威尔逊视角下的全息重归一化群体流动与重归一化原理

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"这篇学术论文探讨了从威尔逊（Wilsonian）视角出发的全息重归一化群体流动和重归一化理论。作者包括J.M. Lizana、T.R. Morris和M. P\'erez-Victoria，分别来自西班牙格拉纳达大学的CAFPE和物理理论与宇宙学系以及英国南安普敦大学的STAG研究中心和物理与天文学学院。文章发表于JHEP03(2016)198，由Springer出版，接受日期为2015年12月23日，接受日期为2016年3月2日，最终于2016年3月30日发布。" 全息重归一化是量子场论和弦理论中的一个重要概念，它涉及将边界理论的无穷大消除，以得到有意义的物理结果。在这个威尔逊的框架中，可重归一化的理论被视为理论空间内的子流形，这些子流形是从特定的固定点开始，随着重归一化群的演化而发展出来的。固定点是理论的临界点，它们在特定的参数组合下使理论的无穷大最小化或消除。论文详细研究了威尔逊作用（Wilson action）满足的汉密尔顿-雅各比方程，这是一种在经典力学中描述系统演化的重要方程。通过对这些方程的分析，作者找出了与固定点相关的不动点和本征变形。本征变形是指在重归一化群作用下保持其结构不变的理论变形。当这些本征变形的演化接近固定点时，它们呈现出对角化的行为，这对于构建重归一化的理论至关重要。作者进一步探讨了全息重归一化与威尔逊形式主义的联系，全息原理指出在AdS/CFT对偶中，引力理论的边界理论可以描述为一个在边界上的量子场论。他们讨论了不同重归一化方案，并指出在运动引力方程的解可以作为参数化重归一化理论的重归一化耦合。这提供了一种将威尔逊方法和非威尔逊方法中的全息重归一化组流相互关联的方式。为了具体展示这些概念，作者通过在AdS空间中具有相互作用的标量理论进行了显式计算。这样的计算有助于直观地理解全息重归一化如何在实际问题中应用，并验证了所提出的理论框架。这篇论文深入研究了威尔逊视角下的全息重归一化群体流动，提供了理论和计算上的见解，为理解和应用这一复杂的理论工具提供了宝贵资料。通过结合重归一化群理论和全息对偶，该研究深化了我们对量子场论在高维引力背景下的理解。

JHEP03(2016)198

Critical Manifold

Figure 1. Points lying in the critical manifold are shown in blue. The bare manifold B cuts the

critical manifold at a single point P . The dashed grey curve illustrates the action of combined RG

evolution and renormalisation as in (2.25), ﬁnishing at a ﬁnite point in R.

for some dimensionless functions of the renormalised couplings C

). We have used (2.12)

and the existence of a limit to recognise that the Λ dependence must take this form.

By comparison with (2.20) we see that the renormalised couplings in the Wilsonian UV

scheme are given by g

= C

). If we choose in particular C

) = c

(a)

, it follows

that the beta functions of the two schemes have the same functional form. Although again

we are displaying equations only for relevant or exactly marginal directions, it is clear from

the discussion below (2.20) that a similar identiﬁcation holds true also for the remaining

case of marginally relevant directions.

Furthermore, expanding the left hand side of (2.30) about h = P , we obtain a Taylor

series with terms of the form

∂

Λ/Λ

(g)/∂g

(∆h)

, where only ∆h = h

/µ

) −

P carries dependence on µ and only the Taylor series coeﬃcient carries dependence on

Λ. Consider ﬁrst the case in which all the eigenvalues of relevant directions fulﬁl the

condition λ

(a)

+ λ

(b)

> λ

(c)

. Without some special tuning, already the ﬁrst order term will

contribute, ﬁxing the µ dependence of ∆h and ensuring that actually only the ﬁrst order

term contributes in the Λ  µ regime:

lim

→∞

Λ/Λ

/µ

)) = g

∗

+ lim

→∞





δf

Λ/Λ

(g)

δg



g=P



/µ

) −P







, (2.31)

At the non-linear level the non-exceptional case case follows the treatment given below (2.22).

– 10 –

JHEP03(2016)198

and that

(x) ·



/µ

) −P



= C

)(xµ/Λ

)





−λ

(a)

as Λ

/µ → ∞, (2.32)

where

(x, y) = lim

→∞





(a)

δf

Λ/Λ

[g(x)]

δg

(y)



g=P

. (2.33)

When we withdraw the condition on the relevant eigenvalues, the second and possibly

higher-order terms can give contributions that are more important than the right-hand

side of (2.31), and also the µ behaviour of the left-hand side is modiﬁed. Then, to obtain

the correct µ dependence of the left-hand side h

/µ

in (2.32) must be corrected. Again

the result is the same for the non-exceptional case, interpreted as the leading terms in an



(a)

expansion. At the non-linear level, schematically,

(x) ·



/µ

) −P



= C

)(xµ/Λ

)





−λ

(a)

)(xµ/Λ

)





