††institutetext: 1 Department of Physics, Indian Institute of Technology
Kharagpur-721302, India††institutetext: 2Centre for Ocean, River, Atmosphere
and Land Sciences (CORAL), Indian Institute of Technology Kharagpur- 721302,
India

# Neumann-Rosochatius system for rotating strings in $AdS_{3}\times
S^{3}\times S^{3}\times S^{1}$ with flux

Adrita Chakraborty1, Rashmi R. Nayak2, Priyadarshini Pandit1, Kamal L.
Panigrahi1<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Strings on $AdS_{3}\times S^{3}\times S^{3}\times S^{1}$ with mixed flux
exhibit exact integrability. We wish to construct an integrable Neumann-
Rosochatius (NR) model of strings starting with the type IIB supergravity
action in $AdS_{3}\times S^{3}\times S^{3}\times S^{1}$ with pure NSNS flux.
We observe that the forms of the Lagrangian and the Uhlenbeck integrals of
motion of the considered system are NR-like with some suitable deformations
which eventually appear due to the presence of flux. We utilize the integrable
framework of the deformed NR model to analyze rigidly rotating spiky strings
moving only in $S^{3}\times S^{1}$. We further present some mathematical
speculations on the rounding-off nature of the spike in the presence of non-
zero angular momentum $J$ in $S^{1}$.

###### Keywords:

AdS/CFT duality, Integrability, Semicalssical strings

##  1 Introduction

The strongly coupled non-perturbative behaviour of gauge theories is
considered to be one of the most tangled long-standing problems in the field
of fundamental research. Since the last two decades, the duality
Maldacena:1997re ; Witten:1998qj ; Gubser:1998bc  between lower dimensional
strongly coupled conformal field theory and higher dimensional weakly coupled
gravity theory in various Anti-de Sitter (AdS) spacetimes has been greatly
felicitated in removing the problem of non-perturbative behaviour of gauge
theories, in the planar limit. The most studied example of the AdS/CFT
correspondence is the duality between $\mathcal{N}=4$ Super Yang-Mills (SYM)
theory and type IIB superstrings in $AdS_{5}\times S^{5}$ background. However,
in the past few years, numerous attempts has been taken in order to establish
the duality in different sub-sectors of the theories on both sides of the
correspondence. Semi-classical approximation being one of the most appropriate
method used extensively in order to find the spectrum in different target
space geometries. In this method we find the dispersion relation between
various conserved charges of semi-classical string dynamics in $AdS$ space
which in turn gives the anomalous dimensions of the dual boundary field theory
operators GKP ; Berenstein:2002jq ; Frolov:2003qc ; Tseytlin:2003ii ;
Engquist:2003rn ; spiky1 . In particular, the study of rigidly rotating string
in the large charge limit moving in some curved background in order to
establish the AdS/CFT holographic realization has gained much attention
Kruczenski:2008bs ; Ishizeki:2008tx ; Jevicki:2008mm ; Freyhult:2009bx ;
Kruczenski:2010xs ; Banerjee:2015nha ; Losi:2010hr ; Banerjee:2019puc ;
Biswas:2020wfn . In order to have a better grasp on the spectrum of
$AdS_{5}\times S^{5}$ superstring theories, various other class of classical
string solutions has also been studied to a great extent.
In establishing the AdS/CFT correspondence, planar integrability has also
played one of the most significant role on both side of the correspondence.
The first evidence in connection with this was found in Minahan:2002ve  where
the anomalous dimensions of one loop dilatation operators in the SU(2) sector
of planar $\mathcal{N}=4$ SYM were matched with the eigenvalues of
diagonalised Hamiltonian acting upon the integrable Heisenberg XXX spin chain
Beisert:2003jj ; Beisert:2003yb . Pioneered by this observation, integrability
helps in learning more powerful spectrum of the local operators in a vastly
simplified manner. The notion of integrability for the holographically
celebrated $AdS_{3}/CFT_{2}$ was initiated from Babichenko:2009dk .
Conventionally, there occur two plausible integrable set-ups in the case of
$AdS_{3}/CFT_{2}$ Sfondrini ; SST  namely the type IIB GS string construction
in $AdS_{3}\times S^{3}\times\mathcal{M}^{4}$, $\mathcal{M}^{4}\equiv T^{4}$
Ads3S3T4A ; BianchiHoare ; BothM4 ; completeworldsheet ; BMN mismatch ; BPS
or $S^{3}\times S^{1}$ Rughoonauth ; Borsato:2015mma ; Abbott ; Abbott:2013mpa
; EberhardtGopakumar ; Tong ; Pittelli:2017spf ; wulff ; review . The
underlying $\mathbb{Z}_{4}$ automorphism in these background $AdS_{3}\times
S^{3}_{1}\times S^{3}_{2}\times S^{1}$ represents a one parameter family of
inequivalent geometries that are labelled by the radii of the two sphere. Here
these radii of curvature of the two spheres are conveniently expressed in
terms of a family of one parameter $\alpha$ asGauntlett:1998fz ,

$\alpha\equiv\cos^{2}{\varphi}=\frac{R_{3}^{2}}{R_{1}^{2}}=1-\frac{R_{3}^{2}}{R_{2}^{2}},~{}~{}\frac{R_{3}^{2}}{R_{2}^{2}}\equiv\sin^{2}\varphi$
(1)

where $R_{1},R_{2}$ and $R_{3}$ are the radii of curvature for the geometries
$S^{3}_{1}$, $S^{3}_{2}$ and $AdS_{3}$ respectively. The parameter $\alpha$
assumes values within $0<\alpha<1$ when a suitable reality condition is
imposed on the exceptional Lie super-algebra
$\mathfrak{d}\left(2,1,;\alpha\right)$ which satisfies exactly the same
triangle equality (1) for its invariant bilinear forms. For this range of
$\alpha$, the bosonic sector requires the action to be that of a sigma model
in a target-space with group manifold $SU(1,1)\times SU(2)\times SU(2)$ which
is nothing but $AdS_{3}\times S^{3}\times S^{3}$. However, there are two
special limits for $\alpha$. One is $\alpha\rightarrow 1$ implying that radius
of one of the spheres blows up and that sphere being a flat space gets
compactified on a $T^{3}$, eventually representing the usual $AdS_{3}\times
S^{3}\times T^{4}$ background. In this case, the symmetry algebra shrinks up
to some abelian factors to $PSU(1,1|2)\times PSU(1,1|2)$. The other limit
demands $\alpha=\frac{1}{2}$, i.e., $\phi=\frac{\pi}{4}$ with which the
spheres $S^{3}_{1}$ and $S^{3}_{2}$ become identical and the algebra merges
into the classical $\mathfrak{osp}(4|2)$ superalgebra.

The existence of classical integrability of the $AdS_{3}\times S^{3}_{1}\times
S^{3}_{2}\times S^{1}$ was observed in Sundin:2012gc  in the presence of mixed
flux. Also, there are some recent findings on the integrability of the same by
using the integrable worldsheet S-matrix realizations, both for pure RR and
generalised mixed flux Borsato:2012ud . However, it is crucial to note that
unlike the $AdS_{3}\times S^{3}\times T^{4}$ background, we still do not have
a clear duality in $AdS_{3}\times S^{3}_{1}\times S^{3}_{2}\times S^{1}$
background. Reason behind such discrepancy is the lack of sufficient
consistency of supergravity approximation. For pure NSNS case, various
backgrounds having an $AdS_{3}$ factor consist of WZW prescription with
explicitly solvable $\mathfrak{sl}(2,\mathbb{R})$ symmetry algebra has been
studied. String theory on $AdS_{3}\times S^{3}_{1}\times S^{3}_{2}\times
S^{1}$ with pure NSNS flux can possess such WZW model description which
involves $\mathcal{N}=1$ affine algebras
$\mathfrak{sl}(2)_{k}\oplus\mathfrak{sl}(2)_{k_{1}}\oplus\mathfrak{su}(2)_{k_{2}}\oplus\mathfrak{u}(1)$deBoer:1999gea
. Here the relevant levels of the algebras are also related to the parameter
$\alpha$ of the relative spherical geometries and the relation goes as

$\alpha=\frac{k}{k_{1}}=\cos^{2}{\varphi},~{}~{}1-\alpha=\frac{k}{k_{2}}=\sin^{2}{\varphi}$

The corresponding dual gauge theory can thus be explored with the help of the
underlying chiral representation theory. In Gukov:2004ym , the protected BPS
spectrum achieved with pure NSNS flux has an indication toward the large
$\mathcal{N}=(4,4)$ superconformal symmetry only when the charge of the
fivebrane wrapped around one of the 3-spheres becomes unity. This is retrieved
by using the relation (1).

Our present work is devoted to the formulation of one-dimensional integrable
NR Model in the geometry of $AdS_{3}\times S^{3}\times S^{3}\times S^{1}$
supported by pure NSNS flux and having the fundamental string as the probe.
The NR model, as an integrable extension of earlier Neumann model, elucidates
the constrained motion of a harmonic oscillator on a unit sphere under the
influence of an extra centrifugal potential of the form $\frac{1}{x_{i}^{2}}$.
Being one of the earliest classical integrable model, NR model has been
studied as an exemplary ground to compute for large class of string solutions
in various integrable 10d $AdS_{d}\times X^{10-d}$ backgrounds with or without
flux by means of some generic NR ansatz Chakraborty:2020las ;
Chakraborty:2019gmt ; Hernandez:2017raj ; Arutyunov:2016ysi ;
Hernandez:2015nba ; Ahn:2008hj ; Hernandez:2014eta . The appearance of NR
model is approved to be quite robust for many deformed backgrounds as well as
for some probe strings other than fundamental ones. We explored here, the NR
construction by assuming the fundamental string as a probe and explicitly
deriving the Lagrangian, Hamiltonian and Uhlenbeck constants. The suitable
values of flux parameters are calculated accordingly from the equations of
motion of the type IIB supergravity and the equation (1).

The organisation of the article is as follows. In section 2 we revisit the
$AdS_{3}\times S^{3}\times S^{3}\times S^{1}$ supergravity solution with pure
NSNS flux and use various parametric relation along with the $\frac{1}{2}$-BPS
supersymmetry preserving relation between the radii of curvature, to verify
whether the flux parameters assume values consistent with those for pure NSNS
case. The geometrical constraints as well as the rotating string ansatz are
also presented in this section. Section 3 includes the Lagrangian and
Hamiltonian formulation for the NR Model along with the construction of
Uhlenbeck integrals of motion. We devote section 4 to the study of a class of
rotating string with explicit derivation of rigidly rotating $N$-spike string
profile moving in $S^{3}\times S^{3}$ Kruczenski:2006pk  in the presence of
non-zero angular momentum $J$ and winding number $m$ in $S^{3}$ Nayak:2021hrd
by solving the integrable NR model Lagrangian. Here we mathematically
speculate the rounding off nature of the spike appearing explicitly due to the
additional angular momentum $J$ in $S^{1}$, unlike the general spiky string
profile where the spikes end in cusps. We conclude in the last section with a
brief future outlook.

