received , . In these papers solutions of IIB supergravity corresponding to non-conformal field theories were constructed. These solutions preserve the $SO(6)$ invariance of the AdS$_{5}\times$S$_{5}$ solution, and thus are similar but not identical to the solution we obtain in the $N_{f}=4$ massless flavors case.

In this section we begin by collecting the known field theory results on the low energy effective action of ${\cal N}=2$ super Yang-Mills theory with gauge group $SU(2)$ and $N_{f}=4$ massless matter multiplets. These are then compared to the result obtained from studying a probe threebrane in supergravity.

The perturbative beta function for ${\cal N}=2$ super Yang-Mills theory with gauge group $SU(N_{c})$ and $N_{f}$ flavor hypermultiplets is proportional to $(2N_{c}-N_{f})$. Thus, for $N_{c}=2$ and $N_{f}=4,$ the perturbative beta function vanishes. If in addition all of the flavors of matter are massless, we obtain a finite conformally invariant theory. The exact effective coupling of the theory has the form

$$\tau=\tau_{1}+i\tau_{2}=\tau_{cl}
+{i\over\pi}\sum_{n=0,2,4,...}c_{n}e^{in\pi\tau_{cl}}
=\tau_{cl}+{i\over\pi}\sum_{n=0,2,4,...}c_{n}q^{n},$$

where $\tau_{cl}$ is the classical coupling of the theory. The coefficient $c_{0}$ is a one loop perturbative correction, which in the Pauli-Villars scheme, has the value $c_{0}=4\log(2)$. The coefficients $c_{n}$ with $n>0$ and even come from nonperturbative (instanton) effects. The two instanton coefficient has 