D-branes at Toric Singularities: Model Building, Yukawa Couplings and Flavour Physics

TL;DR

The authors develop a comprehensive framework for D-branes at toric singularities, using dimer diagrams and Gulotta's algorithm to derive gauge theories and Yukawa couplings directly from toric data. They establish a universal bound on the number of families in toric phases, show that tree-level Yukawa matrices generically possess a massless eigenstate with possible hierarchical nonzero masses, and demonstrate that realistic CKM mixing and CP violation can be realized in explicit toric models (notably via del Pezzo surfaces and D3/D7 sectors). By exploring both D3-D3 and D3-D7 couplings, Seiberg dualities, and Higgsing, the work connects geometric data to fermion masses and mixing, offering a robust, local approach to flavour physics in string phenomenology. The results suggest toric singularities can yield realistic flavour structure while remaining computable and largely independent of global moduli stabilization, motivating further study of non-toric extensions and compact embeddings.

Abstract

We discuss general properties of D-brane model building at toric singularities. Using dimer techniques to obtain the gauge theory from the structure of the singularity, we extract results on the matter sector and superpotential of the corresponding gauge theory. We show that the number of families in toric phases is always less than or equal to three, with a unique exception being the zeroth Hirzebruch surface. With the physical input of three generations we find that the lightest family of quarks is massless and the masses of the other two can be hierarchically separated. We compute the CKM matrix for explicit models in this setting and find the singularities possess sufficient structure to allow for realistic mixing between generations and CP violation.

Figures (32)

• Figure 1: A five-faceted polyhedral cone in $\mathbb{R}^3.$ The normal vectors determine the toric diagram. At the facets of the polyhedral cone the $T^3$ fibre degenerates to $T^2,$ at the edges to $S^1,$ and at the tip $T^3$ degenerates completely.
• Figure 2: Left:$\mathbb{P}_2$ can be described as a $T^2$ fibration over a triangle. At the edges of the triangle the torus $T^2$ degenerates to a circle $S^1$ and at a vertex the fibre degenerates to a point. Right: Blowing up a vertex in $\mathbb{P}_2$ and replacing it with a line generates the first del Pezzo surface.
• Figure 3: The superpotential and quiver gauge theory of the conifold.
• Figure 4: This crossing of zigzag paths corresponds to bi-fundamental matter transforming as $(B,\bar{A}).$
• Figure 5: A part of a dimer showing zigzag paths around two nodes, and the corresponding superpotential terms.