  We initiate a programme to compute curvature corrections to the nonabelian BI
action. This is based on the calculation of derivative corrections to the
abelian BI action, describing a maximal brane, to all orders in F. An exact
calculation in F allows us to apply the SW map, reducing the maximal abelian
point of view to a minimal nonabelian point of view (replacing 1/F with [X,X]
at large F), resulting in matrix model equations of motion. We first study
derivative corrections to the abelian BI action and compute the 2-loop beta
function for an open string in a WZW (parallelizable) background. This beta
function is the first step in the process of computing string equations of
motion, which can be later obtained by computing the Weyl anomaly coefficients
or the partition function. The beta function is exact in F and computed to
orders O(H,H^2,H^3) (H=dB and curvature is R ~ H^2) and O(DF,D^2F,D^3F). In
order to carry out this calculation we develop a new regularization method for
2-loop graphs. We then relate perturbative results for abelian and nonabelian
BI actions, by showing how abelian derivative corrections yield nonabelian
commutator corrections, at large F. We begin the construction of a matrix model
describing \a' corrections to Myers' dielectric effect. This construction is
carried out by setting up a perturbative classification of the relevant
nonabelian tensor structures, which can be considerably narrowed down by the
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.
