Bootstrapping SCFTs with Four Supercharges

We study the constraints imposed by superconformal symmetry, crossing symmetry, and unitarity for theories with four supercharges in spacetime dimension $2\le d\le 4$. We show how superconformal algebras with four Poincar'{e} supercharges can be treated in a formalism applicable to any, in principle continuous, value of $d$ and use this to construct the superconformal blocks for any $d\le 4$. We then use numerical bootstrap techniques to derive upper bounds on the conformal dimension of the first unprotected operator appearing in the OPE of a chiral and an anti-chiral superconformal primary. We obtain an intriguing structure of three distinct kinks. We argue that one of the kinks smoothly interpolates between the $d=2$, $\mathcal{N}=(2,2)$ minimal model with central charge $c=1$ and the theory of a free chiral multiplet in $d=4$, passing through the critical Wess-Zumino model with cubic superpotential in intermediate dimensions.