Instantons and recursion relations in N=2 Susy gauge theory

Abstract

We find the transformation properties of the prepotential ${\cal F}$ of $N=2$ SUSY gauge theory with gauge group $SU(2)$. In particular we show that ${\cal G}(a)=\pi i\left({\cal F}(a)-{1\over 2}a\partial_a{\cal F}(a)\right)$ is modular invariant. This function satisfies the non-linear differential equation $\left(1-{\cal G}^2\right){\cal G}''+{1\over 4}a {{\cal G}'}^3=0$, implying that the instanton contribution are determined by recursion relations. Finally, we find $u=u(a)$ and give the explicit expression of ${\cal F}$ as function of $u$. These results can be extended to more general cases.