aainstitutetext: Indian Institute of Science Education and Research Bhopal,
Bhopal Bypass, Bhopal 462066, India.bbinstitutetext: Indian Institute of
Science Education and Research, Pune 411008, Indiaccinstitutetext: Université
Libre de Bruxelles and International Solvay Institutes,
Campus Plain - CP 231, B-1050 Bruxelles, Belgium

# Dual Theory for maximally $\mathcal{N}$ extended flat Supergravity

Nabamita Banerjee b Arindam Bhattacharjee a Surajit Biswas c and Turmoli Neogi
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Maximally $\mathcal{N}$ extended $2+1$ dimensional flat Supergravity theories
exist for a class of super-Poincaré algebras for arbitrary $\mathcal{N}$. They
have rich asymptotic structures and contain all interesting topological
supergravity solutions in presence of non-trivial holonomy. For the asymptotic
symmetry algebra being a suitable flat limit of the superconformal algebra
$Osp(\mathcal{N}|2;R)$, we have found the $1+1$ dimensional chiral WZW model
as the dual quantum field theory that describes the dynamics of these
solutions. In the Hamiltonian analysis, the reduced phase space resembles a
flat super Liouville theory.

##  1 Introduction and Summary

Gravity in $2+1$ space time dimensions is very interesting. In $2+1$
dimensions it becomes a renormalizable theory and hence one can ask
interesting questions about its quantum aspects (Carlip:1995zj  and the
references therein). Another speciality of $2+1$ dimensions is that gravity
becomes topological in nature and hence does not have any dynamical on-shell
bulk degrees of freedom. All solutions are locally equivalent to the Minkowski
spacetime geometry111In presence of a cosmological constant, depending on the
sign, one gets locally a dS or AdS geometry. Thus we do not have graviton or
gravitational waves in $2+1$ dimensions. Nevertheless, whenever the embedding
spacetime has noncontractible cycles, nontrivial boundary topologies arise and
one ends up getting boundary gravitons. These boundary degrees of freedom play
the most important role in $2+1$ gravity theories. Since the dynamics lies on
the boundary, it is natural to think that gravity in $2+1$ dimensions must
have a dual description in terms of a $1+1$ dimensional theory living on the
boundary of the manifold. The duality is best understood in the Chern-Simons
(CS) formulation of gravity Witten:1988hc ; Achucarro:1986uwr . In $2+1$
dimensions, both gravity and CS theory are topological and they are equivalent
descriptions of each other for the CS gauge group $ISO(2,1)$. In past this
equivalence has been mostly explored for the case of asymptotically AdS
gravities Witten:1983ar . By now for all three interesting asymptotic
structures namely flat, Anti de-sitter (AdS) and de-sitter (dS), the dual
boundary field theory for ordinary Einstein-Hilbert gravity has been found
Witten:1988hf ; Moore:1989yh ; Elitzur:1989nr ; Compere:2014cna . The duality
is classical in nature, where the asymptotic symmetries of the gravity theory
matches with the symmetry of the boundary dual field theory. In this paper we
are interested in asymptotically flat gravity theories, where the asymptotic
symmetry in known as the three dimensional BMS symmetry.

The supersymmetric extensions of BMS3 algebras were derived in Barnich:2015sca
and later they were shown to be the asymptotic symmetries for $2+1$
dimensional supergravity theories by explicit gravity computations. Extended
Supersymmetry brings more interesting features to these gravity theories. One
gets an infinite set of regular 3D topological gravity solutions with non
trivial holonomy that makes them locally different from the ordinary Minkowski
solution. Such purely bosonic solutions, while setting the dynamical boundary
graviton modes to constants, were constructed in Banerjee:2018hbl . It was
found that, in presence of nontrivial $R-symmetry$ in the bulk, these
topological bosonic solutions range over non-supersymmetric, null orbifold
(preserving half supersymmetry), conical defect (preserving full
supersymmetry) to conical surplus types of solutions. The dual boundary field
theories are also known for minimal and higher $\mathcal{N}=2,4$ supergravity
theories for asymptotically flat and AdS geometries Coussaert:1995zp ;
Floreanini:1987as ; article ; SONNENSCHEIN1988752 ; PhysRevLett.62.1817 ;
Chu:1991pn ; Caneschi:1996sr ; Henneaux:1999ib ; Valcarcel:2018kwd ;
Barnich:2015sca ; Barnich:2012rz ; Banerjee:2015kcx . The construction
trivially extends to $\mathcal{N}=8$ supergravity theory as well. For
$\mathcal{N}=8$ supergravity we consider the corresponding R-symmetry group to
be $SU(2)$ and the corresponding generators belong to its fundamental
representation. Recently the dual theory for minimal supersymmetric gravity
theory with dS asymptotic geometry has also been constructed
Bhattacharjee:2020jpq . In all these cases, the dual descriptions turn out to
be a gauged chiral Wess-Zumino-Witten model with $SL(2,R)$ or its appropriate
supersymmetric extensions as the gauge group.

In this paper we look for the dual description of asymptotically flat $2+1$
dimensional maximally $\mathcal{N}$ extended supergravity theory, with
arbitrary $\mathcal{N}$, i.e. the number of supercharges for these theories is
kept arbitrary. The bulk symmetry algebras are just super-Poincaré algebras
with arbitrary $\mathcal{N}$. The asymptotic symmetry for this system and the
generic bosonic gravitational solutions were studied in Banerjee:2018hbl . The
asymptotic symmetry algebra turns out to be a suitable flat limit of the
superconformal algebra (SCA), that we name as $\mathcal{N}$ extended super-
BMS3 algebra. It was found that the asymptotic symmetry forms an algebroid as
it closes with non-linear quadratic terms involving the generators. A large
number of such SCAs of interest were considered in Bina_1997 ; Henneaux:1999ib
and similar interesting feature was also seen in the corresponding
supergravity with asymptotically AdS geometry and was extensively presented in
Henneaux:1999ib . Thus, in this case, even the zero mode sector of the
asymptotic symmetry algebra is centrally extended compared to the bulk super-
Poincaré algebra. In the present paper we find the $1+1$ dimensional dual of
such asymptotically flat $\mathcal{N}$ supergravity theories. Our result
states that a chiral Wess-Zumino-Witten (WZW) model type dual for such
$\mathcal{N}$ supergravity theories is possible to achieve only when its
$\mathcal{N}$ extended super-BMS3 asymptotic symmetry algebra is a suitable
flat limit of SCA $Osp(N|2;R)$. The WZW model has the super-Poincaré algebra
as its gauge group. An appropriate Hamiltonian phase space analysis is
performed and the reduced phase space is found to be a supersymmetric
extension of the Liouville theory in its flat limit.

The paper is organised as follows : In 2 we present the bulk maximally
$\mathcal{N}$ extended Super-Poincaré algebra which is the basis of our
construction. The $2+1$ dimensional supergravity theory that we are interested
in has these superalgebras as its bulk symmetry. We also construct the
supertrace elements of the bulk algebra. In section 3 we present the
supergravity theory in Chern-Simons formulation and find its gauge fixed
classical solutions. This gives us the non-trivial residual boundary modes
that can not be fixed classically. Next in section 4, we present the dual
boundary chiral Wess-Zumino-Witten (WZW) model that governs the dynamics of
the boundary modes. The WZW model has the $\mathcal{N}$ extended Super-
Poincaré group as its gauge group. We also present the gauge version of the
WZW model. We further find the reduced phase space description of the theory
in section 5 and find that the phase space is governed by a flat limit of
super Liouville theory. The paper is concluded in section 6. There are two
appendices A and B that summarize our notations, conventions and useful
identities.

##  2 The 3D maximally $\mathcal{N}$ extended super-Poincaré Algebra

Our aim is to find the two-dimensional theory dual to three dimensional
maximally $\mathcal{N}$-extended asymptotically flat supergravity, for
arbitrary $\mathcal{N}$. A list of such interesting algebras were presented in
Banerjee:2016nio . For this purpose, we consider the most generic
$\mathcal{N}$-extended super-Poincaré algebra with following non zero
elements:

$\displaystyle[J_{a},J_{b}]=\epsilon_{abc}J^{c}\qquad[J_{a},P_{b}]=\epsilon_{abc}P^{c}$
$\displaystyle[T^{\bar{a}},T^{\bar{b}}]=f^{\bar{a}\bar{b}}_{\,\,\,\,\,\,\bar{c}}T^{\bar{c}}\qquad[T^{\bar{a}},Z^{\bar{b}}]=f^{\bar{a}\bar{b}}_{\,\,\,\,\,\,\bar{c}}Z^{\bar{c}}$
$\displaystyle[J_{a},Q^{(1,2),\alpha}_{\hat{p}}]=\frac{1}{2}(\Gamma^{a})^{\hat{q}}_{\,\,\,\hat{p}}Q^{(1,2),\alpha}_{\hat{q}}\quad[T^{\bar{a}},Q^{1,\alpha}_{\hat{p}}]=(\lambda^{\bar{a}})^{\alpha}_{\,\,\,\beta}Q^{1,\beta}_{\hat{p}}\quad[T^{\bar{a}},Q^{2,\alpha}_{\hat{p}}]=-(\lambda^{\bar{a}})^{\alpha}_{\,\,\,\beta}Q^{2,\beta}_{\hat{p}}$
$\displaystyle\left\\{Q^{1,\alpha}_{\hat{p}},Q^{1,\beta}_{\hat{q}}\right\\}=-\frac{1}{2}\eta^{\alpha\beta}(C\Gamma^{a})_{\hat{p}\hat{q}}P_{a}+\frac{1}{6\hat{\alpha}}C_{\hat{p}\hat{q}}(\lambda^{\bar{a}})^{\alpha\beta}Z_{\bar{a}}$
(1)
$\displaystyle\left\\{Q^{2,\alpha}_{\hat{p}},Q^{2,\beta}_{\hat{q}}\right\\}=-\frac{1}{2}\eta^{\alpha\beta}(C\Gamma^{a})_{\hat{p}\hat{q}}P_{a}-\frac{1}{6\hat{\alpha}}C_{\hat{p}\hat{q}}(\lambda^{\bar{a}})^{\alpha\beta}Z_{\bar{a}}.$

Our $2+1$ dimensional asymptotically flat gravity theory in invariant under
the above bulk symmetry algebra. Here the first line is the bosonic spacetime
algebra. The indices $a,b$ runs over values $\\{1,2,3\\}$. The second line is
the automorphism algebra, the barred indices $\bar{a},\bar{b}..$ run over
$\\{1,2,...D\\}$ where $D$ is the dimension of the internal algebra. The
indices $\alpha,\beta$ go from $1,...,d$ and denote the amount of
supersymmetry $\mathcal{N}$. Here $d$ is the dimension of the real
representation $\rho$ of the automprhism, under which the fermions transform.
Indices $\hat{p},\hat{q}..$ run over $\\{\frac{1}{2},-\frac{1}{2}\\}$. The
metric in $(\bar{a},\bar{b})$ space is $\delta_{\bar{a}\bar{b}}$ and in the
internal algebra space $(\alpha\beta)$ is $\eta_{\alpha\beta}$222it is not to
be confused with the Minkowski metric.. The constant $\hat{\alpha}$ is related
to the representation of the automorphism algebra by
$\hat{\alpha}=\frac{C_{\rho}}{3(d-1)}$, where $C_{\rho}$ is the second Casimir
of the internal algebra at representation $\rho$. A specific case when the
superalgebra which is particularly interesting for our study is when the
internal symmetry group is $so(\mathcal{N})$. In this case, $D$ and $d$ are
realted as $D=\frac{d(d-1)}{2}$. Our notations and conventions are further
stated in Appendix A.
Let us make few comments about the internal algebra Bina_1997  in general. The
generators $(\lambda^{\bar{a}})^{\alpha\beta}$ of representation $\rho$ of the
internal algebra are orthogonal and anti-symmetric in their indices. Thus, the
dimension of the internal algebra $d(=\mathcal{N})$ is even. The prime point
is that the anti-commutator of $(\lambda^{\bar{a}})$’s are not constrained and
is kept arbitrary in the algebra. Nevertheless the algebra is highly
constrained by the Jacobi identities. Various important properties of the
generators are listed in Appendix B .

