Abstract
We study the four-point function of the lowest-lying half-BPS operators in the 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 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 N2− 1 and are independent of τ and overline{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 Eleft(frac{3}{2},tau, overline{tau}right) and Eleft(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 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} 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.
Highlights
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
Progress combining various analytic bootstrap techniques [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28] and supersymmetry has allowed the determination of CFT correlators, at strong coupling, in a variety of examples arising from top-down AdS/CFT constructions [17, 29,30,31,32,33,34]
Using a similar strategy to that outlined above in the ’t Hooft limit, we find that the analytic bootstrap constraints combined with the supersymmetric localization constraints coming from (1.1) yield, in the very strong coupling limit, the same Eisenstein series that appear in the low energy expansion of the type IIB four-graviton amplitude
Summary
The rest of this paper is organized as follows. In section 2 we give a brief review of the. 4-point function of the stress tensor superconformal primary operator in N = 4 SYM, its relation to the string theory scattering amplitude, and the supersymmetric localization constraint coming from the mixed derivative (1.1). Let us start with a brief review of the setup of the four-point function of the stress tensor superconformal primary operator in SYM, its relation to the 10d IIB flat space graviton S-matrix and constraints from supersymmetric localization. By either comparing with the (super)graviton four-point scattering amplitude in type IIB string theory in the flat space limit or using the quantity (1.1) (or other similar quantities) derived from supersymmetric localization. Let us first discuss the constraints from the flat space scattering amplitude, and those from supersymmetric localization
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