From quiver diagrams to particle physics

TL;DR

The paper investigates how string theory can realize chiral gauge sectors that resemble the Standard Model by placing D3-branes at Calabi-Yau singularities. By encoding local geometry in quiver diagrams, Uranga systematizes orbifold, non-orbifold, and orientifold constructions and shows that even simple singularities can yield MSSM-like spectra. The work highlights the McKay correspondence, moduli spaces, and geometric transitions as organizing principles, and presents explicit Z3 models that approach realistic particle content, including hypercharge realization. These results demonstrate a concrete path from geometry to phenomenology in string vacua and motivate further study of branes at singularities for realistic model building.

Abstract

Recent scenarios of phenomenologically realistic string compactifications involve the existence of gauge sectors localized on D-branes at singular points of Calabi-Yau threefolds. The spectrum and interactions in these gauge sectors are determined by the local geometry of the singularity, and can be encoded in quiver diagrams. We discuss the physical models arising for the simplest case of orbifold singularities, and generalize to non-orbifold singularities and orientifold singularities. Finally we show that relatively simple singularities lead to gauge sectors surprisingly close to the standard model of elementary particles.

Figures (3)

• Figure 1: Figure a) shows the quiver diagram for a $\bf C^3/(\bf Z_2\times \bf Z_2)$ singularity, which can be defined as the hypersurface $xyz=w^2$ in $\bf C^4$. Partial blow-ups resolve it to the suspended pinch point singularity ($xy=zw^2$), and the conifold ($xy=zw$), whose quiver diagrams are shown in Figures b) and c). Each blow-up is reflected on the quiver as the joining of two nodes (the joint node is denoted by a hat), and the disappearance of certain arrows (depicted with thin heads).
• Figure 2: Figure a) shows the quiver diagram for the $\bf C^3/\bf Z_5$ singularity, with $\bf Z_5$ action $(z_1,z_2,z_3)\to (e^{2\pi i\frac{1}{5}} z_1, e^{2\pi i \frac{1}{5}} z_2, e^{2\pi i\frac{-2}{5}}z_3)$. Figure b) shows the quiver of the singularity after an orientifold projection.
• Figure 3: Quiver diagram of a system of D3- and D7-branes at a $\bf C^3/\bf Z_3$ singularity, reproducing a spectrum close to the (minimal supersymmetric) standard model. The 3-7 sector is triplicated due to the three kinds of D7-branes we have introduced. The dotted node and arrows are present in the generic quiver, but not for our specific choice of Chan-Paton representations. Notice that white nodes correspond to global symmetries of the D3-brane field theory, and that only one of the three $U(1)$ symmetries of the D3-branes survives at low energies.