Instantons and recursion relations in N = 2 SUSY gauge theory

We find the transformation properties of the prepotential F of N = 2 SUSY gauge theory with gauge group SU(2). Next we show that g(a) = πi (F(a) −12a∂aF(a)) is modular invariant. We also show that u = g(a), so that F(〈Φ〉) = 1πi〈tr Φ2〉 + 12〈Φ〉 〈ΦD〉. This implies that g(a) satisfies the non-linear differential equation (1 − g2) g″ + 14ag′3 = 0. We use this equation to derive recursion relations for the instanton contributions. These results can be extended to more general cases.