# More than one Author with different Affiliations

Jitendra Pal<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology Roorkee, Roorkee 247667 Uttarakhand, India  Arnab Mukherjee
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology
Roorkee, Roorkee 247667 Uttarakhand, India  Arindam Lala
<EMAIL_ADDRESS><EMAIL_ADDRESS>Instituto de Física,
Pontificia Universidad Católica de Valparaíso, Casilla 4059, Valparaiso, Chile
Dibakar Roychowdhury<EMAIL_ADDRESS><EMAIL_ADDRESS>Department
of Physics, Indian Institute of Technology Roorkee, Roorkee 247667
Uttarakhand, India

# Analytic (non)integrability of Arutyunov-Bassi-Lacroix model

Jitendra Pal<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology Roorkee, Roorkee 247667 Uttarakhand, India  Arnab Mukherjee
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology
Roorkee, Roorkee 247667 Uttarakhand, India  Arindam Lala
<EMAIL_ADDRESS><EMAIL_ADDRESS>Instituto de Física,
Pontificia Universidad Católica de Valparaíso, Casilla 4059, Valparaiso, Chile
Dibakar Roychowdhury<EMAIL_ADDRESS><EMAIL_ADDRESS>Department
of Physics, Indian Institute of Technology Roorkee, Roorkee 247667
Uttarakhand, India

###### Abstract

We use the notion of the gauge/string duality and discuss the Liouvillian
(non) integrability criteria for string sigma models in the context of
recently proposed Arutyunov-Bassi-Lacroix (ABL) model [JHEP 03 (2021), 062].
Our analysis complements those previous results due to numerical analysis as
well as Lax pair formulation. We consider a winding string ansatz for the
deformed torus $T^{\quantity(\lambda_{1},\lambda_{2},\lambda)}_{k}$ which can
be interpreted as a system of coupled pendulums. Our analysis reveals the
Liouvillian nonintegrablity of the associated sigma model. We also obtain the
_generalized_ decoupling limit and confirm the analytic integrability for the
decoupled sector.

##  1 Introduction and motivation

Integrable systems are very special - they posses an infinite tower of
conserved charges those are in involution. The notion of integrability was
first introduced in the seminal work of Bethe in an attempt to solve the
Heisenberg spin chain model [1]. Subsequently, these techniques were further
developed and applied to models like QCD in [2]. In recent years, following
the pioneering work of [3], there has been a surge in the study of integrable
systems within the framework of gauge/string duality [4]-[27].

In this work, we use the methodology of the AdS/CFT duality [28, 29, 30] in
order to study the dynamics of string solitons over Arutyunov-Bassi-Lacroix
(ABL) background
$\mathcal{R}\times\mathcal{T}^{\quantity(\lambda_{1},\lambda_{2},\lambda)}_{k}$
[31, 32]. Along the way, we explore the classical (non)integrability of the
associated phase space. This ABL background can be considered as a
generalization of the Einsteinian $T^{1,1}$ manifold [33] and is given by [31,
32]

$\displaystyle\differential s^{2}$ $\displaystyle=-\differential
t^{2}+\sum_{i=1}^{2}\lambda_{i}^{2}\left(\differential\theta_{i}^{2}+\sin^{2}\theta_{i}\differential\phi_{i}^{2}\right)+\lambda^{2}\left(\differential\psi+\cos\theta_{1}\differential\phi_{1}+\cos\theta_{2}\differential\phi_{2}\right)^{2}\,,$
(1) $\displaystyle B_{2}$
$\displaystyle=k\left(\differential\psi+\cos\theta_{1}\differential\phi_{1}\right)\wedge\left(\differential\psi+\cos\theta_{2}\differential\phi_{2}\right)\,.$
(2)

Here $\lambda_{1}$, $\lambda_{2}$, $\lambda$ and $k$ are real parameters.
Interestingly, the classical string dynamics is integrable only when
$k=\lambda^{2}$. When $k\neq\lambda^{2}$, the integrability of the system is
lost and it becomes chaotic. This has been confirmed very recently using
numerical techniques [32].

In this paper, we complement all those previous results [31, 32] by taking a
third path which is based on the notion of Kovacic’s algorithm [34, 35]. The
algorithm essentially offers a set of rules in order to verify the Liouvillian
non-integrability criteria for a classical 2$d$ sigma model over general
backgrounds. As our analysis reveals, it establishes a clear compatibility
between the Liouvillian (non)integrability and the Lax pair formulation of
2$d$ sigma model over ABL background.

In our analysis, we consider winding string ansatz where a string is wrapped
around all the isometry directions ($\phi_{1},\phi_{2},\psi$). We observe that
when the modes of the string along the ($\phi_{1},\phi_{2}$) directions are
coupled, the dynamics of the system can be modelled as a system of coupled
pendulums. Subsequently, the application of the Kovacic’s algorithm ensures
non-integrability of this coupled pendulum model. We also observe a
_generalized_ special limit when these pendulums decouple and become
integrable. Interestingly, the decoupling limit studied in [31, 32] can be
seen as a special case of our analysis. While achieving this later limit, we
observe that the string must satisfy certain conditions that have been
discussed in detail in the section 3.

The structure of the paper is as follows: In section 2 we study the string
sigma model in the ABL background (1). Using winding string ansatz, we
explicitly compute the set of differential equations that represents two
coupled harmonic oscillators. We explicitly determined the conditions that
make the Virasoro constraints to satisfy. Furthermore, we apply the Kovacic’s
algorithm to show analytically the non-integrability of this system of coupled
pendulums. In section 3 we analyze the decoupling limits of these coupled
pendulum system and checked the integrability in individual system. Finally,
we conclude in section 4. Additionally, we give a brief account of the
Kovacic’s algorithm in Appendix A in order to make the article self-contained.
Appendix B contains lengthy expressions of certain constant coefficients.

##  2 Classical sigma model on $\mathcal{R}\times
T^{\lambda_{2},\lambda_{1},\lambda}_{k}$

The classical $2d$ string sigma model can be expressed in the conformal gauge
as [36]

$S_{P}=\frac{1}{4\pi\alpha^{\prime}}\int\differential\tau\differential\sigma\quantity(\eta^{\alpha\beta}G_{MN}+\epsilon^{\alpha\beta}B_{MN})\partial_{\alpha}X^{M}\partial_{\beta}X^{N}=\frac{1}{4\pi\alpha^{\prime}}\int\differential\tau\differential\sigma
L_{P}\,.$ (3)

Here $\alpha,\beta$ are string worldsheet coordinates with metric
$\eta_{\alpha\beta}=\text{diag}\quantity(-1,1)$, and the target space
coordinates are labelled as $\displaystyle
X^{\mu}=\quantity(t,\theta_{1},\theta_{2},\phi_{1},\phi_{2},\psi)$.

