TIT/HEP-666

June, 2018

T-duality to Scattering Amplitude and Wilson Loop in Non-commutative Super
Yang-Mills Theory
Song Hea<EMAIL_ADDRESS>, Hongfei Shub<EMAIL_ADDRESS>

aMax Planck Institute for Gravitational Physics (Albert Einstein Institute),
Am Mühlenberg 1, 14476 Golm, Germany
b Department of Physics, Tokyo Institute of Technology, Tokyo, 152-8551, Japan

We first perform bosonic T-duality transformation on one of the marginal TsT
(T-duality, shift, T-duality)-deformed $AdS_{5}\times S_{5}$ spacetime, which
corresponds to 4D $\mathcal{N}=4$ non-commutative super Yang-Mills theory
(NCSYM). We then construct the solution to Killing spinor equations of the
resulting background, and perform the fermionic T-duality transformation. The
final dual geometry becomes the usual $AdS_{5}\times S_{5}$ spacetime but with
a constant NS-NS B-field depending on the non-commutative parameter. As
applications, we study the gluon scattering amplitude and open string (Wilson
loop) solution in the TsT-deformed $AdS_{5}\times S_{5}$ spacetime, which are
dual to the null polygon Wilson loop and the folded string solution
respectively in the final dual geometry.

###### Contents

  1. 1 Introduction
  2. 2 TsT deformed $AdS_{5}\times S^{5}$ spacetime and their T-duality transformations
    1. 2.1 Bosonic T-duality transform along non-radial directions
    2. 2.2 Fermionic T-duality transform
  3. 3 The scattering amplitude in the NCAdS spacetime
    1. 3.1 Open string boundary condition before and after T-dual transformation
    2. 3.2 Null Polygon Wilson loop
  4. 4 Open string solution in NCAdS background
    1. 4.1 Coordinates and folded string solutions in original AdS spacetime
    2. 4.2 Classical solution in NCAdS spacetime dual to folded string
  5. 5 Conclusions and Discussions
  6. A Bosonic T-dual transformation
  7. B Killing spinor equations in original $AdS_{5}\times S^{5}$ spacetime
  8. C Elliptic functions

##  1 Introduction

In the past decades, many perturbative and non-perturbative results have been
achieved in the planar limit of ${\cal N}=4$ super Yang-Mills theory (SYM).
See [1] for reviews about the aspects of spectrum, scattering amplitude and
Wilson loop. One of the significant achievements about scattering amplitude
and Wilson loop is made in [2]. It has been shown that $n$-gluon scattering
amplitude at strong coupling in ${\cal N}=4$ SYM can be calculated from the
minimal area of the surface ending on a null $n$-polygonal Wilson loop at the
boundary of AdS space. This is based on the self-duality of IIB string theory
under a certain combination of bosonic and fermionic T-duality transformations
in $AdS_{5}\times S^{5}$ spacetime [3, 4], which also explains that the
existence of the dual superconformal symmetry [4]. There are many ways to
deform ${\cal N}=4$ SYM. Deformed field theories arising from a new definition
of the product of fields in the Lagrangian, provide an interesting
generalization of the gauge/gravity correspondence [5, 6, 7]. This is also due
to the fact that, on the string theory side, there is a systematic procedure
called the “TsT transformation” (T-duality, shift, T-duality) [8, 9]. The most
well-known examples are non-commutative deformation [10, 11, 12],
$\beta$-deformation [13] and “dipole deformation” [14, 15, 16, 17] of
$AdS_{5}\times S^{5}$ spacetime. A good summary can be found in [18], see also
the reference therein.

We mainly focus on the holography dual of scattering amplitudes and Wilson
loop operators in ${\cal N}=4$ non-commutative super Yang-Mills theory
(NCSYM). In [19], the author has holographically studied the planar gluon
scattering amplitudes in terms of scattering amplitude/Wilson loop duality
333See [20] for a special scattering amplitudes amplitude in the finite
temperature regime of non-commutative Tang-mills theory.. However, the
fermionic T-duality has not been considered, which leads to non-constant
dilaton and complex field strength.

On the other hand, many works about the fermionic T-duality transform and the
scattering amplitude/Wilson loop duality have been done in the past years [21,
22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35]444These works [21, 22,
23, 24, 25, 26, 27, 28, 29] are focus on the ABJM theory and the other works
[30, 31, 32, 33, 34, 35] study the ${\cal N}=4$ SYM theory.. One of the
purposes of the present work is to present the construction of fermionic
T-duality transformation of the NCAdS background to cancel the non-constant
dilaton and the complex field strength, and corresponding string solutions in
the final dual background in more details which are expected to dual to
scattering amplitude. Following the notaton in [19], we have called the
gravity background corresponding to NCSYM as NCAdS spacetime, whose definition
will be shown in (2). More precisely we first perform bosonic T-duality
transformation on the NCAdS background, and construct the solution to Killing
spinor equations of the resulting background. We then perform the fermionic
T-duality transformation, and find the final dual background which is expected
to be equivalent with the NCAdS background. In the final dual background, we
construct the solution with proper boundary conditions which is expected to be
holographic dual of gluon amplitude in NCSYM. Further, a motivation [36] to
explore the relation between closed and open strings in AdS leads us to
construct the open string solution (Wilson loop) ending at the boundary of
NCAdS, which dual to closed string in the final dual background. This would be
important in the study of the closed and open strings relation and the
corresponding observables in NCSYM.

The layout of this paper is as follows. In sec.2, we first review the TsT
transformation of $AdS_{5}\times S^{5}$ spacetime (NCAdS spacetime) which
corresponds to the non-commutative ${\cal N}=4$ Super Yang-Mills theory. We
then perform certain bosonic and fermionic T-dual transformations on the NCAdS
background, and obtain a simplified gravity background. In sec.3, we study the
scattering amplitudes in NCSYM by using the solutions in the simplified
gravity background. In sec.4, we construct the open string solution in the
NCAdS background, which is dual to the folded string solution in the
simplified gravity background. Finally, we devote to the conclusions and
discussions and also mention the future problems. In appendices, we would like
to list some techniques and Elliptic functions which are very useful in our
analysis.

##  2 TsT deformed $AdS_{5}\times S^{5}$ spacetime and their T-duality
transformations

The gravity dual of non-commutative gauge theory in [10, 11] can be generated
from the $AdS_{5}\times S^{5}$ spacetime by using TsT deformation
$(x^{1},x^{2})_{\gamma}$, which stands the T-dualizing $x^{1}\to{x}^{1}_{t}$,
shift $x^{2}$ by $x^{2}+\gamma{x}^{1}_{t}$, then T-dualizing back
${x}^{1}_{t}\to x^{1}$ [8], where ${x}_{t}^{\mu}$ $(\mu=0,1,2,3)$ is the
coordinate T-dual to $x^{\mu}$ and $\gamma$ is the constant deformation
parameter.

By applying the $(x^{1},x^{2})_{\gamma}$ TsT-deformation to $AdS_{5}\times
S^{5}$ spacetime, one obtains the following background:

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\frac{R^{2}}{r^{2}}\left(-dx_{0}^{2}+dx_{3}^{2}+dr^{2}\right)+\frac{R^{2}}{r^{2}}\frac{1}{1+\gamma^{2}\frac{R^{4}}{r^{4}}}\left(dx_{1}^{2}+dx_{2}^{2}\right)+R^{2}ds_{S^{5}}^{2}$
$\displaystyle B$ $\displaystyle=$ $\displaystyle\frac{\gamma
R^{4}}{r^{4}}\frac{1}{1+\gamma^{2}\frac{R^{4}}{r^{4}}}dx^{1}\wedge
dx^{2},\quad\phi=-\frac{1}{2}\log\left(1+\gamma^{2}\frac{R^{4}}{r^{4}}\right),$
(2.1) $\displaystyle F_{1}$ $\displaystyle=$ $\displaystyle 0,\quad
F_{3}=-4\gamma\frac{R^{4}}{r^{5}}dx^{0}\wedge dx^{3}\wedge dr,$ $\displaystyle
F_{5}$ $\displaystyle=$
$\displaystyle-4R^{4}\left(\frac{1}{1+\gamma^{2}\frac{R^{4}}{r^{4}}}\frac{1}{r^{5}}dx^{0}\wedge
dx^{1}\wedge dx^{2}\wedge dx^{3}\wedge dr+\omega_{S^{5}}\right),$

where $ds^{2}$ is the metric of the NCAdS background, $B$ is the NS-NS
B-field, $\phi$ is dilaton and ${F}_{1}$, ${F}_{3}$, ${F}_{5}$ are the R-R
field strengths.

The dual gauge theory of this background (2) is defined on the noncommutative
spacetime with noncommutativity parameter $[x^{1},x^{2}]=i\theta^{12}$. The
constant noncommutativity parameter $\theta^{12}$[37] is associated with the
TsT-deformation parameter $\gamma$ as [10]

$\displaystyle\theta^{12}=2\pi\alpha^{\prime}\left(\left.B_{12}\right|_{r\to
0}\right)^{-1}=2\pi\alpha^{\prime}\gamma.$ (2.2)

Then the so called Seiberg-Witten $\alpha^{\prime}\to 0$ limit has to be taken
with $\alpha^{\prime}\gamma$ fixed.

For large $r$, background (2) reduces to the original $AdS_{5}$$\times S^{5}$
spacetime. Since the large $r$ region corresponds to the IR regime of the
gauge theory, one expects that the non-commutative super Yang-Mills theory
reduces to the original $N=4$ super Yang-Mills theory at IR regime (long
distance). We will study the gluon scattering amplitude at IR regime in
section 3. The background (2) has boundary at $r=0$. Since
$G_{11},G_{22}\propto\frac{r^{2}}{R^{2}}$ in the boundary of background (2),
the physical size of $x_{1}$ and $x_{2}$ directions shrink [11].

