Circling the square: deforming fractional D-branes in type II/Ω ℛ $$ \mathrm{\mathcal{R}} $$ orientifolds

Abstract

We study complex structure deformations of special Lagrangian cycles associated to fractional D-branes at Z 2 singularities in Type II/ΩR orientifold models. By means of solving hypersurface constraints, we show how to compute the volumes of the most simple D-brane configurations. These volumes are given as a function of the deformation parameters depending on the Dbrane position relative to the smoothed out singularity. We observe which cycles keep the special Lagrangian property in various deformation scenarios and what orientifold involutions are allowed. As expected, the volume and thus the tree level value of the gauge coupling hardly change for D-branes not wrapping the exceptional cycle on the deformed singularity, whereas the volume of D-branes passing through the singularity depends on the deformation parameter by some power law.

D6-brane model building onZ2×Z6: MSSM-like and left–right symmetric models

“…Another pressing question thus consists in identifying possible supersymmetry breaking scenarios. While we expect that non-supersymmetric deformations away from the singular orbifold point will predominantly stabilise moduli at the singularity as argued in [68,69], it remains to be seen if the maximal non-Abelian hidden gauge groups SU (4) or SU (3) × SU (3) in the present D6-brane configurations are suitable to generate a gaugino condensate, which breaks supersymmetry, and if so study gauge mediation versus gravity mediation scenarios.      (67)…”

Massless spectra and gauge couplings at one-loop on non-factorisable toroidal orientifolds

“…A step into this landscape can be provided by combining the resolution techniques studied in e.g. [69,1] with generalisations of the deformation methods initiated in [82][83][84]75].…”

Deformations, moduli stabilisation and gauge couplings at one-loop

“…which leads to the following qualitative situations observed first in the context of T 6 /(Z 2 × Z 2 × ΩR) and T 6 /(Z 2 × Z 6 × ΩR) models with discrete torsion in [49,51], see also [82,52,54]:…”

“…However, in the case of Type IIA model building with fractional D6-branes on orbifolds with discrete torsion, the orbifold singularities have to be deformed rather than blown up, which forces us to consider different tools from algebraic geometry: by viewing two-tori as elliptic curves in the weighted projective space P 2 112 , a factorisable toroidal orbifold with discrete torsion can be described as a hypersurface in a weighted projective space, with its topology being a double cover of P 1 × P 1 × P 1 . Building on this hypersurface formalism first sketched in [48] for the T 6 /(Z 2 × Z 2 ) orbifold with discrete torsion and extended to its T 6 /(Z 2 × Z 2 × ΩR) and T 6 /(Z 2 × Z 6 × ΩR) orientifold versions with underlying isotropic square [49,50] or hexagonal [51,52,50] two-tori, respectively, we focus here on the so far most fertile patch in the Type IIA orientifold landscape with rigid D6-branes [2,1,53,54], the T 6 /(Z 2 × Z 6 × ΩR) orientifold with discrete torsion and one rectangular and two hexagonal underlying two-tori. In this case, the Z (1) 2 -twisted sector conceptually differs from the Z (2) 2and Z (3) 2 -twisted sectors, necessitating separate discussions for the respective deformations and making the deformations of this toroidal orbifold more intricate than the other previously discussed orbifolds with discrete torsion.…”

“…This article is organised as follows: in section 2, we briefly review the hypersurface formulation for describing local deformations of T 6 /(Z 2 × Z 2 × ΩR) singularities as discussed in [48,49,51,52] and then go on to discuss additional constraints imposed by the extra Z 3 symmetry of the T 6 /(Z 2 × Z 6 × ΩR) orientifold. Special attention will be devoted to the sLag cycles used for particle physics model building.…”