On the Classification of 6D SCFTs and Generalized ADE Orbifolds
Jonathan J. Heckman, David R. Morrison, Cumrun Vafa
TL;DR
This work provides a geometric classification of 6D SCFTs realized in F-theory, showing that minimal theories correspond to orbifolds $\mathbb{C}^{2}/\Gamma$ with $\Gamma\subset U(2)$ and that the ADE cases arise when $\Gamma\subset SU(2)$. The authors construct a framework from basic building blocks—$(2,0)$ theories, E-strings, and single clusters/orbifolds—to generate all minimal models, then obtain a vast non-minimal landscape by augmenting with E-strings and by decorating base curves with higher-rank gauge algebras, conjecturing a full list. They develop an explicit procedure for the minimal resolutions, prove structural constraints (no quartic vertices, at most one trivalent vertex), and enumerate endpoint theories into generalized A- and D-type families with concrete orbifold data and p/q fractions. The results illuminate the global structure of 6D SCFT moduli spaces and provide a blueprint for exploring flows via blowdowns and for extending the classification to non-minimal models and lower-dimensional compactifications, with potential implications for holography and large-N limits.
Abstract
We study (1,0) and (2,0) 6D superconformal field theories (SCFTs) that can be constructed in F-theory. Quite surprisingly, all of them involve an orbifold singularity C^2 / G with G a discrete subgroup of U(2). When G is a subgroup of SU(2), all discrete subgroups are allowed, and this leads to the familiar ADE classification of (2,0) SCFTs. For more general U(2) subgroups, the allowed possibilities for G are not arbitrary and are given by certain generalizations of the A- and D-series. These theories should be viewed as the minimal 6D SCFTs. We obtain all other SCFTs by bringing in a number of E-string theories and/or decorating curves in the base by non-minimal gauge algebras. In this way we obtain a vast number of new 6D SCFTs, and we conjecture that our construction provides a full list.