###  2.1 Bosonic T-duality transform along non-radial directions

The background (2) is invariant under the the shift isometry $x^{\mu}\to
x^{\mu}+c$, where $\mu=0,1,2,3$, $c$ is a constant. We then perform T-duality
transform along $x^{1},x^{2},x^{3},x^{0}$ in turn by following the Buscher
rule in appendix A. The resulting background becomes

$\displaystyle ds^{\prime}{}^{2}$ $\displaystyle=$
$\displaystyle\frac{r^{2}}{R^{2}}\left(-(d{x}^{\prime 0})^{2}+(d{x}^{\prime
1})^{2}+(d{x}^{\prime 2})^{2}+(d{x}^{\prime
3})^{2}\right)+\frac{R^{2}}{r^{2}}dr^{2}+R^{2}ds_{S^{5}}^{2},$ $\displaystyle
B^{\prime}$ $\displaystyle=$ $\displaystyle-\gamma d{x}^{\prime 1}\wedge
d{x}^{\prime 2},\quad\phi^{\prime}=\log\left(\frac{r^{4}}{R^{4}}\right),$
$\displaystyle{F}^{\prime}_{1}$ $\displaystyle=$
$\displaystyle-4i\frac{R^{4}}{r^{5}}dr,\quad{F}^{\prime}_{3}={F}^{\prime}_{5}=0,$
(2.3)

This background is same as the one obtained from the T-dual transformation of
original $AdS_{5}$ along $x^{1},x^{2},x^{3},x^{0}$ directions but with a
constant $B$-field. We note that the factor $i$ has appeared in
${F}^{\prime}_{1}$ due to T-dualizing along the time direction $x^{0}$ [38].
To cancel the dilaton and the complex field strength, we perform fermionic
T-dual transformation on the background (2.1).

###  2.2 Fermionic T-duality transform

In this subsection, we perform the fermionic T-duality on the background (2.1)
555We follow the notation summarized in [39] .. Given Killing spinors
$(\epsilon_{I},\hat{\epsilon}_{I})$, the fermionic T-duality transform is
generated as [39]

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\epsilon_{I}\gamma^{M}\epsilon_{J}+\hat{\epsilon}_{I}\gamma^{M}\hat{\epsilon}_{J}$
(2.4) $\displaystyle\partial_{M}C_{IJ}$ $\displaystyle=$ $\displaystyle
2i\epsilon_{I}\gamma_{M}\epsilon_{J}$ (2.5) $\displaystyle\tilde{\phi}$
$\displaystyle=$ $\displaystyle\phi^{\prime}+\frac{1}{2}\mbox{Tr}(\log C)$
(2.6)
$\displaystyle\frac{i}{16}e^{{\tilde{\phi}}}{\tilde{F}}^{\alpha\hat{\beta}}$
$\displaystyle=$
$\displaystyle\frac{i}{16}e^{\phi^{\prime}}F^{\prime\alpha\hat{\beta}}-\epsilon_{I}^{\alpha}\hat{\epsilon}_{J}^{\hat{\beta}}(C^{-1})_{IJ},$
(2.7)

where the indices $I$, $J$ are the labels of the different Killing spinors,
and $F^{\alpha\hat{\beta}}$ are the R-R fields in bispinor form

$\displaystyle
F^{\alpha\hat{\beta}}=(\gamma^{M})^{\alpha\hat{\beta}}F_{M}+\frac{1}{3!}(\gamma^{MNP})^{\alpha\hat{\beta}}F_{MNP}+\frac{1}{2}\frac{1}{5!}(\gamma^{MNPQR})^{\alpha\hat{\beta}}F_{MNPQR},\quad
M=0,1,\cdots,9.$ (2.8)

$\tilde{F}^{\alpha\hat{\beta}}$ and $F^{\prime\alpha\hat{\beta}}$ are also
defined in a similar way. Note that the metric and the B-field do not
transform under the above fermionic T-duality. The Killing spinors are not
arbitrary. More precisely, (2.4) is imposed to ensure the fermionic
isometries, i.e.
$\\{\epsilon^{I}Q_{I},\hat{\epsilon}^{\hat{I}}Q_{\hat{I}}\\}=0$ where $Q$ and
$\hat{Q}$ are the supercharges in the supersymmetry algebra. Relaxing (2.4),
fermionic T-duality will lead to field configurations that are not
supergravity solutions. The second equation determines the matrix $C$. In
order to do that, we have to find the unbroken supersymmetry in 10D IIB
supergravity, which is generated by the spinor parameters
$(\epsilon^{\alpha},\hat{\epsilon}^{\hat{\alpha}})$
$(\alpha,\hat{\alpha}=1,\ldots,16)$. Then the parameters must satisfy the
following equations from the supersymmetry transformation of two gravitini
$\psi_{M}$, $\hat{\psi}_{M}$ $(M=0,\ldots,9)$ and two dilatini $\lambda$,
$\hat{\lambda}$:

$\displaystyle\delta\psi_{M}$ $\displaystyle=$
$\displaystyle{e_{\hat{M}}}^{M}\nabla_{M}\epsilon-\frac{e^{\phi^{\prime}}}{8}\gamma^{\hat{N}}F^{\prime}_{\hat{N}}\gamma_{\hat{M}}\hat{\epsilon}=0,$
(2.9) $\displaystyle\delta\hat{\psi}_{\hat{M}}$ $\displaystyle=$
$\displaystyle{e_{\hat{M}}}^{M}\nabla_{M}\hat{\epsilon}+\frac{e^{\phi^{\prime}}}{8}\gamma^{\hat{N}}F^{\prime}_{\hat{N}}\gamma_{\hat{M}}\epsilon=0,$
(2.10) $\displaystyle\delta\lambda$ $\displaystyle=$
$\displaystyle{e_{\hat{M}}}^{M}\gamma^{\hat{M}}\partial_{{M}}\phi^{\prime}\epsilon+e^{\phi^{\prime}}\gamma^{\hat{M}}F^{\prime}_{\hat{M}}\hat{\epsilon}=0,$
(2.11) $\displaystyle\delta\hat{\lambda}$ $\displaystyle=$
$\displaystyle{e_{\hat{M}}}^{M}\gamma^{\hat{M}}\partial_{M}\phi^{\prime}\hat{\epsilon}-e^{\phi^{\prime}}\gamma^{\hat{M}}F^{\prime}_{\hat{M}}\epsilon=0,$
(2.12)

where $\delta$ is the supersymmetry variation, $\nabla_{M}$ is the covariant
derivative and $\gamma^{\hat{M}}$ is the 10D gamma matrices. $\hat{M}$ is the
coordinate of the flat space, and ${e_{\hat{M}}}^{N}$ is the the vielbein.
From the dilatino equations (2.11) and (2.12), $\epsilon$ and $\hat{\epsilon}$
are related with

$\displaystyle\hat{\epsilon}$ $\displaystyle=$ $\displaystyle-i\epsilon.$
(2.13)

Although the supersymmetric parameters in type IIB supergravity are real
Majorana-Weyl spinors, an imaginary unit appears because of the complexified
RR one form (2.1). Therefore, (2.4) is satisfied automatically. Substituting
(2.13) the gravitino equations are simplified as

$\displaystyle{e_{\hat{M}}}^{M}\nabla_{M}\epsilon+i\frac{e^{\phi^{\prime}}}{8}\gamma^{\hat{4}}F^{\prime}_{\hat{4}}\gamma_{\hat{M}}\epsilon=0.$
(2.14)

We can find that the $M=x^{0,1,2,3}$ part of (2.14) is trivial. To solve the
remaining 6D part ($R\times S^{5}$), it is convenient to rewrite the
coordinate as $y^{s}$ ($s=1,2,\cdots,6$) with $|y|=r$. We decompose SO(9,1)
spinor index $\alpha$ into $(a^{\prime}j^{\prime},\dot{a}^{\prime}j^{\prime})$
as in [4], where $a^{\prime},\ \dot{a}^{\prime}=1,2$ are the SO(3,1) spinor
indices and $j^{\prime}=1,\ldots,4$ is the SO(5) spinor index.
$(\sigma^{r})_{j^{\prime}k^{\prime}}$ $(r=1,\ldots,6)$ are the 6D Pauli
matrices. Then (2.14) is solved as

$\displaystyle\epsilon_{aj}^{b^{\prime}l^{\prime}}=\sqrt{\frac{r}{R}}\delta_{a}^{b^{\prime}}M_{j}^{l^{\prime}}(y),\quad\epsilon_{aj}^{\dot{b}^{\prime}l^{\prime}}=0,\quad\hat{\epsilon}_{aj}^{b^{\prime}l^{\prime}}=-i\epsilon_{aj}^{b^{\prime}l^{\prime}}$
(2.15)

where $M_{j}^{l^{\prime}}(y)$ is the SU(4)/SO(5) matrix rotating the point
$(0,0,0,0,0,1)$ on $S^{5}$ to the point
$(y_{1},y_{2},y_{3},y_{4},y_{5},y_{6})/r$. The unprimed indices $a$, $j$ are
regarded as the label of different Killing spinors corresponding to the label
$I$ in (2.5). The main difference between our Killing spinor and the chosen
one in [4] is (2.13). We will show the details of the Killing spinors of
original $AdS_{5}\times S^{5}$ spacetime used in [4] in appendix B. Then one
finds

$\displaystyle C_{aj\leavevmode\nobreak\
bk}=2i\epsilon_{ab}\sigma_{jk}^{r}y_{r},\quad(C^{-1})^{aj\leavevmode\nobreak\
bk}=-\frac{i}{2}\epsilon^{ab}(\sigma^{r})^{jk}\frac{y_{r}}{r^{2}}.$ (2.16)

To determine the transformation of field strength strength, we use

$\displaystyle\epsilon_{aj}^{a^{\prime}j^{\prime}}(C^{-1})^{aj\leavevmode\nobreak\
bk}\hat{\epsilon}_{bk}^{b^{\prime}k^{\prime}}=-\frac{1}{2R}\epsilon^{a^{\prime}b^{\prime}}(\sigma^{6})^{j^{\prime}k^{\prime}}=\frac{i}{2R}(\gamma_{\hat{0}\hat{1}\hat{2}\hat{3}\hat{4}})^{a^{\prime}j^{\prime}\leavevmode\nobreak\
b^{\prime}k^{\prime}}.$ (2.17)

