A hierarchy of WZW models related to super
Poisson-Lie T-duality

Ali<EMAIL_ADDRESS>Adel Rezaei-
<EMAIL_ADDRESS>
Department of Physics, Faculty of Basic Sciences,
Azarbaijan Shahid Madani University, 53714-161, Tabriz, Iran

Motivated by super Poisson-Lie (PL) symmetry of the Wess-Zumino-Witten (WZW)
model based on the $(C^{3}+A)$ Lie supergroup of our previous work [A. Eghbali
et al. JHEP 07 (2013) 134], we first obtain and classify all Drinfeld
superdoubles (DSDs) generated by the Lie superbialgebra structures on the
$({\mathscr{C}}^{3}+{\mathscr{A}})$ Lie superalgebra as a theorem. Then,
introducing a general formulation we find the conditions under which a two-
dimensional $\sigma$-model may be equivalent to a WZW model. With the help of
this formulation and starting the super PL symmetric $(C^{3}+A)$ WZW model, we
get a hierarchy of WZW models related to super PL T-duality, in such a way
that it is different from the super PL T-plurality, because the DSDs are, in
this process, non-isomorphic. The most interesting indication of this work is
that the $(C^{3}+A)$ WZW model does remain invariant under the super PL
T-duality transformation, that is, the model is super PL self-dual.
---

Keywords: Drinfeld superdouble, $\sigma$-model, Super Poisson-Lie symmetry,
WZW model

###### Contents

  1. 1  Introduction
  2. 2  DSDs generated by Manin supertriples with first sub-superalgebra $({\mathscr{C}}^{3}+{\mathscr{A}})$
    1. 2.1  A review of DSDs on the algebraic level
    2. 2.2  Manin supertriples with first sub-superalgebra $({\mathscr{C}}^{3}+{\mathscr{A}})$
  3. 3  A short review of super PL T-duality
  4. 4  Conditions that a $\sigma$-model may be given as a WZW model
  5. 5  A hierarchy of WZW models related to super PL T-duality
    1. 5.1 Super PL symmetry of the $(C^{3}+A)$ WZW model
    2. 5.2 Super PL T-dual $\sigma$-models on the superdouble $\big{(}(C^{3}+A)~{},~{}C^{3}\otimes{A}_{1,1}.ii\big{)}$
    3. 5.3 Super PL symmetry of the $(C^{3}+A).i$ WZW model
    4. 5.4 Super PL T-dual $\sigma$-models on the superdouble $\big{(}(C^{3}+A).i,(2{A}_{1,1}+2{A}\big{)}^{0}.ii\big{)}$
    5. 5.5 Super PL symmetry of the $(C^{3}+A).ii$ WZW model
  6. 6  Summary and concluding remarks
  7. A Isomorphism transformation between the Manin supertriples with the first sub-superalgebra ${\cal G}\equiv({\mathscr{C}}^{3}+{\mathscr{A}})$
    1. A.1 Structure of $DSD1:~{}\big{(}{\cal G},{\cal{I}}_{{}_{(2|2)}}\big{)}\cong\big{(}{\cal G},{\mathscr{C}}^{3}\oplus{{\mathscr{A}}_{{}_{1,1}}}.i\big{)}\cong\big{(}{\cal G},{({\mathscr{C}}^{3}+{\mathscr{A}})_{{}_{k=4}}}^{\epsilon=-1}\big{)}$
    2. A.2 Structure of $DSD4:\big{(}{\cal G},{{\mathscr{C}}_{{}_{p=-1}}}^{1,\epsilon=1}\oplus{{\mathscr{A}}}\big{)}\cong\big{(}{\cal G},{{\mathscr{C}}_{{}_{p=-1}}}^{1,\epsilon=-1}\oplus{{\mathscr{A}}}\big{)}$
    3. A.3 Structure of $DSD5:\big{(}{\cal G},{{\mathscr{C}}_{{}_{p=1}}}^{2,\epsilon=1}\oplus{{\mathscr{A}}_{{}_{1,1}}}\big{)}\cong\big{(}{\cal G},{{\mathscr{C}}_{{}_{p=1}}}^{2,\epsilon=-1}\oplus{{\mathscr{A}}_{{}_{1,1}}}\big{)}\\\ ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\cong\big{(}{\cal G},{{\mathscr{C}}}^{4,\epsilon=1}\oplus{{\mathscr{A}}_{{}_{1,1}}}\big{)}\cong\big{(}{\cal G},{{\mathscr{C}}}^{4,\epsilon=-1}\oplus{{\mathscr{A}}_{{}_{1,1}}}\big{)}$
    4. A.4 Structure of ${DSD22}^{{}^{p}}:\big{(}{\cal G},{{\mathscr{D}}_{{}_{p,p-1}}}^{1,\epsilon=1}\big{)}\cong\big{(}{\cal G},{{\mathscr{D}}_{{}_{1-p,-p}}}^{1,\epsilon=1}\big{)}\cong\big{(}{\cal G},{{\mathscr{D}}_{{}_{p,p-1}}}^{1,\epsilon=-1}\big{)},~{}~{}~{}p\neq 0,1$
    5. A.5 Structure of ${DSD23}^{{}^{k}}:\big{(}{\cal G},{(2{\mathscr{A}}_{{}_{1,1}}+2{\mathscr{A}})^{{}^{0}}}.i\big{)}\cong\big{(}{\cal G},{({{\mathscr{C}}}^{3}+{\mathscr{A}})_{{}_{k-4}}}^{\epsilon=1}\big{)}\\\ ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\cong\big{(}{\cal G},{({{\mathscr{C}}}^{3}+{\mathscr{A}})_{{}_{k}}}^{\epsilon=-1}\big{)},~{}k>4$
    6. A.6 Structure of ${DSD24}^{{}^{k}}:\big{(}{\cal G},{({{\mathscr{C}}}^{3}+{\mathscr{A}})_{{}_{k}}}^{\epsilon=-1}\big{)}\cong\big{(}{\cal G},{{({\mathscr{C}}}^{3}+{\mathscr{A}})_{{}_{4-k}}}^{\epsilon=-1}\big{)},~{}0<k<4$

##  1  Introduction

Over the past four decades, two-dimensional $\sigma$-models with
supermanifolds as target space have emerged in an extensive variety of systems
and their study has become increasingly relevant for some of the challenging
problems of modern physics, ranging from e.g. the condensed matter physics,
superstring models and the quantum Hall effect to the AdS/CFT correspondence
in string theory. In condensed matter, supermanifolds arise mostly in the
study of geometrical problems such as percolation and polymers [1], or in non-
interacting disordered systems [2, 3]. In string theory, supermanifold target
spaces were argued to arise as mirrors of rigid Calabi-Yau (CY) manifolds,
i.e. of CY spaces without complex moduli. The mirror image of such spaces has
no K$\ddot{a}$hler moduli and hence it cannot be a usual CY manifold. Sethi
argued that the dual of a rigid CY is instead given by a CY supermanifold [4].
In superstring theory, the $\sigma$-models with target space supermanifolds
naturally appear in the quantization of superstring theory with Ramond-Ramond
(RR) backgrounds. These models possess manifest target space supersymmetry and
have no worldsheet spinors.

On the other hand, it has been shown that the WZW models on Lie supergroups
are related to local logarithmic conformal field theories [5, 6, 7]. The first
attempt in this direction dates back to about four decades ago [8], where the
flat space Green-Schwarz superstring action was reproduced as a WZW type
$\sigma$-model on the coset superspace (D=10 Poincare supergroup/$SO(9,1))$.
It was shown that the non-manifestly supersymmetric piece of the covariant
Green-Schwarz superstring action is the WZW term associated with mappings of
two-dimensional spacetime SUSY(N = 2)/SO(9,1). Then, this work was extended to
the curved background and shown that [9] type IIB superstring on
$AdS_{5}\times S^{5}$ can be constructed from a $\sigma$-model on the coset
superspace $SU(2,2|4)/SO(4,1)\times SO(5)$. In addition, it was shown that the
$\sigma$-model on the $PSU(1,1|2)$ Lie supergroup can be used to quantize
superstring theory on the $AdS_{3}\times S^{3}$ background with RR flux [10,
11]. In [12], it was given a formulation of string theory on an interesting
four-dimensional background involving RR fields, in such a way that the
$\sigma$-model defined on the coset supermanifold $PSU(1,1|2)/U(1)\times U(1)$
was used for quantizing superstring theory on the $AdS_{2}\times S^{2}$
background with RR flux. In most circumstances, the models of interest are
believed to be more complicated than WZW models on Lie supergroups.
Nevertheless, even the WZW models on Lie supergroups are far from being
understood. Actually, the analysis of WZW model based on Lie supergroups is
notoriously difficult, even for the simplest case of $GL(1|1)$ [13] (see also
[5, 6, 7]).

An important concept that seems to be related to integrability is that of
duality, which plays a fundamental role in physics, relating different
theories in many perspectives. As shown, the integrability [14, 15, 16] and PL
T-duality, introduced in [17, 18], have in particularly revealed to be
strictly connected, the latter being a generalization of the string T-duality
of models with toric-compactified backgrounds. PL T-duality representing a
genuine generalization does not require isometries at all for the target
background. Specifically, symmetry under PL T-duality transformations is based
on the concept of PL dual groups and Drinfeld doubles [19]. Generalization of
T-duality to Lie supergroups [20] and also supermanifolds [21] has been
explored in the context of PL T-duality, originally introduced as an extension
of standard bosonic dualization [17, 18].

For the sake of clarity, let us comment on some of the related previous works.
As the first example of the PL symmetry in the WZW models, it was shown that
[22] the PL T-duality relates the standard SL$(2,\mathbb{R})$ WZW model to a
constrained $\sigma$-model defined on the SL$(2,\mathbb{R})$ group space.
Then, we showed that [23] the WZW model on the Heisenberg Lie group has the PL
symmetry only when its dual pair is the ${A}_{2}\otimes 2{A}_{1}$ Lie group
(see also [24, 25]). Regarding the super PL symmetry of the WZW models based
on Lie supergroups, we first showed that [26] the super PL duality relates the
$GL(1|1)$ WZW model to a $\sigma$-model defined on the $GL(1|1)$ Lie
supergroup when the dual Lie supergroup is isomorphic to the
${{B}\otimes{A}\otimes{A}_{1,1}}$. Then, in Ref. [27] we showed that the WZW
model on the $(C^{3}+A)$ Lie supergroup has also super PL symmetry when the
dual Lie supergroup is considered to be isomorphic to the ${C^{3}\otimes
A_{1,1}}$. Most interesting, we showed that the dual $\sigma$-model on the
${C^{3}\otimes A_{1,1}}$ can be equivalent to a WZW model defined on a Lie
supergroup which is isomorphic to the $(C^{3}+A)$ [27]. There, we announced
that this process can be continued. Following [27], in the present work, we
shall obtain a hierarchy of WZW models related to the super PL T-duality in
such a way that it is different from super PL T-plurality [28]. As shown in
[29], PL T-plurality [30] can construct a chain of supergravity solutions from
a PL symmetric solution. It is pointed out there that the target space of a
WZW model can be used as a seed solution to generate a chain of solutions
through the PL T-plurality. Recently, super non-Abelian T-duality of
$\sigma$-models on group manifolds with the emphasis on the T-dualization of
$\sigma$-models whose target spaces are supermanifolds have received
considerable attention [31] (see also [32]), in such a way that it has been
performed super non-Abelian T-dualization of the principal chiral model based
on the OSP$(1|2)$ Lie supergroup.

As a spin off from this progress, it would be interesting to get a hierarchy
of WZW models related to super PL T-duality. In this work, we first find that
which of the Manin supertriples with the first sub-superalgebra
$({\mathscr{C}}^{3}+{\mathscr{A}})$ represent decomposition of the same (or
more precisely isomorphic) DSD. In Ref. [28], it has been obtained the DSDs
with the first sub-superalgebra $({\mathscr{C}}^{3}+{\mathscr{A}})$ which are
only isomorphic to the semi-Abelian DSD
$(({\mathscr{C}}^{3}+{\mathscr{A}}),{\cal I}_{{}_{(2|2)}})$. Then, using the
super PL T-plurality formulation presented in [28] and starting from
decompositions of the $(({\mathscr{C}}^{3}+{\mathscr{A}}),{\cal
I}_{{}_{(2|2)}})$, it has been found the conformal duality chain of
$2+1$-dimensional cosmological string backgrounds coupling with two fermionic
fields (see also [33]). The main goal of the present work is to show, using
the example of the $(C^{3}+A)$ WZW model [27], that there is a hierarchy of
WZW models related to super PL T-duality. Most importantly, we show that the
WZW model on the $(C^{3}+A)$ does remain invariant under the super PL
T-duality transformation, that is, the model is super PL self-dual. The
classical self-duality of WZW model can be discussed for a general Lie group
$R$. Recently, the extension of the classical self-duality to a quantum one
has been performed by Sakatani et al. [34], in such a way that in order to
discuss the quantum self-duality, they have restricted the Lie group $R$ to be
$SU(2)$.

The paper is organized as follows. In section 2, we determine Manin
supertriples with the first sub-superalgebra
$({\mathscr{C}}^{3}+{\mathscr{A}})$ and show that they can be belonged to the
24 classes of the non-isomorphic DSDs. In section 3, we briefly review the
construction of the super PL T-dualizable $\sigma$-models on supermanifolds.
In section 4, we find the conditions under which a two-dimensional
$\sigma$-model may be equivalent to a WZW model. Section 5 contains the
original results of the work: using the general formulation introduced in
section 4 and then starting the super PL symmetric $(C^{3}+A)$ WZW model, we
get a hierarchy of WZW models related to super PL T-duality. Some concluding
remarks are given in the last section. At the end of the article we added
Appendix A devoted to carry out the DSD isomorphisms among the corresponding
Manin supertriples.

##  2  DSDs generated by Manin supertriples with first sub-superalgebra
$({\mathscr{C}}^{3}+{\mathscr{A}})$

###  2.1  A review of DSDs on the algebraic level

Let us start with the definition of a Lie superalgebra. A Lie superalgebra
${\mathscr{G}}$ is firstly a $Z_{2}$-graded vector space, thus admitting the
decomposition
${\mathscr{G}}={\mathscr{G}}_{{{}_{B}}}\oplus{\mathscr{G}}_{{}_{F}}$ (we refer
to ${\mathscr{G}}_{{{}_{B}}}$ and ${\mathscr{G}}_{{{}_{F}}}$ as the even and
odd subspaces of ${\mathscr{G}}$, respectively333In theoretical physics, the
even elements are sometimes called Bose elements or bosonic, and the odd
elements Fermi elements or fermionic.), secondly there is defined on
${\mathscr{G}}$ a particular kind of binary operation, i.e., a mapping
${\mathscr{G}}\otimes{\mathscr{G}}\rightarrow{\mathscr{G}}$, denoted by
$[.,.]$ satisfying the requirements of super antisymmetry and super Jacobi
identity [35, 36, 37]. One says that the Lie superalgebra ${\mathscr{G}}$ is
of the superdimension $(m|n)$ iff $dim\;{\mathscr{G}}_{{{}_{B}}}=m$ and
$dim~{}{\mathscr{G}}_{{{}_{F}}}=n$. It may be finite or infinite. To evade the
problems with definitions of supergroups we shall define the DSDs only on the
algebraic level.

The DSD $D$ is defined as a Lie supergroup whose Lie superalgebra,
${\mathscr{D}}$, equipped by a supersymmetric ad-invariant non-degenerate
bilinear form $<.~{},~{}.>$ can be decomposed into a pair of maximally
isotropic sub-superalgebras ${\mathscr{G}}$ and ${\tilde{\mathscr{G}}}$, and
${\mathscr{D}}$ as a vector superspace is the direct sum of ${\mathscr{G}}$
and ${\tilde{\mathscr{G}}}$. This ordered triple of Lie superalgebras
$({\mathscr{D}},{\mathscr{G}},{\tilde{\mathscr{G}}})$ is called Manin
supertriple. Moreover, the pair of $({\mathscr{G}},{\tilde{\mathscr{G}}})$ is
called Lie superbialgebra444Notice that we consider ${\tilde{\mathscr{G}}}$ as
a dual ${{\mathscr{G}}^{\ast}}$ to $\mathscr{G}$ and use the Lie superbracket
in ${\tilde{\mathscr{G}}}$ to define the Lie super cobracket in $\mathscr{G}$
[37]. [35, 36, 37]. Note that the dimensions of the sub-superalgebras are
equal and the bases ${T_{{}_{a}}}$ and
${\tilde{T}}^{{}^{a}},a=1,...,dim~{}{\mathscr{G}}$ in the sub-superalgebras
can be chosen so that555Similar to our previous works, here we consider the
standard basis for the supervector spaces so that in writing the basis as a
column matrix, we first present the bosonic bases, then the fermionic ones
(for further details refer to Ref. [38]). We note that $|a|$ denotes the
parity of $a$ where $|a|=0$ for the bosonic coordinates and $|a|=1$ for the
fermionic ones. Throughout the article we use Dewitt’s notation [38] such that
we identify the grading indices by the same indices in the power of $(-1)$,
that is, we use $(-1)^{a}$ instead of $(-1)^{|a|}$.

