We investigate solutions of type II supergravity which have the product R 1,3 × 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 hepth/0008001. In all of these cases, we explicitly construct the complex structures, and the Kähler 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 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.

Kundt spacetimes are of great importance in general relativity in four dimensions and have a number of physical applications in higher dimensions in the context of string theory. The degenerate Kundt spacetimes have many special and unique mathematical properties, including their invariant curvature structure and their holonomy structure. We provide a rigorous geometrical kinematical definition of the general Kundt spacetime in four dimensions; essentially a Kundt spacetime is defined as one admitting a null vector that is geodesic, expansion-free, shear-free and twist-free. A Kundt spacetime is said to be degenerate if the preferred kinematic and curvature null frames are all aligned. The degenerate Kundt spacetimes are the only spacetimes in four dimensions that are not I-non-degenerate, so that they are not determined by their scalar polynomial curvature invariants. We first discuss the nonaligned Kundt spacetimes, and then turn our attention to the degenerate Kundt spacetimes. The degenerate Kundt spacetimes are classified algebraically by the Riemann tensor and its covariant derivatives in the aligned kinematic frame; as an example, we classify Riemann type D degenerate Kundt spacetimes in which ∇(Riem), ∇ (2) (Riem) are also of type D. We discuss other local characteristics of the degenerate Kundt spacetimes. Finally, we discuss degenerate Kundt spacetimes in higher dimensions.

We find that the structure constants 4-form of a metric 3-Lie algebra is the sum of the volume forms of orthogonal 4-planes proving a conjecture in math/0211170. In particular, there is no metric 3-Lie algebra associated to a u(N ) Lie algebra for N > 2. We examine the implication of this result on the existence of a multiple M2-brane theory based on metric 3-Lie algebras.

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We show that the supersymmetric near horizon geometry of heterotic black holes is either an AdS 3 fibration over a 7-dimensional manifold which admits a G 2 structure compatible with a connection with skew-symmetric torsion, or it is a product R 1,1 × S 8 , where S 8 is a holonomy Spin(7) manifold, preserving 2 and 1 supersymmetries respectively. Moreover, we demonstrate that the AdS 3 class of heterotic horizons can preserve 4, 6 and 8 supersymmetries provided that the geometry of the base space is further restricted. Similarly R 1,1 × S 8 horizons with extended supersymmetry are products of R 1,1 with special holonomy manifolds. We have also found that the heterotic horizons with 8 supersymmetries are locally isometric to AdS 3 × S 3 × T 4 , AdS 3 × S 3 × K 3 or R 1,1 × T 4 × K 3 , where the radii of AdS 3 and S 3 are equal and the dilaton is constant.