††institutetext: Istanbul Technical University, Faculty of Science and
Letters, Physics Department,
34469 Maslak, Istanbul,Turkey.

# On the $\mathcal{N}=3$ and $\mathcal{N}=4$ superconformal holographic
dictionary

H. T. Özer<EMAIL_ADDRESS>† , Aytül Filiz<EMAIL_ADDRESS>
###### Abstract

This study presents comprehensive examples of $\mathfrak{osp}(\mathcal{N}|2)$
Chern - Simons supergravity on $AdS_{3}$ for $\mathcal{N}>2$. These
formulations, which include the most general boundary conditions, represent
extensions of previously discovered works (Ozer and Filiz,Eur Phys J C
82(5):472, 2022) for $\mathcal{N}<3$. In our work, we show that under the
loosest set of boundary conditions, the asymptotic symmetry algebras consist
of two copies of the $\mathfrak{osp}(3|2)_{k}$ and $\mathfrak{osp}(4|2)_{k}$
algebras. We subsequently restrict the gauge fields upon the boundary
conditions to achieve supersymmetric extensions of the Brown - Henneaux
boundary conditions. Based on these results, we finally find that the
asymptotic symmetry algebras are two copies of the $\mathcal{N}=3$ and
$\mathcal{N}=4$ superconformal algebras for $\mathcal{N}=(3,3)$ and
$\mathcal{N}=(4,4)$ extended higher - spin supergravity theory in $AdS_{3}$.

##  1 Introduction

The holographic conjecture states that quantum gravity within a $(d+1)$ \-
dimensional spacetime is equivalent to a quantum field theory at the boundary
of the $d$ \- dimensional space. This conjecture is a powerful tool for
studying quantum gravity. An important example of this conjecture is the
holographic duality between $(d+1)$ \- dimensional Anti - de Sitter (AdS)
gravity and conformal field theory in dimension $d$ Maldacena:1997re . This
duality, first proposed by Maldacena in 1997, provides a fundamental framework
for studying the complex interplay between gravitational and quantum field
theoretical phenomena, offering a comprehensive and insightful perspective on
the nature of these fundamental forces.

Conformal Field Theory (CFT) is a fundamental concept in theoretical physics,
with applications across diverse fields, from string theory to condensed
matter theory. The strength of CFTs lies in their symmetries, which impose
significant constraints on the theory’s structure and content. The conformal
symmetry is particularly enhanced in two dimensions, resulting in an infinite
set of generators known as the Virasoro algebra Virasoro:1969zu . Researchers
have explored various extensions of the two - dimensional conformal algebra
beyond the Virasoro algebra, including the Kac - Moody algebra Kac_1968 ;
Moody:1968zz  and the W - algebra Zamolodchikov:1985wn , as well as the
Virasoro algebra itself DiFrancesco:1997nk ; Blumenhagen:2009zz ;
Prochazka:2014gqa ; Ozer:2015dha .

However, extending the Virasoro algebra to its supersymmetric counterpart, the
super Virasoro algebra presents increasing challenges. Despite extensive
investigations into supersymmetric extensions with a limited number of
supersymmetries ($N\leq 4$) Ademollo:1975an ; Ademollo:1976pp ;
Schoutens:1987uu ; Schoutens:1988ig ; Sevrin:1988ew ; Eguchi:1987sm ;
Eguchi:1987wf ; Odake:1988bh ; Odake:1989dm , complete construction of the
super Virasoro algebra remains elusive for $N>4$. The complexities involved in
this endeavor highlight the intricate nature of supersymmetric extensions and
underscore the need for further exploration into the unexplored domain of
higher supersymmetry orders. The Hamiltonian reduction of affine algebras
gives rise to higher spin algebras. These can be described through flat
connections in the Drinfeld - Sokolov form, as extensively documented in
Drinfeld:1984qv . This characterization aligns seamlessly with the overarching
framework of higher spin dualities. These dualities specifically involve three
- dimensional Chern - Simons theory and two - dimensional conformal field
theories (CFTs) featuring $\mathcal{W}$ \- symmetry. Noteworthy instances
include the well - established correlation between $\mathfrak{sl}(N)$ Chern -
Simons theory and CFTs endowed with $\mathcal{W}_{N}$ \- symmetry
Campoleoni:2010zq ; Campoleoni:2011hg . Furthermore, recent and significant
examples include the duality observed between $\mathfrak{hs}(\lambda)$ Chern -
Simons theory and CFTs characterized by $\mathcal{W_{\infty}}(\lambda)$ \-
symmetry Henneaux:2010xg ; Gaberdiel:2011wb . A comprehensive review of this
subject is available in Gaberdiel:2012uj .

In their seminal work, Brown and Henneaux presented a groundbreaking analysis
Brown:1986nw , revealing that the asymptotic symmetries of $\mathrm{AdS_{3}}$
gravity manifest as two copies of the Virasoro algebra. This discovery serves
as a precursor to the $\mathrm{AdS/CFT}$ correspondence Maldacena:1997re  and
embodies the Holographic Principle Susskind:1994vu . Subsequent advancements
in $\mathrm{AdS_{3}}$ holography have expanded this foundation. Noteworthy
extensions include investigations into the asymptotic symmetries of
supergravity and extended supergravity, as documented in Banados:1998pi  and
Henneaux:1999ib , respectively. These studies conclusively established that
the asymptotic symmetry algebra for (extended) supergravity aligns with the
(extended) superconformal algebra Goddard:1986ee ; Schwimmer:1986mf .

The exploration of diverse three - dimensional asymptotic symmetries within
supergravity models remains a subject of considerable interest, as it can
provide insights into the geometric origins of superconformal algebras.
Noteworthy instances in this line of investigation include the asymptotic
analysis of supersymmetric higher - spin gravity Henneaux:2012ny ; Tan:2012xi
; Gaberdiel:2014yla , hypergravity Henneaux:2015ywa ; Fuentealba:2015wza , and
supergravity in the at - space limit Barnich:2014cwa ; Lodato:2016alv ;
Banerjee:2017gzj ; Fuentealba:2017fck ; Basu:2017aqn , as well as
Banerjee:2018hbl . The Chern - Simons formulation of three - dimensional
gravity, introduced by Witten Witten:1988hc , emerges as a crucial tool in
deriving these results. This formulation proves advantageous for a more
straightforward analysis of these symmetries than the metric formulation, as
elucidated in pedagogical reviews such as Banados:1994tn  and Banados:1998sm .

Selecting of fall - off conditions for metric parameters is essential in the
$\mathrm{AdS_{3}}$ holography. Among the widely employed conditions for pure
$\mathrm{AdS_{3}}$ gravity are the Brown - Henneaux boundary conditions
Brown:1986nw . Despite their prevalence, alternative boundary conditions have
been introduced in recent years. Notably, Compere, Song, and Strominger
Compere:2013bya , as well as Troessaert Troessaert:2013fma , have proposed
novel sets, resulting in the emergence of two instances of a warped conformal
algebra with distinct Kac - Moody levels. Avery, Poojary, and Suryanarayana
Avery:2013dja  introduced boundary conditions that yield a semidirect sum of
an $\mathfrak{sl}(2)_{k}$ algebra and a Virasoro algebra. In their
comprehensive study, Grumiller and Riegler Grumiller:2016pqb  investigated the
most general boundary conditions for $\mathrm{AdS_{3}}$ gravity. Consequently,
the asymptotic symmetry algebra was found to be generated by two copies of the
$\mathfrak{sl}(2)_{k}$ algebra. Furthermore, it is noteworthy that the
previously identified boundary conditions were derived by imposing certain
constraints on these most general conditions.

This methodology has been applied across various scenarios, including three -
dimensional AdS space Grumiller:2017sjh , chiral higher spin gravity
Krishnan:2017xct ; Ozer:2017dwk , and (supersymmetric) two - dimensional
gravity Grumiller:2017qao ; Cardenas:2018krd . This paper endeavors to broaden
the scope of the analysis initiated in Grumiller:2016pqb  to three -
dimensional AdS supergravity. We systematically derive the asymptotic symmetry
algebra for the most permissive set of (extended) supergravity boundary
conditions. We reaffirmed some prior findings while presenting novel boundary
conditions linked to warped superconformal algebras. Consequently, this
methodology is an invaluable platform for delving into the intricate
asymptotic structure of (extended) supergravity.

This paper addresses a hitherto unresolved phenomenon by presenting two
candidate solutions for the most general $\mathcal{N}=(3,3)$ and
$\mathcal{N}=(4,4)$ extended higher spin supergravity theories on $AdS_{3}$.
Building upon our previous studies Ozer:2019nkv ; Ozer:2021wtx , it is crucial
to underscore that our inspiration from Grumiller’s seminal work
Grumiller:2016pqb  continues to guide this investigation. The primary
objective of this work is to determine the explicit form of the holographic
dictionary between $\mathfrak{osp}(\mathcal{N}|2)$ Chern - Simons supergravity
on $AdS_{3}$ and two - dimensional $CFT_{2}$ with $\mathcal{N}=3$ and
$\mathcal{N}=4$ superconformal symmetry, respectively. Demonstrating that our
theory aligns with the metric class established by Grumiller:2017sjh , we
observe that the metric formulation can encompass both charge and chemical
potentials present in the Chern - Simons formalism. This offers an alternative
solution to the non - chiral Drinfeld - Sokolov type boundary conditions.

Our primary focus is on the extended $\mathcal{N}=(3,3)$ and
$\mathcal{N}=(4,4)$ Chern - Simons theory based on $\mathfrak{osp}(3|2)$ and
$\mathfrak{osp}(4|2)$ superalgebras, respectively. This results in an
asymptotic symmetry algebra comprising two copies of the
$\mathfrak{osp}(3|2)_{k}$ and $\mathfrak{osp}(4|2)_{k}$ affine algebras. We
impose specific restrictions on the gauge fields under the most general
boundary conditions, leading to supersymmetric extensions of the Brown -
Henneaux Boundary conditions. Furthermore, we establish that the asymptotic
symmetry algebras reduce to two copies of the $\mathcal{N}=3$ and
$\mathcal{N}=4$ superconformal algebras in $AdS_{3}$ supergravity theory.

It is noteworthy that exploring a distinct class of boundary conditions for
(super)gravity, as emerging in the literature (e.g., Compere:2013bya ;
Troessaert:2013fma ; Avery:2013dja ), presents an intriguing problem. Their
higher spin generalization is not as evident as the boundary conditions
proposed by Grumiller and Riegler. In light of these findings, it is clear
that this method provides an excellent laboratory for investigating the rich
asymptotic structure of extended higher - spin supergravity.

This paper is structured as follows: To establish a foundation, we begin with
a concise overview of the fundamental aspects of the $\mathcal{N}=3$ and
$\mathcal{N}=4$ superconformal algebras in the initial sections. Subsequently,
in Section 3, we present a comprehensive formulation of
$\mathfrak{osp}(\mathcal{N}|2)\oplus\mathfrak{osp}(\mathcal{N}|2)$ Chern -
Simons supergravity. Following a brief examination of
$\mathfrak{osp}(\mathcal{N}|2)\oplus\mathfrak{osp}(\mathcal{N}|2)$ Chern -
Simons supergravity on $AdS_{3}$, Sections 3 and 4 are dedicated to
establishing a clear and explicit holographic correspondence between
$\mathfrak{osp}(\mathcal{N}|2)$ Chern - Simons supergravity on $AdS_{3}$ and
two - dimensional $CFT_{2}$ possessing $\mathcal{N}=3$ and $\mathcal{N}=4$
superconformal symmetry. This analysis extends to encompass both affine and
superconformal boundaries. Additionally, we explore the emergence of
asymptotic symmetry and higher spin Ward identities from the bulk equations of
motion coupled to spin currents.

In these sections, we examine the characterization of asymptotic symmetry
algebras as classical duplicates of the $\mathfrak{osp}(\mathcal{N}|2)_{k}$
and affine algebra on the affine boundary. Simultaneously, we establish the
$\mathfrak{so}(\mathcal{N})$ superconformal symmetry algebra for
$\mathcal{N}=3$ and $\mathcal{N}=4$ on the superconformal boundary.
Furthermore, we elaborate on the description of chemical potentials linked to
source fields, manifesting through the temporal components of the connection.

In the concluding section, we summarize our findings and insights, address any
unresolved questions, and propose potential avenues for future research.

##  2 The superconformal algebras:

We will start by going over the basics of the ${\mathcal{N}}=3$ superconformal
algebra. This section is not meant to be a comprehensive survey; instead, it
serves as a summary of the key features relevant to our objectives. The
$\mathcal{N}=3$ superconformal algebra comes in two guises as a linear
Ademollo:1975an ; Ademollo:1976pp  and a non-linear Knizhnik:1986wc ;
Bershadsky:1986ms  versions in the literature. we are first interested in the
linear version version of the algebra.

###  2.1 The $\mathcal{N}=3$ linear - superconformal algebras:

The 8 currents are given by a single spin-$\frac{1}{2}$ current $\Psi(z)$,
three spin-1 currents $J^{a}(z)$, $a=1,2,3$ transforming as a vector
representation under $\mathfrak{so}(3)$, three $\frac{3}{2}$ currents
$G^{i}(z)$, $i=1,2,3$ transforming as a vector representation under
$\mathfrak{so}(3)$, and the spin-2 current $T(z)$. In particular, the
spin-$\frac{1}{2}$ and spin-2 currents are $\mathfrak{so}(3)$ singlets, all
having the standard mode expansions
$T(z)=\sum_{n}L_{n}z^{-n-2}~{},\,G^{i}(z)=\sum_{r}G^{i}_{r}z^{-r-3/2}~{},\,J^{a}(z)=\sum_{n}J^{a}_{n}z^{-n-1}~{},\
$ and $F(z)=\sum_{r}F_{r}z^{-r-1/2}~{}$.

