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March 2014 The Hamiltonian geometry of the space of unitary connections with symplectic curvature
Joel Fine
J. Symplectic Geom. 12(1): 105-123 (March 2014).

Abstract

Let $\mathcal{L} \to \mathcal{M}$ be a Hermitian line bundle over a compact manifold. Write $\mathcal{S}$ for the space of all unitary connections in $\mathcal{L}$ whose curvatures define symplectic forms on $\mathcal{M}$ and $\mathcal{G}$ for the identity component of the group of unitary bundle isometries of $\mathcal{L}$, which acts on $\mathcal{S}$ by pullback. The main observation of this note is that $\mathcal{S}$ carries a $\mathcal{G}$-invariant symplectic structure, there is a moment map for the $\mathcal{G}$-action and that this embeds the components of $\mathcal{S}$ as $\mathcal{G}$-coadjoint orbits. Restricting to the subgroup of $\mathcal{G}$ which covers the identity on $\mathcal{M}$, we see that prescribing the volume form of a symplectic structure can be seen as finding a zero of a moment map. When $\mathcal{M}$ is a Kähler 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.

Citation

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Joel Fine. "The Hamiltonian geometry of the space of unitary connections with symplectic curvature." J. Symplectic Geom. 12 (1) 105 - 123, March 2014.

Information

Published: March 2014
First available in Project Euclid: 29 August 2014

zbMATH: 1304.53082
MathSciNet: MR3194077

Rights: Copyright © 2014 International Press of Boston

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Vol.12 • No. 1 • March 2014
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