##  2 Gravity dual Background for $AdS_{3}\times S^{3}\times S^{3}$ with mixed
flux

The most general form of the bosonic part of type IIB Supergravity action in
the string frame with the fermionic terms dropped consistently, is given by ,

$\begin{split}S_{IIB}=\frac{1}{2\kappa^{2}_{10}}\Big{[}\int
d^{10}x\sqrt{-g}&\Big{(}e^{-2\phi}\left(R+4\partial_{M}\phi\partial^{M}\phi-\frac{1}{2}|H_{3}|^{2}\right)-\frac{1}{2}|F_{(1)}|^{2}-\frac{1}{2}|F_{(3)}|^{2}\\\
&-\frac{1}{4}|F_{(5)}|^{2}\Big{)}-\frac{1}{2}\int C_{(4)}\wedge H_{(3)}\wedge
F_{(3)}\Big{]}\end{split}$ (2)

where $\kappa_{10}^{2}=8\pi G_{10}$ with $G_{10}$ being 10D Newton’s
gravitational constant, $H_{3}$ and $F_{p+2}$ are respectively the 3-form NSNS
and $(p+2)$-form RR fluxes present in the background. These fluxes may be
expressed as,

$H_{3}=dB_{2},~{}~{}~{}F_{p+2}=dC_{p+1},~{}~{}~{}p=-1,1,3,$ (3)

with $B_{2}$ and $C_{p+1}$ are 2-form NSNS and $(p+1)$-form RR fluxes
respectively. In the above equations, the fluxes follow the general expression

$|K_{p}|^{2}=\frac{1}{p!}G^{M_{1}N_{1}}G^{M_{2}N_{2}}.....G^{M_{p}N_{p}}K_{M_{1}...M_{P}}K_{N_{1}...N_{p}}=\frac{1}{p!}K_{p}^{2},$
(4)

where, $K_{p}$ is a $p$-form field. Here we solve type IIB equations of motion
for $AdS_{3}\times S^{3}\times S^{3}\times S^{1}$ background with mixed RR and
NS-NS fluxes, where the subscript 1 and 2 on $S^{3}$ represents two 3 spheres
with different radii, $R_{1}$ and $R_{2}$ respectively.

The metric for $AdS_{3}\times S^{3}_{1}\times S^{3}_{2}\times S^{1}$ in terms
of global coordinates can be written as

$ds^{2}=ds^{2}_{AdS_{3}}+ds^{2}_{1}+ds^{2}_{2}+dw^{2},$ (5)

where $w$ is the coordinate along $S^{1}$ and

$\displaystyle
ds_{AdS_{3}}^{2}=\frac{R_{3}^{2}}{x_{2}^{2}}\left(-d\tau_{p}^{2}+dx_{1}^{2}+dx_{2}^{2}\right),$
(6a) $\displaystyle
ds_{1}^{2}=R_{1}^{2}\left[d\beta_{1}^{2}+\sin^{2}{\beta_{1}}d\beta_{2}^{2}+\cos^{2}{\beta_{1}}d\beta_{3}^{2}\right],$
(6b) $\displaystyle
ds_{2}^{2}=R_{2}^{2}\left[d\beta_{4}^{2}+\sin^{2}{\beta_{4}}d\beta_{5}^{2}+\cos^{2}{\beta_{4}}d\beta_{6}^{2}\right],$
(6c)

respectively represent the metric for a coordinate patch that covers a part of
the $AdS_{3}$ spacetime and the round metrics of the two 3-spheres present in
the background. By using the coordinate transformation as

$\displaystyle\tau_{p}=\frac{R_{3}\cosh{\rho}\sin{t}}{\sinh{\rho}\sin{\phi}-\cosh{\rho}\cos{t}},$
$\displaystyle
x_{1}=\frac{R_{3}\sinh{\rho}\cos{\phi}}{\sinh{\rho}\sin{\phi}-\cosh{\rho}\cos{t}},$
$\displaystyle x_{2}=\frac{R_{3}}{\sinh{\rho}\sin{\phi}-\cosh{\rho}\cos{t}}$

in (6a), a convenient form of the metric of full $AdS_{3}$ spacetime may be
obtained in terms of global cylindrical coordinates $(t,\rho,\phi)$ as

$ds^{2}_{AdS_{3}}=R_{3}^{2}\left[-\cosh^{2}{\rho}dt^{2}+d\rho^{2}+\sinh^{2}{\rho}d\phi^{2}\right].$
(8)

We will continue our work with the global cylindrical $AdS_{3}$ metric. The
associated pure NSNS 3-form background flux is considered as

$H=b_{0}\omega_{0}+b_{1}\omega_{1}+b_{2}\omega_{2},$ (9)

with volume fluxes

$\displaystyle\omega_{0}=vol(AdS_{3})=R_{3}^{3}\cosh{\rho}\sinh{\rho}dt\wedge
d\rho\wedge d\phi,$ (10a)
$\displaystyle\omega_{1}=vol(S^{3}_{1})=R_{1}^{3}\sin{\beta_{1}}\cos{\beta_{1}}d\beta_{1}\wedge
d\beta_{2}\wedge d\beta_{3},$ (10b)
$\displaystyle\omega_{2}=vol(S^{3}_{2})=R_{2}^{3}\sin{\beta_{4}}\cos{\beta_{4}}d\beta_{4}\wedge
d\beta_{5}\wedge d\beta_{6},$ (10c)

where $b_{i}$’s denote the flux coefficients satisfying the constraint

$\frac{1}{6}H_{MNP}H^{MNP}=-{b}_{0}^{2}+b_{1}^{2}+b_{2}^{2},$ (11)

followed from the equation of motion of $\phi$, where $\phi$ is taken to be
zero.
The Ricci scalar is

$\mathcal{R}=6\left(\frac{1}{R_{1}^{2}}+\frac{1}{R_{2}^{2}}-\frac{1}{R_{3}^{2}}\right).$

Using equation (3) the nonzero pure NSNS 2-form flux components associated to
the desired background are calculated from (9) as

$\displaystyle B_{t\phi}=b_{0}R_{3}^{2}\sinh^{2}{\rho},$ (12a) $\displaystyle
B_{\beta_{2}\beta_{3}}=-b_{1}R_{1}^{2}\cos^{2}{\beta_{1}},$ (12b)
$\displaystyle B_{\beta_{5}\beta_{6}}=-b_{2}R_{2}^{2}\cos^{2}{\beta_{4}}.$
(12c)

With all these parametric relations in hand,we are left with

$\frac{1}{R_{3}^{2}}=\frac{b_{0}^{2}}{4},~{}~{}\frac{1}{R_{1}^{2}}=\frac{b_{1}^{2}}{4},~{}~{}\frac{1}{R_{2}^{2}}=\frac{b_{2}^{2}}{4},$
(13)

which quite matches with the condition developed in from Einstein’s equation
by simplifying the stress-energy along with zero Ricci scalar.

Moreover, equation (1) will thus generate a relation between the family of one
parameter $\alpha$ and the flux parameters as

$\alpha=\frac{b_{1}^{2}}{b_{0}^{2}},~{}~{}1-\alpha=\frac{b_{2}^{2}}{b_{0}^{2}}.$
(14)

In the rest of the calculations, we will use this parameter $\alpha$ in
deriving the string solutions and corresponding scaling relations.

###  2.1 Constraints and embeddings for rotating string ansatz

We should have the embeddings for our background to satisfy the constraints
for AdS and spherical geometries. From equation (1) we can find the radii of
the spheres as

$R_{1}^{2}=\frac{R^{2}}{\alpha},~{}~{}R_{2}^{2}=\frac{R^{2}}{1-\alpha},~{}~{}\text{R=Radius
of $AdS_{3}$}$ (15)

Then the geometric constraints assume the forms

$-Y_{0}^{2}+Y_{1}^{2}+Y_{2}^{2}-Y_{3}^{2}=-R^{2},~{}~{}X_{1}^{2}+X_{2}^{2}+X_{3}^{2}+X_{4}^{2}=\frac{R^{2}}{\alpha},~{}~{}W_{1}^{2}+W_{2}^{2}+W_{3}^{2}+W_{4}^{2}=\frac{R^{2}}{1-\alpha}$
(16)

respectively for $AdS_{3}$, $S^{3}_{1}$ and $S^{3}_{2}$. Here $Y_{i}$’s,
$X_{i}$’s and $W_{i}$’s are the conventional notations for the embedding
coordinates of $AdS_{3}$ and the two three spheres $S^{3}_{1}$ and $S^{3}_{2}$
respectively. The followings are the embeddings satisfying the above
geometries

$\displaystyle
Y_{3}+iY_{0}=R\cosh{\rho}e^{it},~{}~{}Y_{1}+iY_{2}=R\sinh{\rho}e^{i\phi}\ ,$
(17a) $\displaystyle
X_{1}+iX_{2}=\frac{R}{\sqrt{\alpha}}\cos{\beta_{1}}e^{i\beta_{2}},~{}~{}X_{3}+iX_{4}=\frac{R}{\sqrt{\alpha}}\sin{\beta_{1}}e^{i\beta_{3}}\
,$ (17b) $\displaystyle
W_{1}+iW_{2}=\frac{R}{\sqrt{1-\alpha}}\cos{\beta_{4}}e^{i\beta_{5}},~{}~{}W_{3}+iW_{4}=\frac{R}{\sqrt{1-\alpha}}\sin{\beta_{4}}e^{i\beta_{6}}$
(17c)

We shall cast our interest in the string dynamics along the coordinates of the
spheres as well as those of $AdS_{3}$. Let us now define the embeddings
$Y_{i}$’s, $X_{i}$’s and $W_{i}$’s in terms of the local coordinates as
follows

$\displaystyle
Y_{3}+iY_{0}=Rz_{0}(\xi)e^{ig_{0}(\xi)+i\omega_{0}\tau},~{}~{}Y_{1}+iY_{2}=Rz_{1}(\xi)e^{ig_{1}(\xi)+i\omega_{1}\tau}\
,$ (18a) $\displaystyle
X_{1}+iX_{2}=\frac{R}{\sqrt{\alpha}}r_{1}(\xi)e^{if_{1}(\xi)+i\omega_{2}\tau},~{}~{}X_{3}+iX_{4}=\frac{R}{\sqrt{\alpha}}r_{2}(\xi)e^{if_{2}(\xi)+i\omega_{3}\tau}\
,$ (18b) $\displaystyle
W_{1}+iW_{2}=\frac{R}{\sqrt{1-\alpha}}r_{3}(\xi)e^{if_{3}(\xi)+i\omega_{4}\tau},~{}~{}W_{3}+iW_{4}=\frac{R}{\sqrt{1-\alpha}}r_{4}(\xi)e^{if_{4}(\xi)+i\omega_{5}\tau}\
.$ (18c)

Here, the local coordinates $z_{0},z_{1}$ and $r_{i}$’s with $i=1,2,3,4$ are
the functions of the parameter $\xi=\alpha_{1}\sigma+\alpha_{2}\tau$, $\tau$
and $\sigma$ being the coordinates of the worldsheet swept out by the motion
of the probe string under consideration while $\alpha_{1}$ and $\alpha_{2}$
are arbitrary constants. The constraints (16) then yield

$z_{0}^{2}-z_{1}^{2}=1$ (19)

for $AdS_{3}$ and

$\sum_{i=1,2}r_{i}^{2}=1,~{}~{}\sum_{j=3,4}r_{j}^{2}=1$ (20)

for two spheres in the desired background. Comparing the relation (17a) with
(18a) we have,

$\cosh{\rho}=z_{0},~{}~{}\sinh{\rho}=z_{1},~{}~{}t=\phi_{0},~{}~{}\phi=\phi_{1};~{}~{}\phi_{i}=g_{i}+\omega_{i}\tau,i=0,1.$
(21)

Similar comparison of relations (17b) and (17c) with (18b) and (18c)
respectively achieve

$\cos{\beta_{1}}=r_{1},~{}~{}\sin{\beta_{1}}=r_{2},~{}~{}\cos{\beta_{4}}=r_{3},~{}~{}\sin{\beta_{4}}=r_{4}$
(22)

satisfying the constraints (16) for the geometries of $S^{3}_{1}$ and
$S^{3}_{2}$. Along with these we also get

$\phi_{2}=f_{1}+\omega_{2}\tau=\beta_{2},~{}\phi_{3}=f_{2}+\omega_{3}\tau=\beta_{3},~{}\phi_{4}=f_{3}+\omega_{4}\tau=\beta_{5},~{}\phi_{5}=f_{4}+\omega_{5}\tau=\beta_{6}.$
(23)

##  3 Construction of Neumann-Rosochatius model

We wish to construct the one-dimensional NR integrable model for a probe
fundamental string in the given $AdS_{3}\times S^{3}_{1}\times S^{3}_{2}\times
S^{1}$ background in the presence of pure NSNS 2-form flux. To accomplish our
aim, we will now explore a systematic formulation of different pieces, like
Lagrangian, Hamiltonian and integrals of motion of our system by implementing
the above embeddings.

