M5-branes全息图与Chern-Simons理论的包裹研究

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"这篇论文是关于物理学中的一个高级主题，主要探讨了包裹的M5膜（M5-branes）在三维超共形场理论（3D SCFTs）中的表现，以及与Chern-Simons理论的关系。研究利用规范/引力对偶（gauge/gravity duality）和3d-3d对应关系，对双曲3空间上的Chern-Simons理论的微扰自由能做出了定量预测。作者指出，微扰展开在大N极限下可能在二圈（two-loop）时终止，这在高维弦理论和量子场论中是一个重要的发现。此外，通过数值检验，他们在多个例子中验证了这一对应关系，并确认了与N的三次幂（N^3 scaling）相关的精确系数。该研究由韩国先进研究所、京畿大学和首尔国立大学的研究人员合作完成，并发表在《Physics Letters B》上。" 这篇论文的核心知识点包括： 1. **包裹的M5膜（Wrapped M5-branes）**：M5-branes是弦理论中的基本对象，具有二维世界体积，可以包裹在特定的空间几何上。它们在理论物理中的作用是复杂的，涉及到超引力、共形场论以及对偶性等概念。 2. **三维超共形场理论（3D Superconformal Field Theories, SCFTs）**：这是一种具有特殊对称性的量子场论，其中包含超对称和共形对称性。这些理论在理解和描述高维物理现象，尤其是黑洞和AdS/CFT对偶性中起着关键作用。 3. **规范/引力对偶（Gauge/Gravity Duality）**：也称为AdS/CFT对偶，是一种将量子场论（如强相互作用的系统）与经典引力理论（如反德西特空间中的弦理论）关联起来的理论工具。它允许在不同的理论框架之间进行计算转换，提供了解决高维问题的新途径。 4. **3d-3d对应关系**：这是近年来提出的一个理论关系，它建立了3维共形场理论与3维Chern-Simons理论之间的联系。这种对应对于理解量子场论和几何之间的相互作用有重要意义。 5. **Chern-Simons理论**：这是一种拓扑量子场论，其作用量依赖于场的外导数的外积，导致理论具有非平凡的拓扑性质。Chern-Simons理论在低维物理学和凝聚态物理中有广泛应用。 6. **双曲3空间（Hyperbolic 3-space）**：这是一个负曲率的三维空间，常用于描述量子场论中的某些背景几何。 7. **微扰自由能**：在量子场论中，自由能是统计力学中的一个概念，通过微扰展开可以计算系统在弱相互作用下的性质。在这里，它被用来研究Chern-Simons理论的行为。 8. **大N极限**：在量子场论中，N通常表示场的自由度数量。大N极限是指当N趋于无穷大时，一些复杂系统的性质可以简化，便于分析。 9. **数值检验**：为了验证理论预测，研究人员通过数值模拟在不同情况下检查了Chern-Simons理论的自由能，这种方法在理论物理中用于验证理论模型的准确性和适用性。 10. **N的三次幂缩放（N^3 scaling）**：这指的是在某些物理量中N的出现方式，它可能是理论中某些特征的数学表示，例如在Chern-Simons理论中，与N相关的系数对自由能的影响。这篇研究展示了理论物理中高深概念的相互作用，不仅深化了我们对包裹的M5膜和Chern-Simons理论的理解，也为未来的研究提供了新的视角和工具。

Physics Letters B 733 (2014) 316–319

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Holography of wrapped M5-branes and Chern–Simons theory

Dongmin Gang

a,∗

,NakwooKim

, Sangmin Lee

c,a

School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea

Department of Physics, Research Institute of Basic Science, Kyung Hee University, Seoul 130-701, Korea

Center for Theoretical Physics, College of Liberal Studies, Seoul National University, Seoul 151-742, Korea

article info abstract

Article history:

Received 14 February 2014

Received in revised form 11 April 2014

Accepted 28 April 2014

Availableonline2May2014

Editor: M. Cveti

We study three-dimensional superconformal ﬁeld theories on wrapped M5-branes. Applying the

gauge/gravity duality and the recently proposed 3d–3d relation, we deduce quantitative predictions

for the perturbative free energy of a Chern–Simons theory on hyperbolic 3-space. Remarkably, the

perturbative expansion is expected to terminate at two-loops in the large N limit. We check the

correspondence numerically in a number of examples, and conﬁrm the N

scaling with precise

coeﬃcients.

