Complex geometry of conifolds and 5-brane wrapped on 2-sphere

We investigate solutions of type II supergravity which have the product R × M6 structure with non-compact M6 factor and which preserve at least four supersymmetries. In particular, we consider various conifolds and the N = 1 supersymmetric “NS5-brane wrapped on 2-sphere” solution recently discussed in hepth/0008001. In all of these cases, we explicitly construct the complex structures, and the Kähler and parallel (3,0) forms of the corresponding M6. In addition, we verify that the above solutions preserve respectively four and eight supersymmetries of the underlying type II theory. We also demonstrate that the ordinary and fractional D3-brane (5-brane wrapped on 2-cycle) solutions on singular, resolved and deformed conifolds, and the (S-dual of) NS5-brane wrapped on 2-sphere can be obtained as special cases from a universal ansatz for the supergravity fields, i.e. from a single 1-d action governing their radial evolution. We show that like the 3-branes on conifolds, the NS5-brane on 2-sphere background can be found as a solution of first order system following from a superpotential. Also at Imperial College, London and Lebedev Institute, Moscow.