Classifying 5d SCFTs via 6d SCFTs: Arbitrary rank

TL;DR

This work extends the geometric framework for classifying $5d$ SCFTs by showing how untwisted circle compactifications of arbitrary-rank $6d$ SCFTs admit a universal parent Calabi–Yau $\tilde{X}_{\mathfrak{T}}$ built by gluing rank-one CY data. The authors develop a graph-based, Gluing-based construction to assemble $\tilde{X}_{\mathfrak{T}}$ from rank-one inputs and provide computational tools (Tate form, blowups, pushforwards) to extract the Coulomb-branch data via triple intersections, Mori cones, and degrees. They address decoupled states with a minimal Mori cone approach and introduce formal gauge algebras to maximize RG flows, culminating in an algorithm to obtain descendant CYs describing all rank-preserving endpoints. The framework links F-theory and M-theory descriptions of Coulomb branches, enabling systematic classification of rank-$n$ $5d$ SCFTs arising from circle-compactified $6d$ SCFTs and sets the stage for explicit computations across arbitrary rank.

Abstract

According to a conjecture, all 5d SCFTs should be obtainable by rank-preserving RG flows of 6d SCFTs compactified on a circle possibly twisted by a background for the discrete global symmetries around the circle. For a 6d SCFT admitting an F-theory construction, its untwisted compactification admits a dual M-theory description in terms of a "parent" Calabi-Yau threefold which captures the Coulomb branch of the compactified 6d SCFT. The RG flows to 5d SCFTs can then be identified with a sequence of flop transitions and blowdowns of the parent Calabi-Yau leading to "descendant" Calabi-Yau threefolds which describe the Coulomb branches of the resulting 5d SCFTs. An explicit description of parent Calabi-Yaus is known for untwisted compactifications of rank one 6d SCFTs. In this paper, we provide a description of parent Calabi-Yaus for untwisted compactifications of arbitrary rank 6d SCFTs. Since 6d SCFTs of arbitrary rank can be viewed as being constructed out of rank one SCFTs, we accomplish the extension to arbitrary rank by identifying a prescription for gluing together Calabi-Yaus associated to rank one 6d SCFTs.

Figures (1)

• Figure 1: Left: Each Kodaira type is associated to an order of vanishing of $f$, $g$ and $\Delta$ which we denote as $(f,g,\Delta)$. We also display the corresponding $6d$ gauge algebras which are different for split, semi-split and non-split cases. Middle: The graph displays the intersection pattern for the rational curves composing the elliptic fiber. The numbers in the nodes of the graph are labels and each edge between two nodes corresponds to a transverse intersection of the corresponding rational curves. The components of type IV fiber all intersect each other transversely at a common point. The $||$ in between the edges for types II and III denote the fact that those intersections are tangential rather than transverse. Right: The elliptic fiber $f$ is written in terms of the rational curves $f^i$ where $i$ is the label appearing in the node.