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arXiv · hep-th/0109094

Flows on quaternionic-Kaehler and very special real manifolds

Dmitri V. Alekseevsky·Vicente Cortés·Chandrashekar Devchand·Antoine Van Proeyen
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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.

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Dmitri V. Alekseevsky, Vicente Cortés, Chandrashekar Devchand, Antoine Van Proeyen. 2001-09-12. Flows on quaternionic-Kaehler and very special real manifolds. https://doi.org/10.1007/s00220-003-0850-4

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