Rigid D6-branes on T6/[Z(2)xZ(2M)xOmegaR] with discrete torsion

TL;DR

This work delivers a comprehensive classification of $T^6/(\mathbb{Z}_2 \times \mathbb{Z}_{2M} \times \Omega\mathcal{R})$ orientifolds with discrete torsion and analyzes rigid D6-branes across factorisable lattices. It unifies SUSY, RR tadpole cancellation, K-theory constraints, and the full massless spectrum (including twisted sectors) for all inequivalent backgrounds, showing that completely rigid D6-branes exist on $\mathbb{Z}_2 \times \mathbb{Z}_6$ and $\mathbb{Z}_2 \times \mathbb{Z}_6'$ cases, which are promising for Standard Model model-building. Conversely, the $\mathbb{Z}_2 \times \mathbb{Z}_4$ background is shown to be incompatible with three-family SM or GUT constructions via simple arguments. The paper provides explicit globally consistent examples, including several rigid-brane realizations, and outlines future work on moduli stabilization, fluxes, and potential LHC-scale signatures of such string vacua. Overall, discrete torsion opens a viable avenue for constructing phenomenologically relevant, fully calculable string vacua with rigid D6-branes.

Abstract

We give a complete classification of T6/[Z(2)xZ(2M)xOmegaR] orientifolds on factorisable tori and rigid D6-branes on them. The analysis includes the supersymmetry, RR tadpole cancellation and K-theory conditions and complete massless open and closed string spectrum (i.e. non-chiral as well as chiral) for fractional or rigid D6-branes for all inequivalent compactification lattices, without and with discrete torsion. We give examples for each orbifold background, which show that on Z(2)xZ(6) and Z(2)xZ(6') there exist completely rigid D6-branes despite the self-intersections of orbifold image cycles. This opens up a new avenue for improved Standard Model building. On the other hand, we show that Standard and GUT model building on the Z(2)xZ(4) background is ruled out by simple arguments.

Figures (4)

• Figure 1:
• Figure 2: The $\mathbb{Z}_4$ (left) and $\mathbb{Z}_3$ (right) invariant lattices. The $\mathbb{Z}_4$ invariant lattice is the root lattice of $B_2=SO(5)$ and has two fixed points (1,2) under $\mathbb{Z}_4$ and two further points that are fixed under the $\mathbb{Z}_2$ subsymmetry, but interchanged by the $\mathbb{Z}_4$ action. The $\mathbb{Z}_3$ (and $\mathbb{Z}_6$) invariant lattice is the root lattice of $A_2=SU(3)$ or equivalently $G_2$ (observe that the lattices coincide even though the simple roots differ). It has fixed points under the $\mathbb{Z}_3$ subsymmetry (1,2,3) and under the $\mathbb{Z}_2$ subsymmetry (1,4,5,6), where $2 \stackrel{\mathbb{Z}_6}{\leftrightarrow} 3$ and $4 \stackrel{\mathbb{Z}_6}{\rightarrow} 5 \stackrel{\mathbb{Z}_6}{\rightarrow} 6$. The A-type lattices have the coordinate orientation $x_{2i-1}$ along the 1-cycle $\pi_{2i-1}$ (depicted in green); for the B-type lattices $x_{2i-1}$ is along $\pi_{2i-1}+\pi_{2i}$ (depicted in yellow).
• Figure 3: The figure on the left hand side shows the 'triangulated' version of a tree channel Klein Bottle amplitude. In difference to a torus the left and right edges are not glued together but to themselves to form crosscaps as indicated in the figure on the right hand side by purple lines.
• Figure 4: The figure on the top shows the tree channel Klein Bottle amplitude (see figure \ref{['Fig:KBtree']}). Topology preserving operations mapping this to the loop channel amplitude are illustrated in the middle: ① cut along the brown line with indicated orientation, ② flip upper rectangle over its right edge, ③ push flipped rectangle down and glue along the green line. Finally, the loop channel diagram is displayed at the bottom. The red and brown line merge into a string closed upon a twist (visible from its tree channel origin). The crosscap map of the blue O6-plane appears as a trace insertion.