††institutetext: Institute of Mathematical Sciences,
Taramani, Chennai 600113, India
&
Homi Bhabha National Institute,
Anushakti Nagar, Mumbai 400094, India

#  All chiral ${\cal W}$-algebra extensions of $\mathfrak{so}(2,3)$

Nishant Gupta and Nemani V. Suryanarayana  nishantg<EMAIL_ADDRESS>
###### Abstract

We show that there are four chiral ${\cal W}$-algebra extensions of
$\mathfrak{so}(2,3)$ algebra and construct them explicitly. We do this by a
simple identification of each of the inequivalent embeddings of a copy of
$\mathfrak{sl}(2,{\mathbb{R}})$ in the $\mathfrak{so}(2,3)$ algebra and the
maximal subalgebra $\mathfrak{h}$ that commutes with it. Then using the
standard 2d chiral CFT techniques we find the corresponding ${\cal W}$-algebra
extensions. Two of the four resultant ${\cal W}$-algebras are new, one of
which may be thought of as the conformal $\mathfrak{bms}_{3}$ algebra valid
for finite values of its central charge.

##  1 Introduction

Symmetries play a crucial role in our understanding of quantum field theories.
More the symmetries, more the control over the physical quantities of that
theory. In the context of 2$d$ conformal field theories it has been well known
that the symmetry algebras are not only infinite dimensional but can also be
non-linear, leading to the study of ${\cal W}$ algebras in physics (starting
from Zamolodchikov:1985wn ; see, for instance, the reviews Bouwknegt:1992wg ;
Blumenhagen:2009zz ). In the context of higher (than two) dimensional
conformal field theories it is not generically expected that the symmetry
algebras are infinite dimensional. This general expectation is corroborated by
the fact that, at least for a CFTd with holographic duals, the asymptotic
symmetry algebra with Brown-Henneaux Brown:1986nw  boundary conditions for
$AdS_{d+1}$ gravity is $\mathfrak{so}(2,d)$ for any $d\geq 3$. However, for
the corresponding ${\mathbb{R}}^{1,d}$ at null-infinities the asymptotic
symmetry algebras are expected to be infinite dimensional – either the
$\mathfrak{bms}_{d+1}$ algebra Bondi:1962px ; Sachs:1962zza ; Sachs:1962wk  or
its various extended versions Barnich:2009se ; Barnich:2010eb ; Barnich:2011mi
; Campiglia:2014yka ; Campiglia:2015yka . These in turn are interpreted to be
the symmetries of the (Euclidean $(d-1)$-dimensional) so called celestial
CFTs. In particular for $d=3$ it has emerged Banerjee:2020zlg ;
Banerjee:2021dlm  that the symmetry algebra relevant includes a copy of chiral
extension of $\mathfrak{iso}(1,3)$ which can be thought of as a chiral
conformal algebra of a 2$d$ CFT. This chiral algebra consisted of the
following six chiral operators: one stress tensor $T(z)$, one
$\mathfrak{sl}(2,{\mathbb{R}})$ current $J_{a}(z)$ ($a=0,\pm 1$) and a doublet
$G_{s}(z)$ ($s=\pm 1/2$) that are conformal (with $h=3/2$) as well as current
algebra primaries (transforming in the doublet representation of
$\mathfrak{sl}(2,{\mathbb{R}})$).111This algebra can be realised as the
asymptotic symmetry algebra of 4$d$ Einstein gravity with an appropriate set
of asymptotically flat boundary conditions Gupta:2021cwo . Inspired to study
this algebra further, in a recent work Gupta:2022mdt  we sought and found the
most general chiral conformal algebra with this spectrum of operators. The
result was a chiral ${\cal W}$-algebra extension of $\mathfrak{so}(2,3)$
algebra. This novel chiral ${\cal W}$-algebra admits a contraction to the
chiral extension of $\mathfrak{iso}(1,3)$ of Banerjee:2020zlg ;
Banerjee:2021dlm  alluded to above.

Even though our motivations in Gupta:2022mdt  came from the holography of 4$d$
flat gravity the end result is expected to play a useful role in
$AdS_{4}/CFT_{3}$. Once we move from $\mathfrak{iso}(1,3)$ to
$\mathfrak{so}(2,3)$ it becomes important to ask if this is the only such
chiral ${\cal W}$-algebra extension of $\mathfrak{so}(2,3)$. In this work we
address the question – and show that the chiral ${\cal W}$-algebra found in
Gupta:2022mdt  is actually one of four such algebras. We identify the
remaining three explicitly completing the list.

We follow the following simple steps towards construction of all the chiral
${\cal W}$-algebra extension of $\mathfrak{so}(2,3)$.

  1. 1.

Identify all possible embeddings of a copy of $\mathfrak{sl}(2,{\mathbb{R}})$
inside $\mathfrak{so}(2,3)$ that are inequivalent upto the action of the group
$O(2,3)$.

  2. 2.

Then identify the maximal subalgbra $\mathfrak{h}\subset\mathfrak{so}(2,3)$
that commutes with the $\mathfrak{sl}(2,{\mathbb{R}})$ of the previous step.

  3. 3.

