全息RG流动与曲面流形上的F定理

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"这篇研究论文探讨了全息RG流动在曲面流形，特别是S3空间中的应用，并与F定理相结合。通过规范重力对偶性，作者J.K.Ghosh、E.Kiritsis、F.Nitti和L.T.Witkowski展示了壳上作用如何在全息RG流动解决方案中定义良好的F函数。该研究揭示了当扰动算子的尺度Δ大于3/2时，F函数与重新归一化的自由能相关；而当Δ小于3/2时，它对应于量子有效电势，即自由能的勒让德变换。此外，他们还讨论了自由费米子和自由玻色子在S3上的特例，以及特定纠缠表面的选择如何允许从纠缠熵定义F函数。" 在这篇论文中，研究人员深入分析了全息重力理论，这是一种理论框架，它将高维的量子场论与低维的引力理论关联起来，以研究场论的性质。特别是在S3空间中，其半径作为RG（Renormalization Group）标度，使得研究者能够观察到不同尺度下的物理现象。RG流动是理解量子场论在不同能量尺度上如何变化的关键工具。 F定理是一个假设，指出在二维共形场论（CFT）的RG流动过程中，一个被称为F函数的量（通常与系统的拓扑自由能或纠缠熵相关）会单调减少。在本研究中，作者通过全息方法在S3背景下的RG流动中验证了这一定理，发现F函数在从紫外线（UV）到红外线（IR）的流动中确实是单调下降的。这表明F函数可以作为描述理论在不同能量尺度上行为的有效度量。论文进一步探讨了不同类型的扰动算子对F函数的影响。当算子的尺度Δ大于3/2时，F函数与重新归一化的自由能相匹配，这提供了一种理解场论动力学的途径。另一方面，如果Δ小于3/2，F函数则被解释为量子有效电势，这是通过勒让德变换从自由能导出的。这种区分显示了F函数如何适应不同类型的场论扰动。在具体的例子中，自由费米子和自由玻色子在S3上的情况被用作测试床，以检验这些理论是否符合全息预测。自由大规模标量的能量问题也得到了解决，这是长期存在的问题之一。同时，通过选择特定的纠缠表面，研究人员还展示了如何从纠缠熵来定义F函数，这些熵与从壳上作用得到的F函数一致，提供了F函数定义的另一种视角。这项工作不仅加深了我们对全息重力理论的理解，还为F定理在曲面流形上的应用提供了实证支持，并在纠缠熵和RG流动之间建立了联系。这项研究的重要性在于它提供了一种新的方法来研究量子场论在不同能量尺度上的行为，这对于理解和探索量子宇宙学、弦理论以及可能的量子信息过程具有重要意义。

JHEP02(2019)055

ﬂat space REE. It would be interesting to understand whether these are related in some

deeper way.

In appendix I we also discuss another monotonic quantity in holography which does not

seem to obviously lead to a universal F -function. It is obtained from the unrenormalized

free energy density by varying the UV cut-oﬀ instead of the curvature scale. It may be

worth investigating whether a good F -function may be obtained in this way.

Another direction for further work is to gather additional evidence that the functions

1,2,3,4

(R) deﬁned in (1.21)–(1.24) are indeed good F -functions. While we checked this

explicitly for a wide range of theories in holography and for free ﬁeld theories, further tests

are desirable.

In the context of holography, here we worked with bottom-up four-dimensional mod-

els, and it would be interesting to check whether our proposal holds in top-down models

from string theories. For three-dimensional ﬁeld theories, there are many examples of

gravity dual holographic RG ﬂows in gauged N = 8, d = 4 supergravity [27–30], and

their M-theory uplifts [31, 32]. One could also consider three-dimensional theories with

ﬂavor, whose RG ﬂows were studied e.g. in [33–35] (in the quenched ﬂavor limit) and [36].

