non-zero metric, dilaton and axion fields. As usual, the metric and axion are combined into a single complex coupling $\tau=\chi+ie^{-\phi}=\tau_{1}+i\tau_{2}$. The coupling $\tau$ is identified with the modular parameter of the elliptic fiber of the F theory compactification. The $(8,9)$ plane is taken to be orthogonal to the sevenbranes. In terms of the complex coordinate $z=x^{8}+ix^{9}$ we make the following ansatz for the metric

The parameter $z$ is to be identified with the Higgs field appearing in the low energy effective action of the ${\cal N}=2$ field theory. With this ansatz, the type IIB supergravity equations of motion reduce to

The complex coupling $\tau$ is identified with the low energy effective coupling of the ${\cal N}=2$ field theory. Supersymmetry constrains the effective coupling of the field theory to be a function of $z$, so that the first equation in  is automatically satisfied. The general solution to the second equation in  is

The functions $F(z)$ and $\bar{F}(\bar{z})$ should be chosen in order that  yields a sensible metric. For the case that we are considering, the explicit form for the metric transverse to the sevenbranes is

where $a$ is the quantity that appears in the Seiberg-Witten solution. This specifies the solution for 