Using (1) and (2), the Lagrangian density $L_{P}$ appearing in (3) can be
written as

$\displaystyle\begin{split}L_{P}&=\quantity(\dot{t}^{2}-t^{\prime
2})+\lambda_{1}^{2}\quantity(\theta_{1}^{\prime
2}-\dot{\theta_{1}}^{2})+\lambda_{2}^{2}\quantity(\theta_{2}^{\prime
2}-\dot{\theta_{2}}^{2})+\lambda^{2}\quantity(\psi^{\prime
2}-\dot{\psi}^{2})+\quantity(\lambda_{1}^{2}\sin^{2}\theta_{1}+\lambda^{2}\cos^{2}\theta_{1})\\\
&\qquad\quantity(\phi_{1}^{\prime
2}-\dot{\phi_{1}}^{2})+\quantity(\lambda_{2}^{2}\sin^{2}\theta_{2}+\lambda^{2}\cos^{2}\theta_{2})\quantity(\phi_{2}^{\prime
2}-\dot{\phi_{2}}^{2})+2\lambda^{2}\cos\theta_{1}\quantity(\psi^{\prime}\phi_{1}^{\prime}-\dot{\psi}\dot{\phi}_{1})\\\
&\qquad+2\lambda^{2}\cos\theta_{2}\quantity(\psi^{\prime}\phi_{2}^{\prime}-\dot{\psi}\dot{\phi}_{2})+2\lambda^{2}\cos\theta_{1}\cos\theta_{2}\quantity(\phi_{1}^{\prime}\phi_{2}^{\prime}-\dot{\phi}_{1}\dot{\phi}_{2})+2k\cos\theta_{1}\quantity(\dot{\phi}_{1}\psi^{\prime}-\dot{\psi}\phi_{1}^{\prime})\\\
&\qquad+2k\cos\theta_{2}\quantity(\dot{\psi}\phi_{2}^{\prime}-\dot{\phi}_{2}\psi^{\prime})+2k\cos\theta_{1}\cos\theta_{2}\quantity(\dot{\phi}_{1}\phi_{2}^{\prime}-\dot{\phi}_{2}\phi_{1}^{\prime})\,.\end{split}$
(4)

The conserved charges associated with the string $\sigma$-model (3) are simply
derived from the above density (4) and are written as

$\displaystyle E$ $\displaystyle=$ $\displaystyle\frac{\partial
L_{P}}{\partial\dot{t}}=2\dot{t}\,,$ (5) $\displaystyle P_{\psi}$
$\displaystyle=$ $\displaystyle\frac{\partial
L_{P}}{\partial\dot{\psi}}=-2\quantity[\lambda^{2}\dot{\psi}+\quantity(k\phi_{1}^{\prime}+\lambda^{2}\dot{\phi_{1}})\cos\theta_{1}-\quantity(k\phi_{2}^{\prime}-\lambda^{2}\dot{\phi_{2}})\cos\theta_{2}]\,,$
(6) $\displaystyle P_{\phi_{1}}$ $\displaystyle=$ $\displaystyle\frac{\partial
L_{P}}{\partial\dot{\phi_{1}}}=-2\big{[}\quantity(\lambda_{1}^{2}\sin^{2}\theta_{1}+\lambda^{2}\cos^{2}\theta_{1})\dot{\phi_{1}}-\quantity(k\psi^{\prime}-\lambda^{2}\dot{\psi})\cos\theta_{1}$
$\displaystyle\qquad\qquad-\quantity(k\phi_{2}^{\prime}-\lambda^{2}\dot{\phi_{2}})\cos\theta_{1}\cos\theta_{2}\big{]}\,,$
$\displaystyle P_{\phi_{2}}$ $\displaystyle=$ $\displaystyle\frac{\partial
L_{P}}{\partial\dot{\phi_{1}}}=-2\big{[}\quantity(\lambda_{2}^{2}\sin^{2}\theta_{2}+\lambda^{2}\cos^{2}\theta_{2})\dot{\phi_{2}}+\quantity(k\psi^{\prime}+\lambda^{2}\dot{\psi})\cos\theta_{2}$
$\displaystyle\qquad\qquad+\quantity(k\phi_{1}^{\prime}+\lambda^{2}\dot{\phi_{1}})\cos\theta_{1}\cos\theta_{2}\big{]}\,.$

Next, we consider the winding string ansatz of the following form [32]

$\displaystyle\begin{split}t={}&\tau\,,\qquad\qquad\qquad\theta_{1}=\theta_{1}\quantity(\tau)\,,\qquad\qquad\qquad\theta_{2}=\theta_{2}\quantity(\tau)\,,\end{split}$
(9)
$\displaystyle\begin{split}\phi_{1}={}&\omega_{1}\tau+\ell_{1}\sigma\,,\qquad\phi_{2}=\omega_{2}\tau+\ell_{2}\sigma\,,\qquad\qquad\psi=\alpha\tau+\ell_{3}\sigma\,,\end{split}$
(10)

where $\ell_{i}$s are the respective winding numbers along the isometries of
the target space.

Substituting (9) and (10) into (4) we find

$\displaystyle\begin{split}\tilde{L}_{P}&=1-\lambda_{1}^{2}\dot{\theta}_{1}^{2}-\lambda_{2}^{2}\dot{\theta}_{2}^{2}+\lambda^{2}\quantity(\ell_{3}^{2}-\alpha^{2})+\quantity(\lambda_{1}^{2}\sin^{2}\theta_{1}+\lambda^{2}\cos^{2}\theta_{1})\quantity(\ell_{1}^{2}-\omega_{1}^{2})\\\\[3.0pt]
&\quad+\quantity(\lambda_{1}^{2}\sin^{2}\theta_{2}+\lambda^{2}\cos^{2}\theta_{2})\quantity(\ell_{2}^{2}-\omega_{2}^{2})+2\lambda^{2}\cos\theta_{1}\quantity(\ell_{1}\ell_{3}-\alpha\omega_{1})\\\\[3.0pt]
&\quad+2\lambda^{2}\cos\theta_{2}\quantity(\ell_{2}\ell_{3}-\alpha\omega_{2})+2\lambda^{2}\cos\theta_{1}\cos\theta_{2}\quantity(\ell_{1}\ell_{2}-\omega_{1}\omega_{2})\\\\[3.0pt]
&\quad+2k\cos\theta_{1}\quantity(\omega_{1}\ell_{3}-\alpha\ell_{1})+2k\cos\theta_{2}\quantity(\alpha\ell_{2}-\omega_{2}\ell_{3})+2k\cos\theta_{1}\cos\theta_{2}\quantity(\omega_{1}\ell_{2}-\omega_{2}\ell_{1}).\end{split}$
(11)

The equations of motion corresponding to $\theta_{i}$ ($i=1,2$) can be derived
from the above Lagrangian density (11) as