Arrange the rest of the generators into finite dimensional irreducible
representations of the subalgebras
$\mathfrak{sl}(2,{\mathbb{R}})\oplus\mathfrak{h}$.

  4. 4.

Postulate a quasi-primary chiral stress tensor $T(z)$ with central charge $c$
for the $\mathfrak{sl}(2,{\mathbb{R}})$ and a chiral current algebra of
$\mathfrak{h}$ of level $\kappa$.

  5. 5.

To the generators in step three associate chiral fields that are both
conformal primaries and the current algebra primaries of appropriate conformal
weights and current algebra representations.

  6. 6.

Use the $\mathfrak{sl}(2,{\mathbb{R}})\oplus\mathfrak{h}$ symmetries to fix
the OPEs among the additional chiral fields of the previous step. Finally
impose the Jacobi identities for the commutators of the modes of these
primaries.

We will see that there are four inequivalent copies of
$\mathfrak{sl}(2,{\mathbb{R}})$ embedded into $\mathfrak{so}(2,3)$. These have
$\mathfrak{h}$ as $\emptyset$, $\mathfrak{so}(1,1)$, $\mathfrak{so}(2)$ and
$\mathfrak{sl}(2,{\mathbb{R}})$. Each of these leads to, following the steps
spelt out above, a chiral ${\cal W}$-algebra extension. The resultant algebra
corresponding to the case with $\mathfrak{h}=\emptyset$ can be identified to
be the ${\cal W}(2,4)$ algebra found long ago Bouwknegt:1988sv . Another,
corresponding to $\mathfrak{h}=\mathfrak{sl}(2,{\mathbb{R}})$, is the one we
found recently in Gupta:2022mdt . We will complete this list by constructing
the remaining two, with $\mathfrak{h}=\mathfrak{so}(1,1)$ and
$\mathfrak{h}=\mathfrak{so}(2)$. The one with
$\mathfrak{h}=\mathfrak{so}(1,1)$ may be called the conformal
$\mathfrak{bms}_{3}$ whose semi-classical limit (that is, valid only for large
values of central charges $c$ and $\kappa$) has been obtained from a 3$d$
conformal gravity computation by Fuentealba:2020zkf .

The rest of this paper is organised as follows. In section 2 we provide a
simple classification of all the inequivalent embedding of
$\mathfrak{sl}(2,{\mathbb{R}})$ inside $\mathfrak{so}(2,3)$. This results in
four different ways to writing the algebra $\mathfrak{so}(2,3)$ which makes
the embedded $\mathfrak{sl}(2,{\mathbb{R}})$ manifest. In section 3 we provide
a derivation of two novel ${\cal W}$-algebra extension of $\mathfrak{so}(2,3)$
completing the list of such algebras along with the two already known from
Bouwknegt:1988sv  and Gupta:2022mdt . In section 4 we conclude with a
discussion and future directions. In Appendix A we provide a similar
derivation of chiral algebra extensions of $\mathfrak{so}(2,2)$ and show that
they are isomorphic to the Brown-Henneaux algebra Brown:1986nw  and the CIG
algebra of Polyakov Polyakov:1987zb  as realised in Avery:2013dja ;
Apolo:2014tua .

##  2 Embedding $\mathfrak{sl}(2,{\mathbb{R}})$ in $\mathfrak{so}(2,3)$

We are interested in promoting the algebra $\mathfrak{so}(2,3)$ to an infinite
dimensional one that contains at least one Virasoro algebra. Each such
Virasoro can in turn be thought of as an infinite dimensional extension of an
$\mathfrak{sl}(2,{\mathbb{R}})$ subalgebra of $\mathfrak{so}(2,3)$. Therefore
we would end up with a potentially distinct chiral ${\cal W}$-algebra
extension of $\mathfrak{so}(2,3)$ depending on which
$\mathfrak{sl}(2,{\mathbb{R}})\subset\mathfrak{so}(2,3)$ is promoted to a
Virasoro. Therefore the first question to answer is: what are all the
inequivalent embeddings of a copy of $\mathfrak{sl}(2,{\mathbb{R}})$ in the
$\mathfrak{so}(2,3)$ algebra?

This can be answered in any (faithful) matrix representation of the algebra
$\mathfrak{so}(2,3)$. We choose to work with the matrix representation of
$\mathfrak{so}(2,3)$ when the representation space is ${\mathbb{R}}^{2,3}$
(with the pseudo-Cartesian coordinates $(x_{0},x_{1},x_{2},x_{3},x_{4})$ where
$x^{0},x^{4}$ are time-like and the rest space-like) on which the
$\mathfrak{so}(2,3)$ algebra elements generate linear homogeneous
transformations $x^{\mu}\rightarrow{\Lambda^{\mu}}_{\nu}x^{\nu}$ that preserve
the line element $ds^{2}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}$, i.e., $\\{\Lambda\in
M_{5}({\mathbb{R}}):{\Lambda^{\mu}}_{\alpha}{\Lambda^{\nu}}_{\beta}\eta_{\mu\nu}=\eta_{\alpha\beta}~{}\&~{}{\rm
det}\,\Lambda=1\\}$ where $\eta_{\mu\nu}={\rm diag}(-1,1,1,1,-1)$.222Another
simple choice would have been to work with the $4\times 4$ real matrix
generators of the $\mathfrak{sp}(4,{\mathbb{R}})$ which is isomorphic to
$\mathfrak{so}(2,3)$. Here the algebra of $\mathfrak{so}(2,3)$ is realised in
terms of $5\times 5$ real matrices $M$ satisfying $M^{\rm T}\eta+\eta M=0$. We
will use the basis $L_{\mu\nu}$

