††institutetext:  Instituto de Física, Universidade de São Paulo,
Rua do Matão Travessa 1371, 05508-090 São Paulo, SP, Brazil

# Novel aspects of integrability for NLSMs in symmetric spaces

Dimitrios Katsinis<EMAIL_ADDRESS>
###### Abstract

We obtained the formal solution of the auxiliary system of Non Linear Sigma
Models, whose target space is a rank 1 symmetric space based on the indefinite
orthogonal group $O(p,q)$, corresponding to an arbitrary solution of the NLSM.
This class includes de Sitter, Anti-de Sitter and Hyperbolic spaces, which are
of interest in view of the AdS/CFT correspondence. The formal solution is
related to the Pohlmeyer reduction of the NLSM, constituting another link
between the NLSM and the reduced theory. Besides deriving the solution, we
also review the Pohlmeyer reduction of such models. Finally, we comment on the
implications for the monodromy matrix and its eigenvalues.

###### Keywords:

Integrable Systems, Bosonic Strings, Long strings, AdS-CFT Correspondence

##  1 Introduction

In the last 50 years there has been a tremendous number of studies, which
point out that integrability is intertwined with high energy physics. This
work concerns the integrability of Non Linear Sigma Models (NLSMs) on
symmetric spaces. The general framework related to 2-dimensional integrable
models was laid down in the 1970s and is still very important even for ongoing
research.

A particular class of such models is the Principal Chiral Model (PCM)
Zakharov:1973pp . The equations of motion of the PCM along with the flatness
of the current are reproduced by the Lax connection

$\mathcal{L}_{\pm}=\frac{\left(\partial_{\pm}g\right)g^{-1}}{1\pm\lambda},$
(1)

where $\lambda\in\mathbb{C}$ is the spectral parameter and $g$ is in general
an element of some coset. The flatness of the latter is equivalent to the
compatibility condition of the so-called auxiliary system, also known as
fundamental linear problem, which reads

$\partial_{\pm}\Psi(\lambda)=\mathcal{L}_{\pm}\Psi(\lambda).$ (2)

Classical integrability amounts to the existence of an infinite tower of
conserved charges. This tower can be constructed for any Lax connection using
the solution of the auxiliary system (2). Defining the monodromy matrix
$T(\sigma_{f},\sigma_{i};\lambda)$ as

$T(\sigma_{f},\sigma_{i};\lambda)=\Psi(\tau,\sigma_{f};\lambda)\Psi^{-1}(\tau,\sigma_{i};\lambda),$
(3)

it follows that its time derivative reads

$\partial_{\tau}T(\sigma_{f},\sigma_{i};\lambda)=\mathcal{L}_{\tau}(\tau,\sigma_{f})T(\sigma_{f},\sigma_{i};\lambda)-T(\sigma_{f},\sigma_{i};\lambda)\mathcal{L}_{\tau}(\tau,\sigma_{i}),$
(4)

where $\mathcal{L}_{\tau}$ is the component of the Lax connection when it is
expressed in terms of the world-sheet coordinates $\sigma$ and $\tau$. In this
case the relevant equation of the auxiliary system is
$\partial_{\tau}\Psi(\tau,\sigma;\lambda)=\mathcal{L}_{\tau}\Psi(\tau,\sigma;\lambda)$.
The constants $\sigma_{i}$ and $\sigma_{f}$ are determined by the specific
solution and its boundary conditions. Depending on the latter, it is either
the trace of the monodromy matrix that is conserved, which is the case for
periodic boundary conditions, or the matrix itself if $\mathcal{L}_{\tau}$
vanishes both at $\sigma_{i}$ and $\sigma_{f}$. For open string boundary
conditions one has to define a boundary monodromy matrix Cherednik:1984vvp ;
Sklyanin:1988yz , which is again related to the solution of auxiliary system.
Expanding the monodromy matrix or its trace with respect to the spectral
parameter one obtains the aforementioned infinite tower of conserved charges.
The existence of all these conserved charges is such a strong contraint even
on the quantized theory so that no particle production is allowed and the
S-matrix of the theory is factorized into two particle S-matrices
Luscher:1977uq .

There is a far wider class of theories, which obviously includes the PCM, the
so-called NLSMs. The action of such theories reads

$S=\int d^{2}\sigma\,G_{\mu\nu}(X)\partial_{+}X^{\mu}\partial_{-}X^{\nu}$ (5)

and their fields, i.e. $X^{\mu}$, are mappings from the 2-dimensional world-
sheet to a manifold $\mathcal{M}$. It turns out that if $\mathcal{M}$ is a
symmetric space, the NLSM is integrable Eichenherr:1979ci ; Eichenherr:1979hz
.111We remind the reader that a symmetric space $\mathcal{M}=F/G$ is defined
as follows. Consider a group $F$ having a subgroup $G$, which correspond to
the Lie algebras $\mathfrak{f}$ and $\mathfrak{g}$, so that $\mathfrak{f}$
admits the canonical decomposition
$\mathfrak{f}=\mathfrak{g}\oplus\mathfrak{p}$. Then, $\mathcal{M}$ is a
symmetric space if the following commutation relations hold
$[\mathfrak{g},\mathfrak{g}]\subset\mathfrak{g},\qquad[\mathfrak{g},\mathfrak{p}]\subset\mathfrak{p},\qquad[\mathfrak{p},\mathfrak{p}]\subset\mathfrak{g}.$
(6) For further information about the integrability of NLSMs see
Zarembo:2017muf ; Driezen:2021cpd  and references therein. Finally, let us
mention that integrability is not restricted to symmetric spaces. Recently
there has been a lot of activity in the construction of integrable NLSMs
Klimcik:2008eq ; Delduc:2013fga ; Delduc:2013qra ; Sfetsos:2013wia ;
Klimcik:2014bta , see also Klimcik:2021bjy ; Hoare:2021dix  and references
therein.

An interesting aspect of NLSMs on symmetric spaces is that besides being
integrable, they are also reducible to integrable systems, which are
multicomponent generalizations of the sine-Gordon equation. The original
examples are the $O(3)$ and $O(4)$ NLSMs, corresponding to the spheres S2 and
S3, which are related to the sine-Gordon and the complex sine-Gordon
equations, respectivelly, via the Pohlmeyer reduction Pohlmeyer:1975nb ;
Lund:1976ze ; Lund:1976xd . Subsequently many more symmetric spaces were
studied including the N-sphere $S^{N}$, i.e. the $O(N+1)$ model
Pohlmeyer:1979ch , and $\mathbb{CP}^{N}$ Eichenherr:1979uk , as well as (Anti)
de Sitter space Barbashov:1980kz ; DeVega:1992xc ; Larsen:1996gn .

Essentially, the reduction amounts to introducing some degrees of freedom,
which are connected non-locally to the fields of the NLSM, and setting the
non-vanishing components of the stress-energy tensor to constants. This is
always possible as the NLSM is conformally invariant at classical level. When
the symmetric space $\mathcal{M}$ is of positive definite signature, which,
for instance, is the case for SN and $\mathbb{CP}^{N}$, it follows that the
constants are necessarily positive, thus $T_{\pm\pm}$ read

$T_{\pm\pm}=G_{\mu\nu}\partial_{\pm}X^{\mu}\partial_{\pm}X^{\nu}=m_{\pm}^{2},$
(7)

in appropriate coordinates. One should keep in mind that the Pohlmeyer
reduction is a many-to-one mapping. In particular, the reduced theory depends
on $m_{+}$ and $m_{-}$ only via their product $m_{+}m_{-}$ and not on their
ratio. We are going to refine this statement, but it is sufficiently accurate
for now. Evidently, even though integrability is preserved, for
$m_{+}m_{-}\neq 0$ conformal invariant is broken by the reduction. Finally, it
worth pointing out that when the action of the NLSM on the symmetric space is
equivalent to free fields on a higher-dimensional space subject to a quadratic
constraint, the equations of motion of the NLSM become linear for a given
solution of the reduced theory.

Using a group theoretical approach the reducibility of NLSM on symmetric
spaces was established in DAuria:1979ham ; DAuria:1980iyh ; DAuria:1980ucx .
This approach is extremely powerful leading to many interesting developments.
These reduced theories are known as Symmetric Space sine-Gordon models
(SSSGs). For a long time their Langrangian formulation was an open problem,
until it was solved for symmetric spaces of rank 1 in Bakas:1995bm , see also
Park:1994bx ; Hollowood:1994vx ; Fernandez-Pousa:1996aoa .222The rank of a
symmetric space $F/G$ is the dimension of the maximal abelian subspaces in the
orthogonal complement of $\mathfrak{g}$ in $\mathfrak{f}$. Specifically, it
was shown that SSSGs are gauged WZW models perturbed by an appropriate
potential, which preserves integrability. For a recent review on the group
theoretical approach of Pohlmeyer reduction, as well as for an exhaustive
analysis of the Lagrangian formulation of SSSGs, see Miramontes:2008wt .

Another intriguing characteristic of integrable NLSMs regards the construction
of new solutions when a solution is already known. This is achieved with the
application of the dressing method Zakharov:1973pp ; Zakharov:1980ty ;
Harnad:1983we . In order to do so one has to solve the auxiliary system (2),
where $g$ corresponding to the known NLSM solution, which in this context is
referred to as the seed solution. Given the solution of the auxiliary system
one can systematically construct an infinite tower of new NLSM solutions.

There is an analogous story for the reduced systems. There are the so-called
Bäcklund transformations for these systems, which are sets of first order non-
linear equations and allow the derivation of new solutions given a known one.
In the case of the sine-Gordon equation these transformations essentially
insert solitons on the background of the known solution. Multiple solutions
corresponding to the same known solution can be combined using addition
formulas, for example see Park:1995np . The Bäcklund transformations are the
counterpart of the dressing transformations. In particular, it has been shown
that a dressing transformation of the NLSM solution, automatically performs a
Bäcklund transformation to its avatar in the reduced theory Hollowood:2009tw .
It is also important that the Bäcklund transformations generate an infinite
tower of conserved charges Pohlmeyer:1975nb . These charges are related to the
ones constructed via the expansion of the monodromy matrix Arutyunov:2003rg ;
Arutyunov:2005nk .

In view of the AdS/CFT correspondence Maldacena:1997re ; Gubser:1998bc ;
Witten:1998qj , which relates planar strongly coupled $\mathcal{N}=4$ Super-
Yang-Mills to classical free IIB string theory in AdS${}_{5}\times$S5, the
interest on the reduced models was revived. In this setup the dynamics of the
superstrings is determined by the Metsaev-Tseytlin action Metsaev:1998it . The
theory is formulated as the super-coset $\frac{PSU(2,2|4)}{SO(1,4)\times
SO(5)}$ and is classically integrable Bena:2003wd . As this coset is a
symmetric space having a $\mathbb{Z}_{4}$ grading, the theory is reducible and
the corresponding models were constructed in Grigoriev:2007bu ;
Mikhailov:2007xr ; Grigoriev:2008jq ; Hoare:2012nx . Moreover the reduced
theory is finite in the UV Roiban:2009vh  and the properties of the classical
charges indicate that the theory is supersymmetric Schmidtt:2011gtt ;
Schmidtt:2011nr . The world-sheet supersymmetry is non-local Goykhman:2011mq ;
Hollowood:2011fq . Similarly, the bosonic reduced model corresponding to
AdS${}_{4}\times\mathbb{CP}^{3}$, which is dual to the ABJM theory
Aharony:2008ug  and is formulated as the $\frac{OSP(2,2|6)}{SO(1,3)\times
U(3)}$ super-coset Arutyunov:2008if ; Stefanski:2008ik , was constructed in
Rashkov:2008rm . The supersymmetric one was presented in Dukalski:2009pr . The
reduced models, maintaining 2-dimensional Lorentz invariance, provide an
alternative approach to the standard light-cone description of the on-shell
degrees of freedom.

Let us be more specific about the reduced models in this context. Equation (7)
may be interpreted as the Virasoro constraint for strings propagating in
$\mathbb{R}_{t}\times\mathcal{M}$, where a linear gauge for the time
coordinate $t$ of the target space has been employed, namely

$t=m_{+}\sigma^{+}+m_{-}\sigma^{-}.$ (8)

This gauge, which is a generalization of the usual static one, i.e. $t=m\tau$,
facilitates the rest of our work. A world-sheet boost with appropriate
velocity turns the linear gauge to the static one. At the level of equations
(7) setting $m_{\pm}\rightarrow c^{\pm 1}m_{\pm}$, the parameter $c$ can be
absorbed via a boosted form of the embedding functions, i.e. by
$X^{\mu}(\sigma^{+},\sigma^{-};m_{+},m_{-})\rightarrow
X^{\mu}(c\,\sigma^{+},c^{-1}\sigma^{-};m_{+},m_{-})$. Notice that the product
$m_{+}m_{-}$ remains invariant. Of course the counterpart in the reduced
theory depends on the boost too. As the boost to the static gauge depends also
on the ratio of $m_{+}$ and $m_{-}$ the Pohlmeyer avatar no longer depends
solely on the product $m_{+}m_{-}$. So, to be more precise the reduced theory
depends only on this product in an appropriate frame.

Many properties of classical string solutions are captured by the reduced
models. Strings, whose world-sheet is infinite, correspond to solitonic
solutions of the reduced system. For instance, the giant magnon Hofman:2006xt
($\mathbb{R}_{t}\times\textrm{S}^{2}$), the dyonic giant magnon Chen:2006gea
($\mathbb{R}_{t}\times\textrm{S}^{3}$) and the single spike Ishizeki:2007we
($\mathbb{R}_{t}\times\textrm{S}^{2}$ and
$\mathbb{R}_{t}\times\textrm{S}^{3}$) correspond to solitonic solutions of the
sine-Gordon and complex sine-Gordon. In an analogous way, the BMN particle
Berenstein:2002jq , which moves at the speed of light on the equator of
$S^{5}$, corresponds to the (stable) vacuum solution of the sine-Gordon
equation. Similar identifications exist for for kink-train solutions of the
sine-Gordon and complex sine-Gordon equation Okamura:2006zv ; Hayashi:2007bq .
See also Katsinis:2018zxi  for a complete classification of elliptic string
solutions in $\mathbb{R}_{t}\times\textrm{S}^{2}$. Similar conclusions also
hold for strings in AdS and dS DeVega:1992xc ; Larsen:1996gn ; Jevicki:2007aa
; Jevicki:2008mm ; Bakas:2016jxp , as well as for dressed elliptic strings in
$\mathbb{R}_{t}\times\textrm{S}^{2}$ Katsinis:2018ewd ; Katsinis:2019oox .

Besides the relation between the string solutions and their counterparts in
the reduced theory, AdS/CFT correspondence is intimately related to
integrability in many more aspects, such as the spectrum of dual theories and
the calculation of correlation functions and Wilson loops, see Beisert:2010jr
for a review. Of particular interest is the spectral problem. Today there are
methods that determine the (quantum mechanically) exact spectrum of
$\mathcal{N}=4$ Super-Yang-Mills, see Gromov:2017blm  and references therein,
but making contact with them is way beyond the scope of this work. The NLSM
and all the aspects of integrability that we mentioned, i.e. Pohlmeyer
reduction, dressing method, Bäcklund transformations, regard only the
classical geometry. In a series of papers Kazakov:2004qf ; Kazakov:2004nh ;
Beisert:2004ag ; Beisert:2005bm ; Beisert:2005di  it was shown that the single
trace operators of planar $\mathcal{N}=4$ Super-Yang-Mills in the
thermodynamic limit, i.e. in the case of infinitely many insertion of fields,
and the NLSM on AdS${}_{5}\times$S5 share a spectral curve. Similarly, the
spectral curve corresponding to the super-coset
$\frac{OSP(2,2|6)}{SO(1,3)\times U(3)}$ was construced in Gromov:2008bz . We
remind the reader that the spectral curve is directly related to the
eigenvalues of the monodromy matrix $T$. This result establishes that both
dual theories, in this specific limit, share their conserved charges.
Nevertheless, even though this result is powerful, the identification of a
specific string solution which is dual to a particular operator not
straightforward. A systematic approach for this problems is elusive.

