On Geometric Classification of 5d SCFTs

TL;DR

The paper develops a geometric framework to classify 5d SCFTs via M-theory on local Calabi–Yau 3-folds, extending rank-1 shrinkable del Pezzo constructions to a finite rank-2 classification. It demonstrates that all rank-2 SCFTs (from smooth CY 3-folds) descend from 6d (1,0) SCFTs compactified on a circle (with possible automorphism twists), and identifies one non-perturbatively excluded family by geometric reasoning. The authors formulate a concrete classification program based on physical equivalence classes of shrinkable 3-folds, establish building blocks (blowups of Hirzebruch surfaces and del Pezzo surfaces) with bounded Mori cones, and encode RG flows via geometric transitions (flops, HW transitions). The work also reveals dualities between gauge-theory descriptions and notes the role of non-perturbative states in refining gauge-theory criteria for UV fixed points, highlighting the deep interplay between geometry and 5d SCFT dynamics. Overall, the results support the conjecture that every 5d SCFT can be obtained from circle compactifications of 6d SCFTs and provide a roadmap for systematic higher-rank classifications using Calabi–Yau geometry.

Abstract

We formulate geometric conditions necessary for engineering 5d superconformal field theories (SCFTs) via M-theory compactification on a local Calabi-Yau 3-fold. Extending the classification of the rank 1 cases, which are realized geometrically as shrinking del Pezzo surfaces embedded in a 3-fold, we propose an exhaustive classification of local 3-folds engineering rank 2 SCFTs in 5d. This systematic classification confirms that all rank 2 SCFTs predicted using gauge theoretic arguments can be realized as consistent theories, with the exception of one family which is shown to be non-perturbatively inconsistent and thereby ruled out by geometric considerations. We find that all rank 2 SCFTs descend from 6d (1,0) SCFTs compactified on a circle possibly twisted with an automorphism together with holonomies for global symmetries around the Kaluza-Klein circle. These results support our conjecture that every 5d SCFT can be obtained from the circle compactification of some parent 6d (1,0) SCFT.

Figures (4)

• Figure 1: Graphical representation of a rank $r$ Kähler surface $S = \cup S_i \subset X$ embedded in local Calabi-Yau 3-fold $X$. The nodes of the graph correspond to 4-cycles $S_i$, while the edges $C_{i,i+1} = S_i \cap S_{i+1}$ correspond to 2-cycles along which the nodes intersect.
• Figure 2: Example of a gluing construction of the Kähler surface $S = \mathbb F_0 \cup \mathbb F_2$. The gluing curves in both surfaces, $C_1, C_2$, are encircled by dashed lines in the left figure. The final geometry (on the right) is the result of identifying these two curves subject to the conditions described in Section \ref{['sec:algorithm']}.
• Figure 3: A local illustration of a flop transition $X \rightarrow X'$ between two CY 3-folds. The red lines in both diagrams correspond to the $-1$ curves in (respectively) $X$ and $X'$.
• Figure 4: