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Supersymmetric Kundt four manifolds and their spinorial evolution flows

Ángel Murcia, C. S. Shahbazi

Abstract

We investigate the differential geometry and topology of four-dimensional Lorentzian manifolds $(M,g)$ equipped with a real Killing spinor $\varepsilon$, where $\varepsilon$ is defined as a section of a bundle of irreducible real Clifford modules satisfying the Killing spinor equation with non-zero real constant. Such triples $(M,g,\varepsilon)$ are precisely the supersymmetric configurations of minimal four-dimensional supergravity and necessarily belong to the class Kundt of space-times, hence we refer to them as supersymmetric Kundt configurations. We characterize a class of Lorentzian metrics on $\mathbb{R}^2\times X$, where $X$ is a two-dimensional oriented manifold, to which every supersymmetric Kundt configuration is locally isometric, proving that $X$ must be an elementary hyperbolic Riemann surface when equipped with the natural induced metric. This yields a class of space-times that vastly generalize the Siklos class of space-times describing gravitational waves in AdS$_4$. Furthermore, we study the Cauchy problem posed by a real Killing spinor and we prove that the corresponding evolution problem is equivalent to a system of differential flow equations, the real Killing spinorial flow equations, for a family of functions and coframes on any Cauchy hypersurface $Σ\subset M$. Using this formulation, we prove that the evolution flow defined by a real Killing spinor preserves the Hamiltonian and momentum constraints of the Einstein equation with negative curvature and is therefore compatible with the latter. Moreover, we explicitly construct all left-invariant evolution flows defined by a Killing spinor on a simply connected three-dimensional Lie group, classifying along the way all solutions to the corresponding constraint equations, some of which also satisfy the constraint equations associated to the Einstein condition.

Supersymmetric Kundt four manifolds and their spinorial evolution flows

Abstract

We investigate the differential geometry and topology of four-dimensional Lorentzian manifolds equipped with a real Killing spinor , where is defined as a section of a bundle of irreducible real Clifford modules satisfying the Killing spinor equation with non-zero real constant. Such triples are precisely the supersymmetric configurations of minimal four-dimensional supergravity and necessarily belong to the class Kundt of space-times, hence we refer to them as supersymmetric Kundt configurations. We characterize a class of Lorentzian metrics on , where is a two-dimensional oriented manifold, to which every supersymmetric Kundt configuration is locally isometric, proving that must be an elementary hyperbolic Riemann surface when equipped with the natural induced metric. This yields a class of space-times that vastly generalize the Siklos class of space-times describing gravitational waves in AdS. Furthermore, we study the Cauchy problem posed by a real Killing spinor and we prove that the corresponding evolution problem is equivalent to a system of differential flow equations, the real Killing spinorial flow equations, for a family of functions and coframes on any Cauchy hypersurface . Using this formulation, we prove that the evolution flow defined by a real Killing spinor preserves the Hamiltonian and momentum constraints of the Einstein equation with negative curvature and is therefore compatible with the latter. Moreover, we explicitly construct all left-invariant evolution flows defined by a Killing spinor on a simply connected three-dimensional Lie group, classifying along the way all solutions to the corresponding constraint equations, some of which also satisfy the constraint equations associated to the Einstein condition.
Paper Structure (14 sections, 37 theorems, 193 equations)

This paper contains 14 sections, 37 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. Four-dimensional supersymmetric Kundt configurations
  3. General theory
  4. Optical invariants of supersymmetric Kundt configurations
  5. Standard supersymmetric Kundt configurations
  6. Real Killing spinorial flows on three-dimensional Cauchy hypersurfaces
  7. The constraint equations
  8. Compatibility with the Einstein condition
  9. Left-invariant real Killing spinorial flows and initial data
  10. Left-invariant Killing Cauchy pairs
  11. The integrability conditions of a left-invariant Killing spinorial flow
  12. Case $\mathrm{E}(1,1)$
  13. Case $\tau_2 \oplus \mathbb{R}$
  14. Case $\tau_{3,\mu}$

Key Result

Theorem 1.1

A triple $(\mathbb{R}^2\times X,g, \varepsilon)$ is a standard supersymmetric Kundt configuration if and only if there exist a family $\left\{\omega_{x_u}\right\}_{x_u\in \mathbb{R}}$ of closed one-forms on $X$ in terms of which the metric $g$ reads: where $\left\{ \beta_{x_u}\right\}_{x_u\in \mathbb{R}}$ is a family of one-forms on $X$ satisfying the following equation: In particular, for every

Theorems & Definitions (80)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • Remark 2.5
  • Lemma 2.6
  • Definition 2.7
  • ...and 70 more