T-duality/plurality of BTZ black hole metric
coupled to two fermionic fields

Ali<EMAIL_ADDRESS>Meysam Hosseinpour-
<EMAIL_ADDRESS>Adel<EMAIL_ADDRESS>

Department of Physics, Faculty of Basic Sciences,
Azarbaijan Shahid Madani University, 53714-161, Tabriz, Iran

We ask the question of classical super (non-)Abelian T-duality for BTZ black
hole metric coupling to two fermionic fields. Our approach is based on super
Poisson-Lie (PL) T-duality in the presence of spectator fields. In order to
study the Abelian T-duality of the metric we dualize over the Abelian Lie
supergroups of the types $(1|2)$ and $(2|2)$, in such a way that it is shown
that both original and dual backgrounds of the models are conformally
invariant up to one-loop order in the presence of field strength. Then, we
study the non-Abelian T-duality of the BTZ vacuum metric coupling to two
fermionic fields. The dualizing is performed on some non-Abelian Lie
supergroups of the type $(2|2)$, in such a way that we are dealing with semi-
Abelian superdoubles which are non-isomorphic as Lie superalgebras in each of
the models. In the non-Abelian T-duality case, it is interesting to mention
that the models can be conformally invariant up to one-loop order in both
cases of the absence and presence of field strength. In addition, starting
from the decomposition of semi-Abelian Drinfeld superdoubles generated by some
of the ${\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}$ Lie superbialgebras we
study the super PL T-plurality of the BTZ vacuum metric coupled to two
fermionic fields. However, our findings are interesting in themselves, but at
a constructive level, can prompt many new insights into supergravity and
manifestly have interesting mathematical relationships with double field
theory.
---

Keywords: BTZ metric, Lie supergroup, $\sigma$-model, String duality

The paper is dedicated to the fond memory of
professor Farhad Darabi who passed away on 13 Sept 2022.

###### Contents

  1. 1 Introduction
  2. 2 A brief review of super PL T-duality with spectators
  3. 3 Abelian T-duality of BTZ metric coupled to two fermionic fields
    1. 3.1 Abelian T-duality with Abelian Lie supergroup of the type $(2|2)$
      1. 3.1.1 The original model
      2. 3.1.2 The dual model
    2. 3.2 Abelian T-duality with Abelian Lie supergroup of the type $(1|2)$
  4. 4 Non-Abelian T-duality of BTZ vacuum metric coupled to two fermionic fields
    1. 4.1 Non-Abelian target space dual starting from the $(C_{1}^{1}+A)$ Lie supergroup
    2. 4.2 Non-Abelian target space dual starting from the $(C^{3}+A)$ Lie supergroup
    3. 4.3 Non-Abelian target space dual starting from the ${C}^{3}\oplus{A}_{1,1}$ Lie supergroup
    4. 4.4 Non-Abelian target space dual starting from the ${D}^{10}_{p=\pm\frac{1}{2}}$ Lie supergroup
    5. 4.5 Non-Abelian target space dual starting from the $(2{A}_{1,1}+2{A})^{0}$ Lie supergroup
  5. 5 Super PL T-plurality of BTZ vacuum metric coupled to two fermionic fields
    1. 5.1 A brief review of super PL T-plurality
    2. 5.2 Super PL T-plurality with respect to the ${C}^{3}\oplus{A}_{1,1}$ Lie supergroup
  6. 6 Conclusion
  7. A Lie superalgebras of the type $(2|2)$
  8. B Solutions of the super Jacobi and mixed super Jacobi identities for the ${\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}$ Lie superalgebra

##  1 Introduction

The BTZ black hole, named after Banados, Teitelboim, and Zanelli (BTZ), is a
black hole solution for $2+1$-dimensional topological gravity with a negative
cosmological constant, mass, angular momentum and charge [1]. The BTZ black
hole is asymptotically anti-de Sitter rather than asymptotically flat, and has
no curvature singularity at the origin, because [2] it is a discrete quotient
of $SL(2,\mathbb{R})$. The line element for the black hole solutions is given
by

$\displaystyle ds^{2}=({M}-{r^{2}\over
l^{2}})dt^{2}-Jdtd\varphi+r^{2}d\varphi^{2}+\big{(}{r^{2}\over
l^{2}}-{M}+{J^{2}\over 4r^{2}}\big{)}^{-1}dr^{2},$ (1.1)

where the radius $l$ is related to the cosmological constant by
$l=(-\Lambda)^{-1/2}.$ Here, the coordinate $\varphi$ is periodic with period
$2\pi$. The constants of motion ${M}$ and $J$ are the mass and angular
momentum of the BTZ black hole, respectively. They are appeared due to the
time translation symmetry and rotational symmetry of the metric, corresponding
to the killing vectors ${\partial/\partial t}$ and
${\partial/\partial\varphi}$, respectively. The line element (1.1) describes a
black hole solution with outer and inner horizons at $r=r_{+}$ and $r=r_{-}$,
respectively,

$\displaystyle r_{\pm}=l\bigl{(}{{M}\over
2}\bigr{)}^{1/2}\biggl{\\{}1\pm\biggl{(}1-{{J^{2}}\over{{M^{2}}l^{2}}}\biggr{)}^{1/2}\biggr{\\}}^{1/2}.$
(1.2)

where the mass and angular momentum are related to $r_{\pm}$ by

$\displaystyle{M}=\frac{r_{+}^{2}+r_{-}^{2}}{l^{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}J=\frac{2r_{+}r_{-}}{l}.$

The region $r_{+}<r<{M}^{1/2}l$ defines an ergosphere, in which the asymptotic
timelike Killing field $\partial/{\partial t}$ becomes spacelike. The
solutions with $-1<{M}<0,\;J=0$ describe point particle sources with naked
conical singularities at $r=0$. The metric with ${M}=-{1},\;J=0$ may be
recognized as that of ordinary anti-de Sitter space; it is separated by a mass
gap from the ${M}=0,\;J=0$ “massless black hole”, whose geometry is discussed
in Ref. [2]. The vacuum state which is regarded as empty space, is obtained by
letting the horizon size go to zero. This amounts to letting ${M}\rightarrow
0$, which requires $J\rightarrow 0$. We have to notice that the metric for the
${M}=J=0$ black hole is not the same as $AdS_{3}$ metric which has negative
mass ${M}=-1$. Locally they are equivalent since there is locally only one
constant curvature metric in three dimensions. However they are inequivalent
globally. Notice that the singularity at the center of the BTZ black hole is
called a conical singularity. However, a conical singularity is not similar to
a canonical singularity because it does not cause the spacetime curvature to
diverge [3].

The BTZ black hole also arises in string theory as a near-horizon region of a
non-extremal system of 1- and 5-branes. In Ref. [4] by using a presentation of
the D-brane counting for extremal black holes, it has been shown that four-
and five-dimensional non-extremal black holes can be mapped to the BTZ black
hole (times a compact manifold) by means of dualities. A slight modification
of this black hole solution yields an exact solution to string theory [5].
There, the BTZ black hole solutions have been considered in the context of the
low energy approximation, then as the exact conformal field theory. In order
to obtain an exact solution to string theory, one must modify the BTZ black
hole solutions by adding an antisymmetric tensor field $H_{{}_{MNP}}$ (field
strength) proportional to the volume form $\varepsilon_{{}_{MNP}}$. It has
been shown that any solution to three-dimensional general relativity with
negative cosmological constant is a solution to low energy string theory with
$H_{{}_{MNP}}=2\epsilon_{{}_{MNP}}/l$, $\Lambda=-1/l^{2}$ and a dilaton field
$\Phi=0$ [5]. In particular, it was observed in [5] that, the two parameter
family of black holes (1.1) along with

$\displaystyle B_{\varphi
t}=\frac{r^{2}}{l},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Phi=0,$ (1.3)

satisfy the low energy string effective action equations. Then, it was
obtained [5] the Abelian dual of this solution by Buscher’s duality
transformations [6]. Indeed, T-duality symmetry is one of the most interesting
properties of string theory that connects seemingly different backgrounds in
which the strings can propagate. The duality is Abelian if it is constructed
on an Abelian isometry group. We refer to Abelian T-duality for stressing the
presence of global Abelian isometries in the target spaces of both the paired
$\sigma$-models, while non-Abelian T-duality [7] refers to the existence of a
non-Abelian isometry on the target space of one of the two $\sigma$-models.
Now that we are discussing the non-Abelian T-duality, it is necessary to
explain that the non-Abelian target space dual of BTZ vacuum solution was
obtained in [8] by making use of the PL T-duality approach in the presence of
spectator fields. There, it has been shown that the BTZ vacuum solution with
no horizon and no curvature singularity is related to a solution with a single
horizon and a curvature singularity. The PL T-duality was proposed by Klimcik
and Severa [9, 10] as a generalization of Abelian [6] and non-Abelian
dualities [7]. In this type of duality, the symmetry does not need to be
realized as an isometry of the initial background; moreover, $\sigma$-models
are formulated on a Drinfeld double group $D$ [11] whose Lie algebra
$\mathscr{D}$ can be decomposed into a pair of maximally isotropic sub-
algebras with respect to a non-degenerate invariant bilinear form on
$\mathscr{D}$, so that the sub-algebras are duals of each other in the usual
sense. In this case it is said that the $\sigma$-models are indeed dual in the
sense of the PL T-duality.

Generalization of T-duality to Lie supergroups and also supermanifolds was
been explored in the context of PL T-duality in [12, 13]. When we deal with a
semi-Abelian Drinfeld superdouble, that is, one of the sub-supergroups of
Drinfeld superdouble is considered to be Abelian, the super PL T-duality
reduces to the super non-Abelian T-duality. Recently, it has been considered
the non-Abelian T-duality of $\sigma$-models on group manifolds, symmetric,
and semi-symmetric spaces with the emphasis on the T-dualization of
$\sigma$-models whose target spaces are supermanifolds [14] (see also [15]).
There, it has been performed super non-Abelian T-dualization of the principal
chiral model based on the $OSP(1|2)$ Lie supergroup.

The first goal of this paper is to find (Abelian)non-Abelian target space
duals of the BTZ metric when is coupled to two fermionic fields. In this
regards, the target spaces of T-dual $\sigma$-models are considered to be the
supermanifolds ${\mathscr{M}}\approx{O}\times{G}$ and
${\tilde{\mathscr{M}}}\approx{O}\times{\tilde{G}}$ where ${G}$ and
${\tilde{G}}$ are (Abelian)non-Abelian Lie supergroups of the type $(2|2)$
acting freely on ${\mathscr{M}}$ and ${\tilde{\mathscr{M}}}$, respectively,
while ${O}$ is the orbit of $G$ in ${\mathscr{M}}$. In the Abelian T-duality
case, we also dualize over the Abelian Lie supergroup of the type $(1|2)$. The
indecomposable four-dimensional Lie superalgebras of the type $(2|2)$ was
first classified by Backhouse in [16]. Unfortunately, there are some omissions
and redundancies in that paper, probably because the author in some cases has
not taken into account all isomorphisms to reduce the list of superalgebras.
We have made some modifications to the Backhouse’s classification; the detail
of the calculations are given in Appendix A. In addition, one can find a
classification of decomposable Lie superalgebras of the type $(2|2)$ in [17].
Based on the (anti-)commutation relations between bases of boson-boson (B-B),
boson-fermion (B-F) and fermion-fermion (F-F) we have classified all
decomposable and indecomposable Lie superalgebras of the type $(2|2)$ into six
disjoint families in Appendix A. Finally, we find super PL T-plurals of the
BTZ vacuum metric coupled to two fermionic fields with respect to the
${C}^{3}\oplus{A}_{1,1}$ Lie supergroup. The general procedure that we shall
apply is a straightforward generalization of the well-known prescription of
purely bosonic PL T-plurality [18] to Lie supergroup case, and we shall refer
to it as super PL T-plurality in this work. Notice that the generalization of
PL T-plurality formulation to Lie supergroup case has been recently explored
in [19].

The outline of this paper is as follows: in Section 2 we introduce our
notations and recall the origin of the super PL T-duality transformations in
the presence of spectator fields at the level of the $\sigma$-model. Abelian
and non-Abelian target space duals of the BTZ metric coupled to two fermionic
fields are investigated in Sections 3 and 4, respectively; the results of non-
Abelian T-duality of the models are summarized in Table 1 at the end of
Section 4. In Section 5, we study super PL T-plurality of the BTZ vacuum
metric coupled to two fermionic fields, in such a way that we start from the
decompositions of semi-Abelian Drinfeld superdoubles generated by the
${\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}$ Lie superbialgebras. Finally, in
Section 6, we present our conclusions and sketch possible developments of this
work. The classification of decomposable and indecomposable Lie superalgebras
of the type $(2|2)$ is left to Appendix A. Appendix B is also devoted to
solutions of the super Jacobi and mixed super Jacobi identities for the
${\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}$ Lie superalgebra.

##  2 A brief review of super PL T-duality with spectators

Before proceeding to review the super PL T-duality on supermanifolds, let us
recall the properties of $\mathbb{Z}_{2}$-graded vector space and also some
definitions related to Lie superalgebras [20]. A super vector space
$\mathbb{V}$ is a ${\mathbb{Z}}_{2}$-graded vector space, i.e., a vector space
over a field $\mathbb{K}$ with a given decomposition of sub-spaces of grade 0
and grade 1, $\mathbb{V}=\mathbb{V}_{{}_{0}}\oplus\mathbb{V}_{{}_{1}}$. The
parity of a nonzero homogeneous element, denoted by $|x|$, is $0$ (even) or
$1$ (odd)444The even elements are sometimes called bosonic, and the odd
elements fermionic. From now on, we use $B$ and $F$ instead of $0$ and $1$,
respectively. according to whether it is in $\mathbb{V}_{0}$ or
$\mathbb{V}_{1}$, namely, $|x|=0$ for any $x\in\mathbb{V}_{{}_{0}}$, while for
any $x\in\mathbb{V}_{{}_{1}}$ we have $|x|=1$. A Lie superalgebra
${\mathscr{G}}$ is a $\mathbb{Z}_{2}$-graded vector space, thus admitting the
decomposition
${\mathscr{G}}={\mathscr{G}}_{{{}_{B}}}\oplus{\mathscr{G}}_{{}_{F}}$, equipped
with a bilinear superbracket structure
$[.,.]:{\mathscr{G}}\otimes{\mathscr{G}}\rightarrow{\mathscr{G}}$ satisfying
the requirements of anti-supersymmetry and super Jacobi identity. If
${\mathscr{G}}$ is finite-dimensional and the dimensions of
${\mathscr{G}}_{{}_{B}}$ and ${\mathscr{G}}_{{}_{F}}$ are $m=\\#B$ and
$n=\\#F$, respectively, then ${\mathscr{G}}$ is said to have dimension
$(m|n)$. We shall identify grading indices by the same indices in the power of
$(-1)$, i.e., we use $(-1)^{x}$ instead of $(-1)^{|x|}$, where $(-1)^{x}$
equals 1 or -1 if the Lie sub-superalgebra element is even or odd,
respectively555 Note that this notation was first used by DeWitt in [21].
Throughout this paper we work with DeWitt’s notation..

Since our main discussion will be related to super PL T-duality, it is worth
mentioning that super PL T-duality is based on the concepts of Drinfeld
superdouble [20, 22, 23], so it is necessary to define Drinfeld superdouble
supergroup $D$. Following [11], a Drinfeld superdouble is simply a Lie
supergroup $D$ whose Lie superalgebra $\mathscr{D}$ admits a decomposition
$\mathscr{D}=\mathscr{G}\oplus{\tilde{\mathscr{G}}}$ into a pair of sub-
superalgebras maximally isotropic with respect to a supersymmetric ad-
invariant non-degenerate bilinear form $<.~{},~{}.>$. The dimension of sub-
superalgebras have to be equal. We furthermore consider $G$ and $\tilde{G}$ as
a pair of maximally isotropic sub-supergroups corresponding to the sub-
superalgebras $\mathscr{G}$ and $\tilde{\mathscr{G}}$, and choose a basis in
each of the sub-superalgebras as $T_{{{}_{a}}}\in\mathscr{G}$ and
${\tilde{T}}^{{{}^{a}}}\in{\tilde{\mathscr{G}}},a=1,...,dim~{}G$, such that

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

The basis of the two sub-superalgebras satisfy the commutation relations

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

where ${f^{c}}_{ab}$ and $\tilde{f}^{ab}_{~{}~{}c}$ are structure constants of
$\mathscr{G}$ and $\tilde{\mathscr{G}}$, respectively. Noted that the Lie
superalgebra structure defined by relation (2.2) is called Drinfeld
superdouble $\mathscr{D}$. The super Jacobi identity on $\mathscr{D}$ relates
the structure constants of the two Lie superalgebras as [22]

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

To continue, let us now consider a two-dimensional non-linear $\sigma$-model
for the $d$ field variables $X^{{}^{M}}=(y^{i},x^{\mu})$, where
$x^{\mu},~{}\mu=1,\cdots,dim\hskip 1.13809ptG$ stand for the coordinates of
Lie supergroup $G$ acting freely from right on the target supermanifold
$\mathscr{M}\approx O\times G$, and $y^{i},~{}i=1,\cdots,d-dim\hskip
1.13809ptG$ are the coordinates labeling the orbit $O$ of $G$ in
${\mathscr{M}}$. Here we work in the standard light-cone variables on the
worldsheet ${\Sigma}$, $\sigma^{\pm}=(\tau\pm\sigma)/{2}$ together with
$\partial_{\pm}=\partial_{\tau}\pm\partial_{\sigma}$. It should be remarked
that the coordinates $y^{i}$ do not participate in the PL T-duality
transformations and are therefore called spectators [24]. The corresponding
action has the form666As we will see, the spectators are considered to be
bosonic (even) coordinates, therefore $(-1)^{i}$ equals 1. [13] (see also [9,
10, 24, 25])

$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{{}_{\Sigma}}d\sigma^{+}d\sigma^{-}\Big{[}(-1)^{a}R_{+}^{a}~{}{{E}_{{}_{ab}}}(g,y^{i})~{}R_{-}^{b}+(-1)^{a}R_{+}^{a}~{}\phi^{{(1)}}_{aj}(g,y^{i})\partial_{-}y^{j}$
(2.4)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+(-1)^{i}\partial_{+}y^{i}\phi^{{(2)}}_{ib}(g,y^{i})R_{-}^{b}+(-1)^{i}~{}\partial_{+}y^{i}\phi_{{}_{ij}}(g,y^{i})\partial_{-}y^{j}\Big{]}.$

where $R_{\pm}^{a}$ are the components of the right-invariant Maurer-Cartan
super one-forms which are constructed by means of an element $g$ of ${G}$ as
follows:

$\displaystyle
R_{\pm}=(-1)^{a}R_{\pm}^{a}~{}T_{a}=(-1)^{a}(\partial_{\pm}gg^{-1})^{a}~{}T_{a},$
(2.5)

in which ${T}_{a}$, $a=1,\cdots,dim~{}G$ are the bases of the Lie superalgebra
${\mathscr{G}}$ of ${G}$. The couplings
${{E}_{{}_{ab}}},\phi^{{(1)}}_{aj},\phi^{{(2)}}_{ib}$ and $\phi_{{}_{ij}}$ may
depend on all variables $x^{\mu}$ and $y^{i}$.

Similarly we introduce another $\sigma$-model for the $d$ field variables
${\tilde{X}}^{{}^{M}}=({\tilde{x}}^{\mu},y^{i})$, where ${\tilde{x}}^{\mu}$’s,
$\mu=1,\cdots,dim\hskip 1.13809ptG$ parameterize an element ${\tilde{g}}$ of
the dual Lie supergroup ${\tilde{G}}$, whose dimension is, however, equal to
that of $G$, and the rest of the variables are the same $y^{i}$’s used in
(2.4). Accordingly, we introduce a different set of bases ${{\tilde{T}}^{a}}$
of the Lie algebra ${\tilde{\mathscr{G}}}$, with $a=1,2,\cdots,dim\hskip
1.13809ptG$. We furthermore consider the components of the right-invariant
Maurer-Cartan super one-forms on ${\tilde{G}}$ in the following form

$\displaystyle(\partial_{\pm}\tilde{g}\tilde{g}^{-1})_{a}={\tilde{R}}_{{\pm}_{a}}=(-1)^{\mu}\partial_{\pm}{\tilde{x}}^{\mu}{\tilde{R}}_{\mu
a}.$ (2.6)

In this case, the corresponding action has the form

$\displaystyle\tilde{S}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{{}_{\Sigma}}d\sigma^{+}d\sigma^{-}\Big{[}(-1)^{b}{\tilde{R}}_{+_{a}}{{{\tilde{E}}}^{{ab}}}({\tilde{g}},{y}^{i})~{}{\tilde{R}}_{-_{b}}+{\tilde{R}}_{+_{a}}{\tilde{\phi}}^{{(1)^{a}}}_{~{}~{}~{}j}({\tilde{g}},{y}^{i})~{}\partial_{-}y^{j}$
(2.7)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+(-1)^{i}\partial_{+}y^{i}{\tilde{\phi}}^{{(2)^{b}}}_{i}({\tilde{g}},{y}^{i})~{}{\tilde{R}}_{-_{b}}+(-1)^{i}\partial_{+}y^{i}{\tilde{\phi}}_{{}_{ij}}({\tilde{g}},{y}^{i})\partial_{-}y^{j}\Big{]}.$

Notice that here one does not require any isometry associated with the Lie
supergroups $G$ and $\tilde{G}$. The $\sigma$-models (2.4) and (2.7) will be
dual to each other in the sense of super PL T-duality [12, 13] if the
associated Lie superalgebras $\mathscr{G}$ and ${\tilde{\mathscr{G}}}$ form a
Drinfeld superdouble which can be decomposed.

