Laplace-difference equation for integrated correlators of operators with general charges in mathcal{N} = 4 SYM

Abstract

We consider the integrated correlators associated with four-point correlation functions leftlangle {mathcal{O}}_2{mathcal{O}}_2{mathcal{O}}_p^{(i)}{mathcal{O}}_p^{(j)}rightrangle in four-dimensional mathcal{N} = 4 supersymmetric Yang-Mills theory (SYM) with SU(N) gauge group, where {mathcal{O}}_p^{(i)} is a superconformal primary with charge (or dimension) p and the superscript i represents possible degeneracy. These integrated correlators are defined by integrating out spacetime dependence with a certain integration measure, and they can be computed via supersymmetric localisation. They are modular functions of complexified Yang-Mills coupling τ. We show that the localisation computation is systematised by appropriately reorganising the operators. After this reorganisation of the operators, we prove that all the integrated correlators for any N, with some crucial normalisation factor, satisfy a universal Laplace-difference equation (with the laplacian defined on the τ-plane) that relates integrated correlators of operators with different charges. This Laplace-difference equation is a recursion relation that completely determines all the integrated correlators, once the initial conditions are given.