Relations between integrated correlators in N = 4 supersymmetric Yang-Mills theory

A bstract Integrated correlation functions in N 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 CN= (Im) ² _ₘ². {ZN|}₌=₀ C N = Im τ 2 ∂ τ ∂ τ ¯ ∂ m 2 log Z N m = 0 and HN=ₘ⁴. {ZN|}₌=₀ H N = ∂ m 4 log Z N m = 0, which are modular invariant functions of the complex coupling τ. While CN C N was recently written in terms of a two-dimensional lattice sum for any N and τ, HN 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 HN 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 HN H N and CN C N to all orders in 1/ N. We also give evidence that the coefficients of the 1/ N expansion of HN 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.