Writing it in the terms of projection operator
$\frac{1}{2}\big{(}(\gamma_{\hat{0}\hat{1}\hat{2}\hat{3}}-i)\gamma_{\hat{4}}\big{)}^{\alpha\hat{\beta}},$
one finds

$\displaystyle e^{\tilde{\phi}}\tilde{F}^{\alpha\hat{\beta}}$
$\displaystyle=-\gamma^{\hat{4}}i\frac{4}{R}-\frac{4}{R}(\gamma_{\hat{0}\hat{1}\hat{2}\hat{3}\hat{4}}-i\gamma_{\hat{4}})^{\alpha\hat{\beta}}=-\frac{4}{R}(\gamma_{\hat{0}\hat{1}\hat{2}\hat{3}\hat{4}})^{\alpha\hat{\beta}}.$
(2.18)

Furthermore, using (2.6) one finds the dilaton vanishes $\tilde{\phi}=0$.
Introducing $z=\frac{R^{2}}{r}$ and $\tilde{x}^{\mu}=x^{\prime\mu}$, we then
obtain the dual background as

$\displaystyle d\tilde{s}^{2}$ $\displaystyle=$
$\displaystyle\frac{R^{2}}{z^{2}}\left(-(d\tilde{x}^{0})^{2}+(d\tilde{x}^{1})^{2}+(d\tilde{x}^{2})^{2}+(d\tilde{x}^{3})^{2}+dz^{2}\right)+R^{2}ds_{S^{5}}^{2},$
$\displaystyle\tilde{B}$ $\displaystyle=$ $\displaystyle-\gamma
d\tilde{x}^{1}\wedge d\tilde{x}^{2},\quad\tilde{\phi}=0$
$\displaystyle\tilde{F}_{1}$ $\displaystyle=$
$\displaystyle\tilde{F}_{3}=0,\quad\tilde{F}_{5}=-4R^{4}\left(\frac{1}{z^{5}}d\tilde{x}^{0}\wedge
d\tilde{x}^{1}\wedge d\tilde{x}^{2}\wedge d\tilde{x}^{3}\wedge
dz+\omega_{S^{5}}\right).$ (2.19)

This is the usual $AdS_{5}\times S^{5}$ background but with the constant
B-field. One may suspect the above fermionic T-duality transform because we
have performed it in complex background. In order to resolve this issue, we
consider the whole duality transform in reversed order. First, starting from
the background (2.2), we perform the fermionic T-duality transformation. Since
the background is real and the constant B-field does not contribute to the
Killing spinor equations, we can use the same spinors
$(\epsilon,\hat{\epsilon})$ as the ones in [4]

$\displaystyle\epsilon_{aj}^{b^{\prime}l^{\prime}}=\sqrt{\frac{r}{R}}\delta_{a}^{b^{\prime}}M_{j}^{l^{\prime}}(y),\quad\epsilon_{aj}^{\dot{b}^{\prime}l^{\prime}}=0,\quad\hat{\epsilon}_{aj}^{b^{\prime}l^{\prime}}=i\epsilon_{aj}^{b^{\prime}l^{\prime}}.$
(2.20)

Then using transformation rules (2.5), (2.6) and (2.7), we find the field
strength and dilaton change to

$\displaystyle
e^{\tilde{\phi}^{\prime}}\tilde{F}^{\prime}=-\frac{4}{R}\gamma_{\hat{0}\hat{1}\hat{2}\hat{3}\hat{4}}+\frac{4}{R}(\gamma_{\hat{0}\hat{1}\hat{2}\hat{3}\hat{4}}-i\gamma_{\hat{4}})=-i\frac{4}{R}\gamma_{\hat{4}},\quad\tilde{\phi}^{\prime}=-4\log(\frac{z}{R}),$
(2.21)

where we have rewritten the equations in [4] in our notations. The NS-NS
metric and B-field do not change under this transformation. Writing by using
the coordinate $r=\frac{R^{2}}{z}$, we can find that the resulting background
coincides with (2.1). Then performing the bosonic T-duality tranform along the
non-radial directions, we can obtain the NCAdS background (2).

We give comment on this result as following: Note that the final dual
background depends on the choice of Killing spinor in (2.5). However, no
matter which set of Killing spinors we choose, the metric and B-field will not
change under the fermionic T-duality transformation. Our transformation can be
regarded as a simplification of the NCAdS background (or NCSYM). If the detail
map from the observables in the NCAdS background (2) to the observables in
(2.2) background is known, one can calculate the observables in (2.2)
background easily. In the following sections, we show two examples of these
observables to show the powerfulness of our result.

##  3 The scattering amplitude in the NCAdS spacetime

We study the IR regime of the gluon scattering amplitude at strong coupling in
NCSYM. We follow the procedure in the original $AdS_{5}\times S^{5}$ spacetime
case [2], and study the open string scattering amplitude on D3-brane near
horizon in NCAdS background (2). We simplify this problem by studying the
object in the final dual background (2.2).

###  3.1 Open string boundary condition before and after T-dual transformation

Just as in the study of gluon scattering amplitude in [2], we consider the
Euclidean worldsheet. The bosonic part of the worldsheet action on background
(2) is

$\displaystyle\frac{1}{4\pi\alpha^{\prime}}\int
d^{2}\sigma\sqrt{g}(g^{ab}G_{\mu\nu}+i\epsilon^{ab}B_{\mu\nu})\partial_{a}x^{\mu}\partial_{b}x^{\nu}.$
(3.1)

The open strings on the D3-brane near horizon satisfy the boundary condition

$\displaystyle\big{(}G_{\mu\nu}\partial_{\sigma}x^{\nu}+iB_{\mu\nu}\partial_{\tau}x^{\nu}\big{)}|_{\partial\Sigma}=0,$
(3.2)

where $\partial\Sigma$ is the boundary of the worldsheet. This is mixed
Neumann-dirichlet boundary conditions in the directions $x^{0},\cdots,x^{3}$.
In components this boundary condition can be written as

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
G_{0\nu}\partial_{\sigma}x^{\nu}|_{\partial\Sigma}$ (3.3) $\displaystyle 0$
$\displaystyle=$ $\displaystyle
G_{3\nu}\partial_{\sigma}x^{\nu}|_{\partial\Sigma}$ (3.4) $\displaystyle 0$
$\displaystyle=$ $\displaystyle
G_{1\nu}\partial_{\sigma}x^{\nu}+iB_{12}\partial_{\tau}x^{2}|_{\partial\Sigma}$
(3.5) $\displaystyle 0$ $\displaystyle=$ $\displaystyle
G_{2\nu}\partial_{\sigma}x^{\nu}+iB_{21}\partial_{\tau}x^{1}|_{\partial\Sigma}.$
(3.6)

We then consider the bosonic T-duality transform along
$x^{1},x^{2},x^{3},x^{0}$ step by step. We will use (A.4) to study the
transformation of the boundary conditions. We label the field $f$ after the
T-duality transformation along direction $x^{a}$ as $f_{(a)}$, where
$a=1,2,3,0$, $f_{(0)}=\tilde{f}$ is the field in background (2.2) 666In this
section, we only consider the field $f$ which is invariant under ferimionic
T-dual transformation. This means the field in (2.1) and (2.2) are same..

We show the first step of T-duality transformation along direction $x^{1}$ for
instance. We start with the boundary condition:

$\displaystyle
G_{\mu\nu}\partial_{\sigma}x^{\nu}|_{\partial\Sigma}=-i\epsilon^{\sigma\tau}B_{\mu\nu}\partial_{\tau}x^{\nu}|_{\partial\Sigma}.$
(3.7)

Taking T-dual transformation along $x^{1}$ direction, (A.4) leads to get

$\displaystyle\partial_{\alpha}x_{(1)}^{1}$ $\displaystyle=$
$\displaystyle-i\epsilon_{\alpha\beta}\big{(}G_{11}\partial_{\beta}x^{1}+G_{1m}\partial_{\beta}x^{m}\big{)}-B_{1m}\partial_{\alpha}x^{m}$
(3.8) $\displaystyle\partial_{\alpha}x_{(1)}^{2}$ $\displaystyle=$
$\displaystyle\partial_{\alpha}x^{2},\leavevmode\nobreak\
\partial_{\alpha}x_{(1)}^{3}=\partial_{\alpha}x^{3},\leavevmode\nobreak\
\partial_{\alpha}x_{(1)}^{0}=\partial_{\alpha}x^{0}$

Resolving these equations, one obtains

$\displaystyle\partial_{a}x^{1}$ $\displaystyle=$ $\displaystyle
i\epsilon_{ba}G_{12(1)}\partial_{b}x_{(1)}^{2}+i\epsilon_{ba}G_{11(1)}\partial_{b}x_{(1)}^{1}$
(3.9) $\displaystyle\partial_{\alpha}x^{2}$ $\displaystyle=$
$\displaystyle\partial_{\alpha}x_{(1)}^{2},\leavevmode\nobreak\
\leavevmode\nobreak\
\partial_{\alpha}x^{3}=\partial_{\alpha}x_{(1)}^{3},\leavevmode\nobreak\
\leavevmode\nobreak\ \partial_{\alpha}x^{0}=\partial_{\alpha}x_{(1)}^{0}$

Then the boundary condition becomes

$\displaystyle\partial_{\tau}x_{(1)}^{1}|_{\partial\Sigma}$ $\displaystyle=$
$\displaystyle 0,\leavevmode\nobreak\ \leavevmode\nobreak\
\partial_{\sigma}x_{(1)}^{3}|_{\partial\Sigma}=0,\qquad\partial_{\sigma}x_{(1)}^{0}|_{\partial\Sigma}=0,$
(3.10) $\displaystyle\partial_{\sigma}x_{(1)}^{2}|_{\partial\Sigma}$
$\displaystyle=$ $\displaystyle-\frac{\gamma
R^{2}}{r^{2}}\big{(}G_{12(1)}\partial_{\sigma}x_{(1)}^{2}+G_{11(1)}\partial_{\sigma}x_{(1)}^{1}\big{)}_{\partial\Sigma}.$
(3.11)

In the same way, we study the boundary condition after the T-duality
transformation along $x^{2},x^{3},x^{0}$ in turn and find the boundary
condition in the final dual coordinates becomes the simple Dirichlet condition

$\displaystyle\partial_{\tau}\tilde{x}^{\mu}|_{\partial\Sigma}$
$\displaystyle=$ $\displaystyle 0,\leavevmode\nobreak\ \leavevmode\nobreak\
\mu=0,1,2,3.$ (3.12)

Since we started with the D3-brane near horizon ($r\to\infty$) in background
(2), the open strings are then fixed at the AdS boundary
$(z=\frac{R^{2}}{r}\to 0)$ in background (2.2).

