${\cal N}=4$ super Yang-Mills, based on the conjectured $SL(2,Z)$ duality of the theory. This duality was used to fix the form of the effective action up to six derivatives. The theory that we are studying is also believed to have an exact $SL(2,Z)$ duality, and under this assumption the analysis of applies.

The unique $SL(2,Z)$ invariant form for the six derivative terms is

Under duality, the second term above mixes with the two and four derivative terms and consequently its coefficient is fixed. The requirement of self duality does not fix $\lambda^{(6)}$, since duality maps this term into itself at lowest order. These are the field theory results that we wish to compare to gravity.

On the gravity side, we will consider a probe moving in a background to be specified below. The probe worldvolume dynamics is captured by a Born-Infeld action. The Born-Infeld action itself is self-dual, but the duality does not act on the separation of the branes. This separation is parametrized by the Higgs fields which belong to the same supermultiplet as the gauge fields. This implies, as pointed out in , that the Higgs fields that realize ${\cal N}=2$ supersymmetry linearly must be related by a 