D-branes, Symplectomorphisms and Noncommutative Gauge Theories

TL;DR

This paper demonstrates that duals of certain supermembranes and D-branes with nontrivial wrapping can be formulated as noncommutative gauge theories using symplectic geometry and Moyal brackets.

Contribution

It introduces a formulation of noncommutative gauge theories derived from supermembrane and D-brane duals using symplectic structures and deformations of Poisson brackets.

Findings

Dual supermembrane and D-brane theories can be expressed as noncommutative gauge theories.

Theories are described via symplectic connections on symplectic fibrations.

The noncommutative structure arises from Moyal deformation of the Poisson bracket.

Abstract

It is shown that the dual of the double compactified D=11 Supermembrane and a suitable compactified D=10 Super 4D-brane with nontrivial wrapping on the target space may be formulated as noncommutative gauge theories. The Poisson bracket over the world-volume is intrinsically defined in terms of the minima of the hamiltonian of the theory, which may be expressed in terms of a non degenerate 2-form. A deformation of the Poisson bracket in terms of the Moyal brackets is then performed. A noncommutative gauge theory in terms of the Moyal star bracket is obtained. It is shown that all these theories may be described in terms of symplectic connections on symplectic fibrations. The world volume being its base manifold and the (sub)group of volume preserving diffeomorphisms generate the symplectomorphisms which preserve the (infinite dimensional) Poisson bracket of the fibration.