On Geometric Classification of 5d SCFTs

TL;DR

This paper develops geometric criteria for engineering 5d superconformal field theories via M-theory on Calabi-Yau 3-folds, providing a systematic classification of rank 2 theories and supporting their origin from 6d SCFTs compactified on a circle.

Contribution

It extends the geometric classification of 5d SCFTs to rank 2 cases, confirming gauge theory predictions and establishing their connection to 6d SCFTs.

Findings

All rank 2 SCFTs predicted by gauge theory are realizable, except one non-perturbatively inconsistent family.

Rank 2 SCFTs originate from 6d (1,0) SCFTs compactified on a circle, possibly with twists and holonomies.

The classification rules out certain theories based on geometric inconsistency.

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.