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

TL;DR

This paper investigates modular invariant functions arising in four-point correlators of ${ m f N}=4$ super-Yang-Mills theory, revealing their structure as Eisenstein series and their relation to string theory amplitudes.

Contribution

It identifies the modular invariants in ${ m f N}=4$ SYM four-point functions as generalized Eisenstein series, providing a new understanding of their mathematical structure and physical implications.

Findings

Modular invariants are linear combinations of Eisenstein series at half-integer orders.

At integer orders, invariants satisfy inhomogeneous Laplace equations.

Results connect ${ m f N}=4$ SYM correlators to string theory amplitude expansions.

Abstract

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. 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 $AdS_5\times S^5$.