$\displaystyle{(L_{\mu\nu})^{\alpha}}_{\beta}=\delta^{\alpha}_{\mu}\eta_{\nu\beta}-\delta^{\alpha}_{\nu}\eta_{\mu\beta},~{}~{}~{}~{}\mu,\nu,\alpha,\beta\cdots\in\\{0,1,2,3,4\\}$
(1)

satisfying

$\displaystyle[L_{\mu\nu},L_{\mu^{\prime}\nu^{\prime}}]=\eta_{\mu\nu^{\prime}}L_{\nu\mu^{\prime}}+\eta_{\nu\mu^{\prime}}L_{\mu\nu^{\prime}}-\eta_{\mu\mu^{\prime}}L_{\nu\nu^{\prime}}-\eta_{\nu\nu^{\prime}}L_{\mu\mu^{\prime}}\,.$
(2)

A general element in the Lie algebra is
$M=\frac{1}{2}\omega^{\mu\nu}L_{\mu\nu}$ for
$\omega^{\mu\nu}=-\omega^{\nu\mu}\in{\mathbb{R}}$, and in the space of $M$ we
would like to identify choices of triplets $(L_{0},L_{\pm 1})$ that form
$\mathfrak{sl}(2,{\mathbb{R}})$ algebra

$\displaystyle[L_{m},L_{n}]=(m-n)\,L_{m+n}\,.$ (3)

We want all such possible choices that are not equivalent under the action of
the group $O(2,3)$ which acts on $M$ as $M\rightarrow\Lambda^{-1}M\Lambda$
where the $5\times 5$ real matrices $\Lambda$ satisfy $\Lambda^{\\!\rm
T}\eta\Lambda=\eta$.333We do not require the $\det\Lambda=1$ condition to keep
$\mathfrak{so}(2,3)$ invariant under $M\rightarrow\Lambda^{-1}M\Lambda$. These
realisations of $L_{n}$ will therefore be a five dimensional representation of
$\mathfrak{sl}(2,{\mathbb{R}})$.

Before proceeding further let us recall some known facts about finite
dimensional representations of $\mathfrak{sl}(2,{\mathbb{R}})$. The
irreducible realisations of the $\mathfrak{sl}(2,{\mathbb{R}})$ are given in
terms of $(2j+1)\times(2j+1)$ dimensional real matrices, one for each
$j\in\frac{1}{2}{\mathbb{N}}$ (along with $j=0$, the singlet). Their matrix
elements may be written as:

$\displaystyle(L_{0})_{qp}$ $\displaystyle=$ $\displaystyle
p\,\delta_{qp},~{}~{}(L_{-1})_{qp}=h(j,p)(p-j)\delta_{q,p+1},$ (4)
$\displaystyle(L_{1})_{qp}$ $\displaystyle=$ $\displaystyle
h(j,q)^{-1}(p+j)\delta_{q+1,p}\,,~{}~{}~{}{\rm
for}~{}~{}~{}p,q\in\\{-j,-j+1,\cdots,j-1,j\\}$ (5)

for any arbitrary non-zero choice of $h(j,p)$ – which can be fixed by
appropriate actions of the group $GL(2,{\mathbb{R}})$. The simplest choices of
$h(j,p)$ include: (i) $h(j,p)=1$ (or any other non-zero real number), (ii)
$h(j,p)=\frac{1}{p-j}$, or $h(j,p)=p+j+1$ etc.444The choice
$h(j,p)=\sqrt{\frac{p+j+1}{p-j}}$ is the one related to the discrete series
unitary representations of $\mathfrak{sl}(2,{\mathbb{R}})$ when $j$ is
analytically continued using the replacement $-j\rightarrow h$ for positive
$h$ Jackiw:1990ka . However this choice of course does not keep $L_{\pm 1}$
real. In all these cases the $L_{0}$ is diagonal with rank $2j$ ($2j+1$) for
$2j$ is even (odd), and with real eigenvalues. Its eigenvalues range over
$-j,\cdots,j$. The only inputs we will take from here is that $L_{0}$ is
diagonalisable with real eigenvalues, whereas $L_{\pm 1}$ are expected to be
nilpotent.

