Trifectas for $T_N$ in 5d

TL;DR

This work provides a unified framework for understanding the 5d $T_N$ SCFTs and their mass-deformed descendants by integrating M-theory geometry, toric techniques, CFDs, box graphs, brane webs, and magnetic quivers. It derives a complete picture of Coulomb-branch geometries, IR gauge-theory descriptions, and Higgs-branch structures, including BPS spectra and 6d uplifts, through a network of CFD transitions, toric flops, and BG-CFD embeddings. The paper also maps descendants to non-Lagrangian theories $B_N^{(i)}$, constructs their magnetic quivers, and presents extensive RG-flow trees and Hasse diagrams, highlighting the deep interconnections among different descriptions of 5d SCFTs. Overall, the results offer a practical, geometrically grounded toolkit to study UV completions, flavor symmetries, and dualities in $T_N$ and related theories with potential for broad applicability in higher-dimensional SCFT classifications.

Abstract

The trinions $T_N$ are a class of 5d $\mathcal{N}=1$ superconformal field theories (SCFTs) realized as M-theory on $\mathbb{C}^3/\mathbb{Z}_N \times \mathbb{Z}_N$. We apply to $T_N$, as well as closely-related SCFTs that are obtained by mass deformations, a multitude of recently developed approaches to studying 5d SCFTs and their IR gauge theory descriptions. Thereby we provide a complete picture of the theories both on the Coulomb branch and Higgs branch, from various geometric points of view - toric and gluing of compact surfaces as well as combined fiber diagrams - to magnetic quivers and Hasse diagrams.

Figures (32)

• Figure 1: An example of a toric Calabi-Yau threefold singularity and its crepant resolution. Each toric divisor is presented as a point on the $(x,y)$-plane, and we label their coordinates in $\mathbb{Z}^3$. Each 2d cone is presented as a line segment between the points and each 3d cone is given by a polygon. On the left hand side, there is a singular toric Calabi-Yau threefold with four non-compact divisors $D_1,\dots,D_4$. On the right hand side, after the crepant resolution, the toric fan is subdivided and each 3d cone (presented as a triangle) has unit volume. There is a new compact divisor $S_1$ in the middle.
• Figure 2: The resolved toric Calabi-Yau threefold singularity corresponds to the $T_5$ theory. The toric fan of the compact toric divisor $S_1$ is constructed including all of its neighboring rays $\bm{v}$, $\bm{v}_i$$(i=1,\dots,6)$.
• Figure 3: The toric fan of each compact surface $S_i$. The subscript $i$ is labelled at the center of each toric fan, and the self-intersection number of each toric curve is written in the brackets. gdP$_n$ refers to the generalized del Pezzo surfaces.
• Figure 4: (a) An example of toric flop of the resolved $T_5$ geometry among the compact divisors. The triangulation in the picture changes, while the rays remain the same. We furthermore indicate how the geometry is glued from the rational surfaces such as in figure \ref{['f:surface-topology']}. The numbers $(-n)$ at the external vertices indicate the self-intersection numbers of curves that are intersections between compact and non-compact divisors and encode the $SU(5)^3$ flavor symmetry. (b) Another presentation of the geometry, where the $i$th internal vertex $i_n^k$ is Bl$_k\mathbb{ F}_n$. In this presentation the flavor symmetry is not manifest.
• Figure 5: The labeling of $T_N$ singularity, with an example of $N=5$.