Modular Invariance in Superstring Theory From ${\cal N} = 4$ Super-Yang-Mills

We study the four-point function of the lowest-lying half-BPS operators in the ${\cal N} =4$ $SU(N)$ 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 $SL(2, \mathbb{Z})$ 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 ${\cal N} = 4$ 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 $\bar \tau$, we find that the terms of order $\sqrt{N}$ and $1/\sqrt{N}$ in the large $N$ expansion of the four-point correlator are proportional to the non-holomorphic Eisenstein series $E({\scriptstyle \frac{3}{2}},\tau,\bar\tau)$ and $E({\scriptstyle \frac{5}{2}},\tau,\bar\tau)$, 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 $N^{\frac{1}{2}-m}$ with integer $m \ge 0$, the expansion of the integrated correlator we study is a linear sum of non-holomorphic Eisenstein series with half-integer index, which are manifestly $SL(2,\mathbb{Z})$ invariant.