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Flows on quaternionic-Kaehler and very special real manifolds

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

TL;DR

The paper analyzes gradient flows on $M\times N$ arising from BPS solutions in 5D supergravity, where $M$ is a quaternionic-Kähler manifold of negative scalar curvature and $N$ is a very special real manifold. It derives a detailed Hessian framework for the energy $f=P^2$, where $P$ is the dressed moment map for gauged isometries, and decomposes the Hessian into $\frak{sp}(1)$ and $\frak{sp}(r)$ contributions to study the spectrum at critical points. For homogeneous quaternionic-Kähler manifolds, it proves the existence of Killing fields with split-signature Hessians, and shows non-degenerate extrema for symmetric cases while non-symmetric spaces can yield degenerate minima; the full dressed Hessian extends the analysis to the $N$-sector and yields a block-structure matrix ${\cal U}$ governing UV/IR attractor behavior. These results illuminate the structure of supersymmetric domain-wall solutions and potential AdS/CFT applications, with broader relevance to other dimensions where quaternionic-Kähler geometry appears.

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.

Flows on quaternionic-Kaehler and very special real manifolds

TL;DR

The paper analyzes gradient flows on arising from BPS solutions in 5D supergravity, where is a quaternionic-Kähler manifold of negative scalar curvature and is a very special real manifold. It derives a detailed Hessian framework for the energy , where is the dressed moment map for gauged isometries, and decomposes the Hessian into and contributions to study the spectrum at critical points. For homogeneous quaternionic-Kähler manifolds, it proves the existence of Killing fields with split-signature Hessians, and shows non-degenerate extrema for symmetric cases while non-symmetric spaces can yield degenerate minima; the full dressed Hessian extends the analysis to the -sector and yields a block-structure matrix governing UV/IR attractor behavior. These results illuminate the structure of supersymmetric domain-wall solutions and potential AdS/CFT applications, with broader relevance to other dimensions where quaternionic-Kähler geometry appears.

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.

Paper Structure

This paper contains 8 sections, 20 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Quaternionic-Kähler moment map
  3. The Hessian of the energy at a critical point
  4. Very special real manifolds
  5. The dressed moment map
  6. The Hessian of the energy
  7. Conclusions
  8. Remarks on the notation

Key Result

Theorem 1

The curvature tensor $R$ of a quaternionic-Kähler manifold of dimension $4r$ is of the form where $R_0$ is the curvature tensor of the quaternionic projective space, $\nu = \frac{scal}{4r(r+2)}$ is the reduced scalar curvature and ${\cal W}$ is an algebraic curvature tensor of type $\hbox{sp} (r)$ (the "Weyl curvature").

Theorems & Definitions (20)

  • Theorem 1
  • Proposition 1
  • Lemma 1
  • Proposition 2
  • Corollary 1
  • Lemma 2
  • Theorem 2
  • Theorem 3
  • Corollary 2
  • Theorem 4
  • ...and 10 more