution

It introduces a Toeplitz operator approach to analyze the deformation of holomorphic sections and establishes a trace formula relevant to Witten's asymptotic conjecture.

Findings

Parallel transport acts like a Toeplitz operator

Derived a semi-classical trace formula for symplectic maps

Applied results to Witten's asymptotic expansion conjecture

Abstract

We use the theory of Berezin-Toeplitz operators of Ma and Marinescu to study the spaces of holomorphic sections of a prequantizing line bundle over compact K\"ahler manifolds under deformations of the complex structure. We show that the parallel transport in the induced vector bundle over the deformation space behaves like a Toeplitz operator, and compute its first coefficient. We then use this result to establish a semi-classical trace formula for the induced quantization of symplectic maps, and give an application to Witten's asymptotic expansion conjecture for the quantum representations of the mapping class group.