Lax pairs for new $\mathbb{Z}_{N}$-symmetric coset $\sigma$-models

and their Yang-Baxter deformations

David Osten

Institute for Theoretical and Mathematical Physics
Lomonosov Moscow State University

Lomonosovsky Avenue, Moscow, 119991, Russia

<EMAIL_ADDRESS>
###### Abstract

Two-dimensional $\sigma$-models with $\mathbb{Z}_{N}$-symmetric homogeneous
target spaces have been shown to be classically integrable when introducing
WZ-terms in a particular way. This article continues the search for new models
of this type now allowing some kinetic terms to be absent, analogously to the
Green-Schwarz superstring $\sigma$-model on $\mathbb{Z}_{4}$-symmetric
homogeneous spaces. A list of such integrable $\mathbb{Z}_{N}$-symmetric
(super)coset $\sigma$-models for $N\leq 6$ and their Lax pairs is presented.
For arbitrary $N$, a big class of integrable models is constructed that
includes both the known pure spinor and Green-Schwarz superstring on
$\mathbb{Z}_{4}$-symmetric cosets.

Integrable Yang-Baxter deformations of this class of
$\mathbb{Z}_{N}$-symmetric (super)coset $\sigma$-models can be constructed in
same way as in the known $\mathbb{Z}_{2}$\- or $\mathbb{Z}_{4}$-cases.
Deformations based on solutions of the modified classical Yang-Baxter
equation, the so-called $\eta$-deformation, require deformation of the
constants defining the Lagrangian and the corresponding Lax pair. Homogeneous
Yang-Baxter deformations (i.e. those based on solutions to the classical Yang-
Baxter equation) leave the equations of motion and consequently the Lax pair
invariant and are expected to be classically equivalent to the undeformed
model.

As an example, the relationship between $\mathbb{Z}_{3}$-symmetric homogeneous
spaces and nearly (para-)Kähler geometries is revisited. Confirming existing
literature it is shown that the integrable choice of WZ-term in the
$\mathbb{Z}_{3}$-symmetric coset $\sigma$-model associated to a nearly Kähler
background gives an imaginary contribution to the action.

###### Contents

  1. 1 Introduction
  2. 2 Review: the pure spinor-type $\mathbb{Z}_{N}$-coset $\sigma$-model
    1. 2.1 Action and Lax Connection
    2. 2.2 Applications
      1. 2.2.1 On the connection of $\mathbb{Z}_{3}$-cosets and nearly (para-)Kähler geometries
      2. 2.2.2 $\mathbb{Z}_{N}$-supercosets as supergravity backgrounds
  3. 3 New integrable $\mathbb{Z}_{N}$-coset $\sigma$-models of hybrid type
    1. 3.1 Overview for $N\leq 6$
    2. 3.2 A class of hybrid $\mathbb{Z}_{N}$-coset $\sigma$-models
  4. 4 Yang-Baxter deformed $\mathbb{Z}_{N}$-coset $\sigma$-models
    1. 4.1 The action
    2. 4.2 $N=3$
    3. 4.3 General $N$
      1. 4.3.1 Deformed pure spinor-type $\mathbb{Z}_{N}$-coset $\sigma$-models
      2. 4.3.2 Deformed hybrid $\mathbb{Z}_{N}$-coset $\sigma$-models with $s_{i}=0$ for $i\leq K$
  5. 5 Outlook

##  1 Introduction

Non-linear $\sigma$-models [1] appear ubiquitously in particle physics,
condensed matter theory and gravity. In two dimension they can be classically
integrable. Prototypical example are the principal chiral model or the
symmetric space $\sigma$-model [2]. Applied to string theory, supersymmetric
versions exist: integrable coset $\sigma$-models with supergravity backgrounds
as target space [3, 4, 5, 6, 7, 8, 9, 10]. These supergravity backgrounds are
very unique spaces – flat space, Lie supergroups, AdS${}_{n}\times$Sn or
AdS${}_{4}\times\mathbb{CP}^{3}$. There are two directions in which one could
go to use the advantages of integrability in more generic models: examples
with less symmetry and examples with (NSNS-)fluxes, in particular for the
application in string theory. Corresponding modifications or deformations,
that one could make at the level of the action, are to introduce WZ-terms or
to perform a Yang-Baxter deformation [11, 12, 13, 14, 15, 16, 17, 18]. This
article will be partly concerned with latter. These deformations are generated
so-called $R$-operators – algebraic objects based on an underlying Lie algebra
structure: solution to the classical or the modified classical Yang-Baxter
equation. In former case these are typically called homogeneous Yang-Baxter
deformation, in latter $\eta$-deformation. Homogeneous Yang-Baxter
deformations are closely connected to abelian and non-abelian $T$-duality
transformations. [19, 20, 21, 22, 23, 24, 25, 26, 27, 28].
$\eta$-deformations, on the other hand, are non-trivial deformations of the
classical dynamics and have novel target space interpretations [29, 30, 31,
32, 33, 34, 35, 36, 28, 37]. It is also possible to combine deformations based
on WZ-terms and seperate Yang-Baxter deformations of the left and right action
of the underlying group into multiparametric integrable deformations [38, 39,
40, 41, 42, 43, 44, 45, 46, 47]. Structures appearing have some similarities
to a problem posed in this paper, the interplay of having WZ-terms and
performing a Yang-Baxter deformation.

A different way to generalise from the setting of a symmetric space
$\sigma$-model and its supersymmetric version, is to consider $\sigma$-models
with $\mathbb{Z}_{N}$-symmetric homogeneous target spaces, generalising the
$\mathbb{Z}_{2}$\- resp. $\mathbb{Z}_{4}$-symmetry of the former. [48, 49, 50,
51, 52, 53, 54, 55, 56, 57, 58, 59]. Based on a $\mathbb{Z}_{N}$-grading of an
underlying Lie algebra $\mathfrak{g}=\bigoplus_{i}\mathfrak{g}^{(i)}$, allows
for existence of certian WZ-terms. These need to be introduced in order to
make the $\sigma$-model on the $\mathbb{Z}_{N}$-symmetric coset space
integrable. Moreover it is has been shown that, in principle,
$\mathbb{Z}_{2N}$-symmetric spaces can be supergravity backgrounds, also for
$N>2$ [60]. The central question of this article is: What other choices of
integrable $\sigma$-models on $\mathbb{Z}_{N}$-symmetric homogeneous spaces
are there and do all of these possess an integrable Yang-Baxter deformation?

##### Key results.

The central object of interest in this article is the action

$S=\frac{1}{2}\int\mathrm{d}^{2}\sigma\
\text{Tr}(j_{+}\mathcal{P}_{-}j_{-})\sim\int\sum_{i=1}^{N-1}\text{Tr}(s_{i}j^{(i)}\wedge\star
j^{(N-i)}+b_{i}j^{(i)}\wedge j^{(N-i)})$ (1.1)

for a group-valued field $g(\sigma)$ and a projector
$\mathcal{P}_{-}=\sum_{i}(s_{i}-b_{i})\mathcal{P}^{(i)}$ with
$\mathcal{P}^{(i)}$ being the projector to the $\mathfrak{g}^{(i)}$-component.
This model is written in terms of its Maurer-Cartan-current
$j=g^{-1}\mathrm{d}g$ and characterised by constants $s_{i}$ and $b_{i}$ in
front of kinetic resp. WZ-terms corresponding to the
$\mathfrak{g}^{(i)}$-component of the $\mathbb{Z}_{N}$-decomposition.
Generalising known results in the literature [48, 49, 59] one choice for these
constants, such that the model is classically integrable, is

$\displaystyle s_{i}$ $\displaystyle=0=s_{N-i}=0,\qquad
b_{i}=-b_{N-i}=-\frac{2i}{N}\qquad\text{for}\quad i=1,...,K,$ (1.2)
$\displaystyle s_{i}$ $\displaystyle=1\qquad
b_{i}=1-\frac{2i}{N}\qquad\text{for}\quad i=K+1,...,N-K-1.$ (1.3)

for a $K\in\\{1,...,M\\}$ where $N=2M+1$ resp $2M+2$. Such different
integrable choices of the $s_{i}\in\\{0,1\\}$ were to be expected, in analogy
to the existence of the pure spinor and Green-Schwarz superstring in the case
of $\mathbb{Z}_{4}$-(super)cosets. Other choices are discussed in section 3.
This setting can be Yang-Baxter deformed, as in certain cases already
demonstrated in [56, 59],

$S=\frac{1}{2}\int\mathrm{d}^{2}\sigma\
\text{Tr}\left(j_{+}\mathcal{P}_{-}\frac{1}{1-\eta
R_{g}\circ\mathcal{P}_{-}}j_{-}\right).$ (1.4)

This model is shown to be classically integrable for certain choices of the
constants $s_{i}$, $b_{i}$. Here $R_{g}=\text{Ad}_{g}^{-1}R\text{Ad}_{g}$. $R$
is a linear operator on $\mathfrak{g}$ and solution of the (modified)
classical Yang-Baxter equation. Such Yang-Baxter deformations of
$\mathbb{Z}_{N}$-cosets are constructed in section 4.

##  2 Review: the pure spinor-type $\mathbb{Z}_{N}$-coset $\sigma$-model

This section reviews the original construction [48] and some applications of
the integrable $\mathbb{Z}_{N}$-cosets $\sigma$-model before generalising it
in several directions in the following sections.

###  2.1 Action and Lax Connection

We consider a $\sigma$-model with target space $M=G/H$, where $G$ and $H$ are
Lie (super)groups with Lie (super)algebras $\mathfrak{g}$ resp.
$\mathfrak{h}\equiv\mathfrak{g}^{(0)}$ that are subject to a
$\mathbb{Z}_{N}$-grading

$\displaystyle\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}$
$\displaystyle=\mathfrak{g}^{(0)}\oplus\mathfrak{g}^{(1)}\oplus...\oplus\mathfrak{g}^{(N-1)}$
(2.1) $\displaystyle\text{with}\qquad[\mathfrak{g}^{(i)},\mathfrak{g}^{(j)}]$
$\displaystyle\subset\mathfrak{g}^{(i+j\ \text{mod}N)}.$ (2.2)

with $\mathfrak{m}=\bigoplus_{i}\mathfrak{g}^{(i)}$. In other words we have a
Lie algebra automorphism $\sigma:\ \mathfrak{g}\rightarrow\mathfrak{g}$ with
$\sigma^{N}=1$ and the $\mathfrak{g}^{(i)}$ being the eigenspaces to that
automorphism.

For a non-Abelian algebra $\mathfrak{g}$ and with $N>2$

$[\mathfrak{h},\mathfrak{h}]\subset\mathfrak{h},\quad[\mathfrak{h},\mathfrak{m}]\subset\mathfrak{m}\quad\text{and}\quad[\mathfrak{m},\mathfrak{m}]\not\subset\mathfrak{h}.$
(2.3)

The last relation makes the difference to a symmetric homogeneous space (a
$\mathbb{Z}_{2}$-coset), explaining names like _non-symmetric, semi-symmetric_
or _$N$ -symmetric homogeneous space_ for these in the literature. In this
paper these spaces are referred to as $\mathbb{Z}_{N}$-cosets for brevity.

These spaces might seem very abstract, but in general they have a close
connection to almost Hermitian manifolds [61, 62] and have been studied in the
mathematical literature. A canonical example is $M=(K\times K\times...\times
K)/K_{diag}=K^{N}/K_{diag}$ for a Lie group $K$ and with the corresponding
$\mathbb{Z}_{N}$-automorphism being the permutation of the $N$ copies of $K$.
An explicit geometric interpretation for $\mathbb{Z}_{3}$-symmetric
homogeneous spaces in terms of nearly (para-)Kähler geometry is given in
section 2.2.

Furthermore, the existence of a non-degenerate Ad-invariant bilinear form on
$\mathfrak{g}$, respecting its $\mathbb{Z}_{N}$-grading, is required in order
to construct a $G$-invariant action:

$\text{Tr}\left(mn\right)=\sum_{i=0}^{N-1}\text{Tr}\left(m^{(i)}n^{(N-i)}\right)\quad\text{with}\quad
m^{(i)},n^{(i)}\in\mathfrak{g}^{(i)}.$ (2.4)

Here it is denoted by ’Tr’, but it can be any bilinear form with these
properties. All calculations in this paper hold as well for Lie superalgebras
with a $\mathbb{Z}_{2N}$-grading, as in [60], where it would be the
supertrace.

