Nonassociative Star Product Deformations for D-brane Worldvolumes in Curved Backgrounds

TL;DR

This work shows that D-brane worldvolume algebras in curved backgrounds can be deformed beyond associativity. By performing disk-level open-string perturbation theory around parallelizable backgrounds, the authors derive Kontsevich-type star products that generalize the flat Moyal case: constant ω yields Moyal, while dω = 0 leads to Kontsevich deformation for curved branes in flat space, and dω ≠ 0 yields a nonassociative Kontsevich product with an A_infty structure controlled by the NS-NS 3-form H. They connect these deformations to open-string parameters, show how the nonassociativity arises from H, and relate the effective action to Matrix theory dielectric effects, including a tachyon condensation interpretation. The framework provides a pathway toward Matrix theory in general curved backgrounds and suggests rich geometric structures—nonassociative manifolds and homotopy algebras—that merit further development for brane dynamics in nontrivial backgrounds.

Abstract

We investigate the deformation of D-brane world-volumes in curved backgrounds. We calculate the leading corrections to the boundary conformal field theory involving the background fields, and in particular we study the correlation functions of the resulting system. This allows us to obtain the world-volume deformation, identifying the open string metric and the noncommutative deformation parameter. The picture that unfolds is the following: when the gauge invariant combination ω= B + F is constant one obtains the standard Moyal deformation of the brane world-volume. Similarly, when dω= 0 one obtains the noncommutative Kontsevich deformation, physically corresponding to a curved brane in a flat background. When the background is curved, H = dω\not= 0, we find that the relevant algebraic structure is still based on the Kontsevich expansion, which now defines a nonassociative star product with an A_\infty homotopy associative algebraic structure. We then recover, within this formalism, some known results of Matrix theory in curved backgrounds. In particular, we show how the effective action obtained in this framework describes, as expected, the dielectric effect of D-branes. The polarized branes are interpreted as a soliton, associated to the condensation of the brane gauge field.