Physics Letters B 737 (2014) 293–297
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Physics Letters B
www.elsevier.com/locate/physletb
Finite-size giant magnons on η-deformed AdS
5
× S
5
Changrim Ahn
∗
, Plamen Bozhilov
1
Department of Physics, Ewha Womans University, DaeHyun 11-1, Seoul 120-750, Republic of Korea
a r t i c l e i n f o a b s t r a c t
Article history:
Received
23 June 2014
Accepted
28 August 2014
Available
online 2 September 2014
Editor:
L. Alvarez-Gaumé
We consider strings moving in the R
t
× S
3
η
subspace of the η-deformed AdS
5
× S
5
and obtain a class of
solutions depending on several parameters. They are characterized by the string energy and two angular
momenta. Finite-size dyonic giant magnon belongs to this class of solutions. Further on, we restrict
ourselves to the case of giant magnon with one nonzero angular momentum, and obtain the leading
finite-size correction to the dispersion relation.
© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP
3
.
1. Introduction
In the recent years important progress has been made in the
field of AdS/CFT duality [1] (for overview see [2]). The main
achievements are due to the discovery of integrable structures on
both sides of the correspondence.
The
most developed case is the correspondence between strings
moving in AdS
5
× S
5
and N = 4SYMin four dimensions. The so-
called
γ -deformation of AdS
5
× S
5
has been proposed in [3]. It
was shown in [4] that this deformation is still integrable for real
γ (known as β-or TsT-deformation).
A
new integrable deformation of the type IIB AdS
5
× S
5
su-
perstring
action, depending on one real parameter η, has been
found recently in [5]. The bosonic part of the superstring sigma
model Lagrangian on this η-deformed background was determined
in [6]. Then the authors of [6] used it to compute the perturbative
S-matrix of bosonic particles in the model.
Interesting
new developments were made in [7]. There the
spectrum of a string moving on η-deformed AdS
5
× S
5
is consid-
ered.
This is done by treating the corresponding worldsheet theory
as integrable field theory.
2
In particular, it was found that the dispersion relation for the
infinite-size giant magnons [9] in this background, in the large
string tension limit g →∞is given by
E =
2g
1 +
˜
η
2
˜
η
arcsinh
˜
η sin
p
2
,
(1.1)
*
Corresponding author.
E-mail
addresses: ahn@ewha.ac.kr (C. Ahn), bozhilov@inrne.bas.bg (P. Bozhilov).
1
On leave from Institute for Nuclear Research and Nuclear Energy, Bulgarian
Academy of Sciences, Bulgaria.
2
See also [8].
where
˜
η is related to the deformation parameter η according to
˜
η =
2η
1 −η
2
. (1.2)
Here, we are going to extend the result (1.1) to the case of finite-
size
giant magnons.
The
paper is organized as follows. In Section 2 we give the
bosonic part of the string Lagrangian on η-deformed AdS
5
× S
5
found in [6] and extract from it the background fields. Then in Sec-
tion 3,
we obtain the exact solutions for the finite-size dyonic giant
magnon coordinates, the corresponding conserved charges and the
angular difference along one of the isometric coordinates on the
deformed sphere S
3
η
.
3
In Section 4 we find the dispersion relation
for the giant magnons with one nonzero angular momentum, in-
cluding
the leading finite-size effect on it. Section 5 is devoted to
our concluding remarks.
2. String Lagrangian and background fields
The bosonic part of the string Lagrangian L on the η-deformed
AdS
5
× S
5
found in [6] is given by a sum of the Lagrangians L
a
and L
s
, for the AdS and sphere subspaces. Since there is nonzero
B-field on both subspaces, which leads to the appearance of Wess–
Zumino
terms, these Lagrangians can be further decomposed as
L
a
= L
g
a
+ L
WZ
a
, L
s
= L
g
s
+ L
WZ
s
, (2.1)
where the superscript “g” is related to the dependence on the
background metric. The explicit expressions for the Lagrangians in
(2.1) are as follows [6]
3
This angular difference is identified with the momentum of the magnon excita-
tions
in the dual spin chain.
http://dx.doi.org/10.1016/j.physletb.2014.08.064
0370-2693/
© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by
SCOAP
3
.