$\displaystyle\ddot{\theta}_{1}$
$\displaystyle+\quantity(1-\frac{\lambda^{2}}{\lambda_{1}^{2}})\quantity(\ell_{1}^{2}-\omega_{1}^{2})\sin\theta_{1}\cos\theta_{1}-\frac{\sin\theta_{1}}{\lambda_{1}^{2}}\quantity\Big[k\quantity(\omega_{1}\ell_{3}-\alpha\ell_{1})+\lambda^{2}\quantity(\ell_{1}\ell_{3}-\alpha\omega_{1})]$
(12)
$\displaystyle\qquad-\frac{\sin\theta_{1}\cos\theta_{2}}{\lambda_{1}^{2}}\quantity\Big[k\quantity(\omega_{1}\ell_{2}-\omega_{2}\ell_{1})+\lambda^{2}\quantity(\ell_{1}\ell_{2}-\omega_{1}\omega_{2})]=0\,,$
$\displaystyle\ddot{\theta}_{2}$
$\displaystyle+\quantity(1-\frac{\lambda^{2}}{\lambda_{2}^{2}})\quantity(\ell_{2}^{2}-\omega_{2}^{2})\sin\theta_{2}\cos\theta_{2}-\frac{\sin\theta_{2}}{\lambda_{2}^{2}}\quantity\Big[k\quantity(\alpha\ell_{2}-\omega_{2}\ell_{3})+\lambda^{2}\quantity(\ell_{2}\ell_{3}-\alpha\omega_{2})]$
(13)
$\displaystyle\qquad-\frac{\sin\theta_{2}\cos\theta_{1}}{\lambda_{2}^{2}}\quantity\Big[k\quantity(\omega_{1}\ell_{2}-\omega_{2}\ell_{1})+\lambda^{2}\quantity(\ell_{1}\ell_{2}-\omega_{1}\omega_{2})]=0\,.$

This looks like an interacting oscillator model. In other words, the sigma
model could be thought of as a model of coupled oscillators which are mutually
interacting.

In addition, the corresponding Virasoro constraints can be written as

$\displaystyle\begin{split}T_{\tau\tau}=T_{\sigma\sigma}&=\frac{1}{2}\Big{[}-1+\lambda_{1}^{2}\dot{\theta_{1}^{2}}+\lambda_{2}^{2}\dot{\theta}_{2}^{2}+\lambda^{2}\quantity(\ell_{3}^{2}+\alpha^{2})+\quantity(\lambda_{1}^{2}\sin^{2}\theta_{1}+\lambda^{2}\cos^{2}\theta_{1})\quantity(\ell_{1}^{2}+\omega_{1}^{2})\\\\[3.0pt]
&\quad+\quantity(\lambda_{1}^{2}\sin^{2}\theta_{2}+\lambda^{2}\cos^{2}\theta_{2})\quantity(\ell_{2}^{2}+\omega_{2}^{2})+2\lambda^{2}\cos\theta_{1}\quantity(\ell_{1}\ell_{3}+\alpha\omega_{1})\\\\[3.0pt]
&\quad+2\lambda^{2}\cos\theta_{2}\quantity(\ell_{2}\ell_{3}+\alpha\omega_{2})+2\lambda^{2}\cos\theta_{1}\cos\theta_{2}\quantity(\ell_{1}\ell_{2}+\omega_{1}\omega_{2})\Big{]}\,,\end{split}$
(14)

$\displaystyle\begin{split}T_{\tau\sigma}=T_{\sigma\tau}&=\lambda^{2}\alpha\ell_{3}+\quantity(\lambda_{1}^{2}\sin^{2}\theta_{1}+\lambda^{2}\cos^{2}\theta_{1})\omega_{1}\ell_{1}+\quantity(\lambda_{1}^{2}\sin^{2}\theta_{2}+\lambda^{2}\cos^{2}\theta_{2})\omega_{2}\ell_{2}\\\\[3.0pt]
&\quad+\lambda^{2}\cos\theta_{1}\quantity(\alpha\ell_{1}+\omega_{1}\ell_{3})+\lambda^{2}\cos\theta_{2}\quantity(\alpha\ell_{2}+\omega_{2}\ell_{3})\\\\[3.0pt]
&\quad+\lambda^{2}\cos\theta_{1}\cos\theta_{2}\quantity(\omega_{1}\ell_{2}+\ell_{1}\omega_{2})\Big{]}\,.\end{split}$
(15)

###  2.1 Consistency requirements

Using the equations of motion (12), (13) we may write

$\displaystyle\begin{split}&\partial_{\tau}T_{\tau\tau}\\\\[3.0pt]
&=\omega_{1}^{2}\quantity(\lambda_{1}^{2}-\lambda^{2})\sin
2\theta_{1}\,\dot{\theta_{1}}+\omega_{2}^{2}\quantity(\lambda_{2}^{2}-\lambda^{2})\sin
2\theta_{2}\,\dot{\theta_{2}}-2\alpha\lambda^{2}\quantity(\omega_{1}\sin\theta_{1}\,\dot{\theta_{1}}+\omega_{2}\sin\theta_{2}\,\dot{\theta_{2}})\\\\[3.0pt]
&\quad-2\lambda^{2}\omega_{1}\omega_{2}\quantity(\sin\theta_{1}\cos\theta_{2}\,\dot{\theta_{1}}+\sin\theta_{2}\cos\theta_{1}\,\dot{\theta_{2}})+k\sin\theta_{1}\,\dot{\theta_{1}}\Big{[}\omega_{1}\quantity(\ell_{3}+\ell_{2}\cos\theta_{2})\\\\[3.0pt]
&\quad-\ell_{1}\quantity(\alpha+\omega_{2}\cos\theta_{2})\Big{]}+k\sin\theta_{2}\,\dot{\theta_{2}}\Big{[}\ell_{2}\quantity(\alpha+\omega_{1}\cos\theta_{1})-\omega_{2}\quantity(\ell_{3}+\ell_{1}\cos\theta_{1})\Big{]}\,,\end{split}$
(16)

$\displaystyle\begin{split}\partial_{\tau}T_{\tau\sigma}&=2\omega_{1}\ell_{1}\sin\theta_{1}\cos\theta_{1}\cdot\dot{\theta}_{1}\quantity(\lambda_{1}^{2}-\lambda^{2})+2\omega_{2}\ell_{2}\sin\theta_{2}\cos\theta_{2}\cdot\dot{\theta}_{2}\quantity(\lambda_{2}^{2}-\lambda^{2})\\\\[3.0pt]
&\quad-\lambda^{2}\quantity(\alpha\ell_{1}+\omega_{1}\ell_{3})\sin\theta_{1}\cdot\dot{\theta}_{1}-\lambda^{2}\quantity(\alpha\ell_{2}+\omega_{2}\ell_{3})\sin\theta_{2}\cdot\dot{\theta}_{2}\\\\[3.0pt]
&\quad-\lambda^{2}\quantity(\omega_{1}\ell_{2}+\omega_{2}\ell_{1})\quantity(\sin\theta_{1}\cos\theta_{2}\cdot\dot{\theta}_{1}+\sin\theta_{2}\cos\theta_{1}\cdot\dot{\theta}_{2})\,.\end{split}$
(17)

