By Pratt S.
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In addition to the topological coupling, the N = 1 supersymmetric action yields a Fayet–Iliopoulos (FI) term of the form: ξa 1 = 2 2 ga (4π α )3 T6 Fa ∧ Fa ∧ Fa − Fa ∧ J ∧ J . 11) The D-flatness condition in the absence of charged scalars requires then that Da = ξa = 0, which is equivalent to Eq. 10). 12) in terms of the magnetic fields Hi along the internal planes defined in Section 7, or equivalently in terms of the angles of D6-branes with respect to the orientifold axis. 3. 13) which can also be understood from a 4d viewpoint as the positivity of the U (1)a gauge coupling ga2 .
1) where gs is the string coupling. It follows that the type I string scale can be chosen hierarchically smaller than the Planck mass [5, 7] at the expense of introducing extra large transverse dimensions felt only by gravity, while keeping the string coupling small [5]. The weakness of 4d gravity compared to gauge interactions (ratio MW /MP ) is then attributed to the largeness of the transverse space R⊥ compared to the string length ls = Ms−1 . An important property of these models is that gravity becomes effectively (4 + n)-dimensional with a strength comparable to those of gauge interactions at the string scale.
Keep supersymmetry by turning on VEVs for charged scalars on the extra brane stacks. 19) φ where ξa is given by Eqs. 14). The sum is extended over all φ scalars φ charged under the a-th U (1)a with charge qa . 10) is modified to: qa va2 T6 (J ∧ J ∧ J − Fa ∧ Fa ∧ J ) = Ms2 T6 (Fa ∧ J ∧ J − Fa ∧ Fa ∧ Fa ). 20) Note that this is valid for small values of va (in string units), since the inclusion of charged scalars in the D-term is in principle valid only perturbatively. Indeed, the model presented above can be implemented by two extra stacks 10 and 11 with diagonal fluxes, presented in the lower part of Table 3, so that all 9- and 5-brane RR tadpoles are cancelled [51].