There remains to relate the couplings
${{E}_{{}_{ab}}},\phi^{{(1)}}_{aj},\phi^{{(2)}}_{ib}$ and $\phi_{{}_{ij}}$ in
(2.4) to
${{{\tilde{E}}}^{{ab}}},{\tilde{\phi}}^{{(1)^{a}}}_{~{}~{}~{}j},{\tilde{\phi}}^{{(2)^{b}}}_{i}$
and ${\tilde{\phi}}_{{}_{ij}}$ in (2.6). It has been shown that [13] the
various couplings in the $\sigma$-model action (2.4) are restricted to be

$\displaystyle{E_{{}_{ab}}}$ $\displaystyle=$
$\displaystyle{\big{(}E^{{-1}}_{0}+\Pi\big{)}_{{ab}}}^{-1},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\phi^{{(1)}}_{ai}=(-1)^{c}~{}{E_{{}_{ab}}}~{}({E^{{-1}}_{0}})^{bc}~{}{F_{~{}ci}}^{{}^{(1)}},$
$\displaystyle\phi^{{(2)}}_{ib}$ $\displaystyle=$
$\displaystyle(-1)^{c}~{}{F_{~{}ia}}^{{}^{(2)}}~{}({E^{{-1}}_{0}})^{ac}~{}{E_{{}_{cb}}},~{}~{}\phi_{{}_{ij}}=F_{{}_{ij}}-(-1)^{e+c}~{}{F_{~{}ia}}^{{}^{(2)}}~{}{\Pi}^{ac}{E_{{}_{cd}}}~{}({E^{{-1}}_{0}})^{de}~{}{F_{~{}ej}}^{{}^{(1)}},~{}~{}~{}$
(2.8)

where the new couplings $E_{0},F^{{}^{(1)}},F^{{}^{(2)}}$ and $F$ maybe
functions of the spectator variables $y^{i}$ only. In equation (2), the
Poisson superbracket $\Pi(g)$ is
$\Pi^{{}^{ab}}(g)=(-1)^{c}~{}b^{{}^{ac}}(g)~{}(a^{-1})_{{}_{c}}^{{}^{~{}b}}(g)$
so that sub-matrices $a_{{}_{a}}^{{}^{~{}b}}(g)$ and $b^{{}^{ab}}(g)$ are
defined in the following way

$\displaystyle g^{-1}T_{{{}_{a}}}~{}g$ $\displaystyle=$
$\displaystyle(-1)^{b}~{}a_{{}_{a}}^{{}^{~{}b}}(g)~{}T_{{{}_{b}}},$
$\displaystyle g^{-1}{\tilde{T}}^{{{}^{a}}}g$ $\displaystyle=$
$\displaystyle(-1)^{c}~{}b^{{}^{ac}}(g)~{}T_{{{}_{c}}}+{(a^{-st})^{{}^{a}}}_{{}_{b}}(g)~{}{\tilde{T}}^{{{}^{b}}}.$
(2.9)

Finally, the relation of the dual action couplings to those of the original
one is given by [13]

$\displaystyle{{\tilde{E}}^{{}^{ab}}}$ $\displaystyle=$
$\displaystyle{\big{(}E_{0}+{\tilde{\Pi}}\big{)}^{-1}}^{ab},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\tilde{\phi}}^{{(1)^{a}}}_{~{}~{}~{}j}=(-1)^{b}~{}{{\tilde{E}}^{{}^{ab}}}~{}{F_{~{}bj}}^{{}^{(1)}},$
$\displaystyle{\tilde{\phi}}^{{(2)^{b}}}_{i}$ $\displaystyle=$
$\displaystyle-{F_{~{}ia}}^{{}^{(2)}}~{}{{\tilde{E}}^{{}^{ab}}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\tilde{\phi}}_{{}_{ij}}=F_{{}_{ij}}-(-1)^{b}~{}{F_{~{}ia}}^{{}^{(2)}}~{}{{\tilde{E}}^{{}^{ab}}}~{}{F_{~{}bj}}^{{}^{(1)}}.$
(2.10)

Analogously, one can define ${\tilde{\Pi}}(\tilde{g})$ and sub-matrices
${\tilde{a}}(\tilde{g}),{\tilde{b}}(\tilde{g})$ by just replacing the untilded
symbols by tilded ones.

Equivalently the action (2.4) can be expressed as

$\displaystyle
S=\frac{1}{2}\int_{{}_{\Sigma}}d\sigma^{+}d\sigma^{-}(-1)^{M}~{}\partial_{+}X^{M}~{}{\cal
E}_{{}_{MN}}(X)~{}\partial_{-}X^{N},$ (2.11)

where ${\cal E}_{{}_{MN}}(X)$ as a second order tensor field on the
supermanifold $\mathscr{M}$ is a composition of the supersymmetric metric
${G}_{{}_{MN}}(X)$ and the anti-supersymmetric torsion potential
${B}_{{}_{MN}}(X)$ ($B$-field)777Note that $G_{{}_{MN}}$ and $B_{{}_{MN}}$ are
the components of the supersymmetric metric $G$ and the anti-supersymmetric
tensor field $B$, respectively,
$G_{{}_{MN}}\;=\;(-1)^{{}^{MN}}\;G_{{}_{NM}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}B_{{}_{MN}}\;=\;-(-1)^{{}^{MN}}\;B_{{}_{NM}}.$
We will assume that the metric ${{}_{{}_{M}}G_{{}_{N}}}$ is superinvertible
and its superinverse is shown by $G^{{}^{MN}}$.. In the absence of spectator
fields, the ${\cal E}_{{}_{MN}}(X)$ reduces to ${\cal E}_{{}_{\mu\nu}}(x)$.
The relationship between ${\cal E}_{{}_{\mu\nu}}(x)$ and ${{E}_{{}_{ab}}}(g)$
is given by the formula

$\displaystyle{\cal
E}_{{}_{\mu\nu}}(x)=(-1)^{a}~{}{R_{{}_{\mu}}}^{~{}a}~{}{{E}_{{}_{ab}}}(g)~{}{{(R^{st})}^{b}}_{{}_{\nu}},$
(2.12)

where the superscript “$st$” stands for supertransposition of the matrix. The
condition of super PL symmetry of $\sigma$-models on the level of the
Lagrangian is given by the formula [12]

$\displaystyle{\cal L}_{{}_{V_{{}_{a}}}}{\cal
E}_{{}_{\mu\nu}}\;=\;(-1)^{{}^{a+\mu a+\rho
c+\sigma}}~{}{{\tilde{f}}^{bc}}_{~{}~{}a}~{}{\cal
E}_{{}_{\mu\rho}}~{}{V_{{}_{c}}}^{{}^{~{}\rho}}~{}{V_{{}_{b}}}^{{}^{~{}\sigma}}~{}{\cal
E}_{{}_{\sigma\nu}},$ (2.13)

where ${{\tilde{f}}^{bc}}_{~{}~{}a}$ are structure coefficients of the dual
Lie superalgebra $\mathscr{G}$ and ${V_{{}_{a}}}^{{}^{~{}\mu}}$ are the
components of left-invariant supervector fields on the Lie supergroup $G$. The
superalgebras $\mathscr{G}$ and ${\tilde{\mathscr{G}}}$ then define the
Drinfeld superdouble that enables to construct tensor ${\cal E}_{{}_{\mu\nu}}$
satisfying (2.13).

##  3 Abelian T-duality of BTZ metric coupled to two fermionic fields

###  3.1 Abelian T-duality with Abelian Lie supergroup of the type $(2|2)$

In this subsection we construct explicitly a pair of Abelian T-dual
$\sigma$-models on $4+1$-dimensional supermanifolds ${\mathscr{M}}$ and
$\tilde{\mathscr{M}}$ as the target superspaces. The original model is built
on the supermanifold ${\mathscr{M}}\approx{O}\times{G}$, where
${G}=I_{{}_{(2|2)}}$ is a four-dimensional Abelian Lie supergroup of the type
$(2|2)$ acting freely on ${\mathscr{M}}$ while ${O}$ is the orbit of $G$ in
${\mathscr{M}}$. The target superspace of the dual model is the supermanifold
$\tilde{\mathscr{M}}\approx{O}\times\tilde{G}$, where ${\tilde{G}}$ is
identical to ${G}$ acting freely on $\tilde{\mathscr{M}}$. In this case, both
the Lie supergroups $G$ and $\tilde{G}$ are Abelian, so we are dealing with a
Abelian Drinfeld superdoubles of the type $(4|4)$. Accordingly, by using (2)
we conclude that both the Poisson superbracket $\Pi^{ab}(g)$ and
${\tilde{\Pi}}_{ab}{(\tilde{g})}$ are vanished. In what follows we shall show
that the original model describes a string propagating in a space with the BTZ
black hole metric coupled to two fermionic fields.

####  3.1.1 The original model

In order to construct the original $\sigma$-model on the
${\mathscr{M}}\approx{O}\times{G}$ we assume that the Abelian Lie supergroup
$G$ is parameterized by an element

$\displaystyle g=e^{tT_{{}_{1}}}e^{\varphi T_{{}_{2}}}e^{\psi
T_{{}_{3}}}e^{\chi T_{{}_{4}}},$ (3.1)

where $(t,\varphi)$ and $(\psi,\chi)$ are the bosonic and fermionic
coordinates, respectively; moreover, $\\{T_{{}_{1}},T_{{}_{2}}\\}$ and
$\\{T_{{}_{3}},T_{{}_{4}}\\}$ stand for the the bosonic and fermionic basis of
Lie superalgebra $\mathscr{G}$ of $G$, respectively. Here, the coordinate of
orbit ${O}$ is represented by one spectator field $y^{i}=\\{r\\}$. So, the
coordinates of ${\mathscr{M}}$ are represented by
$X^{{}^{M}}=(r,x^{a})=(r,t,\varphi;\psi,\chi)$. In this case, the components
of the right-invariant super one-forms are simply obtained to be
$R_{\pm}^{a}=\partial_{\pm}x^{a}$. Now, we need to determine the coupling
matrices of the model. Let us choose the spectator-dependent matrices in the
following form

$\displaystyle{E_{0}}_{{}_{ab}}$ $\displaystyle=$
$\displaystyle\left(\begin{tabular}[]{cccc}$M-\frac{r^{2}}{l^{2}}$&$-\frac{J}{2}-k(r)$&0&0\\\
$-\frac{J}{2}+k(r)$&$r^{2}$&0&0\\\ 0&0&$h_{1}(r)$&$F(r)$\\\
0&0&-$F(r)$&$h_{2}(r)$\\\ \end{tabular}\right),$ (3.6) $\displaystyle F_{ij}$
$\displaystyle=$ $\displaystyle\big{(}{r^{2}\over l^{2}}-{M}+{J^{2}\over
4r^{2}}\big{)}^{-1},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}F^{(1)}_{ai}=F^{(2)}_{ia}=0,$
(3.7)

where $k(r),h_{1}(r),h_{2}(r)$ and $F(r)$ are some arbitrary functions that
depend on the spectator field $\\{r\\}$ only. As mentioned above, since the
$G$ is Abelian, the Poisson superbracket $\Pi^{ab}(g)$ on $G$ is zero;
consequently, the various couplings in the action (2.4) can be reduced to
${{E}}=E_{0},\phi^{{(1)}}=F^{{}^{(1)}},\phi^{{(2)}}=F^{{}^{(2)}}$ and
$\phi=F$. Thus, by making use of the (2.4) and also the above results, the
action of original $\sigma$-model is worked out to be

$\displaystyle S$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int
d\sigma^{+}d\sigma^{-}\Big{[}({M}-{r^{2}\over
l^{2}})\partial_{+}t~{}\partial_{-}t-\big{(}{J\over{2}}+k(r)\big{)}\partial_{+}t~{}\partial_{-}{\phi}-\big{(}{J\over{2}}-k(r)\big{)}\partial_{+}{\phi}~{}\partial_{-}t~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(3.8)
$\displaystyle+~{}{r^{2}}~{}\partial_{+}{\phi}~{}\partial_{-}{\phi}-h_{1}(r)\partial_{+}{\psi}~{}\partial_{-}{\psi}-F(r)~{}\partial_{+}{\psi}~{}\partial_{-}{\chi}+F(r)~{}\partial_{+}{\chi}~{}\partial_{-}{\psi}-h_{2}(r)\partial_{+}{\chi}~{}\partial_{-}{\chi}$
$\displaystyle+~{}({r^{2}\over l^{2}}-{M}+{J^{2}\over
4r^{2}})^{-1}~{}\partial_{+}r~{}\partial_{-}r\Big{]}.$

By considering the definition of line element and $B$-field on a supermanifold
as

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle(-1)^{{}^{MN}}~{}G_{{}_{MN}}~{}dX^{M}~{}dX^{N},$ (3.9)
$\displaystyle B$ $\displaystyle=$
$\displaystyle\frac{1}{2}(-1)^{{}^{MN}}~{}B_{{}_{MN}}~{}dX^{M}\wedge dX^{N},$
(3.10)

and then by comparing the actions (3.8) and (2.11) one gets

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle\big{(}{r^{2}\over
l^{2}}-M+{J^{2}\over 4r^{2}}\big{)}^{-1}dr^{2}+(M-{r^{2}\over
l^{2}})dt^{2}-Jdtd\phi+r^{2}d\phi^{2}-2F(r)d\psi d\chi,$ (3.11) $\displaystyle
B$ $\displaystyle=$ $\displaystyle-k(r)~{}dt\land d\phi-{1\over
2}h_{1}(r)~{}d\psi\land d\psi-{1\over 2}h_{2}(r)~{}d\chi\land d\chi.$ (3.12)

In the $\sigma$-model context, the ultraviolate finiteness of the quantum
version of the model is guaranteed by the conformal invariance of the model.
To achieve this invariance at the one-loop level we must add another term
containing the so-called dilaton field to the Lagrangian of action (2.11)888In
the super Abelian T-duality case, the formula of dual dilaton transformation
is given by
$\displaystyle{\tilde{\Phi}}=\phi^{{}^{(0)}}-\frac{1}{2}\log|s\det({{{}_{a}}({E_{0}})_{b}})|,$
(3.13) where “$s\det$” stands for superdeterminant of the matrix; moreover,
$\phi^{{}^{(0)}}$ is the dilaton that makes the original $\sigma$-model
conformal up to the one-loop order and may depend on both supergroup and
spectator coordinates.. The dilaton field $\Phi$ can be understood as an
additional function on $\mathscr{M}$ that defines the quantum non-linear
$\sigma$-model and couples to scalar curvature of the worldsheet. The
conformal invariance of the model is guaranteed by vanishing of the so-called
beta-function. At the one-loop level the equations for vanishing of the beta-
function on a supermanifold read [13]999 _Notation:_ Suppose that $f$ be a
differentiable function on ${\mathbf{R}}_{c}^{m}\times{\mathbf{R}}_{a}^{n}$
(${\mathbf{R}}_{c}^{m}$ are subset of all real numbers with dimension $m$
while ${\mathbf{R}}_{a}^{n}$ are subset of all odd Grassmann variables with
dimension $n$), then, the relation between the left partial differentiation
and right one is given by
${{}_{{}_{M}}{\overrightarrow{\partial}}}f:=\frac{\overrightarrow{\partial}f}{{\partial}X^{{}^{M}}}\;=\;(-1)^{M(|f|+1)}\;\frac{f\overleftarrow{\partial}}{{\partial}X^{{}^{M}}},$
in which $|f|$ stands for the grading of $f$ [21].

$\displaystyle{\cal
R}_{{}_{MN}}+\frac{1}{4}H_{{}_{MQP}}{H^{{}^{PQ}}}_{{}_{N}}+2{\overrightarrow{\nabla}_{{}_{M}}}{\overrightarrow{\nabla}_{{}_{N}}}\Phi~{}=~{}0,$
(3.14)
$\displaystyle(-1)^{{}^{P}}{\overrightarrow{\nabla}}^{P}\big{(}e^{{}^{-2\Phi}}H_{{}_{PMN}}\big{)}~{}=~{}0,$
(3.15) $\displaystyle 4\Lambda-{\cal
R}-\frac{1}{12}H_{{}_{MNP}}H^{{}^{PNM}}+4{\overrightarrow{\nabla}_{{}_{M}}}{\Phi}\overrightarrow{\nabla}^{{}^{M}}{\Phi}-4{\overrightarrow{\nabla}_{{}_{M}}}\overrightarrow{\nabla}^{{}^{M}}{\Phi}=0,$
(3.16)

where the covariant derivatives ${\overrightarrow{\nabla}_{{}_{M}}}$, Ricci
tensor ${\cal R}_{{}_{MN}}$ and scalar curvature ${\cal R}$ 101010One may
refer to Appendix A of Ref. [19] to find a few relevant details concerning
properties of matrices and tensors on supervector space which feature in the
main text, appear as supertranspose, superdeterminant and supertrace, as well
as the definitions of connection, Riemann tensor field, Ricci tensor and
scalar curvature on supermanifolds (for more details, refer to DeWitt’s book
[21]). are calculated from the metric $~{}G_{{}_{MN}}$ that is also used for
lowering and raising indices, and field strength (torsion) corresponding to
the $B$-field is defined by

$\displaystyle
H_{{}_{MNP}}=(-1)^{{}^{M}}\;\frac{\overrightarrow{\partial}}{\partial
x^{{}^{M}}}B_{{}_{NP}}+(-1)^{{}^{N+M(N+P)}}\;\frac{\overrightarrow{\partial}}{\partial
x^{{}^{N}}}B_{{}_{PM}}+(-1)^{{}^{P(1+M+N)}}\;\frac{\overrightarrow{\partial}}{\partial
x^{{}^{P}}}B_{{}_{MN}}.$ (3.17)

Now we want to obtain conditions under which the background given by (3.11)
and (3.12) satisfies the beta-function equations (3.14)-(3.16). These
equations possess a solution with the metric (3.11) and $B$-field (3.12) and
also a constant dilaton field $\Phi=\phi_{{}_{0}}$ if the following conditions
hold:

$\displaystyle F(r)$ $\displaystyle=$ $\displaystyle
F_{{}_{0}},~{}~{}~{}~{}~{}~{}~{}k(r)={r^{2}\over
l},~{}~{}~{}~{}~{}~{}~{}h_{2}(r)=0,$ $\displaystyle h_{1}(r)$ $\displaystyle=$
$\displaystyle\begin{cases}\frac{-{\cal
C}_{0}}{8(r^{2}-\frac{1}{2}Ml^{2})}~{}~{}~{}~{}~{}~{}&~{}~{}~{}~{}~{}~{}J=Ml,{}{}{}{}{}\\\
\frac{{\cal
C}_{0}}{4l\sqrt{M^{2}l^{2}-J^{2}}}\tanh^{-1}\Big{(}\frac{Ml^{2}-2r^{2}}{l\sqrt{M^{2}l^{2}-J^{2}}}\Big{)}~{}~{}~{}~{}~{}~{}&~{}~{}~{}~{}~{}~{}J^{2}<M^{2}l^{2},\\\
{{{\cal C}_{0}}\over
4l\sqrt{J^{2}-M^{2}l^{2}}}\tan^{-1}\Big{(}\frac{2r^{2}-Ml^{2}}{l\sqrt{J^{2}-M^{2}l^{2}}}\Big{)}~{}~{}~{}~{}~{}~{}&~{}~{}~{}~{}~{}~{}J^{2}>M^{2}l^{2},\end{cases}$
(3.18)

for some constant ${{\cal C}_{0}}$. Notice that here the cosmological constant
is obtained to be $\Lambda=-1/{l^{2}}$. Taking into account the above results,
the scalar curvature of the metric (3.11) is ${\cal R}=-6/l^{2}$, and this is
exactly the same as the case where fermion fields are absent. Furthermore, for
the $B$-field, the corresponding field strength is given by

$\displaystyle H=-\frac{2r}{l}~{}dr\wedge dt\wedge
d\phi+H_{{}_{144}}~{}dr\wedge d\psi\wedge d\psi,$ (3.19)

where

$\displaystyle
H_{{}_{144}}=\begin{cases}\frac{C_{0}r}{(2r^{2}-Ml^{2})^{2}}~{}~{}~{}~{}~{}~{}&~{}~{}~{}~{}~{}~{}J=Ml,{}{}{}{}{}\\\
\frac{C_{0}r}{J^{2}l^{2}-4Ml^{2}r^{2}+4r^{4}}~{}~{}~{}~{}~{}~{}&~{}~{}~{}~{}~{}~{}J^{2}<M^{2}l^{2},~{}J^{2}>M^{2}l^{2}.\\\
\end{cases}$ (3.20)

####  3.1.2 The dual model

In the same way, to construct out the dual $\sigma$-model on the supermanifold
$\tilde{{\mathscr{M}}}\approx O\times\tilde{G}$ we parameterize Abelian Lie
supergroup $\tilde{G}$ with the bosonic coordinates
$(\tilde{t},\tilde{\varphi})$ and fermionic ones $(\tilde{\psi},\tilde{\chi})$
so that its element is defined as (3.1) by replacing untilded quantities with
tilded ones. In order to obtain the dual $\sigma$-model to (3.8), we use the
action (2.7). By taking into account (3.1.1) for (3.7) and then by inserting
the result into (2.10) one can obtain the dual coupling matrices. They are
then read