The quenched ﬂavor case looks particularly treatable, as one does not need to ﬁnd the

curved domain wall solution in the full 11-dimensional bulk theory, but only restrict to the

contribution to the F -functions from the ﬂavor degrees of freedom coming from the D6

branes wrapped on AdS

×S

using curved slicing of AdS

, and use the ﬂavor mass as the

deformation parameter.

Another interesting playground to test our proposal is in the context of conﬁning

theories, which in holography may be realised both in bottom up models (e.g. setups with

exponentially growing potentials, like those studied in [37] or the AdS soliton [38, 39]) and

in top-down constructions like those studied in [38, 40], dual to 3-dimensional conﬁning

gauge theories (in this case the scale is provided by the radius of an internal compact cycle).

In [25] it was observed that boundary ﬁeld theories in holography may undergo phase

transitions when the boundary curvature is adjusted (see also [41]). The change of boundary

curvature is equivalent to the notion of curvature-RG ﬂow explored in this work. An

interesting question to explore is whether the F -theorem holds across a curvature-induced

phase transition. We leave this for future work.

Another open question is to what extent an F -theorem exists in higher odd dimensions,

i.e. d = 5, 7, . . .. Several proposals for F -functions in higher odd dimensions exist [5, 6, 10,

12] and evidence for an F -theorem in d = 5 can be found in [42, 43]. However, so far there

is no proof of monotonicity for odd dimensions with d ≥ 5. The F -functions introduced

in this work, while constructed for d = 3, allow for a straightforward generalisation to any

odd d and are hence suitable for exploring the F -theorem beyond d = 3. We hope to report

on this in the future.

2 Holographic space-times with curved slicing and RG ﬂows

Our focus will be on holographic RG ﬂows of ﬁeld theories on dS

and S

space-times, driven

by a relevant deformation of a UV CFT. These are dual to curved domain wall solutions

– 12 –

JHEP02(2019)055

in d + 1 dimensions. In this section we review these types of solutions in the context of

Einstein gravity coupled to a scalar ﬁeld. This section is in most part a review of [25],

where a systematic study of curved domain walls with maximally-symmetric radial slices

was performed, and the reader is referred to that work for more details. While we present

the setup for general d we will ultimately be interested in odd d and in particular d = 3.

2.1 Ansatz and conventions

The relevant holographic framework is (d + 1)-dimensional Einstein-dilaton gravity. In the

Lorentzian case with signature (− + + . . . +) the corresponding two-derivative action is

given by

S[g, ϕ] = M

d−1

du d

|g|



(g)

−

∂

ϕ∂

ϕ − V (ϕ)



+ S

GHY

, (2.1)

where we also included the Gibbons-Hawking-York term S

GHY

. The Euclidean action can

be obtained from (2.1) by changing the overall sign, S

= −S, and changing the metric to

positive signature.

To study holographic RG ﬂows for ﬁeld theories on dS

or S

we seek solutions of

the following form for the (d + 1)-dimensional metric and the scalar ϕ. In domain wall

coordinates our ansatz is given by

= du

+ e

2A(u)

µν

, ϕ = ϕ(u) , (2.2)

where A(u) is a scale factor that depends on the coordinate u only, and ζ

µν

is a ﬁducial

metric on dS

or S

. To be speciﬁc, we label the radius associated with ζ

µν

by α. Note

that α is unphysical as we can absorb a redeﬁnition of α by an appropriate constant shift

in A(u). Another useful quantity will be the induced metric γ

µν

(u) on a slice of ﬁxed u,

which is given by

µν

(u) ≡ e

2A(u)

µν

. (2.3)

Throughout this paper, we will use the following shorthand notation. We will denote

derivatives with respect to u by a dot while derivatives with respect to ϕ will be abbreviated

to a prime, i.e.:

f(u) ≡

df(u)

, g

(ϕ) ≡

dg(ϕ)

dϕ

. (2.4)

Given our ansatz (2.2) we obtain the following equations of motion:

2(d − 1)

A + ˙ϕ

−2A

(ζ)

= 0 , (2.5)

d(d − 1)

−

˙ϕ

+ V − e

−2A

(ζ)

= 0 , (2.6)

¨ϕ + d

A ˙ϕ −V

= 0 , (2.7)

with R

(ζ)

= d(d−1)/α

the scalar curvature associated with the metric ζ

µν

. Note that these

equations remain unchanged when swapping between Lorentzian and Euclidean signature.

