超重力下不同权重BPS算符的突破性四点函数计算

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本文主要探讨的是超重力逼近（Supergravity Approximation）在计算量子场论中的应用，特别是在弦理论背景下，特别是在AdS/CFT（Anti-de Sitter/Conformal Field Theory）对偶框架下的计算。标题《超重力逼近中的所有不同权重手性主算子的四点函数》揭示了研究者们在强Hooft耦合极限（Strong 't Hooft Coupling Limit）下对N=4超杨梅模型（planar N=4 SYM）中1/2 BPS（半最大超对称BPS）算符的四点函数进行了深入研究。在这一工作中，研究人员首次计算了两个之前未知的四点函数，一个是权重为<2345>的算符，另一个是<3456>的算符。这些权重表示了算符在对称群中的次序，是该领域中的关键物理量。<3456>算符的四点函数尤其引人注目，因为它代表了自强耦合极限理论发展以来计算的下一个、下一个、下一个到最大的相关器，这在理论计算中具有重要意义，因为它突破了以往的计算边界。为了完成这些复杂的计算，研究团队提出了一种新的简化方法，包括对相关函数交换部分和接触部分的直接公式化处理，以及对计算张量C-tensor的算法进行了优化。这些简化不仅提高了计算效率，也为未来类似问题的研究提供了宝贵的参考。作者们利用了有效的IIB型超重力作用（Effective Type-IIB Supergravity Action），这是将弦理论与引力理论结合的一种方式，通过AdS/CFT对偶，将复杂的问题映射到AdS空间中的引力场，使得在强耦合区域的计算成为可能。最后，他们成功验证了最近提出的梅林空间公式（Mellin Space Formula），这是一种在高维理论中计算多点函数的新型工具，它在解析性和效率上超越了传统的方法。这篇工作不仅推进了对N=4 SYM理论的深入理解，也为未来在超重力逼近中的更多计算开辟了新的路径。这篇文章是一个重要的里程碑，它展示了如何在超重力逼近下计算具有不同权重的手性主算子的四点函数，这对于理解超对称量子场论和强耦合物理现象具有深远的影响。

JHEP09(2018)023

C tensors. The N = 4 SYM four-point functions can be represented in many forms

depending on the context. In this paper we will use the language of C tensors to compute

some of the necessary intermediate objects.

The C tensors can be used to track the SO(6)

symmetry instead of the t vectors, by writing (2.1) as

...i

,...,i

, (2.12)

with the tensor C being totally symmetric and traceless in its indices. The vector space

of C tensors with k indices forms a representation space for the traceless symmetric SO(6)

representation with Dynkin labels [0, k, 0]. Choosing a basis in this space amounts to choos-

ing a set of C tensors which we label with an upper index I

that will appear throughout

this paper:

,...,i

, with I

= 1, . . . , dim([0, k, 0]). (2.13)

The possible contractions of four C tensors are characterized by the number of connections

between the diﬀerent C tensors. Denoting the number of connections by a

they should

satisfy a

= a

, a

= 0 and the sum conditions for a’s (2.6), showing that the a partitions

indeed parametrize the possible tensor structures (2.7) for a given correlator (2.2).

3 N = 4 SYM four-point functions from type IIB supergravity

A method to compute the correlation functions from the previous section using the

AdS/CFT correspondence has been known for a long time [3, 4]: chiral primary operators

of 4D N = 4 SYM theory are dual to the Kaluza-Klein modes of type IIB supergravity

on AdS

× S

after compactiﬁcation on S

, which are usually denoted as s

, where as

before the index I

runs over the basis of the corresponding SO(6) representation with

Dynkin labels [0, k, 0]. One computes the on-shell string partition function exp(−S

IIB

) as

a function of the boundary values of the ﬁelds s

(~x) and subsequently interprets it as a

generating functional for correlation functions of CPOs whose sources are s

(~x). Thus, in

practice the correlation functions can be determined from the expression:

(~x

)

(~x

)

(~x

)

(~x

)i =

δs

(~x

)δs

(~x

)δs

(~x

)δs

(~x

)

exp(−S

IIB

)



→0

(3.1)

To compute an n-point function it is necessary to expand the action up to the n-th order in

perturbations of the ﬁelds around their background values. For the four-point case explicit

expressions were derived in [9–11, 27] and the relevant part of the action may be written

in the form:

S =

8π

[dz](L

+ L

). (3.2)

Here we use the Euclidean AdS metric [dz] =

, which leads to a change of the overall

sign for the action. The langrangian splits into a quadratic part L

, a cubic part L

and a

It is because of the necessary appearance of C tensors that we do not follow the notation used in

e.g. [5, 6] that would allow us to write correlators as polynomials in “R symmetry cross ratios” σ and τ :

we feel this would lead to an even bigger proclivity of notations.

– 6 –

JHEP09(2018)023

quartic part L

. The ﬁrst two contribute to the so-called exchange part of the correlator,

because they give rise to exchange interactions captured in Witten diagrams such as in

ﬁgure 2, while L

forms the contact part as they lead to contact interactions (see ﬁgure 1).

We will discuss how one can compute these two parts for a concrete set of weights separately,

beginning with the latter.

3.1 Contact part

The quartic Lagrangian is given by:

= L

(4)

+ L

(2)

+ L

(0)

(4)



(4)

+ A

(4)



∇



∇



(2)



(2)

+ A

(2)



∇

(0)



(0)

+ A

(0)



(3.3)

The repeated indices I

, . . . , I

here imply the summation over all possible [0, k, 0] repre-

sentations as well as summation over the basis of the representation space:

≡

k>2

dim([0,k,0])

. (3.4)

The various couplings S and A [11] depend on contractions of the previously discussed

C tensors, which are Clebsch-Gordan coeﬃcients for the tensor product of SO(6) irreps.

Their explicit expressions depend on the chosen weights and their computation becomes

complicated quite quickly. In fact, this forms the main diﬃculty in the computation of

higher-weight correlators in this formalism. We have found ways to simplify this compu-

tation considerably, but their discussion is fairly technical. We therefore refer the reader

to the discussion in appendix A.

Further simpliﬁcations follow from a closer analysis of the four-derivative terms L

(4)

: it

was shown in [24] that these terms can be made to vanish in the extremal k

= k

+ k

and sub-extremal k

= k

+ k

− 2 cases. Moreover, in [8] the vanishing of the four-

derivative terms was proven for all four-point correlators of 1/2 BPS operators. To show

this, one has to perform integration by parts in the expression for L

(4)

, which produces

contributions to lower-derivative terms L

(4→2)

and L

(4→0)

. However, in the same manner as

in [8], using the reduction formulae (A.19) and (A.22), one can show that the contribution

(4→0)

vanishes identically. Thus, the full contact Lagrangian can be written as

(4→2)

+ L

(2)

+ L

(0)

, (3.5)

where

(4→2)



(4)

+ m

− 4) − 4A

(4)



∇

, (3.6)

where m

denotes the AdS mass of the corresponding scalar ﬁeld.

– 7 –

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