Noncommutativity in weakly curved background by canonical methods

Abstract

Using the canonical method, we investigate the Dp-brane world-volume noncommutativity in a weakly curved background. The term ``weakly curved'' means that, in the leading order, the source of nonflatness is an infinitesimally small Kalb-Ramond field ${B}_{\ensuremath{\mu}\ensuremath{\nu}}$, linear in coordinate, while the Ricci tensor does not contribute, being an infinitesimal of the second order. On the solution of boundary conditions, we find a simple expression for the space-time coordinates in terms of the effective coordinates and momenta. This basic relation helped us to prove that noncommutativity appears only on the world sheet boundary. The noncommutativity parameter has a standard form, but with the infinitesimally small and coordinate-dependent antisymmetric tensor ${B}_{\ensuremath{\mu}\ensuremath{\nu}}$. This result coincides with that obtained on the group manifolds in the limit of the large level $n$ of the current algebra. After quantization, the algebra of the functions on the Dp-brane world volume is represented with the Kontsevich star product instead of the Moyal one in the flat background.