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

We consider the integrated correlators associated with four-point correlation functions (O2O2Op(i) O- p((j))) in four-dimensional N = 4 supersymmetric Yang-Mills theory (SYM) with SU(N) gauge group, where 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 t. 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 t-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.

Automated summary

We consider integrated correlators in N = 4 super Yang-Mills theory with SU(N) gauge group, where operators are properly reorganised. We show that these correlators satisfy a universal Laplace-difference equation relating operators with different charges, which can be computed via supersymmetric localisation and are modular functions of the Yang-Mills coupling. This equation, along with initial conditions, completely determines all the integrated correlators.