In the framework of AdS/CFT, the expectation values of Wilson loops at strong
couppling are calculated by the area of minimal surfaces whose boundary is the
loop in question Maldacena:1998im ; Rey:1998ik . Interestingly enough, null
polygonal Wilson loops are related to gluon scattering amplitudes at strong
coupling Alday:2007hr ; Alday:2007he , see also Alday:2009yn ; Alday:2009dv .
Moreover, by introducing an appropriate contour in the internal space, the
Wilson loop can be made supersymmetric Zarembo:2002an ; Drukker:2007dw .
Implementing localization, the supersymmetric Wilson loops can by calculated
without the use of holography, directly in field theory, exactly (sometimes
even for finite $N$) Pestun:2007rz ; Pestun:2016zxk . The comparison of the
results of both calculations is a highly non-trivial test of AdS/CFT. Finally,
minimal surfaces are also relevant for the calculation of Holographic
Entanglement Entropy (HEE) Ryu:2006bv ; Ryu:2006ef . The HEE corresponding to
a region defined by some entangling surface equals the area of co-dimension 2
minimal surface, which extend in the bulk and whose boundary is the entangling
surface. In the case of AdS4 the minimal surfaces are 2-dimensional, thus
described by a NLSM. All the aforementioned space-like minimal surfaces in AdS
or in the Hyperbolic space are described by NLSMs and also admit a Pohlmeyer
reduction Dorn:2009kq ; Dorn:2009gq ; Jevicki:2009bv ; Burrington:2009bh ;
Dorn:2009hs ; Ishizeki:2011bf ; Kruczenski:2013bsa ; Kruczenski:2014bla ;
Pastras:2016vqu ; He:2017cwd .

The above analysis demonstrates that the auxiliary system (2) is significant
for many reasons. In our previous works we applied the dressing method for
elliptic strings in $\mathbb{R}_{t}\times\textrm{S}^{2}$ Katsinis:2018ewd ,
see also Arutyunov:2005nk , and elliptic minimal surfaces in H3
Katsinis:2020dhe . One could proceed in a systematic manner and the only
equations that had to be solved, were essentially solved by the seed solution
upon altering some parameters. Motivated by this observation in
Katsinis:2020avk  we obtained the formal solution of the auxiliary system of
the $O(3)$ NLSM corresponding to an _arbitrary_ seed. The solution is
constructed by combining appropriately the seed solution, with a “virtual”
one. The latter is obtained from the seed solution by the substitution

$m_{\pm}\rightarrow\frac{1\mp\lambda}{1\pm\lambda}m_{\pm},$ (9)

where $\lambda$ is the spectral parameter. Thus, the virtual solution solves
the equation of motion of the NLSM, has the same avatar in the reduced theory
as the seed one, but as $\lambda\in\mathbb{C}$ it belongs to the
complexification of the coset. A direct consequence of this result is the
existence of a non-linear superposition rule at least for the specific NLSM.
Regarding the dressing method the concept of superposition is already there as
one can construct an infinite tower of NLSM solution by solving the auxiliary
system once. Nevertheless, it turns out that no differential equation has to
be solved. The dressing method is the implementation of a non-linear
superposition in the first place. At the level of the reduced theory this
result implies that there is no need to solve the Bäcklund transformations and
that solitons are inserted for free. These conclusions present a novel aspect
of NLSMs integrability, which expands the framework built in the 1970s and
later on. As the derivation of Katsinis:2020avk  is model specific, the
generalization of these conclusions could be questionable and the structure of
the superposition obscure.

In our recent work Katsinis:2022amo  we were able to obtain the formal
solution the auxiliary system for the whole class of $O(n)$ NLSMs. The
solution relies on the Pohlmeyer reduction. Even though we presented the group
theoretical approach for the study of the reduced models, there exists a
geometric one, which treats the reduction as an embedding problem, introduced
in Lund:1976ze ; Lund:1976xd , see also Barbashov:1980kz ; Barbashov:1983cs ;
Bakas:1996uw . One considers the target space as a sub-manifold of an enhanced
flat space and analyzes the embedding of the NLSM solution in the enhanced
space. In order to do so one introduces a basis in the enhanced space. It
turns out that the solution of the auxiliary system is related to this basis,
as well as on the redefinition of the parameters $m_{\pm}$ given by (9).

The purpose of this work is twofold. In view of AdS/CFT correspondence the
solution of the auxiliary system in the case of AdS space presents a lot of
interest. Of course, it is appealing to find the formal solution of the
auxiliary system for more symmetric spaces. To kill two birds with one stone,
we study the rank 1 symmetric spaces corresponding to the indefinite
orthogonal group $O(p,q)$.

The structure of the paper is as follows. In section 2 we perform the
Pohlmeyer reduction of rank 1 symmetric spaces corresponding to the indefinite
orthogonal group $O(p,q)$. In section 3 we solve the auxiliary system. There
are two appendices. In appendix A we provide additional information for the
Pohlmeyer reduction of AdS. Appendix B contains some details relevant for the
derivation of section 3.

##  2 Pohlmeyer reduction

It is well known that strings propagating in symmetric spaces such as AdS and
dS are related to integrable models, namely sinh-Gordon or cosh-Gordon and
multicomponent generalizations of them, via the Pohlmeyer reduction
Pohlmeyer:1975nb ; Lund:1976ze ; Lund:1976xd . There is extensive work on this
subject in the literature Barbashov:1980kz ; DeVega:1992xc ; Larsen:1996gn .
In this section, we generalize these works, while introducing our own
convention, which will facilitation the second part of the paper. Since the
aforementioned spaces are of indefinite signature, one should distinguish
between different types of reductions. The tangent vectors of the world-sheet
can be space-like, light-like or time-like and consequently the $T_{\pm\pm}$
components of the stress tensor positive, negative or vanishing, see (7). The
reduction is characterized accordingly.

In the context of string theory the various types of reduction correspond to
different geometric setups. Light-like reduction can be interpreted as stings
on the non-compact manifold $\mathcal{M}$. If the reduction is time-like the
geometry is $\mathcal{M}\times\mathcal{M}_{c}$, where $\mathcal{M}_{c}$ is a
compact manifold having positive definite metric. In the case of space-like
reduction the theory is defined in $\mathcal{M}_{nc}\times\mathcal{M}$, where
$\mathcal{M}_{nc}$ is another non-compact manifold. The case
$\mathbb{R}_{t}\times\mathcal{M}$ analyzed in the introduction is a just an
example. Finally, let us mention that the geometry of the last case admits a
time-like reduction too. In the context of the AdS/CFT correspondence light-
like and time-like reduction of non-compact spaces are more natural. Having
analyzed all options, let us introduce our conventions.

It is well known that $\textrm{AdS}_{n+1}$ and $\textrm{dS}_{n+1}$ are sub-
manifolds of $\mathbb{R}^{(2,n)}$ and $\mathbb{R}^{(1,n+1)}$, respectively.
They are both symmetric spaces realized as the
$\textrm{SO}(2,n)/\textrm{SO}(1,n)$ and $\textrm{SO}(1,n+1)/\textrm{SO}(1,n)$
cosets respectively. We can deal with both spaces simultaneously by
introducing the metric tensor $\eta_{\mu\nu}=\textrm{diag}(-1,1,\dots,1,s)$,
where $s=-1$ corresponds to $\textrm{AdS}_{n+1}$ and $s=1$ to
$\textrm{dS}_{n+1}$. Denoting the vectors of the enhanced space as $Y^{\mu}$,
where $Y^{\mu}=(Y^{-1},Y^{0},Y^{1},\dots Y^{n})$, the spaces of interest
correspond to the sub-manifolds

$Y\cdot Y=s\Lambda^{2}.$ (10)

It goes without saying that the inner product is defined as $A\cdot
B=\eta_{\mu\nu}A^{\mu}B^{\nu}$.

As the purpose of this work is to study the relation between the solution of
the auxiliary system and the Pohlmeyer reduction beyond the n-sphere
Katsinis:2022amo , we will consider more general geometries. Rather than
restricting our analysis only on AdS and dS spaces, we will go one step
further and study rank 1 cosets based on the indefinite orthogonal group. In
order to do so we introduce the metric

$\eta_{\mu\nu}=\begin{pmatrix}I_{p,q}&0\\\ 0&s\end{pmatrix},\qquad
I_{p,q}=\begin{pmatrix}-I_{p}&0\\\ 0&I_{q}\end{pmatrix},$ (11)

where $I_{k}$ denotes the $k\times k$ identity matrix and $p+q=n+1$. It the
following we use the notation $I_{q,p}$ for matrices of this specific form.
For $s=+1$ the space is a sub-manifold of $\mathbb{R}^{p,q+1}$, it has
positive constant curvature and corresponds to the coset $SO(p,q+1)/SO(p,q)$.
Similarly, for $s=-1$ the space is a sub-manifold of $\mathbb{R}^{p+1,q}$, it
has negative constant curvature and corresponds to the coset
$SO(p+1,q)/SO(p,q)$. Of course both spaces are related by the the duality
$s\rightarrow-s$ and $p\leftrightarrow q$ Helgason . Finally we remind the
reader that for $s=-1$, $p=0$ and $q=n+1$ we obtain the Hyperbolic space Hn+1.

###  2.1 The NLSM action

Having set the ground, we can discuss the NLSM. The corresponding action reads

$S=\int
d\sigma^{+}d\sigma^{-}\left[\partial_{+}Y\cdot\partial_{-}Y+\lambda\left(Y\cdot
Y-s\Lambda^{2}\right)\right],$ (12)

where $\lambda$ is a Lagrange multiplier. The fields $Y$ are vectors of the
enhanced space $\mathbb{R}^{p,q+1}$ or $\mathbb{R}^{p+1,q}$, depending on the
value of $s$, which are restricted on the sub-manifold (10) via the Lagrange
multiplier. The equations of motion read

$\partial_{+}\partial_{-}Y=\lambda Y.$ (13)

It is straightforward to calculate that the Lagrange multiplier is given by

$\lambda=-\frac{s}{\Lambda^{2}}\partial_{+}Y\cdot\partial_{-}Y,$ (14)

implying that the equations of motion assume the form

$\partial_{+}\partial_{-}Y=-\frac{s}{\Lambda^{2}}\left(\partial_{+}Y\cdot\partial_{-}Y\right)Y.$
(15)

The equations of motion are accompanied by the conservation of the stress
energy tensor. The precise form of the components of the latter depends on the
theory under consideration and more specifically on whether one considers the
target space consisting solely of the aforementioned manifolds, or if the
target space is the direct product of these spaces with another manifold. For
instance, in view of the AdS/CFT correspondence, its very usual to consider
the target spaces $\textrm{AdS}_{5}\times\textrm{S}^{5}$ or
$\textrm{AdS}_{4}\times\mathbb{CP}^{3}$. Keeping an open mind and adopting a
mathematical perspective, we consider the non-vanishing components of the
stress tensor to be

$T_{\pm\pm}=\partial_{\pm}Y\cdot\partial_{\pm}Y=tm^{2}_{\pm}.$ (16)

The case $t=0$ gives rise to light-like reduction, while $t=1$ to space-like
and $t=-1$ to time-light. In the last two cases, string theory interpretation
requires the introduction of another manifold so that the Virasoro constraints
corresponding to the whole target space are satisfied.

###  2.2 The embedding problem

In order to perform the Pohlmeyer reduction, we consider the embedding problem
of the NLSM solution $Y$ in the enhanced space. To do so, we introduce a basis
in the enhanced space. Of course, this basis includes the vectors $Y$,
$\partial_{\pm}Y$, which span the tangent space of the string world-sheet, and
$d-1$ linearly independent vectors. Essentially, we consider the enhanced
space as the direct product of the 3-dimensional space $\mathcal{M}_{3}$,
spanned by $Y$ and $\partial_{\pm}Y$ and the $(n-1)$-dimensional space
$\mathcal{M}_{n-1}$, spanned some vectors $v_{i}$, where
$i=1,\dots,(n-1)$.333Lower-case English letters run from 1 to $n-1$, while
lower-case Greek letters run from 1 to $n+2$. We normalize these vectors as

$v_{i}\cdot v_{j}=s_{i}\delta_{i,j},$ (17)

where $s_{i}=\pm 1$. The careful reader will realize that this definition is
sloppy. For the moment let us comment that since the basis spans either
$\mathbb{R}^{p,q+1}$ or $\mathbb{R}^{p+1,q}$ the total number of $s_{i}$
having the value $+1$ and of the ones having the value $-1$ is constrained. To
make the notation more uniform, we also set

$v_{n+2}=Y,\qquad v_{n+1}=\partial_{+}Y,\qquad v_{n}=\partial_{-}Y.$ (18)

Thus, we have the following basis $\mathcal{V}=\\{v_{1},\dots,v_{d+2}\\}.$
Considering the $\mathcal{M}_{3}$ subspace, the norm of $v_{d+1}$ is fixed by
(10), while the norms of $v_{d}$ and $v_{d+1}$ are fixed by the equations
(16). The geometric constraint also implies that $v_{d}$ and $v_{d+1}$ are
perpendicular to $v_{d+2}$. The only unconstrained degree of freedom is the
angle between the vectors $v_{d}$ and $v_{d+1}$. Thus, the vectors of the
basis obey the following relations:

$v_{i}\cdot v_{j}=s_{i}\delta_{i,j},\quad v_{n}\cdot v_{n}=tm_{-}^{2},\quad
v_{n+1}\cdot v_{n+1}=tm_{+}^{2},\qquad v_{n+2}\cdot v_{n+2}=s\Lambda^{2},\\\
v_{i}\cdot v_{n}=0,\quad v_{i}\cdot v_{n+1}=0,\quad v_{i}\cdot v_{n+2}=0,\quad
v_{n}\cdot v_{n+2}=0,\quad v_{n+1}\cdot v_{n+2}=0.$ (19)

To deal with the inner product $v_{n+1}\cdot v_{n+2}$ we have to be more
careful and take into account the setup of the problem. In other words, we
should make sure that the basis indeed spans the enhanced space. To treat all
cases simultaneously, we define the primary Pohlmeyer field $\varphi$ via the
equation

$\partial_{+}Y\cdot\partial_{-}Y=m_{+}m_{-}f(\varphi).$ (20)

In section A we provide details on the construction of the basis in the
enhanced space and how this affects the definition of the primary Pohlmeyer
field. There are three different options for the function $f$, namely

$f(\varphi)=\begin{cases}\cos\varphi\\\ \cosh\varphi\\\
\exp\varphi\end{cases}\qquad.$ (21)

In order to perform the Pohlmeyer reduction, we calculate the derivatives of
the vectors constituting the basis and expand them on the basis itself, i.e.

