Rigid D6-branes on $ {{{{T^6}}} \left/ {{\left( {{Z_2} \times {Z_{2M}} \times \Omega \mathcal{R}} \right)}} \right.} $ with discrete torsion

Abstract

We give a complete classification of T 6 /(Z 2 × Z 2M × ΩR) 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 × Z 6 and Z 2 × Z 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 × Z 4 background is ruled out by simple arguments.arXiv:1010.6070v3 [hep-th]

RR photons

“…Simplest models include toroidal orbifold compactifications, see e.g., [75][76][77][78][79][80][81]. In this section we consider a type IIA orientifold of the…”

Classification of three-generation models on magnetized orbifolds

“…3 However recently, general structure of these geometries was declared in an exact and analytical way in a "dual" description with operator formalism [51]. In that paper, the authors treated most general cases on magnetized T 2 /Z 2,3,4,6 with nonzero (discretized) Wilson line phases and/or Scherk-Schwarz phases, which were discussed in [52] (see also [53,54,43,[55][56][57][58][59][60][61]). Note that shifted orbifold can be considered on a magnetized torus [62].…”

From Type II string theory toward BSM/dark sector physics

“…In practice, the three types of stringy consistency conditions are straightforwardly computed for factorisable torus backgrounds, T 6 = (T 2 ) 3 , and orbifolds thereof, (T 2 ) 3 /Γ (with Γ = Z N or Z N × Z M with(out) discrete torsion), on which a full classification of three-cycles with typical Betti number b 3 ∼ O(10 − 50) is possible and for which Conformal Field Theory (CFT) methods can be invoked [42][43][44][45] to not only cross-check the RR tadpole cancellation conditions in terms of vacuum amplitudes, but most importantly to determine the set of probe D-branes with U Sp(2) gauge groups for the K-theory constraints. 16,28,[46][47][48] Generalizing these results to so-called non-factorizable tori, e.g. T 6 = (T 3 ) 2 or T 3 × T 1 × T 2 , or orbifolds thereof is possible whenever the sLag cycles can be rewritten in terms of a factorized geometry, as was recently noticed [49][50][51] when extending the first CFT computations 52,53 to intersecting generic D6-branes with chiral spectra on non-factorizable T 6 /(Z 4 ×ΩR) backgrounds.…”

“…The a priori six different choices of orientations of the (T 2 ) 3 lattice under the antiholomorphic involution R can be reduced to two, 16 i.e. T 2 (1) is either of rectangular or of tilted shape, and due to the discrete torsion phase one O6-plane orbit ΩR(Z (i) 2 ) has to be of exotic type, 46,73 with two inequivalent choices, ΩR or ΩRZ (3) 2 , consistent with SUSY D6-branes on the so-called aAA-lattice orientation.…”

“…The computations and resulting non-renormalization are, however, only valid for bulk D6-branes, due to the prefactors trγ Z2 = 0 of the annulus amplitudes used there. All string vacua with chiral matter on fractional 2, 6, 7, 49, 60, [97][98][99][100][101][102][103] or rigid 13,16,18,46,70,73,104,109 D6-branes violate this condition, making the extension of these computations to non-trivial twisted annulus contributions necessary. The second generalization of the above CFT methods consists in considering nonperturbative effects from D2-brane instantons wrapped along compact three-cycles.…”