Non-zero of the $OPEs$ defining the $\mathcal{N}=3$ superconformal algebra
take the standard linear form,

$\displaystyle F(z_{1})\,F(z_{2})$ $\displaystyle\sim$
$\displaystyle\frac{1}{z_{12}}\,\frac{c}{3},$ (1) $\displaystyle
F(z_{1})\,J^{i}(z_{2})$ $\displaystyle\sim$ $\displaystyle 0,$ (2)
$\displaystyle F(z_{1})\,G^{i}(z_{2})$ $\displaystyle\sim$
$\displaystyle\frac{1}{z_{12}}\,J^{i},$ (3) $\displaystyle
J^{i}(z_{1})\,J^{j}(z_{2})$ $\displaystyle\sim$
$\displaystyle\frac{1}{z_{12}^{2}}\,\frac{c}{3}\delta^{ij}+\frac{1}{z_{12}}\,i\epsilon^{ijk}J^{k},$
(4) $\displaystyle J^{i}(z_{1})\,G^{j}(z_{2})$ $\displaystyle\sim$
$\displaystyle\frac{1}{z_{12}^{2}}\,\delta^{ij}F+\frac{1}{z_{12}}\,i\epsilon^{ijk}G^{k},$
(5) $\displaystyle G^{i}(z_{1})\,G^{j}(z_{2})$ $\displaystyle\sim$
$\displaystyle\frac{1}{z_{12}^{3}}\,\frac{2c}{3}\delta^{ij}+\frac{1}{z_{12}^{2}}\,2i\epsilon^{ijk}J^{k}$
(6) $\displaystyle+$
$\displaystyle\frac{1}{z_{12}}\left(2\delta^{ij}T+i\epsilon^{ijk}\partial
J^{k}\right),$ $\displaystyle T(z_{1})\,F(z_{2})$ $\displaystyle\sim$
$\displaystyle\frac{1}{z_{12}^{2}}\,\frac{1}{2}F+\frac{1}{z_{12}}\partial F,$
(7) $\displaystyle T(z_{1})\,J^{i}(z_{2})$ $\displaystyle\sim$
$\displaystyle\frac{1}{z_{12}^{2}}\,J^{i}+\frac{1}{z_{12}}\partial J^{i},$ (8)
$\displaystyle T(z_{1})\,G^{i}(z_{2})$ $\displaystyle\sim$
$\displaystyle\frac{1}{z_{12}^{2}}\,\frac{3}{2}G^{i}+\frac{1}{z_{12}}\partial
G^{i},$ (9) $\displaystyle T(z_{1})\,T(z_{2})$ $\displaystyle\sim$
$\displaystyle\frac{1}{z_{12}^{4}}\,\frac{c}{2}+\frac{1}{z_{12}^{2}}\,2T+\frac{1}{z_{12}}\partial
T.$ (10)

where $c=3k$ is the central charge and $k$ is an arbitrary ‘level’ of the Kac
- Moody subalgebra, $\epsilon^{ijk}$ are the structure constants of
$\mathfrak{so}(3)$ in the fundamental (vector) representation,

The nonlinear version of above ${\mathcal{N}}=3$ superconformal algebra can be
obtained by factoring out the spin-$\frac{1}{2}$ current Knizhnik:1986wc ;
Bershadsky:1986ms  and becomes the result of
$(\ref{qalg11})$-$(\ref{qalg12})$.

###  2.2  The ${\mathcal{N}}=3$ nonlinear - Knizhnik Bershadsky algebra

We have only looked at the ${\mathcal{N}}=3$ linear superconformal algebra so
far. Among the eight currents of the ${\mathcal{N}}=3$ linear superconformal
algebra, We can decouple the spin-$\frac{1}{2}$ current $F(z)$ from the other
seven remaining currents by doing something like Ahn:2013oya ; Ahn:2014via ;
Ahn:2015gba . One get the eight higher spin currents and their OPEs after
decoupling the above spin-$\frac{1}{2}$ current in this section.

The spin-$1$ currents of the ${\mathcal{N}}=3$ superconformal algebra do not
have any singular terms with the spin-$\frac{1}{2}$ current. Then it follows
that the spin-$1$ currents should be used as the previous one $J^{i}(z)$. We
should acquire the new spin-$\frac{3}{2}$ currents and spin-$2$ currents for
the spin-$\frac{3}{2}$ currents and spin-$2$ currents. The new seven currents
are really specified as GS ; Schoutens .

$\displaystyle\hat{T}(z)$ $\displaystyle=$ $\displaystyle
T(z)+\frac{3}{2c}F\partial F(z),$ (11) $\displaystyle\hat{G}^{i}(z)$
$\displaystyle=$ $\displaystyle G^{i}(z)-\frac{3}{c}J^{i}F(z),$ (12)
$\displaystyle\hat{J}^{i}(z)$ $\displaystyle=$ $\displaystyle J^{i}(z).$ (13)

The relative coefficients on the right hand side of (13) can be set by
requiring that the requirements: $F(z_{1})\;\hat{T}(z_{2})\sim
0,\,F(z_{1})\;\hat{G}^{i}(z_{2})\sim 0,\,$ and
$F(z_{1})\;\hat{J}^{i}(z_{2})\sim 0$ be met. Then, using (13), we can compute
the OPEs between the new seven currents, which may be summed as
Knizhnik:1986wc ; Bershadsky:1986ms

$\displaystyle\hat{J}^{i}(z_{1})\,\hat{J}^{j}(z_{2})$ $\displaystyle\sim$
$\displaystyle\frac{1}{z_{12}^{2}}\,\frac{c}{3}\delta^{ij}+\frac{1}{z_{12}}\,i\epsilon^{ijk}\hat{J}^{k},$
(14) $\displaystyle\hat{J}^{i}(z_{1})\,\hat{G}^{j}(z_{2})$ $\displaystyle\sim$
$\displaystyle\frac{1}{z_{12}}\,i\epsilon^{ijk}\hat{G}^{k},$ (15)
$\displaystyle\hat{G}^{i}(z_{1})\,\hat{G}^{j}(z_{1})$ $\displaystyle\sim$
$\displaystyle\frac{1}{z_{12}^{3}}\,\frac{1}{3}(2c-3)\delta^{ij}+\frac{1}{z_{12}^{2}}\,\frac{i(2c-3)}{c}\epsilon^{ijk}\hat{J}^{k}$
(16) $\displaystyle+$
$\displaystyle\frac{1}{z_{12}}\bigg{(}2\delta^{ij}\hat{T}+i\epsilon^{ijk}\partial\hat{J}^{k}-\frac{3}{c}\hat{J}^{i}\hat{J}^{j}\bigg{)},$
$\displaystyle\hat{T}(z_{1})\,\hat{J}^{i}(z_{2})$ $\displaystyle\sim$
$\displaystyle\frac{1}{z_{12}^{2}}\,\hat{J}^{i}+\frac{1}{z_{12}}\partial\hat{J}^{i},$
(17) $\displaystyle\hat{T}(z_{1})\,\hat{G}^{i}(z_{2})$ $\displaystyle\sim$
$\displaystyle\frac{1}{z_{12}^{2}}\,\frac{3}{2}\hat{G}^{i}+\frac{1}{z_{12}}\partial\hat{G}^{i},$
(18) $\displaystyle\hat{T}(z_{1})\,\hat{T}(z_{2})$ $\displaystyle\sim$
$\displaystyle\frac{1}{z_{12}^{4}}\,\frac{1}{4}(2c-1)+\frac{1}{z_{12}^{2}}\,2\hat{T}+\frac{1}{z_{12}}\partial\hat{T}.$
(19)

It is worth noting that the OPE has a nonlinear term between the
spin-$\frac{3}{2}$ currents. The aforementioned $OPEs$
$(\ref{qalg11})$-$(\ref{qalg12})$ easily become the one in Knizhnik:1986wc
(where there is an error in the $OPE$) if one changes
$i\hat{J}^{i}(z)=J_{K}^{i}(z)$ and other currents remain unaltered.

###  2.3 The $\mathcal{N}=4$ linear - superconformal algebra

The $\mathcal{N}=4$ algebra was first given by Ademollo at al.Ademollo:1976wv
, This algebra does not contain the $\mathcal{N}=3$ algebra discussed above.
They both contain an $\mathfrak{su}(2)$ Kac-moody algebra but the supercharges
transform as an $\mathfrak{su}(2)$ triplet in the $\mathcal{N}=3$ algebra and
as a doublet in the $\mathcal{N}=4$ algebra. The $\mathcal{N}=4$
superconformal algebra algebra contains a Virasoro sub-algebra, and
$\mathfrak{su}(2)$ Kac-Moody algebra. The four super charges form a complex
$\mathfrak{su}(2)$ doublet. Therefore, the 8 currents are given by three
spin-1 currents $J^{i}(z)$, $i=1,2,3$ transforming as a complex
$\mathfrak{su}(2)$ doublet, four $\frac{3}{2}$ currents $G^{a}(z)$ and
$\bar{G}^{a}(z)$, $a=1,2$ transforming as an $\mathfrak{su}(2)$ doublets, all
having the standard mode expansions
$T(z)=\sum_{n}L_{n}z^{-n-2}~{},\,G^{a}(z)=\sum_{r}G^{a}_{r}z^{-r-3/2}~{},\,\bar{G}^{a}(z)=\sum_{r}\bar{G}^{a}_{r}z^{-r-3/2}~{},\,$
and $J^{i}(z)=\sum_{n}J^{i}_{n}z^{-n-1}$ .

Non-zero of the $OPEs$ defining the $\mathcal{N}=4$ superconformal algebra
take the standard linear form,

$\displaystyle J^{i}(z_{1})\,J^{j}(z_{2})$ $\displaystyle\sim$
$\displaystyle\frac{1}{z_{12}^{2}}\,\frac{k}{2}\delta^{ij}+\frac{1}{z_{12}}\,i\epsilon^{ijk}J^{k},$
(20) $\displaystyle J^{i}(z_{1})\,G^{a}(z_{2})$ $\displaystyle\sim$
$\displaystyle-\frac{1}{z_{12}}\,\frac{1}{2}(\sigma^{i})_{ab}G^{b},$ (21)
$\displaystyle J^{i}(z_{1})\,\bar{G}^{a}(z_{2})$ $\displaystyle\sim$
$\displaystyle\frac{1}{z_{12}}\,\frac{1}{2}(\sigma^{i})^{*}_{ab}\bar{G}^{b},$
(22) $\displaystyle G^{a}(z_{1})\,\bar{G}^{b}(z_{2})$ $\displaystyle\sim$
$\displaystyle\frac{1}{z_{12}^{3}}\,4k\delta^{ab}\
+\frac{1}{z_{12}^{2}}\,4(\sigma^{i})_{ab}J^{i}$ (23) $\displaystyle+$
$\displaystyle\frac{2}{z_{12}}\left(\delta^{ab}T+(\sigma^{i})_{ab}\partial
J^{i}\right),$ $\displaystyle T(z_{1})\,J^{i}(z_{2})$ $\displaystyle\sim$
$\displaystyle\frac{1}{z_{12}^{2}}\,J^{i}+\frac{1}{z_{12}}\partial J^{i},$
(24) $\displaystyle T(z_{1})\,G^{a}(z_{2})$ $\displaystyle\sim$
$\displaystyle\frac{1}{z_{12}^{2}}\,\frac{3}{2}G^{a}+\frac{1}{z_{12}}\partial
G^{a},$ (25) $\displaystyle T(z_{1})\,\bar{G}^{a}(z_{2})$ $\displaystyle\sim$
$\displaystyle\frac{1}{z_{12}^{2}}\,\frac{3}{2}\bar{G}^{a}+\frac{1}{z_{12}}\partial\bar{G}^{a},$
(26) $\displaystyle T(z_{1})\,T(z_{2})$ $\displaystyle\sim$
$\displaystyle\frac{1}{z_{12}^{4}}\,\frac{c}{2}+\frac{1}{z_{12}^{2}}\,2T+\frac{1}{z_{12}}\partial
T.$ (27)

where $c=6k$ is the central charge and $k$ is an arbitrary ‘level’ of the Kac
- Moody subalgebra, $\epsilon^{ijk}$ are $\mathfrak{su}(2)$ structure
constants and $\sigma^{i}$ are the Pauli matrices.

##  3 The $so(\mathcal{N})$ Holographic Dictionary

After examining some properties of the $\mathcal{N}=3$ and $\mathcal{N}=4$
superconformal algebras, we now examine into their realization in terms of
higher spin fields on $AdS_{3}$. This exploration begins with a concise
overview of Chern - Simons supergravity theory, focusing specifically on
aspects relevant to constructing the holographic dictionary. Subsequently, we
present a detailed derivation of the asymptotic symmetry algebra for the
$so(\mathcal{N})$ case. Following this derivation, we engage in a discussion
of sources and the corresponding holographic Ward identities. These identities
play a crucial role in the examination of higher spin black hole solutions.

###  3.1 Chern-Simons Supergravity in Three Dimensions

This section provides a concise overview of $AdS_{3}$ higher spin supergravity
employing the Chern-Simons formalism. We specifically employ this formalism to
investigate $AdS_{3}$ supergravity in the context of the
$\mathfrak{osp}(\mathcal{N}|2)$ superalgebra basis.