(http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP

1. Introduction

In quantum ﬁeld theories, duality refers to a map between

observables of two seemingly unrelated theories. Duality can be

particularly powerful when one of the two theories is not (yet)

deﬁned rigorously. There are two prominent examples in string

theory: M-theory and holographic gauge/gravity duality [1].While

less well-understood than perturbative string theory, M-theory of-

fers a unifying framework for all string theories. The gauge/gravity

duality relates a quantum ﬁeld theory to a quantum gravity the-

ory in one higher dimensions. Although the gravity theory operates

mostly at the classical level, it often gives powerful predictions for

the quantum ﬁeld theory.

A number of new dualities have been discovered recently

thr

ough compactiﬁcation of M5-branes. Just as M-theory uniﬁes

string theories, M5-branes provide a unifying framework for a large

class of supersymmetric quantum ﬁeld theories. In the simplest

case, the M5-brane theory deﬁnes a 6d conformal ﬁeld theory with

(2, 0) supersymmetry. Wrapping M5-branes on internal manifolds

gives rise to lower dimensional ﬁeld theories with the same or a

smaller number of supersymmetries.

In conventional compactiﬁcations, the compact manifold affects

the

deﬁnition of the lower dimensional ﬁeld theory, but does not

usually bear an independent physical meaning. A novelty in recent

works on M5-branes is that a duality holds between the compacti-

ﬁed ﬁeld theory and a different ﬁeld theory deﬁned on the internal

manifold. For instance, in the celebrated “4d–2d” relation [2] a4d

N = 2 supersymmetric ﬁeld theory is paired with an integrable

ﬁeld theory on a Riemann surface. Similarly, the “3d–3d” relation

Corresponding author.

[3] connects a 3d N = 2 supersymmetric ﬁeld theory with a 3d

Chern–Simons (CS) theory.

The goal of this Letter is to point out and verify a surprising

ediction for the perturbative expansion of CS theory, which is

deduced from a combination of the gauge/gravity duality and the

3d–3d relation. We report on the main results here, and the details

will be published elsewhere [4].

WebeginwithwrappingastackofN M5-br

anes on a hy-

perbolic 3-manifold M. The resulting lower dimensional the-

ory is called T

[M] [5]. One of the fundamental observables of

the theory is the partition function on a squashed three-sphere,

[M]

], with a squashing parameter b, and the associated free

energy F

N,b

=−log |Z

[M]

]|. We will use the dualities to study

properties of F

N,b

without computing it directly from T

[M].

On the one hand, we embed the brane conﬁguration into

the

full M-theory to invoke the gauge/gravity duality. Building

upon the relevant supergravity solution [6] and taking the squash-

ing into account [7], we will show that the gravity computa-

tion gives F

gravity

= N

(b +b

−1

)

vol(M)/12π in the large N limit.

Gauge/gravity duality leads to an equality between the gravity free

energy and ﬁeld theory free energy F

N,b

at large N. On the other

hand, we use the 3d–3d relation to compute F

N,b

from the CS the-

ory. The methods for the computation were developed recently

in [8,5]. A crucial feature of the 3d–3d relation is that the loop-

counting parameter “

h” of the perturbative CS theory is related to

the squashing parameter b as

h = 2π ib

[3,9]. It follows that the

n-th term F

(n)

,deﬁnedas

∞



n=0

(

h/i)

n−1

(n)

+(non-perturbative), (1)

http://dx.doi.org/10.1016/j.physletb.2014.04.051

0370-2693/

SCOAP

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