$\displaystyle<{T_{{}_{a}}},{T_{{}_{b}}}>=0,~{}~{}~{}<{\tilde{T}}^{{}^{a}},{\tilde{T}}^{{}^{b}}>=0,~{}~{}~{}~{}{{\delta}^{{}^{b}}}_{a}=<{\tilde{T}}^{{}^{b}},{T_{{}_{a}}}>=(-1)^{{}^{|a||b|}}<{T_{{}_{a}}},{\tilde{T}}^{{}^{b}}>.$
(2.1)

Due to the ad-invariance of bilinear form $<.,.>$ the algebraic structure of
${\mathscr{D}}$ is determined by the structure of the maximally isotropic sub-
superalgebras because in the bases ${T_{{}_{a}}}$ and ${\tilde{T}}^{{}^{a}}$
the Lie superbrackets are given by

$\displaystyle[{T_{{}_{a}}},{T_{{}_{b}}}]$ $\displaystyle=$
$\displaystyle{f^{c}}_{{}_{ab}}~{}{T_{{}_{c}}},~{}~{}~{}~{}~{}~{}~{}[{\tilde{T}}^{{}^{a}},{\tilde{T}}^{{}^{b}}]={{\tilde{f}}^{ab}}_{\;\;c}~{}{\tilde{T}}^{{}^{c}},$
$\displaystyle{[{T_{{}_{a}}},{\tilde{T}}^{{}^{b}}]}$ $\displaystyle=$
$\displaystyle(-1)^{{}^{b}}{{{\tilde{f}}^{bc}}}_{\;\;a}~{}{T_{{}_{c}}}+(-1)^{{}^{a}}{f^{b}}_{{}_{ca}}~{}{\tilde{T}}^{{}^{c}}.$
(2.2)

The Lie superalgebra structure defined by relation (2.1) is called the DSD
${\mathscr{D}}$. It is worth mentioning that the super Jacobi identity of Lie
superalgebra ${\mathscr{D}}$ imposes the following relation over the structure
constants of Lie superalgebras ${\mathscr{G}}$ and ${\tilde{\mathscr{G}}}$
[37]

$\displaystyle{f^{d}}_{bc}{\tilde{f}^{ae}}_{\;\;\;\;d}={f^{a}}_{dc}{\tilde{f}^{de}}_{\;\;\;\;\;b}+{f^{e}}_{bd}{\tilde{f}^{ad}}_{\;\;\;\;\;c}+(-1)^{be}{f^{a}}_{bd}{\tilde{f}^{de}}_{\;\;\;\;\;c}+(-1)^{ac}{f^{e}}_{dc}{\tilde{f}^{ad}}_{\;\;\;\;\;b}.$
(2.3)

###  2.2  Manin supertriples with first sub-superalgebra
$({\mathscr{C}}^{3}+{\mathscr{A}})$

Since we are interested on the classification of DSDs generated by Manin
supertriples with first sub-superalgebra $({\mathscr{C}}^{3}+{\mathscr{A}})$,
as the first step we define the $({\mathscr{C}}^{3}+{\mathscr{A}})$ Lie
superalgebra. The $({\mathscr{C}}^{3}+{\mathscr{A}})$ is a four-dimensional
Lie superalgebra of the type $(2|2)$ which is spanned by the set of generators
$\\{T_{{}_{1}},T_{{}_{2}};T_{{}_{3}},T_{{}_{4}}\\}$ with gradings;
grade$(T_{{}_{1}})$=grade$(T_{{}_{2}})=0$ and
grade$(T_{{}_{3}})$=grade$(T_{{}_{4}})=1$ 666From now on we consider
$(T_{{}_{1}},T_{{}_{2}})$ and $(T_{{}_{3}},T_{{}_{4}})$ as bosonic and
fermionic generators, respectively.. These generators fulfill the following
non-zero (anti)commutation rules [39]:

$\displaystyle[T_{{}_{1}},T_{{}_{4}}]=T_{{}_{3}},~{}~{}~{}~{}~{}~{}~{}\\{T_{{}_{4}},T_{{}_{4}}\\}=T_{{}_{2}}.$
(2.4)

It is also interesting to note that the $({\mathscr{C}}^{3}+{\mathscr{A}})$ is
itself a two-dimensional Lie superbialgebras, i.e., it is isomorphic to the
$(2|2)$-dimensional DSD $(({\mathscr{A}}_{{}_{1,1}}+{\mathscr{A}}),{\cal
I}_{{}_{(1|1)}})$ [40]. Note that the classification of four-dimensional Lie
superalgebras of the type $(2|2)$ was, for the first time, performed by
Backhouse in [39] (see also [41, 40] for the classification of decomposable
Lie superalgebras of the types $(1|2)$, $(2|1)$ and $(2|2)$).

In Ref. [42] we performed a complete classification of the
$({\mathscr{C}}^{3}+{\mathscr{A}})$ Lie superbialgebra and obtained 31
families of inequivalent Lie superbialgebra structures on
$({\mathscr{C}}^{3}+{\mathscr{A}})$ whose representatives have classified in
Table 1. Based on Table 1, the label of each Manin supertriple, e.g.
$\big{(}({\mathscr{C}}^{3}+{\mathscr{A}}),({\mathscr{C}}^{3}+{\mathscr{A}})_{k}^{\epsilon=\pm
1}\big{)}$, indicates the structure of the first sub-superalgebra
$\mathscr{G}$, e.g. $({\mathscr{C}}^{3}+{\mathscr{A}})$, the structure of the
second sub-superalgebra $\widetilde{\mathscr{G}}$, e.g.
$({\mathscr{C}}^{3}+{\mathscr{A}})_{k}^{\epsilon=\pm 1}$; roman numbers $i$,
$ii$ etc. (if present) distinguish between several possible pairings $<.,.>$
of the sub-superalgebras $\mathscr{G}$ and $\widetilde{\mathscr{G}}$, and the
parameter $k$ indicates the Manin supertriples differing in the rescaling of
$<.,.>$. As we use DeWitt notation here, in the standard basis the structure
constant ${f^{{}^{B}}}_{{}_{FF}}$ (anticommutator of fermion-fermion) must be
pure imaginary. Despite of this we have omitted the coefficient $i=\sqrt{-1}$
from all anticommutation relations in Table 1.

Table 1. Dual Lie superalgebras to the $({\mathscr{C}}^{3}+{\mathscr{A}})$ Lie
superalgebra.
$\tilde{\mathscr{G}}$ Non-zero (anti)commutation relations Comments ${\cal
I}_{(2|2)}$ All of the (anti)commutation relations are zero.
${\mathscr{C}}_{p=1}^{1,\epsilon}\oplus{\mathscr{A}}$
$[{\tilde{T}}^{{}^{1}},{\tilde{T}}^{{}^{2}}]=\epsilon{\tilde{T}}^{{}^{1}},\;\;\;[{\tilde{T}}^{{}^{2}},{\tilde{T}}^{{}^{3}}]=-\epsilon{\tilde{T}}^{{}^{3}}$
${\mathscr{C}}_{p=-1}^{1,\epsilon}\oplus{\mathscr{A}}$
$[{\tilde{T}}^{{}^{1}},{\tilde{T}}^{{}^{2}}]=\epsilon{\tilde{T}}^{{}^{1}},\;\;\;[{\tilde{T}}^{{}^{2}},{\tilde{T}}^{{}^{4}}]=\epsilon{\tilde{T}}^{{}^{4}}$
${\mathscr{C}}_{p=1}^{2,\epsilon}\oplus{\mathscr{A}}_{1,1}$
$[{\tilde{T}}^{{}^{2}},{\tilde{T}}^{{}^{3}}]=\epsilon{\tilde{T}}^{{}^{3}},\;\;\;[{\tilde{T}}^{{}^{2}},{\tilde{T}}^{{}^{4}}]=\epsilon{\tilde{T}}^{{}^{4}}$
${\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}.i$
$[{\tilde{T}}^{{}^{2}},{\tilde{T}}^{{}^{3}}]={\tilde{T}}^{{}^{4}}$
${\mathscr{C}}^{4,\epsilon}\oplus{\mathscr{A}}_{1,1}$
$[{\tilde{T}}^{{}^{2}},{\tilde{T}}^{{}^{3}}]={\tilde{T}}^{{}^{4}},\;\;\;[{\tilde{T}}^{{}^{2}},{\tilde{T}}^{{}^{3}}]=\epsilon{\tilde{T}}^{{}^{3}},\;\;\;[{\tilde{T}}^{{}^{2}},{\tilde{T}}^{{}^{4}}]=\epsilon{\tilde{T}}^{{}^{4}}$
${\mathscr{D}}_{p,p-1}^{1,\epsilon}$
$[{\tilde{T}}^{{}^{1}},{\tilde{T}}^{{}^{2}}]=-\epsilon{\tilde{T}}^{{}^{1}},\;\;\;[{\tilde{T}}^{{}^{2}},{\tilde{T}}^{{}^{3}}]=\epsilon
p{\tilde{T}}^{{}^{3}},\;\;\;[{\tilde{T}}^{{}^{2}},{\tilde{T}}^{{}^{4}}]=\epsilon(p-1){\tilde{T}}^{{}^{4}}$
$p\neq 0,1$
$\big{(}{\mathscr{D}}_{\frac{1}{2},\frac{1}{2}}^{7,\epsilon}\big{)}^{2}$
$[{\tilde{T}}^{{}^{1}},{\tilde{T}}^{{}^{2}}]=-2\epsilon{\tilde{T}}^{{}^{1}},\;\;\;[{\tilde{T}}^{{}^{2}},{\tilde{T}}^{{}^{3}}]=\epsilon{\tilde{T}}^{{}^{3}},\;\;\;[{\tilde{T}}^{{}^{2}},{\tilde{T}}^{{}^{4}}]=\epsilon{\tilde{T}}^{{}^{4}},\;\;\;\\{{\tilde{T}}^{{}^{3}},{\tilde{T}}^{{}^{4}}\\}=2\epsilon{\tilde{T}}^{{}^{1}}$
$\big{(}{\mathscr{D}}_{1-p,p}^{7,\epsilon}\big{)}.i$
$[\tilde{T}^{1},\tilde{T}^{2}]=-\epsilon\tilde{T}^{1},\;\;\;[\tilde{T}^{2},\tilde{T}^{3}]=\epsilon
p\tilde{T}^{3},\;\;\;[\tilde{T}^{2},\tilde{T}^{4}]=-\epsilon(p-1)\tilde{T}^{4},$
$\\{\tilde{T}^{3},\tilde{T}^{4}\\}=-2\epsilon(p-1)\tilde{T}^{1}$
$p<\frac{1}{2},~{}p\neq 0$
$\big{(}{\mathscr{D}}_{1-p,p}^{7,\epsilon}\big{)}.ii$
$[\tilde{T}^{1},\tilde{T}^{2}]=-\epsilon\tilde{T}^{1},\;\;\;[\tilde{T}^{2},\tilde{T}^{3}]=-\epsilon(p-1)\tilde{T}^{3},\;\;\;[\tilde{T}^{2},\tilde{T}^{4}]=\epsilon
p\tilde{T}^{4},$ $\\{\tilde{T}^{3},\tilde{T}^{4}\\}=2\epsilon p\tilde{T}^{1}$
$p<\frac{1}{2},~{}p\neq 0$ $\big{(}{\mathscr{D}}_{0}^{10,\epsilon}\big{)}^{1}$
$[\tilde{T}^{1},\tilde{T}^{2}]=-\epsilon\tilde{T}^{1},\;\;\;[\tilde{T}^{1},\tilde{T}^{3}]=-\frac{\epsilon}{4}\tilde{T}^{4},\;\;\;[\tilde{T}^{2},\tilde{T}^{4}]=\epsilon\tilde{T}^{4},~{}~{}~{}\\{\tilde{T}^{3},\tilde{T}^{3}\\}=\epsilon\tilde{T}^{2},$
$\\{\tilde{T}^{3},\tilde{T}^{4}\\}=2\epsilon\tilde{T}^{1}$
$\big{(}{\mathscr{D}}_{0}^{10,\epsilon}\big{)}^{2}$
$[\tilde{T}^{1},\tilde{T}^{2}]=-\epsilon\tilde{T}^{1},\;\;\;[\tilde{T}^{1},\tilde{T}^{3}]=\frac{\epsilon}{4}\tilde{T}^{4},\;\;\;[\tilde{T}^{2},\tilde{T}^{4}]=\epsilon\tilde{T}^{4},~{}~{}~{}\\{\tilde{T}^{3},\tilde{T}^{3}\\}=-\epsilon\tilde{T}^{2},$
$\\{\tilde{T}^{3},\tilde{T}^{4}\\}=2\epsilon\tilde{X}^{1}$
$\big{(}2{\mathscr{A}}_{1,1}+2{\mathscr{A}}\big{)}^{0}.i$
$\\{{\tilde{T}}^{{}^{3}},{\tilde{T}}^{{}^{3}}\\}={\tilde{T}}^{{}^{1}}$
$\big{(}{\mathscr{C}}_{1}^{1}+{\mathscr{A}}\big{)}^{\epsilon}$
$[{\tilde{T}}^{{}^{1}},{\tilde{T}}^{{}^{2}}]=-\frac{\epsilon}{2}{\tilde{T}}^{{}^{1}},\;\;\;[{\tilde{T}}^{{}^{2}},{\tilde{T}}^{{}^{4}}]=\frac{\epsilon}{2}{\tilde{T}}^{{}^{4}},~{}~{}~{}\\{{\tilde{T}}^{{}^{3}},{\tilde{T}}^{{}^{4}}\\}=\epsilon{\tilde{T}}^{{}^{1}}$
$\big{(}{\mathscr{C}}_{-1}^{2}+{\mathscr{A}}\big{)}^{\epsilon}$
$[{\tilde{T}}^{{}^{2}},{\tilde{T}}^{{}^{3}}]={\epsilon}{\tilde{T}}^{{}^{3}},\;\;\;[{\tilde{T}}^{{}^{2}},{\tilde{T}}^{{}^{4}}]=-{\epsilon}{\tilde{T}}^{{}^{4}},~{}~{}~{}\\{{\tilde{T}}^{{}^{3}},{\tilde{T}}^{{}^{4}}\\}=-2\epsilon{\tilde{T}}^{{}^{1}}$
$\big{(}{\mathscr{C}}^{3}+{\mathscr{A}}\big{)}^{\epsilon}$
$[{\tilde{T}}^{{}^{1}},{\tilde{T}}^{{}^{3}}]=-\frac{\epsilon}{4}{\tilde{T}}^{{}^{4}},\;\;\;\\{{\tilde{T}}^{{}^{3}},{\tilde{T}}^{{}^{3}}\\}=\epsilon{\tilde{T}}^{{}^{2}}$
$\big{(}{\mathscr{C}}^{3}+{\mathscr{A}}\big{)}_{k}^{\epsilon}$
$[{\tilde{T}}^{{}^{2}},{\tilde{T}}^{{}^{3}}]={\epsilon}{\tilde{T}}^{{}^{4}},\;\;\;\\{{\tilde{T}}^{{}^{3}},{\tilde{T}}^{{}^{3}}\\}=k{\tilde{T}}^{{}^{1}}$
$k>0$
${{}^{\ast}}{\epsilon}=\pm 1;$
grade$({\tilde{T}}^{{}^{1}})=$grade$({\tilde{T}}^{{}^{2}})=0$,
grade$({\tilde{T}}^{{}^{3}})=$grade$({\tilde{T}}^{{}^{4}})=1$.

In the following, we consider how many of $(4|4)$-dimensional Lie
superalgebras $\mathscr{D}$ of DSD D where related to the
$({\mathscr{C}}^{3}+{\mathscr{A}})$ Lie superbialgebras of Tables 1 are
isomorphic. To this end, we consider two DSDs isomorphic if they have
isomorphic algebraic structure and there is an isomorphism matrix transforming
one ad-invariant bilinear form to the other. Let $\mathscr{D}$ and
$\mathscr{D}^{\prime}$ be two DSDs with the bases
$X_{{}_{A}}=(T_{{}_{a}},{\tilde{T}}^{{}^{b}})$ and
$X^{\prime}_{{}_{A}}=({T^{\prime}}_{{}_{a}},{\tilde{T^{\prime}}}^{{}^{b}})$,
respectively, so that the bases satisfy equations (2.1) and (2.1) with the
structure constants ${F^{{}^{C}}}_{{}_{AB}}$ and ${F^{\prime^{C}}}_{{}_{AB}}$
of the doubles $\mathscr{D}$ and ${\mathscr{D}^{\prime}}$. By definition, the
DSDs $\mathscr{D}$ and ${\mathscr{D}^{\prime}}$ are isomorphic to each other
iff if there is a linear bijection that transforms grading, bilinear form and
algebraic structure of $\mathscr{D}$ into those of ${\mathscr{D}^{\prime}}$.
This means that in the standard basis there is a block diagonal isomorphism
matrix $\mathbb{C}=$diag$(\mathbb{C}_{{}_{B}},\mathbb{C}_{{}_{F}})$ between
$\mathscr{D}$ and ${\mathscr{D}^{\prime}}$
($\mathscr{D}\overset{\mathbb{C}}{\longrightarrow}\mathscr{D}^{\prime}$) such
that ${\mathscr{D}}_{{}_{B}}\longrightarrow{\mathscr{D}^{\prime}}_{{}_{B}}$
and ${\mathscr{D}}_{{}_{F}}\longrightarrow{\mathscr{D}^{\prime}}_{{}_{F}}$.
Accordingly, the linear map given by

$\displaystyle
X^{\prime}_{{}_{A}}=(-1)^{{}^{B}}~{}{\mathbb{C}_{{}_{A}}}^{{}^{B}}~{}X_{{}_{B}},$
(2.5)

transforms the Lie multiplication of $\mathscr{D}$ into that of
${\mathscr{D}^{\prime}}$ and preserves the canonical form of the bilinear form
$<.,.>$. This is equivalent to777“st” denotes supertransposition.

