Characteristic classes of star products on Marsden-Weinstein reduced symplectic manifolds

TL;DR

This paper explores a quantum reduction scheme for star products on symplectic manifolds, constructing an analogue of the Kirwan map via BRST cohomology, and shows all star products on reduced manifolds are equivalent to reduced star products.

Contribution

It introduces an explicit map on star product equivalence classes using BRST cohomology, extending the understanding of deformation quantization on reduced symplectic manifolds.

Findings

Constructed an explicit map analogous to the Kirwan map in the Cartan model.

Proved that every star product on a reduced manifold is equivalent to a reduced star product.

Abstract

In this note we consider a quantum reduction scheme in deformation quantization on symplectic manifolds proposed by Bordemann, Herbig and Waldmann based on BRST cohomology. We explicitly construct the induced map on equivalence classes of star products which will turn out to be an analogue to the Kirwan map but in the Cartan model of equivariant cohomology. As a byproduct we shall see that every star product on a (suitable) reduced manifold is equivalent to a reduced star product.