Towards quantum dielectric branes: curvature corrections in Abelian beta function and non-Abelian Born–Infeld action

Abstract

We initiate a programme to compute curvature corrections to the non-Abelian Born–Infeld action. This is based on the calculation of derivative corrections to the Abelian Born–Infeld action, describing a maximal brane, to all orders in F = B + 2 π α ′ F . An exact calculation in F allows us to apply the Seiberg–Witten map, reducing the maximal Abelian brane point of view to a minimal non-Abelian brane point of view (replacing 1 / F with [ X , X ] at large F), resulting in matrix model equations of motion in the considered background. We first study derivative corrections to the Abelian Born–Infeld action and compute the two loop beta function for an open bosonic string in a WZW (parallelizable) background. This beta function is the first step in the process of computing open string equations of motion, which can be later obtained by either computing the Weyl anomaly coefficients or the partition function in the given background. The beta function for the gauge field is exact in F and computed to orders O ( H , H 2 , H 3 ) (where H = d B and the curvature is R ∼ H 2 ) and O ( ∇ F , ∇ 2 F , ∇ 3 F ) . In order to carry out this calculation we develop a new regularization method for two loop graphs. We then relate perturbative results for Abelian and non-Abelian Born–Infeld actions, by showing how Abelian derivative corrections yield non-Abelian higher order commutators and vice versa, at large F. We begin the construction of a matrix model describing α ′ corrections to Myers' dielectric effect. This construction is carried out by first setting up a perturbative classification of the relevant non-Abelian tensor structures, which can be considerably narrowed down by the physical constraint of translation invariance in the action and the possibility for generic field redefinitions. The final matrix action is not uniquely determined and depends upon two free parameters. These parameters could be computed via further calculations in the Abelian theory.