Asymptotic expansion of generalized Witten integrals for Hamiltonian circle actions

TL;DR

This paper derives a detailed asymptotic expansion for generalized Witten integrals in Hamiltonian circle actions, linking coefficients to symplectic strata and fixed-point invariants, advancing understanding of symplectic geometry and reduction.

Contribution

It provides a complete asymptotic expansion for Witten integrals on symplectic manifolds with circle actions, including explicit characterization of coefficients involving stratified contributions.

Findings

Coefficients expressed as integrals over symplectic strata

Inclusion of singular contributions from lower-dimensional strata

Connection of coefficients to fixed-point set invariants

Abstract

We derive a complete asymptotic expansion of generalized Witten integrals for Hamiltonian circle actions on arbitrary symplectic manifolds, characterizing the coefficients in the expansion as integrals over the symplectic strata of the corresponding Marsden-Weinstein reduced space and distributions on the Lie algebra. The obtained coefficients involve singular contributions of the lower-dimensional strata related to numerical invariants of the fixed-point set.