Note that the boundary condition of open strings does not change under the
fermionic T-duality transformation. Therefore the scattering amplitude on the
D-brane in NCAdS spacetime is mapped to a T-dual open strings with Dirichlet
boundary condition in the background (2.2). Furthermore since the T-duality
transform does not change the boundary condition of radial direction, the open
string has the Dirichlet condition along $z$ direction.

###  3.2 Null Polygon Wilson loop

We then study which object in background (2.2) dual to the scattering
amplitude in background (2). Following the process in [2], we consider the
scattering of the open strings ending on the IR D3-brane in background (2). We
insert the vertex operator with a momentum $k^{\mu}$ on the boundary of the
worldsheet. At this boundary, $x^{\mu}$ thus carries a momentum $k^{\mu}$,
i.e. the zero mode of field $x^{\mu}$. In background (2.2), this translates to
the condition about the “winding” $\Delta
x^{\mu}=x^{\mu}(\sigma=2\pi)-x^{\mu}(\sigma=0)=\int_{0}^{2\pi}d\sigma\partial_{\sigma}\tilde{x}^{\mu}$.
From (A.4), we find that $\partial_{\sigma}\tilde{x}^{\mu}$ and
$\partial_{\sigma}x^{\mu}$ are related with

$\displaystyle\partial_{\sigma}\tilde{x}^{1}$ $\displaystyle=$
$\displaystyle-i\epsilon_{\sigma\tau}G_{11}\partial_{\tau}x^{1}-B_{12}\partial_{\sigma}x^{2}$
(3.13) $\displaystyle\partial_{\sigma}\tilde{x}^{2}$ $\displaystyle=$
$\displaystyle-i\epsilon_{\sigma\tau}G_{22}\partial_{\tau}x^{2}+B_{12}\partial_{\sigma}x^{1}$
(3.14) $\displaystyle\partial_{\sigma}\tilde{x}^{3}$ $\displaystyle=$
$\displaystyle-i\epsilon_{\sigma\tau}G_{33(2)}\partial_{\tau}x^{3}$ (3.15)
$\displaystyle\partial_{\sigma}\tilde{x}^{0}$ $\displaystyle=$
$\displaystyle-i\epsilon_{\sigma\tau}G_{00(2)}\partial_{\tau}x^{0}.$ (3.16)

Then
$\Delta\tilde{x}^{\mu}=\int_{0}^{2\pi}d\sigma\partial_{\sigma}\tilde{x}^{\mu}$
can be written as

$\displaystyle\Delta\tilde{x}^{1}$ $\displaystyle=$ $\displaystyle
k^{1}-B_{12}x^{2}|_{\sigma=0}^{2\pi}$ (3.17)
$\displaystyle\Delta\tilde{x}^{2}$ $\displaystyle=$ $\displaystyle
k^{2}+B_{12}x^{1}|_{\sigma=0}^{2\pi}$ (3.18)
$\displaystyle\Delta\tilde{x}^{3}$ $\displaystyle=$ $\displaystyle k^{3}$
(3.19) $\displaystyle\Delta\tilde{x}^{0}$ $\displaystyle=$ $\displaystyle
k^{0},$ (3.20)

where $k^{\mu}=k^{\mu}_{prop}\frac{R^{2}}{r^{2}}$ and $k^{\mu}_{prop}$ are the
momentum carried by vertex operator and the proper momentum of string in (2)
respectively. We then use the boundary condition (3.7) to rewrite
$\partial_{\sigma}x^{\mu}$ into $k^{\mu}$

$\displaystyle\Delta\tilde{x}_{i}^{1}$ $\displaystyle=$ $\displaystyle
k_{i}^{1}-\frac{B_{12}B_{21}}{G_{11}G_{22}}k_{i}^{1}$ (3.21)
$\displaystyle\Delta\tilde{x}_{i}^{2}$ $\displaystyle=$ $\displaystyle
k_{i}^{2}-\frac{B_{12}B_{21}}{G_{11}G_{22}}k_{i}^{2}$ (3.22)
$\displaystyle\Delta\tilde{x}_{i}^{3}$ $\displaystyle=$ $\displaystyle
k_{i}^{3}$ (3.23) $\displaystyle\Delta\tilde{x}_{i}^{0}$ $\displaystyle=$
$\displaystyle k_{i}^{0},$ (3.24)

where $i$ is the label of the open strings. Since momentum conservation
($\sum_{i}k_{i}^{\mu}=0$) of the scattering, the segments constructed by
$\Delta\tilde{x}_{i}^{\mu}$ should be always closed. Since we should take $r$
to $\infty$ (horizon), $B_{12}(\sim\frac{R^{4}}{r^{4}})$ is negligible
comparing to $G_{11}\sim\frac{R^{2}}{r^{2}}$. We thus obtain

$\displaystyle\Delta\tilde{x}_{i}^{\mu}=k^{\mu}_{i},$ (3.25)

which shows that the segments are lightlike (null), because all the gluons are
massless. We thus obtain a null polygon Wilson loop at the boundary in (2.2).

We then consider the background (2.2), which is an $AdS_{5}\times S^{5}$
spacetime with constant B-field. The sigma action becomes

$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi\alpha^{\prime}}\int
d^{2}\sigma\left(\partial^{a}\tilde{x}^{\mu}\partial_{a}\tilde{x}^{\nu}\tilde{G}_{\mu\nu}+i\tilde{B}_{\mu\nu}\epsilon^{ab}\partial_{a}\tilde{x}^{\mu}\partial_{b}\tilde{x}^{\nu}\right)$
(3.26) $\displaystyle=$ $\displaystyle\frac{1}{4\pi\alpha^{\prime}}\left(\int
d^{2}\sigma\left(\partial^{a}\tilde{x}^{\mu}\partial_{a}\tilde{x}^{\nu}\tilde{G}_{\mu\nu}\right)+i\int_{\partial\Sigma}\tilde{B}_{\mu\nu}\tilde{x}^{\mu}\partial_{t}\tilde{x}^{\nu}\right).$
(3.27)

where $\partial_{t}$ is a derivative along the worldsheet boundary
$\partial\Sigma$. Since the $\tilde{B}$-field is constant, it does not
contribute to the e.o.m. The second term of (3.27) depends only on the
boundary and just gives the an overall dressing phase

$\displaystyle\Phi=\frac{i}{4\pi\alpha^{\prime}}\int_{\partial\Sigma}d\sigma\gamma\big{(}\tilde{x}^{1}\partial_{\sigma}\tilde{x}^{2}-\tilde{x}^{2}\partial_{\sigma}\tilde{x}^{1}\big{)},$
(3.28)

where the boundary condition (3.12) is used. Since the worldsheet ends on the
null polygon Wilson loop with segments $\Delta\tilde{x}_{i}^{\mu}=k^{\mu}_{i}$
at boundary, we can parametrize the boundary as

$\displaystyle\tilde{x}^{\mu}=\sum_{m}k_{m}^{\mu}\theta(\sigma-\sigma_{m}),$
(3.29)

where $m$ is the label of the segments of Wilson loop, $\sigma_{m}$ is the
location of the cusp. We thus obtain

$\displaystyle\Phi=\frac{i\gamma}{4\pi\alpha^{\prime}}\sum_{m<n}\big{(}k_{m}^{1}k_{n}^{2}-k_{m}^{2}k_{n}^{1}\big{)}.$
(3.30)

This result matches with the discussion from the viewpoint of NCSYM in [19].
One could study the $\alpha^{\prime}$ correction (or $\sqrt{\lambda}$) of
(3.27) by considering the fluctuation around the classical solution
$\tilde{x}^{\mu}$ [40]. Since the boundary of (3.27) should be fixed, the
effect of the constant B-field does not change. This means at any order of
$\alpha^{\prime}$ (or $\sqrt{\lambda}$), the B-field only contributes a phase
factor $\Phi$. This is consistent with our expectation that the non-
commutative super Yang-Mills theory reduce to the original $N=4$ super Yang-
Mills theory at IR regime (long distance)[2].

To close this section, let us show the similarities and differences to the
discussion in [19]. The author in [19] studied the gluon scattering amplitude
from gravity side by using a slight different background 777In [19], the
background corresponds to a $(2,2)$ signature NCSYM., and argued that the
scattering amplitude dual to the null polygon Wilson loop in the T-dual
background. In the present paper, basing on the Buscher rules in appendix A,
we studied how the boundary condition and the momentum of strings transform
under the T-duality. This provides solid proof to show that the scattering
amplitude in background (2) corresponds to the null polygon Wilson loop in
(2.2). We also used the fermionic T-dual transformation to justify the complex
background, which has not been discussed in [19]. Furthermore, we have also
considered the fluctuation around the classical solution, and found the
B-field contribute the same phase factor $\Phi$ at any order of
$\alpha^{\prime}$ (or $\sqrt{\lambda}$) in the planar limit, which is
consistent with the expectation in [19] from the viewpoint of field theory
side.

