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

Understand

We study modular invariants arising in the four-point functions of the stress tensor multiplet operators of the ${\cal N} = 4$ $SU(N)$ 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 ${\cal N} = 2^*$ 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 ${\cal N} = 4$ theory on a round sphere.
• At each order in the $1/N$ expansion, these fourth derivatives are modular invariant functions of $(\tau, \bar \tau)$.
• 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.