aainstitutetext: Centre For Theoretical Studies, Indian Institute of
Technology Kharagpur-721302, Indiabbinstitutetext: Department of Physics,
Indian Institute of Technology Kharagpur-721302, India

# Neumann-Rosochatius system for (m,n) string in $AdS_{3}\times S^{3}$ with
mixed flux

Adrita Chakraborty b and Kamal L. Panigrahi<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

$SL(2,\mathbb{Z})$ invariant action for probe $(m,n)$ string in $AdS_{3}\times
S^{3}\times T^{4}$ with mixed three-form fluxes can be described by an
integrable deformation of an one-dimensional Neumann-Rosochatius (NR) system.
We present the deformed features of the integrable model and study general
class of rotating and pulsating solutions by solving the integrable equations
of motion. For the rotating string, the explicit solutions can be expressed in
terms of elliptic functions. We make use of the integrals of motion to find
out the scaling relation among conserved charges for the particular case of
constant radii solutions. Then we study the closed $(m,n)$ string pulsating in
$R_{t}\times S^{3}$. We find the string profile and calculate the total energy
of such pulsating string in terms of oscillation number $(\cal{N})$ and
angular momentum $(\cal{J})$.

###### Keywords:

AdS/CFT correspondence, Semiclassical string

##  1 Introduction

Planar integrability on both sides of the famous AdS/CFT correspondence
Maldacena:1997re ; Gubser:1998bc ; Witten:1998qj  has been proved to be very
promising technique to unravel a deeper understanding in the study of string
spectrum in different semisymmetric superspaces Zarembo:2010sg . Minahan and
Zarembo in Minahan:2002ve  first established the matching between the one-loop
dilatation operators of the SU(2) sector of $\mathcal{N}=4$ SYM theory with
the Hamiltonian of integrable SO(6) spin chain model Beisert:2003jj ;
Beisert:2003yb . In proving the AdS/CFT duality in the large charge limit,
varieties of rigidly rotating strings have been studied in different
backgrounds to enhance our understanding of the dual states in the gravity
side. Among them special cases on sphere consist of giant magnon Hofman:2006xt
and spiky string Kruczenski:2004wg  solutions which are dual to elementary
excitations with large momentum $p$ and higher twist operators respectively in
the field theory side of the duality. Also in Minahan:2002rc  an exact
correspondence was proposed between string states and some dual field theory
operators by considering spinning as well as pulsating string solutions, the
later first developed in Gubser:2002tv . Despite being more stable solutions
Khan:2005fc  these types of string state have not been explored enough as
compared to the rigidly rotating strings. Pulsating string solutions were
explored earlier in $AdS_{5}\times S^{5}$ background Khan:2003sm ;
Smedback:1998yn  and then in $AdS_{4}\times\mathbb{CP}^{3}$ Chen:2008qq ;
Dimov:2009rd . In some prior works Engquist:2003rn ; Beisert:2003xu ;
Beisert:2003ea  the YM duals for semiclassical spinning and pulsating string
states in $AdS_{5}\times S^{5}$ were analysed in terms of integrable spin
chain description of the field theory. Over the last few years, $AdS_{3}\times
S^{3}\times M^{4}$ geometry supported by mixed three form fluxes (both NS-NS
and R-R) has been studied as a nice tool in the context of classical
integrability in $AdS_{3}/CFT_{2}$ correspondence Cagnazzo:2012se ;
Hoare:2013lja ; Hoare:2013ida ; Hoare:2013pma ; Pittelli:2014ria ;
Pittelli:2017spf ; Baggio:2018gct which is a much explored holographic
extension of the $AdS_{5}/CFT_{4}$ duality proposed earlier. The appearance of
integrability as a symmetry in Green-Schwarz action of type IIB string in
compactified $AdS_{3}\times S^{3}\times M^{4}$ background with mixed R-R and
NS-NS three form flux has put forth a renewed interest to analyse the
$AdS_{3}/CFT_{2}$ duality in the presence of mixed fluxes by means of well
known classical integrable models. Though the explicit form of the dual theory
is yet to explore, such background seems to be a promising laboratory to
perform various general string solutions, consequently giving some ideas of
the respective dual operators.

A general description of the finite gap solutions of classically integrable
string sigma model has been demonstrated in Kazakov:2004qf ; Beisert:2004ag
in terms of solutions of certain integrable models. In this regard, it is
worth emphasizing that a generalised ansatz for string coordinates
Kruczenski:2006pk ; Arutyunov:2003uj ; Arutyunov:2003za  provides a method to
reformulate large class of string configurations in terms of solutions of very
well-known one dimensional integrable Neumann or Neumann-Rosochatius (NR)
system. The Neumann integrable model describes harmonic oscillator restricted
to move on a sphere whereas the NR system is an integrable extension of the
former with an additional centrifugal potential barrier of the form
$\frac{1}{r^{2}}$. These mechanical models have been proved to be very
effective to deal with the problem of geodesics on ellipsoid or equivariant
harmonic maps into sphere Uhlenbeck ; Babelon:2003qtg ; Avan:1991ib ;
Avan:1989dn . Earlier, in the context of classical integrability, classical
giant magnon Kruczenski:2006pk ; Arutyunov:2003uj ; Arutyunov:2003za  and
spiky string Ishizeki:2007we ; Mosaffa:2007ty ; Hayashi:2007bq  solutions on
$AdS_{5}\times S^{5}$ were obtained by using NR model approach. Moreover,
there are several recent works Ahn:2008hj ; Hernandez:2014eta ;
Hernandez:2015nba ; Arutyunov:2016ysi ; Hernandez:2017raj ; Hernandez:2018gcd
; Chakraborty:2019gmt  which reveal that general solutions corresponding to
closed rotating and pulsating string in various 10D AdS/S compact spaces with
pure or mixed flux may be constructed from systematic analysis of the one
dimensional Lagrangian of NR system with integrable deformation.

In a seminal paper, J. H. Schwarz Schwarz:1995dk  had constructed an
$SL(2,{\mathbb{Z}})$ multiplet of string-like solutions in type IIB
supergravity starting from the fundamental string solution. It is also known
that the equations of motion of type IIB theory are invariant under an
$SL(2,{\mathbb{R}})$ rotation group. This enables one to generate new
supergravity solutions in type IIB theory by applying this rotation to known
solutions in supergravity. A discrete subgroup $SL(2,{\mathbb{Z}})$ of this
$SL(2,{\mathbb{R}})$ group has been conjectured later to be the exact symmetry
group of the type IIB string theory. Till date, a lot of work has been devoted
for constructing various string-like as well as five-brane solutions of type
IIB supergravity equations using $SL(2,\mathbb{Z})$ invariance of low energy
effective action of this theory Schwarz:1995dk ; Tseytlin:1996it ; Lu:1998vh ;
Lozano:1997cy ; Cederwall:1997ts ; Kluson:2016dca ; Kluson:2016pxg . In the
near horizon limit, an $SL(2,{\mathbb{Z}})$ transformed bound state solution
of $Q_{5}$ NS5-branes and $Q_{1}$ F-strings, gives rise to $AdS_{3}\times
S^{3}$ background with mixed three form fluxes having integer charges. It has
also been recently shown that the $SL(2,{\mathbb{Z}})$-transformation and the
near horizon limit commute. This allows us to map the $(m,n)$ string action in
$AdS_{3}\times S^{3}$ background with mixed three form fluxes to
$(m^{\prime},n^{\prime})$ string action in $AdS_{3}\times S^{3}$ background
with NS-NS two form flux using the relation

$\textbf{m}^{\prime}=\left(\begin{array}[]{c}m^{\prime}\\\
n^{\prime}\end{array}\right)=\left(\begin{array}[]{cc}d&-b\\\
-c&a\end{array}\right)\left(\begin{array}[]{c}m\\\ n\end{array}\right),$ (1)

where $a,b,c$ and $d$ are the integer entries of $SL(2,\mathbb{Z})$ matrix
with $ad-bc=1$. Interesting analysis of this fact has so far been put forth
with $(m,n)$ string action as natural probe in various backgrounds
Kluson:2016dca ; Kluson:2016pxg ; Banerjee:2016avv ; Barik:2016yav ;
Kluson:2019ifd . In Barik:2016yav , some fascinating observations on giant
magnon and single spike solutions for rigidly rotating string along with the
analysis of pulsating string as probe $(m,n)$ string in mixed flux background
have been performed by mapping it into $(m^{\prime},n^{\prime})$ string in
$AdS_{3}\times S^{3}$ background with NSNS two form flux.

Motivated by such analysis, we cast our focus on the manifest
$SL(2,\mathbb{Z})$ covariant action of probe $(m,n)$ string in $AdS_{3}\times
S^{3}\times T^{4}$ background considering the presence of both the three form
NSNS and RR fluxes using general rotating and pulsating string ansatz in the
light of classical integrability of both sides of the correspondence. In this
article we wish to construct an integrable deformed NR system and utilize the
solutions of integrable equations of motion of that system to obtain the
relations between conserved charges for both rotating and pulsating $(m,n)$
string. The rest of the paper is organized as follows. In section 2, we
present the construction of deformed Lagrangian and Hamiltonian of integrable
NR model for closed $(m,n)$-string rotating in $R_{t}\times S^{3}$ background
in the presence of mixed three form fluxes. Also we discuss for corresponding
Uhlenbeck integrals of motion for such system. We derive elliptic solutions
for rotating $(m,n)$ string which resemble the circular-type string solutions.
We further study of conserved charges and energy-angular momenta dispersion
relation for constant radii case for the closed rotating string. Section 4 is
devoted to the study of $(m,n)$ type pulsating string in mixed flux background
from the solutions of the modified NR model. We study the string profile and
compute small energy correction to the scaling relation in terms of
oscillation number in short string limit. Finally, we conclude our results in
section 4.

##  2 Modified NR model for rotating $(m,n)$ string in $AdS_{3}\times S^{3}$
with mixed flux

We start by writing down the background metric and associated NS-NS and R-R
fluxes for $AdS_{3}\times S^{3}$ background

$\displaystyle
ds^{2}=L^{2}\left[ds^{2}_{\tilde{AdS}_{3}}+ds^{2}_{S^{3}}\right]$ (2a)
$\displaystyle
H=2L^{2}\left(\tilde{\epsilon}_{AdS_{3}}+\epsilon_{S^{3}}\right),~{}~{}L^{2}=r_{5}^{2}$
(2b) $\displaystyle
e^{-2\phi}=\frac{r_{1}^{2}}{g_{s}^{2}r_{5}^{2}},~{}~{}r_{1}^{2}=\frac{16\pi^{4}g_{s}^{2}\alpha^{{}^{\prime}3}Q_{1}}{V_{4}},~{}~{}r_{5}^{2}=Q_{5}\alpha^{{}^{\prime}},~{}~{}V_{4}=(2\pi)^{4}\alpha^{{}^{\prime}2}v$
(2c)

Here $ds^{2}_{\tilde{AdS_{3}}}$ is the line element of the corresponding
background in terms of dimensionless variables. In this section, we wish to
study the dynamics of $(m,n)$ string in general background with mixed three
form flux . On the other hand, the action of $(m,n)$ string in general
background with metric $g_{MN}$ and flux ${\bf B}_{MN}$ is given by

$\displaystyle S_{m,n}=-T_{D1}\int d\tau
d\sigma\left[\sqrt{\textbf{m}^{T}\mathcal{M}^{-1}\textbf{m}}\sqrt{-\det
g_{MN}\partial_{\alpha}x^{M}\partial_{\beta}x^{N}}-\epsilon^{\alpha\beta}\textbf{m}^{T}\textbf{B}_{MN}\partial_{\alpha}x^{M}\partial_{\beta}x^{N}\right],$
(3)

with $g_{MN}$ to be the Einstein frame metric which is invariant under
$SL(2,\mathbb{Z})$ transformations. This action was derived in Kluson:2016pxg
by applying $SL(2,\mathbb{Z})$ invariance of the action of $n$ coincident
D1-branes in general background. The vector B is defined as

$\textbf{B}=\left(\begin{array}[]{c}B^{(2)}\\\ C^{(2)}\end{array}\right)$ (4)

where $B^{(2)}$ and $C^{(2)}$ are NS-NS and R-R two form fluxes respectively.
It transforms under $SL(2,\mathbb{Z})$ transformation as

$\hat{\textbf{B}}=(\Lambda^{T})^{-1}\textbf{B}$ (5)

where $\Lambda$ is given by

$\Lambda=\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right),~{}~{}~{}{\rm
with}~{}~{}~{}~{}ad-bc=1$ (6)

It is convenient to introduce a complex field $\tau=\chi+ie^{-\phi}$
containing both the axion scalar $\chi$ and dilaton $\phi$ corresponding to
NS-NS and R-R sectors respectively. With this complex field, we introduce a
matrix in the following form:

$\mathcal{M}=e^{\phi}\left(\begin{array}[]{cc}\tau\tau^{*}&\chi\\\
\chi&1\end{array}\right)=e^{\phi}\left(\begin{array}[]{cc}\chi^{2}+e^{-2\phi}&\chi\\\
\chi&1\end{array}\right),~{}~{}\det\mathcal{M}=1$ (7)

which transforms under $SL(2,\mathbb{Z})$ transformation as,

$\hat{\mathcal{M}}=\Lambda\mathcal{M}\Lambda^{T}\ .$ (8)

