Towards Geometric D6-Brane Model Building on non-Factorisable Toroidal $\mathbb{Z}_4$-Orbifolds

TL;DR

This paper develops a geometric framework for D-brane model building on non-factorisable toroidal orbifolds, identifying suitable three-cycles for particle physics models, including Pati-Salam models, and connecting geometry with conformal field theory methods.

Contribution

It introduces a geometric approach to D-brane model building on non-factorisable $T^6/\mathbb{Z}_4$ orbifolds, identifying lattice orientations and three-cycles suitable for realistic particle physics models.

Findings

Reduced inequivalent lattice orientations to three and four for two different orbifolds.

Constructed globally consistent models with two and four generations.

Rewrote fractional sLag cycles in a factorised form for CFT analysis.

Abstract

We present a geometric approach to D-brane model building on the non-factorisable torus backgrounds of $T^6/\mathbb{Z}_4$, which are $A_3 \times A_3$ and $A_3 \times A_1 \times B_2$. Based on the counting of `short' supersymmetric three-cycles per complex structure {\it vev}, the number of physically inequivalent lattice orientations with respect to the anti-holomorphic involution ${\cal R}$ of the Type IIA/$\Omega\cal{R}$ orientifold can be reduced to three for the $A_3 \times A_3$ lattice and four for the $A_3 \times A_1 \times B_2$ lattice. While four independent three-cycles on $A_3 \times A_3$ cannot accommodate phenomenologically interesting global models with a chiral spectrum, the eight-dimensional space of three-cycles on $A_3 \times A_1 \times B_2$ is rich enough to provide for particle physics models, with several globally consistent two- and four-generation Pati-Salam models presented here. We further show that for fractional {\it sLag} three-cycles, the compact geometry can be rewritten in a $(T^2)^3$ factorised form, paving the way for a generalisation of known CFT methods to determine the vector-like spectrum and to derive the low-energy effective action for open string states.