Flows on quaternionic-Kaehler and very special real manifolds

Abstract

BPS solutions of 5-dimensional supergravity correspond to certain gradient flows on the product M x N of a quaternionic-Kaehler 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-Kaehler manifolds. For the homogeneous quaternionic-Kaehler 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 in 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-Kaehler 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-Kaehler manifolds we find degenerate local minima.

22 pages

References in corpus (5)

• Hypermultiplets, domain walls and supersymmetric attractors
• On Fermion Masses, Gradient Flows and Potential in Supersymmetric Theories
• Curved BPS domain wall solutions in five-dimensional gauged supergravity
• Domain Walls, Black Holes, and Supersymmetric Quantum Mechanics
• Gaugings and other supergravity tools of p-brane physics

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