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arXiv 1506.09017hep-thgr-qc
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Killing矢量存在时Einstein-Maxwell-（A）dS引力的可积性

On the integrability of Einstein-Maxwell-(A)dS gravity in presence of Killing vectors

Dietmar Klemm, Masato Nozawa, Marco Rabbiosi

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24 pages, uses jheppub.sty. v2: Final version to be published in CQG
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摘要：

研究了在Killing矢量存在下具有非零宇宙学常数的四维Einstein-Maxwell引力的对称性和可积性。首先，我们考虑定常时空，在类时Kaluza-Klein约化和两个向量场的对偶化之后，得到一个与引力耦合的三维非线性西格玛模型，其目标空间是具有SU（2，1）等距群的$\mathbb{C}\text{P}^2$的非紧版本。结果表明，由四维空间中的宇宙学常数引起的标量势破坏了八个SU（2，1）对称性中的三个，对应于广义Ehlers变换和两个Harrison变换。这使得一维海森堡群和平移群$\mathbb{R}^2$的半直积成为剩余对称。我们证明了，在附加的假设下，三维流形是共形的产品空间$\mathbb{R}\timesΣ$，和所有的领域只依赖于沿$\mathbb{R}$的坐标，运动方程是可积的。这将Leigh等人在arXiv：1403.6511中的结果推广到也存在电磁场的情况。在论文的第二部分中，我们考虑纯引力时空允许第二个与类时矢量交换的基灵矢量。我们写下由此产生的二维作用量并讨论其对称性。如果场只依赖于两个坐标中的一个，那么运动方程又是可积的，而解就是克拉辛斯基多年前构造的解。

英文摘要：

We study some symmetry and integrability properties of four-dimensional Einstein-Maxwell gravity with nonvanishing cosmological constant in the presence of Killing vectors. First of all, we consider stationary spacetimes, which lead, after a timelike Kaluza-Klein reduction followed by a dualization of the two vector fields, to a three-dimensional nonlinear sigma model coupled to gravity, whose target space is a noncompact version of $\mathbb{C}\text{P}^2$ with SU(2,1) isometry group. It is shown that the potential for the scalars, that arises from the cosmological constant in four dimensions, breaks three of the eight SU(2,1) symmetries, corresponding to the generalized Ehlers and the two Harrison transformations. This leaves a semidirect product of a one-dimensional Heisenberg group and a translation group $\mathbb{R}^2$ as residual symmetry. We show that, under the additional assumptions that the three-dimensional manifold is conformal to a product space $\mathbb{R}\timesΣ$, and all fields depend only on the coordinate along $\mathbb{R}$, the equations of motion are integrable. This generalizes the results of Leigh et al. in arXiv:1403.6511 to the case where also electromagnetic fields are present. In the second part of the paper we consider the purely gravitational spacetime admitting a second Killing vector that commutes with the timelike one. We write down the resulting two-dimensional action and discuss its symmetries. If the fields depend only on one of the two coordinates, the equations of motion are again integrable, and the solution turns out to be one constructed by Krasinski many years ago.

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