Fake Superpotential for Large and Small Extremal Black Holes
L. Andrianopoli, R. D'Auria, S. Ferrara, M. Trigiante
备注：
30 pages, LaTeX source. Discussion on the radial evolution of the scalar fields, in relation to the symmetries of the W-function, extended. Table 1 added. Typos corrected
We consider the fist order, gradient-flow, description of the scalar fields coupled to spherically symmetric, asymptotically flat black holes in extended supergravities. Using the identification of the fake superpotential with Hamilton's characteristic function we clarify some of its general properties, showing in particular (besides reviewing the issue of its duality invariance) that W has the properties of a Liapunov's function, which implies that its extrema (associated with the horizon of extremal black holes) are asymptotically stable equilibrium points of the corresponding first order dynamical system (in the sense of Liapunov). Moreover, we show that the fake superpotential W has, along the entire radial flow, the same flat directions which exist at the attractor point. This allows to study properties of the ADM mass also for small black holes where in fact W has no critical points at finite distance in moduli space. In particular the W function for small non-BPS black holes can always be computed analytically, unlike for the large black-hole case.
将伪超引力方法应用于5维渐近AdS时空，该时空包含耦合到实标量场和阿贝尔规范场的引力。我们的动机是获得具有R x S^3对称性的体解，以便探索当边界规范理论在R x S^3上时的AdS/CFT对应。得到了费米子场中通过线性序在局域超对称下不变性的伪超引力作用量。规范场使事情比以前的应用更严格的假超引力，允许相当普遍的标量势。这里的超势必须采用W（λ）~ exp（-k λ）+ c exp（2 λ/（3 k））的形式，唯一的自由是常数k的选择。伪超引力的费米子变换规则导致伪Killing旋量方程。从他们的可积性条件，我们得到一阶微分方程，我们解决了解析找到奇异带电的拉格朗日场方程的解决方案。可以加上史瓦西质量项，以产生一个遮蔽奇点的视界。包括“超级明星”在内的解决方案在文献中是已知的。我们计算它们的全息参数。
We develop an effective superpotential formalism for the SU(2)$\times$U(1) invariant sector of $\mathcal{N}=2$ gauged supergravity in five dimensions with a U(1)$^3$ Fayet-Iliopoulos gauging, and determine the exact superpotential that describes all 1/4 BPS solutions in this sector. This includes the Gutowski-Reall black holes, but also a much broader class of solutions with a squashed $S^3$, magnetic flux and vector multiplet sources, as well as complex Euclidean BPS saddles. Some of these solutions are known only numerically, but the exact superpotential allows us to analytically evaluate th…
We consider N=2 supergravity in four dimensions, coupled to an arbitrary number of vector- and hypermultiplets, where abelian isometries of the quaternionic hyperscalar target manifold are gauged. Using a static and spherically or hyperbolically symmetric ansatz for the fields, a one-dimensional effective action is derived whose variation yields all the equations of motion. By imposing a sort of Dirac charge quantization condition, one can express the complete scalar potential in terms of a superpotential and write the action as a sum of squares. This leads to first-order flow equations, that …
