ω形变规范N=8超重力中的奇异黑洞解决方案

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"Blackholesinω-deformedgaugedN=8supergravity" 本文探讨了4维ω形变规范N = 8超重力中的黑洞解。N = 8超重力是理论物理学中的一个关键领域，它涉及具有八重超对称性的引力理论，这种理论在量子场论和弦理论中扮演着重要角色。ω形变是一种特殊的理论修正，它通过引入一个新的参数ω来改变理论的基本结构，从而可能揭示新的物理现象。在本文中，作者Andrés Anabalón、Dumitru Astefanesei等人受到新发现的4维ω形变规范超重力的启发，研究了这个理论在单标量场一致截短条件下的黑洞解决方案。一致截短意味着只考虑部分场的子集，这使得复杂理论的分析变得更为可行。他们构建了具有不同拓扑结构的黑洞解，包括球形、环形（即托尔型）和双曲线层状结构。这些拓扑多样性反映了宇宙中可能存在的不同类型的黑洞几何。标量场在除了曲率奇点之外的所有区域都保持规则，这意味着它没有无定义的行为，这是保证理论物理意义的重要条件。此外，他们发现应力-能量张量满足零能量条件，这是保证这些解在经典引力理论中稳定性的必要条件。有趣的是，当ω参数不为零时，黑洞解的频谱显示出某种简并性。这意味着在特定的边界条件下，可能存在多个解，这些边界条件对应于双场理论中的多轨迹变形。在Anti-de Sitter (AdS)空间的背景下，这些简并性可能与AdS/CFT对应原理有关，这是一个强大的工具，将重力理论与量子场论联系起来。文章指出，保留AdS对称性的边界条件在ω不等于零时导致了简并性，这暗示了在相应的共形场论中可能有丰富的物理现象。这为研究超对称量子场论的性质以及它们与强耦合重力系统的相互作用提供了新的视角。最后，文章经过多次修改后于2014年3月被接受发表，并在M. Cvetiˇc编辑的《Physics Letters B》上发布，这是一本高度认可的物理学期刊，展示了该研究的学术价值和影响力。这篇研究深入探讨了N = 8超重力在ω形变背景下的黑洞解，揭示了新的拓扑结构和潜在的物理特性，对于理解和探索超对称引力理论的复杂性以及其在量子场论中的应用具有重要意义。

Physics Letters B 732 (2014) 137–141

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Black holes in ω-deformed gauged N = 8 supergravity

Andrés Anabalón

a,b,∗

, Dumitru Astefanesei

Departamento de Ciencias, Facultad de Artes Liberales y Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Av. Padre Hurtado 750, Viña delMar,Chile

Université de Lyon, Laboratoire de Physique, UMR 5672, CNRS, École Normale Supérieure de Lyon, 46 allé d’Italie, F-69364 Lyon Cedex 07, France

Instituto de Física, Pontiﬁcia Universidad Católica de Valparaíso, Casilla 4059, Valparaíso, Chile

article info abstract

Article history:

Received 24 January 2014

Received in revised form 9 March 2014

Accepted 18 March 2014

Available online 21 March 2014

Editor: M. Cveti

Motivated by the recently found 4-dimensional ω-deformed gauged supergravity, we investigate the

black hole solutions within the single scalar ﬁeld consistent truncations of this theory. We construct

black hole solutions that have spherical, toroidal, and hyperbolic horizon topologies. The scalar ﬁeld

is regular everywhere outside the curvature singularity and the stress–energy tensor satisﬁes the null

energy condition. When the parameter

ω does not vanish, there is a degeneracy in the spectrum of black

hole solutions for boundary conditions that preserve the asymptotic Anti-de Sitter symmetries. These

boundary conditions correspond to multi-trace deformations in the dual ﬁeld theory.

(http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP

1. Introduction

Supergravity is an effective theory for string theory at low en-

ergies compared to the string energy scale. The Anti-de Sitter/Con-

formal Field Theory (AdS/CFT) duality [1] provides a remarkable

tool for extracting information about strongly coupled gauge the-

ories (in d dimensions) from a dual supergravity description (in

+ 1 dimensions). Within the AdS/CFT duality, the radial coordi-

nate plays the role of the energy scale and so the bulk geometry

has a nice interpretation as a renormalization group (RG) ﬂow of

the dual ﬁeld theory [2] (see, also, the review [3]).

