The Toric SO(10) F-Theory Landscape

TL;DR

This work classifies and analyzes six-dimensional F-theory vacua with gauge group $SO(10)$ and additional Mordell–Weil $U(1)$ or discrete factors using toric hypersurface fibrations. It develops base-independent methods to compute full matter spectra, including neutral singlets, and confirms gauge and gravitational anomaly cancellation, even in the presence of superconformal points (SCPs) tied to non-flat fibers. A key technical advance is unhiggsing to $SO(10)\times U(1)^2$ to count singlets and the use of toric tops to enumerate 36 distinct models, including detailed anomaly-coefficient structure and SCP transitions. The paper also explores connections to four-dimensional physics, showing how some 6d vacua could yield the Standard Model with high-scale SUSY when further compactified with Abelian flux, while noting a sizeable portion of models lie in a toric swampland. Overall, the results provide a comprehensive geometric landscape of 6d $SO(10)$ F-theory vacua with rich abelian structures, SCP phenomena, and transitions that illuminate both formal structure and phenomenological potential.

Abstract

Supergravity theories in more than four dimensions with grand unified gauge symmetries are an important intermediate step towards the ultraviolet completion of the Standard Model in string theory. Using toric geometry, we classify and analyze six-dimensional F-theory vacua with gauge group SO(10) taking into account Mordell-Weil U(1) and discrete gauge factors. We determine the full matter spectrum of these models, including charged and neutral SO(10) singlets. Based solely on the geometry, we compute all matter multiplicities and confirm the cancellation of gauge and gravitational anomalies independent of the base space. Particular emphasis is put on symmetry enhancements at the loci of matter fields and to the frequent appearance of superconformal points. They are linked to non-toric Kähler deformations which contribute to the counting of degrees of freedom. We compute the anomaly coefficients for these theories as well by using a base-independent blow-up procedure and superconformal matter transitions. Finally, we identify six-dimensional supergravity models which can yield the Standard Model with high-scale supersymmetry by further compactification to four dimensions in an Abelian flux background.

Figures (18)

• Figure 1: The polygon $F_3$ describing a torus in the ambient space $dP_1$.
• Figure 2: The elliptic curve $\mathcal{E}$ with two points, $\hat{s}_0$ and $\hat{s}_1$ (see text).
• Figure 3: The extended Dynkin diagram of SO(10) obtained from intersections of divisors $D_I$ and curves $\mathbb{P}^1_J$, with $I,J \in \{0,\ldots,5\}$. Intersections of the torus divisors $[e_1]$ and $[v]$ are also indicated.
• Figure 4: The fibration of the torus $\mathcal{E}$ over $\mathbb{P}^1$ (top left) turns via tuning into a singular $K3$ manifold (top right). Resolution of the singularity generates five $\mathbb{P}^1$s with SO(10) intersections (bottom left); after the Shioda map has been carried out, the U(1) divisor intersects the affine node $\mathbb{P}^1_0$ (bottom right).
• Figure 5: Extended Dynkin diagram of SO(12) at the $d_9=0$ locus. The affine node has an intersection with $[e_1]$. The dashed circles indicate SO(10) nodes before the matter split.