On the other hand, the matrix m in the action (3) is expressed as

$\textbf{m}=\left(\begin{array}[]{c}m\\\ n\end{array}\right)$ (9)

where $m$ and $n$ are the integers counting for the number of fundamental
strings coupled to the NS-NS flux and D1-branes coupled to the R-R flux
respectively. It transforms under $SL(2,\mathbb{Z})$ as
$\hat{\textbf{m}}=\Lambda\textbf{m}$ whereas the $SL(2,\mathbb{Z})$
transformation for $B$ and $\mathcal{M}$ reads as equations (5) and (8)
respectively. Hence this action becomes manifestly $SL(2,\mathbb{Z})$
invariant. Using (4) and (7) we have

$\textbf{m}^{T}\mathcal{M}^{-1}\textbf{m}=\textbf{m}_{i}^{T}(\mathcal{M}^{-1})^{ij}\textbf{m}_{j}=e^{\phi}\left[\left(m-n\chi\right)^{2}+n^{2}e^{-2\phi}\right],$
$\textbf{m}^{T}\textbf{B}=\textbf{m}_{i}\textbf{B}^{i}=mB^{(2)}_{MN}+nC^{(2)}_{MN}$
(10)

Therefore substitution of equation (10) reduces the action as

$\begin{split}S_{m,n}=&-T_{D1}\sqrt{\left(m-n\chi\right)^{2}+n^{2}e^{-2\phi}}\int
d\tau d\sigma\sqrt{-detG_{MN}\partial_{\alpha}x^{M}\partial_{\beta}x^{N}}\\\
&+T_{D1}m\epsilon^{\alpha\beta}\int d\tau d\sigma
B^{(2)}_{MN}\partial_{\alpha}x^{M}\partial_{\beta}x^{N}+T_{D1}n\epsilon^{\alpha\beta}\int
d\tau d\sigma
C^{(2)}_{MN}\partial_{\alpha}x^{M}\partial_{\beta}x^{N}\end{split}$ (11)

It is to be noted that this action is taken in string frame with metric
$G_{MN}$ given by $G^{string}_{MN}=e^{\frac{\phi}{2}}g^{E}_{MN}$. The
nonlinearity of this action due to the presence of the square root of the
determinant results in the equations of motion which are not desirable. To
avoid such discrepancy it may be modified by introducing an auxiliary metric
$h_{\alpha\beta}$. With such consideration, the action becomes

$\begin{split}S_{m,n}=-\frac{\tau_{(m,n)}}{2}\int d\tau
d\sigma\sqrt{-h}h^{\alpha\beta}G_{\alpha\beta}&+q_{(m,n)}\epsilon^{\alpha\beta}\int
d\tau d\sigma B^{(2)}_{MN}\partial_{\alpha}x^{M}\partial_{\beta}x^{N}\\\
&+\tilde{q}_{(m,n)}\epsilon^{\alpha\beta}\int d\tau d\sigma
C^{(2)}_{MN}\partial_{\alpha}x^{M}\partial_{\beta}x^{N}\end{split}$ (12)

where,

$\tau_{(m,n)}=T_{D1}\sqrt{\left(m-n\chi\right)^{2}+n^{2}e^{-2\phi}},~{}q_{(m,n)}=m\frac{T_{F}}{g_{s}},~{}\tilde{q}_{(m,n)}=nT_{D1}$
(13)

Tensions of D1-string and fundamental string, respectively, are

$T_{D1}=\frac{1}{2\pi\alpha^{{}^{\prime}}g_{s}},~{}~{}T_{F}=\frac{1}{2\pi\alpha^{{}^{\prime}}}$
(14)

with $g_{s}$ being the string coupling. It is evident that there is a factor
$\frac{1}{2}$ missing in the flux terms in the action unlike the action for
fundamental string as it is derived from the DBI action of D1 brane instead of
the Polykov action for fundamental string considered in Hoare:2013pma ;
Hoare:2013lja . From this action equation of motion for $h_{\alpha\beta}$ may
be obtained as

$T_{\alpha\beta}=-\frac{2}{\sqrt{-h}}\frac{\partial S_{m,n}}{\partial
h_{\alpha\beta}}=\frac{1}{2}h_{\alpha\beta}h^{\rho\xi}G_{\rho\xi}-G_{\alpha\beta}=0$
(15)

which gives nothing but tracelessness of the stress-energy tensor and can
reproduce the action (11) from (12) by inserting
$h_{\alpha\beta}$=$G_{\alpha\beta}$. Equation of motion for $x^{M}$ can be
easily found as

$\begin{split}&-2\partial_{\alpha}\left[\sqrt{-h}h^{\alpha\beta}G_{MN}\partial_{\beta}x^{N}\right]+\sqrt{-h}h^{\alpha\beta}\partial_{\alpha}x^{K}\partial_{\beta}x^{L}\partial_{M}G_{KL}\\\
&+2q_{(m,n)}H_{MKL}\partial_{\tau}x^{K}\partial_{\sigma}x^{L}+2\tilde{q}_{(m,n)}F_{MKL}\partial_{\tau}x^{K}\partial_{\sigma}x^{L}=0\end{split}$
(16)

where,

$H_{MNK}=\partial_{M}B^{(2)}_{NK}+\partial_{N}B^{(2)}_{KM}+\partial_{K}B^{(2)}_{MN},~{}F_{MNK}=\partial_{M}C^{(2)}_{NK}+\partial_{N}C^{(2)}_{KM}+\partial_{K}C^{(2)}_{MN}$
(17)

In what follows, we will use the metric for $AdS_{3}\times S^{3}$ background
in terms of the global background coordinates,

$ds^{2}=L^{2}\left[-\cosh^{2}\rho dt^{2}+d\rho^{2}+\sinh^{2}\rho
d\phi^{2}+d\theta^{2}+\sin^{2}\theta d\phi_{1}^{2}+\cos^{2}\theta
d\phi_{2}^{2}\right]\ .$ (18)

The accompanying flux components are given by

$\begin{split}&B^{(2)}_{t\phi}=L^{2}q^{2}\sinh^{2}\rho,~{}~{}B^{(2)}_{\phi_{1}\phi_{2}}=-L^{2}q^{2}\cos^{2}\theta\\\
&C^{(2)}_{t\phi}=L^{2}\sqrt{1-q^{2}}\sinh^{2}\rho,~{}~{}C^{(2)}_{\phi_{1}\phi_{2}}=-L^{2}\sqrt{1-q^{2}}\cos^{2}\theta\end{split}$
(19)

where $q$ and $\sqrt{1-q^{2}}=\tilde{q}$, say, are the parameters associated
to field strengths of NS-NS and R-R fluxes respectively. It satisfies, $0\leq
q\leq 1$ and $q^{2}+\tilde{q}^{2}=1$. For $q=0$, it is a case of pure RR flux
and the worldsheet theory can be described in terms of a Green-Schwarz coset
Metsaev:1998it ; Babichenko:2009dk . On the other hand, for $q=1$ it
corresponds to pure NS-NS background, where the theory can be described in
terms of a class of supersymmetric WZW model. For intermediate value of $q$
the theory has not been completely understood till date.

###  2.1 Lagrangian and Hamiltonian formulation

To specify the geometry of $AdS_{3}$ and $S_{3}$ it is convenient to consider
the embedding coordinates to be $Y_{i}$’s and $W_{i}$’s respectively. These
are related to the global coordinates as

$Y_{1}+iY_{2}=\sinh{\rho}e^{i\phi},~{}~{}Y_{3}+iY_{0}=\cosh{\rho}e^{it}$ (20)
$W_{1}+iW_{2}=\sin{\theta}e^{i\phi_{1}},~{}~{}W_{3}+iW_{4}=\cos{\theta}e^{i\phi_{2}}$
(21)

The relations (20) and (21) satisfy the following constraints Arutyunov:2003uj
; Arutyunov:2003za

$-Y_{0}^{2}+Y_{1}^{2}+Y_{2}^{2}-Y_{3}^{2}=-1$ (22)
$W_{1}^{2}+W_{2}^{2}+W_{3}^{2}+W_{4}^{2}=1$ (23)

which precisely define the geometries of $AdS_{3}$ and $S_{3}$ respectively.
As we are interested in the dynamics in $R_{t}\times S^{3}$, we take
$Y_{1}=Y_{2}=0$ so that $Y_{3}+iY_{0}=e^{it}$. Taking
$r_{1}(\xi)=\sin{\theta},r_{2}(\xi)=\cos{\theta},\Phi_{1}(\xi)=\phi_{1}$ and
$\Phi_{2}(\xi)=\phi_{2}$ with $\xi=\alpha\sigma+\beta\tau$, the Lagrangian of
$(m,n)$ string in $R_{t}\times S^{3}$ geometry with mixed three form fluxes
becomes

$\begin{split}&\mathcal{L}=-\frac{\tau_{(m,n)}L^{2}}{2}\left[(\partial_{\tau}t)^{2}-(\partial_{\sigma}t)^{2}+(\partial_{\sigma}r_{1})^{2}-(\partial_{\tau}r_{1})^{2}+(\partial_{\sigma}r_{2})^{2}-(\partial_{\tau}r_{2})^{2}\right]\\\
&-\frac{\tau_{(m,n)}L^{2}}{2}\left[r_{1}^{2}\big{\\{}(\partial_{\sigma}\Phi_{1})^{2}-(\partial_{\tau}\Phi_{1})^{2}\big{\\}}+r_{2}^{2}\big{\\{}(\partial_{\sigma}\Phi_{2})^{2}-(\partial_{\tau}\Phi_{2})^{2}\big{\\}}\right]+\frac{\Lambda}{2}\left(r_{1}^{2}+r_{2}^{2}-1\right)\\\
&-q_{(m,n)}qL^{2}\cos^{2}{\theta}\left[(\partial_{\sigma}\Phi_{1})(\partial_{\tau}\Phi_{2})-(\partial_{\tau}\Phi_{1})(\partial_{\sigma}\Phi_{2})\right]\\\
&-\tilde{q}_{(m,n)}L^{2}\sqrt{1-q^{2}}\cos^{2}\theta\left[(\partial_{\sigma}\Phi_{1})(\partial_{\tau}\Phi_{2})-(\partial_{\tau}\Phi_{1})(\partial_{\sigma}\Phi_{2})\right]\end{split}$
(24)

where $\Lambda$ is the Lagrange multiplier. The same reason as explained in
the case of the action (12) supports for the missing factor $\frac{1}{2}$ in
the flux terms in equation (24) unlike the Lagrangian obtained in
Hoare:2013pma  where fundamental string is taken as a probe. Let us consider
the following parametrization

$t=\kappa\tau,~{}~{}r_{a}=r_{a}(\xi),~{}~{}\Phi_{a}=\Phi_{a}(\xi)=\omega_{a}\tau+f_{a}(\xi),~{}~{}\xi=\alpha\sigma+\beta\tau.$
(25)

where $\kappa,\omega_{a},\alpha$ and $\beta$ are constants. For closed strings
$r_{a}$ and $f_{a}$ should follow periodic conditions

$r_{a}(\xi+2\pi\alpha)=r(\xi),~{}~{}f_{a}(\xi+2\pi\alpha)=f_{a}(\xi)+2\pi\bar{m}_{a},$
(26)

$\bar{m}_{a}$ being integer winding numbers. Using such parametrization for
the string rotating in $R_{t}\times S^{3}$, the Lagrangian becomes,

$\begin{split}\mathcal{L}=&-\frac{\tau_{(m,n)}L^{2}}{2}\left[\kappa^{2}+(\alpha^{2}-\beta^{2})\sum_{a=1}^{2}\left({r_{a}}^{{}^{\prime}2}+{r_{a}}^{2}\left(f_{a}^{{}^{\prime}}-\frac{\beta\omega_{a}}{\alpha^{2}-\beta^{2}}\right)^{2}-\frac{\alpha^{2}\omega_{a}^{2}r_{a}^{2}}{(\alpha^{2}-\beta^{2})^{2}}\right)\right]\\\
&-\left(q_{(m,n)}q+\tilde{q}_{(m,n)}\sqrt{1-q^{2}}\right)L^{2}\alpha^{2}r_{2}^{2}\left(\omega_{2}f_{1}^{{}^{\prime}}-\omega_{1}f_{2}^{{}^{\prime}}\right)+\frac{\Lambda}{2}\left(r_{1}^{2}+r_{2}^{2}-1\right)\
,\end{split}$ (27)

where derivatives with respect to $\xi$ are denoted by prime. Equations of
motion for $f_{a}$’s can be calculated as

$f_{a}^{{}^{\prime}}=\frac{1}{\alpha^{2}-\beta^{2}}\left[\frac{C_{a}}{r_{a}^{2}}+\beta\omega_{a}+\frac{Q\alpha^{2}r_{2}^{2}\omega_{b}}{r_{a}^{2}}\epsilon_{ba}\right]$
(28)

where $\epsilon_{ba}=1$ and $C_{a}$’s are suitable integration constants. Also

$Q=\frac{q_{(m,n)}q+\tilde{q}_{(m,n)}\sqrt{1-q^{2}}}{\tau_{(m,n)}}=\frac{mq+n\sqrt{1-q^{2}}}{\sqrt{(m-n\chi)^{2}+n^{2}e^{-2\phi}}}$
(29)

where we have used (13) and (14). Therefore, the flux parameters are involved
here in the quantity denoted as $Q$ which eventually contains both the
integers $m$ and $n$ along with the axion scalar and dilaton. The special case
with $n=0$ obtains from (29) the flux parameter for probe fundamental string.
Putting this expression of $f_{a}^{{}^{\prime}}$ in the Lagrangian (27) we
achieve,