### The Supertrace Elements

In the context of the CS formulation of gravity theory, an symmetry algebra is
interesting if it admits a non-degenerate invariant bi-linear (quadratic
Casimir form). (Such construction of supertrace elements were considered in
Howe:1995zm ; Giacomini:2006dr ). This condition works for all $\mathcal{N}$
extended superalgebras listed in Banerjee:2016nio . To find the invariant non-
degenerate bi-linear exists, we construct the most generic quadratic
combination of the generators with arbitrary coefficients. For it to be a
Casimir, this quadratic form must commute with every generator of the algebra.
These put conditions on the arbitrary coefficients. If these equations can be
solved consistently, we get a quadratic Casimir in terms of the coefficients
determined by these conditions. Then the supertrace elements come from the
inverse of the above coefficient matrix. The modus operandi is elaborated in
more details in our previous papers Banerjee:2017gzj ; Banerjee:2018hbl ;
Banerjee:2019lrv
We were able to achieve this for our algebra and determine the supertrace
elements. They are a two parameter family given by:

$\displaystyle\left\langle
J_{a},P_{b}\right\rangle=\eta_{ab},\qquad\left\langle
J_{a},J_{b}\right\rangle=\mu_{1}\eta_{ab},\qquad\left\langle
Q^{1,\alpha}_{\hat{p}},Q^{1,\beta}_{\hat{q}}\right\rangle=\left\langle
Q^{2,\alpha}_{\hat{p}},Q^{2,\beta}_{\hat{q}}\right\rangle=2C_{\hat{p}\hat{q}}\eta^{\alpha\beta},$
$\displaystyle\left\langle
T^{\bar{a}},Z^{\bar{b}}\right\rangle=(12\hat{\alpha})\delta^{\bar{a}\bar{b}},\qquad\qquad\left\langle
T^{\bar{a}},T^{\bar{b}}\right\rangle=\mu_{2}(12\hat{\alpha})\delta^{\bar{a}\bar{b}},$
(2)

with $\mu_{1}$ and $\mu_{2}$ being the two arbitrary constants and they can be
set to any value , even zero. As we will see in the later sections that these
parameters bring interesting physics to our gravity system.

##  3 The corresponding Chern-Simons Theory and its Classical Solution

In the Chern-Simons formulation of gravity, a $2+1$ dimensional (super)gravity
theory can be expressed in terms of Chern-Simons theory with a gauge group
identical to that of the (super)Poincare group. The Chern-Simons is given as:

$S=\frac{k}{4\pi}\int_{M}<\mathcal{A},d\mathcal{A}+\frac{2}{3}\mathcal{A}^{2}>.$
(3)

Here the gauge field $A$ is a Lie-algebra-valued one form and $<,>$ represents
metric in the field space that one obtains from the supertrace elements on the
Lie algebra space. $M$ is a three dimensional manifold and $k$ is the level
for the CS theory. For a theory of gravity it takes the value $k=\frac{1}{4G}$
where $G$ is Newton’s constant. We express
$\mathcal{A}=\mathcal{A}^{a}_{\mu}T_{a}dx^{\mu}$ where ${T_{a}}$ are a
particular basis of the Lie-algebra. The equation of motion for the gauge
field $A$ is given as: $F\equiv d\mathcal{A}+\mathcal{A}\wedge\mathcal{A}=0.$

For our supergravity theory, we consider the Lie-algebra as (1) and $M$ as a
three manifold with a boundary, we denote the coordinates as $u,r,\phi$. Here
$u$ is the timelike coordinate and we take $r\rightarrow\infty$ hypersurface
as the boundary of the space time. Thus the coordinates on boundary
hypersurface are $u$ and a periodic direction $\phi$ with a period of $2\pi$ .
Expanding the gauge field in the Lie-algebra basis as

$\displaystyle\mathcal{A}$
$\displaystyle=e^{a}P_{a}+\hat{\omega}^{a}J_{a}+\psi^{\hat{p}}_{1,\alpha}Q^{1,\alpha}_{\hat{p}}++\psi^{\hat{p}}_{2,\alpha}Q^{2,\alpha}_{\hat{p}}+B^{\bar{a}}T_{\bar{a}}+C^{\bar{a}}Z_{\bar{a}}.$
(4)

The action (3) takes the following form (upto total derivatives in
$d(e^{a}\omega_{a})$ and $d(b^{\bar{a}}C_{\bar{a}})$):

$\displaystyle
S=\frac{k}{4\pi}\int[2e^{a}\hat{R_{a}}+\mu_{1}L(\hat{\omega_{a}})$
$\displaystyle-2\bar{\Psi}^{1,{\beta}}\nabla\Psi_{1,\beta}-2\bar{\Psi}^{2,{\beta}}\nabla\Psi_{2,\beta}+12\hat{\alpha}\\{\mu_{2}BdB-2BdC$
$\displaystyle+f^{\bar{a}\bar{b}\bar{c}}(2B_{\bar{a}}B_{\bar{b}}C_{\bar{c}}+\mu_{2}B_{\bar{a}}B_{\bar{b}}B_{\bar{c}})\\}].$
(5)

This is the generalised 3D maximally $\mathcal{N}$ extended supergravity
action with a gravitational anomaly term $L(\hat{\omega_{a}})$. In (4), the
coefficients of each generator have their usual meaning. $e^{a}$ are the
vielbeins and $\hat{\omega^{a}}=\omega^{a}+\gamma e^{a}$ with $\omega^{a}$
being the spin connections. $\psi_{I,\alpha}^{\hat{p}}$ are the gravitini
fields and $B_{\hat{a}},C_{\hat{a}}$ are the bosonic fields corresponding to
automorphism. All these fields are one forms. In particular
$\psi_{I,\alpha}^{\hat{p}}$ are grassmanian one-forms and
$\bar{\psi}_{I,\hat{p}}^{\alpha}=\psi^{\hat{q}}_{I,\beta}C_{\hat{q}\hat{p}}\eta^{\alpha\beta}$
. Various quantities appearing in the action (3) are defined as,

$\displaystyle\hat{R_{a}}=d\hat{\omega_{a}}$
$\displaystyle+\frac{1}{2}\epsilon_{abc}\hat{\omega^{b}}\hat{\omega^{c}},\quad
L=\hat{\omega^{a}}d\hat{\omega_{a}}+\frac{1}{3}\epsilon^{abc}\hat{\omega_{a}}\hat{\omega_{b}}\hat{\omega_{c}},\quad$
$\displaystyle\nabla\Psi_{1,\alpha}=$ $\displaystyle
d\psi_{1,\alpha}^{\hat{p}}+\frac{1}{2}\hat{\omega}^{a}(\Gamma^{a})^{\hat{p}}_{\hat{q}}\psi^{\hat{q}}_{1,\alpha}+B_{\bar{a}}(\lambda^{\bar{a}})^{\beta}_{\alpha}\psi^{\hat{p}}_{1,\beta}$
$\displaystyle\nabla\Psi_{2,\alpha}=$ $\displaystyle
d\psi_{2,\alpha}^{\hat{p}}+\frac{1}{2}\hat{\omega}^{a}(\Gamma^{a})^{\hat{p}}_{\hat{q}}\psi^{\hat{q}}_{2,\alpha}-B_{\bar{a}}(\lambda^{\bar{a}})^{\beta}_{\alpha}\psi^{\hat{p}}_{2,\beta}.$

The equations of motion for various fields take the following forms :

$\displaystyle
de_{a}+\epsilon_{abc}e^{b}\hat{\omega}^{c}-\frac{1}{4}\psi^{\hat{p}}_{1,\alpha}\psi^{\hat{q}}_{1,\beta}\eta^{\alpha\beta}(C\Gamma_{a})_{\hat{p}\hat{q}}-\frac{1}{4}\psi^{\hat{p}}_{1,\alpha}\psi^{\hat{q}}_{1,\beta}\eta^{\alpha\beta}(C\Gamma_{a})_{\hat{p}\hat{q}}=0$
$\displaystyle
d\hat{\omega}_{a}+\frac{1}{2}\epsilon_{abc}\hat{\omega}^{b}\hat{\omega}^{c}=0$
$\displaystyle
d\psi^{\hat{p}}_{1,\alpha}+\frac{1}{2}{\hat{\omega}}^{a}\psi^{\hat{q}}_{1,\alpha}(\Gamma_{a})^{\hat{p}}_{\,\,\,\hat{q}}+\psi^{\hat{p}}_{1,\beta}B^{\bar{a}}(\lambda_{\bar{a}})^{\beta}_{\,\,\,\alpha}=0$
$\displaystyle
d\psi^{\hat{p}}_{2,\alpha}+\frac{1}{2}{\hat{\omega}}^{a}\psi^{\hat{q}}_{2,\alpha}(\Gamma_{a})^{\hat{p}}_{\,\,\,\hat{q}}-\psi^{\hat{p}}_{2,\beta}B^{\bar{a}}(\lambda_{\bar{a}})^{\beta}_{\,\,\,\alpha}=0$
(6) $\displaystyle
dB_{\bar{a}}+\frac{1}{2}f_{\bar{a}\bar{b}\bar{c}}B^{\bar{b}}B^{\bar{c}}=0$
$\displaystyle
dC_{\bar{a}}+f_{\bar{a}\bar{b}\bar{c}}B^{\bar{b}}C^{\bar{c}}+\frac{1}{12\hat{\alpha}}(\psi^{\hat{p}}_{1,\alpha}\psi^{\hat{q}}_{1,\beta}C_{\hat{p}\hat{q}}(\lambda_{\bar{a}})^{\alpha\beta}-\psi^{\hat{p}}_{2,\alpha}\psi^{\hat{q}}_{2,\beta}C_{\hat{p}\hat{q}}(\lambda_{\bar{a}})^{\alpha\beta})=0.$

We can also write down the fermion equation for its upper internal component
as,

$\displaystyle
d\psi^{\hat{p},\alpha}_{1}+{\hat{\omega}}^{\hat{p}}_{\,\,\,\hat{q}}\psi^{\hat{q},\alpha}_{1}-\psi^{\hat{p},\gamma}_{1}B_{\gamma}^{\,\,\,\alpha}=0.$
(7)

###  3.1 Classical Solutions

Even though $2+1$ dimensional (super)gravity is a topological theory, we see
that the classical equations of motion do not fix the classical profile
completely even after gauge fixing. There are residual gauge degrees of
freedoms left unfixed at the boundary. In this section we find the classical
solutions for our theory up to these unfixed residual degrees of freedoms.

We begin with equation for the $B$ field from above (3). Contracting the
equation with $\lambda_{\bar{a}}$ and using the property
$[\lambda_{\bar{a}},\lambda_{\bar{b}}]=f_{\bar{a}\bar{b}}^{\bar{c}}\lambda_{\bar{c}}$
we get:

$dB+B\wedge B=0,$ (8)

where we define $B^{\bar{a}}\lambda_{\bar{a}}=B$ . The solution is given as
$B=A^{-1}dA$, where $A(r,u,\phi)$ is a matrix valued field in the automorphism
algebra space . Similarly the equation for $\omega$ solves to
$\hat{\omega}=\frac{1}{2}\hat{\omega}^{a}\Gamma_{a}=\Lambda^{-1}d\Lambda$.
Here $\Lambda(r,u,\phi)$ is a $SL(2,R)$ group element and $\Gamma_{a}$ are the
generators of $SL(2,R)$. The other equations can be solved using the above two
solutions of $B$ and $\omega$. Below we list the solutions of all the fields
with their explicit index structures.