The gauged N

= 8 supergravity in four dimensions is the maxi-

mal gauged supergravity with spins lower or equal than two and it

can be obtained by Kaluza–Klein reduction of 11-dimensional su-

pergravity [4] on a 7-dimensional sphere [5]. In the holographic

context, much of the interest on the 4-dimensional gauged super-

gravities comes from the utility of the ABJM model [6] in testing

various strongly coupled phenomena in condensed matter physics.

What came as a surprise, recently, is the existence of a con-

tinuous

one-parameter family of inequivalent maximally super-

symmetric gauged supergravities [7] (the critical points were ex-

tensively studied in [8]). These theories are characterized by one

‘angular parameter’,

ω, that generates an electric–magnetic dual-

ity transformation prior to selecting the SO

(8) gauging, and they

are referred to as

ω-deformed gauged N = 8 supergravity. At the

practical level, the

ω parameter can be introduced in the maximal

Corresponding author.

E-mail addresses: andr

es.anabalon@uai.cl (A. Anabalón),

dumitru.astefanesei@ucv.cl (D. Astefanesei).

gauged supergravity by multiplying the 56-bein, V, by a diagonal

matrix containing either e

iω

or e

−iω

[9]. As the moduli potential is

a non-linear function in terms of

V ,theparameterω appears ex-

plicitly in the potential. It is worth noting that, long time ago, this

kind of ‘angles’ was introduced in N

= 4 gauged supergravities in

[10].

In this Letter, we construct e

xact domain wall (see, also, [11,

12]) and black hole solutions in one scalar ﬁeld consistent trunca-

tions of

ω-deformed theories [11], which correspond to RG ﬂows

of 3-dimensional dual quantum ﬁeld theories at zero and ﬁnite

temperatures, respectively. We show that the null energy condi-

tion (which is the relevant energy condition in AdS) is satisﬁed

and construct the c-function using the gravity side of the duality.

For

ω = 0, we obtain domain wall solutions and explicitly write

down the corresponding superpotential. When

ω does not vanish,

there is a degeneracy in the spectrum of black hole solutions. Since

the black holes are interpreted as thermal states in the dual the-

ory, this degeneracy may seem puzzling at ﬁrst sight. However,

this can be understood because the solutions correspond to dif-

ferent boundary conditions which in turn correspond to different

deformations of the dual ﬁeld theory. The degeneracy in the spec-

trum of solutions can then be understood as a sign degeneracy in

the AdS invariant boundary conditions due to a change in the sign

of the scalar ﬁeld. It is also interesting to note that the no-hair

conjecture in asymptotically AdS spacetimes of [14] have as a hy-

pothesis the reality of the superpotential. So, it does not apply to

the

ω-deformed theory, as the superpotential is generically com-

plex [15].

Since the coordinates in which the black hole solutions are

cons

tructed are not very intuitive, we will explicitly provide the

http://dx.doi.org/10.1016/j.physletb.2014.03.035

0370-2693/

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ω形变规范N=8超重力中的奇异黑洞解决方案

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利用 MATLAB 编程设计一个数字带通滤波器， 指标要求如下：通带边缘频率：ωp₁=0.45π, ωP₂=0.65π ,通带：4p≤1dB。阻带边缘频率：ωs₁=0.3π,ωs₂=0.8π,阻带：As≥40dB。

以DTFT定义式 X(e^jω )=∑_(n=-∞)^∞▒〖x(n)e^(-jωn) 〗 为例，定义式中求和符号以及求和项中复指数项e^(-jωn)具有什么意义？为什么傅里叶变换能够求得信号x(n)的频谱？

例: 假设在某地区切片细胞中正常(ω1)和异常(ω2)两类的先验概率分别为P(ω1)=0.9，P(ω2)=0.1。现有一待识别细胞呈现出状态x，由其类条件概率密度分布曲线查得p(x|ω1)=0.2，p(x|ω2)=0.4，试对细胞x进行分类。

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利用 MATLAB 编程设计一个数字带通滤波器，指标要求如下：通带边缘频率：ωp₁=0.45π, ωP₂=0.65π ,通带：4p≤1dB。阻带边缘频率：ωs₁=0.3π,ωs₂=0.8π,阻带：As≥40dB。

以DTFT定义式 X(e^jω )=∑_(n=-∞)^∞▒〖x(n)e^(-jωn) 〗为例，定义式中求和符号以及求和项中复指数项e^(-jωn)具有什么意义？为什么傅里叶变换能够求得信号x(n)的频谱？