LLM几何的激发与新兴规范理论探索

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"这篇学术文章探讨了LLM几何的激发，这是一种由巨引子凝聚体的后效应产生的几何形态。在低能级下，这些激发导致了杨-米尔斯理论的出现，特别是N=4的$$\mathcal{N}=4$$超杨-米尔斯理论。作者通过研究这些新兴规范理论的平面极限动力学，提供了它们是平面N=4 $$\mathcal{N}=4$$超杨-米尔斯理论的证据。文章提到了三个关键观察结果作为支持论据，并讨论了为何出现的应变片理论并非等同于N=4 $$\mathcal{N}=4$$超杨-米尔斯理论。" 在这篇文章中，作者深入研究了LLM几何的激发状态。LLM几何，全称 Lunin-Lin-Maldacena 几何，是AdS/CFT对偶中的一个重要概念，它关联着特定的超引力解与特定的四维共形场论。在这些几何形状中，巨大的引子凝聚体的后效应引发了开放弦的激发，这些开放弦随后在低能级下诱导出一个杨-米尔斯理论。作者专注于新兴规范理论的平面极限动力学，这是量子场论中一种简化处理复杂系统的方法，尤其适用于N=4 $$\mathcal{N}=4$$超杨-米尔斯理论，这是一个具有高度对称性的四维超对称量子场论。他们提出的三个主要观察结果如下： 1. 平面希尔伯特空间的同构性：作者论证了原N=4 $$\mathcal{N}=4$$超杨-米尔斯理论的平面希尔伯特空间与新兴规范理论的平面希尔伯特空间之间存在同构关系。希尔伯特空间是量子力学中的一个核心概念，代表了所有可能量子态的集合，其同构表明两者在量子态层面有相似的结构。 2. OPE系数的消失：OPE（Operator Product Expansion）系数在量子场论中描述了两个场在短距离下的相互作用。作者指出，新兴规范理论在平面极限下的OPE系数趋于零，这一特性是N=4 $$\mathcal{N}=4$$超杨-米尔斯理论的一个显著特征。 3. 异常维度的平面频谱：异常维度是规范理论中的一种特性，它与物理量的尺度变换行为有关。作者认为新兴规范理论的异常维度平面频谱与N=4 $$\mathcal{N}=4$$超杨-米尔斯理论一致，进一步支持了它们的等价性。然而，尽管新兴的应变片理论在平面极限下表现出与N=4 $$\mathcal{N}=4$$超杨-米尔斯理论类似的特性，作者也解释了两者间的差异。这暗示了尽管在某些方面相似，但应变片理论并不等同于N=4 $$\mathcal{N}=4$$超杨-米尔斯理论，可能是因为它们的对称性和物理性质在更广泛的上下文中有所不同。这篇文章通过对LLM几何激发的研究，深化了我们对高能物理和共形场论的理解，特别是关于N=4 $$\mathcal{N}=4$$超杨-米尔斯理论的平面极限行为，这为理解和探索更深层次的物理现象提供了宝贵的洞见。

JHEP07(2018)146

2 Free CFT

Our basic goal is to organize and study excitations of an LLM background, using the

dual N = 4 super Yang-Mills theory. Any LLM geometry is speciﬁed by a boundary

condition, given by coloring the bubbling plane into black and white regions [9]. The LLM

backgrounds we consider have boundary conditions given by concentric annuli, possibly

with a central black disk. The LLM geometry is described by a CFT operator with a

bare dimension of order N

. Concretely, it is a Schur polynomial [13] labeled by a Young

diagram with O(N

) boxes and O(1) corners. Large N correlators of these operators are not

captured by summing only planar diagrams, so we talk about the large N but non-planar

limit of the theory. The excitation is described by adding J boxes to the background, with

 N. Consequently, we can ignore back reaction of the excitation on the LLM geometry.

The CFT operators corresponding to the background and excitation are given by

restricted Schur polynomials [15, 16]. Construction of these operators and their correlators

becomes an exercise in group representation theory. In section 2.1 we discuss elements of

this description, placing an emphasis on if the quantity being considered depends on or is

independent of the collection of branes being excited. This distinction will clarify general

patterns in the CFT computations that follow.

We begin our study in the free ﬁeld theory. The Hilbert space of possible excita-

tions can be written as a direct sum of subspaces. There are subspaces that collect the

excitations localized at the outer or inner edge of a given annulus, or at the outer edge

of the central disk. The excitations are obtained by adding boxes to the Young diagram

describing the background, at a speciﬁc location. They are also localized in the dual grav-

itational description, at a speciﬁc radius on the bubbling plane [21, 22]. Each localized

Hilbert space is labeled by the edge at which it is localized. There are also delocalized

excitations, where the description of the excitation involves adding boxes at diﬀerent lo-

cations on the background Young diagram [21, 22]. We will not have much to say about

delocalized excitations.

The excitations belonging to the localized Hilbert spaces play a central role in our

study. These are the Hilbert spaces of the emergent gauge theories. We give a bijection

between the states belonging to the planar Hilbert space of an emergent gauge theory, and

the states of the planar limit of the original CFT without background. To show that the

bijection takes on a physical meaning, we argue that correlation functions of operators that

are in bijection are related in a particularly simple way, in the large N limit. This result

is signiﬁcant because the basic observables of any quantum ﬁeld theory are its correlation

functions and many properties of the theory can be phrased as statements about correlation

functions. Thanks to the map between correlation functions, any statement about the

planar limit that can be phrased in terms of correlators, immediately becomes a statement

about the planar emergent gauge theories that arise in the large N but non-planar limits

we consider.

2.1 Background dependence

Irreducible representations of the symmetric group S

are labeled by Young diagrams with

n boxes. States in the carrier space of the representation are labeled by standard tableau,

– 6 –

JHEP07(2018)146

in which we populate the boxes with numbers {1, 2, . . . , n} such that the numbers are

decreasing along the rows (from left to right) and along the columns (from top to bottom).

A representation for S

is given by specifying the action of any element σ ∈ S

on the

standard tableau. We will use Young’s orthogonal representation. For example, here is the

action of the two cycle σ = (12) ∈ S

on a speciﬁc tableau

(12)

4 3 1

1 −





4 3 2

(2.1)

The number

that appears in the above equation is counting the number of boxes in the

shortest path from the box labeled 1 to the box labeled 2. The only thing that matters is

the relative position of boxes 1 and 2. Consequently, σ has the same action on all three

states shown below.

4 3 1

(2.2)

For example

(12)

4 3 1

1 −





4 3 2

(2.3)

This demonstrates that the action of the symmetric group on the boxes belonging to the

excitation is background independent. In what follows we will use R (or r) to denote the

Young diagram describing the excitation and +R (or +r) to denote the Young diagram after

it has been placed at an inward pointing corner of the Young diagram for the LLM geometry.

When our excitation has more than one type of ﬁeld, the gauge invariant operator is

constructed by restricting to the subgroup that permutes ﬁelds of a speciﬁc type. For exam-

ple, if we have n Z ﬁelds and m Y ﬁelds, we would start with an irreducible representation

R ` n + m of S

n+m

and restrict to some representation (r, s), r ` n, s ` m, of the S

×S

subgroup. Upon restricting (r, s) may appear more than once, so we need a multiplicity

label α to distinguish the diﬀerent copies. Since we use only the action of the symmetric

group to perform the restrictions, the multiplicity labels are also background independent.

To diagonalize the one loop dilatation operator [42, 43] traded the multiplicity labels for

directed graphs recording how open strings are connected between giant gravitons. These

graphs summarize basic physics coming from the Gauss Law on the brane worldvolume

– 7 –

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