##### Lagrangian.

A non-linear $\sigma$-model on a two-dimensional worldsheet $\Sigma$ with
light-cone coordinates $\sigma_{\pm}=(\tau\pm\sigma)/2$ on such a
$\mathbb{Z}_{N}$-coset $M=G/H$ may be defined by the Lagrangian

$\mathcal{L}=\frac{1}{2}\text{Tr}\left(j_{+}\mathcal{P}_{-}(j_{-})\right)=\frac{1}{2}\text{Tr}\left(\mathcal{P}_{+}(j_{+})j_{-}\right),$
(2.5)

with the Maurer-Cartan currents $j=g^{-1}\mathrm{d}g$ of a group valued field
$g:\ \Sigma\rightarrow G$, and the operators

$\displaystyle\mathcal{P}_{\pm}=\sum_{i=1}^{N-1}(s_{i}\pm
b_{i})\mathcal{P}^{(i)}$ (2.6)

with constants $s_{i}=s_{N-i}$ and $b_{i}=-b_{N-i}$, for $i=1,...,N-1$ and
where $\mathcal{P}^{(i)}$ is the projector to $\mathfrak{g}^{(i)}$. The
constants $s_{i}$ and $b_{i}$ characterise the model. I.e. the Lagrangian can
be written as

$\mathcal{L}\sim\sum_{i}\text{Tr}\left(s_{i}j^{(i)}\wedge\star
j^{(N-i)}+b_{i}j^{(i)}\wedge j^{(N-i)}\right),$ (2.7)

where $s_{i}$ and $b_{i}$ are the factors in front of kinetic resp. WZ-terms.
The existence of these WZ-terms in that form relies on the existence of the
$\mathbb{Z}_{N}$-automorphism. The Lagrangian respects the ’target space’
symmetries of the $\mathbb{Z}_{N}$-cosets in the following way.

  * •

global (right) $G$-symmetry $g\mapsto g_{0}\cdot g$, for (constant) $g_{0}\in
G$, of the Maurer-Cartan forms $j$ itself

  * •

local (left) $H$-symmetry $g\mapsto g\cdot h$, where
$h^{-1}\mathrm{d}h\in\mathfrak{h}$,

$j^{(i)}\mapsto\left(h^{-1}jh+h^{-1}\mathrm{d}h\right)^{(i)}=(h^{-1}jh)^{(i)}=h^{-1}j^{(i)}h$
(2.8)

The last equality followed from
$[\mathfrak{h},\mathfrak{g}^{(i)}]\subset\mathfrak{g}^{(i)}$. With the Ad-
invariance of the bilinear form this implies the invariance of the Lagrangian.

In addition there is a mapping $b_{i}\leftrightarrow-b_{i}$ and
$\mathfrak{g}^{(i)}\leftrightarrow\mathfrak{g}^{(N-i)}$ for all $i$, that will
leave the action invariant.

##### The equations of motion

are given by

$\mathcal{E}=\partial_{+}\left(\mathcal{P}_{-}j_{-}\right)+\partial_{-}\left(\mathcal{P}_{+}j_{+}\right)+[j_{+},\mathcal{P}_{-}j_{-}]+[j_{-},\mathcal{P}_{+}j_{+}]=0.$
(2.9)

These and the Maurer-Cartan condition for $j=g^{-1}\mathrm{d}g$,

$\mathcal{M}=\partial_{+}j_{-}-\partial_{-}j_{+}+[j_{+},j_{-}]=0,$ (2.10)

become, when decomposed in the grading eigenspaces ($i=1,...,N-1$),

$\displaystyle\mathcal{M}^{(0)}$
$\displaystyle=\partial_{+}j_{-}^{(0)}-\partial_{-}j_{+}^{(0)}+[j_{+}^{(0)},j_{-}^{(0)}]+\sum_{j=1}^{N-1}[j_{+}^{(j)},j_{-}^{(N-j)}]=0$
$\displaystyle\mathcal{M}^{(i)}$
$\displaystyle=\partial_{+}j_{-}^{(i)}-\partial_{-}j_{+}^{(i)}+[j_{+}^{(0)},j_{-}^{(i)}]+[j_{+}^{(i)},j_{-}^{(0)}]+\sum_{j\neq
0,i}[j_{+}^{(j)},j_{-}^{(i-j)}]=0$ (2.11)
$\displaystyle(\mathcal{E}^{(i)}+b_{i}\mathcal{M}^{(i)})$
$\displaystyle=s_{i}(\partial_{+}j_{-}^{(i)}+\partial_{-}j_{+}^{(i)}+[j_{+}^{(0)},j_{-}^{(i)}]-[j_{+}^{(i)},j_{-}^{(0)}])+\sum_{j\neq
0,i}\mathcal{D}_{ij}[j_{+}^{(j)},j_{-}^{(i-j)}]=0$

with constants

$\displaystyle\mathcal{D}_{ij}$
$\displaystyle=\left\\{\begin{array}[]{cr}f_{N+j-i}-f_{j}-b_{i}&i>j\\\
f_{j-i}-f_{j}-b_{i}&i<j\end{array}\right..$ (2.14)

Here the combination $f_{i}=s_{i}-b_{i}$ was introduced such that

$\displaystyle\mathcal{P}_{+}=\sum_{i=1}^{N-1}f_{i}\mathcal{P}^{(i)},\qquad\mathcal{P}_{-}=\sum_{i=1}^{N-1}f_{N-i}\mathcal{P}^{(i)}.$

##### The ’pure spinor’-type model and its Lax integrability.

The original work [48] considers the case, in which $s_{i}\neq 0$ for all
$i=1,...,N-1$. In that case the equations of motion can be represented as a
Lax pair. Putting an ansatz for the Lax connection like

$L_{\pm}=\sum_{i=1}^{N-1}l^{(\pm)}_{i}j^{(i)}_{\pm},$ (2.15)

with constants $l^{(\pm)}_{i}$, into the flatness condition
$\partial_{+}L_{-}-\partial_{-}L_{+}+[L_{+},L_{-}]=0$, using the Maurer-Cartan
identity and the equations of motion (2.11), one obtains a set of equations
for the constants $l_{i}^{(\pm)}$. This system becomes underdetermined111There
is also a second solution with opposite sign in that condition. The correspond
model and its Lax connection can be obtained by the previously discussed
mapping $b_{i}\rightarrow-b_{i},\
\mathfrak{g}^{(i)}\rightarrow\mathfrak{g}^{(N-i)}$., when for all
$i,j=1,...,N-1$ with $i\neq j$:

$\displaystyle\mathcal{D}_{ij}$
$\displaystyle=\left\\{\begin{array}[]{cr}s_{i}&i>j\\\
-s_{i}&i<j\end{array}\right.\quad\Rightarrow\quad
0=\left\\{\begin{array}[]{cr}f_{N+j-i}-f_{j}-f_{N-i}&i>j\\\
f_{j-i}-f_{j}+f_{i}&i<j\end{array}\right..$ (2.20)

This way one finds a choice for the constants $f_{i}=s_{i}-b_{i}$ rendering
the corresponding model classically integrable

$\displaystyle f_{i}=s\frac{2i}{N}\quad\Rightarrow\quad s_{i}=s,\
b_{i}=s\left(1-\frac{2i}{N}\right)\ \forall i,$ (2.21)

in the case that all $s_{i}\neq 0$. Consequently, all kinetic terms are
present in the action (2.5). $s$ is a free constant in front of the action and
not interesting for the classical considerations in this paper, and hence we
set $s=1$. One parameterisation of the corresponding Lax connection with
spectral parameter $\lambda$ is given by

$L_{+}(\lambda)=j^{(0)}_{+}+\sum_{i=1}^{N-1}\lambda^{i}j^{(i)}_{+},\qquad
L_{-}(\lambda)=j^{(0)}_{-}+\sum_{i=1}^{N-1}\frac{1}{\lambda^{N-i}}j^{(i)}_{-}.$
(2.22)

As observed in [48, 60], for $N=4$ this Lax pair and the Lagrangian (2.5)
correspond to the ones of the pure spinor-superstring as obtained, for
example, for AdS${}_{5}\times$S5 [8]. For that reason, we will call the case
of $s_{i}=1,\ i=1,...,N-1$ the pure spinor-type $\mathbb{Z}_{N}$-coset
$\sigma$-model in the following.

Another case of an integrable $\sigma$-model on a $\mathbb{Z}_{N}$-coset,
known in the literature, is the $\mathbb{Z}_{4}$-supercoset $\sigma$-model of
Green-Schwarz type [3, 4, 9, 10], meaning that the fermionic components
$j^{(1)}$ and $j^{(3)}$ do not possess kinetic terms. Hence, that case
corresponds to $s_{1}=s_{3}=0$. One generalisation of that to
$\mathbb{Z}_{N}$-(super)cosets with $N>4$ has been given in [49] for $N=4M$,
with for $s_{2M}=1$ and all others $s_{i}=0$. Generalising both of these
classes, new models of hybrid type222hybrid between the pure spinor\- and
Green-Schwarz-type, meaning some fermionic or bosonic kinetic terms are set to
zero. will be introduced in section 3, in which $s_{i}=0$ only for some $i$.
The nomenclature, pure spinor-, Green-Schwarz\- or hybrid type will be used,
although one might not work with a supercoset, for example for odd $N$.

###  2.2 Applications

In this section, besides collecting some basic information on
$\mathbb{Z}_{N}$-cosets as supergravity backgrounds, the connection of
$\mathbb{Z}_{3}$-cosets to homogeneous nearly Kähler- resp. para-Kähler
structures is reviewed. In the mathematical literature this connection has
been pointed out long ago and studied in detail [61, 62, 63]. In the physics
literature on $\sigma$-models this connection and the correspondic
$\sigma$-models have appeared in [53, 54, 55, 57, 58]. In particular, it is
pointed out here in the following that the $\sigma$-models based on nearly
Kähler geometry might be interesting, but that tuning the parameter such that
it the $\sigma$-model is integrable requires an complex action. On para-
complex target spaces one might be able to avoid these issues [58].

####  2.2.1 On the connection of $\mathbb{Z}_{3}$-cosets and nearly
(para-)Kähler geometries

Let $M=G/H$ where $\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}$ is a real
semisimple Lie algebra. The eigenvalues of a $\mathbb{Z}_{3}$-automorphism
$\sigma:\ \mathfrak{g}\rightarrow\mathfrak{g}$, are complex: $1,\mu,\mu^{2}$
with $\mu^{3}$ = 1. The problem, that the eigenspaces are in general subspaces
of $\mathfrak{g}^{\mathbb{C}}$, can be dealt with by observing that the
subspaces $\mathfrak{h}$ and $\mathfrak{m}$ are real:
$\mathfrak{m}=(\mathfrak{m}_{\mu}\oplus\mathfrak{m}_{\mu^{2}})\cap\mathfrak{g}$.
The connection to an almost complex structure $\mathbb{J}$ is that restricted
to $\mathfrak{m}$ one can write the $\mathbb{Z}_{3}$-automorphism
$\sigma^{3}=1$ as

$\sigma|_{\mathfrak{m}}=-\frac{1}{2}\mathbb{1}+\frac{\sqrt{3}}{2}\mathbb{J}\quad\text{with}\quad\mathbb{J}^{2}=-1.$
(2.23)

$\mathfrak{g}^{(1)}$ and $\mathfrak{g}^{(2)}$ are the
holomorphic/antiholomorphic eigenspaces to $\mathbb{J}$, which can be defined
on the (real) algebra $\mathfrak{m}$ (i.e. without referring to
$\mathfrak{g}^{\mathbb{C}}$). Under complex conjugation
$\overline{\mathfrak{g}^{(1)}}=\mathfrak{g}^{(2)}$. With a metric
$\langle\cdot,\cdot\rangle$ on $M$, that can be derived from the Ad-invariant
bilinear form on $\mathfrak{g}$ as for any homogeneous space, one can form the
compatible tripe $(\langle\cdot,\cdot\rangle,\mathbb{J},\omega$ with the
nearly Kähler form $\omega$, with $\mathrm{d}\omega\neq 0$ that gives rise
non-trivial WZ-term in the corresponding $\sigma$-model below. $\mathbb{J}$ is
subject to the following conditions:

$\mathbb{J}[m_{1},m_{2}]|_{\mathfrak{m}}=-[\mathbb{J}m_{1},m_{2}]|_{\mathfrak{m}}=-[m_{1},\mathbb{J}m_{2}]|_{\mathfrak{m}},\qquad[\mathbb{J}m_{1},m_{2}]|_{\mathfrak{h}}=-[m_{1},\mathbb{J}m_{2}]|_{\mathfrak{h}}$
(2.24)

which encodes of the $\mathbb{Z}_{3}$-grading of the algebra
$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}$. The fact that the eigenspaces
of $\mathbb{J}$ do not close by construction hints towards the fact that this
almost complex structure is not integrable. E.g. the torsion $T$ of the
connection of associated almost Hermitan manifold $(M,g,\mathbb{J})$ is given
by exactly that data, characterising the $\mathbb{Z}_{3}$-structure [63]:
$T(m_{1},m_{2})\sim[m_{1},m_{2}]|_{\mathfrak{m}}$.

