Non-Commutative Geometry for D-Branes in Large R-R Field Background

Abstract

We examine the role of non-commutative geometry in D$p$-branes within large R-R field backgrounds. In this context, the background of a significant ($p-1$)-form R-R field can be effectively described using a ($p-1$)-bracket, similar to the method used in the NS-NS case. We begin by recalling how non-commutative geometry arises from the quantization of open string theory. In this framework, the Seiberg-Witten map is a key element that establishes the equivalence between commutative and non-commutative descriptions in the low-energy effective theory. The Poisson bracket characterizes non-commutative structures, with deformation achieved through the Moyal product. Next, we show how the Nambu-Poisson bracket emerges in the context of a single D4-brane with the large R-R field background limit, starting from the BLG model. The generalization to a D$p$-brane leads to the ($p-1$)-bracket description, which reveals a duality web relating NS-NS and R-R field backgrounds via T-duality and S-duality in the low-energy limit. Finally, we extend the single D-brane construction to multiple D-branes by promoting the ordinary product in the bracket to a covariant derivative at the Poisson level.