Next we seek all possible candidates in the algebra $\mathfrak{so}(2,3)$ in
its vector representation (1) that can play the role of $L_{0}$. This requires
us to consider the class of all $M$ that admit only real eigenvalues. We can
take any such matrix $M$, using the freedom of acting with the group $O(2,3)$,
to be a linear combination of any two commuting boost generators
$(L_{0a},L_{b4})$ for $a\neq b$. There are six such choices which are all
equivalent to each other under $O(2,3)$. So we choose:

$\displaystyle L_{0}=\lambda\,L_{01}+\lambda^{\prime}\,L_{34}$ (7)

for $\lambda,\lambda^{\prime}\in{\mathbb{R}}$ with real eigenvalues
$(0,\pm\lambda,\pm\lambda^{\prime})$. One can independently make the exchanges
$\lambda\leftrightarrow-\lambda$,
$\lambda^{\prime}\leftrightarrow-\lambda^{\prime}$ and
$\lambda\leftrightarrow\lambda^{\prime}$ with appropriate $O(2,3)$
transformations.555These matrices are $\Lambda={\rm
diag}(-1,1,-1,1,1),~{}~{}{\rm
diag}(1,1,-1,1,-1),~{}~{}\Lambda_{mn}=\delta_{m+n}$ respectively. So we
further choose $\lambda\geq\lambda^{\prime}\geq 0$. Now taking this as the
general form of $L_{0}$ we can seek candidate $L_{\pm 1}$ such that they form
the algebra (3) of $\mathfrak{sl}(2,{\mathbb{R}})$. There are, depending on
the values of $(\lambda,\lambda^{\prime})$, further $O(2,3)$ transformations
that leave $L_{0}$ invariant and we can use these residual symmetries to put
the candidate $L_{\pm 1}$ into their simplest forms. This exercise can be
carried out systematically and straightforwardly. We find that, at the first
instance, for the existence of non-trivial $L_{\pm 1}$ with $[L_{0},L_{\pm
1}]=\mp L_{\pm 1}$ requires that $(\lambda,\lambda^{\prime})$ have to satisfy
one of the following conditions: (i) $\lambda=1$, (ii) $\lambda^{\prime}=1$,
(iii) $\lambda^{\prime}=1-\lambda$ and (iv) $\lambda^{\prime}=\lambda-1$. We
can then impose the last condition $[L_{1},L_{-1}]=2L_{0}$ in each of these
four cases. We simply list the results of this straightforward analysis:

  1. 1.

$\lambda=1$ leads to three sub-cases (i) $\lambda^{\prime}=1$, (ii)
$0<\lambda^{\prime}<1$ and (iii) $\lambda^{\prime}=0$.

    1. (a)

In the cases $\lambda^{\prime}=1$ and $0<\lambda^{\prime}<1$ there is no
solution for $L_{\pm 1}$.

    2. (b)

When $\lambda=1$ and $\lambda^{\prime}=0$, with $L_{0}=L_{01}$ the most
general $L_{\pm 1}$ are:

$\displaystyle(c_{12}^{2}+c_{13}^{2}-c_{04}^{2})\,L_{1}$ $\displaystyle=$
$\displaystyle
c_{12}(L_{12}-L_{02})+c_{13}(L_{13}-L_{03})+c_{04}(L_{14}-L_{04})$ (8)
$\displaystyle L_{-1}$ $\displaystyle=$ $\displaystyle
c_{12}(L_{02}+L_{12})+c_{13}(L_{03}+L_{13})+c_{04}(L_{04}+L_{14})$ (9)

for $c_{12}^{2}+c_{13}^{2}-c_{04}^{2}\neq 0$. It can be shown that the
residual symmetries ($O(2,3)$ transformations that leave $L_{0}$ invariant)
leave $c_{12}^{2}+c_{13}^{2}-c_{04}^{2}$ invariant up to a positive scale.
Thus depending on the sign of $c_{12}^{2}+c_{13}^{2}-c_{04}^{2}$ we can take
$(c_{12},c_{13},c_{04})$ to be either $(1,0,0)$ or $(0,0,1)$ and these two
will be the only inequivalent choices in this case.

  2. 2.

$\lambda^{\prime}=1$ and $\lambda>1$ leads to two sub cases: (i) $\lambda=2$
and (ii) $\lambda\neq 2$.

    1. (a)

For $\lambda=2$ and $\lambda^{\prime}=1$ with $L_{0}=2L_{01}+L_{34}$, we find:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!b_{04}b_{23}L_{1}$
$\displaystyle=$ $\displaystyle
b_{23}(L_{04}-L_{03}+L_{13}-L_{14})+3b_{04}(L_{23}+L_{24})$ (11)
$\displaystyle L_{-1}$ $\displaystyle=$ $\displaystyle
b_{04}(L_{03}+L_{04})+(L_{13}+L_{14})+b_{23}(L_{23}-L_{24})$ (12)

The residual symmetries can be used again to set $b_{23}=b_{04}=1$.

    2. (b)

For $\lambda>1$ and $\lambda\neq 2$ leads to no solution.

  3. 3.

$\lambda^{\prime}=1-\lambda$ with $\frac{1}{2}\leq\lambda<1$ (since
$\lambda^{\prime}=0$ is already considered) also leads to two sub-cases: (i)
$\lambda=1/2$ and (ii) $1/2<\lambda<1$.