$\partial_{\pm}v_{\alpha}=\left(A_{\pm}\right)_{\alpha\beta}v_{\beta}.$ (22)

This equation can be expressed in matrix form as

$\partial_{\pm}V=A_{\pm}V,$ (23)

where $V$ is the matrix having the vectors of the basis as its rows, i.e. its
matrix elements are $V_{\alpha\beta}=(v_{\alpha})_{\beta}$, see also equation
(143). Using the definitions of $v_{d}$ and $v_{d+1}$, along with the
equations of motion (15), it is straightforward to calculate that

$\partial_{+}v_{n+2}=v_{n+1},\quad\partial_{-}v_{n+2}=v_{n},\quad\partial_{+}v_{n}=\partial_{-}v_{n+1}=-s\frac{m_{+}m_{-}}{\Lambda^{2}}f(\varphi)v_{n+2}.$
(24)

The other derivatives of $v_{d}$ and $v_{d+1}$ are obtained using the
geometric constraint (10), the non-vanishing components of the stress energy
tensor (16) and the equations of motion (15). In particular, these relations
imply that

$\displaystyle Y\cdot\left(\partial_{\pm}^{2}Y\right)$
$\displaystyle=-tm^{2}_{\pm},$ (25)
$\displaystyle\left(\partial_{\pm}Y\right)\cdot\left(\partial_{\pm}^{2}Y\right)$
$\displaystyle=0,$ (26)
$\displaystyle\left(\partial_{\mp}Y\right)\cdot\left(\partial_{\pm}^{2}Y\right)$
$\displaystyle=m_{+}m_{-}f^{\prime}(\varphi)\partial_{\pm}\varphi.$ (27)

Thus, the expansion of the derivatives $\partial_{\pm}^{2}Y$ on the basis is

$\partial^{2}_{\pm}Y=-st\frac{m^{2}_{\pm}}{\Lambda^{2}}Y+\frac{f(\varphi)f^{\prime}(\varphi)}{f^{2}(\varphi)-t^{2}}\partial_{\pm}\varphi\partial_{\pm}Y-t\frac{m_{\pm}}{m_{\mp}}\frac{f^{\prime}(\varphi)}{f^{2}(\varphi)-t^{2}}\partial_{\pm}\varphi\partial_{\mp}Y+m_{\pm}a_{i}^{\pm}v_{i},$
(28)

where $a_{i}^{\pm}$ are $2(n-1)$ unknown parameters. In the following, these
parameters are grouped as the two $(n-1)\times 1$ matrices ${\bf a}_{\pm}$.
Similarly, we can calculate

$\partial_{\pm}v_{i}=t\frac{1}{f^{2}(\varphi)-t^{2}}\frac{s_{i}\,a^{\pm}_{i}}{m_{\pm}}\partial_{\pm}Y-\frac{f}{f^{2}(\varphi)-t^{2}}\frac{s_{i}\,a^{\pm}_{i}}{m_{\mp}}\partial_{\mp}Y+\left(\mathcal{A}_{\pm}\right)_{ij}s_{j}v_{j},$
(29)

where $\mathcal{A}_{\pm}$ are $(n-1)\times(n-1)$ antisymmetric matrices, whose
elements are given by

$\left(\mathcal{A}_{\pm}\right)_{ij}=v_{j}\cdot\partial_{\pm}v_{i}.$ (30)

Putting everything together, the matrices $A^{\pm}_{\alpha\beta}$ are the
following

$\displaystyle A_{+}$
$\displaystyle=\begin{pmatrix}\mathcal{A}_{+}\mathcal{S}&-\frac{f(\varphi)}{f^{2}(\varphi)-t^{2}}\frac{\mathcal{S}{\bf
a}_{+}}{m_{-}}&t\frac{1}{f^{2}(\varphi)-t^{2}}\frac{\mathcal{S}{\bf
a}_{+}}{m_{+}}&{\bf 0}\\\ {\bf
0}^{T}&0&0&-s\frac{m_{+}m_{-}}{\Lambda^{2}}f(\varphi)\\\ m_{+}{\bf
a}_{+}^{T}&-t\frac{m_{+}}{m_{-}}\frac{f^{\prime}(\varphi)}{f^{2}(\varphi)-t^{2}}\partial_{+}\varphi&\frac{f(\varphi)f^{\prime}(\varphi)}{f^{2}(\varphi)-t^{2}}\partial_{+}\varphi&-st\frac{m^{2}_{+}}{\Lambda^{2}}\\\
{\bf 0}^{T}&0&1&0\end{pmatrix},$ (31) $\displaystyle A_{-}$
$\displaystyle=\begin{pmatrix}\mathcal{A}_{-}\mathcal{S}&t\frac{1}{f^{2}(\varphi)-t^{2}}\frac{\mathcal{S}{\bf
a}_{-}}{m_{-}}&-\frac{f(\varphi)}{f^{2}(\varphi)-t^{2}}\frac{\mathcal{S}{\bf
a}_{-}}{m_{+}}&{\bf 0}\\\ m_{-}{\bf
a}_{-}^{T}&\frac{f(\varphi)f^{\prime}(\varphi)}{f^{2}(\varphi)-t^{2}}\partial_{-}\varphi&-t\frac{m_{-}}{m_{+}}\frac{f^{\prime}(\varphi)}{f^{2}(\varphi)-t^{2}}\partial_{-}\varphi&-st\frac{m^{2}_{-}}{\Lambda^{2}}\\\
{\bf 0}^{T}&0&0&-s\frac{m_{+}m_{-}}{\Lambda^{2}}f(\varphi)\\\ {\bf
0}^{T}&1&0&0\end{pmatrix},$ (32)

where column matrices are denoted with bold symbols and $\mathcal{S}$, denotes
the matrix

$\mathcal{S}=\textrm{diag}\left(s_{1},\dots,s_{n-1}\right).$ (33)

The compatibility condition
$\partial_{+}\partial_{-}v_{i}=\partial_{-}\partial_{+}v_{i}$ implies that the
matrices $A_{\pm}$ obey the zero curvature condition

$\partial_{-}A_{+}-\partial_{+}A_{-}+\left[A_{+},A_{-}\right]=0.$ (34)

Explicitly, the equations of motion of the Pohlmeyer fields $\varphi$, ${\bf
a}_{\pm}$ and $\mathcal{A}_{\pm}$ read

$\displaystyle F_{+-}$
$\displaystyle=-\frac{f(\varphi)}{f^{2}(\varphi)-t^{2}}\left[{\bf a}_{-}{\bf
a}_{+}^{T}-{\bf a}_{+}{\bf a}_{-}^{T}\right]\mathcal{S},$ (35)
$\displaystyle\mathcal{D}_{\pm}{\bf a}_{\mp}$
$\displaystyle=t\frac{f^{\prime}(\varphi)\partial_{\mp}\varphi}{f^{2}(\varphi)-t^{2}}{\bf
a}_{\pm},$ (36) $\displaystyle\partial_{+}\partial_{-}\varphi$
$\displaystyle=\frac{{\bf a}_{-}^{T}\mathcal{S}{\bf
a}_{+}}{f^{\prime}(\varphi)}-s\frac{m_{+}m_{-}}{\Lambda^{2}}\frac{f^{2}(\varphi)-t^{2}}{f^{\prime}(\varphi)}-\frac{\partial}{\partial\varphi}\left[\ln\left(\frac{f^{\prime}(\varphi)}{\sqrt{f^{2}(\varphi)-t^{2}}}\right)\right]\partial_{+}\varphi\partial_{-}\varphi,$
(37)

where the covariant derivatives are defined as
$\mathcal{D}_{\pm}=I_{n-1}\partial_{\pm}-\mathcal{S}\mathcal{A}_{\pm}$ and the
field-strength as $F_{+-}=[\mathcal{D}_{+},\mathcal{D}_{-}]$. Notice that the
reduced theory depends only on the product $m_{+}m_{-}$. For our purposes, the
function $f$ is given by (21). Thus, regarding (37), there are two options
$t^{2}=1$ and either $f(\varphi)=\cosh\varphi$ or $f(\varphi)=\cos\varphi$, or
$t=0$ and $f(\varphi)=\exp\varphi$. In both cases the coefficient of
$\partial_{+}\varphi\partial_{-}\varphi$ vanishes. Also notice that
$\left(\mathcal{S}\mathcal{A}_{\pm}\right)^{T}=-\mathcal{S}\left(\mathcal{S}\mathcal{A}_{\pm}\right)\mathcal{S}$,
implying that $\mathcal{S}\mathcal{A}_{\pm}$ are valued in the Lie algebra of
an indefinite orthogonal group $G$. The Killing metric of this group is
$\mathcal{S}$.

In this form, the Pohlmeyer reduced theory has the gauge redundancy

${\bf a}_{\pm}\rightarrow O{\bf
a}_{\pm},\qquad\mathcal{S}\mathcal{A}_{\pm}\rightarrow
O\mathcal{S}\mathcal{A}_{\pm}O^{-1}+\left(\partial_{\pm}O\right)O^{-1},$ (38)

where $O^{-1}=\mathcal{S}O^{T}\mathcal{S}$, i.e. the matrix $O$ is an element
of the indefinite orthogonal group $G$. So far, for sufficiently large $n$, we
have introduced many more fields that the ones of the NLSM.444At this point,
the Pohlmeyer reduced theory depends on the primary Pohlmeyer field -$1$
field-, on ${\bf a}_{\pm}$ -$2(n-1)$ fields- and on $\mathcal{A}_{\pm}$
-$(n-1)(n-2)$ fields-, for a total of $n(n-1)+1$ fields. The gauge redundancy
implies that $(n-1)(n-2)/2$ fields are unphysical, leaving $n(n+1)/2$ degrees
of freedom. The NLSM depends on $(n+1)$ fields. Seemingly we are facing a more
complicated problem than the original NLSM, but this is not the case.

Finally, let us comment that so far by using the symbols $\sigma^{\pm}$ for
the world-sheet coordinates, along with $\partial_{\pm}$ for the derivatives,
we have implicitly assumed that the world-sheet is Minkowksi. Nevertheless,
the above analysis is valid also for Euclidean world-sheet. The main
difference is that for Euclidean world-sheet $\mathcal{A}_{\pm}$ and are
complex and are related by complex conjugation. This is also the case for
${\bf a}_{\pm}$ . Also, the group of transformations in equation (38) is
defined in the field of complex numbers, i.e. it is of the form
$SO(p,q;\mathbb{C})$.

In the following we consider that the world-sheet is Minkwoski and reformulate
the theory, so that it depends only on the physical degrees of freedom. As the
light-like reduced theory is still conformal, whereas conformal invariance is
explicitly broken for time-like and space-like reductions, the reformulation
is different for these two cases. Thus, we have to treat these cases
separately.

###  2.3 Time-like & space-like reductions

For $t^{2}=1$ the Pohlmeyer reduction is very similar to the one of the sphere
Pohlmeyer:1979ch ; Katsinis:2022amo . The idea is to introduce $n\times n$
matrices $\tilde{A}_{\pm}$, which constitute the components of a flat
connection. So, we define $\tilde{A}_{\pm}$ as follows

$\tilde{A}_{+}=\tilde{S}\begin{pmatrix}\mathcal{A}_{+}&\frac{f}{f^{2}-t^{2}}\frac{\mathcal{S}{\bf
a}_{+}}{F(f)}\\\ -\frac{f}{f^{2}-t^{2}}\frac{{\bf
a}^{T}_{+}\mathcal{S}}{F(f)}&0\end{pmatrix},\quad\tilde{A}_{-}=\tilde{S}\begin{pmatrix}\mathcal{A}_{-}&\tilde{s}F(f){\mathcal{S}\bf
a}_{-}\\\ -\tilde{s}F(f){\bf
a}^{T}_{-}\mathcal{S}&0\end{pmatrix},\quad\tilde{S}=\begin{pmatrix}\mathcal{S}&0\\\
0&\tilde{s}\end{pmatrix}$ (39)

where $\tilde{s}=\pm 1$. Henceforth we drop the argument of $f(\varphi)$.
Notice that by definition $\tilde{A}_{\pm}$ obey
$\tilde{A}_{\pm}^{T}=-\tilde{S}\tilde{A}_{\pm}\tilde{S}$, thus they are valued
in the Lie algebra of an indefinite orthogonal. The matrix $\tilde{S}$ is the
Killing metric of this Lie group. We also define $\tilde{\bf{a}}_{\pm}$ and
$\tilde{\bf{Z}}$ via the equations

$\tilde{\bf{a}}_{\pm}=\begin{pmatrix}\bf{a}_{\pm}\\\
0\end{pmatrix},\quad\tilde{\bf{Z}}=\begin{pmatrix}\bf{0}\\\ 1\end{pmatrix}.$
(40)

It is a matter of algebra to show that $\tilde{\bf{a}}_{\pm}$ obey

$\displaystyle\tilde{\mathcal{D}}_{+}\tilde{{\bf a}}_{-}$
$\displaystyle=t\frac{f^{\prime}\partial_{-}\varphi}{f^{2}-t^{2}}{\bf
a}_{+}+\frac{\tilde{s}}{F(f)}\frac{f}{f^{2}-t^{2}}\left(\tilde{\bf{a}}^{T}_{-}\tilde{S}\tilde{\bf{a}}_{+}\right)\tilde{\bf{Z}},$
(41) $\displaystyle\tilde{\mathcal{D}}_{-}\tilde{{\bf a}}_{+}$
$\displaystyle=t\frac{f^{\prime}\partial_{+}\varphi}{f^{2}-t^{2}}{\bf
a}_{-}+F(f)\left(\tilde{\bf{a}}^{T}_{-}\tilde{S}\tilde{\bf{a}}_{+}\right)\tilde{\bf{Z}},$
(42)

where
$\tilde{\mathcal{D}}_{\pm}=I_{n}\partial_{\pm}-\tilde{\mathcal{A}}_{\pm}$,
while (37) assumes the form

$\partial_{+}\partial_{-}\varphi=\frac{\tilde{{\bf
a}}_{-}^{T}\tilde{\mathcal{S}{\bf
a}}_{+}}{f^{\prime}}-s\frac{m_{+}m_{-}}{\Lambda^{2}}\frac{f^{2}-t^{2}}{f^{\prime}}.$
(43)

One can show that the matrices $\tilde{\mathcal{A}}_{\pm}$ indeed obey the
zero curvature condition

$\partial_{-}\tilde{\mathcal{A}}_{+}-\partial_{+}\tilde{\mathcal{A}}_{-}+\left[\tilde{\mathcal{A}}_{+},\tilde{\mathcal{A}}_{-}\right]=0,$
(44)

provided that the function $F$ is given by

$F(f)=\pm\frac{1}{\sqrt{\tilde{s}t\left(1-f^{2}\right)}},\qquad\text{or}\qquad
F(f)=\pm\frac{x}{\sqrt{\tilde{s}t\left(1-f^{2}\right)}}.$ (45)

For a given choice of $f$, which is defined in $\eqref{eq:inner_product}$, and
$t$, the value of $\tilde{s}$ should be appropriate so that the square roots
are well defined. As the $\pm$ signs can be absorbed by redefining ${\bf
a}_{\pm}$, one can select

$F(f)=\frac{1}{\sin\phi}\qquad F(f)=-\cot\phi$ (46)

for $f=\cos\varphi$, which requires $\tilde{s}t=1$, and

$F(f)=\frac{1}{\sinh\phi}\qquad F(f)=-\coth\phi,$ (47)

for $f=\cosh\varphi$, which requires $\tilde{s}t=-1$. Notice that in both
cases for either choices of $F$, appearing directly in $\tilde{A}^{-}$, the
other one appears on the matrix $\tilde{A}^{+}$. Thus, the existence of two
solutions restores the $+\leftrightarrow-$ symmetry.

