aainstitutetext: Indian Institute of Science Education & Research Bhopal,
Bhopal Bypass Road, Bhauri, Bhopal 420 066, Indiabbinstitutetext: Physics
Division, National Center for Theoretical Sciences, Taipei 10617,
Taiwanccinstitutetext: Indian Institute of Science Education & Research
Thiruvananthapuram,
Vithura 695551, Kerala, Indiaddinstitutetext: School of Physics, Institute for
Research in Fundamental Sciences (IPM),
P.O.Box 19395-5531, Tehran, Iran

# Supersymmetrization of deformed BMS algebras

Nabamita Banerjee a,b Arpita Mitra c Debangshu Mukherjee d H. R. Safari
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

$W(a,b)$ and $W(a,b;\bar{a},\bar{b})$ algebras are deformations of
$\mathfrak{bms}_{3}$ and $\mathfrak{bms}_{4}$ algebra respectively. We present
an $\mathcal{N}=2$ supersymmetric extension of $W(a,b)$ and
$W(a,b;\bar{a},\bar{b})$ algebra in presence of $R-$symmetry generators that
rotate the two supercharges. For $W(a,b)$ our construction includes most
generic central extensions of the algebra. In particular we find that
$\mathcal{N}=2$ $\mathfrak{bms}_{3}$ algebra admits a new central extension
that has so far not been reported in the literature. For
$W(a,b;\bar{a},\bar{b})$, we find that an infinite $U(1)_{V}\times U(1)_{A}$
extension of the algebra is not possible with linear and quadratic structure
constants for generic values of the deformation parameters. This implies a
similar constraint for $U(1)_{V}\times U(1)_{A}$ extension of
$\mathcal{N}=2\,\mathfrak{bms}_{4}$ algebra.

††dedication: Last Updated:

IPM/P-2022/01

##  1 Introduction and Summary

For any theory its asymptotic symmetries are of immense physical significance.
The symmetries at the asymptotic boundary of a theory depends on the boundary
fall off of its constituents fields. In most examples, the asymptotic symmetry
is usually enlarged compared to the bulk symmetry of the theory. However, the
bulk symmetry must be contained in the asymptotic symmetry group algebra as a
subalgebra. For a theory in asymptotically flat spacetimes, if one recedes
from sources towards null infinity, at any finite radial distance from the
source one expects the symmetry algebra to be just Poincaré. However at null
infinity, the asymptotic symmetry algebra in the Bondi gauge is enhanced to
the Bondi-Metzner-Sachs or the $\mathfrak{bms}$ algebra generated by infinite
number of generators known as supertranslations BMvdB:1962 ; Sachs:1962 . One
can further extend the $\mathfrak{bms}$ algebra by including superrotations
Barnich:2009se , which manifests itself as a double copy of Virasoro algebra.
The finite dimensional Poincaré algebra is a subalgebra of the extended
$\mathfrak{bms}$ algebra. These infinite dimensional $\mathfrak{bms}$ algebras
have gained a renewed importance due to recent developments on the relations
between soft theorems and asymptotic symmetries in analyzing the vacuua of
gauge theories and gravity Strominger:2013jfa ; He:2014laa ;
Strominger:2014pwa ; Avery:2016zce ; Hamada:2018vrw ; AtulBhatkar:2019vcb . It
is well understood that at any null boundary in two or three dimensional
spacetime one obtain infinite dimensional algebra by constructing the
conserved charges without imposing any specific boundary conditions
Adami:2020ugu . Recently, this is also realized in general dimensions
Adami:2021nnf .

It is interesting to understand how these $\mathfrak{bms}$ algebras get
modified in presence of extended supersymmetries in a theory of gravity.
Furthermore in the presence of internal gauge fields, namely the $R-$ symmetry
fields, the supercharges non trivially rotate among themselves. This brings
interesting dynamics to the system such as the BPS conditions gets modified in
presence of the $R-$charges Ferrara:2010ru ; Fuentealba:2017fck . The effects
of extended supersymmetries and $R-$symmetries have been extensively studied
in the context of asymptotic symmetries of three dimensional supergravity
theories. The supersymmetric deformations of $\mathfrak{bms}_{3}$ algebras and
their consequences have been detailed in Barnich:2014cwa ; Barnich:2015sca ;
Banerjee:2016nio ; Lodato:2016alv ; Banerjee:2017gzj ; Fuentealba:2017fck ;
Banerjee:2018hbl ; Banerjee:2019lrv ; Banerjee:2019epe ; Banerjee:2021uxl . In
particular it has been shown that the $R-$charges also get an infinite
extensions at the null infinity and the space of the asymptotically flat
cosmological solutions gets hugely extended Banerjee:2018hbl ;
Fuentealba:2017fck . A similar study in the context of four dimensional
asymptotically flat extended supergravity theory has not been performed yet.
This brings us to look for deformations of $\mathfrak{bms}_{4}$ algebra.

There are two distinct possible ways of generating new algebras starting from
one, namely deformation and contraction of an algebra. Deformation of a Lie
algebra can be viewed as an inverse procedure of Inönü-Wigner Inonu:1953sp
contraction. While physicists have tackled more with contraction of Lie
algebras, deformations of various well-known Lie algebras in physics have been
recently considered in literature bacry1968possible ; Figueroa-
OFarrill:1989wmj ; Figueroa-OFarrill:2017ycu ; Figueroa-OFarrill:2018ygf ;
Enriquez-Rojo:2021rtv ; Enriquez-Rojo:2021hna . In contraction prescription
one tries to obtain a new non isomorphic algebra through specific limits of a
known algebra, where as in deformation prescription one deforms a Lie algebra
to get new (more stable) algebras by turning on structure constants in some
commutators nijenhuis1967deformations ; levy1967deformation . For instance,
one may take the limit of the Poincaré algebra by sending the speed of light
to infinity (or to zero) to get Galilean (or Carroll) algebra and conversely
the Galilean (or the Carroll) algebra may be deformed into the Poincaré
algebra levy1965nouvelle ; bacry1968possible . In recent works Parsa:2018kys ;
Safari:2019zmc , it has been proven that the three and four dimensional pure
$\mathfrak{bms}$ algebras can be deformed, in their non-ideal part, into two
families of new non-isomorphic infinite dimensional algebras called $W$
algebras. In the context of three space time dimensions, these are known as
$W(a,b)$ algebras, where $\mathfrak{bms}_{3}$ corresponds to $W(0,-1)$ and in
the context of four space time dimensions these are knows as
$W(a,b;\bar{a},\bar{b})$ algebras, where $\mathfrak{bms}_{4}$ corresponds to
$W(-1/2.-1/2;-1/2,-1/2)$. It has been shown that by imposing appropriate
boundary conditions $W(0,b)$ as well as $W(b,b;b,b)$ algebras are obtained as
near horizon symmetry algebras of 3- and 4-dimensional black-holes
Grumiller:2019fmp . Also, $W(b,b;b,b)$ has been obtained as asymptotic
symmetry algebra of flat Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes
Enriquez-Rojo:2021blc . On the other hand, $W(0,1),W(0,0)$ and $W(0,-1)$ have
appeared as asymptotic symmetry algebras in various gravitational theories
Afshar:2021qvi ; Compere:2013bya ; barnich2007classical .

In this paper, our goal is to extend the $W(a,b)$ and $W(a,b;\bar{a},\bar{b})$
algebras with two (fermionic) supercharges. We further consider that the two
supercharges rotate among themselves due to an internal R-symmetry. Our
construction is purely group theoretic with only two inputs: a) we demand the
consistency of the possible extended algebra with the Jacobi identities and b)
we demand that the extended algebra contains Super-Poincaré algebra as its
subalgebra, for particular values of deformation parameters. The explicit
construction goes as follows:

  * •

In three space time dimensions, we first introduce a set of infinite fermionic
generators to grade the known $W(a,b)$ algebras and ensure that the resulting
superalgebra satisfies graded Jacobi identities. Next we perform the similar
construction with infinite bosonic $R-$charge generators. We have further
extended our analysis to include the central charges in the algebras. As we
have stated above $W(0,-1)$ gives the usual $\mathfrak{bms}_{3}$ and various
Supersymmetric extensions of $\mathfrak{bms}_{3}$ algebras are well
investigated Banerjee:2018hbl ,Banerjee:2019lrv . Our construction of
supersymmetric centrally extended $W(a,b)$ algebras in this paper reproduces
the known results for $a=0,b=-1$, although we get a new possible central
extension.

  * •

So far in four space time dimensions we know generic bosonic
$W(a,b;\bar{a},\bar{b})$ algebra with central extensions Safari:2019zmc . As
stated above $\mathfrak{bms}_{4}$ is a special case of these for
$a=b=\bar{a}=\bar{b}=-1/2.$ Furthermore in Fotopoulos:2020bqj  minimal
supersymmetric generalization of $\mathfrak{bms}_{4}$ with 1 supercharge has
been obtained. In this paper, we first extend bosonic $W(a,b;\bar{a},\bar{b})$
with a set of infinite supercharges. Next we perform further extension with
infinite $R-$charges. In this case, the resulting algebra has not been
centrally extended. Interestingly we find that for $\mathfrak{bms}_{4}$ with
two supercharges, one can not infinitely extend the R charge sectors. We have
shown this rigidity for both linear and quadratic dependence of the structure
constants. The Jacobi identities only gets satisfied within the global sector,
i.e. for $\mathcal{N}=2$ Super-Poincaré algebra with global $R-$charges. This
is one of the prime results of this paper.

Here we must mention that both the three and four dimensional algebras
constructed in this paper are purely mathematical. In both cases, the
corresponding Super-Poincaré algebras are embedded in them as subalgebras for
appropriate values of the deformation parameters. Thus, in principle, these
algebras might show up as the asymptotic symmetry algebras for three and four
dimensional asymptotically flat theories. In particular, The ${\cal{N}}=2$
extension of $\mathfrak{bms}_{4}$ is a probable candidate of four dimensional
${\cal{N}}=2$ Supergravity theories with $R-$charges.

Let us summarize the organisation and results of the paper below :

  * •

In section 2, we begin with a brief review of the basic properties of $W(a,b)$
algebra . Next we present new analysis on a $\mathcal{N}=2$ supersymmetric
extension in the presence of $R$ charge which rotate the supercharges among
themselves in the rest of this section. We have completed this section with
the central extension of the supersymmetric $W(a,b)$ algebra. Equations (2.2)
represent the $\mathcal{N}=2,\,W(a,b)$ algebra, where as equations
(LABEL:Central-SUSY-W(a,b)) along with the table below them represent its most
generic central extensions.

  * •

Section 3 discusses the basic properties of $W(a,b,\bar{a},\bar{b})$ algebra
and may be skipped by the experts. We have added it for establishing the
notations used in the later sections.

  * •

In section 4, we extend the $W(a,b,\bar{a},\bar{b})$ algebra to include two
supercharges, albeit any internal symmetry. This section forms the base of the
main results of the paper that have been presented in section 5. Equations
(143) presents the prime results of this section, which is an infinite
extension of $\mathfrak{bms}_{4}$ in presence of two supercharges.

  * •

Section 5 contains the most important results of this paper. In this section
we have introduced two sets of $R$ charges along with other
$\mathcal{N}=2\,\,W(a,b,\bar{a},\bar{b})$ generators and studied the
possibility to find an infinite extension of the algebra. Here we have
considered both the cases for structure constants being linear and non-linear
in its arguments and performed a detailed analysis. We find a non-affirmative
result (unlike the case of section 4) as discussed in the end of the section.

  * •

In section 6, we will conclude with a discussion on main results and possible
future directions.

##  2 3-Dimensional Supersymmetric $W(a,b)$ and $R$-extended $W(a,b)$

Earlier works, such as Parsa:2018kys ; Safari:2019zmc  discussed aspects of
deformation and stability of $\mathfrak{bms}_{3}$ and $\mathfrak{bms}_{4}$
algebras. In this section, we briefly state their results and observations for
$\mathfrak{bms}_{3}$. The centerless $\mathfrak{bms}_{3}$ algebra can be
written as

$\displaystyle\left[J_{m},J_{n}\right]$ $\displaystyle=$
$\displaystyle(m-n)J_{m+n}\ ,$ (1) $\displaystyle\left[J_{m},P_{n}\right]$
$\displaystyle=$ $\displaystyle(m-n)P_{m+n}\ ,$ (2)
$\displaystyle\left[P_{m},P_{n}\right]$ $\displaystyle=$ $\displaystyle 0\ .$
(3)

Physically, the $J_{m}$ s are identified with superrotations while the $P_{n}$
s are supertranslations. This algebra can be deformed into two parameters
family algebra called $W(a,b)$ where $a,b$ are arbitrary real parameters
Safari:2019zmc . Explicitly, the $W(a,b)$ algebra is given by

$\begin{split}&[{J}_{m},{J}_{n}]=(m-n){J}_{m+n}\ ,\\\
&[{J}_{m},{P}_{n}]=-(n+bm+a){P}_{m+n}\ ,\\\ &[{P}_{m},{P}_{n}]=0.\end{split}$
(4)

It is straightforward to see that $W(0,-1)$ corresponds to
$\mathfrak{bms}_{3}$.

###  2.1 Supersymmetric $W(a,b)$ algebra

In this section, we write down a _supersymmetric_ version of the $W(a,b)$
algebra. Subsequently, we will introduce $R$ and $S$ charges and also
determine the central extension to the algebra. Our approach will be somewhat
operational : we start by introducing fermionic generators $G_{s}$ in the
$W(a,b)$ algebra where $s$ runs over half-integers. Our goal would be to write
down an extended algebra starting with the centerless $W(a,b)$ algebra as
given above by demanding consistency of Jacobi identities. For the time being,
unlike $\mathfrak{bms}_{3}$, we do not search for the realization of the
algebra as the asymptotic symmetry algebra of a supersymmetric theory at null
infinity in 3 spacetime dimensions.

Along with the usual $W(a,b)$ algebra as given in (4), we introduce the
following three commutators

$\displaystyle\\{G_{r},G_{s}\\}=P_{r+s}\ ,$ (5)
$\displaystyle[J_{m},G_{s}]=\alpha(m,s)G_{m+s}\ ,$ (6)
$\displaystyle[P_{m},G_{s}]=\beta(m,s)G_{l(m,s)}\ .$ (7)

The above extension is motivated by various super-$\mathfrak{bms}_{3}$
algebras written in Banerjee:2017gzj ; Banerjee:2018hbl ; Banerjee:2019lrv .
We choose to normalize the super-current generators $G_{s}$ in a way such that
the structure constant appearing in (5) is unity. It is expected that any
deformation of the $\mathfrak{bms}_{3}$ algebra by the parameters $a$ and $b$
will not change the index structure appearing on the RHS of (6). For
$\mathfrak{bms}_{3}$, it is known that $[P_{m},G_{r}]=0$. However, it is
possible that a deformation gives a non-trivial commutator between the
supercurrents and supertranslation which vanishes when $a=0,b=-1$111One can
think of the expression $a+b+1$. Clearly for $\mathfrak{bms}_{3}$, this
combination identically vanishes and can be a possible candidate structure
constant in (7). However, if $l(m,s)$ is indeed a linear function of its
argument, we will see such a commutator does not satisfy the Jacobi Identity..
This motivates us to propose (7) where $l(m,s)$ is a linear function in $m$
and $s$. The structure constants $\alpha$ and $\beta$ appearing above are also
assumed to be linear functions of its arguments. Our strategy will be to fix
these structure constants and $l(m,s)$ by demanding consistency of certain
relevant Jacobi identities.

The Jacobi identity involving the generators $G_{r},G_{s}$ and $P_{m}$ is
given by

$[\\{G_{s},G_{r}\\},P_{m}]=\\{G_{s},[G_{r},P_{m}]\\}+\\{G_{r},[G_{s},P_{m}]\\}\
.$ (8)

Using (4) and (5)-(7), we get,

$\beta(m,r)P_{s+l(m,r)}+\beta(m,s)P_{r+l(m,s)}=0$ (9)

Assuming linearity of $l(m,s)$ and the structure constant $\beta(m,s)$ in both
of their arguments, the above equation is satisfied if $l(m,s)=l_{0}+l_{1}m+s$
where $l_{0}$ and $l_{1}$ are constants and $\beta(m,r)=-\beta(m,s)$ for any
$r,s$. This hence yields $\beta(m,s)=0$.

Next, we use Jacobi identities on the operators $J_{m},G_{s}$ and $G_{r}$ to
determine $\alpha(m,s)$. The corresponding Jacobi identity is

$[\\{G_{s},G_{r}\\},J_{m}]=\\{G_{s},[G_{r},J_{m}]\\}+\\{G_{r},[G_{s},J_{m}]\\}\
.$ (10)

Using (4), (5)-(7) we get

$-\alpha(m,r)P_{m+r+s}-\alpha(m,s)P_{m+r+s}=(r+s+bm+a)P_{m+r+s}\ .$ (11)

Clearly, equating the coefficients of $P_{m+r+s}$, we recover,

$\alpha(m,s)=-\left(\frac{bm+a}{2}+s\right)\ .$ (12)

The Jacobi identity applied on $J_{m},J_{n}$ and $G_{s}$ reads as

$[[G_{s},J_{m}],J_{n}]+[[J_{m},J_{n}],G_{s}]+[[J_{n},G_{s}],J_{m}]=0\ .$ (13)

A similar exercise on the above Jacobi identity followed by equating the
coefficient of $G_{m+n+s}$ yields

$\alpha(m,s)\alpha(n,m+s)+(m-n)\alpha(m+s,s)=\alpha(n,s)\alpha(m,n+s)\ .$ (14)

It can be easily seen that the structure constant $\alpha(m,s)$ as determined
in (12) indeed satisfies the above equality. We thus end up with a possible
$\mathcal{N}=1$ supersymmetric extension of $W(a,b)$ algebra given by

$\displaystyle[{J}_{m},{J}_{n}]=(m-n){J}_{m+n},$ (15)
$\displaystyle[{J}_{m},{P}_{n}]=-(n+bm+a){P}_{m+n},$ (16)
$\displaystyle[{P}_{m},{P}_{n}]=0,$ (17)
$\displaystyle\\{G_{r},G_{s}\\}=P_{r+s},$ (18)
$\displaystyle[J_{m},G_{s}]=-\left(\frac{bm+a}{2}+s\right)G_{m+s},$ (19)
$\displaystyle[P_{m},G_{s}]=0$ (20)

Now that we have obtained a possible $\mathcal{N}=1$ extension of the $W(a,b)$
algebra, we can consider including another copy of fermionic supercharges
which we denote by $H_{s}$ where the index $s$ can take half-integer values.
They satisfy the following commutators with the superrotation and
supertranslation generators of the $W(a,b)$ algebra

$\displaystyle\\{H_{r},H_{s}\\}=P_{r+s},\quad[P_{m},H_{s}]=0\ ,$ (21)
$\displaystyle[J_{m},H_{s}]=-\left(\frac{bm+a}{2}+s\right)H_{m+s}\ .$ (22)

The supercurrent generators $G_{s}$ and $H_{s}$ can be used to define the
following linear combinations

$\displaystyle Q_{r}^{1}=\frac{1}{2}\left(G_{r}+iH_{r}\right)\ ,\quad
Q_{r}^{2}=\frac{1}{2}\left(G_{r}-iH_{r}\right)\ .$ (23)

