AdS3商空间中的多边界虫洞与Killing向量研究

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"这篇学术论文探讨了多边界虫洞中的Killing向量构造，特别是针对三边界虫洞的情况，这是在研究AdS/CFT对应关系和量子引力问题时的重要工具。作者Elena Caceres、Arnab Kundu、Ayan Patra和Sanjit Shashi详细阐述了构建带有和不带有旋转的三边界虫洞所需的Killing向量的显式形式，这些形式是通过AdS3的商得到的。他们确保这种方法能够涵盖此类虫洞的所有模空间，并讨论了将这种方法扩展到更复杂的多边界空间（包括高基因型）的可能性。该论文在JHEP期刊上发表，并被标记为开放访问，允许公众自由阅读和引用。" 在物理学中，虫洞是一种理论上的宇宙结构，可能连接两个不同的时空区域。多边界虫洞是虫洞的一种特殊类型，其特征在于拥有多个入口或出口。这些结构在量子引力理论的研究中占据重要地位，因为它们可以提供对AdS（Anti-de Sitter）/CFT（Conformal Field Theory）对应关系的深入理解。AdS/CFT对应是一个强大的工具，它将一个在高维AdS空间中的引力理论与低维边界上的一个规范场论等价起来。 Killing向量是流形上的向量场，其满足 Killing 方程，使得在这个流形上的物理系统具有对称性。在研究黑洞和虫洞时，Killing向量可以帮助我们识别和描述这些对象的不变性质，例如旋转或对称性。在本论文中，作者找到了构建三边界虫洞所需的Killing向量的明确形式，这对于理解这些虫洞的动态行为至关重要。此外，模空间是指一个物理系统的解空间，其中每个解都代表一个可能的配置。确保捕获虫洞的全部模空间意味着研究人员可以全面了解这些虫洞可能的几何形态和动态特性。论文中提到的推广程序到更奇特的多边界空间，意味着这一工作不仅局限于三边界虫洞，还可以应用于更复杂、更多边界的虫洞模型，这可能会揭示更多关于量子引力和AdS/CFT对应的新洞察。关键词如“AdS-CFT对应”、“Black Holes”和“Space-Time”反映了论文的核心内容，即在广义相对论和量子场论的交叉领域进行的深入研究。通过对多边界虫洞的Killing向量进行详细研究，科学家们旨在深化我们对宇宙基本结构和相互作用的理解。

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JHEP02(2020)149

section 3.1, if we restrict our attention to the t = 0 slice, the ﬁxed points will lie outside of

the fundamental domain of the quotient space. Thus, the curvature at every point in the

multiboundary wormhole is well-deﬁned.

Another possible pathology is the manifestation of closed timelike curves (CTCs). If

we consider a Killing vector ξ which is timelike (ξ · ξ < 0) at a particular point, then the

image of that point under the corresponding ﬁnite isometry is timelike-separated. Thus,

we would have a timelike curve connecting the two points, and this curve would become a

CTC upon quotienting. Furthermore, observe that the Killing horizon ξ · ξ = 0 separates

the region with CTCs from the region where ξ ·ξ > 0, so we identify the Killing horizon as

a causal singularity.

This issue of CTCs is overcome simply by excising the causally problematic regions.

Excision eliminates the causal pathologies, but it also results in geodesic incompleteness,

since, in general, there should exist geodesics which go from the ξ · ξ > 0 region into the

ξ·ξ < 0 region. However, although these geodesics are cut-oﬀ, they also all reach the Killing

horizon of ξ, i.e., they all hit the causal singularity. Thus, the fact that these geodesics are

cut-oﬀ prior to performing the quotienting does not matter.

At this point, we require a method to classify diﬀerent quotient spaces of AdS

cor-

responding to multiboundary wormholes. For this, we use the technology of Riemann

surfaces, which provides a simple way of visualizing the diﬀerent possible topologies of

these quotient spaces.

