Coulomb Branch Operator Algebras and Universal Selection Rules for N=2 SCFTs

Coulomb branches of vacua are the most universal moduli spaces that arise in local unitary interacting 4d 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, AC, 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, I₂, that arises from studying the superconformal group, we give I selection rules for AC and, more generally, for arbitrary products in the local operator algebra of any 4d N=2 SCFT. Defining the notion of a "Coulombic" SCFT, we propose explanations for certain phenomena in a 4d/2d correspondence involving 4d N=2 theories and 2d vertex operator algebras (VOAs). Finally, by considering a deformation of I, we explore the case of N=4 SCFTs.