论文标题
超对称昆特四个流形及其旋转进化流量
Supersymmetric Kundt four manifolds and their spinorial evolution flows
论文作者
论文摘要
我们研究了配备了真正的杀伤旋转纺纱子$ \ varepsilon $的四维Lorentzian歧管（M，G）$的差异几何形状和拓扑，其中$ \ varepsilon $被定义为满足杀戮的Spinor平等的不可杀死的Spenoreequeration的一部分。这样的三元组$（m，g，\ varepsilon）$恰恰是最小的四维超级实力的超对称配置，因此必须属于空间时间的kundt级，因此我们将其称为超对称的kundt配置。我们在$ \ mathbb {r}^2 \ times x $上表征了一类Lorentzian指标，其中$ x $是一种面向二维的歧管，每种超对称的kundt配置都是本地等值的，证明$ x $必须是基本的高度riemann表面，当具有自然含量时。这产生了一类空间时间，可以极大地概括了Siklos类的空间时间，描述了广告中的重力波$ _4 $。此外，我们研究了一个真正的杀伤性旋转器所构成的凯奇问题，并证明相应的进化问题等于差分流程方程的系统，真正的杀伤旋转流动方程，用于任何cauchy Hypersurface $σ\ subset $σ\ subset M $。使用这种公式，我们证明由实际杀伤旋转器定义的进化流可保留爱因斯坦方程的Hamiltonian和动量约束，其曲率为负曲率，因此与后者兼容。此外，我们明确构建了由简单连接的三维谎言组杀死旋转器定义的所有左右进化流，并在所有解决方案沿所有解决方案中分类到相应的约束方程，其中一些也满足与爱因斯坦条件相关的约束方程。
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.