2.2 Riemann surfaces: a brief discussion

As the t = 0 slice of AdS

is simply the hyperbolic plane H (modeled by the upper half-

plane), it follows that the corresponding t = 0 slice of a multiboundary wormhole is a

quotient of H by a Fuchsian group, a term for any discrete subgroup of the isometry group

PSL(2, R) of H. In particular, all connected, hyperbolic Riemann surfaces of genus g and

with n boundaries, which we denote by (n, g), can be written as such quotients of H,

meaning that any such Riemann surface can be used to describe the t = 0 slice of some

multiboundary wormhole.

For these particular wormholes, the corresponding Riemann surface will retain some

information about both the topology and the geometry of the total spacetime manifold.

Speciﬁcally, a wormhole with n boundaries and genus g will be a Riemann surface (n, g)

at t = 0. The moduli space of such Riemann surfaces is known to be the Teichm¨uller

space T (n, g), which, when parameterized by Fenchel-Nielsen coordinates, is seen to be

isomorphic to,

T (n, g)

∼

(

, if n = 2, g = 0

3g−3+2n

× R

3g−3+n

, if otherwise

. (2.14)

In other words, (2, 0) has one (positive) geometrical parameter. In all other cases, however,

(n, g) has 6g − 6 + 3n geometrical parameters. While 3g − 3 + 2n of these parameters are

See [15] for a detailed discussion on excision.

– 5 –

JHEP02(2020)149

•

Figure 1. The Riemann surface obtained by quotienting the upper half-plane by dilatation (2.13).

The fundamental domain is bounded by the red semicircles, while the ﬁxed point of (2.13) is the

center of the semicircles. This is the t = 0 slice of the two-sided BTZ.

lengths of minimal (within homotopy classes), non-intersecting, periodic geodesics, the

other 3g − 3 + n parameters are all “twist” angles.

At t = 0, the black hole horizons become minimal periodic geodesics on the Riemann

surface.

With regards to the geometrical parameters, this means that n of the positive

parameters correspond to the lengths (or masses) of the horizons seen from the exterior

regions of the wormhole, while the remaining parameters describe the internal geometry

behind these horizons. If we restrict our attention to static wormholes — that is, those

with zero angular momentum — then the Riemann surface has all of the parameters of the

wormhole geometry, which is made apparent from the metric found by [1].

The constructions of [1] and [2] make heavy use of Riemann surfaces. Especially, [1]

focuses ﬁrst on constructing the (n, g) Riemann surface, then lifting the isometries that they

use from H to AdS

in order to construct the (n, g)-wormhole metric beyond the t = 0 slice.

Physically, this lifting step can be viewed as time-evolving the Riemann surface without

altering the topological data. Thus, the relevant Riemann surface provides a visualization

of the corresponding wormhole. Indeed, this procedure of constructing the Riemann surface

as the t = 0 slice ﬁrst allows us to see that the identiﬁcation of equation (2.13) yields the

two-sided BTZ (ﬁgure 1).

Nevertheless, [1] has a very “piecewise” feature to the construction. For instance, in

the (3, 0) (three-boundary) case, for which the corresponding Riemann surface is a pair

of pants, we know from [2] that the appropriate procedure would be to quotient by two

isometries, using two Killing vectors. Meanwhile, in [1], they cover the pair of pants with

three overlapping cylinders, construct those from subspaces of H, and glue the resulting

three metrics together in order to reconstruct the pair of pants. In short, [1] does not

describe a global quotienting procedure of AdS

This patchy, piecewise method is perfectly ﬁne if we only wish to consider AdS

, since

it relies heavily on the presence of a hyperbolic t = 0 slice. However, in other cases, such as

that of warped AdS

since there is not a clear idea of a suﬃcient hyperbolic slice of the

spacetime, we cannot repeat this procedure. A more global construction is needed should

See [1] for a more detailed discussion of these parameters.

Ref. [1] explicitly shows that the metric outside of these minimal periodic geodesics is precisely that of

the static BTZ.

See [16, 17] for a more detailed discussion of warped AdS

– 6 –

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