– 13 –

JHEP02(2019)055

To apply gauge-gravity duality, we will be exclusively interested in bulk solutions

to (2.5)–(2.7) which are asymptotically AdS

d+1

. In particular, we will seek bulk geometries

with a boundary for u → −∞ at which the bulk space-times asymptotes to AdS

d+1

. This

implies that as we approach the boundary the scale factor has to take the form

A(u) −→

u→−∞



−

sinh

u + c





u→−∞

= ln

2α

−

u + c

− e

2(u+c)/`

+ O(e

4u/`

) , (2.8)

which is the scale factor corresponding to AdS

d+1

space-time in the coordinates of (2.2).

Here ` is the length scale associated with AdS

d+1

, α is the length scale of the ﬁducial metric

µν

and c is an integration constant.

It is this constant c which sets the scalar curvature of the boundary manifold supporting

the QFT. To see this, we write the metric near the boundary in a Feﬀerman-Graham

expansion:

= du



−2u/`

(0)

µν

+ . . .



. (2.9)

It is the metric γ

(0)

µν

which describes the boundary space-time supporting the QFT. Given

our ansatz, we can identify the boundary metric as

(0)

µν



2u/`

2A(u)





u→−∞

µν

4α

−2c/`

µν

. (2.10)

An important quantity in this work will be scalar curvature of the boundary space-time.

Here and in the following this will be denoted by R. Given the expression for the boundary

metric (2.10), the asymptotic form of the scale factor (2.8), the boundary curvature is

given by:

R =

4α

2c/`

(ζ)

4d(d − 1)

2c/`

. (2.11)

The boundary curvature is therefore set by the integration constant c and the unphysical

parameter α does not appear.

At this stage we introduce one further convention which we will adopt throughout this

paper. As the radius α of the ﬁducial metric ζ

µν

is unphysical, we can choose it freely

without aﬀecting any physical results. In the following, we will ﬁnd it convenient to choose

α to be identical to the physical radius associated with the boundary metric γ

(0)

µν

, i.e.

α =

−c/`

. (2.12)

With this choice the following identities will be valid in the remainder of this paper.

(0)

µν

= ζ

µν

, R = R

(ζ)

4d(d − 1)

2c/`

d(d − 1)

. (2.13)

Also, given these conventions, the scale factor A(u) in the vicinity of the boundary (2.8)

becomes:

A(u) =

u→−∞

−

4d(d − 1)

2u/`

+ O



4u/`



. (2.14)

– 14 –

JHEP02(2019)055

2.2 Scalar functions formalism

Rather than working with the second-order equations (2.5)–(2.7), it will be convenient to

rewrite them as a system of ﬁrst-order equations, by introducing an appropriate set of scalar

functions of ϕ. This is always possible locally with the exception of special points at which

˙ϕ = 0 and which we will refer to as bounces, as discussed in [25, 44]. As long as ˙ϕ(u) 6= 0,

we can invert the relation between u and ϕ(u) to deﬁne the following functions of ϕ:

W (ϕ) ≡ −2(d − 1)

A , (2.15)

S(ϕ) ≡ ˙ϕ , (2.16)

T (ϕ) ≡ R e

−2A

d(d − 1)

−2A

, (2.17)

where the expressions on the right hand side are evaluated at u = u(ϕ).

We can then rewrite (2.5)–(2.7) as a set of coordinate-independent, ﬁrst-order non-

linear diﬀerential equations in ﬁeld space:

− SW

T = 0 , (2.18)

2(d − 1)

− S

− 2T + 2V = 0 , (2.19)

−

2(d − 1)

SW − V

= 0 . (2.20)

As these equations are algebraic in T , we can partially solve this system by eliminating T .