$\displaystyle e^{\hat{p}}_{\,\,\,\hat{q}}$
$\displaystyle=(\Lambda^{-1})^{\hat{p}}_{\,\,\,\hat{m}}[\frac{1}{2}(A^{\alpha\beta}\eta_{1,\beta}^{\hat{m}}A_{\alpha\sigma}d\bar{\eta}_{1,\hat{n}}^{\sigma}+\frac{1}{2}A_{\alpha\sigma}d\bar{\eta}_{1,\hat{r}}^{\sigma}A^{\alpha\beta}\eta_{1,\beta}^{\hat{r}}\delta^{\hat{m}}_{\,\,\,\hat{n}})$
$\displaystyle+\frac{1}{2}((A^{-1})^{\alpha\beta}\eta_{1,\beta}^{\hat{m}}(A^{-1})_{\alpha\sigma}d\bar{\eta}_{1,\hat{n}}^{\sigma}+\frac{1}{2}(A^{-1})_{\alpha\sigma}d\bar{\eta}_{1,\hat{r}}^{\sigma}(A^{-1})^{\alpha\beta}\eta_{1,\beta}^{\hat{r}}\delta^{\hat{m}}_{\,\,\,\hat{n}})+(db)^{\hat{m}}_{\,\,\,\hat{n}}](\Lambda)^{\hat{n}}_{\,\,\,\hat{q}}$
$\displaystyle\omega^{\hat{p}}_{\,\,\,\hat{q}}$
$\displaystyle=(\Lambda^{-1}d\Lambda)^{\hat{p}}_{\,\,\,\hat{q}}\qquad
B^{\alpha}_{\,\,\,\beta}=(A^{-1}dA)^{\alpha}_{\,\,\,\beta}$
$\displaystyle\quad C_{\sigma\gamma}$
$\displaystyle=\frac{1}{2}\Big{[}(A^{-1})_{\sigma\beta}\bar{\eta}_{1,\hat{q}}^{\beta}(A^{-1})_{\gamma\xi}d{\eta}^{\hat{q},\xi}_{1}-A_{\sigma\beta}\bar{\eta}_{2,\hat{q}}^{\beta}A_{\gamma\xi}d{\eta}^{\hat{q},\xi}_{2}+(d\tilde{C})_{\sigma\gamma}\Big{]}$
$\displaystyle\quad\psi^{\hat{p},\alpha}_{1}$
$\displaystyle=(A^{-1})^{\alpha}_{\,\,\,\beta}(\Lambda^{-1})^{\hat{p}}_{\,\,\,\hat{q}}d\eta^{\hat{q},\beta}_{1}\qquad\psi_{2,\alpha}^{\hat{p}}=(A)_{\alpha}^{\,\,\,\beta}(\Lambda^{-1})^{\hat{p}}_{\,\,\,\hat{q}}d\eta_{2,\beta}^{\hat{q}}.$
(9)

In the above set of solutions, all fields are functions of all three
coordinates $r,u,\phi$. We further impose a radial gauge condition
$\partial_{\phi}\mathcal{A}_{r}=0$ to remove the redundant gauge freedom of
the solutions and hence a group element splits as
$G(u,\phi,r)=g(u,\phi)h(u,r)$. In this choice of gauge, the field
$\mathcal{A}$ takes following form,

$\mathcal{A}=h^{-1}(a+d)h,\quad
a=g^{-1}dg=a_{u}(u,\phi)du+a_{\phi}(u,\phi)d\phi.$

We choose $h=e^{-rP_{0}}$ asymptotically and hence
$\dot{h}(u,r)=\frac{\partial h(u,r)}{\partial u}=0$ at the boundary
$r\rightarrow\infty$. By this choice the dependence on the radial coordinate
is completely absorbed by the group element $h$ and the boundary can be
assumed to be uniquely located at any arbitrary fixed value of $r=r_{0}$, in
particular to infinity. Thus the boundary describes a two-dimensional time-
like surface, described by coordinated $u,\phi$ with the topology of a
cylinder. The radial gauge choice implies that the solutions can further be
decomposed into,

$\displaystyle\Lambda^{\hat{p}}_{\,\,\,\hat{q}}$
$\displaystyle=(\lambda(u,\phi))^{\hat{p}}_{\,\,\,\hat{m}}(\zeta(u,r))^{\hat{m}}_{\,\,\,\,\hat{q}},~{}~{}(\Lambda^{-1})^{\hat{p}}_{\,\,\,\hat{q}}=(\zeta(u,r)^{-1})^{\hat{p}}_{\,\,\,\hat{m}}(\lambda(u,\phi)^{-1})^{\hat{m}}_{\,\,\,\,\hat{q}}$
$\displaystyle A^{\alpha}_{\,\,\,\beta}$
$\displaystyle=(\gamma(u,\phi))^{\alpha}_{\,\,\,\sigma}(\delta(u,r))^{\sigma}_{\,\,\,\beta},~{}~{}(A^{-1})^{\alpha}_{\,\,\,\beta}=(\delta(u,r)^{-1})^{\alpha}_{\,\,\,\sigma}(\gamma(u,\phi)^{-1})^{\sigma}_{\,\,\,\beta}$
$\displaystyle\eta_{1}^{\hat{p},\alpha}$
$\displaystyle=(\gamma(u,\phi))^{\alpha}_{\,\,\,\beta}\
[(\lambda(u,\phi))^{\hat{p}}_{\,\,\,\hat{q}}\tilde{d}_{1}^{\,\hat{q},\beta}(u,r)+d_{1}^{\,\hat{p},\beta}(u,\phi)]$
$\displaystyle\eta_{2,\alpha}^{\hat{p}}$
$\displaystyle=(\gamma(u,\phi)^{-1})_{\alpha}^{\,\,\,\beta}[(\lambda(u,\phi))^{\hat{p}}_{\,\,\,\hat{q}}\tilde{d}_{2,\beta}^{\,\hat{q}}(u,r)+d_{2,\beta}^{\,\hat{p}}(u,\phi)]$
$\displaystyle\bar{\eta}_{1}^{\hat{p},\alpha}$
$\displaystyle=(\gamma(u,\phi))^{\alpha}_{\,\,\,\beta}[{\bar{\tilde{d}}}_{1}^{\,\hat{q},\beta}(u,r)(\lambda(u,\phi)^{-1})_{\hat{q}}^{\,\,\,\hat{p}}+{\bar{d}}_{1}^{\,\hat{p},\beta}(u,\phi)]$
$\displaystyle\bar{\eta}_{2,\alpha}^{\hat{p}}$
$\displaystyle=(\gamma(u,\phi)^{-1})_{\alpha}^{\,\,\,\beta}[{\bar{\tilde{d}}}_{2,\beta}^{\,\hat{q}}(u,r)(\lambda(u,\phi)^{-1})_{\hat{q}}^{\,\,\,\hat{p}}+{\bar{d}}_{2,\beta}^{\,\hat{p}}(u,\phi)]$
$\displaystyle\tilde{C}_{\,\alpha,\beta}$
$\displaystyle=c(u,\phi)_{\alpha,\beta}+\tilde{c}(u,r)_{\alpha,\beta}$
$\displaystyle-\kappa\bar{d}_{1,\hat{p},\alpha}(u,\phi)(\lambda(u,\phi))^{\hat{p}}_{\,\,\,\hat{q}}\tilde{d}_{1,\beta}^{\,\hat{q}}(u,r)+\frac{1}{\kappa}\bar{d}_{2,\hat{p},\alpha}(u,\phi)(\lambda(u,\phi))^{\hat{p}}_{\,\,\,\hat{q}}\tilde{d}_{2,\beta}^{\,\hat{q}}(u,r)$
$\displaystyle b^{\,\hat{p}}_{\,\,\,\hat{q}}$
$\displaystyle=F(u,\phi)^{\hat{p}}_{\,\,\,\hat{q}}+(\lambda(u,\phi))^{\hat{p}}_{\,\,\,\hat{m}}(E(u,r))^{\hat{m}}_{\,\,\,\hat{n}}(\lambda(u,\phi)^{-1})^{\hat{n}}_{\,\,\,\hat{q}}$
$\displaystyle-\kappa\big{[}d_{1,\alpha}^{\hat{p}}(u,\phi)(\lambda^{-1}(u,\phi))_{\hat{q}}^{\,\,\,\hat{m}}{\bar{\tilde{d}}}_{1,\hat{m}}^{\,\alpha}(u,r)+(\lambda(u,\phi)^{-1})_{\hat{r}}^{\,\,\,\hat{m}}{\bar{\tilde{d}}}_{1,\hat{m}}^{\,\alpha}(u,r)d_{1,\alpha}^{\,\,\,\hat{r}}\delta^{\hat{p}}_{\,\,\,\hat{q}}\big{]}$
$\displaystyle-\frac{1}{\kappa}\big{[}d_{2,\alpha}^{\hat{p}}(u,\phi)(\lambda^{-1}(u,\phi))_{\hat{q}}^{\,\,\,\hat{m}}{\bar{\tilde{d}}}_{2,\hat{m}}^{\,\alpha}(u,r)+(\lambda(u,\phi)^{-1})_{\hat{r}}^{\,\,\,\hat{m}}{\bar{\tilde{d}}}_{2,\hat{m}}^{\,\alpha}(u,r)d_{2,\alpha}^{\,\,\,\hat{r}}\delta^{\hat{p}}_{\,\,\,\hat{q}}\big{]}.$
(10)

The constant $\kappa$ appearing in the above equations can be found as follows
: since $A$ is a matrix in the internal algebra space, it can be expanded in
terms of generators
$(\lambda^{\bar{a}})^{\alpha}_{\,\,\,\gamma}:\,A^{\alpha}_{\,\,\,\beta}=\sum_{\bar{b}}X_{\bar{b}}(\lambda^{\bar{b}})^{\alpha}_{\,\,\,\beta}.$
Similarly we have
$(A^{-1})^{\alpha}_{\,\,\,\beta}=\sum_{\bar{a}}Y_{\bar{a}}(\lambda^{\bar{a}})^{\alpha}_{\,\,\,\beta}$.
Now demanding $(A^{-1}A)^{\alpha}_{\,\,\,\beta}=\delta^{\alpha}_{\beta}$ we
get

$\displaystyle\sum_{\bar{a},\bar{b}}Y_{\bar{a}}X_{\bar{b}}[f^{\bar{a}\bar{b}\bar{c}}(\lambda_{\bar{c}})^{\alpha}_{\,\,\,\beta}+c_{1}\nabla^{\bar{a}\bar{b}}\delta^{\alpha}_{\beta}]=\delta^{\alpha}_{\beta},$
(11)

$c_{1}$ is a constant of the group and $\nabla^{\bar{a}\bar{b}}$ is the
commutator of the $\lambda^{\bar{a}}$ matrices. To solve the above constraint
we need $Y_{\bar{a}}=\kappa X_{\bar{a}}$ which implies $A^{-1}=\kappa A$, for
some constant $\kappa$.

In the above set of equations (3.1), the $u-$ derivative of all the fields
$\zeta,\delta,\tilde{d}_{1,2},E,\tilde{c}$ are set to zero value at the
boundary $r\rightarrow\infty$ . Thus, at the boundary these field are fixed
and have no dynamics. On the other hand $\lambda,\gamma,d_{1,2},F$ and $c$ are
the residual degrees that remain unfixed by classical eoms. In the next
section we present the theory that governs the dynamics of these residual
fields.

