构建正方形形式因子：AdS-CFT对与积分场论

193 浏览量更新于2024-07-16 收藏 529KB PDF 举报

本文主要探讨了将圆柱体通过一种创新的方法切割成正方形，从而引入了一个新的概念——“正方形形状因数”（Square Form Factor）。这种方法是在AdS/CFT对偶（AdS-CFT Correspondence）框架下，结合六边形提议（Hexagon Proposal）的思路提出的，这是一种在理论物理，特别是量子场论（Quantum Field Theories）中的重要工具，尤其是在可积场论（Integrable Field Theories）的研究中。在切割正方形的过程中，每个正方形代表两个平方，这一特性被利用来构建一种严格限制正方形形状因数的一致性条件。这种条件对于理解和计算物理系统的性质至关重要，因为它能够提供一种有效的数学框架来处理复杂的多粒子相互作用。文章的核心内容涉及对Gaudin模型的重构，Gaudin模型是理论物理学中一种描述多个粒子系统相互作用的重要模型。通过利用定义中的行列式森林扩展（Determinant Forest Expansion），作者能够将这个一致性条件转化为实际操作，进而恢复出Gaudin模型的结构。此外，论文还展示了如何运用这一方法来计算非平衡态的Bethe状态规范（Norm of Off-Shell Bethe States）。在某些简化的情况下，这种方法提供了计算这些复杂物理量的实用途径，这对于理解量子系统的行为具有重要意义。关键词包括AdS/CFT对偶、SU(2)代数以及可积场论，这些都是研究者探索强相互作用领域（如量子色动力学）的关键概念。文章发表在《Journal of High Energy Physics》（JHEP）上，是开放获取的，表明其研究成果可供学术界广泛查阅和讨论。整个研究工作得到了SCOAP3项目的资金支持，并被引用为arXiv预印本1810.11430。这篇论文不仅提出了一种新颖的数学工具，而且还展示了其在实际物理问题中的应用潜力，对于进一步推动AdS/CFT对偶中的理论研究和技术发展具有重要价值。

JHEP03(2019)097

the formulas

f({¯α}, {α}) =

∈α

∈¯α

f(v

, u

) , f

({α}, {α}) =

∈α

6=u

f(u

, u

) , a({¯α}) =

∈¯α

a(v

) .

3 From the Gaudin determinant to the square and back

In this section we propose the square form factor and a consistency condition for it. After

proposing this consistency equation, we verify that the Gaudin norm fulﬁls it as a check for

our proposal. Reversing the argument, we are able to reconstruct the Gaudin determinant

from the consistency condition and some initial conditions and analytic considerations.

3.1 The square bootstrap and the Gaudin determinant

Inspired by the BKV idea of cutting the three-point correlation function into two hexagon

form factors [23], we apply the same procedure to two-point functions, i.e. a cylinder

worldsheet. In this case we will obtain two squares after cutting two times, each with two

physical sides and two mirror sides, being the last the new sides created by this process.

We decided to deﬁne our square by cutting two times the cylinder instead of one time

because in the later case the two mirror sides of the square are constrained to have the

same number of excitations (see ﬁgure 1).

The most general square is characterized by four sets of rapidities, one for each of

the physical sides and one for each of the mirror sides. For the most part of this article

the two mirror sides will be empty, so we only have to indicate the content of the two

physical sides. Therefore we denote the square by S

({u} → {w}), where the subindex

indicate the length of the physical sides. If both sets of rapidities are equal, we will just

write S

({u}) = S

({u} → {u}) to alleviate notation. With this notation the relationship

between the scalar product of Bethe states and the square is given by



C(v

)

B(u

)



= S

({u} → {v}) + mirror corrections . (3.1)

The superposition of two Bethe states gives us the two-point function of an operator at

zero with an operator at inﬁnity, i.e., the normalization of the two-point functions. Despite

this being an unphysical quantity, it is important to compute it to correctly normalize our

states. In this article we will consider the L  1 limit, so we can ignore mirror corrections.

In appendix E we will comment about how to include mirror corrections.