    1. (a)

$\lambda=1/2$ with $L_{0}=\frac{1}{2}(L_{01}+L_{34})$ leads to a solution

$\displaystyle 4\,a_{04}\,L_{-1}$ $\displaystyle=$ $\displaystyle-
L_{03}+L_{04}-L_{13}+L_{14}$ (13) $\displaystyle~{}~{}a_{04}^{-1}\,L_{1}$
$\displaystyle=$ $\displaystyle L_{03}+L_{04}-L_{13}-L_{14}$ (14)

again we can use the residual symmetries of $L_{0}$ to set $a_{04}=1/2$.

    2. (b)

$1/2<\lambda<1$ – leads to no solution.

  4. 4.

The final case is $\lambda^{\prime}=\lambda-1$. This means $\lambda\geq 1$. We
have already covered the cases of $\lambda=1$ and $\lambda=2$. So we restrict
to $\lambda>1$ and not equal to $2$. And we can show that there is no solution
in this case.

Thus we have arrived at the result that there are precisely four inequivalent
embeddings of $\mathfrak{sl}(2,{\mathbb{R}})$ into $\mathfrak{so}(2,3)$: (i)
$(\lambda,\lambda^{\prime})=(1,0)$ with $(c_{12},c_{13},c_{04})=(1,0,0)$, (ii)
$(\lambda,\lambda^{\prime})=(1,0)$ with $(c_{12},c_{13},c_{04})=(0,0,1)$,
(iii) $(\lambda,\lambda^{\prime})=(2,1)$, and (iv)
$(\lambda,\lambda^{\prime})=(1/2,1/2)$. As the next step, we will write down
the algebra of $\mathfrak{so}(2,3)$ from the perspective of each of these
embeddings. We choose to characterise these four avatars of
$\mathfrak{so}(2,3)$ by their maximal subalgebras $\mathfrak{h}$ that commute
with the embedded $\mathfrak{sl}(2,{\mathbb{R}})$.

###  2.1 Four avatars of $\mathfrak{so}(2,3)$

#### $\mathfrak{h}=\mathfrak{so}(1,1)$

This case corresponds to $(\lambda,\lambda^{\prime})=(1,0)$ &
$(c_{12},c_{13},c_{04})=(1,0,0)$ with the generators

$\displaystyle L_{-1}$ $\displaystyle=$ $\displaystyle
L_{02}+L_{12},~{}~{}L_{0}=L_{01},~{}~{}L_{1}=-L_{02}+L_{12},~{}~{}H=L_{34},$
(15) $\displaystyle P^{0}_{-1}$ $\displaystyle=$ $\displaystyle-
L_{04}-L_{14},~{}~{}P^{0}_{0}=L_{24},~{}~{}P^{0}_{1}=-L_{04}+L_{14}\,,$ (16)
$\displaystyle P^{1}_{-1}$ $\displaystyle=$ $\displaystyle-
L_{03}-L_{13},~{}~{}P^{0}_{0}=L_{23},~{}~{}P^{0}_{1}=-L_{03}+L_{13}\,,\,$ (17)

satisfying

$\displaystyle[L_{m},L_{n}]$ $\displaystyle=$ $\displaystyle(m-n)L_{m+n}\,,$
(18) $\displaystyle[L_{m},P^{a}_{n}]$ $\displaystyle=$
$\displaystyle(m-n)P^{a}_{m+n}\,,$ (19) $\displaystyle[P^{a}_{m},P^{b}_{n}]$
$\displaystyle=$
$\displaystyle-\eta^{ab}(m-n)L_{m+n}-\epsilon^{ab}\eta_{mn}\,H\,,$ (20)
$\displaystyle[H,P^{a}_{m}]$ $\displaystyle=$ $\displaystyle
P^{b}_{m}\,{\epsilon_{b}}^{a}\,,$ (21)

for $m,n,\cdots\in\\{0,\pm 1\\}$ and $a,b,\cdots\in\\{0,1\\}$. Further
$\epsilon^{01}=-\epsilon^{10}=1$, $\eta_{ab}={\rm diag}(-1,1)=\eta^{ab}$ and
${\epsilon_{b}}^{a}=\eta_{bc}\epsilon^{ca}$, and
$\eta_{mn}=(3m^{2}-1)\delta_{m+n}=m^{2}+n^{2}-mn-1$. This avatar is related to
the way $\mathfrak{so}(2,3)$ is realised as the conformal symmetry of
${\mathbb{R}}^{1,2}$. Clearly, in this case the subalgebra $\mathfrak{h}$ that
commutes with the subalgebra $\mathfrak{sl}(2,{\mathbb{R}})$ is generated by
the boost generator $H=L_{34}$ and hence is $\mathfrak{so}(1,1)$.