##  4 Open string solution in NCAdS background

The AdS/CFT correspondence enables us to study the strong coupling gauge
theory by using the classical solution in gravity side. In this section we
study the classical open solution in NCAdS background (2), which is dual to
the folded string [41] solution in background (2.2).

If the string configuration does not depend on directions $x^{1}$ and $x^{2}$,
the classical solution of this type string configuration should be described
by the same classical solution 888Some of those string configurations have
been given by [36]. as in the original $AdS_{5}\times S^{5}$ spacetime. We
will study non-trivial classical solution which depends on $x^{1}$ and $x^{2}$
directions. However due to the appearance of B-field and the complicated
metric in (2), it is not an easy work to construct this kind classical
solution. Indeed in [11], the authors studied the Wilson line in (2), and
suggest that the open strings cannot be localized near the boundary in
background (2). This is one of the difficulties to study the non-commutative
super Yang-Mills theory from gravity side. In this section, we show one
procedure to solve this problem.

We start with the classical solution in background (2.2), and use the Buscher
rule to find the dual classical solution in NCAdS spacetime (2). More
precisely, we start with the folded string [41] in original $AdS_{5}\times
S^{5}$ spacetime, which plays an important role in the study of AdS/CFT
correspondence. Our calculation in this section can be regarded as the NCAdS
spacetime version of the folded string studied in [36].

Following the usual notation for folded string in literature, we use the
Lorentizian signature worldsheet action

$\displaystyle S=-\frac{1}{4\pi\alpha^{\prime}}\int d\tau
d\sigma\big{(}\eta^{\alpha\beta}\partial_{\alpha}x^{\mu}\partial_{\beta}x^{\nu}G_{\mu\nu}-\epsilon^{\alpha\beta}B_{\mu\nu}\partial_{\alpha}x^{\mu}\partial_{\beta}x^{\nu}\big{)},$
(4.1)

where $\epsilon^{\tau\sigma}=1$, and $\eta^{\alpha\beta}$ is the metric of
worldsheet in Lorentization signature. Note that the background (2.2) does not
change by varying the signature of worldsheet. However, the transformation of
coordinate becomes (A.10). For the Euclidean signature worldsheet, the
T-duality transformation will map the real solutions to the complex ones (see
[42] for recent developments). Here we are considering the Lorentizian
signature, such that the T-duality will map the real solutions to the real
ones.

###  4.1 Coordinates and folded string solutions in original AdS spacetime

We first consider the folded string solution in the global coordinate of
original $AdS_{5}\times S^{5}$ spacetime. To fix the notation, we should
describe the coordinates that was used from now. The embedding coordinates of
original $AdS_{5}\times S^{5}$ spacetime are defined by

$\displaystyle ds_{{}_{AdS_{5}}}^{2}=dX_{M}dX^{M},\quad-
X_{M}X^{M}=X_{-1}^{2}+X_{0}^{2}-X_{1}^{2}-X_{2}^{2}-X_{3}^{2}-X_{4}^{2}=1,$
(4.2)

where we have set $R=1$ for simplicity. The global
coordinates$(t,\rho,\Omega_{3})$ is given by

$\displaystyle ds^{2}$ $\displaystyle=-\cosh^{2}\rho
dt^{2}+d\rho^{2}+\sinh^{2}\rho(d\phi^{2}+\cos^{2}\phi
d\theta_{1}^{2}+\sin^{2}\phi d\theta_{2}^{2}).$ (4.3)

with

$\displaystyle X_{0}+iX_{-1}$ $\displaystyle=\cosh\rho e^{it},\quad
X_{1}+iX_{2}=\sinh\rho\cos\phi e^{i\theta_{1}},\quad
X_{3}+iX_{4}=\sinh\rho\sin\phi e^{i\theta_{2}}.$ (4.4)

The Poincare coordinates are defined by

$\displaystyle z$ $\displaystyle=\frac{1}{X_{-1}-X_{4}},\quad
x^{0}=\frac{X_{0}}{X_{-1}-X_{4}},\quad x^{i}$
$\displaystyle=\frac{X^{i}}{X_{-1}-X_{4}},\quad i=1,2,3,$ (4.5)

whose metric is given by

$\displaystyle ds^{2}=\frac{dz^{2}+dx_{\mu}dx^{\mu}}{z^{2}}.$ (4.6)

The folded string is solved under the ansatz in global coordinate of AdS5
space

$\displaystyle t$
$\displaystyle=\kappa\tau,\quad\theta_{1}=\kappa\omega\tau,\quad\phi=\theta_{2}=0.$

In the conformal gauge, the e.o.m. and Virasoro constrains become

$\displaystyle\rho^{\prime\prime}+\kappa^{2}(\omega^{2}-1)\sinh\rho\cosh\rho$
$\displaystyle=0$ $\displaystyle\rho^{\prime
2}-\kappa^{2}(\cosh^{2}\rho-\omega^{2}\sinh^{2}\rho)$ $\displaystyle=0.$

This is solved by 999Our notation for is different with the one used in [36].

$\displaystyle\sinh(\rho)=\frac{1}{\omega}\frac{\mbox{sn}(\kappa\omega\sigma|\frac{1}{\omega^{2}})}{\mbox{dn}(\kappa\omega\sigma|\frac{1}{\omega^{2}})},\quad\cosh(\rho)=\frac{1}{\mbox{dn}(\kappa\omega\sigma|\frac{1}{\omega^{2}})},$
(4.7)

where sn and dn are the Jacobi Elliptic functions. See Appendix C for the
definition. Here we have focused on the case $\omega^{2}>1$. The folded string
rotates with the angular velocity $\frac{d\theta_{1}}{dt}=\omega$. The radial
coordinate $\rho(\sigma)$ varies in the range
$(0,\mbox{arctanh}(\frac{1}{\omega}))$,which is fixed by the condition
$\frac{d\rho}{d\sigma}|_{\rho\to\rho_{0}}=0$. Since the constant B-field does
not effect the e.o.m. and Virasoro constraints, the classical folded string
solution in original $AdS_{5}\times S^{5}$ spacetime is also a solution in
background (2.2). Writing the solution (4.7) in the background (2.2), we
obtain the classical folded string solution

$\displaystyle{z}$
$\displaystyle=\frac{\mbox{dn}(\kappa\omega\sigma|\frac{1}{\omega^{2}})}{\sin(\kappa\tau)},\quad\tilde{x}^{0}=\frac{\cos(\kappa\tau)}{\sin(\kappa\tau)}$
(4.8) $\displaystyle\tilde{x}^{1}$
$\displaystyle=\frac{1}{\omega}\mbox{sn}(\kappa\omega\sigma|\frac{1}{\omega^{2}})\frac{\cos(\kappa\omega\tau)}{\sin(\kappa\tau)},\quad\tilde{x}^{2}=\frac{1}{\omega}\mbox{sn}(\kappa\omega\sigma|\frac{1}{\omega^{2}})\frac{\sin(\kappa\omega\tau)}{\sin(\kappa\tau)}.$

###  4.2 Classical solution in NCAdS spacetime dual to folded string

Our task is to find the corresponding T-dual(back) solution in background (2),
which is the background before the T-dual transformation. By definition, $r$
can be obtained by

$\displaystyle r$ $\displaystyle=$
$\displaystyle\frac{1}{\tilde{z}}=\frac{\sin(\kappa\tau)}{\mbox{dn}(\kappa\omega\sigma|\frac{1}{\omega^{2}})}.$
(4.9)

The Buscher rule for coordinates (A.10) leads to

$\displaystyle\partial_{\beta}\tilde{x}^{1}$ $\displaystyle=$
$\displaystyle\epsilon_{\beta\alpha}\partial^{\alpha}x^{1}\frac{{\cal
M}}{r^{2}}-\partial_{\beta}x^{2}\frac{{\cal M\gamma}}{r^{4}}$
$\displaystyle\partial_{\beta}\tilde{x}^{2}$ $\displaystyle=$
$\displaystyle\epsilon_{\beta\alpha}\partial^{\alpha}x^{2}\frac{1}{r^{2}}+\epsilon_{\beta\alpha}\eta^{\alpha\gamma}\Big{(}\epsilon_{\gamma\delta}\partial^{\delta}x^{1}\frac{{\cal
M}\gamma}{r^{4}}-\partial_{\gamma}x^{2}\frac{{\cal
M}\gamma^{2}}{r^{6}}\Big{)}$ $\displaystyle\partial_{\beta}\tilde{x}^{3}$
$\displaystyle=$
$\displaystyle\epsilon_{\beta\alpha}\partial^{\alpha}x^{3}\frac{1}{r^{2}}$
(4.10) $\displaystyle\partial_{\beta}\tilde{x}^{0}$ $\displaystyle=$
$\displaystyle-\epsilon_{\beta\alpha}\partial^{\alpha}x^{0}\frac{1}{r^{2}}.$

where ${\cal M}^{-1}=1+\frac{\gamma^{2}R^{4}}{r^{4}}$, and $x^{\mu}$ means the
coordinate in background (2). Resolving $\partial_{\alpha}x^{\mu}$ by using
$\partial_{\beta}\tilde{x}^{\mu}$, we find

$\displaystyle\partial_{\sigma}x^{0}$
$\displaystyle=r^{2}\partial_{\tau}\tilde{x}^{0},\quad\partial_{\tau}x^{0}=r^{2}\partial_{\sigma}\tilde{x}^{0},$
$\displaystyle\partial_{\sigma}\big{(}x^{1}+ix^{2}\big{)}$
$\displaystyle=-i\gamma\partial_{\sigma}\big{(}\tilde{x}^{1}+i\tilde{x}^{2}\big{)}-r^{2}\partial_{\tau}\big{(}\tilde{x}^{1}+i\tilde{x}^{2}\big{)},$
(4.11) $\displaystyle\partial_{\tau}\big{(}x^{1}+ix^{2}\big{)}$
$\displaystyle=-i\gamma\partial_{\tau}\big{(}\tilde{x}^{1}+i\tilde{x}^{2}\big{)}-r^{2}\partial_{\sigma}\big{(}\tilde{x}^{1}+i\tilde{x}^{2}\big{)}.$

