The type IIA flux potential, 4-forms and Freed-Witten anomalies

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 rho(A)rho(AB)(B), where the rho(A) are in one-to-one correspondence with the 4-form fluxes of the 4d effective theory. The rho(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 rho(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.