$\displaystyle(-1)^{{}^{AB+ED}}~{}\mathbb{C}~{}{\cal
Y}^{{}^{E}}~{}\mathbb{C}^{{}^{st}}$ $\displaystyle=$ $\displaystyle{\cal
Y^{\prime}}^{{}^{C}}~{}{{\mathbb{C}}_{{}_{C}}}^{{}^{E}},$ (2.6)
$\displaystyle(-1)^{{}^{A}}~{}\mathbb{C}~{}\mathbb{B}~{}\mathbb{C}^{{}^{st}}$
$\displaystyle=$ $\displaystyle\mathbb{B},$ (2.7)

where $({\cal Y}^{{}^{E}})_{{}_{AB}}=-{F^{{}^{E}}}_{{}_{AB}}$ and $({\cal
Y^{\prime}}^{{}^{E}})_{{}_{AB}}=-{F^{\prime^{E}}}_{{}_{AB}}$ are the matrix
representations of the Lie superalgebras $\mathscr{D}$ and
${\mathscr{D}^{\prime}}$, respectively. In equation (2.6), the indices $A$ and
$B$ correspond to the row and column of matrix ${\cal Y}^{{}^{E}}$,
respectively, and $D$ denotes the column of matrix $\mathbb{C}^{{}^{st}}$. In
equation (2.7), the index $A$ denotes the row of matrix $\mathbb{B}$ so that
$\mathbb{B}$ is the block matrix of the bilinear form $<.,.>$. In the standard
basis given by (2.1) it is then read

$\displaystyle\mathbb{B}_{{}_{AB}}=<X_{{}_{A}},X_{{}_{B}}>=\left(\begin{tabular}[]{cc|cc}0&${\bf
1}_{{}_{m}}$&0&0\\\ ${\bf 1}_{{}_{m}}$&0&0&0\\\ \hline\cr 0&0&0&-${\bf
1}_{{}_{n}}$\\\ 0&0&${\bf 1}_{{}_{n}}$&0\\\ \end{tabular}\right),$ (2.12)

where ${\bf 1}_{{}_{m(n)}}$ is the identity matrix of dimension $m(n)$. It is
obvious that the superdimension of these DSDs is $(2m|2n)$.

As was mentioned earlier, all Manin supertriples with the first sub-
superalgebra $({\mathscr{C}}^{3}+{\mathscr{A}})$ have listed in Table 1. We
are going to classify the corresponding DSDs by looking for the classes of
Manin supertriples that are isomorphic as DSDs. In this way, we apply the
method mentioned above to determine non-isomorphic DSDs as well as the
transformation between Manin supertriples that are isomorphic. The DSD
isomorphisms among the corresponding Manin supertriples are given in Appendix
A. The final distinction between non-isomorphic DSDs and their decomposition
into Manin supertriples provides the following theorem.
Theorem 1: Any DSD generated by Manin supertriples with the first sub-
superalgebra ${\cal G}\equiv({\mathscr{C}}^{3}+{\mathscr{A}})$ belongs just to
one of the following 24 classes and allows decomposition into non-isomorphic
Manin supertriples listed in the class and their duals

$DSD1:$ | $\big{(}{\cal G},{\cal{I}}_{{}_{(2|2)}}\big{)}\cong\big{(}{\cal G},{{\mathscr{C}}}^{3}\oplus{{\mathscr{A}}_{{}_{1,1}}}.i\big{)}\cong\big{(}{\cal G},{({{\mathscr{C}}}^{3}+{\mathscr{A}})_{{}_{k=4}}}^{\epsilon=-1}\big{)}$
---|---
$DSD2:$ | $\big{(}{\cal G},{{\mathscr{C}}_{{}_{p=1}}}^{1,\epsilon=1}\oplus{{\mathscr{A}}}\big{)}$
$DSD3:$ | $\big{(}{\cal G},{{\mathscr{C}}_{{}_{p=1}}}^{1,\epsilon=-1}\oplus{{\mathscr{A}}}\big{)}$
$DSD4:$ | $\big{(}{\cal G},{{\mathscr{C}}_{{}_{p=-1}}}^{1,\epsilon=1}\oplus{{\mathscr{A}}}\big{)}\cong\big{(}{\cal G},{{\mathscr{C}}_{{}_{p=-1}}}^{1,\epsilon=-1}\oplus{{\mathscr{A}}}\big{)}$
$DSD5:$ | $\big{(}{\cal G},{{\mathscr{C}}_{{}_{p=1}}}^{2,\epsilon=1}\oplus{{\mathscr{A}}_{{}_{1,1}}}\big{)}\cong\big{(}{\cal G},{{\mathscr{C}}_{{}_{p=1}}}^{2,\epsilon=-1}\oplus{{\mathscr{A}}_{{}_{1,1}}}\big{)}\cong\big{(}{\cal G},{{\mathscr{C}}}^{4,\epsilon=1}\oplus{{\mathscr{A}}_{{}_{1,1}}}\big{)}\cong\big{(}{\cal G},{{\mathscr{C}}}^{4,\epsilon=-1}\oplus{{\mathscr{A}}_{{}_{1,1}}}\big{)}$
$DSD6:$ | $\big{(}{\cal G},({{\mathscr{D}}_{{}_{\frac{1}{2},\frac{1}{2}}}}^{7,\epsilon=1})^{2}\big{)}$
$DSD7:$ | $\big{(}{\cal G},({{\mathscr{D}}_{{}_{\frac{1}{2},\frac{1}{2}}}}^{7,\epsilon=-1})^{2}\big{)}$
${DSD8}^{{}^{p}}$ : | $\big{(}{\cal G},({{\mathscr{D}}_{{}_{1-p,p}}}^{7,\epsilon=1}).i\big{)},~{}~{}~{}~{}~{}~{}~{}p\neq 0,~{}p<\frac{1}{2}$
${DSD9}^{{}^{p}}$ : | $\big{(}{\cal G},({{\mathscr{D}}_{{}_{1-p,p}}}^{7,\epsilon=-1}).i\big{)},~{}~{}~{}~{}~{}p\neq 0,~{}p<\frac{1}{2}$
${DSD10}^{{}^{p}}$ : | $\big{(}{\cal G},({{\mathscr{D}}_{{}_{1-p,p}}}^{7,\epsilon=1}).ii\big{)},~{}~{}~{}~{}~{}~{}p\neq 0,~{}p<\frac{1}{2}$
${DSD11}^{{}^{p}}$ : | $\big{(}{\cal G},({{\mathscr{D}}_{{}_{1-p,p}}}^{7,\epsilon=-1}).ii\big{)},~{}~{}~{}~{}p\neq 0,~{}p<\frac{1}{2}$
${DSD12}$: | $\big{(}{\cal G},({{\mathscr{D}}_{{}_{0}}}^{10,\epsilon=1})^{1}\big{)}$
${DSD13}$: | $\big{(}{\cal G},({{\mathscr{D}}_{{}_{0}}}^{10,\epsilon=-1})^{1}\big{)}$
${DSD14}$: | $\big{(}{\cal G},({{\mathscr{D}}_{{}_{0}}}^{10,\epsilon=1})^{2}\big{)}$

${DSD15}$: | $\big{(}{\cal G},({{\mathscr{D}}_{{}_{0}}}^{10,\epsilon=-1})^{2}\big{)}$
---|---
${DSD16}$: | $\big{(}{\cal G},({{{\mathscr{C}}_{{}_{1}}}^{1}+{\mathscr{A}})}^{\epsilon=1}\big{)}$
${DSD17}$: | $\big{(}{\cal G},({{{\mathscr{C}}_{{}_{1}}}^{1}+{\mathscr{A}})}^{\epsilon=-1}\big{)}$
${DSD18}$: | $\big{(}{\cal G},({{{\mathscr{C}}_{{}_{-1}}}^{2}+{\mathscr{A}})}^{\epsilon=1}\big{)}$
${DSD19}$: | $\big{(}{\cal G},({{{\mathscr{C}}_{{}_{-1}}}^{2}+{\mathscr{A}})}^{\epsilon=-1}\big{)}$
${DSD20}$: | $\big{(}{\cal G},({{{\mathscr{C}}}^{3}+{\mathscr{A}})}^{\epsilon=1}\big{)}$
${DSD21}$: | $\big{(}{\cal G},({{{\mathscr{C}}}^{3}+{\mathscr{A}})}^{\epsilon=-1}\big{)}$
${DSD22}^{{}^{p}}:$ | $\big{(}{\cal G},{{\mathscr{D}}_{{}_{p,p-1}}}^{1,\epsilon=1}\big{)}\cong\big{(}{\cal G},{{\mathscr{D}}_{{}_{1-p,-p}}}^{1,\epsilon=1}\big{)}\cong\big{(}{\cal G},{{\mathscr{D}}_{{}_{p,p-1}}}^{1,\epsilon=-1}\big{)},~{}~{}~{}p\neq 0,1$
${DSD23}^{{}^{k}}:$ | $\big{(}{\cal G},{(2{\mathscr{A}}_{{}_{1,1}}+2{\mathscr{A}})^{{}^{0}}}.i\big{)}\cong\big{(}{\cal G},{({{\mathscr{C}}}^{3}+{\mathscr{A}})_{{}_{k-4}}}^{\epsilon=1}\big{)}\cong\big{(}{\cal G},{({{\mathscr{C}}}^{3}+{\mathscr{A}})_{{}_{k}}}^{\epsilon=-1}\big{)},~{}~{}~{}k>4$
${DSD24}^{{}^{k}}:$ |  $\big{(}{\cal G},{({{\mathscr{C}}}^{3}+{\mathscr{A}})_{{}_{k}}}^{\epsilon=-1}\big{)}\cong\big{(}{\cal G},{{({\mathscr{C}}}^{3}+{\mathscr{A}})_{{}_{4-k}}}^{\epsilon=-1}\big{)},~{}~{}~{}0<k<4$.

One can see that for many DSDs there are several decompositions into Manin
supertriples. For example, the structure of DSD1 contains more than one pair
of non-isomorphic dual Manin supertriples. In Ref. [28] it has shown that this
structure gives us the possibility to apply the super PL T-plurality to
$\sigma$-models whose target spaces are Lie supergroups. Then, it has been
obtained a conformal duality chain of $2+1$-dimensional cosmological string
backgrounds coupling with two fermionic fields.

##  3  A short review of super PL T-duality

In this section we briefly review the construction of the super PL
T-dualizable $\sigma$-models on supermanifolds [20]. Let $\mathscr{M}$ be a
$(d_{{}_{B}}|d_{{}_{F}})$-dimensional target supermanifold 888The
superdimension of the supermanifold $\mathscr{M}$ is written as ($\\#$ bosons
$|$ $\\#$ fermions)=$(d_{{}_{B}}|d_{{}_{F}})$; because of the invertibility of
the metric $G_{{}_{MN}}$, $d_{{}_{F}}$ must be even [38]. and consider two-
dimensional $\sigma$-model on $\mathscr{M}$ given by action

$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{{}_{\Sigma}}\\!d\sigma^{+}d\sigma^{-}~{}(-1)^{{}^{M}}~{}\partial_{{}_{+}}\phi^{{}^{M}}~{}{\cal
E}_{{}_{MN}}~{}\partial_{{}_{-}}\phi^{{}^{N}},$ (3.1)

where $\partial_{{}_{\pm}}$ are the derivatives with respect to the standard
lightcone variables $\sigma^{\pm}=\tau\pm\sigma$ on the worldsheet $\Sigma$.
The functions $\phi^{{}^{M}}$ include the bosonic coordinates $x^{\mu}$ and
the fermionic ones $\theta^{i}$, and the labels $M,N$ run over
$\mu=0,\cdots,d_{{}_{B}}-1$ and $i=1,\cdots,d_{{}_{F}}$. The tensor field
${\cal E}_{{}_{MN}}$ can be understood as a sum of the supersymmetric metric
${G}_{{}_{MN}}$ and torsion potential ${B}_{{}_{MN}}$ defining the geometric
properties of the supermanifold $\mathscr{M}$.

If the Noether’s current one-forms corresponding to the right action of the
Lie supergroup $G$ on the target supermanifold $\mathscr{M}$ are not closed
and satisfy the super Maurer-Cartan equation [38] on the extremal surfaces, we
say that the $\sigma$-model (3.1) has the super PL symmetry with respect to
the Lie supergroup $\tilde{G}$ (the dual Lie supergroup to $G$ with the same
superdimension $G$). It is a condition that is given by the following relation
[20]

$\displaystyle{\cal L}_{{}_{V_{{}_{a}}}}{\cal
E}_{{}_{MN}}\;=\;(-1)^{{}^{a+Ma+Qc+P}}~{}{{\tilde{f}}^{bc}}_{~{}~{}a}~{}{\cal
E}_{{}_{MQ}}~{}{V_{{}_{c}}}^{{}^{~{}Q}}~{}{V_{{}_{b}}}^{{}^{~{}P}}~{}{\cal
E}_{{}_{PN}},$ (3.2)

where ${\cal L}_{{}_{V_{{}_{a}}}}$ stands for the Lie derivative corresponding
to the left invariant supervector fields
${V_{{}_{a}}}={V_{{}_{a}}}^{{}^{M}}{{{\overrightarrow{\partial}}/{\partial{\phi^{{}^{M}}}}}}$
(defined with left derivative) satisfying
$[{V_{{}_{a}}},{V_{{}_{b}}}]={f^{c}}_{ab}~{}{V_{{}_{c}}}$, and
${{\tilde{f}}^{bc}}_{~{}~{}a}$ are the structure constants of the dual Lie
superalgebra $\widetilde{\mathscr{G}}$ of $\tilde{G}$. By direct calculation,
one can obtain

$\displaystyle{\cal L}_{{}_{V_{{}_{a}}}}{\cal E}_{{}_{MN}}$ $\displaystyle=$
$\displaystyle(-1)^{{}^{Ma+M+P}}\;{{\frac{\overrightarrow{\partial}}{\partial{\phi^{{}^{M}}}}}}\;{V_{{}_{a}}}^{{}^{P}}\;{\cal
E}_{{}_{PN}}+{V_{{}_{a}}}^{{}^{P}}\;{{\frac{\overrightarrow{\partial}}{\partial{\phi^{{}^{P}}}}}}\;{\cal
E}_{{}_{MN}}$ (3.3)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+(-1)^{{}^{MN+MP+Na+N+P}}\;{{\frac{\overrightarrow{\partial}}{\partial{\phi^{{}^{N}}}}}}\;{V_{{}_{a}}}^{{}^{P}}\;{\cal
E}_{{}_{MP}}.$

The integrability condition on the Lie derivative, $[{\cal
L}_{{}_{V_{{}_{a}}}},{\cal L}_{{}_{V_{{}_{b}}}}]={\cal
L}_{{}_{[V_{{}_{a}},V_{{}_{b}}]}}$, then implies the mixed super Jacobi
identities (2.3) showing that this construction leads naturally to the DSD.

Both the original and dual geometries of the super PL T-dualizable
$\sigma$-models are derived from the so-called DSD which is a Lie supergroup.
In the following, we shall consider T-dual $\sigma$-model on a Lie supergroup
$G$. To this end, suppose now that $G$ acts transitively and freely on the
supermanifold $\mathscr{M}$; then the target can be identified with the
supergroup $G$. In this case, the $\sigma$-model having target space in the
Lie supergroup $G$ is given by the following action [20]

$\displaystyle
S\;=\;\frac{1}{2}\int_{{}_{\Sigma}}\\!d\sigma^{+}d\sigma^{-}\;(-1)^{{}^{a}}{{R_{{}_{+}}}}^{a}\;{E_{{}_{ab}}}(g)\;{{R_{{}_{-}}}}^{b},$
(3.4)

where ${{R_{{}_{\pm}}}}^{a}$ are the components of the left invariant super
one-forms with left derivative which are defined by means of an element
$g:\Sigma\rightarrow G$ in the following formula

$\displaystyle\partial_{{}_{\pm}}g~{}g^{-1}=(-1)^{a}~{}{R_{{}_{\pm}}^{a}}~{}T_{{{}_{a}}}=(-1)^{a}~{}\partial_{{}_{\pm}}\phi^{{}^{M}}\;{{{}_{{}_{M}}}R}^{{}^{a}}~{}T_{{{}_{a}}},$
(3.5)

in which $\phi^{{}^{M}}$ are coordinates of the elements of $G$. The
${E_{{}_{ab}}}(g)$ is a certain bilinear form on the ${\mathscr{G}}$ and is
defined by 999Here one must use the superinverse formula introduced in [38].

