The Hamiltonian geometry of the space of unitary connections with symplectic curvature

TL;DR

This paper explores the geometric structure of the space of unitary connections with symplectic curvature, revealing a symplectic form, a moment map, and connections to the Calabi conjecture on Kähler manifolds.

Contribution

It introduces a Hamiltonian geometric framework for the space of unitary connections with symplectic curvature, linking it to coadjoint orbits and the Calabi conjecture.

Findings

The space of connections admits a G-invariant symplectic structure.

A moment map characterizes the space and relates to volume-prescribing problems.

Special cases connect to the Calabi conjecture on Kähler manifolds.

Abstract

Let L->M be a Hermitian line bundle over a compact manifold. Write S for the space of all unitary connections in L whose curvatures define symplectic forms on M and G for the group of unitary bundle isometries of L, which acts on S by pull-back. The main observation of this note is that S carries a G-invariant symplectic structure, there is a moment map for the G-action and that this embeds the components of S as G_0-coadjoint orbits (where G_0 is the component of the identity). Restricting to the subgroup of G which covers the identity on M, we see that prescribing the volume form of a symplectic structure can be seen as finding a zero of a moment map. When M is a K\"ahler manifold, this gives a moment-map interpretation of the Calabi conjecture. We also describe some directions for future research based upon the picture outlined here.