###  3.1 Target space Lagrangian and Hamiltonian

To start with, let us consider the Polyakov action of fundamental string with
WZ interaction term given as

$\displaystyle S=-\frac{T}{2}\int d\tau
d\sigma\sqrt{-\gamma}\gamma^{\alpha\beta}G_{MN}\partial_{\alpha}X^{M}\partial_{\beta}X^{N}+\frac{T}{2}\int
d\sigma
d\tau\epsilon^{\alpha\beta}B_{MN}\partial_{\alpha}X^{M}\partial_{\beta}X^{N}\
,$ (24)

where $T$ is the string tension, $\gamma_{\alpha\beta}$ is the auxiliary
metric which represents the intrinsic geometry of the background and can be
written in conformal gauge as
$\gamma_{\alpha\beta}=\eta_{\alpha\beta}$=diag(-1,+1) and
$\epsilon^{\tau\sigma}=-\epsilon^{\sigma\tau}=1$. The 10-dimensional
background coordinates are denoted by the indices $M$, $N$ and the
2-dimensional worldsheet coordinates are denoted by the indices $\alpha$,
$\beta$. $G_{MN}$ and $B_{MN}$ represent the induced metric of the background
and the anti-symmetric pullback of the finite flux respectively.

$B_{t\phi}=b_{0}R^{2}z_{1}^{2},~{}~{}B_{\beta_{2}\beta_{3}}=-b_{1}\frac{R^{2}}{\alpha}r_{2}^{2},~{}~{}B_{\beta_{5}\beta_{6}}=-b_{2}\frac{R^{2}}{(1-\alpha)}r_{4}^{2}.$
(25)

The target space Lagragian obtained from the Polyakov action may be written as
(24)

$\begin{split}\mathcal{L}=&-\frac{T}{2}\Big{[}-\partial_{a}Y_{0}\partial^{a}Y_{0}^{*}+\partial_{a}Y_{1}\partial^{a}Y_{1}^{*}+\partial_{a}X_{1}\partial^{a}X_{1}^{*}+\partial_{a}X_{2}\partial^{a}X_{2}^{*}+\partial_{a}W_{1}\partial^{a}W_{1}^{*}\\\
&+\partial_{a}W_{2}\partial^{a}W_{2}^{*}\Big{]}-\frac{\Lambda}{2}\left(|Y_{0}|^{2}-|Y_{1}|^{2}-1\right)-\frac{\tilde{\Lambda}_{1}}{2}\left(|X_{1}|^{2}+|X_{2}|^{2}-1\right)\\\
&-\frac{\tilde{\Lambda}_{2}}{2}\left(|W_{1}|^{2}+|W_{2}|^{2}-1\right)+\frac{T}{2}\epsilon^{ab}B_{MN}\partial_{a}X^{M}\partial_{b}X^{N}.\end{split}$
(26)

Using equations (21), (22) and (23) and subsituting for the derivatives of the
coordinates in the Lagrangian (26), we get

$\begin{split}&\mathcal{L}=-\frac{TR^{2}}{2}\Big{[}(\alpha_{2}^{2}-\alpha_{1}^{2})(z_{0}^{\prime
2}-z_{1}^{\prime
2})+z_{0}^{2}(\alpha_{2}^{2}-\alpha_{1}^{2})\left(g_{0}^{\prime}+\frac{\alpha_{2}\omega_{0}}{\alpha_{2}^{2}-\alpha_{1}^{2}}\right)^{2}-\frac{\alpha_{1}^{2}z_{0}^{2}\omega_{0}^{2}}{\alpha_{2}^{2}-\alpha_{1}^{2}}+\frac{\alpha_{1}^{2}z_{1}^{2}\omega_{1}^{2}}{\alpha_{2}^{2}-\alpha_{1}^{2}}\\\
&-z_{1}^{2}(\alpha_{2}^{2}-\alpha_{1}^{2})\left(g_{1}^{\prime}+\frac{\alpha_{2}\omega_{1}}{\alpha_{2}^{2}-\alpha_{1}^{2}}\right)^{2}\Big{]}-\frac{\tilde{\Lambda}}{2}\left(g^{ab}z_{a}z_{b}+1\right)-\frac{TR^{2}}{2\alpha}\Big{[}(\alpha_{1}^{2}-\alpha_{2}^{2})(r_{1}^{\prime
2}+r_{2}^{\prime 2})\\\
&+\sum_{i=1}^{2}r_{i}^{2}(\alpha_{1}^{2}-\alpha_{2}^{2})\left(f_{i}^{\prime}-\frac{\alpha_{2}\omega_{i+1}}{(\alpha_{1}^{2}-\alpha_{2}^{2})}\right)^{2}-\sum_{i=1}^{2}\frac{\alpha_{1}^{2}r_{i}^{2}\omega_{i+1}^{2}}{(\alpha_{1}^{2}-\alpha_{2}^{2})}\Big{]}-\frac{\Lambda_{1}}{2}(r_{1}^{2}+r_{2}^{2}-1)\\\
&-\frac{TR^{2}}{2(1-\alpha)}\Big{[}(\alpha_{1}^{2}-\alpha_{2}^{2})(r_{3}^{\prime
2}+r_{4}^{\prime
2})+\sum_{p=3}^{4}r_{p}^{2}(\alpha_{1}^{2}-\alpha_{2}^{2})\left(f_{p}^{\prime}-\frac{\alpha_{2}\omega_{p+1}}{(\alpha_{1}^{2}-\alpha_{2}^{2})}\right)^{2}-\sum_{p=1}^{2}\frac{\alpha_{1}^{2}r_{p}^{2}\omega_{p+1}^{2}}{(\alpha_{1}^{2}-\alpha_{2}^{2})}\Big{]}\\\
&-\frac{\Lambda_{2}}{2}(r_{3}^{2}+r_{4}^{2}-1)+TR^{2}\alpha_{1}\bigg{[}-b_{1}\frac{r_{2}^{2}}{\alpha}\left(\omega_{2}f_{2}^{{}^{\prime}}-\omega_{3}f_{1}^{{}^{\prime}}\right)-b_{2}\frac{r_{4}^{2}}{(1-\alpha)}\left(\omega_{4}f_{4}^{{}^{\prime}}-\omega_{5}f_{3}^{{}^{\prime}}\right)\\\
&+b_{0}z_{1}^{2}\left(\omega_{0}g_{1}^{{}^{\prime}}-\omega_{1}g_{0}^{{}^{\prime}}\right)\bigg{]}.\end{split}$
(27)

We will now derive the equations of motion corresponding to various local
coordinates present in the target-space Lagragian (27) by means of Euler-
Lagrange equation.

Equations of motion for the embeddings for $AdS_{3}$ :

The equations of motion for $z_{a}$ for $a=0,1$ can be derived as

$\left(\alpha_{1}^{2}-\alpha_{2}^{2}\right)z_{0}^{{}^{\prime\prime}}=\left[\left(\alpha_{1}^{2}-\alpha_{2}^{2}\right)g_{0}^{{}^{\prime}2}-2\alpha_{2}\omega_{0}g_{0}^{{}^{\prime}}-\omega_{0}^{2}+\frac{\tilde{\Lambda}}{TR^{2}}\right]z_{0},$
(28)

and

$\left(\alpha_{1}^{2}-\alpha_{2}^{2}\right)z_{1}^{{}^{\prime\prime}}=\left[\left(\alpha_{1}^{2}-\alpha_{2}^{2}\right)g_{1}^{{}^{\prime}2}-2\alpha_{2}\omega_{1}g_{1}^{{}^{\prime}}-\omega_{1}^{2}+2\alpha_{1}b_{0}\left(\omega_{0}g_{1}^{{}^{\prime}}-\omega_{1}g_{0}^{{}^{\prime}}\right)-\frac{\tilde{\Lambda}}{TR^{2}}\right]z_{1},$
(29)

where the angular coordinates $g_{a}$’s satisfy the relations

$g_{0}^{{}^{\prime}}=\frac{1}{\alpha_{1}^{2}-\alpha_{2}^{2}}\left[\frac{C_{0}}{z_{0}^{2}}+\alpha_{2}\omega_{0}+\alpha_{1}\omega_{1}b_{0}\frac{z_{1}^{2}}{z_{0}^{2}}\right],$
(30)

and

$g_{1}^{{}^{\prime}}=\frac{1}{\alpha_{1}^{2}-\alpha_{2}^{2}}\left[\alpha_{2}\omega_{1}+\alpha_{1}b_{0}\omega_{0}-\frac{C_{1}}{z_{1}^{2}}\right],$
(31)

$C_{0}$ and $C_{1}$ being the integration constants. Substituting the
expressions of $g_{0}^{{}^{\prime}}$ and $g_{1}^{{}^{\prime}}$ in the
Lagragian we get

$\begin{split}&\mathcal{L}_{AdS_{3}}=-\frac{TR^{2}}{2}\left(\alpha_{1}^{2}-\alpha_{2}^{2}\right)\Bigg{[}-z_{0}^{{}^{\prime}2}+z_{1}^{{}^{\prime}2}-\frac{\left(C_{0}+\alpha_{1}\omega_{1}b_{0}z_{1}^{2}\right)^{2}}{\left(\alpha_{1}^{2}-\alpha_{2}^{2}\right)^{2}z_{0}^{2}}+\frac{\left(\alpha_{1}\omega_{0}b_{0}z_{1}^{2}-C_{1}\right)^{2}}{\left(\alpha_{1}^{2}-\alpha_{2}^{2}\right)^{2}z_{1}^{2}}\\\
&+\frac{\alpha_{1}^{2}}{\left(\alpha_{1}^{2}-\alpha_{2}^{2}\right)^{2}}{\left(\omega_{0}^{2}z_{0}^{2}-\omega_{1}^{2}z_{1}^{2}\right)}\Bigg{]}.+\frac{TR^{2}\alpha_{1}b_{0}z_{1}^{2}}{\left(\alpha_{1}^{2}-\alpha_{2}^{2}\right)^{2}}\Big{(}\alpha_{1}b_{0}\omega_{0}^{2}-\frac{\omega_{0}C_{1}}{z_{1}^{2}}-\frac{\omega_{1}C_{0}}{z_{0}^{2}}-\frac{\alpha_{1}b_{0}\omega_{1}^{2}z_{1}^{2}}{z_{0}^{2}}\Big{)}\\\
&-\frac{\tilde{\Lambda}}{2}\left(g^{ab}z_{a}z_{b}+1\right)\end{split}$ (32)

