Symplectic Surgery and the Spin-C Dirac operator

TL;DR

This paper investigates how the equivariant index of twisted Spin-C Dirac operators behaves under symplectic cutting and proves the Guillemin-Sternberg conjecture relating quantization and symplectic reduction.

Contribution

It establishes gluing properties of the equivariant index for Hamiltonian G-spaces and proves the Guillemin-Sternberg conjecture for non-abelian groups.

Findings

Proved the Guillemin-Sternberg conjecture for compact connected Lie groups.

Established gluing formulas for the equivariant index under symplectic cutting.

Extended previous abelian group results to non-abelian cases.

Abstract

Let $G$ be a compact connected Lie group, and $M$ a compact Hamiltonian $G$-space, with moment map $J$. For each $G$-equivariant Hermitian vector bundle $E$ over $M$, one has an associated twisted Spin-C Dirac operator, whose equivariant index is a symplectic invariant of $E$. In the present paper, we study gluing properties of the equivariant index under "symplectic cutting" operations. Our main application is a proof of the Guillemin-Sternberg conjecture, which says that if $E=L$ is a quantizing line bundle and $0$ a regular value of $J$, the multiplicity of the trivial representation in the equivariant index is equal to the Riemann-Roch number of the symplectic quotient. This generalizes previous results for the case that $G=T$ is abelian.