2 + 1 Abelian ‘ Gauge Theory ’ Inspired by Ideal Hydrodynamics

We study a possibly integrable model of abelian gauge fields on a two-dimensional surface M , with volume form μ . It has the same phase space as ideal hydrodynamics, a coadjoint orbit of the volume-preserving diffeomorphism group of M . Gauge field Poisson brackets differ from the Heisenberg algebra, but are reminiscent of Yang-Mills theory on a null surface. Enstrophy invariants are Casimirs of the Poisson algebra of gauge invariant observables. Some symplectic leaves of the Poisson manifold are identified. The Hamiltonian is a magnetic energy, similar to that of electrodynamics, and depends on a metric whose volume element is not a multiple of μ . The magnetic field evolves by a quadratically non-linear ‘Euler’ equation, which may also be regarded as describing geodesic flow on SDiff(M,μ). Static solutions are obtained. For uniform μ , an infinite sequence of local conserved charges beginning with the hamiltonian are found. The charges are shown to be in involution, suggesting integrability. Besides being a theory of a novel kind of ideal flow, this is a toy-model for Yang-Mills theory and matrix field theories, whose gauge-invariant phase space is conjectured to be a coadjoint orbit of the diffeomorphism group of a non-commutative space.