Toeplitz operators and Hamiltonian torus action

TL;DR

This paper investigates the semi-classical properties of Toeplitz operators under Hamiltonian torus actions on Kähler manifolds, showing how they descend to reduced spaces and extending results to orbifolds with spectral density estimates.

Contribution

It proves that the Guillemin-Sternberg isomorphism is a Fourier integral operator and extends Toeplitz operator descent to orbifolds, with spectral density estimates.

Findings

Guillemin-Sternberg isomorphism is a Fourier integral operator

Toeplitz operators descend to symplectic quotients and orbifolds

Spectral density of reduced Toeplitz operators estimated

Abstract

This paper is devoted to semi-classical aspects of symplectic reduction. Consider a compact prequantizable Kahler manifold M with a Hamiltonian torus action. Guillemin and Sternberg introduced an isomorphism between the invariant part of the quantum space associated to M and the quantum space associated to the symplectic quotient of M, provided this quotient is non-singular. We prove that this isomorphism is a Fourier integral operator and that the Toeplitz operators of M descend to Toeplitz operators of the reduced phase space. We also extend these results to the case where the symplectic quotient is an orbifold and estimate the spectral density of a reduced Toeplitz operator, a result related to the Riemann-Roch-Kawazaki theorem.