兰道极与全息重力理论的探索

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"这篇研究论文探讨了全息照相技术在紫外线不完整量规理论中的应用，特别是关于d = 4，N = 4的$$\mathcal{N} = 4$$超级杨-米尔斯理论，该理论与N f夸克香精耦合，并在紫外（UV）尺度ΛLP下存在兰道极点。文章指出，双重重力解在沿着全息方向的有限适当距离处表现出紫外线奇点。尽管存在这样的奇点，但通过解析连续性，可以完全实现全息重归一化，从而处理紫外线截止。" 在本文中，作者详细分析了当能量E远小于ΛLP时，系统的行为符合有效场理论的预期，没有出现病态现象。然而，当能量E接近或小于ΛLP的标度时，观察到引力势能转变为排斥力，同时在温度T低于ΛLP时，系统的比热变为负值。这些发现表明，兰道极点的存在对系统性质有显著影响，特别是在高能或高温区域。文章强调，尽管研究主要集中在N = 4的$$\mathcal{N} = 4$$超级杨-米尔斯理论，但这些定性的结果可能适用于更广泛的一类理论，只要这些理论的紫外线奇点具有违反指数的超尺度度量，即θ > d - 1。这扩展了研究的适用范围，并暗示了对于理解其他具有类似特征的理论的全息对应性可能有重要启示。此外，这项工作发表于JHEP02(2017)047，由Springer为SISSA出版，于2016年12月2日提交，2017年1月16日接受，2月8日发布。作者包括Antón F. Faedo、David Mateos、Christiana Pantelidou和Javier Tarrío，分别来自巴塞罗那大学、ICREA和布鲁塞尔自由大学等机构。总结起来，这篇研究揭示了全息照相在处理紫外线不完整理论中的潜力，特别是在理解和模拟具有兰道极点的系统中，提供了新的洞察。同时，它还强调了全息重归一化的有效性，以及理论中紫外线奇点如何影响物理性质，尤其是引力相互作用和热力学行为。这些发现有助于深化我们对量子场论和弦理论的理解，尤其是在高能量尺度下的行为。

JHEP02(2017)047

3 Supersymmetric solution

In this section we will use the ten-dimensional parameterization of the metric given in (2.10)

with b = c = 1 and g traded in favor of χ as in (2.12). The D3-D7 system with smeared

D7-branes preserves N = 1 supersymmetry and the corresponding BPS equations are [6]

∂

φ = −Q

, ∂

χ = −3(1 − e

2χ

) +

∂

f = −3 + 2e

2χ

, ∂

h = e

4(χ−f)

(3.1)

When Q

= 0 this has the well known AdS

× M

solution, with ∂

φ = 0 and χ = 0. In

the ﬂavored case the solution reads

1 + Q

(% − %

)

1/2

1 + Q

(% − %

)

6 + Q

[1 + 6(% − %

) + c

6/(Q

)

6(%−%

)

]

−1/(Q

)

1/6

1/2

−(%−%

)

1 + Q

(% − %

)



6 + Q

[1 + 6(% − %

) + c

6/(Q

)

6(%−%

)

]



1/3

h = c

4χ(ˆ%)

−4f(ˆ%)

dˆ% ,

(3.2)

where φ

≡ φ(%

) is the value of the dilaton at an arbitrary reference point %

, and c

1,2,3

are constants of integration. In order for the dilaton to be real the radial coordinate must

satisfy

< % (3.3)

where

≡ %

−

. (3.4)

At precisely %

the dilaton diverges, in agreement with the presence of a Landau pole at

which the coupling constant diverges. Using (3.4) one may rewrite the solution in terms of

alone as

(% − %

)

= 6

1/2

% − %

1 + 6(% − %

) + c

6(%−%

)

= c

1/2

−(%−%

)

√

% − %

[1 + 6(% − %

) + c

6(%−%

)

]

1/3

(3.5)

with h as in (3.2). We see that by choosing the Landau pole as the reference point the

solution takes a particularly simple form. By an additional shift of the radial coordinate

we will set %

= 0 without loss of generality.

Let us now discuss the role of the integration constants. As noted in [6], if the integra-

tion constant c

in (3.2) is non-zero then the proper radial distance from any point to the

– 6 –

JHEP02(2017)047

IR of the theory is ﬁnite, whereas if c

= 0 there is an inﬁnite throat. We will therefore

set c

= 0 hereafter, since otherwise there would be an unphysical IR cut-oﬀ in the gauge

theory. This means, in particular, that the range of the radial coordinate is 0 < % < ∞.

We will refer to the region % → ∞ as the IR part of the geometry. In this region the dilaton

approaches zero, as expected from the fact that the dual gauge theory is IR-free [27], and

the curvature invariants diverge, as we discuss below. This divergence is absent if the quark

mass is non-zero [7], as we discuss in section 8.

The constant c

can be set to any convenient value through the following rescalings of

and of the Minkowski coordinates

→ x

−1

, c

→ c

−4

. (3.6)

Two particularly convenient choices are c

= 1 and c

= Q

1/6

. With the former choice the

solution takes a particularly simple form in the UV, as we will see below, whereas with the

latter the ﬂavorless limit Q

→ 0 in (3.2) becomes manifestly smooth.

Setting c

= 0 and redeﬁning the last integration constant c

appropriately, the warp

factor takes the form

h = c



−1



2/3

−2/3





, −

− 4%



− Γ



, −



, (3.7)

where Γ[s, x] is the incomplete Gamma function. Since the arguments of the incomplete

Gamma functions are always negative and



(−1)

2/3



, −x



√





if x > 0 , (3.8)

it follows that h is real if c

is real. For the ﬂavored case h vanishes linearly as % → 0 if

= 0 and it approaches a constant if c

6= 0. Following [6] hereafter we will set c

= 0.

This is analogous to “dropping the 1” in the usual warp factor of D3-branes. It would be

interesting to extend our analysis to the case in which c

6= 0, which would correspond to

studying smeared D7-branes in the asymptotically ﬂat background sourced by D3-branes.

3.1 IR and UV asymptotics

Having ﬁxed the solution, let us now examine its IR and UV limits. In the IR region we

have %  1 and the metric components asymptote to

IR: e

, e

' 1 , e

' c

1/6

−%

, h '

−2/3

4 · 6

2/3

. (3.9)

The asymptotic form of the solution is most transparently understood in terms of a new

radial coordinate z deﬁned through

% = log

. (3.10)

In this coordinate the IR region is at z → ∞ and the asymptotic form of the ﬁve-dimensional

metric (2.14) is

8/3



4 · 6

1/3







log



1/3

µν

+ L



. (3.11)

– 7 –

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