D=5, N=2欧几里得超重力的多中心解：保持超对称性的稳定配置

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"这篇论文探讨了欧几里得超重力理论中的多中心解，特别是在五维、N=2的超导重力框架下，如何构建保持超对称性的多中心解。研究背景源于不变的超重力理论，其中BPS（Bogomol'nyi-Prasad-Sommerfield）状态的无作用力条件是稳定静态多中心解存在的关键。Kastor和Traschen的工作开创性地展示了正宇宙常数的Einstein-Maxwell理论中，可存在描述任意数量带电黑洞的解决方案。伦敦进一步扩展了这一理论至五维及以上维度。论文主要内容包括： 1. **BPS状态和无作用力条件**：在不变的超重力理论中，BPS状态的特性使得这些系统在量子力学层面上保持稳定。无作用力条件确保了这些解在经典和量子层次上都是稳定的，因为它排除了物体之间的引力相互作用。 2. **Kastor-Traschen解**：Kastor和Traschen的贡献在于他们找到了第一组Einstein-Maxwell理论的解，这些解在正宇宙常数背景下描述了多个带电黑洞。这些解揭示了宇宙常数如何影响多黑洞系统的结构和稳定性。 3. **London的推广**：伦敦的工作将这种理解扩展到了五维以及更高的维度，这为理解多维度宇宙中的黑洞行为提供了新的视角。 4. **D=5, N=2欧几里得超导重力的多中心解**：论文的核心成果是构造了五维、N=2超导重力与任意数量矢量多重偶合的多中心解。这些解保持了超对称性，这在超重力理论中是至关重要的，因为它关系到解的稳定性。 5. **时间依赖的多中心解**：尽管超重力理论中的时间依赖多中心解尚未被完全构建，但本文的进展为未来的探索铺平了道路，尤其是在考虑动态系统时。 6. **开放获取**：该论文标记为“Open Access”，意味着公众可以免费获取和阅读，促进了科学知识的传播和共享。通过这些研究，科学家们更深入地理解了多中心黑洞系统的性质，以及超重力理论在不同维度下的表现。这些发现对于黑洞物理、弦理论和高维宇宙模型的构建都有重要意义。" 这篇论文的发布不仅增加了我们对超重力理论的理解，还为未来的研究提供了新的工具和方向，尤其是对于那些探索多中心黑洞解的物理学家来说，它提供了一个重要的理论基础。

Physics Letters B 767 (2017) 253–257

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Euclidean supergravity and multi-centered solutions

W.A. Sabra

Centre for Advanced Mathematical Sciences and Physics Department, American University of Beirut, Lebanon

a r t i c l e i n f o a b s t r a c t

Article history:

Received

21 January 2017

Accepted

27 January 2017

Available

online 2 February 2017

Editor:

M. Cveti

In ungauged supergravity theories, the no-force condition for BPS states implies the existence of

stable static multi-centered solutions. The ﬁrst solutions to Einstein–Maxwell theory with a positive

cosmological constant describing an arbitrary number of charged black holes were found by Kastor

and Traschen. Generalisations to ﬁve and higher dimensional theories were obtained by London.

Multi-centered solutions in gauged supergravity, even with time-dependence allowed, have yet to be

constructed. In this letter we construct supersymmetry-preserving multi-centered solutions for the case

of D = 5, N = 2Euclidean gauged supergravity coupled to an arbitrary number of vector multiplets.

Higher dimensional Einstein–Maxwell multi-centered solutions are also presented.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

In Newtonian gravity, one can obtain a system of point par-

ticles,

each having a charge equal to its mass, in static equilib-

rium

by balancing the mutual attractive gravitational and repul-

sive

electrostatic forces. In Einstein theory of general relativity and

quite surprisingly the analogous situation ﬁrst appeared in the

early work of Weyl, Majumdar and Papapetrou [1]. The Majumdar–

Papapetrou

(MP) solutions were found to describe a system of

multi-centered extremal Reissner–Nordström black holes in ther-

mal

and mechanical equilibrium [2]. The MP metrics are the static

limits of the Israel–Wilson–Perjés (IWP) solutions [3]. If one con-

siders

Einstein–Maxwell theory as the bosonic sector of the theory

of N = 2, D = 4supergravity, then the MP metrics turn out to be

solutions admitting half of the supersymmetry [4]. Asystematic

classiﬁcation by Tod [5] also demonstrated that the IWP metrics

are the unique solutions with time-like Killing vector admitting

supercovariantly constant spinors. Analogues of the MP solutions

were also found for black holes with a dilaton ﬁeld in [6]. In [7]

general

half-supersymmetric solutions, which can be considered as

generalisation of the MP and IWP metrics, to the theories N = 2,

D = 4 supergravity with vector multiplets were found. BPS solu-

tions

in ﬁve-dimensional Einstein–Maxwell theory were considered

in [8]. The metric in this case is of the Tanghelini form [9]. More-

over,

electric and magnetic BPS solutions breaking half of the su-

persymmetry

in ungauged ﬁve-dimensional supergravity coupled

to vector multiplets were constructed in [10].

E-mail address: ws00@aub.edu.lb.

The ﬁrst multi-centered solutions asymptotic to de Sitter space

were obtained in four dimensions in [11]. These are non-static

solutions to the Einstein–Maxwell equations in the presence of a

positive cosmological constant. They describe an arbitrary number

of charged black holes in motion due to the positive cosmologi-

cal

constant. As should be expected, these solutions reduce to MP

solutions in the limit of vanishing cosmological constant. Multi-

centered

solutions to d-dimensional Einstein–Maxwell theory with

a positive cosmological constant were given in [12]. In the ﬁve-

dimensional

case and with an imaginary coupling g (with the

cosmological constant being proportional to −g

), this theory may

be viewed as the bosonic sector of pure de Sitter D = 5, N = 2su-

pergravity.

Within this fake supergravity framework, it was shown

in [12] that the multi-centered solutions preserve some super-

symmetry

through the explicit construction of the corresponding

Killing spinors. Multi-centered solutions to D = 5, N = 2gauged

supergravity coupled to an arbitrary number of vector multiplets

were considered in [13,14]. In the fake de Sitter supergravity case,

one obtains rotating multi-centered solutions. However, in the

standard Anti-de Sitter cases, the multi-centered solutions have a

complex space–time metric.

this letter, we will be mainly concerned with the gauged ver-

sion

of the ﬁve-dimensional Euclidean supergravity theory which

was recently constructed in [15]. It will be demonstrated through

the analysis of the Killing spinor equations that this theory ad-

mits

multi-centered solutions with real space–time metric and real

ﬁelds unlike in the Lorentzian theory where the solution is com-

plex.

The results obtained may be useful as many investigations of

the AdS/CFT conjecture [16] have in fact been performed in Eu-

clidean

space.

http://dx.doi.org/10.1016/j.physletb.2017.01.067

0370-2693/

SCOAP

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