Equations of motion for the embeddings for $S^{3}_{1}$ :

The equations of motion for $r_{i}$ for $i=1,2$ can be derived as

$r_{1}^{{}^{\prime\prime}}=\left[f_{1}^{{}^{\prime}2}-\frac{\left(\omega_{2}^{2}+2\alpha_{2}\omega_{2}f_{1}^{{}^{\prime}}\right)}{(\alpha_{1}^{2}-\alpha_{2}^{2})}+\frac{\alpha\Lambda_{1}}{TR^{2}(\alpha_{1}^{2}-\alpha_{2}^{2})}\right]r_{1},$
(33)
$r_{2}^{{}^{\prime\prime}}=\left[f_{2}^{{}^{\prime}2}-\frac{\left(\omega_{3}^{2}+2\alpha_{2}\omega_{3}f_{2}^{{}^{\prime}}\right)}{(\alpha_{1}^{2}-\alpha_{2}^{2})}+\frac{2\alpha_{1}b_{1}}{(\alpha_{1}^{2}-\alpha_{2}^{2})}\left(\omega_{2}f_{2}^{{}^{\prime}}-\omega_{3}f_{1}^{{}^{\prime}}\right)+\frac{\alpha\Lambda_{1}}{TR^{2}\left(\alpha_{1}^{2}-\alpha_{2}^{2}\right)}\right]r_{2}.$
(34)

Where the angular coordinates $f_{i}$’s satisfy the relations

$f_{1}^{{}^{\prime}}=\frac{1}{(\alpha_{1}^{2}-\alpha_{2}^{2})}\left[\frac{v_{1}}{r_{1}^{2}}+\alpha_{2}\omega_{2}+\frac{\alpha_{1}b_{1}\omega_{3}r_{2}^{2}}{r_{1}^{2}}\right]$
(35)

and

$f_{2}^{{}^{\prime}}=\frac{1}{(\alpha_{1}^{2}-\alpha_{2}^{2})}\left[\frac{v_{2}}{r_{2}^{2}}+\alpha_{2}\omega_{3}-\alpha_{1}b_{1}\omega_{2}\right]$
(36)

Here $v_{i}$’s are suitable constants for integration. Substituting
$f_{1}^{{}^{\prime}}$ and $f_{2}^{{}^{\prime}}$ will give us the Lagrangian
for $S^{3}_{1}$ as

$\begin{split}\mathcal{L}_{S^{3}_{1}}=&-\frac{TR^{2}}{2\alpha}(\alpha_{1}^{2}-\alpha_{2}^{2})\sum_{i=1}^{2}\left[r_{i}^{{}^{\prime}2}+\frac{r_{i}^{-2}}{(\alpha_{1}^{2}-\alpha_{2}^{2})^{2}}\left(v_{i}+b_{1}\alpha_{1}r_{2}^{2}\epsilon^{ij}\omega_{j+1}\right)^{2}-\frac{\alpha_{1}^{2}\omega^{2}_{i+1}r_{i}^{2}}{(\alpha_{1}^{2}-\alpha_{2}^{2})^{2}}\right]\\\
&+\frac{TR^{2}\alpha_{1}r_{2}^{2}b_{1}}{\alpha(\alpha_{1}^{2}-\alpha_{2}^{2})}\sum_{i=1}^{2}\left(\epsilon^{ij}\frac{v_{i}\omega_{j+1}}{r_{i}^{2}}+\frac{\alpha_{1}b_{1}\omega_{3}^{2}r_{2}^{2}}{r_{1}^{2}}+b_{1}\alpha_{1}\omega_{2}^{2}\right)-\frac{\Lambda_{1}}{2}\left(\sum_{i=1}^{2}r_{i}^{2}-1\right)\end{split}$
(37)

Equations of motion for the embeddings for $S^{3}_{2}$ :

The equations of motion for $r_{p}$ for $p=3,4$ can be derived as

$r_{3}^{{}^{\prime\prime}}=\left[f_{3}^{{}^{\prime}2}-\frac{\left(\omega_{4}^{2}+2\alpha_{2}\omega_{4}f_{3}^{{}^{\prime}}\right)}{(\alpha_{1}^{2}-\alpha_{2}^{2})}+\frac{(1-\alpha)\Lambda_{2}}{TR^{2}(\alpha_{1}^{2}-\alpha_{2}^{2})}\right]r_{3}$
(38)

and

$r_{4}^{{}^{\prime\prime}}=\left[f_{4}^{{}^{\prime}2}-\frac{\left(\omega_{5}^{2}+2\alpha_{2}\omega_{5}f_{4}^{{}^{\prime}}\right)}{(\alpha_{1}^{2}-\alpha_{2}^{2})}+\frac{2\alpha_{1}b_{2}}{(\alpha_{1}^{2}-\alpha_{2}^{2})}\left(\omega_{4}f_{4}^{{}^{\prime}}-\omega_{5}f_{3}^{{}^{\prime}}\right)+\frac{(1-\alpha)\Lambda_{2}}{TR^{2}(\alpha_{1}^{2}-\alpha_{2}^{2})}\right]r_{4}$
(39)

whereas the angular coordinates $f_{i}$’s satisfy the relations

$f_{3}^{{}^{\prime}}=\frac{1}{(\alpha_{1}^{2}-\alpha_{2}^{2})}\left[\frac{v_{3}}{r_{3}^{2}}+\alpha_{2}\omega_{4}+\frac{\alpha_{1}b_{2}\omega_{5}r_{4}^{2}}{r_{3}^{2}}\right]$
(40)

and

$f_{4}^{{}^{\prime}}=\frac{1}{(\alpha_{1}^{2}-\alpha_{2}^{2})}\left[\frac{v_{4}}{r_{4}^{2}}+\alpha_{2}\omega_{5}-\alpha_{1}b_{2}\omega_{4}\right]$
(41)

The Lagrangian for $S^{3}_{2}$ is obtained with these expressions of $f_{p}$’s
as,

$\begin{split}\mathcal{L}_{S^{3}_{2}}&=-\frac{TR^{2}}{2(1-\alpha)}(\alpha_{1}^{2}-\alpha_{2}^{2})\sum_{p=3}^{4}\left[r_{p}^{{}^{\prime}2}+\frac{r_{p}^{-2}}{(\alpha_{1}^{2}-\alpha_{2}^{2})^{2}}\left(v_{p}+b_{2}\alpha_{1}r_{4}^{2}\epsilon^{pq}\omega_{q+1}\right)^{2}-\frac{\alpha_{1}^{2}\omega^{2}_{p+1}r_{p}^{2}}{(\alpha_{1}^{2}-\alpha_{2}^{2})^{2}}\right]\\\
&+\frac{TR^{2}b_{2}\alpha_{1}r_{4}^{2}}{(1-\alpha)(\alpha_{1}^{2}-\alpha_{2}^{2})}\sum_{p=3}^{4}\left(\epsilon^{pq}\frac{v_{p}\omega_{q+1}}{r_{p}^{2}}+\frac{b_{2}\alpha_{1}\omega_{5}^{2}r_{4}^{2}}{r_{3}^{2}}+b_{2}\alpha_{1}\omega_{4}^{2}\right)-\frac{\Lambda_{2}}{2}\left(\sum_{p=3}^{4}r_{p}^{2}-1\right)\end{split}$
(42)

The total Lagrangian is (32), (37) and (42) as

$\mathcal{L}=\mathcal{L}_{AdS_{3}}+\mathcal{L}_{S^{3}_{1}}+\mathcal{L}_{S^{3}_{2}}.$
(43)

Here it is to be noted that all the finite fluxes turned over on different
geometries of the background are incorporated in the corresponding Lagrangians
in those geometries. Evidently, the terms along with those due to fluxes in
the Lagragian involves either $x^{2}$-type harmonic oscillator potential or
$\frac{1}{x^{2}}$-type centrifugal potential barrier, $x$ denoting the
position coordinates in respective geometries. Hence, it is obvious that the
total Lagrangian of a probe fundamental string in $AdS_{3}\times
S^{3}_{1}\times S^{3}_{2}$ background having pure NSNS 2-form flux quite
resembles with the Lagrangian of one-dimensional integrable NR model, except
having some integrable deformations added to it due to the finite flux.

###  3.2 Integrals of motion

One of the necessary conditions that an integrable system should consists of
is the presence of infinite tower of conserved quantities in involution,
namely the integrals of motion. K. Uhlenbeck first put forth the generalised
form of the integrals of motion along with some deformations for the one-
dimensional classical NR integrable system containing N number of harmonic
oscillators and these are expressed as

$I_{i}=\alpha_{1}^{2}x_{i}^{2}+\sum_{j\neq
i}\frac{1}{\omega_{i}^{2}-\omega_{j}^{2}}\left[\left(x_{i}x_{j}^{{}^{\prime}}-x_{j}x_{i}^{{}^{\prime}}\right)^{2}+\pi_{\alpha_{i}}^{2}\frac{x_{j}^{2}}{x_{i}^{2}}+\pi_{\alpha_{j}}^{2}\frac{x_{i}^{2}}{x_{j}^{2}}\right],\
$ (44)

which satisfy the relations
$\left\\{I_{i},I_{j}\right\\}=0~{}\forall~{}i,j\in\left\\{1,2,......,N\right\\}$
and $\sum_{i=1}^{N}I_{i}=\alpha_{1}^{2}$. In our study we will also check the
existence of such integrals of motion for our system to validate the
construction of integrable NR model. It is to be noted that the Uhlenbeck
integrals of motion would contain some deformations caused by the finite
fluxes in different geometry of our background. But these deformations should
be in such a way that the deformed integrals of motion should be conserved
quantities following the necessary conditions. To calculate the deformation we
write the extended expression of the integrals of motion as

$\bar{I}_{i}=\alpha_{1}^{2}x_{i}^{2}+\sum_{j\neq
i}\frac{1}{\omega_{i}^{2}-\omega_{j}^{2}}\left[\left(x_{i}x_{j}^{{}^{\prime}}-x_{j}x_{i}^{{}^{\prime}}\right)^{2}+\pi_{\alpha_{i}}^{2}\frac{x_{j}^{2}}{x_{i}^{2}}+\pi_{\alpha_{j}}^{2}\frac{x_{i}^{2}}{x_{j}^{2}}+2f\right],$
(45)

where $x_{i}$’s takes the form
$x_{i}(\xi)=r_{i}(\xi)e^{if_{i}(\xi)+i\omega_{i}\tau}$ in polar coordinates,
$\pi_{i}$’s are corresponding canonically conjugate momenta and $f$ is the
deformation which is independent of any terms like $x_{i}^{\prime}$ but
depends on the flux parameter $b_{0},b_{1},b_{2}$. We can calculate this
function by imposing the condition that $\bar{I}_{i}^{\prime}=0$. After some
quick algebra along with using the constraints and some equations of motions,
we find the the function to appear as total derivatives, integrating which we
can immediately get the deformed Uhlenbeck constant. Like the constants in the
absence of the flux, the deformed constants also satisfy the condition
$\sum_{i=1}^{N}\bar{I}_{i}=\alpha_{1}^{2}(1-q^{2})$.

