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 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.

Geometric engineering, mirror symmetry and 6 d 1 0 → 4 d N = 2 $$ 6{\mathrm{d}}_{\left(1,0\right)}\to 4{\mathrm{d}}_{\left(\mathcal{N}=2\right)} $$

“…The classification of 6d SCFTs [3,4] is based on their geometric engineering in F-theory. The corresponding F-theory geometry giving rise to a 6d SCFT involves elliptic CY 3-folds with local singularities.…”

“…We adopt the notation of [4,23], where the structure of the tensorial Coulomb branch of a given 6d SCFT is represented as follows:…”

“…g encodes the type of 7-brane giving rise to the 6d gauge group. The matter content can be determined from this datum using 6d gauge anomaly cancellation [4,[42][43][44][45][46][47][48].…”

“…G = {(α 2 ; α −1 , α −1 )} and α ∈ Z n=2, 3,4,6,8,12 These cases were originaly studied in [49] and they correspond, after the blow up which gets rid of the singularity, to a single P 1 with negative self-intersections 2,3,4,6,8 and 12, respectively. Moreover the elliptic fibration, except for the Z 2 case which leads to the A 1 (2,0) theory, has some singularity leading to gauge symmetry on them.…”

“…Similarly, the S duality of four dimensional N = 2 class S theories could be derived from compactifying 6d (2, 0) theory on a punctured Riemann surface [2]. Recently, a classification of 6d (1,0) theories has been proposed which is a surprisingly rich set [3,4] (see also [5]). It is natural to ask what kind of 4d theory we can get and what kind of interesting 4d dynamics we can learn from their compactification.…”