On the Geometry of N=2 Minkowski Vacua of Gauged N=2 Supergravity Theories in Four Dimensions

Gauging isometries of four-dimensional N=2 supergravity theories yields an N=2 supersymmetric theory with a scalar potential. In this note, we study the well-known constraints for four-dimensional N=2 Minkowski vacua of such theories. We propose that classically a projective special K\"ahler submanifold of the projective K\"ahler target space of the ungauged theory describes the moduli space of the complex scalar fields of massless vector multiplets for N=2 Minkowski vacua configurations, which then receives quantum corrections from integrating out massive fields. Subloci of projective special K\"ahler manifolds appear as supersymmetric flux vacua in the context of type IIB Calabi-Yau threefold compactifications with background fluxes as well. While these flux vacua equations arise from the critical locus of an N=1 superpotential, we show that these equations can also be obtained from the N=2 supersymmetric Minkowski vacuum equations of gauged N=2 supergravity theories upon gauging suitable isometries in the semi-classical universal hypermultiplet sector of type IIB string Calabi-Yau threefold compactifications. Thus, we give an intrinsic N=2 supersymmetric interpretation to the flux vacua equations.