Fission, Fusion, and 6D RG Flows

TL;DR

This work develops a unified, algebraic framework for 6D SCFTs built from F-theory, showing that nearly all known theories can be obtained from a small set of UV progenitors via two operations: fission (tensor-branch deformation followed by homplex Higgs deformations) and fusion (gauging a common flavor symmetry and adding a tensor multiplet). Higgs-branch data are encoded by continuous $ obreak \\mathfrak{su}(2) ightarrow obreak \\mathfrak{g}_{flav}$ and discrete $ obreak \\\Gamma_{ADE} ightarrow E_8$ homomorphisms, which induce a natural partial order corresponding to RG flows. The authors show that almost all theories are fission products of rank-$k$ orbi-instantons $(E_8,G_{ADE})$, with a streamlined labeling by $g_{max}$ and nilpotent data; a single fusion step then accounts for remaining outliers. They provide explicit constructions, refinements for long $A$-type endpoints, and a scheme to classify 6D RG flows via group-theoretic data, while outlining the role of semi-simple deformations and potential extensions to holography and compactifications. These results offer a coherent, algorithmic path to the complete landscape of 6D SCFTs and their RG networks.

Abstract

We show that all known 6D SCFTs can be obtained iteratively from an underlying set of UV progenitor theories through the processes of "fission" and "fusion." Fission consists of a tensor branch deformation followed by a special class of Higgs branch deformations characterized by discrete and continuous homomorphisms into flavor symmetry algebras. Almost all 6D SCFTs can be realized as fission products. The remainder can be constructed via one step of fusion involving these fission products, whereby a single common flavor symmetry of decoupled 6D SCFTs is gauged and paired with a new tensor multiplet at the origin of moduli space, producing an RG flow "in reverse" to the UV. This leads to a streamlined labeling scheme for all known 6D SCFTs in terms of a few pieces of group theoretic data. The partial ordering of continuous homomorphisms $\mathfrak{su}(2) \rightarrow \mathfrak{g}_{\text{flav}}$ for $\mathfrak{g}_{\text{flav}}$ a flavor symmetry also points the way to a classification of 6D RG flows.

Figures (1)

• Figure 1: Depiction of fission and fusion for 6D SCFTs. Fission consists of first performing a tensor branch deformation, which is then followed by a specialized Higgs branch deformation associated with either a continuous $\mathfrak{su}(2) \rightarrow \mathfrak{g}_{\text{flav}}$ homomorphism or a homomorphism $\Gamma_{ADE} \rightarrow E_8$, with $\Gamma_{ADE}$ a finite subgroup of $SU(2)$. In F-theory, these specify a restricted class of complex structure deformations which we refer to as "homplex deformations." Fusion corresponds to gauging a flavor symmetry of at least one, but possibly several decoupled 6D SCFTs and pairing this gauge symmetry with a tensor multiplet. A new SCFT is generated by tuning this new tensor multiplet to the origin of moduli space. Quite surprisingly, all 6D SCFTs can be obtained from a single fission step or as the fusion of fission products obtained from a small number of UV progenitor theories. Moreover, in a sense we will make precise below, almost all theories can be generated by fission alone.