Besides classifying the compact, six-dimensional nearly Kähler geometries, it
has been shown in [63] as well that the almost complex structures on these
examples, $S^{3}\times
S^{3}=\frac{\text{SU}(2)\times\text{SU}(2)\times\text{SU}(2)}{\text{SU}(2)}$,
$S^{6}=\frac{G_{2}}{\text{SU}(3)}$,
$\mathbb{CP}^{3}=\frac{\text{Sp}(2)}{\text{SU}(2)\text{U}(1)}$ and the flag
manifold $\frac{\text{SU}(3)}{\text{U}(1)\times\text{U}(1)}$, come from the
$\mathbb{Z}_{3}$-symmetry.

Another possibility is, to have an almost para-complex resp. nearly para-
Kähler structure on $M$. The difference to the almost complex case is mainly
that here the eigenspaces $\mathfrak{g}^{(1)}$ and $\mathfrak{g}^{(2)}$ are
subspaces to $\mathfrak{g}$ but the $\mathbb{Z}_{3}$-automorphism is only
defined on $\mathfrak{g}^{\mathbb{C}}$. All formulas for that can be obtained
from the ones above by $\mathbb{J}=i\mathbb{K}$ with an para-complex structure
$\mathbb{K}^{2}=1$. Relations like (2.24) hold as well. An example for a para-
complex $\mathbb{Z}_{N}$-coset and the integrable $\sigma$-model therein has
been given recently in [58].

##### The integrable $\sigma$-model in terms of geometric data.

In terms of an operator $\mathcal{O}$ with $\mathcal{O}^{2}=\mathbb{1}$, which
come either from a para-complex structure $\mathbb{K}$ or an almost complex
structure $\mathbb{J}$ on $\mathfrak{m}$, with $\mathcal{O}=\mathbb{K}$ resp.
$i\mathbb{J}$. In any case, $\mathcal{O}j^{(1)}=-j^{(1)}$ and
$\mathcal{O}j^{(2)}=j^{(2)}$. Then, one can rewrite the $\mathbb{Z}_{3}$-coset
action for a $\mathfrak{g}$-valued current $j=g^{-1}\mathrm{d}g$ from (2.5) as

$\displaystyle L$
$\displaystyle=\frac{1}{2}\text{Tr}\left((s+b)j^{(1)}_{+}j^{(2)}_{-}+(s-b)j^{(2)}_{+}j^{(1)}_{-}\right)=\frac{1}{2}\text{Tr}\left(m_{+}(s+b\mathcal{O})m_{-}\right)$
$\displaystyle\sim(sg+b\omega)(m_{+},m_{-})$ (2.25)

for $m=j|_{\mathfrak{m}}$ and $h=j^{(0)}=j|_{\mathfrak{h}}$. The equations of
motion $\mathcal{E}=0$ and the Maurer-Cartan identity $\mathcal{M}=0$ are
computed as:

$\displaystyle\mathcal{E}$
$\displaystyle=\partial_{+}m_{-}+\partial_{-}m_{+}+[h_{+},m_{-}]-[m_{+},h_{-}]-3b\mathcal{O}[m_{+},m_{-}]|_{\mathfrak{m}}=0$
$\displaystyle\mathcal{M}|_{\mathfrak{h}}$
$\displaystyle=\partial_{+}h_{-}-\partial_{-}h_{+}+[h_{+},h_{-}]+[m_{+},m_{-}]|_{\mathfrak{h}}=0$
(2.26) $\displaystyle\mathcal{M}|_{\mathfrak{m}}$
$\displaystyle=\partial_{+}m_{-}-\partial_{-}m_{+}+[h_{+},m_{-}]+[m_{+},h_{-}]+[m_{+},m_{-}]|_{\mathfrak{m}}=0,$

using the conditions (2.24). Analogously to the previous section, the
integrable choices for $s$ and $b$ is $b=\pm s/3$ and the Lax pair can be
given only in terms of the decomposition $j=h+m$ and the structure
$\mathcal{O}$:

$\displaystyle L_{+}(\lambda)$
$\displaystyle=h_{+}+\frac{1}{2}(\lambda^{2}+\lambda)m_{+}+\frac{1}{2}(\lambda^{2}-\lambda)\mathcal{O}m_{+},$
$\displaystyle L_{-}(\lambda)$
$\displaystyle=h_{-}+\frac{1}{2}(\frac{1}{\lambda^{2}}+\frac{1}{\lambda})m_{-}+\frac{1}{2}(\frac{1}{\lambda}-\frac{1}{\lambda^{2}})\mathcal{O}m_{-}.$

The problem with the case of an almost complex structure is that the WZ-term
$j^{(1)}\wedge j^{(2)}$ is imaginary due to $j^{(1)}=\overline{j^{(2)}}$.
Hence, for the integrable $\sigma$-model, where the constant in front of that
term is fixed, the Lagrangian corresponds to the pullback of some kind of
hermitian metric $g+i\omega$ to the world-sheet. Such problems with reality
were already pointed out in [48]. For the para-complex case this problem does
not occur and it is possible to write down a real action for integrable
$\mathbb{Z}_{3}$-/$\mathbb{Z}_{N}$-models.

Of course, one might be able to define other kind of integrable models on
these nearly Kähler geometries. I.e. all of the earlier mentioned compact six-
dimensional geometries admitting a $\mathbb{Z}_{3}$-grading corresponding to
an almost complex structure – $S^{3}\times S^{3}$, $S^{6}$, $\mathbb{CP}^{3}$
and the flag manifold $\mathbb{F}^{3}$ – are also $\mathbb{Z}_{2}$-symmetric
homogeneous spaces. Hence, we can define $\mathbb{Z}_{2}$-cosets
$\sigma$-models on those. Also, instead basing the construction on the
$\mathbb{Z}_{3}$-grading (hence an almost complex structure), some of these
also admit complex structures based on which interesting integrable models
where defined in [53, 55, 53, 57, 64].

##### Example.

Let us study the case of the coset $G\times G\sim G\times G\times G/G_{diag}$,
with a Lie (super)group $G$, to which also the case of $S^{3}\times S^{3}$
belongs for $G=$ SU$(2)$. The $\mathbb{Z}_{3}$-outer automorphism $\sigma$ is
simply the permutation of the three copies of $G$. One possible
parameterisation of $G\times G\times G$ is $(g_{1}h,g_{2}h,h)$ for $g_{i},h\in
G$, such that $h$ parameterises $G_{diag}$ and can be subsequently chosen to
be 1.

$\displaystyle M=G\times G\hookrightarrow G\times G\times G,\
(g_{1},g_{2})\rightarrow(g_{1},g_{2},1)$ (2.27)

This corresponds to the embedding
$\mathfrak{g}\times\mathfrak{g}=\mathfrak{m}\hookrightarrow\mathfrak{g}\times\mathfrak{g}\times\mathfrak{g}$:

$j=(j_{1},j_{2})=(g_{1}^{-1}\mathrm{d}g_{1},g_{2}^{-1}\mathrm{d}g_{2})\rightarrow(j_{1},j_{2},-j_{1}-j_{2})$
(2.28)

on the level of the algebra. The currents $j_{1},j_{2}\in\mathfrak{g}$, which
are Maurer-Cartan forms on the two copies of $G$, should not be confused with
the projections to the $\mathbb{Z}_{3}$-components $j^{(1)}$, $j^{(2)}$. A
bilinear form on $\mathfrak{m}$ is inherited from the canonical Ad-invariant
bilinear form on $\mathfrak{g}\times\mathfrak{g}\times\mathfrak{g}$ by that
embedding, denoted by ’Tr’,

$\left\langle(m_{1},m_{2}),(n_{1},n_{2})\right\rangle=\text{Tr}\left(m_{1}n_{1}+m_{2}n_{2}+(m_{1}+m_{2})(n_{1}+n_{2})\right).$
(2.29)

With that, the metric part of the Lagrangian becomes:

$\displaystyle L=\frac{1}{2}\left\langle
j_{+},j_{-}\right\rangle=\frac{1}{2}\left\langle(j_{1+},j_{2+}),(j_{1-},j_{2-})\right\rangle\quad\Rightarrow\quad
S\sim\int\text{Tr}\left(j_{1}\wedge\star j_{1}+j_{2}\wedge\star
j_{2}+j_{1}\wedge\star j_{2}\right).$

In particular this does not correspond to the bi-invariant metric on $G\times
G$ anymore, because of the coupling term. But, as expected, it is invariant
under the global action an element $(G_{1},G_{2},G_{3})\in G\times G\times G$:

$(G_{1},G_{2},G_{3}).(g_{1},g_{2})=(G_{1}g_{1}G_{3},G_{2}g_{2}G_{3}),\quad(G_{1},G_{2},G_{3}).(j_{1},j_{2})=(G_{3}^{-1}j_{1}G_{3},G_{3}^{-1}j_{2}G_{3})$
(2.30)

by the Ad$(G)$-invariance of the trace. The local symmetry $g_{i}\rightarrow
g_{i}h(x)$ was fixed already by choosing the representatives on the currents
$j\in\mathfrak{m}$ as above. This construction generalises to the construction
of metrics on $G^{N}/G$-cosets with an $\mathbb{Z}_{N}$-automorphism [65].

From (2.23) one can derive a choice of almost complex structure $\mathbb{J}$
on $\mathfrak{m}$:

$\displaystyle\mathbb{J}=\frac{1}{\sqrt{3}}\left(\begin{array}[]{cc}1&2\\\
-2&-1\end{array}\right),\qquad\mathbb{J}^{2}=-1.$

Coupling the $\sigma$-model to corresponding 2-form of the compatible triple,
$\omega=\left\langle\cdot,\mathbb{J}(\cdot)\right\rangle$:

$L=\frac{1}{2}\left\langle
j_{+},(\mathbb{1}+c\mathbb{J})(j_{-})\right\rangle\quad\Rightarrow\quad
S\sim\int\text{Tr}\left(j_{1}\wedge\star j_{1}+j_{2}\wedge\star
j_{2}+j_{1}\wedge\star j_{2}+\sqrt{3}cj_{1}\wedge j_{2}\right).$ (2.31)

This is a potentially interesting model of two interacting
$\mathfrak{g}$-valued currents. But, as mentioned in the previous paragraph,
the integrable choice of parameter of this model is imaginary
$c=\pm\frac{i}{3}$. As a consequence, such a WZ-term gives an imaginary
contribution to the action.

####  2.2.2 $\mathbb{Z}_{N}$-supercosets as supergravity backgrounds

From the point of view of string theory the question is, of course, whether
$\mathbb{Z}_{2M}$-(super)-cosets could (for $M>2$) give rise some new,
probably quite exotic 333due to the presence of $H$-flux, supergravity
backgrounds, potentially with AdS-factors for application in AdS/CFT.
Compactifications based on nearly Kähler geometry have been discussed for
example in [66].

