arXiv 1002.42232010-02hep-thgr-qcmath.RA

N=8超引力中积分和连续U对偶轨道的观测

Observations on Integral and Continuous U-duality Orbits in N=8 Supergravity

L. Borsten, D. Dahanayake, M. J. Duff, S. Ferrara, A. Marrani, W. Rubens

备注：: 43 pages, 8 tables; minor corrections, references added; version to appear in Class. Quantum Grav
发表信息：: Class.Quant.Grav.27:185003,2010
DOI：: 10.1088/0264-9381/27/18/185003

摘要：

人们常常想知道，当两个先验不同的极值黑p膜解实际上是U-对偶相关的。在经典超引力极限中，一大类理论的答案已经知道了一段时间。然而，在完整的量子理论中，U-对偶群被分解为一个离散子群，在这种情况下，U-对偶轨道的问题是一个微妙的问题。在目前的工作中，我们解决这个问题的背景下，N=8超引力在四维，五维和六维。本说明的目的是介绍和澄清目前对这些离散轨道的了解，同时补充一些尚未在文献中出现的细节。为此，我们利用的数学框架的整体约旦代数和Freudenthal三元系。D=6中的双子黑弦的电荷向量是SO（5，5;Z），它与一个由两个算术U-对偶不变量唯一指定的双电荷约化标准形有关。类似地，D=5中的黑洞（弦）电荷矢量是E_{6(6)}（Z），等价于一个三电荷标准型，同样由一组三个算术U-对偶不变量唯一固定。四维空间的情况就不那么清楚了：虽然保持1/8以上超对称性的黑洞可以用已知的算术E_{7(7)}（Z）不变量完全分类，但1/8-BPS和非BPS的黑洞产生越来越微妙的轨道结构，这些结构仍有待于正确理解。然而，对于非常特殊的投射黑洞子类，我们知道一个完整的分类。所有的射影黑洞都是E_{7(7)}（Z），它与一个由已知的算术U-对偶不变量集唯一确定的四电荷或五电荷标准形有关。此外，E_{7(7)}（Z）传递地作用于具有给定首阶熵的黑洞的电荷矢量。

英文摘要：

One would often like to know when two a priori distinct extremal black p-brane solutions are in fact U-duality related. In the classical supergravity limit the answer for a large class of theories has been known for some time. However, in the full quantum theory the U-duality group is broken to a discrete subgroup and the question of U-duality orbits in this case is a nuanced matter. In the present work we address this issue in the context of N=8 supergravity in four, five and six dimensions. The purpose of this note is to present and clarify what is currently known about these discrete orbits while at the same time filling in some of the details not yet appearing in the literature. To this end we exploit the mathematical framework of integral Jordan algebras and Freudenthal triple systems. The charge vector of the dyonic black string in D=6 is SO(5,5;Z) related to a two-charge reduced canonical form uniquely specified by a set of two arithmetic U-duality invariants. Similarly, the black hole (string) charge vectors in D=5 are E_{6(6)}(Z) equivalent to a three-charge canonical form, again uniquely fixed by a set of three arithmetic U-duality invariants. The situation in four dimensions is less clear: while black holes preserving more than 1/8 of the supersymmetries may be fully classified by known arithmetic E_{7(7)}(Z) invariants, 1/8-BPS and non-BPS black holes yield increasingly subtle orbit structures, which remain to be properly understood. However, for the very special subclass of projective black holes a complete classification is known. All projective black holes are E_{7(7)}(Z) related to a four or five charge canonical form determined uniquely by the set of known arithmetic U-duality invariants. Moreover, E_{7(7)}(Z) acts transitively on the charge vectors of black holes with a given leading-order entropy.

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