The newly defined generators $Q^{1}_{r}$ and $Q^{2}_{r}$ satisfy

$\displaystyle\\{Q_{r}^{1},Q_{s}^{2}\\}=P_{r+s}\
,\quad\\{Q_{r}^{1},Q_{s}^{1}\\}=0\ ,\quad\\{Q_{r}^{2},Q_{s}^{2}\\}=0\ ,$
$\displaystyle[J_{m},Q^{i}_{s}]=-\left(\frac{bm+a}{2}+s\right)Q^{i}_{m+s}\
,\quad[P_{m},Q^{i}_{s}]=0\ .$ (24)

###  2.2 R-extension of Supersymmetric $W(a,b)$ algebra

Our next aim is to write the generalized algebra in the presence of
$R$-charges. Physically speaking, the $R$-charge generators rotate the
supercharge generators and thus brings in an additional non trivialities in
the algebra. It is known that introduction of $R$ charge generators
necessitates the introduction of $S$-charge generators Howe:1995zm  in the
context of $\mathfrak{bms}_{3}$. Motivated by the $\mathcal{N}=2$
$\mathfrak{bms}_{3}$ algebra as discussed in Banerjee:2019lrv  we begin our
analysis by proposing the following relations involving the $R$-charge and
$S$-charge generators

$\displaystyle[R_{n},Q_{r}^{1}]=\beta(n,r)~{}Q_{n+r}^{1},$
$\displaystyle[R_{n},Q_{r}^{2}]=-\beta^{\prime}(n,r)Q_{n+r}^{2}\ ,$ (25)
$\displaystyle{[P_{n},R_{m}]}=\sigma(n,m)S_{n+m},$
$\displaystyle\\{Q_{r}^{1},Q_{s}^{2}\\}=P_{r+s}+\eta(r,s)S_{r+s}\ ,$ (26)
$\displaystyle{[R_{n},J_{m}]}=\gamma(n,m)R_{n+m},$
$\displaystyle[S_{n},J_{m}]=\kappa(n,m)S_{n+m}\ .$ (27)

In writing the above ansatz, we have assumed that an $R-$deformation of the
$\mathcal{N}=2$ super-$W(a,b)$ algebra will not affect the index structure of
the undeformed algebra. The Jacobi identity for the operators
$Q_{m}^{1},Q_{n}^{2}$ and $R_{s}$ is given by,

$[\\{Q_{r}^{1},Q_{s}^{2}\\},R_{m}]=\\{Q_{r}^{1},[Q_{s}^{2},R_{m}]\\}+\\{Q_{s}^{2},[Q_{r}^{1},R_{m}]\\}\
.$ (28)

Using (25) and (26) in the above, we get,

$\displaystyle{[P_{r+s}+\eta(r,s)S_{r+s},R_{m}]}$
$\displaystyle=(\beta^{\prime}(m,s)-\beta(m,r))P_{r+s+m}$ (29)
$\displaystyle\hskip
14.22636pt+(\beta^{\prime}(m,s)\eta(r,m+s)-\beta(m,r)\eta(m+r,s))S_{m+r+s}\ .$

Further, noting that $[S_{m},R_{n}]=0$ implies that the LHS of (29) is
independent of the translation generator $P_{m}$. In order to make this Jacobi
identity consistent the coefficient of the translation generator on RHS must
vanish identically implying

$\beta^{\prime}(m,s)=\beta(m,r)\ .$ (30)

Since the above has to be true for arbitrary half-integer values of $r,s$ and
integer values of $m$ and also both $\beta$ and $\beta^{\prime}$ must be
linear, we conclude that both $\beta^{\prime}(m,s)$ and $\beta(m,r)$ depend
only on $m$. For simplicity, we denote the structure constant appearing in
(25) as $\beta(m)$ since,

$\beta^{\prime}(m)=\beta(m)\ .$ (31)

Demanding further consistency of (29) yields,

$\displaystyle\sigma(r+s,m)=\beta(m)(\eta(r,s+m)-\eta(r+m,s))\ .$ (32)

The two Jacobi identities involving $Q_{r}^{1},J_{n}$ and $R_{m}$ and,
$R_{l},J_{m}$ and $J_{n}$,

$\displaystyle[[Q_{r}^{1},J_{n}],R_{m}]+[[J_{n},R_{m}],Q_{r}^{1}]+[[R_{m},Q_{r}^{1}],J_{n}]$
$\displaystyle=$ $\displaystyle 0\ ,$ (33)
$\displaystyle{[[R_{l},J_{m}],J_{n}]}+[[J_{m},J_{n}],R_{l}]+[[J_{n},R_{l}],J_{m}]$
$\displaystyle=$ $\displaystyle 0\ ,$ (34)

lead to

$\displaystyle m\beta(m)=\gamma(m,n)\beta(m+n)\ ,$ (35)
$\displaystyle\gamma(l,m)\gamma(l+m,n)-(m-n)\gamma(l,m+n)-\gamma(l,n)\gamma(l+n,m)$
$\displaystyle=0\ .$ (36)

Assuming the structure constants $\gamma(m,n)$ and $\beta(m)$ to be linear in
$m$ and $n$ we explicitly take them to be of the form,

$\displaystyle\beta(m)$ $\displaystyle=$ $\displaystyle\beta_{0}+m\beta_{1}\
,$ (37) $\displaystyle\gamma(m,n)$ $\displaystyle=$
$\displaystyle\gamma_{0}+\gamma_{1}m+\gamma_{2}n\ .$ (38)

The above ansatz for $\gamma(m,n)$ along with (36) ensures

$\displaystyle\gamma_{1}=1\ .$ (39)

(35) then gives us five equations satisfied by four parameters
$\beta_{0},\beta_{1},\gamma_{0}$ and $\gamma_{2}$ are

$\displaystyle\gamma_{0}\beta_{0}=0\ ,$
$\displaystyle\beta_{0}\gamma_{2}+\beta_{1}\gamma_{0}=0\ ,$ (40)
$\displaystyle\gamma_{0}\beta_{1}=0\ ,$
$\displaystyle\beta_{1}(\gamma_{2}+1)=0\ ,$ (41)
$\displaystyle\gamma_{2}\beta_{1}$ $\displaystyle=$ $\displaystyle 0\ .$ (42)

Clearly the above set of equations are over constrained but admits the
following consistent solution

$\gamma_{0}=0\ ,\quad\gamma_{2}=0\ ,\quad\beta_{1}=0\ ,$ (43)

while $\beta_{0}$ is a non-zero constant that cannot be fixed further. One can
easily check that this is also consistent with the Jacobi identity for
$R_{m},R_{n}$ and $Q^{i}_{r}$. Thus, we can write the following commutation
relations

$[R_{n},Q^{1}_{r}]=\beta_{0}Q^{1}_{n+r}\
,\quad[R_{n},Q^{2}_{r}]=-\beta_{0}Q^{2}_{n+r}\ ,\quad[R_{n},J_{m}]=nR_{n+m}\
.$ (44)

The Jacobi identity involving $J_{m},J_{n}$ and $S_{l}$,

$[[J_{m},J_{n}],S_{l}]+[[J_{n},S_{l}],J_{m}]+[[S_{l},J_{m}],J_{n}]=0$ (45)

yields

$\kappa(l,m)\kappa(l+m,n)-(m-n)\kappa(l,m+n)-\kappa(l,n)\kappa(n+l,m)=0\ .$
(46)

Similar to the ansatz for $\gamma(m,n)$, we assume the following anstaz for
$\kappa(m,n)$

$\kappa(m,n)=\kappa_{0}+\kappa_{1}m+\kappa_{2}n.$ (47)

Plugging the above ansatz into (46) we get

$\kappa_{1}=1\ .$ (48)

The Jacobi identity for the operators $J_{l},P_{m}$ and $R_{n}$,

$\displaystyle[[J_{l},P_{m}],R_{n}]+[[P_{m},R_{n}],J_{l}]+[[R_{n},J_{l}],P_{m}]=0\
,$ (49)

leads to the relation

$\displaystyle-(m+bl+a)\sigma(m+l,n)+\sigma(m,n)\kappa(m+n,l)-n\sigma(m,n+l)=0\
.$ (50)

Assuming a linear ansatz for $\sigma(m,n)$ i.e.

$\sigma(m,n)=\sigma_{0}+\sigma_{1}m+\sigma_{2}n\ ,$ (51)

one can substitute it back into (50) to get the set of seven relations:

$\displaystyle b\sigma_{1}=0\ ,$ $\displaystyle\sigma_{0}(\kappa_{0}-a)=0\ ,$
(52) $\displaystyle\sigma_{1}(\kappa_{2}-b-1)=0\ ,$
$\displaystyle\sigma_{2}(\kappa_{2}-b-1)=0\ ,$ (53)
$\displaystyle\sigma_{1}(\kappa_{0}-a)=0\ ,$
$\displaystyle\sigma_{2}(\kappa_{0}-a)=0\ ,$ (54)
$\displaystyle\sigma_{0}(\kappa_{2}-b)-a\sigma_{1}$ $\displaystyle=$
$\displaystyle 0\ .$ (55)

The above system of equations have two consistent solution sets (detailed in
appendix A):

  * •

Case I: $\sigma_{0}=\sigma_{1}=0$; $\kappa_{0}=a$, $\kappa_{2}=b+1$ while
$\sigma_{2}$ is arbitrary.

  * •

Case II: $\sigma_{1}=\sigma_{2}=0$; $\kappa_{0}=a$, $\kappa_{2}=b$ while
$\sigma_{0}$ is arbitrary.

In light of the above, we can rewrite (26) and (27) as

Case I: $\displaystyle\quad[P_{n},R_{m}]=\sigma_{2}mS_{n+m}\
,\quad[S_{n},J_{m}]=(a+n+(b+1)m)S_{n+m}\ ;$ (56) Case II:
$\displaystyle\quad[P_{n},R_{m}]=\sigma_{0}S_{n+m}\
,\quad[S_{n},J_{m}]=(a+n+bm)S_{n+m}\ .$

Finally, we need to find the structure constant $\eta(r,s)$ appearing in
$\\{Q_{r}^{1},Q_{s}^{2}\\}$ in (26). Assuming a linear form of $\eta(r,s)$
i.e.

$\eta(r,s)=\eta_{0}+\eta_{1}r+\eta_{2}s\ ,$ (57)

we use (32) to see that we must have a relation of the form

$\sigma(r+s,m)=m\beta_{0}(\eta_{2}-\eta_{1})\ .$ (58)

It is quite evident that a choice of parameters as defined in Case II in our
preceding analysis is inconsistent with the above equation since the LHS is a
constant and independent of $m$. However, from the structure constants of Case
I, we arrive at the relation

$\sigma_{2}=\beta_{0}(\eta_{2}-\eta_{1})\ .$ (59)

Finally, the Jacobi identity for $Q_{r}^{1},Q_{s}^{2}$ and $J_{m}$ can be
written as

$[\\{Q_{r}^{1},Q_{s}^{2}\\},J_{m}]=\\{Q_{r}^{1},[Q_{s}^{2},J_{m}]\\}+\\{Q^{2}_{s},[Q_{r}^{1},J_{m}]\\}\
.$ (60)

Using (26) and (27), the above gives rise to,

$\eta(r,s)\kappa(r+s,m)=\left(\frac{bm+a}{2}+s\right)\eta(r,s+m)+\left(\frac{bm+a}{2}+r\right)\eta(r+m,s)\
.$ (61)

Assuming, $\kappa(m,n)=a+m+(b+1)n$, the above equation will be satisfied for
an ansatz of the form (57) provided,

$\eta_{0}=0\quad\text{and}\quad\eta_{1}=-\eta_{2}\ .$ (62)

Thus, the $R$ and $S$-charge sector algebra under a deformation reads as
222$\eta_{1}=0$ or $\beta_{0}=0$ are also viable choices of parameters that
satisfy the Jacobi identities. But for interpreting $R_{0}$ as the
$R-$symmetry generator we must consider non-zero values of those parameters.:

$\displaystyle[R_{n},Q_{r}^{1}]=\beta_{0}Q_{n+r}^{1},$
$\displaystyle[R_{n},Q_{r}^{2}]=-\beta_{0}Q_{n+r}^{2}\ ,$ (63)
$\displaystyle{[P_{n},R_{m}]}=-2\beta_{0}\eta_{1}mS_{n+m},$
$\displaystyle\\{Q_{r}^{1},Q_{s}^{2}\\}=P_{r+s}+\eta_{1}(r-s)S_{r+s}\ ,$ (64)
$\displaystyle{[R_{n},J_{m}]}=nR_{n+m},$
$\displaystyle[S_{n},J_{m}]=(a+n+(b+1)m)S_{n+m}\ .$ (65)

Redefining $S_{n}\rightarrow\mathcal{S}_{n}/\eta_{1}$ and
$R_{n}\rightarrow\beta_{0}\mathcal{R}_{n}$, we arrive at,

$\displaystyle[\mathcal{R}_{n},Q_{r}^{1}]=Q_{n+r}^{1},$
$\displaystyle[\mathcal{R}_{n},Q_{r}^{2}]=-Q_{n+r}^{2}\ ,$ (66)
$\displaystyle{[P_{n},\mathcal{R}_{m}]}=-2m\mathcal{S}_{n+m},$
$\displaystyle\\{Q_{r}^{1},Q_{s}^{2}\\}=P_{r+s}+(r-s)\mathcal{S}_{r+s}\ ,$
(67) $\displaystyle{[\mathcal{R}_{n},J_{m}]}=n\mathcal{R}_{n+m},$
$\displaystyle[\mathcal{S}_{n},J_{m}]=(a+n+(b+1)m)\mathcal{S}_{n+m}\ .$ (68)

The full $W(a,b)$ algebra including $R$ and $S$ charge generators takes the
form

$\displaystyle[{J}_{m},{J}_{n}]=(m-n){J}_{m+n},\quad[{J}_{m},{P}_{n}]=-(n+bm+a){P}_{m+n},$
$\displaystyle[J_{m},Q^{1}_{r}]=-\left(\frac{bm+a}{2}+r\right)Q^{1}_{m+r},\quad[J_{m},Q^{2}_{r}]=-\left(\frac{bm+a}{2}+r\right)Q^{2}_{m+r},$
$\displaystyle[J_{m},\mathcal{R}_{n}]=-n\mathcal{R}_{n+m},$
$\displaystyle[J_{m},\mathcal{S}_{n}]=-(a+n+(b+1)m)\mathcal{S}_{n+m},$
$\displaystyle[\mathcal{R}_{m},Q_{r}^{1}]=Q_{m+r}^{1},\quad[\mathcal{R}_{m},Q_{r}^{2}]=-Q_{m+r}^{2},$
$\displaystyle\\{Q_{r}^{1},Q_{s}^{2}\\}=P_{r+s}+(r-s)\mathcal{S}_{r+s},\quad[P_{m},\mathcal{R}_{n}]=-2nS_{n+m}.$
(69)

where indices $\\{m,n,p,q\\}\in\mathbb{Z}$ while
$\\{r,s\\}\in\mathbb{Z}+\frac{1}{2}$ and $i\in\\{1,2\\}$. All other
commutators vanish. The conformal weight of the generators $P_{m}$ and $S_{m}$
are $-b+1$ and $-b$ respectively, while the weight of $Q^{1},Q^{2}$ is
$-\frac{b}{2}+1$. For the specific case $a=0$ and $b=-1$ which corresponds to
supersymmetric-$\mathfrak{bms}_{3}$, we recover the same algebra as given in
Banerjee:2019lrv  333There is a typo in Eq. 3.19 and Eq. 8.59 of
Banerjee:2019lrv . The structure constant in $[M_{n},R_{m}]$ commutator will
be $-2m$ instead of $-4m$ otherwise the $({\cal G}_{r}^{1},{\cal
G}_{s}^{2},{R}_{m})$ will not be satisfied..

###  2.3 Central extensions of supersymmetric $W(a,b)$

One can show that the $W(a,b)$ algebra for generic values of its parameters
just admits one central term in its Witt part but for certain specific values
of $a$ and $b$, it admits various central extensions which were classified in
gao2011low . The most general centrally extended supersymmetric $W(a,b)$
algebra can be written as,

$\displaystyle[{J}_{m},{J}_{n}]=(m-n){J}_{m+n}+u(m,n),$ (70)
$\displaystyle[{J}_{m},{P}_{n}]=-(n+bm+a){P}_{m+n}+v(m,n),$ (71)
$\displaystyle[J_{m},Q^{1}_{r}]=-\left(\frac{bm+a}{2}+r\right)Q^{1}_{m+r}+x_{1}(m,r),$
(72)
$\displaystyle[J_{m},Q^{2}_{r}]=-\left(\frac{bm+a}{2}+r\right)Q^{2}_{m+r}+x_{2}(m,r),$
(73) $\displaystyle[J_{m},\mathcal{R}_{n}]=-n\mathcal{R}_{n+m}+y(m,n),$ (74)
$\displaystyle[J_{m},\mathcal{S}_{n}]=-(a+n+(b+1)m)\mathcal{S}_{n+m}+z(m,n),$
(75)
$\displaystyle[\mathcal{R}_{m},Q_{r}^{1}]=Q_{m+r}^{1}+g_{1}(m,r),\quad[\mathcal{R}_{m},Q_{r}^{2}]=-Q_{m+r}^{2}+g_{2}(m,r),$
(76)
$\displaystyle\\{Q_{r}^{1},Q_{s}^{2}\\}=P_{r+s}+(r-s)\mathcal{S}_{r+s}+f(r,s),\quad[P_{m},\mathcal{R}_{n}]=-2n\mathcal{S}_{n+m}+h(m,n),$
(77)
$\displaystyle\\{Q_{r}^{1},Q_{s}^{1}\\}=w_{1}(r,s),\quad\\{Q_{r}^{2},Q_{s}^{2}\\}=w_{2}(r,s),$
(78) $\displaystyle[{P}_{m},{P}_{n}]=t_{1}(m,n),\
[\mathcal{R}_{m},\mathcal{R}_{n}]=w(m,n),$ (79)
$\displaystyle[\mathcal{R}_{m},\mathcal{S}_{n}]=k(m,n),\
[\mathcal{S}_{m},\mathcal{S}_{n}]=s(m,n)$ (80)
$\displaystyle[P_{m},\mathcal{S}_{n}]=t_{2}(m,n),\
[P_{m},Q_{r}^{i}]=h^{i}(m,r),\ [\mathcal{S}_{n},Q_{r}^{i}]=f^{i}(n,r)\ ,$ (81)

where the Jacobi identity between the generators are expected put constraints
on unknown functions $u,v,x_{i},y,z,w,g_{i},f,h,w_{i},t_{i},k,s,h^{i},f^{i}$,
that denote the possible central extensions.

The central term in the commutator $[{J}_{m},{J}_{n}]$ which we denoted as
$u(m,n)$, is an arbitrary anti symmetric function. The Jacobi identity

$[J_{m},[J_{n},J_{l}]]+[J_{n},[J_{l},J_{m}]]+[J_{l},[J_{m},J_{n}]]=0,$ (82)

leads to the relation,

$(n-l)u(m,n+l)+(l-m)u(n,m+l)+(m-n)u(l,n+m)=0\ ,$ (83)

which has the nontrivial solution
$u(m,n)=C_{jj}^{(1)}\,(m^{3}-m)\,\delta_{m+n,0}$. This, as expected, is of the
form of the usual Virasoro central charge. Other Jacobi identities do not put
any new constraint on $u(m,n)$. A redefinition of $J_{m}\rightarrow
J_{m}+A\delta_{m,0}$ with an appropriate choice of $A$ can be used to absorb
the linear term in $m$.

