Modular invariance in superstring theory from N=4 super-Yang-Mills

We study the four-point function of the lowest-lying half-BPS operators in the N = 4 SU(N) super-Yang-Mills theory and its relation to the flat-space four-graviton amplitude in type IIB superstring theory. We work in a large-N expansion in which the complexified Yang-Mills coupling tau is fixed. In this expansion, non-perturbative instanton contributions are present, and the SL(2; Z) duality invariance of correlation functions is manifest. Our results are based on a detailed analysis of the sphere partition function of the mass-deformed SYM theory, which was previously computed using supersymmetric localization. This partition function determines a certain integrated correlator in the undeformed N = 4 SYM theory, which in turn constrains the four-point correlator at separated points. In a normalization where the two-point functions are proportional to N-2 - 1 and are independent of tau and (tau) over bar, we find that the terms of order root N and 1/root N in the large N expansion of the four-point correlator are proportional to the non-holomorphic Eisenstein series E(3/2, tau, (tau) over bar) and E(5/2, tau, (tau) over bar), respectively. In the flat space limit, these terms match the corresponding terms in the type IIB S-matrix arising from R-4 and D-4 R-4 contact interactions, which, for the R-4 case, represents a check of AdS/CFT at finite string coupling. Furthermore, we present striking evidence that these results generalize so that, at order N1/2- m with integer m >= 0, the expansion of the integrated correlator we study is a linear sum of non-holomorphic Eisenstein series with half-integer index, which are manifestly SL(2, Z) invariant.