Abstract
We study modular invariants arising in the four-point functions of the stress tensor multiplet operators of the \( \mathcal{N} \) = 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 \( \mathcal{N} \) = 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 \( \mathcal{N} \) = 4 theory on a round sphere. At each order in the 1/N expansion, these fourth derivatives are modular invariant functions of (τ, \( \overline{\tau} \)). 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.
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Chester, S.M., Green, M.B., Pufu, S.S. et al. New modular invariants in \( \mathcal{N} \) = 4 Super-Yang-Mills theory. J. High Energ. Phys. 2021, 212 (2021). https://doi.org/10.1007/JHEP04(2021)212
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DOI: https://doi.org/10.1007/JHEP04(2021)212