We consider type II string compactifications on Calabi-Yau orientifolds with fluxes and D-branes, and analyse the F-term scalar potential that simultaneously involves closed and open string modes. In type IIA models with D6-branes this potential can be directly computed by integrating out Minkowski three-forms. The result shows a multibranched structure along the space of lifted open string moduli, in which discrete shifts in special Lagrangian and Wilson line deformations are compensated by changes in the RR flux quanta. The same sort of discrete shift symmetries are present in the superpotential and constrain the Kähler potential. As for the latter, inclusion of open string moduli breaks the factorisation between complex structure and Kähler moduli spaces. Nevertheless, the 4d Kähler metrics display a set of interesting relations that allow to rederive the scalar potential analytically. Similar results hold for type IIB flux compactifications with D7-brane Wilson lines.

We compute the full classical 4d scalar potential of type IIA Calabi-Yau orientifolds in the presence of fluxes and D6-branes. We show that it can be written as a bilinear form V = Z AB ρ A ρ B , where the ρ A are in one-to-one correspondence with the 4-form fluxes of the 4d effective theory. The ρ A only depend on the internal fluxes, the axions and the topological data of the compactification, and are fully determined by the Freed-Witten anomalies of branes that appear as 4d string defects. The quadratic form Z AB only depends on the saxionic partners of these axions. In general, the ρ A can be seen as the basic invariants under the discrete shift symmetries of the 4d effective theory, and therefore the building blocks of any flux-dependent quantity.All these polynomials may be obtained by derivation from one of them, associated to a universal 4-form. The standard N = 1 supergravity flux superpotential is uniquely determined from this master polynomial, and vice versa. arXiv:1802.05771v2 [hep-th] 3 Sep 2018