No abstract

We introduce the notion of a special complex manifold: a complex manifold (M,J) with a flat torsionfree connection \nabla such that (\nabla J) is symmetric. A special symplectic manifold is then defined as a special complex manifold together with a \nabla-parallel symplectic form \omega . This generalises Freed's definition of (affine) special K\"ahler manifolds. We also define projective versions of all these geometries. Our main result is an extrinsic realisation of all simply connected (affine or projective) special complex, symplectic and K\"ahler manifolds. We prove that the above three types of special geometry are completely solvable, in the sense that they are locally defined by free holomorphic data. In fact, any special complex manifold is locally realised as the image of a holomorphic 1-form \alpha : C^n \to T^* C^n. Such a realisation induces a canonical \nabla-parallel symplectic structure on M and any special symplectic manifold is locally obtained this way. Special K\"ahler manifolds are realised as complex Lagrangian submanifolds and correspond to closed forms \alpha. Finally, we discuss the natural geometric structures on the cotangent bundle of a special symplectic manifold, which generalise the hyper-K\"ahler structure on the cotangent bundle of a special K\"ahler manifold.Comment: 24 pages, latex, section 3 revised (v2), modified Abstract and Introduction, version to appear in J. Geom. Phy

A class of Z 2 -graded Lie algebra and Lie superalgebra extensions of the pseudoorthogonal algebra of a spacetime of arbitrary dimension and signature is investigated. They have the form g = g 0 + g 1 , with g 0 = so(V ) + W 0 and g 1 = W 1 , where the algebra of generalized translations W = W 0 + W 1 is the maximal solvable ideal of g, W 0 is generated by W 1 and commutes with W . Choosing W 1 to be a spinorial so(V )-module (a sum of an arbitrary number of spinors and semispinors), we prove that W 0 consists of polyvectors, i.e. all the irreducible so(V )-submodules of W 0 are submodules of ∧V . We provide a classification of such Lie (super)algebras for all dimensions and signatures. The problem reduces to the classification of so(V )-invariant ∧ k V -valued bilinear forms on the spinor module S.

BPS solutions of 5-dimensional supergravity correspond to certain gradient flows on the product M × N of a quaternionic-Kähler manifold M of negative scalar curvature and a very special real manifold N of dimension n ≥ 0. Such gradient flows are generated by the "energy function" f = P 2 , where P is a (bundle-valued) moment map associated to n + 1 Killing vector fields on M . We calculate the Hessian of f at critical points and derive some properties of its spectrum for general quaternionic-Kähler manifolds. For the homogeneous quaternionic-Kähler manifolds we prove more specific results depending on the structure of the isotropy group. For example, we show that there always exists a Killing vector field vanishing at a point p ∈ M such that the Hessian of f at p has split signature. This generalizes results obtained recently for the complex hyperbolic plane (universal hypermultiplet) in the context of 5-dimensional supergravity. For symmetric quaternionic-Kähler manifolds we show the existence of non-degenerate local extrema of f , for appropriate Killing vector fields. On the other hand, for the non-symmetric homogeneous quaternionic-Kähler manifolds we find degenerate local minima. This work was supported by the priority programme "String Theory"of the Deutsche Forschungsgemeinschaft.

A supermanifold M is canonically associated to any pseudo Riemannian spin manifold (M 0 , g 0 ). Extending the metric g 0 to a field g of bilinear forms g(p) on T p M , p ∈ M 0 , the pseudo Riemannian supergeometry of (M, g) is formulated as G-structure on M , where G is a supergroup with even part G 0 ∼ = Spin(k, l); (k, l) the signature of (M 0 , g 0 ). Killing vector fields on (M, g) are, by definition, infinitesimal automorphisms of this G-structure. For every spinor field s there exists a corresponding odd vector field X s on M . Our main result is that X s is a Killing vector field on (M, g) if and only if s is a twistor spinor. In particular, any Killing spinor s defines a Killing vector field X s .