$\begin{split}\mathcal{L}=&-\frac{\tau_{(m,n)}L^{2}}{2}\left[\kappa^{2}+\left(\alpha^{2}-\beta^{2}\right)\sum_{a=1}^{2}r_{a}^{{}^{\prime}2}+\frac{1}{\left(\alpha^{2}-\beta^{2}\right)}\sum_{a=1}^{2}\left(\frac{C_{a}^{2}+Q^{2}\alpha^{4}r_{2}^{4}\omega_{b}^{2}}{r_{a}^{2}}\right)\right]\\\
&+\frac{\tau_{(m,n)}L^{2}}{2}\left[\frac{\alpha^{2}}{\alpha^{2}-\beta^{2}}\sum_{a=1}^{2}\left(\omega_{a}^{2}r_{a}^{2}+2C_{a}Q\omega_{b}r_{2}^{2}\epsilon_{ba}\right)\right]+\frac{\Lambda}{2}\left(r_{1}^{2}+r_{2}^{2}-1\right)\\\
&-Q\tau_{(m,n)}\frac{\alpha^{2}L^{2}r_{2}^{2}}{\alpha^{2}-\beta^{2}}\left[\frac{\omega_{2}C_{1}}{r_{1}^{2}}-\frac{C_{2}\omega_{1}}{r_{2}^{2}}+\frac{Q\alpha^{2}(\omega_{1}^{2}r_{1}^{2}+\omega_{2}^{2}r_{2}^{2})}{r_{1}^{2}}\right]\
.\end{split}$ (30)

The Lagrangian contains both $r_{a}^{2}$ and $\frac{1}{r_{a}^{2}}$-terms
representing harmonic oscillator type potential and centrifugal potential
barrier respectively. Hence it can be identified as the Lagrangian of one
dimensional integrable Neumann-Rosochatius system with extra terms due to
presence of mixed flux. We can get the equations of motion for $r_{1}$ and
$r_{2}$ as

$r_{1}^{{}^{\prime\prime}}=r_{1}\left[\left(f_{1}^{{}^{\prime}}-\frac{\beta\omega_{1}}{\alpha^{2}-\beta^{2}}\right)^{2}-\frac{\alpha^{2}\omega_{1}^{2}}{(\alpha^{2}-\beta^{2})^{2}}\right]+Ar_{1}\
,$ (31)

and

$r_{2}^{{}^{\prime\prime}}=r_{2}\left[\left(f_{2}^{{}^{\prime}}-\frac{\beta\omega_{2}}{\alpha^{2}-\beta^{2}}\right)^{2}-\frac{\alpha^{2}\omega_{2}^{2}}{(\alpha^{2}-\beta^{2})^{2}}\right]+Ar_{2}+\frac{2Q\alpha^{2}r_{2}}{(\alpha^{2}-\beta^{2})^{2}}\left(\omega_{2}f_{1}^{{}^{\prime}}-\omega_{1}f_{2}^{{}^{\prime}}\right)\
,$ (32)

where $A=\frac{\Lambda}{\tau_{(m,n)}L^{2}}$ is a constant. Putting the
expressions of $f_{a}$’s from (28) in equations (31) and (32) we achieve,

$\begin{split}\left(\alpha^{2}-\beta^{2}\right)r_{1}^{{}^{\prime\prime}}+&\frac{1}{\left(\alpha^{2}-\beta^{2}\right)}\left(\frac{C_{1}^{2}}{r_{1}^{3}}+\frac{Q^{2}\alpha^{4}r_{2}^{4}\omega_{2}^{2}}{r_{1}^{3}}\right)+\frac{\alpha^{2}}{\left(\alpha^{2}-\beta^{2}\right)}\omega_{1}^{2}r_{1}-Ar_{1}\\\
&+\frac{2Q\alpha^{2}r_{2}^{2}}{\left(\alpha^{2}-\beta^{2}\right)}\left[\frac{\omega_{2}C_{1}}{r_{1}^{3}}+\frac{Q\alpha^{2}r_{2}^{2}\omega_{2}^{2}}{r_{1}^{3}}\right]=0\
,\end{split}$ (33)

and

$\begin{split}&\left(\alpha^{2}-\beta^{2}\right)r_{2}^{{}^{\prime\prime}}-\frac{2Q^{2}\alpha^{4}r_{2}^{3}\omega_{2}^{2}}{r_{1}^{2}\left(\alpha^{2}-\beta^{2}\right)}-\frac{Q^{2}\alpha^{4}r_{2}\omega_{1}^{2}}{\left(\alpha^{2}-\beta^{2}\right)}+\frac{1}{\left(\alpha^{2}-\beta^{2}\right)}\frac{C_{2}^{2}}{r_{2}^{3}}+\frac{\alpha^{2}}{\left(\alpha^{2}-\beta^{2}\right)}\omega_{2}^{2}r_{2}\\\
&+\frac{2Q\alpha^{2}}{\left(\alpha^{2}-\beta^{2}\right)}\left(\omega_{2}C_{1}-\omega_{1}C_{2}\right)r_{2}-Ar_{2}-\frac{2Q\alpha^{2}}{\left(\alpha^{2}-\beta^{2}\right)}\left[\frac{\omega_{2}C_{1}r_{2}}{r_{1}^{2}}+\frac{2Q\alpha^{2}r_{2}^{3}\omega_{2}^{2}}{r_{1}^{2}}+Q\alpha^{2}\omega_{1}^{2}r_{2}\right]=0\
.\end{split}$ (34)

Above two equations can also be obtained from the Euler-Lagrange equations of
the following equivalent Lagrangian of an integrable NR system

$\begin{split}\mathcal{L}=&\frac{\left(\alpha^{2}-\beta^{2}\right)}{2}\sum_{a=1}^{2}r_{a}^{{}^{\prime}2}-\frac{1}{2\left(\alpha^{2}-\beta^{2}\right)}\sum_{a=1}^{2}\left(\frac{C_{a}^{2}+Q^{2}\alpha^{4}r_{2}^{4}\omega_{b}^{2}}{r_{a}^{2}}\right)\\\
&+\left[\frac{\alpha^{2}}{2\left(\alpha^{2}-\beta^{2}\right)}\sum_{a=1}^{2}\left(\omega_{a}^{2}r_{a}^{2}+2C_{a}Q\omega_{b}r_{2}^{2}\epsilon_{ba}\right)\right]+\frac{A}{2}\left(r_{1}^{2}+r_{2}^{2}\right)\\\
&-\frac{Q\alpha^{2}r_{2}^{2}}{2\left(\alpha^{2}-\beta^{2}\right)}\left[\frac{\omega_{2}C_{1}}{r_{1}^{2}}-\frac{C_{2}\omega_{1}}{r_{2}^{2}}+\frac{Q\alpha^{2}(\omega_{1}^{2}r_{1}^{2}+\omega_{2}^{2}r_{2}^{2})}{r_{1}^{2}}\right]\,\end{split}$
(35)

for general coordinates $r_{1}$ and $r_{2}$. The Virasoro constraints may be
expressed as

$\displaystyle
T_{\tau\tau}+T_{\sigma\sigma}=0,~{}~{}~{}~{}~{}~{}T_{\sigma\tau}=T_{\tau\sigma}=0\
.$ (36a)

With the embeddings (22) and (23) and using the ansatz (25), the constraints
reduce to

$(\alpha^{2}+\beta^{2})(r_{1}^{{}^{\prime}2}+r_{2}^{{}^{\prime}2})+r_{1}^{2}\left[\alpha^{2}f_{1}^{{}^{\prime}2}+(\omega_{1}+\beta
f_{1}^{{}^{\prime}})^{2}\right]+r_{2}^{2}\left[\alpha^{2}f_{2}^{{}^{\prime}2}+(\omega_{2}+\beta
f_{2}^{{}^{\prime}})^{2}\right]=\kappa^{2}\,$ (37)

and

$\alpha\beta(r_{1}^{{}^{\prime}2}+r_{2}^{{}^{\prime}2})+\alpha\left[\omega_{1}r_{1}^{2}f_{1}^{{}^{\prime}}+\omega_{2}r_{2}^{2}f_{2}^{{}^{\prime}}\right]+\alpha\beta\left[f_{1}^{{}^{\prime}2}r_{1}^{2}+f_{2}^{{}^{\prime}2}r_{2}^{2}\right]=0$
(38)

respectively. The Hamiltonian of such a system can be written as

$\begin{split}&\mathcal{H}=\frac{\left(\alpha^{2}-\beta^{2}\right)}{2}\sum_{a=1}^{2}r_{a}^{{}^{\prime}2}+\frac{1}{2\left(\alpha^{2}-\beta^{2}\right)}\sum_{a=1}^{2}\left(\frac{C_{a}^{2}+Q^{2}\alpha^{4}r_{2}^{4}\omega_{b}^{2}}{r_{a}^{2}}\right)\\\
&-\left[\frac{\alpha^{2}}{2\left(\alpha^{2}-\beta^{2}\right)}\sum_{a=1}^{2}\left(\omega_{a}^{2}r_{a}^{2}+2C_{a}Q\omega_{b}r_{2}^{2}\epsilon_{ba}\right)\right]+\\\
&\frac{Q\alpha^{2}r_{2}^{2}}{2\left(\alpha^{2}-\beta^{2}\right)}\left[\frac{\omega_{2}C_{1}}{r_{1}^{2}}-\frac{C_{2}\omega_{1}}{r_{2}^{2}}+\frac{Q\alpha^{2}(\omega_{1}^{2}r_{1}^{2}+\omega_{2}^{2}r_{2}^{2})}{r_{1}^{2}}\right]\end{split}$
(39)

This again takes the form of the Hamiltonian of integrable one dimensional NR
model. Hence it is obvious that classically integrable NR model may represent
a nice tool to study for probe $(m,n)$ string in the desired background, even
in the presence of flux.

###  2.2 Integrals of Motion

Any classically integrable system contains infinite number of conserved
quantities, also known as integrals of motion, in involution, i.e., they must
Poisson commute each other. The integrability of Neumann-Rosochatius system
demands the existence of a set of integrals of motion, named as Uhlenbeck
constants, in involution. These were first introduced by K. Uhlenbeck
Uhlenbeck . In the case of closed rotating string in $S^{3}$ there are two
integrals of motion $I_{1}$ and $I_{2}$ constrained by the relation
$I_{1}+I_{2}=1$. A general form of Uhlenbeck integrals of motion for the case
of closed rotating string can be written as

$I_{a}=\alpha^{2}x_{a}x_{a}^{{}^{\prime}}+\sum_{b\neq
a}\frac{|\bar{x_{b}}p_{a}-x_{a}\bar{p_{b}}|^{2}}{\omega_{a}^{2}-\omega_{b}^{2}}\
,$ (40)

which eventually gives

$I_{a}=\alpha^{2}r_{a}^{2}+\left(\alpha^{2}-\beta^{2}\right)^{2}\sum_{b\neq
a}\frac{\left(r_{a}^{{}^{\prime}}r_{b}-r_{a}r_{b}^{{}^{\prime}}\right)^{2}}{\omega_{a}^{2}-\omega_{b}^{2}}+\sum_{b\neq
a}\frac{1}{\omega_{a}^{2}-\omega_{b}^{2}}\left(\frac{C_{a}^{2}r_{b}^{2}}{r_{a}^{2}}+\frac{C_{b}^{2}r_{a}^{2}}{r_{b}^{2}}\right)\
,$ (41)

for some arbitrary values of constants $C_{a}$’s. Here we have used the
embeddings (22),
(23) and ansatz (25) for closed string rotating on $S^{3}$ with
$x_{a}(\xi)=r_{a}e^{f_{a}(\xi)}$ Kruczenski:2006pk . To find out the integrals
of motion in the deformed background, let us proceed with the methodology
described in Hernandez:2014eta ; Hernandez:2015nba  where a term
$g=g(r_{1},r_{2},Q)$ representing the deformation due to fluxes is added
without affecting the integrable properties of $I_{a}$ such that,

$\begin{split}\bar{I}_{a}=\alpha^{2}r_{a}^{2}+\left(\alpha^{2}-\beta^{2}\right)^{2}\sum_{b\neq
a}\frac{\left(r_{a}^{{}^{\prime}}r_{b}-r_{a}r_{b}^{{}^{\prime}}\right)^{2}}{\omega_{a}^{2}-\omega_{b}^{2}}+&\sum_{b\neq
a}\frac{1}{\omega_{a}^{2}-\omega_{b}^{2}}\left(\frac{C_{a}^{2}r_{b}^{2}}{r_{a}^{2}}+\frac{C_{b}^{2}r_{a}^{2}}{r_{b}^{2}}\right)\\\
&+\sum_{b\neq a}\frac{2g}{\omega_{a}^{2}-\omega_{b}^{2}}\ .\end{split}$ (42)

The function $g$ may be derived by setting $\bar{I_{a}^{{}^{\prime}}}=0$. This
will yield,

$g=(\omega_{1}^{2}-\omega_{2}^{2})r_{1}^{2}\alpha^{2}-(\alpha^{2}-\beta^{2})(r_{1}^{{}^{\prime}2}+r_{2}^{{}^{\prime}2})-\left(\frac{C_{1}^{2}}{r_{1}^{2}}+\frac{C_{2}^{2}r_{1}^{2}}{r_{2}^{2}}\right)$
(43)

Now, by using the constraints

$r_{1}^{2}+r_{2}^{2}=1,~{}~{}r_{1}r_{1}^{{}^{\prime}}+r_{2}r_{2}^{{}^{\prime}}=0,~{}~{}r_{1}r_{1}^{{}^{\prime\prime}}+r_{1}^{{}^{\prime}2}+r_{2}^{{}^{\prime}2}+r_{2}r_{2}^{{}^{\prime\prime}}=0$
(44)

and the equations (31), (32) and (28), we get the integral of motion in the
following form:

$\begin{split}\bar{I_{1}}=&\frac{\alpha^{2}-\beta^{2}}{\omega_{1}^{2}-\omega_{2}^{2}}\left(r_{1}r_{2}^{{}^{\prime}}-r_{1}^{{}^{\prime}}r_{2}\right)^{2}+\frac{2}{\omega_{1}^{2}-\omega_{2}^{2}}\left[\left(\frac{C_{1}-Q\alpha^{2}r_{2}^{2}\omega_{2}}{r_{1}}\right)^{2}+\left(\frac{C_{2}+Q\alpha^{2}r_{2}^{2}\omega_{1}}{r_{2}}\right)^{2}\right]\\\
&-\frac{2\alpha^{2}}{\omega_{1}^{2}-\omega_{2}^{2}}\left[\left(1+\frac{2Q^{2}\alpha^{2}r_{2}^{2}}{r_{1}^{2}}\right)\left(\omega_{1}^{2}r_{1}^{2}+\omega_{2}^{2}r_{2}^{2}\right)+2Qr_{2}^{2}\left(\frac{C_{1}\omega_{2}}{r_{1}^{2}}-\frac{C_{2}\omega_{1}}{r_{2}^{2}}\right)\right]\\\
&+\frac{1}{\omega_{1}^{2}-\omega_{2}^{2}}\left(\frac{C_{1}^{2}}{r_{1}^{2}}+\frac{C_{2}^{2}r_{1}^{2}}{r_{2}^{2}}\right)\
.\end{split}$ (45)

It is henceforth clear that in the absence of flux, the deformed constants
satisfy the relation $\sum_{a=1}^{2}\bar{I}_{a}=1$.

