强耦合下的η形变AdS 5×S 5：有限尺寸效应与Lüscher校正

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"η形变的AdS5×S5在强耦合下的有限尺寸效应" 在物理学领域，尤其是高能物理和弦理论中，阿德斯/康沃尔-贾法里对应（AdS/CFT对偶）是一个重要的理论工具，它建立了超引力理论在反德西特空间（AdS）和特定的量子场论（CFT）之间的关系。这篇由Changrim Ahn发表在Physics Letters B 767 (2017) 上的文章研究了在强耦合环境下，η形变的AdS5×S5空间中的巨型磁振子的Lüscher修正，并对比了相应的弦态的有限尺寸效应。 η形变是一种对称性扩展，它将AdS5×S5背景的对称性扩展到su(2|2)q不变的S-矩阵。这种变形对于理解弦理论在非平凡背景下的行为至关重要，因为它允许研究更复杂且可能更接近实际物理系统的模型。文章中，作者首先计算了在强耦合条件下，η形变的AdS5×S5空间中巨型磁振子的Lüscher校正。Lüscher校正是量子散射理论中用于描述在有限体积内粒子能量修正的一个技术，它可以揭示系统在远离无限体积极限时的特性。在这个过程中，作者使用了su(2|2)q不变的S-矩阵，这是一个在强耦合下保持不变的数学对象，它描述了世界表征（worldsheet）上的散射过程。随后，作者将这些Lüscher修正的结果与之前导出的对应弦态的有限尺寸效应进行了比较。这两者的匹配表明，su(2|2)q不变的S-矩阵成功地捕捉到了η形变背景下的世界表征激发。这不仅是对AdS/CFT对偶的进一步验证，也是对η形变理论的实证支持，因为它们的预测与弦理论在特定背景下的行为相一致。文章还强调，这是在强耦合下研究η形变的重要一步，因为在这一极限下，许多物理现象变得更加显著，而弱耦合方法可能不再适用。通过这种研究，我们可以更深入地理解AdS5×S5空间的复杂结构以及其在强相互作用下的行为，这对理解量子场论和弦理论的性质，特别是在凝聚态物理和核物理等领域的应用具有重要意义。此外，这篇文章是开放获取的，这意味着其研究成果可以被广泛阅读和使用，无需订阅或付费。这促进了科学知识的传播和合作，有助于加快科学研究的步伐。该研究得到了SCOAP3（可持续开放访问出版计划）的支持，这是全球科研界共同努力推动开放科学的一个实例。这项工作深化了我们对AdS/CFT对偶的理解，特别是在强耦合和有限尺寸效应下的η形变AdS5×S5空间。通过su(2|2)q不变的S-矩阵，研究人员能够探索更复杂的物理场景，为未来的研究提供了坚实的基础。

Physics Letters B 767 (2017) 121–125

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Finite-size effect of η-deformed AdS

× S

at strong coupling

Changrim Ahn

Department of Physics, Ewha Womans University, DaeHyun 11-1, Seoul 120-750, South Korea

a r t i c l e i n f o a b s t r a c t

Article history:

Received

5 December 2016

Received

in revised form 22 January 2017

Accepted

25 January 2017

Available

online 1 February 2017

Editor:

M. Cveti

We compute Lüscher corrections for a giant magnon in the η-deformed (AdS

× S

)

using the

su(2|2)

-invariant S-matrix at strong coupling and compare with the ﬁnite-size effect of the corre-

sponding

string state, derived previously. We ﬁnd that these two results match and conﬁrm that the

su(2|2)

-invariant S-matrix is describing world-sheet excitations of the η-deformed background.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

AdS/CFT duality [1], a correspondence between string theo-

ries

in AdS background with certain supersymmetric and confor-

mal

Yang–Mills theories on the boundary space-time of the AdS

space, has been a hot topic for theoretical researches and pro-

duced

many important quantitative results and applications (for

overview see [2]). In these developments, integrability has played a

crucial role on both sides of the correspondence. Two-dimensional

world-sheet actions for the string theory moving in the back-

ground

are described by nonlinear sigma models on coset group

manifolds which are classically integrable. Aspects of quantum in-

tegrable

structure of supersymmetric Yang–Mills theories appear

in Bethe ansatz equations and related exact integrable machineries

which can determine conformal dimensions of the CFTs. Quantum

S-matrices of the world-sheet actions provide integrable frame-

work

which interpolates from the strong to weak coupling limits.

important direction of research is to ﬁnd new AdS/CFT pairs

which show novel integrability structures. One such string theory,

which has been studied recently, is the type IIB superstring theory

in the η-deformed target space (AdS

× S

)

for a real parameter

η [3]. The classical integrability of nonlinear sigma model is pro-

vided

by solutions of the classical Yang–Baxter equation [4]. (See

[5–7] for related issues.) It has been conjectured in [3] that full

quantum S-matrix of the deformed sigma model is given by the

R-matrix of the q-deformed Hubbard model which has been pro-

posed

much earlier in [8]. When q is a complex phase, the dressing

phase of the S-matrix and bound-states have been analyzed in [9].

Scattering amplitudes of bosonic excitations for small values of the

world-sheet momentum have been computed and shown to agree

with the q-deformed S-matrix in the large string tension (strong

E-mail address: ahn@ewha.ac.kr.

coupling) limit for real q with explicit relation with η [10]. Based

on the exact S-matrix, thermodynamic Bethe ansatz equations for

ground states and dressing phase for real q have been studied

in [11].

pertinent issue which should be mentioned is that the de-

formed

sigma model is not a fully consistent string theory at quan-

tum

level. It has been found that this η-deformed sigma model

does not solve the type IIB supergravity equations of motion [12],

but solves, instead, a generalization of them [13]. These general-

ized

ones allow only scale invariance but not full Weyl invariance

at one-loop [14]. The Weyl invariance can be restored if the de-

formation

is generalized by some modiﬁed solutions of the Yang–

Baxter

equation [15]. This suggests that one should pay attention

while treating the η-deformed theory at quantum level.

this letter, we provide another evidence for the q-deformed

S-matrix to describe the string theory on the η-deformed geom-

etry.

For this purpose, we consider ﬁnite-size effects of a giant

magnon state, a classical string conﬁguration living on a subspace

of the (AdS

× S

)

[16]. These corrections have been computed

for the undeformed AdS

× S

in [17,18] and for the γ -deformed

AdS

× S

in [19,20] from both directions of string solutions and

world-sheet S-matrices. For the η-deformed case, this effect has

been studied from only string theory side in [21], which will be

reviewed in sect. 2. Exact q-deformed S-matrix and related for-

mula

will be presented in sect. 3. We present our computation of

the Lüscher corrections for a giant magnon based on q-deformed

S-matrix in sect. 4 along with a conjecture on the deformed dress-

ing

phase in sect. 5. In sect. 6, we conclude with a short summary

and comments.

2. Finite-size effect of a giant magnon in (AdS

× S

)

In this section, we give a brief review on computing the en-

ergy

of a giant magnon using Neumann–Rosochatius ansatz fol-

http://dx.doi.org/10.1016/j.physletb.2017.01.063

0370-2693/

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