Noncommutativity in weakly curved background by canonical methods

TL;DR

This paper investigates how weakly curved backgrounds induce noncommutativity on Dp-branes using canonical methods, revealing boundary-specific noncommutativity characterized by a coordinate-dependent antisymmetric tensor and a Kontsevich star product.

Contribution

It provides a canonical analysis of Dp-brane noncommutativity in weakly curved backgrounds, deriving explicit relations and showing the emergence of a Kontsevich star product after quantization.

Findings

Noncommutativity appears only on the world-sheet boundary.

The noncommutativity parameter is a coordinate-dependent antisymmetric tensor.

Quantization leads to the Kontsevich star product replacing the Moyal product.

Abstract

Using canonical method we investigate $Dp-$brane world-volume noncommutativity in weakly curved background. The term weakly curved means that in the leading order, the source of non-flatness is infinitesimally small Kalb-Ramond field $B_{\mu\nu}$ linear in coordinate, while the Ricci tensor does not contribute being the infinitesimal of the second order. On the solution of boundary conditions, we find 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 infinitesimally small and coordinate dependent antisymmetric tensor $B_{\mu\nu}$. This result coincides with that obtained on the group manifolds in the limit of the large level $n$ of current algebra. After quantization the algebra of functions on Dp-brane world-volume is represented with the Kontsevich star product instead of the Moyal one in the flat background.