One can fix the central term $v(m,n)$ of the $[{J}_{m},{P}_{n}]$ commutator in
the following way. The Jacobi identity between $J_{m},J_{n}$ and $P_{l}$ leads
to

$-(a+bn+l)v(m,n+l)+(a+bm+l)v(n,m+l)-(m-n)v(n+m,l)=0\ .$ (84)

Specific values of $a$ and $b$, yield even more non-trivial solutions. We
systematically tabulate all the cases below

  1. 1.

$a=b=0$ where, $v(m,n)=(C^{(1)}_{jp}m^{2}+C^{(2)}_{jp}m)\delta_{m+n,0}$ ,

  2. 2.

$a=0$, $b=1$ where, $v(m,n)=(C^{(3)}_{jp}m+C^{(4)}_{jp})\delta_{m+n,0}$ ,

  3. 3.

$a=0$, $b=-1$ where, $v(m,n)=(C^{(5)}_{jp}m^{3}+C^{(6)}_{jp}m)\delta_{m+n,0}$
,

  4. 4.

$a=0$, $b\neq 0,1,-1$ where, $v(m,n)=C_{jp}^{(7)}m\delta_{m+n,0}$ ,

  5. 5.

$a\neq 0$ and $b$ is arbitrary where,
$v(m,n)=C_{jp}^{(8)}\left(1+\frac{b-1}{a}m\right)\delta_{m+n,0}$ .

Here the subscript $jp$ denotes the central extension in $[J,P]$ commutator.
Out of the 8 central terms appearing in the above five scenarios only
$C^{(1)}_{jp},C^{(3)}_{jp},C^{(4)}_{jp}$ and $C^{(5)}_{jp}$ are the non
trivial ones. Other central terms can be absorbed by a simple redefinition of
${P}_{m}\rightarrow{P}_{m}+B\delta_{m,0}$ and choosing the constant $B$
subsequently in an appropriate manner. Thus we drop the remaining central
terms $C^{(2)}_{jp},C^{(6)}_{jp},C^{(7)}_{jp}$ and $C^{(8)}_{jp}$ for the
remaining analysis.

The above analysis demonstrates that there may be certain values for the
parameters $a$ and $b$ for which certain central terms will be allowed in the
algebra. This opens up a host of possibilities in the central extension. We
will focus on the most general extension that is admissible for arbitrary
values of $a$ and $b$.

The commutator $[J_{m},Q^{1}_{r}]$ may admit a central term given by,

$[{J}_{m},Q^{1}_{r}]=-\left(\frac{bm+a}{2}+r\right)Q^{1}_{m+r}+x_{1}(m,r),$
(85)

where $x_{1}(m,r)$ is an arbitrary function. The Jacobi identity between
$J_{m},J_{n}$ and $Q^{1}_{r}$ gives us,

$-\left(\frac{bn+a}{2}+r\right)x_{1}(m,n+r)+\left(\frac{bm+a}{2}+r\right)x_{1}(n,m+r)-(m-n)x_{1}(n+m,r)=0\
.$ (86)

The $x_{1}$ central term appearing in the $[J_{m},,Q^{1}_{r}]$ commutator is
identically zero since a central term proportional to $\delta_{m+r,0}$ is
identically zero as $m$ is an integer and $r$ is a half-integer. An
identically similar argument is true for $x_{2}(m,r)$ which is the central
extension in the $[J_{m},Q^{2}_{r}]$ commutator as given in (73).

The central term in $[J_{m},S_{n}]$ commutator is denoted by $z(m,n)$ and the
full commutator is written as

$[J_{m},\mathcal{S}_{n}]=-(a+n+(b+1)m)\mathcal{S}_{n+m}+z(m,n)\ .$ (87)

The Jacobi identity of $J_{m},J_{n}$ and $\mathcal{S}_{l}$ yields,

$-(a+(b+1)n+l)z(m,n+l)+(a+(b+1)m+l)z(n,m+l)-(m-n)z(n+m,l)=0,$ (88)

which admits the following nontrivial solutions

  1. 1.

$a\neq 0$ and $b$ is arbitrary where
$z(m,n)=C^{(0)}_{js}\left(1+\frac{b}{a}m\right)\delta_{m+n,0}$ ,

  2. 2.

$a=0$ and $b\neq 0,-1,-2$ where $z(m,n)=C_{js}^{(1)}m\delta_{m+n,0}$ ,

  3. 3.

$a=0$ and $b=-1$ where
$z(m,n)=(C_{js}^{(2)}m+C_{js}^{(3)}m^{2})\delta_{m+n,0}$ ,

  4. 4.

$a=0$ and $b=-2$ where
$z(m,n)=(C_{js}^{(4)}m+C_{js}^{(5)}m^{3})\delta_{m+n,0}$ ,

  5. 5.

$a=b=0$, where $z(m,n)=(C^{(6)}_{js}+C^{(7)}_{js}m)\delta_{m+n,0}$ .

Again, performing the shift
$\mathcal{S}_{m}\rightarrow\mathcal{S}_{m}+S\delta_{m,0}$ will remove some of
the constants appearing above with an appropriate choice of $S$. A detailed
analysis reveals that we can drop $C_{js}^{(0)},C_{js}^{(1)},C_{js}^{(2)}$ and
$C_{js}^{(4)}$.

The central term $y(m,n)$,

$[J_{m},\mathcal{R}_{n}]=-n\mathcal{R}_{n+m}+y(m,n),$ (89)

can be determined from the Jacobi identity of $J_{m},J_{n}$ and
$\mathcal{R}_{l}$ which gives,

$-ly(m,n+l)+ly(n,m+l)-(m-n)y(n+m,l)=0\ .$ (90)

This has the nontrivial solution $y(m,n)=C^{(0)}_{jr}m^{2}\delta_{m+n,0}$.
This was discussed in earlier works gao2011low ; Safari:2019zmc .

The anticommutator $\\{Q_{r}^{1},Q_{s}^{2}\\}$ may have the possible central
term $f(r,s)$ and is given by,

$\displaystyle\\{Q_{r}^{1},Q_{s}^{2}\\}=P_{r+s}+(r-s)\mathcal{S}_{r+s}+f(r,s)\
.$ (91)

The Jacobi identity of $Q^{1}_{r},Q^{2}_{s}$ and $J_{m}$ gives,

$\left(\frac{bm+a}{2}+s\right)f(r,s+m)+\left(\frac{bm+a}{2}+r\right)f(r+m,s)=-v(m,r+s)-(r-s)z(m,r+s)\
.$ (92)

The above equation needs to be dealt with care in case-by-case basis. We
tabulate all possible solutions for various values of the deformation
parameter $a$ and $b$

  1. 1.

When $a=b=0$, we get the solution as $f(r,s)=C_{qq}^{(0)}r\delta_{r+s,0}$.
Also, for consistency of the above equation, we must have
$C_{jp}^{(1)}=C_{js}^{(6)}=C_{js}^{(7)}=0$. This in turn ensures that for
$a=b=0$, $v(m,n)=z(m,n)=0$. Note that the linear term appearing in $f(r,s)$
cannot be absorbed in the shift of generators. A possible absorbing of the
central term can be performed by shifting the supertranslation generators
$P_{n}$. This was already performed earlier to ensure $C_{jp}^{(2)}$ drops out
in the expression of the central charge. Thus, there is no more freedom to
absorb this piece in the generators.

  2. 2.

When $a=0$ and $b=1$, we get the solution $f(r,s)=C_{qq}^{(1)}\delta_{r+s,0}$.
Again, consistency demands us to set $C_{jp}^{(3)}=C_{jp}^{(4)}=0$, again
ensuring that $v(m,n)$ vanishes for this case.

  3. 3.

When $a=0$ and $b=-1$, we get the solution
$f(r,s)=2C_{jp}^{(5)}r^{2}\delta_{r+s,0}$ along with the constraint that
$C_{js}^{(3)}=0$. This implies for $a=0$ and $b=-1$, we have $z(m,n)=0$.

  4. 4.

When $a=0$ and $b=-2$, we get the solution to be
$f(r,s)=C_{js}^{(5)}r^{3}\delta_{r+s,0}$.

  5. 5.

When $a=0$ and $b=2$, we get the solution
$f(r,s)=C_{qq}^{(2)}r^{2}\delta_{r+s,0}$ while for $a=0$ and
$b\neq-2,-1,0,1,2$, $f(r,s)$ must vanish identically.

  6. 6.

When $a\neq 0$ and $b$ is arbitrary, we recover we get $f(r,s)=0$ identically.

$g_{1}(m,n)$ is an arbitrary symmetric function which denotes the central term
in the $[\mathcal{R}_{m},Q_{r}^{1}]$ commutator and is given by,

$\displaystyle[\mathcal{R}_{m},Q_{r}^{1}]=Q_{m+r}^{1}+g_{1}(m,r)\ .$ (93)

For $g_{1}(m,r)\propto\delta_{m+r,0}$, we can easily conclude that this will
be zero identically since $m$ is an integer and $r$ is a half-integer.

The commutator of $[P_{m},\mathcal{R}_{n}]$ may admit a central term $h(m,n)$
which appears as follows,

$\displaystyle[P_{m},\mathcal{R}_{n}]=-2n\mathcal{S}_{m+n}+h(m,n)\ .$ (94)

The Jacobi identity of $J_{m},\mathcal{R}_{n}$ and $P_{l}$ leads to

$(bm+a+l)h(l+m,n)+nh(l,m+n)=2nz(m,n+l)\ .$ (95)

Clearly, the RHS of the above equation depends crucially on the values of $a$
and $b$. However, dealing case-by-case it turns out that $h(m,n)=0$ for all
values of $a$ and $b$ along with the constraint $C_{js}^{(5)}=0$. This in turn
implies $z(m,n)$ vanishes for $a=0$ and $b=-2$.

The commutator of $[\mathcal{R}_{m},\mathcal{S}_{n}]$ can admit a central
extension given by

$[\mathcal{R}_{m},\mathcal{S}_{n}]=k(m,n)\ .$ (96)

The Jacobi identity of $Q^{1}_{r},Q^{2}_{s}$ and $\mathcal{R}_{m}$ gives

$(r-s)k(m,r+s)+f(r,m+s)-f(m+r,s)=0\ .$ (97)

Depending on the form of $f(r,s)$, we will have different solutions for
$k(m,n)$. We list the possible solutions as follows:

  1. 1.

For $a=b=0$, the above equation simplifies to

$(r-s)k(m,r+s)=C_{qq}^{(0)}m\delta_{m+r+s,0}\ .$ (98)

For the above equation to be consistent we must have $C_{qq}^{(0)}=0$ which
further implies for this case, $f(r,s)=k(m,n)=0$.

  2. 2.

For $a=0,b=1$ we simply recover $(r-s)k(m,r+s)=0$ which immediately implies
$k(m,n)=0$.

  3. 3.

For $a=0$, $b=-1$, we see that the equation for $k(m,n)$ is satisfied provided
$k(m,n)=2C_{jp}^{(5)}m\delta_{m+n,0}$.

  4. 4.

For $a=0$, $b=-2$, consistency demands us to set $C_{js}^{(5)}=0$ which inturn
ensures $z(m,n)=f(r,s)=k(m,n)=0$.

  5. 5.

For $a=0$, $b=2$, we see the solution of $k(m,n)=C_{qq}^{(2)}m\delta_{r+s,0}$.

  6. 6.

For $a\neq 0$ and arbitrary $b$, we must have $k(r,s)=0$.

A quick glance at (70)-(81) tells us that the central extension to the
commutator $[\mathcal{R}_{m},\mathcal{R}_{n}]$ will affect the Jacobi identity
between $J_{m}$, $\mathcal{R}_{n}$ and $\mathcal{R}_{p}$. Denoting the central
extension in this case as

$[\mathcal{R}_{m},\mathcal{R}_{n}]=w(m,n)\ ,$ (99)

the $J_{m}$, $\mathcal{R}_{n}$, $\mathcal{R}_{p}$ Jacobi identity leads to

$pw(n,p+m)=nw(p,m+n)\ .$ (100)

A little algebra shows that the solution to the above functional equation is
given by $w(m,n)=C_{rr}m\delta_{m+n,0}$.

The central term in the $[S_{m},S_{n}]$ commutator is denoted as $s(m,n)$ and
can be explicitly written as

$[S_{m},\mathcal{S}_{n}]=s(m,n)\ .$ (101)

The Jacobi identity between $P_{m},\mathcal{R}_{n}$ and $\mathcal{S}_{l}$
gives

$ns(l,m+n)=0\ ,$ (102)

which naturally implies $s(m,n)=0$ identically. The reader can easily verify
that the Jacobi identities of ($\mathcal{R}_{m},\mathcal{S}_{n},Q^{i}_{r}$),
($\mathcal{R}_{m},P_{n},Q^{i}_{r}$) and ($Q^{1}_{r},Q^{2}_{s},P_{m}$) implies
$f^{i}(m,n)=h^{i}(m,n)=t_{2}(m,n)=0$. The supertranslation commutator
$[P_{m},P_{n}]$ also does not admit any central term, as discussed in further
details in an earlier work Parsa:2018kys  by one of the authors.

Finally, we consider the anticommutator $\\{Q_{r}^{1},Q_{s}^{1}\\}$ which may
admit a central term as

$\displaystyle\\{Q_{r}^{1},Q_{s}^{1}\\}=w_{1}(r,s)\ .$ (103)

It is clear that $w_{1}(r,s)$ must be symmetric in its arguments. The Jacobi
identity between $Q^{1}_{r},Q^{1}_{s}$ and $R_{m}$ gives

$w_{1}(r,s+m)+w_{1}(s,m+r)=0\ .$ (104)

For $m=0$, we see that $w_{1}(m,n)$ should be antisymmetric which is clearly a
contradiction. Thus, $w_{1}(m,n)=0$ identically and a similar argument
involving $Q^{2}_{r}$ yields $w_{2}(m,n)=0$. This completes a full description
of the central extension for the $W(a,b)$ algebra which clearly depends on the
values of the parameters $a$ and $b$.

Here, we tabulate the $W(a,b)$ for $a=0$ and $b=-1$, which is the same as the
super-$\mathfrak{bms}_{3}$ algebra

$\displaystyle[{J}_{m},{J}_{n}]=(m-n){J}_{m+n}+C_{jj}^{(1)}\,m^{3}\,\delta_{m+n,0},$
(105)
$\displaystyle[{J}_{m},{P}_{n}]=-(n-m){P}_{m+n}+C_{jp}^{(5)}\,m^{3}\,\delta_{m+n,0},$
$\displaystyle[J_{m},Q^{1}_{r}]=-\left(-\frac{m}{2}+r\right)Q^{1}_{m+r},\quad[J_{m},Q^{2}_{r}]=-\left(-\frac{m}{2}+r\right)Q^{2}_{m+r},$
$\displaystyle[J_{m},\mathcal{R}_{n}]=-n\mathcal{R}_{n+m}+C_{jr}^{(0)}\,m^{2}\,\delta_{m+n,0},$
$\displaystyle[J_{m},\mathcal{S}_{n}]=-n\mathcal{S}_{n+m},$
$\displaystyle[\mathcal{R}_{m},Q_{r}^{1}]=Q_{m+r}^{1},\quad[\mathcal{R}_{m},Q_{r}^{2}]=-Q_{m+r}^{2},$
$\displaystyle\\{Q_{r}^{1},Q_{s}^{2}\\}=P_{r+s}+(r-s)\mathcal{S}_{r+s}+2C_{jp}^{(5)}\,r^{2}\,\delta_{r+s,0},\quad[P_{m},\mathcal{R}_{n}]=-2nS_{n+m},$
$\displaystyle[\mathcal{R}_{m},\mathcal{R}_{n}]=C_{rr}\,m\,\delta_{m+n,0}\ ,\
[\mathcal{R}_{m},\mathcal{S}_{n}]=2C_{jp}^{(5)}m\delta_{m+n,0}\ .$

The above is largely in agreement with the results of Banerjee:2019lrv  except
for the $[J_{m},\mathcal{R}_{n}]$ case, where we find a new central term. It
is interesting to understand the source of this central term in the three
dimensional asymptotically flat bulk supergravity theory. We have elaborated
on the importance of this new central term in the discussion session.

The above analysis gives a complete classification of all possible central
extensions to the $W(a,b)$ algebra for arbitrary values of $a$ and $b$.
Although, our ansatze (70)-(81) was very general, we eventually ended up with
only a few non-zero central extensions for certain specific values of $a$ and
$b$. Certain central extension such as $z(m,n)$, defined in (87) and following
(88) did yield non-trivial solutions. However, demanding consistency with
subsequent Jacobi indentities led us to conclude $z(m,n)=0$ identically for
all values of $a$ and $b$. To summarize our findings, we tabulate the non-
trivial central charges obtained for other specific domain of the deformation
parameters $a$ and $b$ in the following :

Central extensions | $a=0,b=-1$ | $a=0,b=1$ | $a=0,b=2$ | $a\neq 0,b\neq 0$
---|---|---|---|---
$v(m,n)$ | $C_{jp}^{(5)}m^{3}\delta_{m+n,0}$ | 0 | 0 | 0
$f(r,s)$ | $2C_{jp}^{(5)}r^{2}\delta_{r+s,0}$ | $C_{qq}^{(1)}\delta_{r+s,0}$ | $C_{qq}^{(2)}r^{2}\delta_{r+s,0}$ | 0
$k(m,n)$ | $2C_{jp}^{(5)}m\delta_{m+n,0}$ | 0 | $C_{qq}^{(2)}m\delta_{m+n,0}$ | 0
Table 1: Central Extensions

Note that in the above we have not included the central extension in the
$[J_{m},J_{n}]$ commutator which is the usual Virasoro central extension
$C_{jj}m^{3}\delta_{m+n}$ that is present for any values of $a$ and $b$. In
addition the central terms in the $[J_{m},{\cal R}_{n}]$ and $[{\cal
R}_{m},{\cal R}_{n}]$ commutator also exists for all values of $a$ and $b$ and
is given by $y(m,n)=C_{jr}^{(0)}m^{2}\delta_{m+n,0}$ and
$w(m,n)=C_{rr}m\delta_{m+n,0}$ respectively. For all other cases other than
the above ones, the central extensions vanish. The above forms of the central
extensions along with the commutators given in equations (70)-(81) constitute
the most generic centrally extended $\mathcal{N}=$2 supersymmetric $W(a,b)$
algebra in presence of $R-$symmetry.

##  3  $\mathfrak{bms}_{4}$ group and $W(a,b;\bar{a},\bar{b})$ algebra

Having established a realization of the extended $W(a,b)$ algebra, we now move
on to generalize the above analysis in four spacetime dimensions. Like before,
our starting point will be the $\mathfrak{bms}_{4}$ algebra which can be
thought of as a special case of the more general $W(a,b;\bar{a},\bar{b})$
algebra Safari:2019zmc ; Safari:2020pje . In the early 1960s, BMvdB:1962 ;
Sachs:1962  attempted to understand and study the radiation that will be
detected by a distant observer. Interestingly, they found that the full set of
symmetries for an asymptotically flat spacetime444There are various equivalent
ways in which one can specify asymptotic behaviour of spacetimes. For a
detailed exposition, the reader is urged to consult Frauendiener:2000mk ;
Ashtekar:2014zfa  and references therein. is an infinite dimensional group
spanned by the so-called supertranslation and superrotations generators which
is dubbed the $\mathfrak{bms}_{4}$ group.