$\displaystyle{\tilde{E}}^{{ab}}$ $\displaystyle=$
$\displaystyle\left(\begin{tabular}[]{cccc}$\frac{4r^{2}}{4Mr^{2}-J^{2}}$&$\frac{2Jl+4r^{2}}{l(4Mr^{2}-J^{2})}$&0&0\\\
$\frac{2Jl-4r^{2}}{l(4Mr^{2}-J^{2})}$&$\frac{4(Ml^{2}-r^{2})}{l^{2}(4Mr^{2}-J^{2})}$&0&0\\\
0&0&$0$&$\frac{-1}{F_{0}}$\\\
0&0&$\frac{1}{F_{{}_{0}}}$&$\frac{h_{1}(r)}{F_{{}_{0}}^{2}}$\\\
\end{tabular}\right),$ (3.25) $\displaystyle{\tilde{\phi}}_{{}_{ij}}$
$\displaystyle=$ $\displaystyle\big{(}{r^{2}\over l^{2}}-{M}+{J^{2}\over
4r^{2}}\big{)}^{-1},~{}~{}~{}~{}~{}~{}~{}~{}~{}{\tilde{\phi}}^{{(1)^{a}}}_{~{}~{}~{}j}={\tilde{\phi}}^{{(2)^{b}}}_{i}=0,$
(3.26)

where $h_{1}(r)$ is given by relation (3.1.1). Finally, by using (3.26) and
(2.7) one can construct out the action of dual model. The line element and
anti-supersymmetric field corresponding to the dual action may be expressed as

$\displaystyle d{\tilde{s}}^{2}$ $\displaystyle=$
$\displaystyle\big{(}{r^{2}\over l^{2}}-M+{J^{2}\over
4r^{2}}\big{)}^{-1}dr^{2}+\big{(}\frac{1}{Mr^{2}-\frac{J^{2}}{4}}\big{)}\Big{[}r^{2}d\tilde{t}^{2}+Jd\tilde{t}d\tilde{\phi}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+(M-\frac{r^{2}}{l^{2}})d\tilde{\phi}^{2}\Big{]}-\frac{2}{F_{0}}~{}d\tilde{\psi}d\tilde{\chi},$
$\displaystyle{\tilde{B}}$ $\displaystyle=$
$\displaystyle\frac{{4r^{2}}}{l(4Mr^{2}-{J^{2}})}~{}d\tilde{t}\land
d\tilde{\phi}-{h_{1}(r)\over{2F_{0}^{2}}}~{}d\tilde{\chi}\land d\tilde{\chi}.$
(3.27)

The components of metric are ill defined at the region
$r=\frac{J}{2}M^{\frac{-1}{2}}$. We can test whether there is true singularity
by calculating the scalar curvature, which is

$\displaystyle{\tilde{\cal
R}}~{}=~{}\frac{16M^{2}l^{2}(4Mr^{2}-{J^{2}})-2J^{2}(3J^{2}+16Mr^{2})}{l^{2}(4Mr^{2}-{J^{2}})^{2}}.$
(3.28)

It is clear that $r=\frac{J}{2}M^{\frac{-1}{2}}$ is a true curvature
singularity, so the difficult at this region cannot be removed by a change of
coordinates; moreover, the horizons of the dual metric are at the same
location as the metric of original model. We also find that the field strength
corresponding to the $\tilde{B}$-field is

$\displaystyle\tilde{H}=\frac{-8rJ^{2}}{l(J^{2}-4Mr^{2})^{2}}~{}dr\wedge
d\tilde{t}\wedge d\tilde{\phi}+\frac{H_{{}_{144}}}{F_{{}_{0}}^{2}}~{}dr\wedge
d\tilde{\chi}\wedge d\tilde{\chi},$ (3.29)

where $H_{{}_{144}}$ is given by relation (3.20). The dual dilaton field that
makes the model conformal is obtained by using equation (3.13) to be

$\displaystyle\tilde{\Phi}=\phi_{0}-{1\over 2}\log({Mr^{2}-\frac{J^{2}}{4}}).$
(3.30)

Thus, the dual metric with the field strength (3.29) and dilaton field (3.30)
satisfy one-loop beta-function equations (3.14)-(3.16) in a way that the
cosmological constant must be $\tilde{\Lambda}=-{1}/{l^{2}}$.

###  3.2 Abelian T-duality with Abelian Lie supergroup of the type $(1|2)$

In order to study Abelian T-duality of the BTZ black hole metric coupled to
two fermionic fields, there is a possibility that we can perform the dualizing
on Abelian Lie supergroups $G$ and $\tilde{G}$ of the type $(1|2)$; moreover,
the spectator fields are considered to be $y^{i}=(r,t)$. As we will see, the
original model will be the same as (3.8). Here we find a new Abelian target
space dual for the action (3.8). In fact, the result of duality will be
different from those of (3.27) and also (3.28).

In order to construct out the original $\sigma$-model, we parameterize an
element of $G$ as

$\displaystyle g=e^{\varphi T_{{}_{1}}}e^{\psi T_{{}_{2}}}e^{\chi
T_{{}_{3}}},$ (3.31)

where $\varphi$ is only the bosonic coordinate, while $(\psi,\chi)$ denote
fermionic coordinates. Also, $\\{T_{{}_{1}}\\}$ and
$\\{T_{{}_{2}},T_{{}_{3}}\\}$ are bosonic and fermionic generators of $G$,
respectively. In this case, the dualizing is performed on the coordinates
$(\varphi;\psi,\chi)$, so it is more appropriate to choose the spectator-
dependent matrices as follows:

$\displaystyle{E_{0}}_{{}_{ab}}$ $\displaystyle=$
$\displaystyle\left(\begin{tabular}[]{ccc}$r^{2}$&0&0\\\
0&$h_{1}(r)$&$F_{0}$\\\ 0&-$F_{0}$&$0$\\\
\end{tabular}\right),~{}~{}~{}~{}~{}F^{(1)}_{ai}=\left(\begin{tabular}[]{cc}0&$\frac{-J}{2}+\frac{r^{2}}{l}$\\\
$0$&$0$\\\ $0$&$0$\\\ \end{tabular}\right),$ (3.38) $\displaystyle
F^{(2)}_{ia}$ $\displaystyle=$
$\displaystyle\left(\begin{tabular}[]{ccc}0&$0$&$0$\\\
$\frac{-J}{2}+\frac{r^{2}}{l}$&$0$&$0$\\\
\end{tabular}\right),~{}~{}~{}~{}~{}~{}F_{ij}=\left(\begin{tabular}[]{cc}$(\frac{r^{2}}{l^{2}}-M+\frac{J^{2}}{4r^{2}})^{-1}$&$0$\\\
0&$M-\frac{r^{2}}{l^{2}}$\\\ \end{tabular}\right).$ (3.43)

where $h_{1}(r)$ is given by relation (3.1.1). Then, using the fact that
$\Pi(g)=0$ and utilizing formulae (2) together with (2.4), the original
$\sigma$-model is obtained to be the same as (3.8) provided that one also
considers relation (3.1.1).

The corresponding coupling matrices to the dual model can be obtained by
making use of relations (2.10) and (3.38). They are then read

$\displaystyle{\tilde{E}}^{{ab}}$ $\displaystyle=$
$\displaystyle\left(\begin{tabular}[]{ccc}$\frac{1}{r^{2}}$&0&0\\\
0&0&$\frac{1}{F_{0}}$\\\ 0&$\frac{-1}{F_{0}}$&$\frac{-h_{1}(r)}{F_{0}^{2}}$\\\
\end{tabular}\right),~{}~{}~{}~{}~{}~{}{\tilde{\phi}}^{{(1)^{a}}}_{~{}~{}~{}j}=\left(\begin{tabular}[]{cc}0&$\frac{-J}{2r^{2}}+\frac{1}{l}$\\\
$0$&$0$\\\ $0$&$0$\\\ \end{tabular}\right),$ (3.50)
$\displaystyle{\tilde{\phi}}^{{(2)^{b}}}_{i}$ $\displaystyle=$
$\displaystyle\left(\begin{tabular}[]{ccc}0&$0$&$0$\\\
$\frac{J}{2r^{2}}+\frac{1}{l}$&$0$&$0$\\\
\end{tabular}\right),~{}~{}~{}~{}~{}~{}~{}{\tilde{\phi}}_{{}_{ij}}=\left(\begin{tabular}[]{cc}$(\frac{r^{2}}{l^{2}}-M+\frac{J^{2}}{4r^{2}})^{-1}$&$0$\\\
0&$M-\frac{J^{2}}{4r^{2}}$\\\ \end{tabular}\right).$ (3.55)

Thus, employing (2.7) the dual background can be cast in the form

$\displaystyle d\tilde{s}^{2}$ $\displaystyle=$
$\displaystyle\big{(}{r^{2}\over l^{2}}-{M}+{J^{2}\over
4r^{2}}\big{)}^{-1}dr^{2}+(M-\frac{J^{2}}{4r^{2}})dt^{2}+{2\over
l}dtd\tilde{\phi}+{1\over
r^{2}}d\tilde{\phi}^{2}-\frac{2}{F_{0}}d\tilde{\psi}d\tilde{\chi},$
$\displaystyle\tilde{B}$ $\displaystyle=$
$\displaystyle\frac{J}{2r^{2}}~{}dt\land
d\tilde{\phi}+{h_{1}(r)\over{2F_{0}^{2}}}~{}d\tilde{\chi}\land d\tilde{\chi},$
(3.56)

where $h_{1}(r)$ is given by relation (3.1.1). Let us now enhance and clarify
the structure of the dual spacetime. As it is seen, the dual metric has an
apparent singularity. Calculating the scalar curvature, which is

$\displaystyle{\tilde{\cal R}}~{}=~{}\frac{8Mr^{2}-7J^{2}}{2r^{4}},$ (3.57)

one should test whether there is a true singularity, and we thus see that
$r=0$ is a true singularity. In addition to the above, one immediately
verifies the field equations (3.14)-(3.16) for the dual background with the
dilaton field $\tilde{\Phi}=\phi_{{}_{0}}-\log r$; moreover, to satisfy the
equations we must have a cosmological constant as in the original model, i.e.,
$\tilde{\Lambda}=-{1}/{l^{2}}$. Notice that the dual dilaton field satisfies
formula (3.13).

In summary, we obtained two Abelian duals for the BTZ metric coupled to two
fermionic fields. When the dualizing was implemented by the Abelian Lie
supergroup of the type $(2|2)$, we found that the singularity of the dual
metric is appeared at the region $r=\frac{J}{2}M^{\frac{-1}{2}}$, whereas when
we deal with a $(1|2)$-dimensional Abelian Lie supergroup, we encounter a
singularity at the origin. Moreover, in both cases of the dual metrics,
horizons are at the same location as the metric of original model.

##  4 Non-Abelian T-duality of BTZ vacuum metric coupled to two fermionic
fields

Our goal in this section is that to calculate non-Abelian target space duals
(here as super PL T-duality on a semi-Abelian superdouble) of the BTZ vacuum
metric when is coupled to two fermionic fields. The Lie supergroup acting
freely on the original target supermanifold of $T$-dual $\sigma$-models is
considered to be one of the non-Abelian Lie supergroups of the type $(2|2)$
whose Lie superalgebras were classified by Backhouse in [16] as mentioned in
Introduction. The classification of decomposable and indecomposable Lie
superalgebras of the type $(2|2)$ has been displayed into six disjoint
families in Tables 2 to 7 of Appendix A. In this section we perform the
duality on some Lie supergroups whose Lie superalgebras are considered to be
the $({\mathscr{C}}_{1}^{1}+{\mathscr{A}})$ of Family $I$,
$({\mathscr{C}}^{3}+{\mathscr{A}})$ of Family $II$,
${\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}$ of Family $III$,
${\mathscr{D}}^{10}_{p=\pm\frac{1}{2}}$ of Family $IV$ and
$(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})^{0}$ of Family $V$. These Lie
superalgebras possess two bosonic generators $\\{T_{{}_{1}},T_{{}_{2}}\\}$
along with two fermionic ones $\\{T_{{}_{3}},T_{{}_{4}}\\}$111111From now on
we consider $\\{T_{{}_{1}},T_{{}_{2}}\\}$ and $\\{T_{{}_{3}},T_{{}_{4}}\\}$ as
bosonic and fermionic generators, respectively.. Note that in order to satisfy
the dualizability conditions, the Lie supergroup of dual target supermanifold
must be chosen Abelian. In fact, we are dealing with semi-Abelian superdoubles
which are non-isomorphic as Lie superalgebras in each of the models.

###  4.1 Non-Abelian target space dual starting from the $(C_{1}^{1}+A)$ Lie
supergroup

In what follows we shall obtain the BTZ vacuum metric coupled to two fermionic
fields from a $T$-dualizable $\sigma$-model constructing on a five-dimensional
target supermanifold ${\mathscr{M}}\approx O\times G$ where $G$ as the first
sub-supergroup of Drinfeld superdouble is considered to be the $(C_{1}^{1}+A)$
acting freely on ${\mathscr{M}}$, while $O$ as the orbit of ${G}$ in
${\mathscr{M}}$ is a one-dimensional space with the coordinate
$y^{i}=\\{y\\}$. The second sub-supergroup, ${\tilde{G}}={I}_{{}_{(2|2)}}$,
acting freely on the dual supermanifold ${\tilde{\mathscr{M}}}\approx
O\times{\tilde{G}}$ is assumed to be Abelian of the type $(2|2)$. Hence, the
super PL T-duality reduces to the super non-Abelian T-duality. One can find
the (anti-)commutation relations of the
$({\mathscr{C}}^{1}_{1}+{\mathscr{A}})$ Lie superalgebra in Table 2 of
Appendix A. As mentioned in Introduction section, having Drinfeld superdoubles
one can construct super PL T-dual $\sigma$-models on them. The Lie
superalgebra of the Drinfeld superdouble which we refer to as
$\big{(}({\mathscr{C}}^{1}_{1}+{\mathscr{A}})\;,\;{\cal
I}_{{}_{(2|2)}}\big{)}$ is defined by the following nonzero (anti-)commutation
relations121212Notice that in Ref. [26], the equations of motion of a super
non-Abelian T-dual $\sigma$-model on the $(C_{1}^{1}+A)$ Lie supergroup in the
curved background have been explicitly solved by making use of the
transformation between the supergroup coordinates of the model living in the
flat background and its flat supercoordinates.:

$\displaystyle{[T_{{}_{1}},T_{{}_{2}}]}$ $\displaystyle=$ $\displaystyle
T_{{}_{2}},~{}~{}~{}~{}~{}~{}~{}~{}[T_{{}_{1}},T_{{}_{3}}]=T_{{}_{3}},~{}~{}~{}~{}~{}~{}~{}\\{T_{{}_{3}},T_{{}_{4}}\\}=T_{{}_{2}},$
$\displaystyle{[T_{{}_{1}},{\tilde{T}}^{2}]}$ $\displaystyle=$
$\displaystyle-{\tilde{T}}^{2},~{}~{}~{}~{}~{}[T_{{}_{1}},{\tilde{T}}^{3}]=-{\tilde{T}}^{3},~{}~{}~{}~{}[T_{{}_{2}},{\tilde{T}}^{2}]={\tilde{T}}^{1},$
$\displaystyle{[T_{{}_{3}},{\tilde{T}}^{2}]}$ $\displaystyle=$
$\displaystyle-{\tilde{T}}^{4},~{}~{}~{}~{}~{}[T_{{}_{4}},{\tilde{T}}^{2}]=-{\tilde{T}}^{3},~{}~{}~{}~{}\\{T_{{}_{3}},{\tilde{T}}^{3}\\}=-{\tilde{T}}^{1},$
(4.1)

where $\\{T_{{}_{1}},T_{{}_{2}};T_{{}_{3}},T_{{}_{4}}\\}$ generate the
$({\mathscr{C}}^{1}_{1}+{\mathscr{A}})$, and
$\\{{\tilde{T}}^{1},{\tilde{T}}^{2}\\}$ and
$\\{{\tilde{T}}^{3},{\tilde{T}}^{4}\\}$ are the respective bosonic and
fermionic generators of the ${\cal I}_{{}_{(2|2)}}$. In order to calculate the
components of the right-invariant super one-forms $R_{\pm}^{a}$ on the
$(C_{1}^{1}+A)$ Lie supergroup we parameterize an element of the supergroup as

$\displaystyle
g~{}=~{}e^{x_{{}_{1}}T_{{}_{1}}}~{}e^{x_{{}_{2}}T_{{}_{2}}}~{}e^{\psi
T_{{}_{3}}}~{}e^{\chi T_{{}_{4}}},$ (4.2)

where $(x_{{}_{1}},x_{{}_{2}})$ are bosonic fields, while $(\psi,\xi)$ stand
for fermionic ones. From now on, we will use this notation for bosonic and
fermionic fields. Now, one may use equation (2.5) to get

$\displaystyle R_{\pm}^{1}$ $\displaystyle=$
$\displaystyle\partial_{\pm}x_{{}_{1}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}R_{\pm}^{2}~{}=~{}\partial_{\pm}x_{{}_{2}}e^{x_{{}_{1}}}+\partial_{\pm}\chi~{}\psi
e^{x_{{}_{1}}},$ $\displaystyle R_{\pm}^{3}$ $\displaystyle=$
$\displaystyle-\partial_{\pm}\psi~{}e^{x_{{}_{1}}},~{}~{}~{}~{}~{}~{}~{}~{}~{}R_{\pm}^{4}~{}=~{}-\partial_{\pm}\chi.$
(4.3)

Let us now choose the spectator-dependent background matrices as

$\displaystyle{E_{0}}_{{}_{ab}}=\left(\begin{tabular}[]{cccc}$0$&$\frac{1}{2}e^{-2y}$&0&0\\\
$\frac{1}{2}e^{-2y}$&$0$&0&0\\\ 0&0&$0$&$h(y)$\\\
0&0&-$h(y)$&$\alpha_{{}_{0}}n(y)$\\\
\end{tabular}\right),~{}~{}~{}~{}~{}F^{(1)}_{ai}=F^{(2)}_{ia}=0,~{}~{}~{}~{}F_{ij}=l^{2},$
(4.8)

where $\alpha_{{}_{0}}$ is an arbitrary constant. Also, $h(y)$ and $n(y)$ are
functions of the variable $\\{y\\}$ only. In the case of the super non-Abelian
T-duality, it can be deduced from formula (2) that the Poisson superbracket
$\Pi(g)$ vanishes provided that the dual Lie supergroup is Abelian. Thus, by
using these and also employing (2.4) one can get the action of original model
such that its corresponding line element and $B$-field are given by

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
l^{2}dy^{2}+e^{{x_{{}_{1}}}-2y}~{}dx_{{}_{1}}dx_{{}_{2}}-\psi
e^{{x_{{}_{1}}}-2y}~{}dx_{{}_{1}}d\chi-2h(y)e^{x_{{}_{1}}}~{}d\psi d\chi,$
(4.9) $\displaystyle B$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\alpha_{{}_{0}}n(y)~{}d\chi\wedge d\chi.$ (4.10)

The scalar curvature of the metric is

$\displaystyle{\cal
R}=-\frac{6}{l^{2}}-\frac{1}{2l^{2}h^{2}(y)}\Big{[}8h(y)h^{\prime}(y)+5{h^{\prime}}^{2}(y)-4h(y)h^{\prime\prime}(y)\Big{]},$
(4.11)

where prime denotes differentiation with respect to the argument $\\{y\\}$. It
is straightforward to verify that the only non-zero component of the strength
field corresponding to $B$-field (4.10) is
$H{{}_{y\chi\chi}}=\alpha_{{}_{0}}n^{\prime}(y)$. We are now interested in the
satisfaction of the conformal invariance conditions of the model. In order to
guarantee the conformal invariance of the model, at least at the one-loop
level, one must show that the background of model including (4.9) and (4.10)
satisfies the vanishing of the beta-function equations. Before we proceed to
investigate these conditions further, let us briefly comment on the formula of
dilaton transformation for the case of PL T-duality. In [18], von Unge showed
that the duality transformation must be supplemented by a correction that
comes from integrating out the fields on the dual group in path-integral
formulation so that it can be absorbed at the one-loop level into the
transformation of the dilaton field. Following [18], in the super PL T-duality
case, these transformations may be expressed as

$\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\phi^{{}^{(0)}}+\frac{1}{2}\log\big{|}s\det\big{(}{{}_{{}_{a}}}{E_{{}_{b}}}(g,y^{i})\big{)}\big{|}-\frac{1}{2}\log\big{|}s\det\big{(}{{}_{{}_{a}}}{{E_{0}}_{{}_{b}}}(y^{i})\big{)}\big{|}-\frac{1}{2}\log\big{|}s\det\big{(}{{}_{{}_{a}}}{a^{{}^{b}}}{\tiny(g)}\big{)}\big{|},~{}~{}~{}$
(4.12) $\displaystyle{\tilde{\Phi}}$ $\displaystyle=$
$\displaystyle\phi^{{}^{(0)}}+\frac{1}{2}\log|s\det\big{(}{{{\tilde{E}}^{ab}}}(\tilde{g},y^{i})\big{)}|-\frac{1}{2}\log|s\det\big{(}{{{\tilde{a}}^{a}}_{{}_{~{}}b}({\tilde{g}})}\big{)}|,$
(4.13)

where $\phi^{{}^{(0)}}$ may depend on both supergroup and spectator
coordinates. Note that in the case of super non-Abelian T-duality, formula
(4.12) becomes simpler, because we have $E={E_{0}}$. In the present example,
sub-matrix $a_{{}_{a}}^{{}^{~{}b}}(g)$ can be obtained from the first equation
of (2), then, one can easily find that
$s\det\big{(}{{}_{{}_{a}}}{a^{{}^{{}^{b}}}}{\tiny(g)}\big{)}=1$. Consequently,
it follows from (4.12) that $\Phi=\phi^{{}^{(0)}}$ such that the beta-function
equations are satisfied with a constant dilaton field,
$\Phi=\phi^{{}^{(0)}}={\cal C}_{{}_{0}}$, and vanishing cosmological constant
if functions $h(y)$ and $n(y)$ obey the following relations