Supercosets, where the isometry algebra is a Lie superalgebra with zero
Killing form, and the parameters $b_{i}$ are chosen such that the
$\sigma$-model is integrable (for $s_{i}=1$), are supergravity backgrounds –
in the sense of them being conformally invariant at 1-loop. [60]. This result
is generalising the analysis for supergroup principal chiral models and
$\mathbb{Z}_{2}$\- and $\mathbb{Z}_{4}$-coset [6, 5].

The classical Lie superalgebras with zero Killing form are [67], the basic Lie
superalgebras $A(n,n)=\text{PSL}(n|n)$, $D(n+1|n)=\text{OSp}(2(n+1)|2n)$, the
exceptional one $D(2|1;\alpha)$, and the strange Lie superalgebras $P(n)$,
$Q(n)$ (both subalgebras of $\text{PSL}(n|n)$) and their real forms. For a
review of their construction see [68]. In principle, the possible outer
$\mathbb{Z}_{N}$-automorphisms of these have been studied in [69, 70] and
should be scanned for potentially interesting cases that are applicable for
supergravity backgrounds.

In addition to supersymmetric versions of the examples in the previous
section, which were motivated by geometry, there other more group-theoretical
types of gradings. Such are known in the construction of the strange Lie
superalgebra $P(n)$ [67], in the construction of harmonic superspace [71], or
for several superconformal algebras several $\mathbb{Z}$-gradings appear, see
for example [72]. In these examples the grading is basically a consequence of
the Cartan decomposition and the grading comes from an inner automorphism (the
adjoint action with an Cartan element h) $\text{ad}(h)$. Take
$\mathfrak{sl}(2)$ as an example with generators $(h,e_{\pm})$ with
$[h,e_{\pm}]=\pm e_{\pm}$, $[e_{+},e_{-}]=h$. With identifications
$\mathfrak{g}^{(0)}=\text{span}(h)$ and
$\mathfrak{g}^{(1,2)}=\text{span}(e_{\pm})$ there is a natural, but slightly
trivial, $\mathbb{Z}_{3}$-grading due to
$[\mathfrak{g}^{(i)},\mathfrak{g}^{(i)}]=0$. The above mentioned grading on
Lie superalgebras are generalisations of this structure.

##  3 New integrable $\mathbb{Z}_{N}$-coset $\sigma$-models of hybrid type

So far, the literature on $\mathbb{Z}_{N}$-cosets has mainly centered around
the pure spinor-type (super)string $\sigma$-models, reviewed in the previous
section [48, 60, 73, 53, 54, 58]. But, Green-Schwarz superstring
$\sigma$-models are important as generalisation of the Polyakov action on
superspace, e.g. on AdS${}_{5}\times$S5. Furthermore, as has been shown in
[49] for $\mathbb{Z}_{4M}$-cosets in [59] for general $\mathbb{Z}_{N}$-soets,
certain Green-Schwarz like $\mathbb{Z}_{N}$-cosets $\sigma$-models exist for
$N>4$, too.

The task is to explore this space of theories and to find constants $s_{i}$
(some of which are $0$) and $b_{i}$ such that the model (2.5) possesses a Lax
representation. These are referred to as hybrid $\mathbb{Z}_{N}$-coset models
in the following. The ansatz444chosen such that it automatically implies the
Maurer-Cartan identity for the Lax connection

$L_{+}(\lambda)=j^{(0)}_{+}+\sum_{i=1}^{N-1}l_{i}(\lambda)j^{(i)}_{+},\qquad
L_{-}=j^{(0)}_{-}+\sum_{i=1}^{N-1}\frac{1}{l_{N-i}(\lambda)}j^{(i)}_{-}$ (3.1)

leads to the following equations for the constants $l_{i}$ when imposing both
the flatness on $L$ and the equations of motion for the current
$j=\mathfrak{g}^{-1}\mathrm{d}g$ (2.11):

$\displaystyle i\text{ s.t. }s_{i}$ $\displaystyle=0:\qquad
l_{N-i}=\frac{1}{l_{i}},\quad
F_{i}(\lambda)\mathcal{D}_{ij}=\left\\{\begin{array}[]{cr}l_{j}/l_{N+j-i}-l_{i}&i>j\\\
l_{j}/l_{j-i}-l_{i}&i<j\end{array}\right.$ (3.4) $\displaystyle i\text{ s.t.
}s_{i}$ $\displaystyle\neq
0:\qquad\frac{1}{2}\left(\frac{1}{l_{N-i}}-l_{i}\right)\frac{\mathcal{D}_{ij}}{s_{i}}=\left\\{\begin{array}[]{cr}l_{j}/l_{N+j-i}-\frac{1}{2}\left(\frac{1}{l_{N-i}}+l_{i}\right)&i>j\\\
l_{j}/l_{j-i}-\frac{1}{2}\left(\frac{1}{l_{N-i}}+l_{i}\right)&i<j\end{array}\right.$
(3.7)

with $\mathcal{D}_{ij}$ from (2.14) given in terms of the constants $s_{i}$
and $b_{i}$. When this system is underdetermined the above ansatz is a
spectral parameter dependent Lax connection. Because the ’equations of motion’
for the $\mathfrak{g}^{(i)}$-component is a constraint when $s_{i}=0$ the Lax
condition has to give a constraint, as well. There can be an arbitrary
spectral parameter dependent function of proportionality between these two
constraints. This denoted above $F_{i}(\lambda)$. The freedom to choose those
functions and the necessity for case separation makes the search for
integrable choices of $s_{i}$ and $b_{i}$ difficult to handle. For that
reason, only models for $N\leq 6$ are scanned in generality in the next
section 3.1, as $\mathbb{Z}_{6}$-cosets would potentially be interesting as
supersymmetric version of the previously discussed $\mathbb{Z}_{3}$-cosets. A
big class of models for arbitrary $N$, for which the constants $f_{i}$ can be
given in a closed form, is presented in section 3.2.

###  3.1 Overview for $N\leq 6$

In table 1 the integrable hybrid $\mathbb{Z}_{N}$-coset $\sigma$-models have
been collected. The corresponding Lax connection is given there by specifying
the constants $l_{i}$ in the ansatz (4.42) as functions of the spectral
parameter $\lambda$.

$\begin{array}[]{c|cc|c}\text{grading}&\\{s_{i}\\}&\\{b_{i}\\}&\\{l_{i}(\lambda)\\}\\\
\hline\cr\hbox{\multirowsetup$\mathbb{Z}_{3}$}&(0,0)&(\frac{2}{3},-\frac{2}{3})&(\lambda,\frac{1}{\lambda})\\\
&(1,1)&(-\frac{1}{3},\frac{1}{3})&(\lambda,\lambda^{2})\\\
\hline\cr\hbox{\multirowsetup$\mathbb{Z}_{4}$}&(0,0,0)&\hbox{\multirowsetup$(-b,0,b)$}&\hbox{\multirowsetup
no Lax pair found}\\\ &(1,0,1)&&\\\
&(0,1,0)&(-\frac{1}{2},0,\frac{1}{2})&(\lambda,\lambda^{2},\frac{1}{\lambda})\\\
&(1,1,1)&(-\frac{1}{2},0,\frac{1}{2})&(\lambda,\lambda^{2},\lambda^{3})\\\
\hline\cr\hbox{\multirowsetup$\mathbb{Z}_{5}$}&(0,0,0,0)&(-\frac{2}{5},-\frac{4}{5},\frac{2}{5},\frac{4}{5})&(\lambda,\lambda^{2},\frac{1}{\lambda^{2}},\frac{1}{\lambda})\\\
&(0,1,1,0)&(-\frac{2}{5},\frac{1}{5},-\frac{1}{5},\frac{2}{5})&(\lambda,\lambda^{2},\lambda^{3}\frac{1}{\lambda})\\\
&(1,0,0,1)&(-\frac{1}{5},-\frac{2}{5},\frac{2}{5},\frac{1}{5})&(\lambda^{3},\lambda,\frac{1}{\lambda},\lambda^{2})\\\
&(1,1,1,1)&(-\frac{3}{5},-\frac{1}{5},\frac{1}{5},\frac{3}{5})&(\lambda,\lambda^{2},\lambda^{3},\lambda^{4})\\\
\hline\cr\hbox{\multirowsetup$\mathbb{Z}_{6}$}&(0,0,0,0,0)&\hbox{\multirowsetup$(-b_{1},-b_{2},0,b_{2},b_{1})$}&\hbox{\multirowsetup
no Lax pair found}\\\ &(0,1,0,1,0)&&\\\ &(1,0,1,0,1)&&\\\ &(1,0,0,0,1)&&\\\
&(0,0,1,0,0)&(-\frac{1}{3},-\frac{2}{3},0,\frac{2}{3},\frac{1}{3})&(\lambda,\lambda^{2},\lambda^{3},\frac{1}{\lambda^{2}},\frac{1}{\lambda})\\\
&(0,1,1,1,0)&(-\frac{1}{3},\frac{1}{3},0,-\frac{1}{3},\frac{1}{3})&(\lambda,\lambda^{2},\lambda^{3},\lambda^{4},\frac{1}{\lambda})\\\
&(1,1,0,1,1)&(-\frac{1}{3},\frac{1}{3},0,-\frac{1}{3},\frac{1}{3})&(\lambda^{2},\lambda,1,\lambda^{2},\lambda)\\\
&(1,1,1,1,1)&(-\frac{2}{3},-\frac{1}{3},0,\frac{1}{3},\frac{2}{3})&(\lambda,\lambda^{2},\lambda^{3},\lambda^{4},\lambda^{5})\end{array}$
Table 1: A, not necessarily complete, list for the data $\\{s_{i},b_{i}\\}$ of
$\mathbb{Z}_{N}$-coset $\sigma$-models for $N\leq 6$ and, if integrable and
found by the author, the coefficients $\\{l_{i}\\}$ in the ansatz (3.1) of the
corresponding Lax connection

All the integrable choices for $s_{i}$ and $b_{i}$, found there, have in
common that all the non-zero $s_{i}$ are equal. As total factors in front of
the action are neglected here, we choose $s_{i}=s_{N-i}=1$ for the non-
vanishing $s_{i}$. This equality might seem mysterious but it makes sense
geometrically. Let $\tilde{\mathfrak{m}}$ denote $\sum_{i\ \text{with}\
s_{i}\neq 0}\mathfrak{g}^{(i)}$ then the kinetic term, coupling to some kind
of metric, of the hybrid $\mathbb{Z}_{N}$-coset $\sigma$-model simply becomes

$\mathcal{L}\sim\text{Tr}\left(j_{+}\mathcal{P}_{\tilde{\mathfrak{m}}}(j_{-})\right),$
(3.8)

which corresponds to the reduced metric to some kind of non-homogeneous
quotient of $G$, as $(\mathfrak{g}-\tilde{\mathfrak{m}})$ does not correspond
a subalgebra of $\mathfrak{g}$ in general. The maybe expected Green-Schwarz
models for $s_{i}=\left\\{\begin{array}[]{cc}1&i\text{ even}\\\ 0&i\text{
odd}\end{array}\right.$ do not seem to exist for arbitrary $N$.

Furthermore, commenting on the found integrable choices for $b_{i}$, there
might be more possibilities. For example, to all models we can find a second
integrable model with $b_{i}\rightarrow-b_{i}$ and the corresponding Lax
connection when exchanging $\mathfrak{g}^{(i)}$ and $\mathfrak{g}^{(N-i)}$.