Finally, let us comment that by promoting $\tilde{\bf{Z}}$ to dynamical field,
the reduced system, as described by (41) - (44), has the following gauge
redundancy

$\tilde{\bf a}_{\pm}\rightarrow\tilde{O}\tilde{\bf
a}_{\pm},\qquad\mathcal{S}\mathcal{A}_{\pm}\rightarrow\tilde{O}\tilde{\mathcal{S}}\tilde{\mathcal{A}}_{\pm}\tilde{O}^{T}+\left(\partial_{\pm}\tilde{O}\right)\tilde{O}^{T},\qquad\tilde{\bf
Z}\rightarrow\tilde{O}\tilde{\bf Z},$ (48)

where $\tilde{O}$ belongs to the indefinite orthogonal Lie group having
$\tilde{S}$ as Killing metric. Nevertheless, since $\tilde{\bf{Z}}$ in not
dynamical, we can interpret the theory as a gauge fixed version of a theory
originally having this symmetry. This is expected by the Lagrangian
formulation of SSSGs as gauge fixed WZW models defined on an appropriate
coset, which are perturberbed by an integrable potential Bakas:1995bm ;
Miramontes:2008wt . The group of the aforementioned symmetry coincides with
the subgroup used to define the coset of the Lagrangian formulation.

In order to continue the derivation, we use equation (44) to solve for
$\tilde{\bf a}_{\pm}$. More specifically, if follows that
$\tilde{A}^{\pm}=\left(\partial_{\pm}\tilde{O}\right)\tilde{S}\tilde{O}^{T}\tilde{S}$,
which implies that $\tilde{\bf a}_{\pm}$ are given by

$\tilde{\bf{a}}_{+}=F(f)\frac{1-f^{2}}{f}\mathcal{O}\partial_{+}{\bf
Z},\qquad\tilde{\bf{a}}_{-}=-\frac{\tilde{s}}{F(f)}\mathcal{O}\partial_{-}{\bf
Z},\qquad{\bf Z}=\tilde{S}\mathcal{O}^{T}\tilde{S}\tilde{\bf{Z}},$ (49)

where the norm of ${\bf Z}$ is ${\bf Z}^{T}\tilde{S}{\bf Z}=\tilde{s}$. The
equations of the reduced theory, taking (45) into account, assume the form

$\displaystyle\partial_{+}\partial_{-}\varphi=$
$\displaystyle\tilde{s}\frac{f^{2}-1}{ff^{\prime}}\partial_{-}{\bf
Z}^{T}\tilde{\mathcal{S}}\partial_{+}{\bf
Z}-s\frac{m_{+}m_{-}}{\Lambda^{2}}\frac{f^{2}-1}{f^{\prime}},$ (50)
$\displaystyle\partial_{+}\partial_{-}{\bf Z}=$
$\displaystyle\frac{f}{\tilde{F}^{2}}\frac{f^{\prime}}{1-f^{2}}\partial_{+}\varphi\partial_{-}{\bf
Z}+\frac{\tilde{F}^{2}}{f}\frac{f^{\prime}}{1-f^{2}}\partial_{-}\varphi\partial_{+}{\bf
Z}-\tilde{s}\left(\partial_{-}{\bf Z}^{T}\tilde{\mathcal{S}}\partial_{+}{\bf
Z}\right){\bf Z},$ (51)

where $\tilde{F}=1$ or $\tilde{F}=f$ and $\tilde{\mathcal{S}}$ is given by
(39). The two choices for $F$, which are given in (45), and equivalently of
$\tilde{F}$, restore the symmetry $\sigma^{+}\leftrightarrow\sigma^{-}$. In
the Lagrangian formulation of the theory these correspond to vector and axial
gauging of the WZW model, see Miramontes:2008wt . We remind the reader that
the choice of $f$, $\tilde{s}$ and $t$ are correlated so that $F$ is real. As
the norm of ${\bf Z}$ is constraint, the reduced theory depends on $n$ fields.
Thus, the reduction eliminates one field from the NLSM. Moreover, as
advertised the reduced theory depends only on the product $m_{+}m_{-}$.
Finally, notice that unless the enhanced space is 3-dimensional, so that the
reduced theory consists only of the primary Pohlmeyer field, the equations we
derived are non-Lagrangian. Non-local field redefinitions are required to give
a Lagrangian description to the reduced theory, see Grigoriev:2007bu  and
references therein. This topic is also discussed in the section A.

###  2.4 Light-like reduction

Let us proceed to the study of the light-like reduction. Our approach is
motivated by Dorn:2009kq ; Hoare:2012nx . The light-like Pohlmeyer reduction
is fundamentally different than the time-like and space-like ones. The basic
difference is that equations (36) are source free. This prevents us from using
the approach of the previous subsection. Equations (45) are not defined in for
$t=0$. As $\mathcal{S}^{2}=I_{n-1}$ and
$\mathcal{A}_{\pm}^{T}=-\mathcal{A}_{\pm}$ it follows that

$\partial_{\pm}\left({\bf a}_{\mp}^{T}\mathcal{S}{\bf a}_{\mp}\right)=0,$ (52)

thus the norms of ${\bf a}_{\pm}$ depend on a sole world-sheet coordinate,
i.e.

${\bf a}_{\pm}^{T}\tilde{S}{\bf a}_{\pm}=f_{\pm}\left(\sigma^{\pm}\right).$
(53)

Implementing the gauge transformation (38) we can simplify the problem
choosing the light-cone gauge $\mathcal{A}_{-}=0$. Then, equations (35) and
(36) assume the form

$\mathcal{S}\partial_{-}\mathcal{A}_{+}=-\exp\left(-\varphi\right)\left[{\bf
a}_{-}{\bf a}_{+}^{T}-{\bf a}_{+}{\bf a}_{-}^{T}\right]\mathcal{S}$ (54)

and

$\partial_{+}{\bf a}_{-}=\mathcal{S}\mathcal{A}_{+}{\bf
a}_{-},\qquad\partial_{-}{\bf a}_{+}=0.$ (55)

These equations have the residual chiral gauge redundancy

${\bf a}_{\pm}\rightarrow O_{+}{\bf
a}_{\pm},\qquad\mathcal{S}\mathcal{A}_{\pm}\rightarrow
O_{+}\mathcal{S}\mathcal{A}_{\pm}O_{+}^{-1}+\left(\partial_{\pm}O_{+}\right)O_{+}^{-1},$
(56)

where $O_{+}=O_{+}(\sigma^{+})$ and $O_{+}^{-1}=\mathcal{S}O_{+}\mathcal{S}$,
implying $O_{+}\in G$. As ${\bf a}_{+}={\bf a}_{+}(\sigma_{+})$, we can always
use an appropriate transformation in order to set $\left({\bf
a}_{+}\right)_{i}=u_{+}(\sigma^{+})\delta_{i,(n-1)}$ if its norm is positive
or negative. In the degenerate case of ${\bf a}_{+}$ being null, we can set
$\left({\bf
a}_{+}\right)_{i}=u_{+}(\sigma^{+})\left(\delta_{i,1}+\delta_{i,(n-1)}\right)$,
where $s_{1}s_{n-1}=-1$, see (33). From now on we assume that the norm is non-
vanishing, thus we define

${\bf a}_{+}=\begin{pmatrix}{\bf 0}\\\ u_{+}\end{pmatrix},\qquad
u_{+}=u_{+}(\sigma^{+}),$ (57)

where ${\bf 0}$ is a $(n-2)\times 1$ column matrix consisting of zeros. For
notational convenience, we also drop the argument of $u_{+}(\sigma^{+})$. As
equation (57) breaks the symmetry group G to its maximal subgroup, we
decompose ${\bf a}_{-}$ and $\mathcal{S}$, which is defined in equation (33),
in a similar manner as

${\bf a}_{-}=\begin{pmatrix}{\bf v}\\\
v\end{pmatrix},\qquad\mathcal{S}=\begin{pmatrix}\bar{\mathcal{S}}&{\bf 0}\\\
{\bf 0}^{T}&s_{n-1}\end{pmatrix},$ (58)

where ${\bf v}$ is a $(n-2)\times 1$ column matrix and $\bar{\mathcal{S}}$ is
a $(n-2)\times(n-2)$ matrix. Then, equation (54) implies that
$\mathcal{A}_{+}$ has the structure

$\mathcal{A}_{+}=\begin{pmatrix}{\bf 0}_{n-2}&{\bf\Lambda}\\\
-{\bf\Lambda}^{T}&0\end{pmatrix},$ (59)

where ${\bf 0}_{n-2}$ is a $(n-2)\times(n-2)$ matrix consisting of zeros.
Moreover, it follows that ${\bf\Lambda}$ satisfies the equation

$\partial_{-}{\bf\Lambda}=-\exp\left(-\varphi\right)s_{n-1}u_{+}\bar{\mathcal{S}}{\bf
v}.$ (60)

Finally, equation (55) implies

${\bf\Lambda}=\frac{1}{v}\bar{\mathcal{S}}\partial_{+}{\bf
v},\qquad\partial_{+}v=-s_{n-1}{\bf\Lambda}^{T}{\bf v}.$ (61)

The second equation is equivalent to the fact that ${\bf
a}_{-}^{T}\tilde{S}{\bf a}_{-}=f_{-}\left(\sigma^{-}\right)$. Putting
everything together, the equations of the reduced theory read

$\displaystyle\partial_{-}\left(\frac{1}{v}\partial_{+}{\bf v}\right)$
$\displaystyle=-u_{+}u_{-}\exp\left(-\varphi\right){\bf v},$ (62)
$\displaystyle\partial_{+}\partial_{-}\varphi$
$\displaystyle=u_{+}u_{-}v\exp\left(-\varphi\right)-s\frac{m_{+}m_{-}}{\Lambda^{2}}\exp\left(\varphi\right),$
(63)

where we discarded the sign $s_{n-1}$ (it can be absorbed into the definition
of $v$) and rescaled ${\bf v}$ and $v$ with $u_{-}(\sigma^{-})$ so that

${\bf a}_{-}=u_{-}\begin{pmatrix}{\bf v}\\\ v\end{pmatrix},\qquad
u_{-}=u_{-}(\sigma^{-}),$ (64)

where ${\bf v}^{T}\bar{\mathcal{S}}{\bf v}+s_{n-1}v^{2}=\bar{s}$ and
$\bar{s}=\pm 1$. Notice that for $t=0$ equation (28) implies that

$\left(\partial^{2}_{\pm}Y\right)\cdot\left(\partial^{2}_{\pm}Y\right)=m^{2}_{\pm}{\bf
a}_{\pm}^{T}\tilde{S}{\bf a}_{\pm},$ (65)

thus

$\left(\partial^{2}_{+}Y\right)\cdot\left(\partial^{2}_{+}Y\right)=s_{n-1}m^{2}_{+}u_{+}^{2},\qquad\left(\partial^{2}_{-}Y\right)\cdot\left(\partial^{2}_{-}Y\right)=\bar{s}m^{2}_{-}u_{-}^{2}.$
(66)

Using a world-sheet diffeomorphism acting as $\sigma^{\pm}\rightarrow
f_{\pm}(\sigma^{\pm})$ so that $f^{\prime 2}u^{4}_{\pm}=1$, we set
$\left(\partial^{2}_{\pm}Y\right)\cdot\left(\partial^{2}_{\pm}Y\right)$ to
constants.555Notice that no cross-terms are introduced as
$\left(\partial_{\pm}Y\right)\cdot\left(\partial_{\pm}Y\right)=0$ and
$\left(\partial_{\pm}Y\right)\cdot\left(\partial_{\pm}^{2}Y\right)=0$.
Finally, equations (62) and (63) assume the form

$\displaystyle\partial_{-}\left(\frac{1}{v}\partial_{+}{\bf v}\right)$
$\displaystyle=\mu\exp\left(-\varphi\right){\bf v},$ (67)
$\displaystyle\partial_{+}\partial_{-}\varphi$
$\displaystyle=\mu\left[v\exp\left(-\varphi\right)-s_{\mu}\exp\left(\varphi\right)\right],$
(68)

where we set $\varphi\rightarrow\varphi+\ln\frac{\sqrt{|u_{+}||u_{-}|}}{\mu}$,
$\mu=\left|\frac{m_{+}m_{-}}{\Lambda^{2}}\right|^{1/2}$ and
$s_{\mu}=s\frac{m_{+}m_{-}}{\mu^{2}\Lambda^{2}}$. We also discarded the
factors $\frac{u_{+}u_{-}}{|u_{+}||u_{-}|}$ since the theory is invariant
under $v\rightarrow-v$.

##  3 The solution of the auxiliary system

As we presented the Pohlmeyer reduction of the rank 1 cosets related to the
indefinite orthogonal group, we proceed to the the solution of the auxiliary
system. Initially we discuss the mapping connecting the vectors of enhanced
space with the elements of coset. Then, we derive the formal solution of the
auxiliary system.

###  3.1 The mapping

In order to assign an element of the coset to each vector of the enhanced
space we use the mapping

$g=\theta\left(I_{n+2}-2s\frac{YY^{T}\eta}{\Lambda^{2}}\right),\qquad\theta=\left(I_{n+2}-2s\frac{Y_{0}Y_{0}^{T}\eta}{\Lambda^{2}}\right),$
(69)

where $I_{n}$ denotes the $n\times n$ identity matrix and $\eta$ is the metric
of the enhanced space, see (11). Recall that the parameter $s$ is defined via
the equation $Y^{T}\eta Y=s\Lambda^{2}$ and gets the values $\pm 1$. Finally,
$Y_{0}$ is a constant vector of the enhanced space, which implies that it also
obeys $Y_{0}^{T}\eta Y_{0}=s\Lambda^{2}$. It is straightforward to show that
$\theta$ obeys $\theta^{2}=I_{n+2}$, as well as $\theta^{T}\eta\theta=\eta$.
Similarly, $g$ has the following properties

$\bar{g}=g,\qquad g^{T}\eta g=\eta\qquad g\theta g\theta=I_{n+2}.$ (70)

Thus $g$ is an element of the desired coset.