###  3.2 Connection to
$\mathfrak{osp}(\mathcal{N}|2)\oplus\mathfrak{osp}(\mathcal{N}|2)$ Chern-
Simons Supergravity

In three dimensions, the Einstein-Hilbert action for supergravity with a
negative cosmological constant can be equivalently formulated using Chern-
Simons theory over a spacetime manifold $\mathcal{M}$:

$S=S_{CS}[\Gamma]-S_{CS}[\bar{\Gamma}],$ (28)

where

$S_{CS}[\Gamma]=\frac{k}{4\pi}\int_{\mathcal{M}}\mathfrak{str}\bigg{(}\Gamma\wedge\mathrm{d}\Gamma+\frac{2}{3}\,\Gamma\wedge\Gamma\wedge\Gamma\bigg{)},$
(29)

first introduced by Achucarro and Townsend Achucarro:1986uwr  and further
developed by Witten Witten:1988hc .

The Chern-Simons level $k$ is eventually related to the ratio of the $AdS_{3}$
radius $l$ and Newton’s constant $G$, as well as the central charge $c$ of the
superconformal field theory.

It is important to note that the 1-forms $(\Gamma,\bar{\Gamma})$ take values
in the gauge group of the $\mathfrak{osp}(\mathcal{N}|2)$ superalgebra. The
supertrace $\mathfrak{str}$, representing a metric on the
$\mathfrak{osp}(\mathcal{N}|2)$ Lie superalgebra, is taken over the
superalgebra generators. The $\mathfrak{osp}(\mathcal{N}|2)$ superalgebra is
defined as the even and odd parts, respectively. To describe (extended)
supergravity, one employs the even part as
$\mathfrak{sl}(2)\oplus\mathfrak{so}(\mathcal{N})$ with generators $L_{I}$
($I=0,\pm 1$) and $T_{AB}=-T_{BA}$ ($A,B=1,2,...,\mathcal{N}$). The odd part
$\mathfrak{g}_{1}$ has generators $G^{\alpha}_{A}$ ($\alpha=\pm\frac{1}{2}$).
These generators satisfy the following algebraic relations:

$\displaystyle\left[L_{I},L_{J}\right]$ $\displaystyle=$
$\displaystyle(I-J)L_{I+J},$ (30)
$\displaystyle\left[L_{I},G^{A}_{\alpha}\right]$ $\displaystyle=$
$\displaystyle\left(\frac{I}{2}-\alpha\right)G^{A}_{I+\alpha},$ (31)
$\displaystyle\left[G^{A}_{\alpha},T^{AB}\right]$ $\displaystyle=$
$\displaystyle\delta^{AB}G^{C}_{\alpha}-\delta^{AC}G^{B}_{\alpha},$ (32)
$\displaystyle\left[T^{AB},T^{CD}\right]$ $\displaystyle=$
$\displaystyle\delta^{AD}T^{BC}+\delta^{BC}T^{AD}$ (33) $\displaystyle-$
$\displaystyle\delta^{AC}T^{BD}-\delta^{BD}T^{AC},$
$\displaystyle\left\\{G^{A}_{\alpha},G^{B}_{\beta}\right\\}$ $\displaystyle=$
$\displaystyle-2L_{\alpha+\beta}\delta^{AB}-\left(\alpha-\beta\right)T^{AB}.$
(34)

Here, we present some of the most notable relations.

  * •

$\mathfrak{osp}(1|2)$:
For $\mathcal{N}=1$, the generators $T^{AB}$ are no longer available, and one
can drop the $\mathfrak{so}(\mathcal{N})$ indices. Therefore, the
$\mathfrak{osp}(1|2)$ algebra is given by:

$\displaystyle[L_{I},L_{J}]$
$\displaystyle=(I-J)L_{I+J},\quad[L_{I},G_{\alpha}]=\left(\frac{I}{2}-\alpha\right)G_{I+\alpha},\quad\\{G_{\alpha},G_{\beta}\\}=-2L_{\alpha+\beta}.$
(35)

  * •

$\mathfrak{osp}(2|2)$:
For $\mathcal{N}=2$, due to the antisymmetry of the $\mathfrak{so}(2)$
generators, we can write $T^{AB}=\epsilon^{AB}{\tt J}$. Therefore, the algebra
reduces to the $\mathfrak{osp}(2|2)$ superalgebra. We denote the bosonic
generators as ${\tt J}$, ${\tt L}_{I}$ $(I=\pm 1,0)$, and the fermionic ones
as ${\tt G}_{\alpha}^{A}$ $(\alpha=\pm\frac{1}{2},A=1,2)$. Their commutation
relations read:

$\displaystyle[L_{I},L_{J}]$
$\displaystyle=(I-J)L_{I+J},\quad[L_{I},G_{\alpha}^{A}]=\left(\frac{I}{2}-\alpha\right)G_{I+\alpha}^{A},$
(36) $\displaystyle[G_{\alpha}^{A},{\tt J}]$
$\displaystyle=\epsilon^{AB}G_{\alpha}^{B},\quad\\{G_{\alpha}^{A},G_{\beta}^{B}\\}=-2L_{\alpha+\beta}\delta^{AB}-(\alpha-\beta)\epsilon^{AB}{\tt
J}.$ (37)

  * •

$\mathfrak{osp}(3|2)$:
For $\mathcal{N}=3$, owing to the antisymmetry of the $\mathfrak{so}(3)$
generators, we can express $T^{ABC}=\epsilon^{ABC}{\tt J}^{C}$. Consequently,
the algebra reduces to the $\mathfrak{osp}(3|2)$ superalgebra. We denote the
bosonic generators by ${\tt J}^{A}$ ($A=1,2,3$), ${\tt L}_{I}$ ($I=\pm 1,0$),
and the fermionic generators by ${\tt G}_{\alpha}^{A}$
($\alpha=\pm\frac{1}{2}$, $A=1,2,3$). Their commutation relations are given by

$\displaystyle\left[{\tt L}_{I},{\tt L}_{J}\right]$
$\displaystyle=\left(I-J\right){\tt L}_{I+J},$ (38) $\displaystyle\left[{\tt
L}_{I},{\tt G}_{\alpha}^{A}\right]$
$\displaystyle=\left(\frac{I}{2}-\alpha\right){\tt G}_{I+\alpha}^{A},$ (39)
$\displaystyle\left[{\tt J}^{A},{\tt J}^{B}\right]$
$\displaystyle=\epsilon^{ABC}{\tt J}^{C},$ (40) $\displaystyle\left[{\tt
J}^{A},{\tt G}_{\alpha}^{B}\right]$ $\displaystyle=\epsilon^{ABC}{\tt
G}_{\alpha}^{C},$ (41) $\displaystyle\left\\{{\tt G}_{\alpha}^{A},{\tt
G}_{\beta}^{B}\right\\}$ $\displaystyle=2\,\delta^{AB}{\tt L}_{\alpha+\beta}$
$\displaystyle-\left(\alpha-\beta\right)\delta_{\alpha,-\beta}\epsilon^{ABC}{\tt
J}^{C},$ (42)

except for zero commutators.
It is important to note that there are variants with different spin contents,
referred to as the $\mathfrak{osp}(3|2)$ superalgebra in the literature
Ozer:2019nkv . Specifically, a copy of the $\mathcal{N}=2$ superalgebra arises
when the even-graded sector of the superalgebra is decomposed into a spin-2
component, the $\mathfrak{sl}(2)$ generators ${\tt L}_{i}$ ($i=\pm 1,0$), a
spin-1 element ${\tt J}$, a spin-2 multiplet ${\tt A}_{i}$ ($i=\pm 1,0$), and
a spin-3 multiplet ${\tt W}_{i}$ ($i=\pm 2,\pm 1,0$). All these generators
together span the bosonic sub-algebra
$\mathfrak{sl}(3)\oplus\mathfrak{sl}(2)\oplus\mathfrak{u}(1)$. Furthermore,
the odd-graded elements decompose into two spin-$\frac{3}{2}$ multiplets ${\tt
G}_{r}^{M}$ ($r=\pm\frac{1}{2}$, $M=\pm$) and two spin-$\frac{5}{2}$
multiplets ${\tt S}_{r}^{M}$ ($r=\pm\frac{3}{2},\pm\frac{1}{2}$, $M=\pm$).

  * •

$\mathfrak{osp}(4|2)$:
For the case $\mathcal{N}=4$ where
$\mathfrak{so}(4)\simeq\mathfrak{su}(2)\oplus\mathfrak{su}(2)$, special
treatment is required. We denote the bosonic generators as ${\tt J}^{A}$
$(A=1,2,3)$, ${\tt L}_{I}$ $(I=\pm 1,0)$, and the fermionic generators as
${\tt G}_{\alpha}^{a}$, $\bar{{\tt G}}_{\alpha}^{a}$
$(\alpha=\pm\frac{1}{2},\,a=1,2)$. Their commutation relations are given by

$\displaystyle[{\tt L}_{I},{\tt L}_{J}]$ $\displaystyle=(I-J){\tt L}_{I+J},$
(43) $\displaystyle[{\tt L}_{I},{\tt G}_{\alpha}^{a}]$
$\displaystyle=\left(\frac{I}{2}-\alpha\right){\tt G}_{I+\alpha}^{a},$ (44)
$\displaystyle[{\tt L}_{I},\bar{{\tt G}}_{\alpha}^{a}]$
$\displaystyle=\left(\frac{I}{2}-\alpha\right)\bar{{\tt G}}_{I+\alpha}^{a},$
(45) $\displaystyle[{\tt J}^{A},{\tt J}^{B}]$
$\displaystyle=\epsilon^{ABC}{\tt J}^{C},$ (46) $\displaystyle[{\tt
J}^{A},{\tt G}_{\alpha}^{a}]$ $\displaystyle=(\sigma^{A})_{ab}{\tt
G}_{\alpha}^{b},$ (47) $\displaystyle[{\tt J}^{A},\bar{{\tt G}}_{\alpha}^{a}]$
$\displaystyle=(\sigma^{A})^{*}_{ab}\bar{{\tt G}}_{\alpha}^{b},$ (48)
$\displaystyle\\{{\tt G}_{\alpha}^{a},\bar{{\tt G}}_{\beta}^{b}\\}$
$\displaystyle=2\delta^{ab}{\tt L}_{\alpha+\beta}$
$\displaystyle-(\alpha-\beta)\delta_{\alpha,-\beta}(\sigma^{c})_{ab}{\tt
J}^{c},$ (49)

except for zero commutators. Here, $\epsilon^{ijk}$ represents the structure
constants of $\mathfrak{su}(2)$, and $\sigma^{i}$ denotes the Pauli matrices.

The equations of motion for Chern-Simons theory, often referred to as the
flatness conditions, correspond to the vanishing of the field strengths, i.e.,
$F=\bar{F}=0$, where

$F=\mathrm{d}\Gamma+\Gamma\wedge\Gamma=0,\quad\bar{F}=\mathrm{d}\bar{\Gamma}+\bar{\Gamma}\wedge\bar{\Gamma}=0.$
(50)

This condition is equivalent to Einstein’s equations. The connection to
Einstein’s equations is established by expressing Lie algebra-valued
generalizations of the vielbein and spin connection in terms of the gauge
connections. Consequently, the metric $g_{\mu\nu}$ can be obtained from the
vielbein $e=\frac{\ell}{2}(\Gamma-\bar{\Gamma})$ as follows:

$g_{\mu\nu}=\frac{1}{2}\,\mathfrak{str}(e_{\mu}e_{\nu}).$ (51)

By adopting the radial gauge, asymptotically $AdS_{3}$ connections can be
represented as

$\displaystyle\Gamma$ $\displaystyle=$ $\displaystyle
b^{-1}a\left(t,\phi\right)b+b^{-1}\mathrm{d}b,\quad\bar{\Gamma}=b\bar{a}\left(t,\phi\right)b^{-1}+b\mathrm{d}b^{-1},$
(52)

where the state-independent group element $b(\rho)$ is given by

$\displaystyle b(\rho)=e^{{\tt L}_{-1}}e^{\rho{\tt L}_{0}},$ (53)

enabling a more general metric that encompasses all
$\mathfrak{osp}(\mathcal{N}|2)$ charges. It is crucial to note that, as long
as $\delta b=0$, the choice of $b$ is irrelevant for asymptotic symmetries.
This freedom allows for a more general metric, and it is essential to choose
boundary conditions that preserve this generality for supergravity.

Furthermore, in the radial gauge, the connections $a\left(t,\phi\right)$ and
$\bar{a}\left(t,\phi\right)$ are $\mathfrak{osp}(3|2)$ Lie superalgebra-valued
fields that are independent of the radial coordinate:

$\displaystyle a\left(t,\varphi\right)$ $\displaystyle=$ $\displaystyle
a_{t}\left(t,\varphi\right)\mathrm{d}t+a_{\varphi}\left(t,\varphi\right)\mathrm{d}\varphi,$
(54) $\displaystyle\bar{a}\left(t,\varphi\right)$ $\displaystyle=$
$\displaystyle\bar{a}_{t}\left(t,\varphi\right)\mathrm{d}t+\bar{a}_{\varphi}\left(t,\varphi\right)\mathrm{d}\varphi.$
(55)

Hereafter, we focus only on the unbarred sector since the analysis of the
barred sector works in complete analogy, yielding the same outcomes. The same
algorithm can be applied to the barred sector due to the procedure used.

