Journal of Symplectic Geometry

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

Joel Fine

Full-text: Open access

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.

Article information

Source
J. Symplectic Geom., Volume 12, Number 1 (2014), 105-123.

Dates
First available in Project Euclid: 29 August 2014

Permanent link to this document
https://projecteuclid.org/euclid.jsg/1409317365

Mathematical Reviews number (MathSciNet)
MR3194077

Zentralblatt MATH identifier
1304.53082

Citation

Fine, Joel. The Hamiltonian geometry of the space of unitary connections with symplectic curvature. J. Symplectic Geom. 12 (2014), no. 1, 105--123. https://projecteuclid.org/euclid.jsg/1409317365


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