###  3.2 The formal solution of the auxiliary system

Having set the ground, in this section we derive the solution of the auxiliary
system (2). Initially, we define $\Psi=g\hat{\Psi}$, so that the auxiliary
system assumes the form

$\partial_{\pm}\hat{\Psi}=\pm\frac{2\lambda}{1\pm\lambda}\hat{j}_{\pm}\hat{\Psi},$
(71)

where the currents $\hat{j}_{\pm}$ are defined as
$\hat{j}_{\pm}=-\frac{1}{2}g^{-1}\partial_{\pm}g$. Their explicit form is

$\hat{j}_{\pm}=\frac{s}{\Lambda^{2}}\left[\left(\partial_{\pm}Y\right)Y^{T}-Y\partial_{\pm}Y^{T}\right]\eta.$
(72)

Interestingly enough, equation (71) implies that $\hat{\Psi}(0)$ is constant.
This is a crucial remark in order to gain intuition about the form of the
stucture of the solution of the auxiliary system. The only linear system of
equation that is somehow related to the problem is (23). Its natural to guess
that the solution depends linearly on $V$, which is $\lambda$ independent, and
the rest of the solution reduced to $V^{-1}$ for $\lambda=0$. This is close,
but its not correct. It turns out the correct ansatz is

$\hat{\Psi}=V^{-1}\Delta^{-1}\tilde{\Psi},$ (73)

where the matrix $\Delta$ is defined as

$\Delta=\begin{pmatrix}I_{n-1}&0&0&0\\\ 0&\frac{1+\lambda}{1-\lambda}&0&0\\\
0&0&\frac{1-\lambda}{1+\lambda}&0\\\ 0&0&0&1\end{pmatrix},$ (74)

while the matrix $V$ is constructed by the vectors of the basis used in the
Pohlmeyer reduction, see (23). Taking equations (22) into account, the
auxiliary system assumes the form

$\partial_{\pm}\tilde{\Psi}=\Delta\left[A^{\pm}\pm\frac{2\lambda}{1\pm\lambda}V\hat{j}_{\pm}V^{-1}\right]\Delta^{-1}\tilde{\Psi}.$
(75)

It is a matter of algebra to show that the above pair of equations assumes the
form

$\partial_{\pm}\tilde{\Psi}=\left[A^{\pm}|_{m_{\pm}\rightarrow\frac{1\mp\lambda}{1\pm\lambda}m_{\pm}}\right]\tilde{\Psi}.$
(76)

For completeness we provide some details of this calculation in appendix B.
Comparing equations (76) and (23) it follows that

$\tilde{\Psi}=V|_{m_{\pm}\rightarrow\frac{1\mp\lambda}{1\pm\lambda}m_{\pm}}.$
(77)

Putting everything together, the solution of the auxiliary system reads

$\Psi(\lambda)=gV^{-1}\Delta^{-1}\left(V|_{m_{\pm}\rightarrow\frac{1\mp\lambda}{1\pm\lambda}m_{\pm}}\right)$
(78)

and coincides with the result of Katsinis:2022amo , which is expected as the
$O(n)$ NLSM is a special case of the models under consideration. Let us also
mention that the solutions was actually guessed using the result of the $O(3)$
NLSM Katsinis:2020avk , which was expressed in terms of the embedding
functions.

###  3.3 Properties of the solution

Having solved the auxiliary system, we present some basic properties of the
solution. To begin with, it is straightforward to show that the solution is
normalized so that

$\Psi(0)=g.$ (79)

Of course, the set of solutions of the auxiliary system is of the form
$\Psi(\lambda)C(\lambda)$ where $C(\lambda)$ is a constant, in general
$\lambda$-dependent, matrix.

As $g$, belonging to a coset, obeys the relations in (69), it turns out that
there are some consistency conditions for the solution of the auxiliary system
(2). Since $g$ is real it follows that for Minkowski world-sheet
$\Psi(\lambda)$ has to satisfy

$\bar{\Psi}(\bar{\lambda})=\Psi(\lambda).$ (80)

Given that the NLSM solution $Y$ is real function of the real parameters
$m_{\pm}$, the solution (78) satisfies this constraint. In the case of
Euclidean world-sheet, since $\partial_{\pm}$ are interchanged by complex
conjugation, the analogous equation is

$\bar{\Psi}(\bar{\lambda})=\Psi(-\lambda).$ (81)

The NLSM solution $Y$ is function of the complex parameters $m_{\pm}$. Notice
that $m_{\pm}$ are related by complex conjugation, thus the constraint is
indeed satisfied.

Since $g^{T}\eta g=\eta$ it follows that

$\Psi\eta\Psi^{T}=\Psi^{T}\eta\Psi=\eta.$ (82)

Using equations (143) and (144), along with equations (147)-(149) and the
properties of the vectors $v_{\alpha}$, which constitute the basis, it is
straightforward to show that

$\Delta^{-1}\left(V|_{m_{\pm}\rightarrow\frac{1\mp\lambda}{1\pm\lambda}m_{\pm}}\right)\eta\left(V^{T}|_{m_{\pm}\rightarrow\frac{1\mp\lambda}{1\pm\lambda}m_{\pm}}\right)\Delta^{-1}\left(V^{-1}\right)^{T}=V\eta,$
(83)

which proves that $\Psi\eta\Psi^{T}=\eta$. One can prove the other equality in
a similar manner.

Finally, as $\theta g\theta g=I_{n+2}$ it follows that
$g\theta\Psi(1/\lambda)\theta$ belongs to the set of solutions of the
auxiliary system. Thus, one has to impose the constraint

$g\theta\Psi(1/\lambda)\theta=\Psi(\lambda)M(\lambda),$ (84)

where $M(\lambda)$ is a constant, in general $\lambda$-dependent, matrix.
Using equation (73) along with $\Psi=g\hat{\Psi}$ we obtain

$\mathcal{I}\tilde{\Psi}(1/\lambda)\theta=\tilde{\Psi}(\lambda)M(\lambda),\quad\mathcal{I}=\begin{pmatrix}I_{n-1}&0\\\
0&-I_{3}\end{pmatrix}.$ (85)

Equation (77) implies that the above constraint is equivalent to the fact that
the following equation

$\mathcal{I}V|_{m_{\pm}\rightarrow-m_{\pm}}=VM(\lambda),$ (86)

is true for any $m_{\pm}$ for some constant matrix $M$.

##  4 Discussion

In this work we demonstrated that all rank 1 symmetric spaces related to
indefinite orthogonal groups have a property that has been unnoticed for more
than 4 decades. One can obtain the formal solution of the auxiliary system,
which imprints the integrability of the models, corresponding to an arbitrary
NLSM solution. The solution of the auxiliary system is closely related to the
Pohlmeyer reduction, revealing another link between the NLSM and its reduced
theory. In view of this result the dressing method is essentially the
implementation of a non-linear superposition principle. The seed solution is
combined appropriately with a “virtual” one in order to obtain a new solution.
The “virtual” solution is constructed systematically from the seed one by the
substitution (9). This implies that the “virtual” solution solves the
equations of motion, shares the same Pohlmeyer counterpart with the seed
solution, but belongs to the complexification of the coset. Even though we
focus on theories having a Minkowski world-sheet, our results apply for
Euclidean world-sheet too.

Since the relation between the NLSM and the reduced theory, along with their
relevance for AdS/CFT correspondence, is analyzed extensively in the
introduction, in the following we mostly present some future directions and
ideas which to our opinion worth exploring.

First of all, it would be interesting to study more extensively the Pohlmeyer
reduced. The study of the Bäcklund transformations of the reduced theory
following the approach of Katsinis:2020avk  is of particular interest. More
specifically, one could show explicitly that the Pohlmeyer reduced theory
corresponding to the dressed solution, which is constructed using the simplest
dressing factor, is related with a Bäcklund transformation to the Pohlmeyer
reduced theory of the seed solution. As a byproduct, the relation between the
location of the poles of the dressing factor will be related to the parameter
of the Bäcklund transformation. Finally, a more extended study of the reduced
theories in the case of Euclidean world-sheet would be compelling.

Our results introduce a new perspective on long strings, i.e. strings in the
decompactification limit of their world-sheet. Considering them as dressed
versions of short strings, this work implies that the dressed strings are
superpositions of the short strings. Thus, their properties are inherited from
the short strings. This point of view is closely related to the study of
properties of dressed strings along the lines of Katsinis:2019oox ;
Katsinis:2019sdo . These works relate the stability of classical strings to
the dressed version of these strings. It turns out that the configurations are
unstable whenever superluminal solitons can propagate on the background of the
Pohlmeyer counterpart of them. Nevertheless, one should notice that the
boundary conditions are crucial for such investigations, therefore an abstract
study may be very challenging. On the contrary our results make the stability
analysis of specific string solutions pretty straightforward.

As this work indicates that the non-linear superposition concerns a wide class
of symmetric spaces, its natural to wonder whether this could be the case for
even more theories. Before considering a generic symmetric space, a prime
candidate would be $\mathbb{CP}^{n}$. In view of recent developments regarding
the construction of integrable 2-dimensional theories, the study of $\lambda$
Sfetsos:2013wia , Yang-Baxter / $\eta$ Klimcik:2008eq ; Delduc:2013fga ;
Delduc:2013qra , bi-Yang-Baxter Klimcik:2014bta  deformations and
generalizations of them is definitely is interesting. Regarding the
$\lambda$-deformations, the Pohlmeyer reductions has been presented in
Hollowood:2014rla , providing a starting point for the aforementioned study.

The original and the reduced theory are closely related at classical level. It
is natural to wonder what happens when quantum corrections are taken into
account. In the case of $O(3)$ NLSM and the sine-Gordon equations, the quantum
theories are very different. As the conformal symmetry of the NLSM is
anomalous, the discrepancy is not surprising. Turning this argument on its
head, it could be the case that a theory, which is conformal at quantum level,
such as superstrings on AdS${}_{5}\times$S5 is somehow related to the
quantized reduced theory. The natural regime for such an investigation is the
BMN particle Berenstein:2002jq , which is related to the vacuum of the reduced
theory. One can show that the S-matrices of both theories are related
Hoare:2009fs ; Hoare:2010fb ; Hoare:2011fj . The same is true for the
partition functions too Hoare:2009rq ; Iwashita:2010tg ; Iwashita:2011ha .

These results motivate various studies related to this work. As supersymmetry
seems to bind together the NLSM and the reduced theory, generalizing the
presented construction in the case of symmetric spaces related to supercosets
would certainly be interesting. First of all, supercosets have only been
studied group theoretically. Even establishing such a reduction as an
embedding problem is intriguing. The supersymmetric extension of the non-
linear superposition could be used to further probe the spectral problem of
AdS/CFT at classical level by constructing explicitly the quasi-momenta and
the spectral curve of Beisert:2005bm .

Given a seed solution, the superposition creates an infinite tower of
classical vacua of the theory. Of course, perturbation theory around the seed
solution fails to describe these vacua, thus the presented construction is
essential in order describe them. In the spirit of the previous paragraph one
could study the semi-classical quantization of the the theory around different
vacua, which essentially correspond to different instanton charge. This could
also shed light on the relation between the NLSM and the reduced theory in the
quantum regime. A related subject concerns the study of admissible
configuations of the path integral Krichever:2020tgp .

Finally, although we mentioned the spectral curve Beisert:2005bm  in order to
motivate the introduction of supersymmetry, even without supersymmetry the
spectral curve deserves a thorough study. A direct consequence of the
construction we presented is that we can calculate explicitly the monodromy
matrix. As a first step, it would be interesting to calculate explicitly the
quasi-momenta corresponding to a specific solution. Equation (3) implies that
the monodromy matrix reads

$T(\sigma_{f},\sigma_{i};\lambda)=g(\tau,\sigma_{f})V^{-1}(\tau,\sigma_{f})\Delta^{-1}(\lambda)\left(V(\tau,\sigma_{f})|_{m_{\pm}\rightarrow\frac{1\mp\lambda}{1\pm\lambda}m_{\pm}}\right)\\\
\cdot\left(V^{-1}(\tau,\sigma_{i})|_{m_{\pm}\rightarrow\frac{1\mp\lambda}{1\pm\lambda}m_{\pm}}\right)\Delta(\lambda)V(\tau,\sigma_{i})g^{-1}(\tau,\sigma_{i}).$
(87)

Notice that $\tau$ is the world-sheet temporal coordinate, which for a general
manifold is non-trivially connected to the target space temporal coordinate
$t$. Considering the $O(n)$ model, i.e. stings on
$\mathbb{R}_{t}\times\textrm{S}^{n-1}$, the target space time $t$ is given by
$t=m_{+}\sigma^{+}+m_{-}\sigma^{-}$. In this case one can boost the solution
in order to achieve $t\sim\tau$.

Leaving this subtlety aside and assuming periodic boundary conditions, the
eigenvalues of the monodromy matrix coincide with the eigenvalues of the
matrix

$\mathcal{T}=\left(V(\tau,\sigma_{f})V^{-1}(\tau,\sigma_{i})\right)|_{m_{\pm}\rightarrow\frac{1\mp\lambda}{1\pm\lambda}m_{\pm}}.$
(88)

The point to make is that even though the NLSM solution and correspondingly
the coset element $g$ and the matrix $V$ satisfy periodic boundary conditions
for $m_{\pm}$, the rescaling of these parameters spoils the periodic boundary
conditions. The non-trivial monodromy emerges because the NLSM solution is
periodic for specific $m_{\pm}$. Of course, one should perceive the notation
of (88) as rescaling $m_{\pm}$ first and then multiplying the two terms. In
any case, the quasi-momenta are directly linked to the embedding of the NLSM
solution in the enhanced space. It would be interesting to calculate them
explicitly in terms of the embedding functions or relate them with the reduced
theory.

Naturally, identifying in field theory the analogue of the construction we
presented is very interesting. Even though it seems far-fetched, AdS/CFT
correspondence motivates the existence of a similar structure as the one that
was presented in the work.

###### Acknowledgements.

The work of the author was supported by FAPESP grant 2021/01819-0.

##  Appendix A On the definition of the basis in the enhanced space

In this appendix we apply the formalism developed in section 2. We restrict
our analysis to setups which are relevant for AdS/CFT. First of all we have to
clarify the definition of the primary Pohlmeyer field along with the
construction of the basis in the enhanced space. In order to do so we have to
distinguish between time-like world-sheets and space-like ones, for instance
see Dorn:2009kq , which depends on the world-sheet metric. The situation is
more complicated when the space is of indefinite signature.

As mentioned the type of reduction affects the construction of the basis in
the enhanced space. One should make sure that the basis indeed spans the
enhanced space and that the inner product of $\partial_{+}Y$ and
$\partial_{-}Y$ is appropriately defined. The basis naturally splits the
enhanced space as the direct product $\mathcal{M}_{3}\times\mathcal{M}_{n-1}$,
where $\mathcal{M}_{3}$ is spanned by $Y$ and $\partial_{\pm}Y$. In the case
of AdS the enhanced space is $\mathbb{R}^{(2,n)}$, while in the case of dS it
is $\mathbb{R}^{(1,n+1)}$. Thus, in the former case the basis should consist
of 2 time-like vectors and $n$ space-like, while in the latter of $1$ time-
like and $n+1$ space-like ones. In the following we analyse the definition of
the primary Pohlmeyer field for each type of reduction in the case of AdS.
Similar analysis also applies to dS space, as well as on all other rank 1
symmetric spaces based on indefinite orthogonal groups. As a warm up we first
analyze the case of sphere for both choices of world-sheet signatures.

For Minkowski world-sheets we use the following conventions

$\sigma^{\pm}=\frac{1}{2}\left(\sigma\pm\tau\right),\quad\partial_{\pm}=\partial_{\sigma}\pm\partial_{\tau},$
(89)

while for Euclidean ones

$\sigma^{\pm}=\frac{1}{2}\left(\sigma\pm
i\tau\right),\quad\partial_{\pm}=\partial_{\sigma}\mp i\partial_{\tau}.$ (90)

###  A.1 Sphere with Minkowski world-sheet

We begin our presentation by discussing the basis in the case the target space
is a sphere and the world-sheet is Minkowski. The inner product
$\partial_{+}X\cdot\partial_{-}X$ satisfies the Cauchy-Schwartz inequality

$\partial_{+}X\cdot\partial_{-}X\leq\left|\partial_{+}X\right|\left|\partial_{-}X\right|=|m_{+}m_{-}|.$
(91)

Thus, the only consistent definition of the Pohlmeyer field is

$\partial_{+}X\cdot\partial_{-}X=m_{+}m_{-}\cos\varphi.$ (92)

Of course, the norms of all vectors spanning $\mathcal{M}_{n-1}$ are positive.
It is straightforward to calculate that

$\displaystyle\partial_{\sigma}X\cdot\partial_{\sigma}X$
$\displaystyle=\frac{m_{+}^{2}+m_{-}^{2}}{4}+\frac{m_{+}m_{-}}{2}\cos\varphi,$
(93) $\displaystyle\partial_{\tau}X\cdot\partial_{\tau}X$
$\displaystyle=\frac{m_{+}^{2}+m_{-}^{2}}{4}-\frac{m_{+}m_{-}}{2}\cos\varphi,$
(94) $\displaystyle\partial_{\sigma}X\cdot\partial_{\tau}X$
$\displaystyle=\frac{m_{+}^{2}-m_{-}^{2}}{4}.$ (95)

Both vectors $\partial_{\sigma}X$ and $\partial_{\tau}X$ have positive
definite norm as required.

