Rank-two 5d SCFTs from M-theory at isolated toric singularities: a systematic study

TL;DR

<3-5 sentence high-level summary> This work systematically analyzes rank-two 5d N=1 SCFTs engineered by M-theory on isolated toric CY3 singularities. By combining fiberwise M-theory/IIA duality with toric and graph-theoretic techniques, the authors derive explicit Type IIA descriptions for all isolated rank-two CY3 singularities, map geometric prepotentials to gauge-theory data, and classify mass deformations into gauge and non-gauge phases. They provide detailed accounts of BPS spectra, magnetic walls, and RG flows across ten rank-two cases, including parity analyses and decoupling limits that connect to non-Lagrangian sectors. The results establish a comprehensive geometric-to-field-theory dictionary and offer a scalable framework for higher-rank generalizations in future work.

Abstract

We carry out a detailed exploration of the deformations of rank-two five-dimensional superconformal field theories (SCFTs) $\mathcal{T}_{\mathbf{X}}$, which are geometrically engineered by M-theory on the space transverse to isolated toric Calabi-Yau (CY) threefold singularities $\mathbf{X}$. Deformations of 5d $\mathcal{N}=1$ SCFTs can lead to "gauge-theory phases," but also to "non-gauge-theoretic phases," which have no known Lagrangian interpretation. In previous work, a technique relying on fiberwise M-theory/type IIA duality was developed to associate a type IIA background to any resolution of $\mathbf{X}$ which admits a suitable projection of its toric diagram. The type IIA background consists of an A-type ALE space fibered over the real line, with stacks of coincident D6-branes wrapping 2-cycles in the ALE resolution. In this work, we combine that technique with some elementary ideas from graph theory, to analyze mass deformations of $\mathcal{T}_{\mathbf{X}}$ when $\mathbf{X}$ is a isolated toric CY$_3$ singularity of rank-two (that is, it has two compact divisors). We explicitly derive type IIA descriptions of all isolated rank-two CY$_3$ toric singularities. We also comment on the renormalization group flows in the extended parameter spaces of these theories, which frequently relate distinct geometries by flowing to theories with lower flavor symmetries, including those that describe non-gauge-theoretic phases.

Figures (30)

• Figure 1: RG flows connecting the different rank-two toric singularities discussed in this paper, the corresponding gauge-theory phases, and their relations under parity (denoted by $\bm{\mathcal{P}})$. Non-gauge-theoretic phases are not shown in the figure, but do arise as discussed in the main text. Top: Rank-two theories stemming from the $E_{3}{}^{2,0}$ ("beetle") singularity. Bottom: Rank-two theories stemming from the $E_{3}{}^{2,1}$ singularity. There is a $\bm{\mathcal{P}}$-transformed version of this flow, which is not shown here.
• Figure 2: Left: An example of an allowed vertical reduction of a 2d toric diagram, which gives rise to a resolved $A_{1}$ singularity visualized by the 1d toric diagram below. Right: An example of a disallowed vertical reduction, due to the presence of an edge that would collide with either one of the vertices along the vertical direction under such a reduction.
• Figure 3: The 10 $SL(2,\mathbb{Z})$-inequivalent isolated toric rank-2 singularities.
• Figure 4: The two crepant resolutions of the $E_{1}{}^{2,2}$ singularity. Resolution (a) admits a vertical reduction.
• Figure 5: Resolution (a) of the $E_{1}{}^{2,2}$ singularity and its vertical reduction.