$S^1$-equivariant local index and transverse index for non-compact symplectic manifolds

TL;DR

This paper introduces an $S^1$-equivariant index for non-compact symplectic manifolds with Hamiltonian $S^1$-action, using Dirac-type operator perturbation, and proves a related quantization conjecture.

Contribution

It defines a new $S^1$-equivariant index for non-compact symplectic manifolds and proves the quantization conjecture in this setting.

Findings

Established the $S^1$-equivariant index for non-compact symplectic manifolds.

Proved the quantization conjecture for this index.

Discussed the relation to transverse elliptic operator index.

Abstract

We define an $S^1$-equivariant index for non-compact symplectic manifolds with Hamiltonian $S^1$-action. We use the perturbation by Dirac-type operator along the $S^1$-orbits. We give a formulation and a proof of quantization conjecture for this $S^1$-equivariant index. We also give comments on the relation between our $S^1$-equivariant index and the index of transverse elliptic operators.