on the $y$ variables and working in the large $|a|$ region (which corresponds to the semi-classical regime of the field theory) we find

$$\Big[{\partial\over\partial a}{\partial\over\partial \bar{a}}
-k^{2}{N\over\lambda}
\Big({8\pi\over g_{cl}^{2}}+{6\over\pi}+{2\over\pi}
\log \Big({a\bar{a}\over\Lambda^{2}}\Big)\Big)
\Big]f=0.$$

We have not been able to solve this equation exactly. However an approximate solution in the large $|a|$ region is given by

$$f\approx Ne^{-{2k\sqrt{N}|a|\over\sqrt{\lambda\pi}}
\sqrt{{2\pi^{2}\over g_{cl}^{2}}+{1\over 2}+\log(|a|)}},$$

Using this approximate solution we obtain

$$f(y=0,a,\bar{a})=\int d^{4}k f(k,a,\bar{a})=
{12\pi^{2}\lambda^{2}\over 16N(a\bar{a})^{2}
\Big(\log\Big[{\sqrt{a\bar{a}}\over\Lambda}exp({2\pi^{2}\over g_{cl}^{2}}+{1\over 2})\Big]
\Big)^{2}}.$$

This result determines the coefficient of the four derivative terms

$$\tau_{2}^{2}f(y=0,a,\bar{a})\sim {N\over (a\bar{a})^{2}}+
O\Big({1\over (a\bar{a})^{2}log|a|}\Big).$$

This is exactly the same behaviour as obtained from the field theory analysis.

The complex coupling $\tau$ in this supergravity background has a logarithmic dependence on $a\bar{a}$ corresponding to the logarithmic dependence of the field theory coupling on the energy scale. Gravity solutions that have couplings with this logarithmic dependence have been constructed in type-0 theories.

We should now address the validity of this computation. There are two potential sources of corrections to the supergravity background - string loop effects and curvature corrections. At large $N$ and large ’t Hooft coupling both of these types of corrections are small and supergravity is a reliable description of the background. As the ’t 