Quantum moment map and obstructions to the existence of closed Fedosov star products

TL;DR

This paper investigates the properties of Fedosov star products in symplectic and Kähler manifolds, revealing cohomological invariants and obstructions to their existence, with implications for quantum geometry.

Contribution

It introduces new cohomological obstructions to the existence of closed Fedosov star products and relates them to quantum moment maps and symplectic invariants.

Findings

Normalized trace depends only on cohomology path component

Identifies obstructions to closed Fedosov star products

Re-discover previous obstruction in Kähler case

Abstract

It is shown that the normalized trace of Fedosov star product for quantum moment map depends only on the path component in the cohomology class of the symplectic form and the cohomology class of the closed formal 2-form required to define Fedosov connections (Theorem 1.3). As an application we obtain a family of obstructions to the existence of closed Fedosov star products naturally attached to symplectic manifolds (Theorem 1.5) and K\"ahler manifolds (Theorem 1.6). These obstructions are integral invariants depending only on the path component of the cohomology class of the symplectic form. Restricted to compact K\"ahler manifolds we re-discover an obstruction found earlier in [29].