##  4 The dual Wess-Zumino-Witten Model

In order to write the dual WZW theory at the boundary we reduce the above
Chern-Simons action in its Hamiltonian form. For a well defined variational
principle, actions generically require a boundary term to be added. In this
case the term looks like,

$\delta I_{bdy}=-\frac{k}{2\pi}\int du\wedge d\phi\wedge
dr\,\partial_{r}\left\langle a_{u},\delta a_{u}\right\rangle.$ (12)

In principle there would be another term which we omit by choosing radial
gauge as sataed earlier : $\mathcal{A}(u,r,\phi)=h^{-1}(r)(a(u,\phi)+d)h(r)$.
To evaluate this boundary term we first need to fix the asymptotic structure
of the CS gauge field. The bosonic part of the field $\mathcal{A}$ is fixed by
demanding that we have asymptotically flat metric at the boundary. The
asymptotic form of the residual gauge field is:

$\displaystyle a$
$\displaystyle=\sqrt{2}(J_{1}+\mathcal{P}J_{0}+\mathcal{J}P_{0}-\frac{\pi}{k}\psi_{1,\alpha}Q^{1,\alpha}_{+}-\frac{\pi}{k}\psi_{2,\alpha}Q^{2,\alpha}_{+}-\frac{2\pi}{k}\mathcal{Z}_{a}T^{a}-\frac{2\pi}{k}\mathcal{T}_{a}Z^{a})\,d\phi$
$\displaystyle+(\sqrt{2}P_{1}+\frac{8\pi}{k}\mathcal{Z}_{a}Z^{a}+\mathcal{P}_{a}P^{a})du.$
(13)

The above form implies the following constraints on the asymptotic fields:

$\displaystyle e^{a}_{u}=\omega^{a}_{\phi}\hskip 20.0pt\omega^{a}_{u}=0\hskip
20.0pt\psi^{I,\alpha}_{u}=0$ $\displaystyle B_{u}=0\hskip
20.0ptC^{\bar{a}}_{u}=-4B^{\bar{a}}_{\phi}.$ (14)

The $\phi$ component of the gauge field (4) is further constrained as,

$\displaystyle\hat{\omega}_{\phi}^{1}=\sqrt{2}~{};~{}\omega_{\phi}^{2}=0~{};~{}\psi_{\phi,\alpha}^{1-}=\psi_{\phi,\alpha}^{2-}=0~{};~{}e_{\phi}^{1}=e_{\phi}^{2}=0.$
(15)

With these the surface term becomes,

$I_{bdy}=-\frac{k}{4\pi}\int
du\,d\phi[\omega^{a}_{\phi}\omega_{a\phi}-4B^{\bar{a}}_{\phi}B_{\bar{a}\phi}]$
(16)

and the total action takes the following form:

$\displaystyle I_{tot}$ $\displaystyle=\frac{k}{4\pi}\int
du\,d\phi\left\langle a_{\phi},a_{u}\right\rangle-\frac{k}{4\pi}\int
du\,d\phi[\omega^{a}_{\phi}\omega_{a\phi}-4B^{\bar{a}}_{\phi}B_{\bar{a}\phi}]+\frac{1}{3}\int\left\langle
A,A^{2}\right\rangle$ $\displaystyle=$ $\displaystyle\frac{k}{4\pi}\Big{[}\int
du\,d\phi~{}\big{[}\omega^{a}_{\phi}e_{a,u}+e^{a}_{\phi}\omega_{a,u}+\mu_{1}\omega^{a}_{u}\omega_{a,\phi}+2C_{\hat{p}\hat{q}}\psi^{\alpha,\hat{p}}_{I,u}\psi^{\hat{q}}_{I,\alpha,\phi}$
$\displaystyle+12\hat{\alpha}(C^{\bar{a}}_{\phi}B_{\bar{a},u}+B^{\bar{a}}_{\phi}C_{\bar{a},u}+\mu_{2}B^{\bar{a}}_{\phi}B_{\bar{a},u})-\omega^{a}_{\phi}\omega_{a\phi}+4B^{\bar{a}}_{\phi}B_{\bar{a}\phi}\big{]}$
$\displaystyle\,\,+\frac{1}{6}\int\big{[}3\epsilon_{abc}\omega^{a}\omega^{b}e^{c}+\mu_{1}\epsilon_{abc}\omega^{a}\omega^{b}\omega^{c}+\frac{3}{2}\omega^{a}(C\Gamma_{a})_{\hat{p}\hat{q}}\psi^{\alpha,\hat{p}}_{I}\psi^{\hat{q}}_{I,\alpha}$
$\displaystyle+\hskip
10.0pt6(\lambda^{\bar{a}})^{\alpha\beta}B^{\bar{a}}C_{\hat{p}\hat{q}}(\psi^{\hat{p}}_{1,\alpha}\psi^{\hat{q}}_{1,\beta}-\psi^{\hat{p}}_{2,\alpha}\psi^{\hat{q}}_{2,\beta})\nolinebreak+12\hat{\alpha}f_{\bar{a}\bar{b}\bar{c}}(3B^{\bar{a}}B^{\bar{b}}C^{\bar{c}}+\mu_{2}B^{\bar{a}}B^{\bar{b}}B^{\bar{c}})\big{]}\Big{]}.$
(17)

We shall evaluate the above action on the classical solutions obtained in the
last section. The last two terms of (17) simplify as follows,

$\displaystyle
6(B^{\bar{a}}(\lambda^{\bar{a}})^{\alpha\beta}C_{\hat{p}\hat{q}}(\psi^{\hat{p}}_{1,\alpha}\psi^{\hat{q}}_{1,\beta}-\psi^{\hat{p}}_{2,\alpha}\psi^{\hat{q}}_{2,\beta})$
$\displaystyle=-6~{}Tr[(A^{-1})^{\alpha}_{\,\,\,\sigma}(dA)^{\sigma}_{\,\,\,\gamma}((A^{-1})^{\gamma}_{\,\,\,\xi}d\bar{\eta}_{1,\hat{q}}^{\xi}(A^{-1})_{\alpha}^{\,\,\,\zeta}d\eta_{1,\zeta}^{\hat{q}}$
$\displaystyle-A^{\gamma}_{\,\,\,\xi}d\bar{\eta}_{2,\hat{q}}^{\xi}A_{\alpha}^{\,\,\,\zeta}d\eta_{2,\zeta}^{\hat{q}})]$

and

$\displaystyle 12\hat{\alpha}f_{\bar{a}\bar{b}\bar{c}}\times
3B^{\bar{a}}B^{\bar{b}}C^{\bar{c}}=$
$\displaystyle~{}6\times\Big{(}\frac{2D}{d(d-1)}\Big{)}Tr[d\\{(A^{-1})^{\alpha}_{\,\,\,\sigma}(dA)^{\sigma}_{\,\,\,\gamma}((A^{-1})^{\gamma}_{\,\,\,\xi}\bar{\eta}_{1,\hat{q}}^{\xi}(A^{-1})_{\alpha}^{\,\,\,\zeta}d\eta_{1,\zeta}^{\hat{q}}$
$\displaystyle-A^{\gamma}_{\,\,\,\xi}\bar{\eta}_{2,\hat{q}}^{\xi}A_{\alpha}^{\,\,\,\zeta}d\eta_{2,\zeta}^{\hat{q}})\\}]$
$\displaystyle+6\times\Big{(}\frac{2D}{d(d-1)}\Big{)}Tr[(A^{-1})^{\alpha}_{\,\,\,\sigma}(dA)^{\sigma}_{\,\,\,\gamma}((A^{-1})^{\gamma}_{\,\,\,\xi}d\bar{\eta}_{1,\hat{q}}^{\xi}(A^{-1})_{\alpha}^{\,\,\,\zeta}d\eta_{1,\zeta}^{\hat{q}}$
$\displaystyle-A^{\gamma}_{\,\,\,\xi}d\bar{\eta}_{2,\hat{q}}^{\xi}A_{\alpha}^{\,\,\,\zeta}d\eta_{2,\zeta}^{\hat{q}})].$

Finally the on-shell action after possible simplifications evaluates to,

$\displaystyle I_{tot}=$ $\displaystyle\frac{k}{4\pi}\Bigg{[}\int
dud\phi\Big{[}2\mu_{1}(\Lambda^{-1}\dot{\Lambda})^{\hat{p}}_{\,\,\,\hat{m}}(\Lambda^{-1}\Lambda^{\prime})^{\hat{m}}_{\,\,\,\,\hat{p}}+2((A^{-1})^{\alpha}_{\,\,\,\beta}(\bar{\eta}_{1,\hat{p}}^{\beta})^{\prime}(A^{-1})_{\alpha}^{\,\,\,\sigma}\dot{\eta}^{\hat{p}}_{1,\sigma}+A^{\alpha}_{\,\,\,\beta}(\bar{\eta}_{2,\hat{p}}^{\beta})^{\prime}A_{\alpha}^{\,\,\,\sigma}\dot{\eta}^{\hat{p}}_{2,\sigma})$
$\displaystyle+4(\dot{\Lambda})^{\hat{p}}_{\,\,\,\hat{m}}(\Lambda^{-1})^{\hat{m}}_{\,\,\,\,\hat{q}}\Big{(}\frac{1}{2}((A^{-1})^{\alpha\beta}\eta^{\hat{q}}_{1,\beta}(A^{-1})_{\alpha\sigma}({\bar{\eta}}^{\sigma}_{1,\hat{p}})^{\prime}+\frac{1}{2}(A^{-1})_{\alpha\sigma}(\bar{\eta}^{\sigma}_{1,\hat{n}})^{\prime}(A^{-1})^{\alpha\beta}\eta^{\hat{n}}_{1,\beta}\delta^{\hat{q}}_{\hat{p}})$
$\displaystyle+\frac{1}{2}(A^{\alpha\beta}\eta^{\hat{q}}_{2,\beta}A_{\alpha\sigma}({\bar{\eta}}^{\sigma}_{2,\hat{p}})^{\prime}+\frac{1}{2}A_{\alpha\sigma}(\bar{\eta}^{\sigma}_{2,\hat{n}})^{\prime}A^{\alpha\beta}\eta^{\hat{n}}_{2,\beta}\delta^{\hat{q}}_{\hat{p}})+(b^{\prime})^{\hat{q}}_{\,\,\,\hat{p}}\Big{)}-\frac{8D}{d(d-1)}\times(A^{-1}\dot{A})^{\alpha}_{\,\,\,\sigma}({\tilde{C}}^{\prime})^{\sigma}_{\,\,\,\alpha}$
$\displaystyle-2(\Lambda^{-1}\Lambda^{\prime})^{\hat{p}}_{\,\,\,\hat{m}}(\Lambda^{-1}\Lambda^{\prime})^{\hat{m}}_{\,\,\,\hat{p}}+4(A^{-1}A^{\prime})^{\alpha}_{\,\,\,\sigma}(A^{-1}A^{\prime})^{\sigma}_{\,\,\,\alpha}-2\mu_{2}\times\frac{2D}{d(d-1)}\times(A^{-1}A^{\prime})^{\alpha}_{\,\,\,\sigma}(A^{-1}\dot{A})^{\sigma}_{\,\,\,\alpha}\Big{]}$
$\displaystyle+\frac{1}{6}\int
drdud\phi\Big{[}6\Big{(}\frac{2D}{d(d-1)}-1\Big{)}~{}(A^{-1})^{\alpha}_{\,\,\,\sigma}(dA)^{\sigma}_{\,\,\,\gamma}((A^{-1})^{\gamma}_{\,\,\,\xi}d\bar{\eta}_{1,\hat{q}}^{\xi}(A^{-1})_{\alpha}^{\,\,\,\zeta}d\eta_{1,\zeta}^{\hat{q}}-A^{\gamma}_{\,\,\,\xi}d\bar{\eta}_{2,\hat{q}}^{\xi}A_{\alpha}^{\,\,\,\zeta}d\eta_{2,\zeta}^{\hat{q}})$
$\displaystyle+6\times\Big{(}\frac{2D}{d(d-1)}\Big{)}d\\{(A^{-1})^{\alpha}_{\,\,\,\sigma}(dA)^{\sigma}_{\,\,\,\gamma}((A^{-1})^{\gamma}_{\,\,\,\xi}\bar{\eta}_{1,\hat{q}}^{\xi}(A^{-1})_{\alpha}^{\,\,\,\zeta}d\eta_{1,\zeta}^{\hat{q}}-A^{\gamma}_{\,\,\,\xi}\bar{\eta}_{2,\hat{q}}^{\xi}A_{\alpha}^{\,\,\,\zeta}d\eta_{2,\zeta}^{\hat{q}})\\}\Big{]}$
$\displaystyle+\frac{2\mu_{1}}{3}\int
Tr\big{[}(\Lambda^{-1}d\Lambda)^{3}\big{]}-4\mu_{2}\times\frac{2D}{d(d-1)}\times\int
Tr\big{[}(A^{-1}dA)^{3}\big{]}.\Bigg{]}$ (18)

We have used the property
$\overline{(\Lambda^{-1})^{\hat{p}}_{\,\,\,\hat{q}}\psi^{\hat{q}}_{1,\alpha}}=\overline{\psi^{\hat{q}}_{1,\alpha}}~{}\Lambda_{\hat{q}}^{\,\,\,\hat{p}}$,
where $\psi^{\hat{p}}_{1,\alpha}$ is a fermion and
$\Lambda^{\hat{p}}_{\,\,\,\hat{q}}$ is a $SL(2,R)$ group element. From the
index structure of the above expression, it is clear that they form a trace.
So far our analysis holds true for all maximally $\mathcal{N}$ extended super-
Poincaré algebras of section 2. As it is evident, the first four lines on the
right hand side of (4) are two dimensional terms whereas the remaining ones
are three dimensional. Among the three dimensional terms, the second one is a
total derivative and hence, can be written down as a two dimensional term up
to a boundary contribution, that we ignore eventually. The last two terms are
genuine three dimensional terms and they signify that, in presence of
arbitrary constants $\mu_{1},\mu_{2}$, they survive. But due to their WZW
structure, their variations are purely two dimensional. Only problematic term
is the first one. It is a combination of two expressions which are neither
total derivatives nor exact WZW types. We see that for the special case of the
$so(\mathcal{N})$ internal algebra, where $d$ and $D$ are related as
$D=\frac{d(d-1)}{2}$, these two expressions cancel with each other, leaving
only the total derivative term. However, this simplification is not possible
for generic groups, where this relation does not hold. Thus for
$so(\mathcal{N})$ automorphism algebra the above action simplifies to