###  3.2 A class of hybrid $\mathbb{Z}_{N}$-coset $\sigma$-models

It seems illusory to find a closed form for all integrable choices of $b_{i}$
for arbitrary $N$, but one class of hybrid $\mathbb{Z}_{N}$-coset can be
found, extrapolating from table 1. Let $N=2M+1$ resp. $2M+2$, then a class of
hybrid $\mathbb{Z_{N}}$-coset $\sigma$-models is characterised for each $N$ by
a number $K=1,...,M$. The $\sigma$-model is defined by the choice of
constants:

$\displaystyle s_{i}$ $\displaystyle=0=s_{N-i}=0,\qquad
b_{i}=-b_{N-i}=-\frac{2i}{N}\qquad\text{for}\quad i=1,...,K$ $\displaystyle
s_{i}$ $\displaystyle=1.\qquad b_{i}=1-\frac{2i}{N}\qquad\text{for}\quad
i=K+1,...,N-K-1.$

So, the components $j^{(i)}$ and $j^{(N-i)}$ do not have kinetic terms in the
action for $i=1,...,K$. A choice of Lax connection that is equivalent to the
Maurer-Cartan identity and equations of motion is:

$\displaystyle L_{+}(\lambda)$
$\displaystyle=j_{+}^{(0)}+\sum_{i=1}^{N-K-1}\lambda^{i}j_{+}^{(i)}+\sum_{i=N-K}^{N-1}\frac{1}{\lambda^{N-i}}k_{+}^{(i)}$
(3.9) $\displaystyle L_{-}(\lambda)$
$\displaystyle=j_{-}^{(0)}+\sum_{i=1}^{K1}\lambda^{i}j_{-}^{(i)}+\sum_{i=K+1}^{N-1}\frac{1}{\lambda^{N-i}}k_{+}^{(i)}$

For $K=0$ the action and original Lax connection of the pure spinor-type
models [48] is reproduced, for $N=4L$ and $K=M$ the Green-Schwarz action of
[49]. The $K=M$ case for arbitrary $N$ has been constructed in [59] already.
As discussed there, it is noteworthy that for odd $N$ (as can also be seen for
the $\mathbb{Z}_{3}$\- and $\mathbb{Z}_{5}$-cases in table 1), the $K=M$-case
of the above class of models doesn’t have any kinetic terms. Its dynamic is
only governed by constraints. By means of the Lax connection (3.9) we have a
Lax representation of this constraint structure. For the
$\mathbb{Z}_{3}$-cosets based on an almost complex structure from section 2.2,
the action of that model could avoid the issues with reality of the action.

##  4 Yang-Baxter deformed $\mathbb{Z}_{N}$-coset $\sigma$-models

###  4.1 The action

As the Yang-Baxter deformation of the $\mathbb{Z}_{N}$-model, we define

$L=\frac{1}{2}\mathrm{Tr}\left(j_{+}\mathcal{P}_{-}\frac{1}{1-\eta
R_{g}\circ\mathcal{P}_{-}}j_{-}\right)$ (4.1)

generalising the one of the principal chiral model, the $\mathbb{Z}_{2}$\- and
the $\mathbb{Z}_{4}$-(super)cosets. The deformation is characterised by a
skewsymmetric555meaning $\mathrm{Tr}\left(m\
R(n)\right)=-\mathrm{Tr}\left(R(m)\ n\right)$. This is also true in case of a
Lie superalgebra if, as is typically the case, the $R$-operator is chosen to
be even. operator $R:\ \mathfrak{g}\rightarrow\mathfrak{g}$ via the
combination $R_{g}=\mathrm{Ad}_{g}^{-1}\circ R\circ\mathrm{Ad}_{g}$. $\eta$ is
the deformation parameter, and the current $j$ and the projectors
$\mathcal{P}_{\pm}$ are defined as in section 2.

The dynamics can be conveniently phrased in terms of the deformed currents

$k_{\pm}=\frac{1}{1\pm\eta R_{g}\circ\mathcal{P}_{\pm}}j_{\pm}.$ (4.2)

The equations of motion are

$\mathcal{E}=\partial_{+}\left(\mathcal{P}_{-}k_{-}\right)+\partial_{-}\left(\mathcal{P}_{+}k_{+}\right)+[k_{+},\mathcal{P}_{-}k_{-}]+[k_{-},\mathcal{P}_{+}k_{+}]=0$
(4.3)

and, in terms of the current $k$, the Maurer-Cartan equation for the current
$j$ becomes

$\mathcal{M}=\partial_{+}k_{-}-\partial_{-}k_{+}+[k_{+},k_{-}]+\eta^{2}c^{2}[\mathcal{P}_{+}k_{+},\mathcal{P}_{-}k_{-}]-\eta
R_{g}(\mathcal{E})=0,$ (4.4)

if the $R$-operator (and hence also the operator $R_{g}$) fulfils the modified
classical Yang-Baxter equation:

$[R(m),R(n)]-R\left([R(m),n]-[m,R(n)]\right)=c^{2}[m,n].$ (4.5)

All of these calculations work exactly analogously to [13] for any choice of
grading and any choice of projectors $\mathcal{P}_{\pm}$ (hence for any choice
of constants $s_{i}$ and $b_{i}$). For two choices of $s_{i}$ and $b_{i}$, the
Yang-Baxter deformation was already introduced in [59].

##### Homogeneous Yang-Baxter deformations.

We observe that, in case $R$ is a solution of the classical Yang-Baxter
equation ((4.5) for $c=0$), the equations of motion (hence also a potential
Lax connection) will the same as in the undeformed case (as then $c\eta=0$).
In order to show that both models are really (classically) equivalent one has
to check that the transformation $j\rightarrow k$ is a canonical
transformation.

From the form of the action, resp. from the kind of redefinition (4.2)
connecting deformed and undeformed model, we see that the deformation takes
the form of a $\beta$-shift, generalising the deformation of the principal
chiral model and the $\mathbb{Z}_{2}$\- and $\mathbb{Z}_{4}$-case. In case
$c=0$, it is expected that this transformation is connected to abelian/non-
abelian $T$-duality [19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. The homogeneous
Yang-Baxter deformation for the Green-Schwarz $\mathbb{Z}_{4M}$-(super)coset
$\sigma$-model had already been constructed [56].

In the following this article will only be concerned with inhomogeneous Yang-
Baxter (often called $\eta$-)deformations.

###  4.2 $N=3$

Let us the exemplify the construction of the Lax connection for the case of
the $\mathbb{Z}_{3}$-coset. Decomposed into the components of the grading the
Maurer-Cartan equation (4.3) and the equations of motion (4.4) become

$\displaystyle\mathcal{M}^{(0)}$
$\displaystyle=\partial_{+}k_{-}^{(0)}-\partial_{-}k_{+}^{(0)}+[k_{+}^{(0)},k_{-}^{(0)}]+\mathcal{A}_{1}[k_{+}^{(1)},k_{-}^{(2)}]+\mathcal{A}_{2}[k_{+}^{(2)},k_{-}^{(1)}]=0$
$\displaystyle\mathcal{M}^{(1)}$
$\displaystyle=\partial_{+}k_{-}^{(1)}-\partial_{-}k_{+}^{(1)}+[k_{+}^{(0)},k_{-}^{(1)}]+[k_{+}^{(1)},k_{-}^{(0)}]+\mathcal{C}[k_{+}^{(2)},k_{-}^{(2)}]=0$
$\displaystyle\mathcal{M}^{(2)}$
$\displaystyle=\partial_{+}k_{-}^{(2)}-\partial_{-}k_{+}^{(2)}+[k_{+}^{(0)},k_{-}^{(2)}]+[k_{+}^{(2)},k_{-}^{(0)}]+\mathcal{C}[k_{+}^{(1)},k_{-}^{(1)}]=0$
(4.6) $\displaystyle(\mathcal{E}^{(1)}+b\mathcal{M}^{(1)})/s$
$\displaystyle=\partial_{+}k_{-}^{(1)}+\partial_{-}k_{+}^{(1)}+[k_{+}^{(0)},k_{-}^{(1)}]-[k_{+}^{(1)},k_{-}^{(0)}]-\mathcal{D}[k_{+}^{(2)},k_{-}^{(2)}]=0$
$\displaystyle(\mathcal{E}^{(2)}-b\mathcal{M}^{(2)})/s$
$\displaystyle=\partial_{+}k_{-}^{(2)}+\partial_{-}k_{+}^{(2)}+[k_{+}^{(0)},k_{-}^{(2)}]-[k_{+}^{(2)},k_{-}^{(0)}]+\mathcal{D}[k_{+}^{(1)},k_{-}^{(1)}]=0$

with

$\displaystyle\mathcal{A}_{1}$
$\displaystyle=1+\epsilon^{2}(s-b)^{2},\quad\mathcal{A}_{2}=1+\epsilon^{2}(s+b)^{2},$
$\displaystyle\mathcal{C}$
$\displaystyle=1+\epsilon^{2}(s^{2}-b^{2})\quad\text{and}\quad\mathcal{D}=\frac{b}{s}\left(3+\epsilon(s^{2}-b^{2})\right)=\frac{b}{s}\left(2-\mathcal{C}\right).$

where we defined $\epsilon=c\eta$ and Let us take the following ansatz for the
Lax connection:

$L_{\pm}=k_{\pm}^{(0)}+l_{\pm,1}k_{\pm}^{(1)}+l_{\pm,2}k_{\pm}^{(2)}.$

One ends up with the following four equations for coefficients $l_{\pm,i}$

$\displaystyle l_{+,1}\ l_{-,1}$
$\displaystyle=\frac{l_{-,2}+l_{+,2}}{2}\cdot\mathcal{C}+\frac{l_{-,2}-l_{+,2}}{2}\cdot\mathcal{D},\qquad
l_{+,1}\ l_{-,2}=\mathcal{A}_{1}$ $\displaystyle l_{+,2}\ l_{-,2}$
$\displaystyle=\frac{l_{-,1}+l_{+,1}}{2}\cdot\mathcal{C}-\frac{l_{-,1}-l_{+,1}}{2}\cdot\mathcal{D},\qquad
l_{+,2}\ l_{-,1}=\mathcal{A}_{2}$

when imposing the Lax condition
$\partial_{+}L_{-}-\partial_{-}L_{+}+[L_{+},L_{-}]=0$. In order to make that
system underdetermined, it is sufficient to impose the following two
conditions on $\mathcal{A}_{1}$, $\mathcal{A}_{2}$, $\mathcal{C}$ and
$\mathcal{D}$, which in turn allow to determine the deformation of the
constants $s$ and $b$ in terms of $\epsilon$:

$\displaystyle\mathcal{C}=\mathcal{D}\qquad\text{and}\qquad\mathcal{A}_{2}=\mathcal{C}^{2}.$
(4.7)

Both equations are solved simultaneously by

$\epsilon^{2}=\frac{3b-s}{(s-b)^{2}(s+b)}$ (4.8)

allowing to eliminate $\mathcal{C}$ and $\mathcal{D}$, and determine constants
$s$ and $b$ in terms of $\epsilon$. There will be a more complicated relation
between $s$ and $b$ than $b=\pm s/3$ in the undeformed model. Still, there is
a remaining degree of freedom for the choice of $s$ and $b$, which is
analogous to the freedom to rescale both $s$ and $b$ in the undeformed model.
Potentially, this remaining degree of freedom will be fixed in a Hamiltonian
analysis, when requiring that the associated charges are in involution.

With the above conditions, we get that
$l_{+,2}=\sqrt{\mathcal{A}_{2}}(l_{+,1}/\sqrt{\mathcal{A}_{1}})^{2}$. The Lax
connection with spectral parameter $\lambda$ takes the form:

$L_{+}(\lambda)=k_{+}^{(0)}+\lambda\sqrt{\mathcal{A}_{1}}k_{+}^{(1)}+\lambda^{2}\sqrt{\mathcal{A}_{2}}k_{+}^{(2)},\qquad
L_{-}(\lambda)=k_{-}^{(0)}+\frac{1}{\lambda^{2}}\sqrt{\mathcal{A}_{2}}k_{-}^{(1)}+\frac{1}{\lambda}\sqrt{\mathcal{A}_{1}}k_{+}^{(2)}$
(4.9)

with the constants $s$, $b$ in $\mathcal{A}_{i}$ (partially) fixed by equation
(4.8). This form will motivate the ansatz for the deformed
$\mathbb{Z}_{N}$-models.666Let us note that there is a second set of
sufficient conditions:
$\displaystyle\mathcal{C}=-\mathcal{D}\qquad\text{and}\qquad\mathcal{A}_{1}=\mathcal{C}^{2}.$
This leads to a second integrable model, analogous to the undeformed model.
The corresponding identifications are still $b_{i}\leftrightarrow
b_{N-i}=-b_{i}$ and $\mathfrak{g}^{(i)}\leftrightarrow\mathfrak{g}^{(N-i)}$.