##  4 $\mathcal{N}=(3,3)$ $\mathfrak{osp}(3|2)\oplus\mathfrak{osp}(3|2)$
Higher-Spin Chern-Simons Supergravity

We initiate the study of asymptotically $AdS_{3}$ boundary conditions for
$\mathfrak{osp}(3|2)\oplus\mathfrak{osp}(3|2)$ Chern-Simons theory in the
affine case. We illustrate how the procedure outlined in Grumiller:2016pqb
can be employed to assess the asymptotic symmetry algebra. Based on the
obtained results, the most general solution of Einstein’s equations that is
asymptotically $AdS_{3}$ can be expressed using the following general metric
form:

$\displaystyle\mathrm{d}s^{2}$ $\displaystyle=$
$\displaystyle\mathrm{d}\rho^{2}+2\left[e^{\rho}N^{(0)}_{i}+N^{(1)}_{i}+e^{-\rho}N^{(2)}_{i}+\mathcal{O}\left(e^{-2\rho}\right)\right]\mathrm{d}\rho\mathrm{d}x^{i}$
(56) $\displaystyle+$
$\displaystyle\left[e^{2\rho}g^{(0)}_{ij}+e^{\rho}g^{(1)}_{ij}+g^{(2)}_{ij}+\mathcal{O}\left(e^{-\rho}\right)\right]\mathrm{d}x^{i}\mathrm{d}x^{j}.$

Therefore, it is crucial to define the most general $\mathcal{N}=(3,3)$
supergravity boundary conditions that preserve this form of the metric.

###  4.1 Affine Boundary Conditions

The objective of this section is to construct $\mathcal{N}=(3,3)$ extended
higher-spin $AdS_{3}$ supergravity as
$\mathfrak{osp}(3|2)\oplus\mathfrak{osp}(3|2)$ Chern-Simons gauge theory on
the affine boundary. We proceed with our calculations to elucidate the
asymptotic symmetry algebra for the loosest set of boundary conditions.
Therefore, we consider the principal embedding of $\mathfrak{osp}(1|2)$ into
$\mathfrak{osp}(3|2)$ as a sub-algebra, giving rise to the asymptotic
symmetry.

We start by proposing the $\mathfrak{osp}(3|2)$ Lie superalgebra valued
$a_{\varphi}$ component of the gauge connection in the form:

$\displaystyle a_{\varphi}$ $\displaystyle=$
$\displaystyle\rho_{A}\mathcal{J}^{A}{\tt J}^{A}+\gamma_{i}\mathcal{L}^{i}{\tt
L}_{i}+\sigma_{M}^{p}\mathcal{G}_{A}^{p}{\tt G}_{p}^{A}$ (57)

where $\rho_{A}=\frac{2}{k}$, $\gamma_{0}=-2\gamma_{\pm 1}=-\frac{4}{k}$, and
$\sigma_{A}^{-\frac{1}{2}}=-\sigma_{A}^{\frac{1}{2}}=-\frac{1}{k}$ are scaling
parameters to be identified later. We have twelve state-dependent functions
consisting of six bosonic $\left(\mathcal{J}^{A},\mathcal{L}^{i}\right)$ and
six fermionic $\mathcal{G}_{A}^{p}$, usually called charges. The time
component $a_{t}$ of the connection $a\left(t,\varphi\right)$ can be given as:

$\displaystyle a_{t}$ $\displaystyle=$ $\displaystyle\eta_{A}{\tt
J}^{A}+\mu^{i}{\tt L}_{i}+\nu_{A}^{p}{\tt G}_{p}^{A}.$ (58)

In this case, we have twelve independent functions
$(\eta_{A},\mathcal{\mu}^{i},\nu_{A}^{p})$, as chemical potentials, which are
not allowed to vary, $\delta a_{t}=0$.

Using the flatness conditions $(\ref{flatness})$, the equations of motion for
fixed chemical potentials impose the following additional conditions on the
charges $(\mathcal{J}^{A},\mathcal{L}^{i},\mathcal{G}_{A}^{p})$:

$\displaystyle\partial_{t}\mathcal{J}^{A}$ $\displaystyle=$
$\displaystyle\frac{k}{2}\partial_{\varphi}\eta_{A}+\epsilon^{ABC}(\mathcal{J}^{B}\eta_{C}-\frac{1}{2}(\mathcal{G}_{A}^{\frac{1}{2}}\nu_{C}^{-\frac{1}{2}}+\mathcal{G}_{A}^{-\frac{1}{2}}\nu_{C}^{\frac{1}{2}})$
(59) $\displaystyle 2\partial_{t}\mathcal{L}^{0}$ $\displaystyle=$
$\displaystyle-\frac{k}{4}\partial_{\varphi}\mu^{0}-\mathcal{L}^{+1}\mu^{-1}+\mathcal{L}^{-1}\mu^{+1}-\frac{1}{2}(\mathcal{G}_{A}^{\frac{1}{2}}\nu_{A}^{-\frac{1}{2}}-\mathcal{G}_{A}^{-\frac{1}{2}}\nu_{A}^{\frac{1}{2}})$
(60) $\displaystyle\partial_{t}\mathcal{L}^{\pm 1}$ $\displaystyle=$
$\displaystyle\frac{k}{2}\partial_{\varphi}\mu^{\mp 1}+\mathcal{L}^{0}\mu^{\mp
1}\mp\mathcal{L}^{\mp
1}\mu^{0}\mp\mathcal{G}_{A}^{\mp\frac{1}{2}}\nu_{A}^{\mp\frac{1}{2}},$ (61)
$\displaystyle\partial_{t}\mathcal{G}_{A}^{\mp\frac{1}{2}}$ $\displaystyle=$
$\displaystyle\mp
k\partial_{\varphi}\nu_{A}^{\mp\frac{1}{2}}\mp\frac{1}{2}\mu^{0}\mathcal{G}_{A}^{\mp\frac{1}{2}}\mp\mu^{-1}\mathcal{G}_{A}^{\pm\frac{1}{2}}+2\mathcal{L}^{0}\nu_{A}^{\mp\frac{1}{2}}+2\mathcal{L}^{\mp}\nu_{A}^{\pm\frac{1}{2}}$
(62) $\displaystyle-$
$\displaystyle\epsilon^{ABC}(\eta_{{}_{B}}\mathcal{G}_{C}^{\mp\frac{1}{2}}+2\mathcal{J}^{B}\nu_{C}^{\mp\frac{1}{2}})$
(63)

representing the temporal evolution of the eight state-dependent source
fields.

We aim to derive the asymptotic symmetry algebra for the most general boundary
conditions through canonical analysis. To achieve this, we consider all gauge
transformations given by

$\delta_{\lambda}\Gamma=\mathrm{d}\lambda+\left[\Gamma,\lambda\right],$ (64)

which preserve the most general boundary conditions. At this stage, it is
appropriate to express the gauge parameter in terms of the
$\mathfrak{osp}(3|2)$ Lie superalgebra basis:

$\lambda=b^{-1}\bigg{[}\varrho_{A}{\tt J}^{A}+\epsilon^{i}{\tt
L}_{i}+\zeta_{A}^{p}{\tt G}_{p}^{A}\bigg{]}b.$ (65)

Note that the gauge parameter includes four bosonic variables $\varrho$,
$\epsilon^{i}$, and four fermionic variables $\zeta_{M}^{p}$, which are
arbitrary functions of boundary coordinates.

Now, let’s focus on the gauge parameters satisfying (64). The infinitesimal
gauge transformations are determined as follows:

$\displaystyle\partial_{t}\mathcal{J}^{A}$
$\displaystyle=\frac{k}{2}\partial_{\varphi}\varrho_{A}+\epsilon^{ABC}(\mathcal{J}^{B}\varrho_{C}-\frac{1}{2}(\mathcal{G}_{A}^{\frac{1}{2}}\zeta_{C}^{-\frac{1}{2}}+\mathcal{G}_{A}^{-\frac{1}{2}}\zeta_{C}^{\frac{1}{2}})),$
(66) $\displaystyle 2\partial_{t}\mathcal{L}^{0}$
$\displaystyle=-\frac{k}{4}\partial_{\varphi}\epsilon^{0}-\mathcal{L}^{+1}\epsilon^{-1}+\mathcal{L}^{-1}\epsilon^{+1}-\frac{1}{2}(\mathcal{G}_{A}^{\frac{1}{2}}\zeta_{A}^{-\frac{1}{2}}-\mathcal{G}_{A}^{-\frac{1}{2}}\zeta_{A}^{\frac{1}{2}}),$
(67) $\displaystyle\partial_{t}\mathcal{L}^{\pm 1}$
$\displaystyle=\frac{k}{2}\partial_{\varphi}\epsilon^{\mp
1}+\mathcal{L}^{0}\epsilon^{\mp 1}\mp\mathcal{L}^{\mp
1}\epsilon^{0}\mp\mathcal{G}_{A}^{\mp\frac{1}{2}}\zeta_{A}^{\mp\frac{1}{2}},$
(68) $\displaystyle\partial_{t}\mathcal{G}_{A}^{\mp\frac{1}{2}}$
$\displaystyle=\mp
k\partial_{\varphi}\zeta_{A}^{\mp\frac{1}{2}}\mp\frac{1}{2}\epsilon^{0}\mathcal{G}_{A}^{\mp\frac{1}{2}}\mp\epsilon^{-1}\mathcal{G}_{A}^{\pm\frac{1}{2}}+2\mathcal{L}^{0}\zeta_{A}^{\mp\frac{1}{2}}+2\mathcal{L}^{\mp}\zeta_{A}^{\pm\frac{1}{2}}$
$\displaystyle-\epsilon^{ABC}(\varrho_{{}_{B}}\mathcal{G}_{C}^{\mp\frac{1}{2}}+2\mathcal{J}^{B}\zeta_{C}^{\mp\frac{1}{2}}).$
(69)

Constraints for the chemical potentials can also be derived analogously:

$\displaystyle\partial_{t}\eta^{A}$
$\displaystyle=\frac{k}{2}\partial_{\varphi}\varrho_{A}+\epsilon^{ABC}(\eta^{B}\varrho_{C}-\frac{1}{2}(\mathcal{G}_{A}^{\frac{1}{2}}\zeta_{C}^{-\frac{1}{2}}+\mathcal{G}_{A}^{-\frac{1}{2}}\zeta_{C}^{\frac{1}{2}})),$
(70) $\displaystyle 2\partial_{t}\mu^{0}$
$\displaystyle=-\frac{k}{4}\partial_{\varphi}\epsilon^{0}-\mu^{+1}\epsilon^{-1}+\mu^{-1}\epsilon^{+1}-\frac{1}{2}(\nu_{A}^{\frac{1}{2}}\zeta_{A}^{-\frac{1}{2}}-\nu_{A}^{-\frac{1}{2}}\zeta_{A}^{\frac{1}{2}}),$
(71) $\displaystyle\partial_{t}\mu^{\pm 1}$
$\displaystyle=\frac{k}{2}\partial_{\varphi}\epsilon^{\mp
1}+\mu^{0}\epsilon^{\mp 1}\mp\mu^{\mp
1}\epsilon^{0}\mp\nu_{A}^{\mp\frac{1}{2}}\zeta_{A}^{\mp\frac{1}{2}},$ (72)
$\displaystyle\partial_{t}\nu_{A}^{\mp\frac{1}{2}}$ $\displaystyle=\mp
k\partial_{\varphi}\zeta_{A}^{\mp\frac{1}{2}}\mp\frac{1}{2}\epsilon^{0}\nu_{A}^{\mp\frac{1}{2}}\mp\epsilon^{-1}\nu_{A}^{\pm\frac{1}{2}}+2\mu^{0}\zeta_{A}^{\mp\frac{1}{2}}+2\mu^{\mp}\zeta_{A}^{\pm\frac{1}{2}}$
$\displaystyle-\epsilon^{ABC}(\varrho_{{}_{B}}\nu_{C}^{\mp\frac{1}{2}}+2\eta^{B}\zeta_{C}^{\mp\frac{1}{2}}).$
(73)

As a final step, the canonical boundary charge $\mathcal{Q[\lambda]}$ that
generates the transformations (66) - (69) can be defined. The variation of the
canonical boundary charge $\mathcal{Q[\lambda]}$ Banados:1994tn  leading to
the asymptotic symmetry algebra is given by

$\delta_{\lambda}\mathcal{Q}=\frac{k}{2\pi}\int\mathrm{d}\varphi\;\mathfrak{str}\left(\lambda\delta\Gamma_{\varphi}\right).$
(74)

Therefore, the variation of the canonical boundary charge
$\mathcal{Q[\lambda]}$ can be functionally integrated to yield

$\mathcal{Q[\lambda]}=\int\mathrm{d}\varphi\;\left[\mathcal{J}^{A}\varrho_{A}+\mathcal{L}^{i}\epsilon^{-i}+\mathcal{G}_{A}^{p}\zeta_{A}^{-p}\right].$
(75)

After determining both the infinitesimal transformations and the canonical
boundary charge, we are now positioned to derive the asymptotic symmetry
algebra using the standard method Blgojevic:2002 . This can be obtained
through the following relation:

$\delta_{\lambda}\digamma=\\{\digamma,\mathcal{Q}[\lambda]\\}$ (76)

for any phase space functional $\digamma$. The charges $\mathcal{L}^{i}(z)$,
$\mathcal{J}^{A}(z)$, and $\mathcal{G}_{M}^{p}(z)$ are identified as the
generators of the asymptotic symmetry algebra. Finally, the operator product
algebra can be expressed as follows:

$\displaystyle\mathcal{L}^{i}(z_{1})\mathcal{L}^{j}(z_{2})$
$\displaystyle\sim$
$\displaystyle\frac{\frac{k}{2}\eta^{ij}}{z_{12}^{2}}+\frac{(i-j)}{z_{12}}\mathcal{L}^{i+j}(z_{2}),$
(77) $\displaystyle\mathcal{L}^{i}(z_{1})\mathcal{G}_{A}^{p}(z_{2})$
$\displaystyle\sim$ $\displaystyle\frac{({i\over
2}-p)}{z_{12}}\mathcal{G}_{A}^{i+p}(z_{2}),$ (78)
$\displaystyle\mathcal{J}^{A}(z_{1})\mathcal{J}^{B}(z_{2})$
$\displaystyle\sim$
$\displaystyle\frac{\frac{k}{2}\eta}{z_{12}^{2}}\delta^{AB}+\frac{\epsilon^{ABC}}{z_{12}}\mathcal{J}^{C}(z_{2}),$
(79) $\displaystyle\mathcal{J}^{A}(z_{1})\mathcal{G}_{B}^{p}(z_{2})$
$\displaystyle\sim$
$\displaystyle\frac{\epsilon^{ABC}}{z_{12}}\mathcal{G}_{C}^{p}(z_{2}),$ (80)
$\displaystyle\mathcal{G}_{A}^{p}(z_{1})\mathcal{G}_{B}^{q}(z_{2})$
$\displaystyle\sim$
$\displaystyle\frac{\frac{k}{2}}{z_{12}^{2}}\eta_{AB}^{pq}+\frac{2}{z_{12}}\delta^{AB}\mathcal{L}^{p+q}(z_{2})+\delta_{p,-q}(p-q)\frac{\epsilon^{ABC}}{z_{12}}\mathcal{J}^{C}(z_{2}).$
(81)

Here, $z_{12}=z_{1}-z_{2}$, $\eta^{ij}=\mathfrak{str}({\tt L}_{i}{\tt
L}_{j})$, and $\eta^{pq}=\mathfrak{str}({\tt G}_{p}^{A}{\tt G}_{q}^{A})$ are
the bilinear forms in the fundamental representation of the
$\mathfrak{osp}(3|2)$ Lie superalgebra. The operator product algebra can also
be expressed in a more compact form:

$\displaystyle\mathfrak{\mathcal{\mathfrak{J}}}^{A}(z_{1})\mathfrak{\mathcal{\mathfrak{J}}}^{B}(z_{2})$
$\displaystyle\sim$
$\displaystyle\frac{\frac{k}{2}\eta^{AB}}{z_{12}^{2}}+\frac{\mathfrak{\mathcal{\mathfrak{f}}}^{AB}_{~{}~{}~{}C}\mathfrak{\mathcal{\mathfrak{J}}}^{C}(z_{2})}{z_{12}}.$
(82)

Note that $\eta^{AB}$ is the supertrace matrix, and
$\mathfrak{\mathcal{\mathfrak{f}}}^{AB}_{~{}~{}C}$ are the structure constants
of the related algebra with $(A,B=0,\pm 1,\pm\frac{1}{2})$, i.e.,
$\eta^{ip}=0$ and $\mathfrak{\mathcal{\mathfrak{f}}}^{ij}_{~{}~{}i+j}=(i-j)$.
Lastly, by repeating the same analysis for the barred sector, the asymptotic
symmetry algebra of $\mathcal{N}=(3,3)$ supergravity for the loosest set of
boundary conditions is given by two copies of the affine
$\mathfrak{osp}(3|2)_{k}$ algebra.

###  4.2 Superconformal Boundary Conditions

In this section, our objective is to examine the asymptotic symmetry algebra
for the supersymmetric extension of the Brown-Henneaux boundary conditions. We
commence by imposing the Drinfeld-Sokolov highest weight gauge condition on
the $\mathfrak{osp}(3|2)$ Lie superalgebra-valued connection (57) to further
constrain the coefficients. Thus, the Drinfeld-Sokolov reduction sets the
fields as follows:

$\displaystyle\mathcal{L}^{0}=\mathcal{G}_{A}^{+\frac{1}{2}}=0,\quad\mathcal{L}^{-1}=\mathcal{L},\quad\mathcal{G}_{A}^{-\frac{1}{2}}=\mathcal{G}_{A},\quad\gamma_{+1}\mathcal{L}^{+1}=1.$
(83)

It is noteworthy that the superconformal boundary conditions are the
supersymmetric extension of the well-known Brown-Henneaux boundary conditions
proposed in Brown:1986nw  for $AdS_{3}$ supergravity. Consequently, the
supersymmetric gauge connection takes the form:

$\displaystyle a_{\varphi}$ $\displaystyle=$ $\displaystyle{\tt
L}_{+1}+\gamma_{-1}\mathcal{L}{\tt L}_{-1}+\rho_{A}\mathcal{J}^{A}{\tt
J}^{A}+\sigma_{A}^{-{\frac{1}{2}}}\mathcal{G}_{A}{\tt
G}_{-{\frac{1}{2}}}^{A}.$ (84)

Here, $\gamma_{-1}=-\frac{1}{k}$, $\rho=\frac{1}{2k}$,
$\sigma_{+}^{-{\frac{1}{2}}}=\frac{1}{2k}$, and
$\sigma_{-}^{-{\frac{1}{2}}}=-\frac{1}{2k}$ are scaling parameters. We
introduce four functions: two bosonic ($\mathcal{J},\mathcal{L}$) and two
fermionic ($\mathcal{G}_{\pm}$) charges. After implementing these steps, we
are on the verge of obtaining the superconformal asymptotic symmetry algebra.
Following the results implied by the Drinfeld-Sokolov reduction, the gauge
parameter $\lambda$ has only four independent functions
$(\varrho_{A},\mathcal{\epsilon}\equiv\epsilon^{+1},\,\zeta_{A}\equiv\zeta_{A}^{+\frac{1}{2}})$,
given by:

$\displaystyle\lambda$ $\displaystyle=$ $\displaystyle
b^{-1}\left[\epsilon{\tt L}_{1}-\epsilon^{\prime}{\tt
L}_{0}+\Bigg{(}\frac{6\mathcal{L}\epsilon}{c}+\frac{3\mathcal{G}_{A}\zeta_{A}}{c}+\frac{\epsilon^{\prime\prime}}{2}\Bigg{)}{\tt
L}_{-1}\right.$ (85) $\displaystyle\left.+\varrho_{A}{\tt J}^{A}+\zeta_{A}{\tt
G}_{\frac{1}{2}}^{A}+\left(\frac{3\epsilon^{ABC}\mathcal{J}^{B}\zeta_{C}}{c}-\frac{3\mathcal{G}_{A}\epsilon}{c}-\zeta_{A}^{\prime}\right){\tt
G}_{-\frac{1}{2}}^{A}\right]b.$

Substituting this gauge parameter into the expression for the transformation
of the fields (64), we obtain the infinitesimal gauge transformations:

$\displaystyle\delta_{\lambda}\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{c}{12}\epsilon^{{}^{\prime\prime\prime}}+\epsilon\mathcal{L}^{\prime}+2\mathcal{L}\epsilon^{\prime}+\frac{3}{2}\mathcal{G}_{A}\zeta_{A}^{\prime}+\frac{1}{2}\zeta_{A}\mathcal{G}_{A}^{\prime}+\frac{3\epsilon^{ABC}\zeta_{A}\mathcal{J}^{B}\mathcal{G}_{C}}{c}$
(86) $\displaystyle\delta_{\lambda}\mathcal{J}^{A}$ $\displaystyle=$
$\displaystyle-\frac{c}{3}\varrho_{A}^{\prime}+\epsilon^{ABC}\bigg{(}\mathcal{J}^{B}\varrho_{C}+\mathcal{G}_{B}\zeta_{C}\bigg{)}$
(87) $\displaystyle\delta_{\lambda}\mathcal{G}^{A}$ $\displaystyle=$
$\displaystyle\frac{c}{3}\zeta_{A}^{\prime\prime}+2\bigg{(}\mathcal{L}-\frac{3}{2c}\mathcal{J}^{B}\mathcal{J}^{B}\bigg{)}\zeta_{A}+\frac{3\mathcal{J}^{A}\mathcal{J}^{B}\zeta_{B}}{c}+\frac{3\mathcal{G}_{A}\epsilon^{\prime}}{2}+\epsilon\mathcal{G}_{A}^{\prime}$
(88) $\displaystyle+$
$\displaystyle\epsilon^{ABC}\Bigg{(}\bigg{(}\varrho_{C}+\frac{3}{c}\epsilon\mathcal{J}^{C}\bigg{)}\mathcal{G}_{B}+\zeta_{B}{\mathcal{J}^{C}}^{\prime}-2\mathcal{J}^{B}\zeta_{C}^{\prime}\Bigg{)}$

For the given boundary conditions, the boundary charge $\mathcal{Q}[\lambda]$
is integrable, and we identify $\mathcal{L}$, $\mathcal{G}^{A}$, and
$\mathcal{J}^{A}$ as charges. However, these charges do not generate an
algebra due to some nonlinear terms $(\mathcal{J}^{B}\mathcal{G}_{C})$ in
$(\ref{ebc086})$. Moreover, to obtain the correct superconformal algebra
$(\ref{qalg11})$-$(\ref{qalg12})$, we need to perform a shift on $\mathcal{L}$
and a redefinition of the gauge parameter $\varrho_{A}$ as:

$\mathcal{L}\rightarrow\mathcal{L}+\frac{3}{2c}\mathcal{J}^{A}\mathcal{J}^{A},\quad\varrho_{A}\rightarrow\varrho_{A}-\frac{3}{c}\epsilon\mathcal{J}^{A}.$
(89)

The new gauge parameter $\lambda$ is given as:

$\displaystyle\lambda$ $\displaystyle=$ $\displaystyle
b^{-1}\left[\epsilon{\tt L}_{1}-\epsilon^{\prime}{\tt
L}_{0}+\Bigg{(}\frac{6}{c}\bigg{(}\mathcal{L}+\frac{3}{2c}\mathcal{J}^{A}\mathcal{J}^{A}\bigg{)}\epsilon+\frac{3\mathcal{G}_{A}\zeta_{A}}{c}+\frac{\epsilon^{\prime\prime}}{2}\Bigg{)}{\tt
L}_{-1}\right.$ (90)
$\displaystyle\left.+\left(\varrho_{A}-\frac{3\mathcal{J}^{A}\epsilon}{c}\right){\tt
J}^{A}+\zeta_{A}{\tt
G}_{\frac{1}{2}}^{A}+\left(-\frac{3\mathcal{G}_{A}\epsilon}{c}+\frac{3\epsilon^{ABC}\mathcal{J}^{B}\zeta_{C}}{c}-\zeta_{A}^{\prime}\right){\tt
G}_{-\frac{1}{2}}^{A}\right]b.$

Substituting this new gauge parameter into the expression
$(\ref{boundarycond})$, we obtain the new infinitesimal gauge transformations:

$\displaystyle\delta_{\lambda}\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{c}{12}\epsilon^{{}^{\prime\prime\prime}}+\epsilon\mathcal{L}^{\prime}+2\mathcal{L}\epsilon^{\prime}+\frac{3}{2}\mathcal{G}_{A}\zeta_{A}^{\prime}+\frac{1}{2}\zeta_{A}\mathcal{G}_{A}^{\prime}+\frac{3\epsilon^{ABC}\zeta_{A}\mathcal{J}^{B}\mathcal{G}_{C}}{c},$
(91) $\displaystyle\delta_{\lambda}\mathcal{J}^{A}$ $\displaystyle=$
$\displaystyle-\frac{c}{3}\varrho_{A}^{\prime}+\epsilon^{ABC}\bigg{(}\mathcal{J}^{B}\varrho_{C}+\mathcal{G}_{B}\zeta_{C}\bigg{)},$
(92) $\displaystyle\delta_{\lambda}\mathcal{G}^{A}$ $\displaystyle=$
$\displaystyle\frac{c}{3}\zeta_{A}^{\prime\prime}+2\mathcal{L}\zeta_{A}+\frac{3\mathcal{J}^{A}\mathcal{J}^{B}\zeta_{B}}{c}+\frac{3\mathcal{G}_{A}\epsilon^{\prime}}{2}+\epsilon\mathcal{G}_{A}^{\prime}$
(93)
$\displaystyle+\epsilon^{ABC}\Bigg{(}\varrho_{C}\mathcal{G}_{B}+\zeta_{B}{\mathcal{J}^{C}}^{\prime}-2\mathcal{J}^{B}\zeta_{C}^{\prime}\Bigg{)}.$

In fact, these gauge transformations provide insight into the asymptotic
symmetry algebra Banados:1998pi . By examining the asymptotic symmetries, one
can integrate the variation of the canonical boundary charges, i.e.,
$\delta_{\lambda}\mathcal{Q}$ in expression $(\ref{Qvar})$, such that

$\mathcal{Q}[\lambda]=\int\mathrm{d}\varphi\;\left[\mathcal{L}\epsilon+\mathcal{G}_{A}\zeta_{A}+\mathcal{J}^{A}\varrho_{A}\right].$
(94)