$\displaystyle I_{tot}=$ $\displaystyle\frac{k}{4\pi}\Bigg{[}\int
dud\phi~{}Tr\Big{[}2\mu_{1}(\Lambda^{-1}\dot{\Lambda})(\Lambda^{-1}\Lambda^{\prime})+2(A^{-1}\bar{\eta_{1}}^{\prime}A^{-1}\dot{\eta_{1}}+A\bar{\eta_{2}}^{\prime}A\dot{\eta_{2}})$
$\displaystyle+2\dot{\Lambda}\Lambda^{-1}\Big{(}A^{-1}\eta_{1}A^{-1}\bar{\eta}_{1}^{\prime}+A^{-1}\bar{\eta}_{1}^{\prime}A^{-1}\eta_{1}\text{I}+A\eta_{2}A\bar{\eta}_{2}^{\prime}+A\bar{\eta}_{2}^{\prime}A\eta_{2}\text{I}+b^{\prime}\Big{)}$
$\displaystyle-2\Lambda^{-1}\Lambda^{\prime}\Lambda^{-1}\Lambda^{\prime}+4A^{-1}A^{\prime}A^{-1}A^{\prime}-2\mu_{2}A^{-1}A^{\prime}A^{-1}\dot{A}-4A^{-1}\dot{A}{\tilde{C}}^{\prime}$
$\displaystyle+A^{-1}dA(A^{-1}\bar{\eta}_{1}A^{-1}d\eta_{1}-A\bar{\eta}_{2}Ad\eta_{2})\Big{]}$
$\displaystyle+\frac{2\mu_{1}}{3}\int
Tr\big{[}(\Lambda^{-1}d\Lambda)^{3}\big{]}-4\mu_{2}\int
Tr\big{[}(A^{-1}dA)^{3}\big{]}\Bigg{]}.$ (19)

Although the WZW term is a closed form, its variation being zero, the
reduction to a total derivative exact form is not possible in general due to
topological obstructions Barnich:2013yka ; Nappi:1993ie ; Bernstein:1988zd ;
Salomonson:1988mk ; Stone:1989vg . This is why we have the last two explicit
three dimensional terms in the above action. However their variations are
purely two dimensional. For the specific case superalgebra, with
$so(\mathcal{N})$ automorphism every other terms of the action reduces to
purely 2D terms, as we have shown here. For this particular group the constant
$\kappa$ of equation (3.1) is a pure phase and can be set to unity without any
loss of generality333When the internal symmetry algebra is $so(\mathcal{N})$,
the matrix $A$ has unit determinant, thus $\kappa=e^{2\pi i/d},d=\mathcal{N}$.
It can be absorbed in the fermions as overall phase.. In the next section, we
will hence construct the Liouville like theory for this particular case.

Further using the gauge decomposed forms of the solutions and neglecting total
derivatives terms the above action rightly simplifies to,

$\displaystyle I_{tot}$ $\displaystyle=\frac{k}{4\pi}\bigg{[}\int
dud\phi~{}Tr\Big{[}2\mu_{1}\lambda^{-1}\dot{\lambda}\lambda^{-1}\lambda^{\prime}+2\Big{(}\,\gamma\,{\bar{d}_{1}}^{\prime}\,\gamma\dot{d}_{1}+\gamma^{-1}\,{\bar{d}_{2}}^{\prime}\,\gamma^{-1}\,\dot{d}_{2}\Big{)}$
$\displaystyle+2\dot{\lambda}\lambda^{-1}\Big{(}\,\gamma\,d_{1}\gamma\,\bar{d}_{1}^{\prime}+\gamma^{-1}\,d_{2}\,\gamma^{-1}\,\bar{d}_{2}^{\prime}+F^{\prime}\Big{)}-2\lambda^{-1}\lambda^{\prime}\lambda^{-1}\lambda^{\prime}$
$\displaystyle+4\gamma^{-1}\gamma^{\prime}\gamma^{-1}\gamma^{\prime}-2\mu_{2}\gamma^{-1}\gamma^{\prime}\gamma^{-1}\dot{\gamma}-\gamma^{-1}\gamma^{\prime}\Big{(}\gamma{\bar{d}}_{1}\gamma\dot{d}_{1}-\gamma^{-1}{\bar{d}}_{2}\gamma^{-1}\dot{d}_{2}\Big{)}$
$\displaystyle+\frac{1}{2}\gamma^{-1}\gamma^{\prime}{\dot{\lambda}}\lambda^{-1}\Big{(}\gamma
d_{1}\gamma\bar{d}_{1}+\gamma^{-1}d_{2}\gamma^{-1}\bar{d}_{2}\Big{)}-\gamma^{-1}\dot{\gamma}c^{\prime}\Big{]}$
$\displaystyle+\frac{2\mu_{1}}{3}\int
Tr\big{[}(\Lambda^{-1}d\Lambda)^{3}\big{]}-4\mu_{2}\int
Tr\big{[}(A^{-1}dA)^{3}\big{]}\bigg{]}.$ (20)

The above $1+1$ dimensional action (4) describes the dual theory of $2+1$
dimensional asymptotically flat supergravity theory with $so(\mathcal{N})$ as
its bulk automorphism symmetry for arbitrary $\mathcal{N}$. It is a chiral WZW
model with two WZW terms, that belong to $SL(2,R)$ and the internal
automorphism group $so(\mathcal{N})$ of the theory. Following Barnich:2015sca
and Banerjee:2018hbl , one can check that the boundary theory has a global and
a gauge symmetry. The global symmetry corresponds to $2+2D$ types of bosonic
transformations generators and $2d$ types of fermionic transformation
generators. The Noether currents corresponding to these symmetry
transformations are only non-trivial along the $u-$ direction and hence they
directly give the corresponding Noether charges. These global symmetry charges
give rise to affine extended super-Poincare symmetry, identical to the bulk
symmetry of the higher dimensional supergravity theory. Furthermore using
usual CFT techniques of modified Sugawara constructions, the current algebra
can be used to find the infinite dimensional mode algebra. Since the boundary
theory is a gauge theory one has to take the suitable gauge invariant
combinations of charges for this purpose. This analysis is straightforward but
algebraically tedious and we do not present the details in this work. The
resultant infinite dimensional mode algebra is identical to the particular
$\mathcal{N}-$extended superBMS3 algebra, a suitable flat limit of SCA
$Osp(\mathcal{N}|2,R)$. This is the asymptotic symmetry algebra of the
corresponding $2+1$ dimensional supergravity theory and is given as,

$\displaystyle[\hat{\mathfrak{J}}_{n},\hat{\mathfrak{J}}_{m}]$
$\displaystyle=(n-m)\hat{\mathfrak{J}}_{n+m}+\frac{c_{J}}{12}n^{3}\delta_{n+m,0}\;,\quad[\hat{\mathfrak{J}}_{n},\hat{\mathfrak{M}}_{m}]=(n-m)\hat{\mathfrak{M}}_{n+m}+\frac{c_{M}}{12}n^{3}\delta_{n+m,0}\;,$
$\displaystyle[\hat{\mathfrak{J}}_{n},\psi^{(1,2),\alpha}_{m}]$
$\displaystyle=\left(\frac{n}{2}-m\right)\psi^{(1,2),\alpha}_{n+m}\;,\qquad[\hat{\mathfrak{J}}_{n},R^{a}_{m}]=-mR^{a}_{n+m}\;,\qquad\quad[\hat{\mathfrak{J}}_{n},S^{a}_{m}]=-mS^{a}_{n+m}\;,$
$\displaystyle[R^{a}_{n},R^{b}_{m}]$
$\displaystyle=n\,\hat{\alpha}\,c_{R}\delta^{ab}\delta_{n+m,0}+if^{abc}R^{c}_{n+m}\;,\quad[R^{a}_{n},S^{b}_{m}]=n\,\hat{\alpha}\,c_{M}\delta^{ab}\delta_{n+m,0}+if^{abc}S^{c}_{n+m}\;,$
$\displaystyle[R^{a}_{n},r^{1,\alpha}_{p}]$
$\displaystyle=i(\lambda^{a})^{\alpha}_{\beta}r^{1,\beta}_{n+p}\;,\qquad\qquad\qquad\qquad\;\;[R^{a}_{n},r^{2,\alpha}_{p}]=-i(\lambda^{a})^{\alpha}_{\beta}r^{2,\beta}_{n+p}$
$\displaystyle\\{\psi^{1,\alpha}_{n},\psi^{1,\beta}_{m}\\}$
$\displaystyle=\frac{c_{M}}{6}n^{2}\delta_{n+m,0}\eta^{\alpha\beta}+\hat{\mathfrak{M}}_{n+m}\eta^{\alpha\beta}-\frac{i}{6\hat{\alpha}}(n-m)(\lambda^{a})^{\alpha\beta}S^{a}_{n+m}$
$\displaystyle-\frac{1}{48\hat{\alpha}}(S^{a}S^{a})_{n+m}\eta^{\alpha\beta}-\frac{1}{144\hat{\alpha}^{2}}\\{\lambda^{a},\lambda^{b}\\}^{\alpha\beta}\frac{1}{4}(S^{a}S^{b})_{n+m}\;,$
$\displaystyle\\{\psi^{2,\alpha}_{n},\psi^{2,\beta}_{m}\\}$
$\displaystyle=\frac{c_{M}}{6}n^{2}\delta_{n+m,0}\eta^{\alpha\beta}+\hat{\mathfrak{M}}_{n+m}\eta^{\alpha\beta}+\frac{i}{6\hat{\alpha}}(n-m)(\lambda^{a})^{\alpha\beta}S^{a}_{n+m}\;,$
$\displaystyle-\frac{1}{48\hat{\alpha}}(S^{a}S^{a})_{n+m}\eta^{\alpha\beta}-\frac{1}{144\hat{\alpha}^{2}}\\{\lambda^{a},\lambda^{b}\\}^{\alpha\beta}\frac{1}{4}(S^{a}S^{b})_{n+m}\;.$
(21)

In the above algebra, $\hat{\mathfrak{J}}_{n}$ are modes of stress tensor,
$\hat{\mathfrak{M}}_{n}$ are modes of a spin two current,
$R^{a}_{n},S^{a}_{n}$ are modes of two independent D numbers of spin one
currents and $\psi^{1,\alpha}_{n},\psi^{2,\beta}_{n}$ are modes of two
independent $d(=\mathcal{N})$ numbers of spin half fermionic currents of the
$1+1$ dimensional boundary theory.

