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

Integrated correlation functions in 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 tau)(2)partial derivative(tau)partial derivative(tau)<overline>partial derivative(2)(m)log Z(N)|(m=0) and H-N = partial derivative(4)(m)logZ(N)|(m=0), which are modular invariant functions of the complex coupling tau. While C-N was recently written in terms of a two-dimensional lattice sum for any N and tau, 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 to any desired order in 1/N and finite tau. 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 and C-N to all orders in 1/N. We also give evidence that the coefficients of the 1/N expansion of H-N can be written as lattice sums to all orders. Lastly, these large N and finite tau results are used to accurately estimate the integrated correlators at finite N and finite tau.