N=4 SYM Wilson环精确关联器：定位、缺陷CFT与可积性解析

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本文主要探讨了在N=4超 Yang-Mills (SYM) 理论中的精确相关器在1/8 BPS Wilson环上的应用。作者利用超对称定位、operator product expansion (OPE) 和 Gram-Schmidt正交化方法，计算了一类特殊类型的运算符插入相关函数。这些相关函数展现出显著的特征：它们具有行列式的简单结构，不受位置的影响，且属于拓扑子部门，即它们不依赖于理论的't Hooft耦合和群论等级。在1/2 BPS的圆周或直线Wilson环路的应用中，这些结果揭示了一个丰富的无限精确缺陷CFT（缺陷共形场论）数据集，特别是关于在环路上插入任意长度的受保护缺陷原色的结构常数，这对于理解理论的非平凡物理行为至关重要。在强耦合极限下，作者展示了与AdS2弦世界表的直接计算高度一致，这进一步验证了计算的精度。文章还涉及了与之前通过可积性技术得出的“广义Bremsstrahlung函数”的联系。在平直空间的极限中，作者的计算结果能够复现已知的结论，并且提供了一种有限的N（阿贝尔规范群的大小）推广。此外，值得注意的是，较大的N处的相关器可以通过“量子光谱曲线”方法中的多项式乘积（Q函数）表示为简单的积分形式，这揭示了定位、缺陷CFT和理论的可积分性质之间深刻的相互作用。这项研究不仅深化了我们对N=4 SYM理论中精确计算的理解，而且通过不同理论工具的交叉应用，展示了理论的丰富结构和潜在的统一性。这一工作对于探索量子场论的更深层次性质以及可能的物理应用具有重要意义。

JHEP05(2018)109

these quantities are nontrivial functions of λ and N. Note that, although we often set the

normalization of the two-point function to be unity in conformal ﬁeld theories, for special

operators the normalization itself can have physical meaning.

For instance, the length-1

operator (Y ·

Φ) is related to the displacement operator and has a canonical normalization

which is related to the Bremsstrahlung function [22].

Now, if we go to the topological conﬁguration by setting the vectors Y

to be

= (cos τ

, sin τ

, 0, i, 0, 0) , (2.16)

we get

topological



−



× n

topological



−



× c

(2.17)

This shows that the topological correlators compute the normalization and the structure

constant in the defect CFT up to trivial overall factors. Alternatively, one can consider

the ratio

)

1/2



topological

)

1/2

, (2.18)

and get rid of the overall factors. The quantity which appears on the right hand side

of (2.18) is a structure constant in the standard CFT normalization; namely the normal-

ization in which the two-point function becomes unity.

Note that, for higher-point functions, there is no such a direct relation between the

general correlators and the topological correlators: the general higher-point correlators are

nontrivial functions of the cross ratios while the topological correlators do not depend at

all on the positions. Thus for higher-point functions, one cannot reconstruct the general

correlators just from the topological correlators.

3 Computation of the correlators

We now compute the correlators on the 1/8 BPS Wilson loop

(τ

)

(τ

) ···

(τ

)i, (3.1)

using the results from localization. We ﬁrst discuss the correlators on the 1/2 BPS Wilson

loop from the OPE perspective and then present a general method that applies also to the

1/8 BPS Wilson loop.

3.1 Correlators on the 1/2 BPS Wilson loop from OPE

When all the operators are length-1 and the Wilson loop is circular (or equivalently 1/2

BPS), the correlators (3.1) were already computed in [32]. Let us ﬁst brieﬂy review their

Other examples are the stress-energy tensor and the conserved currents, whose two-point functions are

related to C

and C

– 7 –

JHEP05(2018)109

computation: by performing localization, one can reduce the computation of the 1/8 BPS

Wilson loop in N = 4 SYM to the computation of the Wilson loop in two-dimensional

Yang-Mills theory in the zero instanton sector [13]. Under this reduction, the insertion of

the position-dependent scalar

Φ is mapped to the insertion of the dual ﬁeld strength ∗F

Φ ⇔ i ∗ F

. (3.2)

Using this correspondence,

one can insert

Φ’s on the circular Wilson loop by diﬀerentiating

its expectation value with respect to the area A:

Φ ···

circle

∂

hWi

(∂A)



A=2π

. (3.3)

Using the expression (3.3), one can compute arbitrary correlation functions of single-letter

insertions

Φ.

To study more general BPS correlators, we also need to know how to insert operators

of longer length,

with L > 1. The ﬁrst guess might be to relate it simply to the L-th

derivative of the Wilson loop,

∼

∂

hW[C]i

(∂A)

? (3.4)

This guess, however, turns out to be incorrect. To see why it is so, let us consider

as an

example. We know that the second derivative of hWi corresponds to the insertion of two

Φ’s on the Wilson loop. Since the correlator we are studying is topological, one can bring

the two

Φ’s close to each other without aﬀecting the expectation value and rewrite them

using the operator product expansion. This procedure does produce the length-2 operator

as we wanted, but the problem is that it also produces other operators:

OPE

Φ =

+ c

Φ + c

1 . (3.5)

Here c

’s are some numerical coeﬃcients and 1 is the identity operator. Thus, to really get

the length-2 operator, one has to subtract these unnecessary OPE terms from

Φ:

: =

Φ − c

1 . (3.6)

Here

Φ on the right hand side denotes two single-letter insertions at separate points while

: is a length-2 operator inserted at a single point. Since this subtraction procedure is

conceptually similar to the normal ordering, we hereafter put the normal-ordering symbol

:∗: to the operator obtained in this way.

The coeﬃcients c

’s are nothing but the OPE coeﬃcients of the topological OPE (3.5).

They are thus related to the following three-point functions:

∝ h

Φi|

circle

, c

∝ h

Φ1i|

circle

. (3.7)

At weak coupling, this correspondence was checked by the direct perturbative computation on both

sides in [21].

Owing to the representation theory of SO(5), the OPE does not produce higher-charge operators,

with k > 2.

– 8 –

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