###  2.3 Elliptic solutions for $r_{1}$ and $r_{2}$

In this section we present the solutions for such system of $(m,n)$ string
rotating in $S^{3}$ with two different angular momenta. In this context we can
introduce three parameters on a sphere like $\zeta_{1},\zeta_{2}$ and
$\zeta_{3}$ to express the roots of the equation in ellipsoidal coordinate
$\zeta$

$\frac{r_{1}^{2}}{\zeta-\omega_{1}^{2}}+\frac{r_{2}^{2}}{\zeta-\omega_{2}^{2}}=0\
.$ (46)

With angular frequencies $\omega_{1}<\omega_{2}$, this equation is defined on
a sphere while being invariant under $r_{a}\rightarrow\lambda r_{a}$
Babelon:1992rb . The range of ellipsoidal coordinate is
$\omega_{1}^{2}\leq\zeta\leq\omega_{2}^{2}$ for which $\zeta$ covers
$\frac{1}{4}$th of a sphere corresponding to $r_{i}\geq 0$ and the whole
sphere may be thought of as a covering of the domain of $\zeta$ having
branches along its boundary. If we enter the expressions of $r_{1}^{2}$ and
$r_{2}^{2}$ in terms of $\zeta$ into the equations of motion we may get a set
of second order differential equations for $\zeta$. But for the sake of
simplicity, we inject this ellipsoidal coordinate $\zeta$ into the Uhlenbeck
integrals of motion Arutyunov:2003uj ; Arutyunov:2003za  and can have a first
order differential equation for $\zeta$. Using the constraint
$r_{1}^{2}+r_{2}^{2}=1$ corresponding to spherical geometry in equation (46)
we find

$r_{1}^{2}=\frac{\omega_{1}^{2}-\zeta}{\omega_{1}^{2}-\omega_{2}^{2}},~{}~{}r_{2}^{2}=\frac{\zeta-\omega_{2}^{2}}{\omega_{1}^{2}-\omega_{2}^{2}},~{}~{}\left(r_{1}r_{2}^{{}^{\prime}}-r_{1}^{{}^{\prime}}r_{2}\right)^{2}=\frac{\zeta^{{}^{\prime}2}}{4(\omega_{1}^{2}-\zeta)(\zeta-\omega_{2}^{2})}\
.$ (47)

The integral of motion (45) in terms of the ellipsoidal coordinate takes the
following form

$\begin{split}\bar{I}_{1}=&\frac{\alpha^{2}-\beta^{2}}{\omega^{2}_{1}-\omega_{2}^{2}}\frac{\zeta^{{}^{\prime}2}}{4(\omega_{1}^{2}-\zeta)(\zeta-\omega_{2}^{2})}+\frac{2Q^{2}\alpha^{4}(\zeta-\omega_{2}^{2})(\zeta+\omega_{1}^{2}+\omega_{2}^{2})}{(\omega_{1}^{2}-\omega_{2}^{2})(\omega_{1}^{2}-\zeta)}+\\\
&\frac{2\alpha^{2}(\omega_{1}^{2}-\zeta)}{(\omega_{1}^{2}-\omega_{2}^{2})}+\frac{2\alpha^{2}\omega_{2}^{2}Q}{(\omega_{1}^{2}-\zeta)}+\frac{2\alpha^{2}}{\omega_{1}^{2}-\omega_{2}^{2}}\left[Q\omega_{1}^{2}+\omega_{2}^{2}(1-Q)\right]\
.\end{split}$ (48)

Solving for $\zeta^{{}^{\prime}2}$ from this expression of deformed Uhlenbeck
constant we explore that

$\zeta^{{}^{\prime}2}=-4P_{3}(\zeta),~{}~{}P_{3}(\zeta)=\frac{2\alpha^{2}(1-Q)}{\alpha^{2}-\beta^{2}}\prod_{i=1}^{3}\left(\zeta-\zeta_{i}\right)\
.$ (49)

Here $P_{3}(\zeta)$ is evidently a third order polynomial defining an elliptic
curve
$s^{2}+P_{3}(\zeta)=0$, $s_{i}$ and $\zeta_{i}$ being the general coordinates
on that elliptic curve. Now changing the variables as

$\zeta=\zeta_{2}+(\zeta_{3}-\zeta_{2})\eta^{2}$ (50)

and substituting for $\eta$ in equation (49) we get

$\eta(\xi)=\hbox{cn}\left(\alpha\xi\sqrt{\frac{2\left(1-Q\right)\left(\zeta_{3}-\zeta_{1}\right)}{\alpha^{2}-\beta^{2}}}+\xi_{0},k\right)\
,$ (51)

where $k=\frac{\zeta_{3}-\zeta_{2}}{\zeta_{3}-\zeta_{1}}$ is the elliptic
modulus and $\xi_{0}$ is the integration constant which may be set to zero by
a rotation. This gives the expression for $r_{1}$ from equation (46) as

$r_{1}^{2}(\xi)=\frac{\zeta_{3}-\omega_{1}^{2}}{\omega_{2}^{2}-\omega_{1}^{2}}+\frac{\zeta_{2}-\zeta_{3}}{\omega_{2}^{2}-\omega_{1}^{2}}\hbox{sn}^{2}\left(\alpha\xi\sqrt{\frac{2\left(1-Q\right)\left(\zeta_{3}-\zeta_{1}\right)}{\alpha^{2}-\beta^{2}}},k\right)$
(52)

and similarly the expression for $r_{2}$ is

$r_{2}^{2}(\xi)=\frac{\omega_{2}^{2}-\zeta_{3}}{\omega_{2}^{2}-\omega_{1}^{2}}-\frac{\zeta_{2}-\zeta_{3}}{\omega_{2}^{2}-\omega_{1}^{2}}\hbox{sn}^{2}\left(\alpha\xi\sqrt{\frac{2\left(1-Q\right)\left(\zeta_{3}-\zeta_{1}\right)}{\alpha^{2}-\beta^{2}}},k\right)$
(53)

It is obvious from these expressions that the constraint
$r_{1}^{2}+r_{2}^{2}=1$ is satisfied. The range of elliptic modulus $k$ should
be $0<k<1$ which eventually needs $\zeta_{1}<\zeta_{2}<\zeta_{3}$. Also we can
achieve circular type solution for
$\omega_{1}^{2}\leq\zeta_{2,3}\leq\omega_{2}^{2}$ in the codomain of equations
(52) and (53) between $0$ and $1$ provided that no such restriction is needed
for $\zeta_{1}$.

###  2.4 Conserved charges and dispersion relation

In this section we are interested in finding out the scaling relation among
various conserved charges. The conserved energy and angular momenta for this
system are expressed as

$E=-\int{d\sigma\frac{\partial\mathcal{L}}{\partial(\partial_{\tau}t)}},~{}~{}J_{a}=\int{d\sigma\frac{\partial\mathcal{L}}{\partial(\partial_{\tau}\Phi_{a})}},$
(54)

with $a=1,2$ Therefore using the Lagrangian (24), conserved quantities may be
expressed as

$\displaystyle E=2\pi\tau_{(m,n)}L^{2}\kappa\ ,$ (55a) $\displaystyle
J_{1}=\frac{1}{\alpha}\int_{0}^{2\pi}d\xi\tau_{(m,n)}L^{2}\left[r_{1}^{2}\left(\omega_{1}^{2}+\beta
f_{1}^{{}^{\prime}}\right)+Q\alpha r_{2}^{2}f_{2}^{{}^{\prime}}\right]\ ,$
(55b) $\displaystyle
J_{2}=\frac{1}{\alpha}\int_{0}^{2\pi}d\xi\tau_{(m,n)}L^{2}\left[r_{2}^{2}\left(\omega_{2}^{2}+\beta
f_{2}^{{}^{\prime}}\right)+Q\alpha r_{2}^{2}f_{1}^{{}^{\prime}}\right]\ .$
(55c)

###  2.5 Constant radii solutions

A convenient method of deriving the energy-angular momenta dispersion relation
is to employ the equations (52) and (53) in (55b) and (55c) and to write the
energy in terms of winding numbers $\bar{m}_{i}$’s, angular momenta $J$’s and
Uhlenbeck constants $\bar{I}_{i}$’s. However, there are some important limits
of the parameters $\zeta_{i}$’s for which the computation of dispersion
relation can be simplified. The corresponding choices of the parameters may be
considered such that the discriminant of $P_{3}(\zeta)$ reduces to zero. The
possible choices are : (i) constant radii case which is achieved with the
limit $\zeta_{2}\rightarrow\zeta_{3}$, (ii) $\zeta_{1}=\zeta_{2}$ which gives
$k=1$ and (iii) $\zeta_{1}=\zeta_{2}=\zeta_{3}$ which can only be solved for
equal angular frequencies $\omega_{1}=\omega_{2}$. We are interested in
deriving the constant radii solutions for the conserved energy and momenta
which can be obtained by taking the limit $\zeta_{2}\rightarrow\zeta_{3}$ in
the equations (52) and (53). These type of solutions have been constructed by
using Neumann-Rosochatius integrable system in Hernandez:2014eta  for circular
strings without incorporating the Weierstrass elliptic functions. With this
limit, the radial coordinates become constant such as
$r_{1}=\sqrt{\frac{\zeta_{3}-\omega_{1}^{2}}{\omega_{2}^{2}-\omega_{1}^{2}}}=a_{1}$,
say and
$r_{2}=\sqrt{\frac{\omega_{2}^{2}-\zeta_{3}}{\omega_{2}^{2}-\omega_{1}^{2}}}=a_{2}$,
say. Then the derivatives of the angles become constant, thereby giving

$f_{a}=\bar{m}_{a}\xi+f_{0a}\ ,$ (56)

where the integration constants $f_{0a}$ can be chosen to be zero through a
rotation and
$\bar{m}_{a}=\frac{1}{\alpha^{2}-\beta^{2}}\left[\frac{C_{a}}{a_{a}^{2}}+\beta\omega_{a}+\frac{Q\alpha^{2}a_{2}^{2}\omega_{b}}{a_{a}^{2}}\epsilon_{ba}\right]$
are assumed to be the constant integer windings of the string satisfying the
closed string periodicity conditions. Also the Virasoro constraints (37), (38)
and currents (55b), (55c) yield

$a_{1}^{2}=\frac{\bar{m}_{2}(\omega_{2}+\beta\bar{m}_{2})}{\bar{m}_{2}(\omega_{2}+\beta\bar{m}_{2})-\bar{m}_{1}(\omega_{1}+\beta\bar{m}_{1})},~{}~{}a_{2}^{2}=\frac{\bar{m}_{1}(\omega_{1}+\beta\bar{m}_{1})}{\bar{m}_{1}(\omega_{1}+\beta\bar{m}_{1})-\bar{m}_{2}(\omega_{2}+\beta\bar{m}_{2})}\
,$ (57)

and

$\bar{m}_{1}J_{1}+\bar{m}_{2}J_{2}=0\ .$ (58)

The above relations along with the conserved quantities yield from (37),

$\begin{split}\frac{E^{2}}{\alpha^{2}}=\left(J_{1}+J_{2}\right)^{2}+\frac{(1-w)^{2}}{w}J_{1}J_{2}&-2QT_{(m,n)}\bar{m}_{1}\left(J_{2}+J_{1}w\right)\\\
&+T_{(m,n)}^{2}\left(\bar{m}_{1}\bar{m}_{2}-Q^{2}\bar{m}_{1}^{2}w\right)\left(\frac{\bar{m}_{1}-\bar{m}_{2}w}{\bar{m}_{2}-\bar{m}_{1}w}\right)\
,\end{split}$ (59)

where we have used $T_{(m,n)}=2\pi\tau_{(m,n)}L^{2}$ and
$w=\frac{\omega_{1}+\beta\bar{m}_{1}}{\omega_{2}+\beta\bar{m}_{2}}$. To derive
the dispersion relation it would be convenient to find out $w$ in terms of
windings $\bar{m}_{a}$ and angular momenta $J_{a}$. To do this let us write
the reduced equations of motion for constant radii solutions as

