Geometry of Killing spinors in neutral signature. (24th September 2015)

• Record Type: Journal Article
• Authors: Klemm, Dietmar; Nozawa, Masato
• Is Part Of: Classical and quantum gravity. Volume 32:Number 18(2015:Sep.)
• Page Start: 421
• Page End: 94
• Publication Date: 2015-09-24
• Subjects: Quantum gravity -- Periodicals; Gravitation -- Periodicals; Relativity (Physics) -- Periodicals; Space and time -- Periodicals; Periodicals; 521.1
• DOI: 10.1088/0264-9381/32/18/185012
• Languages: English
• ISSNs: 0264-9381

Abstract

<abstract> <title>Abstract</title> <p>We classify the supersymmetric solutions of minimal <italic>N</italic> = 2 gauged supergravity in four dimensions with neutral signature. They are distinguished according to the sign of the cosmological constant and whether the vector field constructed as a bilinear of the Killing spinor is null or non-null. In neutral signature the bilinear vector field can be spacelike, which is a new feature not arising in Lorentzian signature. In the <inline-formula><tex-math><?CDATA ${\rm{\Lambda }}\lt 0$?></tex-math><?MML <mml:math> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:mo><</mml:mo> <!--MPSinvTimes--> <mml:mn>0</mml:mn> </mml:math>?><inline-graphic xlink:href="ark:/27927/pgj2j4sr36t" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /></inline-formula> non-null case, the canonical form of the metric is described by a fibration over a three-dimensional base space that has <inline-formula><tex-math><?CDATA ${\rm{U}}(1)$?></tex-math><?MML <mml:math> <mml:mi mathvariant="normal">U</mml:mi> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math>?><inline-graphic xlink:href="ark:/27927/pgj2j4sr335" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /></inline-formula> holonomy with torsion. We find that a generalized monopole equation determines the twist of the bilinear Killing field, which is reminiscent of an Einstein–Weyl structure. If, moreover, the electromagnetic field strength is self-dual, one gets the Kleinian signature analogue of the Przanowski–Tod class of metrics, namely a pseudo-hermitian spacetime determined by solutions of the continuous Toda equation, conformal to a scalar-flat pseudo-Kähler manifold, and admitting in addition a charged conformal Killing spinor. In the <inline-formula><tex-math><?CDATA ${\rm{\Lambda }}\lt 0$?></tex-math><?MML <mml:math> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:mo><</mml:mo> <!--MPSinvTimes--> <mml:mn>0</mml:mn> </mml:math>?><inline-graphic xlink:href="ark:/27927/pgj2j4sr4f6" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /></inline-formula> null case, the supersymmetric solutions define an integrable null Kähler structure. In the <inline-formula><tex-math><?CDATA ${\rm{\Lambda }}\gt 0$?></tex-math><?MML <mml:math> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:mo>></mml:mo> <!--MPSinvTimes--> <mml:mn>0</mml:mn> </mml:math>?><inline-graphic xlink:href="ark:/27927/pgj2j4sr4dn" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /></inline-formula> non-null case, the manifold is a fibration over a Lorentzian Gauduchon–Tod base space. Finally, in the <inline-formula><tex-math><?CDATA ${\rm{\Lambda }}\gt 0$?></tex-math><?MML <mml:math> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:mo>></mml:mo> <!--MPSinvTimes--> <mml:mn>0</mml:mn> </mml:math>?><inline-graphic xlink:href="ark:/27927/pgj2j4sr4c3" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /></inline-formula> null class, the metric is contained in the Kundt family, and it turns out that the holonomy is reduced to <inline-formula><tex-math><?CDATA $\mathrm{Sim}(1)\times \mathrm{Sim}(1)$?></tex-math><?MML <mml:math> <mml:mi>Sim</mml:mi> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> <mml:mo>×</mml:mo> <!--MPSinvTimes--> <mml:mi>Sim</mml:mi> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math>?><inline-graphic xlink:href="ark:/27927/pgj2j4sr4bj" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /></inline-formula>. There appear no self-dual solutions in the null class for either sign of the cosmological constant. </p> </abstract>