Atomic Classification of 6D SCFTs

TL;DR

This paper classifies all six-dimensional superconformal field theories using F-theory, revealing a linear chain structure of tensor branches and a geometric realization of ADE-type quiver theories.

Contribution

It introduces a novel classification method for 6D SCFTs via an atomic approach and provides a geometric realization of ADE-type quivers in F-theory.

Findings

Tensor branches form linear chains of intersecting spheres.

Classification of homomorphisms from ADE groups into E8.

F-theory realization of E8 instanton probing ADE singularities.

Abstract

We use F-theory to classify possibly all six-dimensional superconformal 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 seven-branes 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 non-trivial examples involving E8 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 E8 in purely geometric terms, largely matching results obtained in the mathematics literature from an intricate group theory analysis.