Abstract
We study the four-point function of the lowest-lying half-BPS operators in the $$ \mathcal{N} $$
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 τ 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 $$ \mathcal{N} $$
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 N2− 1 and are independent of τ and $$ \overline{\tau} $$
τ
¯
, we find that the terms of order $$ \sqrt{N} $$
N
and $$ 1/\sqrt{N} $$
1
/
N
in the large N expansion of the four-point correlator are proportional to the non-holomorphic Eisenstein series $$ E\left(\frac{3}{2},\tau, \overline{\tau}\right) $$
E
3
2
τ
τ
¯
and $$ E\left(\frac{5}{2},\tau, \overline{\tau}\right) $$
E
5
2
τ
τ
¯
, respectively. In the flat space limit, these terms match the corresponding terms in the type IIB S-matrix arising from R4 and D4R4 contact inter-actions, which, for the R4 case, represents a check of AdS/CFT at finite string coupling. Furthermore, we present striking evidence that these results generalize so that, at order $$ {N}^{\frac{1}{2}-m} $$
N
1
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.
Correlation functions at the bulk point singularity from the gravitational eikonal S-matrix