In the next step we demand the invariance of the conserved charges
$\mathcal{Q}_{i}$ associated with the reduced sigma model (11). This allows us
to compute the following constraint equations111 Here we have used the
following standard definition of the conserved charge: $\displaystyle
J_{i}=\frac{1}{2\pi\alpha^{\prime}}\int_{0}^{2\pi}\differential\sigma P_{i}$,
$i=\quantity(\psi,\phi_{1},\phi_{2})$ and the $P_{i}$s are given in (6)-(2).:

$\displaystyle\partial_{\tau}J_{\psi}$ $\displaystyle=$
$\displaystyle\quantity(k\ell_{1}+\lambda^{2}\omega_{1})\sin\theta_{1}\cdot\dot{\theta}_{1}-\quantity(k\ell_{2}-\lambda^{2}\omega_{2})\sin\theta_{2}\cdot\dot{\theta}_{2}=0\,,$
(18) $\displaystyle\partial_{\tau}J_{\phi_{1}}$ $\displaystyle=$
$\displaystyle\quantity[2\omega_{1}\quantity(\lambda_{1}^{2}-\lambda^{2})\sin\theta_{1}\cos\theta_{1}+\sin\theta_{1}\quantity(k\ell_{3}-\alpha\lambda^{2})+\sin\theta_{1}\cos\theta_{2}\quantity(k\ell_{2}-\omega_{2}\lambda^{2})]\cdot\dot{\theta}_{1}$
(19)
$\displaystyle+\quantity(k\ell_{2}-\omega_{2}\lambda^{2})\sin\theta_{2}\cos\theta_{1}\cdot\dot{\theta}_{2}=0\,,$
$\displaystyle\partial_{\tau}J_{\phi_{2}}$ $\displaystyle=$
$\displaystyle\quantity[2\omega_{2}\quantity(\lambda_{2}^{2}-\lambda^{2})\sin\theta_{2}\cos\theta_{2}-\sin\theta_{2}\quantity(k\ell_{3}+\alpha\lambda^{2})-\sin\theta_{2}\cos\theta_{1}\quantity(k\ell_{1}+\omega_{1}\lambda^{2})]\cdot\dot{\theta}_{2}$
(20)
$\displaystyle-\quantity(k\ell_{1}+\omega_{1}\lambda^{2})\sin\theta_{1}\cos\theta_{2}\cdot\dot{\theta}_{1}=0\,.$

The above set of equations (18)-(20) implies

$\Pi_{\theta_{1}}=\dot{\theta}_{1}=0\,,\qquad\qquad\quad\Pi_{\theta_{2}}=\dot{\theta}_{2}=0\,,$
(21)

where $\Pi_{\theta_{i}}$ are the momenta conjugate to $\theta_{i}$.

Using (21), it is now trivial to check that

$\partial_{\tau}T_{\tau\tau}=0=\partial_{\tau}T_{\tau\sigma}\,,$ (22)

which satisfy the consistency requirements for the Virasoro constraints.

###  2.2 The coupled pendulum model

Next, in order to study the interacting two (gravitational) pendulum system
described by (12), (13) we choose the following invariant plane in the phase
space [9]:

$\theta_{2}\sim 0\,,\qquad\Pi_{\theta_{2}}\sim\dot{\theta}_{2}\sim 0\,.$ (23)

Using (23) the equation of motion for $\theta_{1}$ can be written as

$\ddot{\theta}_{1}+A\sin\theta_{1}\cos\theta_{1}-\quantity(B+D)\sin\theta_{1}=0\,,$
(24)

where

$\displaystyle A$
$\displaystyle=\quantity(1-\frac{\lambda^{2}}{\lambda_{1}^{2}})\quantity(\ell_{1}^{2}-\omega_{1}^{2})\,;$
(25) $\displaystyle B$
$\displaystyle=\frac{1}{\lambda_{1}^{2}}\left[k\quantity(\omega_{1}\ell_{3}-\alpha\ell_{1})+\lambda^{2}\quantity(\ell_{1}\ell_{3}-\alpha\omega_{1})\right]\,;$
(26) $\displaystyle D$
$\displaystyle=\frac{1}{\lambda_{1}^{2}}\left[k\quantity(\omega_{1}\ell_{2}-\omega_{2}\ell_{1})+\lambda^{2}\quantity(\ell_{1}\ell_{2}-\omega_{1}\omega_{2})\right]\,.$
(27)

Let $\bar{\theta}_{1}$ be the solution to (24). In the next step, we shall
consider small fluctuations ($\eta$) about the invariant plane $\theta_{2}\sim
0$, $\Pi_{\theta_{2}}\sim 0$ which results in the normal variational equation
(NVE) of the following form:

$\ddot{\eta}+\quantity(F-G)\eta-\tilde{D}\cos\bar{\theta}_{1}\cdot\eta=0\,,$
(28)

where

$\displaystyle F$
$\displaystyle=\quantity(1-\frac{\lambda^{2}}{\lambda_{2}^{2}})\quantity(\ell_{2}^{2}-\omega_{2}^{2})\,;$
(29) $\displaystyle G$
$\displaystyle=\frac{1}{\lambda_{2}^{2}}\left[k\quantity(\alpha\ell_{2}-\omega_{2}\ell_{3})+\lambda^{2}\quantity(\ell_{2}\ell_{3}-\alpha\omega_{2})\right]\,;$
(30) $\displaystyle\tilde{D}$
$\displaystyle=\frac{1}{\lambda_{2}^{2}}\left[k\quantity(\omega_{1}\ell_{2}-\omega_{2}\ell_{1})+\lambda^{2}\quantity(\ell_{1}\ell_{2}-\omega_{1}\omega_{2})\right]\,.$
(31)

In order to study the NVE (28) it will be convenient to introduce the variable
$z$ such that [9]

$\cos\bar{\theta}_{1}=z\,.$ (32)

With this change in variable, (28) can be expressed as

$\eta^{\prime\prime}(z)+\frac{f^{\prime}(z)\eta^{\prime}(z)}{2f(z)}+\quantity(F-G-\tilde{D}z)\frac{\eta(z)}{f(z)}=0\,.$
(33)

The above equation (33) is similar to a second order linear homogeneous
differential equation, known as the Lamé equation [9], with

$f(z)=\dot{\bar{\theta}}_{1}^{2}\,\sin^{2}\bar{\theta}_{1}=\left[2E^{2}-A\quantity(1-z^{2})-2\quantity(B+D)z\right]\quantity(1-z^{2}).$
(34)

