New modular invariants in N = 4 Super-Yang-Mills theory

Shai M. Chester, Michael B. Green, Silviu S. Pufu, Yifan Wang, Congkao Wen

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39 Scopus citations


We study modular invariants arising in the four-point functions of the stress tensor multiplet operators of the N = 4 SU(N) super-Yang-Mills theory, in the limit where N is taken to be large while the complexified Yang-Mills coupling τ is held fixed. The specific four-point functions we consider are integrated correlators obtained by taking various combinations of four derivatives of the squashed sphere partition function of the N = 2 theory with respect to the squashing parameter b and mass parameter m, evaluated at the values b = 1 and m = 0 that correspond to the N = 4 theory on a round sphere. At each order in the 1/N expansion, these fourth derivatives are modular invariant functions of (τ,τ¯). We present evidence that at half-integer orders in 1/N , these modular invariants are linear combinations of non-holomorphic Eisenstein series, while at integer orders in 1/N, they are certain “generalized Eisenstein series” which satisfy inhomogeneous Laplace eigenvalue equations on the hyperbolic plane. These results reproduce known features of the low-energy expansion of the four-graviton amplitude in type IIB superstring theory in ten-dimensional flat space and have interesting implications for the structure of the analogous expansion in AdS5× S5.

Original languageEnglish (US)
Article number212
JournalJournal of High Energy Physics
Issue number4
StatePublished - Apr 2021

All Science Journal Classification (ASJC) codes

  • Nuclear and High Energy Physics


  • 1/N Expansion
  • AdS-CFT Correspondence
  • Conformal Field Theory
  • Scattering Amplitudes


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