Mirror supersymmetry in the space of (super) extended Poincaré algebras p(p,q)

We classify extended Poincaré Lie superalgebras and Lie algebras of any signature (p, q), i.e. Lie superalgebras and ₂-graded Lie algebras g = g₀ + g₁, where g₀ = s₀(V) + V is the (generalized) Poincaré Lie algebra of the pseudo Euclidean vector space V = ^{p, q} of signature (p, q) and g₁ is a spin 1/2 s₀(V)-module extended to a s₀-module with kernel V.As a result of the classification, we obtain, if g₁ = S is the spinor module, the numbers L ⁺(n, s) (resp. L ^–(n, s)) of independent such Lie super algebras (resp. Lie algebras), which are periodic functions of the dimension n=p+q (mod 8) and the signature s=p–q (mod 8) and satisfy: L ⁺(–n, s)=L ^– (n, s).Supported by Max-Planck-Institut für Mathematik (Bonn).Supported by the Alexander von Humboldt Foundation, MSRI (Berkeley) and SFB 256 (Bonn University).     

Reflection groups on Riemannian manifolds

We investigate discrete groups G of isometries of a complete connected Riemannian manifold M which are generated by reflections, in particular those generated by disecting reflections. We show that these are Coxeter groups, and that the orbit space M/G is isometric to a Weyl chamber C which is a Riemannian manifold with corners and certain angle conditions along intersections of faces. We can also reconstruct the manifold and its action from the Riemannian chamber and its equipment of isotropy group data along the faces. We also discuss these results from the point of view of Riemannian orbifolds. Mathematics Subject Classification Primary 51F15, 53C20, 20F55, 22E40     

On the geometry of spaces of oriented geodesics

Let M be either a simply connected pseudo-Riemannian space of constant curvature or a rank one Riemannian symmetric space, and consider the space L(M) of oriented geodesics of M. The space L(M) is a smooth homogeneous manifold and in this paper we describe all invariant symplectic structures, (para)complex structures, pseudo-Riemannian metrics and (para)Kähler structure on L(M).     

Conification of Kähler and Hyper-Kähler Manifolds

Given a Kähler manifold M endowed with a Hamiltonian Killing vector field Z, we construct a conical Kähler manifold ${\hat{M}}$ such that M is recovered as a Kähler quotient of ${\hat{M}}$ . Similarly, given a hyper-Kähler manifold (M, g, J ₁, J ₂, J ₃) endowed with a Killing vector field Z, Hamiltonian with respect to the Kähler form of J ₁ and satisfying ${\mathcal{L}_ZJ_2 = -2J_3}$ , we construct a hyper-Kähler cone ${\hat{M}}$ such that M is a certain hyper-Kähler quotient of ${\hat{M}}$ . In this way, we recover a theorem by Haydys. Our work is motivated by the problem of relating the supergravity c-map to the rigid c-map. We show that any hyper-Kähler manifold in the image of the c-map admits a Killing vector field with the above properties. Therefore, it gives rise to a hyper-Kähler cone, which in turn defines a quaternionic Kähler manifold. Our results for the signature of the metric and the sign of the scalar curvature are consistent with what we know about the supergravity c-map.     

Monge-Ampère Equations on (Para-)K?hler Manifolds: from Characteristic Subspaces to Special Lagrangian Submanifolds

We present the basic notions and results of the geometric theory of second order PDEs in the framework of contact and symplectic manifolds including characteristics, formal integrability, existence and uniqueness of formal solutions of non-characteristic Cauchy problems. Then, we focus our attention to Monge-Ampère equations (MAEs) and discuss a natural class of MAEs arising in K?hler and para-K?hler geometry whose solutions are special Lagrangian submanifolds.     

Quaternionic and para-quaternionic <Emphasis Type="Italic">CR</Emphasis> structure on (4n+3)-dimensional manifolds

We define notion of a quaternionic and para-quaternionic CR structure on a (4n+3)-dimensional manifold M as a triple (ω₁,ω₂,ω₃) of 1-forms such that the corresponding 2-forms satisfy some algebraic relations. We associate with such a structure an Einstein metric on M and establish relations between quaternionic CR structures, contact pseudo-metric 3-structures and pseudo-Sasakian 3-structures. Homogeneous examples of (para)-quaternionic CR manifolds are given and a reduction construction of non homogeneous (para)-quaternionic CR manifolds is described.     

Compatible complex structures on almost quaternionic manifolds

On an almost quaternionic manifold we study the integrability of almost complex structures which are compatible with the almost quaternionic structure . If , we prove that the existence of two compatible complex structures forces to be quaternionic. If , that is is an oriented conformal 4-manifold, we prove a maximum principle for the angle function of two compatible complex structures and deduce an application to anti-self-dual manifolds. By considering the special class of Oproiu connections we prove the existence of a well defined almost complex structure on the twistor space of an almost quaternionic manifold and show that is a complex structure if and only if is quaternionic. This is a natural generalization of the Penrose twistor constructions.

     

Flows on Quaternionic-Kähler and Very Special Real Manifolds

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 n0. Such gradient flows are generated by the ``energy function' f=P<sup>2</sup>, 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.     

Choosing roots of polynomials smoothly

We clarify the question whether for a smooth curve of polynomials one can choose the roots smoothly and related questions. Applications to perturbation theory of operators are given.