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

• Published in: The journal of high energy physics Vol. 2020; no. 11
• Main Authors: Chester, Shai M., Green, Michael B., Pufu, Silviu S., Wang, Yifan, Wen, Congkao
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 06-11-2020; Springer Nature B.V
• Subjects: Classical and Quantum Gravitation; Correlation analysis; Coupling; Elementary Particles; Gravitons; High energy physics; Instantons; Integers; Invariance; Mathematical analysis; Operators (mathematics); Partitions (mathematics); Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Regular Article - Theoretical Physics; Relativity Theory; S matrix theory; Series (mathematics); String Theory; Yang-Mills theory; AdS-CFT Correspondence; Scattering Amplitudes; Conformal Field Theory; 1/N Expansion
• ISSN: 1029-8479
• DOI: 10.1007/JHEP11(2020)016

A
bstract
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
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
= 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
¯
, we find that the terms of order
N
and
1
/
N
in the large
N
expansion of the four-point correlator are proportional to the non-holomorphic Eisenstein series
E
3
2
τ
τ
¯
and
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
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
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.