$\bar{m}_{1}^{2}-\frac{2\beta\omega_{1}\bar{m}_{1}}{\alpha^{2}-\beta^{2}}-\frac{\omega_{1}^{2}}{\alpha^{2}-\beta^{2}}+A=0\
,$ (60)
$\bar{m}_{2}^{2}-\frac{2\beta\omega_{2}\bar{m}_{2}}{\alpha^{2}-\beta^{2}}-\frac{\omega_{2}^{2}}{\alpha^{2}-\beta^{2}}+A+\frac{2Q\alpha^{2}}{(\alpha^{2}-\beta^{2})^{2}}\left(\omega_{2}\bar{m}_{1}-\omega_{1}\bar{m}_{2}\right)=0\
.$ (61)

Now adding equations (55b) and (55c), subtracting equation (61) from (60) and
solving the resulting system of equations we get the relation

$\begin{split}&\bar{m}_{1}^{2}-\bar{m}_{2}^{2}+\frac{2\beta}{\alpha^{2}-\beta^{2}}\left[\frac{\bar{m}_{2}}{w}(\omega_{1}+\beta\bar{m}_{1})-\beta\bar{m}_{2}^{2}-\omega_{1}\bar{m}_{1}\right]\\\
&+\frac{\left[\frac{1}{w}(\omega_{1}+\beta\bar{m}_{1})-\beta\bar{m}_{2}\right]^{2}-\omega_{1}^{2}}{\alpha^{2}-\beta^{2}}-\frac{2Q\alpha^{2}}{(\alpha^{2}-\beta^{2})^{2}}\left[\frac{\bar{m}_{1}}{w}(\omega_{1}+\beta\bar{m}_{1})-\beta\bar{m}_{2}\bar{m}_{1}-\omega_{1}\bar{m}_{2}\right]=0\
,\end{split}$ (62)

where $\omega_{1}$ is expressed as

$\omega_{1}=\frac{\bar{m}_{1}w\left[J-T_{(m,n)}Q\alpha\left(\bar{m}_{1}-\bar{m}_{2}\right)\right]-\bar{m}_{2}J}{T_{(m,n)}(\bar{m}_{1}-\bar{m}_{2})}-\beta\bar{m}_{1}\
,$ (63)

Here $J=J_{1}+J_{2}$ is the total angular momentum. Eliminating $\omega_{1}$
from these two relations we are left with a quartic equation of $w$ given by

$\begin{split}&\left(1-w^{2}\right)\left(\bar{m}_{1}w-\bar{m}_{2}\right)^{2}J^{2}-4QT_{(m,n)}\bar{m}_{1}w\left(\bar{m}_{1}w-\bar{m}_{2}\right)\left(\bar{m}_{1}-\bar{m}_{2}\right)\left(1-w^{2}\right)\\\
&+\frac{w^{2}Q^{2}T_{(m,n)}^{2}}{J^{2}}\left(\bar{m}_{1}-\bar{m}_{2}\right)\left[4\bar{m}_{1}^{2}\left(1-w^{2}\right)\left(\bar{m}_{1}-\bar{m}_{2}\right)+\bar{m}_{2}\left(\bar{m}_{2}-w\right)\right]\\\
&+\frac{w^{2}T_{(m,n)}^{2}}{J^{2}}\left(\bar{m}_{1}-\bar{m}_{2}\right)^{3}\left(\bar{m}_{1}+\bar{m}_{2}\right)\left(1-Q^{2}\right)=0\
.\end{split}$ (64)

Instead of solving this equation explicitly, it is convenient to get an
approximate solution as a power series expansion in large $J$, i.e., small
$\frac{T_{(m,n)}}{J}$ which is given by

$w=1+\frac{QT_{(m,n)}}{J}\frac{\bar{m}_{2}}{\bar{m}_{1}}\left(\bar{m}_{1}-\bar{m}_{2}\right)-\frac{Q^{2}T_{(m,n)}^{2}}{J^{2}}\frac{\bar{m}_{1}}{2\bar{m}_{2}^{2}}\left(\bar{m}_{1}-\bar{m}_{2}\right)^{2}+\mathcal{O}\left(\frac{T_{(m,n)}}{J}\right)^{3}\
.$ (65)

where we have neglected the other higher order terms in the series expansion.
Substituting equation (65) in equation (59) we get a relation between energy
and angular momenta as follows

$\displaystyle
E^{2}=J^{2}-2QT_{(m,n)}\bar{m}_{1}J+T_{(m,n)}^{2}\frac{\bar{m}_{2}^{2}}{\bar{m}_{1}^{2}}\left(\bar{m}_{1}^{2}-\bar{m}_{2}^{2}\right)\left(1-Q^{2}\right)-\frac{2Q^{2}T_{(m,n)}^{2}}{J}J_{1}\bar{m}_{1}\times$
$\displaystyle\times\left(\bar{m}_{1}-\bar{m}_{2}\right)\left[\frac{\bar{m}_{2}}{\bar{m}_{1}}-\frac{QT_{(m,n)}}{J}\frac{\bar{m}_{1}}{2\bar{m}_{2}^{2}}\left(\bar{m}_{1}-\bar{m}_{2}\right)\right]\
,$

where we have considered $\alpha=1$. For two equal angular momentum
$J_{1}=J_{2}$ (which sets $\bar{m}_{1}=-\bar{m}_{2}=\bar{m}$) , one gets

$E^{2}=J^{2}-4\pi\tau_{(m,n)}L^{2}Q\bar{m}J+2\pi^{2}\tau^{2}_{(m,n)}L^{4}Q^{2}\bar{m}^{2}\,$
(67)

Again the solution with constant and equal radii demands $\omega=1$ from the
expressions (57) and (58). This eventually yields the dispersion relation as

$E=J-2\pi\tau_{(m,n)}L^{2}Q\bar{m}+\frac{\pi^{2}\tau^{2}_{(m,n)}L^{4}\left(1-Q^{2}\right)\bar{m}^{2}}{J}$
(68)

Here, $T_{(m,n)}=2\pi\tau_{(m,n)}L^{2}$. The dispersion relation obtained for
the equal and constant radii case quite matches with those for rigid circular
string rotating in $R\times S^{3}$ with fundamental string as the natural
probe Hoare:2013lja ; Hoare:2013ida  except the flux parameter which in this
case is given as $Q$ described in (29). For large $J$, the above expression
reduces to $E\approx J$. Such linear dependence of $E$ on $J$ stems for the
identification of the string state with the gauge theory operator
corresponding to the maximal-spin state of XXX spin chain.

##  3 NR system for pulsating (m,n) strings in $AdS_{3}\times S^{3}$ with
mixed flux

In this section, we study the circular closed string pulsating in $R_{t}\times
S^{3}$ by using a deformed integrable one dimensional Neumann-Rosochatius
model. We use the following embedding

$\displaystyle Y_{3}+iY_{0}=\cosh{\rho}e^{it}=z_{0}(\tau)e^{ih_{0}(\tau)}\ ,$
(69a) $\displaystyle
W_{1}+iW_{2}=\sin{\theta}e^{i\phi_{1}}=r_{1}(\tau)e^{i(f_{1}(\tau)+\bar{m}_{1}\sigma)}\
,$ (69b) $\displaystyle
W_{3}+iW_{4}=\cos{\theta}e^{i\phi_{2}}=r_{2}(\tau)e^{i(f_{2}(\tau)+\bar{m}_{2}\sigma)}\
.$ (69c)

These will follow the relations between global and local coordinates as

$\displaystyle\cosh{\rho}=z_{0}(\tau),~{}~{}t=h_{0}(\tau)\ ,$
$\displaystyle\sin{\theta}=r_{1}(\tau),~{}~{}\phi_{1}=\Phi_{1}(\tau,\sigma)=f_{1}(\tau)+\bar{m}_{1}\sigma\
,$
$\displaystyle\cos{\theta}=r_{2}(\tau),~{}~{}\phi_{2}=\Phi_{2}(\tau,\sigma)=f_{2}(\tau)+\bar{m}_{2}\sigma\
.$ (70)

Winding numbers $\bar{m}_{a}$’s are assumed along the $\sigma$ direction only
to make the time-direction single-valued. With this embedding, the Lagrangian
can now be expressed as,

$\begin{split}&\mathcal{L}=\frac{\tau_{(m,n)}L^{2}}{2}\left[-\left(\dot{z}_{0}^{2}+z_{0}^{2}\dot{h}_{0}^{2}\right)+\sum_{a=1}^{2}\left(\dot{r}_{a}^{2}+r_{a}^{2}\dot{f}_{a}^{2}-r_{a}^{2}\bar{m}_{a}^{2}\right)\right]\\\
&+\left(q_{(m,n)}q+\tilde{q}_{(m,n)}\sqrt{1-q^{2}}\right)L^{2}r_{2}^{2}\left(\bar{m}_{2}\dot{f}_{1}-\bar{m}_{1}\dot{f}_{2}\right)-\frac{\Lambda}{2}\left(\sum_{a=1}^{2}r_{a}^{2}-1\right)-\frac{\tilde{\Lambda}}{2}\left(z_{0}^{2}+1\right)\
,\end{split}$ (71)

where the derivative with respect to $\tau$ is denoted by dots. $\Lambda$ and
$\tilde{\Lambda}$ are suitable Lagrange multipliers.

###  3.1 Lagrangian and Hamiltonian formulation

Euler-Lagrange equations of motion for $z_{0}$ and $f_{a}$’s can be derived
from the Lagrangian (71) as

$\displaystyle\ddot{z}_{0}-\frac{C_{0}^{2}}{4z_{0}^{3}}+\tilde{A}z_{0}=0$
(72a)
$\displaystyle\dot{f}_{1}=\frac{C_{1}}{r_{1}^{2}}-\frac{Qr_{2}^{2}\bar{m}_{2}}{r_{1}^{2}}$
(72b) $\displaystyle\dot{f}_{2}=\frac{C_{2}}{r_{2}^{2}}+Q\bar{m}_{1}\ ,$ (72c)

where ,$\dot{h}_{0}=\frac{C_{0}}{2z_{0}^{2}}$ and
$\tilde{A}=\frac{\tilde{\Lambda}}{\tau_{(m,n)}L^{2}}$. Putting the expressions
for $\dot{f}_{1}$ and $\dot{f}_{2}$ in terms of $r_{1}$ and $r_{2}$ in the
Lagrangian (71) we achieve,

$\begin{split}&\mathcal{L}=\frac{\tau_{(m,n)}L^{2}}{2}\left[-\left(\dot{z}_{0}^{2}+z_{0}^{2}\dot{h}_{0}^{2}\right)+\sum_{a=1}^{2}\left(\dot{r}_{a}^{2}+\frac{\left(C_{a}+Q\bar{m}_{b}r_{2}^{2}\epsilon_{ba}\right)^{2}}{r_{a}^{2}}-r_{a}^{2}\bar{m}_{a}^{2}\right)\right]\\\
&-Q\tau_{(m,n)}L^{2}r_{2}^{2}\left(\frac{\bar{m}_{1}C_{2}}{r_{2}^{2}}-\frac{\bar{m}_{2}C_{1}}{r_{1}^{2}}+\frac{Q\bar{m}_{1}^{2}r_{2}^{2}}{r_{2}^{2}}+\frac{Q\bar{m}_{2}^{2}r_{2}^{2}}{r_{1}^{2}}\right)-\frac{\Lambda}{2}\left(\sum_{a=1}^{2}r_{a}^{2}-1\right)-\frac{\tilde{\Lambda}}{2}\left(z_{0}^{2}+1\right)\
.\end{split}$ (73)

We can get the equations of motion for $r_{1}$ and $r_{2}$ as

$\ddot{r}_{1}+\frac{\left(C_{1}-Q\bar{m}_{2}r_{2}^{2}\right)^{2}}{r_{1}^{3}}+\bar{m}_{1}^{2}r_{1}+\frac{2Q(\bar{m}_{2}C_{1}-Q\bar{m}_{2}r_{2}^{2})r_{2}^{2}}{r_{1}^{3}}+Ar_{1}=0\
,$ (74)

and

$\begin{split}&\ddot{r}_{2}+\frac{Q\bar{m}_{2}r_{2}\left(C_{1}-Q\bar{m}_{2}r_{2}^{2}\right)}{r_{1}^{2}}-\frac{Q\bar{m}_{1}r_{2}\left(C_{2}+Q\bar{m}_{1}r_{2}^{2}\right)}{r_{2}^{2}}+\frac{\left(C_{2}+Q\bar{m}_{1}r_{2}^{2}\right)^{2}}{r_{2}^{3}}+\bar{m}_{2}^{2}r_{2}\\\
&+2Q\left(Q\bar{m}_{1}^{2}r_{2}-\frac{\bar{m}_{2}C_{1}r_{2}}{r_{1}^{2}}+\frac{2Q\bar{m}_{2}^{2}r_{2}^{3}}{r_{1}^{2}}\right)+Ar_{2}=0\
.\end{split}$ (75)

These three equations of motion for $z_{0}$, $r_{1}$ and $r_{2}$ can also be
obtained from the following Lagrangian

