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

TL;DR

The article analyzes type II NS⊗NS backgrounds with a non-compact M^6 factor, focusing on conifolds (singular, resolved, deformed) and the NS5$_{S^2}$ solution. It provides explicit complex structures, Kähler forms, and parallel (3,0)-forms, clarifying the supersymmetry content (8 SUSY for conifolds, 4 for NS5$_{S^2}$) and demonstrating a universal one-dimensional radial action that unifies D3-brane on conifold solutions and NS5$_{S^2}$; this action admits a superpotential, yielding first-order BPS equations. For conifolds, the work yields explicit Calabi–Yau metrics across all three variants via 1-parameter families constrained by closed holomorphic 3-forms. The NS5$_{S^2}$ background is shown to possess a carefully constructed complex structure and a corresponding $\nabla^+$-parallel (3,0)-form, confirming its supersymmetric nature, with a detailed interpolation framework to connect wrapped 5-branes and 3-brane on conifold solutions.

Abstract

We investigate solutions of type II supergravity which have the product R^4 x M^6 structure with non-compact M^6 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 hep-th/0008001. In all of these cases, we explicitly construct the complex structures, and the Kaehler and parallel (3,0) forms of the corresponding M^6. In addition, we verify that the above solutions preserve, respectively, eight and four supersymmetries of type II theory. We also demonstrate that the ordinary and fractional D3-brane 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 and 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.