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, matching results obtained in the mathematics literature from an intricate group theory analysis.

Abstract: We study renormalization group flows between six-dimensional superconformal field theories (SCFTs) using their geometric realizations as singular limits of F-theory compactified on elliptically fibered Calabi-Yau threefolds. There are two general types of flows: one corresponds to giving expectation values to scalars in the tensor multiplets (tensor branch flow) realized as resolving the base of the geometry. The other corresponds to giving expectation values to hypermultiplets (Higgs branch flow) realized as complex structure deformations of the geometry. To corroborate this physical picture we calculate the change in the anomaly polynomial for these theories, finding strong evidence for a flow from a UV fixed point to an IR fixed point. Moreover, we find evidence against non-trivial dualities for 6D SCFTs. In addition we find non-trivial RG flows for theories realizing small E8 instantons on ALE spaces.

We study how to couple a 6D superconformal field theory (SCFT) to gravity. In F-theory, the models in question are obtained working on the supersymmetric background R^{5,1} x B where B is the base of a compact elliptically fibered Calabi-Yau threefold in which two-cycles have contracted to zero size. When the base has orbifold singularities, we find that the anomaly polynomial of the 6D SCFTs can be understood purely in terms of the intersection theory of fractional divisors: the anomaly coefficient vectors are identified with elements of the orbifold homology. This also explains why in certain cases, the SCFT can appear to contribute a "fraction of a hypermultiplet" to the anomaly polynomial. Quantization of the lattice of string charges also predicts the existence of additional light states beyond those captured by such fractional divisors. This amounts to a refinement to the lattice of divisors in the resolved geometry. We illustrate these general considerations with explicit examples, focusing on the case of F-theory on an elliptic Calabi-Yau threefold with base P^2 / Z_3.

Little string theories (LSTs) are UV complete non-local 6D theories decoupled from gravity in which there is an intrinsic string scale. In this paper we present a systematic approach to the construction of supersymmetric LSTs via the geometric phases of F-theory. Our central result is that all LSTs with more than one tensor multiplet are obtained by a mild extension of 6D superconformal field theories (SCFTs) in which the theory is supplemented by an additional, non-dynamical tensor multiplet, analogous to adding an affine node to an ADE quiver, resulting in a negative semidefinite Dirac pairing. We also show that all 6D SCFTs naturally embed in an LST. Motivated by physical considerations, we show that in geometries where we can verify the presence of two elliptic fibrations, exchanging the roles of these fibrations amounts to T-duality in the 6D theory compactified on a circle.