###  4.1 Gauging the Chiral WZW Model

The fields of the boundary chiral WZW model are not independent. The
asymptotic boundary condition (15) of the bulk supergravity theory imposes
following constraints on them,

$\displaystyle(\lambda^{-1}\lambda^{\prime})^{1}=\sqrt{2},\quad(\lambda^{-1}\frac{\tilde{\alpha}}{2}\lambda)^{1}=0,$
$\displaystyle(\lambda^{-1}d_{1}^{\prime}+\gamma^{-1}\gamma^{\prime}\lambda^{-1}d_{1})^{-}_{\alpha}=0,$
$\displaystyle(\lambda^{-1}d_{2}^{\prime}-\gamma^{\prime}\gamma^{-1}\lambda^{-1}d_{2})^{-}_{\alpha}=0,$
(22)

where

$\displaystyle\tilde{\alpha}^{\hat{p}}_{\,\,\,\hat{q}}$
$\displaystyle=2(F^{\prime})^{\hat{p}}_{\,\,\,\hat{q}}+\gamma_{\alpha}^{\,\,\,\sigma}d^{\hat{p}}_{1,\sigma}\gamma^{\alpha}_{\,\,\,\delta}\bar{d}^{\delta\,\prime}_{1,\hat{q}}+(\gamma^{-1})_{\alpha}^{\,\,\,\sigma}d^{\hat{p}}_{2,\sigma}(\gamma^{-1})^{\alpha}_{\,\,\,\delta}\bar{d}^{\delta\,\prime}_{1,\hat{q}}$
(23)
$\displaystyle+(\gamma^{\prime}\gamma^{-1})^{\alpha}_{\,\,\,\beta}d^{\hat{p}}_{1,\alpha}\bar{d}^{\beta}_{1,\hat{q}}-(\gamma^{-1}\gamma^{\prime})^{\alpha}_{\,\,\,\beta}d^{\hat{p}}_{2,\alpha}\bar{d}^{\beta}_{2,\hat{q}}.$

Here we have listed only the $2d+2$ number of first class constraints. The
above constraints (4.1) can also be expressed in terms the global charges,
although the explicit relation is not important for our analysis. It sets two
charges corresponding to two bosonic transformations along $\Gamma_{0}$ to
constant values and $2d$ number of fermionic charges correcponding to $2d$
fermionic generators with lower positive component to zero values.We need to
introduce $2d+2$ number of gauge fields to to gauge the corresponding
summetries. The details of the gauging prescription for a WZW model can be
found in FEHER19921 ; Henneaux:1999ib ; Barnich_2013 ; Barnich:2015sca . Here
we just present the gauged action as,

$I[\lambda,F,\gamma,c,d_{1},d_{2},\Psi_{1},\Psi_{2},\hat{A}_{\mu},\tilde{A}_{\mu}]=I[\lambda,F,\gamma,c,d_{1},d_{2},\Psi_{1},\Psi_{2}]+I_{gauge},$
(24)

where $I[\lambda,F,\gamma,c,d_{1},d_{2},\Psi_{1},\Psi_{2}]$ is the boundary
chiral WZW action as in (4) and the local gauge fixing term $I_{gauge}$ is
given as,

$\displaystyle I_{gauge}$ $\displaystyle=\frac{k}{\pi}\int
dud\phi\Big{[}\hat{A}(\lambda^{-1}\frac{\tilde{\alpha}}{2}\lambda)+\tilde{A}(\lambda^{-1}\lambda^{\prime}-\mu_{M})$
(25)
$\displaystyle+(\lambda^{-1}d_{1}^{\prime}+\gamma^{-1}\gamma^{\prime}\lambda^{-1}d_{1})^{-}_{\alpha}\Psi_{1}^{\alpha}+(\lambda^{-1}d_{2}^{\prime}-\gamma^{\prime}\gamma^{-1}\lambda^{-1}d_{2})^{-}_{\alpha}\Psi_{2}^{\alpha}\Big{]}.$

Here $\hat{A},\tilde{A},\Psi_{i}^{\alpha},i=1,2$ are the $2d+2$ local gauge
fields. More over $\hat{A},\tilde{A}$ are along $\Gamma_{0}$ generator and
$\Psi_{i,+}^{\alpha}=0$ . The constant $\mu_{M}=\mu\Gamma_{1}$ is introduced
so that the charge can be set to the required constant value.

##  5 The Reduced Phasespace and the Liouville-like theory

In order to find the reduced phase space description of the WZW model of (4)
we follow the standard procedure of Barnich:2015sca ; Banerjee:2019epe . Thus
we expand the fields in the Chevalley-Serre basis of the corresponding gauge
group. This is also known as Gauss Decomposition of fields. In our case, we
only know the basis generstors for SL(2, R) explicitly, where as the same for
the internal automorphism group it is unknown. Thus for the present case our
decomposition is,

$\displaystyle\lambda=e^{\frac{\sigma\Gamma_{1}}{2}}e^{-\frac{\varphi\Gamma_{2}}{2}}e^{\tau\Gamma_{0}},\quad
F=-(\frac{\eta}{2}\Gamma_{0}+\frac{\theta}{2}\Gamma_{2}+\frac{\zeta}{2}\Gamma_{1}),$
(26)

where $\sigma,\varphi,\tau,\eta,\theta$ are scalar fields and are functions of
both $u,\phi$. Other fields are scalars with respect to $SL(2,R)$. The
Gaussian decomposition is useful as in this decompostion one part of the 3D
bulk part of the WZW model simplifies to a total derivative term as,

$\displaystyle\frac{2}{3}Tr[(d\Lambda\Lambda^{-1})^{3}]=drdud\phi~{}\epsilon^{\nu\gamma\delta}\partial_{\nu}(e^{-\phi}~{}\partial_{\gamma}\tau~{}\partial_{\delta}\sigma).$
(27)

Thus using Stoke’s formula the bulk term can be reduced to a 2D term. Two
product operators that are mostly used are given by,

$\displaystyle\lambda^{-1}\lambda=\begin{bmatrix}-\sigma^{\prime}\tau
e^{-\phi}-\phi^{\prime}/2&-\sqrt{2}\sigma^{\prime}\tau^{2}e^{-\phi}+\sqrt{2}\tau^{\prime}-\sqrt{2}\tau\phi^{\prime}\\\
\frac{\sigma^{\prime}}{\sqrt{2}}e^{-\phi}&\tau\sigma^{\prime}e^{-\phi}+\phi^{\prime}/2\\\
\end{bmatrix}$ (28)

and

$\displaystyle\dot{\lambda}\lambda^{-1}=\begin{bmatrix}-\frac{\dot{\sigma}}{2}-\sigma\dot{\tau}e^{-\phi}&\sqrt{2}\dot{\tau}e^{-\phi}\\\
\frac{\dot{\sigma}}{\sqrt{2}}-\frac{\sigma}{\sqrt{2}}\dot{\phi}-\frac{\sigma^{2}}{\sqrt{2}}\dot{\tau}e^{-\phi}&\frac{\dot{\phi}}{2}+\sigma\dot{\tau}e^{-\phi}\\\
\end{bmatrix}.$ (29)

Putting the above data, the reduced action (4) can be the written as,

$\displaystyle I=$ $\displaystyle\frac{k}{4\pi}\bigg{[}\int
dud\phi\Big{[}\mu_{1}\varphi^{\prime}\dot{\varphi}-2\dot{\varphi}\theta^{\prime}-4\sigma\dot{\tau}e^{-\varphi}\theta^{\prime}+4\dot{\tau}\zeta^{\prime}e^{-\varphi}+2\dot{\sigma}\eta^{\prime}-2\sigma\dot{\varphi}\eta^{\prime}-2\sigma^{2}\dot{\tau}e^{-\varphi}\eta^{\prime}-{\varphi^{\prime}}^{2}$
$\displaystyle+\Big{\\{}2(\gamma^{\alpha}_{\,\,\,\sigma}({\bar{d}_{1,\hat{p}}^{\sigma}})^{\prime}\gamma_{\alpha}^{\,\,\,\delta}\dot{d}_{1,\delta}^{\hat{p}}+(\gamma^{-1})^{\alpha}_{\,\,\,\sigma}({\bar{d}_{2,\hat{p}}^{\sigma}})^{\prime}(\gamma^{-1})_{\alpha}^{\,\,\,\delta}\dot{d}_{2,\delta}^{\hat{p}})$
$\displaystyle+4(\dot{\lambda})^{\hat{p}}_{\,\,\,\hat{m}}(\lambda^{-1})^{\hat{m}}_{\,\,\,\hat{q}}(\frac{1}{2}\gamma^{\alpha}_{\,\,\,\sigma}d^{\hat{q},\sigma}_{1}\gamma_{\alpha}^{\,\,\,\delta}(\bar{d}_{1,\hat{p},\delta})^{\prime}+\frac{1}{2}(\gamma^{-1})^{\alpha}_{\,\,\,\sigma}d^{\hat{q},\sigma}_{2}(\gamma^{-1})_{\alpha}^{\,\,\,\delta}(\bar{d}_{2,\hat{p},\delta})^{\prime}+(F^{\prime})^{\hat{q}}_{\,\,\,\hat{p}})$
$\displaystyle+4(\gamma^{-1}\gamma^{\prime})^{\alpha}_{\,\,\,\beta}(\gamma^{-1}\gamma^{\prime})^{\beta}_{\,\,\,\alpha}-2\mu_{2}(\gamma^{-1}\gamma^{\prime})^{\alpha}_{\,\,\,\beta}(\gamma^{-1}\dot{\gamma})^{\beta}_{\,\,\,\alpha}-(\gamma^{-1}\dot{\gamma})^{\alpha}_{\,\,\,\beta}(c^{\prime})^{\beta}_{\,\,\,\alpha}$
$\displaystyle-(\gamma^{-1}\gamma^{\prime})^{\alpha}_{\,\,\,\beta}(\gamma^{\beta}_{\,\,\,\sigma}{\bar{d}}^{\hat{p},\sigma}_{1}\gamma_{\alpha}^{\,\,\,\delta}\dot{d}_{1,\hat{p},\delta}-(\gamma^{-1})^{\beta}_{\,\,\,\sigma}{\bar{d}}^{\hat{p},\sigma}_{2}(\gamma^{-1})_{\alpha}^{\,\,\,\delta}\dot{d}_{2,\hat{p},\delta})$
$\displaystyle+(\gamma^{-1}\gamma^{\prime})^{\alpha}_{\,\,\,\beta}({\dot{\lambda}}\lambda^{-1})^{\hat{p}}_{\,\,\,\hat{q}}(\frac{1}{2}\gamma^{\beta}_{\,\,\,\sigma}d^{\hat{q},\sigma}_{1}\gamma_{\alpha}^{\,\,\,\delta}\bar{d}_{1,\hat{p},\delta}+\frac{1}{2}(\gamma^{-1})^{\beta}_{\,\,\,\sigma}d^{\hat{q},\sigma}_{2}(\gamma^{-1})_{\alpha}^{\,\,\,\delta}\bar{d}_{2,\hat{p},\delta})\Big{\\}}\Big{]}$
$\displaystyle-4\mu_{2}\int Tr[(A^{-1}dA)^{3}]\bigg{]}.$ (30)

The explicit index structure of the above equation finally gives a trace. The
first class constraints (4.1) can be written in terms of these newly defined
fields. Let us first look at $(\lambda^{-1}\lambda^{\prime})^{1}=\sqrt{2}$
condition. It reduces as,

$\displaystyle(\lambda^{-1}\lambda^{\prime})^{1}=\sqrt{2}\Rightarrow\sigma^{\prime}=\sqrt{2}e^{\varphi}.$
(31)

Next we look at fermionic current constraints in equation (4.1). They are
given as,

$\displaystyle(\lambda^{-1}d_{1}^{\prime}+\gamma^{-1}\gamma^{\prime}\lambda^{-1}d_{1})^{-}_{\alpha}=0\Rightarrow(\lambda^{-1}\gamma^{-1}{d_{1}^{N}}^{\prime})^{-}_{\alpha}=0,$
$\displaystyle(\lambda^{-1}d_{2}^{\prime}-\gamma^{\prime}\gamma^{-1}\lambda^{-1}d_{2})^{-}_{\alpha}=0\Rightarrow(\lambda^{-1}\gamma~{}{d_{2}^{N}}^{\prime})^{-}_{\alpha}=0.$

Redefining new fermionic parameters as
$d^{\hat{p}}_{1,\alpha}=(\gamma^{-1})_{\alpha}^{\,\,\,\beta}d_{1,\beta}^{N,\hat{p}}$
and
$d^{\hat{p}}_{2,\alpha}=\gamma_{\alpha}^{\,\,\,\beta}d_{2,\beta}^{N,\hat{p}}$,
the two above conditions can be written in a compact form as,

$\displaystyle(d_{1,\alpha}^{N-})^{\prime}=\frac{\sigma}{\sqrt{2}}(d_{1,\alpha}^{N+})^{\prime},~{}(d_{2,\alpha}^{N-})^{\prime}=\frac{\sigma}{\sqrt{2}}(d_{2,\alpha}^{N+})^{\prime}.$
(33)

Finally from the last constrain equation
$(\lambda^{-1}\frac{\hat{\alpha}}{2}\lambda)^{1}=0$ we get,

$\displaystyle\eta^{\prime}\sigma^{2}-2\zeta^{\prime}+2\sigma\theta^{\prime}=0.$
(34)

Further, using the redefinition of fermions and (31),(33),(34), the reduced
action can be written as,

