Global embeddings for branes at toric singularities

TL;DR

The paper develops an algorithmic framework to embed local toric singularities, including Toric Lego constructions, into globally consistent Calabi–Yau compactifications, using hypersurfaces in toric varieties classified by Kreuzer–Skarke polytopes. It systematically incorporates D3/D7 branes and orientifolds, and shows how to uplift to F-theory, providing concrete examples with the (dP0)^3 singularity and its global embeddings. A detailed analysis of the brane content, tadpole cancellation, and spectra is paired with a landscape survey of singularities in the KS dataset, revealing that highly singular toric configurations are common and structurally rich. The results illuminate how local model-building sectors can be embedded in realistic global geometries, with implications for moduli stabilization, supersymmetry breaking, and the broader exploration of string vacua using toric and algorithmic methods.

Abstract

We describe how local toric singularities, including the Toric Lego construction, can be embedded in compact Calabi-Yau manifolds. We study in detail the addition of D-branes, including non-compact flavor branes as typically used in semi-realistic model building. The global geometry provides constraints on allowable local models. As an illustration of our discussion we focus on D3 and D7-branes on (the partially resolved) (dP0)^3 singularity, its embedding in a specific Calabi-Yau manifold as a hypersurface in a toric variety, the related type IIB orientifold compactification, as well as the corresponding F-theory uplift. Our techniques generalize naturally to complete intersections, and to a large class of F-theory backgrounds with singularities.

Figures (13)

• Figure 1: The polytope for $dP_0$, or equivalently, the toric diagram for the (resolved) $\mathbb{C}^3/\mathbb{Z}_3$. The particular assignment of labels to the vertices follows from the ones in the global embedding.
• Figure 2: $Y^{3,0}$ resolved into two ${dP_0}$'s separated by a $\mathbb{P}^1$. We have skewed the perspective slightly to ease visualization. The actual position of the points in the two-dimensional plane can be obtained by forgetting the last two coordinates in the polytope in eq. \ref{['eq:Y30-polytope']}.
• Figure 3: Local model for ${\left(dP_0\right)^3}$.
• Figure 4: A toy model for the MSSM, from Aldazabal:2000sa. The filled dark dots denote gauge groups, while the white dots denote global symmetry groups, coming from non-compact $D7$ branes. The labels on the arrows denote with which MSSM field they should be identified.
• Figure 5: Dimer model for branes at the $\mathbb{C}^3/\mathbb{Z}_3$ singularity, given by the periodic honeycomb lattice (we have only shown a few cells). The labels on the faces of the dimer model indicate which gauge factor in \ref{['fig:dp0mssm']} they correspond to.