$\displaystyle{E(g)}\;=\;\big{(}{E_{{}_{0}}}^{-1}+\Pi(g)\big{)}^{-1},~{}~{}~{}~{}~{}~{}~{}~{}~{}\Pi(g)=b(g)a^{-1}(g),$
(3.6)

where ${E_{{}_{0}}}$ is a constant matrix, and $\Pi(g)$ defines the super
Poisson structure on the $G$. The $a(g)$ and $b(g)$ are sub-matrices of the
adjoint representation of the supergroup $G$ on $\mathscr{D}$ in the basis
$(T_{{{}_{a}}},{\tilde{T}}^{{{}^{a}}})$

$\displaystyle g^{-1}T_{{{}_{a}}}~{}g$ $\displaystyle=$
$\displaystyle(-1)^{c}~{}a_{{}_{a}}^{{}^{~{}c}}(g)~{}T_{{{}_{c}}},$
$\displaystyle g^{-1}{\tilde{T}}^{{{}^{a}}}g$ $\displaystyle=$
$\displaystyle(-1)^{c}~{}b^{{}^{ac}}(g)~{}T_{{{}_{c}}}+{(a^{{}^{-st}})^{{~{}a}}}_{c}(g)~{}{\tilde{T}}^{{{}^{c}}}.$
(3.7)

Notice that both bases $T_{{{}_{a}}}\in\mathscr{G}$ and
${\tilde{T}}^{{{}^{a}}}\in{\widetilde{\mathscr{G}}}$ satisfy relations
(2.1)-(2.3). We expect that there exists an equivalent T-dual $\sigma$-model
in which the roles of ${\mathscr{G}}$ and $\widetilde{\mathscr{G}}$ are
exchanged. Accordingly, one can repeat all steps of the previous construction
to end up with the following dual $\sigma$-model

$\displaystyle\tilde{S}\;=\;\frac{1}{2}\int_{{}_{\Sigma}}\\!d\sigma^{+}d\sigma^{-}\;(-1)^{{}^{b}}{{\tilde{R}_{{}_{+}a}}}\;{{\tilde{E}}^{{}^{ab}}}(\tilde{g})\;{{\tilde{R}_{{}_{-}b}}},$
(3.8)

where
${{\tilde{R}_{{}_{\pm}a}}}=(\partial_{{}_{\pm}}\tilde{g}~{}{\tilde{g}}^{{}^{-1}})_{{}_{a}}=(-1)^{{}^{M}}\partial_{{}_{\pm}}{\tilde{\phi}}^{{}^{M}}{{\tilde{R}_{Ma}}}$
are the components of the right invariant super one-forms with
${\tilde{\phi}}^{{}^{M}}$ being local coordinates of $\tilde{g}\in\tilde{G}$,
and the matrix ${{\tilde{E}}^{{}^{ab}}}(\tilde{g})$ is obtained by formula

$\displaystyle{{\tilde{E}}}(\tilde{g})\;=\;\big{(}{E_{{}_{0}}}+{\tilde{\Pi}}(\tilde{g})\big{)}^{-1},$
(3.9)

in which ${\tilde{\Pi}}(\tilde{g})$ is defined as in (3.6) by replacing
untilded quantities with tilded ones. Both equations of motion of the
lagrangian systems of the models (3.4) and (3.8) can be reduced from equation
of motion on the whole DSD, not depending on the choice of Manin supertriple:

$\displaystyle<\partial_{\pm}l~{}l^{-1},\varepsilon^{\mp}>=0,$ (3.10)

where $l\in D$ 101010One writes $l=g{\tilde{h}};g\in
G,{\tilde{h}}\in{\tilde{G}}$ (such decomposition of supergroup elements exists
at least at the vicinity of the unit element); moreover, one may write
$l={\tilde{g}}h$ where $h\in G,{\tilde{g}}\in{\tilde{G}}.$, and subspaces
$\varepsilon^{+}=span\\{T_{{}_{a}}+{E_{{}_{0}}}_{{}_{ab}}{\tilde{T}}^{{}^{b}}\\}$
and
$\varepsilon^{-}=span\\{T_{{}_{a}}-{(E_{{}_{0}}^{{}^{st}})}_{{}_{ab}}{\tilde{T}}^{{}^{b}}\\}$
are orthogonal with respect to the $<.,.>$ and span the whole Lie superalgebra
${\mathscr{D}}$.

##  4  Conditions that a $\sigma$-model may be given as a WZW model

In what follows we shall obtain the conditions under which the $\sigma$-model
(3.1) may be equivalent to a WZW model. Let us first consider the standard
form of WZW model based on a Lie supergroup $G$

$\displaystyle S_{{}_{WZW}}(g)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\Sigma}d\sigma^{+}d\sigma^{-}<g^{-1}\partial_{+}g,\;g^{-1}\partial_{-}g>$
(4.1)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{1}{12}\int_{B_{3}}<g^{-1}dg\;\hat{,}\;[g^{-1}dg\;\hat{,}\;g^{-1}dg]>,$

where $g^{-1}~{}\partial_{\pm}g$ are the components of the left invariant
super one-forms with left derivative which may be expressed as111111The
relationship between the super one-forms $L^{a}$ and supervector fields
$V_{{}_{b}}$ is given in the usual way:
$(-1)^{b}L^{a}V_{{}_{b}}={\delta}^{a}_{~{}b}$.

$\displaystyle
g^{-1}~{}\partial_{{}_{\pm}}g=(-1)^{a}~{}{L_{{}_{\pm}}^{a}}~{}T_{{{}_{a}}}=(-1)^{a}~{}\partial_{{}_{\pm}}\phi^{{}^{M}}\;{{{}_{{}_{M}}}L}^{{}^{a}}~{}T_{{{}_{a}}}.$
(4.2)

In the second term of (4.1) the integration is over a $3$-dimensional
(super)manifold $B_{3}$ which has $\Sigma$ as a boundary. To define a WZW
model, one needs a bilinear form
$\Omega_{{}_{ab}}=<T_{{{}_{a}}},T_{{{}_{b}}}>$ in the generators $T_{{}_{a}}$,
which is non-degenerate supersymmetric ad-invariant metric on $\mathscr{G}$
and satisfies in the following relation [43] (see also [26, 27])

$\displaystyle{f^{c}}_{ab}\;\Omega_{{}_{cd}}+(-1)^{bd}{f^{c}}_{ad}\;\Omega_{{}_{cb}}\;=\;0.$
(4.3)

Accordingly, the terms that are being integrated over in (4.1) can be written
as

$\displaystyle<g^{-1}\partial_{+}g,\;g^{-1}\partial_{-}g>$ $\displaystyle=$
$\displaystyle(-1)^{a}~{}\partial_{{}_{+}}\phi^{{}^{M}}{{{}_{{}_{M}}}L}^{{}^{a}}~{}\Omega_{{}_{ab}}~{}{({L^{st})}^{{}^{a}}}_{{}_{N}}\partial_{{}_{-}}\phi^{{}^{N}},$
(4.4) $\displaystyle<g^{-1}dg\;\hat{,}\;[g^{-1}dg\;\hat{,}\;g^{-1}dg]>$
$\displaystyle=$ $\displaystyle
d^{3}\sigma~{}\varepsilon^{{}^{\gamma\alpha\beta}}(-1)^{{}^{a+bP}}\partial_{{}_{\gamma}}\phi^{{}^{M}}{{{}_{{}_{M}}}L}^{{}^{a}}\Omega_{{}_{ad}}$
(4.5)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times{f^{d}}_{bc}~{}{({L^{st})}^{{}^{c}}}_{{}_{P}}~{}{({L^{st})}^{{}^{b}}}_{{}_{N}}~{}\partial_{{}_{\alpha}}\phi^{{}^{N}}\partial_{{}_{\beta}}\phi^{{}^{P}}.$

Hence the domain of $L^{a}$ is implicitly generalized to $B_{3}$ with
$\alpha=\tau,\sigma$ and $\xi$ in the second term of (4.1), where the extra
direction is labeled by $\xi$. By regarding the action (4.1) as a
$\sigma$-model action of the form

$\displaystyle\bar{S}=\frac{1}{2}\int_{{}_{\Sigma}}\\!d\sigma^{+}d\sigma^{-}~{}\partial_{{}_{+}}\bar{\phi}^{{}^{M}}~{}{{{}_{{}_{M}}}\bar{G}}_{{}_{N}}~{}\partial_{{}_{-}}\bar{\phi}^{{}^{N}}+\frac{1}{12}\int_{{}_{B}}\\!d^{3}\sigma~{}\varepsilon^{{}^{\gamma\beta\alpha}}\partial_{{}_{\gamma}}\bar{\phi}^{{}^{M}}{{{}_{{}_{M}}}\bar{H}}_{{}_{PN}}\partial_{{}_{\alpha}}\bar{\phi}^{{}^{N}}\partial_{{}_{\beta}}\bar{\phi}^{{}^{P}},$
(4.6)

in which

$\displaystyle\bar{H}_{{}_{MNP}}=(-1)^{{}^{M}}\;\frac{\overrightarrow{\partial}}{\partial\bar{\phi}^{{}^{M}}}\bar{B}_{{}_{NP}}+(-1)^{{}^{N+M(N+P)}}\;\frac{\overrightarrow{\partial}}{\partial\bar{\phi}^{{}^{N}}}\bar{B}_{{}_{PM}}+(-1)^{{}^{P(1+M+N)}}\;\frac{\overrightarrow{\partial}}{\partial\bar{\phi}^{{}^{P}}}\bar{B}_{{}_{MN}},$
(4.7)

is the torsion of the field $\bar{B}_{{}_{MN}}$, and then by using equations
(4.4) and (4.5) and finally from comparing (4.1) with (4.6) we can read off
the background of $\sigma$-model in terms of the elements of WZW model as
follows:

$\displaystyle{\bar{G}}_{{}_{MN}}$ $\displaystyle=$
$\displaystyle(-1)^{a}~{}{L_{{}_{M}}}^{{}^{a}}~{}\Omega_{{}_{ab}}~{}{({L^{st})}^{{}^{a}}}_{{}_{N}},$
(4.8) $\displaystyle\bar{H}_{{}_{MPN}}$ $\displaystyle=$
$\displaystyle-(-1)^{{}^{a+bP}}{L_{{}_{M}}}^{{}^{a}}~{}\Omega_{{}_{ad}}{f^{d}}_{bc}~{}{({L^{st})}^{{}^{c}}}_{{}_{P}}~{}{({L^{st})}^{{}^{b}}}_{{}_{N}}.$
(4.9)

In the following we find the equation establishing a relationship between the
background of $\sigma$-model (including ${\bar{G}}_{{}_{MN}}$ and
$\bar{B}_{{}_{MN}}$) and the left invariant super one-forms of Lie supergroup
$G$ that the WZW model is constructed on. For this purpose one may first use
the super Maurer-Cartan equation [38]

$\displaystyle
d{L}^{{}^{a}}=-\frac{1}{2}(-1)^{{}^{bc}}{f^{a}}_{bc}~{}{L}^{{}^{b}}\wedge{L}^{{}^{c}},$
(4.10)

to obtain

$\displaystyle{({L^{st})}^{{}^{a}}}_{{}_{M}}\frac{\overleftarrow{\partial}}{\partial\phi^{{}^{N}}}-(-1)^{{}^{MN}}{({L^{st})}^{{}^{a}}}_{{}_{N}}\frac{\overleftarrow{\partial}}{\partial\phi^{{}^{M}}}=(-1)^{{}^{N(b+M)}}{f^{a}}_{bc}~{}{({L^{st})}^{{}^{c}}}_{{}_{N}}{({L^{st})}^{{}^{b}}}_{{}_{M}}.$
(4.11)

Then, by inserting (4.11) into (4.9) and employing (4.8) we obtain the
following master equation (ME)

$\displaystyle{({L^{st})}^{{}^{a}}}_{{}_{Q}}~{}\bar{G}^{{}^{QM}}{{}_{{}_{M}}\bar{H}}_{{}_{NP}}={({L^{st})}^{{}^{a}}}_{{}_{N}}\frac{\overleftarrow{\partial}}{\partial\phi^{{}^{P}}}-(-1)^{{}^{NP}}{({L^{st})}^{{}^{a}}}_{{}_{P}}\frac{\overleftarrow{\partial}}{\partial\phi^{{}^{N}}}.$
(4.12)

This formula will be useful in the next section. We will solve the ME (4.12)
for a given $\sigma$-model ($T$-dual models on the DSDs generated by the
$(C^{3}+A)$) including the metric ${\bar{G}}_{{}_{MN}}$ and torsion potential
${\bar{B}}_{{}_{MN}}$. If there is an answer, we then obtain the left
invariant super one-forms corresponding to a Lie supergroup ${G}$ whose
structure constants are still unknown. One can then use the super Maurer-
Cartan equation (4.11) together with (4.3) to determine the supergroup
structure and corresponding bilinear form which is necessary for defining a
WZW model.

##  5  A hierarchy of WZW models related to super PL T-duality

As mentioned in the introduction section, in Ref. [27] we showed that the
super PL duality relates the $(C^{3}+A)$ WZW model to a $\sigma$-model defined
on the $(C^{3}+A)$ Lie supergroup when the dual Lie supergroup is isomorphic
to the ${C^{3}\otimes{A}_{{}_{1,1}}}$. Then we showed that the dual model is
equivalent to a WZW model based on the $(C^{3}+A).i$ Lie supergroup whose Lie
superalgebra is isomorphic to the $({\mathscr{C}}^{3}+{\mathscr{A}})$. Finally
we announced that this process can be continued. This means that one can
obtain a hierarchy of WZW models related to the super PL T-duality in such a
way that it is different from super PL T-plurality [28]; because as we will
show below, the DSDs involved in the process are non-isomorphic. To this end,
we use the formulation presented in preceding section. Before proceeding to
obtain this hierarchy, let us give a concise review of the $(C^{3}+A)$ WZW
model and its super PL symmetry.

###  5.1 Super PL symmetry of the $(C^{3}+A)$ WZW model

As mentioned in section 4, a key ingredient in defining the WZW model is the
bilinear form $\Omega_{{}_{ab}}$. A bilinear form on the
$({\mathscr{C}}^{3}+{\mathscr{A}})$ Lie superalgebra defined by
(anti)commutation relations (2.4) can be obtained by (4.3), giving

$\displaystyle\Omega_{{}_{ab}}=\left(\begin{tabular}[]{cccc}$0$&$1$&0&$0$\\\
$1$&0&0&$0$\\\ $0$&0&0&$1$\\\ $0$&$0$&$-1$&0\\\ \end{tabular}\right).$ (5.5)

Let us introduce a supergroup element represented by

$\displaystyle g=e^{\chi T_{4}}~{}e^{yT_{1}}~{}e^{xT_{2}}~{}e^{\psi T_{3}},$
(5.6)

where $x(\tau,\sigma)$ and $y(\tau,\sigma)$ denote bosonic fields while
$\psi(\tau,\sigma)$ and $\chi(\tau,\sigma)$ stand for fermionic fields. To
calculate the left invariant super one-forms we use (2.4) and (4.2) together
with (5.6). Then we find that

$\displaystyle{L_{{}_{\pm}}^{1}}$ $\displaystyle=$
$\displaystyle\partial_{\pm}y,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{L_{{}_{\pm}}^{2}}=\partial_{\pm}x-\partial_{\pm}\chi\frac{\chi}{2},$
$\displaystyle{L_{{}_{\pm}}^{3}}$ $\displaystyle=$
$\displaystyle-\partial_{\pm}\psi+\partial_{\pm}\chi~{}y,~{}~{}~{}~{}~{}{L_{{}_{\pm}}^{4}}=-\partial_{\pm}\chi.$
(5.7)

Hence, by using (4.1) and (4.4) together with (4.5), the WZW action on the
$(C^{3}+A)$ Lie supergroup is written as [27]

$\displaystyle S_{{}_{WZW}}(g)=\frac{1}{2}\int
d\sigma^{+}d\sigma^{-}\;\Big{[}\partial_{+}y\partial_{-}x+\partial_{+}x\partial_{-}y-\partial_{+}\psi\;\partial_{-}\chi+\partial_{+}\chi\;\partial_{-}\psi+\partial_{+}y~{}{\chi}~{}\partial_{-}\chi\Big{]}.$
(5.8)

In order to investigate the super PL symmetry of the WZW model (5.8) one has
to employ equation (3.2). First, we need to find the left invariant
supervector fields on the $(C^{3}+A)$. By applying (5.1) in relation
$(-1)^{b}{L}^{a}{{V_{b}}}=\delta^{a}_{~{}b}$ we obtain

$\displaystyle V_{1}=\frac{\overrightarrow{\partial}}{\partial
y},~{}~{}~{}~{}~{}~{}~{}V_{2}=\frac{\overrightarrow{\partial}}{\partial
x},~{}~{}~{}~{}~{}~{}~{}~{}V_{3}=\frac{\overrightarrow{\partial}}{\partial\psi},~{}~{}~{}~{}~{}~{}~{}V_{4}=\frac{\chi}{2}\frac{\overrightarrow{\partial}}{\partial
x}+{y}\frac{\overrightarrow{\partial}}{\partial\psi}+\frac{\overrightarrow{\partial}}{\partial\chi}.$
(5.9)

Then, by comparing actions (3.1) and (5.8), one obtains the background matrix
${\cal E}_{{}_{MN}}$ corresponding to the action (5.8). Now, by substituting
the resulting background matrix and also the supervector fields (5.9) on the
right hand side of (3.2) and then by direct calculation of Lie derivative of
${\cal E}_{{}_{MN}}$ with respect to $V_{a}$, one can obtain the structure
constants of the dual Lie superalgebra to the
$({\mathscr{C}}^{3}+{\mathscr{A}})$ in such a way that only non-zero
commutation relation of the dual pair is [27]

$\displaystyle[{\tilde{T}}^{2},{\tilde{T}}^{3}]=-\frac{1}{2}{\tilde{T}}^{4},$
(5.10)

in which grade$({\tilde{T}}^{1})$=grade$({\tilde{T}}^{2})=0$ and
grade$({\tilde{T}}^{3})$=grade$({\tilde{T}}^{4})=1$. According to the
classification of decomposable Lie superalgebras of the type $(2|2)$ [40], the
Lie superalgebra defined by (5.10) is isomorphic to the
${\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}$, which we denote by
${\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}.ii$ 121212Note that the
${\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}.ii$ is also isomorphic to the
${\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}.i$ of Table 1.. Note that both
sets of generators of (2.4) and (5.10) are maximally isotropic with respect to
the bilinear form defined by the brackets (2.1). Nevertheless, the
$(({\mathscr{C}}^{3}+{\mathscr{A}})~{},~{}{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}.ii)$
as a Lie superbialgebra satisfies mixed super Jacobi identities (2.3).