−λ

as Λ

/µ → ∞, (2.34)

where the exponents λ

are sums of eigenvalues with λ

< Max{λ

(a)

} and the coeﬃcients

depend on C

3 Holographic Wilsonian description of renormalisable theories

3.1 Hamilton-Jacobi evolution

Let us consider an asymptotically-AdS space in d + 1 Euclidean dimensions. In some

neighbourhood of the boundary, we can use Feﬀerman-Graham coordinates and write the

metric as

+ h

µν

(z, x)dx

. (3.1)

The boundary is located at z = 0. In this paper we consider a ﬁxed metric, i.e. we neglect

the backreaction of other ﬁelds on the geometry. Furthermore, below we will specialise to

AdS space. These are strong restrictions and we comment on the complete treatment with

a ﬂuctuating geometry in the ﬁnal discussion.

In agreement with holography and the UV/IR connection [4], all the information about

the ultraviolet of the gauge theory is encoded in the dual picture near the boundary of the

asymptotically-AdS space. A natural way to do it is by enforcing boundary conditions

on the degrees of freedom of the gravity theory. On the other hand, as in ﬁeld theory,

a regularisation is required to make quantities such as correlation functions well-deﬁned.

The standard regularisation used in the literature of gauge/gravity duality is to cut the

space oﬀ close and parallel to the boundary [30]. Then, the boundary conditions must be

imposed at the new boundary, i.e. the cutoﬀ position. More generally, we can consider

placing this cutoﬀ boundary (which we keep parallel to the conformal boundary) at larger

values of the radial coordinate z, which correspond to a lower UV cutoﬀ in the ﬁeld theory.

– 11 –

JHEP03(2016)198

A consistent way of imposing these boundary conditions is to add an action that depends

on the ﬁelds restricted to the UV boundary.

It is then natural to identify the cut-oﬀ

partition function in (2.2), for some boundary action S

, with

1/l

[g] =

[Dφ]

−S

(g)[φ(l)]−S

[φ]

, (3.2)

where [Dφ]

indicates functional integration in the ﬁelds φ of the gravity theory, with sup-

port restricted to z ≥ l, and S

is the classical gravity action. The space-time integrals

inside the functional integral are always understood to be restricted to the support of the

ﬁelds. The boundary action S

is a diﬀerentiable functional of the ﬁelds restricted to the

boundary and of the gauge-theory couplings. We will often not display explicitly the argu-

ments of S

. For deﬁniteness, the ﬁelds φ are assumed to be l-independent dimensionless

functions of the dimensionful space-time variables. This can be achieved with some dimen-

sionful constant of the gravity theory, such as the AdS curvature. Then, the l dependence

of S

is dictated by dimensional analysis, similarly to (2.5). It is useful to distinguish the

boundary values of the ﬁelds and write (3.2) in the following equivalent form:

1/l

[g] =

Dϕ e

−S

(g)[ϕ]

[Dφ]

l,ϕ

−S

[φ]

. (3.3)

Here, [Dφ]

l,ϕ

indicates a path-integral measure for ﬁelds with support z ≥ l and such that

φ(l, x) = ϕ(x). The usage of (3.2) (or (3.3)) entails a particular deﬁnition of the cutoﬀ

procedure in the gauge theory [18, 20, 21]. It is not clear at all that this regularisation

can be formulated, for arbitrary l, in an independent form in terms of the ﬁeld-theory

degrees of freedom, and we will not attempt here to ﬁnd such a correspondence. At any

rate, (3.2) allows to formulate holographically all the ﬁeld-theoretical Wilsonian formalism

reviewed in the previous section in terms of the dual gravity theory. The relation between

the gravity boundary action at l and its corresponding Wilson action at cutoﬀ 1/l will be

examined below.

To any pair (l

, g

), RG invariance associates a ﬂow g

= f

) given, for l

< l, by

−S

)[ϕ]

[Dφ]

l,ϕ

−S

)[φ(l

)]−S

[φ]

. (3.4)

Here, [Dφ]

l,ϕ

indicates a measure for ﬁelds φ(z) with support l

≤ z ≤ l such that φ(l) = ϕ.

In the following we work in the large N limit (with ﬁxed large ’t Hooft coupling), which

is dual to gravity in the classical ﬁeld-theory approximation and allows for a saddle-point

calculation of the path integrals.

In this limit, the gravity action can be written in terms

of a local Lagrangian S

[φ] =

dzd

xL(φ(z, x), ∂φ(z, x), z), and the path integrals in (3.2)

or (3.3) are obtained by extremizing the exponent, subject to the UV boundary conditions

(

(l, x) =

δS

(g)[ϕ]

δϕ

(x)



ϕ=φ(l)

)

, (3.5)

We always employ ﬁeld-theoretical language, which should be appropriately translated to string-

theory analogues in precise formulations beyond the low-energy ﬁeld-theory approximation, valid at large

’t Hooft coupling.

The N global factors in the actions and in the normalisation of the operators that are necessary for a

well-deﬁned large-N limit are implicit in our equations.

– 12 –

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