N=4非手性超空间中光滑Wilson环：AdS/CFT对偶的基石

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本文主要探讨的是4维N=4非手性超空间中的平滑Wilson环（Smooth Wilson Loops）在超 Yang-Mills理论中的应用。这种特殊类型的Wilson环是AdS/CFT对偶（Anti-de Sitter/Conformal Field Theory correspondence）框架下AdS_5×S^5超弦理论中的一个关键元素。它超越了传统的Bosonic 1/2 BPS马尔达西纳-威尔逊环（Maldacena-Wilson Loop），后者仅涉及光锥方向的运动。 N=4非手性超空间是指在保持部分N=4超对称性的同时，允许更广泛路径的超几何结构，与传统的N=4超对称性中的限制有所不同。这个Wilson环算子不仅具有全局超保形性，还具有局部κ对称性，这表明它不仅受到整体超群的保护，而且在路径选择上也具有一定的灵活性。作者们在文章中详细分析了这一算子的性质，并特别关注其真空期望值（Vacuum Expectation Value, VEV）在量子场论中的行为。他们计算了在非手性超空间中，特别是在反交换坐标系和特定度规下的路径级别的VEV，直至四阶。这种方法确保了路径的连续性和光滑性，即使在路径相交时，路径导数的变化也是连续的。文章的重要贡献在于它处理了之前文献中关于非手性超空间内平滑路径的空白，通过二次样条函数构建的循环，这些循环在连接路径时保证了物理量的连续性。这一工作对于理解AdS/CFT对偶中的非传统对象，如非光锥方向的量子效应，具有重要意义。这篇论文提供了一个深入研究N=4非手性超空间中平滑Wilson环的关键见解，对于超对称理论、AdS/CFT对偶以及弦理论等领域有着重要的理论价值和实际应用潜力。由于是开放获取（Open Access）资源，这篇研究成果便于学术界广泛的查阅和进一步研究。

JHEP12(2015)140

where Q are the supercharges ac ti ng on the ﬁelds. This is equivalent to the familiar

translation property

O(x) = e

O(0)e

−xP

, (2.16)

with the diﬀerence that the translations in superspace do not commute in general. Using

this we can deﬁne the operator at an arbitrary point in superspace by

O(x, θ) = e

xP+θQ

O(0)e

−xP−θ Q

. (2.17)

The supersymmetry transformations of a local gauge-invariant operator are given by

ǫQ

O(x, θ)e

−ǫQ

= O(x − ǫΓθ, θ + ǫ). (2.18)

Looking at the linear term s in ǫ we ﬁnd that

ˆα

, O(x, θ)] = −Q

ˆα

O(x, θ), (2.19)

where Q

ˆα

is the representation of the supercharge acting on the coordinates . Expl icitly,

we have Q

ˆα

= −∂

ˆα

+ (Γ

ˆµ

θ)

ˆα

∂

ˆµ

. Similarly, we obtain that P

ˆµ

= −∂

ˆµ

and [P

ˆµ

, O(x)] =

−P

ˆµ

O(x) = ∂

ˆµ

O(x). The representation on the coord i nate s of course satisﬁes the same

graded commutation relations as the representation on the ﬁelds, i.e. {Q

ˆα

, Q

} = 2Γ

ˆµ

ˆα

ˆµ

It may happen that a particul ar combination of Poincar´e supercharges annihilates

an operator [ǫQ, O(x)] = 0 for some spinor ǫ. Then the supersymmetric version O(x, θ)

depends only on some θ’s and the r ef or e the Graßmann expansion is shorter. These are

BPS operators. The condition [ǫQ, O(x)] = 0 on the bottom component gets translated into

another condi ti on on its supersymmetric version O(x, θ). Working to ﬁrst order i n ǫ, we ﬁnd

O(x, θ) = e

θQ

ǫQ

O(x)e

−ǫQ

−θQ

= e

(θ+ǫ) Q+(ǫΓ

ˆµ

θ)P

ˆµ

O(x)e

−(θ+ ǫ)Q−(ǫΓ

ˆµ

θ)P

ˆµ

= O(x + ǫΓθ, θ + ǫ) = O(x, θ) + (ǫD)O(x, θ) + O(ǫ

). (2.20)

Here D

ˆα

is the super s y mme tr y covariant derivative (

2.1). The conclusion of this compu-

tation is that [Q

ˆα

, O(x)] = 0 implies D

ˆα

O(x, θ) = 0. This concludes the discussion of local

operators.