$\displaystyle
h(y)=-\frac{1}{2}e^{-2y},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}n(y)=1-\frac{1}{4}e^{-4y}.$
(4.14)

It can be useful to comment on the fact that the metric (4.9) along with
$h(y)$ obtained in (4.14) is flat in the sense that its scalar curvature and
Ricci tensor vanish. On the other hand, one may use the coordinate
transformation

$\displaystyle
e^{y}=\frac{l}{r},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}e^{x_{{}_{1}}}=\frac{(t-l\varphi)}{l},~{}~{}~{}~{}~{}~{}~{}~{}~{}{x_{{}_{2}}}=-l(t+l\varphi),$
(4.15)

then, the background of model, that is conformally invariant up to one-loop
order, reduces to

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\frac{l^{2}}{r^{2}}dr^{2}-\frac{r^{2}}{l^{2}}dt^{2}+r^{2}d\varphi^{2}+\frac{r^{2}}{l^{2}}\big{(}-\frac{\psi}{l}dtd\chi+\psi
d\varphi d\chi+d\psi d\chi\big{)},$ $\displaystyle H$ $\displaystyle=$
$\displaystyle-\frac{\alpha_{{}_{0}}r^{3}}{2l^{4}}~{}dr\wedge d\chi\wedge
d\chi,$ $\displaystyle\Phi$ $\displaystyle=$ $\displaystyle{\cal C}_{{}_{0}}.$
(4.16)

Indeed, the above metric is nothing but the BTZ vacuum metric which has been
coupled to two fermionic fields $(\psi,\chi)$. A remarkable point about the
metric in (4.16) is that unlike $2+1$-dimensional BTZ vacuum metric (the
$J=M=0$ case of (1.1)) which has a constant negative scalar curvature, it has
vanishing scalar curvature. In fact, by coupling the fermionic fields to the
metric, the scalar curvature undergoes a change. It is then straightforward to
compute the corresponding dual background. By considering the coupling
matrices in (4.8) and by employing (2.10) one can show that the dual couplings
take the following form

$\displaystyle{{\tilde{E}}^{{}^{ab}}}={\big{(}E_{0}+{\tilde{\Pi}}\big{)}^{-1}}^{ab},~{}~{}~{}~{}~{}~{}{\tilde{\phi}}^{{(1)^{a}}}_{~{}~{}~{}j}={\tilde{\phi}}^{{(2)^{b}}}_{i}=0,~{}~{}~{}~{}~{}~{}~{}~{}{\tilde{\phi}}_{{}_{ij}}=F_{{}_{ij}}=l^{2},$
(4.17)

in which the Poisson superbracket ${\tilde{\Pi}}_{ab}$ can be obtained from
equation (2) by replacing untilded quantities with tilded ones. To this end,
we parameterize the dual Lie supergroup ${I}_{{}_{(2|2)}}$ with coordinates
${\tilde{x}}_{a}=({\tilde{x}}_{{}_{1}},{\tilde{x}}_{{}_{2}};{\tilde{\psi}},{\tilde{\xi}})$
so that its elements are defined as in (4.2). We then find

$\displaystyle{\tilde{\Pi}}_{ab}=\left(\begin{tabular}[]{cccc}$0$&-${\tilde{x}}_{{}_{2}}$&-${\tilde{\psi}}$&0\\\
${\tilde{x}}_{{}_{2}}$&$0$&0&0\\\
${\tilde{\psi}}$&0&$0$&-${\tilde{x}}_{{}_{2}}$\\\
0&0&-${\tilde{x}}_{{}_{2}}$&0\\\ \end{tabular}\right).$ (4.22)

Putting (4.22) and $E_{0}$ from (4.8) together into (4.17) and using the fact
that the components of the right-invariant super one-forms on
${I}_{{}_{(2|2)}}$ are
${\tilde{R}}_{\pm_{a}}=\partial_{\pm}{{\tilde{x}}_{a}}$, the background of
dual $\sigma$-model is obtained to be

$\displaystyle{\tilde{d}s^{2}}$ $\displaystyle=$
$\displaystyle{l^{2}}dy^{2}+\frac{1}{\Delta}\Big{[}-e^{-2y}d{\tilde{x}}_{{}_{1}}d{\tilde{x}}_{{}_{2}}+\frac{\alpha_{{}_{0}}}{\Delta}(1-\frac{1}{4}e^{-4y}){\tilde{\psi}}d{\tilde{x}}_{{}_{2}}d{\tilde{\psi}}$
(4.23)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{2}{\Delta}e^{-2y}{\tilde{x}}_{{}_{2}}{\tilde{\psi}}d{\tilde{x}}_{{}_{2}}d{\tilde{\chi}}-e^{-2y}~{}d{\tilde{\psi}}d{\tilde{\chi}}\Big{]},$
$\displaystyle{\tilde{B}}$ $\displaystyle=$
$\displaystyle\frac{1}{\Delta}\Big{[}{\tilde{x}}_{{}_{2}}~{}d{\tilde{x}}_{{}_{1}}\wedge
d{\tilde{x}}_{{}_{2}}-{\tilde{x}}_{{}_{2}}~{}d{\tilde{\psi}}\wedge
d{\tilde{\chi}}-\frac{\alpha_{{}_{0}}}{2}(1-\frac{1}{4}e^{-4y})d{\tilde{\psi}}\wedge
d{\tilde{\psi}}$ (4.24)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+\frac{\alpha_{{}_{0}}}{\Delta}(1-\frac{1}{4}e^{-4y}){\tilde{x}}_{{}_{2}}{\tilde{\psi}}d{\tilde{x}}_{{}_{2}}\wedge
d{\tilde{\psi}}+\frac{1}{\Delta}({{\tilde{x}}_{{}_{2}}}^{2}+\frac{1}{4}e^{-4y}){\tilde{\psi}}d{\tilde{x}}_{{}_{2}}\wedge
d{\tilde{\chi}}\Big{]},$

where $\Delta={{\tilde{x}}_{{}_{2}}}^{2}-\frac{1}{4}e^{-4y}$. The metric
(4.23) is flat in the sense that its scalar curvature vanishes. Looking at the
field equations (3.14)-(3.16) one verifies the conformal invariance of dual
model with a constant dilaton field and vanishing cosmological constant.
Notice that the dilaton field making the dual model conformal can be also
obtained from (4.13) which gives a constant value. Finally, it is seen that in
this case, the super PL T-duality transforms rather extensive and complicated
background given by equations (4.23) and (4.24) to much simpler form such as
(4.16).

Before closing this subsection let us highlight some point concerning these
models. As discussed in Introduction, the BTZ metric (1.1) (even for the
vacuum state) satisfies the one-loop beta-function equations with a non-
vanishing field strength, while here, as can be clearly seen, the original
model with background (4.16) satisfies the one-loop beta-function equations in
both cases of absence $(\alpha_{{}_{0}}=0)$ and presence $(\alpha_{{}_{0}}\neq
0)$ of the field strength.

###  4.2 Non-Abelian target space dual starting from the $(C^{3}+A)$ Lie
supergroup

The $({\mathscr{C}}^{3}+{\mathscr{A}})$ Lie superalgebra131313In Ref. [27], it
has been shown that the super PL duality relates the WZW model based on the
$(C^{3}+A)$ Lie supergroup to a $\sigma$-model defined on the $(C^{3}+A)$ when
Lie superalgebra of the dual Lie supergroup is
${{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{{}_{1,1}}}$. is in turn interesting in
the sense that its commutator of B-B is zero. Its (anti-)commutation relations
are given in Table 3 of Appendix A. The Lie superalgebra of semie-Abelian
Drinfeld superdouble $\big{(}({\mathscr{C}}^{3}+{\mathscr{A}})\;,\;{\cal
I}_{{}_{(2|2)}}\big{)}$ obeys the following (anti-)commutation relations

$\displaystyle{[T_{{}_{1}},T_{{}_{4}}]}$ $\displaystyle=$ $\displaystyle
T_{{}_{3}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\\{T_{{}_{4}},T_{{}_{4}}\\}=T_{{}_{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\\{T_{{}_{1}},{\tilde{T}}^{3}\\}=-{\tilde{T}}^{4},$
$\displaystyle{[T_{{}_{4}},{\tilde{T}}^{2}]}$ $\displaystyle=$
$\displaystyle-{\tilde{T}}^{4},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\\{T_{{}_{4}},{\tilde{T}}^{3}\\}=-{\tilde{T}}^{1},$
(4.25)

where $T_{{}_{a}}$’s generate the $({\mathscr{C}}^{3}+{\mathscr{A}})$, while
${\tilde{T}}^{a}$’s stand for the ${\cal I}_{{}_{(2|2)}}$. The expression for
a generic element of the $(C^{3}+A)$ Lie supergroup can be written as

$\displaystyle g~{}=~{}e^{\psi
T_{{}_{3}}}~{}e^{x_{{}_{1}}T_{{}_{1}}}~{}e^{x_{{}_{2}}T_{{}_{2}}}~{}e^{\chi
T_{{}_{4}}}.$ (4.26)

In order to construct the original $\sigma$-model we need to calculate the
right-invariant one-forms on the $(C^{3}+A)$. To this end, one should employ
(4.2) and (4.26) and then use equation (2.5). The result is

$\displaystyle R_{\pm}^{1}$ $\displaystyle=$
$\displaystyle\partial_{\pm}x_{{}_{1}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}R_{\pm}^{2}~{}=~{}\partial_{\pm}x_{{}_{2}}+\partial_{\pm}\chi~{}\frac{\chi}{2},$
$\displaystyle R_{\pm}^{3}$ $\displaystyle=$
$\displaystyle-\partial_{\pm}\psi-\partial_{\pm}\chi~{}x_{{}_{1}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}R_{\pm}^{4}~{}=~{}-\partial_{\pm}\chi.$
(4.27)

In what follows, the dualizing is performed on the $(C^{3}+A)$ Lie supergroup,
and the spectator field is still displayed by $\\{y\\}$. Here we choose the
spectator-dependent background matrices in the following form

$\displaystyle{E_{0}}_{{}_{ab}}=\left(\begin{tabular}[]{cccc}$a_{{}_{1}}(y)$&0&0&0\\\
0&$a_{{}_{2}}(y)$&0&0\\\ 0&0&$0$&$a_{{}_{3}}(y)$\\\
0&0&-$a_{{}_{3}}(y)$&$b(y)$\\\
\end{tabular}\right),~{}~{}~{}~{}~{}F^{(1)}_{ai}=F^{(2)}_{ia}=0,~{}~{}~{}~{}F_{ij}=\frac{l^{2}}{4},$
(4.32)

where $a_{{}_{i}}(y),~{}i=1,2,3$ and $b(y)$ are some functions that may depend
on the coordinate $\\{y\\}$. Now, by applying formulae (2.4) and (2) one can
obtain the background of the original $\sigma$-model, giving us

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\frac{l^{2}}{4}dy^{2}+a_{{}_{1}}(y)~{}{dx_{{}_{1}}}^{2}+a_{{}_{2}}(y)~{}{dx_{{}_{2}}}^{2}-a_{{}_{2}}(y)\chi
dx_{{}_{2}}d\chi-2a_{{}_{3}}(y)~{}d\psi d\chi,$ (4.33) $\displaystyle B$
$\displaystyle=$ $\displaystyle-\frac{1}{2}b(y)~{}d\chi\wedge d\chi.$ (4.34)

In order to obtain dilaton field that supports the background of the model one
may apply formula (4.12). First, by using the first equation of (2) and
(anti-)commutation relations of the $({\mathscr{C}}^{3}+{\mathscr{A}})$ given
in (4.2) one immediately concludes that
$s\det\big{(}{{}_{{}_{a}}}{a^{{}^{{}^{b}}}}{\tiny(g)}\big{)}=1$. Hence, from
(4.12) we find that $\Phi=\phi^{{}^{(0)}}$. As we will see, the dilaton field
that makes the original $\sigma$-model conformal is found to be $\Phi={\cal
C}_{{}_{0}}+y$, so we choose $\phi^{{}^{(0)}}={\cal C}_{{}_{0}}+y$. Looking at
the one-loop conformal invariance conditions, the field equations
(3.14)-(3.16) are then satisfied with the metric (4.33), $B$-field (4.34) and
a zero cosmological constant together with the aforementioned dialton field if
one considers

$\displaystyle
a_{{}_{1}}(y)=-a_{{}_{2}}(y)=e^{y},~{}~{}~{}~{}~{}~{}~{}a_{{}_{3}}(y)=e^{-y},~{}~{}~{}~{}~{}~{}b(y)=\beta_{{}_{0}}(1-\frac{1}{2}e^{-2y}),$
(4.35)

for some constant $\beta_{{}_{0}}$. Inserting (4.35) into (4.33) one
immediately finds that the scalar curvature of the metric is non-zero, and it
has a negative constant value as ${\cal R}=-16/l^{2}$. It can be easily shown
that under the coordinate transformation

$\displaystyle
e^{y}={r}^{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{x_{{}_{1}}}=\varphi,~{}~{}~{}~{}~{}~{}~{}~{}~{}{x_{{}_{2}}}=\frac{1}{l}t,$
(4.36)

the background becomes

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\frac{l^{2}}{r^{2}}dr^{2}-\frac{r^{2}}{l^{2}}dt^{2}+r^{2}d\varphi^{2}+\frac{r^{2}}{l}\chi
dtd\chi-\frac{2}{r^{2}}d\psi d\chi,$ $\displaystyle H$ $\displaystyle=$
$\displaystyle-\frac{\beta_{{}_{0}}}{r^{5}}~{}dr\wedge d\chi\wedge d\chi,$
$\displaystyle\Phi$ $\displaystyle=$ $\displaystyle{\cal C}_{{}_{0}}+2\log r.$
(4.37)

Again, the above metric is nothing but the BTZ vacuum metric coupling to two
fermionic fields. Note that here the terms related to the coupling of fermions
to the metric are different from those of metric in (4.16). Analogously, we
here can have a conformal background in both cases of absence
$(\beta_{{}_{0}}=0)$ and presence ($\beta_{{}_{0}}\neq 0$) of the field
strength.

To continue, we obtain a new dual background for the BTZ vacuum metric coupled
to two fermionic fields. Employing (2.10), (4.32) and (4.35) one can get the
dual couplings. In this case, the Poisson superbracket and non-zero background
matrices read

$\displaystyle{\tilde{\Pi}}_{ab}=\left(\begin{tabular}[]{cccc}0&0&0&-${\tilde{\psi}}$\\\
0&$0$&0&0\\\ 0&0&$0$&0\\\ ${\tilde{\psi}}$&0&0&-${\tilde{x}}_{{}_{2}}$\\\
\end{tabular}\right),~{}~{}{\tilde{E}}^{{ab}}=\left(\begin{tabular}[]{cccc}$e^{-y}$&0&-${\tilde{\psi}}$&0\\\
0&-$e^{-y}$&0&0\\\ ${\tilde{\psi}}$&0&${\tilde{x}}_{{}_{2}}e^{2y}$&$e^{y}$\\\
0&0&-$e^{y}$&0\\\
\end{tabular}\right),~{}~{}~{}{\tilde{\phi}}_{{}_{ij}}=\frac{l^{2}}{4},$
(4.46)

where we have here assumed that $\beta_{{}_{0}}=0$. Finally, with the help of
(2.7) the line element and the tensor field ${\tilde{B}}$ take the following
forms

$\displaystyle{\tilde{d}s^{2}}$ $\displaystyle=$
$\displaystyle\frac{l^{2}}{4}dy^{2}+e^{-y}\big{(}{d{\tilde{x}}_{{}_{1}}}^{2}-{d{\tilde{x}}_{{}_{2}}}^{2}\big{)}-2e^{y}~{}d{\tilde{\psi}}d{\tilde{\chi}},$
(4.47) $\displaystyle{\tilde{B}}$ $\displaystyle=$
$\displaystyle{\tilde{\psi}}~{}{d{\tilde{x}}_{{}_{1}}}\wedge
d{\tilde{\psi}}-\frac{1}{2}{\tilde{x}}_{{}_{2}}e^{2y}~{}d{\tilde{\psi}}\wedge
d{\tilde{\psi}}.$ (4.48)

The scalar curvature of the dual metric is also constant and equal to the same
value from the metric of original model. Before proceeding to investigate the
conformal invariance conditions of the dual background, let us evaluate the
dual dilaton. First we find that $s\det({\tilde{E}}^{{ab}})=-e^{-4y}$ and
$s\det({{\tilde{a}}^{{}^{a}}}_{~{}b}({\tilde{g}}))=1$, then it follows from
(4.13) that ${\tilde{\Phi}}=\phi^{{}^{(0)}}-2y$, and hence we get the dilaton
by remembering that $\phi^{{}^{(0)}}={\cal C}_{{}_{0}}+y$ which gives the
final result

$\displaystyle{\tilde{\Phi}}={\cal C}_{{}_{0}}-y.$ (4.49)

One can check that the field equations (3.14)-(3.16) are satisfied for the
metric (4.47), the tensor field (4.48) and the dilaton field (4.49) together
with a zero cosmological constant. This means that the dual background is also
conformally invariant up to one-loop order. In order to get more insight of
the dual metric one may use the transformation

$\displaystyle
e^{y}=\frac{1}{{r}^{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{{\tilde{x}}_{{}_{1}}}=\varphi,~{}~{}~{}~{}~{}~{}~{}~{}~{}{{\tilde{x}}_{{}_{2}}}=\frac{1}{l}t,$
(4.50)

then, the dual background takes the following form

$\displaystyle{\tilde{d}s^{2}}$ $\displaystyle=$
$\displaystyle\frac{l^{2}}{r^{2}}dr^{2}-\frac{r^{2}}{l^{2}}dt^{2}+r^{2}d\varphi^{2}-\frac{2}{r^{2}}d{\tilde{\psi}}d{\tilde{\chi}},$
$\displaystyle{\tilde{B}}$ $\displaystyle=$
$\displaystyle{\tilde{\psi}}~{}d\varphi\wedge
d{\tilde{\psi}}-\frac{1}{2lr^{4}}t~{}d{\tilde{\psi}}\wedge d{\tilde{\psi}},$
$\displaystyle{\tilde{\Phi}}$ $\displaystyle=$ $\displaystyle{\cal
C}_{{}_{0}}+2\log r.$ (4.51)

As can be clearly seen, the bosonic part of the metric remains unchanged under
the non-Abelian T-duality. This means that the dual model metric also
describes the BTZ vacuum metric coupled to two fermionic fields.

Starting with the superdouble $\big{(}({\mathscr{C}}^{3}+{\mathscr{A}}),{\cal
I}_{{}_{(2|2)}}\big{)}$ we thus illustrated a concrete example of the super PL
T-duality and found another non-Abelian target space dual for the BTZ vacuum
metric coupled to two fermionic fields in the form of equations (4.47)-(4.49).

###  4.3 Non-Abelian target space dual starting from the
${C}^{3}\oplus{A}_{1,1}$ Lie supergroup

Here the Lie supergroup of target space supermanifold ${{\mathscr{M}}}\approx
O\times G$ is considered to be ${C}^{3}\oplus{A}_{1,1}$ whose Lie superalgebra
is decomposable and it belongs to Family III. According to Table 4 of Appendix
A, only the commutator of B-F for this superalgebra is non-zero. The Lie
superalgebra of semie-Abelian Drinfeld superdouble
$\big{(}{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}\;,\;{\cal
I}_{{}_{(2|2)}}\big{)}$ is defined by the following (anti-)commutation
relations

$\displaystyle{[T_{{}_{1}},T_{{}_{4}}]}=T_{{}_{3}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{[T_{{}_{1}},{\tilde{T}}^{3}]}=-{\tilde{T}}^{4},~{}~{}~{}~{}~{}~{}~{}~{}\\{T_{{}_{4}},{\tilde{T}}^{3}\\}=-{\tilde{T}}^{1}.$
(4.52)

The parametrization of a general element of ${C}^{3}\oplus{A}_{1,1}$ we choose
as in (4.26), giving us

$\displaystyle R_{\pm}^{1}$ $\displaystyle=$
$\displaystyle\partial_{\pm}x_{{}_{1}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}R_{\pm}^{2}~{}=~{}\partial_{\pm}x_{{}_{2}},$
$\displaystyle R_{\pm}^{3}$ $\displaystyle=$
$\displaystyle-\partial_{\pm}\psi-\partial_{\pm}\chi~{}x_{{}_{1}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}R_{\pm}^{4}~{}=~{}-\partial_{\pm}\chi.$
(4.53)

By a convenient choice of the spectator-dependent matrices in the form of

$\displaystyle{E_{0}}_{{}_{ab}}=\left(\begin{tabular}[]{cccc}$e^{y}$&0&0&0\\\
0&-$e^{y}$&0&0\\\ 0&0&$0$&$e^{y}$\\\ 0&0&-$e^{y}$&0\\\
\end{tabular}\right),~{}~{}~{}~{}~{}F^{(1)}_{ai}=F^{(2)}_{ia}=0,~{}~{}~{}~{}F_{ij}=\frac{l^{2}}{4},$
(4.58)

and then with the help of relations (2.4) and (2) one can construct the action
of original $\sigma$-model on the ${C}^{3}\oplus{A}_{1,1}$. The corresponding
background including the line element and zero $B$-field is given by

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\frac{l^{2}}{4}dy^{2}+e^{y}\Big{[}{dx_{{}_{1}}}^{2}-~{}{dx_{{}_{2}}}^{2}-2~{}d\psi
d\chi\Big{]},$ $\displaystyle B$ $\displaystyle=$ $\displaystyle 0.$ (4.59)

The metric is flat in the sense that both its scalar curvature and Ricci
tensor vanish. The bosonic part of the above metric (the first three terms) is
similar to that of metric (4.33), therefore, one may use the transformation
(4.36) to conclude that the metric in (4.59) is nothing but the BTZ vacuum
metric coupled to two fermionic fields. In addition, the dilaton field that
supports the background is found from (4.12) to be $\Phi=\phi^{{}^{(0)}}={\cal
C}_{{}_{0}}$. Using these, one verifies the one-loop beta-function equations
with zero cosmological constant.

