6D SCFTs and Phases of 5D Theories

TL;DR

The paper develops a geometric framework linking 6D SCFTs realized in F-theory to 5D SCFTs via circle reduction, using M-theory on the same elliptic Calabi–Yau threefold to unify tensor-branch and direct reductions. It shows that each 6D SCFT with minimal SUSY reduces to 1–4 decoupled 5D SCFTs, and that reduction of the tensor branch generally yields 5D generalized quivers; these two 5D phases are connected by flop transitions within the extended Kähler cone. The analysis provides a concrete description of the resulting 5D fixed points, their emergent flavor symmetries, and the role of canonical singularities in M-theory, with explicit illustrations from non-Higgsable clusters, rigid A-type theories, and M5-brane probe constructions. This work broadens the landscape of 5D SCFTs and clarifies their geometric and gauge-theoretic origins, potentially informing holographic duals and index computations. The results offer a cohesive path from 6D classifications to a rich 5D fixed-point sector, revealing how geometry drives phase structure and dualities in higher-dimensional quantum field theories.

Abstract

Starting from 6D superconformal field theories (SCFTs) realized via F-theory, we show how reduction on a circle leads to a uniform perspective on the phase structure of the resulting 5D theories, and their possible conformal fixed points. Using the correspondence between F-theory reduced on a circle and M-theory on the corresponding elliptically fibered Calabi--Yau threefold, we show that each 6D SCFT with minimal supersymmetry directly reduces to a collection of between one and four 5D SCFTs. Additionally, we find that in most cases, reduction of the tensor branch of a 6D SCFT yields a 5D generalization of a quiver gauge theory. These two reductions of the theory often correspond to different phases in the 5D theory which are in general connected by a sequence of flop transitions in the extended Kahler cone of the Calabi--Yau threefold. We also elaborate on the structure of the resulting conformal fixed points, and emergent flavor symmetries, as realized by M-theory on a canonical singularity.

Figures (7)

• Figure 1: Depiction of the phase structure for 6D theories reduced on a circle. Reducing a $(1,0)$ 6D SCFT leads to a 5D SCFT, as indicated on the right. A sequence of flop transitions in the extended Kähler cone of the Calabi--Yau threefold connects this chamber of moduli space to the one obtained by dimensional reduction of the generalized 6D quiver. This leads to a generalized 5D quiver, which need not possess a fixed point in this chamber of moduli space.
• Figure 2: Geometry of the $-3$ theory. upper left: Reduction of the tensor branch over $S^1$; upper center: flop phase transition; upper right: reduction of the 6D SCFT over $S^1$; lower left: gauge symmetry enhanced to $SU(3)$; lower right: strong coupling limit of $SU(3)$ theory. In the 5D limit, $C'$ and $\mathbb{P}^2$ decompactify.
• Figure 3: Geometry of the $-4$ theory. upper left: Reduction of the tensor branch over $S^1$; upper center: flop phase transition; upper right: reduction of the 6D SCFT over $S^1$lower left: gauge symmetry enhanced to $SO(8)$; lower right: strong coupling limit of $SO(8)$ theory. In the 5D limit, the $A_1$ locus and $\mathbb{P}^2_{[1,1,2]}$ decompactify.
• Figure 4: Geometry of the $-6$ theory. The base of the elliptic fibration is the noncompact surface $B$. For each $\mathbb{P}^1$ with a non-trivial self-intersection number inside a given surface, the latter is indicated within the corresponding surface. In the 5D limit, the $A_1$ locus and $\mathbb{P}^2_{[1,1,4]}$ decompactify.
• Figure 5: Schematic structure of the geometries of certain NHCs as Dynkin graphs: the nodes correspond to surfaces while the links correspond to intersections.