###  5.2 Super PL T-dual $\sigma$-models on the superdouble
$\big{(}(C^{3}+A)~{},~{}C^{3}\otimes{A}_{1,1}.ii\big{)}$

We shall construct a pair of super PL T-dual $\sigma$-models which is
associated with the DSD
$(({\mathscr{C}}^{3}+{\mathscr{A}})~{},~{}{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}.ii)$.
Using relations (2.1), (2.4) and (5.10) we conclude that the Lie superalgebra
of the superdouble
$(({\mathscr{C}}^{3}+{\mathscr{A}})~{},~{}{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}.ii)$
obeys the following set of non-trivial (anti)commutation relations131313It can
be easily shown that the Lie superalgebra
$(({\mathscr{C}}^{3}+{\mathscr{A}})~{},~{}{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}.ii)$
with the (anti)commutation relations (5.11) is isomorphic with each of the
DSDs belonging to family DSD1 of Theorem 1. [27]

$\displaystyle[T_{{}_{1}},T_{{}_{4}}]$ $\displaystyle=$ $\displaystyle
T_{{}_{3}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\\{T_{{}_{4}},T_{{}_{4}}\\}=T_{{}_{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}[{\tilde{T}}^{{}^{2}},{\tilde{T}}^{{}^{3}}]=-\frac{1}{2}{\tilde{T}}^{{}^{4}},$
$\displaystyle{[T_{{}_{1}},{\tilde{T}}^{{}^{3}}]}$ $\displaystyle=$
$\displaystyle-{\tilde{T}}^{{}^{4}},~{}~{}~{}~{}~{}~{}~{}~{}[T_{{}_{4}},{\tilde{T}}^{{}^{2}}]=-\frac{1}{2}T_{{}_{3}}-{\tilde{T}}^{{}^{4}},~{}~{}~{}~{}\\{T_{{}_{4}},{\tilde{T}}^{{}^{3}}\\}=-\frac{1}{2}T_{{}_{2}}-{\tilde{T}}^{{}^{1}}.$
(5.11)

Below, we show that the original $\sigma$-model on the $(C^{3}+A)$ is the same
as the WZW model obtained in (5.8). It can also be interesting to show that
the dual model is itself equivalent to a WZW model.

$\bullet$ Original model as WZW model based on the $(C^{3}+A)$.
In order to construct the original $\sigma$-model with the $(C^{3}+A)$ Lie
supergroup as the target space, we use the same parametrization as (5.6).
Using (3.5) we then find that

$\displaystyle{R_{{}_{\pm}}^{1}}$ $\displaystyle=$
$\displaystyle\partial_{\pm}y,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{R_{{}_{\pm}}^{2}}=\partial_{\pm}x+\partial_{\pm}\chi\frac{\chi}{2},$
$\displaystyle{R_{{}_{\pm}}^{3}}$ $\displaystyle=$
$\displaystyle\partial_{\pm}y~{}\chi-\partial_{\pm}\psi,~{}~{}~{}~{}~{}~{}~{}{R_{{}_{\pm}}^{4}}=-\partial_{\pm}\chi.$
(5.12)

In addition, using (3) and the second formula of (3.6) together with (5.6) and
(5.11), one can obtain the super Poisson structure, giving us

$\displaystyle{\Pi}^{ab}(g)=\left(\begin{tabular}[]{cccc}$0$&$0$&$0$&$0$\\\
$0$&$0$&-$\frac{\chi}{2}$&$0$\\\ $0$&$\frac{\chi}{2}$&$0$&$0$\\\
$0$&$0$&$0$&$0$\\\ \end{tabular}\right).$ (5.17)

It is more appropriate to choose the $\sigma$-model constant matrix as
follows:

$\displaystyle{E_{0}}_{{}_{ab}}$ $\displaystyle=$
$\displaystyle\left(\begin{tabular}[]{cccc}0&1&0&0\\\ 1&0&0&0\\\ 0&0&0&1\\\
0&0&-1&0\\\ \end{tabular}\right).$ (5.22)

With these relations at hand and employing (3.4) together with the first
formula of (3.6), one can get the original $\sigma$-model which will be
nothing but the WZW model (5.8).

$\bullet$ Dual model as WZW model based on the $(C^{3}+A).i$.
We parameterize the dual Lie supergroup $C^{3}\otimes{A}_{1,1}.ii$ with
bosonic coordinates $({\tilde{y}},{\tilde{x}})$ and fermionic ones
$({\tilde{\psi}},{\tilde{\chi}})$ so that its elements are defined as in (5.6)
by replacing untilded quantities with tilded ones. In the same way, to
construct out the dual $\sigma$-model one needs to calculate the right
invariant super one-forms
${{\tilde{R}_{{}_{\pm}a}}}=(\partial_{{}_{\pm}}\tilde{g}~{}{\tilde{g}}^{{}^{-1}})_{{}_{a}}$
on the $C^{3}\otimes{A}_{1,1}.ii$. By utilizing (5.10) we obtain

$\displaystyle{\tilde{R}}_{\pm_{1}}={\partial_{\pm}}{\tilde{y}},~{}~{}~{}~{}~{}~{}~{}{\tilde{R}}_{\pm_{2}}={\partial_{\pm}}{\tilde{x}},~{}~{}~{}~{}{\tilde{R}}_{\pm_{3}}={\partial_{\pm}}{\tilde{\psi}},~{}~{}~{}~{}~{}~{}~{}{\tilde{R}}_{\pm_{4}}=-{\partial_{\pm}}{\tilde{\psi}}\frac{\tilde{x}}{2}+{\partial_{\pm}}{\tilde{\chi}}.$
(5.23)

As mentioned in section 3, to obtain the dual super Poisson structure one must
use equation (3) by replacing untilded quantities with tilded ones. Thus, by
employing (5.11), we then obtain

$\displaystyle{\tilde{\Pi}}_{ab}(\tilde{g})=\left(\begin{tabular}[]{cccc}$0$&$0$&$0$&-${\tilde{\psi}}$\\\
$0$&$0$&$0$&$0$\\\ $0$&$0$&$0$&$0$\\\
${\tilde{\psi}}$&$0$&$0$&-${\tilde{x}}$\\\ \end{tabular}\right).$ (5.28)

Finally, inserting (5.22) and (5.28) into (3.9) and then using (5.23) and
(3.8), the dual action on the $C^{3}\otimes{A}_{1,1}.ii$ are worked out [27]

$\displaystyle{\tilde{S}}=\frac{1}{2}\int
d\sigma^{+}d\sigma^{-}\;\Big{[}\partial_{+}{\tilde{y}}\partial_{-}{\tilde{x}}+\partial_{+}{\tilde{x}}\partial_{-}{\tilde{y}}-\partial_{+}{\tilde{\psi}}\partial_{-}{\tilde{\chi}}+\partial_{+}{\tilde{\chi}}\partial_{-}{\tilde{\psi}}-3{\tilde{x}}\partial_{+}{\tilde{\psi}}\;\partial_{-}{\tilde{\psi}}\Big{]}.$
(5.29)

The next step in this procedure is to show that the dual $\sigma$-model
constructed out on the $C^{3}\otimes{A}_{1,1}.ii$ can be equivalent to a WZW
model based on $\hat{G}=(C^{3}+A).i$ Lie supergroup whose Lie superalgebra is
isomorphic to the $({\mathscr{C}}^{3}+\mathscr{A})$. First, we assume that Lie
supergroup $\hat{G}$ is unknown. To begin with we determine the metric and
$\tilde{B}$-field corresponding to the action (5.29). By identifying the
action (5.29) with the $\sigma$-model of the form (3.1) we find that only the
non-zero components of the metric and ${\tilde{B}}$-field are
${\tilde{G}}_{{}_{{\tilde{y}}{\tilde{x}}}}={\tilde{G}}_{{}_{{\tilde{\psi}}{\tilde{\chi}}}}=1$
and ${\tilde{B}}_{{}_{{\tilde{\psi}}{\tilde{\psi}}}}=3{\tilde{x}}$,
respectively. One quickly finds that the only non-zero component of
${\tilde{H}}$ is
${\tilde{H}}_{{}_{{\tilde{x}}{\tilde{\psi}}{\tilde{\psi}}}}=2$. Inserting
these into the ME (4.12), we can obtain the left invariant super one-forms of
Lie supergroup $\hat{G}$. It results141414Here we have parameterized the
$\hat{G}$ with the coordinates
$({\hat{y}},{\hat{x}};{\hat{\psi}},{\hat{\chi}})$.

$\displaystyle{\hat{L}_{{}_{\pm}}^{1}}$ $\displaystyle=$
$\displaystyle\partial_{\pm}\hat{y}-\frac{3}{2}\partial_{\pm}\hat{\psi}~{}\hat{\psi},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\hat{L}_{{}_{\pm}}^{2}}=\partial_{\pm}\hat{x},$
$\displaystyle{\hat{L}_{{}_{\pm}}^{3}}$ $\displaystyle=$
$\displaystyle-\partial_{\pm}\hat{\psi},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\hat{L}_{{}_{\pm}}^{4}}=\frac{3}{2}\partial_{\pm}\hat{x}~{}\hat{\psi}-\frac{3}{2}\partial_{\pm}\hat{\psi}~{}\hat{x}-\partial_{\pm}\hat{\chi}.$
(5.30)

With this result at hand, one may employ the super Maurer-Cartan equation
(4.10) to obtain the structure constants of Lie superalgebra
$\hat{\mathscr{G}}$ of the $\hat{G}$. Then, the $\hat{\mathscr{G}}$ is defined
by the following non-zero (anti)commutation relations

$\displaystyle[\hat{T}_{2},\hat{T}_{3}]=-3\hat{T}_{4},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\\{\hat{T}_{3},\hat{T}_{3}\\}=3\hat{T}_{1}.$
(5.31)

It can be easily shown that the $\hat{\mathscr{G}}$ is isomorphic to the
$({\mathscr{C}}^{3}+{\mathscr{A}})$ under transformation
$\hat{T}_{1}\rightarrow{T}_{2}/3,\hat{T}_{2}\rightarrow-3{T}_{1},\hat{T}_{3}\rightarrow{T}_{4},\hat{T}_{4}\rightarrow{T}_{3}$,
and hence we denote it by $({\mathscr{C}}^{3}+{\mathscr{A}}).i$. It is worth
noting that the bilinear form on the $({\mathscr{C}}^{3}+{\mathscr{A}}).i$ as
a non-degenerate solution to (4.3) is obtained to be the same as (5.5). On the
other hand, the parametrization of a general element of $({C}^{3}+{A}).i$ in
the form of

$\displaystyle\hat{g}=e^{(\hat{\chi}+\frac{3}{2}\hat{x}\hat{\psi})\hat{T}_{4}}~{}e^{\hat{y}\hat{T}_{1}}~{}e^{\hat{x}\hat{T}_{2}}~{}e^{\hat{\psi}\hat{T}_{3}},$
(5.32)

leads to the super one-forms (5.2). Finally, employing (4.1) and using the
above results, one can get the WZW action on the $({C}^{3}+{A}).i$, giving us

$\displaystyle{\hat{S}}_{{}_{WZW}}(\hat{g})=\frac{1}{2}\int
d\sigma^{+}d\sigma^{-}\Big{[}\partial_{+}{\hat{y}}\partial_{-}{\hat{x}}+\partial_{+}{\hat{x}}\partial_{-}{\hat{y}}-\partial_{+}{\hat{\psi}}\partial_{-}{\hat{\chi}}+\partial_{+}{\hat{\chi}}\partial_{-}{\hat{\psi}}-3{\hat{x}}\partial_{+}{\hat{\psi}}\;\partial_{-}{\hat{\psi}}\Big{]}.$
(5.33)

which is nothing but (5.29). Thus, our results show that the dual
$\sigma$-model to the $(C^{3}+A)$ WZW model is itself a WZW model. In fact, we
showed that the $(C^{3}+A)$ WZW model does remain invariant under the super PL
T-duality transformation, that is, the model is super PL self-dual.

###  5.3 Super PL symmetry of the $(C^{3}+A).i$ WZW model

Here we shall investigate super PL symmetry in the $(C^{3}+A).i$ WZW model
describing by action (5.33). From the comparison of action (5.33) with (3.1),
one can write the background matrix of the model as follows:

$\displaystyle{\hat{\cal
E}}_{{}_{MN}}=\left(\begin{tabular}[]{cccc}$0$&$1$&$0$&$0$\\\
$1$&$0$&$0$&$0$\\\ $0$&$0$&$3{\hat{x}}$&$1$\\\ $0$&$0$&-$1$&$0$\\\
\end{tabular}\right).$ (5.38)

Furthermore, we need to calculate the left-invariant supervector fields on the
$(C^{3}+A).i$. Using (5.2) one gets

$\displaystyle\hat{V}_{1}=\frac{\overrightarrow{\partial}}{\partial\hat{y}},~{}~{}~{}~{}~{}~{}~{}\hat{V}_{2}=\frac{\overrightarrow{\partial}}{\partial\hat{x}}+\frac{3}{2}\hat{\psi}\frac{\overrightarrow{\partial}}{\partial\hat{\chi}},~{}~{}~{}~{}~{}~{}~{}~{}\hat{V}_{3}=\frac{3}{2}\hat{\psi}\frac{\overrightarrow{\partial}}{\partial\hat{y}}+\frac{\overrightarrow{\partial}}{\partial\hat{\psi}}-\frac{3}{2}\hat{x}\frac{\overrightarrow{\partial}}{\partial\hat{\chi}},~{}~{}~{}~{}~{}\hat{V}_{4}=\frac{\overrightarrow{\partial}}{\partial\hat{\chi}}.$
(5.39)

Substituting the above relations in formula (3.2) one can obtain the structure
constants of the dual pair to the $(\mathscr{C}^{3}+\mathscr{A}).i$ Lie
superalgebra, giving

$\displaystyle\\{{\tilde{\hat{T}}}^{4},{\tilde{\hat{T}}}^{4}\\}=3{\tilde{\hat{T}}}^{2},$
(5.40)

The Lie superalgebra defined by (5.40) is isomorphic to the
$(2{\mathscr{A}}_{1,1}+2{\mathscr{A}}\big{)}^{0}.i$ Lie superalgebra of Table
1. Based on this, we denote it by
$(2{\mathscr{A}}_{1,1}+2{\mathscr{A}}\big{)}^{0}.ii$. Both sets of generators
of (2.4) and (5.40) are maximally isotropic with respect to the bilinear form
defined by the brackets (2.1). Nevertheless, the
$(({\mathscr{C}}^{3}+{\mathscr{A}}).i,(2{\mathscr{A}}_{1,1}+2{\mathscr{A}}\big{)}^{0}.ii)$
as a Lie superbialgebra satisfies mixed super Jacobi identities (2.3).

###  5.4 Super PL T-dual $\sigma$-models on the superdouble
$\big{(}(C^{3}+A).i,(2{A}_{1,1}+2{A}\big{)}^{0}.ii\big{)}$

The Lie superalgebra of the superdouble
$(({\mathscr{C}}^{3}+{\mathscr{A}}).i,(2{\mathscr{A}}_{1,1}+2{\mathscr{A}}\big{)}^{0}.ii)$
obeys the following set of non-trivial (anti)commutation relations151515Notice
that the Lie superalgebra
$(({\mathscr{C}}^{3}+{\mathscr{A}}).i,(2{\mathscr{A}}_{1,1}+2{\mathscr{A}}\big{)}^{0}.ii)$
with the (anti)commutation relations (5.41) is isomorphic with each of the
DSDs belonging to family ${DSD23}^{{}^{k}}$ of Theorem 1. [27]

$\displaystyle[\hat{T}_{{}_{2}},\hat{T}_{{}_{3}}]$ $\displaystyle=$
$\displaystyle-3\hat{T}_{{}_{4}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\\{\hat{T}_{{}_{3}},\hat{T}_{{}_{3}}\\}=3\hat{T}_{{}_{1}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}[{\tilde{\hat{T}}}^{{}^{4}},{\tilde{\hat{T}}}^{{}^{4}}]=3{\tilde{\hat{T}}}^{{}^{2}},$
$\displaystyle{[\hat{T}_{{}_{2}},{\tilde{\hat{T}}}^{{}^{4}}]}$
$\displaystyle=$ $\displaystyle
3(-\hat{T}_{{}_{4}}+{\tilde{\hat{T}}}^{{}^{3}}),~{}~{}~{}~{}~{}~{}~{}~{}[\hat{T}_{{}_{3}},{\tilde{\hat{T}}}^{{}^{1}}]=-3{\tilde{\hat{T}}}^{{}^{3}},~{}~{}~{}~{}~{}~{}~{}~{}~{}\\{\hat{T}_{{}_{3}},{\tilde{\hat{T}}}^{{}^{4}}\\}=3{\tilde{\hat{T}}}^{{}^{2}}.$
(5.41)

Below, we construct $T$-dual $\sigma$-models on the DSD defined by (5.41). As
expected, the original $\sigma$-model will be equivalent to the WZW model
based on the $(C^{3}+A).i$, i.e., action (5.33). The interesting thing about
the dual model is that it will also be equivalent to a WZW model.

$\bullet$ Original model as WZW model based on the $(C^{3}+A).i$.
In order to construct the original $\sigma$-model with the $(C^{3}+A).i$ Lie
supergroup as the target superspace, we use the same parametrization as
(5.32). Using (3.5) we then find that

$\displaystyle{\hat{R}_{{}_{\pm}}^{1}}$ $\displaystyle=$
$\displaystyle\partial_{\pm}\hat{y}+\frac{3}{2}\partial_{\pm}\hat{\psi}~{}\hat{\psi},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\hat{R}_{{}_{\pm}}^{2}}=\partial_{\pm}\hat{x},$
$\displaystyle{\hat{R}_{{}_{\pm}}^{3}}$ $\displaystyle=$
$\displaystyle-\partial_{\pm}\hat{\psi},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\hat{R}_{{}_{\pm}}^{4}}=-\frac{3}{2}\partial_{\pm}\hat{x}~{}\hat{\psi}+\frac{3}{2}\partial_{\pm}\hat{\psi}~{}\hat{x}-\partial_{\pm}\hat{\chi}.$
(5.42)

It follows from (3) and the second formula of (3.6) that only non-zero
component of the super Poisson structure is
${\hat{\Pi}}^{{}^{\hat{\chi}\hat{\chi}}}(\hat{g})=-3\hat{x}$. By considering
the $\sigma$-model constant matrix in the form of (5.22) and by employing
(3.4) together with the first formula of (3.6), one can get the action of
$\sigma$-model on the $(C^{3}+A).i$ which will be nothing but the WZW model
(5.33).