Substituting (4.8) to (4.11), we obtain

$\displaystyle r$
$\displaystyle=\frac{\sin(\kappa\tau)}{\mbox{dn}(\kappa\omega\sigma|\frac{1}{\omega^{2}})},\quad
x^{0}=-\frac{\frac{\text{cn}\left(\omega\kappa\sigma\left|\frac{1}{\omega^{2}}\right.\right)\text{sn}\left(\omega\kappa\sigma\left|\frac{1}{\omega^{2}}\right.\right)}{\omega^{2}\text{dn}\left(\omega\kappa\sigma\left|\frac{1}{\omega^{2}}\right.\right)}-\mbox{E}\left(\text{am}\left(\omega\kappa\sigma\left|\frac{1}{\omega^{2}}\right.\right)|\frac{1}{\omega^{2}}\right)}{\left(\frac{1}{\omega^{2}}-1\right)\omega},$
(4.12) $\displaystyle x^{1}+ix^{2}$
$\displaystyle=-i\gamma\frac{e^{i\kappa\omega\tau}}{\sin(\kappa\tau)}\mbox{sn}(\kappa\omega\sigma|\frac{1}{\omega^{2}})-\frac{e^{i\kappa\omega\tau}\text{cn}\left(\omega\kappa\sigma\left|\frac{1}{\omega^{2}}\right.\right)(\cos(\kappa\tau)-i\omega\sin(\kappa\tau))}{(\omega^{2}-1)\text{dn}\left(\omega\kappa\sigma\left|\frac{1}{\omega^{2}}\right.\right)}.$

Using Mathematica, one can check that the solution (4.12) satisfy the e.o.m.
in the background (2)

$\displaystyle
r(x^{0})^{\prime\prime}-r\ddot{x}^{0}+2\dot{r}\dot{x}^{0}-2r^{\prime}(x^{0})^{\prime}=0$
(4.13)
$\displaystyle\big{(}(x^{2})^{\prime\prime}-\ddot{x}^{2}\big{)}\left(\gamma^{2}r+r^{5}\right)+(x^{2})^{\prime}\left(2\gamma^{2}r^{\prime}-2r^{4}r^{\prime}\right)+\dot{x}^{2}\left(2r^{4}\dot{r}-2\gamma^{2}\dot{r})\right)+4\gamma
r^{2}r^{\prime}\dot{x}^{1}-4\gamma r^{2}\dot{r}(x^{1})^{\prime}=0$
$\displaystyle\big{(}(x^{1})^{\prime\prime}-\ddot{x}^{1}\big{)}\left(\gamma^{2}r+r^{5}\right)+(x^{1})^{\prime}\left(2\gamma^{2}r^{\prime}-2r^{4}r^{\prime}\right)+\dot{x}^{1}\left(2r^{4}\dot{r}-2\gamma^{2}\dot{r}\right)+4\gamma
r^{2}\dot{r}(x^{2})^{\prime}-4\gamma r^{\prime}r^{2}\dot{x}^{2}=0$
$\displaystyle\big{(}r^{\prime\prime}-\ddot{r}\big{)}\left(\gamma^{4}r+2\gamma^{2}r^{5}+r^{9}\right)+\big{(}\dot{r}^{2}-(r^{\prime})^{2}\big{)}\left(2\gamma^{2}r^{4}+r^{8}+\gamma^{4}\right)+\big{(}(\dot{x}^{0})^{2}-((x^{0})^{\prime})^{2}\big{)}\left(2\gamma^{2}r^{4}+r^{8}+\gamma^{4}\right)$
$\displaystyle+\big{(}((x^{1})^{\prime})^{2}-(\dot{x}^{1})^{2}\big{)}\left(r^{8}-\gamma^{2}r^{4}\right)+4\gamma
r^{6}\big{(}(x^{1})^{\prime}\dot{x}^{2}-\dot{x}^{1}(x^{2})^{\prime}\big{)}+\big{(}(\dot{x}^{2})^{2}-((x^{2})^{\prime})^{2}\big{)}\left(\gamma^{2}r^{4}-r^{8}\right)=0.$

and the Virasoro constraints

$\displaystyle T_{\tau\tau}$ $\displaystyle=$ $\displaystyle
T_{\sigma\sigma}=\frac{1}{2}\big{(}\dot{x}^{\mu}\dot{x}_{\mu}+(x^{\mu})^{\prime}(x_{\mu})^{\prime}\big{)}=0$
(4.14) $\displaystyle T_{\tau\sigma}$ $\displaystyle=$ $\displaystyle
T_{\sigma\tau}=\dot{x}^{\mu}(x_{\mu})^{\prime}=0,$ (4.15)

where $\dot{}$ and ′ mean the derivative about $\tau$ and $\sigma$
respectively101010In background (2.2), the e.o.m. are  $\displaystyle
r^{\prime\prime}+\frac{-(r^{\prime})^{2}+\dot{r}^{2}-(x^{0\prime})^{2}+(\dot{x}^{0})^{2}+(x^{1\prime})^{2}-(\dot{x}^{1})^{2}+(x^{2\prime})^{2}-(\dot{x}^{2})^{2}}{r}$
$\displaystyle=$ $\displaystyle\ddot{r},$
$\displaystyle(x^{0})^{\prime\prime}+\frac{2\dot{r}\dot{x}^{0}-2r^{\prime}x^{0\prime}}{r}$
$\displaystyle=$ $\displaystyle\ddot{x}^{0},$
$\displaystyle(x^{1})^{\prime\prime}+\frac{2\dot{r}\dot{x}^{1}-2r^{\prime}x^{1\prime}}{r}$
$\displaystyle=$ $\displaystyle\ddot{x}^{1}.$ (4.16) . Therefore, (4.12) is a
classical solution in NCAdS background (2). One could also follow the
procedure in [36], and interchange $\tau$ and $\sigma$ to have an open string
interpretation for this classical solution

$\displaystyle r$
$\displaystyle=\frac{\sin(\kappa\sigma)}{\mbox{dn}(\kappa\omega\tau|\frac{1}{\omega^{2}})},\quad
x^{0}=-\frac{\frac{\text{cn}\left(\omega\kappa\tau\left|\frac{1}{\omega^{2}}\right.\right)\text{sn}\left(\omega\kappa\tau\left|\frac{1}{\omega^{2}}\right.\right)}{\omega^{2}\text{dn}\left(\omega\kappa\tau\left|\frac{1}{\omega^{2}}\right.\right)}-\mbox{E}\left(\text{am}\left(\omega\kappa\tau\left|\frac{1}{\omega^{2}}\right.\right)|\frac{1}{\omega^{2}}\right)}{\left(\frac{1}{\omega^{2}}-1\right)w},$
(4.17) $\displaystyle x^{1}+ix^{2}$
$\displaystyle=-i\gamma\frac{e^{i\kappa\omega\sigma}}{\sin(\kappa\sigma)}\mbox{sn}(\kappa\omega\tau|\frac{1}{\omega^{2}})-\frac{e^{i\kappa\omega\sigma}\text{cn}\left(\omega\kappa\tau\left|\frac{1}{\omega^{2}}\right.\right)(\cos(\kappa\sigma)-i\omega\sin(\kappa\sigma))}{(\omega^{2}-1)\text{dn}\left(\omega\kappa\tau\left|\frac{1}{\omega^{2}}\right.\right)}.$

At the boundary of NCAdS background, i.e. $r=0$,
$\sigma=0,\frac{\pi}{\kappa}$. The worldsheet (4.12) ends on two curves
corresponding to $\sigma=0,\frac{\pi}{\kappa}$ respectively

$\displaystyle x^{0}$ $\displaystyle=$
$\displaystyle-\frac{\frac{\text{cn}\left(\omega\kappa\tau\left|\frac{1}{\omega^{2}}\right.\right)\text{sn}\left(\omega\kappa\tau\left|\frac{1}{\omega^{2}}\right.\right)}{\omega^{2}\text{dn}\left(\omega\kappa\tau\left|\frac{1}{\omega^{2}}\right.\right)}-\mbox{E}\left(\text{am}\left(\omega\kappa\tau\left|\frac{1}{\omega^{2}}\right.\right)|\frac{1}{\omega^{2}}\right)}{\left(\frac{1}{\omega^{2}}-1\right)w}$
$\displaystyle x^{1}+ix^{2}$ $\displaystyle=$
$\displaystyle-i\gamma\frac{1}{\sin(\kappa\sigma=0)}\mbox{sn}(\kappa\omega\tau|\frac{1}{\omega^{2}})-\frac{\text{cn}\left(\omega\kappa\tau\left|\frac{1}{\omega^{2}}\right.\right)}{(w^{2}-1)\text{dn}\left(\omega\kappa\tau\left|\frac{1}{\omega^{2}}\right.\right)}.$
(4.18)

and

$\displaystyle x^{0}$
$\displaystyle=-\frac{\frac{\text{cn}\left(\omega\kappa\tau\left|\frac{1}{\omega^{2}}\right.\right)\text{sn}\left(\omega\kappa\tau\left|\frac{1}{\omega^{2}}\right.\right)}{\omega^{2}\text{dn}\left(\omega\kappa\tau\left|\frac{1}{\omega^{2}}\right.\right)}-\mbox{E}\left(\text{am}\left(\omega\kappa\tau\left|\frac{1}{\omega^{2}}\right.\right)|\frac{1}{\omega^{2}}\right)}{\left(\frac{1}{\omega^{2}}-1\right)w},$
$\displaystyle x^{1}+ix^{2}$
$\displaystyle=-i\gamma\frac{e^{i\pi\omega}}{\sin(\kappa\sigma=\pi)}\mbox{sn}(\kappa\omega\tau|\frac{1}{\omega^{2}})+\frac{e^{i\pi\omega}\text{cn}\left(\omega\kappa\tau\left|\frac{1}{\omega^{2}}\right.\right)}{(\omega^{2}-1)\text{dn}\left(\omega\kappa\tau\left|\frac{1}{\omega^{2}}\right.\right)}.$
(4.19)

The two curves (4.18) and (4.19) at the boundary are related by the spatial
rotation $-e^{i\kappa\omega\sigma}|_{\sigma=\frac{\pi}{\kappa}}$. One may
wonder weather it makes sense or not as $x^{1}+ix^{2}\to\infty$. It is easy to
show $x^{1}+ix^{2}$ gives a finite value as it couple to the metric in (2).
Note that (4.12) can be regarded as a periodic open string $\tau\in(0,2\pi)$
and $\sigma\in(0,\frac{\pi}{\kappa})$.