###  4.3 General $N$

The Maurer-Cartan equation (4.4) and the equations of motion (4.3) become

$\displaystyle\mathcal{M}^{(0)}$
$\displaystyle=\partial_{+}k_{-}^{(0)}-\partial_{-}k_{+}^{(0)}+[k_{+}^{(0)},k_{-}^{(0)}]+\sum_{j=1}^{N-1}\mathcal{A}_{j}[k_{+}^{(j)},k_{-}^{(N-j)}]=0$
$\displaystyle\mathcal{M}^{(i)}$
$\displaystyle=\partial_{+}k_{-}^{(i)}-\partial_{-}k_{+}^{(i)}+[k_{+}^{(0)},k_{-}^{(i)}]+[k_{+}^{(i)},k_{-}^{(0)}]+\sum_{j\neq
0,i}\mathcal{C}_{ij}[k_{+}^{(j)},k_{-}^{(i-j)}]=0$ (4.10)
$\displaystyle(\mathcal{E}^{(i)}+b_{i}\mathcal{M}^{(i)})$
$\displaystyle=s_{i}(\partial_{+}k_{-}^{(i)}+\partial_{-}k_{+}^{(i)}+[k_{+}^{(0)},k_{-}^{(i)}]-[k_{+}^{(i)},k_{-}^{(0)}])+\sum_{j\neq
0,i}\mathcal{D}_{ij}[k_{+}^{(j)},k_{-}^{(i-j)}]=0$

when decomposed in the grading eigenspaces ($i=1,...,N-1$). The constants
$\mathcal{A}_{j}$, $\mathcal{C}_{ij}$ and $\mathcal{D}_{ij}$ are:

$\displaystyle\mathcal{A}_{i}$ $\displaystyle=1+\epsilon
f_{i}^{2},\quad\mathcal{C}_{ij}=1+\epsilon
f_{j}f_{j-i},\quad\mathcal{D}_{ij}=f_{j-i}-f_{j}-b_{i}\mathcal{C}_{ij}$ (4.11)

where again $\epsilon=c\eta$ was introduced and the indices on $s$ and $b$ are
to be understood as ’mod $N$’, so e.g. $b_{-i}=b_{N-i}=-b_{i}$ for simplicity
of notation.

####  4.3.1 Deformed pure spinor-type $\mathbb{Z}_{N}$-coset $\sigma$-models

Motivated by the ’pure spinor’ $\mathbb{Z}_{3}$-case, we make the following
ansatz for the Lax connection in case $s_{i}\neq 0$.

$L_{+}(\lambda)=k_{+}^{(0)}+\sum_{i=1}^{N-1}\lambda^{i}\sqrt{\mathcal{A}_{i}}k_{+}^{(i)},\qquad
L_{-}(\lambda)=k_{-}^{(0)}+\sum_{i=1}^{N-1}\frac{1}{\lambda^{N-i}}\sqrt{\mathcal{A}_{N-i}}k_{-}^{(i)}$
(4.12)

Indeed the flatness condition to this Lax connection is equivalent to the
equations of motion and Maurer-Cartan identity above (4.10), if the following
equations hold:

$\displaystyle\mathcal{C}_{ij}^{2}=\left\\{\begin{array}[]{cr}\frac{\mathcal{A}_{j}\mathcal{A}_{N+j-i}}{\mathcal{A}_{N-i}}&i>j\\\
\frac{\mathcal{A}_{j}\mathcal{A}_{j-i}}{\mathcal{A}_{i}}&i<j\end{array}\right.$
(4.15)
$\displaystyle\mathcal{D}_{ij}=\left\\{\begin{array}[]{cr}s_{i}\mathcal{C}_{ij}&i>j\\\
-s_{i}\mathcal{C}_{ij}&i<j\end{array}\right.$ (4.18)

These equations are solved when the constants $f_{i}=s_{i}-b_{i}$ solve the
following equations

$-\epsilon^{2}=\left\\{\begin{array}[]{cr}\frac{f_{N+j-i}-f_{j}+f_{N-i}}{f_{N+j-i}f_{j}f_{N-i}}&i>j\\\
\frac{f_{j-i}-f_{j}+f_{i}}{f_{j-i}f_{j}f_{i}}&i<j\end{array}\right..$ (4.19)

A solution to this condition is

$f_{i}=\frac{1}{\epsilon}\tan\left(\sigma(\epsilon)\epsilon\frac{2i}{N}\right)\qquad\overset{\epsilon\rightarrow
0}{\longrightarrow}\quad\sigma(0)\frac{2i}{N}$ (4.20)

with $\epsilon=c\eta$ and some regular function $\sigma(\epsilon)$ of the
deformation parameter. We observe that this gives the expected form for
$f_{i}=s_{i}-b_{i}$ in the undeformed limit,
$f_{i}(\epsilon=0)=\sigma(0)\frac{2i}{N}$, with $\sigma(0)$ playing the role a
total scale there. The freedom to choose this function is the same freedom
that was observed in the $\mathbb{Z}_{3}$-model. This function would be
expected to be fixed when working in the Hamiltonian formalism, analogously to
the $\mathbb{Z}_{4}$-case [30, 74]. In the scope of this paper, we will
determine it below by comparing it to the known Green-Schwarz
$\mathbb{Z}_{4}$-case [13].

Resultantly, (4.12) is a Lax connection if $f_{i}$ is given by (4.20). Let us
give explicit expressions for $\mathcal{A}_{i}$ and $s_{i}$ in terms of the
deformation parameter that will be useful in the following:

$\displaystyle\mathcal{A}_{i}$
$\displaystyle=\frac{1}{\cos^{2}\left(\sigma(\epsilon)\epsilon\frac{2i}{N}\right)},\qquad
s_{i}=\frac{1}{2\epsilon}\sin\left(2\epsilon\sigma(\epsilon)\right)\sqrt{\mathcal{A}_{i}\mathcal{A}_{N-i}}.$
(4.21)

####  4.3.2 Deformed hybrid $\mathbb{Z}_{N}$-coset $\sigma$-models with
$s_{i}=0$ for $i\leq K$

As in section 3 one generalisation of the above in detail is discussed in
detail. We use the same conditions as there. But, analogously to the Yang-
Baxter-deformed ’pure spinor’-type $\mathbb{Z}_{N}$-model, the constants
$s_{i}$, $b_{i}$ defining the models need to be deformed, too.

$\displaystyle f_{\bar{I}}$
$\displaystyle=-f_{N-\bar{I}}=\frac{1}{\epsilon}\tan\left(\sigma(\epsilon)\epsilon\frac{2\bar{I}}{N}\right),\qquad
f_{I}=\frac{1}{\epsilon}\tan\left(\sigma(\epsilon)\epsilon\frac{2I}{N}\right)$
(4.22)

This corresponds to:

$\displaystyle\mathcal{A}_{\bar{I}}$
$\displaystyle=\mathcal{A}_{N-\bar{I}}=\frac{1}{\cos\left(\sigma(\epsilon)\epsilon\frac{2\bar{I}}{N}\right)},\qquad\mathcal{A}_{I}=\frac{1}{\cos\left(\sigma(\epsilon)\epsilon\frac{2\bar{I}}{N}\right)}$
(4.23) $\displaystyle s_{\bar{I}}$ $\displaystyle=s_{N-\bar{I}}=0,\qquad
s_{I}=\frac{1}{2\epsilon}\sin\left(2\epsilon\sigma(\epsilon)\right)\sqrt{\mathcal{A}_{I}\mathcal{A}_{N-I}}$
(4.24)

A Lax pair, combining the ideas of the deformed pure spinor-type
$\mathbb{Z}_{N}$-model and the one of hybrid hybrid $\mathbb{Z}_{N}$-coset
models, is this one

$\displaystyle L_{+}(\lambda)$
$\displaystyle=k_{+}^{(0)}+\sum_{i=1}^{N-K-1}\lambda^{i}\sqrt{\mathcal{A}_{i}}k_{+}^{(i)}+\sum_{i=N-K}^{N-1}\frac{\sqrt{\mathcal{A}_{i}}}{\lambda^{N-i}}k_{+}^{(i)}$
$\displaystyle L_{-}(\lambda)$
$\displaystyle=k_{-}^{(0)}+\sum_{i=1}^{K}\lambda^{i}\sqrt{\mathcal{A}_{N-i}}k_{-}^{(i)}+\sum_{i=K+1}^{N-1}\frac{\sqrt{\mathcal{A}_{N-i}}}{\lambda^{N-i}}k_{-}^{(i)}$
(4.25)
$\displaystyle=k_{-}^{(0)}+\sum_{i=1}^{N-K-1}\frac{\sqrt{\mathcal{A}_{i}}}{\lambda^{i}}k_{-}^{(N-i)}+\sum_{i=N-K-1}^{N-1}\lambda^{N-i}\sqrt{\mathcal{A}_{i}}k_{-}^{(N-i)}.$
(4.26)

Inserting this into the flatness condition
$\partial_{+}L_{-}-\partial_{-}L_{+}+[L_{+},L_{-}]=0$ and using equation of
motion and Maurer-Cartan equations (4.10) results in the following conditions
on the constants $\mathcal{A}_{i}$, $\mathcal{C}_{ij}$ and $\mathcal{D}_{ij}$
from (4.11) for the $\mathfrak{g}^{(i)}$-component of the flatness condition:

  * •

$i\leq K$ or $i\geq N-K$: In two cases a non-trivial condition arises

$\displaystyle
F_{i}(\lambda)\mathcal{D}_{ij}=-\left\\{\begin{array}[]{lc}\lambda^{i}\left(\sqrt{\mathcal{A}_{j}\mathcal{A}_{N+j-i}}/\lambda^{N}-\mathcal{C}_{ij}\sqrt{A}_{i}\right)&\quad
i\leq K,\ j\geq N-K,\ j-i\leq N-K-1\\\
\lambda^{i}\left(\sqrt{\mathcal{A}_{j}\mathcal{A}_{j-i}}-\mathcal{C}_{ij}\sqrt{A}_{i}/\lambda^{N}\right)&\quad
i,j\geq N-K,\ i-j\leq K\end{array}\right.$ (4.29)

In both cases the corresponding functions of proportionality are

$F_{i}(\lambda)=\frac{2\epsilon}{\sin\left(2\epsilon\sigma(\epsilon)\right)}\frac{\cos\left(\sigma(\epsilon)\epsilon\frac{2(N-i)}{N}\right)\lambda^{N}-\cos\left(\sigma(\epsilon)\epsilon\frac{2i}{N}\right)}{2\lambda^{N-i}}.$
(4.30)

Again, as in the undeformed case, this insures that constraint structure
arising from the flatness of the Lax connection is the same one coming from
the equation of motion, as again the $\mathfrak{g}^{(i)}$-component of the
equation of motion (4.10) for $s_{i}=0$ is a constraint. These constraints
are:

$0\approx\Phi^{i}=\left\\{\begin{array}[]{lc}\sum_{j=N-K}^{N-K-1+i}\mathcal{C}_{ij}\sqrt{\mathcal{A}_{i}}[k_{+}^{(j)},k_{-}^{(N+i-j)}]&\quad
i=\bar{I}\leq K\\\
\sum_{j=i-K-1}^{N-K-1}\mathcal{C}_{ij}\sqrt{\mathcal{A}_{i}}[k_{+}^{(j)},k_{-}^{(i-j)}]&\quad
i=N-\bar{I}\geq N-K\end{array}\right.$ (4.31)

The concrete form of the $F_{i}(\lambda)$ is, of course, not important here
but it is important to show that it is exists and it is also notworthy that
similar functions appeared in the Hamiltonian analysis of the $\mathbb{Z}_{4}$
case with constraints [75, 76]. All other contributions to the constraints
vanish when the following conditions are fulfilled

$\displaystyle\mathcal{D}_{ij}=0,\qquad\mathcal{C}^{2}_{ij}=\left\\{\begin{array}[]{cr}\mathcal{A}_{j}\mathcal{A}_{N+j-i}/\mathcal{A}_{i}&i>j\\\
\mathcal{A}_{j}\mathcal{A}_{j-i}/\mathcal{A}_{i}&i<j\end{array}\right.,$

keeping in mind, that here $\mathcal{A}_{i}=\mathcal{A}_{N-i}$.

