1/2-BPS Wilson环与AdS2/CFT1：强耦合下的算子插值与AdS维度振动
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更新于2024-07-16
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本文主要探讨的是"半BPS Wilson循环与AdS 2 / CFT 1"的关系，这是在N = 4超级杨-米尔斯理论的背景下进行的研究。1/2-BPS Wilson线是一种在该理论中具有特定对称性的物体，它保留了一维超共形对称性。在这个框架内，研究的是线上的局部算子插入的相关函数，这些函数的性质受到这个超对称性的严格制约，定义了一个缺陷CFT 1，即一种在1/2-BPS Wilson线上存在的局域量子场论。
在强耦合极限下，理论中的一个关键发现是，一组具有受保护的尺度维度的基本算符插入，对应于AdS 5 × S 5 中的双重基本弦的波动。这些波动最终会在AdS 5 的边界处以直线形式结束，而且可以被视作在AdS 2 世界面上传播的轻型场。通过AdS / CFT对偶，研究人员使用了AdS空间/共形场论的技术来计算这些算子插入的四点函数在树级水平的AdS 2 维滕图。这些图形提供了强耦合条件下双算子插入的物理描述。
此外，文章还涉及了OPE（Operator Product Expansion，运算符乘积展开）技术，通过这种技术，作者能够提取出基于基本激发的“双粒子”算子的异常尺度的领先强耦合修正。这些修正反映了量子效应对经典理论的修正，对于理解理论的精细结构至关重要。
特别地，当讨论到圆形Wilson环时，研究者将特殊类型的标量插入的四点函数结果与二维杨-米尔斯理论的定位预测进行了比较，展示了AdS / CFT对偶的有效性和理论一致性。
这篇论文深入研究了1/2-BPS Wilson线的物理性质及其与AdS 2 / CFT 1之间的联系，提供了一个从强耦合视角计算量子场论相关函数的有力工具，同时也验证了AdS / CFT对偶在处理复杂理论问题时的适用性。这项工作对于理解弦理论、量子引力以及强相互作用等领域有着重要的理论价值。
504 S. Giombi et al. / Nuclear Physics B 922 (2017) 499–527
stress that compared to the gravitational AdS
2
models in [41–45] our bulk action (2.4) is defined
in fixed AdS
2
background, i.e. does not contain gravity: before fixing the static gauge
7
the string
action (2.1) is reparametrization invariant, but gravity never becomes dynamical in critical super-
string theory. In line with this, the boundary theory has no analog of the pseudo-Goldstone mode
[42] related to the (spontaneously broken) reparametrizations.
2. AdS
5
× S
5
string action in static gauge as AdS
2
bulk theory action
The bosonic part of the superstring action in AdS
5
×S
5
has the standard form
S
B
=
1
2
T
d
2
σ
√
hh
μν
1
z
2
∂
μ
x
r
∂
ν
x
r
+∂
μ
z∂
ν
z
+
∂
μ
y
a
∂
ν
y
a
(1 +
1
4
y
2
)
2
,T=
√
λ
2π
, (2.1)
where σ
μ
= (t, s) are Euclidean world-sheet coordinates, r = (0, i) = (0, 1, 2, 3) label coordi-
nates of the Euclidean 4-boundary and a = 1, ..., 5are S
5
labels. The minimal surface corre-
sponding to the straight Wilson line at the boundary is described by
z = s, x
0
= t, x
i
= 0 ,y
a
= 0 . (2.2)
The corresponding induced metric is that of AdS
2
, i.e. g
μν
dσ
μ
dσ
ν
=
1
s
2
(dt
2
+ds
2
).
We will study correlators of small fluctuations of “transverse” string coordinates (x
i
, y
a
) near
this minimal surface that will thus propagate in the induced AdS
2
metric. The resulting global
symmetry of the bosonic action will thus be SO(2, 1) ×[SO(3) ×SO(6)]. To make the SO(2, 1)
symmetry (which will be the conformal symmetry at the corresponding 1d boundary theory)
manifest it is useful to choose the AdS
2
adapted coordinates and fix the static gauge in which
z and x
0
do not fluctuate. The relevant embedding of AdS
2
into AdS
5
is described by (x
2
≡
x
i
x
i
, i = 1, 2, 3)
ds
2
5
=
(1 +
1
4
x
2
)
2
(1 −
1
4
x
2
)
2
ds
2
2
+
dx
i
dx
i
(1 −
1
4
x
2
)
2
,ds
2
2
=
1
z
2
(dx
2
0
+dz
2
). (2.3)
Starting with the Nambu action and fixing the static gauge by the conditions on x
0
and z as in
(2.2) we get
S
B
= T
d
2
σ
det
(1 +
1
4
x
2
)
2
(1 −
1
4
x
2
)
2
g
μν
(σ ) +
∂
μ
x
i
∂
ν
x
i
(1 −
1
4
x
2
)
2
+
∂
μ
y
a
∂
ν
y
a
(1 +
1
4
y
2
)
2
≡ T
d
2
σ
√
gL
B
, (2.4)
where g
μν
=
1
s
2
δ
μν
is the background AdS
2
metric. This action can be interpreted as that of a
straight fundamental string in AdS
5
× S
5
stretched along z, i.e. from the boundary towards the
center of AdS
5
. It may be also viewed as a 2d field theory of 3 +5 scalars in AdS
2
geometry
with manifest symmetry SO(2, 1) ×[SO(3) × SO(6)]. Interpreted as a 2d bulk AdS
2
theory, it
7
Defining the Wilson loop expectation value in string theory in conformal gauge where one has two more (compared
to physical static gauge) dynamical coordinates and ghosts one would end effectively with an integral over boundary
reparametrizations (see [46–48]). In this case the identification between the operators on the Wilson line on the gauge
theory side and the string excitations appears to become more intricate. This question deserves further study.
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