$\bullet$ Dual model as WZW model based on the $(C^{3}+A).ii$.
In order to calculate the components of the right invariant super one-forms
${{\tilde{\hat{R}}_{{}_{\pm}a}}}$ on the $(2{A}_{1,1}+2{A}\big{)}^{0}.ii$ Lie
supergroup we parameterize an element of the supergroup as

$\displaystyle{\tilde{\hat{g}}}~{}=~{}e^{\tilde{\hat{\chi}}{\tilde{\hat{T}}}^{{}^{4}}}e^{\tilde{\hat{y}}{\tilde{\hat{T}}}^{{}^{1}}}~{}e^{\tilde{\hat{x}}{\tilde{\hat{T}}}^{{}^{2}}}~{}e^{\tilde{\hat{\psi}}{\tilde{\hat{T}}}^{{}^{3}}}.$
(5.43)

Now, one may use equation (3.5) to get ${\tilde{\hat{R}}}_{\pm_{a}}$’s on the
$(2{A}_{1,1}+2{A}\big{)}^{0}.ii$, giving

$\displaystyle{\tilde{\hat{R}}}_{\pm_{1}}={\partial_{\pm}}{\tilde{\hat{y}}},~{}~{}~{}~{}~{}~{}~{}{\tilde{\hat{R}}}_{\pm_{2}}={\partial_{\pm}}{\tilde{\hat{x}}}+\frac{3}{2}{\partial_{\pm}}{\tilde{\hat{\chi}}}~{}{\tilde{\hat{\chi}}},~{}~{}~{}~{}{\tilde{\hat{R}}}_{\pm_{3}}={\partial_{\pm}}{\tilde{\hat{\psi}}},~{}~{}~{}~{}~{}~{}~{}{\tilde{\hat{R}}}_{\pm_{4}}={\partial_{\pm}}{\tilde{\hat{\chi}}}.$
(5.44)

To obtain the dual super Poisson structure one must employ equations (5.41)
and (5.43) in formula (3) and the second formula of (3.6). It then results

$\displaystyle{\tilde{\hat{\Pi}}}_{ab}(\tilde{\hat{g}})=\left(\begin{tabular}[]{cccc}$0$&$0$&$0$&$0$\\\
$0$&$0$&$3{\tilde{\hat{\chi}}}$&$0$\\\
$0$&$-3{\tilde{\hat{\chi}}}$&$-3{\tilde{\hat{y}}}$&$0$\\\ $0$&$0$&$0$&$0$\\\
\end{tabular}\right).$ (5.49)

Thus, one may insert (5.22) and (5.49) into (3.9) and then use (5.44) and
(3.8) to find the dual action on the $(2{A}_{1,1}+2{A}\big{)}^{0}.ii$. Using
integrating by parts, the action becomes

$\displaystyle{\tilde{\hat{S}}}=\frac{1}{2}\int
d\sigma^{+}d\sigma^{-}\;\Big{[}\partial_{+}{\tilde{\hat{y}}}\partial_{-}{\tilde{\hat{x}}}+\partial_{+}{\tilde{\hat{x}}}\partial_{-}{\tilde{\hat{y}}}+3\partial_{+}{\tilde{\hat{y}}}{\tilde{\hat{\chi}}}\partial_{-}{\tilde{\hat{\chi}}}+6\partial_{+}{\tilde{\hat{\chi}}}~{}{\tilde{\hat{\chi}}}\partial_{-}{\tilde{\hat{y}}}-\partial_{+}{\tilde{\hat{\psi}}}\partial_{-}{\tilde{\hat{\chi}}}+\partial_{+}{\tilde{\hat{\chi}}}\partial_{-}{\tilde{\hat{\psi}}}\Big{]}.~{}~{}$
(5.50)

In order to complete the chain of WZW models related to super PL T-duality, we
show that the dual $\sigma$-model on the $(2{A}_{1,1}+2{A}\big{)}^{0}.ii$ can
be also equivalent to a WZW model which is built on $\bar{G}=(C^{3}+A).ii$ Lie
supergroup whose Lie superalgebra is isomorphic to the
$({\mathscr{C}}^{3}+\mathscr{A})$. Similarly to the previous case, assume that
Lie supergroup $\bar{G}$ is unknown. To determine the structure of $\bar{G}$
Lie supergroup, first we find the super one-forms corresponding to $\bar{G}$.
To this end, one must insert the inverse of metric and the field strength
corresponding to the dual action (5.50) into the the ME (4.12). From the
comparison of action (5.50) with $\sigma$-model (3.1), we find that the
inverse of metric is

$\displaystyle{\tilde{\hat{G}}}^{{MN}}=\left(\begin{tabular}[]{cccc}$0$&$1$&$0$&$0$\\\
$1$&$0$&-$\frac{3}{2}{\tilde{\hat{\chi}}}$&$0$\\\
$0$&-$\frac{3}{2}{\tilde{\hat{\chi}}}$&$0$&$1$\\\ $0$&$0$&-$1$&$0$\\\
\end{tabular}\right),$ (5.55)

and as the only non-zero component of ${\tilde{\hat{B}}}$-field is
${\tilde{\hat{B}}}_{{}_{{\tilde{\hat{y}}}{\tilde{\hat{\chi}}}}}=9{\tilde{\hat{\chi}}}/2$,
the only non-zero component of field strength $\tilde{\hat{H}}$ is
${\tilde{\hat{H}}}_{{}_{{\tilde{\hat{y}}}{\tilde{\hat{\chi}}}{\tilde{\hat{\chi}}}}}=9$.
Inserting these into the ME (4.12), the left invariant super one-forms on the
$\bar{G}$ are worked out to be of the form

$\displaystyle{\bar{L}_{{}_{\pm}}^{1}}$ $\displaystyle=$
$\displaystyle\partial_{\pm}\bar{y},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\bar{L}_{{}_{\pm}}^{2}}=\partial_{\pm}\bar{x}-\frac{9}{2}\partial_{\pm}\bar{\chi}~{}\bar{\chi},$
$\displaystyle{\bar{L}_{{}_{\pm}}^{3}}$ $\displaystyle=$
$\displaystyle-6\partial_{\pm}\bar{y}~{}\bar{\chi}-\partial_{\pm}\bar{\psi}+3\partial_{\pm}\bar{\chi}~{}\bar{y},~{}~{}~{}~{}~{}~{}~{}~{}{\bar{L}_{{}_{\pm}}^{4}}=-\partial_{\pm}\bar{\chi}.$
(5.56)

Here we have parameterized the $\bar{G}$ by the coordinates
$({\bar{y}},{\bar{x}};{\bar{\psi}},{\bar{\chi}})$. By substituting the super
one-forms (5.4) into equation of the super Maurer-Cartan (4.10) one can obtain
the structure constants of Lie superalgebra $\bar{\mathscr{G}}$ whose non-zero
superbrackets read

$\displaystyle[\bar{T}_{1},\bar{T}_{4}]=9\bar{T}_{3},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\\{\bar{T}_{4},\bar{T}_{4}\\}=9\bar{T}_{2}.$
(5.57)

Obviously, the $\bar{\mathscr{G}}$ is isomorphic to the
$({\mathscr{C}}^{3}+{\mathscr{A}})$ and hence we denote it by
$({\mathscr{C}}^{3}+{\mathscr{A}}).ii$. By using (4.3) and (5.57), it can be
shown that the bilinear form on the $({\mathscr{C}}^{3}+{\mathscr{A}}).ii$ is
nothing but (5.5). Notice that now that the structure of the Lie superalgebra
$({\mathscr{C}}^{3}+{\mathscr{A}}).ii$ is known, one can obtain the super one-
forms (5.4) by a convenient parametrization of the supergroup element in the
following form

$\displaystyle\bar{g}=e^{\bar{\chi}\bar{T}_{4}}~{}e^{\bar{y}\bar{T}_{1}}~{}e^{\bar{x}\bar{T}_{2}}~{}e^{(\bar{\psi}+6\bar{y}\bar{\chi})\bar{T}_{3}}.$
(5.58)

Finally, using these results one can construct the WZW model based on the
$({C}^{3}+{A}).ii$ whose action is

$\displaystyle{\bar{S}}_{{}_{WZW}}(\bar{g})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int
d\sigma^{+}d\sigma^{-}\;\Big{[}\partial_{+}{{\bar{y}}}\partial_{-}{\
{\bar{x}}}+\partial_{+}{{\bar{x}}}\partial_{-}{{\bar{y}}}+3\partial_{+}{{\bar{y}}}{{\bar{\chi}}}\partial_{-}{{\bar{\chi}}}+6\partial_{+}{\bar{\chi}}~{}{{\bar{\chi}}}\partial_{-}{{\bar{y}}}$
(5.59)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\partial_{+}{{\bar{\psi}}}\partial_{-}{{\bar{\chi}}}+\partial_{+}{{\bar{\chi}}}\partial_{-}{{\bar{\psi}}}\Big{]}.$

As it can be seen, the above action is exactly equal to the action of
$\sigma$-model (5.50). In fact, we showed that the dual $\sigma$-model to the
$(C^{3}+A).i$ WZW model is itself a WZW model.

###  5.5 Super PL symmetry of the $(C^{3}+A).ii$ WZW model

In order to investigate the super PL symmetry of the $(C^{3}+A).ii$ WZW model,
we employ formula (3.2) to find the structure constants of the dual Lie
supergroup to the $(C^{3}+A).ii$. In this way, we need to specify the
background matrix of the $(C^{3}+A).ii$ WZW model, as well as the left-
invariant supervector fields corresponding to the $(C^{3}+A).ii$. By
identifying the action (5.59) with the $\sigma$-model of the form (3.1) we can
read off the background matrix ${\bar{\cal E}}_{{}_{MN}}$ in the coordinate
base $(d\bar{y},d\bar{x};d\bar{\psi},d\bar{\chi})$ as

$\displaystyle{\bar{\cal
E}}_{{}_{MN}}=\left(\begin{tabular}[]{cccc}$0$&$1$&$0$&$3{{\bar{\chi}}}$\\\
$1$&$0$&$0$&$0$\\\ $0$&$0$&$0$&$1$\\\ -$6{{\bar{\chi}}}$&$0$&-$1$&$0$\\\
\end{tabular}\right).$ (5.64)

Furthermore, by applying (5.4) in relation
$(-1)^{b}{L}^{a}{{V_{b}}}=\delta^{a}_{~{}b}$ one can obtain the left-invariant
supervector fields on the $(C^{3}+A).ii$, giving us

$\displaystyle\bar{V}_{1}=\frac{\overrightarrow{\partial}}{\partial\bar{y}}-6{{\bar{\chi}}}\frac{\overrightarrow{\partial}}{\partial\bar{\psi}},~{}~{}~{}~{}~{}~{}~{}\bar{V}_{2}=\frac{\overrightarrow{\partial}}{\partial\bar{x}},~{}~{}~{}~{}~{}~{}~{}~{}\bar{V}_{3}=\frac{\overrightarrow{\partial}}{\partial\bar{\psi}},~{}~{}~{}~{}~{}\bar{V}_{4}=\frac{9}{2}{{\bar{\chi}}}\frac{\overrightarrow{\partial}}{\partial\bar{x}}+3{{\bar{y}}}\frac{\overrightarrow{\partial}}{\partial\bar{\psi}}+\frac{\overrightarrow{\partial}}{\partial\bar{\chi}}.$
(5.65)

Now, by substituting relations (5.64) and (5.65) in formula (3.2) one can
obtain the structure constants of the dual pair to the
$(\mathscr{C}^{3}+\mathscr{A}).ii$ Lie superalgebra in such a way that only
non-zero commutation relation of the dual pair is

$\displaystyle[{\tilde{\bar{T}}}^{2},{\tilde{\bar{T}}}^{3}]=-\frac{9}{2}{\tilde{\bar{T}}}^{4}.$
(5.66)

Actually, the resulting Lie superalgebra is isomorphic to the
${\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}.i$ of Table 1, and hence we denote
this dual pair by ${{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}}.iii$. Indeed,
two sets of generators (5.57) and (5.66) are dual to each other in the sense
of (2.1). Despite this, one can check that the
$(\mathscr{C}^{3}+\mathscr{A}).ii$ Lie superalgebra and its dual pair, the
${{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}}.iii$, satisfy mixed super Jacobi
identities (2.3). The Lie superalgebra of the DSD which we refer to as the
$\left((\mathscr{C}^{3}+\mathscr{A}).ii,{{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}}.iii\right)$
is defined by the following non-zero (anti)commutation relations

$\displaystyle{[\bar{T}_{1},\bar{T}_{4}]}$ $\displaystyle=$ $\displaystyle
9\bar{T}_{3},~{}~{}~{}~{}~{}~{}\\{\bar{T}_{4},\bar{T}_{4}\\}=9\bar{T}_{2},~{}~{}~{}~{}[{\tilde{\bar{T}}}^{2},{\tilde{\bar{T}}}^{3}]=-\frac{9}{2}{\tilde{\bar{T}}}^{4},~{}~{}~{}~{}~{}[\bar{T}_{1},{\tilde{\bar{T}}}^{3}]=-9{\tilde{\bar{T}}}^{4},$
$\displaystyle{[\bar{T}_{{}_{4}},{\tilde{\bar{T}}}^{{}^{2}}]}$
$\displaystyle=$
$\displaystyle-\frac{9}{2}\bar{T}_{{}_{3}}-9{\tilde{\bar{T}}}^{{}^{4}},~{}~{}~{}~{}\\{\bar{T}_{{}_{4}},{\tilde{\bar{T}}}^{{}^{3}}\\}=-\frac{9}{2}\bar{T}_{{}_{2}}-9{\tilde{\bar{T}}}^{{}^{1}}.$
(5.67)

It is interesting to comment on the fact that the DSDs
$\left((\mathscr{C}^{3}+\mathscr{A}).ii,{{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}}.iii\right)$
and
$\big{(}(\mathscr{C}^{3}+\mathscr{A})$,${{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}}.ii\big{)}$
as Lie superalgebras are isomorphic, and so one can find an isomorphism that
preserves also the bilinear form $<.\;,\;.>$, so that they belong to the same
DSD. The isomorphism of Manin supertriples between
$\left((\mathscr{C}^{3}+\mathscr{A}).ii,{{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}}.iii\right)$
and
$\left((\mathscr{C}^{3}+\mathscr{A}),{{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}}.ii\right)$
is given by the following transformations

$\displaystyle\bar{T}_{1}\rightarrow
9{T}_{1},~{}~{}~{}~{}~{}~{}\bar{T}_{2}\rightarrow\frac{1}{9}T_{2},~{}~{}~{}~{}~{}~{}\bar{T}_{3}\rightarrow
T_{3},~{}~{}~{}~{}~{}~{}\bar{T}_{4}\rightarrow T_{4},$
$\displaystyle\tilde{\bar{T}}_{1}\rightarrow\frac{1}{9}{\tilde{T}}_{1},~{}~{}~{}~{}~{}~{}\tilde{\bar{T}}_{2}\rightarrow{9}{\tilde{T}}_{2},~{}~{}~{}~{}~{}~{}\tilde{\bar{T}}_{3}\rightarrow{\tilde{T}}_{3},~{}~{}~{}~{}~{}~{}\tilde{\bar{T}}_{4}\rightarrow{\tilde{T}}_{4}.$
(5.68)

According to the above result, the story of hierarchy of WZW models related to
super PL T-duality will be again repeated.

$WZW$ model on $(C^{3}+A)$ Dual pair: $C^{3}\otimes A_{1,1}.ii$ SPL symmetry
$DSD1\cong\big{(}(C^{3}+A),C^{3}\otimes A_{1,1}.ii\big{)}$ Original
$\sigma$-model on $D$ Dual model as $WZW$ model on $(C^{3}+A).i$ Dual pair:
$(2A_{1,1}+2A)^{0}.ii$ Dual $\sigma$-model on $D$SPL symmetry
$DSD^{k}_{23}\cong\big{(}(C^{3}+A).i,(2A_{1,1}+2A)^{0}.ii\big{)}$ Original
$\sigma$-model on $D$ Dual model as $WZW$ model on $(C^{3}+A).ii$ Dual pair:
$C^{3}\otimes A_{1,1}.iii$ Dual $\sigma$-model on $D$SPL symmetry
$DSD1\cong\big{(}(C^{3}+A).ii,C^{3}\otimes A_{1,1}.iii\big{)}$ Original
$\sigma$-model on $D$ Figure 1: Scheme of a hierarchy of WZW models related to
super PL T-duality.

In summary, starting the $(C^{3}+A)$ WZW model we showed that the dual
$\sigma$-model to the $(C^{3}+A)$ WZW model was itself a WZW model, which was
built on the $(C^{3}+A).i$. Then, investigating the super PL symmetry in the
$(C^{3}+A).i$ WZW model, we showed that the dual $\sigma$-model to the
$(C^{3}+A).i$ WZW model was again a WZW model, which was built this time on
the $(C^{3}+A).ii$. Thus, we found a hierarchy of WZW models related to super
PL T-duality so that it was different from super PL T-plurality; because as we
showed, the DSDs involved in the process were non-isomorphic. For the sake of
clarity, the results obtained in this section are depicted as a flowchart in
Figure 1.

