Lower order asymptotics for Szeg\"{o} and Toeplitz kernels under Hamiltonian circle actions

TL;DR

This paper investigates the detailed asymptotic behavior of Szeg"{o} and Toeplitz kernels in a modified Berezin-Toeplitz quantization framework on compact K"{a}hler manifolds with Hamiltonian circle actions, linking geometry and quantization.

Contribution

It extends previous work by analyzing lower order asymptotics of kernels under Hamiltonian circle actions, connecting them to the moment map and symplectic quotient geometry.

Findings

Derived asymptotic formulas relating kernels to the moment map

Established connections between quantization and symplectic geometry

Generalized previous results to non-trivial circle actions

Abstract

We consider a natural variant of Berezin-Toeplitz quantization of compact K\"{a}hler manifolds, in the presence of a Hamiltonian circle action lifting to the quantizing line bundle. Assuming that the moment map is positive, we study the diagonal asymptotics of the associated Szeg\"{o} and Toeplitz operators, and specifically their relation to the moment map and to the geometry of a certain symplectic quotient. When the underlying action is trivial and the moment map is taken to be identically equal to one, this scheme coincides with the usual Berezin-Toeplitz quantization. This continues previous work on near-diagonal scaling asymptotics of equivariant Szeg\"{o} kernels in the presence of Hamiltonian torus actions.