A Lefschetz fixed point formula for symplectomorphisms

TL;DR

This paper develops an asymptotic Lefschetz fixed point formula for automorphisms of prequantum bundles on compact Kähler manifolds, with applications to quantum representations of the mapping class group.

Contribution

It introduces a new asymptotic formula generalizing the Lefschetz fixed point theorem for prequantum bundle automorphisms in geometric quantization.

Findings

Derived an asymptotic character formula for automorphisms

Generalized Lefschetz fixed point formula in the quantum setting

Applicable to quantum representations of the mapping class group

Abstract

Consider a compact K\"ahler manifold endowed with a prequantum bundle. Following the geometric quantization scheme, the associated quantum spaces are the spaces of holomorphic sections of the tensor powers of the prequantum bundle. In this paper we construct an asymptotic representation of the prequantum bundle automorphism group in these quantum spaces. We estimate the characters of these representations under some transversality assumption. The formula obtained generalizes in some sense the Lefschetz fixed point formula for the automorphisms of the prequantum bundle preserving its holomorphic structure. Our results will be applied in two forthcoming papers to the quantum representation of the mapping class group.