##  6  Summary and concluding remarks

We have classified $(4|4)$-dimensional DSDs generated by the Manin
supertriples with the first sub-superalgebra
$({\mathscr{C}}^{3}+{\mathscr{A}})$, i.e., Lie superalgebras provided with
bilinear non-degenerate supersymmetric and super ad-invariant form such that
they can be decomposed into a pair of maximally isotropic sub-superalgebras.
We claimed that there are just 24 classes of the non-isomorphic DSDs. The
result has been summarized in Theorem 1. In section 4 we introduced a general
formulation to find the conditions under which a two-dimensional
$\sigma$-model may be equivalent to a WZW model. With the help of this
formulation, we have found a chain of WZW models related to the super PL
T-duality: we showed that the super PL T-duality relates the $(C^{3}+A)$ WZW
model to a $\sigma$-model defined on the $(C^{3}+A)$ Lie supergroup when the
dual Lie supergroup is ${C^{3}\otimes{A}_{{}_{1,1}}}.ii$. Then we showed that
the dual model is equivalent to a WZW model based on the $(C^{3}+A).i$ Lie
supergroup whose Lie superalgebra is isomorphic to the
$({\mathscr{C}}^{3}+{\mathscr{A}})$ [27]. Accordingly, we concluded that the
$(C^{3}+A)$ WZW model did remain invariant under the super PL T-duality
transformation, that is, the model was super PL self-dual. As the same way, we
have shown that the super PL T-duality relates the $(C^{3}+A).i$ WZW model to
a $\sigma$-model defined on the dual Lie supergroup $(2A_{1,1}+2A)^{0}.ii$. We
have also stressed that the dual model built on the $(2A_{1,1}+2A)^{0}.ii$ is
equivalent to the WZW model on isomorphic Lie supergroup $(C^{3}+A).ii$.
Finally, we described a pair of super PL T-dual $\sigma$-models which was
associated with the $DSD1\cong\big{(}(C^{3}+A).ii,C^{3}\otimes
A_{1,1}.iii\big{)}$. Note that in this process, the DSDs
$DSD1\cong\big{(}(C^{3}+A),C^{3}\otimes A_{1,1}.ii\big{)}$ and
$DSD^{k}_{23}\cong\big{(}(C^{3}+A).i,(2A_{1,1}+2A)^{0}.ii\big{)}$ were non-
isomorphic, as mentioned in Theorem 1. Accordingly, this process was different
from super PL T-plurality. Because of being isomorphism the DSDs
$\big{(}(C^{3}+A),C^{3}\otimes A_{1,1}.ii\big{)}$ and
$\big{(}(C^{3}+A).ii,C^{3}\otimes A_{1,1}.iii\big{)}$, the process of
calculating chain of WZW models was repeated again.

As mentioned in the introduction section, we have already obtained the dual
$\sigma$-model to the $GL(1|1)$ WZW model [26], as well as to the Heisenberg
WZW model [23] and the $GL(2,\mathds{R})$ WZW model [25]. In all cases, the
dual $\sigma$-models were conformally invariant up to one-loop order and in
some cases even up to two-loops, but they were not equivalent to a WZW model.
Therefore, finding an example like the $(C^{3}+A)$ WZW model is extremely
rare.

### Acknowledgements

This work was supported by Iran National Science Foundation: INSF under
research fund No. 94/s/41258.

##  Appendix A Isomorphism transformation between the Manin supertriples with
the first sub-superalgebra ${\cal G}\equiv({\mathscr{C}}^{3}+{\mathscr{A}})$

In this appendix we give isomorphism transformation between the Manin
supertriples with the first sub-superalgebra ${\cal
G}\equiv({\mathscr{C}}^{3}+{\mathscr{A}})$ of Table 1 that are isomorphic as
Lie superalgebras. The DSDs $\mathscr{D}$ and $\mathscr{D}^{\prime}$ with
these special bases
$X_{{}_{A}}=(T_{{}_{1}},T_{{}_{2}},{\tilde{T}}^{{}^{1}},{\tilde{T}}^{{}^{2}};T_{{}_{3}},T_{{}_{4}},{\tilde{T}}^{{}^{3}},{\tilde{T}}^{{}^{4}}),X^{\prime}_{{}_{A}}=(T^{\prime}_{{}_{1}},T^{\prime}_{{}_{2}},{\tilde{T}}^{\prime^{1}},{\tilde{T}}^{\prime^{2}};T^{\prime}_{{}_{3}},T^{\prime}_{{}_{4}},\\\
{\tilde{T}}^{\prime^{3}},{\tilde{T}}^{\prime^{4}})$ are isomorphic iff there
is a superinvertible $8\times 8$ transformation matrix
${\mathbb{C}_{{}_{A}}}^{{}^{B}}$ such that the linear map given by equation
(2.5) transforms the Lie multiplication of $\mathscr{D}$ into that of
$\mathscr{D}^{\prime}$ and preserves the bilinear form (2.7) as well as
equation (2.6). As we will show, the isomorphism transformations between the
Manin supertriples of Table 1 are not unique; they contain several free
parameters. One can give them in a simple form setting the parameters zero or
one.

###  A.1 Structure of $DSD1:~{}\big{(}{\cal
G},{\cal{I}}_{{}_{(2|2)}}\big{)}\cong\big{(}{\cal
G},{\mathscr{C}}^{3}\oplus{{\mathscr{A}}_{{}_{1,1}}}.i\big{)}\cong\big{(}{\cal
G},{({\mathscr{C}}^{3}+{\mathscr{A}})_{{}_{k=4}}}^{\epsilon=-1}\big{)}$

$\bullet$ The isomorphism of Manin supertriples between $\big{(}{\cal
G},{\cal{I}}_{{}_{(2|2)}}\big{)}$ and $\big{(}{\cal
G},{\mathscr{C}}^{3}\oplus{{\mathscr{A}}_{{}_{1,1}}}.i\big{)}$ is given by the
following transformation

$\displaystyle X^{\prime}_{{}_{1}}$ $\displaystyle=$
$\displaystyle\frac{1}{ab}X_{{}_{1}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{5}}=-\frac{1}{b}X_{{}_{5}},$
$\displaystyle X^{\prime}_{{}_{2}}$ $\displaystyle=$ $\displaystyle
a^{{}^{2}}X_{{}_{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{6}}=-aX_{{}_{6}},$
$\displaystyle X^{\prime}_{{}_{3}}$ $\displaystyle=$ $\displaystyle
a^{{}^{2}}X_{{}_{2}}+abX_{{}_{3}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{7}}=-cX_{{}_{5}}-bX_{{}_{7}},$
$\displaystyle X^{\prime}_{{}_{4}}$ $\displaystyle=$
$\displaystyle-\frac{1}{ab}X_{{}_{1}}+\frac{1}{a^{{}^{2}}}X_{{}_{4}},~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{8}}=-\frac{1}{a}X_{{}_{8}},$
(A.1)

where $(X_{{}_{1}},\cdots,X_{{}_{8}})$ and
$(X^{\prime}_{{}_{1}},\cdots,X^{\prime}_{{}_{8}})$ are generators of the Manin
supertriples $\big{(}{\cal G},{\cal{I}}_{{}_{(2|2)}}\big{)}$ and $\big{(}{\cal
G},{\mathscr{C}}^{3}\oplus{{\mathscr{A}}_{{}_{1,1}}}.i\big{)}$, respectively.

$\bullet\bullet$ The isomorphism of Manin supertriples between $\big{(}{\cal
G},{\cal{I}}_{{}_{(2|2)}}\big{)}$ and $\big{(}{\cal
G},{({\mathscr{C}}^{3}+{\mathscr{A}})_{{}_{k=4}}}^{\epsilon=-1}\big{)}$ is
given by the following transformation

$\displaystyle X^{\prime}_{{}_{1}}$ $\displaystyle=$
$\displaystyle\frac{1}{ab}X_{{}_{1}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{5}}=-\frac{1}{b}X_{{}_{5}},$
$\displaystyle X^{\prime}_{{}_{2}}$ $\displaystyle=$ $\displaystyle
a^{{}^{2}}X_{{}_{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{6}}=-cX_{{}_{5}}-aX_{{}_{6}},$
$\displaystyle X^{\prime}_{{}_{3}}$ $\displaystyle=$ $\displaystyle
a^{{}^{2}}X_{{}_{2}}+abX_{{}_{3}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{7}}=2aX_{{}_{6}}-bX_{{}_{7}}+\frac{bc}{a}X_{{}_{8}},$
$\displaystyle X^{\prime}_{{}_{4}}$ $\displaystyle=$
$\displaystyle-\frac{1}{ab}X_{{}_{1}}+\frac{1}{a^{{}^{2}}}X_{{}_{4}},~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{8}}=\frac{2}{b}X_{{}_{5}}-\frac{1}{a}X_{{}_{8}},$
(A.2)

where $(X_{{}_{1}},\cdots,X_{{}_{8}})$ and
$(X^{\prime}_{{}_{1}},\cdots,X^{\prime}_{{}_{8}})$ are generators of the Manin
supertriple $\big{(}{\cal G},{\cal{I}}_{{}_{(2|2)}}\big{)}$ and
$\big{(}{\cal G},{({\mathscr{C}}^{3}+\\\
{\mathscr{A}})_{{}_{k=4}}}^{\epsilon=-1}\big{)}$, respectively. Similarly, one
can find an isomorphism transformation between the DSDs $\big{(}{\cal
G},{\mathscr{C}}^{3}\oplus{{\mathscr{A}}_{{}_{1,1}}}.i\big{)}$ and
$\big{(}{\cal
G},{({\mathscr{C}}^{3}+{\mathscr{A}})_{{}_{k=4}}}^{\epsilon=-1}\big{)}$ that
preserves also equations (2.6) and (2.7).

###  A.2 Structure of $DSD4:\big{(}{\cal
G},{{\mathscr{C}}_{{}_{p=-1}}}^{1,\epsilon=1}\oplus{{\mathscr{A}}}\big{)}\cong\big{(}{\cal
G},{{\mathscr{C}}_{{}_{p=-1}}}^{1,\epsilon=-1}\oplus{{\mathscr{A}}}\big{)}$

The isomorphism of Manin supertriples between $\big{(}{\cal
G},{{\mathscr{C}}_{{}_{p=-1}}}^{1,\epsilon=1}\oplus{{\mathscr{A}}}\big{)}$ and
$\big{(}{\cal
G},{{\mathscr{C}}_{{}_{p=-1}}}^{1,\epsilon=-1}\oplus{{\mathscr{A}}}\big{)}$ is
given by the following transformation

$\displaystyle X^{\prime}_{{}_{1}}$ $\displaystyle=$
$\displaystyle\frac{1}{ab}(X_{{}_{1}}+dX_{{}_{2}}),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{5}}=-\frac{1}{b}X_{{}_{5}},$
$\displaystyle X^{\prime}_{{}_{2}}$ $\displaystyle=$ $\displaystyle-
X_{{}_{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{6}}=-a(cX_{{}_{5}}+X_{{}_{6}})+\frac{a^{{}^{2}}+1}{2a}X_{{}_{8}},$
$\displaystyle X^{\prime}_{{}_{3}}$ $\displaystyle=$ $\displaystyle
ab(cX_{{}_{2}}+X_{{}_{3}}),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{7}}=b(-X_{{}_{7}}+cX_{{}_{8}}),$
$\displaystyle X^{\prime}_{{}_{4}}$ $\displaystyle=$ $\displaystyle
cX_{{}_{1}}+cdX_{{}_{2}}+dX_{{}_{3}}-X_{{}_{4}},~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{8}}=-\frac{1}{a}X_{{}_{8}},$
(A.3)

where $(X_{{}_{1}},\cdots,X_{{}_{8}})$ denotes the basis in the Manin
supertriple $\big{(}{\cal
G},{{\mathscr{C}}_{{}_{p=-1}}}^{1,\epsilon=1}\oplus{{\mathscr{A}}}\big{)}$,
and $(X^{\prime}_{{}_{1}},\cdots,X^{\prime}_{{}_{8}})$ is the basis in the
Manin supertriple $\big{(}{\cal
G},{{\mathscr{C}}_{{}_{p=-1}}}^{1,\epsilon=-1}\oplus{{\mathscr{A}}}\big{)}$.

###  A.3 Structure of $DSD5:\big{(}{\cal
G},{{\mathscr{C}}_{{}_{p=1}}}^{2,\epsilon=1}\oplus{{\mathscr{A}}_{{}_{1,1}}}\big{)}\cong\big{(}{\cal
G},{{\mathscr{C}}_{{}_{p=1}}}^{2,\epsilon=-1}\oplus{{\mathscr{A}}_{{}_{1,1}}}\big{)}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\cong\big{(}{\cal
G},{{\mathscr{C}}}^{4,\epsilon=1}\oplus{{\mathscr{A}}_{{}_{1,1}}}\big{)}\cong\big{(}{\cal
G},{{\mathscr{C}}}^{4,\epsilon=-1}\oplus{{\mathscr{A}}_{{}_{1,1}}}\big{)}$

The isomorphism transformation of Manin supertriples:
$~{}\bullet\\{X_{{}_{A}}\\}\in\big{(}{\cal
G},{{\mathscr{C}}_{{}_{p=1}}}^{2,\epsilon=1}\oplus{{\mathscr{A}}_{{}_{1,1}}}\big{)}\longrightarrow\\{X^{\prime}_{{}_{A}}\\}\in\big{(}{\cal
G},{{\mathscr{C}}_{{}_{p=1}}}^{2,\epsilon=-1}\oplus{{\mathscr{A}}_{{}_{1,1}}}\big{)}$

$\displaystyle X^{\prime}_{{}_{1}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2ab}X_{{}_{1}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{5}}=-\frac{1}{2b}X_{{}_{8}},$
$\displaystyle X^{\prime}_{{}_{2}}$ $\displaystyle=$ $\displaystyle
X_{{}_{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{6}}=-\frac{1}{2a}X_{{}_{5}}-aX_{{}_{7}}-cX_{{}_{8}},$
$\displaystyle X^{\prime}_{{}_{3}}$ $\displaystyle=$
$\displaystyle-2abX_{{}_{3}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{7}}=-\frac{2bc}{a}X_{{}_{5}}+2bX_{{}_{6}}-bX_{{}_{8}},$
$\displaystyle X^{\prime}_{{}_{4}}$ $\displaystyle=$ $\displaystyle
X_{{}_{4}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{8}}=\frac{1}{a}X_{{}_{5}}.$
(A.4)

$~{}\bullet\bullet\\{X_{{}_{A}}\\}\in\big{(}{\cal
G},{{\mathscr{C}}_{{}_{p=1}}}^{2,\epsilon_{{}_{1}}}\oplus{{\mathscr{A}}_{{}_{1,1}}}\big{)}\longrightarrow\\{X^{\prime}_{{}_{A}}\\}\in\big{(}{\cal
G},{{\mathscr{C}}}^{4,\epsilon_{{}_{2}}}\oplus{{\mathscr{A}}_{{}_{1,1}}}\big{)}$

$\displaystyle X^{\prime}_{{}_{1}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2\epsilon_{{}_{1}}ab}X_{{}_{1}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{5}}=-\frac{1}{2\epsilon_{{}_{1}}b}X_{{}_{8}},$
$\displaystyle X^{\prime}_{{}_{2}}$ $\displaystyle=$
$\displaystyle-\epsilon_{{}_{1}}\epsilon_{{}_{2}}X_{{}_{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{6}}=\frac{\epsilon_{{}_{2}}}{2a}X_{{}_{5}}-aX_{{}_{7}},$
$\displaystyle X^{\prime}_{{}_{3}}$ $\displaystyle=$
$\displaystyle-\epsilon_{{}_{1}}\epsilon_{{}_{2}}X_{{}_{2}}-2\epsilon_{{}_{1}}abX_{{}_{3}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{7}}=2\epsilon_{{}_{1}}bX_{{}_{6}}-bX_{{}_{8}},$
$\displaystyle X^{\prime}_{{}_{4}}$ $\displaystyle=$
$\displaystyle\frac{1}{2\epsilon_{{}_{1}}ab}X_{{}_{1}}-\frac{1}{\epsilon_{{}_{1}}\epsilon_{{}_{2}}}X_{{}_{4}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{8}}=\frac{1}{a}X_{{}_{5}},$
(A.5)

where $\epsilon_{{}_{1}},\epsilon_{{}_{2}}=\pm 1$.
$~{}\bullet\bullet\bullet\\{X_{{}_{A}}\\}\in\big{(}{\cal
G},{{\mathscr{C}}}^{4,\epsilon=1}\oplus{{\mathscr{A}}_{{}_{1,1}}}\big{)}\longrightarrow\\{X^{\prime}_{{}_{A}}\\}\in\big{(}{\cal
G},{{\mathscr{C}}}^{4,\epsilon=-1}\oplus{{\mathscr{A}}_{{}_{1,1}}}\big{)}$

$\displaystyle X^{\prime}_{{}_{1}}$ $\displaystyle=$
$\displaystyle\frac{1}{ab}X_{{}_{1}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{5}}=-\frac{1}{b}X_{{}_{5}},$
$\displaystyle X^{\prime}_{{}_{2}}$ $\displaystyle=$ $\displaystyle-
X_{{}_{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{6}}=-aX_{{}_{6}}+\frac{1+a^{2}}{2a}X_{{}_{8}},$
$\displaystyle X^{\prime}_{{}_{3}}$ $\displaystyle=$
$\displaystyle-(1+ab)X_{{}_{2}}+abX_{{}_{3}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{7}}=-bX_{{}_{7}},$
$\displaystyle X^{\prime}_{{}_{4}}$ $\displaystyle=$
$\displaystyle-\frac{1+ab}{ab}X_{{}_{1}}-X_{{}_{4}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{8}}=-\frac{1}{a}X_{{}_{8}}.$
(A.6)

