| Summary: | Abstract We study the four-point function of the lowest-lying half-BPS operators in the N $$ \mathcal{N} $$ = 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 N $$ \mathcal{N} $$ = 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 τ and τ ¯ $$ \overline{\tau} $$ , we find that the terms of order N $$ \sqrt{N} $$ and 1 / N $$ 1/\sqrt{N} $$ in the large N expansion of the four-point correlator are proportional to the non-holomorphic Eisenstein series E 3 2 τ τ ¯ $$ E\left(\frac{3}{2},\tau, \overline{\tau}\right) $$ and E 5 2 τ τ ¯ $$ E\left(\frac{5}{2},\tau, \overline{\tau}\right) $$ , 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 inter-actions, 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 1 2 − m $$ {N}^{\frac{1}{2}-m} $$ 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.
|
|---|