Coulomb Branch Operator Algebras and Universal Selection Rules for $\mathcal{N}=2$ SCFTs

Coulomb branches of vacua are the most universal moduli spaces that arise in local unitary interacting 4d $\mathcal{N}=2$ superconformal field theories (SCFTs). In these theories, $1/2$-BPS primaries parameterize the Coulomb branches and form (anti-)chiral rings. We define the notion of a Coulomb branch operator algebra, $\mathcal{A}_{\mathcal{C}}$, that contains these chiral and anti-chiral rings along with infinitely many more operators and products that are less protected by supersymmetry. Using a universal symmetry, $\mathcal{I}\cong\mathbb{Z}_2$, that arises from studying the superconformal group, we give $\mathcal{I}$ selection rules for $\mathcal{A}_{\mathcal{C}}$ and, more generally, for arbitrary products in the local operator algebra of any 4d $\mathcal{N}=2$ SCFT. Defining the notion of a "Coulombic" SCFT, we propose explanations for certain phenomena in a 4d/2d correspondence involving 4d $\mathcal{N}=2$ theories and 2d vertex operator algebras. Finally, by considering deformations of $\mathcal{I}$, we explore the case of $\mathcal{N}>2$ SCFTs.