The infinite dimensional centerless asymptotic symmetry algebra of four
dimensional flat spacetime, conventionally known as the $\mathfrak{bms}_{4}$
algebra Barnich:2011mi ; Safari:2019zmc , is given by

$\displaystyle[\mathcal{L}_{m},\mathcal{L}_{n}]$
$\displaystyle=(m-n)\mathcal{L}_{m+n}$
$\displaystyle\left[\bar{\mathcal{L}}_{m},\bar{\mathcal{L}}_{n}\right]$
$\displaystyle=(m-n)\bar{\mathcal{L}}_{m+n}$
$\displaystyle\left[\mathcal{L}_{m},\bar{\mathcal{L}}_{n}\right]$
$\displaystyle=0$ $\displaystyle\left[\mathcal{L}_{m},T_{p,q}\right]$
$\displaystyle=\left(\frac{m+1}{2}-p\right)T_{p+m,q}$
$\displaystyle\left[\bar{\mathcal{L}}_{m},T_{p,q}\right]$
$\displaystyle=\left(\frac{m+1}{2}-q\right)T_{p,q+m}$
$\displaystyle\left[T_{p,q},T_{k,l}\right]$ $\displaystyle=0$ (106)

where the indices $m,n,p,q,k,l\in\mathbb{Z}$. The generators $\mathcal{L}_{m}$
and $\bar{\mathcal{L}}_{m}$ forming two independent copies of the Witt algebra
are known to correspond to _superrotations_ while the generators $T_{p,q}$ are
known to correspond to _supertranslations_. Following Caroca:2018obf ;
Safari:2019zmc , we briefly state the map between the global sector of
$\mathfrak{bms}_{4}$ algebra and Poincaré algebra.

Denoting the Lorentz generators as $M_{\mu\nu}$ and the translations as
$P_{\mu}$, we known that in four spacetime dimensions, they satisfy the
algebra

$\displaystyle\left[M_{\mu\nu},M_{\rho\sigma}\right]$ $\displaystyle=$
$\displaystyle
i(\eta_{\mu\rho}M_{\nu\sigma}+\eta_{\sigma\mu}M_{\rho\nu}-\eta_{\nu\rho}M_{\mu\sigma}-\eta_{\sigma\nu}M_{\rho\mu})$
(107) $\displaystyle\left[M_{\mu\nu},P_{\sigma}\right]$ $\displaystyle=$
$\displaystyle i(\eta_{\sigma\mu}P_{\nu}-\eta_{\sigma\nu}P_{\mu})$ (108)
$\displaystyle\left[P_{\mu},P_{\nu}\right]$ $\displaystyle=$ $\displaystyle 0$
(109)

where the indices $\mu,\nu,\rho,\sigma\in\\{0,1,2,3\\}$ and
$\eta_{\mu\nu}\equiv\mbox{diag}(-1,+1,+1,+1)$ is the flat Minkowski metric. We
can define the generator of rotations and boosts as

$J_{i}=\frac{1}{2}\epsilon_{ijk}M^{jk}\quad\mbox{and}\quad K_{i}=M^{0i}$ (110)

respectively, where $\epsilon_{ijk}$ is the Levi-Civita tensor and the indices
$i,j,k\in\\{1,2,3\\}$. Further, we define the quantities

$\displaystyle\mathcal{L}_{\pm 1}$ $\displaystyle=$ $\displaystyle iS_{1}\pm
S_{2}$ (111) $\displaystyle\bar{\mathcal{L}}_{\pm 1}$ $\displaystyle=$
$\displaystyle iR_{1}\pm R_{2}$ (112) $\displaystyle\mathcal{L}_{0}$
$\displaystyle=$ $\displaystyle S_{3}$ (113)
$\displaystyle\bar{\mathcal{L}}_{0}$ $\displaystyle=$ $\displaystyle R_{3}$
(114)

where

$R_{i}=\frac{1}{2}(J_{i}+iK_{i})\quad\mbox{and}\quad
S_{i}=\frac{1}{2}(J_{i}-iK_{i}).$ (115)

It can be easily verified that the set of operators $\\{\mathcal{L}_{\pm
1},\mathcal{L}_{0},\bar{\mathcal{L}}_{\pm 1},\bar{\mathcal{L}}_{0}\\}$
satisfies the algebra

$\displaystyle\left[\mathcal{L}_{m},\mathcal{L}_{n}\right]$ $\displaystyle=$
$\displaystyle(m-n)\mathcal{L}_{m+n}$ (116)
$\displaystyle\left[\bar{\mathcal{L}}_{m},\bar{\mathcal{L}}_{n}\right]$
$\displaystyle=$ $\displaystyle(m-n)\bar{\mathcal{L}}_{m+n}$ (117)
$\displaystyle\left[\mathcal{L}_{m},\bar{\mathcal{L}}_{n}\right]$
$\displaystyle=$ $\displaystyle 0$ (118)

for $(m,n)\in\\{\pm 1,0\\}$ thus showing that the set of operators defined in
(111)-(114) indeed correspond to the global part of the infinite dimensional
$\mathfrak{bms}_{4}$ algebra.

The translation generators $P_{\mu}$ can be mapped to linear combinations of
$T_{p,q}$ where $(p,q)\in\\{0,1\\}$ as follows:

$\displaystyle P^{0}$ $\displaystyle=$ $\displaystyle H=(T_{1,0}-T_{0,1})$
(119) $\displaystyle P^{1}$ $\displaystyle=$
$\displaystyle(-i)(T_{1,1}+T_{0,0})$ (120) $\displaystyle P^{2}$
$\displaystyle=$ $\displaystyle T_{1,1}-T_{0,0}$ (121) $\displaystyle P^{3}$
$\displaystyle=$ $\displaystyle T_{1,0}+T_{0,1}$ (122)

This demonstrates that appropriate combinations of the global part of
$\mathfrak{bms}_{4}$ algebra consisting of the operators $\\{\mathcal{L}_{\pm
1},\mathcal{L}_{0},\bar{\mathcal{L}}_{\pm
1},\bar{\mathcal{L}}_{0},T_{1,0},T_{0,1},T_{0,0},T_{1,1}\\}$ can be suitably
repackaged to give the Poincaré algebra. It is noteworthy that the
$T_{0,1},T_{1,0},T_{0,0}\ \mbox{and}\ T_{1,1}$ gives rise to the translation
generators while a certain combination of
$(\mathcal{L}_{m},\bar{\mathcal{L}}_{m})$ where $(m,n)\in\\{\pm 1,0\\}$ gives
rise to the Lorentz generators. In some sense, this gives us further intuition
to associate $T_{p,q}$ with _supertranslations_ while associating
$\mathcal{L}_{m}$ and $\bar{\mathcal{L}}_{m}$ with _superrotations_.

Similarly, for four spacetime dimensions it has been proved that
$\mathfrak{bms}_{4}$ algebra is not rigid and can be deformed into four
parameters family algebra called $W(a,b;\bar{a},\bar{b})$ algebra with
commutators as

$\begin{split}&[\mathcal{L}_{m},\mathcal{L}_{n}]=(m-n)\mathcal{L}_{m+n},\\\
&[\bar{\mathcal{L}}_{m},\bar{\mathcal{L}}_{n}]=(m-n)\bar{\mathcal{L}}_{m+n},\\\
&[\mathcal{L}_{m},\bar{\mathcal{L}}_{n}]=0.\\\
&[\mathcal{L}_{m},T_{p,q}]=-(a+bm+p)T_{m+p,q},\\\
&[\bar{\mathcal{L}}_{n},T_{p,q}]=-(\bar{a}+\bar{b}n+q)T_{p,n+q},\\\
&[T_{p,q},T_{k,l}]=0,\end{split}$ (123)

where $a,b,\bar{a}$ and $\bar{b}$ are arbitrary real parameters. In this way,
$\mathfrak{bms}_{4}$ algebra (106) can be viewed as
$W(-\frac{1}{2},-\frac{1}{2};-\frac{1}{2},-\frac{1}{2})$. This algebra can be
viewed as two copies of $W(a,b)$ and $W(\bar{a},\bar{b})$ with the
identification of $T_{p,q}$ as a product of supertranslation generators of
both the algebras. However, as we see in the next section, this structure does
not extend in the supersymmetric extensions of the algebra. Another
interesting case is $W(0,0;0,0)$, which represents an infinite dimensional
algebra of the symmetries of the near horizon geometry of nonextremal black
holes Donnay:2019zif . The algebra with $a=b=\bar{a}=\bar{b}=-\frac{1+s}{2}$
for $0<s<1$ describes the asymptotic symmetry algebra of decelerating FLRW
spacetime Enriquez-Rojo:2021blc .

##  4 Supersymmetric $W(a,b;\bar{a},\bar{b})$ algebra from $\mathcal{N}=2$
super-$\mathfrak{bms}_{4}$

In this section we write down a supersymmetric extension of
$W(a,b;\bar{a},\bar{b})$ algebra with two supercharges. To get to this, like
the $\mathfrak{bms}_{3}$ algebra, our first goal is to write the
supersymmetrized $\mathfrak{bms}_{4}$ algebra. The first attempt towards the
construction of a super-$\mathfrak{bms}$ algebra was carried out in
AWADA198652 , which however did not consider superrotation generators.
Avery:2015iix , further explored asymptotic fermionic charges in ${\cal N}=1$
supergravity on four dimensional asymptotically flat background 555There also
exists a realization of super-$\mathfrak{bms}$ algebra at spatial infinity
Henneaux:2020ekh ; Fuentealba:2020aax . Fotopoulos:2019vac ;
Fotopoulos:2020bqj  have derived such an algebra by analysing OPEs of
appropriate operators of Einstein-Yang-Mills theory at the celestial sphere.
However, their convention for indices on the supertranslation generators is
different from ours. This changes the index structure that appears in the
commutators. We fix the index structure of the supersymmetrized
$\mathfrak{bms}_{4}$ algebra by demanding consistency between its global part
and the four dimensional super-Poincaré algebra which along with (107)-(109)
now also contains

$\left\\{\mathcal{Q}_{A},\bar{\mathcal{Q}}_{\dot{B}}\right\\}=2(\sigma^{\mu})_{A\dot{B}}P_{\mu}\
\ ,\ \left[M^{\mu\nu},\mathcal{Q}_{A}\right]=i(\sigma^{\mu\nu})_{A}^{\
B}\mathcal{Q}_{B}\ \ ,\
[M^{\mu\nu},\bar{\mathcal{Q}}^{\dot{A}}]=i(\bar{\sigma}^{\mu\nu})^{\dot{A}}_{\
\dot{B}}\bar{\mathcal{Q}}^{\dot{B}}\ .$ (124)

Our starting point in the current context is the algebra stated at (106).
However, as mentioned earlier, we need to determine the index structure once
we include the super-current generators which we denote by $Q^{i}_{r}$ and
$\bar{Q}^{i}_{r}$ where $i=1,2$ while $r\in\mathbb{Z}+\frac{1}{2}$. We begin
with the global algebra described in the earlier section 3 to determine the
indices. Hence, we propose the following ansatz involving the supertranslation
and superrotation generators with the fermionic supercurrent generators

$\\{Q_{r}^{i},\bar{Q}_{s}^{j}\\}=\delta^{ij}T_{f(r,s),g(r,s)}\ \ ;\ \
[\mathcal{L}_{m},Q^{i}_{r}]=\alpha(m,r)Q^{i}_{h(m,r)}\ \ ;\ \
[\bar{\mathcal{L}}_{m},\bar{Q}^{i}_{r}]=\bar{\alpha}(m,r)\bar{Q}^{i}_{\bar{h}(m,r)}\
,$ (125)

and all other elements in the super-algebra are zero. We must note here that a
priori, it is not necessary that the other possible (anti)-commutators are
zero for a four dimensional asymptotically flat supergravity theory, however
we consider this simplified deformation for the purpose of this paper. Our
approach is pragmatic–we simply want to write a possible supersymmetric
extension of $\mathfrak{bms}_{4}$ algebra, such that its global part coincides
with the super-Poincaré algebra, without bothering about the fact if it can be
realized from the asymptotic symmetry analysis of a physical supergravity
theory. The OPE analysis of Fotopoulos:2019vac ; Fotopoulos:2020bqj  found
that the only non-zero commutators in super-$\mathfrak{bms}_{4}$ algebra are
the ones mentioned above in (125). This motivates us to propose the ansatz as
written above. We further make the simplifying assumption that the functions
parametrizing the indices $f,g,h,\bar{h}$ are all linear in their arguments
666This structure does not hold for certain symmetry algebras such as the one
discussed in Fuentealba:2021xhn ..

We _demand_ the map between supercurrent modes and the fermionic generators of
the super-Poincaré algebra to be

$\displaystyle\mathcal{Q}^{i}_{1}\rightarrow Q^{i}_{+\frac{1}{2}}\quad$ ,
$\displaystyle\quad\mathcal{Q}^{i}_{2}\rightarrow Q^{i}_{-\frac{1}{2}}\ ,$
(126)
$\displaystyle\bar{\mathcal{Q}}^{i}_{\dot{1}}\rightarrow\bar{Q}^{i}_{+\frac{1}{2}}\quad$
,
$\displaystyle\quad\bar{\mathcal{Q}}^{i}_{\dot{2}}\rightarrow\bar{Q}^{i}_{-\frac{1}{2}}\
.$ (127)

Linearity of the indices implies

$\displaystyle f(r,s)$ $\displaystyle=f_{0}+f_{1}r+f_{2}s\ ,$ (128)
$\displaystyle g(r,s)$ $\displaystyle=g_{0}+g_{1}r+g_{2}s\ .$

where $f_{i}$ and $g_{i}$ are constants. Using (124), along with (119)-(122),
we must have

$\displaystyle\\{\mathcal{Q}^{i}_{1},\bar{\mathcal{Q}}^{j}_{\dot{1}}\\}=-2(P_{0}-P_{3})\delta^{ij}=4T_{1,0}\delta^{ij}\
,\
\\{\mathcal{Q}^{i}_{1},\bar{\mathcal{Q}}^{j}_{\dot{2}}\\}=2(P_{1}-iP_{2})\delta^{ij}=-4iT_{1,1}\delta^{ij}\
,$ (129)
$\displaystyle\\{\mathcal{Q}^{i}_{2},\bar{\mathcal{Q}}^{j}_{\dot{1}}\\}=2(P_{1}+iP_{2})\delta^{ij}=-4iT_{0,0}\delta^{ij}\
,\
\\{\mathcal{Q}^{i}_{2},\bar{\mathcal{Q}}^{j}_{\dot{2}}\\}=-2(P_{0}+P_{3})\delta^{ij}=-4T_{0,1}\delta^{ij}\
.$

Thus, the mapping (126) requires the functions $f(r,s)$ and $g(r,s)$ to
satisfy

$\displaystyle f\left(+\frac{1}{2},+\frac{1}{2}\right)=1\
,f\left(+\frac{1}{2},-\frac{1}{2}\right)=1\
,f\left(-\frac{1}{2},+\frac{1}{2}\right)=0\
,f\left(-\frac{1}{2},-\frac{1}{2}\right)=0\ ,$ (130) $\displaystyle
g\left(+\frac{1}{2},+\frac{1}{2}\right)=0\
,g\left(+\frac{1}{2},-\frac{1}{2}\right)=1\
,g\left(-\frac{1}{2},+\frac{1}{2}\right)=0\
,g\left(-\frac{1}{2},-\frac{1}{2}\right)=1\ .$

Thus, we need to solve for the six unknowns $f_{i},g_{i}(i=0,1,2)$ appearing
in (128) from the above eight equations. As it turns out, (detailed in
appendix B) there does exist a consistent solution to the above system given
by

$f_{0}=g_{0}=+\frac{1}{2},f_{1}=-g_{2}=1,f_{2}=g_{1}=0\ .$ (131)

Thus, we eventually recover

$\\{Q_{r}^{i},\bar{Q}_{s}^{j}\\}=\delta^{ij}T_{r+\frac{1}{2},-s+\frac{1}{2}}\
,\ \\{Q^{i}_{r},Q^{j}_{s}\\}=\\{\bar{Q}^{i}_{r},\bar{Q}^{j}_{s}\\}=0$ (132)

The exact map (126) can be seen to be

$\mathcal{Q}^{i}_{1}=2Q^{i}_{+\frac{1}{2}}\ ,\
\bar{\mathcal{Q}}^{i}_{\dot{1}}=2\bar{Q}^{i}_{+\frac{1}{2}}\ ,\
\mathcal{Q}_{2}^{i}=-2iQ^{i}_{-\frac{1}{2}}\ ,\
\bar{\mathcal{Q}}^{i}_{\dot{2}}=-2i\bar{Q}^{i}_{-\frac{1}{2}}\ .$ (133)

Using the above map, along with the map described in Sec. 3, we get,

$\displaystyle[\mathcal{L}_{-1},Q^{i}_{+\frac{1}{2}}]=-Q^{i}_{-\frac{1}{2}}\
,\ [\mathcal{L}_{0},Q^{i}_{+\frac{1}{2}}]=-\frac{1}{2}Q^{i}_{+\frac{1}{2}}\ ,\
[\mathcal{L}_{+1},Q^{i}_{+\frac{1}{2}}]=0\ ,$ (134)
$\displaystyle[\mathcal{L}_{-1},Q^{i}_{-\frac{1}{2}}]=0\ ,\
[\mathcal{L}_{0},Q^{i}_{-\frac{1}{2}}]=\frac{1}{2}Q^{i}_{-\frac{1}{2}}\ ,\
[\mathcal{L}_{+1},Q^{i}_{-\frac{1}{2}}]=Q^{i}_{+\frac{1}{2}}\ .$

Further assuming linearity of the structure constant $\alpha(m,r)$ and
$h(m,r)$ appearing in (125), we see that the above global sector is consistent
provided one has

$[\mathcal{L}_{m},Q^{i}_{r}]=\left(\frac{m}{2}-r\right)Q^{i}_{m+r}\ .$ (135)

An identical exercise on the “barred” sector first leads us to the relations

$\displaystyle[\bar{\mathcal{L}}_{-1},\bar{Q}^{i}_{+\frac{1}{2}}]=0\ ,\
[\bar{\mathcal{L}}_{0},\bar{Q}^{i}_{+\frac{1}{2}}]=\frac{1}{2}\bar{Q}^{i}_{+\frac{1}{2}}\
,\
[\bar{\mathcal{L}}_{+1},\bar{Q}^{i}_{+\frac{1}{2}}]=\bar{Q}^{i}_{-\frac{1}{2}}\
,$ (136)
$\displaystyle[\bar{\mathcal{L}}_{-1},\bar{Q}^{i}_{-\frac{1}{2}}]=-\bar{Q}^{i}_{+\frac{1}{2}}\
,\
[\bar{\mathcal{L}}_{0},\bar{Q}^{i}_{-\frac{1}{2}}]=-\frac{1}{2}\bar{Q}^{i}_{-\frac{1}{2}}\
,\ [\bar{\mathcal{L}}_{+1},\bar{Q}^{i}_{-\frac{1}{2}}]=0\ .$

This shows that we must have

$[\bar{\mathcal{L}}_{m},\bar{Q}^{j}_{s}]=\left(\frac{m}{2}+s\right)\bar{Q}^{j}_{-m+s}\
.$ (137)