  * •

$K+1\leq i\leq N-K-1$: The two relevant sets of equation, that arise here when
in comparing the flatness of the Lax connection to equations of motion and
Maurer-Cartan equation (4.10) are:

$\displaystyle 0$
$\displaystyle=\left\\{\begin{array}[]{lc}\mathcal{D}_{ij}+s_{i}\mathcal{C}_{ij}=f_{N+j-i}-f_{j}+f_{i}\mathcal{C}_{ij}&\quad
0<i-j\leq K,\ j\leq N-K-1\\\
\mathcal{D}_{ij}-s_{i}\mathcal{C}_{ij}=f_{N+j-i}-f_{j}-f_{N-i}\mathcal{C}_{ij}&\quad
K+1\leq i-j\leq N-K-1,\ j\leq N-K-1\\\
\mathcal{D}_{ij}+s_{i}\mathcal{C}_{ij}=f_{j-i}-f_{j}+f_{i}\mathcal{C}_{ij}&\quad
j\leq N-K-1,\ 0<j-i\leq N-K-1\\\
\mathcal{D}_{ij}-s_{i}\mathcal{C}_{ij}=f_{j-i}-f_{j}+f_{N-i}\mathcal{C}_{ij}&\quad
j\geq N-K,\ 0<j-i\leq N-K-1\end{array}\right.$ (4.36)

and

$\displaystyle\mathcal{C}_{ij}^{2}=\left\\{\begin{array}[]{cl}\mathcal{A}_{j}\mathcal{A}_{N+j-i}/\mathcal{A}_{N-i}&\quad
0<i-j\leq K,\ j\leq N-K-1\\\
\mathcal{A}_{j}\mathcal{A}_{N+j-i}/\mathcal{A}_{i}&\quad K+1\leq i-j\leq
N-K-1,\ j\leq N-K-1\\\ \mathcal{A}_{j}\mathcal{A}_{j-i}/\mathcal{A}_{i}&\quad
j\leq N-K-1,\ 0<j-i\leq N-K-1\\\
\mathcal{A}_{j}\mathcal{A}_{j-i}/\mathcal{A}_{N-i}&\quad j\geq N-K,\ 0<j-i\leq
N-K-1\end{array}\right.$ (4.41)

With choice of constants $f_{i}$ in (4.22), and resultantly for the
$\mathcal{A}_{j}$ in, (4.24) all the above conditions are fulfilled. Hence,
(4.25) is the Lax representation of the Yang-Baxter deformed partly Green-
Schwarz type $\mathbb{Z}_{N}$-graded (super)coset $\sigma$-model

##### Fixing the function $\sigma(\epsilon)$.

The $N=4$, $K=1$-model is the Green-Schwarz type $\mathbb{Z}_{4}$-supercoset.
Let us compare the constants to the Lax connection of Delduc, Magro and Vicedo
[13]:

$L^{\text{(DMV)}}_{\pm}(\lambda)=k_{\pm}^{(0)}+\lambda\sqrt{1+\epsilon^{2}}k_{\pm}^{(1)}+\lambda^{\pm
2}\frac{1+\epsilon^{2}}{1-\epsilon^{2}}k_{\pm}^{(1)}+\frac{1}{\lambda}\sqrt{1+\epsilon^{2}}k_{\pm}^{(3)}$
(4.42)

and the projector
$\mathcal{P}_{-}=-\mathcal{P}^{(1)}+\frac{2}{1-\epsilon^{2}}\mathcal{P}^{(2)}+\mathcal{P}^{(3)}$.
The corresponding identifications are

$\displaystyle\sqrt{\mathcal{A}_{2}}$
$\displaystyle=\frac{1}{\cos\left(\sigma(\epsilon)\epsilon\right)}\overset{!}{=}\frac{1+\epsilon^{2}}{1-\epsilon^{2}}\quad\Rightarrow\quad\sigma(\epsilon)=\frac{1}{\epsilon}\arccos\left(\frac{1-\epsilon^{2}}{1+\epsilon^{2}}\right)$

which implies
$\sin(\epsilon\sigma(\epsilon))=\frac{2\epsilon}{1+\epsilon^{2}}$,
$\sin(2\epsilon\sigma(\epsilon))=4\epsilon\frac{1-\epsilon^{2}}{(1+\epsilon^{2})^{2}}$
and, as wished,

$\displaystyle\sqrt{\mathcal{A}_{1}}$
$\displaystyle=\sqrt{\mathcal{A}_{3}}=\frac{1}{\cos\left(\sigma(\epsilon)\epsilon/2\right)}=\sqrt{1+\epsilon^{2}},$
$\displaystyle s_{2}$
$\displaystyle=\frac{1}{\epsilon}\tan(\epsilon\sigma(\epsilon))=\frac{2}{1-\epsilon^{2}},\qquad
b_{1}=-\frac{1}{\epsilon}\tan(\epsilon\sigma(\epsilon)/2)=-1,$

showing that indeed (4.25) is a generalisation of (4.42) for arbitrary
deformed $\mathbb{Z}_{N}$-(super)-coset $\sigma$-models, valid for each
$K=1,...,M$. This choice of function $\sigma(\epsilon)$ should also reproduce
the found actions for $K=0$ and $K=M$ from [59].

But let it be noted that it is not necessary to fix the function
$\sigma(\epsilon)$ from the point of view of equivalence of the flatness of
the Lax connection to the equations of motion and Maurer-Cartan equation. It
might be necessary to fix it to ensure that the associated conserved charges
are in involution. Also, in the Lax formulation in terms of the deformed
currents (and typically also in the associated deformation of the Poisson
structure) only the combination $\epsilon^{2}=c^{2}\eta^{2}$ appears. So, in
principle, both non-split ($c=1$) and split solutions ($c=i$) to the modified
classical Yang-Baxter deformation seem possible in defining a deformation and
keeping things real. In case of $c=i$, one can set
$\epsilon^{2}\rightarrow-\eta^{2}$. $c$ only appears linearly in the arguments
of the expressions for $\mathcal{A}_{i}$, $f_{i}$, etc., in every instance as
argument of a trigonometric function.

##  5 Outlook

Several new models were proposed in this article together with their Lax
representations. In order to complete the proof of classical integrability of
these models a Hamiltonian analysis should be performed. It is expected that
one needs to invoke $r-s$-formalism and that both the undeformed and deformed
$\mathbb{Z}_{N}$-models fall into the class of models with twist function [50,
51, 52, 74, 58]. This would show the involution of the conserved charges
associated to these Lax pairs. Moreover, in case of the hybrid models a
thorough constraint analysis is needed to show Hamiltonian integrability, i.e.
involution of the conserved charges and all the constraints, of this class of
models. In the $\mathbb{Z}_{4}$-supercoset Green-Schwarz $\sigma$-model the
first class part of the (fermionic) constraints corresponding to the non-
existent ’kinetic’ term for $j^{(1)}$ and $j^{(3)}$ is the generator for the
(fermionic) $\kappa$-symmetry [76], raising the question what happens when
$s_{i}$ is a bosonic direction. Also, taking analogy from the Green-Schwarz
$\mathbb{Z}_{4}$-supercoset $\sigma$-model where the WZ-term allows for
$\kappa$-symmetry, which in turn allows get rid of some fermionic degrees of
freedom, one would assume that this $\sigma$-model has some kind of effective
target space, in which some degrees of freedom associated to the components
$j^{(i)}$ for which $s_{i}=0$ do not contribute.

This might also clarify the relationship between different types of
$\mathbb{Z}_{N}$-coset $\sigma$-model for the same $N$ and the same coset
$M=G/H$ but different choice of constants $s_{i}$. In analogy to the pure
spinor and Green-Schwarz superstring on $\mathbb{Z}_{4}$-cosets, which are
different formulations of the same theory, one might expect some relation
between such models. Another question is what the relationship between two
$\sigma$-models on $M=G/H$ resp. $M=\tilde{G}/\tilde{H}$, where $M$ can be
respresented by two different cosets with $\mathbb{Z}_{N}$ symmetry, but with
different $N$. For some cases it has been shown in [54] that these models
differ only by a topological term in the action and hence are classically
equivalent. It would be interesting to investigate that further.

In the end the answer to the question, how useful these integrable models
could be in the study of string theory, depends on whether one finds some
homogeneous supergravity backgrounds that possess $\mathbb{Z}_{N}$-symmetry.
These seem to be quite exotic but related geometries, like nearly Kähler
geometries have been discussed in the context AdS-flux compactifications [66]
supergroup cosets leading to supergravity solutions, i.e. if the isometry
group of the homogeneous space is a supergroup with vanishing Killing form
[60] might be possible. Analogously to [34] it should be also check whether
for the Yang-Baxter deformation some kind of unimodularity condition on the
operator $R$ ensures is sufficient such that the deformed background is still
a solution of supergravity.

### Acknowledgements

The author thanks Dmitri Bykov, Ben Hoare, Nat Levine and Maxime Trepanier for
discussions and Arkady Tseytlin for comments on the draft.

