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

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