标量变形下的AdS/dS压缩全息研究

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"标量冷凝水压扁全息" 这篇研究论文探讨了在存在标量变形情况下的全息原理，特别是在处理两个参数球体的压缩问题时。文章中提到的"标量冷凝水压扁全息"是指将标量场的概念应用于全息对偶理论中的一个特定场景，即球体的压缩过程。全息原理是现代物理学中的一个重要概念，它提出一个高维引力理论可以被低维边界上的量子场论完全描述。研究人员评估了自由和相互作用的O（N）向量模型的分区函数。O（N）模型是一种理论工具，常用于理解连续对称性自发破缺的现象，其中N代表自由度的数量。在这种情况下，模型中的标量场被用于模拟球体的压缩，以及可能的形变。文章指出，他们发现了在爱因斯坦引力框架下，与标量场耦合的AdS（Anti-de Sitter）和dS（de Sitter）空间的正规解。AdS空间因其与共形场论（CFT）的全息对应而著名，而dS空间则与膨胀的宇宙模型有关。双挤压边界几何关系在这两者之间建立了联系，这为理解和研究这两种空间提供了新的视角。特别地，AdS解决方案的热力学性质显示出与具有真实质量变形的自由O（N）模型预测的行为一致。这意味着，尽管引入了复杂性，但基本的物理规律依然保持了一致性。而在dS空间中，这些解决方案定义了一种半经典的“无边界”度量，这与膨胀的渐近de Sitter空间的各向异性变形相匹配。通过dS/CFT对应，作者进一步利用相互作用的O（N）模型的分区函数来构建一个无边界度量的全息玩具模型。这个模型产生的概率分布具有定性的相似性，能够归一化并在第三个圆形球体上达到全局峰值，尽管在强各向异性条件下振幅较低。这种分布的特性揭示了全息原理在处理非均匀性问题时的潜力。这项工作深化了我们对全息原理的理解，尤其是在考虑非均匀性和标量场作用时的复杂性。它不仅展示了理论模型如何与引力解相协调，还提供了研究de Sitter空间各向异性的新方法。这些发现对理解宇宙学的某些方面，如早期宇宙的膨胀阶段，以及探索量子引力理论的边界条件具有重要意义。

JHEP09(2018)068

Notice that we have implemented the gauge l

(r) = 1. The next terms in the UV expansion

of the solutions read

(r) = A

+ A

(1−k)r

, l

(r) = B

+ B

(1−k)r

, l

(r) = C

+ C

(1−k)r

, (2.5)

Φ(r) =

)

1/3

−r

)

2/3

−2r

+ D

−(2+k)r

, (2.6)

where the sum over k goes over all positive integers.

We plug the series expansions (2.5)–(2.6) into the Einstein equations and solve them

order by order in powers of e

. The results of this procedure are summarized in appendix A.

The important upshot is that the UV expansion is controlled by seven independent param-

eters {A

, B

, C

, A

, B

, α, β}. It turns out that the Einstein equations are invariant under

constant shifts of r which we use to eliminate one of the parameters, setting A

. Com-

paring the asymptotic form of the metric with the metric (1.1) on the double squashed

sphere one can ﬁnd the following relation between the squashing parameters A and B and

the leading order coeﬃcients B

and C

A =

− 1 , B =

− 1 . (2.7)

The leading coeﬃcients B

, C

and α specify the asymptotic values of metric and ﬁeld.

As we discuss in appendix A the values of the subleading coeﬃcients (A

, B

and β) are

ﬁxed by imposing regularity conditions (either on a NUT or a Bolt) in the bulk of the full

solution of the nonlinear equations of motion.

In practice we use the IR expansions (cf. (A.4) and (A.7)) as initial conditions to inte-

grate the equations of motion numerically to the UV. This yields a three-parameter family

of solutions that are controlled by two coeﬃcients specifying the IR behavior of the scale

factors l

(r) and by the initial value Φ

of the scalar ﬁeld. There are two distinct classes of

solutions. The ﬁrst class consists of regular solutions for which the metric functions l

(r)

grow exponentially, the scalar ﬁeld gradually decays and the boundary metric is a sphere

with two non-trivial squashing parameters as in (1.1). A representative example of a NUT

solution of this kind is shown in ﬁgure 2. We also ﬁnd a class of singular solutions for

which one or more of the metric functions l

(r) vanish at some ﬁnite value of r, leading

to a curvature singularity. We will ignore the second class of solutions since they do not

contribute to the wave function in the large three-volume regime.

The Bolt solutions only exist for suﬃciently large, positive squashings. In this regime

there is often more than one combination of IR parameters that yields the same values of

the leading asymptotic parameters A, B and α.

The regularity condition on the scalar ﬁeld in the interior yields a relation β(α) between

the coeﬃcients of its UV proﬁle which depends on the squashings and encodes information

about the scalar potential. In section 3 we will compare our results with the free O(N)-

model using the AdS/CFT duality. Under the holographic dictionary this relation can be

translated to a relation between the source and vev of the dual theory. To do so we match

the conformal dimensions of the deformations on both sides. Because on the CFT side the

conformal dimension of the source is two, we have to use the alternate quantization of AdS,

– 5 –

JHEP09(2018)068

1 2 3 4 5

0.1

0.2

0.3

0.4

0.5

1 2 3 4 5

Figure 2. A typical solution with a NUT in the IR and with a non-trivial scalar proﬁle in the

radial direction. The geometry asymptotes to a double squashed sphere in the UV.

which means we ﬁx β on the boundary instead of α by applying a Legendre transform [27],

i.e. we have Neumann boundary conditions. The exact procedure requires an analysis of

the action using holographic renormalization [28–31] of which the precise details can be

found in appendix A. This immediately gives the relation between α in the bulk and the

vev on the boundary for a source J = β,

hOi = α . (2.8)

The potential (2.2) is special in the sense that, at least for suﬃciently small squashings,

β tends to a constant when α (or equivalently Φ

) is taken larger. This property depends

delicately on the large ﬁeld regime of the potential. From a dual viewpoint this means

there is a critical deformation β at which the expectation value α of the operator dual to

Φ diverges. This is also a feature of the vector model we consider in section 3 below [32],

which serves to justify the bulk boundary comparison we explore there.

Figure 3 shows the relation β(α), or in holographic notation J(hOi), for B = 0 and

for three diﬀerent values of the squashing parameter A. The third panel indicates that the

behavior of β(α) is qualitatively diﬀerent for suﬃciently negative squashings. Speciﬁcally,

we ﬁnd there is a phase transition at A = −3/4, precisely where the Ricci scalar on the

boundary changes sign, such that for A ≤ −3/4 the parameter β no longer converges. A

second solution with α 6= 0 comes into play in this regime even at β = 0. The new solution

is thermodynamically subdominant, as we will see below, but may nevertheless contribute

to certain observables [32, 33]. A similar behavior of β(α) is found in the entire region

of conﬁguration space (A, B) where the Ricci scalar of the boundary geometry is negative

(cf. ﬁgure 1). We note also that the relation β(α) associated with the generalized Bolt

solutions, shown in the ﬁrst panel in ﬁgure 3, is reminiscent of the constant temperature

relation found for black holes with scalar hair in this theory, as expected [34, 35].

The thermodynamic behavior of our set of solutions can be studied by evaluating their

regularized, Euclidean on-shell action. Since we do not have analytic solutions we evaluate

the regularized on-shell action numerically following the accurate procedure developed

in [26] and summarized in appendix A. Figure 4 shows the resulting free energy for a

number of representative slices of constant β through the three-dimensional phase space of

– 6 –

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