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Laplace-difference equation for integrated correlators of operators with general charges in $\mathcal{N}=4$ SYM (2303.13195v3)

Published 23 Mar 2023 in hep-th

Abstract: 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) with $SU(N)$ gauge group, where $\mathcal{O}{(i)}_p$ 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 $\tau$. 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 $\tau$-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.

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