Similar to the construction of dual $\sigma$-models in the previous
subsections, the Lie supergroup $G$ of dual supermanifold is here assumed to
be Abelian. With this in mind, one finds that the only non-zero component of
the super Poisson structure on the dual Lie supergroup is
${\tilde{\Pi}}_{{{\tilde{x}}_{{}_{1}}{\tilde{\chi}}}}=-{\tilde{\psi}}$.
Moreover, using the first equation of (2.10) one gets
$s\det({{{\tilde{E}}^{ab}}})=-1$. We will use this in calculating the dual
dilaton field in the next. The corresponding elements to the dual model can be
obtained by making use of relations (4.58) and (2.10). They are then read

$\displaystyle{\tilde{d}s^{2}}$ $\displaystyle=$
$\displaystyle\frac{l^{2}}{4}dy^{2}+e^{-y}\Big{[}{d{\tilde{x}}_{{}_{1}}}^{2}-{d{\tilde{x}}_{{}_{2}}}^{2}-2d{\tilde{\psi}}d{\tilde{\chi}}\Big{]},$
$\displaystyle{\tilde{B}}$ $\displaystyle=$
$\displaystyle{\tilde{\psi}}e^{-2y}~{}{d{\tilde{x}}_{{}_{1}}}\wedge
d{\tilde{\psi}}.$ (4.60)

By transforming coordinate $y\rightarrow-y$ (without changing the rest of the
coordinates) it can be easily shown that the dual metric turns into the same
metric as the original model in (4.59). Hence, it is said that the metric is
self-dual. In order to investigate the conformal invariance conditions of the
dual background one may use (4.13) to obtain ${\tilde{\Phi}}={\cal
C}_{{}_{0}}$. Using these, one verifies the field equations (3.14)-(3.16) for
dual background (4.60) with zero cosmological constant.

###  4.4 Non-Abelian target space dual starting from the
${D}^{10}_{p=\pm\frac{1}{2}}$ Lie supergroup

There is a possibility that we can also perform the dualizing on the
${D}^{10}_{p=\pm\frac{1}{2}}$ Lie supergroup to construct other non-Abelian
target space dual for the BTZ vacuum metric coupling to two fermionic fields.
The corresponding Lie superalgebra to this supergroup belongs to Familiy IV of
Table 5, where only the commutator of F-F among all commutators is zero. In
this subsection we deal with T-dual $\sigma$-models constructing out on the
semie-Abelian Drinfeld superdouble $\big{(}{\mathscr{D}}^{10}_{p},{\cal
I}_{{}_{(2|2)}}\big{)}$ whose Lie superalgebra is defined by the following
non-zero Lie superbrackets

$\displaystyle{[T_{{}_{1}},T_{{}_{2}}]}=T_{{}_{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}[T_{{}_{1}},T_{{}_{3}}]=(p+1)T_{{}_{3}},~{}~{}~{}~{}~{}~{}[T_{{}_{1}},T_{{}_{4}}]=pT_{{}_{4}},~{}~{}~{}~{}~{}~{}~{}[T_{{}_{2}},T_{{}_{4}}]=T_{{}_{3}},~{}~{}~{}~{}~{}~{}$
$\displaystyle{[T_{{}_{1}},{\tilde{T}}^{2}]}=-{\tilde{T}}^{2},~{}~{}~{}~{}~{}[T_{{}_{1}},{\tilde{T}}^{3}]=-(p+1){\tilde{T}}^{3},~{}~{}~{}~{}[T_{{}_{1}},{\tilde{T}}^{4}]=-p{\tilde{T}}^{4},~{}~{}~{}~{}[T_{{}_{2}},{\tilde{T}}^{2}]={\tilde{T}}^{1},~{}~{}~{}~{}~{}$
$\displaystyle{[T_{{}_{2}},{\tilde{T}}^{3}]}=-{\tilde{T}}^{4},~{}~{}~{}~{}\\{T_{{}_{3}},{\tilde{T}}^{3}\\}=-(p+1){\tilde{T}}^{1},~{}~{}~{}\\{T_{{}_{4}},{\tilde{T}}^{3}\\}=-{\tilde{T}}^{2},~{}~{}~{}~{}\\{T_{{}_{4}},{\tilde{T}}^{4}\\}=-p{\tilde{T}}^{1}.~{}$
(4.61)

To obtain the right-invariant super one-forms we parameterize an element of
the ${D}^{10}_{p}$ as in (4.26). We then find

$\displaystyle R_{\pm}^{1}$ $\displaystyle=$
$\displaystyle\partial_{\pm}x_{{}_{1}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}R_{\pm}^{2}~{}=~{}e^{x_{{}_{1}}}~{}\partial_{\pm}x_{{}_{2}},$
$\displaystyle R_{\pm}^{3}$ $\displaystyle=$
$\displaystyle\partial_{\pm}x_{{}_{1}}(p+1)\psi-\partial_{\pm}\psi-\partial_{\pm}\chi~{}x_{{}_{2}}e^{(p+1)x_{{}_{1}}},~{}~{}~{}R_{\pm}^{4}~{}=~{}-\partial_{\pm}\chi~{}e^{px_{{}_{1}}}.$
(4.62)

In this way, the BTZ vacuum metric coupled to two fermionic fields may be
yielded from the original $\sigma$-model on the superdouble
$\big{(}{\mathscr{D}}^{10}_{p},{\cal I}_{{}_{(2|2)}}\big{)}$ if one considers
the spectator-dependent matrices as in (4.8) by setting $\alpha_{{}_{0}}=0$.
Using these and employing (2.4) we then get

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
l^{2}dy^{2}+e^{{x_{{}_{1}}}-2y}~{}dx_{{}_{1}}dx_{{}_{2}}+2(p+1)h(y)\psi
e^{p{x_{{}_{1}}}}~{}dx_{{}_{1}}d\chi-2h(y)e^{px_{{}_{1}}}~{}d\psi d\chi,$
$\displaystyle B$ $\displaystyle=$ $\displaystyle 0.$ (4.63)

Notice that the bosonic part of the metric is also similar to that of metric
(4.9). Here the field strength is absent, and hence with a constant dilton
field one guarantees the conformal invariance of the background (4.63)
provided that we have $h(y)=e^{-2y}$ as well as $p=\pm 1/2$. It follows from
this result that the metric is flat in the sense that its scalar curvature
vanishes. In the following we will focus on the $p=-1/2$ case of the
superdouble only.

Before proceeding to construct the dual background, let us calculate the dual
couplings. Utilizing relation (2) for untilded quantities and also employing
(4.4) we get

$\displaystyle{\tilde{\Pi}}_{ab}=\left(\begin{tabular}[]{cccc}0&-${\tilde{x}}_{{}_{2}}$&-$\frac{1}{2}{\tilde{\psi}}$&$\frac{1}{2}{\tilde{\chi}}$\\\
${\tilde{x}}_{{}_{2}}$&$0$&0&-${\tilde{\psi}}$\\\
$\frac{1}{2}{\tilde{\psi}}$&0&$0$&0\\\
-$\frac{1}{2}{\tilde{\chi}}$&${\tilde{\psi}}$&0&0\\\ \end{tabular}\right),$
(4.68)

then, the dual coupling matrices are worked out

$\displaystyle{\tilde{E}}^{{ab}}=\left(\begin{tabular}[]{cccc}0&$\frac{1}{\Delta_{{}_{+}}}$&-$\frac{{\tilde{\psi}}e^{2y}}{\Delta_{{}_{+}}}$&0\\\
-$\frac{1}{\Delta_{{}_{-}}}$&$\frac{{\tilde{\psi}}{\tilde{\chi}}e^{2y}}{2\Delta}$&-$\frac{{\tilde{\chi}}e^{2y}}{2\Delta_{{}_{-}}}$&-$\frac{{\tilde{\psi}}e^{2y}}{2\Delta_{{}_{-}}}$\\\
-$\frac{{\tilde{\psi}}e^{2y}}{\Delta_{{}_{-}}}$&-$\frac{{\tilde{\chi}}e^{2y}}{2\Delta_{{}_{+}}}$&-$\frac{{{\tilde{x}}_{{}_{2}}}{\tilde{\psi}}{\tilde{\chi}}e^{4y}}{\Delta}$&$e^{2y}$\\\
0&-$\frac{{\tilde{\psi}}e^{2y}}{2\Delta_{{}_{+}}}$&-$e^{2y}$&0\\\
\end{tabular}\right),~{}~{}~{}{\tilde{\phi}}^{{(1)^{a}}}_{~{}~{}~{}j}={\tilde{\phi}}^{{(2)^{b}}}_{i}=0,~{}~{}~{}~{}~{}~{}~{}~{}{\tilde{\phi}}_{{}_{ij}}=l^{2},$
(4.73)

where $\Delta_{{}_{\pm}}={{\tilde{x}}_{{}_{2}}}\pm\frac{1}{2}e^{-2y}$ and
$\Delta=\Delta_{{}_{+}}\Delta_{{}_{-}}$. We need to have the superdeterminant
of matrix ${\tilde{E}}^{{ab}}$. One immediately finds that
$s\det({\tilde{E}}^{{ab}})=e^{-4y}/\Delta$. Finally, the corresponding
elements to the dual model can be obtained by making use of relations (4.68)
and (4.73) together with (2.7), giving us

$\displaystyle{\tilde{d}s^{2}}$ $\displaystyle=$
$\displaystyle{l^{2}}dy^{2}+\frac{1}{\Delta}\Big{[}\frac{1}{2}e^{2y}{\tilde{\psi}}{\tilde{\chi}}~{}{d{\tilde{x}}_{{}_{2}}}^{2}-e^{-2y}~{}d{\tilde{x}}_{{}_{1}}d{\tilde{x}}_{{}_{2}}+2{\tilde{x}}_{{}_{2}}{\tilde{\psi}}e^{2y}~{}d{\tilde{x}}_{{}_{1}}d{\tilde{\psi}}$
(4.74)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+e^{2y}{\tilde{x}}_{{}_{2}}{\tilde{\chi}}~{}d{\tilde{x}}_{{}_{2}}d{\tilde{\psi}}+e^{2y}{\tilde{x}}_{{}_{2}}{\tilde{\psi}}~{}d{\tilde{x}}_{{}_{2}}d{\tilde{\chi}}\Big{]}-2e^{2y}~{}d{\tilde{\psi}}d{\tilde{\chi}},$
$\displaystyle{\tilde{B}}$ $\displaystyle=$
$\displaystyle\frac{1}{\Delta}\Big{[}{\tilde{x}}_{{}_{2}}~{}d{\tilde{x}}_{{}_{1}}\wedge
d{\tilde{x}}_{{}_{2}}-\frac{1}{2}{\tilde{\psi}}~{}d{\tilde{x}}_{{}_{1}}\wedge
d{\tilde{\psi}}+\frac{1}{4}{\tilde{\chi}}~{}d{\tilde{x}}_{{}_{2}}\wedge
d{\tilde{\psi}}+\frac{1}{4}{\tilde{\psi}}~{}d{\tilde{x}}_{{}_{2}}\wedge
d{\tilde{\chi}}$ (4.75)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{1}{2}{\tilde{x}}_{{}_{2}}e^{4y}{\tilde{\psi}}{\tilde{\chi}}~{}d{\tilde{\psi}}\wedge
d{\tilde{\psi}}\Big{]}.$

The scalar curvature of the dual metric is also zero. It seems that under the
non-Abelian T-duality the scalar curvature has been restored from the dual
model to the original one. From equation (4.13) and also from the
superdeterminant of matrix ${\tilde{E}}^{{ab}}$ that found above, one gets the
dual dilaton field, giving ${\tilde{\Phi}}={\cal
C}_{{}_{0}}-2y-\frac{1}{2}\log|\Delta|$. Unfortunately, the field equations
(3.14)-(3.16) do not satisfy with the metric (4.74), $B$-field (4.75) and the
above dialton. In fact, unlike the original model, the dual model cannot be
conformally invariant up to one-loop order.

Let us discuss below the reason for not preserving conformal invariance under
non-Abelian T-duality for this example. In Ref. [24], Sfetsos proved the
classical canonical equivalence to the $\sigma$-models related by PL
T-duality. The canonical transformations are essentially classical and their
equivalence can hold in some special cases but it fails in most cases.
However, checking the equivalence by studying conformal invariance up to one-
loop order is important. On the other hand, it has been shown that [28] a
sufficient condition for the invariance of the reduced string effective action
under PL T-duality is the vanishing of the traces of the structure constants
of each Lie algebra constituting the Drinfeld double. Notice that the field
equations (3.14)-(3.16) can be interpreted as field equations for
${G}_{{}_{MN}}$, ${B}_{{}_{MN}}$ and $\Phi$ of the low energy string effective
action. The string tree level effective action on a $d$-dimensional
supermanifold $\mathscr{M}$ for these background fields is given by [13]

$\displaystyle S_{eff}$ $\displaystyle=$ $\displaystyle\int
d^{{}^{d}}X\;\sqrt{-G}e^{-2\Phi}{\cal L}_{{}_{eff}}$ (4.76) $\displaystyle=$
$\displaystyle\int d^{{}^{d}}X\;\sqrt{-G}e^{-2\Phi}\Big{[}{\cal
R}+4{\overrightarrow{\nabla}_{{}_{M}}}{\Phi}\overrightarrow{\nabla}^{{}^{M}}{\Phi}+\frac{1}{12}H_{{}_{MNP}}H^{{}^{PNM}}-4\Lambda\Big{]},$

where $G$ stands for the superdeterminant of ${}_{{}_{M}}{G}_{{}_{N}}$. In the
case of the above example, one first finds the supertrace (str) of adjoint
representations ${{({\cal X}_{b})}_{c}}^{a}=-{f^{a}}_{bc}$ of
${\mathscr{D}}^{10}_{p=-\frac{1}{2}}$, giving us, str${({\cal X}_{1})}=-1$,
str${({\cal X}_{i})}=0,~{}i=2,3,4$. Then, by calculating the effective
Lagrangians corresponding to both original and dual models, we find that the
dual Lagrangian is not invariant under super PL T-duality transformation and
therefore both Lagrangians are not equal. Obviously, this anomaly is due to
the non-vanishing traces of the structure constants of the superdouble
$\big{(}{\mathscr{D}}^{10}_{p},{\cal I}_{{}_{(2|2)}}\big{)}$. Moreover, in
this example one can show that the integration weights $\sqrt{-G}e^{-\Phi}$
and $\sqrt{-\tilde{G}}e^{-\tilde{\Phi}}$ are not equal. The reason behind this
can be due to the particularity of our model. Because, there is a possibility
of absorbing the anomalous terms into dilaton shift which is the same as a
diffeomorphism transformation. We note that the equations (4.12) and (4.13)
are only transformations which lead to a proportionality between the
integration weights $\sqrt{-G}e^{-\Phi}$ and
$\sqrt{-\tilde{G}}e^{-\tilde{\Phi}}$ [28].

###  4.5 Non-Abelian target space dual starting from the
$(2{A}_{1,1}+2{A})^{0}$ Lie supergroup

As the last example of this section we shall consider the
$(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})^{0}$ Lie superalgebra of Familiy V
which is defined by one non-zero superbracket only as presented in Table 6.
First of all, let us define the Lie superalgebra of semie-Abelian Drinfeld
superdouble $\big{(}(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})^{0},{\cal
I}_{{}_{(2|2)}}\big{)}$. The generators of
$(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})^{0}$ and ${\cal I}_{{}_{(2|2)}}$ are
denoted, respectively, $T_{a}$ and ${\tilde{T}}^{a}$, $a=1,...,4$ and satisfy
the (anti-)commutation relations

$\displaystyle\\{T_{{}_{3}},T_{{}_{3}}\\}=T_{{}_{1}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}[T_{{}_{3}},{\tilde{T}}^{1}]=-{\tilde{T}}^{3}.$
(4.77)

Choosing a parametrization as in (4.2) for the elements of the
$(2{A}_{1,1}+2{A})^{0}$ Lie supergroup leads to the following right-invariant
super one-forms

$\displaystyle
R_{\pm}^{1}=\partial_{\pm}x_{{}_{1}}+\partial_{\pm}\psi~{}\frac{\psi}{2},~{}~{}~{}~{}~{}~{}R_{\pm}^{2}=\partial_{\pm}x_{{}_{2}},~{}~{}~{}~{}~{}~{}R_{\pm}^{3}=-\partial_{\pm}\psi,~{}~{}~{}~{}~{}R_{\pm}^{4}=-\partial_{\pm}\chi.$
(4.78)

Here we choose the spectator-dependent matrices as in (4.32) by setting
$a_{{}_{1}}(y)=-a_{{}_{2}}(y)=e^{y},~{}a_{{}_{3}}(y)=e^{-y}$ and $b(y)=0$.
Thus, the corresponding elements to the original $\sigma$-model including the
line element and $B$-field may be expressed as

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\frac{l^{2}}{4}dy^{2}+e^{y}\Big{[}{dx_{{}_{1}}}^{2}-~{}{dx_{{}_{2}}}^{2}-\psi~{}{dx_{{}_{1}}}d\psi\Big{]}-2e^{-y}~{}d\psi
d\chi,$ $\displaystyle B$ $\displaystyle=$ $\displaystyle 0.$ (4.79)

The metric is flat in the sense that its scalar curvature is ${\cal
R}=-16/l^{2}$. Moreover, one can easily show that the background (4.79) is
conformally invariant up to one-loop order with zero cosmological constant and
dilaton $\Phi={\cal C}_{{}_{0}}+y$. From comparison with relation (4.12), we
conclude that $\phi^{{}^{(0)}}={\cal C}_{{}_{0}}+y$. Again one can use
coordinate transformation (4.36) to rewrite the metric in (4.79) in the form
of the BTZ vacuum metric coupling to two fermionic fields.

As already mentioned, the super Poisson structure on the dual Lie supergroup
can be obtained by making use of the dual version of equation (2). Using
(anti-)commutation relations of the superdouble in (4.77), one finds that the
only non-zero component of the dual super Poisson structure is
${\tilde{\Pi}}_{{{\tilde{\psi}}{\tilde{\psi}}}}=-{{\tilde{x}}_{{}_{1}}}$.
Finally, we can obtain the dual background corresponding to (4.79), in such a
way that the line element and the tensor field ${\tilde{B}}$ take the
following forms

$\displaystyle{\tilde{d}s^{2}}$ $\displaystyle=$
$\displaystyle\frac{l^{2}}{4}dy^{2}+e^{-y}\big{(}{d{\tilde{x}}_{{}_{1}}}^{2}-{d{\tilde{x}}_{{}_{2}}}^{2}\big{)}-2e^{y}~{}d{\tilde{\psi}}d{\tilde{\chi}},$
(4.80) $\displaystyle{\tilde{B}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}{\tilde{x}}_{{}_{1}}e^{2y}~{}d{\tilde{\chi}}\wedge
d{\tilde{\chi}}.$ (4.81)

The scalar curvature of the metric is ${\tilde{\cal R}}=-16/l^{2}$ just as the
original model; moreover, one finds that the only non-zero components of field
strength are
${\tilde{H}}_{{}_{y{\tilde{\chi}}{\tilde{\chi}}}}=2{\tilde{x}}_{{}_{1}}e^{2y},~{}{\tilde{H}}_{{}_{{\tilde{x}}_{{}_{1}}{\tilde{\chi}}{\tilde{\chi}}}}=e^{2y}$.
Also, the dual dilaton is obtained from (4.13) to be ${\tilde{\Phi}}={\cal
C}_{{}_{0}}-y$. Using these, one verifies the field equations (3.14)-(3.16)
with a zero cosmological constant.

For the sake of clarity the results obtained in this section are summarized in
Table 1; we display non-Abelian target space duals of the BTZ vacuum metric
coupled to two fermionic fields together with background of original models
and their transformed.