Thus, a particular realization of the $\mathcal{N}=2$
super-$\mathfrak{bms}_{4}$ algebra can be written as

$\begin{split}&[\mathcal{L}_{m},\mathcal{L}_{n}]=(m-n)\mathcal{L}_{m+n},\\\
&[\bar{\mathcal{L}}_{m},\bar{\mathcal{L}}_{n}]=(m-n)\bar{\mathcal{L}}_{m+n},\\\
&[\mathcal{L}_{m},T_{p,q}]=\left(\frac{m+1}{2}-p\right)T_{m+p,q},\\\
&[\bar{\mathcal{L}}_{m},T_{p,q}]=\left(\frac{m+1}{2}-q\right)T_{p,m+q},\\\
&\\{Q^{i}_{r},\bar{Q}^{j}_{s}\\}=\delta^{ij}T_{r+1/2,-s+1/2},\\\
&[\mathcal{L}_{m},Q^{i}_{r}]=\left(\frac{m}{2}-r\right)Q^{i}_{m+r},\\\
&[\bar{\mathcal{L}}_{m},\bar{Q}^{i}_{r}]=\left(\frac{m}{2}+r\right)\bar{Q}^{i}_{-m+r}\
.\end{split}$ (138)

while the other (anti)-commutators are identically zero. One can easily check
that all the Jacobi identities are satisfied for the above written algebra.
Now, that we have fixed the indices in the super-$\mathfrak{bms}_{4}$ algebra,
we assume that deformations do not change that and thus will carry over to the
$W(a,b;\bar{a},\bar{b})$ algebra. Thus, we now propose the following
supersymmetrized $W(a,b;\bar{a},\bar{b})$ algebra

$\begin{split}&[\mathcal{L}_{m},\mathcal{L}_{n}]=(m-n)\mathcal{L}_{m+n},\\\
&[\bar{\mathcal{L}}_{m},\bar{\mathcal{L}}_{n}]=(m-n)\bar{\mathcal{L}}_{m+n},\\\
&[\mathcal{L}_{m},T_{p,q}]=-\left(a+bm+p\right)T_{m+p,q},\\\
&[\bar{\mathcal{L}}_{m},T_{p,q}]=-\left(\bar{a}+\bar{b}m+q\right)T_{p,m+q},\\\
&\\{Q^{i}_{r},\bar{Q}^{j}_{s}\\}=\delta^{ij}T_{r+1/2,-s+1/2},\\\
&[\mathcal{L}_{m},Q^{i}_{r}]=\alpha(m,r)Q^{i}_{m+r},\\\
&[\bar{\mathcal{L}}_{m},\bar{Q}^{i}_{r}]=\bar{\alpha}(m,r)\bar{Q}^{i}_{-m+r}\
.\end{split}$ (139)

Strictly speaking, the most general extension would require us to choose all
such as
$[\mathcal{L}_{m},\bar{Q}^{i}_{r}],[\bar{\mathcal{L}}_{m},{Q}^{i}_{r}]$ to be
non-zero and demand consistency of Jacobi identities. However we have chosen
the above as a possible extension to verify if it satisfies all Jacobi
identities and thus making it a consistent graded Lie algebra. The Jacobi
identity for $Q^{i}_{r},\bar{Q}^{j}_{s}$ and $\mathcal{L}_{m}$ is given by

$[\\{Q_{r}^{i},\bar{Q}_{s}^{j}\\},\mathcal{L}_{m}]=\\{Q_{r}^{i},[\bar{Q}_{s}^{j},\mathcal{L}_{m}]\\}+\\{\bar{Q}_{s}^{j},[Q_{r}^{i},\mathcal{L}_{m}]\\}$
(140)

Using (139), we get,

$\alpha(m,r)=-\left(a+bm+r+\frac{1}{2}\right)\ .$ (141)

The Jacobi identities of $Q^{i}_{r},\bar{Q}^{j}_{s}$ and
$\bar{\mathcal{L}}_{m}$ similarly gives

$\bar{\alpha}(m,r)=-\left(\bar{a}+\bar{b}m-r+\frac{1}{2}\right)\ .$ (142)

Thus, a ${\mathcal{N}}=2$ supersymmetric $W(a,b;\bar{a},\bar{b})$ algebra is
given by

$\begin{split}&[\mathcal{L}_{m},\mathcal{L}_{n}]=(m-n)\mathcal{L}_{m+n},\\\
&[\bar{\mathcal{L}}_{m},\bar{\mathcal{L}}_{n}]=(m-n)\bar{\mathcal{L}}_{m+n},\\\
&[\mathcal{L}_{m},T_{p,q}]=-\left(a+bm+p\right)T_{m+p,q},\\\
&[\bar{\mathcal{L}}_{m},T_{p,q}]=-\left(\bar{a}+\bar{b}m+q\right)T_{p,m+q},\\\
&\\{Q^{i}_{r},\bar{Q}^{j}_{s}\\}=\delta^{ij}T_{r+1/2,-s+1/2},\\\
&[\mathcal{L}_{m},Q^{i}_{r}]=-\left(a+bm+r+\frac{1}{2}\right)Q^{i}_{m+r},\\\
&[\bar{\mathcal{L}}_{m},\bar{Q}^{i}_{r}]=-\left(\bar{a}+\bar{b}m-r+\frac{1}{2}\right)\bar{Q}^{i}_{-m+r}\
,\end{split}$ (143)

and all other commutators are zero. It satisfies all the Jacobi identities.
The above algebra is a simple extension of bosonic $W(a,b;\bar{a},\bar{b})$
algebra with two supercurrent generators, as it does not contain any rotation
among the supercurrent generators. In the context of super-Poincaré algebra,
the $R$-charge rotates the SUSY generators amongst themselves. To be
mathematically precise, the $R-$extension is the largest subgroup of the
automorphism group of the supersymmetry algebra which commutes with the
Lorentz group. In the next section we perform the $R$-extension of the above
${\mathcal{N}}=2$ supersymmetric $W(a,b;\bar{a},\bar{b})$ algebra, by further
introducing the R-charge generators to rotate the two supercurrent generators.

##  5 $R$-extended supersymmetric $W(a,b;\bar{a},\bar{b})$ algebra

Before going to $R$-extension of the $W(a,b;\bar{a},\bar{b})$ algebra, let us
briefly recall some important points about super-Poincare algebras in presence
of $R-$symmetry. As discussed in Hori:2003ic  generic $\mathcal{N}=2$ super-
Poincare algebra contains two species of $R$-symmetry generators–vectorial and
axial which act on the supercharges as

$\displaystyle[Q^{i}_{\pm\frac{1}{2}},R_{0}]=Q^{i}_{\pm\frac{1}{2}}\ $ ,
$\displaystyle\
[\bar{Q}^{i}_{\pm\frac{1}{2}},R_{0}]=-\bar{Q}^{i}_{\pm\frac{1}{2}}\ ,$ (144)
$\displaystyle{[Q^{i}_{\pm\frac{1}{2}},\bar{R}_{0}]}=\pm
Q^{i}_{\pm\frac{1}{2}}\ $ , $\displaystyle\
[\bar{Q}^{i}_{\pm\frac{1}{2}},\bar{R}_{0}]=\mp\bar{Q}^{i}_{\pm\frac{1}{2}}$
(145)

As a matter of fact theories with extended supersymmetries are extremely rich
precisely due to the presence of these two kinds of supercharges. These
$R$-symmetries can in fact be used as a powerful tool to define various
_twistings_ in a theory (A-type or B-type) resulting in what is known as
Topological Field Theories whose correlators happen to be independent of the
background metric. Although conventionally theories with either one kind of
$R$-symmetry are considered, in general the full theory does contain both
kinds of R-symmetries. In the current context since we are specifically
interested in $\mathcal{N}=2$ SUSY, the $R$-symmetry generator in fact
generates the group $U(1)_{V}\times U(1)_{A}$. Following Hori:2003ic , we will
associate $R_{0}$ with the vectorial $R$-symmetry while $\bar{R}_{0}$ will be
associated as the axial $R$-symmetry. Another important aspect of super-
Poincare algebras is the fact that the $R-$charges commute with the bosonic
Poincare generators. We utilize these facts in our constructions. To be
precise, in the following we demand that our $R$-extended
$W(a,b;\bar{a},\bar{b})$ algebra must have vanishing commutators of the
$R-$charge generators with other bosonic generators in the global sector for
the deformation $W(-1/2,-1/2;-1/2,-1/2)$777$W(-1/2,-1/2;-1/2,-1/2)$ is the
$\mathfrak{bms}_{4}$ algebra whose global sector coincides with the
$\mathcal{N}=2$ super-Poincare algebra.

In order to make the extension of $R$-charges in the context of
$W(a,b;\bar{a},\bar{b})$ algebra, we will start with a general ansatz,

$\begin{split}&[Q_{r}^{i},R_{n}]=\beta^{(i)}(r,n)Q^{i}_{\theta_{1}(r,n)}\ ,\
[\bar{Q}_{r}^{i},R_{n}]=\bar{\beta}^{(i)}(r,n)\bar{Q}^{i}_{\bar{\theta}_{1}(r,n)}\
,\\\ &{[Q_{r}^{i},\bar{R}_{n}]}=\kappa^{(i)}(r,n)Q^{i}_{\theta_{2}(r,n)}\ ,\
[\bar{Q}_{r}^{i},\bar{R}_{n}]=\bar{\kappa}^{(i)}(r,n)\bar{Q}^{i}_{\bar{\theta}_{2}(r,n)}\
.\end{split}$ (146)

It must be noted that in the above ansatz, although $i$ is a repeated index on
the RHS, there is no sum over $i$. Although we will be primarily interested in
$\mathcal{N}=2$ but the discussion in this section is valid for arbitrary
value of $\mathcal{N}$. Thus we consider $i=1,2,..,\mathcal{N}$. Furthermore,
since we are working in a very general setting, we assume that $\beta^{i}$ is
different for $i=1,2,..,\mathcal{N}$. This is of course not the most general
extension one can think of but rather a simpler starting point which is also
consistent with the global sub-sector (144)-(145).

In the following analysis, we will assume that the indices appearing in the
above ansatz are linear in its arguments, which is a basic feature of most
algebras. Consistency with the global subsector i.e. (144)-(145) fixes the
form of the indices as

$\begin{split}&[Q_{r}^{i},R_{n}]=\beta^{(i)}(r,n)Q^{i}_{cn+r}\ ,\
[\bar{Q}_{r}^{i},R_{n}]=\bar{\beta}^{(i)}(r,n)\bar{Q}^{i}_{\bar{c}n+r}\\\
&{[Q_{r}^{i},\bar{R}_{n}]}=\kappa^{(i)}(r,n)Q^{i}_{kn+r}\ ,\
[\bar{Q}_{r}^{i},\bar{R}_{n}]=\bar{\kappa}^{(i)}(r,n)\bar{Q}^{i}_{\bar{k}n+r}\
.\end{split}$ (147)

where $c,\bar{c},k$ and $\bar{k}$ are integers and the structure constants are
non-zero at least in the global subsector i.e. when $r=\pm\frac{1}{2}$ and
$n=0$. We have to find their form away from the global sector. Subsequently,
we will concentrate on the commutators $[R_{n},T_{p,q}]$ and
$[\bar{R}_{n},T_{p,q}]$. Consider the Jacobi identity for the operators
$Q^{i}_{r},\bar{Q}^{j}_{s}$ and $R_{n}$ which gives us

$[\\{Q_{r}^{i},\bar{Q}_{s}^{j}\\},R_{n}]=\\{Q_{r}^{i},[\bar{Q}_{s}^{j},R_{n}]\\}+\\{\bar{Q}_{s}^{j},[Q_{r}^{i},R_{n}]\\}\
.$ (148)

Using (147), in the above, we get,

$\delta^{ij}{[T_{p,q},R_{n}]}=\delta^{ij}\bigg{[}\bar{\beta}^{(j)}\left(-q+\frac{1}{2},n\right)T_{p,-\bar{c}n+q}+\beta^{(i)}\left(p-\frac{1}{2},n\right)T_{cn+p,q}\bigg{]}.$

In the above equation $i,j$ are free indices. In particular for
$\mathcal{N}=2$ SUSY , we see that,

$\displaystyle{[T_{p,q},R_{n}]}$
$\displaystyle=\bar{\beta}^{(1)}\left(-q+\frac{1}{2},n\right)T_{p,-\bar{c}n+q}+\beta^{(1)}\left(p-\frac{1}{2},n\right)T_{cn+p,q}$
(149)
$\displaystyle=\bar{\beta}^{(2)}\left(-q+\frac{1}{2},n\right)T_{p,-\bar{c}n+q}+\beta^{(2)}\left(p-\frac{1}{2},n\right)T_{cn+p,q}\
.$

which also implies $\beta^{(1)}(r,n)=\beta^{(2)}(r,n)$ and
$\bar{\beta}^{(1)}(r,n)=\bar{\beta}^{(2)}(r,n)$. An identical exercise with
$Q^{i}_{r},\bar{Q}^{j}_{s}$ and $\bar{R}_{n}$ gives

$\displaystyle{[T_{p,q},\bar{R}_{n}]}$
$\displaystyle=\bar{\kappa}^{(1)}\left(-q+\frac{1}{2},n\right)T_{p,-\bar{k}n+q}+\kappa^{(1)}\left(p-\frac{1}{2},n\right)T_{kn+p,q}$
(150)
$\displaystyle=\bar{\kappa}^{(2)}\left(-q+\frac{1}{2},n\right)T_{p,-\bar{k}n+q}+\kappa^{(2)}\left(p-\frac{1}{2},n\right)T_{kn+p,q}\
.$

leading to the condition that $\kappa^{(1)}(r,n)=\kappa^{(2)}(r,n)$ and
$\bar{\kappa}^{(1)}(r,n)=\bar{\kappa}^{(2)}(r,n)$. Given the relation between
the structure constants for $i=1,2$, we see that the index is extraneous and
hence we will simply be dropping it from our notation subsequently. Thus, we
may write more simply

$\displaystyle[Q_{r}^{i},R_{n}]=\beta(r,n)Q^{i}_{cn+r}\ ,\
[\bar{Q}_{r}^{i},R_{n}]=\bar{\beta}(r,n)\bar{Q}^{i}_{\bar{c}n+r}\ ,$ (151)
$\displaystyle{[Q_{r}^{i},\bar{R}_{n}]}=\kappa(r,n)Q^{i}_{kn+r}\ ,\
[\bar{Q}_{r}^{i},\bar{R}_{n}]=\bar{\kappa}(r,n)\bar{Q}^{i}_{\bar{k}n+r}\ ,$
$\displaystyle{[T_{p,q},R_{n}]}=\bar{\beta}\left(-q+\frac{1}{2},n\right)T_{p,-\bar{c}n+q}+\beta\left(p-\frac{1}{2},n\right)T_{cn+p,q}\
,$
$\displaystyle{[T_{p,q},\bar{R}_{n}]}=\bar{\kappa}\left(-q+\frac{1}{2},n\right)T_{p,-\bar{k}n+q}+\kappa\left(p-\frac{1}{2},n\right)T_{kn+p,q}\
.$

Note that the above commutation relations also ensure that the Jacobi
identities between $T_{p,q},T_{m,n},R_{l}$ as well as
$T_{p,q},Q^{i}_{r},R_{n}$ (and it’s corresponding counterparts with
$Q^{i}_{r}$ replaced with $\bar{Q}^{j}_{s}$ and $R_{n}$ replaced with
$\bar{R}_{n}$) are also satisfied. Now, that we have identified $R_{n}$ and
$\bar{R}_{n}$ to the vectorial and axial $R$-supercurrents, we make a further
assumption that

$[R_{n},\bar{R}_{m}]=0\ .$ (152)

The above assumptions applied to the Jacobi identity of $R_{n},\bar{R}_{m}$
and $Q^{i}_{r}$ leads to a relation between the structure constants

$\kappa(cn+r,m)\beta(r,n)=\kappa(r,m)\beta(km+r,n)\ ,$ (153)

and analogously the Jacobi identity of $R_{n},\bar{R}_{m}$ and
$\bar{Q}^{i}_{r}$ gives rise to,

$\bar{\kappa}(\bar{c}n+r,m)\bar{\beta}(r,n)=\bar{\kappa}(r,m)\bar{\beta}(\bar{k}m+r,n)\
.$ (154)

The above two relations seems to put certain constraints on the free
parameters $c,\bar{c},k$ and $\bar{k}$. However, we will explore this
subsequently.

Finally, we need to fix the algebra between the $R$-supercurrents and the
superrotations $\mathcal{L}_{m}$ and $\bar{\mathcal{L}}_{m}$. For this purpose
we consider a set of Jacobi identities and fixing the form of the commutators
between the $R$-charge supercurrents and the superrotation generators.

  1. 1.

Jacobi identity for $\mathcal{L}_{m},Q^{i}_{r},R_{n}$ and
$\mathcal{L}_{m},\bar{Q}^{j}_{s},R_{n}$
We start with the Jacobi identity for $\mathcal{L}_{m},Q^{i}_{r}$ and $R_{n}$
which is given by

$[[\mathcal{L}_{m},Q^{i}_{r}],R_{n}]+[[Q^{i}_{r},R_{n}],\mathcal{L}_{m}]+[[R_{n},\mathcal{L}_{m}],Q^{i}_{r}]=0\
.$ (155)

Using (143) and (LABEL:eq:QRT-simplified), we can simplify the above equation
to obtain

$[[R_{n},\mathcal{L}_{m}],Q^{i}_{r}]+[\alpha(m,r)\beta(m+r,n)-\beta(r,n)\alpha(m,cn+r)]Q^{i}_{cn+m+r}=0\
.$ (156)

Clearly, looking at the above, on very general grounds, one can schematically
write,

$[R_{n},\mathcal{L}_{m}]=w_{1}(n,m)\mathcal{L}_{t_{1}(n,m)}+h_{1}(n,m)R_{u_{1}(n,m)}+\bar{h}_{1}(n,m)\bar{R}_{v_{1}(n,m)}\
.$ (157)

The Jacobi identity for $\mathcal{L}_{m},\bar{Q}^{j}_{s}$ and $R_{n}$ along
with (143) and (147) leads to

$[[R_{n},\mathcal{L}_{m}],\bar{Q}^{j}_{s}]=0\ .$ (158)

Now, if we consider the $[R_{n},\mathcal{L}_{m}]$ to be of the form as (157),
we easily see that the part $[R_{u(n,m)},\bar{Q}^{j}_{s}]$ and
$[\bar{R}_{v(n,m)},\bar{Q}^{j}_{s}]$ will be generically non-zero
_individually_ for arbitrary values of $n,m$ and $s$ however a linear
combination with specific forms of $h_{1}$ and $\bar{h}_{1}$ might presumably
ensure that the expression vanishes. Note that (157) has certain features
which puts it in stark contrast to its 3 dimensional $W(a,b)$ analog. Firstly,
the first term appearing on the RHS in the above equation has no analog for
the $W(a,b)$ algebra as stated explicitly in (2.2). We can also see,
commutators similar to the $W(a,b)$ algebra i.e.

$[R_{n},\mathcal{L}_{m}]\sim R_{u(n,m)}\ \text{and}\
[\bar{R}_{n},\mathcal{L}_{m}]\sim\bar{R}_{v(n,m)}$ (159)

is clearly inconsistent with (158) for arbitrary values of the indices $m$ and
$n$.

  2. 2.

