Atomic classification of 6D SCFTs

Abstract

We use F-theory to classify possibly all six-dimensional super-conformal field theories (SCFTs). This involves a two step process: We first classify all possible tensor branches allowed in F-theory (which correspond to allowed collections of contractible spheres) and then classify all possible configurations of sevenbranes wrapped over them. We describe the first step in terms of "atoms" joined into "radicals" and "molecules," using an analogy from chemistry. The second step has an interpretation via quiver-type gauge theories constrained by anomaly cancellation. A very surprising outcome of our analysis is that all of these tensor branches have the structure of a linear chain of intersecting spheres with a small amount of possible decoration at the two ends. The resulting structure of these SCFTs takes the form of a generalized quiver consisting of ADE-type nodes joined by conformal matter. A collection of highly nontrivial examples involving E-8 small instantons probing an ADE singularity is shown to have an F-theory realization. This yields a classification of homomorphisms from ADE subgroups of SU(2) into E-8 in purely geometric terms, matching results obtained in the mathematics literature from an intricate group theory analysis.