Table 1. Non-Abelian target space duals of the BTZ vacuum metric
coupled to two fermionic fields  Superdouble Original background Transformed
background Dual background
${\mathscr{D}}=({\mathscr{G}},{\tilde{\mathscr{G}}})$ of original model
$ds^{2}=l^{2}dy^{2}+e^{{x_{{}_{1}}}-2y}\big{[}dx_{{}_{1}}dx_{{}_{2}}$
$ds^{2}=\frac{l^{2}}{r^{2}}dr^{2}-\frac{r^{2}}{l^{2}}dt^{2}+r^{2}d\varphi^{2}$
${\tilde{d}s^{2}}={l^{2}}dy^{2}+\frac{1}{\Delta}\Big{[}-e^{-2y}d{\tilde{x}}_{{}_{1}}d{\tilde{x}}_{{}_{2}}$
$-\psi~{}dx_{{}_{1}}d\chi+d\psi d\chi\big{]},$
$+\frac{r^{2}}{l^{2}}\Big{[}-\frac{\psi}{l}dtd\chi+\psi d\varphi d\chi$
$+\frac{\alpha_{{}_{0}}}{\Delta}(1-\frac{1}{4}e^{-4y}){\tilde{\psi}}d{\tilde{x}}_{{}_{2}}d{\tilde{\psi}}$
$\big{(}({\mathscr{C}}^{1}_{1}+{\mathscr{A}}),{\cal I}_{{}_{(2|2)}}\big{)}$
$+d\psi d\chi\Big{]},$
$+\frac{2}{\Delta}e^{-2y}\big{(}{\tilde{x}}_{{}_{2}}{\tilde{\psi}}d{\tilde{x}}_{{}_{2}}d{\tilde{\chi}}-d{\tilde{\psi}}d{\tilde{\chi}}\big{)}\Big{]},$
$B=-\frac{\alpha_{{}_{0}}}{2}(1-\frac{1}{4}e^{-4y})d\chi\wedge d\chi,$
$B=-\frac{\alpha_{{}_{0}}}{2}(1-\frac{r^{4}}{4l^{4}})d\chi\wedge d\chi,$
${\tilde{B}}=\frac{1}{\Delta}\Big{[}{\tilde{x}}_{{}_{2}}d{\tilde{x}}_{{}_{1}}\wedge
d{\tilde{x}}_{{}_{2}}-{\tilde{x}}_{{}_{2}}d{\tilde{\psi}}\wedge
d{\tilde{\chi}}$ $-\frac{\alpha_{{}_{0}}(4-{e^{-4y}})}{8}d{\tilde{\psi}}\wedge
d{\tilde{\psi}}+d{\tilde{x}}_{{}_{2}}\wedge d\eta\Big{]},$ $\Phi={\cal
C}_{{}_{0}}$ $\Phi={\cal C}_{{}_{0}}$ ${\tilde{\Phi}}={\cal C}_{{}_{0}}$
$ds^{2}=\frac{l^{2}}{4}dy^{2}+e^{y}\big{(}{dx_{{}_{1}}}^{2}-{dx_{{}_{2}}}^{2}$
$ds^{2}=\frac{l^{2}}{r^{2}}dr^{2}-\frac{r^{2}}{l^{2}}dt^{2}+r^{2}d\varphi^{2}$
${\tilde{d}s^{2}}=\frac{l^{2}}{4}dy^{2}+e^{-y}~{}{d{\tilde{x}}_{{}_{1}}}^{2}$
$+\chi dx_{{}_{2}}d\chi\big{)}-2e^{-y}~{}d\psi d\chi,$ $+\frac{r^{2}}{l}\chi
dtd\chi-\frac{2}{r^{2}}d\psi d\chi,$
$-e^{-y}~{}{d{\tilde{x}}_{{}_{2}}}^{2}-2e^{y}~{}d{\tilde{\psi}}d{\tilde{\chi}},$
$\big{(}({\mathscr{C}}^{3}+{\mathscr{A}}),{\cal I}_{{}_{(2|2)}}\big{)}$
${\tilde{B}}={\tilde{\psi}}~{}{d{\tilde{x}}_{{}_{1}}}\wedge d{\tilde{\psi}}$
$B=-\frac{1}{2}b(y)~{}d\chi\wedge d\chi,$
$B=-\frac{\beta_{{}_{0}}}{2}(1-\frac{1}{2r^{4}})d\chi\wedge d\chi,$
$-\frac{1}{2}{\tilde{x}}_{{}_{2}}e^{2y}~{}d{\tilde{\psi}}\wedge
d{\tilde{\psi}},$ $\Phi={\cal C}_{{}_{0}}+y$ $\Phi={\cal C}_{{}_{0}}+2\log r$
${\tilde{\Phi}}={\cal C}_{{}_{0}}-y$
$ds^{2}=\frac{l^{2}}{4}dy^{2}+e^{y}\big{(}{dx_{{}_{1}}}^{2}-{dx_{{}_{2}}}^{2}$
$ds^{2}=\frac{l^{2}}{r^{2}}dr^{2}-\frac{r^{2}}{l^{2}}dt^{2}+r^{2}d\varphi^{2}$
${\tilde{d}s^{2}}=\frac{l^{2}}{4}dy^{2}+e^{-y}\big{[}{d{\tilde{x}}_{{}_{1}}}^{2}$
$-2d\psi d\chi\big{)},$ $-{2}{r^{2}}d\psi d\chi,$
$-{d{\tilde{x}}_{{}_{2}}}^{2}-2d{\tilde{\psi}}d{\tilde{\chi}}\big{]},$
$\big{(}{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1},{\cal
I}_{{}_{(2|2)}}\big{)}$ $B=0,$ $B=0,$
${\tilde{B}}={\tilde{\psi}}e^{-2y}~{}{d{\tilde{x}}_{{}_{1}}}\wedge
d{\tilde{\psi}},$ $\Phi={\cal C}_{{}_{0}}$ $\Phi={\cal C}_{{}_{0}}$
${\tilde{\Phi}}={\cal C}_{{}_{0}}$
$ds^{2}=l^{2}dy^{2}+e^{{x_{{}_{1}}}-2y}dx_{{}_{1}}dx_{{}_{2}}$
$ds^{2}=\frac{l^{2}}{r^{2}}dr^{2}-\frac{r^{2}}{l^{2}}dt^{2}+r^{2}d\varphi^{2}$
${\tilde{d}s^{2}}={l^{2}}dy^{2}+\frac{1}{\Delta}\Big{[}\frac{1}{2}e^{2y}{\tilde{\psi}}{\tilde{\chi}}{d{\tilde{x}}_{{}_{2}}}^{2}$
$+e^{-(\frac{{x_{{}_{1}}}}{2}+2y)}\big{(}\psi dx_{{}_{1}}d\chi$
$+{r^{2}}\Big{(}l(t-l\varphi)\Big{)}^{-\frac{3}{2}}\Big{[}dt\psi d\chi$
$-e^{-2y}d{\tilde{x}}_{{}_{1}}d{\tilde{x}}_{{}_{2}}+2{\tilde{x}}_{{}_{2}}{\tilde{\psi}}e^{2y}~{}d{\tilde{x}}_{{}_{1}}d{\tilde{\psi}}$
$-2d\psi d\chi\big{)},$ $-ld\varphi\psi d\chi\Big{]}$
$+e^{2y}{\tilde{x}}_{{}_{2}}{\tilde{\chi}}d{\tilde{x}}_{{}_{2}}d{\tilde{\psi}}+e^{2y}{\tilde{x}}_{{}_{2}}{\tilde{\psi}}d{\tilde{x}}_{{}_{2}}d{\tilde{\chi}}\Big{]}$
$\big{(}{\mathscr{D}}^{10}_{p=-\frac{1}{2}},{\cal I}_{{}_{(2|2)}}\big{)}$
$-2\frac{r^{2}}{l}\big{[}l(t-l\varphi)\big{]}^{-\frac{1}{2}}d\psi d\chi,$
$-2e^{2y}d{\tilde{\psi}}d{\tilde{\chi}},$ $B=0,$ $B=0,$
${\tilde{B}}=\frac{1}{\Delta}\Big{[}{\tilde{x}}_{{}_{2}}d{\tilde{x}}_{{}_{1}}\wedge
d{\tilde{x}}_{{}_{2}}-\frac{{\tilde{\psi}}}{2}d{\tilde{x}}_{{}_{1}}\wedge
d{\tilde{\psi}}$ $+\frac{1}{4}{\tilde{\chi}}~{}d{\tilde{x}}_{{}_{2}}\wedge
d{\tilde{\psi}}+\frac{1}{4}{\tilde{\psi}}~{}d{\tilde{x}}_{{}_{2}}\wedge
d{\tilde{\chi}}$
$+\frac{1}{2}{\tilde{x}}_{{}_{2}}e^{4y}{\tilde{\psi}}{\tilde{\chi}}~{}d{\tilde{\psi}}\wedge
d{\tilde{\psi}}\Big{]},$ $\Phi={\cal C}_{{}_{0}}$ $\Phi={\cal C}_{{}_{0}}$
${\tilde{\Phi}}={\cal C}_{{}_{0}}-2y-\frac{1}{2}\log|\Delta|$
$ds^{2}=\frac{l^{2}}{4}dy^{2}+e^{y}\big{(}{dx_{{}_{1}}}^{2}-{dx_{{}_{2}}}^{2}$
$ds^{2}=\frac{l^{2}}{r^{2}}dr^{2}-\frac{r^{2}}{l^{2}}dt^{2}+r^{2}d\varphi^{2}$
${\tilde{d}s^{2}}=\frac{l^{2}}{4}dy^{2}+e^{-y}~{}{d{\tilde{x}}_{{}_{1}}}^{2}$
$-\psi dx_{{}_{1}}d\psi\big{)}-2e^{-y}~{}d\psi d\chi,$ $-{r^{2}}\psi d\varphi
d\psi-\frac{2}{r^{2}}d\psi d\chi,$
$-e^{-y}~{}{d{\tilde{x}}_{{}_{2}}}^{2}-2e^{y}~{}d{\tilde{\psi}}d{\tilde{\chi}},$
$\big{(}(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})^{0}$ $,{\cal
I}_{{}_{(2|2)}}\big{)}$ $B=0$ $B=0$
${\tilde{B}}=-\frac{1}{2}{\tilde{x}}_{{}_{1}}e^{2y}~{}d{\tilde{\chi}}\wedge
d{\tilde{\chi}},$ $\Phi={\cal C}_{{}_{0}}+y$ $\Phi={\cal C}_{{}_{0}}+2\log r$
${\tilde{\Phi}}={\cal C}_{{}_{0}}-y$

$b(y)={\beta_{{}_{0}}}(1-\frac{1}{2}e^{-2y})$,
$\Delta={{\tilde{x}}_{{}_{2}}}^{2}-\frac{1}{4}e^{-4y}$,
$d\eta=\frac{\alpha_{{}_{0}}}{\Delta}(1-\frac{1}{4}e^{-4y}){\tilde{x}}_{{}_{2}}{\tilde{\psi}}d{\tilde{\psi}}+\frac{1}{\Delta}({{\tilde{x}}_{{}_{2}}}^{2}+\frac{1}{4}e^{-4y}){\tilde{\psi}}d{\tilde{\chi}}$.

##  5 Super PL T-plurality of BTZ vacuum metric coupled to two fermionic
fields

In the preceding section, we studied the non-Abelian T-dualization of the BTZ
vacuum metric coupling to two fermionic fields, in such a way that we worked
with semi-Abelian superdoubles which were non-isomorphic as Lie superalgebras
in each of the models. In this section we first show that for the
${\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}$ Lie superalgebra there are
several Manin supertriples and the possibility of embedding these Manin
supertriples into the corresponding Drinfeld superdoubles. We also obtained a
non-Abelian dual for the BTZ vacuum metric coupled to two fermionic fields by
constructing mutually T-dual $\sigma$-models on the Manin supertriple
$({\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1},{\cal I}_{{}_{(2|2)}})$. Here we
shall obtain the conformal duality chain of cosmological string backgrounds
with non-vanishing torsion on the other Manin supertriples in corresponding
Drinfeld superdoubles.

###  5.1 A brief review of super PL T-plurality

To set our notation let us briefly review the construction of the super PL
T-plural $\sigma$-models on Lie supergroups [19]. In what follows we shall
consider untransformed $\sigma$-model on $G$ of the superdouble
$D=(G,{\tilde{G}})$ in the following form

$\displaystyle
S=\frac{1}{2}\int_{{}_{\Sigma}}d\sigma^{+}d\sigma^{-}\Big{[}(-1)^{a}R_{+}^{a}(g)~{}{{E}_{{}_{ab}}}(g,y^{i})~{}R_{-}^{b}(g)+\partial_{+}y^{i}F_{{}_{ij}}(y^{i})\partial_{-}y^{j}\Big{]}.$
(5.1)

We start with the assumption that there are several decompositions of a
Drinfeld superdouble. Let
$\mathbb{X}_{{}_{A}}=\\{T_{{}_{a}},{\tilde{T}}^{{}^{b}}\\},~{}a,b=1,...,dim\hskip
1.13809ptG$ be generators of Lie sub-superalgebras $\mathscr{G}$ and
$\widetilde{\mathscr{G}}$ of a Drinfeld superdouble $\mathscr{D}$ associated
with the $\sigma$-model (5.1), and consider
$\mathbb{X}^{\prime}_{{}_{A}}=\\{U_{{}_{a}},{\tilde{U}}^{{}^{b}}\\}$ as
generators of other Manin supertriple
$({\mathscr{G}}_{{}_{U}},{\widetilde{\mathscr{G}}}_{{}_{U}})$ in the same
Drinfeld superdouble ($\mathscr{D}\cong{\mathscr{D}}_{{}_{U}}$). Both
$\mathscr{D}$ and ${\mathscr{D}}_{{}_{U}}$ satisfy equations (2.1) and (2.2).
It is said that $\mathscr{D}$ and ${\mathscr{D}}_{{}_{U}}$ are isomorphic to
each other iff there is an invertible supermatrix ${C_{{}_{A}}}^{{}^{B}}$ such
that the linear map given by
$\mathbb{X}_{{}_{A}}=(-1)^{{}^{B}}~{}{C_{{}_{A}}}^{{}^{B}}~{}\mathbb{X}^{\prime}_{{}_{B}}$
transforms the Lie multiplication of $\mathscr{D}$ into that of
${\mathscr{D}}_{{}_{U}}$ and preserves the canonical form of the bilinear form
$<.,.>$. One may define the isomorphism transformation between the bases of
$\mathscr{D}$ and ${\mathscr{D}}_{{}_{U}}$ in explicit components form as
follows [19]:

$\displaystyle T_{{}_{a}}$ $\displaystyle=$
$\displaystyle(-1)^{{}^{c}}~{}{F_{{}_{a}}}^{{}^{c}}~{}U_{{}_{c}}+G_{{}_{ac}}~{}{\tilde{U}}^{{}^{c}},$
$\displaystyle{\tilde{T}}^{{}^{b}}$ $\displaystyle=$
$\displaystyle(-1)^{{}^{c}}~{}H^{{}^{bc}}~{}U_{{}_{c}}+{K^{{}^{b}}}_{{}_{c}}~{}{\tilde{U}}^{{}^{c}}.$
(5.2)

Now we can define a $\sigma$-model over the Lie supergroup associated with the
generators $U_{{}_{a}}$. In fact, the transformed $\sigma$-model is given by
the same form as (5.1) but with $E(g,y^{i})$ replaced by
${{E}_{{}_{ab}}}(g_{{}_{U}},y^{i})$. It is then read

$\displaystyle
S_{{}_{U}}=\frac{1}{2}\int_{{}_{\Sigma}}d\sigma^{+}d\sigma^{-}\Big{[}(-1)^{a}R_{+}^{a}(g_{{}_{U}})~{}{{E}_{{}_{ab}}}(g_{{}_{U}},y^{i})~{}R_{-}^{b}(g_{{}_{U}})+\partial_{+}y^{i}F_{{}_{ij}}(y^{i})\partial_{-}y^{j}\Big{]},$
(5.3)

where

$\displaystyle
E(g_{{}_{{}_{U}}})=\big{(}NM^{{}^{-1}}+\Pi(g_{{}_{{}_{U}}})\big{)}^{{}^{-1}},$
(5.4)

in which the matrices $M,N$ and $\Pi(g_{{}_{{}_{U}}})$ in the form of their
components may be expressed as

$\displaystyle M_{{}_{ab}}$ $\displaystyle=$
$\displaystyle(-1)^{{}^{c}}~{}{{(K^{st})}_{{}_{a}}}^{{c}}~{}{E_{{}_{0}}}_{{}_{cb}}-{(G^{st})}_{{}_{ab}},$
(5.5) $\displaystyle{N^{{}^{a}}}_{{}_{b}}~{}$ $\displaystyle=$
$\displaystyle{(F^{st})^{a}}_{{}_{b}}-(-1)^{{}^{c}}~{}{(H^{st})}^{{}^{ac}}~{}{E_{{}_{0}}}_{{}_{cb}},$
(5.6) $\displaystyle\Pi^{{}^{ab}}(g_{{}_{{}_{U}}})$ $\displaystyle=$
$\displaystyle(-1)^{{}^{c}}~{}b^{{}^{ac}}(g_{{}_{{}_{U}}})~{}{{(a^{-1})}_{{}_{c}}}^{{}^{b}}(g_{{}_{{}_{U}}}).$
(5.7)

Notice that the sub-matrices $a(g_{{}_{{}_{U}}})$ and $b(g_{{}_{{}_{U}}})$ is
defined as in (2).

In Ref. [18], von Unge showed that the plurality transformation must be
supplemented by a correction that comes from integrating out the fields on the
dual group $\tilde{G}$ in path-integral formulation. In some cases it can be
absorbed at the one-loop level into the transformation of the dilaton field.
Following [18], the transformation of dilaton on Lie supergroup can be written
as follows:

$\displaystyle\Phi_{{}_{{}_{U}}}=\phi^{{}^{(0)}}+\frac{1}{2}\log\big{|}s\det\big{(}{{}_{{}_{a}}}{E_{{}_{b}}}(g_{{}_{{}_{U}}})\big{)}\big{|}-\frac{1}{2}\log\big{|}s\det\big{(}{{}_{{}_{a}}}{M_{{}_{b}}}\big{)}\big{|}+\frac{1}{2}\log\Big{|}s\det\Big{(}{{}_{{}_{a}}}{(a^{-1})^{{}^{b}}}{\tiny(g_{{}_{{}_{U}}})}\Big{)}\Big{|},$
(5.8)

in which $\phi^{{}^{(0)}}$ is the dilaton making the original $\sigma$-model
conformal and may depend on the coordinates of $G_{{}_{U}}$. Since the super
PL T-duality is a special case of the super plurality, one may consider
$F=K={\bf 1},G=H=0$ in (5.2) to obtain
${N^{{}^{a}}}_{{}_{b}}={\delta^{{}^{a}}}_{{}_{b}},M_{{}_{ab}}={E_{{}_{0}}}_{ab}$.
Plugging this into (5.8) we arrive at (4.12). In the following we employ the
above formulation to study super PL T-plurality of the BTZ vacuum metric
coupled to two fermionic fields.

###  5.2 Super PL T-plurality with respect to the ${C}^{3}\oplus{A}_{1,1}$ Lie
supergroup

So far, there has been no classification of the Lie superbialgebras structures
on the ${\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}$. We have presented the
general form of the structure constants related to the dual Lie superalgebras
to ${\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}$ in Appendix B. Here we shall
present only those occurring in this work, i.e., only first class including
the following four isomorphic Manin supertriples:

$\displaystyle~{}~{}\big{(}{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1},{\cal
I}_{{}_{(2|2)}}\big{)}\cong\big{(}{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1},{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}.i\big{)}\cong\big{(}{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1},{(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})^{0}}.i\big{)}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\cong\big{(}{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1},(({\mathscr{A}}_{1,1}+2{\mathscr{A}})^{2}\oplus{\mathscr{A}}_{1,1}).i\big{)}.$
(5.9)

The isomorphism transformation between these superdoubles will be presented in
the next. Referring to solutions of the super Jacobi and mixed super Jacobi
identities listed in Appendix B, if one sets $\alpha=\gamma=0,\beta\neq 0$ in
case (1) or case (2) of the solutions, then it is isomorphic to the
${\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}$. We denote this isomorphic Lie
superalgebra by ${\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}.i$. To obtain the
dual Lie superalgebra ${(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})^{0}}.i$ of the
Manin supertriples
$\big{(}{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1},{(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})^{0}}.i\big{)}$
one may set $\alpha=\beta=0,\gamma\neq 0$ in case (2) or
$\alpha=\gamma=0,\beta\neq 0$ in case (3); moreover, this superalgebra can be
obtained from case (4) by putting $\beta=\gamma=\lambda=0,\alpha\neq 0$ or
$\alpha=\beta=0,\gamma\neq 0$ in case (5). Finally, the
$(({\mathscr{A}}_{1,1}+2{\mathscr{A}})^{2}\oplus{\mathscr{A}}_{1,1}).i$ is
obtained from case (3) by putting $\alpha=0,\beta\neq 0,\gamma\neq 0$;
moreover, it can be obtained from case (4) by putting
$\alpha=\beta=\lambda=0,\gamma\neq 0$.