###  A.2 Sphere with Euclidean world-sheet

Even though we will not discuss Euclidean world-sheets further, we present the
basis used in the reduction when the target space is a hypersphere. In this
case it follows that

$\partial_{+}X\cdot\partial_{-}X=\partial_{\sigma}X\cdot\partial_{\sigma}X+\partial_{\tau}X\cdot\partial_{\tau}X\geq
0$ (96)

We define the Pohlmeyer field as

$\partial_{+}X\cdot\partial_{-}X=m_{+}m_{-}\cosh\varphi.$ (97)

The right-hand-side is positive definite since $m_{\pm}=m_{R}\pm im_{I}$,
where $m_{R}$ and $m_{I}$ are real. Again, it is straightforward to calculate
that

$\displaystyle\partial_{\sigma}X\cdot\partial_{\sigma}X$
$\displaystyle=\frac{m_{+}m_{-}}{2}\cosh\varphi+\frac{m_{+}^{2}+m_{-}^{2}}{4}=m_{R}^{2}\cosh^{2}\frac{\varphi}{2}+m_{I}^{2}\sinh^{2}\frac{\varphi}{2},$
(98) $\displaystyle\partial_{\tau}X\cdot\partial_{\tau}X$
$\displaystyle=\frac{m_{+}m_{-}}{2}\cosh\varphi-\frac{m_{+}^{2}+m_{-}^{2}}{4}=m_{R}^{2}\sinh^{2}\frac{\varphi}{2}+m_{I}^{2}\cosh^{2}\frac{\varphi}{2},$
(99) $\displaystyle\partial_{\sigma}X\cdot\partial_{\tau}X$
$\displaystyle=\frac{m_{-}^{2}-m_{+}^{2}}{4i}=-m_{R}m_{I}.$ (100)

Both vectors $\partial_{\sigma}X$ and $\partial_{\tau}X$ have positive
definite norm as required.

###  A.3 AdS time-like reduction & Minkowski world-sheet

In this section we analyze the time-like reduction of AdS spaces. We remind
the reader that this case corresponds to $t=-1$, i.e.
$\partial_{\pm}Y\cdot\partial_{\pm}Y=-m^{2}_{\pm}$. Consider an arbitrary
vector $X$ in the sub-manifold spanned by $\partial_{\pm}Y$. It will be of the
form

$X=a\partial_{+}Y+b\partial_{-}Y.$ (101)

It follows that

$X\cdot
X=-\left(|am_{+}|-|bm_{-}|\right)^{2}-2|am_{+}||bm_{-}|\left(1-\frac{ab\partial_{+}Y\cdot\partial_{-}Y}{|am_{+}||bm_{-}|}\right).$
(102)

If the last parenthesis is positive, all vectors in $\mathcal{M}_{3}$ are of
negative norm, implying that for $f=\cos\varphi$ it follows that
$\mathcal{M}_{3}=\mathbb{R}^{(3,0)}$. Thus, for $f=\cos\varphi$ the basis
doesn’t span the enhanced space $\mathbb{R}^{(2,n)}$. In this case it is
necessary to use $f=\cosh\varphi$, i.e.

$\partial_{+}Y\cdot\partial_{-}Y=m_{+}m_{-}\cosh\varphi,$ (103)

so that $\mathcal{M}_{3}$ can be $\mathbb{R}^{(2,1)}$. We could have
introduced another sign, such as $t$ in this definition, but we opt to take
advantage of the transformation $\alpha\rightarrow\alpha+i\pi$ in order to
deal with this sign. Also notice that

$\displaystyle\partial_{\sigma}Y\cdot\partial_{\sigma}Y$
$\displaystyle=-\frac{m_{+}^{2}+m_{-}^{2}}{4}+\frac{m_{+}m_{-}}{2}\cosh\varphi,$
(104) $\displaystyle\partial_{\tau}Y\cdot\partial_{\tau}Y$
$\displaystyle=-\frac{m_{+}^{2}+m_{-}^{2}}{4}-\frac{m_{+}m_{-}}{2}\cosh\varphi,$
(105) $\displaystyle\partial_{\sigma}Y\cdot\partial_{\tau}Y$
$\displaystyle=-\frac{m_{+}^{2}-m_{-}^{2}}{4}.$ (106)

The consistency of the basis requires

$\cosh\varphi\geq\frac{m_{+}^{2}+m_{-}^{2}}{2|m_{+}m_{-}|},$ (107)

so that one of the vectors $\partial_{\sigma}Y$ and $\partial_{\tau}Y$ is
time-like, while the other one is space-like

In this case, it follows that $\mathcal{S}=I_{n-1}$, i.e. the Killing metric
of $SO(n-1)$. As $f=\cosh\varphi$, it it required that $\tilde{s}t=-1$. For
the time-like reduction $t=-1$, thus $\tilde{s}=1$. As a result
$\tilde{\mathcal{S}}=I_{n}$, which is the Killing metric of $SO(n)$. The
reduced theory is formulated as the $SO(1,n)/SO(n)$ perturbed WZW model, see
section 5.2 of Miramontes:2008wt .

For completeness, we also present the equation of the reduced theory, which
read

$\displaystyle\partial_{+}\partial_{-}\varphi=$
$\displaystyle\tanh\varphi\,\partial_{-}{\bf
Z}^{T}\tilde{\mathcal{S}}\partial_{+}{\bf
Z}+\frac{m_{+}m_{-}}{\Lambda^{2}}\sinh\varphi,$ (108)
$\displaystyle\partial_{+}\partial_{-}{\bf Z}=$
$\displaystyle-\frac{\partial_{+}\varphi\partial_{-}{\bf
Z}}{\tanh\varphi}-\frac{\partial_{-}\varphi\partial_{+}{\bf
Z}}{\sinh\varphi\cosh\varphi}-\left(\partial_{-}{\bf
Z}^{T}\tilde{\mathcal{S}}\partial_{+}{\bf Z}\right){\bf Z}.$ (109)

The complementary form of equation (109) reads

$\partial_{+}\partial_{-}{\bf Z}=-\frac{\partial_{+}\varphi\partial_{-}{\bf
Z}}{\sinh\varphi\cosh\varphi}-\frac{\partial_{-}\varphi\partial_{+}{\bf
Z}}{\tanh\varphi}-\left(\partial_{-}{\bf
Z}^{T}\tilde{\mathcal{S}}\partial_{+}{\bf Z}\right){\bf Z}.$ (110)

Equation (108) and (109) appear in exact the same form in Grigoriev:2007bu ;
Rashkov:2008rm . Regarding Miramontes:2008wt , in the case of AdS2 the
equations obviously coincide, nevertheless this is not the case for AdS3. The
different form of the equations is due to the parametrization.

Let us show how the different parametrizations are related in the case of
(109). We parametrize ${\bf Z}$ as ${\bf Z}=\begin{pmatrix}\cos u\\\ \sin
u\end{pmatrix}$, while $\tilde{\mathcal{S}}=I_{2}$. It is a matter of algebra
to show that the equations of motion read

$\displaystyle\partial_{+}$
$\displaystyle\partial_{-}\varphi=\tanh\varphi\,\partial_{-}u\partial_{+}u+\frac{m_{+}m_{-}}{\Lambda^{2}}\sinh\varphi,$
(111) $\displaystyle\partial_{+}$
$\displaystyle\left[\left(c+\textrm{sech}\varphi\right)\partial_{-}u\right]-\partial_{-}\left[\left(c+\cosh\varphi\right)\partial_{+}u\right]=0,$
(112)

where $c$ is an free parameter. Implementing the non-local field redefinition

$\partial_{-}u=2\frac{\tanh^{2}\frac{\phi}{2}}{c+\textrm{sech}\varphi}\partial_{-}\theta,\qquad\partial_{+}u=-2\frac{\tanh^{2}\frac{\phi}{2}}{c+\cosh\varphi}\partial_{+}\theta$
(113)

and setting $c=-1$, the equations of motion assume the form

$\displaystyle\partial_{+}$
$\displaystyle\partial_{-}\varphi=\frac{\sinh\varphi}{\cosh^{4}\frac{\phi}{2}}\partial_{-}\theta\partial_{+}\theta+\frac{m_{+}m_{-}}{\Lambda^{2}}\sinh\varphi,$
(114) $\displaystyle\partial_{+}$
$\displaystyle\left[\tanh^{2}\frac{\phi}{2}\,\partial_{-}\theta\right]+\partial_{-}\left[\tanh^{2}\frac{\phi}{2}\,\partial_{+}\theta\right]=0,$
(115)

which is the one appearing in Miramontes:2008wt . Finally, let us mention that
these equations of motion follow from the Lagrangian density

$\mathcal{L}=\frac{1}{4}\partial_{+}\varphi\partial_{-}\varphi+\tanh^{2}\frac{\phi}{2}\partial_{-}\theta\partial_{+}\theta+m_{+}m_{-}\sinh^{2}\frac{\varphi}{2}.$
(116)

Regarding the case of (110), using the same parametrization and the non-local
field redefinition

$\partial_{-}u=-2\frac{\coth^{2}\frac{\phi}{2}}{c+\cosh\varphi}\partial_{-}\theta,\qquad\partial_{+}u=2\frac{\coth^{2}\frac{\phi}{2}}{c+\textrm{sech}\varphi}\partial_{+}\theta$
(117)

for $c=1$ it follows that the equations of motion assume the form

$\displaystyle\partial_{+}$
$\displaystyle\partial_{-}\varphi=-\frac{\sinh\varphi}{\sinh^{4}\frac{\phi}{2}}\partial_{-}\theta\partial_{+}\theta+\frac{m_{+}m_{-}}{\Lambda^{2}}\sinh\varphi,$
(118) $\displaystyle\partial_{+}$
$\displaystyle\left[\coth^{2}\frac{\phi}{2}\,\partial_{-}\theta\right]+\partial_{-}\left[\coth^{2}\frac{\phi}{2}\,\partial_{+}\theta\right]=0,$
(119)

These equations of motion follow from the Lagrangian density

$\mathcal{L}=\frac{1}{4}\partial_{+}\varphi\partial_{-}\varphi+\coth^{2}\frac{\phi}{2}\partial_{-}\theta\partial_{+}\theta+m_{+}m_{-}\sinh^{2}\frac{\varphi}{2}.$
(120)

Finally, in order to point out the relation between the reduced systems
corresponding to time-like and space-like reduction for any $n$, let us
mention that for any $n$ the column ${\bf Z}$ can be parametrized as

${\bf Z}=\begin{pmatrix}\cos u\\\ \sin u\,\hat{n}\end{pmatrix},$ (121)

where $\hat{n}$ is a $(n-1)\times 1$ column matrix of unit norm.

###  A.4 AdS Light-like reduction & Minkowski world-sheet

In this section we analyze light-like reduction of AdS spaces. We remind the
reader that this case corresponds to $t=0$, i.e.
$\partial_{\pm}Y\cdot\partial_{\pm}Y=0$. It is customary to define the
Pohlmeyer field via the equation

$\partial_{+}Y\cdot\partial_{-}Y=m_{+}m_{-}\exp\varphi.$ (122)

In this case, given two light-like vectors we can always define a time-like
and a space-like one. Thus, $\mathcal{M}_{3}=\mathbb{R}^{(2,1)}$, which
implies that $\mathcal{M}_{n-1}=\mathbb{R}^{(0,n-1)}$. It is a matter of
algebra to show that

$\displaystyle\partial_{\sigma}Y\cdot\partial_{\sigma}Y$
$\displaystyle=+\frac{m_{+}m_{-}}{2}\exp\varphi,$ (123)
$\displaystyle\partial_{\tau}Y\cdot\partial_{\tau}Y$
$\displaystyle=-\frac{m_{+}m_{-}}{2}\exp\varphi,$ (124)
$\displaystyle\partial_{\sigma}Y\cdot\partial_{\tau}Y$ $\displaystyle=0.$
(125)

Obviously, one of the vector $\partial_{\sigma}Y$ and $\partial_{\tau}Y$ is
space-like, while the other vector one time-like, as required.

Let is also comment on the reduced system. In the case of AdS3, equation (67)
is satisfies identically as ${\bf v}={\bf 0}$, and (68) reduces to sinh- or
cosh-Gordon. In the degenerate case of ${\bf a}_{-}$ being null, which was
omitted, the equation reduces to the Liouville equation and the configuration
actually lives in AdS2. For AdS4 parametrizing $v=\cos\beta$ and ${\bf
v}=(\sin\beta)$, the reduced system reads

$\displaystyle\partial_{-}\partial_{+}\beta$
$\displaystyle=\mu\exp\left(-\varphi\right)\sin\beta,$ (126)
$\displaystyle\partial_{+}\partial_{-}\varphi$
$\displaystyle=\mu\left[\cos\beta\exp\left(-\varphi\right)-s_{\mu}\exp\left(\varphi\right)\right],$
(127)

which matches the results of Rashkov:2008rm .

###  A.5 AdS Space-like reduction & Minkowski world-sheet

In this section we analyze the space-like reduction of AdS spaces. We remind
the reader that this case corresponds to $t=1$, i.e.
$\partial_{\pm}Y\cdot\partial_{\pm}Y=m^{2}_{\pm}$. Contrary to the time-like
case, we define the inner product as

$\partial_{+}Y\cdot\partial_{-}Y=m_{+}m_{-}\cos\varphi,$ (128)

it follows that $\mathcal{M}_{3}=\mathbb{R}^{(1,2)}$. It is a matter of
algebra to show that

$\displaystyle\partial_{\sigma}Y\cdot\partial_{\sigma}Y$
$\displaystyle=\frac{m_{+}^{2}+m_{-}^{2}}{4}+\frac{m_{+}m_{-}}{2}\cos\varphi,$
(129) $\displaystyle\partial_{\tau}Y\cdot\partial_{\tau}Y$
$\displaystyle=\frac{m_{+}^{2}+m_{-}^{2}}{4}-\frac{m_{+}m_{-}}{2}\cos\varphi,$
(130) $\displaystyle\partial_{\sigma}Y\cdot\partial_{\tau}Y$
$\displaystyle=\frac{m_{+}^{2}-m_{-}^{2}}{4}.$ (131)

Thus, both vectors $\partial_{\sigma}Y$ and $\partial_{\tau}Y$ are space-like
as required.

In this case, it follows that $\mathcal{S}=I_{1,n-2}$, i.e. the Killing metric
of $SO(1,n-2)$. As $f(\varphi)=\cos\varphi$, it it required that
$\tilde{s}t=1$. For the space-like reduction $t=1$, thus $\tilde{s}=1$. As a
result $\tilde{\mathcal{S}}=I_{1,n-1}$, which is the Killing metric of
$SO(1,n-1)$. Again, the reduced theory is formulated as the
$SO(1,n)/SO(1,n-1)$ perturbed WZW model, see section 5.1 of Miramontes:2008wt
. In order to make contact with the results of this work, once more we need to
implement non-local field redefinitions.