$\displaystyle I$ $\displaystyle=\frac{k}{4\pi}\bigg{[}\int
dud\phi\Big{[}\mu_{1}\varphi^{\prime}\dot{\varphi}-2\dot{\varphi}\theta^{\prime}+2\dot{\sigma}\eta^{\prime}-2\sigma\dot{\varphi}\eta^{\prime}-{\varphi^{\prime}}^{2}$
$\displaystyle-2e^{\varphi}((\dot{d}_{1,\alpha}^{N+})(d_{1}^{N+,\alpha})+(\dot{d}_{2,\alpha}^{N+})(d_{2}^{N+,\alpha}))+\dot{\varphi}((d_{1,\alpha}^{N+})(d_{1}^{N+,\alpha})^{\prime}+(d_{2,\alpha}^{N+})(d_{2}^{N+,\alpha})^{\prime})$
$\displaystyle+Tr\Big{\\{}-2({\bar{d}}_{1}^{N})^{\prime}\dot{\gamma}\gamma^{-1}d_{1}^{N}-2{\bar{d}}_{1}^{N}\gamma^{\prime}\gamma^{-1}\dot{\gamma}\gamma^{-1}d_{1}^{N}+2{\bar{d}}_{1}^{N}\gamma^{\prime}\gamma^{-1}{\dot{d}}_{1}^{N}$
$\displaystyle+2({\bar{d}}_{2}^{N})^{\prime}\gamma^{-1}\dot{\gamma}d_{2}^{N}-2{\bar{d}}_{2}^{N}\gamma^{-1}\gamma^{\prime}\gamma^{-1}\dot{\gamma}d_{2}^{N}-2{\bar{d}}_{2}^{N}\gamma^{-1}\gamma^{\prime}{\dot{d}}_{2}^{N}$
$\displaystyle+4\gamma^{-1}\gamma^{\prime}\gamma^{-1}\gamma^{\prime}-2\mu_{2}\gamma^{-1}\gamma^{\prime}\gamma^{-1}\dot{\gamma}-\gamma^{-1}\dot{\gamma}c^{\prime}-\gamma^{-1}\gamma^{\prime}(\gamma{\bar{d}}_{1}\gamma\dot{d}_{1}-\gamma^{-1}{\bar{d}}_{2}\gamma^{-1}\dot{d}_{2})$
$\displaystyle+\gamma^{-1}\gamma^{\prime}{\dot{\lambda}}\lambda^{-1}(\frac{1}{2}\gamma
d_{1}\gamma\bar{d}_{1}+\frac{1}{2}\gamma^{-1}d_{2}\gamma^{-1}\bar{d}_{2})\Big{\\}}\Big{]}-4\mu_{2}\int
Tr[(A^{-1}dA)^{3}]\bigg{]}.$

The above equation is the reduced Hamiltonian form of the boundary theory. For
further simplification let us consider the two fermion terms with explicit
indices,

$\displaystyle I_{FF}$ $\displaystyle=(\frac{k}{4\pi})\int
dud\phi[\sqrt{2}\dot{\sigma}d_{1,\alpha}^{N+}(d_{1}^{N+,\alpha})^{\prime}+\sqrt{2}\dot{\sigma}d_{2,\alpha}^{N+}(d_{2}^{N+,\alpha})^{\prime}$
$\displaystyle-\sqrt{2}\sigma(\bar{d}_{1,\alpha}^{N+})^{\prime}\dot{d}_{1}^{N+,\alpha}+2(\bar{d}_{1,\alpha}^{N+})^{\prime}\dot{d}_{1}^{N-,\alpha}-\sqrt{2}\sigma(\bar{d}_{2,\alpha}^{N+})^{\prime}\dot{d}_{2}^{N+,\alpha}+2(\bar{d}_{2,\alpha}^{N+})^{\prime}\dot{d}_{2}^{N-,\alpha}].$
(35)

The last two terms of the above expression are same as 3rd and 4th terms up to
total $\phi,u$ derivatives. Thus we get,

$\displaystyle I_{FF}$ $\displaystyle=(\frac{k}{4\pi})\int
dud\phi[\sqrt{2}\dot{\sigma}d_{1,\alpha}^{N+}(d_{1}^{N+,\alpha})^{\prime}+\sqrt{2}\dot{\sigma}d_{2,\alpha}^{N+}(d_{2}^{N+,\alpha})^{\prime}$
$\displaystyle-2\sqrt{2}\sigma(\bar{d}_{1,\alpha}^{N+})^{\prime}\dot{d}_{1}^{N+,\alpha}-2\sqrt{2}\sigma(\bar{d}_{2,\alpha}^{N+})^{\prime}\dot{d}_{2}^{N+,\alpha}].$
(36)

The above four terms can be further simplified up to total derivatives and
reduction condition as,

$\displaystyle I_{FF}$ $\displaystyle=(\frac{k}{4\pi})\int
dud\phi[-2e^{\phi}(\dot{d}_{1,\alpha}^{N+}d_{1}^{N+,\alpha}+\dot{d}_{2,\alpha}^{N+}d_{2}^{N+,\alpha})].$
(37)

We redefine new fields as,
$\xi=2(\theta+\sigma\eta)+(d_{1,\alpha}^{N+}d_{1}^{N+,\alpha}+d_{2,\alpha}^{N+}d_{2}^{N+,\alpha})$
and $\chi_{i,\alpha}=e^{\phi/2}d_{i,\alpha}^{N+}$, where $i=1,2$. Finally we
can write the action as,

$\displaystyle I$ $\displaystyle=\frac{k}{4\pi}\bigg{[}\int
dud\phi\Big{[}\mu_{1}\varphi^{\prime}\dot{\varphi}-\dot{\varphi}\xi^{\prime}-{\varphi^{\prime}}^{2}-2(\dot{\chi}_{1}\chi_{1}+\dot{\chi}_{2}\chi_{2})$
$\displaystyle+Tr\Big{\\{}-2({\bar{d}}_{1}^{N})^{\prime}\dot{\gamma}\gamma^{-1}d_{1}^{N}-2{\bar{d}}_{1}^{N}\gamma^{\prime}\gamma^{-1}\dot{\gamma}\gamma^{-1}d_{1}^{N}+2{\bar{d}}_{1}^{N}\gamma^{\prime}\gamma^{-1}{\dot{d}}_{1}^{N}$
$\displaystyle+2({\bar{d}}_{2}^{N})^{\prime}\gamma^{-1}\dot{\gamma}d_{2}^{N}-2{\bar{d}}_{2}^{N}\gamma^{-1}\gamma^{\prime}\gamma^{-1}\dot{\gamma}d_{2}^{N}-2{\bar{d}}_{2}^{N}\gamma^{-1}\gamma^{\prime}{\dot{d}}_{2}^{N}$
$\displaystyle+4\gamma^{-1}\gamma^{\prime}\gamma^{-1}\gamma^{\prime}-2\mu_{2}\gamma^{-1}\gamma^{\prime}\gamma^{-1}\dot{\gamma}-\gamma^{-1}\dot{\gamma}c^{\prime}-\gamma^{-1}\gamma^{\prime}(\gamma{\bar{d}}_{1}\gamma\dot{d}_{1}-\gamma^{-1}{\bar{d}}_{2}\gamma^{-1}\dot{d}_{2})$
$\displaystyle+\gamma^{-1}\gamma^{\prime}{\dot{\lambda}}\lambda^{-1}(\frac{1}{2}\gamma
d_{1}\gamma\bar{d}_{1}+\frac{1}{2}\gamma^{-1}d_{2}\gamma^{-1}\bar{d}_{2})\Big{\\}}\Big{]}-4\mu_{2}\int
Tr[(A^{-1}dA)^{3}]\bigg{]}.$ (38)

In the above equation (5), the ${\dot{\lambda}}\lambda^{-1}$ needs to be
expressed as in (29). We have not put in the explicit form for simplicity. (5)
is the flat limit of a super Liouville theory.

##  6 Conclusion and Outlook

In this paper we have constructed the $1+1$ dimensional dual field theory for
the three-dimensional maximally $\mathcal{N}$-extended supergravity. We have
explicitly shown that the dual action can be expressed in a chiral WZW form
for the special case, where the internal automorphism group of the bulk
gravity superalgebra is $so(\mathcal{N})$ and its asymptotic symmetry algebra
is a suitable flat limit of particular SCA, $Osp(\mathcal{N}|2;R)$. Finally we
have performed the Hamiltonian reduction of the system and found that the
reduced phase space description of it is given by a supersymmetric Liouville
like theory in its flat limit.

As was shown in Banerjee:2018hbl , there are a few more $\mathcal{N}$-extended
supergravity theories where the asymptotic symmetry groups are different than
$Osp(\mathcal{N}|2;R)$. Our construction of the gravity dual as given in this
paper does not hold for these theories. Let us briefly discuss the reasoning
here. In our search for a gravity dual to asymptotically flat gravity
theories, we proceed by drawing a parallel with the AdS/CFT correspondence.
Thus we look for a two dimensional field theory that has the same symmetry as
the asymptotic symmetry of the three dimensional bulk gravity theory. The
prescription that we have followed for construction of such a boundary dual is
based on the 3D gravity-Chern-Simons theory-WZW model duality. We demand that
the two dimensional WZW theory must have an infinite dimensional symmetry same
as the asymptotic symmetry of the bulk gravity theory. This imposes certain
constraints on the WZW fields. When we take these constraints into account, we
find that the corresponding WZW model has the same (super)algebra as that of
the bulk (super)gravity theory as its current algebra. From this current
algebra, one can construct an infinite dimensional mode algebra, where the
modes are that of a dimension 2 stress tensor, and its super-partner , a spin
$3/2$ operator of the WZW theory. Following a modified Sugawara pescription,
these boundary operators are constructed by taking bilinears of the current
algebra generators. It had been explicitly shown in Bina_1997  that
construction of a spin $3/2$ operator as a bilinear form is only possible for
SCA $Osp(\mathcal{N}|2;R)$. Our construction in the current paper justifies
the same result.

Thus, for the other $\mathcal{N}$-extended supergravity theories of
Banerjee:2018hbl , the dual construction has to be different and the dual
theory would not be a WZW model. It remains an interesting open problem to
find these duals.

## Acknowledgements

We would like to thank Glenn Barnich for many fruitful discussions about this
project. We would also like to thank Ranveer Singh for a very useful
communication during the course of this work. TN would like to thank IISER
Bhopal for kind hospitality during the start of this project. AB would like to
thank IISER Bhopal from where this work has been done. SB would like to thank
IISER Bhopal for the financial support and the facilities they have provided
during the course of this work. For TN, the work is partially supported by the
ERC Advanced Grant “High-Spin-Grav” and by FNRS-Belgium (convention IISN
4.4503.15). The work of NB is partially supported by SERB ECR grant. Finally
we thank the people of India for their generous support to fundamental
research.

##  Appendix A Conventions and Notations

We shall present our conventions for the various types of indices in this
appendix. It is also important to note the metric in each of these vector
spaces the whole algebra spans. In what follows, we denote these different
spaces by representative indices of that space.

The metric in the $ab$ space (spanned by bosonic generators) is given by:

$\displaystyle\eta_{ab}=\begin{pmatrix}-1&0&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix}$

In the $\hat{p}\hat{q}$ space, the inherent metric is 444The counting here
starts with negative indices

$\displaystyle C_{\hat{p}\hat{q}}=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}$

The range of the different kinds of indices are:

$\displaystyle a,b$ $\displaystyle\rightarrow\\{1,2,3\\}$
$\displaystyle\bar{a},\bar{b}$ $\displaystyle\rightarrow\\{1,2,...,D\\}$
$\displaystyle\alpha,\beta$ $\displaystyle\rightarrow\\{1,2,...,d\\}$
$\displaystyle\hat{p},\hat{q}$
$\displaystyle\rightarrow\\{\frac{1}{2},-\frac{1}{2}\\}$

where $d=\mathcal{N},$ is the amount of supersymmetry and $D$ is the dimension
of the internal algebra.

