Terminal Singularities, Milnor Numbers, and Matter in F-theory

TL;DR

This paper explores the physical implications of certain singularities in F-theory compactifications, linking geometric properties like Milnor numbers to uncharged matter content and ensuring anomaly-free spectra.

Contribution

It systematically analyzes non-crepant terminal singularities in F-theory, relating Milnor numbers to uncharged matter and providing explicit anomaly-free six-dimensional spectra.

Findings

Identified a class of terminal singularities with computable uncharged matter.

Linked Milnor numbers to local complex deformations and matter count.

Demonstrated anomaly-free spectra in various models.

Abstract

We initiate a systematic investigation of F-theory on elliptic fibrations with singularities which cannot be resolved without breaking the Calabi-Yau condition, corresponding to $\mathbb Q$-factorial terminal singularities. It is the purpose of this paper to elucidate the physical origin of such non-crepant singularities in codimension two and to systematically analyse F-theory compactifications containing such singularities. The singularities reflect the presence of localised matter states from wrapped M2-branes which are not charged under any massless gauge potential. We identify a class of $\mathbb Q$-factorial terminal singularities on elliptically fibered Calabi-Yau threefolds for which we can compute the number of uncharged localised hypermultiplets in terms of their associated Milnor numbers. These count the local complex deformations of the singularities. The resulting six-dimensional spectra are shown to be anomaly-free. We exemplify this in a variety of cases, including models with non-perturbative gauge groups with both charged and uncharged localised matter. The underlying mathematics will be discussed further in a forthcoming publication.