It is difficult to solve the NVE (33) exactly. However, we can expand the
coefficients of (33) for small values of the variable $|z|$. The resulting
solution can be expressed in terms of special functions and can be written as

$\displaystyle\begin{split}\eta(z)&=\exp\quantity[\frac{\tilde{D}z}{\Upsilon}\quantity(\frac{A}{2}-1)+\frac{z}{\Upsilon}\quantity(B+D)\quantity(F-G)]\left\\{\mathcal{C}_{1}\,\,\,\texttt{Hermite}\quantity[\frac{\chi_{1}}{8\Upsilon^{3}},\frac{\chi_{2}}{2\Upsilon^{\frac{3}{2}}}+\frac{\Upsilon^{\frac{1}{2}}z}{A-2}]\right.\\\\[4.0pt]
&\left.+\,\mathcal{C}_{2}\,\,\,{}_{1}\texttt{F}_{1}\quantity[-\frac{\chi_{1}}{16\Upsilon^{3}},\frac{1}{2},\quantity(\frac{\chi_{2}}{2\Upsilon^{\frac{3}{2}}}+\frac{\Upsilon^{\frac{1}{2}}z}{A-2})^{2}]\right\\}\,,\end{split}$
(35)

where $\mathcal{C}_{1}$, $\mathcal{C}_{2}$ are constants of integration,

$\Upsilon=A^{2}+\quantity(B+D)^{2}-3A+2\,,$ (36)

and the detail expressions for $\chi_{1}$ and $\chi_{2}$ are provided in
appendix B. Here $\texttt{Hermite}\quantity[n,z]$ and
${}_{1}\texttt{F}_{1}\quantity[a;b;z]$ are the Hermite polynomial function and
Kummer confluent hypergeometric function, respectively. Clearly, this form of
the solution is not Liouvillian indicating the non-integrability of the system
[9, 20, 21, 22, 25, 27].

Furthermore, using the change of variable [20, 21, 22, 25, 27]

$\eta(z)=\exp\quantity[\int\quantity(w(z)-\frac{\mathcal{B}(z)}{2})\differential
z]$ (37)

we can recast (33) in the Schrödinger form

$w^{\prime}(z)+w^{2}(z)=\mathcal{V}(z)\equiv\frac{2\mathcal{B}^{\prime}(z)+\mathcal{B}^{2}(z)-4\mathcal{A}(z)}{4}\,,$
(38)

where

$\mathcal{A}(z)=\frac{F-G-\tilde{D}z}{f(z)}\,,\qquad\mathcal{B}(z)=\frac{f^{\prime}(z)}{2f(z)}\,.$
(39)

Using (39) in (38) we can compute the function $\mathcal{V}(z)$ as

$\displaystyle\begin{split}\mathcal{V}(z)&=-\frac{3}{4\quantity(z^{2}-1)^{2}}+\frac{\quantity(\beta_{1}+z\tilde{\beta}_{1})}{4\quantity(z^{2}-1)}-\frac{3\beta_{2}}{4\quantity(Az^{2}-2\quantity(B+D)z-\quantity(A-2))^{2}}\\\
&\quad+\frac{\quantity(\beta_{3}+z\tilde{\beta}_{3})}{4\quantity(Az^{2}-2\quantity(B+D)z-\quantity(A-2))}\,,\end{split}$
(40)

where

$\displaystyle\beta_{1}$
$\displaystyle=\frac{1}{\beta_{0}}\quantity(1-A+2\tilde{D}\quantity(B+D)-2\quantity(F-G))\,;\qquad\beta_{0}=\quantity(B+D)^{2}-1\,,$
(41) $\displaystyle\tilde{\beta}_{1}$
$\displaystyle=\frac{1}{\beta_{0}}\quantity(\quantity(1-A)\quantity(B+D)-2\quantity(B+D)\quantity(F-G))\,,$
(42) $\displaystyle\beta_{2}$
$\displaystyle=A\quantity(A-2)+\quantity(B+D)^{2}\,,$ (43)
$\displaystyle\begin{split}\beta_{3}&=\frac{1}{\beta_{0}}\Big{(}A^{2}-2AB^{2}-4ABD-2AB\tilde{D}-2AD^{2}-2AD\tilde{D}+2AF-2AG-A\\\
\quad&-4B^{2}F+4B^{2}G+2B^{2}-8BDF+8BDG+4BD+4B\tilde{D}-4D^{2}F+4D^{2}G\\\
\quad&+2D^{2}+4D\tilde{D}\Big{)}\,,\end{split}$ (44)
$\displaystyle\begin{split}\tilde{\beta}_{3}&=\frac{1}{\beta_{0}}\quantity(A\quantity(A-1)\quantity(B+D)-2A\tilde{D}+2A\quantity(F-G)\quantity(B+D))\,.\end{split}$
(45)

Notice that, in writing (40) we have set the constant of integration $E=1$ in
(34).

Clearly, the function $\mathcal{V}(z)$ has poles of order $2$ at

$z=\pm 1\,,\qquad\quad
z=\frac{1}{A}\quantity[\quantity(B+D)\pm\sqrt{A\quantity(A-2)+\quantity(B+D)^{2}}]\,,$
(46)

and the order of $\mathcal{V}(z)$ at infinity can be computed to be $1$.
Hence, $\mathcal{V}(z)$ does not belong to any of the three cases discussed in
the Appendix A. Therefore, the form of the solution to equation (38) must be
non-Liouvillian.

In fact, for small values of $|z|$ we can series expand $\mathcal{V}(z)$ in
(40) and can find the solution to (38) as

$\displaystyle\begin{split}w(z)&=\frac{\chi_{3}}{\Pi_{2}}\cdot\frac{\texttt{Bi}^{\prime}\quantity(\Pi_{1})+\mathcal{C}_{3}\,\texttt{Ai}^{\prime}\quantity(\Pi_{1})}{\texttt{Bi}\quantity(\Pi_{1})+\mathcal{C}_{3}\,\texttt{Ai}\quantity(\Pi_{1})}\,,\end{split}$
(47)

where $\mathcal{C}_{3}$ is a constant of integration,

$\displaystyle\Pi_{1}$ $\displaystyle=\frac{\chi_{3}z+\chi_{4}}{\Pi_{2}}\,,$
$\displaystyle\Pi_{2}$
$\displaystyle=2^{\frac{2}{3}}\quantity(A-2)\chi_{3}^{\frac{2}{3}}\,,$ (48)

$\displaystyle\begin{split}\chi_{3}&=8A^{2}B+8A^{2}D-4A^{2}\tilde{D}-8ABF+8ABG-12AB-8ADF+8ADG-12AD\\\
&\quad+16A\tilde{D}+12B^{3}+36B^{2}D+36BD^{2}+16BF-16BG-8B+12D^{3}+16DF\\\
&\quad-16DG-8D-16\tilde{D}\,,\end{split}$ (49)
$\displaystyle\begin{split}\chi_{4}&=-4A^{3}+4A^{2}F-4A^{2}G+20A^{2}-3AB^{2}-6ABD-3AD^{2}-16AF+16AG-32A\\\
&\quad+6B^{2}+12BD+6D^{2}+16F-16G+16\,.\end{split}$ (50)

As it is clear from the above expression (47), the solution to (38) is written
in terms of Airy functions $\texttt{Ai}(z)$, $\texttt{Bi}(z)$ and their
derivatives $\texttt{Ai}^{\prime}(z)$, $\texttt{Bi}^{\prime}(z)$ thereby
making it non-Liouvillian. This reassures the non-integrability of the system.

