洛伦兹AdS2/CFT1对偶中的曲率探测：运动空间方法与保形块对应

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本文探讨了洛伦兹AdS$_2$/CFT$_1$对应中的曲率探测。在这个框架下，作者利用运动空间方法，从整个体积点的重建中建立了一种全新的对应关系。AdS$_2$是阿斯图里亚斯-德西特空间（Anti-de Sitter space）的一种特殊形式，它是广义相对论中的一个关键概念，用于研究量子引力和弦理论中的宇宙学模型。CFT$_1$则是1维 conforme field theory的缩写，它是拓扑量子场论的一种，常被用于描述强相互作用下的量子系统。在研究中，作者将OPE块（Operator Product Expansion Block，一种在量子场论中用于描述两个算符靠近时的相互作用的行为的构建）与CFT$_1$中的保形块（Weyl-invariant blocks）进行了公式化。这些块是理论中的基本对象，它们在解析连续性和对称性方面扮演着核心角色。通过对非相互作用标量场理论的分析，作者揭示了散装传播子（Bulk Propagator）与CFT世界中的保形块之间的精确映射，这有助于理解空间-时间结构的量子效应。当考虑引力中的基本度量——应力张量时，这一对应关系能够探测到AdS$_2$空间的曲率变化。这种探测能力源自于CFT$_1$的物理性质，它能捕捉到背景几何的微小扰动。作者借鉴了从二维Dilaton引力理论到Schwarzian理论的推导过程，展示了如何通过重新参数化来提取整个时空的渐近边界信息。最后，作者通过一系列的一致性检查，成功地找到了AdS$_2$的黎曼曲率张量，这是对广义相对论中的曲率度量的直接体现。这项工作不仅深化了我们对AdS/CFT对偶的理解，而且提供了从CFT侧计算引力物理现象的新途径，对于理解量子引力、黑洞信息悖论以及宇宙学等问题具有重要的理论价值。这一研究为未来在更高维度的AdS/CFT框架下探索复杂时空结构的可能性开辟了新的视角。

Physics Letters B 798 (2019) 134936

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

The probe of curvature in the Lorentzian AdS

/CFT

correspondence

Xing Huang

a,b

, Chen-Te Ma

c,d,e,∗

Institute of Modern Physics, Northwest University, Xi’an 710069, China

Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi’an 710069, China

Institute of Quantum Matter, School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou 510006, Guangdong, China

The Laboratory for Quantum Gravity and Strings, Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag, Rondebosch

7700, South Africa

Department of Physics and Center for Theoretical Sciences, National Taiwan University, Taipei 10617, Taiwan, ROC

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

Article history:

Received

11 July 2019

Received

in revised form 25 August 2019

Accepted

10 September 2019

Available

online 12 September 2019

Editor:

N. Lambert

Keywords:

Curvature

Lorentzian

AdS

/CFT

correspondence

OPE

block

Conformal

block

We establish the Lorentzian AdS

/CFT

correspondence from a reconstruction of all bulk points through

the kinematic-space approach. The OPE block is exactly a bulk local operator. We formulate the

correspondence between the bulk propagator in the non-interacting scalar ﬁeld theory and the conformal

block in CFT

. When we consider the stress tensor, the variation probes the variation of AdS

metric.

The reparameterization provides the asymptotic boundary of the bulk spacetime as in the derivation of

the Schwarzian theory from two-dimensional dilaton gravity theory. Finally, we ﬁnd the AdS

Riemann

curvature tensor based on the above consistent check.

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

1. Introduction

The holographic correspondence states that the physical de-

grees

of freedom in quantum gravity theory has an equivalent

description from the boundary. Nowadays, the only perturbatively

computable quantum gravity theory without any known incon-

sistency

is string theory. String theory lives on two-dimensional

worldsheet, and the ﬂuctuation of target space without conformal

anomaly provides a theory containing Einstein gravity theory [1].

Based on the implication of string theory, the four-dimensional

conformal ﬁeld theory (CFT

), N = 4 supersymmetric Yang-Mills

theory is dual to string theory on the ﬁve-dimensional anti-de Sit-

ter

spacetime times a ﬁve-dimensional sphere manifold (AdS

) [2]. This motivates the more generic and testable holographic-

conjecture,

AdS/CFT correspondence.

The

quantum ﬂuctuation in Einstein gravity theory has the

known issue in the renormalizability. Therefore, the ultraviolet

(UV) information cannot be probed so far. Through the AdS/CFT

correspondence, the AdS quantum gravity theory can be deﬁned

by CFT. Since CFT is scale-invariant, the UV divergence is not prob-

Corresponding author at: Institute of Quantum Matter, School of Physics

and Telecommunication Engineering, South China Normal University, Guangzhou

510006, Guangdong, China.

E-mail

addresses: xingavatar@gmail.com (X. Huang), yefgst@gmail.com (C.-T. Ma).

lematic. Therefore, AdS/CFT correspondence or CFT becomes an

important direction in the study of quantum gravity theory. Be-

cause

the techniques of CFT are widely used in condensed matter

physics, the correspondence is also useful in the application of

strongly correlated condensed matter system [3].

probe the bulk physics from the boundary theory, the un-

avoidable

study is the operator dictionary in the AdS/CFT corre-

spondence.

To reach the goal, we should ﬁrst consider the pairs

of operator insertions or bi-local operators. We can use the op-

erator

product expansion (OPE) to organize the CFT bi-local op-

erators.

Through the geodesic Witten diagrams, people obtained

the conformal block for the spacelike-separated points [4]. For

the time-like separated points, the OPE block corresponds to the

codimension-two surface operator [5,6], which is localized on the

bulk codimension-two minimum surface on a given time slice that

serves as the boundary of casual wedge speciﬁed by two points.

Each conformal block for spacelike-separated points is equivalent

to the correlator of two OPE-blocks [5,6]. The conformal kinematics

of a scalar OPE block can be written as the Klein-Gordon equation

in the kinematic space. Here we continue to use the term kine-

matic

space to refer to the spacetime speciﬁed by a pair of points.

The OPE block of the stress tensor also exactly corresponds to the

modular Hamiltonian [7]for the natural study of holographic en-

tanglement

entropy, which obeys the Liouville and Toda equations

in the gravity theory and its spin-3 extension [6].

https://doi.org/10.1016/j.physletb.2019.134936

0370-2693/

SCOAP

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