It is interesting to calculate the on-shell action which is equal to the
minimal area of the surface ending on the Wilson loop. Substituting (4.17) to
(4.1), we find

$\displaystyle-4\pi\alpha^{\prime}S=$
$\displaystyle\int_{0}^{2\pi}d\tau\int_{0}^{\pi/\kappa}d\sigma
2\kappa^{2}\Big{(}\gamma\csc^{2}(\kappa\sigma)\text{cn}\left(\kappa\omega\tau\left|\frac{1}{\omega^{2}}\right.\right)\text{dn}\left(\kappa\omega\tau\left|\frac{1}{\omega^{2}}\right.\right)\text{sn}\left(\kappa\omega\tau\left|\frac{1}{\omega^{2}}\right.\right)$
$\displaystyle-\csc^{2}(\kappa\sigma)+\text{sn}\left(\kappa\omega\tau\left|\frac{1}{\omega^{2}}\right.\right)^{2}\Big{)}$
$\displaystyle=$ $\displaystyle
4\pi^{2}\kappa\omega^{2}-2\pi\omega\mbox{E}\left(\text{am}\left(2\pi\omega\kappa\left|\frac{1}{\omega^{2}}\right.\right)|\frac{1}{\omega^{2}}\right)+S_{div},$
(4.20)

where $S_{div}$ is the divergent pieces given by

$\displaystyle
S_{div}=\left.\left(4\pi\kappa\cot(\kappa\sigma)+\frac{\gamma\cot(\kappa\sigma)\left(\text{cn}\left(2\pi\omega\kappa\left|\frac{1}{\omega^{2}}\right.\right)^{2}-1\right)}{\omega}\right)\right|_{\sigma=0}^{\sigma=\pi/\kappa}.$
(4.21)

The source of the divergence in the integration comes from the singular
behavior of the integrand near the boundary. We perform the regularization by
shifting the boundary from $r=0$ to $r=\epsilon$ with a small $\epsilon$. For
a given $\tau$, the range of $\sigma$ change from $(0,\frac{\pi}{\kappa})$ to
$(\frac{\arcsin\left(\epsilon\text{dn}\left(\omega\kappa\tau\left|\frac{1}{\omega^{2}}\right.\right)\right)}{\kappa},\frac{\pi-\arcsin\left(\epsilon\text{dn}\left(\omega\kappa\tau\left|\frac{1}{\omega^{2}}\right.\right)\right)}{\kappa})$.
We first integrate over $\sigma$ with a fixed $\tau$ and find

$\displaystyle-4\pi\alpha^{\prime}S$ $\displaystyle=\int_{0}^{2\pi}d\tau
2\kappa\left(\frac{2\gamma\text{cn}\left(\kappa\omega\tau\left|\frac{1}{\omega^{2}}\right.\right)\text{sn}\left(\kappa\omega\tau\left|\frac{1}{\omega^{2}}\right.\right)\sqrt{1-\epsilon^{2}\text{dn}\left(\kappa\omega\tau\left|\frac{1}{\omega^{2}}\right.\right)^{2}}}{\epsilon}\right.$
(4.22)
$\displaystyle\left.+\text{sn}\left(\kappa\omega\tau\left|\frac{1}{\omega^{2}}\right.\right)^{2}\left(\pi-2\sin^{-1}\left(\epsilon\text{dn}\left(\kappa\omega\tau\left|\frac{1}{\omega^{2}}\right.\right)\right)\right)-\frac{2\sqrt{1-\epsilon^{2}\text{dn}\left(\kappa\omega\tau\left|\frac{1}{\omega^{2}}\right.\right)^{2}}}{\epsilon\text{dn}\left(\kappa\omega\tau\left|\frac{1}{\omega^{2}}\right.\right)}\right)$

Expanding the integrand by using $\epsilon$, one finds

$\displaystyle-4\pi\alpha^{\prime}S$
$\displaystyle=\int_{0}^{2\pi}d\tau\frac{4\gamma\kappa\text{cn}\left(\tau\omega\kappa\left|\frac{1}{\omega^{2}}\right.\right)\text{sn}\left(\tau\omega\kappa\left|\frac{1}{\omega^{2}}\right.\right)-\frac{4\kappa}{\text{dn}\left(\kappa\omega\tau\left|\frac{1}{\omega^{2}}\right.\right)}}{\epsilon}+2\pi\kappa\text{sn}\left(\kappa\omega\tau\left|\frac{1}{\omega^{2}}\right.\right)^{2}$
(4.23)
$\displaystyle-2\epsilon\left(\kappa\text{dn}\left(\kappa\omega\tau\left|\frac{1}{\omega^{2}}\right.\right)\left(\gamma\text{cn}\left(\kappa\omega\tau\left|\frac{1}{\omega^{2}}\right.\right)\text{dn}\left(\kappa\omega\tau\left|\frac{1}{\omega^{2}}\right.\right)\text{sn}\left(\kappa\omega\tau\left|\frac{1}{\omega^{2}}\right.\right)+2\text{sn}\left(\kappa\omega\tau\left|\frac{1}{\omega^{2}}\right.\right)^{2}-1\right)\right)$
$\displaystyle+O\left(\epsilon^{3}\right).$

Then integrating over $\tau$, one obtains

$\displaystyle-4\pi\alpha^{\prime}S$
$\displaystyle=\frac{4\gamma\omega\left[1-\text{dn}\left(2\pi\kappa\omega\left|\frac{1}{\omega^{2}}\right.\right)\right]-\frac{4}{\sqrt{\omega^{2}-1}}\arccos\left(\frac{\text{cn}\left(2\pi\kappa\omega\left|\frac{1}{\omega^{2}}\right.\right)}{\text{dn}\left(2\pi\kappa\omega\left|\frac{1}{\omega^{2}}\right.\right)}\right)}{\epsilon}$
$\displaystyle+4\pi^{2}\kappa\omega^{2}-2\pi\omega\mbox{E}\left(\text{am}\left(2\pi\kappa\omega\left|\frac{1}{\omega^{2}}\right.\right)|\frac{1}{\omega^{2}}\right)$
$\displaystyle+\epsilon\left(\frac{2\text{cn}\left(2\pi\kappa\omega\left|\frac{1}{\omega^{2}}\right.\right)\text{sn}\left(2\pi\kappa\omega\left|\frac{1}{\omega^{2}}\right.\right)}{\omega}+\frac{2}{3}\gamma\omega\text{dn}\left(2\pi\kappa\omega\left|\frac{1}{\omega^{2}}\right.\right)^{3}-\frac{2\gamma\omega}{3}\right).$
(4.24)

Note that the parameter $\gamma$ appears in the divergence term, but does not
appear in the $\epsilon^{0}$ term. The divergence in (4.24) has the form of
$\frac{1}{\epsilon}$, which is the same form as the one of quark-antiquark
Wilson loop case but different from the divergent behavior
($\log\frac{1}{\epsilon}$) in cusp Wilson loop case [43]. The divergence in
(4.24) can be interpreted as a self energy of heavy quark pairs in NCSYM.

The subtracted on-shell action is

$\displaystyle(-4\pi\alpha^{\prime}S)_{reg}$
$\displaystyle=(1+\epsilon\frac{\partial}{\partial\epsilon})(-4\pi\alpha^{\prime}S)=4\pi^{2}\kappa\omega^{2}-2\pi\omega\mbox{E}\left(\text{am}\left(2\pi\kappa\omega\left|\frac{1}{\omega^{2}}\right.\right)|\frac{1}{\omega^{2}}\right),$

where we have dropped the $\epsilon$ term. Here we just make use of minimal
substraction to regular the action. We notice that the subtracted on-shell
action does not depend on the parameter $\gamma$.

##  5 Conclusions and Discussions

In this paper, we have performed certain bosonic T-duality and fermionic
T-duality transformations on the NCAdS background, and found the final dual
background is the usual $AdS_{5}\times S^{5}$ background but with a constant
NS-NS B-field depending on the non-commutative parameter. Our transformation
can be regarded as the simplification of the NCAdS background, which is very
useful to study the physics in non-commutative super Yang-Mills theory. As
application, we have studied the gluon scattering amplitudes and Wilson loop
in the NCSYM holographically by using the simplified final dual background. In
the final dual background, we found the worldsheet ending on the null polygon
Wilson loop dual to the gluon scattering amplitudes in the NCSYM theory, which
extends the scattering amplitude/Wilson loop duality for the NCSYM. We found
that the non-commutative deformation will contribute to the gluon scattering
amplitude as overall dressing phase phase factor shown in section 3, which is
valid even for the finite ${\lambda}$. Furthermore, motivated by the relation
between closed and open strings [36], we started with the folded string in the
final dual background, and constructed the periodic open string (Wilson loop)
solution in the NCAdS background. We have also calculated the on-shell action
of the open string solution, which describes the minimal area of the ending on
the Wilson loop. The divergence of the on-shell action appears in the form of
$\frac{1}{\epsilon}$, where $\epsilon$ is the small regularization parameter.
We also noticed that the subtracted on-shell action does not depend on the
non-commutative parameter.