For completeness, we also present the equation of the reduced theory, which
read

$\displaystyle\partial_{+}\partial_{-}\varphi=$
$\displaystyle\tan\varphi\,\partial_{-}{\bf
Z}^{T}\tilde{\mathcal{S}}\partial_{+}{\bf
Z}+\frac{m_{+}m_{-}}{\Lambda^{2}}\sinh\varphi,$ (132)
$\displaystyle\partial_{+}\partial_{-}{\bf Z}=$
$\displaystyle-\frac{\partial_{+}\varphi\partial_{-}{\bf
Z}}{\tan\varphi}-\frac{\partial_{-}\varphi\partial_{+}{\bf
Z}}{\sin\varphi\cos\varphi}-\left(\partial_{-}{\bf
Z}^{T}\tilde{\mathcal{S}}\partial_{+}{\bf Z}\right){\bf Z}.$ (133)

The complementary form of equation (133) reads

$\partial_{+}\partial_{-}{\bf Z}=-\frac{\partial_{+}\varphi\partial_{-}{\bf
Z}}{\sin\varphi\cos\varphi}-\frac{\partial_{-}\varphi\partial_{+}{\bf
Z}}{\tan\varphi}-\left(\partial_{-}{\bf
Z}^{T}\tilde{\mathcal{S}}\partial_{+}{\bf Z}\right){\bf Z}.$ (134)

We parametrize ${\bf Z}$ as ${\bf Z}=\begin{pmatrix}\hat{n}\cosh u\\\ \sinh
u\end{pmatrix}$, where $\hat{n}$ is a $(n-1)\times 1$ column matrix of unit
norm. Consequently, the metric $\tilde{\mathcal{S}}$ reads
$\tilde{\mathcal{S}}=\begin{pmatrix}I_{n-1}&0\\\ 0&-1\end{pmatrix}$. It is
evident that equations (132)-(134) are obtained from equations (108)-(110) via
the double analytic continuation $u\rightarrow iu+\pi/2$ and
$\varphi\rightarrow i\varphi$.

There is also an alternative choice for the inner product. We can define it as

$\partial_{+}Y\cdot\partial_{-}Y=m_{+}m_{-}\cosh\varphi,$ (135)

so that $\mathcal{M}_{3}$ can be $\mathbb{R}^{(2,1)}$. It is a matter of
algebra to show that

$\displaystyle\partial_{\sigma}Y\cdot\partial_{\sigma}Y$
$\displaystyle=\frac{m_{+}^{2}+m_{-}^{2}}{4}+\frac{m_{+}m_{-}}{2}\cosh\varphi,$
(136) $\displaystyle\partial_{\tau}Y\cdot\partial_{\tau}Y$
$\displaystyle=\frac{m_{+}^{2}+m_{-}^{2}}{4}-\frac{m_{+}m_{-}}{2}\cosh\varphi,$
(137) $\displaystyle\partial_{\sigma}Y\cdot\partial_{\tau}Y$
$\displaystyle=\frac{m_{+}^{2}-m_{-}^{2}}{4}.$ (138)

The consistency of the basis requires

$\cosh\varphi\geq\frac{m_{+}^{2}+m_{-}^{2}}{2|m_{+}m_{-}|},$ (139)

so that one of the vectors $\partial_{\sigma}Y$ and $\partial_{\tau}Y$ is
time-like and the other one space-like.

In this case, it follows that $\mathcal{S}=I_{n-1}$, i.e. the Killing metric
of $SO(n-1)$. As $f(\varphi)=\cosh\varphi$, it it required that
$\tilde{s}t=-1$. For the space-like reduction $t=1$, thus $\tilde{s}=-1$. As a
result $\tilde{\mathcal{S}}=I_{1,n-1}$, which is the Killing metric of
$SO(1,n-1)$. The reduced theory is formulated as the $SO(1,n)/SO(1,n-1)$
perturbed WZW model, see section 5.1 of Miramontes:2008wt .

For completeness, we also present the equation of the reduced theory, which
read

$\displaystyle\partial_{+}\partial_{-}\varphi=$
$\displaystyle-\tanh\varphi\,\partial_{-}{\bf
Z}^{T}\tilde{\mathcal{S}}\partial_{+}{\bf
Z}+\frac{m_{+}m_{-}}{\Lambda^{2}}\sinh\varphi,$ (140)
$\displaystyle\partial_{+}\partial_{-}{\bf Z}=$
$\displaystyle-\frac{\partial_{+}\varphi\partial_{-}{\bf
Z}}{\tanh\varphi}-\frac{\partial_{-}\varphi\partial_{+}{\bf
Z}}{\sinh\varphi\cosh\varphi}+\left(\partial_{-}{\bf
Z}^{T}\tilde{\mathcal{S}}\partial_{+}{\bf Z}\right){\bf Z}.$ (141)

The complementary form of equation (141) reads

$\partial_{+}\partial_{-}{\bf Z}=-\frac{\partial_{+}\varphi\partial_{-}{\bf
Z}}{\sinh\varphi\cosh\varphi}-\frac{\partial_{-}\varphi\partial_{+}{\bf
Z}}{\tanh\varphi}+\left(\partial_{-}{\bf
Z}^{T}\tilde{\mathcal{S}}\partial_{+}{\bf Z}\right){\bf Z}.$ (142)

Notice that the sign of the terms involving $\partial_{-}{\bf
Z}^{T}\tilde{\mathcal{S}}\partial_{+}{\bf Z}$ as well as the metric
$\tilde{\mathcal{S}}$ are different with respect the time-like case. We
parametrize ${\bf Z}$ as ${\bf Z}=\begin{pmatrix}\cosh u\\\ \hat{n}\sinh
u\end{pmatrix}$, where $\hat{n}$ is a $(n-1)\times 1$ column matrix of unit
norm. Consequently, the metric $\tilde{\mathcal{S}}$ reads
$\tilde{\mathcal{S}}=\begin{pmatrix}-1&0\\\ 0&I_{n-1}\end{pmatrix}$. It is
evident that equations (140)-(142) are obtained from equations (108)-(110) via
the analytic continuation $u\rightarrow iu$.

##  Appendix B Details of the solution of the auxiliary system

In this section we provide some details on the final part of the derivation of
section 3.2. To begin with the matrix $V$, which appears in (23), has the
structure

$V=\begin{pmatrix}v_{1}^{T}\\\ \vdots\\\ v_{n+2}^{T}\\\ \end{pmatrix},$ (143)

while the inverse matrix has the structure

$V^{-1}=\begin{pmatrix}v_{1}^{\prime}&\dots&v_{n+2}^{\prime}\end{pmatrix},$
(144)

where, obviously, the vectors $v_{\alpha}^{\prime}$ obey
$v_{\alpha}^{T}v_{\beta}^{\prime}=\delta_{\alpha\beta}$. On a similar token
the currents $\hat{j}_{\pm}$ have the following form

$\hat{j}_{+}=\frac{s}{\Lambda^{2}}\left[v_{n+1}v_{n+2}^{T}-v_{n+2}v_{n+1}^{T}\right],\qquad\hat{j}_{-}=\frac{s}{\Lambda^{2}}\left[v_{n}v_{n+2}^{T}-v_{n+2}v_{d}^{T}\right].$
(145)

It is a matter of algebra to show that

$V\hat{j}_{+}V^{-1}=\begin{pmatrix}{\bf 0}_{n-1}&{\bf 0}&{\bf 0}&{\bf 0}\\\
{\bf 0}^{T}&0&0&s\frac{m_{+}m_{-}}{\Lambda^{2}}f(\varphi)\\\ {\bf
0}^{T}&0&0&st\frac{m_{+}^{2}}{\Lambda^{2}}\\\ {\bf
0}^{T}&0&-1&0\end{pmatrix},\thickspace V\hat{j}_{-}V^{-1}=\begin{pmatrix}{\bf
0}_{n-1}&{\bf 0}&{\bf 0}&{\bf 0}\\\ {\bf
0}^{T}&0&0&st\frac{m_{+}^{2}}{\Lambda^{2}}\\\ {\bf
0}^{T}&0&0&s\frac{m_{+}m_{-}}{\Lambda^{2}}f(\varphi)\\\ {\bf
0}^{T}&-1&0&0\end{pmatrix},$ (146)

where ${\bf 0}_{n-1}$ is the $(n-1)\times(n-1)$ zero matrix and ${\bf 0}$ is
the $(n-1)\times 1$ zero matrix.

The explicit form of the vectors $v_{\alpha}^{\prime}$ is

$v_{i}^{\prime}=s_{i}\eta v_{i},\qquad
v_{n+2}^{\prime}=\frac{s}{\Lambda^{2}}\eta v_{n+2}$ (147)

where $i=1,\dots,n-1$ and $\eta$ is the metric of the enhanced space. The rest
of the columns read

$\displaystyle v^{\prime}_{n}$
$\displaystyle=\frac{1}{m_{-}^{2}\left(t^{2}-f^{2}\right)}\eta\left[tv_{n}-\frac{m_{-}}{m_{+}}fv_{n+1}\right],$
(148) $\displaystyle v^{\prime}_{n+1}$
$\displaystyle=\frac{1}{m_{+}^{2}\left(t^{2}-f^{2}\right)}\eta\left[tv_{n+1}-\frac{m_{+}}{m_{-}}fv_{n}\right].$
(149)

