Additivity of Spin^c Quantization under Cutting

TL;DR

This paper proves that the spin^c quantization of a manifold with a group action is additive under a cutting operation, using Kostant-type formulas to relate the original and cut spaces.

Contribution

It introduces a cutting construction for S^1-equivariant spin^c manifolds and demonstrates the additivity of their quantizations, extending previous understanding of geometric quantization.

Findings

Quantization of the original manifold equals the sum of quantizations of cut spaces.

The proof employs Kostant-type formulas based on local data around fixed points.

The construction applies to G-equivariant spin^c structures with group actions.

Abstract

A G-equivariant spin^c structure on a manifold gives rise to a virtual representation of the group G, called the spin^c quantization of the manifold. We present a cutting construction for S^1-equivariant spin^c manifolds, and show that the quantization of the original manifold is isomorphic to the direct sum of the quantizations of the cut spaces. Our proof uses Kostant-type formulas, which express the quantization in terms of local data around the fixed point set of the S^1-action.