The three dimensional Dirac matrices satisfy the usual commutation relation:
$\\{\Gamma_{a},\Gamma_{b}\\}=2\eta_{ab}$. They also satisfy following useful
identities:

$\displaystyle\Gamma_{a}\Gamma_{b}$
$\displaystyle=\epsilon_{abc}\Gamma^{c}+\eta_{ab}\bf{I}$ (39)
$\displaystyle(\Gamma)^{\alpha}_{\beta}(\Gamma)^{\gamma}_{\delta}$
$\displaystyle=2\delta^{\alpha}_{\delta}\delta^{\gamma}_{\beta}-\delta^{\alpha}_{\beta}\delta^{\gamma}_{\delta}$
(40)

The explicit form of the Dirac matrices are chosen as,

$\displaystyle\Gamma_{0}=\sqrt{2}\begin{pmatrix}0&1\\\
0&0\end{pmatrix},~{}~{}~{}\Gamma_{1}=\sqrt{2}\begin{pmatrix}0&0\\\
1&0\end{pmatrix},~{}~{}~{}\Gamma_{2}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$
(41)

Furthermore we have ,

$C^{T}=-C\,,\hskip 22.76228ptC\Gamma C^{-1}=-\Gamma^{T}\,,\hskip
22.76228ptC_{\alpha\beta}C^{\beta}_{\gamma}=-\delta^{\alpha\gamma}$ (42)

###  A.1 Convention of Grading

In our work involving superalgebras, we have both Grassmann forms as well as
differential forms. Both involve crucial minus signs when exchanging
quantities, and there are more than one conventions in the literature on how
to implement the combined sign rule, taking into account both the Grassmann
and the form gradings (WinNT  and the references therein). In this paper we
have used the Bernstein’s convention (the super odd sign rule) which is as
follows: For algebraic terms $\alpha_{i}$ and $\alpha_{j}$ with homological
degrees $n_{i}$ and $n_{j}$, and super-degrees $\sigma_{i}$ and $\sigma_{j}$
respectively, we take

$\alpha_{i}.\alpha_{j}=(-1)^{(n_{i}+\sigma_{i}).(n_{j}+\sigma_{j})}\alpha_{j}.\alpha_{i}$

##  Appendix B Useful identities and relations

In this appendix, we list some of the useful relations and simplifications
used in our calculations.

### Useful relations for the internal symmetry group generators

$\displaystyle(\lambda^{\bar{a}})^{\alpha\beta}=-(\lambda^{\bar{a}})^{\beta\alpha}\,$
$\displaystyle,\hskip
14.22636pt(\lambda^{\bar{a}}\lambda^{\bar{b}})^{\alpha\beta}=(\lambda^{\bar{b}}\lambda^{\bar{a}})^{\beta\alpha}\,,\hskip
14.22636pt[\lambda_{\bar{a}},\lambda_{\bar{b}}]=f_{\bar{a}\bar{b}\bar{c}}\lambda^{\bar{c}}$
(43)
$\displaystyle(\lambda^{\bar{a}})^{\beta\gamma}(\lambda^{\bar{a}})^{\alpha}_{\delta}+(\lambda^{\bar{a}})^{\alpha\gamma}(\lambda^{\bar{a}})^{\beta}_{\delta}$
$\displaystyle=\frac{C_{\rho}}{d-1}(2\eta^{\alpha\beta}\delta^{\gamma}_{\delta}-\eta^{\alpha\gamma}\delta^{\beta}_{\delta}-\eta^{\beta\gamma}\delta^{\alpha}_{\delta})$
(44)
$\displaystyle(\lambda^{\bar{a}})^{\alpha\beta}(\lambda^{\bar{a}})_{\sigma\gamma}$
$\displaystyle=\frac{C_{\rho}}{d-1}(\delta^{\alpha}_{\sigma}\delta^{\beta}_{\gamma}-\delta^{\alpha}_{\gamma}\delta^{\beta}_{\sigma})$
(45)
$\displaystyle(\lambda^{\bar{a}})_{\alpha\beta}(\lambda^{\bar{a}})^{\beta\gamma}$
$\displaystyle=-C_{\rho}\delta^{\gamma}_{\alpha}\,,\hskip
22.76228pt\lambda^{\bar{a}}\lambda^{\bar{a}}=-C_{\rho}I$ (46)
$\displaystyle(\lambda_{\bar{a}})^{\alpha}_{\beta}(\lambda^{\bar{b}})^{\beta}_{\gamma}(\lambda_{\bar{a}})^{\gamma}_{\sigma}$
$\displaystyle=\Big{(}i_{\rho}l^{2}-\frac{6}{\sigma_{0}}\Big{)}(\lambda^{\bar{b}})^{\alpha}_{\sigma}$
(47) $\displaystyle\frac{C_{\rho}}{(d-1)}$ $\displaystyle=3\hat{\alpha},\qquad
f^{\bar{a}\bar{b}\bar{c}}=-\frac{2D}{dC_{\rho}}Tr[\lambda^{\bar{a}}\lambda^{\bar{b}}\lambda^{\bar{c}}]$
(48) $\displaystyle Tr[\lambda^{\bar{a}}\lambda^{\bar{b}}]$
$\displaystyle=-\frac{d}{D}C_{\rho}\delta^{\bar{a}\bar{b}}$ (49)

## Simplification of terms in the total action:

The origin of the various terms in the simplified form of the action are as
follows:

$\displaystyle\mu_{1}\omega^{a}_{u}\omega_{a,\phi}\rightarrow$ $\displaystyle
Tr[2\mu_{1}(\Lambda^{-1}\dot{\Lambda}\Lambda^{-1}\Lambda^{\prime})]$
$\displaystyle
2C_{\hat{p}\hat{q}}\psi^{\hat{p}}_{I,\alpha,u}\psi^{\hat{q}}_{I,\alpha,\phi}\rightarrow$
$\displaystyle
Tr[2(A^{-1}\bar{\eta_{1}}^{\prime}A^{-1}\dot{\eta_{1}}+A\bar{\eta_{2}}^{\prime}A\dot{\eta_{2}})]$
$\displaystyle\frac{1}{6}3\epsilon_{abc}\omega^{a}\omega^{b}e^{c}~{}and~{}\omega^{a}_{\phi}e^{a}_{u}+e^{a}_{\phi}\omega^{a}_{u}\rightarrow$
$\displaystyle
Tr[2\dot{\Lambda}\Lambda^{-1}(A^{-1}\eta_{1}A^{-1}\bar{\eta}_{1}^{\prime}+A^{-1}\bar{\eta}_{1}^{\prime}A^{-1}\eta_{1}\text{I}$
$\displaystyle+A\eta_{2}A\bar{\eta}_{2}^{\prime}+A\bar{\eta}_{2}^{\prime}A\eta_{2}\text{I}+b^{\prime})]$
$\displaystyle
C^{\bar{a}}_{\phi}B^{\bar{a}}_{u}+B^{\bar{a}}_{\phi}C^{\bar{a}}_{u}+\mu_{2}B^{\bar{a}}_{\phi}B^{\bar{a}}_{u}\rightarrow$
$\displaystyle
Tr[2\mu_{2}A^{-1}A^{\prime}A^{-1}\dot{A}+4A^{-1}\dot{A}{\tilde{C}}^{\prime}]$
$\displaystyle\omega^{a}_{\phi}\omega_{a\phi}\rightarrow$ $\displaystyle
Tr[2(\Lambda^{-1}\Lambda^{\prime}\Lambda^{-1}\Lambda^{\prime})]$
$\displaystyle 4B^{\bar{a}}_{\phi}B_{\bar{a}\phi}\rightarrow$ $\displaystyle
Tr[4(A^{-1}A^{\prime}A^{-1}A^{\prime})]$
$\displaystyle\mu_{1}\epsilon_{abc}\omega^{a}\omega^{b}\omega^{c}\rightarrow$
$\displaystyle Tr[\mu_{1}(A^{-1}dA)^{3}]~{}(WZW~{}term~{}1)$
$\displaystyle\mu_{2}f_{\bar{a}\bar{b}\bar{c}}B^{\bar{a}}B^{\bar{b}}B^{\bar{c}}\rightarrow$
$\displaystyle Tr[\mu_{2}(\Lambda^{-1}d\Lambda)^{3}]~{}(WZW~{}term~{}2)$
$\displaystyle
6(\lambda^{\bar{a}})^{\alpha\beta}B^{\bar{a}}C_{\hat{p}\hat{q}}(\psi^{\hat{p}}_{1,\alpha}\psi^{\hat{q}}_{1,\beta}-\psi^{\hat{p}}_{2,\alpha}\psi^{\hat{q}}_{2,\beta})~{}and~{}B^{\bar{a}}C^{\bar{b}}B^{\bar{c}}\rightarrow$
$\displaystyle
Tr[A^{-1}dA(A^{-1}\bar{\eta}_{1}A^{-1}d\eta_{1}-A\bar{\eta}_{2}Ad\eta_{2})]$

###  Terms in Hamiltonian Reduced Action

$\displaystyle
Tr[2\mu_{1}(\lambda^{-1}\dot{\lambda}\lambda^{-1}\lambda^{\prime})]$
$\displaystyle=\mu\varphi^{\prime}\dot{\varphi}+2\mu(\dot{\sigma}\tau^{\prime}e^{-\varphi})+2\mu(\sigma^{\prime}\dot{\tau}e^{-\varphi})$
$\displaystyle Tr[4\dot{\lambda}\lambda^{-1}F^{\prime}]$
$\displaystyle=4[-\frac{\dot{\varphi}\theta^{\prime}}{2}+\sigma\zeta^{\prime}e^{-\varphi}+\frac{\dot{\sigma}\eta^{\prime}}{2}-\frac{\sigma\dot{\varphi}\eta^{\prime}}{2}-\frac{\sigma^{2}\dot{\tau}e^{-\varphi}\eta^{\prime}}{2}]$
$\displaystyle Tr[-2(\lambda^{-1}\lambda^{\prime})^{2}]$
$\displaystyle=-2(2\sigma^{\prime}\tau^{\prime}e^{-\varphi}+\frac{{\varphi^{\prime}}^{2}}{2})$
$\displaystyle\frac{2\mu}{3}\int[(d\Lambda\Lambda^{-1})^{3}]$
$\displaystyle=\int
dud\phi(-2\mu(\dot{\sigma}\tau^{\prime}e^{-\varphi})+2\mu(\sigma^{\prime}\dot{\tau}e^{-\varphi}))$
$\displaystyle
Tr[2(\gamma({\bar{d}}_{1})^{\prime}\gamma\dot{d}_{1}+\gamma^{-1}({\bar{d}}_{2})^{\prime}\gamma^{-1}\dot{d}_{2})]$
$\displaystyle=Tr[2((\bar{d}_{1}^{N})^{\prime}\dot{d}_{1}^{N}+(\bar{d}_{2}^{N})^{\prime}\dot{d}_{2}^{N})$
$\displaystyle+2(-({\bar{d_{1}}}^{N})^{\prime}\dot{\gamma}\gamma^{\prime}d_{1}^{N}-\bar{d_{1}}^{N}\gamma^{\prime}\gamma^{-1}\dot{\gamma}\gamma^{-1}d_{1}^{N}+\bar{d_{1}}^{N}\gamma^{\prime}\gamma^{-1}\dot{d_{1}}^{N})$
$\displaystyle+2((\bar{d_{2}}^{N})^{\prime}\gamma^{\prime}\dot{\gamma}d_{2}^{N}-\bar{d_{2}}^{N}\gamma^{-1}\gamma^{\prime}\gamma^{-1}\dot{\gamma}d_{2}^{N}-\bar{d_{2}}^{N}\gamma^{-1}\gamma^{\prime}\dot{d_{2}}^{N})]$
$\displaystyle Tr[2\dot{\lambda}\lambda^{-1}\gamma
d_{1}\gamma(\bar{d}_{1})^{\prime}]$
$\displaystyle=\sqrt{2}\dot{\sigma}d_{1,\alpha}^{N+}(d_{1}^{N+,\alpha})^{\prime}+\dot{\varphi}d_{1,\alpha}^{N+}(d_{1}^{N+,\alpha})^{\prime}$
$\displaystyle
Tr[2\dot{\lambda}\lambda^{-1}\gamma^{-1}d_{2}\gamma^{-1}(\bar{d}_{2})^{\prime}]$
$\displaystyle=\sqrt{2}\dot{\sigma}d_{2,\alpha}^{N+}(d_{2}^{N+,\alpha})^{\prime}+\dot{\varphi}d_{2,\alpha}^{N+}(d_{2}^{N+,\alpha})^{\prime}$

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