Jacobi identity for $\bar{\mathcal{L}}_{m},\bar{Q}^{j}_{s},R_{n}$ and
$\bar{\mathcal{L}}_{m},Q^{i}_{r}$ and $R_{n}$
The Jacobi identity of $\bar{\mathcal{L}}_{m},\bar{Q}^{j}_{s}$ and $R_{n}$
leads us to

$\displaystyle[[R_{n},\bar{\mathcal{L}}_{m}],\bar{Q}^{j}_{s}]$ (160)
$\displaystyle+\left(\bar{\beta}(s,n)\left(\bar{a}+\bar{b}m-\bar{c}n-s+\frac{1}{2}\right)-\bar{\beta}(-m+s,n)\left(\bar{a}+\bar{b}m-s+\frac{1}{2}\right)\right)\bar{Q}^{j}_{\bar{c}n+s-m}=0\
.$

The above along with the Jacobi identity for $\bar{\mathcal{L}}_{m},Q^{i}_{r}$
which simplifies to

$[[R_{n},\bar{\mathcal{L}}_{m}],Q^{i}_{r}]=0\ ,$ (161)

suggests of a relation of the form

$[R_{n},\bar{\mathcal{L}}_{m}]=w_{2}(n,m)\bar{\mathcal{L}}_{t_{2}(n,m)}+h_{2}(n,m)R_{u_{2}(n,m)}+\bar{h}_{2}(n,m)\bar{R}_{v_{2}(n,m)}\
.$ (162)

  3. 3.

Jacobi identity for $\mathcal{L}_{m},Q^{i}_{r},\bar{R}_{n}$ and
$\mathcal{L}_{m},\bar{Q}^{j}_{s},\bar{R}_{n}$
The Jacobi identity for $\mathcal{L}_{m},Q^{i}_{r}$ and $\bar{R}_{n}$ leads to

$\displaystyle[[\bar{R}_{m},\mathcal{L}_{m}],Q^{i}_{r}]+$ (163)
$\displaystyle\left(\kappa(r,n)\left(a+bm+kn+r+\frac{1}{2}\right)-\kappa(m+r,n)\left(a+bm+r+\frac{1}{2}\right)\right)Q^{i}_{kn+m+r}=0\
,$

while the Jacobi identity for $\mathcal{L}_{m},\bar{Q}^{j}_{s}$ and
$\bar{R}_{n}$ gives

$[[\bar{R}_{n},\mathcal{L}_{m}],\bar{Q}^{j}_{s}]=0\ .$ (164)

This leads us to propose

$[\bar{R}_{n},\mathcal{L}_{m}]=w_{3}(n,m)\mathcal{L}_{t_{3}(n,m)}+h_{3}(n,m)R_{u_{3}(n,m)}+\bar{h}_{3}(n,m)\bar{R}_{v_{3}(n,m)}\
.$ (165)

  4. 4.

Jacobi identity for $\bar{\mathcal{L}}_{m},Q^{i}_{r},\bar{R}_{n}$ and
$\bar{\mathcal{L}}_{m},\bar{Q}^{j}_{s},\bar{R}_{n}$
The Jacobi identity for $\bar{\mathcal{L}}_{m},Q^{i}_{r},\bar{R}_{n}$
simplifies to

$[[\bar{R}_{n},\bar{\mathcal{L}}_{m}],Q^{i}_{r}]=0\ $ (166)

and the $\bar{\mathcal{L}}_{m},\bar{Q}^{j}_{s},\bar{R}_{n}$ Jacobi identity
leads to

$\displaystyle[[\bar{R}_{n},\bar{\mathcal{L}}_{m}],\bar{Q}^{j}_{s}]$ (167)
$\displaystyle+\left(\bar{\kappa}(s,n)\left(\bar{a}+\bar{b}m-\bar{k}n-s+\frac{1}{2}\right)-\bar{\kappa}(-m+s,n)\left(\bar{a}+\bar{b}m-s+\frac{1}{2}\right)\right)\bar{Q}^{j}_{\bar{k}n+s-m}=0\
.$

The above two equations lead us to the ansatz

${[\bar{R}_{n},\bar{\mathcal{L}}_{m}]}=w_{4}(n,m)\bar{\mathcal{L}}_{t_{4}(n,m)}+h_{4}(n,m)R_{u_{4}(n,m)}+\bar{h}_{4}(n,m)\bar{R}_{v_{4}(n,m)}\
.$ (168)

  5. 5.

Jacobi identity for $\mathcal{L}_{m},T_{p,q},R_{n}$ and
$\bar{\mathcal{L}}_{m},T_{p,q},R_{n}$
The Jacobi identity for the operators $\mathcal{L}_{m},T_{p,q}$ and $R_{n}$ is
given by

$[[\mathcal{L}_{m},T_{p,q}],R_{n}]+[[T_{p,q},R_{n}],\mathcal{L}_{m}]+[[R_{n},\mathcal{L}_{m}],T_{p,q}]=0\
,$ (169)

which upon using (143) and (LABEL:eq:QRT-simplified) leads us to the relation

$\displaystyle[[R_{n},\mathcal{L}_{m}],T_{p,q}]$ (170)
$\displaystyle+\left((a+bm+cn+p)\beta\left(p-\frac{1}{2},n\right)-(a+bm+p)\beta\left(m+p-\frac{1}{2},n\right)\right)T_{cn+m+p,q}=0\
,$

which is consistent with the ansatz (157). Further, the Jacobi identity for
$\bar{\mathcal{L}}_{m},T_{p,q},R_{n}$ gives

$\displaystyle[[R_{n},\bar{\mathcal{L}}_{m}],T_{p,q}]$ (171)
$\displaystyle+\left(\bar{\beta}\left(-q+\frac{1}{2},n\right)(\bar{a}+\bar{b}m-\bar{c}n+q)-\bar{\beta}\left(-m-q+\frac{1}{2},n\right)(\bar{a}+\bar{b}m+q)\right)T_{p,-\bar{c}n+m+q}=0\
,$

which is consistent with the ansatz (162).

  6. 6.

Jacobi identity for $\mathcal{L}_{m},T_{p,q},\bar{R}_{n}$ and
$\bar{\mathcal{L}}_{m},T_{p,q},\bar{R}_{n}$
The Jacobi identity for $\mathcal{L}_{m},T_{p,q},\bar{R}_{n}$ leads to

$\displaystyle[[\bar{R}_{n},\mathcal{L}_{m}],T_{p,q}]$ (172)
$\displaystyle+\left({\kappa}\left(p-\frac{1}{2},n\right)(a+bm+kn+p)-\kappa\left(m+p-\frac{1}{2},n\right)(a+bm+p)\right)T_{kn+m+p,q}=0\
,$

while for the other tuple, namely $\bar{\mathcal{L}}_{m},T_{p,q},\bar{R}_{n}$,
we have,

$\displaystyle[[\bar{R}_{n},\bar{\mathcal{L}}_{m}],T_{p,q}]$ (173)
$\displaystyle+\left(\bar{\kappa}\left(-q+\frac{1}{2},n\right)(\bar{a}+\bar{b}m-\bar{k}n+q)-\bar{\kappa}\left(-m-q+\frac{1}{2},n\right)(\bar{a}+\bar{b}m+q)\right)T_{p,-\bar{k}n+m+q}=0\
.$

One can easily see that the above two equations are indeed consistent with the
proposed ansatze (162) and (168) respectively.

Armed with a series of ansatze and a number of Jacobi identities, we enumerate
below the index structure and which equations they follows from.

  * •

Plugging ansatz (157) in (156), we get,

$\displaystyle-w_{1}(n,m)\alpha(t_{1},r)Q^{i}_{t_{1}+r}$ $\displaystyle-
h_{1}(n,m)\beta(r,u_{1})Q^{i}_{cu_{1}+r}-\bar{h}_{1}(n,m)\kappa(r,v_{1})Q^{i}_{kv_{1}+r}$
(174)
$\displaystyle+[\alpha(m,r)\beta(m+r,n)-\beta(r,n)\alpha(m,cn+r)]Q^{i}_{cn+m+r}=0\
.$

The above equation must be satisfied for arbitrary permissible values of $m,n$
and $r$. Also, by definition, $\beta$ and $\kappa$ cannot be identically zero.
This implies,

$\displaystyle u_{1}(n,m)$ $\displaystyle=$ $\displaystyle n+\frac{m}{c}$
$\displaystyle v_{1}(n,m)$ $\displaystyle=$
$\displaystyle\frac{c}{k}n+\frac{m}{k}\ .$

Since clearly the indices must be integers, we see both $k$ and $c$ must
divide every integer which naturally implies that both can takes values $\pm
1$. Using (158), we further get,

$\frac{\bar{c}}{c}=\frac{\bar{k}}{k}=\xi\ (\text{say})\ .$ (175)

where $\xi$ is some real number. Further looking at (174), we can have either
$w_{1}$ to be identically zero or $t_{1}=cn+m$. Further, (170) gives

$\displaystyle\left(-w_{1}(n,m)(a+bcn+bm+p)-h_{1}(n,m)\beta\left(p-\frac{1}{2},n+\frac{m}{c}\right)\right.$
(176)
$\displaystyle\left.-\bar{h}_{1}(n,m)\kappa\left(p-\frac{1}{2},\frac{c}{k}n+\frac{m}{k}\right)+(a+bm+cn+p)\beta\left(p-\frac{1}{2},n\right)\right.$
$\displaystyle\left.-(a+bm+p)\beta\left(m+p-\frac{1}{2},n\right)\right)T_{cn+m+p,q}$
$\displaystyle+\left(-h_{1}(n,m)\bar{\beta}\left(-q+\frac{1}{2},n+\frac{m}{c}\right)-\bar{h}_{1}(n,m)\bar{\kappa}\left(-q+\frac{1}{2},\frac{c}{k}n+\frac{m}{k}\right)\right)T_{p,-\bar{c}n-\xi
m+q}=0\ .$

In the above expression, each of the coefficients of the $T_{cn+m+p,q}$ and
$T_{p,-\bar{c}n-\xi m+q}$ has to vanish individually.

  * •

Ansatz (162) along with (160) implies $t_{2}(n,m)=-\bar{c}n+m$ provided
$w_{2}$ does not vanish identically. Along similar arguments as before, we
also conclude

$\displaystyle u_{2}$ $\displaystyle=$ $\displaystyle n-\frac{m}{\bar{c}}$
$\displaystyle v_{2}$ $\displaystyle=$
$\displaystyle\frac{\bar{c}}{\bar{k}}n-\frac{m}{\bar{k}}.$

Again, since both $u_{2}$ and $v_{2}$ must be integers for all values of $m$
and $n$, this leads us to conclude that $\bar{c}$ and $\bar{k}$ can only take
values $\pm 1$. Thus, $\xi$ appearing in (175) can only be $\pm 1$. Now, since
$c^{2}=k^{2}=\bar{c}^{2}=\bar{k}^{2}=\xi^{2}=1$, we can rewrite

$u_{1}=n+mc\ ,\ v_{1}=kcn+km\ ,\ u_{2}=n-\xi cm\ ,\ v_{2}=kcn-\xi km\ .$ (177)

Further, using (171), we get,

$\displaystyle-
w_{2}(n,m)(\bar{a}-\bar{b}\bar{c}n+\bar{b}m+q)-h_{2}(n,m)\bar{\beta}\left(-q+\frac{1}{2},n-\xi\frac{m}{c}\right)$
(178)
$\displaystyle-\bar{h}_{2}(n,m)\bar{\kappa}\left(-q+\frac{1}{2},\frac{c}{k}n-\xi\frac{m}{k}\right)+\bar{\beta}\left(-q+\frac{1}{2},n\right)(\bar{a}+\bar{b}m-\bar{c}n+q)$
$\displaystyle\hskip
142.26378pt-\bar{\beta}\left(-m-q+\frac{1}{2},n\right)(\bar{a}+\bar{b}m+q)=0\
,$

and

$h_{2}(n,m)\beta\left(p-\frac{1}{2},n-\xi\frac{m}{c}\right)+\bar{h}_{2}(n,m)\kappa\left(p-\frac{1}{2},\frac{c}{k}n-\xi\frac{m}{k}\right)=0\
.$ (179)

  * •

Ansatz (165) along with (163) tells us that we must have

$\displaystyle u_{3}$ $\displaystyle=$
$\displaystyle\frac{k}{c}n+\frac{m}{c}\equiv kcn+cm$ $\displaystyle v_{3}$
$\displaystyle=$ $\displaystyle n+\frac{m}{k}\equiv n+km\ ,$

while $w_{3}$ identically vanishes or $t_{3}=kn+m$. The Jaocib identity (172)
implies

$\displaystyle-
w_{3}(n,m)(a+bkn+bm+p)-h_{3}(n,m)\beta\left(p-\frac{1}{2},kcn+cm\right)$ (180)
$\displaystyle-\bar{h}_{3}(n,m)\kappa\left(p-\frac{1}{2},n+km\right)+{\kappa}\left(p-\frac{1}{2},n\right)(a+bm+kn+p)$
$\displaystyle\hskip
113.81102pt-\kappa\left(m+p-\frac{1}{2},n\right)(a+bm+p)=0\ ,$

and

$\displaystyle
h_{3}(n,m)\bar{\beta}\left(-q+\frac{1}{2},kcn+cm\right)+\bar{h}_{3}(n,m)\bar{\kappa}\left(-q+\frac{1}{2},n+km\right)=0\
.$ (181)

  * •

Ansatz (168) along with (167) tells us,

$\displaystyle u_{4}$ $\displaystyle=$
$\displaystyle\frac{k}{c}n-\xi\frac{m}{c}\equiv kcn-\xi cm$ $\displaystyle
v_{4}$ $\displaystyle=$ $\displaystyle n-\xi\frac{m}{k}\equiv n-\xi km\ ,$

along with $t_{4}=-\bar{k}n+m$ provided if $w_{4}$ is non-zero. The Jacobi
identity (173) gives

$\displaystyle-w_{4}(n,m)(\bar{a}+\bar{b}m-\xi
k\bar{b}n+q)-h_{4}(n,m)\bar{\beta}\left(-q+\frac{1}{2},kcn-\xi cm\right)$
(182) $\displaystyle-\bar{h}_{4}(n,m)\bar{\kappa}\left(-q+\frac{1}{2},n-\xi
km\right)+\bar{\kappa}\left(-q+\frac{1}{2},n\right)(\bar{a}+\bar{b}m-\bar{k}n+q)$
$\displaystyle\hskip
113.81102pt-\bar{\kappa}\left(-m-q+\frac{1}{2},n\right)(\bar{a}+\bar{b}m+q)=0\
,$

and

$h_{4}(n,m)\beta\left(p-\frac{1}{2},kcn-\xi
cm\right)+\bar{h}_{4}(n,m)\kappa\left(p-\frac{1}{2},n-\xi km\right)=0\ .$
(183)

Thus we see that simply imposing Jacobi identities in a systematic manner we
have obtained the following simplified algebra involving the $R$-supercurrents

$\displaystyle[Q_{r}^{i},R_{n}]=\beta(r,n)Q^{i}_{cn+r}\ ,\
[\bar{Q}_{r}^{i},R_{n}]=\bar{\beta}(r,n)\bar{Q}^{i}_{\xi{c}n+r}\ ,$ (184)
$\displaystyle{[Q_{r}^{i},\bar{R}_{n}]}=\kappa(r,n)Q^{i}_{kn+r}\ ,\
[\bar{Q}_{r}^{i},\bar{R}_{n}]=\bar{\kappa}(r,n)\bar{Q}^{i}_{\xi{k}n+r}\ ,$
$\displaystyle{[T_{p,q},R_{n}]}=\bar{\beta}\left(-q+\frac{1}{2},n\right)T_{p,-\xi{c}n+q}+\beta\left(p-\frac{1}{2},n\right)T_{cn+p,q}\
,$
$\displaystyle{[T_{p,q},\bar{R}_{n}]}=\bar{\kappa}\left(-q+\frac{1}{2},n\right)T_{p,-\xi{k}n+q}+\kappa\left(p-\frac{1}{2},n\right)T_{kn+p,q}\
,$
$\displaystyle{[R_{n},\mathcal{L}_{m}]}=w_{1}(n,m)\mathcal{L}_{cn+m}+h_{1}(n,m)R_{n+cm}+\bar{h}_{1}(n,m)\bar{R}_{kcn+km}\
,$
$\displaystyle[R_{n},\bar{\mathcal{L}}_{m}]=w_{2}(n,m)\bar{\mathcal{L}}_{-\xi
cn+m}+h_{2}(n,m)R_{n-\xi cm}+\bar{h}_{2}(n,m)\bar{R}_{kcn-\xi km}\ ,$
$\displaystyle[\bar{R}_{n},\mathcal{L}_{m}]=w_{3}(n,m)\mathcal{L}_{kn+m}+h_{3}(n,m)R_{kcn+cm}+\bar{h}_{3}(n,m)\bar{R}_{n+km}\
,$
$\displaystyle[\bar{R}_{n},\bar{\mathcal{L}}_{m}]=w_{4}(n,m)\bar{\mathcal{L}}_{-\xi
kn+m}+h_{4}(n,m)R_{kcn-\xi cm}+\bar{h}_{4}(n,m)\bar{R}_{n-\xi km}\ .$

There is of course a number of equations constraining the form of the
structure constants appearing in the above algebra. We will come back to
analyse them subsequently.

Using the above, we further list down the following Jacobi identities.

  1. 1.

Jacobi identity for $R_{m},R_{n}$ and $\mathcal{L}_{p}$ leads to

$\displaystyle w_{1}(n,p)w_{1}(m,cn+p)=w_{1}(m,p)w_{1}(n,cm+p)\ ,$ (185)
$\displaystyle h_{1}(m,cn+p)w_{1}(n,p)=h_{1}(n,cm+p)w_{1}(m,p)\ ,$
$\displaystyle\bar{h}_{1}(m,cn+p)w_{1}(n,p)=\bar{h}_{1}(n,cm+p)w_{1}(m,p)\ .$

  2. 2.

Jacobi identity for $R_{m},R_{n}$ and $\bar{\mathcal{L}}_{p}$ leads to

$\displaystyle w_{2}(n,p)w_{2}(m,-\xi cn+p)=w_{2}(m,p)w_{2}(n,-\xi cm+p)\ ,$
(186) $\displaystyle h_{2}(m,-\xi cn+p)w_{2}(n,p)=h_{2}(n,-\xi
cm+p)w_{2}(m,p)\ ,$ $\displaystyle\bar{h}_{2}(m,-\xi
cn+p)w_{2}(n,p)=\bar{h}_{2}(n,-\xi cm+p)w_{2}(m,p)\ .$

  3. 3.

Jacobi identity for $\bar{R}_{m},\bar{R}_{n}$ and ${\mathcal{L}}_{p}$ leads to

$\displaystyle w_{3}(n,p)w_{3}(m,kn+p)=w_{3}(m,p)w_{3}(n,km+p)\ ,$ (187)
$\displaystyle h_{3}(m,kn+p)w_{3}(n,p)=h_{3}(n,km+p)w_{3}(m,p)\ ,$
$\displaystyle\bar{h}_{3}(m,kn+p)w_{3}(n,p)=\bar{h}_{3}(n,km+p)w_{3}(m,p)\ .$

  4. 4.

