AdS/CFT与锥形缺陷：运动空间与测地线分析

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"这篇学术文章‘锥形缺陷的运动空间’深入探讨了AdS/CFT对应中的一个关键概念——运动空间，并特别关注了在存在圆锥形缺陷时空情况下的运动空间定义与应用。作者Jesse C. Cresswell和Amanda W. Peet来自于多伦多大学的物理和数学系，他们在2017年发表的这篇文章中揭示了非最小测地线在这种特殊背景下的重要性。" 文章详细讨论了运动空间如何作为AdS/CFT对应（Anti-de Sitter/Conformal Field Theory对应）的一个桥梁，这种对应是理论物理学中一种强大的工具，用于理解和连接量子场论和广义相对论。运动空间通常被定义为在AdS边界上锚定的测地线的空间，或者作为CFT（共形场论）点对的空间。然而，当bulk（即AdS空间的内部）不再全局类似于AdS3时，由于非最小测地线的存在，这两种定义可能会变得模糊。本文以圆锥形缺陷时空为研究对象，这种时空结构在凝聚态物理和弦理论中有重要应用，因为它们能够模拟物质的不完美和局部扰动。作者指出，从AdS3的运动空间中，可以通过特定的商操作得到圆锥形缺陷的运动空间，这个商操作同时也导致了AdS3到缺陷的映射。由此产生的运动空间是多种等效的基本区域之一。边界视角的分析进一步深化了理解。通过将Operator Product Expansion（OPE，算子乘积展开）块分解为单个块的测地线贡献，可以解析出圆锥形缺陷的运动空间。作者建立了一个对偶关系，即部分OPE块与通过单个测地线（无论是最小的还是非最小的）上的体场积分之间存在对应。文章还指出，这种对偶性有助于解析非最小测地线的物理意义，特别是在计算CFT中的某些不变量时。通过这种方法，物理学家可以更准确地理解和计算在圆锥形缺陷背景下CFT的性质，这对于研究高维量子系统和黑洞物理等领域具有重要意义。这篇工作提供了对AdS/CFT对应中复杂几何结构的深刻洞察，尤其是在面对非标准几何结构如圆锥形缺陷时空时。它不仅深化了我们对运动空间的理解，也为未来研究非平凡时空背景下的量子场论提供了宝贵的理论工具。

JHEP11(2017)155

The embedding equations

U = sinh t, X = cosh t cos θ, Y = cosh t sin θ, (2.9)

lead to the dS

metric in global coordinates,

= −dt

+ cosh

t dθ

. (2.10)

Conformal or “kinematic” coordinates (α, θ) will be used often in this paper as they nat-

urally ﬁt with the description of kinematic space as the space of geodesics in AdS

. The

transformation cosh t = 1/ sin α, where now α ∈ [0, π], leads to the dS

metric

−dα

+ dθ

sin

. (2.11)

With these conventions laid out, the remainder of this section brieﬂy recounts the derivation

of the kinematic space geometry for pure AdS

, then details two methods of obtaining the

kinematic space for conical defects from the bulk.

2.2 Kinematic space from diﬀerential entropy

In [8] a deﬁnition of kinematic space for constant time slices of AdS

in terms of diﬀerential

entropy was derived from integral geometry. Each interval of the boundary, denoted by

an ordered pair of points (u, v), corresponds to a point in kinematic space covered by null

coordinates (u, v). The kinematic space metric in these coordinates was found to be

∂

S(u, v)

∂u∂v

dudv, (2.12)

where S(u, v) was the length of the shortest oriented geodesic connecting the ends of the

interval (u, v) through the bulk. Since the length of a minimal geodesic is holographically

interpreted as the entanglement entropy of the interval it subtends, the quantity ∂

S/∂u∂v

was dubbed diﬀerential entropy [13, 14, 23, 24]. However, many interesting spacetimes in-

cluding the conical defects and BTZ black holes have multiple geodesics connecting pairs

of spacelike separated boundary points. Non-minimal geodesics do not correspond to en-

tanglement between spatial regions, but have been conjectured to describe correlations

between internal degrees of freedom [19]. Because of this potential interest, and their im-

portance in the geodesic approximation for correlation functions [25, 26], in this paper the

diﬀerential entropy deﬁnition will be expanded to include non-minimal geodesics.

For the constant time slice of AdS

, there is a unique oriented geodesic connecting

each ordered pair of boundary points so the issue of non-minimal geodesics in eq. (2.12)

does not arise. Geodesics can be labelled by their half-opening angle α and centre angle θ,

and have length

S(α) =

log

2 sin α

, (2.13)

where µ serves as a gravitational infrared cutoﬀ [27]. By transforming between kinematic

coordinates and null coordinates using u = θ −α, and v = θ + α, eq. (2.12) can be applied

– 5 –

JHEP11(2017)155

2π

∼

2π/N

n=0

n=1

n=2

n=1

n=2

n=0

Figure 2. The Penrose diagram for the dS

kinematic space for pure AdS

is shown as the full

rectangular region, with θ = θ +2π. For the conical defect case the diﬀerential entropy deﬁnition of

kinematic space produces a vertical strip subregion. The same angular identiﬁcation which gives the

conical defect from AdS

also gives the kinematic space. N = 3 is shown for illustration throughout

most of this paper.

to ﬁnd

sin

[(v − u)/2]

dudv

−dα

+ dθ

sin

(2.14)

Thus, the kinematic space of a constant time slice of AdS

is dS

according to the diﬀerential

entropy deﬁnition, as shown in ﬁgure 2. The α < π/2 and α > π/2 halves are mapped

into one another under orientation reversal which acts as α → π − α and θ → θ + π. The

geodesics with α = π/2 cut straight across the Poincar´e disk and have maximal length.

Now consider a constant time slice of the conical defect geometry eq. (2.7). Since

the total angle around the boundary is 2π/N, the centre angle of a geodesic will now be

denoted

θ ∈ [0, 2π/N]. Once again, for any pair of boundary points there is a unique

minimal geodesic connecting them through the bulk. Minimal geodesics have half-opening

angles in the domain α ∈ [0, π/2N ], and by reversing orientations with α → π − α and

θ →

θ + π/N , also the domain α ∈ [(2N−1)π/2N, π]. Minimal geodesics cover the top and

bottom regions of kinematic space in ﬁgure 2.

In contrast to AdS

, there can be non-minimal geodesics connecting pairs of boundary

points. It will be useful to label geodesics and their corresponding regions in kinematic

space by the number of times they wind around the defect, n. The cases of integer and

non-integer N will be treated separately for clarity.

For integer N there are N −1 non-minimal geodesics connecting each pair of boundary

points, with winding numbers 1 ≤ n ≤ N −1. Geodesics with winding number n ﬁll in the

regions of kinematic space

α ∈



nπ

(n + 1)π



, α ∈



(2N −n − 1)π

(2N −n)π



, (2.15)

where these domains are related by orientation reversal. The upper and lower halves of

kinematic space are divided by geodesics with α = π/2 which touch the conical defect. On

– 6 –

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