Branes with fluxes wrapped on spheres

Abstract

Following an eight-dimensional gauged supergravity approach we construct the most general solution describing D6-branes wrapped on a Kähler four-cycle taken to be the product of two spheres of different radii. Our solution interpolates between a Calabi–Yau four-fold and the spaces S^(2) × S^(2) × S^(2) × IR^(2) or S^(2) × S^(2) × IR^(4) , depending on generic choices for the parameters. Then we turn on a background four-form field strength, corresponding to D2-branes, and show explicitly how our solution is deformed. For a particular choice of parameters it represents a flow from a Calabi–Yau four-fold times the three-dimensional Minkowski space-time in the ultraviolet, to the space-time AdS_(4) × Q^(1,1,1) in the infrared. In general, the solution in the infrared has a singularity which within type-IIA supergravity corresponds to the near horizon geometry of the solution for the D2-D6 system. Finally, we uncover the relation with work done in the eighties on Freund–Rubin type compactifications.