Instantons and recursion relations in N=2 Susy gauge theory

Abstract

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

8 pages, LaTex

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