Local models for intersecting brane worlds

TL;DR

The paper develops local, non-compact Calabi-Yau threefolds with compact 3-cycles intersecting at points to realize intersecting D6-brane worlds, providing calculable four-dimensional chiral gauge theories decoupled from gravity in the local limit. It introduces simple rules for constructing 3-cycles, computing spectra, RR tadpoles, and supersymmetry via special Lagrangian calibrations, and presents explicit examples with Standard Model-like gauge groups and three generations, including mirrors to D3/D7 at singularities. Generalizations to more intricate fibrations (multiple C^* degenerations, double elliptic fibrations, higher genus), orientifolds, and alternative non-compact geometries are outlined, broadening the landscape of local chiral sectors. The work highlights the potential for low string scales in local models, connections to mirror symmetry and M-theory G2 lifts, and sets the stage for systematic searches for SM-like spectra within a controllable, gravity-decoupled framework.

Abstract

We describe the construction of configurations of D6-branes wrapped on compact 3-cycles intersecting at points in non-compact Calabi-Yau threefolds. Such constructions provide local models of intersecting brane worlds, and describe sectors of four-dimensional gauge theories with chiral fermions. We present several classes of non-compact manifolds with compact 3-cycles intersecting at points, and discuss the rules required for model building with wrapped D6-branes. The rules to build 3-cycles are simple, and allow easy computation of chiral spectra, RR tadpoles and the amount of preserved supersymmetry. We present several explicit examples of these constructions, some of which have Standard Model like gauge group and three quark-lepton generations. In some cases, mirror symmetry relates the models to other constructions used in phenomenological D-brane model building, like D-branes at singularities. Some simple N=1 supersymmetric configurations may lead to relatively tractable G_2 manifolds upon lift to M-theory, which would be non-compact but nevertheless yield four-dimensional chiral gauge field theories.

Figures (24)

• Figure 1: Bottom-up approach to embedding the gauge theory sectors from intersecting D-branes in string compactification. In a first step a) one considers local configurations of D6-branes wrapped on compact 3-cycles in a non-compact space $\bf W$. The local configuration may be subsequently embedded in a global compactification, step b). Many features of the chiral gauge field theory on the D6-branes are not sensitive to the details of the compactification and depend only on the local geometry $\bf W$.
• Figure 2: Schematic depiction of the fibration structure of the threefold $\bf W$. At certain points on the $z$-plane, the $\bf C^*$ or the elliptic fiber degenerate.
• Figure 3: Schematic depiction of the intersection between two compact 3-cycles. The two 3-cycles touch at $z=0$ in the $z$-plane, where they also touch in the $\bf C^*$ fiber since the $\bf S^1$ shrinks to zero. On the elliptic fiber, the 3-cycles intersect according to their $(p,q)$ labels.
• Figure 4: Two examples of different compact 3-cycles. Fig a) shows a compact 3-cycles obtained by fibering over a segment in the $z$-plane a $(p,q)=(-1,2)$ 1-cycle in the elliptic fiber and the $\bf S^1$ in the $\bf C^*$ fibration. Fig b) shows a compact 3-cycle obtained by fibering suitable $(p,q)$ 1-cycles over segments over a network in the $z$-plane. Note the conservation of $(p,q)$ wrapping numbers across the junction.
• Figure 5: The prong creation process shows the existence of junction 3-cycles in our geometries.