Jacobi identity for $\bar{R}_{m},\bar{R}_{n}$ and $\bar{\mathcal{L}}_{p}$
leads to

$\displaystyle w_{4}(n,p)w_{4}(m,-\xi kn+p)=w_{4}(m,p)w_{4}(n,-\xi km+p)\ ,$
(188) $\displaystyle h_{4}(m,-\xi kn+p)w_{4}(n,p)=h_{4}(n,-\xi
km+p)w_{4}(m,p)\ ,$ $\displaystyle\bar{h}_{4}(m,-\xi
kn+p)w_{4}(n,p)=\bar{h}_{4}(n,-\xi km+p)w_{4}(m,p)\ .$

  5. 5.

Jacobi identity for ${R}_{m},\bar{R}_{n}$ and ${\mathcal{L}}_{p}$ leads to

$\displaystyle w_{3}(n,p)w_{1}(m,kn+p)=w_{1}(m,p)w_{3}(n,cm+p)\ ,$ (189)
$\displaystyle h_{1}(m,kn+p)w_{3}(n,p)=h_{3}(n,cm+p)w_{1}(m,p)\ ,$
$\displaystyle\bar{h}_{1}(m,kn+p)w_{3}(n,p)=\bar{h}_{3}(n,cm+p)w_{1}(m,p)\ .$

  6. 6.

Jacobi identity for ${R}_{m},\bar{R}_{n}$ and $\bar{\mathcal{L}}_{p}$ leads to

$\displaystyle w_{4}(n,p)w_{2}(m,-\xi kn+p)=w_{2}(m,p)w_{4}(n,-\xi cm+p)\ ,$
(190) $\displaystyle h_{2}(m,-\xi kn+p)w_{4}(n,p)=h_{4}(n,-\xi
cm+p)w_{2}(m,p)\ ,$ $\displaystyle\bar{h}_{2}(m,-\xi
kn+p)w_{4}(n,p)=\bar{h}_{4}(n,-\xi cm+p)w_{2}(m,p)\ .$

Finally, we have another family of Jacobi identities involving two
superrotation generator and one $R$-supercurrent generator. The Jacobi
identities for $\mathcal{L}_{m},\bar{\mathcal{L}}_{n},R_{n}$ and
$\mathcal{L}_{m},\bar{\mathcal{L}}_{n},\bar{R}_{n}$ are trivially satisfied.
We list down the equations that follow from the non-trivial Jacobi identities
systematically

  1. 1.

Jacobi identity for $\mathcal{L}_{m},\mathcal{L}_{n}$ and $R_{p}$ implies

$\displaystyle(m-n)w_{1}(p,m+n)+w_{1}(p,m)(n-cp-m)-w_{1}(p,n)(m-cp-n)$ (191)
$\displaystyle+h_{1}(p,n)w_{1}(p+cn,m)-h_{1}(p,m)w_{1}(p+cm,n)$
$\displaystyle+\bar{h}_{1}(p,n)w_{3}(kcp+kn,m)-\bar{h}_{1}(p,m)w_{3}(kcp+km,n)=0\
,$ $\displaystyle(m-n)h_{1}(p,m+n)+$ $\displaystyle
h_{1}(p,n)h_{1}(p+cn,m)-h_{1}(p,m)h_{1}(p+cm,n)=$ (192)
$\displaystyle\bar{h}_{1}(p,m)h_{3}(kcp+km,n)-\bar{h}_{1}(p,n)h_{3}(kcp+kn,m)\
,$ $\displaystyle(m-n)\bar{h}_{1}(p,m+n)+$ $\displaystyle
h_{1}(p,n)\bar{h}_{1}(p+cn,m)-h_{1}(p,m)\bar{h}_{1}(p+cm,n)=$ (193)
$\displaystyle\bar{h}_{1}(p,m)\bar{h}_{3}(kcp+km,n)-\bar{h}_{1}(p,n)\bar{h}_{3}(kcp+kn,m)\
.$

  2. 2.

Jacobi identity for $\mathcal{L}_{m},\mathcal{L}_{n}$ and $\bar{R}_{p}$
implies

$\displaystyle(m-n)w_{3}(p,m+n)+w_{3}(p,m)(n-kp-m)-w_{3}(p,n)(m-kp-n)$ (194)
$\displaystyle+h_{3}(p,n)w_{1}(kcp+cn,m)-h_{3}(p,m)w_{1}(kcp+cm,n)$
$\displaystyle+\bar{h}_{3}(p,n)w_{3}(p+kn,m)-\bar{h}_{3}(p,m)w_{3}(p+km,n)=0\
,$ $\displaystyle(m-n)h_{3}(p,m+n)+$ $\displaystyle
h_{3}(p,n)h_{1}(kcp+cn,m)-h_{3}(p,m)h_{1}(kcp+cm,n)=$ (195)
$\displaystyle\bar{h}_{3}(p,m)h_{3}(p+km,n)-\bar{h}_{3}(p,n)h_{3}(p+kn,m)\ ,$
$\displaystyle(m-n)\bar{h}_{3}(p,m+n)+$ $\displaystyle
h_{3}(p,n)\bar{h}_{1}(kcp+cn,m)-h_{3}(p,m)\bar{h}_{1}(kcp+cm,n)=$ (196)
$\displaystyle\bar{h}_{3}(p,m)\bar{h}_{3}(p+km,n)-\bar{h}_{3}(p,n)\bar{h}_{3}(p+kn,m)\
.$

  3. 3.

Jacobi identity for $\bar{\mathcal{L}}_{m},\bar{\mathcal{L}}_{n}$ and
${R}_{p}$ implies

$\displaystyle(m-n)w_{2}(p,m+n)+w_{2}(p,m)(n+\xi cp-m)-w_{2}(p,n)(m+\xi cp-n)$
(197) $\displaystyle+h_{2}(p,n)w_{2}(p-\xi cn,m)-h_{2}(p,m)w_{2}(p-\xi cm,n)$
$\displaystyle+\bar{h}_{2}(p,n)w_{4}(kcp-\xi
kn,m)-\bar{h}_{2}(p,m)w_{4}(kcp-\xi km,n)=0\ ,$
$\displaystyle(m-n)h_{2}(p,m+n)+$ $\displaystyle h_{2}(p,n)h_{2}(p-\xi
cn,m)-h_{2}(p,m)h_{2}(p-\xi cm,n)=$ (198)
$\displaystyle\bar{h}_{2}(p,m)h_{4}(kcp-\xi
km,n)-\bar{h}_{2}(p,n)h_{4}(kcp-\xi kn,m)\ ,$
$\displaystyle(m-n)\bar{h}_{2}(p,m+n)+$ $\displaystyle
h_{2}(p,n)\bar{h}_{2}(p-\xi cn,m)-h_{2}(p,m)\bar{h}_{2}(p-\xi cm,n)=$ (199)
$\displaystyle\bar{h}_{2}(p,m)\bar{h}_{4}(kcp-\xi
km,n)-\bar{h}_{2}(p,n)\bar{h}_{4}(kcp-\xi kn,m)\ .$

  4. 4.

Jacobi identity for $\bar{\mathcal{L}}_{m},\bar{\mathcal{L}}_{n}$ and
$\bar{R}_{p}$ implies

$\displaystyle(m-n)w_{4}(p,m+n)+w_{4}(p,m)(n-\xi kp-m)-w_{4}(p,n)(m-\xi kp-n)$
(200) $\displaystyle+h_{4}(p,n)w_{2}(kcp-\xi cn,m)-h_{4}(p,m)w_{2}(kcp-\xi
cm,n)$ $\displaystyle+\bar{h}_{4}(p,n)w_{4}(p-\xi
kn,m)-\bar{h}_{4}(p,m)w_{4}(p-\xi km,n)=0\ ,$
$\displaystyle(m-n)h_{4}(p,m+n)+$ $\displaystyle h_{4}(p,n)h_{2}(kcp-\xi
cn,m)-h_{4}(p,m)h_{2}(kcp-\xi cm,n)=$ (201)
$\displaystyle\bar{h}_{4}(p,m)h_{4}(p-\xi km,n)-\bar{h}_{4}(p,n)h_{4}(p-\xi
kn,m)\ ,$ $\displaystyle(m-n)\bar{h}_{4}(p,m+n)+$ $\displaystyle
h_{4}(p,n)\bar{h}_{2}(kcp-\xi cn,m)-h_{4}(p,m)\bar{h}_{2}(kcp-\xi cm,n)=$
(202) $\displaystyle\bar{h}_{4}(p,m)\bar{h}_{4}(p-\xi
km,n)-\bar{h}_{4}(p,n)\bar{h}_{4}(p-\xi kn,m)\ .$

###  5.1 Algebra with linear structure constants

In our analysis so far, the structure constants have been kept arbitrary.
However the above equations are severely constraining in determining the form
of the structure constants appearing above and solving them generally seems
quite difficult. Hence, at this point, we make two powerful simplifying
assumptions, which will somewhat reduce the complexity of the equations above.
The assumptions are

  * •

The structure constants appearing in the proposed algebra are linear in its
arguments.

  * •

The global sub-sector of $W(-1/2,-1/2;-1/2,-1/2)$ algebra must coincide with
the $R$-extended $\mathcal{N}=2$ super-Poincarë algebra.

This leads us to propose an ansatz of the form

$\mu_{i}(n,p)=\omega_{i0}+\omega_{i1}n+\omega_{i2}p\ \quad\text{(for
$i=1,2,3,4)$}\ ,$ (203)

where $\mu_{i}$ denotes the structure constants $w(n,p),h(n,p)$ or
$\bar{h}(n,p)$. Since, for the global subsector we must have
$[R_{0},\mathcal{L}_{m}]=[\bar{R}_{0},\mathcal{L}_{m}]=[R_{0},\bar{\mathcal{L}}_{m}]=[\bar{R}_{0},\bar{\mathcal{L}}_{m}]=0$,
it implies that $\mu_{i}(0,0)=\mu_{i}(0,1)=\mu_{i}(0,-1)=0$. This along with
the proposed ansatz leads us to conclude that these structure constants must
be of the form

$\mu_{i}(n,p)=\omega n\ .$ (204)

where $\omega$ is an arbitrary constant. We also note that the first equation
of each of the set (185), (186), (187), (188) now becomes trivial.

The global sector of the $R$-supercurrent i.e. $R_{0}$ and $\bar{R}_{0}$ is
supposed to commute with the generators of the Poincarè algebra
$M_{\mu\nu},P_{\mu}$ while, its commutator with the SUSY generators
$Q^{i}_{\pm\frac{1}{2}},\bar{Q}^{i}_{\pm\frac{1}{2}}$ must be nonzero.
Specifically, imposing this constraint on the translation generators
$P_{\mu}$, and using the identification (119)-(122), we get,

$[T_{0,0},R_{0}]=[T_{0,1},R_{0}]=[T_{1,0},R_{0}]=[T_{1,1},R_{0}]=0\ .$ (205)

An identical set of relations is true for $\bar{R}_{0}$. The above implies
that the structure constants are related by

$\bar{\beta}\left(+\frac{1}{2},0\right)=\bar{\beta}\left(-\frac{1}{2},0\right)=-{\beta}\left(+\frac{1}{2},0\right)=-\beta\left(-\frac{1}{2},0\right)\neq
0\ .$ (206)

Now, assuming that both $\beta(r,n)$ and $\bar{\beta}(r,n)$ are linear in its
argument, the above relation immediately implies that they must be of the form

$\beta(r,n)=\beta_{1}n+\beta_{0}\quad\text{and}\quad\bar{\beta}(r,n)=\bar{\beta}_{1}n-{\beta}_{0}\
,$ (207)

where $\beta_{1},\beta_{2},\bar{\beta}_{1}$ and $\bar{\beta}_{2}$ are
constants. One can repeat the same exercise with the structure constants
$\kappa(r,n)$ and $\bar{\kappa}(r,n)$ to see that they are also independent of
the first index $r$ and hence can be written as

$\kappa(r,n)=\kappa_{1}n+\kappa_{0}\quad\text{and}\quad\bar{\kappa}(r,n)=\bar{\kappa}_{1}n-{\kappa}_{0}\
.$ (208)

One immediate consequence of the above is that $\eqref{eq:RRbarQJacobi}$ and
$\eqref{eq:RRbarQbarJacobi}$ is now trivially satisfied. With all of this
conclusions, we have a more simplified algebra, given by,

$\displaystyle[Q_{r}^{i},R_{n}]=\beta(n)Q^{i}_{cn+r}\ ,\
[\bar{Q}_{r}^{i},R_{n}]=\bar{\beta}(n)\bar{Q}^{i}_{\xi{c}n+r}\ ,$ (209)
$\displaystyle{[Q_{r}^{i},\bar{R}_{n}]}=\kappa(n)Q^{i}_{kn+r}\ ,\
[\bar{Q}_{r}^{i},\bar{R}_{n}]=\bar{\kappa}(n)\bar{Q}^{i}_{\xi{k}n+r}\ ,$
$\displaystyle{[T_{p,q},R_{n}]}=\bar{\beta}\left(n\right)T_{p,-\xi{c}n+q}+\beta\left(n\right)T_{cn+p,q}\
,$
$\displaystyle{[T_{p,q},\bar{R}_{n}]}=\bar{\kappa}\left(n\right)T_{p,-\xi{k}n+q}+\kappa\left(n\right)T_{kn+p,q}\
,$
$\displaystyle{[R_{n},\mathcal{L}_{m}]}=w_{1}(n)\mathcal{L}_{cn+m}+h_{1}(n)R_{n+cm}+\bar{h}_{1}(n)\bar{R}_{kcn+km}\
,$ $\displaystyle[R_{n},\bar{\mathcal{L}}_{m}]=w_{2}(n)\bar{\mathcal{L}}_{-\xi
cn+m}+h_{2}(n)R_{n-\xi cm}+\bar{h}_{2}(n)\bar{R}_{kcn-\xi km}\ ,$
$\displaystyle[\bar{R}_{n},\mathcal{L}_{m}]=w_{3}(n)\mathcal{L}_{kn+m}+h_{3}(n)R_{kcn+cm}+\bar{h}_{3}(n)\bar{R}_{n+km}\
,$
$\displaystyle[\bar{R}_{n},\bar{\mathcal{L}}_{m}]=w_{4}(n)\bar{\mathcal{L}}_{-\xi
kn+m}+h_{4}(n)R_{kcn-\xi cm}+\bar{h}_{4}(n)\bar{R}_{n-\xi km}\ .$

Now, we will focus on (176) specifically which yields two equations given by

$\displaystyle
w_{1}(n)(a+bcn+bm+p)-h_{1}(n)\beta\left(n+cm\right)-\bar{h}_{1}(n)\kappa\left(ckn+km\right)+cn\beta\left(n\right)=0\
,$ (210) $\displaystyle-
h_{1}(n)\bar{\beta}\left(n+cm\right)-\bar{h}_{1}(n)\bar{\kappa}\left(ckn+km\right)=0\
.$ (211)

The first equation written above must be true for arbitrary integral values of
$m,n$ and $p$. However, it contains a term of the form $w_{1}(n)p$ which must
vanish, implying that $w_{1}(n)=0$ identically. A similar argument can be
applied to the Jacobi identities (178), (180) and (182) to reach the
conclusion that $w_{2}=w_{3}=w_{4}=0$ too. Plugging in the linear forms of
$h_{1}(n),\bar{h}_{1}(n)$ and $\beta(n)$ in the equations (210) and (211), we
get,

$\displaystyle c\beta_{1}-\omega_{1}\beta_{1}-ck\kappa_{1}\bar{\omega}_{1}$
$\displaystyle=0\ ,$ (212)
$\displaystyle-c\omega_{1}\beta_{1}-k\kappa_{1}\bar{\omega}_{1}$
$\displaystyle=0\ ,$ $\displaystyle
c\beta_{0}-\omega_{1}\beta_{0}-\kappa_{0}\bar{\omega}_{1}$ $\displaystyle=0\
.$

The above set of equations imply $\beta_{1}=0$. Applying the exact same
argument to (178), (180) and (182), yields,
$\bar{\beta}_{1}=\kappa_{1}=\bar{\kappa}_{1}=0$. Essentially, this shows
irrespective of the arguments, $\beta,\bar{\beta},\kappa$ and $\bar{\kappa}$
are constants and related as $\beta=-\bar{\beta}$ and $\kappa=-\bar{\kappa}$.
However, (210) and (211) reduces to simply

$-\omega_{1}\beta_{0}-\bar{\omega}_{1}\kappa_{0}+c\beta_{0}=0\quad\text{and}\quad\omega_{1}\beta_{0}+\bar{\omega}_{1}\kappa_{0}=0\
,$ (213)

which leads us to conclude that the structure constant $\beta$ must be
identically zero which is clearly in contradiction with (206). (178), (180).
Similarly (182) leads to the conclusion that $\bar{\beta},\kappa$ and
$\bar{\kappa}$ must also be vanishing. Thus we essentially find that an
infinite dimensional extension of $R$-charges in
$\mathcal{N}=2\,\mathfrak{bms}_{4}$ algebra is impossible with linear
structure constants. Therefore we conclude that a generic $\mathcal{N}=2$
$W(a,b;\bar{a},\bar{b})$ algebra (143) of section 4 can not have infinite
$R-$extension with linear structure constants .

