New Modular Invariants in $\mathcal{N}=4$ Super-Yang-Mills Theory

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 $\tau$ 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 ${\cal N} = 2^*$theorywithrespecttothesquashingparameter$b$andmassparameter$m$, 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 $1\mathrm{/}N$ expansion, these fourth derivatives are modular invariant functions of $(\tau ,\bar{\tau })$. We present evidence that at half-integer orders in $1\mathrm{/}N$, these modular invariants are linear combinations of non-holomorphic Eisenstein series, while at integer orders in $1\mathrm{/}N$, 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 $Ad{S}_5\times S^5$.