Let us start with inserting the expression for $x_{i}$ and $p_{i}$ in equation
(45) and finding the integrals of motion for $AdS_{3}$. In this case we have
two integrals $\bar{I}_{1}$ and $\bar{I}_{2},$ but due to the constraint they
satisfy i.e, $\bar{I}_{1}-\bar{I}_{0}=-\alpha_{1}^{2}(1-q^{2})$ (for
$AdS_{3}$), we are left with a single independent constant,

$\bar{I}_{1}=\alpha_{1}^{2}z_{0}^{2}+\frac{1}{\omega_{1}^{2}-\omega_{0}^{2}}\left[(\alpha_{1}^{2}-\alpha_{2}^{2})^{2}(z_{1}z_{0}^{{}^{\prime}}-z_{0}z_{1}^{{}^{\prime}})^{2}+\frac{C_{0}^{2}z_{1}^{2}}{z_{0}^{2}}+\frac{C_{1}^{2}z_{0}^{2}}{z_{1}^{2}}+2f\right]$
(46)

We will now determine $f$ by imposing $\bar{I}_{1}^{{}^{\prime}}=0$ and using
the constraint $z_{1}^{2}-z_{0}^{2}=-1$ together with the equation
(28),(29),(30) and (31) as follows

$f^{{}^{\prime}}+\alpha_{1}^{2}b_{0}^{2}(\omega_{1}^{2}-\omega_{0}^{2})z_{0}z_{0}^{{}^{\prime}}+\frac{z_{0}^{{}^{\prime}}\left(2\alpha_{1}b_{0}\omega_{1}C_{0}-\alpha_{1}^{2}b_{0}^{2}\omega_{1}^{2}\right)}{z_{0}^{3}}=0$
(47)

We can see from the above expression that the differential equation in $f$ is
a total derivative which can be immediately integrated to give the form of the
deformed Uhlenbeck integrals of motion as

$\begin{split}\bar{I}_{1}=\alpha_{1}^{2}~{}(1-b_{0}^{2})~{}z_{0}^{2}+\frac{1}{\omega_{1}^{2}-\omega_{0}^{2}}&\Bigg{[}(\alpha_{1}^{2}-\alpha_{2}^{2})^{2}(z_{1}z_{0}^{{}^{\prime}}-z_{0}z_{1}^{{}^{\prime}})^{2}+\frac{(C_{0}-\alpha_{1}b_{0}\omega_{1})^{2}z_{1}^{2}}{z_{0}^{2}}+\frac{C_{1}^{2}z_{0}^{2}}{z_{1}^{2}}\Bigg{]},\end{split}$
(48)

where we have neglected the constant term appearing due to the presence of
flux, in order to write the expression in terms of complete square. Proceeding
in the similar fashion for the two spheres, we find the deformed Uhlenbeck
integrals of motion satisfying the constraint
$\bar{I}_{2}+\bar{I}_{3}=\alpha_{1}^{2}(1-q^{2})$ and
$\bar{I}_{3}+\bar{I}_{4}=\alpha_{1}^{2}(1-q^{2})$ respectively.

$\begin{split}\bar{I}_{2}=\alpha_{1}^{2}~{}(1-b_{1}^{2})~{}r_{1}^{2}+\frac{1}{\omega_{2}^{2}-\omega_{3}^{2}}&\Bigg{[}(\alpha_{1}^{2}-\alpha_{2}^{2})^{2}(r_{1}r_{2}^{{}^{\prime}}-r_{2}r_{1}^{{}^{\prime}})^{2}+\frac{(v_{1}+\alpha_{1}b_{1}\omega_{3})^{2}r_{2}^{2}}{r_{1}^{2}}+\frac{v_{2}^{2}r_{1}^{2}}{r_{2}^{2}}\Bigg{]},\end{split}$
(49)

and

$\begin{split}\bar{I}_{4}=\alpha_{1}^{2}~{}(1-b_{2}^{2})~{}r_{3}^{2}+\frac{1}{\omega_{4}^{2}-\omega_{5}^{2}}&\Bigg{[}(\alpha_{1}^{2}-\alpha_{2}^{2})^{2}(r_{3}r_{4}^{{}^{\prime}}-r_{4}r_{3}^{{}^{\prime}})^{2}+\frac{(v_{3}^{2}+\alpha_{1}b_{2}\omega_{5})^{2}r_{4}^{2}}{(r_{3}^{2}}+\frac{v_{4}^{2}r_{3}^{2}}{r_{4}^{2}}\Bigg{]}.\end{split}$
(50)

We can see in the above expressions for the deformed integrals of motions that
they satisfy the constraint i.e, $\sum_{i}I_{i}=\alpha_{1}^{2}$, only for pure
RR case. It may also be noted that the Hamiltonian associated with the NR
model can be expressed as the linear combination of integrals of motion of the
system. The background in our case have three different independent integrals
of motions. To be precise, for each geometry i.e.,
$AdS_{3},S_{3}^{1},S_{3}^{2}$, we have two coordinates, two phases, their
corresponding conjugate momenta and two constraints. So in order to solve the
system under consideration, classical Liouville integrability demands 3
Uhlenbeck constants to be in involution. Among these the canonically conjugate
momenta of two cyclic phases in the Lagrangian naturally gives two conserved
quantities. The remaining 2 are the deformed Uhlenbeck integrals of motion
found above. But the two Uhlenback integrals of motion satisfy the following
conditions, $\bar{I}_{1}-\bar{I}_{0}=-\alpha_{1}^{2}(1-q^{2})$ for $AdS_{3}$
and $\sum\bar{I}_{i}=\alpha_{1}^{2}(1-q^{2})$ for the two spheres. This
clearly tells that one of the two quantities is independent for each geometry.

##  4 Rotating string solutions

In this section we will consider the spiky string in $S_{1}^{3}$ with non-zero
angular momentum $J$ and winding $m$ in $S^{1}\subset S_{2}^{3}$. We start
with the Lagrangian for the whole background and then consider the part which
deals with the dynamics of the string. Here the differentiation is with
respect to $\xi$ which is a function of $\sigma,\tau$ in the form
$\xi=\alpha_{1}\sigma+\alpha_{2}\tau$. In the embedding coordinates, the
Lagrangian takes the following form:

$\begin{split}\mathcal{L}=&-\frac{T}{2}\Big{[}-\partial_{a}Y_{0}\partial^{a}Y_{0}^{*}+\partial_{a}Y_{1}\partial^{a}Y_{1}^{*}+\partial_{a}X_{1}\partial^{a}X_{1}^{*}+\partial_{a}X_{2}\partial^{a}X_{2}^{*}+\partial_{a}W_{1}\partial^{a}W_{1}^{*}\\\
&+\partial_{a}W_{2}\partial^{a}W_{2}^{*}+\Lambda\left(|Y_{0}|^{2}-|Y_{1}|^{2}-1\right)+\tilde{\Lambda}_{1}\left(|X_{1}|^{2}-|X_{2}|^{2}-1\right)\\\
&+\tilde{\Lambda}_{2}\left(|W_{1}|^{2}-|W_{2}|^{2}-1\right)\Big{]}+\frac{T}{2}\epsilon^{ab}B_{MN}\partial_{a}X^{M}\partial_{b}X^{N}\end{split}$
(51)

Using (18a,18b,18c) the lagrangian can be written as

$\begin{split}\mathcal{L}&=-\frac{TR^{2}}{2}\Big{[}(\alpha_{2}^{2}-\alpha_{1}^{2})(z_{0}^{\prime
2}-z_{1}^{\prime
2})+z_{0}^{2}(\alpha_{2}^{2}-\alpha_{1}^{2})\left(g_{0}^{\prime}+\frac{\alpha_{2}\omega_{0}}{\alpha_{2}^{2}-\alpha_{1}^{2}}\right)^{2}-\frac{\alpha_{1}^{2}z_{0}^{2}\omega_{0}^{2}}{\alpha_{2}^{2}-\alpha_{1}^{2}}+\frac{\alpha_{1}^{2}z_{1}^{2}\omega_{1}^{2}}{\alpha_{2}^{2}-\alpha_{1}^{2}}\\\
&-z_{1}^{2}(\alpha_{2}^{2}-\alpha_{1}^{2})\left(g_{1}^{\prime}+\frac{\alpha_{2}\omega_{1}}{\alpha_{2}^{2}-\alpha_{1}^{2}}\right)^{2}+\frac{\tilde{\Lambda}}{2}\left(g^{ab}z_{a}z_{b}+1\right)\Big{]}-\frac{TR^{2}}{2\alpha}\Big{[}(\alpha_{1}^{2}-\alpha_{2}^{2})(r_{1}^{\prime
2}+r_{2}^{\prime 2})\\\
&+\sum_{i=1}^{2}r_{i}^{2}(\alpha_{1}^{2}-\alpha_{2}^{2})\left(f_{i}^{\prime}-\frac{\alpha_{2}\omega_{i+1}}{(\alpha_{1}^{2}-\alpha_{2}^{2})}\right)^{2}-\sum_{i=1}^{2}\frac{\alpha_{1}^{2}r_{i}^{2}\omega_{i+1}^{2}}{(\alpha_{1}^{2}-\alpha_{2}^{2})}+\frac{\Lambda_{1}}{2}(r_{1}^{2}+r_{2}^{2}-1)\Big{]}\\\
&-\frac{TR^{2}}{2(1-\alpha)}\Big{[}(\alpha_{1}^{2}-\alpha_{2}^{2})(r_{3}^{\prime
2}+r_{4}^{\prime
2})+\sum_{p=3}^{4}r_{p}^{2}(\alpha_{1}^{2}-\alpha_{2}^{2})\left(f_{p}^{\prime}-\frac{\alpha_{2}\omega_{p+1}}{(\alpha_{1}^{2}-\alpha_{2}^{2})}\right)^{2}-\sum_{p=1}^{2}\frac{\alpha_{1}^{2}r_{p}^{2}\omega_{p+1}^{2}}{(\alpha_{1}^{2}-\alpha_{2}^{2})}\\\
&+\frac{\Lambda_{1}}{2}(r_{3}^{2}+r_{4}^{2}-1)\Big{]}+TR^{2}\alpha_{1}\Big{[}-b_{1}\frac{r_{2}^{2}}{\alpha}\left(\omega_{2}f_{2}^{{}^{\prime}}-\omega_{3}f_{1}^{{}^{\prime}}\right)-b_{2}\frac{r_{4}^{2}}{1-\alpha}\left(\omega_{4}f_{4}^{{}^{\prime}}-\omega_{5}f_{3}^{{}^{\prime}}\right)\\\
&+b_{0}z_{1}^{2}\left(\omega_{0}g_{1}^{{}^{\prime}}-\omega_{1}g_{0}^{{}^{\prime}}\right)\Big{]}.\end{split}$
(52)

The conformal constraints are

$-|\dot{Y_{0}}|^{2}+|\dot{Y}_{1}|^{2}+|\dot{X}_{i}|^{2}+|\dot{W}_{i}|^{2}-|{Y_{0}}^{\prime}|^{2}+|Y_{1}^{\prime}|^{2}+|X_{i}^{\prime}|^{2}+|W_{i}^{\prime}|^{2}=0,~{}~{}~{}~{}~{}~{}~{}~{}i=1,2$

and

$-\dot{Y}_{0}Y_{0}^{\prime*}+\dot{Y}_{1}Y_{1}^{\prime*}+\dot{X}_{i}X_{i}^{\prime*}+\dot{W}_{i}W_{i}^{\prime*}+c.c=0.$
(53)

