Testing string vacua in the lab: from a hidden CMB to dark forces in flux compactifications

We perform a detailed analysis of the phenomenological properties of hidden Abelian gauge bosons with a kinetic mixing with the ordinary photon within type IIB flux compactications. We study the interplay between moduli stabilisation and the Green-Schwarz mechanism that gives mass to the hidden photon paying particular attention to the role of D-terms. We present two generic classes of explicit Calabi-Yau examples with an isotropic and an anisotropic shape of the extra dimensions showing how the last case turns out to be very promising to make contact with current experiments. In fact, anisotropic compactications lead naturally to a GeV-scale hidden photon ("dark forces" that can be searched for in beam dump experiments) for an intermediate string scale; or, allowing for some fine-tuning, even to an meV-scale hidden photon (which could lead to a "hidden CMB" and can be tested by light-shining-through-a-wall experiments) in the case of TeV-scale strings.