Atomic Classification of 6D SCFTs

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.

Figures (9)

• Figure 1: The general structure of the base geometry of a 6D SCFT. This geometry is constructed from a number of curves (i.e., circles / nodes) which always have at least a D/E-type gauge symmetry supported over them. These nodes are joined together by "links" which are composed of sequences of curves which minimally support no D/E-type gauge symmetry. These links are also SCFTs, so that a base can even have no nodes at all. A striking consequence of our classification is that these bases always have the structure of a single line, with only a small amount of decoration on the two leftmost and rightmost nodes.
• Figure 7: Allowed fiber decorations of the left half of the $\overset{3,3}\bigcirc \simeq 1315131$ interior link. The right half is simply the mirror image.
• Figure 8: Allowed fiber decorations of the left half of the $\overset{4,4}\oplus \simeq 123151321$ interior link. The right half is simply the mirror image.
• Figure 9: Allowed fiber decoration of the left half of the $\overset{5,5}\oplus \simeq 12231513221$ interior link. The right half is simply the mirror image.
• Figure 10: Allowed fiber decoration of the $\overset{4,2}\oplus \simeq 12231$ interior link.