##  3 Analytic integrability of the decoupled systems

In this section we shall discuss the analytic integrability of the two
pendulum system in the decoupling limit. It is easy to check from (12), (13)
that the two pendula decouples from each other in the limit

$\frac{k}{\lambda^{2}}=\frac{\quantity(\omega_{1}\omega_{2}-\ell_{1}\ell_{2})}{\quantity(\omega_{1}\ell_{2}-\omega_{2}\ell_{1})}\,,$
(51)

and the equations of motion (12) and (13) simplify to

$\displaystyle\ddot{\theta}_{1}+A\sin\theta_{1}\cos\theta_{1}-B\sin\theta_{1}$
$\displaystyle=0\,,$ (52)
$\displaystyle\ddot{\theta}_{2}+F\sin\theta_{2}\cos\theta_{2}-G\sin\theta_{2}$
$\displaystyle=0\,,$ (53)

where $A$, $B$, $F$ and $G$ have been defined earlier.

We notice that, when

$\omega_{1}\leftrightarrow-\ell_{1}\,,\qquad\text{or}\qquad\omega_{2}\leftrightarrow\ell_{2}\,,$
(54)

the decoupling limit (51) reduces to what was studied in [31, 32]; namely,
$\displaystyle k=\lambda^{2}$. The constraint (54) implies that either the
string must wind anti-clockwise along the isometry direction $\phi_{1}$, or it
winds in the clockwise direction along $\phi_{2}$. Thus we may consider the
limit $\displaystyle k=\lambda^{2}$ as a special case of (51).

Next, in order to check the integrability of the individual systems, we study
their dynamics in the corresponding phase spaces
$\\{\theta_{i},\Pi_{\theta_{i}}\\}$ ($i=1,2$). For the first pendulum we
choose the invariant plane given by

$\theta_{1}=\dot{\theta}_{1}=0\,,$ (55)

and consider the fluctuation $\delta\theta_{1}\sim\sigma(\tau)$ around this
plane to compute the NVE as

$\ddot{\sigma}+\quantity(A-B)\sigma=0\,,$ (56)

which is nothing but simple harmonic motion with frequency $\sim\sqrt{A-B}$,
hence trivially integrable as per differential Galois theory [9].

In a similar manner we can choose an invariant plane
$\theta_{2}=\dot{\theta}_{2}=0$ in the phase space of the second pendulum.
Subsequently, the Kovacic’s algorithm ensures the integrability of this system
as well.

##  4 Conclusions

In this paper, we have established classical Liouvillian non-integrability of
string sigma model in the recently proposed Arutyunov-Bassi-Lacroix (ABL)
background (1) which is a generalization of the Einsteinian $T^{1,1}$ manifold
[31]. We use the Kovacic’s algorithm [34, 35] in order to perform our
computations. Our analysis complements the claim made in [31, 32] and is
compatible with the Lax pair formulation of [31].

We observe that, if we consider a string that winds around the deformed torus
$T^{\quantity(\lambda_{1},\lambda_{2},\lambda)}_{k}$, the system can be
described by two coupled harmonic oscillators. We analyse the corresponding
coupled differential equation. By appropriately choosing an invariant plane in
the phase space of the system, we analyse the corresponding normal variational
equation (NVE). The solution to this equation turns out to be non-Liouvillian
which therefore establishes the non-integrability. We further recast the NVE
in the Schrödinger form (38) and carefully analyze the polynomial function
(40) arising from it. We observe incompatibility of this function with the
Kovacic’s classification (see Appendix A), thereby establishing the non-
integrability.

Next, we proceed to study the limit in which the coupled oscillators decouple
and turn out to be integrable. As a matter of fact, for the particular example
of winding string, we find a _generalized_ decoupling limit and, under special
conditions (54), it gives the decoupling limit studied in [31, 32]. This
analysis reveals that, in transiting to this later limit in [31, 32], the
string winds (anti-)clockwise along the isometry direction ($\phi_{1}$)
$\phi_{2}$.

The Kovacic’s algorithm has been proven to be an excellent mathematical tool
to (dis)prove analytic integrability over general backgrounds [8],[9],
[20]-[27]. As has been shown in the present work, one can reliably use this
formalism as it complements the existing methodologies in the literature [31,
32]. It will be interesting to apply this formulation, along with the other
analytic methods, to explore integrable structures of other systems in future.

## Acknowledgments

J.P., A.M. and D.R. is indebted to the authorities of IIT Roorkee for their
unconditional support towards researches in basic sciences. A.M. acknowledges
The Science and Engineering Research Board (SERB), India for financial
support. D.R. would also like to acknowledge The Royal Society, UK for
financial assistance, and acknowledges the Grant (No. SRG/2020/000088)
received from The Science and Engineering Research Board (SERB), India. The
work of A.L. is supported by the Chilean _National Agency for Research and
Development_ (ANID)/ FONDECYT / POSTDOCTORADO BECAS CHILE / Project No.
3190021.

##  Appendix A The Kovacic’s algorithm

The Kovacic’s algorithm is a systematic procedure to check the Liouvillian
non-integrability of a dynamical system [34, 35]. This algorithm is
implemented in order to realize whether the second order linear homogeneous
differential equation of the type

$\eta^{\prime\prime}(z)+\mathcal{M}(z)\eta^{\prime}(z)+\mathcal{N}(z)\eta(z)=0$
(A1)

with polynomial coefficients $\mathcal{M}(z)$, $\mathcal{N}(z)$ are integrable
in the Liouvillian sense. We look for solutions of (A1) in the Liouvillian
form; namely, those solutions which can be expressed in terms of simple
algebraic polynomial functions, exponentials, or trigonometric functions. As a
matter of fact, this algorithm is a variant of the Galois theory of
differential equations (the Piccard-Vessoit theory [37]) and utilizes the
$SL\quantity(2,\mathbb{C})$ group of symmetries of the differential equation
(A1) in its analysis.