These canonical boundary charges provide a convenient asymptotic operator
product algebra for $\mathcal{N}=(3,3)$ superconformal boundaries:

$\displaystyle\mathcal{L}(z_{1})\mathcal{L}(z_{2})$ $\displaystyle\sim$
$\displaystyle\frac{{c/2}}{{z_{12}^{4}}}+\frac{2\mathcal{L}}{{z_{12}^{2}}}+\frac{\mathcal{L}^{\prime}}{z_{12}},$
(95) $\displaystyle\mathcal{L}(z_{1})\mathcal{J}(z_{2})$ $\displaystyle\sim$
$\displaystyle\frac{\mathcal{J}}{z_{12}^{2}}+\frac{\mathcal{J}^{\prime}}{z_{12}},$
(96) $\displaystyle\mathcal{L}(z_{1})\mathcal{G}_{A}(z_{2})$
$\displaystyle\sim$
$\displaystyle\frac{{\frac{3}{2}\mathcal{G}_{A}}}{z_{12}^{2}}+\frac{{\mathcal{G}_{A}}^{\prime}}{z_{12}},$
(97) $\displaystyle\mathcal{J}^{A}(z_{1})\mathcal{J}^{B}(z_{2})$
$\displaystyle\sim$
$\displaystyle\frac{\frac{c}{3}}{z_{12}^{2}}\delta^{AB}+\frac{\epsilon^{ABC}}{z_{12}}\mathcal{J}^{C},$
(98) $\displaystyle\mathcal{J}^{A}(z_{1})\mathcal{G}_{A}(z_{2})$
$\displaystyle\sim$
$\displaystyle\frac{\epsilon^{ABC}}{z_{12}}\mathcal{G}_{C},$ (99)
$\displaystyle\mathcal{G}_{A}(z_{1})\mathcal{G}_{B}(z_{2})$
$\displaystyle\sim$
$\displaystyle\delta^{AB}\bigg{(}\frac{{2c/3}}{{z_{12}^{3}}}+\frac{2\mathcal{L}}{z_{12}}\bigg{)}$
(100)
$\displaystyle+2\epsilon^{ABC}\bigg{(}\frac{\mathcal{J}^{C}}{z_{12}^{2}}+\frac{{\frac{1}{2}\partial\mathcal{J}^{C}}}{z_{12}}\bigg{)}+\frac{{\frac{3}{c}\mathcal{J}^{A}}\mathcal{J}^{B}}{z_{12}}.$

Upon repeating the same analysis for the barred sector, it is observed that
the asymptotic symmetry algebra for the loosest set of boundary conditions of
$\mathcal{N}=(3,3)$ supergravity consists of two copies of the super-Virasoro
algebra with central charge $c=3k$.

##  5 $\mathcal{N}=(4,4)$ $\mathfrak{osp}(4|2)\oplus\mathfrak{osp}(4|2)$
Higher-Spin Chern-Simons Supergravity

After laying the groundwork with the canonical analysis of the
$\mathfrak{osp}(3|2)$ case, we present the extended $\mathcal{N}=(4,4)$
higher-spin Chern-Simons supergravity theory based on the
$\mathfrak{osp}(4|2)_{k}$ superalgebra.

###  5.1 Affine Boundary Conditions

The objective of this section is to construct $\mathcal{N}=(4,4)$ extended
higher-spin $AdS_{3}$ supergravity as a
$\mathfrak{osp}(4|2)\oplus\mathfrak{osp}(4|2)$ Chern-Simons gauge theory on
the affine boundary. We proceed with our calculations to elucidate the
asymptotic symmetry algebra for the loosest set of boundary conditions. As
stated in the previous section, we consider the principal embedding of
$\mathfrak{osp}(1|2)$ into $\mathfrak{osp}(4|2)$ as a sub-algebra.

We can now formulate the most general boundary conditions for asymptotically
$AdS_{3}$ spacetimes. To achieve this, it is useful to define the gauge
connection as follows,

$\displaystyle a_{\varphi}$ $\displaystyle=\rho_{A}\mathcal{J}^{A}{\tt
J}_{A}+\gamma_{i}\mathcal{L}^{i}{\tt
L}_{i}+\sigma_{A}^{p}\mathcal{G}_{A}^{p}{\tt
G}_{p}^{A}+\bar{\sigma}_{A}^{p}\bar{\mathcal{G}}_{A}^{p}\bar{{\tt G}}_{p}^{A}$
(101) $\displaystyle a_{t}$ $\displaystyle=\eta_{A}{\tt J}_{A}+\mu^{i}{\tt
L}_{i}+\nu_{A}^{p}{\tt G}_{p}^{A}+\bar{\nu}_{A}^{p}\bar{{\tt G}}_{p}^{A}$
(102)

where

$\displaystyle\rho$
$\displaystyle=\sigma_{M}^{\frac{1}{2}}=-\sigma_{M}^{\frac{1}{2}}=\frac{1}{k},\quad
2\gamma_{1}=2\gamma_{-1}-2\gamma_{0}=\frac{4}{k}$ (103)

are scaling parameters. Consequently, we have fourteen functions: six bosonic
$\left(\mathcal{J}^{A},\mathcal{L}^{i}\right)$ and eight
$\left(\mathcal{G}_{A}^{p},\bar{\mathcal{G}}_{A}^{p}\right)$ as charges.
Additionally, we have a total of fourteen independent functions
$\left(\eta_{A},\mu^{i},\nu_{A}^{p},\bar{\nu}_{A}^{p}\right)$ as chemical
potentials for the time component. In the presence of the loosest set of
boundary conditions, thanks to the flatness conditions $(\ref{flatness})$, the
equations of motion for fixed chemical potentials impose additional conditions
as the temporal evolution of the fourteen independent source fields
$\left(\mathcal{J}^{A},\mathcal{L}^{i},\mathcal{G}_{A}^{p},\bar{\mathcal{G}}_{A}^{p}\right)$
as calculated below:

$\displaystyle\partial_{t}\mathcal{J}^{A}$
$\displaystyle=\frac{k}{2}\partial_{\varphi}\eta_{A}+\epsilon^{ABC}\mathcal{J}^{B}\eta_{C}$
$\displaystyle-(\sigma^{A})_{BC}\bigg{(}\nu_{B}^{\frac{1}{2}}\bar{\mathcal{G}}_{C}^{-\frac{1}{2}}+\nu_{B}^{-\frac{1}{2}}\bar{\mathcal{G}}_{C}^{\frac{1}{2}}-\bar{\nu}_{B}^{\frac{1}{2}}{\mathcal{G}}_{C}^{-\frac{1}{2}}-\bar{\nu}_{B}^{-\frac{1}{2}}\mathcal{G}_{C}^{\frac{1}{2}}\bigg{)}$
(104) $\displaystyle 2\partial_{t}\mathcal{L}^{0}$
$\displaystyle=-\frac{k}{4}\partial_{\varphi}\mu^{0}-\mathcal{L}^{+1}\mu^{-1}+\mathcal{L}^{-1}\mu^{+1}$
$\displaystyle+\frac{1}{2}\bigg{(}\bar{\mathcal{G}}_{A}^{\frac{1}{2}}\nu_{A}^{-\frac{1}{2}}-\bar{\mathcal{G}}_{A}^{-\frac{1}{2}}\nu_{A}^{\frac{1}{2}}-\mathcal{G}_{A}^{-\frac{1}{2}}\nu_{A}^{\frac{1}{2}}+\mathcal{G}_{A}^{\frac{1}{2}}\nu_{A}^{-\frac{1}{2}}\bigg{)}$
(105) $\displaystyle\partial_{t}\mathcal{L}^{\mp 1}$
$\displaystyle=\frac{k}{2}\partial_{\varphi}\mu^{\pm
1}+2\mathcal{L}^{0}\mu^{\pm 1}\pm\mathcal{L}^{\pm
1}\mu^{0}\pm\bar{\mathcal{G}}_{A}^{\pm\frac{1}{2}}\nu_{A}^{\pm\frac{1}{2}}\pm\mathcal{G}_{A}^{\pm\frac{1}{2}}\bar{\nu}_{A}^{\pm\frac{1}{2}},$
(106) $\displaystyle\partial_{t}\mathcal{G}_{A}^{\pm\frac{1}{2}}$
$\displaystyle=\pm
k\partial_{\varphi}\nu_{A}^{\pm\frac{1}{2}}\pm\frac{1}{2}\mu^{0}\mathcal{G}_{A}^{\pm\frac{1}{2}}\pm\mu^{\mp}\mathcal{G}_{A}^{\pm\frac{1}{2}}+2\mathcal{L}^{0}\nu_{A}^{\pm\frac{1}{2}}+2\mathcal{L}^{\mp}\nu_{A}^{\pm\frac{1}{2}}$
$\displaystyle+\frac{i}{2}(\sigma^{B})^{*}_{AC}\bigg{(}\eta_{B}\mathcal{G}_{C}^{\pm\frac{1}{2}}+2\mathcal{J}^{B}\nu_{C}^{\pm\frac{1}{2}}\bigg{)}$
(107) $\displaystyle\partial_{t}\bar{\mathcal{G}}_{A}^{\pm\frac{1}{2}}$
$\displaystyle=\pm
k\partial_{\varphi}\nu_{A}^{\pm\frac{1}{2}}\pm\frac{1}{2}\mu^{0}\bar{\mathcal{G}}_{A}^{\pm\frac{1}{2}}\pm\mu^{\mp}\bar{\mathcal{G}}_{A}^{\pm\frac{1}{2}}+2\mathcal{L}^{0}\bar{\nu}_{A}^{\pm\frac{1}{2}}+2\mathcal{L}^{\mp}\bar{\nu}_{A}^{\pm\frac{1}{2}}$
$\displaystyle+\frac{i}{2}(\sigma^{B})^{*}_{AC}\bigg{(}\eta_{B}\bar{\mathcal{G}}_{C}^{\pm\frac{1}{2}}-2\mathcal{J}^{B}\bar{\nu}_{C}^{\pm\frac{1}{2}}\bigg{)}$
(108)

Thus, the consequences of our calculations to derive the relevant superalgebra
for the loosest set of boundary conditions can now be evaluated through a
canonical analysis. We now consider the boundary-preserving gauge
transformations (encompassing all) (64) generated by the $\mathfrak{osp}(4|2)$
Lie superalgebra-valued gauge parameter $\lambda$, which we choose as

$\lambda=b^{-1}\bigg{[}\varrho_{A}{\tt J}_{A}+\epsilon^{i}{\tt
L}_{i}+\zeta_{A}^{p}{\tt G}_{p}^{A}+\bar{\zeta}_{A}^{p}\bar{{\tt
G}}_{p}^{A}\bigg{]}b.$ (109)

Note that there are in total fourteen arbitrary functions on the boundary,
consisting of six bosonic $(\varrho_{A},\,\epsilon^{i})$ and eight fermionic
$(\mathcal{\zeta}_{A}^{p},\,\bar{\mathcal{\zeta}}_{A}^{p})$. Inserting this
expression into (64) imposes the gauge transformations as:

$\displaystyle\partial_{t}\mathcal{J}^{A}$
$\displaystyle=\frac{k}{2}\partial_{\varphi}\varrho_{A}+\epsilon^{ABC}\mathcal{J}^{B}\varrho_{C}$
$\displaystyle\quad-(\sigma^{A})_{BC}\bigg{(}\zeta_{B}^{\frac{1}{2}}\bar{\mathcal{G}}_{C}^{-\frac{1}{2}}+\zeta_{B}^{-\frac{1}{2}}\bar{\mathcal{G}}_{C}^{\frac{1}{2}}$
$\displaystyle\quad-\bar{\zeta}_{B}^{\frac{1}{2}}{\mathcal{G}}_{C}^{-\frac{1}{2}}-\bar{\zeta}_{B}^{-\frac{1}{2}}\mathcal{G}_{C}^{\frac{1}{2}}\bigg{)},$
(110) $\displaystyle 2\partial_{t}\mathcal{L}^{0}$
$\displaystyle=-\frac{k}{4}\partial_{\varphi}\epsilon^{0}-\mathcal{L}^{+1}\epsilon^{-1}+\mathcal{L}^{-1}\epsilon^{+1}$
$\displaystyle\quad+\frac{1}{2}\bigg{(}\bar{\mathcal{G}}_{A}^{\frac{1}{2}}\zeta_{A}^{-\frac{1}{2}}-\bar{\mathcal{G}}_{A}^{-\frac{1}{2}}\zeta_{A}^{\frac{1}{2}}$
$\displaystyle\quad-\mathcal{G}_{A}^{-\frac{1}{2}}\zeta_{A}^{\frac{1}{2}}+\mathcal{G}_{A}^{\frac{1}{2}}\zeta_{A}^{-\frac{1}{2}}\bigg{)},$
(111) $\displaystyle\partial_{t}\mathcal{L}^{\mp 1}$
$\displaystyle=\frac{k}{2}\partial_{\varphi}\epsilon^{\pm
1}+2\mathcal{L}^{0}\epsilon^{\pm 1}$ $\displaystyle\quad\pm\mathcal{L}^{\pm
1}\epsilon^{0}\pm\bar{\mathcal{G}}_{A}^{\pm\frac{1}{2}}\zeta_{A}^{\pm\frac{1}{2}}$
$\displaystyle\quad\pm\mathcal{G}_{A}^{\pm\frac{1}{2}}\bar{\zeta}_{A}^{\pm\frac{1}{2}},$
(112) $\displaystyle\partial_{t}\mathcal{G}_{A}^{\pm\frac{1}{2}}$
$\displaystyle=\pm
k\partial_{\varphi}\zeta_{A}^{\pm\frac{1}{2}}\pm\frac{1}{2}\epsilon^{0}\mathcal{G}_{A}^{\pm\frac{1}{2}}$
$\displaystyle\quad\pm\epsilon^{\mp}\mathcal{G}_{A}^{\pm\frac{1}{2}}+2\mathcal{L}^{0}\zeta_{A}^{\pm\frac{1}{2}}+2\mathcal{L}^{\mp}\zeta_{A}^{\pm\frac{1}{2}}$
$\displaystyle\quad+\frac{i}{2}(\sigma^{B})^{*}_{AC}\bigg{(}\varrho_{B}\mathcal{G}_{C}^{\pm\frac{1}{2}}+2\mathcal{J}^{B}\zeta_{C}^{\pm\frac{1}{2}}\bigg{)},$
(113) $\displaystyle\partial_{t}\bar{\mathcal{G}}_{A}^{\pm\frac{1}{2}}$
$\displaystyle=\pm
k\partial_{\varphi}\zeta_{A}^{\pm\frac{1}{2}}\pm\frac{1}{2}\epsilon^{0}\bar{\mathcal{G}}_{A}^{\pm\frac{1}{2}}$
$\displaystyle\quad\pm\epsilon^{\mp}\bar{\mathcal{G}}_{A}^{\pm\frac{1}{2}}+2\mathcal{L}^{0}\bar{\zeta}_{A}^{\pm\frac{1}{2}}$
$\displaystyle\quad+2\mathcal{L}^{\mp}\bar{\zeta}_{A}^{\pm\frac{1}{2}}+\frac{i}{2}(\sigma^{B})^{*}_{AC}\bigg{(}\varrho_{B}\bar{\mathcal{G}}_{C}^{\pm\frac{1}{2}}$
$\displaystyle\quad-2\mathcal{J}^{B}\bar{\zeta}_{C}^{\pm\frac{1}{2}}\bigg{)}$
(114)

