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Supersymmetric Kundt four manifolds and their spinorial evolution flows

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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 nonzero real constant. Such triples \((M,g,\varepsilon )\) are precisely the supersymmetric configurations of minimal AdS 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 \(\Sigma \subset M\). 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.

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Notes

  1. Observe that the associated real Killing spinor \(\varepsilon \) can be completely reconstructed from \(\left\{ \beta _t, \mathfrak {e}^t\right\} _{t\in \mathcal {I}}\), although its explicit expression is not of relevance for us.

  2. The notation \(\tau _2 \oplus \mathbb {R}\) and \(\tau _{3,\mu }\) is adopted from [34].

  3. The third column of the Table should be understood as the additional condition that a Killing Cauchy pair has to satisfy in order to be constrained Einstein.

  4. We shall denote \(\left\{ \beta _t \right\} _{t\in \mathcal {I}}\) equivalently as \(\left\{ \beta ^t \right\} _{t\in \mathcal {I}}\) to unify notation along the section.

  5. Note that \( {}_2F_1 \left( \frac{1}{2}, \frac{1}{6}; \frac{7}{6};0 \right) =\frac{ \sqrt{\pi } \, \Gamma \left( 7/6\right) }{\Gamma \left( 2/3 \right) }\).

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Acknowledgements

We would like to thank Bernardo Araneda, Diego Conti, Antonio F. Costa and Romeo Segnan Dalmasso for useful discussions, Calin Lazaroiu for reading a preliminary version of the draft and proposing several improvements and Patrick Meessen for various insightful remarks and pointers. The work of Á.M. was funded by the Spanish FPU Grant No. FPU17/04964 and by the Istituto Nazionale di Fisica Nucleare (INFN), through the INFN Call No. 23590. Á.M. also received additional support from the MCIU/AEI/FEDER UE grant PID2021-125700NB-C21. CSS’s research was funded by the María Zambrano Excellency Program 468806966 of the Kingdom of Spain.

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  1. Istituto Nazionale di Fisica Nucleare, Sezione di Padova, Padua, Italy

    Ángel Murcia

  2. Departamento de Matemáticas, UNED, Madrid, Spain

    C. S. Shahbazi

  3. Fachbereich Mathematik, Universität Hamburg, Hamburg, Germany

    C. S. Shahbazi

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  1. Ángel Murcia
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Correspondence to Ángel Murcia.

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Murcia, Á., Shahbazi, C.S. Supersymmetric Kundt four manifolds and their spinorial evolution flows. Lett Math Phys 113, 106 (2023). https://doi.org/10.1007/s11005-023-01728-1

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