Now we want the string profile for the spiky string spinning in $S_{1}^{3}$
along with the additional non-zero angular momentum $J$ in $S^{1}\subset
S_{2}^{3}$. This can be done by taking $Y_{1}$ and $W_{i}$ to be constant. The
radius of the two spheres are related by a parameter $\alpha$ as discussed
earlier. Note that the radius of $AdS_{3},S_{1}$ and $S_{2}$ are
$R,\frac{R}{\sqrt{\alpha}}$ and $\frac{R}{\sqrt{1-\alpha}}$ respectively. The
metric of the system under consideration is then given by

$ds^{2}=-dY_{0}d\bar{Y}_{0}+\sum_{i=1}^{2}dX_{i}d\bar{X}_{i}+dZd\bar{Z}.$ (54)

We consider the following NR ansatz

$\begin{split}Y_{0}+&iY_{3}=Re^{i\omega_{0}\tau},~{}~{}~{}~{}X_{1}+iX_{2}=\frac{R}{\sqrt{\alpha}}r_{1}(\xi)e^{i(f_{1}(\xi)+\omega_{2}\tau)},\\\
&X_{3}+iX_{4}=\frac{R}{\sqrt{\alpha}}r_{2}(\xi)e^{i(f_{2}(\xi)+\omega_{3}\tau)},~{}~{}~{}~{}Z_{1}+iZ_{2}=\frac{R}{\sqrt{1-\alpha}}e^{i(\psi(\xi)+\nu\tau)}.\end{split}$
(55)

With this ansatz, the string Lagrangian reduces to the following 1-dimensional
integrable system,

$\begin{split}\mathcal{L}=&-\frac{TR^{2}}{2}\omega_{0}^{2}-\frac{TR^{2}}{2\alpha}\Bigg{[}(\alpha_{1}^{2}-\alpha_{2}^{2})(r_{1}^{\prime
2}+r_{2}^{\prime
2})+\sum_{i=1}^{2}r_{i}^{2}(\alpha_{1}^{2}-\alpha_{2}^{2})\left(f_{i}^{\prime}-\frac{\alpha_{2}\omega_{i+1}}{(\alpha_{1}^{2}-\alpha_{2}^{2})}\right)^{2}\\\
&-\sum_{i=1}^{2}\frac{\alpha_{1}^{2}r_{i}^{2}\omega_{i+1}^{2}}{(\alpha_{1}^{2}-\alpha_{2}^{2})}+\frac{\Lambda_{1}}{2}(r_{1}^{2}+r_{2}^{2}-1)\Bigg{]}-\frac{TR^{2}}{2(1-\alpha)}\Bigg{[}-\frac{\alpha_{1}^{2}\nu^{2}}{(\alpha_{1}^{2}-\alpha_{2}^{2})}\\\
&+(\alpha_{1}^{2}-\alpha_{2}^{2})\left(\psi^{\prime}-\frac{\alpha_{2}\nu}{(\alpha_{1}^{2}-\alpha_{2}^{2})}\right)^{2}\Bigg{]}.\end{split}$
(56)

Here we have considered pure RR case. Also it can be seen from the ansatz
that, $t=\omega_{0}\tau$. We impose the periodicity conditions on the
parameters to find solutions of the closed string as

$r_{i}(\xi)=r_{i}(\xi+2\pi\alpha_{1}),\psi(\xi)=\psi(\xi+2\pi\alpha_{1})-2\pi
m,f_{i}(\xi)=f_{i}(\xi+2\pi\alpha_{1})-2\pi m_{i},~{}~{}i=1,2.$ (57)

where $m,m_{1}$ and $m_{2}$ are taken to be integers. The Euler-Lagrange
equations of motion for $f_{i}$ and $\psi$ are given by

$f_{1}^{{}^{\prime}}=\frac{1}{(\alpha_{1}^{2}-\alpha_{2}^{2})}\left[\frac{v_{1}}{r_{1}^{2}}+\alpha_{2}\omega_{2}\right],~{}f_{2}^{{}^{\prime}}=\frac{1}{(\alpha_{1}^{2}-\alpha_{2}^{2})}\left[\frac{v_{2}}{r_{2}^{2}}+\alpha_{2}\omega_{3}\right],~{}\psi^{{}^{\prime}}=\frac{v+\alpha_{2}\nu}{\alpha_{1}^{2}-\alpha_{2}^{2}}$
(58)

where $v_{1},v_{2},v$ are the constants of integration. Let us write the angle
in $S^{1}\subset S^{3}$ as $\theta$, so we can write

$Z=e^{i\theta},~{}~{}~{}~{}~{}~{}\theta=\psi+\nu\tau=\frac{v+\alpha_{2}\nu}{\alpha_{1}^{2}-\alpha_{2}^{2}}\xi.$
(59)

Now we define the winding number in $\theta$ to be of the form

$2\pi
m=\int_{0}^{2\pi\alpha_{1}}d\xi\psi{{}^{\prime}}=\frac{v+\alpha_{2}\nu}{\alpha_{1}^{2}-\alpha_{2}^{2}}\int
d\xi.$ (60)

We do not take any winding in $\phi_{3}$ direction. This gives the following
condition

$2\pi m_{2}=\int_{0}^{2\pi\alpha_{1}}d\xi
f_{2}^{{}^{\prime}}=\int_{0}^{2\pi\alpha_{1}}d\xi\left(\frac{v_{2}}{r_{2}^{2}}+\alpha_{2}\omega_{3}\right)=0.$
(61)

The expression for lagrangian with parameters $r_{1},r_{2}$ can be expressed
as

$\begin{split}L=-&\frac{TR^{2}}{2}\omega_{0}^{2}-\frac{TR^{2}}{2\alpha}\Bigg{[}(\alpha_{1}^{2}-\alpha_{2}^{2})(r_{1}^{\prime
2}+r_{2}^{\prime
2})+\frac{r_{1}^{-2}v_{1}^{2}}{\alpha_{1}^{2}-\alpha_{2}^{2}}-\sum_{i=1}^{2}\frac{\alpha_{1}^{2}r_{i}^{2}\omega_{i+1}^{2}}{(\alpha_{1}^{2}-\alpha_{2}^{2})}\\\
&+\frac{r_{2}^{-2}v_{2}^{2}}{\alpha_{1}^{2}-\alpha_{2}^{2}}+\frac{\Lambda_{1}}{2}(r_{1}^{2}+r_{2}^{2}-1)\Bigg{]}-\frac{TR^{2}}{2(1-\alpha)}\Bigg{[}\frac{v^{2}}{\alpha_{1}^{2}-\alpha_{2}^{2}}-\frac{\alpha_{1}^{2}\nu^{2}}{\alpha_{1}^{2}-\alpha_{2}^{2}}\Bigg{]}.\end{split}$
(62)

The corresponding Hamiltonian then becomes

$\begin{split}H=-\frac{TR^{2}}{2\alpha}&\Bigg{[}(\alpha_{1}^{2}-\alpha_{2}^{2})(r_{1}^{\prime
2}+r_{2}^{\prime
2})-\frac{r_{1}^{-2}v_{1}^{2}}{\alpha_{1}^{2}-\alpha_{2}^{2}}+\sum_{i=1}^{2}\frac{\alpha_{1}^{2}r_{i}^{2}\omega_{i+1}^{2}}{(\alpha_{1}^{2}-\alpha_{2}^{2})}-\frac{r_{2}^{-2}v_{2}^{2}}{\alpha_{1}^{2}-\alpha_{2}^{2}}\Bigg{]}\\\
&-\frac{TR^{2}}{2(1-\alpha)}\Bigg{[}-\frac{v^{2}}{\alpha_{1}^{2}-\alpha_{2}^{2}}+\frac{\alpha_{1}^{2}\nu^{2}}{\alpha_{1}^{2}-\alpha_{2}^{2}}\Bigg{]}.\end{split}$
(63)

Calculating the first and second Virasoro constraints we get,

$\begin{split}(\alpha_{1}^{2}+\alpha_{2}^{2})&\left(r_{1}^{\prime
2}+r_{2}^{\prime
2}\right)+(\alpha_{1}^{2}+\alpha_{2}^{2})\left(r_{1}^{2}f_{1}^{\prime
2}+r_{2}^{2}f_{2}^{\prime 2}+\psi^{\prime
2}\frac{\alpha}{1-\alpha}\right)+\omega_{2}^{2}r_{1}^{2}+\omega_{3}^{2}r_{2}^{2}\\\
&+\nu^{2}\frac{\alpha}{1-\alpha}+2\alpha_{2}\left(\omega_{2}f_{1}^{\prime}r_{1}^{2}+\omega_{3}f_{2}^{\prime}r_{2}^{2}+\nu\psi^{\prime}\frac{\alpha}{1-\alpha}\right)=\omega_{0}^{2}\alpha,\end{split}$
(64)

and

$\begin{split}\alpha_{2}\left(r_{1}^{\prime 2}+r_{2}^{\prime
2}+r_{1}^{2}f_{1}^{\prime 2}+r_{2}^{2}f_{2}^{\prime 2}+\frac{\psi^{\prime
2}\alpha}{1-\alpha}\right)=-\left(\omega_{2}f_{1}^{\prime}r_{1}^{2}+\omega_{3}f_{2}^{\prime}r_{2}^{2}+\nu\psi^{\prime}\frac{\alpha}{1-\alpha}\right).\end{split}$
(65)

Simplifying (65) and substituting in (64) we get,

$\begin{split}(\alpha_{1}^{2}-\alpha_{2}^{2})\left(r_{1}^{\prime
2}+r_{2}^{\prime 2}+r_{1}^{2}f_{1}^{\prime 2}+r_{2}^{2}f_{2}^{\prime
2}+\frac{\psi^{\prime
2}\alpha}{1-\alpha}\right)+\omega_{2}^{2}r_{1}^{2}+\omega_{3}^{2}r_{2}^{2}+\frac{\nu^{2}\alpha}{1-\alpha}=\omega_{0}^{2}\alpha\end{split}$
(66)

and

$\omega_{2}v_{1}+\omega_{3}v_{2}+\nu
v\frac{\alpha}{1-\alpha}=-\alpha_{2}\omega_{0}^{2}\alpha.$ (67)

The first constraint leads to the form of the Hamiltonian to be

$H=\omega_{0}^{2}\alpha\Big{(}\frac{\alpha_{1}^{2}+\alpha_{2}^{2}}{\alpha_{1}^{2}-\alpha_{2}^{2}}\Big{)}.$
(68)