## References

  * [1] M. Gell-Mann and M. Levy, The axial vector current in beta decay, Nuovo Cim. 16 (1960) 705.
  * [2] H. Eichenherr and M. Forger, On the Dual Symmetry of the Nonlinear Sigma Models, Nucl. Phys. B 155 (1979) 381–393.
  * [3] M. Henneaux and L. Mezincescu, A Sigma Model Interpretation of Green-Schwarz Covariant Superstring Action, Phys. Lett. B 152 (1985) 340–342.
  * [4] R. Metsaev and A. A. Tseytlin, Type IIB superstring action in AdS${}_{5}\times$S5 background, Nucl. Phys. B 533 (1998) 109–126, [hep-th/9805028].
  * [5] N. Berkovits, M. Bershadsky, T. Hauer, S. Zhukov, and B. Zwiebach, Superstring theory on AdS(2) x S**2 as a coset supermanifold, Nucl. Phys. B 567 (2000) 61–86, [hep-th/9907200].
  * [6] M. Bershadsky, S. Zhukov, and A. Vaintrob, PSL(n$|$n) sigma model as a conformal field theory, Nucl. Phys. B 559 (1999) 205–234, [hep-th/9902180].
  * [7] I. Bena, J. Polchinski, and R. Roiban, Hidden symmetries of the $AdS_{5}\times S^{5}$ superstring, Phys. Rev. D 69 (2004) 046002, [hep-th/0305116].
  * [8] B. C. Vallilo, Flat currents in the classical AdS${}_{5}\times$S5 pure spinor superstring, JHEP 03 (2004) 037, [hep-th/0307018].
  * [9] G. Arutyunov and S. Frolov, Superstrings on AdS${}_{4}\times$CP3 as a Coset Sigma-model, JHEP 09 (2008) 129, [arXiv:0806.4940].
  * [10] B. Stefanski, jr, Green-Schwarz action for Type IIA strings on AdS${}_{4}\times$CP3, Nucl. Phys. B 808 (2009) 80–87, [arXiv:0806.4948].
  * [11] C. Klimčík, Yang-Baxter sigma models and dS/AdS T duality, JHEP 12 (2002) 051, [hep-th/0210095].
  * [12] C. Klimčík, On integrability of the Yang-Baxter sigma-model, J. Math. Phys. 50 (2009) 043508, [arXiv:0802.3518].
  * [13] F. Delduc, M. Magro, and B. Vicedo, An integrable deformation of the AdS${}_{5}\times$S5 superstring action, Phys. Rev. Lett. 112 (2014), no. 5 051601, [arXiv:1309.5850].
  * [14] T. J. Hollowood, J. L. Miramontes, and D. M. Schmidtt, Integrable Deformations of Strings on Symmetric Spaces, JHEP 11 (2014) 009, [arXiv:1407.2840].
  * [15] I. Kawaguchi, T. Matsumoto, and K. Yoshida, A Jordanian deformation of AdS space in type IIB supergravity, JHEP 06 (2014) 146, [arXiv:1402.6147].
  * [16] C. Klimčík, $\eta$ and $\lambda$ deformations as $\mathcal{E}$-models, Nucl. Phys. B900 (2015) 259–272, [arXiv:1508.05832].
  * [17] T. Matsumoto and K. Yoshida, Yang–Baxter sigma models based on the CYBE, Nucl. Phys. B893 (2015) 287–304, [arXiv:1501.03665].
  * [18] B. Hoare and S. J. van Tongeren, On jordanian deformations of AdS5 and supergravity, J. Phys. A49 (2016), no. 43 434006, [arXiv:1605.03554].
  * [19] T. Matsumoto and K. Yoshida, Lunin-Maldacena backgrounds from the classical Yang-Baxter equation - towards the gravity/CYBE correspondence, JHEP 06 (2014) 135, [arXiv:1404.1838].
  * [20] D. Osten and S. J. van Tongeren, Abelian Yang–Baxter deformations and TsT transformations, Nucl. Phys. B915 (2017) 184–205, [arXiv:1608.08504].
  * [21] B. Hoare and A. A. Tseytlin, Homogeneous Yang-Baxter deformations as non-abelian duals of the AdS5 sigma-model, J. Phys. A49 (2016), no. 49 494001, [arXiv:1609.02550].
  * [22] R. Borsato and L. Wulff, Integrable Deformations of $T$-Dual $\sigma$ Models, Phys. Rev. Lett. 117 (2016), no. 25 251602, [arXiv:1609.09834].
  * [23] R. Borsato and L. Wulff, On non-abelian T-duality and deformations of supercoset string sigma-models, JHEP 10 (2017) 024, [arXiv:1706.10169].
  * [24] J. J. Fernandez-Melgarejo, J.-i. Sakamoto, Y. Sakatani, and K. Yoshida, $T$-folds from Yang-Baxter deformations, JHEP 12 (2017) 108, [arXiv:1710.06849].
  * [25] J.-i. Sakamoto, Y. Sakatani, and K. Yoshida, Homogeneous Yang-Baxter deformations as generalized diffeomorphisms, J. Phys. A50 (2017), no. 41 415401, [arXiv:1705.07116].
  * [26] B. Hoare and F. K. Seibold, Poisson-Lie duals of the $\eta$ deformed symmetric space sigma model, JHEP 11 (2017) 014, [arXiv:1709.01448].
  * [27] D. Lüst and D. Osten, Generalised fluxes, Yang-Baxter deformations and the O(d,d) structure of non-abelian T-duality, JHEP 05 (2018) 165, [arXiv:1803.03971].
  * [28] B. Hoare and F. K. Seibold, Supergravity backgrounds of the $\eta$-deformed AdS${}_{2}\times$S${}^{2}\times$T6 and AdS${}_{5}\times$S5 superstrings, JHEP 01 (2019) 125, [arXiv:1811.07841].
  * [29] F. Delduc, M. Magro, and B. Vicedo, On classical $q$-deformations of integrable sigma-models, JHEP 11 (2013) 192, [arXiv:1308.3581].
  * [30] F. Delduc, M. Magro, and B. Vicedo, Derivation of the action and symmetries of the $q$-deformed $AdS_{5}\times S^{5}$ superstring, JHEP 10 (2014) 132, [arXiv:1406.6286].
  * [31] B. Hoare and A. A. Tseytlin, Type IIB supergravity solution for the T-dual of the $\eta$-deformed AdS${}_{5}\times$ S5 superstring, JHEP 10 (2015) 060, [arXiv:1508.01150].
  * [32] G. Arutyunov, S. Frolov, B. Hoare, R. Roiban, and A. A. Tseytlin, Scale invariance of the $\eta$-deformed AdS${}_{5}\times$S5 superstring, T-duality and modified type II equations, Nucl. Phys. B903 (2016) 262–303, [arXiv:1511.05795].
  * [33] B. Hoare and S. J. van Tongeren, Non-split and split deformations of ${\mathrm{AdS}}_{5}$, J. Phys. A49 (2016), no. 48 484003, [arXiv:1605.03552].
  * [34] R. Borsato and L. Wulff, Target space supergeometry of $\eta$ and $\lambda$-deformed strings, JHEP 10 (2016) 045, [arXiv:1608.03570].
  * [35] D. Orlando, S. Reffert, J.-i. Sakamoto, and K. Yoshida, Generalized type IIB supergravity equations and non-Abelian classical r-matrices, J. Phys. A49 (2016), no. 44 445403, [arXiv:1607.00795].
  * [36] R. Borsato and L. Wulff, Marginal deformations of WZW models and the classical Yang–Baxter equation, J. Phys. A52 (2019), no. 22 225401, [arXiv:1812.07287].
  * [37] S. J. van Tongeren, Unimodular jordanian deformations of integrable superstrings, SciPost Phys. 7 (2019) 011, [arXiv:1904.08892].
  * [38] C. Klimčík, Integrability of the bi-Yang-Baxter sigma-model, Lett. Math. Phys. 104 (2014) 1095–1106, [arXiv:1402.2105].
  * [39] F. Delduc, M. Magro, and B. Vicedo, Integrable double deformation of the principal chiral model, Nucl. Phys. B 891 (2015) 312–321, [arXiv:1410.8066].
  * [40] B. Hoare, Towards a two-parameter q-deformation of AdS${}_{3}\times$S${}^{3}\times$M4 superstrings, Nucl. Phys. B891 (2015) 259–295, [arXiv:1411.1266].
  * [41] F. Delduc, S. Lacroix, M. Magro, and B. Vicedo, On the Hamiltonian integrability of the bi-Yang-Baxter sigma-model, JHEP 03 (2016) 104, [arXiv:1512.02462].
  * [42] C. Klimcik, Yang-Baxter $\sigma$-model with WZNW term as ${\mathcal{E}}$-model, Phys. Lett. B 772 (2017) 725–730, [arXiv:1706.08912].
  * [43] F. Delduc, B. Hoare, T. Kameyama, and M. Magro, Combining the bi-Yang-Baxter deformation, the Wess-Zumino term and TsT transformations in one integrable $\sigma$-model, JHEP 10 (2017) 212, [arXiv:1707.08371].
  * [44] F. Delduc, B. Hoare, T. Kameyama, S. Lacroix, and M. Magro, Three-parameter integrable deformation of $\mathbb{Z}_{4}$ permutation supercosets, JHEP 01 (2019) 109, [arXiv:1811.00453].
  * [45] C. Klimčik, Dressing cosets and multi-parametric integrable deformations, JHEP 07 (2019) 176, [arXiv:1903.00439].
  * [46] F. K. Seibold, Two-parameter integrable deformations of the $AdS_{3}\times S^{3}\times T^{4}$ superstring, JHEP 10 (2019) 049, [arXiv:1907.05430].
  * [47] B. Hoare and S. Lacroix, Yang–Baxter deformations of the principal chiral model plus Wess–Zumino term, J. Phys. A 53 (2020), no. 50 505401, [arXiv:2009.00341].
  * [48] C. A. S. Young, Non-local charges, Z(m) gradings and coset space actions, Phys. Lett. B632 (2006) 559–565, [hep-th/0503008].
  * [49] S.-M. Ke, K.-J. Shi, and C. Wang, Flat currents of a Green-Schwarz sigma model on supercoset targets with Z(4m) grading, Int. J. Mod. Phys. A 23 (2008) 4219–4243.
  * [50] S.-M. Ke, X.-Y. Li, C. Wang, and R.-H. Yue, Classical exchange algebra of the nonlinear sigma model on a supercoset target with Z(2n) grading, Chin. Phys. Lett. 28 (2011) 101101.
  * [51] S.-M. Ke, W.-L. Yang, C. Wang, and Z.-Y. Wang, Hamiltonian analysis of a Green-Schwarz sigma model on a supercoset target with Z(4m) grading, Chin. Phys. C 35 (2011) 1004–1008.
  * [52] S.-M. Ke, W.-L. Yang, C. Wang, K.-X. Jiang, and K.-J. Shi, The classical exchange algebra of a Green-Schwarz sigma model on supercoset target space with Z(4m) grading, J. Math. Phys. 52 (2011) 083511.
  * [53] D. Bykov, Integrable properties of sigma-models with non-symmetric target spaces, Nucl. Phys. B 894 (2015) 254–267, [arXiv:1412.3746].
  * [54] D. V. Bykov, Cyclic gradings of lie algebras and lax pairs for $\sigma$-models, Theoretical and Mathematical Physics 189 (2016), no. 3 1734–1741.
  * [55] D. Bykov, Complex structures and zero-curvature equations for $\sigma$-models, Phys. Lett. B 760 (2016) 341–344, [arXiv:1605.01093].
  * [56] S.-M. Ke, W.-L. Yang, K.-X. Jang, C. Wang, X.-M. Shuai, Z.-Y. Wang, and G. Shi, Yang-Baxter deformations of supercoset sigma models with $\mathbb{Z}$(4m) grading, Chin. Phys. C 41 (2017), no. 11 113101.
  * [57] D. Bykov, Sigma Models with Complex, Graded and $\eta$-Deformed Target Spaces, Phys. Part. Nucl. 49 (2018), no. 5 963–965.
  * [58] F. Delduc, T. Kameyama, S. Lacroix, M. Magro, and B. Vicedo, Ultralocal Lax connection for para-complex $\mathbb{Z}_{T}$-cosets, arXiv:1909.00742.
  * [59] B. Hoare, Integrable Deformations of Sigma Models, arXiv:2109.14284.
  * [60] D. Kagan and C. A. S. Young, Conformal sigma-models on supercoset targets, Nucl. Phys. B745 (2006) 109–122, [hep-th/0512250].
  * [61] J. A. Wolf and A. Gray, Homogeneous spaces defined by lie group automorphisms. i,ii, Journal of Differential Geometry 2 (1968) 115–159.
  * [62] A. Gray, Nearly Kähler manifolds, Journal of Differential Geometry 4 (1970), no. 3 283 – 309.
  * [63] J.-B. Butruille, Classification des variété approximativement kähleriennes homogénes, Annals of Global Analysis and Geometry 27 (2005), no. 3 201–225.
  * [64] D. Bykov, Flag manifold $\sigma$-models: The $\frac{1}{N}$-expansion and the anomaly two-form, Nucl. Phys. B 941 (2019) 316–360, [arXiv:1901.02861].
  * [65] A. J. Ledger and M. Obata, Affine and Riemannian $s$-manifolds, Journal of Differential Geometry 2 (1968), no. 4 451 – 459.
  * [66] G. Lopes Cardoso, G. Curio, G. Dall’Agata, D. Lust, P. Manousselis, and G. Zoupanos, NonKahler string backgrounds and their five torsion classes, Nucl. Phys. B 652 (2003) 5–34, [hep-th/0211118].
  * [67] V. Kac, Lie superalgebras, Advances in Mathematics 26 (1977), no. 1 8 – 96.
  * [68] L. Frappat, P. Sorba, and A. Sciarrino, Dictionary on Lie superalgebras, hep-th/9607161.
  * [69] V. G. Kac, Automorphisms of finite order of semisimple Lie algebras, Funct. Anal. Appl. 3 (1969) 252–254.
  * [70] V. V. Serganova, Automorphisms of simple Lie superalgebras, Math. USSR, Izv. 24 (1985) 539–551.
  * [71] A. S. Galperin, E. A. Ivanov, V. I. Ogievetsky, and E. S. Sokatchev, Harmonic superspace.  Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2007.
  * [72] V. G. Kac, M. Wakimoto, et al., Quantum reduction and representation theory of superconformal algebras, Advances in Mathematics 185 (2004), no. 2 400–458.
  * [73] J. Kluson, Hamiltonian Analysis of Non-Linear Sigma Model on Supercoset Target, JHEP 10 (2006) 046, [hep-th/0608146].
  * [74] S. Lacroix, Integrable models with twist function and affine Gaudin models, PhD-thesis, arXiv:1809.06811.
  * [75] M. Magro, The Classical Exchange Algebra of $AdS_{5}\times S^{5}$, JHEP 01 (2009) 021, [arXiv:0810.4136].
  * [76] B. Vicedo, Hamiltonian dynamics and the hidden symmetries of the $AdS_{5}\times S^{5}$ superstring, JHEP 01 (2010) 102, [arXiv:0910.0221].