Without going into detailed mathematical analysis which is rather involved, we
limit ourselves to find the relation between the coefficients
$\mathcal{M}(z)$, $\mathcal{M}^{\prime}(z)$ and $\mathcal{N}(z)$ that makes
(A1) integrable. In order to do so, we perform the following change of
variable:

$\eta(z)=\exp\quantity[\int\quantity(w(z)-\frac{\mathcal{M}(z)}{2})\differential
z].$ (A2)

This allows us to rewrite (A1) as

$w^{\prime}(z)+w^{2}(z)=\mathcal{V}(z)\equiv\frac{\mathcal{M}^{\prime}(z)+\mathcal{M}^{2}(z)-4\mathcal{N}(z)}{4}\,,$
(A3)

where ‘$\,{}^{\prime}\,$’ denotes differentiation w.r.to $z$.

The group of symmetry transformations of the solutions of (A1) is in fact a
subgroup $G$ of $SL\quantity(2,\mathbb{C})$ which can be classified as
follows: (i) For any $a,\,b\in\mathbb{C}$, if $G$ is generated by the matrix
$M=\quantity(\\{a,\,0\\},\\{b,\,1/a\\})$ then $w(z)$ is a rational polynomial
function of degree $1$; (ii) When $M=\quantity(\\{a,\,0\\},\\{0,\,1/a\\})$,
$M=\quantity(\\{0,\,a\\},\\{-1/a,\,0\\})$ $w(z)$ is a function of degree $2$;
(iii) When $G$ is a finite group and not generated by a matrix of the form
mentioned in (i) and (ii) above, $w(z)$ is a function of degree $4$, $6$, or
$12$; (iv) When $G=SL\quantity(2,\mathbb{C})$, any non-vanishing solution
$w(z)$ is not Liouvillian.

The Kovacic’s algorithm also proposes a set of three necessary but not
sufficient conditions for the rational polynomial function $\mathcal{V}(z)$
that must be compatible with the aforementioned group theoretical analysis
[34]. These are as follows: (i) $\mathcal{V}(z)$ has pole of order $1$, or
$2n$ ($n\in\mathbb{Z}^{+}$). Also, the order of $\mathcal{V}(z)$ at infinity,
defined as the highest power of the denominator minus that of the numerator,
is either $2n$ or greater than 2; (ii) $\mathcal{V}(z)$ either has pole of
order $2$, or poles of order $2n+1$ greater than $2$; (iii) $\mathcal{V}(z)$
has poles not greater than $2$ and the order of $\mathcal{V}(z)$ at infinity
is at least $2$. If none of these conditions are met, the solution to (A1) is
non-Liouvillian and ensures the non-integrability of (A1). On the other hand,
fulfilment of any one of the above conditions makes us eligible to apply the
Kovacic’s algorithm to the ODE (A1). It is then necessary to determine whether
$w(z)$ is a polynomial function of degree $1$, $2$, $4$, $6$, or $12$ in which
case (A1) is integrable.

##  Appendix B Expressions for the constants $\chi_{1}$ and $\chi_{2}$ in (35)

$\displaystyle\begin{split}\chi_{1}&=-4FA^{5}+4GA^{5}+\tilde{D}^{2}A^{4}+2B\tilde{D}A^{4}+2D\tilde{D}A^{4}+32FA^{4}-32GA^{4}-8\tilde{D}^{2}A^{3}-14B\tilde{D}A^{3}\\\
&\quad-14D\tilde{D}A^{3}-4B^{2}FA^{3}-4D^{2}FA^{3}-8BDFA^{3}+4B\tilde{D}FA^{3}+4D\tilde{D}FA^{3}-100FA^{3}+4B^{2}GA^{3}\\\
&\quad+4D^{2}GA^{3}+8BDGA^{3}-4B\tilde{D}GA^{3}-4D\tilde{D}GA^{3}+100GA^{3}+24\tilde{D}^{2}A^{2}+4B^{2}F^{2}A^{2}+4D^{2}F^{2}A^{2}\\\
&\quad+8BDF^{2}A^{2}+4B^{2}G^{2}A^{2}+4D^{2}G^{2}A^{2}+8BDG^{2}A^{2}+2B^{3}\tilde{D}A^{2}+2D^{3}\tilde{D}A^{2}+6BD^{2}\tilde{D}A^{2}\\\
&\quad+36B\tilde{D}A^{2}+6B^{2}D\tilde{D}A^{2}+36D\tilde{D}A^{2}+20B^{2}FA^{2}+20D^{2}FA^{2}+40BDFA^{2}-24B\tilde{D}FA^{2}\\\
&\quad-24D\tilde{D}FA^{2}+152FA^{2}-20B^{2}GA^{2}-20D^{2}GA^{2}-40BDGA^{2}+24B\tilde{D}GA^{2}+24D\tilde{D}GA^{2}\\\
&\quad-8B^{2}FGA^{2}-8D^{2}FGA^{2}-16BDFGA^{2}-152GA^{2}-32\tilde{D}^{2}A-16B^{2}F^{2}A-16D^{2}F^{2}A\\\
&\quad-32BDF^{2}A-16B^{2}G^{2}A-16D^{2}G^{2}A-32BDG^{2}A-8B^{3}\tilde{D}A-8D^{3}\tilde{D}A-24BD^{2}\tilde{D}A\\\
&\quad-40B\tilde{D}A-24B^{2}D\tilde{D}A-40D\tilde{D}A-32B^{2}FA-32D^{2}FA-64BDFA+48B\tilde{D}FA\\\
&\quad+48D\tilde{D}FA-112FA+32B^{2}GA+32D^{2}GA+64BDGA-48B\tilde{D}GA-48D\tilde{D}GA+32B^{2}FGA\\\
&\quad+32D^{2}FGA+64BDFGA+112GA+16\tilde{D}^{2}+16B^{2}F^{2}+16D^{2}F^{2}+32BDF^{2}+16B^{2}G^{2}+16D^{2}G^{2}\\\
&\quad+32BDG^{2}+8B^{3}+32BDG^{2}+8D^{3}\tilde{D}+24BD^{2}\tilde{D}+16B\tilde{D}+24B^{2}D\tilde{D}+16D\tilde{D}\\\
&\quad+16B^{2}F+16D^{2}F+32BDF-32B\tilde{D}F-32D\tilde{D}F+32F-16B^{2}G-16D^{2}G-32BDG+32B\tilde{D}G\\\
&\quad+32D\tilde{D}G-32B^{2}FG-32D^{2}FG-64BDFG-32G.\end{split}$ (B1)

$\displaystyle\begin{split}\chi_{2}&=-A^{2}B-A^{2}D-A^{2}\tilde{D}-2ABF+2ABG+3AB-2ADF+2ADG+3AD\\\
&\quad+4A\tilde{D}-B^{3}-3B^{2}D-3BD^{2}+4BF-4BG-2B-D^{3}+4DF-4DG-2D-4\tilde{D}.\end{split}$
(B2)

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before the command that updates the counter it’s supposed to be using, the
label will pick up the last counter to be cross referenced instead. In
particular, a `\label` command should not go before the caption of a figure or
a table.