$\begin{split}L_{NR}=&\frac{1}{2}\left(\dot{z}_{0}^{2}+\dot{r}_{1}^{2}+\dot{r}_{2}^{2}\right)-\frac{1}{2}\frac{\left(C_{1}-Q\bar{m}_{2}r_{2}^{2}\right)^{2}}{r_{1}^{2}}-\frac{1}{2}\frac{\left(C_{2}+Q\bar{m}_{1}r_{2}^{2}\right)^{2}}{r_{2}^{2}}-\frac{C_{0}^{2}}{8z_{0}^{2}}+\frac{1}{2}\left(\bar{m}_{1}^{2}r_{1}^{2}+\bar{m}_{2}^{2}r_{2}^{2}\right)\\\
&+\frac{A}{2}\left(r_{1}^{2}+r_{2}^{2}-1\right)+\frac{\tilde{A}}{2}z_{0}^{2}+\frac{1}{2}Qr_{2}^{2}\left(\frac{\bar{m}_{1}C_{2}}{r_{2}^{2}}-\frac{\bar{m}_{2}C_{1}}{r_{1}^{2}}+\frac{Q\bar{m}_{1}^{2}r_{2}^{2}}{r_{2}^{2}}+\frac{Q\bar{m}_{2}^{2}r_{2}^{2}}{r_{1}^{2}}\right)\
.\end{split}$ (76)

Therefore the Hamiltonian of the system is

$\begin{split}H_{NR}=&\frac{1}{2}\left(\dot{z}_{0}^{2}+\dot{r}_{1}^{2}+\dot{r}_{2}^{2}\right)+\frac{1}{2}\frac{\left(C_{1}-Q\bar{m}_{2}r_{2}^{2}\right)^{2}}{r_{1}^{2}}+\frac{1}{2}\frac{\left(C_{2}+Q\bar{m}_{1}r_{2}^{2}\right)^{2}}{r_{2}^{2}}+\frac{C_{0}^{2}}{8z_{0}^{2}}-\frac{1}{2}\left(\bar{m}_{1}^{2}r_{1}^{2}+\bar{m}_{2}^{2}r_{2}^{2}\right)\\\
&-\frac{A}{2}\left(r_{1}^{2}+r_{2}^{2}-1\right)-\frac{\tilde{A}}{2}z_{0}^{2}-\frac{1}{2}Qr_{2}^{2}\left(\frac{\bar{m}_{1}C_{2}}{r_{2}^{2}}-\frac{\bar{m}_{2}C_{1}}{r_{1}^{2}}+\frac{Q\bar{m}_{1}^{2}r_{2}^{2}}{r_{2}^{2}}+\frac{Q\bar{m}_{2}^{2}r_{2}^{2}}{r_{1}^{2}}\right)\
.\end{split}$ (77)

From these two expressions it is obvious that the forms of both the Lagrangian
and Hamiltonian are in agreement with those of the one-dimensional Neumann-
Rosochatius system, only with extra terms due to the presence of mixed flux in
the background.

The Uhlenbeck integrals of motion may be obtained in a similar way as used in
the case of the spinning string ansatz. These yield the expressions as

$\begin{split}\bar{I_{a}}=&\frac{1}{\bar{m}_{1}^{2}-\bar{m}_{2}^{2}}\left(r_{1}\dot{r}_{2}-\dot{r}_{1}r_{2}\right)^{2}+\frac{2}{\bar{m}_{1}^{2}-\bar{m}_{2}^{2}}\left[\left(\frac{C_{1}-Qr_{2}^{2}\bar{m}_{2}}{r_{1}}\right)^{2}+\left(\frac{C_{2}+Qr_{2}^{2}\bar{m}_{1}}{r_{2}}\right)^{2}\right]\\\
&-\frac{2}{\bar{m}_{1}^{2}-\bar{m}_{2}^{2}}\left[\left(1+\frac{2Q^{2}r_{2}^{2}}{r_{1}^{2}}\right)\left(\bar{m}_{1}^{2}r_{1}^{2}+\bar{m}_{2}^{2}r_{2}^{2}\right)+2Qr_{2}^{2}\left(\frac{C_{1}\bar{m}_{2}}{r_{1}^{2}}-\frac{C_{2}\bar{m}_{1}}{r_{2}^{2}}\right)\right]\\\
&+\frac{1}{\bar{m}_{1}^{2}-\bar{m}_{2}^{2}}\left(\frac{C_{1}^{2}}{r_{1}^{2}}+\frac{C_{2}^{2}r_{1}^{2}}{r_{2}^{2}}\right)\
.\end{split}$ (78)

The two Virasoro constraints $G_{\tau\tau}+G_{\sigma\sigma}=0$ and
$G_{\tau\sigma}=G_{\sigma\tau}=0$ can be calculated as

$\displaystyle\dot{r}_{1}^{2}+\dot{r}_{2}^{2}+\dot{f}_{1}^{2}r_{1}^{2}+\dot{f}_{2}^{2}r_{2}^{2}+\bar{m}_{1}^{2}r_{1}^{2}+\bar{m}_{2}^{2}r_{2}^{2}=\dot{z}_{0}^{2}+z_{0}^{2}\dot{h}_{0}^{2}\
,$ (79a)
$\displaystyle\bar{m}_{1}r_{1}^{2}\dot{f}_{1}+\bar{m}_{2}r_{2}\dot{f}_{2}=0$
(79b)

respectively.

###  3.2 Solutions for $r_{1}$ and $r_{2}$ and string profile

Here we use the same procedure for finding the pulsating solutions to this
integrable system by choosing ellipsoidal coordinates keeping the conditions
as mentioned in section 2.3 intact. In case of pulsating string the
ellipsoidal coordinates are taken to be the functions of $\tau$ only and the
roots of the equation

$\frac{r_{1}^{2}}{\zeta-\bar{m}_{1}^{2}}+\frac{r_{2}^{2}}{\zeta-\bar{m}_{2}^{2}}=0\
,$ (80)

may be found in terms of $\zeta$. The time derivative of this equation gives

$\left(r_{1}\dot{r}_{2}-\dot{r}_{1}r_{2}\right)^{2}=\frac{\dot{\zeta}^{2}}{4(\bar{m}_{1}^{2}-\zeta)(\zeta-\bar{m}_{2}^{2})}\
.$ (81)

and also gives $r_{1}^{2}$ and $r_{2}^{2}$ as

$r_{1}^{2}=\frac{\bar{m}_{1}^{2}-\zeta}{\bar{m}_{1}^{2}-\bar{m}_{2}^{2}},~{}~{}r_{2}^{2}=\frac{\zeta-\bar{m}_{2}^{2}}{\bar{m}_{1}^{2}-\bar{m}_{2}^{2}}\
.$ (82)

Solving equation (78) for $\dot{\zeta}^{2}$ we can get a similar solution
containing 3rd order polynomial exactly like the case for rotating string and
it is given as

$\dot{\zeta}^{2}=-4P_{3}(\zeta),~{}~{}P_{3}(\zeta)=2(1-Q)\prod_{i=1}^{3}\left(\zeta-\zeta_{i}\right)\
.$ (83)

Using the same change of variables as given in equation (50) and with $\eta$
taken to be a function of $\tau$ only we can get

$\eta(\tau)=\hbox{cn}\left(\tau\sqrt{2\left(1-Q\right)\left(\zeta_{3}-\zeta_{1}\right)}+\tau_{0},k\right)\
$ (84)

We may choose integration constant $\tau_{0}$ and the elliptic modulus
$k=\frac{\zeta_{3}-\zeta_{2}}{\zeta_{3}-\zeta_{1}}$. From equation (80) we get
the expressions for $r_{1}^{2}$ and $r_{2}^{2}$ as

$r_{1}^{2}(\tau)=\frac{\zeta_{3}-\bar{m}_{1}^{2}}{\bar{m}_{2}^{2}-\bar{m}_{1}^{2}}+\frac{\zeta_{2}-\zeta_{3}}{\bar{m}_{2}^{2}-\bar{m}_{1}^{2}}\hbox{sn}^{2}\left(\tau\sqrt{2\left(1-Q\right)\left(\zeta_{3}-\zeta_{1}\right)},k\right)\
,$ (85)

and

$r_{2}^{2}(\tau)=\frac{\bar{m}_{2}^{2}-\zeta_{3}}{\bar{m}_{2}^{2}-\bar{m}_{1}^{2}}-\frac{\zeta_{2}-\zeta_{3}}{\bar{m}_{2}^{2}-\bar{m}_{1}^{2}}\hbox{sn}^{2}\left(\tau\sqrt{2\left(1-Q\right)\left(\zeta_{3}-\zeta_{1}\right)},k\right)\
.$ (86)

Here also the hidden constraint for spherical geometry is satisfied by the
expressions (85) and (86). To restrict the elliptic modulus $k$ within the
fundamental domain, i.e, $0<k<1$, we should choose the order of the
$\zeta_{i}$’s as $\zeta_{1}<\zeta_{2}<\zeta_{3}$. Similar to the case for
rotating string, here also $\zeta$ is assumed to cover $\frac{1}{4}$th of the
sphere with $r_{i}^{2}\geq 0$. This consequently binds the range of $\zeta$ as
$\bar{m}_{1}^{2}\leq\zeta\leq\bar{m}_{2}^{2}$ .

###  3.3 Conserved Charges and Dispersion Relation

The conserved quantities, in the case of pulsating string may be found from
the target space Lagrangian in the given background, as follows

$E=\frac{\partial\mathcal{L}}{\partial(\partial_{\tau}t)}=-\tau_{(m,n)}L^{2}z_{0}^{2}\dot{h}_{0},~{}~{}J_{a}=\frac{\partial\mathcal{L}}{\partial(\partial_{\tau}\phi_{a})}=\tau_{(m,n)}L^{2}\left(r_{a}^{2}\dot{f}_{a}+Qr_{2}^{2}m_{b}\epsilon_{ab}\right)\
.$ (87)

Here, by taking $z_{0}=1$ we get
$\dot{z}_{0}^{2}+z_{0}^{2}\dot{h}_{0}^{2}=\frac{E}{\tau_{(m,n)}L^{2}}$ with
$\dot{h}_{0}^{2}=\frac{\mathcal{E}^{2}}{z_{0}^{4}}$. We may express the
Virasoro constraint (79a) as a quartic equation of $r_{1}$ by substituting for
$\dot{h}_{0}$ and $\dot{f}_{a}$ in terms of conserved energy
$\mathcal{E}=\frac{E}{\tau_{(m,n)L^{2}}}$ and angular momenta
$\mathcal{J}_{a}=\frac{J_{a}}{\tau_{(m,n)L^{2}}}$. To make it simple, let us
consider the case $\mathcal{J}_{1}=\mathcal{J}_{2}$ and
$\bar{m}_{1}=-\bar{m}_{2}$. With the above, we get

$r_{1}\dot{r}_{1}=\sqrt{\left[\bar{m}_{1}^{2}(1+Q)-\mathcal{E}^{2}\right]r_{1}^{4}+\left[\mathcal{E}^{2}-\bar{m}_{1}^{2}(1+2Q)+2Q\bar{m}_{1}\mathcal{J}_{1}\right]r_{1}^{2}-(\mathcal{J}_{1}^{2}-\bar{m}_{1}^{2}+2Q\bar{m}_{1}\mathcal{J}_{1})}$
(88)

It is obvious from the above equation that the expression $\dot{r}_{1}^{2}$
assumes infinite values for both $r_{1}\rightarrow 0$ and
$r_{1}\rightarrow\infty$ and oscillates in between. Therefore it must have
some minimum value which can be derived to be
$\left[\mathcal{E}^{2}-\bar{m}_{1}^{2}(1+2Q)+2Q\bar{m}_{1}\mathcal{J}_{1}\right]$
at
$r_{1}^{2}=\pm\sqrt{\frac{\mathcal{J}_{1}^{2}-\bar{m}_{1}^{2}+2Q\bar{m}_{1}\mathcal{J}_{1}}{\bar{m}_{1}^{2}(1+Q)-\mathcal{E}^{2}}}$.
This yields the quantum oscillation number Park:2005kt  as

$\begin{split}&\mathcal{N}=\frac{N}{\tau_{(m,n)L^{2}}}=\oint
r_{1}\dot{r}_{1}dr_{1}\\\
&=\int_{0}^{\sqrt{a_{+}}}dr_{1}\sqrt{\left[\bar{m}_{1}^{2}(1+Q)-\mathcal{E}^{2}\right]r_{1}^{4}+\left[\mathcal{E}^{2}-\bar{m}_{1}^{2}(1+2Q)+2Q\bar{m}_{1}\mathcal{J}_{1}\right]r_{1}^{2}-(\mathcal{J}_{1}^{2}-\bar{m}_{1}^{2}+2Q\bar{m}_{1}\mathcal{J}_{1})}\\\
&=\int_{0}^{\sqrt{a_{+}}}dr_{1}\sqrt{\left(r_{1}^{2}-a_{-}\right)(a_{+}-r_{1}^{2})}\
,\end{split}$ (89)

where $a_{\pm}$ are the roots of the biquadratic function under the square
root. This quantum number being a large adiabatic quantity determines the
string level and it is generally a convenient way to compute the dispersion
relation for pulsating string in terms of $\mathcal{N}$ and $\mathcal{J}$ as
done in Minahan:2002rc ; Park:2005kt ; Pradhan:2013sja . Taking partial
derivative of the above equation with respect to $m_{1}$ we get

$\frac{\partial\mathcal{N}}{\partial\bar{m}_{1}}=\int_{0}^{\sqrt{a_{+}}}dr_{1}\left[\frac{\bar{m}_{1}(1+Q)r_{1}^{4}-\left[\bar{m}_{1}(1+2Q)-Q\mathcal{J}_{1}\right]r_{1}^{2}+(\bar{m}_{1}-Q\mathcal{J}_{1})}{\sqrt{\left(r_{1}^{2}-a_{-}\right)(a_{+}-r_{1}^{2})}}\right]\
.$ (90)

Expressing the integrals in terms of standard elliptic integrals we are left
with