#### $\mathfrak{h}=\mathfrak{so}(2)$

This case corresponds to $(\lambda,\lambda^{\prime})=(1,0)$ &
$(c_{12},c_{13},c_{04})=(0,0,1)$ with the generators

$\displaystyle L_{0}$ $\displaystyle=$ $\displaystyle
L_{01},~{}~{}L_{1}=L_{04}-L_{14},~{}~{}L_{-1}=L_{04}+L_{14},~{}~{}R=L_{23},~{}~{}$
(22) $\displaystyle P^{1}_{-1}$ $\displaystyle=$ $\displaystyle-
L_{03}-L_{13},~{}~{}P^{1}_{0}=L_{34},~{}~{}P^{1}_{1}=L_{03}-L_{13},$ (23)
$\displaystyle P^{2}_{-1}$ $\displaystyle=$ $\displaystyle-
L_{02}-L_{12},~{}~{}P^{2}_{0}=L_{24},~{}~{}P^{2}_{1}=L_{02}-L_{12}\,$ (24)

satisfying

$\displaystyle[L_{m},L_{n}]$ $\displaystyle=$ $\displaystyle(m-n)L_{m+n},$
(25) $\displaystyle[L_{m},P^{a}_{n}]$ $\displaystyle=$
$\displaystyle(m-n)P^{a}_{m+n},$ (26) $\displaystyle[P^{a}_{m},P^{b}_{n}]$
$\displaystyle=$
$\displaystyle\delta^{ab}(m-n)L_{m+n}+\epsilon^{ab}\eta_{mn}\,R\,,$ (27)
$\displaystyle[R,P^{a}_{m}]$ $\displaystyle=$
$\displaystyle\epsilon^{ab}P^{b}_{m}\,,$ (28)

for $m,n,\cdots\in\\{0,\pm 1\\}$ and $a,b,\cdots\in\\{1,2\\}$. Further
$\epsilon^{12}=1=-\epsilon^{21}$. Contrary to the previous case
($\mathfrak{h}=\mathfrak{so}(1,1)$) the subalgebra $\mathfrak{h}$ is generated
by the rotation generator $L_{23}$ and hence is $\mathfrak{so}(2)$. Thus, even
though this case is very similar, it is not equivalent to the one with
$\mathfrak{h}=\mathfrak{so}(1,1)$.

#### $\mathfrak{h}=\emptyset$

This case corresponds to $(\lambda,\lambda^{\prime})=(2,1)$. The generators
are

$\displaystyle L_{0}$ $\displaystyle=$ $\displaystyle 2L_{01}+L_{34},$ (29)
$\displaystyle L_{1}$ $\displaystyle=$ $\displaystyle-
L_{03}+L_{04}+L_{13}-L_{14}+3L_{23}+3L_{24}\,,$ (30) $\displaystyle L_{-1}$
$\displaystyle=$ $\displaystyle L_{03}+L_{04}+L_{13}+L_{14}+L_{23}-L_{24},$
(31) $\displaystyle W_{-3}$ $\displaystyle=$
$\displaystyle\beta(L_{03}-L_{04}+L_{13}-L_{14}),$ (32) $\displaystyle W_{-2}$
$\displaystyle=$ $\displaystyle\beta(L_{02}+L_{12}),$ (33) $\displaystyle
W_{-1}$ $\displaystyle=$
$\displaystyle\beta\frac{3}{5}(-L_{03}-L_{04}-L_{13}-L_{14}+\frac{2}{3}(L_{23}-L_{24})),$
(34) $\displaystyle W_{0}$ $\displaystyle=$
$\displaystyle\beta\frac{3}{5}(-L_{01}+2L_{34}),$ (35) $\displaystyle W_{1}$
$\displaystyle=$
$\displaystyle\beta\frac{3}{5}(L_{03}-L_{04}-L_{13}+L_{14}+2(L_{23}+L_{24})),$
(36) $\displaystyle W_{2}$ $\displaystyle=$ $\displaystyle
3\beta(L_{02}-L_{12}),$ (37) $\displaystyle W_{3}$ $\displaystyle=$
$\displaystyle 9\beta(-L_{03}-L_{04}+L_{13}+L_{14}),$ (38)

satisfying

$\displaystyle[L_{m},L_{n}]$ $\displaystyle=$ $\displaystyle(m-n)\,L_{m+n},$
(39) $\displaystyle[L_{m},W_{n}]$ $\displaystyle=$
$\displaystyle(3m-n)\,W_{m+n},$ (40) $\displaystyle[W_{m},W_{n}]$
$\displaystyle=$
$\displaystyle\frac{\beta^{2}}{100}\,A(m,n)L_{m+n}+\frac{\beta}{10}\,B(m,n)W_{m+n},$
(41)

where

$\displaystyle A(m,n)$ $\displaystyle=$
$\displaystyle(m-n)\Big{(}3(m^{4}+n^{4})-2mn(m-n)^{2}-39(m^{2}+n^{2})+20mn+108\Big{)},$
(42) $\displaystyle B(m,n)$ $\displaystyle=$
$\displaystyle(m-n)(n^{2}+m^{2}-mn-7)\,.$ (43)

We can fix $\beta$ to be any non-zero real number by a residual $O(2,3)$
transformation – and we choose $\beta=10/\sqrt{1680}$ for later convenience.
This avatar of $\mathfrak{so}(2,3)$ is special as the corresponding embedding
of $\mathfrak{sl}(2,{\mathbb{R}})$ in it is what can be called the principal
embedding – as the rest of the generators form a single 5-dimensional
irreducible representation of $\mathfrak{sl}(2,{\mathbb{R}})$.