On the other hand, an open-closed string map for the TsT deformation has been
argued in a series of papers [44, 45, 46, 47, 48, 49, 50, 51]. Using the open-
closed string map on the TsT deformation background, one finds the open string
metric and coupling go back to the original background. The information about
TsT deformation only appear in the non-commutative parameter. The series
T-duality transformation in our present work also map the TsT deformation
background to the original background but with a inverse radial coordinate in
the metric. It is an interesting problem to explore the relations between our
works and the open-closed string map.

It would be interesting to study the bosonic and fermionic T-duality
transformations on other type of TsT deformation of the $AdS_{5}\times S^{5}$
background, which may lead to a simplification in the same way. We also would
like to construct the fermionic T duality of deformed ABJM theory [18]. Unlike
$AdS_{5}\times S_{5}$, the holographic background of ABJM is not self-dual
background by fermionic T-dual, it will be very interesting and highly non-
trivial to holographically investigate the corresponding string solutions
which correspond to gluon scattering amplitude, Wilson Loop operator and
integrability structures [52] of anomalous dimension of local operators in
various deformed ABJM theories in the future.

## Acknowledgement

We would like to thank K. Ito, H. Nakajima and J. Sakamoto for the useful
discussions. We also thank E. O. Colgain and M. M. Sheikh-Jabbari for very
useful comments on this manuscript. S.H. are grateful to the Mainz institute
for Theoretical Physics (MITP) and Les Houches workshop on structures in local
quantum field at Ecole de Physique des Houches for its hospitality and its
partial supports during the completion of this work. S.H. is supported from
Max-Planck fellowship in Germany, the German-Israeli Foundation for Scientific
Research and Development. The work of H.S. are supported in part by JSPS
Research Fellowship for Young Scientists, from the Japan Ministry of
Education, Culture, Sports, Science and Technology

##  Appendix A Bosonic T-dual transformation

We summarize the Buscher rule that is used in this paper. One can also see
[53, 18] for review. We start with the Euclidean worldsheet action as
following

$\displaystyle S=\frac{1}{4\pi\alpha^{\prime}}\int
d^{2}\sigma(\sqrt{h}h^{ab}g_{mn}+i\epsilon^{ab}b_{mn})\partial_{a}x^{m}\partial_{b}x^{n}.$
(A.1)

and assume that the background field $g_{mn}$ and $b_{mn}$ are invariant under
the shift isometry:

$\displaystyle x^{1}\to x^{1}+c,\leavevmode\nobreak\ \leavevmode\nobreak\
x^{\hat{m}}\to x^{\hat{m}},$ (A.2)

where $c$ is a constant and $\hat{m}\neq 1$. We denote the field $f$
($f=g_{mn},b_{mn},\phi,x^{m}$) after the T-dual transformation as
$f^{\prime}$. We do the T dual operation along $x^{1}$ and then the Buscher
rule can be summarized as

$\displaystyle{g}_{11}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{g_{11}},\quad{g}^{\prime}_{1i}=\frac{b_{1i}}{g_{11}},\quad{g}^{\prime}_{ij}=g_{ij}-\frac{g_{1i}g_{1j}-b_{1i}b_{1j}}{g_{11}}$
$\displaystyle{b}_{1i}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{g_{1i}}{g_{11}},\quad{b}^{\prime}_{ij}=b_{ij}-\frac{g_{1i}b_{1j}-b_{1i}g_{1j}}{g_{11}},$
(A.3) $\displaystyle\phi^{\prime}$ $\displaystyle=$
$\displaystyle\phi-\frac{1}{2}\log|g_{11}|.$

The coordinates transform as

$\displaystyle\partial_{a}{x^{\prime}}^{1}$ $\displaystyle=$
$\displaystyle-i\epsilon_{ab}[g_{11}\partial_{b}{x}^{1}+g_{1\hat{m}}\partial_{b}x^{\hat{m}}]-b_{1\hat{m}}\partial_{a}x^{\hat{m}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
{x^{\prime}}^{\hat{m}}=x^{\hat{m}}.$ (A.4)

We then summarize the transformation of RR fields. Given a $p-$form
$\omega_{p}$, we decompose it as

$\displaystyle\omega_{p}=\bar{\omega}_{p}+\omega_{p[y]}\wedge dy,$ (A.5)

where
$\bar{\omega}_{p}=\frac{1}{p!}\omega_{\alpha_{1}\cdots\alpha_{p}}dx^{\alpha_{1}}\wedge\cdots\wedge
dx^{\alpha_{p}}$ does not contain $dy$ component. $\omega_{p[y]}$ is a $(p-1)$
form as $(\omega_{p[y]})_{\alpha_{1}\cdots\alpha_{p-1}y}$. For convention, we
define the two one-form fields $j$ and $b$ as:

$\displaystyle j$ $\displaystyle=$ $\displaystyle\frac{G_{\alpha
y}}{G_{yy}}dx^{\alpha},\quad b=B_{[y]}+dy.$ (A.6)

The T-duality rules for the R-R potential are then given by

$\displaystyle C_{p}^{\prime}$ $\displaystyle=$ $\displaystyle
C_{p+1[y]}+\bar{C}_{p-1}\wedge b+C_{p-1[y]}\wedge b\wedge j.$ (A.7)

In the modified field strength ${\cal F}_{p}=F_{p}+H\wedge C_{p-3}$, they
change as

$\displaystyle{\cal F}_{p}^{\prime}$ $\displaystyle=$ $\displaystyle{\cal
F}_{p+1[y]}+\bar{{\cal F}}_{p-1}\wedge b+{\cal F}_{p-1[y]}\wedge b\wedge j$
(A.8)

where we used $db=H_{[y]}$.

Sometimes, it is conventional to use Lorentizian worldsheet action, e.g. the
study of GKP string:

$\displaystyle S=-\frac{1}{4\pi\alpha^{\prime}}\int d\tau
d\sigma[\eta^{\alpha\beta}\partial_{a}x^{m}\partial_{\beta}x^{n}g_{mn}-\epsilon^{\alpha\beta}b_{mn}\partial_{\alpha}x^{m}\partial_{\beta}x^{n}].$
(A.9)

In this case, (A.4) becomes

$\displaystyle\epsilon^{\alpha\beta}\partial_{\beta}{x}^{\prime
1}=\eta^{\alpha\beta}\partial_{\beta}x^{m}G_{1m}-\epsilon^{\alpha\beta}\partial_{\beta}x^{m}B_{1m}.$
(A.10)

##  Appendix B Killing spinor equations in original $AdS_{5}\times S^{5}$
spacetime

We consider the Killing spinor equations in the original spacetime

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\frac{R^{2}}{r^{2}}[dx^{2}+dr^{2}]+R^{2}ds_{5}^{2}=\frac{R^{2}}{r^{2}}[dx^{2}+\sum^{6}_{j=1}dy_{j}dy_{j}],$
(B.1) $\displaystyle F_{5}$ $\displaystyle=$
$\displaystyle-4R^{4}\left(\frac{1}{r^{5}}dt\wedge dx_{1}\wedge dx_{2}\wedge
dx_{3}\wedge dr+\omega_{S^{5}}\right),$ (B.2)

where $|y|=r$. Then the Killing spinor equations become

$\displaystyle\delta\psi_{\hat{M}}$ $\displaystyle=$
$\displaystyle{e_{\hat{M}}}^{M}\nabla_{M}\epsilon+\frac{1}{2}\frac{1}{R}\gamma^{\hat{0}\hat{1}\hat{2}\hat{3}\hat{4}}\gamma_{\hat{M}}\hat{\epsilon}=0,$
(B.3) $\displaystyle\delta\hat{\psi}_{\hat{M}}$ $\displaystyle=$
$\displaystyle{e_{\hat{M}}}^{M}\nabla_{M}\hat{\epsilon}-\frac{1}{2}\frac{1}{R}\gamma^{\hat{0}\hat{1}\hat{2}\hat{3}\hat{4}}\gamma_{\hat{M}}\epsilon=0.$
(B.4)

We are mainly interested on the Killing spinor in [4], where
$\hat{\epsilon}=i\epsilon$ and $\epsilon$ is independent of the the
coordinates $M=x^{0,1,2,3}$. This leads to the relation

$\displaystyle i\gamma^{\hat{0}\hat{1}\hat{2}\hat{3}\hat{4}}\epsilon$
$\displaystyle=\gamma_{\hat{4}}\epsilon.$ (B.5)

Then the Killing spinor equations become

$\displaystyle{e_{\hat{j}}}^{j}\nabla_{j}\epsilon-\frac{1}{2R}\gamma_{\hat{j}}\gamma_{\hat{4}}\epsilon$
$\displaystyle=0,$ (B.6)

where $j$ the coordinate of the remaining 6D part ($R\times S^{5}$). Since the
remaining 6D part does not transform under our deformation. This leads to the
Killing spinors (2.20) used in [4]. It is easy to find (B.6) has the same form
as with (2.14). This is the reason why we choose the same $\epsilon$ as [4] to
perform fermionic T-dual transformation in sec.2. The difference between our
Killing spinor and the chosen one in [4] is the signature of $\hat{\epsilon}$.

##  Appendix C Elliptic functions

In this appendix, we summarize the Elliptic Integrals and Jacobi Elliptic
functions that were used in sec.4. We main follow the notations in [54]. The
elliptic integrals E and $F$ are defined by:

$\mbox{E}(\phi|m)=\int_{0}^{\phi}\sqrt{1-m\sin^{2}\theta}d\theta,\quad\mbox{F}(\phi|m)=\int_{0}^{\phi}\frac{d\theta}{\sqrt{1-m\sin^{2}\theta}}.$
(C.1)

Using $\phi$ in $u=\mbox{F}(\phi|m)$, we define the Jacobi Elliptic functions
by

$\displaystyle\mbox{sn}(u|m)=\sin\phi,\quad\mbox{cn}(u|m)=\cos\phi,\quad\mbox{dn}(u|m)=\sqrt{1-m\sin^{2}\phi},\quad\phi=\mbox{am}(u|m).$
(C.2)

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