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The Superconformal Xing Equation

Crossing symmetry provides a powerful tool to access the non-perturbative dynamics of conformal and superconformal field theories. Here we develop the mathematical formalism that allows to construct the crossing equations for arbitrary four-point functions in theories with superconformal symmetry of type I, including all superconformal field theories in $d=4$ dimensions. Our advance relies on a supergroup theoretic construction of tensor structures that generalizes an approach which was put forward in \cite{Buric:2019dfk} for bosonic theories. When combined with our recent construction of the relevant superblocks, we are able to derive the crossing symmetry constraint in particular for four-point functions of arbitrary long multiplets in all 4-dimensional superconformal field theories.