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

TL;DR

This paper analyzes F-theory on elliptically fibered Calabi–Yau threefolds with ${\mathbb{Q}}$-factorial terminal singularities that cannot be crepantly resolved. It shows that such codimension-two singularities host localised uncharged hypermultiplets whose multiplicities are governed by Milnor and Tyurina numbers, and establishes a precise relation between complex structure deformations and the localised spectrum. The authors derive general formulas for the neutral hypermultiplet content, including the split into localised and non-localised parts, and demonstrate anomaly-free six-dimensional spectra across several explicit models, including cases with trivial and non-trivial gauge groups. They provide a detailed mathematical framework to count localised states via versal deformations and discuss how the topological Euler characteristic encodes the global spectrum. The work paves the way for further mathematical and physical study of terminal singularities in higher-dimensional F-theory compactifications.

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.

Figures (6)

• Figure 1: Our notation for an elliptically fibered Calabi-Yau manifold with $\Delta = \Sigma_1 \cup \Sigma_0$.
• Figure 2: Some Kodaira fiber types appearing in this paper.
• Figure 3: Origin of uncharged hyper multiplets in six-dimensional F-theory compactifications.
• Figure 4: Affine Dynkin diagram of the partially resolved fiber over $P_1: z_1 = f_0 =0$. The red cross denotes the intersection with the zero-section $z=0$ of the Weierstrass model and the dashed lines symbolize the deleted $\mathbb P^1$s in the standard Kodaira fiber which are not realized in the actual fiber. The resolution was obtained as resolution of a Tate model realising the vanishing orders (\ref{['IImodelWeier1']}).
• Figure 5: Affine Dynkin diagram of the resolved $\{z_1\}\cap \{\tilde{a}_3\}$ locus. The red cross denotes the intersection with the zero-section $z=0$ of the Weierstrass model. The blue and red colour indicates the splitting of $\mathbb{P}^1_{A,B,C}$.