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arXiv 0903.2777hep-th
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对称超引力理论中黑洞解的完全积分

The full integration of black hole solutions to symmetric supergravity theories

W. Chemissany, J. Rosseel, M. Trigiante, T. Van Riet

备注：
27 pages, typos corrected, references added, 1 figure added, Discussion on black holes and the generating geodesic significantly extended. Statement about the relation between the D=3 geodesics from BPS and non-BPS extreme black holes made explicit by defining the Wick rotation mapping the corresponding orbits
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摘要：

我们证明了所有的静态和球对称的黑洞解的理论与对称的目标空间是可积的，我们提供了一个显式的积分方法。这种精确积分是基于描述的黑洞解作为测地线曲线上的模空间的理论时，减少了时间的方向。这些测地线运动方程可以重写为一个特定的拉克斯对方程，数学家们已经提供了积分算法时，初始条件描述的对角化拉克斯矩阵。另一方面，由幂零Lax矩阵描述的解，它起源于极端的规则（小）D = 4黑洞，可以作为在可对角化的情况下获得的解的适当限制而获得，正如我们在对应于规则（和小）D = 4黑洞的生成测地线（即D = 3模型的最一般的测地线模全球对称性）上所示。作为我们分析的副产品，我们给出了显式形式的威克旋转连接的轨道的BPS和非BPS的解决方案，在最大超对称超引力及其STU截断。

英文摘要：

We prove that all stationary and spherical symmetric black hole solutions to theories with symmetric target spaces are integrable and we provide an explicit integration method. This exact integration is based on the description of black hole solutions as geodesic curves on the moduli space of the theory when reduced over the time-like direction. These geodesic equations of motion can be rewritten as a specific Lax pair equation for which mathematicians have provided the integration algorithms when the initial conditions are described by a diagonalizable Lax matrix. On the other hand, solutions described by nilpotent Lax matrices, which originate from extremal regular (small) D = 4 black holes can be obtained as suitable limits of solutions obtained in the diagonalizable case, as we show on the generating geodesic (i.e. most general geodesic modulo global symmetries of the D = 3 model) corresponding to regular (and small) D = 4 black holes. As a byproduct of our analysis we give the explicit form of the Wick rotation connecting the orbits of BPS and non-BPS solutions in maximally supersymmetric supergravity and its STU truncation.

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