Nonassociative Star Product Deformations for D -Brane World-Volumes in Curved 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 = 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 ∞ 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.Noncommutative quantum field theoretic limits of string theory have received considerable attention in the recent literature, and have been studied in a variety of papers (see, e.g., [1,2,3,4,5,6] and references therein). The attention is focused on a specific scaling limit, where the effects of large magnetic backgrounds are translated into Moyal noncommutative deformations of the D-brane world-volume algebra of functions. The open string physics is therefore captured within a quantum field theory (which is renormalizable, despite appearances [7,8]). A common point to most previous investigations is that the background (sigma model) fields are taken to be constant and that, as a consequence, the target space is flat. One may then ask the natural question of what happens if the background is curved, i.e., if the background fields are no longer constant? This question received some attention in a couple of recent papers [9,10,11,12], but there is no general answer to it (other papers of interest with some relation to this subject are, e.g., [13,14,15,16]). Our goal in this work is to address this problem in the context of a simple model with weakly curved backgrounds, which can be on one side connected to the known flat background framework, and on the other hand can be related to formal results of brane physics in WZW models, which can be analyzed exactly with conformal field theory techniques [10,11,17].More concretely, the aim of this paper is to understand how the presence of a nontrivial background field affects the world-volume deformation of a D-brane. It is known that, in the presence of a constant background B-field, the physics can be exactly described either by a sigma model approach [18,19,20,21,22,23,24,25], or alternatively, by trans...

Non(anti)commutative superspace

“…In [19] the deformation of D-brane world-volumes in the presence of NS-NS curved backgrounds was investigated in a nonsupersymmetric context. It was shown that, if both the brane and the background are curved, i.e.…”

“…H ≡ d(B + F ) = 0, then the deformation of the world-volume is a Kontsevich deformation which defines a nonassociative, noncommutative product. Noncommutativity is governed by the usual NS-NS B-field, whereas nonassociativity arises from the NS-NS field strength H. This suggests that our supergeometries might naturally appear, if an analysis similar to [19] were to be performed in a manifestly supersymmetric formalism (i.e. working with a Green-Schwarz [20] or Berkovits [21] string) for backgrounds with nonvanishing super p-form field strengths.…”

Matrix theory origins of non-geometric fluxes

“…Taking into account the algebra (3.5) and the commutation relation among the operators U i , we obtain 70) namely the present situation is already associative. This is expected, since the f flux does not arise from a p-form source, but it is a metric flux.…”

“…Let us note that such a term was also obtained in Refs. [70,71] from the expansion of the Dirac-Born-Infeld action. The second and the third H-dependent terms appear as (parts of) the Chern-Simons action.…”

Poisson Manifolds, Lie Algebroids, Modular Classes: a Survey

“…To each of these theories, there corresponds, more generally, a twisted version. Twisted Poisson manifolds were first considered in some quantization problems in string theory [22,105], then they were defined as geometric objects [62,113], and finally viewed as particular cases of Lie algebroids with twisted Poisson structures [111,72]. Just as a Lie algebra with a triangular r-matrix gives rise to a Lie bialgebra in the sense of Drinfeld [29], a Lie algebroid with a Poisson structure gives rise to a Lie bialgebroid in the sense of Mackenzie and Xu [95,64], while a twisted Poisson structure gives rise to a quasi-Lie-bialgebroid 2 .…”