Relations between integrated correlators in N $$ \mathcal{N} $$ = 4 supersymmetric Yang-Mills theory

Abstract Integrated correlation functions in N $$ \mathcal{N} $$ = 4 supersymmetric Yang-Mills theory with gauge group SU(N) can be expressed in terms of the localised S 4 partition function, Z N , deformed by a mass m. Two such cases are C N = Im τ 2 ∂ τ ∂ τ ¯ ∂ m 2 log Z N m = 0 $$ {\mathcal{C}}_N={\left(\operatorname{Im}\tau \right)}^2{\partial}_{\tau }{\partial}_{\overline{\tau}}{\partial}_m^2\log {\left.{Z}_N\right|}_{m=0} $$ and H N = ∂ m 4 log Z N m = 0 $$ {\mathcal{H}}_N={\partial}_m^4\log {\left.{Z}_N\right|}_{m=0} $$ , which are modular invariant functions of the complex coupling τ. While C N $$ {\mathcal{C}}_N $$ was recently written in terms of a two-dimensional lattice sum for any N and τ, H N $$ {\mathcal{H}}_N $$ has only been evaluated up to order 1/N 3 in a large-N expansion in terms of modular invariant functions with no known lattice sum realisation. Here we develop methods for evaluating H N $$ {\mathcal{H}}_N $$ to any desired order in 1/N and finite τ. We use this new data to constrain higher loop corrections to the stress tensor correlator, and give evidence for several intriguing relations between H N $$ {\mathcal{H}}_N $$ and C N $$ {\mathcal{C}}_N $$ to all orders in 1/N. We also give evidence that the coefficients of the 1/N expansion of H N $$ {\mathcal{H}}_N $$ can be written as lattice sums to all orders. Lastly, these large N and finite τ results are used to accurately estimate the integrated correlators at finite N and finite τ.