###  5.2 Algebra with non-linear structure constants

We now relax the criteria of the linearity of structure constants and assume
further that they can be at best quadratic in the arguments. Continuing to
refer to the structure constants $w_{i},h_{i}$ and $\bar{h}_{i}$ as $\mu_{i}$,
we write down an ansatz of the form

$\mu_{i}(n,p)=\omega_{i0}+\omega_{i1}n+\omega_{i2}p+\omega_{i3}np+\omega_{i4}n^{2}+\omega_{i5}p^{2}\
.$ (214)

Since, $\mu_{i}(0,p)$ must vanish for $p=0,\pm 1$, we can write

$\mu_{i}(n,p)=n(\omega_{i1}+\omega_{i3}p+\omega_{i4}n)\ .$ (215)

Demanding (206), we see that $\beta$ and $\kappa$ (and similarly $\bar{\beta}$
and $\bar{\kappa}$) must take the form

$\displaystyle\beta(r,n)$ $\displaystyle=$
$\displaystyle\beta_{0}+\beta_{2}n+\beta_{3}r^{2}+\beta_{4}n^{2}+\beta_{5}rn\
,$ (216) $\displaystyle\kappa(r,n)$ $\displaystyle=$
$\displaystyle\kappa_{0}+\kappa_{2}n+\kappa_{3}r^{2}+\kappa_{4}n^{2}+\kappa_{5}rn$
(217)

Unlike the linear case, (153) and (154) are not trivially satisfied now.
Focussing specifically on (154), we recover the following equations

$\displaystyle\beta_{0}\kappa_{3}=\beta_{3}\kappa_{0}=\beta_{2}\kappa_{3}=\beta_{3}\kappa_{2}=\beta_{3}\kappa_{4}=\beta_{4}\kappa_{3}=0\
,$ (218)
$\displaystyle\beta_{5}\kappa_{2}=\beta_{2}\kappa_{5}=\beta_{5}\kappa_{4}=\beta_{4}\kappa_{5}=\beta_{3}\kappa_{3}=\beta_{5}\kappa_{5}=0\
,$ $\displaystyle k\beta_{5}\kappa_{3}=c\beta_{3}\kappa_{5}\ ,\
k\beta_{5}\kappa_{0}=c\beta_{0}\kappa_{5}\ ,$ $\displaystyle
k\beta_{3}\kappa_{3}+2\beta_{3}\kappa_{5}=c\beta_{3}\kappa_{3}+2\beta_{5}\kappa_{3}=c\beta_{5}\kappa_{3}+2\beta_{4}\kappa_{3}=k\beta_{3}\kappa_{5}+2\beta_{3}\kappa_{4}=0\
.$

We will get an analogous set of equations following from (154). The above set
of equations implies

$\beta_{3}=\kappa_{3}=\beta_{5}=\kappa_{5}=0\ ,$ (219)

thus making the structure constants $\beta(r,n)$ (and $\bar{\beta}(r,n)$) and
$\kappa(r,n)$ (and $\bar{\kappa}(r,n)$) independent of $r$. Like before, (206)
implies

$\beta_{0}=-\bar{\beta}_{0}\quad(\text{and similarly}\
\kappa_{0}=-\bar{\kappa}_{0})$ (220)

Now, armed with the above results from (176) we see we again recover equations
very close to that of (210) and (211) which again leads us to conclude
$w_{i}=0$ identically. Thus, in this case, the two equations following from
(176) simplifies to

$\displaystyle-
h_{1}(n,m)\beta(n+cm)-\bar{h}_{1}(n,m)\kappa(kcn+km)+cn\beta(n)$
$\displaystyle=$ $\displaystyle 0\ ,$ (221) $\displaystyle-
h_{1}(n,m)\bar{\beta}(n+cm)-\bar{h}_{1}(n,m)\bar{\kappa}(kcn+km)$
$\displaystyle=$ $\displaystyle 0\ .$ (222)

Plugging in the forms of the ansatze for
$h_{1}(n,m),\bar{h}_{1}(n,m),\beta(n)$ and $\kappa(n)$, in (221) and equating
the various coefficients of gives

$\displaystyle c\beta_{0}-\omega_{1}\beta_{0}-\bar{\omega}_{1}\kappa_{0}$
$\displaystyle=$ $\displaystyle 0\ ,$ (223)
$\displaystyle\omega_{3}\beta_{0}+c\omega_{1}\beta_{2}+\bar{\omega}_{3}\kappa_{0}+k\bar{\omega}_{1}\kappa_{2}$
$\displaystyle=$ $\displaystyle 0\ ,$ (224)
$\displaystyle\omega_{4}\beta_{0}-c\beta_{2}+\omega_{1}\beta_{2}+\bar{\omega}_{4}\kappa_{0}+ck\bar{\omega}_{1}\kappa_{2}$
$\displaystyle=$ $\displaystyle 0\ ,$ (225)
$\displaystyle\omega_{4}\beta_{2}-c\beta_{4}+\omega_{1}\beta_{4}+ck\bar{\omega}_{4}\kappa_{2}+\bar{\omega}_{1}\kappa_{4}$
$\displaystyle=$ $\displaystyle 0\ ,$ (226) $\displaystyle
c\omega_{3}\beta_{2}+\omega_{1}\beta_{4}+k\bar{\omega}_{3}\kappa_{2}+\bar{\omega}_{1}\kappa_{4}$
$\displaystyle=$ $\displaystyle 0\ ,$ (227)
$\displaystyle\omega_{3}\beta_{2}+c\omega_{4}\beta_{2}+2c\omega_{1}\beta_{4}+ck\bar{\omega}_{3}\kappa_{2}+k\bar{\omega}_{4}\kappa_{2}+2c\bar{\omega}_{1}\kappa_{4}$
$\displaystyle=$ $\displaystyle 0\ ,$ (228)
$\displaystyle\omega_{4}\beta_{4}+\bar{\omega}_{4}\kappa_{4}$ $\displaystyle=$
$\displaystyle 0\ ,$ (229)
$\displaystyle\omega_{3}\beta_{4}+\bar{\omega}_{3}\kappa_{4}$ $\displaystyle=$
$\displaystyle 0\ ,$ (230) $\displaystyle
2c\omega_{3}\beta_{4}+\omega_{4}\beta_{4}+2c\bar{\omega}_{3}\kappa_{4}+\bar{\omega}_{4}\kappa_{4}$
$\displaystyle=$ $\displaystyle 0\ ,$ (231)
$\displaystyle\omega_{3}\beta_{4}+2c\omega_{4}\beta_{4}+\bar{\omega}_{3}\kappa_{4}+2c\bar{\omega}_{4}\kappa_{4}$
$\displaystyle=$ $\displaystyle 0\ .$ (232)

The same exercise with (222) gives the following equations

$\displaystyle\omega_{1}\beta_{0}+\bar{\omega}_{1}\kappa_{0}$ $\displaystyle=$
$\displaystyle 0\ ,$ (233)
$\displaystyle\omega_{4}\beta_{0}-\omega_{1}\bar{\beta}_{1}+\bar{\omega}_{4}\kappa_{0}-ck\bar{\omega}_{1}\bar{\kappa}_{2}$
$\displaystyle=$ $\displaystyle 0\ ,$ (234)
$\displaystyle\omega_{3}\beta_{0}-c\omega_{1}\bar{\beta}_{1}+\bar{\omega}_{3}\kappa_{0}-k\bar{\omega}_{1}\bar{\kappa}_{2}$
$\displaystyle=$ $\displaystyle 0\ ,$ (235)
$\displaystyle\omega_{1}\bar{\beta}_{4}+\omega_{4}\bar{\beta}_{1}+ck\bar{\omega}_{4}\bar{\kappa}_{2}+\bar{\omega}_{1}\bar{\kappa}_{4}$
$\displaystyle=$ $\displaystyle 0\ ,$ (236)
$\displaystyle\omega_{1}\bar{\beta}_{4}+c\omega_{3}\bar{\beta}_{1}+k\bar{\omega}_{3}\bar{\kappa}_{2}+\bar{\omega}_{1}\bar{\kappa}_{4}$
$\displaystyle=$ $\displaystyle 0\ ,$ (237) $\displaystyle
2c\omega_{1}\bar{\beta}_{4}+\omega_{3}\bar{\beta}_{1}+c\omega_{4}\bar{\beta}_{1}+ck\bar{\omega}_{3}\bar{\kappa}_{2}+k\bar{\omega}_{4}\bar{\kappa}_{2}+2c\bar{\omega}_{1}\bar{\kappa}_{4}$
$\displaystyle=$ $\displaystyle 0\ ,$ (238)
$\displaystyle\omega_{4}\bar{\beta}_{4}+\bar{\omega}_{4}\bar{\kappa}_{4}$
$\displaystyle=$ $\displaystyle 0\ ,$ (239)
$\displaystyle\omega_{3}\bar{\beta}_{4}+\bar{\omega}_{3}\bar{\kappa}_{4}$
$\displaystyle=$ $\displaystyle 0\ ,$ (240) $\displaystyle
2c\omega_{3}\bar{\beta}_{4}+\omega_{4}\bar{\beta}_{4}+2c\bar{\omega}_{3}\bar{\kappa}_{4}+\bar{\omega}_{4}\bar{\kappa}_{4}$
$\displaystyle=$ $\displaystyle 0\ ,$ (241)
$\displaystyle\omega_{3}\bar{\beta}_{4}+2c\omega_{4}\bar{\beta}_{4}+\bar{\omega}_{3}\bar{\kappa}_{4}+2c\bar{\omega}_{4}\bar{\kappa}_{4}$
$\displaystyle=$ $\displaystyle 0\ .$ (242)

The above system of equations are highly non-trivial to solve in general. But
from (223) and (233) we easily see that

$\beta_{0}=0\ ,$ (243)

which implies $\beta(0)=0$ and similarly we find
$\bar{\beta}(0)={\kappa}(0)=\bar{\kappa}(0)=0$. This is in contradiction with
the constraint from the global subalgebra (206). Thus, akin to the case of
linear structure constants, even for quadratic structure constants, an
extension by $U(1)_{V}\times U(1)_{A}$ realised as infinitely many generators
$R_{n}$ and $\bar{R}_{n}$ ($R$-symmetry generators) of
$W(a,b,\bar{a},\bar{b})$ algebra turns out to be _impossible_.

##  6 Conclusion

As mentioned earlier, symmetry algebras are powerful tools which severely
constrain the dynamics and vacua of gauge and gravity theories. In this work,
we have concentrated on supersymmetric $W(a,b)$ and $W(a,b;\bar{a},\bar{b})$
algebras which are deformations of the asymptotic symmetry algebra of
supergravity theories in three and four spacetime dimensions respectively.
Earlier works Parsa:2018kys ; Safari:2019zmc  has established that generic
deformations of $\mathfrak{bms}_{3}$ algebras involve two parameters $a$ and
$b$ while generic deformations of $\mathfrak{bms}_{4}$ algebras involve
$a,b,\bar{a}$ and $\bar{b}$. The $a=0,b=-1$ centerless $R$-extension of
supersymmetric $W(a,b)$ algebra, given by (2.2) indeed matches with earlier
results of Banerjee:2019lrv  where the authors performed an asymptotic
symmetry analysis to obtain the super-$\mathfrak{bms}_{3}$ algebra. We also
classified and wrote down possible central extension of supersymmetric,
$R$-extended $W(a,b)$ algebra. We observed interesting and novel central
charges appearing in the $\\{Q^{1}_{r},Q^{2}_{s}\\}$ anti-commutator, denoted
by $f(r,s)$ for various values of $a$ and $b$. We also find that
$[J_{m},\mathcal{R}_{n}]$ admits a quadratic central charge which has not been
realized through the asymptotic symmetry analysis performed in
Banerjee:2019lrv  although it seems to be present for arbitrary values of $a$
and $b$. From the analysis of Banerjee:2019lrv , it is certain that such a
central term would not arise with the usual Barnich-Compere boundary
conditions. Nevertheless, the study of Afshar:2015wjm  clearly indicates that
with an asymptotically Rindler like behavior will modify the asymptotic
symmetry algebra with such central extensions. Thus it remains an interesting
open problem to find the importance of this central term in the context of
three dimensional asymptotically flat supergravity theory in more generic
contexts. As mentioned earlier, the $W(0,0)$ and $W(0,1)$ algebras have also
appeared as asymptotic symmetry algebras of gravity theories Compere:2013bya ;
Afshar:2021qvi , so it is worth exploring appropriate boundary/falloff
conditions to obtain supersymmetric $W(0,0)$ and $W(0,1)$ algebras as
asymptotic symmetry algebras in some supergravity theory. To conclude, the
analysis of the present work, being mathematically rigorous, provides new
asymptotic algebras and hence opens up the possibilities of finding new
boundary conditions for supergravity fields. We hope to report on these
possibilities in future works.

The construction of the $R$-extended supersymmetric $W(a,b;\bar{a},\bar{b})$
algebra turned out to be more involved. Physically, the $R$-charge generators
are supposed to rotate the global SUSY-generators which motivates us to
propose an ansatz of the form (146). To be mathematically precise, we
essentially tried to extend the super-$W(a,b;\bar{a},\bar{b})$ algebra
(written explicitly in (143)) by a $U(1)_{V}\times U(1)_{A}$ group where each
sector is represented by infinitely many generators. One of the sectors of the
$U(1)_{V}$ symmetry can be thought of as _vectorial_ $R$-symmetry while the
other copy of the $U(1)_{A}$ symmetry can be thought of as _axial_
$R$-symmetry. We considered two primary guiding principles to fix the algebra

  * •

For $a=b=\bar{a}=\bar{b}=-\frac{1}{2}$, the global subalgebra must be
identical to the $R$-extended super-Poincaré algebra.

  * •

The indices appearing in all the proposed commutators involving the
$R$-charges must be linear in its arguments.

In order to simplify our calculations, we considered linear as well as
quadratic structure constants. In either cases, we realized that having
infinitely many $R$-charges is in contradiction with one or more Jacobi
identities that must be followed by such a graded Lie algebra. Essentially, it
seems there is an obstruction in the $u(1)\times u(1)$ extension of
super-$W(a,b;\bar{a},\bar{b})$ algebra–which will naturally hinder the
construction of an $R$-extended super-$\mathfrak{bms}_{4}$ algebra. Recent
work Banerjee:2021uxe  has carried out $u(1)$ and $u(N)$ extensions of
$\mathfrak{bms}_{4}$ by analysing celestial amplitudes of Einstein-Maxwell and
Einstein-Yang-Mills theories and have indeed obtained non-trivial asymptotic
symmetry algebras at the boundary which does include infinitely many
generators parametrizing the $u(1)$ or $u(N)$ symmetry. This is however, not
in contradiction with our results. Our demand on the behaviour of $R$-charges
i.e. it must non-trivially rotate the global SUSY generators forces us to
demand (206) which ensures the $[Q^{i}_{r},R_{m}]$ commutator to be non-zero
for the global sector. Such a constraint need not be followed for the $u(1)$
or $u(N)$ gauge groups that enter the analysis of Banerjee:2021uxe . Relaxing
(206) in our current work does indeed recover the symmetry algebras derived in
Banerjee:2021uxe . Finally, the methodology of our construction by throughly
analysing all possible Jacobi identities along with imposing consistency with
the global subsector is quite general. It is possible to adapt this algorithm
to construct $u(1)$ or $u(N)$ extension for other exotic symmetry algebras. It
is worth emphasizing that although we have studied $R$-extended super-$W(a,b)$
and super-$W(a,b;\bar{a},\bar{b})$ algebras in this papers, it is only for
specific values of $a$ and $b$, we are aware of physical theories of gravity
or supergravity where these are realized as boundary symmetry algebras. It
will be interesting to explore what kind of supergravity theories give rise to
these wide range of $W$-algebras for more generic values of $a,b,\bar{a}$ and
$\bar{b}$.

We would also like to mention that the construction of the present paper is
not exhaustive. Cases that have not been considered here are technically
difficult ones to address and there is no other clarifications/reasoning to
not consider them. It will be good to find an exhaustive construction of all
possible supersymmetrization of deformed $\mathfrak{bms}$ algebras and we hope
to report on them in future.

Let us conclude the paper with the importance of the study of supersymmetric
extensions of the deformations of $\mathfrak{bms}$ algebras. As is well
understood, $\mathfrak{bms}$ algebras are symmetries of asymptotically flat
gravity theories at their null boundaries. Some of their deformations have
also been realized as the symmetry algebra at the horizon of certain black
hole backgrounds. In the context of three spacetime dimensions, the presence
of extended supersymmetries and internal $R-$symmetries plays crucial roles in
characterizing the soft hair modes (that gives non trivial cosmological
solutions) and their thermodynamics Fuentealba:2017fck ; Afshar:2013vka ;
Barnich:2012xq ; Banerjee:2018hbl . The similar study has not been performed
for four space time dimensions, where non trivial black hole and gravitational
wave solutions exist. In the context of $\mathfrak{bms}_{4}$, the soft hair
modes contributes to Black Hole entropy, albeit they do not correspond to the
entire microscopic degeneracy. The microscopic degeneracy for a class of four
dimensional $\mathcal{N}=2,4,8$ supersymmetric BPS black holes are very well
understood David:2006ji ; David:2006yn ; David:2006ru ; David:2006ud ;
Banerjee:2008ky ; Sen:2009md . It would be interesting to understand how much
of this entropy is contributed by the soft hairs. Our present results suggest
that, for Black Holes appearing in $\mathcal{N}=2$ spergravity theory, where
the internal R-symmetry (that only scales the supercharges) will not have any
contributions to the soft hairs. The similar study for other supergravity
theories with exotic internal symmetries remain an open problem to study in
the future.

## Acknowledgements

We would like to thank M. M. Sheikh-Jabbari for collaboration and useful
discussions during the initial course of this work and for his comments on the
manuscript. DM would like to thank IISER Bhopal for kind hospitality during
the end of this project. The work of NB is partially supported by SERB ECR
grant ECR/2018/001255. The work of DM is supported in part by the SERB core
research grant CRG/2018/002373. Finally we thank the people of India for their
generous support to fundamental research. HRS acknowledges the support of
Saramadan grants ISEF/M/99407.

##  Appendix A Solution to structure constants $\sigma(m,n)$ and $\kappa(m,n)$
for $W(a,b)$ algebra

We briefly provide the details of solution for the system of equations as
given in (52)-(55). We will assume that the deformation parameter $b\neq 0$.
Clearly, from the first equation of (52), it follows $\sigma_{1}=0$. This
reduces the effective system of equations to the following:

$\displaystyle\sigma_{0}(\kappa_{0}-a)=0\ ,\ $ $\displaystyle\
\sigma_{2}(\kappa_{2}-b-1)=0\ ,$ (244)
$\displaystyle\sigma_{0}(\kappa_{2}-b)=0\ ,\ $ $\displaystyle\
\sigma_{2}(\kappa_{0}-a)=0\ .$ (245)

From the first equation of (244) we can conclude

$\text{either}\ \sigma_{0}=0\ \text{or}\ \kappa_{0}=a\ .$ (246)

In the former case when $\sigma_{0}=0$, we end up with two effective
equations, namely,

$\sigma_{2}(\kappa_{2}-b-1)=\sigma_{2}(\kappa_{0}-a)=0\ .$ (247)

For the case when $\sigma_{0}=\sigma_{1}=0$, a non-zero structure constant
$\sigma(m,n)$ implies $\sigma_{2}\neq 0$ but arbitrary while we must have

$\kappa_{0}=a\ \text{and}\ \kappa_{2}=b+1\ .$ (248)

For the other case, when $\kappa_{0}=a$, the effective system of equation is

$\sigma_{0}(\kappa_{2}-b)=\sigma_{2}(\kappa_{2}-b-1)=0\ .$ (249)

For a non-zero $\sigma_{0}$, we must have $\kappa_{2}=b$ and $\sigma_{2}=0$.
These cases exhaust all possible solutions to the system of equations
(52)-(55).

##  Appendix B Solution to index structure of the fermionic supercurrent
anticommutator

In order to fix the index structure as per the ansatz (128), the index
structure must satisfy the set of equations given by (130) in order to be
consistent with super-Poincaré algebra. We have eight equations governing six
quantities, namely $f_{i},g_{i}$ for $i=0,1,2$. Although this system is
overconstrained, we can indeed find a consistent set of solutions for the
above. Using the condition
$f\left(+\frac{1}{2},+\frac{1}{2}\right)=f\left(+\frac{1}{2},-\frac{1}{2}\right)=1$,
we end up with $f_{2}=0$ while the condition
$g\left(+\frac{1}{2},-\frac{1}{2}\right)=g\left(-\frac{1}{2},-\frac{1}{2}\right)=1$
gives $g_{1}=0$. The above four conditions essentially gives us two equations

$f_{0}+\frac{f_{1}}{2}=g_{0}-\frac{g_{2}}{2}=1\ .$ (250)

The two other conditions on $f(r,s)$ given by
$f\left(-\frac{1}{2},+\frac{1}{2}\right)=f\left(-\frac{1}{2},-\frac{1}{2}\right)=0$
results in a single equation

$f_{0}-\frac{f_{2}}{2}=0\ ,$ (251)

which finally yields

$f_{0}=\frac{1}{2}\quad\text{and}\quad f_{1}=1\ .$ (252)

The conditions
$g\left(+\frac{1}{2},+\frac{1}{2}\right)=g\left(-\frac{1}{2},+\frac{1}{2}\right)=0$
results in the equation

$g_{0}+\frac{g_{2}}{2}=0\ ,$ (253)

which finally gives

$g_{0}=\frac{1}{2}\quad\text{and}\quad g_{2}=-1\ .$ (254)

Thus, we see although it seems this might be an over constrained system of
equations, we can find a consistent solution set which we report in (131).

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