AdS3/CFT2中的旋转OPE块：新表示与AdS2几何关联
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本文主要探讨了在AdS/cft对偶框架下，对CFT_2中的旋转原色OPE块（Operator Product Expansion block）进行深入研究。作者们在[48]和[49]的基础上，扩展了之前的工作，旨在获取CFT_2中用于旋转原色的OPE块的积分表达式。关键发现是，当OPE块由保守的旋量原色构成时，积分过程表现出显著的特点：它转化为了AdS_2测地线上两个加权场的乘积，而非简单的AdS_3测地线运算。
这一结果揭示了在CFT_2的背景下，保守的旋量OPE块实际上以一种不同于AdS_3测地线算符的方式被表示，这提示了在低维AdS空间（AdS_2）中对这些算符的新理解。作者强调，这种表示方式对于理解四点旋转共形块具有重要意义，因为它们可以被解析为两个AdS_2测地线Witten图的乘积，从而提供了更为直观的计算工具。
此外，文章还讨论了如何通过HKLL散射场重构方法，将AdS_2上的这种表示形式与AdS_3中的无质量高自旋场相联系。这种方法允许作者将CFT_2中的保守OPE块与其在AdS_3背景下的物理意义建立连接，这对于理论物理学家来说是一个重要的进展，因为它不仅加深了对AdS/cft对偶的理解，也为高维引力理论和量子信息等领域提供了新的见解。
总结来说，这篇论文的核心贡献是提供了一个新颖的视角来处理CFT_2中的旋转OPE块，并通过具体的数学公式和AdS几何结构，展示了它们在不同维度AdS空间中的不同表现形式及其物理含义。这对于推动AdS/cft对偶的进一步研究和相关领域的实证验证具有重要意义。
Physics Letters B 792 (2019) 397–405
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Comments on spinning OPE blocks in AdS
3
/CFT
2
Suchetan Das
Ramakrishna Mission Vivekananda Educational and Research Institute, Belur Math, Howrah-711202, West Bengal, India
a r t i c l e i n f o a b s t r a c t
Article history:
Received
6 February 2019
Received
in revised form 25 March 2019
Accepted
28 March 2019
Available
online 2 April 2019
Editor:
M. Cveti
ˇ
c
We extend the work of [48], [49], to obtain an integral expression of OPE blocks for spinning primaries
in CFT
2
. We observe, when the OPE blocks are made out of conserved spinning primaries, the integral
becomes a product of two copies of weighted AdS
2
fields, smeared along geodesics. In this way, conserved
current OPE blocks in CFT
2
have a different representation in terms of AdS
2
geodesic operators, in stead
of viewing them as AdS
3
geodesic operators. We also show, how this representation can be related
to AdS
3
massless higher spin fields through HKLL bulk field reconstruction. Using this picture, we
consistently obtain the closed form expression of four point spinning conformal block as a product of
two AdS
2
Geodesic Witten diagrams.
© 2019 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
Recently, using embedding space formalism [1–6], a general-
ization
of scalar Geodesic Witten diagram (GWD) [7]to spinning
geodesic Witten diagram has been proposed [25–32].
1
This com-
putes
conformal partial waves (CPW) ([8–10]) with spinning ex-
change
operator as well as spinning external operators in CFT
d
([11–24]). For scalar GWD we have:
W
.0
(x
1
, x
2
, x
3
, x
4
) ∝
γ
12
dλ
γ
34
dλ
G
b∂
(y(λ), x
1
)G
b∂
(y(λ), x
2
) ×
G
bb
(y(λ), y(λ
); )G
b∂
(y(λ
), x
3
)G
b∂
(y(λ
), x
4
) (1)
where G
b∂
, the bulk to boundary propagator and G
bb
, the bulk to
bulk propagator in AdS
d+1
are integrated over two geodesics γ
12
and γ
34
. This prescription could be understood from a more funda-
mental
identification between the so called ‘OPE blocks’ B
ij
k
(x
1
, x
2
)
(building block of an OPE) in the CFT and ‘Geodesic operators’ in
the bulk [36], [37], [39].
2
In CFT
2
[44–46], the expression for spinning conformal block
involves a product of two CFT
1
conformal blocks k
h
(z) [47].
W
h
i
,
¯
h
i
h,
¯
h
(z,
¯
z) = k
h
(z)k
¯
h
(
¯
z) + k
¯
h
(z)k
h
(
¯
z), (2)
E-mail address: suchetan1993@gmail.com.
1
In the first order Hilbert-Palatini formalism of gravity, the holographic dual con-
formal
blocks are also studied using open Wilson networks [33–35].
2
See [40–43]for dual description of OPE blocks in states other than vacuum.
z =
z
12
z
34
z
13
z
24
; k
h
(z) = z
h
2
F
1
(
h − h
12
, h + h
34
; 2h; z
)
(3)
This (anti)holomorphic factorization implies that in 2D CFT, in
stead of using AdS
3
GWD, one can express any spinning CPW as
a product of two AdS
2
GWD. A related question is to ask how
the spinning OPE blocks in this case are related in terms of AdS
2
Geodesic operators. The main point of this note is to explore this
connection between spinning OPE blocks and AdS
2
geodesic oper-
ators.
Our
starting point is the observation that in 2D CFT, the method
of finding an integral expression for scalar OPE block discussed in
[48], [49], can be easily extended to the case of spinning OPE block,
constructed out of spinning primary field. This type of simple ex-
tension
is special in CFT
2
, where all spinning primaries are labeled
by
two real numbers (h,
¯
h), which is unavailable in higher dimen-
sions.
3
For instance, using such simplification, available only in 2D,
Osborn has derived [46]the closed form expression of spinning
conformal block (2)by a simple generalization from the spinless
case in CFT
2
, without having to use the embedding space for-
malism,
which is necessary for deriving expressions for conformal
blocks for four point function of non-zero spin operators in higher
dimensions.
3
In D=2, the conformal symmetry is infinite dimensional and so the OPE’s can
be organized into sum of blocks, governed by the full Virasoro symmetry, but here
we consider only global OPE blocks, which are governed by the global conformal
symmetry. In general, closed form expression of Virasoro conformal block is not
known yet, except in a particular limit where the Virasoro block reduces to the
global block [50]and hence, one can use GWD prescription to compute the block
from AdS
3
[51–54].
https://doi.org/10.1016/j.physletb.2019.03.058
0370-2693/
© 2019 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.
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