Kappa-symmetry. Let us now consider a Wilson loop on a light-like path in ten dimen-

sions. The conne c ti on can be written as A(τ) = dτ ˙x

ˆµ

. Using the inﬁnitesimal action of

the supercharges on the fundamental ﬁelds A and ψ yields

ˆµ

(x) = [ǫQ, A

ˆµ

(x)] = −ǫΓ

ˆµ

ψ(x), (2.21)

ψ(x) = [ǫQ, ψ(x)] =

(

ˆµˆν

ǫ)F

ˆµˆν

(x), (2.22)

and we ﬁnd

[ǫQ, A] = −dτ(ǫ ˙x

ˆµ

ψ). (2.23)

The minus s ig n in the equation below is crucia l. It ensures the invariance of a superﬁeld under the

combined ﬁeld and coordinate t ra n s fo rm a tio n s .

– 8 –

JHEP12(2015)140

If ˙x is a null vector, this variation vanishes if ǫ = κ ˙x

ˆµ

, where κ is a spinor and

Γ are

the te n-di m ens i onal Pauli matrices with upper indices (see appendix

A.2). This is a local

condition since ˙x is in general not constant, so diﬀerent supersy mme t r i es are preserved at

diﬀerent points around the Wil s on lo op contour. It is a 1/2 BPS condition , as the spinor

ˆα

has 8 degrees of freedom as ˙x

ˆµ

for null ˙x

ˆµ

has an 8 dimensional kernel.

Just like for local operators, the invariance of the bosonic Wilson loop under some

local worldline supers y mme tr y transformati ons lifts to the invariance of the super Wilson

loop under some local sup e r -di ﬀ eom orp hi sms . These super-diﬀeomorphisms are obtained

by replacing ǫ → κp

ˆµ

and Q → D, where p

ˆµ

= ˙x

ˆµ

+ θΓ

ˆµ

θ is the super-momentum, a

supersymmetrization of ˙x

ˆµ

The kappa-symmetry transfor mations are just the transformations of the su perspace

coordinates under the sup e r- di ﬀe omor phi s m κ

ˆα

ˆµ

ˆα

ˆµ

≡ ǫ(κ)

ˆα

. For example, we ﬁnd

ˆµ

ǫ(κ)

ˆα

, p

ˆµ

= 2ǫ(κ)Γ

ˆµ

θ, δ

ˆα

ǫ(κ)

, θ

ˆα

= ǫ(κ)

ˆα

. (2.24)

We should note that out of the 16 components of κ, only 8 appear in the transformations,

while the other 8 are projected out when acted upon by the matrix p

ˆµ

. Using the light-

like condition on p it is easy to show that there is an eq ui valence relation κ ∼ κ + p ·Γξ on

the spinors κ. That is, κ and κ + p · Γξ produce the same kappa-transformation.

Let us now show that the full 10d super Wilson loop (

2.10) is invariant under the

kappa-symmetry transformat i ons. The kappa-transformat i ons can, as explained above, be

seen as diﬀeomorphis ms in super s pac e generated by the vector ﬁeld

ˆα

∂

ˆα

+ (δ

θΓ

ˆµ

θ)∂

ˆµ

= δ

ˆα

=: ζ

with ζ

= (0, δ

ˆα

) , (2.25)

where A = (ˆµ, ˆα) is a collective superspace ind ex . Under a general diﬀeomorphism ζ

the super-connection transforms as

A =





dτ p

dτ



, ζ

i

, (2.26)

where we have introduced the abbreviation p

= (p

ˆµ

ˆα

). Note that the latter term

in eq uat i on (

2.26) r e pr e se nts an inﬁnitesimal, ﬁeld-depende nt gauge transformation and

can ther e for e be neglected if δ

is appli e d to a gauge invariant operator. By applying

equation (

2.26) to the case of kappa-transformati ons one ﬁnds the following expression for

the variat i on of the super Wilson loop ope r ator

W =

tr P



exp





δA



tr P



exp





dτ



ˆµ

ˆα

ˆαˆµ

ˆα





. (2.27)

With the constraints F

ˆα

= 0 (

2.5) and F

ˆµ ˆα

= Γ

ˆα

β,ˆµ

as discussed in section 2.1 we see

that the above vanishes for p

ˆµ

= 0 as

ˆα

ˆµ

ˆα

β,ˆµ

= (κ

ˆν

ˆµ

Ψ) p

ˆν

ˆµ

∝ p

ˆµ

= 0. (2.28)

– 9 –

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