N=4超杨-米尔斯理论中颜色不变量的威尔逊环

83 浏览量更新于2024-07-16 收藏 822KB PDF 举报

"这篇学术论文详细探讨了威尔逊循环（Wilson loops）在N=4超杨-米尔斯理论（$$ \mathcal{N} = 4 $$ super Yang-Mills theory）中的1/2 BPS（Bogomol'nyi-Prasad-Sommerfield）圆环路径的真空期望值（VEV）。研究者通过颜色不变性方法得到了适用于任何群G的任意表示R的VEV表达式。这个表达式揭示了不同表示间的精确关系，并在扰动理论中将颜色不变量简化到七阶，同时证明了对于<W> R的对数比<W> R本身具有更简洁的形式。在SU(N)对称和反对称表示的特殊情况下，发现<W> R的对数满足二次Casimir因子分解，这一性质至少在七阶是成立的。最后，论文中还推导了对于任意但固定的SU(N)表示，<W> R的大N展开，直到1/N^2项。" 这篇由Bartomeu Fiol、Jairo Martínez-Montoya和Alan Rios Fukelman合作完成的研究发表在JHEP期刊的2019年5月刊上，研究编号为JHEP05(2019)202。文章深入研究了量子场论中的一个重要概念——威尔逊循环，它是理解和计算强相互作用物理过程的关键工具。N=4超杨-米尔斯理论是一种特殊的量子场论模型，具有高对称性和强大的数学结构，常用于理论物理学的高级研究。在本文中，作者首先提出了一种新的技术，用颜色不变量来描述1/2 BPS Wilson循环的VEV，这种方法对于分析不同表示下的VEV关系非常有用。颜色不变性是描述量子色动力学（QCD）中夸克和胶子相互作用的关键概念，它确保了物理量在颜色空间中的不变性。通过对颜色不变量进行扰动理论计算，作者简化了这些不变量，从而得到更易于处理的表达式。此外，他们在SU(N)群的对称和反对称表示下，观察到一个有趣的特性：<W> R的对数可以分解为二次Casimir因子，这表明存在一种深刻的对称性。Casimir操作符在代数结构中扮演着重要角色，其平方项通常与群的表示的维度有关。这种分解对于理解Wilson循环的性质和简化计算具有重要意义。最后，论文进行了大N展开的计算，这是在处理无穷维矩阵模型或大N极限时常用的技术。通过这种方式，他们能够解析地探索在N趋于无穷时Wilson循环的行为，这对于理解和预测强相互作用系统的行为至关重要。这项工作为N=4超杨-米尔斯理论中的Wilson循环提供了新的见解，对高能物理和量子场论的理论研究产生了深远的影响。通过颜色不变性的视角，不仅深化了我们对理论结构的理解，也提供了更有效的计算手段。

JHEP05(2019)202

2 hW i

in terms of color invariants

In this section we revisit the evaluation of hW i

for N = 4 super Yang Mills, for an

arbitrary representation R of a generic Lie algebra G. Thanks to supersymmetric localiza-

tion [21] this problem reduces to a Gaussian matrix model, and it has been solved exactly,

for various choices of gauge group G and representation R [25–27]. As mentioned in the

introduction, typically this is done by ﬁrst reducing the matrix integral to an integral over

the Cartan subalgebra, as in (1.9). While this procedure allows to obtain compact expres-

sions for hW i

for some choices of R, this has to be done in a case by case basis, and it

obscures the dependence on the choice of G and R. Since in this work we are particularly

interested in expressing hW i

in terms of color invariants, we will follow a diﬀerent route.

We will instead carry out the integrals over the full Lie algebra. Speciﬁcally,

hW i



2πM



dim R

dM tr

2πM

−

8π

tr M

dM e

−

8π

tr M

(2.1)

If we denote by m

the coeﬃcients of the matrix M in the Lie algebra, the two-point

function in this Gaussian matrix model is

8π

a, b = 1, . . . , d

(2.2)

To compute the vev of the normalized Wilson loop, we expand the exponent insertion

in (2.1), use the two-point function (2.2) and apply Wick’s theorem,

hW i

∞

k=0

(2π)

(2k)!

. . . m

itr T

. . . T

∞

k=0

...a

(2.3)

where g = g

/4 and d

...a

are the symmetrized traces deﬁned in (1.2). This expression

for hW i

N =4

is exact — recall that there are no instanton corrections for hWi

in N = 4 [21]

— and valid for any G and any R. It encompasses and uniﬁes all the known results for

particular choices of G and R [25–27].

Now it is a matter of evaluating d

...a

, the fully symmetrized traces (1.2) with

pairwise contracted indices. At every order, the outcome is a combination of lower order

color invariants, involving the original representation R, and the adjoint representation A.

At low orders, it’s easy enough to evaluate them by hand, using the techniques detailed

in [7]. For instance,

= tr T

= c

aabb



+ T





−



– 5 –

JHEP05(2019)202

To push the evaluation to higher orders, we use FormTracer [28]. Up to order g

we obtain

hW i

= 1 + c

g +



−





−





− c

−

abcd





−

432



−



abcd



−

288

−

+ c

abcd

cdef

efab



−

180



abcd



−



abcd

−

abcdef

−

− c

) +

175

−

385

288

72757

326592



−



abcd



−

817

3240



abcd

691

18900

abcd

cdef

efab

−

abcdef

abcg

defg

abcdef

abcg

defg

− c

)

abcd

cdef

efab

− c

)

abcdef

+ . . . (2.4)

By construction, every color invariant in this expansion involves an even number of indices.

Since d

...a

= 0 for k odd, for every color invariant the adjoint representation contributes

an even number of indices, and thus the representation R also contributes an even number.

Up to the order computed, no color invariants involving d

...a

with an odd number of

indices appear in the expansion. They would necessarily involve more than one d

...a

, e.g.

abc

ade

bcde

. It is not clear to us whether such color invariants will appear at higher orders.

The reader that feels intimidated by the expansion (2.4) might ﬁnd some comfort

in the fact that, as we will show in section 4, the perturbative expansion of ln hW i

considerably simpler.

Besides the possibility of evaluating hW i

order by order in g

for all G and R at

once, the result (2.3) allows to derive some general exact relations among vevs of 1/2 BPS

Wilson loops in diﬀerent representations. The ﬁrst identity that we will point out is rather

obvious. For a generic representation R, recall that the complex conjugate representation

– 6 –

剩余30页未读，继续阅读

weixin_38608693

粉丝: 2
资源: 907

N=4超杨-米尔斯理论中颜色不变量的威尔逊环

matlab曲线的颜色代码-FisherLuDAlessandroWilson_AnalysisCode:Fisher等人的分析代码。2019

威尔逊循环5d N = 1 $$ \ mathcal {N} = 1 $$理论和S对偶

自交叉威尔逊循环模拟双部分散射的所有阶次结果

弗兰克导联和威尔逊导联区别

中心差分法与纽马克和威尔逊法对比

matlab威尔逊方程

r语言绘制向量威布尔函数

C语言程序运行时，从键盘输入正整数N，输出大于等于N的最小整数x，要求x是一个素数，且x的各个数字之和也是素数。

如何利用Volterra/Wiener方法分析非线性系统的输入/输出特性？

在项目实战中，如何运用Volterra/Wiener方法对非线性系统进行输入/输出特性分析，并举例说明如何构建Volterra级数模型？

最新资源