On the supergravity formulation of mirror symmetry in generalized Calabi–Yau manifolds

We derive the complete supergravity description of the N=2N=2 scalar potential which realizes a generic flux-compactification on a Calabi–Yau manifold (generalized geometry). The effective potential Veff=V(∂ZV=0)Veff=V(∂ZV=0), obtained by integrating out the massive axionic fields of the special quaternionic manifold, is manifestly mirror symmetric, i.e. invariant with respect to Sp(2h2+2)×Sp(2h1+2)Sp(2h2+2)×Sp(2h1+2) and their exchange, being h1h1, h2h2 the complex dimensions of the underlying special geometries. VeffVeff has a manifestly N=1N=1 form in terms of a mirror symmetric superpotential W proposed, some time ago, by Berglund and Mayr.