Do not use `\nonumber` inside the `{array}` environment. It will not stop
equation numbers inside `{array}` (there won’t be any anyway) and it might
stop a wanted equation number in the surrounding equation.

###  III-E Some Common Mistakes

  * •

The word “data” is plural, not singular.

  * •

The subscript for the permeability of vacuum $\mu_{0}$, and other common
scientific constants, is zero with subscript formatting, not a lowercase
letter “o”.

  * •

In American English, commas, semicolons, periods, question and exclamation
marks are located within quotation marks only when a complete thought or name
is cited, such as a title or full quotation. When quotation marks are used,
instead of a bold or italic typeface, to highlight a word or phrase,
punctuation should appear outside of the quotation marks. A parenthetical
phrase or statement at the end of a sentence is punctuated outside of the
closing parenthesis (like this). (A parenthetical sentence is punctuated
within the parentheses.)

  * •

A graph within a graph is an “inset”, not an “insert”. The word alternatively
is preferred to the word “alternately” (unless you really mean something that
alternates).

  * •

Do not use the word “essentially” to mean “approximately” or “effectively”.

  * •

In your paper title, if the words “that uses” can accurately replace the word
“using”, capitalize the “u”; if not, keep using lower-cased.

  * •

Be aware of the different meanings of the homophones “affect” and “effect”,
“complement” and “compliment”, “discreet” and “discrete”, “principal” and
“principle”.

  * •

Do not confuse “imply” and “infer”.

  * •

The prefix “non” is not a word; it should be joined to the word it modifies,
usually without a hyphen.

  * •

There is no period after the “et” in the Latin abbreviation “et al.”.

  * •

The abbreviation “i.e.” means “that is”, and the abbreviation “e.g.” means
“for example”.

An excellent style manual for science writers is [7].

###  III-F Authors and Affiliations

The class file is designed for, but not limited to, six authors. A minimum of
one author is required for all conference articles. Author names should be
listed starting from left to right and then moving down to the next line. This
is the author sequence that will be used in future citations and by indexing
services. Names should not be listed in columns nor group by affiliation.
Please keep your affiliations as succinct as possible (for example, do not
differentiate among departments of the same organization).

###  III-G Identify the Headings

Headings, or heads, are organizational devices that guide the reader through
your paper. There are two types: component heads and text heads.

Component heads identify the different components of your paper and are not
topically subordinate to each other. Examples include Acknowledgments and
References and, for these, the correct style to use is “Heading 5”. Use
“figure caption” for your Figure captions, and “table head” for your table
title. Run-in heads, such as “Abstract”, will require you to apply a style (in
this case, italic) in addition to the style provided by the drop down menu to
differentiate the head from the text.

Text heads organize the topics on a relational, hierarchical basis. For
example, the paper title is the primary text head because all subsequent
material relates and elaborates on this one topic. If there are two or more
sub-topics, the next level head (uppercase Roman numerals) should be used and,
conversely, if there are not at least two sub-topics, then no subheads should
be introduced.

###  III-H Figures and Tables

#### Positioning Figures and Tables

Place figures and tables at the top and bottom of columns. Avoid placing them
in the middle of columns. Large figures and tables may span across both
columns. Figure captions should be below the figures; table heads should
appear above the tables. Insert figures and tables after they are cited in the
text. Use the abbreviation “Fig. 1”, even at the beginning of a sentence.

TABLE I: Table Type Styles Table | Table Column Head
---|---
Head | Table column subhead | Subhead | Subhead
copy | More table copya |  |
aSample of a Table footnote.

Figure 1: Example of a figure caption.

Figure Labels: Use 8 point Times New Roman for Figure labels. Use words rather
than symbols or abbreviations when writing Figure axis labels to avoid
confusing the reader. As an example, write the quantity “Magnetization”, or
“Magnetization, M”, not just “M”. If including units in the label, present
them within parentheses. Do not label axes only with units. In the example,
write “Magnetization (A/m)” or “Magnetization {A[m(1)]}”, not just “A/m”. Do
not label axes with a ratio of quantities and units. For example, write
“Temperature (K)”, not “Temperature/K”.

## Acknowledgment

The preferred spelling of the word “acknowledgment” in America is without an
“e” after the “g”. Avoid the stilted expression “one of us (R. B. G.) thanks
$\ldots$”. Instead, try “R. B. G. thanks$\ldots$”. Put sponsor acknowledgments
in the unnumbered footnote on the first page.

## References

Please number citations consecutively within brackets [1]. The sentence
punctuation follows the bracket [2]. Refer simply to the reference number, as
in [3]—do not use “Ref. [3]” or “reference [3]” except at the beginning of a
sentence: “Reference [3] was the first $\ldots$”

Number footnotes separately in superscripts. Place the actual footnote at the
bottom of the column in which it was cited. Do not put footnotes in the
abstract or reference list. Use letters for table footnotes.

Unless there are six authors or more give all authors’ names; do not use “et
al.”. Papers that have not been published, even if they have been submitted
for publication, should be cited as “unpublished” [4]. Papers that have been
accepted for publication should be cited as “in press” [5]. Capitalize only
the first word in a paper title, except for proper nouns and element symbols.

For papers published in translation journals, please give the English citation
first, followed by the original foreign-language citation [6].

## References

  * [1] G. Eason, B. Noble, and I. N. Sneddon, “On certain integrals of Lipschitz-Hankel type involving products of Bessel functions,” Phil. Trans. Roy. Soc. London, vol. A247, pp. 529–551, April 1955.
  * [2] J. Clerk Maxwell, A Treatise on Electricity and Magnetism, 3rd ed., vol. 2. Oxford: Clarendon, 1892, pp.68–73.
  * [3] I. S. Jacobs and C. P. Bean, “Fine particles, thin films and exchange anisotropy,” in Magnetism, vol. III, G. T. Rado and H. Suhl, Eds. New York: Academic, 1963, pp. 271–350.
  * [4] K. Elissa, “Title of paper if known,” unpublished.
  * [5] R. Nicole, “Title of paper with only first word capitalized,” J. Name Stand. Abbrev., in press.
  * [6] Y. Yorozu, M. Hirano, K. Oka, and Y. Tagawa, “Electron spectroscopy studies on magneto-optical media and plastic substrate interface,” IEEE Transl. J. Magn. Japan, vol. 2, pp. 740–741, August 1987 [Digests 9th Annual Conf. Magnetics Japan, p. 301, 1982].
  * [7] M. Young, The Technical Writer’s Handbook. Mill Valley, CA: University Science, 1989.