Branes with fluxes wrapped on spheres

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 × S × S × IR or S × S × IR, 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 AdS4 × Q 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. Branes wrapped on supersymmetric cycles provide a natural path to obtain gravity duals of field theories with low supersymmetry. These field theories are twisted since preserving some supersymmetry after wrapping the brane, requires the identification (expressed better, the relation) of the spin connection on the cycle and some external R-symmetry gauge fields [1]. Therefore, the dual supergravity solutions can be naturally constructed in an appropriate gauged supergravity, and are eventually lifted to ten or eleven dimensions. This approach was started in [2], and has been further developed for a wide variety of branes wrapped on diverse supersymmetric cycles [3]-[21]. The case of D6-branes is of special interest because they lift to pure geometry in eleven dimensions. This fact allows to argue how compactifications of M-theory on manifolds with reduced holonomy arise as the local eleven dimensional description of D6-branes wrapped on supersymmetric cycles in manifolds of lower dimension with a different holonomy group [22]. This extends the work of [23], where D6-branes wrapping the three-cycle in the deformed conifold were shown to be described in eleven dimensions as a compactification on a seven manifold with G2 holonomy. These lifts to eleven dimensions for D6-branes wrapping various cycles were explicitly constructed using eight dimensional gauged supergravity [24] in [7, 11, 13, 19]. However these purely gravitational geometries are deformed when background fluxes are included. In [13] M-theory on a Calabi-Yau four-fold was shown to arise as the eleven dimensional description of D6-branes wrapped on Kähler four-cycles inside Calabi-Yau three-folds. The deformation of this background by a four-form field strength along the unwrapped coordinates was recently considered in [21], where it was shown to induce a flow from E2,1 × CY4 at ultraviolet to AdS4 ×Q1,1,1 in the infrared limit. The four-cycle in [13, 21] was taken to be a product of two two-spheres of the same radius so that the metric was Einstein. In this paper we will eliminate the Einstein condition on the four-cycle allowing the spheres to have different radii and will also introduce a four-form flux. When lifted to eleven dimensions, and in the absence of flux, our solution will represent M-theory on a Calabi–Yau four-fold. We will find a three parameter family of metrics in which the conical singularity of the four-fold is generically resolved by being replaced by a bolt or nut singularity which is removable [25]. Then we turn on a background four-form field strength, corresponding to D2-branes, which provides another mechanism for resolving the singularity. After determining the most general supersymmetry preserving solution we discuss its behavior for various choices for the parameters. A special choice