## References

  * (1) V.E. Zakharov and A.V. Mikhailov, _Relativistically Invariant Two-Dimensional Models in Field Theory Integrable by the Inverse Problem Technique. (In Russian)_ , _Sov. Phys. JETP_ 47 (1978) 1017\.
  * (2) I.V. Cherednik, _“Factorizing Particles on a Half Line and Root Systems”_ , _Theor. Math. Phys._ 61 (1984) 977.
  * (3) E.K. Sklyanin, _“Boundary Conditions for Integrable Quantum Systems”_ , _J. Phys. A_ 21 (1988) 2375.
  * (4) M. Luscher, _Quantum Nonlocal Charges and Absence of Particle Production in the Two-Dimensional Nonlinear Sigma Model_ , _Nucl. Phys. B_ 135 (1978) 1.
  * (5) H. Eichenherr and M. Forger, _On the Dual Symmetry of the Nonlinear Sigma Models_ , _Nucl. Phys. B_ 155 (1979) 381.
  * (6) H. Eichenherr and M. Forger, _More about non-linear sigma models on symmetric spaces_ , _Nucl. Phys. B_ 164 (1980) 528.
  * (7) K. Zarembo, _“Integrability in Sigma-Models”_ , 1712.07725.
  * (8) S. Driezen, _Modave Lectures on Classical Integrability in $2d$ Field Theories_, 2112.14628.
  * (9) C. Klimcik, _On integrability of the Yang-Baxter sigma-model_ , _J. Math. Phys._ 50 (2009) 043508 [0802.3518].
  * (10) F. Delduc, M. Magro and B. Vicedo, _On classical $q$-deformations of integrable sigma-models_, _JHEP_ 11 (2013) 192 [1308.3581].
  * (11) F. Delduc, M. Magro and B. Vicedo, _An integrable deformation of the $AdS_{5}\times S^{5}$ superstring action_, _Phys. Rev. Lett._ 112 (2014) 051601 [1309.5850].
  * (12) K. Sfetsos, _Integrable interpolations: From exact CFTs to non-Abelian T-duals_ , _Nucl. Phys. B_ 880 (2014) 225 [1312.4560].
  * (13) C. Klimcik, _Integrability of the bi-Yang-Baxter sigma-model_ , _Lett. Math. Phys._ 104 (2014) 1095 [1402.2105].
  * (14) C. Klimčík, _Brief lectures on duality, integrability and deformations_ , _Rev. Math. Phys._ 33 (2021) 2130004 [2101.05230].
  * (15) B. Hoare, _Integrable Deformations of Sigma Models_ , 2109.14284.
  * (16) K. Pohlmeyer, _Integrable Hamiltonian Systems and Interactions Through Quadratic Constraints_ , _Commun. Math. Phys._ 46 (1976) 207.
  * (17) F. Lund and T. Regge, _Unified Approach to Strings and Vortices with Soliton Solutions_ , _Phys. Rev. D_ 14 (1976) 1524.
  * (18) F. Lund, _Note on the Geometry of the Nonlinear Sigma Model in Two-Dimensions_ , _Phys. Rev. D_ 15 (1977) 1540.
  * (19) K. Pohlmeyer and K.-H. Rehren, _Reduction of the Two-dimensional O( $n$) Nonlinear Sigma Model_, _J. Math. Phys._ 20 (1979) 2628.
  * (20) H. Eichenherr and J. Honerkamp, _Reduction of the CP N Nonlinear $\sigma$ Model_, _J. Math. Phys._ 22 (1981) 374.
  * (21) B.M. Barbashov and V.V. Nesterenko, _Relativistic String Model in a Space-time of a Constant Curvature_ , _Commun. Math. Phys._ 78 (1981) 499.
  * (22) H.J. De Vega and N.G. Sanchez, _Exact integrability of strings in D-Dimensional De Sitter space-time_ , _Phys. Rev. D_ 47 (1993) 3394.
  * (23) A.L. Larsen and N.G. Sanchez, _Sinh-Gordon, cosh-Gordon and Liouville equations for strings and multistrings in constant curvature space-times_ , _Phys. Rev. D_ 54 (1996) 2801 [hep-th/9603049].
  * (24) R. D’Auria, T. Regge and S. Sciuto, _A General Scheme for Bidimensional Models With Associate Linear Set_ , _Phys. Lett. B_ 89 (1980) 363.
  * (25) R. D’Auria, T. Regge and S. Sciuto, _Group Theoretical Construction of Two-dimensional Models With Infinite Set of Conservation Laws_ , _Nucl. Phys. B_ 171 (1980) 167.
  * (26) R. D’Auria and S. Sciuto, _Group Theoretical Construction of Two-dimensional Supersymmetric Models_ , _Nucl. Phys. B_ 171 (1980) 189.
  * (27) I. Bakas, Q.-H. Park and H.-J. Shin, _Lagrangian formulation of symmetric space sine-Gordon models_ , _Phys. Lett. B_ 372 (1996) 45 [hep-th/9512030].
  * (28) Q.-H. Park, _Deformed coset models from gauged WZW actions_ , _Phys. Lett. B_ 328 (1994) 329 [hep-th/9402038].
  * (29) T.J. Hollowood, J.L. Miramontes and Q.-H. Park, _Massive integrable soliton theories_ , _Nucl. Phys. B_ 445 (1995) 451 [hep-th/9412062].
  * (30) C.R. Fernandez-Pousa, M.V. Gallas, T.J. Hollowood and J.L. Miramontes, _The Symmetric space and homogeneous sine-Gordon theories_ , _Nucl. Phys. B_ 484 (1997) 609 [hep-th/9606032].
  * (31) J.L. Miramontes, _Pohlmeyer reduction revisited_ , _JHEP_ 10 (2008) 087 [0808.3365].
  * (32) V.E. Zakharov and A.V. Mikhailov, _“ON THE INTEGRABILITY OF CLASSICAL SPINOR MODELS IN TWO-DIMENSIONAL SPACE-TIME”_ , _Commun. Math. Phys._ 74 (1980) 21.
  * (33) J.P. Harnad, Y. Saint Aubin and S. Shnider, _“Backlund Transformations for Nonlinear $\sigma$ Models With Values in Riemannian Symmetric Spaces”_, _Commun. Math. Phys._ 92 (1984) 329.
  * (34) Q.-H. Park and H.J. Shin, _Duality in complex sine-Gordon theory_ , _Phys. Lett. B_ 359 (1995) 125 [hep-th/9506087].
  * (35) T.J. Hollowood and J.L. Miramontes, _“Magnons, their Solitonic Avatars and the Pohlmeyer Reduction”_ , _JHEP_ 04 (2009) 060 [0902.2405].
  * (36) G. Arutyunov and M. Staudacher, _Matching higher conserved charges for strings and spins_ , _JHEP_ 03 (2004) 004 [hep-th/0310182].
  * (37) G. Arutyunov and M. Zamaklar, _Linking Backlund and monodromy charges for strings on AdS(5) x S**5_ , _JHEP_ 07 (2005) 026 [hep-th/0504144].
  * (38) J.M. Maldacena, _The Large N limit of superconformal field theories and supergravity_ , _Adv. Theor. Math. Phys._ 2 (1998) 231 [hep-th/9711200].
  * (39) S.S. Gubser, I.R. Klebanov and A.M. Polyakov, _Gauge theory correlators from noncritical string theory_ , _Phys. Lett. B_ 428 (1998) 105 [hep-th/9802109].
  * (40) E. Witten, _Anti-de Sitter space and holography_ , _Adv. Theor. Math. Phys._ 2 (1998) 253 [hep-th/9802150].
  * (41) R.R. Metsaev and A.A. Tseytlin, _Type IIB superstring action in AdS(5) x S**5 background_ , _Nucl. Phys. B_ 533 (1998) 109 [hep-th/9805028].
  * (42) I. Bena, J. Polchinski and R. Roiban, _Hidden symmetries of the AdS(5) x S**5 superstring_ , _Phys. Rev. D_ 69 (2004) 046002 [hep-th/0305116].
  * (43) M. Grigoriev and A.A. Tseytlin, _Pohlmeyer reduction of AdS(5) x S**5 superstring sigma model_ , _Nucl. Phys. B_ 800 (2008) 450 [0711.0155].
  * (44) A. Mikhailov and S. Schafer-Nameki, _Sine-Gordon-like action for the Superstring in AdS(5) x S**5_ , _JHEP_ 05 (2008) 075 [0711.0195].
  * (45) M. Grigoriev and A.A. Tseytlin, _On reduced models for superstrings on AdS(n) x S**n_ , _Int. J. Mod. Phys. A_ 23 (2008) 2107 [0806.2623].
  * (46) B. Hoare and A.A. Tseytlin, _Pohlmeyer reduction for superstrings in AdS space_ , _J. Phys. A_ 46 (2013) 015401 [1209.2892].
  * (47) R. Roiban and A.A. Tseytlin, _UV finiteness of Pohlmeyer-reduced form of the AdS(5) x S**5 superstring theory_ , _JHEP_ 04 (2009) 078 [0902.2489].
  * (48) D.M. Schmidtt, _Supersymmetry Flows, Semi-Symmetric Space Sine-Gordon Models And The Pohlmeyer Reduction_ , _JHEP_ 03 (2011) 021 [1012.4713].
  * (49) D.M. Schmidtt, _Integrability vs Supersymmetry: Poisson Structures of The Pohlmeyer Reduction_ , _JHEP_ 11 (2011) 067 [1106.4796].
  * (50) M. Goykhman and E. Ivanov, _Worldsheet Supersymmetry of Pohlmeyer-Reduced $AdS_{n}xS^{n}$ Superstrings_, _JHEP_ 09 (2011) 078 [1104.0706].
  * (51) T.J. Hollowood and J.L. Miramontes, _The $AdS_{5}xS_{5}$ Semi-Symmetric Space Sine-Gordon Theory_, _JHEP_ 05 (2011) 136 [1104.2429].
  * (52) O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, _N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals_ , _JHEP_ 10 (2008) 091 [0806.1218].
  * (53) G. Arutyunov and S. Frolov, _Superstrings on AdS(4) x CP**3 as a Coset Sigma-model_ , _JHEP_ 09 (2008) 129 [0806.4940].
  * (54) B. Stefanski, jr, _Green-Schwarz action for Type IIA strings on AdS(4) x CP**3_ , _Nucl. Phys. B_ 808 (2009) 80 [0806.4948].
  * (55) R.C. Rashkov, _A Note on the reduction of the AdS(4) x CP**3 string sigma model_ , _Phys. Rev. D_ 78 (2008) 106012 [0808.3057].
  * (56) M. Dukalski and S.J. van Tongeren, _On fermionic reductions of the AdS(4) x CP**3 superstring_ , _Phys. Rev. D_ 80 (2009) 046005 [0906.0706].
  * (57) D.M. Hofman and J.M. Maldacena, _Giant Magnons_ , _J. Phys. A_ 39 (2006) 13095 [hep-th/0604135].
  * (58) H.-Y. Chen, N. Dorey and K. Okamura, _Dyonic giant magnons_ , _JHEP_ 09 (2006) 024 [hep-th/0605155].
  * (59) R. Ishizeki and M. Kruczenski, _Single spike solutions for strings on S**2 and S**3_ , _Phys. Rev. D_ 76 (2007) 126006 [0705.2429].
  * (60) D.E. Berenstein, J.M. Maldacena and H.S. Nastase, _Strings in flat space and pp waves from N=4 superYang-Mills_ , _JHEP_ 04 (2002) 013 [hep-th/0202021].
  * (61) K. Okamura and R. Suzuki, _A Perspective on Classical Strings from Complex Sine-Gordon Solitons_ , _Phys. Rev. D_ 75 (2007) 046001 [hep-th/0609026].
  * (62) H. Hayashi, K. Okamura, R. Suzuki and B. Vicedo, _Large Winding Sector of AdS/CFT_ , _JHEP_ 11 (2007) 033 [0709.4033].
  * (63) D. Katsinis, I. Mitsoulas and G. Pastras, _Elliptic string solutions on $\mathbb{R}\times\hbox{S}^{2}$ and their pohlmeyer reduction_, _Eur. Phys. J. C_ 78 (2018) 977 [1805.09301].
  * (64) A. Jevicki, K. Jin, C. Kalousios and A. Volovich, _Generating AdS String Solutions_ , _JHEP_ 03 (2008) 032 [0712.1193].
  * (65) A. Jevicki and K. Jin, _Solitons and AdS String Solutions_ , _Int. J. Mod. Phys. A_ 23 (2008) 2289 [0804.0412].
  * (66) I. Bakas and G. Pastras, _On elliptic string solutions in AdS 3 and dS3_, _JHEP_ 07 (2016) 070 [1605.03920].
  * (67) D. Katsinis, I. Mitsoulas and G. Pastras, _Dressed elliptic string solutions on $\mathbb{R}\times\hbox{S}^{2}$_, _Eur. Phys. J. C_ 78 (2018) 668 [1806.07730].
  * (68) D. Katsinis, I. Mitsoulas and G. Pastras, _Salient features of dressed elliptic string solutions on $\mathbb{R}\times\hbox{S}^{2}$_, _Eur. Phys. J. C_ 79 (2019) 869 [1903.01408].
  * (69) N. Beisert et al., _Review of AdS/CFT Integrability: An Overview_ , _Lett. Math. Phys._ 99 (2012) 3 [1012.3982].
  * (70) N. Gromov, _Introduction to the Spectrum of $N=4$ SYM and the Quantum Spectral Curve_, 1708.03648.
  * (71) V.A. Kazakov, A. Marshakov, J.A. Minahan and K. Zarembo, _Classical/quantum integrability in AdS/CFT_ , _JHEP_ 05 (2004) 024 [hep-th/0402207].
  * (72) V.A. Kazakov and K. Zarembo, _Classical / quantum integrability in non-compact sector of AdS/CFT_ , _JHEP_ 10 (2004) 060 [hep-th/0410105].
  * (73) N. Beisert, V.A. Kazakov and K. Sakai, _Algebraic curve for the SO(6) sector of AdS/CFT_ , _Commun. Math. Phys._ 263 (2006) 611 [hep-th/0410253].
  * (74) N. Beisert, V.A. Kazakov, K. Sakai and K. Zarembo, _The Algebraic curve of classical superstrings on AdS(5) x S**5_ , _Commun. Math. Phys._ 263 (2006) 659 [hep-th/0502226].
  * (75) N. Beisert, V.A. Kazakov, K. Sakai and K. Zarembo, _Complete spectrum of long operators in N=4 SYM at one loop_ , _JHEP_ 07 (2005) 030 [hep-th/0503200].
  * (76) N. Gromov and P. Vieira, _The AdS(4) / CFT(3) algebraic curve_ , _JHEP_ 02 (2009) 040 [0807.0437].
  * (77) J.M. Maldacena, _Wilson loops in large N field theories_ , _Phys. Rev. Lett._ 80 (1998) 4859 [hep-th/9803002].
  * (78) S.-J. Rey and J.-T. Yee, _Macroscopic strings as heavy quarks in large N gauge theory and anti-de Sitter supergravity_ , _Eur. Phys. J. C_ 22 (2001) 379 [hep-th/9803001].
  * (79) L.F. Alday and J.M. Maldacena, _Gluon scattering amplitudes at strong coupling_ , _JHEP_ 06 (2007) 064 [0705.0303].
  * (80) L.F. Alday and J. Maldacena, _Comments on gluon scattering amplitudes via AdS/CFT_ , _JHEP_ 11 (2007) 068 [0710.1060].
  * (81) L.F. Alday and J. Maldacena, _Null polygonal Wilson loops and minimal surfaces in Anti-de-Sitter space_ , _JHEP_ 11 (2009) 082 [0904.0663].
  * (82) L.F. Alday, D. Gaiotto and J. Maldacena, _Thermodynamic Bubble Ansatz_ , _JHEP_ 09 (2011) 032 [0911.4708].
  * (83) K. Zarembo, _Supersymmetric Wilson loops_ , _Nucl. Phys. B_ 643 (2002) 157 [hep-th/0205160].
  * (84) N. Drukker, S. Giombi, R. Ricci and D. Trancanelli, _More supersymmetric Wilson loops_ , _Phys. Rev. D_ 76 (2007) 107703 [0704.2237].
  * (85) V. Pestun, _Localization of gauge theory on a four-sphere and supersymmetric Wilson loops_ , _Commun. Math. Phys._ 313 (2012) 71 [0712.2824].
  * (86) V. Pestun et al., _Localization techniques in quantum field theories_ , _J. Phys. A_ 50 (2017) 440301 [1608.02952].
  * (87) S. Ryu and T. Takayanagi, _Holographic derivation of entanglement entropy from AdS/CFT_ , _Phys. Rev. Lett._ 96 (2006) 181602 [hep-th/0603001].
  * (88) S. Ryu and T. Takayanagi, _Aspects of Holographic Entanglement Entropy_ , _JHEP_ 08 (2006) 045 [hep-th/0605073].
  * (89) H. Dorn, G. Jorjadze and S. Wuttke, _On spacelike and timelike minimal surfaces in AdS(n)_ , _JHEP_ 05 (2009) 048 [0903.0977].
  * (90) H. Dorn, _Some comments on spacelike minimal surfaces with null polygonal boundaries in AdS(m)_ , _JHEP_ 02 (2010) 013 [0910.0934].
  * (91) A. Jevicki and K. Jin, _Series Solution and Minimal Surfaces in AdS_ , _JHEP_ 03 (2010) 028 [0911.1107].
  * (92) B.A. Burrington and P. Gao, _Minimal surfaces in AdS space and Integrable systems_ , _JHEP_ 04 (2010) 060 [0911.4551].
  * (93) H. Dorn, N. Drukker, G. Jorjadze and C. Kalousios, _Space-like minimal surfaces in AdS x S_ , _JHEP_ 04 (2010) 004 [0912.3829].
  * (94) R. Ishizeki, M. Kruczenski and S. Ziama, _Notes on Euclidean Wilson loops and Riemann Theta functions_ , _Phys. Rev. D_ 85 (2012) 106004 [1104.3567].
  * (95) M. Kruczenski and S. Ziama, _Wilson loops and Riemann theta functions II_ , _JHEP_ 05 (2014) 037 [1311.4950].
  * (96) M. Kruczenski, _Wilson loops and minimal area surfaces in hyperbolic space_ , _JHEP_ 11 (2014) 065 [1406.4945].
  * (97) G. Pastras, _Static elliptic minimal surfaces in AdS 4_, _Eur. Phys. J. C_ 77 (2017) 797 [1612.03631].
  * (98) Y. He, C. Huang and M. Kruczenski, _Minimal area surfaces in $AdS_{n+1}$ and Wilson loops_, _JHEP_ 02 (2018) 027 [1712.06269].
  * (99) D. Katsinis, D. Manolopoulos, I. Mitsoulas and G. Pastras, _Dressed minimal surfaces in AdS 4_, _JHEP_ 11 (2020) 128 [2007.10922].
  * (100) D. Katsinis, I. Mitsoulas and G. Pastras, _The Dressing Method as Non Linear Superposition in Sigma Models_ , _JHEP_ 03 (2021) 024 [2011.04610].
  * (101) D. Katsinis, _Solving formally the Auxiliary System of $O(N)$ Non Linear Sigma Model_, 2201.02493.
  * (102) B.M. Barbashov, V.V. Nesterenko and A.M. Chervyakov, _REDUCTION IN THE RELATIVISTIC STRING MODEL FOR THE d-DIMENSIONAL SPACE-TIME_ , _Theor. Math. Phys._ 59 (1984) 458.
  * (103) I. Bakas and K. Sfetsos, _Universal aspects of string propagation on curved backgrounds_ , _Phys. Rev. D_ 54 (1996) 3995 [hep-th/9604195].
  * (104) S. Helgason, _Differential geometry, Lie groups, and symmetric spaces_ , Academic Press, New York (1978).
  * (105) D. Katsinis, I. Mitsoulas and G. Pastras, _Stability Analysis of Classical String Solutions and the Dressing Method_ , _JHEP_ 09 (2019) 106 [1903.01412].
  * (106) T.J. Hollowood, J.L. Miramontes and D.M. Schmidtt, _Integrable Deformations of Strings on Symmetric Spaces_ , _JHEP_ 11 (2014) 009 [1407.2840].
  * (107) B. Hoare and A.A. Tseytlin, _Tree-level S-matrix of Pohlmeyer reduced form of AdS(5) x S**5 superstring theory_ , _JHEP_ 02 (2010) 094 [0912.2958].
  * (108) B. Hoare and A.A. Tseytlin, _On the perturbative S-matrix of generalized sine-Gordon models_ , _JHEP_ 11 (2010) 111 [1008.4914].
  * (109) B. Hoare and A.A. Tseytlin, _Towards the quantum S-matrix of the Pohlmeyer reduced version of $AdS_{5}\times S^{5}$ superstring theory_, _Nucl. Phys. B_ 851 (2011) 161 [1104.2423].
  * (110) B. Hoare, Y. Iwashita and A.A. Tseytlin, _Pohlmeyer-reduced form of string theory in AdS(5) x S(5): Semiclassical expansion_ , _J. Phys. A_ 42 (2009) 375204 [0906.3800].
  * (111) Y. Iwashita, _One-loop corrections to AdS 5 x S5 superstring partition function via Pohlmeyer reduction_, _J. Phys. A_ 43 (2010) 345403 [1005.4386].
  * (112) Y. Iwashita, R. Roiban and A.A. Tseytlin, _Two-loop corrections to partition function of Pohlmeyer-reduced theory for $AdS_{5}xS^{5}$ superstring_, _Phys. Rev. D_ 84 (2011) 126017 [1109.5361].
  * (113) I. Krichever and N. Nekrasov, _Towards Lefschetz Thimbles in Sigma Models, I_ , _J. Exp. Theor. Phys._ 132 (2021) 734 [2010.15575].