Abstract
Using an off-shell Killing spinor analysis we perform a systematic investigation of the supersymmetric background and black hole solutions of the ${\cal N}=(1,1)$ Cosmological New Massive Gravity model. The solutions with a null Killing vector are the same pp-wave solutions that one finds in the ${\cal N}=1$ model but we find new solutions with a time-like Killing vector that are absent in the ${\cal N}=1$ case. An example of such a solution is a Lifshitz spacetime. We also consider the supersymmetry properties of the so-called rotating hairy BTZ black holes and logarithmic black holes in an $AdS_3$ background. Furthermore, we show that under certain assumptions there is no supersymmetric Lifshitz black hole solution.
Highlights
Background configurations of both N = 1 TMG and N = 1 NMG are severely restricted due to the spinor structure of the N = 1 supersymmetry, which allows only planar-wave type solutions with a null Killing vector as well as maximally supersymmetric AdS3 and Minkowski backgrounds [11, 12]
As we will show in this paper, the merit of the N = (1, 1) theory is that the spinors of the theory are Dirac instead of Majorana spinors, which allows a larger variety of supersymmetric background solutions than in the N = 1 case [15, 16]
The main aim of this paper is to study the supersymmetric backgrounds as well as black hole solutions of the N = (1, 1) cosmological NMG, or shortly N = 1 cosmological New Massive Gravity (CNMG), theory [13] using the off-shell Killing spinor analysis
Summary
We first consider the case that the function f introduced in eq (2.9) is zero everywhere, i.e. f = 0. This implies that Kμ is a null Killing vector. Contracting this equation with Kμ we find that (3.2). Choose coordinates (u, v, x) such that v is an affine parameter along these geodesics, i.e. By virtue of our choice for Kμ the metric components further simplify to guv = P (u, x), gvv = gxv = 0 ,. Where P = P (u, x) since we demand the null direction to be along the v direction All these choices yield a metric of the following generic form ds2 = hij(x, u) dxi dxj + 2P (x, u) du dv ,.
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