The conserved charges can be written as

$E=TR^{2}\int\frac{\omega_{0}}{\alpha_{1}}d\xi~{},~{}~{}J=\frac{TR^{2}}{1-\alpha}\int\frac{d\xi}{\alpha_{1}}\left(\frac{v\alpha_{2}}{\alpha_{1}^{2}-\alpha_{2}^{2}}+\frac{\alpha_{1}^{2}\nu}{\alpha_{1}^{2}-\alpha_{2}^{2}}\right),$
(69)
$J_{1}=\frac{TR^{2}}{\alpha}\int\frac{d\xi}{\alpha_{1}}\Bigg{(}\frac{v_{1}\alpha_{2}}{\alpha_{1}^{2}-\alpha_{2}^{2}}+\frac{\omega_{2}\alpha_{1}^{2}r_{1}^{2}}{\alpha_{1}^{2}-\alpha_{2}^{2}}\Bigg{)},J_{2}=\frac{TR^{2}}{\alpha}\int\frac{d\xi}{\alpha_{1}}\Bigg{(}\frac{v_{2}\alpha_{2}}{\alpha_{1}^{2}-\alpha_{2}^{2}}+\frac{\omega_{3}\alpha_{1}^{2}r_{2}^{2}}{\alpha_{1}^{2}-\alpha_{2}^{2}}\Bigg{)},$
(70)

Now we will use the condition, $H$ being conserved along with the constraint
(68) and the relation $r_{1}^{2}+r_{2}^{2}=1$ in (63), we get

$\begin{split}\omega_{0}^{2}\alpha\left(\frac{1+\alpha_{2}^{2}}{1-\alpha_{2}^{2}}\right)=(1-\alpha_{2}^{2})&\frac{r_{2}^{\prime
2}}{1-r_{2}^{2}}-\frac{1}{1-\alpha_{2}^{2}}\left(\frac{v_{1}^{2}}{1-r_{2}^{2}}+\frac{v_{2}^{2}}{r_{2}^{2}}+\frac{\alpha
v^{2}}{1-\alpha}\right)\\\
&+\frac{1}{1-\alpha_{2}^{2}}\left(\omega_{2}^{2}+(\omega_{3}^{2}-\omega_{2}^{2})r_{2}^{2}+\frac{\alpha\nu^{2}}{1-\alpha}\right),\end{split}$
(71)

which can also be written as

$\begin{split}(1-\alpha_{2}^{2})^{2}r_{2}^{\prime
2}=\frac{1}{r_{2}^{2}}&\Bigg{[}\left((1+\alpha_{2}^{2})\omega_{0}^{2}\alpha-\omega_{2}^{2}-\frac{\alpha(\nu^{2}-v^{2})}{1-\alpha}\right)r_{2}^{2}(1-r_{2}^{2})\\\
&-(\omega_{3}^{2}-\omega_{2}^{2})r_{2}^{4}(1-r_{2}^{2})+v_{2}^{2}+(v_{1}^{2}-v_{2}^{2})r_{2}^{2}\Bigg{]}.\end{split}$
(72)

It can be seen above that the differential equation in $r_{2}$ has three
zeroes corresponding to the turning points $(r_{2}^{{}^{\prime}}=0)$. We want
to look for spiky string solution which demands solution with two turning
points. At large $r_{2}$ limit, the above equation describes a string which
does not reach the boundary under the condition
$\omega_{3}^{2}<\omega_{2}^{2}$. Looking at the expression for roots, we can
take one of the roots to be $r_{2}=1$ resulting in fixing the value of one of
the constants of motion i.e, $v_{1}=0$. Substituting the value of $v_{1}$, the
above equation reduces to

$(1-\alpha_{2}^{2})^{2}r_{2}^{\prime
2}=\frac{1-r_{2}^{2}}{r_{2}^{2}}\Bigg{[}\left((1+\alpha_{2}^{2})\omega_{0}^{2}\alpha-\omega_{2}^{2}-\frac{\alpha(\nu^{2}-v^{2})}{1-\alpha}\right)r_{2}^{2}-(\omega_{3}^{2}-\omega_{2}^{2})r_{2}^{4}+v_{2}^{2}\Bigg{]}.$
(73)

Here we can see that again we get two zeroes, where $r_{2}=1$ is a double
zero. Substituting this value of $r_{2}$ in the above equation, the right hand
side of the equation reduces to
$(1+\alpha_{2}^{2})\omega_{0}^{2}\alpha=\omega_{3}^{2}+\cfrac{\alpha(\nu^{2}-v^{2})}{1-\alpha}-v_{2}^{2}$
and using $v_{1}=0$ in equation (67), we find the value of $\alpha_{2}$ to be
$\left(\cfrac{-\omega_{3}v_{2}-\nu
v\frac{\alpha}{1-\alpha}}{\alpha\omega_{0}^{2}}\right)$. Putting the value of
$\alpha_{2}$ in the above equation we get a quadratic equation in
$\alpha\omega_{0}^{2}$, the roots of which is complicated and does not give
simplified solution to our equation. Hence we will drop this line of action
and take a different approach in order to solve the problem. We want to
express (71) in terms of variable $y$, where $y$ can be defined as

$y=\frac{1}{1-2r_{2}^{2}}=\frac{1}{\cos 2\beta_{1}}.$ (74)

Here we have used $r_{2}=\sin\beta_{1}$ where $\beta_{1}$ is the global
coordinate of $S^{3}$. Substituting the value of $r_{2}$, we get the equation
of the form

$\begin{split}(1-\alpha_{2}^{2})^{2}&y^{{}^{\prime}2}=2y\Bigg{[}4y^{2}v_{1}^{2}(y-1)+4y^{2}v_{2}^{2}(y+1)-\omega_{3}^{2}(y-1)^{2}(y+1)\\\
&-\omega_{2}^{2}(y+1)^{2}(y-1)+2y(y^{2}-1)\left(\alpha\omega_{0}^{2}(1+\alpha_{2}^{2})-\frac{\alpha(\nu^{2}-v^{2})}{1-\alpha}\right)\Bigg{]},\end{split}$
(75)

which can be written in terms of Polynomial as

$y^{\prime}=\frac{\sqrt{2yP(y)}}{1-\alpha_{2}^{2}},$ (76)

where the expression for the polynomial $P(y)$ is given as

$\begin{split}&P(y)=y^{3}\left[4v_{1}^{2}+4v_{2}^{2}-\omega_{3}^{2}-\omega_{2}^{2}+2\left(\alpha\omega_{0}^{2}(1+\alpha_{2}^{2})-\frac{\alpha(\nu^{2}-v^{2})}{1-\alpha}\right)\right]+\omega_{2}^{2}-\omega_{3}^{2}\\\
&+y^{2}\left[-4v_{1}^{2}+4v_{2}^{2}+\omega_{3}^{2}-\omega_{2}^{2}\right]+y\left[\omega_{2}^{2}+\omega_{3}^{2}-2\left(\alpha\omega_{0}^{2}(1+\alpha_{2}^{2})-\frac{\alpha(\nu^{2}-v^{2})}{1-\alpha}\right)\right]\end{split}$
(77)
$\equiv\left[4v_{1}^{2}+4v_{2}^{2}-\omega_{3}^{2}-\omega_{2}^{2}+2\alpha\left(\omega_{0}^{2}(1+\alpha_{2}^{2})-\frac{(\nu^{2}-v^{2})}{1-\alpha}\right)\right](y-y_{1})(y-y_{2})(y-y_{3}),$
(78)

where $y_{1},y_{2}$ and $y_{3}$ are considered as the three roots of the
polynomial $P(y)=0$. Consistent string solution demands all the roots of the
polynomial to be real. Here we can reduce the number of constants by
eliminating some of them in terms of the other using the constraint equations.
Let us take the polynomial $P(y)$ to have one negative and two positive real
roots. From the algebra of the third order polynomial, we can write the
product of the roots determined by the parameter as

$a\equiv
4v_{1}^{2}+4v_{2}^{2}-\omega_{3}^{2}-\omega_{2}^{2}+2\alpha\left(\omega_{0}^{2}(1+\alpha_{2}^{2})-\frac{(\nu^{2}-v^{2})}{1-\alpha}\right)=\frac{\omega_{3}^{2}-\omega_{2}^{2}}{y_{1}y_{2}y_{3}}$
(79)

Here we can see that $`a^{\prime}$ is negative provided some of the constants
in the expression i.e, $v_{1},v_{2},v,\alpha,\alpha_{2}$ are small. This
condition results in the polynomial $P(y)$ being positive between the two
positive roots taken. Now, we have to fix the two physical constants $y_{1}$
and $y_{2}$, in between which our polynomial is positive.

$P(y)=\frac{\omega_{3}^{2}-\omega_{2}^{2}}{y_{1}y_{2}y_{3}}(y-y_{1})(y-y_{2})(y-y_{3})=8v_{2}^{2}\frac{(y-y_{1})(y-y_{2})(y-y_{3})}{(1-y_{1})(1-y_{2})(1-y_{3})}.$
(80)

By using above expression, we can write $y_{1}$ in terms of $y_{2}$ and
$y_{3}$ as,

$y_{1}=-\frac{y_{2}y_{3}}{y_{2}+y_{3}+y_{2}y_{3}\frac{\omega_{2}^{2}+\omega_{3}^{2}-2\left(\alpha\omega_{0}^{2}(1+\alpha_{2}^{2})-\frac{\alpha(\nu^{2}-v^{2})}{1-\alpha}\right)}{\omega_{2}^{2}-\omega_{3}^{2}}}.$
(81)

The expressions for $v_{1}$ and $v_{2}$ in terms of the roots
$y_{1},y_{2},y_{3}$ are,

$v_{2}^{2}=\frac{\omega_{3}^{2}-\omega_{2}^{2}}{8}\frac{(1-y_{1})(1-y_{2})(1-y_{3})}{y_{1}y_{2}y_{3}},~{}~{}v_{1}^{2}=\frac{\omega_{3}^{2}-\omega_{2}^{2}}{8}\frac{(1+y_{1})(1+y_{2})(1+y_{3})}{y_{1}y_{2}y_{3}}$
(82)

Here we clearly see that for the range of the solution $-1\leq y_{1}\leq 0\leq
y_{2}\leq y_{3}\leq 1$, we get $v_{1}^{2}>0$ and $v_{2}^{2}>0$ which confirms
the consistency of our roots for the polynomial $P(y)$. Here in this article,
since we want to study the $N$-spike string, we need to glue together $2n$
number of integrals between $y_{2}$ and $y_{3}$. So we replace $\int d\xi$ by

$\int
d\xi=2n\int_{y_{2}}^{y_{3}}\frac{dy}{y^{\prime}}=\frac{2n(1-\alpha_{2}^{2})}{\sqrt{-2a}}F_{1},$
(83)

where $F_{1}$ is the integral which has been calculated in the appendix. Now
we will find the winding number $m$ defined in (60) as,

$m=\frac{v+\alpha_{2}\nu}{\pi\sqrt{-2a}}nF_{1}.$ (84)

We want to calculate the equation for $\nu$ which can be obtained by solving
for $v$ and using the second virasoro constraint (67) as

$\omega_{2}v_{1}+\omega_{3}v_{2}+\frac{\nu\alpha}{1-\alpha}\left(\frac{\pi
m\sqrt{-2a}}{nF_{1}}-\alpha_{2}\nu\right)=-\alpha_{2}\omega_{0}^{2}\alpha$
(85)

We can eliminate one of the constant by using the condition given in (61) as

$-2v_{2}F_{6}+\alpha_{2}\omega_{3}F_{1}=0.$ (86)

The above relation can be used to calculate $\alpha_{2}$ which in turn can be
substituted to find the winding number done later in this section.
The expression for conserved charges in terms of Jacobi Elliptic integrals are
as follows

