Toroidal compactification and symmetry breaking in open-string theories

Discrete deformations in Type I vacua

“…After an S transformation to the transverse tree-channel, these lead to new tadpoles whose cancellation requires (different types of) D7-branes intersecting Chan-Paton groups of reduced rank. While for the standard Ω projection this phenomenon may be related to the presence of a quantized background for the NS-NS antisymmetric tensor (both in toroidal [1] and in orbifold [3] compactifications), a similar understanding is missing for the ΩR orientifolds of [4]. Finding an appropriate description of the observed rank reduction is the main motivation of the present letter.…”

“…survive the projection whereas, up to the identification B ′ ≡ B ′ + Z, the antisymmetric tensor is frozen to the two possible values B ′ = 0 and B ′ = 1 2 [1]. A T-duality along the x 1 direction leads to the orientifold…”

“…Since the early work of Bianchi, Pradisi and Sagnotti [1], it is known that Type I string vacua involve both continuous and discrete moduli. Thus, in the case of ordinary orientifolds based on the world-sheet parity Ω [2], discrete internal backgrounds for the NS-NS two-form, B ab , are still allowed, although the corresponding (continuous) deformations are projected out.…”

Type I strings with F- and B-flux

“…In this paper we continue to elaborate on type I strings with magnetic flux and resolve the first subtlety by introducing discrete NSNS two-form flux [17], as well. Via T-duality this corresponds to a tilt of the dual torus.…”

On gauge couplings and thresholds in Type I Gepner models and otherwise

“…Although their systematization was already achieved in the early 90's [8]- [16], including the possibilities of minimally coupling R-R p-form potentials and reducing the rank of the Chan-Paton group by turning on a quantized NS-NS antisymmetric tensor background [15,16], the geometric description in terms of D-branes and Ω-planes [20,21], pioneered in [22,23] has definitely consecrated this framework as the most promising one to embed Particle Physics in String Theory. Simple instances of chiral model based on toroidal orbifolds [24]- [29] with or without intersecting branes [30]- [33], that are T-dual to magnetized branes [34]- [38], represent a useful guidance for more sophisticated and hopefully realistic constructions that may require inter alia (non) commuting open string Wilson lines or their closed string dual constructions [16]- [19].…”