We are then left with the following system of equations:

2(d − 1)

+ (d −1)S

− dSW

+ 2V = 0 , (2.21)

−

2(d − 1)

SW − V

= 0 . (2.22)

2.3 Holographic RG ﬂows for ﬁeld theories on dS

or S

Holographic RG ﬂows correspond to solutions for A(u) and ϕ(u) where e

A(u)

evolves mono-

tonically in u from the UV ﬁxed point at which e

→ ∞ to the IR end point at which

→ 0. At the UV ﬁxed point the scalar takes a value ϕ

which coincides with a

maximum of the potential V . The scalar then changes along the ﬂow reaching a value ϕ

at the IR end point, with ϕ

some generic point. Note that, in contrast to the ﬂow in u,

the change in ϕ(u) along the ﬂow does not have to be monotonic. However, in this work it

will be suﬃcient to consider solutions where ϕ(u) is monotonic. In this case ϕ can be used

as a variable describing progress along the ﬂow.

First, we will review the solutions in the

vicinity of UV and IR loci before describing complete RG ﬂows.

If ϕ(u) reverses direction as a function of u, then ϕ can still be used as a parameter along the ﬂow,

albeit piecewise along the various branches.

– 15 –

JHEP02(2019)055

UV ﬁxed points. UV ﬁxed points are associated with extrema of the scalar potential

V (ϕ) with V (ϕ

ext

) < 0. For maxima of V the RG ﬂow solutions describe a relevant

deformation away from the UV ﬁxed point. In the vicinity of the UV, the potential can

hence be expanded as

V = −

d(d − 1)

(ϕ − ϕ

)

+ O



(ϕ − ϕ

)



. (2.23)

For a UV maximum,

we require m

to be negative and larger than the BF bound,

−d

/4 < m

< 0.

The geometry in the UV takes the form of an asymptotically AdS

d+1

space-time with

AdS length `. The solutions for A(u) and ϕ(u) in the vicinity of the UV will take the form

of a near-boundary expansion for the scale factor and a scalar in AdS

d+1

space-time:

A(u) = −

−

4d(d − 1)

2u/`

+ . . . (2.24)

ϕ(u) = ϕ

+ ϕ

−

∆

−

∆

−

u/`

+ ϕ

∆

u/`

+ . . . , (2.25)

as u → −∞, where

∆

≡



d ±

+ 4`



. (2.26)

For a maximum, we have 0 ≤ ∆

−

≤ d/2 and d/2 ≤ ∆

≤ d.

The standard holographic dictionary assigns dimension ∆

= ∆

to the scalar oper-

ator O dual to the bulk ﬁeld ϕ. The parameter ϕ

−

is then identiﬁed with the source j of

O which parametrises the deformation of the CFT away from the UV as in equation (1.1),

while ϕ

is related to the vev of O,

j = ϕ

−

, hOi = (2∆

− d) ϕ

. (2.27)

In the restricted range of parameters

d − 2

< ∆

−

, or, equivalently

< ∆

d + 2

, (2.28)

one can use a diﬀerent holographic dictionary, called ‘alternative quantisation’ [45], in

which the operator dimension is identiﬁed with ∆

−

, and the source j and the vev hOi in

equation (2.27) are interchanged. Generally, we will be using the standard dictionary, with

∆

= ∆

, unless explicitly stated.

We now turn to the corresponding solutions in terms of the scalar functions W, S, T .

In the vicinity of a maximum of V the functions W and S can be expanded in powers of

The parameter ` appearing in (2.23) is deﬁned via V (ϕ

) = −d(d−1)/`

. To avoid confusion with `

deﬁned later, we will sometimes also denote ` by `

. However, when possible, we will drop the subscript

‘UV’ to remove clutter.

This is the generic situation for a UV ﬁxed point. A minimum of the potential may be a UV ﬁxed

point, but then the RG ﬂow is driven by an irrelevant operator acquiring a vev. In this case, m

> 0 in

equation (2.23).

– 16 –

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