$\bullet$
$\mathscr{D}={\bf\big{(}{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1},{\cal
I}_{{}_{(2|2)}}\big{)}}$: In order to obtain a possible conformal duality
chain on the isomorphic Drinfeld superdoubles of (5.2) we begin with
$\mathscr{D}=({\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1},{\cal
I}_{{}_{(2|2)}})$ whose Lie superalgebra has been given in (4.52). The
corresponding right-invariant super one-forms and constant matrix $E_{{}_{0}}$
illustrated in subsection 4.3. Accordingly, the action of $\sigma$-model on
the ${C}^{3}\oplus{A}_{1,1}$ in the form (5.1) is given by

$\displaystyle
S=\frac{1}{2}\int_{{}_{\Sigma}}d\sigma^{+}d\sigma^{-}\Big{[}\frac{l^{2}}{4}~{}\partial_{+}y\partial_{-}y+e^{y}~{}\big{(}\partial_{+}{x_{{}_{1}}}\partial_{-}{x_{{}_{1}}}-\partial_{+}{x_{{}_{2}}}\partial_{-}{x_{{}_{2}}}+\partial_{+}{\psi}\partial_{-}{\chi}-\partial_{+}{\chi}\partial_{-}{\psi}\big{)}\Big{]}.~{}~{}$
(5.10)

Notice that under the transformation (4.36), the background of action
(equation (4.59)) describes the BTZ vacuum metric coupling to two fermionic
fields as was mentioned in subsection 4.3. In addition, it was shown there
that the dilton field guaranteeing the conformal invariance of the model is
$\Phi=\phi^{{}^{(0)}}={\cal C}_{{}_{0}}$.

$\bullet$
${\mathscr{D}}_{{}_{U}}={\bf\big{(}{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1},{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}.i\big{)}}$:
The Lie superalgebra of the superdouble
${\mathscr{D}}_{{}_{U}}=\big{(}{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1},{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}.i\big{)}$
obeys the following set of non-trivial (anti-)commutation relations:

$\displaystyle[U_{{}_{1}},U_{{}_{4}}]$ $\displaystyle=$ $\displaystyle
U_{{}_{3}},~{}~{}~{}~{}~{}~{}~{}~{}~{}[{\tilde{U}}^{{}^{2}},{\tilde{U}}^{{}^{3}}]=\beta{\tilde{U}}^{{}^{4}},~{}~{}~{}~{}~{}~{}~{}~{}~{}{[U_{{}_{1}},{\tilde{U}}^{{}^{3}}]}=-{\tilde{U}}^{{}^{4}},$
$\displaystyle{[U_{{}_{4}},{\tilde{U}}^{{}^{2}}]}$ $\displaystyle=$
$\displaystyle\beta
U_{{}_{3}},~{}~{}~{}~{}~{}~{}~{}\\{U_{{}_{4}},{\tilde{U}}^{{}^{3}}\\}=\beta
U_{{}_{2}}-{\tilde{U}}^{{}^{1}},$ (5.11)

where $U_{{}_{a}}$’s generate the
${\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}$, while the bases
${\tilde{U}}^{a}$’s stand for the
${\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}.i$. Also, $\beta$ is a non-zero
constant. Here and henceforth we denote the bosonic bases by
$(U_{{}_{1}},U_{{}_{2}},{\tilde{U}}^{{}^{1}},{\tilde{U}}^{{}^{2}})$, and
fermionic ones by
$(U_{{}_{3}},U_{{}_{4}},{\tilde{U}}^{{}^{3}},{\tilde{U}}^{{}^{4}})$. The
isomorphism between the superdoubles
$\mathscr{D}=({\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1},{\cal
I}_{{}_{(2|2)}})$ and
${\mathscr{D}}_{{}_{U}}=({\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1},{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}.i)$
is given by the following transformation:

$\displaystyle{T_{{}_{1}}}$ $\displaystyle=$ $\displaystyle
U_{{}_{1}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\tilde{T}}^{{}^{1}}=-\beta
U_{{}_{2}}+{\tilde{U}}^{{}^{1}},$ $\displaystyle{T_{{}_{2}}}$ $\displaystyle=$
$\displaystyle
U_{{}_{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\tilde{T}}^{{}^{2}}=\beta
U_{{}_{1}}+{\tilde{U}}^{{}^{2}},$ $\displaystyle{T_{{}_{3}}}$ $\displaystyle=$
$\displaystyle-
U_{{}_{3}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\tilde{T}}^{{}^{3}}=-{\tilde{U}}^{{}^{3}},$
$\displaystyle{T_{{}_{4}}}$ $\displaystyle=$ $\displaystyle-
U_{{}_{4}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\tilde{T}}^{{}^{4}}=-{\tilde{U}}^{{}^{4}}.$
(5.12)

Comparing relations (5.2) and (5.12) one can find the sub-matrices
${F_{{}_{a}}}^{{}^{c}},G_{{}_{ac}},H^{{}^{bc}}$ and ${K^{{}^{b}}}_{{}_{c}}$.
Then, using these and constant matrix $E_{0}$ of (4.58) together with formulae
(5.5) and (5.6) we can find the matrices $M_{{}_{ab}}$ and
${N^{{}^{a}}}_{{}_{b}}$ leading to

$\displaystyle NM^{{}^{-1}}=\left(\begin{array}[]{cccc}e^{-y}&-\beta&0&0\\\
\beta&-e^{-y}&0&0\\\ 0&0&0&e^{-y}\\\ 0&0&-e^{-y}&0\\\ \end{array}\right).$
(5.17)

We use the parametrization of a general group element of
${C}^{3}\oplus{A}_{1,1}$ as in (4.26), but with $T_{{}_{a}}$’s replaced by
$U_{{}_{a}}$’s. Then, one utilizes formula (2) to calculate the matrices
$a(g_{{}_{U}})$ and $b(g_{{}_{U}})$ for this decomposition, giving us

$\displaystyle
a_{{}_{a}}^{{}^{{}^{~{}b}}}(g_{{}_{U}})=\left(\begin{array}[]{cccc}1&0&-\chi&0\\\
0&1&0&0\\\ 0&0&-1&0\\\ 0&0&{x_{{}_{1}}}&-1\\\
\end{array}\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}b^{{}^{ab}}(g_{{}_{U}})=\left(\begin{array}[]{cccc}0&0&0&0\\\
0&0&\beta\chi&0\\\ 0&-\beta\chi&0&0\\\ 0&0&0&0\\\ \end{array}\right).$ (5.26)

Plugging (5.26) into (5.7) one gets the Poisson superbracket
${\Pi}^{{}^{ab}}(g_{{}_{U}})$, which the result is exactly similar to sub-
matrix $b^{{}^{ab}}(g_{{}_{U}})$ of equation (5.26). These results give us the
background

$\displaystyle{E}_{{}_{ab}}(g_{{}_{U}})=\left(\begin{array}[]{cccc}-\frac{e^{{}^{-y}}}{\Delta(y)}&\frac{\beta}{\Delta(y)}&0&\frac{\beta^{2}\chi
e^{{}^{y}}}{\Delta(y)}\\\
-\frac{\beta}{\Delta(y)}&\frac{e^{{}^{-y}}}{\Delta(y)}&0&\frac{\beta\chi}{\Delta(y)}\\\
0&0&0&e^{{}^{y}}\\\ \frac{\beta^{2}\chi
e^{{}^{y}}}{\Delta(y)}&-\frac{\beta\chi}{\Delta(y)}&-e^{{}^{y}}&0\\\
\end{array}\right),$ (5.31)

where $\Delta(y)=\beta^{2}-e^{-2y}$. In order to calculate the corresponding
dilaton field we find that
$s\det(_{a}{{E}_{{}_{b}}(g_{{}_{U}})})=e^{-2y}/\Delta(y),~{}s\det(_{a}{{M}_{{}_{b}}(g_{{}_{U}})})=-1$
and $s\det(a_{{}_{a}}^{{}^{{}^{~{}b}}}(g_{{}_{U}}))=1$. Then, the dilaton is
obtained by making use of (5.8), and by remembering that
$\phi^{{}^{(0)}}={\cal C}_{{}_{0}}$ one gets the final result in the form

$\displaystyle\Phi_{{}_{U}}={\cal
C}_{{}_{0}}-\frac{1}{2}\log\big{|}\beta^{2}e^{2y}-1\big{|}.$ (5.32)

Using (5.31) and the right-invariant super one-forms on the
${C}^{3}\oplus{A}_{1,1}$ (equation (4.3)) one can write the action of model on
the superdouble
$({\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1},{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}.i)$
in the form of (5.3). Finally, background including the line element and
$B$-field is, in the coordinate basis, read off

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\frac{l^{2}}{4}dy^{2}+\frac{1}{\Delta(y)}\Big{[}e^{-y}\big{(}-d{x_{{}_{1}}}^{2}+d{x_{{}_{2}}}^{2}\big{)}-2{\beta}^{2}e^{y}~{}d{x_{{}_{1}}}\chi
d\chi\Big{]}-2e^{y}~{}d\psi d\chi,$ $\displaystyle B$ $\displaystyle=$
$\displaystyle\frac{\beta}{\Delta(y)}\Big{[}d{x_{{}_{1}}}\wedge
d{x_{{}_{2}}}-\chi~{}d{x_{{}_{2}}}\wedge d\chi\Big{]}.$ (5.33)

By taking into consideration the scalar curvature of the metric which is

$\displaystyle{\cal
R}=-\frac{8\beta^{2}(2\beta^{2}+5e^{-2y})}{l^{2}\Delta^{{}^{2}}(y)},$ (5.34)

one verifies the one-loop conformal invariance conditions of the background
(5.33) with dilaton field (5.32) and a zero cosmological constant. Note that
the metric of (5.33) is ill defined at the region $y=-\ln|\beta|$. This
singularity also appears in the scalar curvature, and hence, $y=-\ln|\beta|$
is a true singularity. In fact, we have shown that the solution of BTZ vacuum
metric coupled to two fermionic fields with no curvature singularity is, under
the T-plurality, related to a solution with a curvature singularity. This
situation was not seen in the previous section regarding to super non-Abelian
T-duality.

$\bullet$
${\mathscr{D}}_{{}_{U}}={\bf\big{(}{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1},{(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})^{0}}.i\big{)}}$:
From the Manin supertriples of (5.2) we see that
$\mathscr{D}=({\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1},{\cal
I}_{{}_{(2|2)}})$ and
${\mathscr{D}}_{{}_{U}}={\big{(}{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1},{(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})^{0}}.i\big{)}}$
as Lie superalgebras are isomorphic, and so one can find an isomorphism
transforming the Lie multiplication of $\mathscr{D}$ into that of
${\mathscr{D}}_{{}_{U}}$ and preserving the canonical form of the bilinear
form $<.,.>$ such that they belong to the same Drinfeld superdouble. The
isomorphism transformation of Manin supertriples between
$({\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1},{\cal I}_{{}_{(2|2)}})$ and
${\big{(}{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1},{(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})^{0}}.i\big{)}}$
is given by

$\displaystyle{T_{{}_{1}}}$ $\displaystyle=$ $\displaystyle
U_{{}_{1}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\tilde{T}}^{{}^{1}}={\tilde{U}}^{{}^{1}},$
$\displaystyle{T_{{}_{2}}}$ $\displaystyle=$ $\displaystyle
U_{{}_{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\tilde{T}}^{{}^{2}}={\tilde{U}}^{{}^{2}},$
$\displaystyle{T_{{}_{3}}}$ $\displaystyle=$ $\displaystyle-
U_{{}_{3}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\tilde{T}}^{{}^{3}}=-\frac{\gamma}{2}U_{{}_{4}}-{\tilde{U}}^{{}^{3}},$
$\displaystyle{T_{{}_{4}}}$ $\displaystyle=$ $\displaystyle-
U_{{}_{4}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\tilde{T}}^{{}^{4}}=-\frac{\gamma}{2}U_{{}_{3}}-{\tilde{U}}^{{}^{4}}.$
(5.35)

Here, $\\{U_{{}_{a}},{\tilde{U}}^{{}^{b}}\\}$ are elements of bases of the
superdouble
${\big{(}{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1},{(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})^{0}}.i\big{)}}$
whose Lie superalgebra is defined by the following non-zero (anti-)commutation
relations:

$\displaystyle[U_{{}_{1}},U_{{}_{4}}]=U_{{}_{3}},~{}~{}~{}~{}~{}~{}\\{{\tilde{U}}^{{}^{3}},{\tilde{U}}^{{}^{3}}\\}=\gamma{\tilde{U}}^{{}^{1}},~{}~{}~{}~{}{[U_{{}_{1}},{\tilde{U}}^{{}^{3}}]}=-\gamma
U_{{}_{3}}-{\tilde{U}}^{{}^{4}},~{}~{}~{}~{}~{}\\{U_{{}_{4}},{\tilde{U}}^{{}^{3}}\\}=-{\tilde{U}}^{{}^{1}},$
(5.36)

where $\gamma$ is a non-zero constant. In this case, the matrix
$a_{{}_{a}}^{{}^{{}^{~{}b}}}(g_{{}_{U}})$ is the same form as in (5.26).
Calculating the matrices $M_{{}_{ab}}$, ${N^{{}^{a}}}_{{}_{b}}$ and
${\Pi}^{{}^{ab}}(g_{{}_{U}})$ for the decomposition (5.35) we then get

$\displaystyle NM^{{}^{-1}}=\left(\begin{array}[]{cccc}e^{-y}&0&0&0\\\
0&-e^{-y}&0&0\\\ 0&0&0&(e^{-y}-\frac{\gamma}{2})\\\
0&0&-(e^{-y}+\frac{\gamma}{2})&0\\\
\end{array}\right),~{}~{}{\Pi}(g_{{}_{U}})=\left(\begin{array}[]{cccc}0&0&0&0\\\
0&0&0&0\\\ 0&0&-\gamma x_{{}_{1}}&0\\\ 0&0&0&0\\\
\end{array}\right),~{}~{}~{}$ (5.45)

leading to a background

$\displaystyle{E}_{{}_{ab}}(g_{{}_{U}})$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}e^{y}&0&0&0\\\ 0&-e^{y}&0&0\\\
0&0&0&\frac{1}{(e^{-y}+\frac{\gamma}{2})}\\\
0&0&-\frac{1}{(e^{-y}-\frac{\gamma}{2})}&\frac{\gamma
x_{{}_{1}}}{(e^{-2y}-\frac{\gamma^{2}}{4})}\\\ \end{array}\right).$ (5.50)

In order to evaluate the total dilaton contribution we have to use
$\phi^{{}^{(0)}}={\cal C}_{{}_{0}}$ which gives the final result

$\displaystyle\Phi_{{}_{U}}={\cal
C}_{{}_{0}}+\frac{1}{2}\log\big{|}\frac{\gamma^{2}}{4}e^{2y}-1\big{|}.$ (5.51)

Since the first sub-superalgebra is
${\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}$, the right-invariant super one-
forms are the same forms as in (4.3). Finally, using (5.3) one can construct
the action of the model on the superdouble
$\big{(}{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1},{(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})^{0}}.i\big{)}$
whose background is

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\frac{l^{2}}{4}dy^{2}+e^{y}\big{(}d{x_{{}_{1}}}^{2}-d{x_{{}_{2}}}^{2}\big{)}-\frac{2e^{-y}}{(e^{-2y}-\frac{\gamma^{2}}{4})}~{}d\psi
d\chi,$ $\displaystyle B$ $\displaystyle=$
$\displaystyle\frac{\gamma}{2(e^{-2y}-\frac{\gamma^{2}}{4})}~{}d\psi\wedge
d\chi.$ (5.52)

One immediately finds that the scalar curvature of the metric is

$\displaystyle{\cal
R}=\frac{\gamma^{2}(10e^{-2y}-\gamma^{2})}{l^{2}(e^{-2y}-\frac{\gamma^{2}}{4})^{2}}.$
(5.53)

Again we encounter a true singularity that appears here at the region
$y=-\ln|{\gamma}/{2}|$. Looking at the one-loop beta-function equations one
verifies the conformal invariance conditions of the background (5.52) with
dilaton field (5.51) and a zero cosmological constant.

$\bullet$
${\mathscr{D}}_{{}_{U}}={\bf\big{(}{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1},\big{(}({\mathscr{A}}_{1,1}+2{\mathscr{A}})^{2}\oplus{\mathscr{A}}_{1,1}\big{)}.i\big{)}}$:
As the last example of this section we ask the question of super PL
T-plurality of the BTZ vacuum metric coupled to two fermionic fields with
respect to the superdouble
$\big{(}{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1},\big{(}({\mathscr{A}}_{1,1}+2{\mathscr{A}})^{2}\oplus{\mathscr{A}}_{1,1}\big{)}.i\big{)}$
whose Lie superalgebra is defined by the following non-zero Lie superbrackets:

$\displaystyle[U_{{}_{1}},U_{{}_{4}}]$ $\displaystyle=$ $\displaystyle
U_{{}_{3}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\\{{\tilde{U}}^{{}^{3}},{\tilde{U}}^{{}^{4}}\\}=\gamma{\tilde{U}}^{{}^{1}},~{}~{}~{}~{}~{}~{}~{}~{}~{}{[U_{{}_{1}},{\tilde{U}}^{{}^{3}}]}=-\gamma
U_{{}_{4}}-{\tilde{U}}^{{}^{4}},$
$\displaystyle{[U_{{}_{1}},{\tilde{U}}^{{}^{4}}]}$ $\displaystyle=$
$\displaystyle-\gamma
U_{{}_{3}},~{}~{}~{}~{}~{}~{}~{}\\{U_{{}_{4}},{\tilde{U}}^{{}^{3}}\\}=-{\tilde{U}}^{{}^{1}},$
(5.54)

where $\gamma$ is a non-zero constant. The isomorphism transforming the Lie
multiplication of
$\mathscr{D}=({\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1},{\cal
I}_{{}_{(2|2)}})$ into that of
${\mathscr{D}}_{{}_{U}}=\big{(}{\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1},\big{(}({\mathscr{A}}_{1,1}+2{\mathscr{A}})^{2}\oplus{\mathscr{A}}_{1,1}\big{)}.i\big{)}$
and preserving the canonical form of the bilinear form $<.,.>$, is given by

$\displaystyle{T_{{}_{1}}}$ $\displaystyle=$ $\displaystyle
U_{{}_{1}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\tilde{T}}^{{}^{1}}={\tilde{U}}^{{}^{1}},$
$\displaystyle{T_{{}_{2}}}$ $\displaystyle=$ $\displaystyle
U_{{}_{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\tilde{T}}^{{}^{2}}={\tilde{U}}^{{}^{2}},$
$\displaystyle{T_{{}_{3}}}$ $\displaystyle=$ $\displaystyle-
U_{{}_{3}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\tilde{T}}^{{}^{3}}=-{\tilde{U}}^{{}^{3}},$
$\displaystyle{T_{{}_{4}}}$ $\displaystyle=$ $\displaystyle-
U_{{}_{4}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\tilde{T}}^{{}^{4}}=-{\gamma}U_{{}_{4}}-{\tilde{U}}^{{}^{4}}.$
(5.55)

After calculating the sub-matrices
${F_{{}_{a}}}^{{}^{c}},G_{{}_{ac}},H^{{}^{bc}}$ and ${K^{{}^{b}}}_{{}_{c}}$ we
employ formulae (5.5) and (5.6) to obtain the matrices $M_{{}_{ab}}$ and
${N^{{}^{a}}}_{{}_{b}}$, in such a way that one must also use constant matrix
$E_{0}$ of (4.58). In addition, the matrix ${\Pi}(g_{{}_{U}})$ is obtained for
this decomposition by using (2) and (5.7). It then follows from (5.4) that

$\displaystyle{E}_{{}_{ab}}(g_{{}_{U}})$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}e^{y}&0&0&0\\\ 0&-e^{y}&0&0\\\
0&0&-\gamma e^{2y}&\gamma x_{{}_{1}}e^{2y}-e^{y}\\\ 0&0&\gamma
x_{{}_{1}}e^{2y}+e^{y}&-\gamma{x_{{}_{1}}}^{2}e^{2y}\\\ \end{array}\right).$
(5.60)

Finally, background including the line element and $B$-field takes the
following form

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\frac{l^{2}}{4}dy^{2}+e^{y}\Big{[}{dx_{{}_{1}}}^{2}-~{}{dx_{{}_{2}}}^{2}-2~{}d\psi
d\chi\Big{]},$ $\displaystyle B$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\gamma~{}e^{2y}~{}d\psi\wedge d\psi.$ (5.61)

Unlike the previous two examples, in this case the metric is flat in the sense
that both its scalar curvature and Ricci tensor vanish. In addition, it is
interesting to note that the above metric is exactly the same as the metric of
action (5.10) on the superdouble
$({\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1},{\cal I}_{{}_{(2|2)}})$. Here,
the dilaton obtaining from equation (5.8) is ${\Phi}=\phi^{{}^{(0)}}$. Finally
we get the dilaton by remembering that $\phi^{{}^{(0)}}={\cal C}_{{}_{0}}$
which gives the final result ${\Phi}={\cal C}_{{}_{0}}$.

##  6 Conclusion

In summary, we have studied Abelian T-duality, non-Abelian T-duality and
T-plurality of the BTZ metric coupled to two fermionic fields. In studying
Abelian T-duality, we have obtained two duals for the BTZ metric when is
coupled to two fermionic fields. When the dualizing was implemented by the
Abelian Lie supergroup of the type $(2|2)$, we found that the singularity of
the dual metric is appeared at the region $r=\frac{J}{2}M^{\frac{-1}{2}}$,
whereas when we dealt with a $(2|1)$-dimensional Abelian Lie supergroup, we
encountered a singularity at the origin. Moreover, in both cases of the dual
metrics, horizons were at the same location as the metric of original model.