Analogously, the gauge transformations for the chemical potentials are
calculated as:

$\displaystyle\partial_{t}\eta^{A}$
$\displaystyle=\frac{k}{2}\partial_{\varphi}\varrho_{A}+\epsilon^{ABC}\eta^{B}\varrho_{C}$
$\displaystyle\quad-(\sigma^{A})_{BC}\bigg{(}\zeta_{B}^{\frac{1}{2}}\bar{\nu}_{C}^{-\frac{1}{2}}+\zeta_{B}^{-\frac{1}{2}}\bar{\nu}_{C}^{\frac{1}{2}}$
$\displaystyle\quad-\bar{\zeta}_{B}^{\frac{1}{2}}{\nu}_{C}^{-\frac{1}{2}}-\bar{\zeta}_{B}^{-\frac{1}{2}}\nu_{C}^{\frac{1}{2}}\bigg{)},$
(115) $\displaystyle 2\partial_{t}\mu^{0}$
$\displaystyle=-\frac{k}{4}\partial_{\varphi}\epsilon^{0}-\mu^{+1}\epsilon^{-1}+\mu^{-1}\epsilon^{+1}$
$\displaystyle\quad+\frac{1}{2}\bigg{(}\bar{\nu}_{A}^{\frac{1}{2}}\zeta_{A}^{-\frac{1}{2}}-\bar{\nu}_{A}^{-\frac{1}{2}}\zeta_{A}^{\frac{1}{2}}$
$\displaystyle\quad-\nu_{A}^{-\frac{1}{2}}\zeta_{A}^{\frac{1}{2}}+\nu_{A}^{\frac{1}{2}}\zeta_{A}^{-\frac{1}{2}}\bigg{)},$
(116) $\displaystyle\partial_{t}\mu^{\mp 1}$
$\displaystyle=\frac{k}{2}\partial_{\varphi}\epsilon^{\pm
1}+2\mu^{0}\epsilon^{\pm 1}$ $\displaystyle\quad\pm\mu^{\pm
1}\epsilon^{0}\pm\bar{\nu}_{A}^{\pm\frac{1}{2}}\zeta_{A}^{\pm\frac{1}{2}}$
$\displaystyle\quad\pm\nu_{A}^{\pm\frac{1}{2}}\bar{\zeta}_{A}^{\pm\frac{1}{2}},$
(117) $\displaystyle\partial_{t}\nu_{A}^{\pm\frac{1}{2}}$ $\displaystyle=\pm
k\partial_{\varphi}\zeta_{A}^{\pm\frac{1}{2}}\pm\frac{1}{2}\epsilon^{0}\nu_{A}^{\pm\frac{1}{2}}$
$\displaystyle\quad\pm\epsilon^{\mp}\nu_{A}^{\pm\frac{1}{2}}+2\mu^{0}\zeta_{A}^{\pm\frac{1}{2}}+2\mu^{\mp}\zeta_{A}^{\pm\frac{1}{2}}$
$\displaystyle\quad+\frac{i}{2}(\sigma^{B})^{*}_{AC}\bigg{(}\varrho_{B}\nu_{C}^{\pm\frac{1}{2}}+2\eta^{B}\zeta_{C}^{\pm\frac{1}{2}}\bigg{)},$
(118) $\displaystyle\partial_{t}\bar{\nu}_{A}^{\pm\frac{1}{2}}$
$\displaystyle=\pm
k\partial_{\varphi}\zeta_{A}^{\pm\frac{1}{2}}\pm\frac{1}{2}\epsilon^{0}\bar{\nu}_{A}^{\pm\frac{1}{2}}$
$\displaystyle\quad\pm\epsilon^{\mp}\bar{\nu}_{A}^{\pm\frac{1}{2}}+2\mu^{0}\bar{\zeta}_{A}^{\pm\frac{1}{2}}$
$\displaystyle\quad+2\mu^{\mp}\bar{\zeta}_{A}^{\pm\frac{1}{2}}+\frac{i}{2}(\sigma^{B})^{*}_{AC}\bigg{(}\varrho_{B}\bar{\nu}_{C}^{\pm\frac{1}{2}}$
$\displaystyle\quad-2\eta^{B}\bar{\zeta}_{C}^{\pm\frac{1}{2}}\bigg{)}$ (119)

Following a methodology analogous to the previous section, we proceed to
identify the canonical boundary charges $\mathcal{Q}[\lambda]$ responsible for
generating the transformations $(\ref{trans231})$-$(\ref{trans232})$. As is
well-known, it is convenient to express the variation of the canonical
boundary charge $\delta_{\lambda}\mathcal{Q}$ (see $(\ref{Qvar})$) to deduce
the asymptotic symmetry algebra Banados:1994tn . Therefore, the canonical
boundary charge $\mathcal{Q}[\lambda]$ is derived as follows:

$\mathcal{Q}[\lambda]=\int\mathrm{d}\varphi\;\left[\mathcal{J}^{A}\varrho_{A}+\mathcal{L}^{i}\epsilon^{-i}+\mathcal{G}_{A}^{p}\zeta^{-p}+\bar{\mathcal{G}}_{A}^{p}\bar{\zeta}^{-p}\right].$
(120)

The subsequent step towards establishing the asymptotic symmetry algebra
involves calculating the Poisson bracket algebra using the standard method
Blgojevic:2002 , as obtained from the relation $(\ref{Qvar2})$, which holds
for any phase space functional $\digamma$.

In light of the aforementioned, the operator product algebra for the bosonic
sector is then expressed as follows:

$\displaystyle\mathcal{L}^{i}(z_{1})\mathcal{L}^{j}(z_{2})\,$
$\displaystyle\sim\,\frac{\frac{k}{2}\eta^{ij}}{z_{12}^{2}}\,+\,\frac{(i-j)}{z_{12}}\mathcal{L}^{i+j},$
(121) $\displaystyle\mathcal{L}^{i}(z_{1})\mathcal{G}_{A}^{p}(z_{2})\,$
$\displaystyle\sim\,\frac{({i\over 2}-p)}{z_{12}}\mathcal{G}_{A}^{i+p},$ (122)
$\displaystyle\mathcal{L}^{i}(z_{1})\bar{\mathcal{G}}_{A}^{p}(z_{2})\,$
$\displaystyle\sim\,\frac{({i\over 2}-p)}{z_{12}}\bar{\mathcal{G}}_{A}^{i+p},$
(123) $\displaystyle\mathcal{J}^{A}(z_{1})\mathcal{J}^{B}(z_{2})\,$
$\displaystyle\sim\,\frac{\frac{k}{2}\delta^{AB}}{z_{12}^{2}}+\,\frac{\epsilon^{ABC}}{z_{12}}\mathcal{J}^{C},$
(124) $\displaystyle\mathcal{J}^{A}(z_{1})\mathcal{G}_{B}^{p}(z_{2})\,$
$\displaystyle\sim\,\frac{(\sigma^{A})_{BC}}{z_{12}}\bar{\mathcal{G}}_{C}^{p},$
(125) $\displaystyle\mathcal{J}^{A}(z_{1})\bar{\mathcal{G}}_{B}^{p}(z_{2})\,$
$\displaystyle\sim\,\frac{(\sigma^{A})_{BC}}{z_{12}}\mathcal{G}_{C}^{p},$
(126) $\displaystyle\mathcal{G}_{A}^{p}(z_{1})\mathcal{G}_{B}^{q}(z_{2})\,$
$\displaystyle\sim\,\delta^{AB}\left(\frac{\frac{k}{2}}{z_{12}^{2}}\eta_{AB}^{pq}\,+\,\frac{2}{z_{12}}\mathcal{L}^{p+q}\right)$
$\displaystyle-i\delta_{p,-q}(p-q)\frac{(\sigma^{C})_{AB}}{z_{12}}\mathcal{J}^{C}.$
(127)

where $z_{12}=z_{1}-z_{2}$, or in a more concise form,

$\mathfrak{\mathcal{\mathfrak{J}}}^{A}(z_{1})\mathfrak{\mathcal{\mathfrak{J}}}^{B}(z_{2})\,\sim\,\frac{\frac{k}{2}\eta^{AB}}{z_{12}^{2}}\,+\,\frac{\mathfrak{\mathcal{\mathfrak{f}}}^{AB}_{~{}~{}~{}C}\mathfrak{\mathcal{\mathfrak{J}}}^{C}(z_{2})}{z_{12}}.$
(128)

Note that $\eta^{AB}$ is the supertrace matrix, and
$\mathfrak{\mathcal{\mathfrak{f}}}^{AB}_{~{}~{}C}$’s are the structure
constants of the related algebra with $(A,B=0,\pm 1,\pm\frac{1}{2},0,\pm
1,\pm\frac{1}{2},\pm\frac{3}{2})$, i.e., $\eta^{ip}=0$ and
$\mathfrak{\mathcal{\mathfrak{f}}}^{ij}_{~{}~{}i+j}=(i-j)$.

Since the barred sector is entirely analogous, identical results are obtained.
Consequently, it is deduced that the asymptotic symmetry algebra for the
loosest set of boundary conditions of $\mathcal{N}=(4,4)$ supergravity is two
copies of the affine $\mathfrak{osp}(4|2)_{k}$ algebra.

###  5.2 Superconformal Boundary Conditions

As discussed previously, it is convenient to note that superconformal boundary
conditions are the supersymmetric extension of the well-known Brown-Henneaux
boundary conditions presented in Brown:1986nw  for $AdS_{3}$ supergravity. In
this section, our primary objective is to construct the asymptotic symmetry
algebra for the most general boundary conditions, serving as the
supersymmetric extension of the Brown-Henneaux conditions. To achieve this, we
initiate our discussion by imposing the Drinfeld-Sokolov highest weight gauge
condition on the $\mathfrak{osp}(3|2)$ Lie superalgebra-valued connection
(101), setting the fields as follows:

$\displaystyle\mathcal{L}^{0}$ $\displaystyle=$
$\displaystyle\mathcal{G}_{M}^{+\frac{1}{2}}=\bar{\mathcal{G}}_{M}^{+\frac{1}{2}}=0,$
$\displaystyle\mathcal{L}^{-1}$ $\displaystyle=$
$\displaystyle\mathcal{L},\quad\mathcal{G}_{M}^{-\frac{1}{2}}=\mathcal{G}_{M},\quad\bar{\mathcal{G}}_{M}^{-\frac{1}{2}}=\bar{\mathcal{G}}_{M},\quad\gamma_{+1}\mathcal{L}^{+1}=1.$
(129)

Correspondingly, the supersymmetric gauge connection is given by:

$\displaystyle a_{\varphi}$ $\displaystyle=$ $\displaystyle{\tt
L}_{1}+\gamma_{-1}\mathcal{L}{\tt L}_{-1}+\rho_{A}\mathcal{J}^{A}{\tt J}_{A}$
(130) $\displaystyle+\sigma_{M}^{-{\frac{1}{2}}}\mathcal{G}_{M}{\tt
G}_{-{\frac{1}{2}}}^{M}+\bar{\sigma}_{M}^{-{\frac{1}{2}}}\bar{\mathcal{G}}_{M}\bar{{\tt
G}}_{-{\frac{1}{2}}}^{M},$ $\displaystyle a_{t}$ $\displaystyle=$
$\displaystyle\eta_{A}{\tt J}_{A}+\mu{\tt L}_{1}+\nu_{M}{\tt
G}_{{+\frac{1}{2}}}^{M}+\bar{\nu}_{M}\bar{{\tt G}}_{{+\frac{1}{2}}}^{M}$ (131)
$\displaystyle+\sum_{i=-1}^{0}\mu^{i}{\tt L}_{i}+\nu_{M}^{-{\frac{1}{2}}}{\tt
G}_{-{\frac{1}{2}}}^{M}+\bar{\nu}_{M}^{-{\frac{1}{2}}}\bar{{\tt
G}}_{-{\frac{1}{2}}}^{M},$

where $\eta_{A}$, $\mu\equiv\mu^{+1}$,
$\nu_{{}_{M}}\equiv\nu_{{}_{M}}^{+\frac{1}{2}}$, and
$\bar{\nu}_{{}_{M}}\equiv\bar{\nu}_{{}_{M}}^{+\frac{1}{2}}$ are interpreted as
the independent chemical potentials.

