Futaki invariant for Fedosov's star products

TL;DR

This paper introduces a Futaki-type invariant that obstructs the existence of closed Fedosov's star products on K"ahler manifolds, linking geometric invariants to deformation quantization.

Contribution

It constructs a new obstruction, analogous to the Futaki invariant, for the existence of closed Fedosov's star products on K"ahler manifolds.

Findings

Obstruction derived from a Futaki-type invariant.

Connection between moment map zeros and star product closure.

Extension of Futaki invariant concept to Fedosov's quantization.

Abstract

We study obstructions to the existence of closed Fedosov's star products on a given K\"ahler manifold. In our previous paper, we proved that the Levi-Civita connection of a K\"ahler manifold will produce a closed (in the sense of Connes-Flato-Sternheimer) Fedosov's star product only if it is a zero of the Cahen-Gutt moment map on the space of symplectic connections. By analogy with the Futaki invariant obstructing the existence of cscK metrics, we build an obstruction for the existence of zero of the moment map and hence for the existence of closed Fedosov's star products on a K\"ahler manifold.