$\begin{split}\frac{\partial\mathcal{N}}{\partial\bar{m}_{1}}=&\left(\bar{m}_{1}-Q\mathcal{J}_{1}\right)\frac{1}{\sqrt{a_{+}}}\textbf{K}(\epsilon)-\left[\bar{m}_{1}(1+2Q)-Q\mathcal{J}_{1}\right]\sqrt{a_{+}}\left[\textbf{K}(\epsilon)-\textbf{E}(\epsilon)\right]\\\
&-\frac{\bar{m}_{1}(1+Q)}{3}\sqrt{a_{+}}\left[a_{-}+2\left(a_{+}+a_{-}\right)\textbf{E}(\epsilon)-\left(a_{-}+2a_{+}\right)\textbf{K}(\epsilon)\right]\
,\end{split}$ (91)

where we have used $\epsilon=\frac{a_{-}}{a_{+}}$. The standard expansions of
elliptic integrals of first kind and second kind are respectively

$\displaystyle\textbf{K}\left(\epsilon\right)=\frac{\pi}{2}+\frac{\pi\epsilon}{8}+\frac{9\pi\epsilon^{2}}{128}+\frac{25\pi\epsilon^{3}}{512}+\frac{1225\pi\epsilon^{4}}{32768}+\mathcal{O}[\epsilon]^{5}\
,$ (92a)
$\displaystyle\textbf{E}\left(\epsilon\right)=\frac{\pi}{2}-\frac{\pi\epsilon}{8}-\frac{3\pi\epsilon^{2}}{128}-\frac{5\pi\epsilon^{3}}{512}-\frac{175\pi\epsilon^{4}}{32768}+\mathcal{O}[\epsilon]^{5}\
.$ (92b)

Substituting for these expansions in equation (90) and taking the short string
limit, we achieve

$\begin{split}&\frac{\partial\mathcal{N}}{\partial\bar{m}_{1}}=\sqrt{2}(\bar{m}_{1}-Q\mathcal{J}_{1})\left[\frac{89\pi}{128}+\frac{123\pi}{128}\frac{Q\mathcal{J}_{1}}{\bar{m}_{1}(1+Q)}+\mathcal{O}\left(\frac{\mathcal{J}_{1}^{2}}{\bar{m}_{1}^{2}}\right)\right]+\\\
&\frac{1}{\sqrt{2}}\\{\bar{m}_{1}(1+2Q)-Q\mathcal{J}_{1}\\}\left[\frac{11\pi}{32}+\frac{11\pi}{32}\frac{Q\mathcal{J}_{1}}{\bar{m}_{1}(1+Q)}+\mathcal{O}\left(\frac{\mathcal{J}_{1}^{2}}{\bar{m}_{1}^{2}}\right)\right]\\\
&+\bar{m}_{1}(1+Q)\left[\frac{19\pi}{48}-\frac{35\pi}{48}\frac{Q\mathcal{J}_{1}}{\bar{m}_{1}(1+Q)}+\mathcal{O}\left(\frac{\mathcal{J}_{1}^{2}}{\bar{m}_{1}^{2}}\right)\right]+\\\
&\sqrt{2}(\bar{m}_{1}-Q\mathcal{J}_{1})\left[\frac{89\pi}{256}\frac{Q}{\bar{m}_{1}^{2}(1+Q)^{2}}-\frac{267\pi}{128}\frac{Q^{3}\mathcal{J}_{1}}{\bar{m}_{1}^{3}(1+Q)^{4}}+\mathcal{O}\left(\frac{\mathcal{J}_{1}^{2}}{\bar{m}_{1}^{4}}\right)\right]\mathcal{E}^{2}+\\\
&\frac{1}{\sqrt{2}}\\{\bar{m}_{1}(1+2Q)-Q\mathcal{J}_{1}\\}\left[\frac{11\pi}{64}\frac{Q}{\bar{m}_{1}^{2}(1+Q)^{2}}+\frac{37\pi}{32}\frac{Q^{2}\mathcal{J}_{1}}{\bar{m}_{1}^{3}(1+Q)^{3}}+\mathcal{O}\left(\frac{\mathcal{J}_{1}^{2}}{\bar{m}_{1}^{4}}\right)\right]\mathcal{E}^{2}\\\
&+\bar{m}_{1}(1+Q)\left[-\frac{57\pi}{96}\frac{Q}{\bar{m}_{1}^{2}(1+Q)^{2}}-\frac{7\pi}{12}\frac{Q^{2}\mathcal{J}_{1}}{\bar{m}_{1}^{3}(1+Q)^{3}}+\mathcal{O}\left(\frac{\mathcal{J}_{1}^{2}}{\bar{m}_{1}^{4}}\right)\right]\mathcal{E}^{2}\\\
&+\sqrt{2}(\bar{m}_{1}-Q\mathcal{J}_{1})\left[-\frac{15\pi}{32}\frac{Q^{2}}{\bar{m}_{1}^{4}(1+Q)^{4}}+\frac{45\pi}{128}\frac{Q^{3}\mathcal{J}_{1}}{\bar{m}_{1}^{5}(1+Q)^{5}}+\mathcal{O}\left(\frac{\mathcal{J}_{1}^{2}}{\bar{m}_{1}^{6}}\right)\right]\mathcal{E}^{4}+\\\
&\frac{1}{\sqrt{2}}\\{\bar{m}_{1}(1+2Q)-Q\mathcal{J}_{1}\\}\left[\frac{5\pi}{16}\frac{Q^{2}}{\bar{m}_{1}^{4}(1+Q)^{4}}+\frac{9\pi}{32}\frac{Q^{3}\mathcal{J}_{1}}{\bar{m}_{1}^{5}(1+Q)^{5}}+\mathcal{O}\left(\frac{\mathcal{J}_{1}^{2}}{\bar{m}_{1}^{6}}\right)\right]\mathcal{E}^{4}\\\
&+\bar{m}_{1}(1+Q)\left[\frac{\pi}{12}\frac{Q^{2}}{\bar{m}_{1}^{4}(1+Q)^{4}}-\frac{11\pi}{48}\frac{Q^{3}\mathcal{J}_{1}}{\bar{m}_{1}^{5}(1+Q)^{5}}+\mathcal{O}\left(\frac{\mathcal{J}_{1}^{2}}{\bar{m}_{1}^{6}}\right)\right]\mathcal{E}^{4}+\mathcal{O}\left(\mathcal{E}^{6}\right)\
.\end{split}$ (93)

Integrating the equation (93) with respect to $\bar{m}_{1}$ and then expanding
$\mathcal{E}$ as a function of oscillation number $\mathcal{N}$ and angular
momentum $\mathcal{J}_{1}$ for small values of $\mathcal{J}_{1}$ we achieve

$\begin{split}\mathcal{E}^{2}=&\left(25.6704+1.8333\mathcal{N}\right)+\left(24.7749+1.2526\mathcal{N}\right)\mathcal{J}_{1}+\left(6.0259+0.0082\mathcal{N}\right)\mathcal{J}_{1}^{2}\\\
&-\left(0.311+0.0827\mathcal{N}\right)\mathcal{J}_{1}^{3}+\mathcal{O}[\mathcal{J}_{1}^{4}]\
,\end{split}$ (94)

where we have taken $Q=1$ and considered the limit for $\bar{m}_{1}\rightarrow
1$. With such limit in hand, small angular momenta, i.e.,
$\mathcal{J}_{1}\rightarrow 0$ yields from equation (94)

$\mathcal{E}\approx 1.354\sqrt{\mathcal{N}}\ .$

This result resembles the energy-oscillation number relation with small energy
limit for fundamental string pulsating in background with pure NSNS flux which
can be achieved by assuming $m=1,n=0$ which yields $q=1$ in equation (29).
Also it matches with the energy and oscillation number expansion upto its
leading order term in short string limit in Pradhan:2013sja  for fundamental
string pulsating in one plane.

##  4 Conclusion

We have investigated, in this paper, the features of $AdS_{3}\times S^{3}$
background in the presence of mixed flux in the light of classical integrable
model by considering $(m,n)$ string as a natural probe which is a suitable
near horizon $SL(2,\mathbb{Z})$ bound state of $NS5$-branes and fundamental
strings. We produced a plausible description of the dynamics of $(m,n)$ string
rotating and pulsating in our desired background by constructing the well-
known 1d Neumann-Rosochatius integrable system. The Lagrangian, Hamiltonian
and integrals of motion we computed are similar to those of integrable NR
models along with some deformations added as a consequence of the finite flux
in the background. It is henceforth a nice understanding of $(m,n)$ string, as
a natural probe, credibly supported by integrable deformation of NR mechanical
system. We have also elucidated the periodic circular-type string profile for
both the cases of closed $(m,n)$ strings rotating and pulsating with two
different angular momenta in $\mathbb{R}_{t}\times S^{3}$. The reduction of
Uhlenbeck integrals of motion into some first order differential equations has
provided such circular-type solutions of the deformed NR integrable model and
this in turn makes solving integrable equations of motion of the system rather
simplified. The systematic approach for Neumann-Rosochatius model has digged a
concrete way to end up with some general class of dispersion relations among
various conserved charges for both the pulsating and rotating $(m,n)$ string.
As an immediate solution we have computed the scaling relation for rotating
string with constant radii and two equal angular momenta as an expansion of
large $J$ for some arbitrary flux parameter. Also the small energy correction
to the scaling relation for pulsating string solution has been found out in
terms of oscillation number. Since our results show similar form as that of
fundamental string, an intriguing idea of a counterpart of our solutions
arises in the spirit of Pohlmeyer Reduced theory as achieved with the bosonic
part of the action for fundamental string moving in $R\times S^{3}$ in
Hoare:2013lja . The WZ interaction term in the $(m,n)$ string action includes
an overall flux parameter as $\left(mq+n\sqrt{1-q^{2}}\right)$ instead of $q$
which leads us to speculate that the Pohlmeyer Reduced theory for $(m,n)$
string in $R_{t}\times S^{3}$ might have a mass parameter $\mu$ deformed
accordingly, which for $n=0$ reduces to the case of F1 string. Moreover the
fact that $(m,n)$ string action in $AdS_{3}\times S^{3}$ with mixed three form
fluxes can be mapped to $(m^{{}^{\prime}},n^{{}^{\prime}})$ string action in
$AdS_{3}\times S^{3}$ with pure NSNS two form fluxes stems for the deformation
$\mu\rightarrow\mu\left(1-m^{{}^{\prime}2}q^{2}\right)$ in the corresponding
Pohlmeyer Reduced model. It would be interesting to study such PR counterparts
of string solutions with $(m,n)$ string as a natural probe to generalise the
analysis to the Sine-Gordon and Complex Sine-Gordon models.

Another promising idea arising as a future direction from our study is to
construct an NR model for rigidly rotating and pulsating $(m,n)$ probe string
in $AdS_{3}\times S^{3}\times S^{3}\times S^{1}$ background with various mixed
fluxes, which is another classically integrable counterpart of $AdS_{3}\times
S^{3}\times M^{4}$ background and has been previously studied by using Yangian
hidden symmetry. A general class of string solutions can also be studied by
reducing the $q$-deformed $AdS_{3}\times S^{3}$ background into 1d NR model.
We wish to return to these problems in future.