$\begin{split}\frac{\alpha\pi\mathcal{J}_{1}}{n}&=\frac{v_{1}\alpha_{2}}{\sqrt{-2a}}F_{1}+\frac{\omega_{2}}{2\sqrt{-2a}}F_{3},~{}~{}~{}~{}~{}\frac{\alpha\pi\mathcal{J}_{2}}{n}=\frac{v_{2}\alpha_{2}}{\sqrt{-2a}}-\frac{\omega_{3}}{2\sqrt{-2a}}F_{2}\\\
&\frac{\pi\mathcal{E}}{n}=\frac{\omega_{0}(1-\alpha_{2}^{2})}{\sqrt{-2a}}F_{1},~{}~{}~{}~{}~{}\frac{(1-\alpha)\pi\mathcal{J}}{n}=\frac{(v\alpha_{2}+\nu)}{\sqrt{-2a}}F_{1}.\end{split}$
(87)

where $F_{1},F_{2},F_{3}$ are the elliptic integrals defined in the appendix
and

$E=2\pi T\mathcal{E},~{}~{}J=2\pi T\mathcal{J},~{}~{}J_{1}=2\pi
T\mathcal{J}_{1},~{}~{}J_{2}=2\pi T\mathcal{J}_{2}$

has been used in the above equations. Now we would like to speculate the
configuration of the rigid $N$-spike string when we generalise the motion of
the string to the addition of angular momentum $J$ and winding in $S^{3}$.
This can be checked by evaluating the derivative at maximum value of
$\beta_{1}$ or minimum value of $y=y_{2}$ at fixed time. At a given fixed time
we can also calculate the number of spikes using the relation
$\Delta{\phi_{2}}=\frac{2\pi}{2n}$, where $\Delta\phi_{2}$ is the angle
between a valley or a minimum and a maximum. We get $\phi_{2}$ from the
initial coordinate to be,

$X_{1}=\frac{R}{\sqrt{\alpha}}\cos{\beta_{1}}e^{i\phi_{2}},~{}~{}~{}\phi_{2}=\omega_{2}\tau+\int
d\xi f_{1}^{{}^{\prime}}.$ (88)

which in turn gives the expression for $\Delta\phi_{2}$ as

$\Delta\phi_{2}=\int d\phi_{2}=\frac{1}{\alpha_{1}^{2}-\alpha_{2}^{2}}\int
d\xi\left(\frac{v_{1}}{1-r_{2}^{2}}-\frac{\omega_{2}}{\omega_{3}}\frac{v_{2}}{r_{2}^{2}}\right)=\frac{2}{\sqrt{-2a}}\left(v_{1}F_{5}+\frac{\omega_{2}}{\omega_{3}}v_{2}F_{6}\right).$
(89)

Now we will evaluate the derivative at the minimum value of $y=y_{2}$. This
gives us the derivative at the maximum value of $\beta_{1}$ at a fixed time

$\frac{d\beta_{1}}{d\phi_{2}}\bigg{|}_{y=y_{2}}=\frac{\beta_{1}^{{}^{\prime}}d\xi}{\omega_{2}d\tau+f_{1}^{{}^{\prime}}d\xi}\bigg{|}_{y=y_{2}}$
(90)

Using that for fixed time i.e., $dt=\omega_{0}d\tau=0$, gives the the
following

$\frac{d\beta_{1}}{d\phi_{2}}\bigg{|}_{y=y_{2}}=\frac{\beta_{1}^{{}^{\prime}}}{f_{1}^{{}^{\prime}}}\bigg{|}_{y=y_{2}}$
(91)

Solving the above equation we get,

$\frac{d\beta_{1}}{d\phi_{2}}\bigg{|}_{y=y_{2}}=\frac{P(y)}{{\sqrt{2}y^{\frac{3}{2}}\sqrt{y-1}}}\frac{\sqrt{y+1}}{2v_{1}+\alpha_{2}\omega_{2}}\frac{1}{\left(\cfrac{\alpha_{2}\omega_{2}}{2v_{1}+\alpha_{2}\omega_{2}}+y\right)}\Bigg{|}_{y=y_{2}}.$
(92)

We can clearly see here that for $y=y_{2}$, the polynomial $P(y_{2})$ which is
there in the numerator vanishes but in general the denominator does not. The
denominator vanishes only under the condition
$y_{2}=-\frac{\alpha_{2}\omega_{2}}{2v_{1}+\alpha_{2}\omega_{2}}$ but this
corresponds to motion only in $S^{3}$, also this gives negative $y_{2}$ which
is not a consistent choice. Thus we observe that in general when we extend the
motion of the string to $S^{3}$, the spikes get rounded off at $y=y_{2}$ and
not end in cusps. Similar argument has been presented in Ishizeki:2008tx .
We will now calculate the expression for the winding number. This can be done
by solving for $\alpha_{2}$ in (86) and replacing in (84), we get

$m=\frac{\nu}{2\pi\omega_{3}}nF_{6}\sqrt{(1-y_{1})(1-y_{2})(y_{3}-1)}$ (93)

It can be seen from the expression that the winding number depends on four
independent parameters $n,y_{1},y_{2},y_{3}$. All other parameters can be
expressed in terms of these the independent parameters. Given the number of
independent parameters, the energy can be written as

$\mathcal{E}=\mathcal{E}(\mathcal{J}_{1},\mathcal{J}_{2},\mathcal{J},n,m)$

Due to the complicated form of the roots further assumptions needs to be taken
to solve this. We can check the similar nature of the rounding off of the
spikes when we introduce some flux to the background.

##  5 Conclusion and outlook

We have elucidated the emergence of classically integrable NR model underlying
in the two dimensional string-sigma model that describes the mixed flux-
supported $AdS_{3}\times S^{3}_{1}\times S^{3}_{2}\times S^{1}$ probed with
fundamental string. We have emphasized on the generic rotating string ansatz
during the NR formalism in our present scenario. The parameter $\alpha$ of the
relative geometry of the spheres in the background has played considerably
significant role in the overall construction of the integrable model. In
support of our construction, we reproduced the Lagrangian and the independent
Uhlenbeck constants for our system and showed that all these pieces
justifiably assume quite similar forms as those of the undeformed general NR
system. What is exclusive here is that the Lagrangian includes some extra
terms which involve the flux parameters as well as $\alpha$. Nevertheless, the
integrals of motion are not affected anyway by the parameter $\alpha$ and they
individually satisfy the constraints of the geometries in $AdS_{3}\times
S^{3}_{1}\times S^{3}_{2}$. Moreover, the condition for Liouville
integrability is also satisfied by the integrals of motion of the system we
developed. After looking at the integrable structure of the system, we study
the rigidly rotating spiky string solutions with energy $E$ and angular
momentum $J$. The string here is rotating in pure $S^{3}\times S^{1}$ with the
addition of non-zero angular momentum $J$ in $S^{1}\subset S^{3}$. We find the
expression for the conserved charges as well as winding number $m$, in terms
of elliptic integrals. Finally, we mathematically speculate the configuration
of the spiky string profile. We establish that, due to the addition of non-
zero angular momentum $J$ in $S^{1}$, the spikes of the rigidly rotating $N$
spike string, get rounded off and not end in cusp. The generalization of the
NR solutions to a class of pulsating string should be interesting as the
pulsating strings present a more stable family of solutions than the usual
rotating ones. It is well-known that the stability of string solutions for
definite spin values increases with the increase in the number of
pulsationsKhan:2005fc . We expect that string pulsating in both $S^{3}_{1}$
and $S^{3}_{2}$ simultaneously with two linearly dependent oscillation numbers
in each sphere can acquire improved stability than those in only one sphere
for a specific chosen value of spin along AdS directions. We wish to come back
to this problem in near future.

## Appendix A

Some of the useful integrals used are summarized below

$\begin{split}F_{1}=\int_{v_{2}}^{v_{3}}&\frac{dy}{\sqrt{-y(y-y_{1})(y-y_{2})(y-y_{3})}},F_{2}=\int_{v_{2}}^{v_{3}}\frac{dy(1-y)}{y\sqrt{-y(y-y_{1})(y-y_{2})(y-y_{3})}}\\\
&F_{3}=\int_{v_{2}}^{v_{3}}\frac{dy(1+y)}{y\sqrt{-y(y-y_{1})(y-y_{2})(y-y_{3})}}\end{split}$
(94)

The above integrals takes the form of the elliptic integrals as follows

$\begin{split}&F_{1}=\frac{2}{\sqrt{y_{3}(y_{2}-y_{1})}}K\left[\sqrt{\frac{y_{1}(y_{2}-y_{3})}{y_{3}(y_{2}-y_{1})}}\right],\\\
&F_{2}=\frac{-2}{y_{1}y_{2}}\sqrt{\frac{y_{2}-y_{1}}{y_{3}}}E\left[\sqrt{\frac{y_{1}(y_{2}-y_{3})}{y_{3}(y_{2}-y_{1})}}\right]+\frac{2(1-y_{1})}{y_{1}\sqrt{y_{3}(y_{2}-y_{1})}}K\left[\sqrt{\frac{y_{1}(y_{2}-y_{3})}{y_{3}(y_{2}-y_{1})}}\right],\\\
&F_{3}=\frac{-2}{y_{1}y_{2}}\sqrt{\frac{y_{2}-y_{1}}{y_{3}}}E\left[\sqrt{\frac{y_{1}(y_{2}-y_{3})}{y_{3}(y_{2}-y_{1})}}\right]+\frac{2(1+y_{1})}{y_{1}\sqrt{y_{3}(y_{2}-y_{1})}}K\left[\sqrt{\frac{y_{1}(y_{2}-y_{3})}{y_{3}(y_{2}-y_{1})}}\right]\end{split}$
(95)

Some other integrals used in the manuscript are:

$F_{4}(y_{2},y)=\int_{y_{2}}^{y}dy\frac{y}{\sqrt{-y(y-y_{1})(y-y_{2})(y-y_{3})}}$
(96)
$F_{5}(y_{2},y)=\int_{y_{2}}^{y}dy\frac{y}{(1+v)\sqrt{-y(y-y_{1})(y-y_{2})(y-y_{3})}}$
(97)
$F_{6}(y_{2},y)=\int_{y_{2}}^{y}dy\frac{y}{(1-y)\sqrt{-y(y-y_{1})(y-y_{2})(y-y_{3})}}$
(98)

On solving the above integrals we get elliptic functions

$F_{4}(y_{2},y_{3})=\frac{2y_{2}}{\sqrt{y_{3}(y_{2}-y_{1})}}\Pi\left[\frac{y_{3}-y_{2}}{y_{3}},\sqrt{\frac{y_{1}(y_{2}-y_{3})}{y_{3}(y_{2}-y_{1})}}\right]$
(99)
$F_{5}(y_{2},y_{3})=\frac{2y_{2}}{(1+y_{2})\sqrt{y_{3}(y_{2}-y_{1})}}\Pi\left[\frac{y_{3}-y_{2}}{y_{3}(1+y_{2})},\sqrt{\frac{y_{1}(y_{2}-y_{3})}{y_{3}(y_{2}-y_{1})}}\right]$
(100)
$F_{6}(y_{2},y_{3})=\frac{2y_{2}}{(1-y_{2})\sqrt{y_{3}(y_{2}-y_{1})}}\Pi\left[\frac{y_{3}-y_{2}}{y_{3}(1-y_{2})},\sqrt{\frac{y_{1}(y_{2}-y_{3})}{y_{3}(y_{2}-y_{1})}}\right]$
(101)

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