#### $\mathfrak{h}=\mathfrak{sl}(2,{\mathbb{R}})$

This case corresponds to $(\lambda,\lambda^{\prime})=(1/2,1/2)$. The
generators are

$\displaystyle L_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(L_{01}+L_{34}),~{}~{}L_{1}=\frac{1}{2}(L_{03}+L_{04}-L_{13}-L_{14}),$
(44) $\displaystyle L_{-1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(-L_{03}+L_{04}-L_{13}+L_{14}),~{}~{}J_{0}=\frac{1}{2}(L_{01}-L_{34}),$
(45) $\displaystyle J_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(-L_{03}-L_{14}+L_{04}+L_{13}),~{}~{}J_{-1}=\frac{1}{2}(L_{03}+L_{14}+L_{04}+L_{13}),$
(46) $\displaystyle G_{\frac{1}{2},\frac{1}{2}}$ $\displaystyle=$
$\displaystyle\alpha(-L_{02}+L_{12}),~{}~{}G_{-\frac{1}{2},\frac{1}{2}}=\alpha(-L_{23}-L_{24}),$
(47) $\displaystyle G_{\frac{1}{2},-\frac{1}{2}}$ $\displaystyle=$
$\displaystyle\alpha(L_{23}-L_{24}),~{}~{}G_{-\frac{1}{2},-\frac{1}{2}}=\alpha(L_{02}+L_{12}),$
(48)

satisfying

$\displaystyle[L_{m},L_{n}]$ $\displaystyle=$
$\displaystyle(m-n)L_{m+n},~{}~{}[J_{a},J_{b}]=(a-b)J_{a+b},$ (49)
$\displaystyle[J_{a},G_{sr}]$ $\displaystyle=$
$\displaystyle\frac{1}{2}(a-2s)\,G_{a+s,r},~{}~{}[L_{m},G_{sr}]=\frac{1}{2}(m-2r)\,G_{s,m+r},$
(50) $\displaystyle[G_{sr},G_{s^{\prime}r^{\prime}}]$ $\displaystyle=$
$\displaystyle-2\alpha^{2}(\epsilon_{rr^{\prime}}J_{s+s^{\prime}}+\epsilon_{ss^{\prime}}L_{r+r^{\prime}})\,$
(51)

where $m,n,\cdots\in\\{0,\pm 1\\}$, $a,b,\cdots\in\\{0,\pm 1\\}$ and
$r,s,\cdots\in\\{\pm 1/2\\}$, with $\epsilon_{-\frac{1}{2},\frac{1}{2}}=1$.
Again we can choose $\alpha$ to be any non-zero real number and we fix it to
be unity. This completes the exercise of finding the inequivalent embeddings
of $\mathfrak{sl}(2,{\mathbb{R}})$ inside $\mathfrak{so}(2,3)$ and writing the
$\mathfrak{so}(2,3)$ algebra that makes the corresponding
$\mathfrak{sl}(2,{\mathbb{R}})$ manifest. Each such embedding is based on the
identification of a subalgebra
$\mathfrak{sl}(2,{\mathbb{R}})\oplus\mathfrak{h}\subset\mathfrak{so}(2,3)$ –
which are also in one-to-one correspondence with the maximal subalgebras of
$\mathfrak{so}(2,3)$ listed in deGraaf:2019dni  that contain at least one copy
of $\mathfrak{sl}(2,{\mathbb{R}})$ in them.

Next we would like to promote each of these avatars (18, 25, 39, 49) of
$\mathfrak{so}(2,3)$ into an infinite dimensional algebra that consists of a
Virasoro. We will show in the next section that the first two avatars above
admit minimal chiral ${\cal W}$-algebra extensions that are novel. The third
avatar leads to the known ${\cal W}(2,4)$ algebra, and the extension of the
fourth one is what was done in our earlier paper Gupta:2022mdt .

##  3 The four ${\cal W}$-algebra extensions

We now would like to find the minimal infinite dimensional extensions of the
four avatars of $\mathfrak{so}(2,3)$ found in section 2 such that they contain
modes of a chiral stress tensor. We follow the following set of rules for
this:

  * •

For every copy of $\mathfrak{sl}(2,{\mathbb{R}})$ with generators $L_{n}$ (for
$n=0,\pm 1$) we postulate a chiral quasi-primary stress tensor $T(z)$ with
dimension $h=2$ and central charge $c$, and modes $L_{n}$ (for
$n\in{\mathbb{Z}}$).

  * •

For the generators of the subalgebra $\mathfrak{h}$ we postulate a
level-$\kappa$ current with $h=1$.

  * •

For every set of generators that form a non-trivial finite dimensional
representation of $\mathfrak{sl}(2,{\mathbb{R}})$ of dimension $k$ we
postulate a primary of dimension $h=\frac{1}{2}(k+1)$.

  * •

We write down the most general OPEs among the chiral operators as obtained
above imposing the global symmetries at hand.

  * •

Finally we fix the undermined coefficients in these OPEs using the OPE
associativity - implemented in terms of commutators of the modes and their
Jacobi identities explicitly.

