D-branes at Singularities, Compactification, and Hypercharge

Abstract

We report on progress towards the construction of SM-like gauge theories on the world-volume of D-branes at a Calabi-Yau singularity. In particular, we work out the topological conditions on the embedding of the singularity inside a compact CY threefold, that select hypercharge as the only light U(1) gauge factor. We apply this insight to the proposed open string realization of the SM of hep-th/0508089, based on a D3-brane at a dP_8 singularity, and present a geometric construction of a compact Calabi-Yau threefold with all the required topological properties. We comment on the relevance of D-instantons to the breaking of global U(1) symmetries.

Figures (9)

• Figure 1: The MSSM-like quiver gauge theory obtained in MH. Each line represents three generations of bi-fundamentals. In the text below we will identify the geometric condition that isolates the $U(1)_Y$ hypercharge as the only surviving massless $U(1)$ gauge symmetry.
• Figure 2: Our proposed D3-brane realization of the MSSM involves a $dP_8$ singularity embedded inside a CY manifold, such that two of its 2-cycles, $\alpha_1$ and $\alpha_2$, develop an $A_2$ singularity which forms part of a curve of $A_2$ singularities on the CY, and all remaining 2-cycles except $\alpha_4$ are non-trivial within the full CY.
• Figure 3: Starting point of our construction of a CY threefold with the desired topology. The curves $f_i$ and $g_i$ are fibers in the two rulings on $S_1$.
• Figure 4: The CY threefold after flopping the curves $C_{45}, C_6, C_7$ and $C_8$. The curves $f_{45}$, $g_{45}$, $C_{45}$, $C_6$, $C_7$, and $C_8$ are all $(-1)$-curves on $S_5$.
• Figure 5: The CY threefold after flopping $C_1$. The curves $f_1$, $g_1$, and $C_1$ are additional $(-1)$-curves on $S_6$.