correction to the prepotential was computed in . The one instanton corrected prepotential is

$${\cal F}={1\over 2}\tau_{cl}A^{2}-{i\tau_{cl}\over 2\pi}
   {q\over A^{2}}m_{1}m_{2}m_{3}m_{4}.$$

This corresponds to a low energy effective coupling

The one loop correction to the real function ${\cal H}$ is $${\cal H}= {3\over 256\pi^{2}}\log^{2}\Big(
   {A\bar{A}\over\langle A\rangle \langle \bar{A}\rangle}\Big),$$

and the one instanton correction is given by

$${\cal H}(\varphi,\bar{\varphi})={-qm_{1}m_{2}m_{3}m_{4}\over
   8\pi^{2}\varphi^{4}} \log\bar{\varphi}.$$

The one anti-instanton contribution is given by the complex conjugate of the one instanton correction. The pure scalar two and four derivative terms appearing in the low energy effective action, after performing the field redefinition needed to compare to the brane result, are easily obtained by using the formulas quoted in ,. The results are

where

$$K_{\varphi\bar{\varphi}}\equiv Im\Big(
    {\partial^{2}{\cal F}\over\partial\varphi^{2}}\Big) 
    =\tau_{2}={4\pi^{2}\over g_{cl}^{2}}-
{6\pi\over g_{cl}^{2}}m_{1}m_{2}m_{3}m_{4}\Big[
{q\over a^4}+{\bar{q}\over \bar{a}^4}\Big]$$

and

$${\eqalign{\tilde{\cal H}_{\varphi\varphi\bar{\varphi}\bar{\varphi}}&=
  16\Big({\partial^{4}{\cal H}\over \partial\varphi\partial\varphi
  \partial\bar{\varphi}\partial\bar{\varphi}}-
  {\partial^{3}{\cal H}\over \partial\varphi\partial\varphi
  \partial\bar{\varphi}}
  (K_{\bar{\varphi}\varphi})^{-1}{\partial 
  K_{\varphi\bar{\varphi}}\over\partial\bar{\varphi}}-
  {\partial K_{\varphi\bar{\varphi}}\over \partial\varphi}
  (K_{\bar{\varphi}\varphi})^{-1}
  {\partial^{3}{\cal H}\over \partial\varphi
  \partial\bar{\varphi}\partial\bar{\varphi}}\cr
  &+2{\partial K_{\varphi\bar{\varphi}}\over\partial\varphi}
  (K_{\bar{\varphi}\varphi})^{-1}
  {\partial^{2}{\cal H}\over \partial\varphi\partial\bar{\varphi}}
  (K_{\bar{\varphi}\varphi})^{-1}{\partial K_{\varphi\bar{\varphi}}
  \over\partial\bar{\varphi}}\Big)\cr
  &={3\over 8\pi^{2}}
  {1\over\varphi^{2}\bar{\varphi}^{2}}
  +{40 m_{1}m_{2}m_{3}m_{4}\over\pi^{2}}
  \Big[{q\over\bar{\varphi}^{2}\varphi^{6}}
  +{\bar{q}\over\varphi^{2}\bar{\varphi}^{6}}\Big].}}$$

Notice that for large $\varphi$ the fall off of the four derivative terms is like $|\varphi|^{-4}$. This has an interesting supergravity interpretation.

The first step in the supergravity analysis entails solving  for the background geometry, with the 