###  A.4 Structure of ${DSD22}^{{}^{p}}:\big{(}{\cal
G},{{\mathscr{D}}_{{}_{p,p-1}}}^{1,\epsilon=1}\big{)}\cong\big{(}{\cal
G},{{\mathscr{D}}_{{}_{1-p,-p}}}^{1,\epsilon=1}\big{)}\cong\big{(}{\cal
G},{{\mathscr{D}}_{{}_{p,p-1}}}^{1,\epsilon=-1}\big{)},~{}~{}~{}p\neq 0,1$

The isomorphism transformation of Manin supertriples:

$~{}\bullet\\{X_{{}_{A}}\\}\in\big{(}{\cal
G},{{\mathscr{D}}_{{}_{p,p-1}}}^{1,\epsilon=1}\big{)}\longrightarrow\\{X^{\prime}_{{}_{A}}\\}\in\big{(}{\cal
G},{{\mathscr{D}}_{{}_{p^{\prime},p^{\prime}-1}}}^{1,\epsilon=1}\big{)}$

$\displaystyle X^{\prime}_{{}_{1}}$ $\displaystyle=$ $\displaystyle
aX_{{}_{1}}-acX_{{}_{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{5}}=\frac{1}{2bp^{\prime}}X_{{}_{8}},$
$\displaystyle X^{\prime}_{{}_{2}}$ $\displaystyle=$ $\displaystyle
X_{{}_{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{6}}=\frac{abp^{\prime}}{p^{\prime}-1}X_{{}_{5}}-\frac{1}{2abp^{\prime}}X_{{}_{7}},$
$\displaystyle X^{\prime}_{{}_{3}}$ $\displaystyle=$
$\displaystyle\frac{1}{a}X_{{}_{3}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{7}}=-2bp^{\prime}X_{{}_{6}}-bX_{{}_{8}},$
$\displaystyle X^{\prime}_{{}_{4}}$ $\displaystyle=$ $\displaystyle
cX_{{}_{3}}+X_{{}_{4}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{8}}=2abp^{\prime}X_{{}_{5}},$
(A.7)

where $p^{\prime}=1-p$.
$~{}\bullet\bullet\\{X_{{}_{A}}\\}\in\big{(}{\cal
G},{{\mathscr{D}}_{{}_{p,p-1}}}^{1,\epsilon=1}\big{)}\longrightarrow\\{X^{\prime}_{{}_{A}}\\}\in\big{(}{\cal
G},{{\mathscr{D}}_{{}_{p,p-1}}}^{1,\epsilon=-1}\big{)}$

$\displaystyle X^{\prime}_{{}_{1}}$ $\displaystyle=$ $\displaystyle
aX_{{}_{1}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{5}}=-\frac{1}{b}X_{{}_{5}},$
$\displaystyle X^{\prime}_{{}_{2}}$ $\displaystyle=$ $\displaystyle-
X_{{}_{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{6}}=-\frac{1}{ab}X_{{}_{6}}-cX_{{}_{8}},$
$\displaystyle X^{\prime}_{{}_{3}}$ $\displaystyle=$
$\displaystyle\frac{1}{a}X_{{}_{3}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{7}}=-bX_{{}_{7}},$
$\displaystyle X^{\prime}_{{}_{4}}$ $\displaystyle=$ $\displaystyle-
X_{{}_{4}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{8}}=-abX_{{}_{8}},$
(A.8)

where $c=\frac{1+a^{2}b^{2}}{2ab(1-p)}$.

###  A.5 Structure of ${DSD23}^{{}^{k}}:\big{(}{\cal
G},{(2{\mathscr{A}}_{{}_{1,1}}+2{\mathscr{A}})^{{}^{0}}}.i\big{)}\cong\big{(}{\cal
G},{({{\mathscr{C}}}^{3}+{\mathscr{A}})_{{}_{k-4}}}^{\epsilon=1}\big{)}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\cong\big{(}{\cal
G},{({{\mathscr{C}}}^{3}+{\mathscr{A}})_{{}_{k}}}^{\epsilon=-1}\big{)},~{}k>4$

The isomorphism transformation of Manin supertriples:
$~{}\bullet\\{X_{{}_{A}}\\}\in\big{(}{\cal
G},{(2{\mathscr{A}}_{{}_{1,1}}+2{\mathscr{A}})^{{}^{0}}}.i\big{)}\longrightarrow\\{X^{\prime}_{{}_{A}}\\}\in\big{(}{\cal
G},{({{\mathscr{C}}}^{3}+{\mathscr{A}})_{{}_{k}}}^{\epsilon=-1}\big{)}$

$\displaystyle X^{\prime}_{{}_{1}}$ $\displaystyle=$
$\displaystyle\frac{1}{ab}X_{{}_{1}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{5}}=-\frac{1}{b}X_{{}_{5}},$
$\displaystyle X^{\prime}_{{}_{2}}$ $\displaystyle=$ $\displaystyle
a^{2}X_{{}_{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{6}}=-aX_{{}_{6}},$
$\displaystyle X^{\prime}_{{}_{3}}$ $\displaystyle=$
$\displaystyle\frac{ac^{2}}{b-2c}X_{{}_{2}}+abX_{{}_{3}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{7}}=-cX_{{}_{6}}-bX_{{}_{7}},$
$\displaystyle X^{\prime}_{{}_{4}}$ $\displaystyle=$
$\displaystyle\frac{c^{2}}{a^{2}b(2c-b)}X_{{}_{1}}+\frac{1}{a^{2}}X_{{}_{4}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{8}}=-\frac{c}{ab}X_{{}_{5}}-\frac{1}{a}X_{{}_{8}},$
(A.9)

where $c(c-b)=a(b-2c)$ and $k=\frac{b-2c}{a}>4$.
$~{}\bullet\bullet\\{X_{{}_{A}}\\}\in\big{(}{\cal
G},{({{\mathscr{C}}}^{3}+{\mathscr{A}})_{{}_{k-4}}}^{\epsilon=1}\big{)}\longrightarrow\\{X^{\prime}_{{}_{A}}\\}\in\big{(}{\cal
G},{({{\mathscr{C}}}^{3}+{\mathscr{A}})_{{}_{k}}}^{\epsilon=-1}\big{)}$

$\displaystyle X^{\prime}_{{}_{1}}$ $\displaystyle=$
$\displaystyle\frac{1}{a^{2}}X_{{}_{1}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{5}}=-\frac{1}{a}X_{{}_{5}},$
$\displaystyle X^{\prime}_{{}_{2}}$ $\displaystyle=$ $\displaystyle
a^{2}X_{{}_{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{6}}=2bX_{{}_{5}}-aX_{{}_{6}}-bX_{{}_{8}},$
$\displaystyle X^{\prime}_{{}_{3}}$ $\displaystyle=$ $\displaystyle
a^{2}X_{{}_{3}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{7}}=2aX_{{}_{6}}-aX_{{}_{7}},$
$\displaystyle X^{\prime}_{{}_{4}}$ $\displaystyle=$
$\displaystyle\frac{1}{a^{2}}X_{{}_{4}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{8}}=\frac{2}{a}X_{{}_{5}}-\frac{1}{a}X_{{}_{8}}.$
(A.10)

###  A.6 Structure of ${DSD24}^{{}^{k}}:\big{(}{\cal
G},{({{\mathscr{C}}}^{3}+{\mathscr{A}})_{{}_{k}}}^{\epsilon=-1}\big{)}\cong\big{(}{\cal
G},{{({\mathscr{C}}}^{3}+{\mathscr{A}})_{{}_{4-k}}}^{\epsilon=-1}\big{)},~{}0<k<4$

The isomorphism of Manin supertriples between $\big{(}{\cal
G},{({{\mathscr{C}}}^{3}+{\mathscr{A}})_{{}_{k}}}^{\epsilon=-1}\big{)}$ and
$\big{(}{\cal
G},{{({\mathscr{C}}}^{3}+{\mathscr{A}})_{{}_{4-k}}}^{\epsilon=-1}\big{)}$ is
given by the following transformation

$\displaystyle X^{\prime}_{{}_{1}}$ $\displaystyle=$
$\displaystyle-\frac{1}{a^{2}}X_{{}_{1}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{5}}=-\frac{1}{a}X_{{}_{5}},$
$\displaystyle X^{\prime}_{{}_{2}}$ $\displaystyle=$ $\displaystyle
a^{2}X_{{}_{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{6}}=-2bX_{{}_{5}}+aX_{{}_{6}}-bX_{{}_{8}},$
$\displaystyle X^{\prime}_{{}_{3}}$ $\displaystyle=$
$\displaystyle-a^{2}X_{{}_{3}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{7}}=-2aX_{{}_{6}}-aX_{{}_{7}},$
$\displaystyle X^{\prime}_{{}_{4}}$ $\displaystyle=$
$\displaystyle\frac{1}{a^{2}}X_{{}_{4}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X^{\prime}_{{}_{8}}=\frac{2}{a}X_{{}_{5}}+\frac{1}{a}X_{{}_{8}},$
(A.11)

where $(X_{{}_{1}},\cdots,X_{{}_{8}})$ denotes the basis in the Manin
supertriple $\big{(}{\cal
G},{({{\mathscr{C}}}^{3}+{\mathscr{A}})_{{}_{k}}}^{\epsilon=-1}\big{)}$, and
$(X^{\prime}_{{}_{1}},\cdots,X^{\prime}_{{}_{8}})$ is the basis in the Manin
supertriple $\big{(}{\cal
G},{{({\mathscr{C}}}^{3}+{\mathscr{A}})_{{}_{4-k}}}^{\epsilon=-1}\big{)}$.

## References

  * [1] G. Parisi and N. Sourlas, Self avoiding walk and supersymmetry, J. Phy. Lett. (Paris) 41 (1980) L403.
  * [2] K. B. Efetov, Supersymmetry and theory of disordered metals, Adv. Phys.  32 (1983) 53-127.
  * [3] H. A. Weidenmüller and M. Zirnbauer, Instanton approximation to the graded non linear sigma model for the integer quantum hall effect, Nucl. Phys. B  305 (1988) 339-366.
  * [4] S. Sethi, Supermanifolds, rigid manifolds and mirror symmetry, Nucl. Phys. B 430 (1994) 31-50, arXiv:hep-th/9404186.
  * [5] V. Schomerus and H. Saleur, The GL(1|1) WZW model: From supergeometry to logarithmic CFT, Nucl. Phys. B 734 (2006) 221-245, arXiv:hep-th/0510032.
  * [6] G. Gotz, T. Quella and V. Schomerus, The WZW model on $PSU(1,1|2)$, J. High Energy Phys.  03 (2007) 003, arXiv:hep-th/0610070.
  * [7] H. Saleur and V. Schomerus, On the SU(2|1) WZNW model and its statistical mechanics applications, Nucl. Phys. B  775 (2007) 312-340, arXiv:hep-th/0611147.
  * [8] M. Henneaux, L. Mezincescu, A $\sigma$-model interpretation of Green-Schwarz covariant superstring action, Phys. Lett. B 152 (1985) 340-342.
  * [9] R. R. Metsaev, A. A. Tseytlin, Type IIB superstring action in $AdS_{5}\times S^{5}$ background, Nucl. Phys. B 533 (1998) 109-126, arXiv:hep-th/9805028.
  * [10] N. Berkovits, C. Vafa and E. Witten, Conformal field theory of AdS background with Ramond-Ramond flux, J. High Energy Phys. 03 (1999) 018, arXiv:hep-th/9902098.
  * [11] M. Bershadsky, S. Zhukov and A. Vaintrob, $PSL(n|n)$ sigma model as a conformal field theory, Nucl. Phys. B 559 (1999) 205-234, arXiv:hep-th/9902180.
  * [12] N. Berkovits, M. Bershadsky, T. Hauer, S. Zhukov and B. Zwiebach, Superstring theory on $AdS_{2}\times S^{2}$ as a coset supermanifold, Nucl. Phys. B  567 (2000) 61-86, arXiv:hep-th/9907200.
  * [13] L. Rozansky and H. Saleur, Quantum field theory for the multivariable Alexander-Conway polynomial, Nucl. Phys. B 376 (1992) 461-509.
  * [14] K. Sfetsos, Integrable interpolations: From exact CFTs to non-Abelian T-duals, Nucl. Phys. B 880 (2014) 225-246, arXiv:1312.4560.
  * [15] C. Klimcik, On integrability of the Yang-Baxter sigma-model, J. Math. Phys. 50 (2009) 043508, arXiv:0802.3518.
  * [16] 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, arXiv:1309.5850.
  * [17] C. Klimcik and P. Severa, Dual nonAbelian duality and the Drinfeld double, Phys. Lett. B 351 (1995) 455-462, hep-th/9502122.
  * [18] C. Klimcik, Poisson-Lie T duality, Nucl. Phys. B (Proc. Suppl.) 46 (1996) 116-121, hep-th/9509095.
  * [19] V. G. Drinfeld, Quantum groups, in Proc. Intern. Cong. Math., Berkeley (1986) vol. 1, Amer. Math. Soc. (1987), pp. 798-820.
  * [20] A. Eghbali and A. Rezaei-Aghdam, Poisson-Lie T-dual sigma models on supermanifolds, J. High Energy Phys. 09 (2009) 094, arXiv:0901.1592 [hep-th].
  * [21] A. Eghbali and A. Rezaei-Aghdam, String cosmology from Poisson-Lie T-dual sigma models on supermanifolds, J. High Energy Phys. 01 (2012) 151, arXiv:1107.2041 [hep-th].
  * [22] A. Alekseev, C. Klimcik and A. Tseytlin, Quantum Poisson-Lie T-duality and WZNW model, Nucl. Phys. B 458 (1996) 430-444, arXiv:hep-th/9509123.
  * [23] A. Eghbali and A. Rezaei-Aghdam, Poisson Lie symmetry and D-branes in WZW model on the Heisenberg Lie group $H_{4}$, Nucl. Phys. B 899 (2015) 165, arXiv:1506.06233 [hep-th].
  * [24] A. Eghbali, L. Mehran-nia and A. Rezaei-Aghdam, BTZ black hole from Poisson-Lie T-dualizable sigma models with spectators, Phys. Lett. B 772 (2017) 791, arXiv:1705.00458 [hep-th].
  * [25] A. Eghbali, Exact conformal field theories from mutually T-dualizable $\sigma$-models Phys. Rev. D 99 (2018) 026001, arXiv:1812.07664 [hep-th].
  * [26] A. Eghbali and A. Rezaei-Aghdam, Super Poisson-Lie symmetry of the $GL(1|1)$ WZNW model and worldsheet boundary conditions, Nucl. Phys. B 866 (2013) 26, arXiv:1207.2304 [hep-th].
  * [27] A. Eghbali and A. Rezaei-Aghdam, WZW models as mutual super Poisson-Lie T-dual sigma models, J. High Energy Phys. 07 (2013) 134, arXiv:1303.4069 [hep-th].
  * [28] A. Eghbali, Cosmological string backgrounds from super Poisson-Lie T-plurality, Nucl. Phys. B 958 (2020) 115110 , arXiv:2003.11160 [hep-th].
  * [29] Y. Sakatani, Poisson-Lie T-plurality for WZW backgrounds Prog. Theor. Exp. Phys. 103B03 (2021), arXiv:2102.01069 [hep-th].
  * [30] R. von Unge, Poisson-Lie T-plurality, J. High Energy Phys. 07 (2002) 014, arXiv:hep-th/0205245.
  * [31] D. Bielli, S. Penati, D. Sorokin and M. Wolf, Super non-Abelian T-duality, Nucl. Phys. B 983 (2022) 115904, arXiv:2112.12168 [hep-th].
  * [32] D. Bielli, Non-Abelian T-duality in superspace, Ph.D. thesis, University of Milano-Bicocca, Italy, 2023.
  * [33] A. Eghbali, M. Hosseinpour-Sadid, and A. Rezaei-Aghdam, T-duality/plurality of BTZ black hole metric coupled to two fermionic fields, arXiv:2309.14543 [hep-th].
  * [34] Y. Sakatani and Y. Satoh, On quantum Poisson-Lie T-duality of WZNW models arXiv:2311.18530 [hep-th].
  * [35] N. Andruskiewitsch, Lie superbialgebras and Poisson-Lie supergroups, Abh. Math. Sem. Univ. Hamburg 63 (1993) 147.
  * [36] C. Juszczak and J. T. Sobczyk, Classification of low dimentional Lie super-bialgebras, J. Math. Phys. 39 (1998) 4982.
  * [37] A. Eghbali, A. Rezaei-Aghdam and F. Heidarpour, Classification of two and three dimensional Lie super-bialgebras, J. Math. Phys. 51 (2010) 073503, arXiv:0901.4471 [math-ph].
  * [38] B. DeWitt, Supermanifolds, Cambridge University Press (1992).
  * [39] N. Backhouse, A classification of four-dimensional Lie superalgebras, J. Math. Phys. 19 (1978) 2400.
  * [40] A. Eghbali and A. Rezaei-Aghdam, The $gl(1|1)$ Lie superbialgebras, J. Geom. Phys.  65 (2013) 7, arXiv:1112.0652 [math-ph].
  * [41] A. Eghbali, A. Rezaei-Aghdam and F. Heidarpour, Classification of four and six dimensional Drinfel’d superdoubles, J. Math. Phys.  51 (2010) 103503, arXiv:0901.4471 [math-ph].
  * [42] A. Eghbali and A. Rezaei-Aghdam, Lie superbialgebra structures on the Lie superalgebra $({\cal C}^{3}+{\cal A})$ and deformation of related integrable Hamiltonian systems, J. Math. Phys. 58 (2017) 063514, arXiv:1606.04332 [math-ph].
  * [43] C. R. Nappi and E. Witten, A WZW model based on a non-semi-simple group, Phys. Rev. Lett. 71 (1993) 3751.