## References

  * (1) J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Int. J. Theor. Phys. 38, 1113-1133 (1999), [arXiv:hep-th/9711200 [hep-th]].
  * (2) S. Gubser, I. R. Klebanov and A. M. Polyakov, “Gauge theory correlators from noncritical string theory,” Phys. Lett. B 428, 105-114 (1998) [arXiv:hep-th/9802109 [hep-th]].
  * (3) E. Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. 2, 253-291 (1998) [arXiv:hep-th/9802150 [hep-th]].
  * (4) K. Zarembo, “Strings on Semisymmetric Superspaces,” JHEP 05, 002 (2010) [arXiv:1003.0465 [hep-th]].
  * (5) J. Minahan and K. Zarembo, “The Bethe ansatz for N=4 superYang-Mills,” JHEP 03, 013 (2003) [arXiv:hep-th/0212208 [hep-th]].
  * (6) N. Beisert, “The complete one loop dilatation operator of N=4 superYang-Mills theory,” Nucl. Phys. B 676, 3-42 (2004) [arXiv:hep-th/0307015 [hep-th]].
  * (7) N. Beisert and M. Staudacher, “The N=4 SYM integrable super spin chain,” Nucl. Phys. B 670, 439-463 (2003) [arXiv:hep-th/0307042 [hep-th]].
  * (8) A. Cagnazzo and K. Zarembo, “B-field in AdS(3)/CFT(2) Correspondence and Integrability,” JHEP 11, 133 (2012) [arXiv:1209.4049 [hep-th]].
  * (9) B. Hoare, A. Stepanchuk and A. Tseytlin, “Giant magnon solution and dispersion relation in string theory in $AdS_{3}$x$S^{3}$x$T^{4}$ with mixed flux,” Nucl. Phys. B 879, 318-347 (2014) [arXiv:1311.1794 [hep-th]].
  * (10) B. Hoare and A. Tseytlin, “Massive S-matrix of $AdS_{3}\times S^{3}\times T^{4}$ superstring theory with mixed 3-form flux,” Nucl. Phys. B 873, 395-418 (2013) [arXiv:1304.4099 [hep-th]].
  * (11) B. Hoare and A. Tseytlin, “On string theory on $AdS_{3}\times S^{3}\times T^{4}$ with mixed 3-form flux: tree-level S-matrix,” Nucl. Phys. B 873, 682-727 (2013) [arXiv:1303.1037 [hep-th]].
  * (12) M. Baggio and A. Sfondrini, “Strings on NS-NS Backgrounds as Integrable Deformations,” Phys. Rev. D 98, no.2, 021902 (2018) [arXiv:1804.01998 [hep-th]].
  * (13) A. Pittelli, A. Torrielli and M. Wolf, “Secret symmetries of type IIB superstring theory on $AdS_{3}\times S^{3}\times M^{4}$,” J. Phys. A 47, no.45, 455402 (2014) [arXiv:1406.2840 [hep-th]].
  * (14) A. Pittelli, “Yangian Symmetry of String Theory on $AdS_{3}\times S^{3}\times S^{3}\times S^{1}$ with Mixed 3-form Flux,” Nucl. Phys. B 935, 271-289 (2018) [arXiv:1711.02468 [hep-th]].
  * (15) D. M. Hofman and J. M. Maldacena, “Giant Magnons,” J. Phys. A 39, 13095-13118 (2006) [arXiv:hep-th/0604135 [hep-th]].
  * (16) M. Kruczenski, “Spiky strings and single trace operators in gauge theories,” JHEP 08, 014 (2005) [arXiv:hep-th/0410226 [hep-th]].
  * (17) J. A. Minahan, “Circular semiclassical string solutions on AdS(5) x S(5),” Nucl. Phys. B 648, 203-214 (2003) [arXiv:hep-th/0209047 [hep-th]].
  * (18) S. Gubser, I. Klebanov and A. M. Polyakov, “A Semiclassical limit of the gauge / string correspondence,” Nucl. Phys. B 636, 99-114 (2002) [arXiv:hep-th/0204051 [hep-th]].
  * (19) A. Khan and A. L. Larsen, “Improved stability for pulsating multi-spin string solitons,” Int. J. Mod. Phys. A 21, 133-150 (2006) [arXiv:hep-th/0502063 [hep-th]].
  * (20) A. Khan and A. L. Larsen, “Spinning pulsating string solitons in AdS(5) x S**5,” Phys. Rev. D 69, 026001 (2004) [arXiv:hep-th/0310019 [hep-th]]. Cite Article
  * (21) M. Smedback, “Pulsating strings on AdS(5) x S**5,” JHEP 07, 004 (2004) [arXiv:hep-th/0405102 [hep-th]].
  * (22) B. Chen and J. B. Wu, “Semi-classical strings in AdS(4) x CP**3,” JHEP 09, 096 (2008) [arXiv:0807.0802 [hep-th]].
  * (23) H. Dimov and R. C. Rashkov, “On the pulsating strings in AdS(4) x CP**3,” Adv. High Energy Phys. 2009, 953987 (2009) [arXiv:0908.2218 [hep-th]].
  * (24) J. Engquist, J. A. Minahan and K. Zarembo, “Yang-Mills duals for semiclassical strings on AdS(5) x S(5),” JHEP 11, 063 (2003) [arXiv:hep-th/0310188 [hep-th]].
  * (25) N. Beisert, J. A. Minahan, M. Staudacher and K. Zarembo, “Stringing spins and spinning strings,” JHEP 09, 010 (2003) [arXiv:hep-th/0306139 [hep-th]].
  * (26) N. Beisert, S. Frolov, M. Staudacher and A. A. Tseytlin, “Precision spectroscopy of AdS / CFT,” JHEP 10, 037 (2003) [arXiv:hep-th/0308117 [hep-th]].
  * (27) V. Kazakov, A. Marshakov, J. Minahan and K. Zarembo, “Classical/quantum integrability in AdS/CFT,” JHEP 05, 024 (2004) [arXiv:hep-th/0402207 [hep-th]].
  * (28) N. Beisert, V. Kazakov and K. Sakai, “Algebraic curve for the SO(6) sector of AdS/CFT,” Commun. Math. Phys. 263, 611-657 (2006) [arXiv:hep-th/0410253 [hep-th]].
  * (29) M. Kruczenski, J. Russo and A. A. Tseytlin, “Spiky strings and giant magnons on S**5,” JHEP 10, 002 (2006) [arXiv:hep-th/0607044 [hep-th]].
  * (30) G. Arutyunov, S. Frolov, J. Russo and A. A. Tseytlin, “Spinning strings in AdS(5) x S**5 and integrable systems,” Nucl. Phys. B 671, 3-50 (2003) [arXiv:hep-th/0307191 [hep-th]].
  * (31) G. Arutyunov, J. Russo and A. A. Tseytlin, “Spinning strings in AdS(5) x S**5: New integrable system relations,” Phys. Rev. D 69, 086009 (2004) [arXiv:hep-th/0311004 [hep-th]].
  * (32) K. Uhlenbeck, “Equivariant harmonic maps into spheres”, Lect. Notes Math. 949 39, (1982).
  * (33) O. Babelon, D. Bernard and M. Talon, “Introduction to Classical Integrable Systems,” doi:10.1017/CBO9780511535024
  * (34) J. Avan and M. Talon, “Alternative lax structures for the classical and quantum Neumann model,” Phys. Lett. B 268, 209-216 (1991)
  * (35) J. Avan and M. Talon, “Poisson Structure and Integrability of the Neumann-moser-uhlenbeck Model,” Int. J. Mod. Phys. A 5, 4477-4488 (1990)
  * (36) R. Ishizeki and M. Kruczenski, “Single spike solutions for strings on S**2 and S**3,” Phys. Rev. D 76, 126006 (2007) [arXiv:0705.2429 [hep-th]].
  * (37) A. Mosaffa and B. Safarzadeh, “Dual spikes: New spiky string solutions,” JHEP 08, 017 (2007) [arXiv:0705.3131 [hep-th]].
  * (38) H. Hayashi, K. Okamura, R. Suzuki and B. Vicedo, “Large Winding Sector of AdS/CFT,” JHEP 11, 033 (2007) [arXiv:0709.4033 [hep-th]].
  * (39) C. Ahn, P. Bozhilov and R. Rashkov, “Neumann-Rosochatius integrable system for strings on AdS(4) x CP**3,” JHEP 09, 017 (2008) [arXiv:0807.3134 [hep-th]].
  * (40) R. Hernández and J. M. Nieto, “Spinning strings in $AdS_{3}\times S^{3}$ with NS–NS flux,” Nucl. Phys. B 888, 236-247 (2014) [arXiv:1407.7475 [hep-th]].
  * (41) R. Hernandez and J. M. Nieto, “Elliptic solutions in the Neumann–Rosochatius system with mixed flux,” Phys. Rev. D 91, no.12, 126006 (2015) [arXiv:1502.05203 [hep-th]].
  * (42) G. Arutyunov, M. Heinze and D. Medina-Rincon, “Integrability of the $\eta$-deformed Neumann–Rosochatius model,” J. Phys. A 50, no.3, 035401 (2017) [arXiv:1607.05190 [hep-th]].
  * (43) R. Hernandez and J. M. Nieto, “Spinning strings in the $\eta$-deformed Neumann-Rosochatius system,” Phys. Rev. D 96, no.8, 086010 (2017) [arXiv:1707.08032 [hep-th]].
  * (44) R. Hernández, J. M. Nieto and R. Ruiz, “Pulsating strings with mixed three-form flux,” JHEP 04, 078 (2018) [arXiv:1803.03078 [hep-th]].
  * (45) A. Chakraborty and K. L. Panigrahi, “Neumann-Rosochatius system for strings in ABJ Model,” JHEP 12, 024 (2019) [arXiv:1909.12632 [hep-th]].
  * (46) J. H. Schwarz, “An SL(2,Z) multiplet of type IIB superstrings,” Phys. Lett. B 360, 13-18 (1995) [arXiv:hep-th/9508143 [hep-th]].
  * (47) A. A. Tseytlin, “Selfduality of Born-Infeld action and Dirichlet three-brane of type IIB superstring theory,” Nucl. Phys. B 469, 51-67 (1996) [arXiv:hep-th/9602064 [hep-th]].
  * (48) J. Lu and S. Roy, “An SL(2,Z) multiplet of type IIB super five-branes,” Phys. Lett. B 428, 289-296 (1998) [arXiv:hep-th/9802080 [hep-th]].
  * (49) Y. Lozano, “D-brane dualities as canonical transformations,” Phys. Lett. B 399, 233-242 (1997) [arXiv:hep-th/9701186 [hep-th]].
  * (50) M. Cederwall and P. Townsend, “The Manifestly Sl(2,Z) covariant superstring,” JHEP 09, 003 (1997) [arXiv:hep-th/9709002 [hep-th]].
  * (51) J. Klusoň, “(p, q)-five brane and (p, q)-string solutions, their bound state and its near horizon limit,” JHEP 06, 002 (2016) [arXiv:1603.05196 [hep-th]].
  * (52) J. Kluson, “(m, n)-String in (p, q)-string and (p, q)-five-brane background,” Eur. Phys. J. C 76, no.11, 582 (2016) [arXiv:1602.08275 [hep-th]].
  * (53) A. Banerjee, S. Biswas and R. R. Nayak, “D1 string dynamics in curved backgrounds with fluxes,” JHEP 04, 172 (2016) [arXiv:1601.06360 [hep-th]].
  * (54) S. P. Barik, M. Khouchen, J. Klusoň and K. L. Panigrahi, “$SL(2,Z)$ invariant rotating $(m,n)$ strings in $AdS_{3}\times S^{3}$ with mixed flux,” Eur. Phys. J. C 77, no.5, 298 (2017) [arXiv:1610.03402 [hep-th]].
  * (55) J. Klusoň, “$(m,n)$-String and D1-Brane in Stringy Newton-Cartan Background,” JHEP 04, 163 (2019) [arXiv:1901.11292 [hep-th]].
  * (56) R. Metsaev and A. A. Tseytlin, “Type IIB superstring action in AdS(5) x S**5 background,” Nucl. Phys. B 533, 109-126 (1998) [arXiv:hep-th/9805028 [hep-th]].
  * (57) A. Babichenko, B. Stefanski, Jr. and K. Zarembo, “Integrability and the AdS(3)/CFT(2) correspondence,” JHEP 03, 058 (2010) [arXiv:0912.1723 [hep-th]].
  * (58) O. Babelon and M. Talon, “Separation of variables for the classical and quantum Neumann model,” Nucl. Phys. B 379, 321-339 (1992) [arXiv:hep-th/9201035 [hep-th]].
  * (59) I. Y. Park, A. Tirziu and A. A. Tseytlin, “Semiclassical circular strings in AdS(5) and ’long’ gauge field strength operators,” Phys. Rev. D 71, 126008 (2005) [arXiv:hep-th/0505130 [hep-th]].
  * (60) P. M. Pradhan and K. L. Panigrahi, “Pulsating Strings With Angular Momenta,” Phys. Rev. D 88, no.8, 086005 (2013) [arXiv:1306.0457 [hep-th]].


(?) initialise their mappings.

  * •

The $\mathsf{Hyper}$ rule can be applied to a DL-clause of type DL1. All other
clauses participating in the inference are of the form (101), so the result is
of the form (101). Such an inference corresponds to the completion rules CR1
or CR2.

  * •

The $\mathsf{Hyper}$ rule can be applied to a DL-clause of type DL2. The other
clause participating in the inference is of the form (101), so the result is
of the form (102) or (103). Moreover, function symbol $f$ occurs in
$\mathcal{O}$ in exactly one pair of clauses DL2, and the $\mathsf{Hyper}$
rule is applied eagerly; thus, whenever $f$ occurs in a context in a clause of
the form (102), it also occurs in a clause of the form (103). Now the
$\mathsf{Succ}$ rule can be applied to the function symbol $f$, in which case
the cautious strategy thus returns a context whose core is of the form $B(x)$.
All of these inferences correspond to the completion rule CR3.

  * •

The $\mathsf{Hyper}$ rule can be applied to a DL-clause of type DL3. The two
other clauses participating in the inference are of the form (104) and (101),
so the result is of the form (104); the $\mathsf{Pred}$ rule can then be
applied to the latter clause, producing a clause of the form (101). Such a
pair of inferences corresponds to the completion rule CR4.

  * •

The $\mathsf{Hyper}$ rule can be applied to a DL-clause of type DL5. The other
clause participating in the inference is of the form (104), so the result is
of the form (104) as well; the $\mathsf{Pred}$ rule can then be applied to the
latter clause, producing a clause of the form (102). Such a pair of inferences
corresponds to the completion rule CR10.

One can show in an analogous way that each inference of the calculus by ? (?)
corresponds to one or more inferences of our calculus. Furthermore, it is
clear that our algorithm runs in polynomial time.

We next consider applying algorithm A1–A4 to $\mathcal{O}$ with the eager
strategy. One can show that the core of each context is of the form $A(x)$,
$R(y,x)$, or ${A(x)\wedge R(y,x)}$, and that context can contain clauses of
the form 101–107.

$\displaystyle\top$ $\displaystyle\rightarrow S_{2}(y,x)$ (106)
$\displaystyle\top$ $\displaystyle\rightarrow B(y)$ (107)

The proof is analogous to the case of the cautions strategy (without
correspondence to the completion rules) so we omit the details for the sake of
brevity. The only minor difference is that, if an application of the the
$\mathsf{Pred}$ to contexts $u$ and $v$ introduces a clause of the form (102)
in $u$, then the $\mathsf{Succ}$ rule does not become applicable to $u$ since
the precondition of the $\mathsf{Succ}$ rule is still satisfied by $v$. Thus,
the $\mathsf{Succ}$ rule never introduces contexts whose cores contain
conjunctions of binary atoms. Thus, if $\mathcal{O}$ contains $k_{1}$ unary
and $k_{2}$ binary predicates, the number of contexts is bounded by
$O(k_{1}\cdot k_{2})$, and each context can contain at most
${k_{1}+k_{2}+k_{1}\cdot k_{2}}$ clauses. All rules can be applied in
polynomial time, so the algorithm runs in polynomial time. ∎