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

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

2011-01-12

arxiv.org/abs/1101.2420v1

Mathematics

Symplectic Geometry

17 pages

Scientific paper

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.

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