Although the construction of the hexagon form factor from symmetry and crossing

arguments is very simple and elegant, a similar construction for the square presents some

obstructions. Since here we are addressing the problem from the weak coupling limit, we

do not have access to the crossing transformation. This transformation entails shifting the

rapidity of an excitation in such a way that it crosses some particular cuts in the rapidity

plane. However these cuts have a width of 4g, thus they close at weak coupling and the

mirror and crossing transformations are inaccessible perturbatively [36]. Nevertheless, an

– 5 –

JHEP03(2019)097

all-loop construction of the square should behave nicely under crossing transformations,

e.g. we expect that S

({u

cross

} → {w

cross

}) = S

({w} → {u}).

Among the several diﬀerences that exist between this square and the hexagon form

factor, we are interested in the one concerning the cutting and the gluing of form factors.

The cutting operation applied to the square remains well deﬁned after we employ it once,

which means that we can continue cutting a square into smaller ones. We can reverse the

argument and state that gluing two squares reconstructs one square. This idea can be used

as a “bootstrap procedure” to construct such square. If we consider only the asymptotic

part of the square (which means directly gluing them without adding excitations to the

mirror edges), the reconstruction formula will be given by

({u} → {v}) =

α∪¯α={u}

β∪

β={v}

w({α}, {¯α}, {β}, {

β})S

({α} → {β})S

({¯α} → {

β}) , (3.2)

where the weight w({α}, {¯α}, {β}, {

β}), which we will call breaking factor, takes into ac-

count the cutting of the physical states into two. We have also deﬁned L = l

+ l

. In this

article we will work in the L ∼ l

∼ l

 1, so we will not consider any potential problems

that could arise from the discrete nature of the states when we cut them. As we said before,

this limit also allow us to ignore the mirror corrections as they are exponentially suppressed.

To construct the breaking factors we proceed in a similar manner as [38], i.e. we write

the monodromy matrix as the product of two monodromy matrices, each associated to a

subchain of the original chain. Using the deﬁnition of the monodromy matrix given in

equation (2.2) we can rewrite the product of L Lax matrices into two separate products

with l

and l

Lax matrices each. Equating the entries of the auxiliary space we can

prove that

(L)

(u) = T

)

(u) ⊗ T

)

(u) =⇒

(

(u) = B

(u) ⊗ D

(u) + A

(u) ⊗ B

(u)

(u) = C

(u) ⊗ A

(u) + D

(u) ⊗ C

(u)

. (3.3)

These formulas provide us with the recipe to break the creation and annihilation operators

for the case of one excitation. For the general square we need the expression of several

creation and annihilation operators. As the RTT relations involve at most two rapidities,

it is easy to argue that the structure of the weight functions can be ﬁxed from the breaking

factor for two excitations. For the case of creation operators we have

B(u)B(v) = B

(u)B

(v) ⊗ D

(u)D

(v) + B

(u)A

(v) ⊗ D

(u)B

(v)

+ A

(u)B

(v) ⊗ B

(u)D

(v) + A

(u)A

(v) ⊗ B

(u)B

(v)

= B

(u)B

(v) ⊗ D

(u)D

(v) + f(u, v)B

(u)A

(v) ⊗ B

(v)D

(u)

− g(u, v)B

(u)A

(v) ⊗ B

(u)D

(v) + f(v, u)B

(v)A

(u) ⊗ B

(u)D

(v)

+ g(u, v)B

(u)A

(v) ⊗ B

(u)D

(v) + A

(u)A

(v) ⊗ B

(u)B

(v) .

Although this is true for cutting the square vertically, horizontal cuts should behave diﬀerently. We

will not address that kind of cuts in this article.

– 6 –

剩余31页未读，继续阅读

weixin_38642897

粉丝: 3
资源: 895

构建正方形形式因子：AdS-CFT对与积分场论

苏教版六年级数学下册第一次月考试卷精选.doc

人教版六年级数学上册期中考试题及答案【通用】.pdf

入门，从一个按钮和灯开始学PLC吧！（三）

param-1.12.2-py2.py3-none-any.whl

财务系统学生更改密码和绑卡操作方法.zip

基于MATLAB实现旅行推销员问题(TSP)代码+项目说明(课程大作业)+测试数据.zip

Python360翻译网页版GUI工具

志汇叮咚超级外卖6.4.3全开源完整包.zip

UE4环境安装.txt

pycares-3.1.1-cp36-cp36m-win_amd64.whl

最新资源