AdS/dCFT中的两点函数与边界保形自举方程：N=4 SYM缺陷理论的计算
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本文主要探讨了AdS/dCFT框架下的两点函数及其在边界保形自举方程中的应用，焦点在于N=4超级杨-米尔斯（SYM）理论的一个缺陷版本。这个缺陷理论与D5-D3探针-布雷系统相对应，其中包含k个单位的背景 gauge 通量。在这个背景下，研究者计算了受保护的标量算子之间的连结两点函数，这些函数在没有缺陷的N=4 SYM理论中通常是不存在的，例如不等角维算子的两点函数。
计算过程涉及到对理论中的关键性质的理解，包括不同类型的两点函数在缺陷存在时的行为，以及它们如何在理论的对称性和约束条件下变化。作者通过运用算子乘积展开（OPE）和边界算子展开（BOE），这两种工具是边界保形自举方程的核心，来深入探究缺陷CFT的对称性和拓扑性质。通过OPE，他们能够提取出N=4 SYM理论中的一些结构常数，这需要对单点函数的非平凡多项式k相关性有深入了解，这部分已经被明确证明。同时，对SU(2)后代运算符的单点函数的研究也是必不可少的，这是另一个明确确定的组成部分。
论文的贡献不仅限于理论计算，还在于对边界耦合的约束，这些耦合关系是通过BOE得到的。OPE的使用依赖于对缺陷理论一点和两点函数的精确知识，而BOE则提供了另一种分析工具，它有助于揭示边界条件下的额外物理信息。整个研究过程展示了AdS/dCFT对称性的强大之处，以及如何通过这种对称性来探索深层次的量子场论性质。
这篇发表在JHEP08(2017)020上的文章，通过对AdS/dCFT中两点函数的精细分析，深化了我们对N=4 SYM理论缺陷版本和其边界行为的理解，同时也促进了边界保形自举方程这一前沿领域的发展。作者们的工作为理论物理学家提供了宝贵的洞察，特别是在量子场论与几何背景相互作用的复杂系统中。
JHEP08(2017)020
A CFT with a boundary or defect has a richer structure. Here, also one-point functions
of composite operators O
i
can be non-vanishing. Conformal symmetry and the scaling
dimension ∆
i
of the operator fix the one-point functions up to a constant a
i
[1]:
7
hO
i
(x)i =
a
i
(2x
3
)
∆
i
. (2.8)
Thus, one-point functions in a dCFT exhibit a complexity similar to three-point functions
in a CFT. Two-point functions in a dCFT can be non-vanishing also for operators of
unequal scaling dimensions and are fixed to be of the form
hO
i
(x)O
j
(y)i =
f(ξ)
(2x
3
)
∆
i
(2y
3
)
∆
j
, (2.9)
where f(ξ) is a function of the conformal ratio (2.6).
The two-point function tends to the one in the absence of the defect if the distance to
the defect is large compared to the distance between the points:
lim
z
3
→∞
hO
i
(x + z)O
j
(y + z)i =
M
ij
|x − y|
∆
i
+∆
j
. (2.10)
Then, since the OPE expansion is around the point ξ = 0, the OPE of the operators in the
bulk (2.7) is unchanged. Using this OPE, we can express the two-point function in terms
of ∆, λ and a as [2]
f(ξ) = ξ
−
∆
i
+∆
j
2
"
M
ij
+
X
k
λ
ij
k
a
k
F
bulk
(∆
k
, ∆
i
− ∆
j
, ξ)
#
, (2.11)
where the bulk conformal block is given by
F
bulk
(∆, δ∆, ξ) = ξ
∆
2
2
F
1
(
1
2
(∆ + δ∆),
1
2
(∆ − δ∆); ∆ − 1; −ξ) . (2.12)
Thus, two-point functions in the dCFT exhibit a complexity similar to the one of four-point
functions in a usual CFT.
A further feature of dCFTs is the existence of boundary or defect operators. Since
the theory on the defect is a usual CFT in one dimension less, the space-time dependence
of the boundary-boundary two-point functions is completely determined by the scaling
dimensions. Moreover, the boundary operators posses an OPE fixed by structure constants
ˆ
λ, which allows to construct all higher-point functions of boundary operators.
8
However, we can also have non-vanishing two-point functions between a bulk and a
boundary operator. Via conformal symmetry, these are fixed to be of the form
hO
i
(x)
ˆ
O
j
(y)i =
µ
ij
(2x
3
)
∆
i
−∆
j
|x − (y, 0)|
2∆
j
, (2.13)
7
We adopt here the convention usually used in the boundary bootstrap program, which involves an
explicit factor of 2 in the space-time dependence.
8
Strictly speaking, one can even eliminate the bulk structure constants λ from the list of conformal data
by expressing them in terms of the boundary structure constants
ˆ
λ and the bulk-to-boundary couplings µ
defined below, see for instance [3, 4]. This is, however, not the approach we will be taking here.
– 6 –
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