Infinite dimensional moment map geometry and closed Fedosov's star products

TL;DR

This paper explores the geometry of symplectic connections and moment maps on infinite dimensional spaces, linking deformation quantization with Fedosov star products and their geometric properties on Kähler manifolds.

Contribution

It introduces a new connection between trace densities in star products and moment map geometry, providing a geometric characterization of Fedosov star products close to being closed.

Findings

Finite dimensional submanifold structure of zeroes of the functional 

Link between trace densities and moment map geometry

Characterization of Fedosov star products that are nearly closed

Abstract

We study the Cahen-Gutt moment map on the space of symplectic connections of a symplectic manifold. On a K\"ahler manifold, we define a Calabi-type functional $\mathscr{F}$ on the space of K\"ahler metrics in a given K\"ahler class. We study the zeroes of $\mathscr{F}$. Given a zero of $\mathscr{F}$ with non-negative Ricci tensor, we show the space of zeroes around the given one has the structure of a finite dimensional embedded submanifold. We give a new motivation, coming from deformation quantisation, for the study of moment maps on infinite dimensional spaces. More precisely, we establish a strong link between trace densities for star products (obtained from Fedosov's type methods) and moment map geometry on infinite dimensional spaces. As a consequence, we provide, on certain K\"ahler manifolds, a geometric characterization of a space of Fedosov's star products that are closed up to order 3 in $\nu$.