Flows on Quaternionic-K�hler and Very Special Real Manifolds (original) (raw)
2003, Communications in Mathematical Physics
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.
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