黑洞的幂零轨道与Tits Satake普适类
Black Hole Nilpotent Orbits and Tits Satake Universality Classes
摘要：
本文研究了基于齐次对称特殊几何的N = 2超引力模型的类时降维中得到的伪四元数陪集流形U/H* 的幂零轨道的分类问题。在D=3的方法中，这种分类相当于对超引力的正则和奇异极值黑洞解的分类。我们表明，这种轨道的模式是一个普遍的属性，只取决于所考虑的模型的Tits-Satake普适性类，这样的类的数量是5。利用U的Tits Satake子代数U_{TS}的Weyl群W及其子群W_H的本质性质，给出了一种新的分类和构造普适类幂零轨道的算法.证明了全群U的轨道关于稳定子群H* 分裂为子轨道的过程受离散陪集W/W_H的结构控制。对于普适类SO（4，5）/[SO（2，3）x SO（2，2）]的情形，我们导出了包含37个元素的幂零轨道的完整列表。我们还展示了如何普遍的轨道定期嵌入类的所有成员是无限的数量。作为检验，我们将新算法应用于Tits Satake类G_（2，2）/[SL（2）x SL（2）]，证实了以前得到的包含7个幂零轨道的结果。展望未来的发展所获得的结果的基础上进行了概述。
英文摘要：
In this paper we consider the problem of classification of nilpotent orbits for the pseudo-quaternionic coset manifolds U/H* obtained in the time-like dimensional reduction of N = 2 supergravity models based on homogeneous symmetric special geometries. Within the D=3 approach this classification amounts to a classification of regular and singular extremal black hole solutions of supergravity. We show that the pattern of such orbits is a universal property depending only on the Tits-Satake universality class of the considered model, the number of such classes being five. We present a new algorithm for the classification and construction of the nilpotent orbits for each universality class which is based on an essential use of the Weyl group W of the Tits Satake subalgebra U_{TS} of U and on a certain subgroup W_H thereof. The splitting of orbits of the full group Uinto suborbits with respect to the stability subgroup H* is shown to be governed by the structure of the discrete coset W/W_H. For the case of the universality class SO(4,5) /[SO(2,3) x SO(2,2)] we derive the complete list of nilpotent orbits which happens to contain 37 elements. We also show how the universal orbits are regularly embedded in all the members of the class that are infinite in number. As a matter of check we apply our new algorithm also to the Tits Satake class G_(2,2)/[SL(2)x SL(2)] confirming the previously obtained result encompassing 7 nilpotent orbits. Perspectives for future developments based on the obtained results are outlined.
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