Explicit Modular Invariant Two-Loop Superstring Amplitude Relevant for R^4

Abstract

In this note we derive an explicit modular invariant formula for the two loop 4-point amplitude in superstring theory in terms of a multiple integral (7 complex integration variables) over the complex plane which is shown to be convergent. We consider in particular the case of the leading term for vanishing momenta of the four graviton amplitude, which would correspond to the two-loop correction of the R^4 term in the effective Action. The resulting expression is not positive definite and could be zero, although we cannot see that it vanishes.