Using super PL T-duality approach in the presence of spectator fields we have
constructed some non-Abelian T-dualizable $\sigma$-models on some of the semi-
Abelian Drinfeld superdoubles, in such a way that we have dealt with
$(2|2)$-dimensional non-Abelian Lie superalgebras
$({\mathscr{C}}_{1}^{1}+{\mathscr{A}})$, $({\mathscr{C}}^{3}+{\mathscr{A}})$,
${\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}$,
${\mathscr{D}}^{10}_{p=\pm{1/2}}$, and
$(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})^{0}$ as the first sub-superalgebras of
the superdoubles. By a convenient choice of the spectator-dependent background
matrices we showed that the background of original $\sigma$-models describes a
string propagating in a target space with the BTZ vacuum metric coupling to
two fermionic fields, in such a way that a new family of the solutions to
supergravity equations was found in both cases of absence and presence of the
field strength. Accordingly, we think that the choice of spectator-dependent
matrices plays a key role in the structure of our models. In some cases we
showed that the dual backgrounds also yields the BTZ vacuum metric coupled to
two fermionic fields.

Starting from the decomposition of semi-Abelian Drinfeld superdouble
$({\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1},{\cal I}_{{}_{(2|2)}})$ we could
calculate the super PL T-plural of the BTZ vacuum metric coupled to two
fermionic fields with respect to the ${C}^{3}\oplus{A}_{1,1}$ Lie supergroup.
In fact, we obtained a conformally invariant duality chain of
$2+1$-dimensional cosmological string backgrounds coupled to two fermionic
fields in the form of relations (5.10), (5.33), (5.52) and (5.61). The most
interesting feature of our super PL T-plural $\sigma$-models is the existence
of true singularities in some of the metrics of the models. This situation was
not seen in the case of super non-Abelian T-duals. Our current goal was to get
better understanding of super PL T-plurality through investigation of examples
presented in section 5. We hope that the discussed super PL T-plurality can
yield insights to supergravity.

The findings of our study showed that the target supermanifold
$\mathscr{M}\approx O\times G$ with non-Abelian Lie supergroups of the type
$(2|2)$ is wealthy. In addition to super PL symmetric backgrounds constructed
out in this paper one can study the super non-Abelian T-dulaization of
Schwarzschild and Godel metrics when are coupled to two fermionic fields. We
intend to address this problem in the future.

##  Appendix A Lie superalgebras of the type $(2|2)$

The indecomposable Lie superalgebras of the type $(2|2)$ was first presented
by Backhouse in [16]. In that classification, Lie superalgebras are divided
into two types: trivial and nontrivial Lie superalgebras for which the anti-
commutations of F-F are, respectively, zero or non-zero. The decomposable Lie
superalgebras of the type $(2|2)$ have been recently obtained in [17]. As
mentioned in Introduction section, there are some omissions and redundancies
in the Backhouse’s classification, probably because the author in some cases
has not taken into account all isomorphisms to reduce the list of
superalgebras. In this Appendix we make some modifications to the Backhouse’s
classification. Before proceeding to this end, it should be noted that Lie
superalgebras of the type $(2|2)$ possess two bosonic generators
$\\{T_{{}_{1}},T_{{}_{2}}\\}$ and two fermionic ones
$\\{T_{{}_{3}},T_{{}_{4}}\\}$ as was mentioned in section 4.

Below one can find the Lie superalgebras of Backhouse’s classification that
are isomorphic together

$\displaystyle(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})^{1}\cong(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})^{2}\cong{(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})}_{0<p<1/2}^{3}\cong{(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})}_{p>0}^{4},~{}~{}~{}~{}$
(A.1)
$\displaystyle{(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})}_{p>1/2}^{3}\cong{(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})}_{p=1}^{3}.$
(A.2)

However, one can check that the following real transformations guarantee the
isomorphism between these Lie superalgebras:
$\bullet$ One can simply show that under the transformation

$\displaystyle T^{\prime}_{{}_{1}}$ $\displaystyle=$
$\displaystyle{a^{2}}T_{{}_{1}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}T^{\prime}_{{}_{3}}=-aT_{{}_{3}},$
$\displaystyle T^{\prime}_{{}_{2}}$ $\displaystyle=$
$\displaystyle{a^{2}}T_{{}_{1}}+b^{{}^{2}}T_{{}_{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}T^{\prime}_{{}_{4}}=-aT_{{}_{3}}-bT_{{}_{4}},$
(A.3)

for some constants $a,b$, the Lie superalgebras
$(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})^{1}$ and
$(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})^{2}$ are isomorphic together, such that
$T_{{}_{a}}$’s, $a=1,...,4$ are the bases of the
$(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})^{1}$, while $T^{\prime}_{{}_{a}}$’s,
$a=1,...,4$ stand for the $(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})^{2}$.
$\bullet\bullet$ The isomorphism of between the Lie superalgebras
$(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})^{1}$ and
${(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})}_{p>0}^{4}$ is given by the following
transformation

$\displaystyle T^{\prime}_{{}_{1}}$ $\displaystyle=$
$\displaystyle\lambda_{{}_{\pm}}^{2}~{}T_{{}_{1}}+T_{{}_{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}T^{\prime}_{{}_{3}}=\lambda_{{}_{\pm}}T_{{}_{3}}-T_{{}_{4}},$
$\displaystyle T^{\prime}_{{}_{2}}$ $\displaystyle=$ $\displaystyle
T_{{}_{1}}+\lambda_{{}_{\pm}}^{2}~{}T_{{}_{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}T^{\prime}_{{}_{4}}=-T_{{}_{3}}-\lambda_{{}_{\pm}}T_{{}_{4}},$
(A.4)

where $\lambda_{{}_{\pm}}=\frac{1}{2p}(-1\pm\sqrt{1+4p^{2}})$. Here,
$T_{{}_{a}}$’s, $a=1,...,4$ and $T^{\prime}_{{}_{a}}$’s, $a=1,...,4$ are
generators of the $(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})^{1}$ and
${(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})}_{p>0}^{4}$, respectively.
$\bullet\bullet\bullet$ One may use the transformation

$\displaystyle T^{\prime}_{{}_{1}}$ $\displaystyle=$
$\displaystyle(\frac{\delta}{p}-2p\delta-1)T_{{}_{1}}+(1-2p\delta)T_{{}_{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}T^{\prime}_{{}_{3}}=\delta
T_{{}_{3}}-T_{{}_{4}},$ $\displaystyle T^{\prime}_{{}_{2}}$ $\displaystyle=$
$\displaystyle(1-2p\delta)T_{{}_{1}}+(\frac{\delta}{p}-2p\delta-1)T_{{}_{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}T^{\prime}_{{}_{4}}=-T_{{}_{3}}+\delta
T_{{}_{4}},$ (A.5)

to show that the Lie superalgebras
${(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})}_{0<p<1/2}^{3}$ and
$(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})^{1}$ are isomorphic together. Here, we
have used the bases $\\{T_{{}_{a}}\\}$ and $\\{T^{\prime}_{{}_{a}}\\}$ to
define the ${(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})}_{0<p<1/2}^{3}$ and
$(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})^{1}$, respectively. Note that the
parameter $\delta$ in (A) is given by
$\delta=\frac{1}{2p}(1+\sqrt{1-4p^{2}})$.
$\bullet\bullet\bullet\hskip 2.84526pt\bullet$ The isomorphism transforming
the Lie superalgebra of ${(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})}_{p=1}^{3}$
into that of ${(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})}_{p>1/2}^{3}$ is
expressed as

$\displaystyle T^{\prime}_{{}_{1}}$ $\displaystyle=$
$\displaystyle\Big{[}-1+3p^{2}+(1-2p^{2})\gamma_{{}_{\pm}}\Big{]}T_{{}_{1}}+p^{2}(1-2\gamma_{{}_{\pm}})T_{{}_{2}},~{}~{}~{}T^{\prime}_{{}_{3}}=-\gamma_{{}_{\pm}}T_{{}_{3}}+pT_{{}_{4}},$
$\displaystyle T^{\prime}_{{}_{2}}$ $\displaystyle=$
$\displaystyle(1-4p^{2})T_{{}_{1}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}T^{\prime}_{{}_{4}}=-T_{{}_{3}}+2pT_{{}_{4}},$
(A.6)

where $\gamma_{{}_{\pm}}=\frac{1}{2}\big{[}1\pm\sqrt{3(4p^{2}-1)}\big{]}$.
Here, the Lie superalgebras
${(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})}_{p>1/2}^{3}$ and
${(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})}_{p=1}^{3}$ are generated by means of
the bases $\\{T_{{}_{a}}\\}$ and $\\{T^{\prime}_{{}_{a}}\\}$, respectively.

We should also pay attention to the fact that Lie superalgebra
$({\mathscr{D}}^{7}_{1-p\;p})$ for the value of $p=0$ is isomorphic to that of
$({\mathscr{C}}^{1}_{1}+{\mathscr{A}})$; moreover, it is isomorphic to the
${({\mathscr{D}}^{7}_{\frac{1}{2}\;\frac{1}{2}})}^{2}$ if one sets $p=1/2$. We
have included the above modifications to the Backhouse’s classification, and
have classified all decomposable and indecomposable Lie superalgebras of the
type $(2|2)$ into six disjoint families in Tables 2 to 7. Our classification
is based on the (anti-)commutation relations between bases of B-B, B-F and
F-F. Note that in Tables 2 to 7, only Lie superalgebras having non-zero
(anti-)commutation relations have been listed. Accordingly, the Abelian Lie
superalgebra ${\cal{I}}_{{}_{(2|2)}}$ is absent in these Tables.

Table 2. Familiy I: (B-B, B-F, F-F)  $\mathscr{G}$ Non-zero (anti-)commutation
relations Comments ${({\mathscr{D}}^{7}_{\frac{1}{2}\;\frac{1}{2}})}^{1}$
$[T_{1},T_{2}]=X_{2},\;\;[T_{1},T_{3}]=\frac{1}{2}T_{3},\;\;[T_{1},T_{4}]=\frac{1}{2}T_{4},\;\;\\{T_{3},T_{3}\\}=T_{2},$
$\\{T_{4},T_{4}\\}=X_{2}$
${({\mathscr{D}}^{7}_{\frac{1}{2}\;\frac{1}{2}})}^{2}$
$[T_{1},T_{2}]=T_{2},\;\;[T_{1},T_{3}]=\frac{1}{2}T_{3},\;\;[T_{1},T_{4}]=\frac{1}{2}T_{4},\;\;\\{T_{3},T_{3}\\}=T_{2},$
$\\{T_{4},T_{4}\\}=-T_{2}$
${({\mathscr{D}}^{7}_{\frac{1}{2}\;\frac{1}{2}})}^{3}$
$[T_{1},T_{2}]=X_{2},\;\;[T_{1},T_{3}]=\frac{1}{2}T_{3},\;\;[T_{1},T_{4}]=\frac{1}{2}T_{4},\;\;\\{T_{3},T_{3}\\}=T_{2}$
$({\mathscr{D}}^{7}_{1-p\;p})$
$[T_{1},T_{2}]=T_{2},\;\;[T_{1},T_{3}]=pT_{3},\;\;[T_{1},T_{4}]=(1-p)T_{4},\;\\{T_{3},T_{4}\\}=T_{2}$
$p\neq 0,~{}p<\frac{1}{2}$ $({\mathscr{D}}^{8}_{\frac{1}{2}})$
$[T_{1},T_{2}]=T_{2},\;\;[T_{1},T_{3}]=\frac{1}{2}T_{3},\;\;[T_{1},T_{4}]=T_{3}+\frac{1}{2}T_{4},\;\\{T_{4},T_{4}\\}=T_{2}$
$({\mathscr{D}}^{9}_{\frac{1}{2}\;p})$
$[T_{1},T_{2}]=T_{2},\;\;[T_{1},T_{3}]=\frac{1}{2}T_{3}-pT_{4},\;\;[T_{1},T_{4}]=pT_{3}+\frac{1}{2}T_{4},$
$p>0$ $\\{T_{3},T_{3}\\}=T_{2},\;\;\\{T_{4},T_{4}\\}=T_{2}$
$({\mathscr{D}}^{10}_{0})^{1}$
$[T_{1},T_{2}]=T_{2},\;\;[T_{1},T_{3}]=T_{3},\;\;[T_{2},T_{4}]=T_{3},\;\;\\{T_{4},T_{4}\\}=T_{1},$
$\\{T_{3},T_{4}\\}=-\frac{1}{2}T_{2}$ $({\mathscr{D}}^{10}_{0})^{2}$
$[T_{1},T_{2}]=T_{2},\;\;[T_{1},T_{3}]=T_{3},\;\;[T_{2},T_{4}]=T_{3},\;\;\\{T_{4},T_{4}\\}=-T_{1},$
$\\{T_{3},T_{4}\\}=\frac{1}{2}T_{2}$ $({\mathscr{C}}^{1}_{1}+{\mathscr{A}})$
$[T_{1},T_{2}]=T_{2},\;\;[T_{1},T_{3}]=T_{3},\;\;\\{T_{3},T_{4}\\}=T_{2}$
$({\mathscr{C}}^{1}_{\frac{1}{2}}+{\mathscr{A}})$
$[T_{1},T_{2}]=T_{2},\;\;[T_{1},T_{3}]=\frac{1}{2}T_{3},\;\;\\{T_{3},T_{3}\\}=T_{2}$
$\equiv({\mathscr{C}}^{1}_{\frac{1}{2}})\oplus{\mathscr{A}}$

Table 3. Familiy II: (B-F, F-F)  $\mathscr{G}$ Non-zero (anti-)commutation
relations Comments $({\mathscr{C}}^{2}_{-1}+{\mathscr{A}})$
$[T_{1},T_{3}]=T_{3},\;\;[T_{1},T_{4}]=-T_{4},\;\;\\{T_{3},T_{4}\\}=T_{2}$
Jordan-Winger quantization $({\mathscr{C}}^{3}+{\mathscr{A}})$
$[T_{1},T_{4}]=T_{3},\;\;\\{T_{4},T_{4}\\}=T_{2}$ Nilpotent
$({\mathscr{C}}^{5}_{0}+{\mathscr{A}})$
$[T_{1},T_{3}]=-T_{4},\;[T_{1},T_{4}]=T_{3},\;\\{T_{3},T_{3}\\}=T_{2},\;\\{T_{4},T_{4}\\}=T_{2}$
$\big{(}{\mathscr{B}}+({\mathscr{A}}_{1,1}+{\mathscr{A}})\big{)}$
$[T_{1},T_{3}]=T_{3},\;\;\\{T_{4},T_{4}\\}=T_{2}$

Table 4. Familiy III: (B-F)  $\mathscr{G}$ Non-zero (anti-)commutation
relations Comments ${\mathscr{D}}^{5}~{}~{}$
$[T_{1},T_{3}]=T_{3},\;\;[T_{1},T_{4}]=T_{4},\;\;[T_{2},T_{4}]=T_{3}$
${\mathscr{D}}^{6}$
$[T_{1},T_{3}]=T_{3},\;\;[T_{1},T_{4}]=T_{4},\;\;[T_{2},T_{3}]=-T_{4},\;\;[T_{2},T_{4}]=T_{3}$
${\mathscr{B}}\oplus{\mathscr{B}}$
$[T_{1},T_{3}]=T_{3},\;\;[T_{2},T_{4}]=T_{4}$
${\mathscr{B}}\oplus{\mathscr{A}}\oplus{\mathscr{A}}_{1,1}$
$[T_{1},T_{3}]=T_{3}$ $\equiv{\mathscr{C}}^{2}_{p=0}\oplus{\mathscr{A}}_{1,1}$
${\mathscr{C}}^{2}_{p}\oplus{\mathscr{A}}_{1,1}$
$[T_{1},T_{3}]=T_{3},~{}~{}[T_{1},T_{4}]=pT_{4}$ $0<|p|\leq 1$
${\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}$ $[T_{1},T_{4}]=T_{3}$ Nilpotent
${\mathscr{C}}^{4}\oplus{\mathscr{A}}_{1,1}$
$[T_{1},T_{3}]=T_{3},\;\;[T_{1},T_{4}]=T_{3}+T_{4}$
${\mathscr{C}}^{5}_{p}\oplus{\mathscr{A}}_{1,1}$
$[T_{1},T_{3}]=pT_{3}-T_{4},~{}~{}\;~{}~{}~{}[T_{1},T_{4}]=T_{3}+pT_{4}$
$p\geq 0$

Table 5. Familiy IV: (B-B, B-F)  $\mathscr{G}$ Non-zero (anti-)commutation
relations Comments ${\mathscr{D}}^{1}_{pq}$
$[T_{1},T_{2}]=T_{2},\;\;[T_{1},T_{3}]=pT_{3},\;\;[T_{1},T_{4}]=qT_{4}$
$pq\neq 0,\;\;p\geq q$ ${\mathscr{D}}^{8}_{p}$
$[T_{1},T_{2}]=T_{2},\;\;[T_{1},T_{3}]=pT_{3},\;\;[T_{1},T_{4}]=T_{3}+pT_{4}$
$p\neq 0$ ${\mathscr{D}}^{9}_{pq}$
$[T_{1},T_{2}]=T_{2},\;\;[T_{1},T_{3}]=pT_{3}-qT_{4},\;\;[T_{1},T_{4}]=qT_{3}+pT_{4}$
$q>0$ ${\mathscr{D}}^{10}_{p}$
$[T_{1},T_{2}]=T_{2},\;\;[T_{1},T_{3}]=(p+1)T_{3},\;\;[T_{1},T_{4}]=pT_{4},\;\;[T_{2},T_{4}]=T_{3}$
${\mathscr{C}}^{1}_{p}\oplus{\mathscr{A}}$
$[T_{1},T_{2}]=X_{2},\;\;[T_{1},T_{3}]=pT_{3}$ $p\neq 0$

Table 6. Familiy V: (F-F)  $\mathscr{G}$ Non-zero (anti-)commutation relations
Comments ${(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})}^{3}_{p=1}$
$\\{T_{3},T_{3}\\}=T_{1},~{}\\{T_{4},T_{4}\\}=T_{2},~{}\\{T_{3},T_{4}\\}=T_{1}+T_{2}$
Nilpotent ${(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})}^{3}_{p=\frac{1}{2}}$
$\\{T_{3},T_{3}\\}=T_{1},~{}\\{T_{4},T_{4}\\}=T_{2},~{}\\{T_{3},T_{4}\\}=\frac{1}{2}(T_{1}+T_{2})$
Nilpotent $(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})^{0}$
$\\{T_{3},T_{3}\\}=T_{1}$ $\equiv({\mathscr{A}}_{1,1}+{\mathscr{A}})$
$\oplus{\mathscr{A}}\oplus{\mathscr{A}}_{1,1}$, Nilpotent
$(2{\mathscr{A}}_{1,1}+2{\mathscr{A}})^{1}$
$\\{T_{3},T_{3}\\}=T_{1},\;\;\;\\{T_{4},T_{4}\\}=T_{2}$
$\equiv({\mathscr{A}}_{1,1}+{\mathscr{A}})$
$\oplus({\mathscr{A}}_{1,1}+{\mathscr{A}})$, Nilpotent
$\big{(}({\mathscr{A}}_{1,1}+2{\mathscr{A}})^{1}\oplus{\mathscr{A}}_{1,1}\big{)}$
$\\{T_{3},T_{3}\\}=T_{1},\;\;\;\\{T_{4},T_{4}\\}=T_{1}$ Nilpotent
$\big{(}({\mathscr{A}}_{1,1}+2{\mathscr{A}})^{2}\oplus{\mathscr{A}}_{1,1}\big{)}$
$\\{T_{3},T_{3}\\}=T_{1},\;\;\;\\{T_{4},T_{4}\\}=-T_{1}$ Nilpotent

Table 7. Familiy VI: (B-B)
$\mathscr{G}$ Non-zero (anti-)commutation relations Comments
${\mathscr{C}}^{1}_{p=0}\oplus{\mathscr{A}}~{}~{}~{}~{}~{}$
$[T_{1},T_{2}]=T_{2}~{}~{}~{}~{}~{}~{}~{}~{}$

##  Appendix B Solutions of the super Jacobi and mixed super Jacobi identities
for the ${\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}$ Lie superalgebra

In order to obtain the Lie superbialgebra structures on the
$\mathscr{G}={\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}$ Lie superalgebra one
must solve the mixed super Jacobi identities (2.3) and also the following
super Jacobi identity (on the dual Lie superalgebra ${\tilde{\mathscr{G}}}$)

$(-1)^{a(b+c)}{\tilde{f}^{bd}}_{\;\;\;\;e}{\tilde{f}^{ca}}_{\;\;\;\;d}+{\tilde{f}^{ad}}_{\;\;\;\;e}{\tilde{f}^{bc}}_{\;\;\;\;d}+(-1)^{c(a+b)}{\tilde{f}^{cd}}_{\;\;\;\;e}{\tilde{f}^{ab}}_{\;\;\;\;d}=0.$
(B.1)

Because of tensorial form of the identities (2.3) and (B.1), it is not easy to
work with them. So, we suggest writing these equations as matrix forms using
the adjoint representations as presented in [22]. The initial steps of the
analysis for solving the equations are made using a computer. Thus, by solving
equations (2.3) and (B.1) we obtain the general form of the structure
constants related to the dual Lie superalgebras to