The Importance of Being Rigid: D6-Brane Model Building on T6/Z2xZ6' with Discrete Torsion

TL;DR

The work develops a comprehensive framework for constructing particle-physics models with rigid D6-branes on $T^6/(Z_2 imesZ_6' imesOmega ext{R})$ with discrete torsion. It proves pairwise lattice identifications (${f AAA} ightleftharpoons{f ABB}$ and ${f AAB} ightleftharpoons{f BBB}$) that drastically reduce model-search space and provides a complete classification of rigid D6-branes with and without adjoint, symmetric, or antisymmetric matter. Globally consistent five-stack Pati-Salam models are found, with three-generation chiral content and a constrained spectrum of exotics; local MSSM and left-right symmetric attempts fail to cancel twisted RR tadpoles globally, highlighting the importance of global consistency. The analysis shows one-loop gauge coupling unification for nearly isotropic tori and the possibility of a low string scale in highly anisotropic setups, suggesting testable implications for TeV-scale physics and exotic $Z'$ or dark-photon candidates. Overall, the paper maps rigorous geometric and string-theoretic constraints into viable beyond-Standard-Model constructions within a controlled, discretized moduli regime.

Abstract

Model building with rigid D6-branes on the Type IIA orientifold on T6/Z2xZ6' with discrete torsion is considered. The systematic search for models of particle physics is significantly reduced by proving new symmetries among different lattice orientations. Suitable rigid D6-branes without matter in adjoint and symmetric representations are classified, and SO(2N) and USp(2N) gauge factors on orientifold invariant D6-branes are distinguished in terms of their discrete Wilson line and displacement parameters. Constraints on the non-existence of exotic matter prohibit global completions of local MSSM and left-right symmetric models, while globally defined supersymmetric Pati-Salam models are found. For the latter, only one particle generation possesses perturbative Yukawa couplings. Masses for the mild amount of exotic matter and the role of Abelian symmetries are briefly discussed. Last but not least, it is shown that for all three two-torus volumes of about the same order of magnitude, gauge coupling unification at one-loop can be achieved, while for highly unisotropic choices a low string scale in the TeV range is compatible with the observed strengths of gauge and gravitational couplings.

Figures (2)

• Figure 1: The $SU(3)^3$ compactification lattice of the $T^6/\mathbb{Z}_2 \times \mathbb{Z}_6'$ orbifold defined in equation (\ref{['Eq:Z2Z6p-def']}). The $\mathbb{Z}_2$ fixed points $1\stackrel{\omega}{\circlearrowleft},4\stackrel{\omega}{\to}5\stackrel{\omega}{\to}6\stackrel{\omega}{\to}4$ on each two-torus are depicted in red, the $\mathbb{Z}_3$ fixed points $1\stackrel{\theta,\omega}{\circlearrowleft},2\stackrel{\theta,\omega}{\leftrightarrow} 3$ in blue. The anti-holomorphic involution eq. (\ref{['Eq:Z2Z6p-involution']}) is consistent with two choices of orientations A (for which $4 \stackrel{\cal R}{\circlearrowleft}, 5 \stackrel{\cal R}{\leftrightarrow} 6$) and B (for which $4 \stackrel{\cal R}{\leftrightarrow} 5, 6 \stackrel{\cal R}{\circlearrowleft}$) of each $SU(3)$ root lattice.
• Figure 2: The three types of one-loop contributions to the gauge coupling as a function of the Kähler modulus $v$ : $-\frac{1}{4\pi^2} \ln(\eta(iv))$ (orange), $-\frac{1}{4\pi^2} \ln \left(\frac{\vartheta_1(\frac{1}{2}, iv)}{\eta (i v)}\right)$ (blue), $-\frac{1}{4\pi^2} \ln \left(e^{-\pi v /4}\frac{|\vartheta_1(- i\frac{v}{2}, iv)|}{\eta (i v)}\right)$ (green). The left plot shows the regime where the geometric approximation is valid ($v>1$), while the right plot gives a closer look at the region $1<v<2$. The red, dotted curves in the right plot correspond to the large $v$-behaviour of the one-loop contributions, showing that the contributions asymptote very fast to linear contributions in terms of the Kähler modulus $v$.