爱因斯坦-AdS空间的重整化体积与全息Rényi熵

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"这篇文章探讨了四维和六维渐近抗德西斯特（AAdS）爱因斯坦流形的重新归一化体积与重新归一化的欧几里得体积重力作用之间的等价关系。研究扩展至更高偶数维度，并提出了一种计算方法。此外，文章还涉及全息Rényi熵的计算，特别是在考虑有限张力的宇宙麸（codimension-2）的影响下。" 在量子重力和弦理论中，爱因斯坦-AdS（Anti-de Sitter）空间的行为是一个关键的概念，因为它与共形场论（CFT）之间存在着全息对应关系，即著名的AdS/CFT对应。这个对应原理表明，一个高维AdS空间的引力理论可以被低维边界上的无粒子CFT完全描述。在这种背景下，理解和量化AdS空间中的几何属性对于理解对应的CFT的性质至关重要。本研究中提到的重新归一化体积是处理量子引力中无限大的背景量时的一个重要技术。由于引力理论在量子尺度上是无穷大的，因此需要通过添加拓扑项或使用其他手段进行重整化，以便得到有意义的物理结果。在这里，作者展示了一个单个拓扑项如何实现这种重整化，并且这个重整化后的体积与引力作用之间存在等价关系。此外，Rényi熵是一种用于量化量子系统复杂性的工具，特别是对于CFT。在全息设置中，Rényi熵可以通过在AdS空间中计算特定几何对象（如宇宙麸的面积）来获得。通过复制品技巧，将时空切片成多个副本，可以构造出一个具有圆锥形奇异性的几何体，其引力作用对应于Rényi熵。作者指出，当考虑这些宇宙麸的有限张力时，Rényi熵会包含几何形状的反作用效应。文章中提出的等价性和公式对于计算奇数维全息CFT的Rényi熵尤其有用，因为它们考虑了宇宙麸的归一化面积和复制品指数的关系。这为理解和计算复杂系统的信息理论特性提供了新的视角和数学工具。总结来说，这项工作深化了我们对AdS/CFT对应的理解，特别是关于几何量的重整化及其与边界CFT中的熵量之间的联系。它不仅提供了新的计算方法，也揭示了在量子引力背景下，几何属性与信息理论量之间的微妙关系。这对于进一步探索黑洞物理、量子信息和宇宙学等领域具有重要意义。

JHEP08(2018)136

renormalized ERE is computed in the same way as in eq. (1.6). Also, a renormalized

version of the total action can be derived by evaluating the renormalized Einstein-AdS

action on the conically singular bulk manifold that considers the brane as a singular source

in the Riemann curvature. Then, the contribution due to the cosmic brane can be rewritten

as a renormalized version of the NG action.

The organization of the paper is as follows: in section 2, we explicitate the relation

between renormalized volume and renormalized Einstein-AdS action, making contact with

the mathematical literature in the four [4] and six [5] dimensional cases. We also give

our conjecture for the general relation in the 2n−dimensional case, relating it to Albin’s

prescription [7]. In section 3, we exhibit the emergence of the renormalized total action, in

agreement with Dong’s cosmic brane prescription [35], from evaluating the I

ren

action on

the conically singular manifold considering the brane as a source. We also comment that the

inclusion of the renormalized NG action for the brane can be considered as a one-parameter

family of deformations in the deﬁnition of the renormalized bulk volume. In section 4, we

obtain the renormalized modular entropy

ren

and the renormalized ERE S

ren

starting

from the renormalized total action, and we compare the resulting modular entropy with

the existing literature for renormalized areas of minimal surfaces [6]. In section 5, we

consider the computation of S

ren

for the particular case of a ball-like entangling region

at the CFT

2n−1

, and we check that the usual result for S

ren

is recovered in the m → 1

limit. Finally, in section 6, we comment on the physical applications of our conjectured

renormalized volume formula, on our topological procedure for computing renormalized

EREs and on future generalizations thereof.

2 Renormalized Einstein-AdS action is renormalized volume

The standard EH action, when evaluated on an AAdS Einstein manifold, is proportional

to the volume of the manifold, which is divergent. We propose that for 2n−dimensional

spacetimes, the renormalized Einstein-AdS action I

ren

is also proportional to the corre-

sponding renormalized volume of the bulk manifold. To motivate this conjecture, we ﬁrst

introduce I

ren

, and we then compare it with known formulas for the renormalized volume

of AAdS Einstein manifolds in four and six-dimensions. Finally, we give a concrete formula

for the renormalized volume in the general 2n−dimensional case, and we comment on its

properties.

There are diﬀerent but equivalent prescriptions for renormalizing the action. For ex-

ample, the standard Holographic Renormalization scheme [14–20] and the Kounterterms

procedure [23–27]. The equivalence between the two renormalization schemes, for the case

of Einstein-AdS gravity, was explicitly discussed in refs. [26, 27], so using either one or

the other is a matter of convenience. However, as discussed in the introduction, we con-

sider the Kounterterms-renormalized action as it can be readily compared with the existing

renormalized volume formulas.

We consider the 2n− dimensional Einstein-AdS action I

ren

as derived using the Koun-

terterms prescription [23], which is given by

ren

] =

16πG

√

G (R − 2Λ) +

16πG

∂M

2n−1

, (2.1)

– 4 –

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