When summarizing our next steps towards deriving the asymptotic symmetry
algebra in one paragraph, it is appropriate to provide the following brief
overview. All functions except the chemical potentials can be determined by
the flatness conditions (50) in the usual manner. The equations of motion for
the fixed chemical potentials can also be obtained conventionally as the time
evolution of the canonical boundary charges. However, the extension to
$\mathfrak{osp}(4|2)$ introduces technical complexities, which, though
overcome, are omitted here due to space constraints. We choose to allocate
space for the calculations of the gauge parameter $\lambda$ and further
developments.

Based on the results to be obtained, we can now derive the superconformal
asymptotic symmetry algebra. Using the Drinfeld-Sokolov reduction, we have
only six independent parameters: $\mathcal{\varrho_{A}}$,
$\mathcal{\epsilon}\equiv\mathcal{\epsilon}^{+1}$,
$\mathcal{\zeta}_{{}_{M}}\equiv\mathcal{\zeta}_{{}_{M}}^{+{\frac{1}{2}}}$, and
$\bar{\mathcal{\zeta}}_{{}_{M}}\equiv\bar{\mathcal{\zeta}}_{{}_{M}}^{+{\frac{1}{2}}}$.
It is now possible to compute the gauge transformations by considering all
transformations (64) that preserve the boundary conditions with the
$\mathfrak{osp}(4|2)$ Lie superalgebra-valued gauge parameter $\lambda$.

$\displaystyle\lambda$ $\displaystyle=$ $\displaystyle
b^{-1}\left[\epsilon{\tt L}_{1}-\epsilon^{\prime}{\tt
L}_{0}+\left(\frac{6}{c}\mathcal{L}\epsilon+\frac{3\bar{\mathcal{G}}_{A}\zeta_{A}}{c}+\frac{3\mathcal{G}_{A}\bar{\zeta_{A}}}{c}+\frac{\epsilon^{\prime\prime}}{2}\right){\tt
L}_{-1}\right.$ (132) $\displaystyle\left.+\varrho_{A}{\tt
J}^{A}+\zeta_{A}{\tt G}_{\frac{1}{2}}^{A}+\bar{\zeta}_{A}\bar{{\tt
G}}_{\frac{1}{2}}^{A}+\left(\frac{6(\sigma^{A})^{*}_{BC}\mathcal{J}^{A}\zeta_{C}}{c}-\frac{3\mathcal{G}_{A}\epsilon}{c}-\zeta_{A}^{\prime}\right){\tt
G}_{-\frac{1}{2}}^{A}\right.$
$\displaystyle\left.-\left(\frac{6(\sigma^{A})_{BC}\mathcal{J}^{A}\bar{\zeta_{C}}}{c}+\frac{3\mathcal{G}_{A}\epsilon}{c}+\bar{\zeta_{A}}^{\prime}\right)\bar{{\tt
G}}_{-\frac{1}{2}}^{A}\right]b.$

Since we are dealing with asymptotic symmetries, it is natural to demand
obtaining the canonical boundary charges $\mathcal{Q}[\lambda]$. So, the
variation of the canonical boundary charge, i.e.,
$\delta_{\lambda}\mathcal{Q}$ (Eq. (74)), can be integrated to yield

$\mathcal{Q}[\lambda]=\int\mathrm{d}\varphi\;\left[\mathcal{J}^{A}\varrho_{A}+\mathcal{L}\epsilon+\mathcal{G}_{{}_{M}}\zeta_{{}_{M}}+\bar{\mathcal{G}}_{{}_{M}}\bar{\zeta}_{{}_{M}}\right].$
(133)

However, these canonical boundary charges do not yield a convenient asymptotic
operator product algebra in the complex coordinates when using Eq. (76) for
$\mathcal{N}=(4,4)$ superconformal boundaries. This is due to the fact that
some boundary charges do not transform like a primary conformal field:

$\displaystyle\mathcal{L}(z_{1})\mathcal{L}(z_{2})$ $\displaystyle\sim$
$\displaystyle\frac{\frac{c}{2}}{z_{12}^{4}}+\frac{2\mathcal{L}}{z_{12}^{2}}+\frac{\mathcal{L}^{\prime}+\frac{6}{c}\mathcal{G}_{A}\bar{\mathcal{G}}_{A}}{z_{12}}$
(134) $\displaystyle\mathcal{L}(z_{1})\mathcal{J}^{A}(z_{2})$
$\displaystyle\sim$ $\displaystyle 0$ (135)
$\displaystyle\mathcal{L}(z_{1})\mathcal{G}_{A}(z_{2})$ $\displaystyle\sim$
$\displaystyle\frac{\frac{3}{2}\mathcal{G}_{A}}{z_{12}^{2}}+\frac{\mathcal{G}_{A}^{\prime}-\frac{6}{c}(\sigma^{B})^{*}_{BC}\mathcal{J}^{B}\mathcal{G}_{C}}{z_{12}}$
(136) $\displaystyle\mathcal{L}(z_{1})\bar{\mathcal{G}}_{A}(z_{2})$
$\displaystyle\sim$
$\displaystyle\frac{\frac{3}{2}\mathcal{G}_{A}}{z_{12}^{2}}+\frac{\mathcal{G}_{A}^{\prime}+\frac{6}{c}(\sigma^{B})_{BC}\mathcal{J}^{B}\mathcal{G}_{C}}{z_{12}}$
(137) $\displaystyle\mathcal{J}^{A}(z_{1})\mathcal{J}^{B}(z_{2})$
$\displaystyle\sim$
$\displaystyle\frac{\frac{c}{12}}{z_{12}^{2}}+\,\frac{i\epsilon^{ABC}}{z_{12}}\mathcal{J}^{C}$
(138) $\displaystyle\mathcal{J}^{A}(z_{1})\mathcal{G}_{B}(z_{2})$
$\displaystyle\sim$
$\displaystyle\frac{\frac{i}{2}(\sigma^{B})_{BC}}{z_{12}}\mathcal{G}_{C}$
(139) $\displaystyle\mathcal{J}^{A}(z_{1})\bar{\mathcal{G}}_{B}(z_{2})$
$\displaystyle\sim$
$\displaystyle\frac{-\frac{i}{2}(\sigma^{B})_{BC}}{z_{12}}\bar{\mathcal{G}}_{C}$
(140) $\displaystyle\mathcal{G}_{A}(z_{1})\mathcal{G}_{B}(z_{2})\,$
$\displaystyle\sim$ $\displaystyle\,\delta^{AB}\left({{2c\over
3}\over{z_{12}^{3}}}\,+\,{{2(\mathcal{L}+\frac{6}{c}\mathcal{J}^{C}\mathcal{J}^{C})}\over{z_{12}}}\right)$
(141)
$\displaystyle-4(\sigma^{C})_{AB}\left({{\mathcal{J}^{C}}\over{z_{12}^{2}}}\,+\,{{\frac{1}{2}{\mathcal{J}^{C}}^{\prime}}\over{z_{12}}}\right)$

Additionally, there are nonlinear terms such as $(\mathcal{J^{A}J^{A}})(z)$,
$(\mathcal{G}_{A}\bar{\mathcal{G}}_{A})(z)$, and
$(\mathcal{J}^{A}\mathcal{G}_{B})(z)$, as discussed in the previous section
(Eq. (4.2)). Therefore, it is necessary to consider some redefinitions on the
boundary charges and gauge parameters as

$\mathcal{L}\rightarrow\mathcal{L}+\frac{3}{2c}\mathcal{J}^{A}\mathcal{J}^{A},\,~{}~{}~{}~{}\varrho_{A}\rightarrow\varrho_{A}-\frac{3}{c}\epsilon\mathcal{J}^{A}.$
(142)

It is crucial to highlight that these novel variables do not influence the
boundary charges. This observation ultimately results in operator product
expansions of a suitably chosen asymptotic symmetry algebra for
$\mathcal{N}=(4,4)$ superconformal boundaries. The set of conformal generators
$\mathcal{G}_{A}$ is transformed as
$\mathcal{G}_{A}\rightarrow\mathcal{G}_{1}\pm\mathcal{G}_{2}$, and
$\bar{\mathcal{G}}_{A}\rightarrow\bar{\mathcal{G}}_{1}\pm\bar{\mathcal{G}}_{2}$
in the complex coordinates, utilizing Eq. (76). By applying the identical
procedure for the barred sector, it can be inferred that the asymptotic
symmetry algebra for the most liberal set of boundary conditions of
$\mathcal{N}=(4,4)$ supergravity comprises two copies of the $\mathcal{N}=4$
Superconformal algebra with a central charge $c=6k$.

##  6 Concluding Remarks

In conclusion, our study has examined the comprehensive examples of the
$\mathcal{N}=(3,3)$ and $\mathcal{N}=(4,4)$ formulations of
$\mathfrak{osp}(N|2)$ Chern-Simons supergravity on $AdS_{3}$, inspired by the
$\mathcal{N}<3$ cases studied previouslyOzer:2021wtx ; Ozer:2019nkv . The
focus on $\mathcal{N}=(3,3)$ and $\mathcal{N}=(4,4)$ cases has been
instrumental, shedding light on the asymptotic structure of these extended
supergravity theories. Our findings reveal that, for the loosest set of
boundary conditions, the asymptotic symmetry algebras of these formulations
are two copies of the $\mathfrak{osp}(3|2)_{k}$ and $\mathfrak{osp}(4|2)_{k}$
affine algebras, respectively. Additionally, by constraining the connection
fields on the boundary conditions, we successfully obtained the supersymmetric
extensions of the Brown-Henneaux boundary conditions.

Building upon these results, we have conclusively identified that the
asymptotic symmetry algebras for $\mathcal{N}=(3,3)$ and $\mathcal{N}=(4,4)$
extended higher-spin supergravity theories in$AdS_{3}$ are two copies of the
$\mathcal{N}=3$ and $\mathcal{N}=4$ superconformal algebras, respectively.
This significant contribution enhances our understanding of the asymptotic
structure of these extended supergravity theories and provides a valuable
foundation for future researchers exploring AdS supergravity models.

Furthermore, our study hints at a potential relationship between higher spin
theory and string theory, as suggested in our prior work on $\mathcal{N}=2$.
While we briefly touched upon this connection, we emphasize the importance of
examining the physical implications of integrating higher spin theories into
the broader framework of string theory, particularly for theories with reduced
supersymmetry. Our illustrative example involving the $\mathcal{N}=4$ higher
spin theory in the limit $\lambda\rightarrow 0$ within superstring theory on
$AdS_{3}\times S^{3}\times T^{4}$ serves as a starting point. The groundwork
for this understanding was initially laid out in Gaberdiel:2014cha , and more
recently, the relevant string theory has been meticulously developed, as
documented in Gaberdiel:2018rqv ; Eberhardt:2018ouy  and Eberhardt:2019ywk .

However, we acknowledge that a comprehensive exploration of such relationships
requires further investigation, posing questions like "Can we pinpoint the
higher spin theory corresponding to the theory presented in Datta:2017ert ?"
and "Can we discern the string theory that aligns with higher spin theories?"
This exploration holds the potential to uncover specific junctures where our
higher spin theory may converge to a limit, building upon the foundation laid
in the previous study and leaving room for future discoveries. Our conclusions
open avenues for future research, offering insights into the interplay between
higher spin theories and string theory within AdS supergravity.

As a result, this work endeavors to deepen our understanding of three-
dimensional supersymmetry and fields with spin greater than two in Einstein’s
supergravity theories, while emphasizing their connection with string theory.
The primary objective is to comprehensively develop these theories and
explicitly determine their asymptotic symmetries under the most general
boundary conditions, as outlined in the introduction.

One focal point of the research is understanding the asymptotic behavior of
established supersymmetric theories and identifying their asymptotic
symmetries under the most general boundary conditions. Asymptotic symmetries
play a crucial role in comprehending theories’ behavior at large distances,
offering significant insights into their general behavior. Additionally, the
work highlights the process of associating the results of the developed
Einstein supergravity theories with string theory. However, detailed
discussions on the straightforward aspects of this relationship are deferred
to a subsequent study. By deferring the straightforward details of the
relationship with string theory to a future work, this work aims to pave the
way for further exploration. The intention is to provide a comprehensive
foundation for the mathematical structure of supersymmetric theories, leaving
the explicit exploration of the intricate details of the interplay between
these theories and string theory for subsequent research endeavors.

##  7 Acknowledgments

The authors are thankful to Jan de Boer, Matthias R. Gaberdiel,Carlos
E.Valcárcel Flores , Changhyun Ahn and Lorenz Eberhardt for interest to this
work and useful comments. The work of A.F is supported by TUBITAK Grant
123F255. H.T.Ö is supported by TUBITAK Grant 123F255.

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