Bootstrapping $\mathcal{N}=3$ superconformal theories

We initiate the bootstrap program for $\mathcal{N}=3$ superconformal field theories (SCFTs) in four dimensions. The problem is considered from two fronts: the protected subsector described by a $2d$ chiral algebra, and crossing symmetry for half-BPS operators whose superconformal primaries parametrize the Coulomb branch of $\mathcal{N}=3$ theories. With the goal of describing a protected subsector of a family of $\mathcal{N}=3$ SCFTs, we propose a new $2d$ chiral algebra with super Virasoro symmetry that depends on an arbitrary parameter, identified with the central charge of the theory. Turning to the crossing equations, we work out the superconformal block expansion and apply standard numerical bootstrap techniques in order to constrain the CFT data. We obtain bounds valid for any theory but also, thanks to input from the chiral algebra results, we are able to exclude solutions with $\mathcal{N}=4$ supersymmetry, allowing us to zoom in on a specific $\mathcal{N}=3$ SCFT.