Following these rules we consider one copy of a chiral Stress Tensor: $T(z)$ –
a quasi-primary with $h=2$ and central charge $c$ for each
$\mathfrak{sl}(2,{\mathbb{R}})$ subalgebra we identified in the previous
section with

$\displaystyle
T(z)T(w)\sim\frac{c/2}{(z-w)^{4}}+\frac{2T(w)}{(z-w)^{2}}+\frac{\partial
T(w)}{z-w}\,.$ (52)

Then each of the four avatars of $\mathfrak{so}(2,3)$ lead to the following
sets of chiral operators.

  1. 1.

In the $\mathfrak{h}=\mathfrak{so}(1,1)$ case we have, along with $T(z)$,

    1. (a)

An $\mathfrak{so}(1,1)$ current $H(z)$ which is a conformal primary with $h=1$
and level-$\kappa$.

$\displaystyle T(z)H(w)$ $\displaystyle\sim$
$\displaystyle\frac{H(w)}{(z-w)^{2}}+\frac{\partial H(w)}{z-w},$ (53)
$\displaystyle H(z)H(w)$ $\displaystyle\sim$
$\displaystyle\frac{\kappa}{(z-w)^{2}}\,.$ (55)

    2. (b)

A pair of chiral operators $P^{a}(z)$ that are conformal primaries with $h=2$
as well as a doublet of $\mathfrak{so}(1,1)$ current $H(z)$.666One could have
postulated that $P^{a}(z)$ are only quasi-primaries at this stage. It turns
out, however, that consistency of the final algebra with associativity does
not allow central terms in $T(z)P^{a}(w)$ OPE and hence we preclude them.

$\displaystyle T(z)P^{a}(w)$ $\displaystyle\sim$
$\displaystyle\frac{2P^{a}(w)}{(z-w)^{2}}+\frac{\partial P^{a}(w)}{z-w}\,,$
(56) $\displaystyle H(z)P^{a}(w)$ $\displaystyle\sim$
$\displaystyle\frac{P^{b}(w)\,{\epsilon_{b}}^{a}\,}{z-w}\,.$ (58)

This leaves the OPE $P^{a}(z)P^{b}(w)$ to be determined. We will refer to the
resultant algebra to be ${\cal W}(2;2^{2},1)$.

  2. 2.

In the $\mathfrak{h}=\mathfrak{so}(2)$ avatar we have, along with $T(z)$,

    1. (a)

An $\mathfrak{so}(2)$ current $R(z)$ which is a conformal primary with $h=1$
and level-$\kappa$.

$\displaystyle T(z)R(w)$ $\displaystyle\sim$
$\displaystyle\frac{R(w)}{(z-w)^{2}}+\frac{\partial R(w)}{z-w},$ (59)
$\displaystyle R(z)R(w)$ $\displaystyle\sim$
$\displaystyle\frac{\kappa}{(z-w)^{2}}\,.$ (61)

    2. (b)

A pair of chiral operators $P^{a}(z)$ that are conformal primaries with $h=2$
as well as a doublet of $\mathfrak{so}(2)$ current $R(z)$.

$\displaystyle T(z)P^{a}(w)$ $\displaystyle\sim$
$\displaystyle\frac{2P^{a}(w)}{(z-w)^{2}}+\frac{\partial P^{a}(w)}{z-w}\,,$
(62) $\displaystyle R(z)P^{a}(w)$ $\displaystyle\sim$
$\displaystyle\frac{\epsilon^{ab}P^{b}(w)\,}{z-w}\,.$ (64)

The OPE $P^{a}(z)P^{b}(w)$ will be determined later. We will refer to the
resultant algebra to be $\widetilde{\cal W}(2;2^{2},1)$.

  3. 3.

In the case with $\mathfrak{h}=\emptyset$ we have a stress tensor $T(z)$ and

    1. (a)

A chiral operator $W(z)$ that is a conformal primary of dimension $h=4$.

$\displaystyle T(z)W(w)\sim\frac{4W(w)}{(z-w)^{2}}+\frac{\partial W(w)}{z-w}$
(65)

The $W(z)W(w)$ will be determined later. We will identify the resultant
algebra with the known ${\cal W}(2,4)$ algebra of Bouwknegt:1988sv .

  4. 4.

In the case with $\mathfrak{h}=\mathfrak{sl}(2,{\mathbb{R}})$ we have a $T(z)$
and

    1. (a)

A triplet $J_{a}(z)$ with $a=0,\pm 1$, which are conformal primaries with
$h=1$

$\displaystyle T(z)J_{a}(w)$ $\displaystyle\sim$
$\displaystyle(z-w)^{-2}J_{a}(w)+(z-w)^{-1}\partial J_{a}(w)\,,$ (66)

and among themselves form a level-$\kappa$ $\mathfrak{sl}(2,{\mathbb{R}})$
algebra,

$\displaystyle J_{a}(z)J_{b}(w)$ $\displaystyle\sim$
$\displaystyle-\frac{\kappa}{2}\eta_{ab}(z-w)^{-2}+(z-w)^{-1}{f_{ab}}^{c}J_{c}(w)\,.$
(67)

    2. (b)

A pair $G_{s}(z)$ for $s=\pm 1/2$ that are conformal primaries with $h=3/2$

$\displaystyle T(z)G_{s}(w)$ $\displaystyle\sim$
$\displaystyle(3/2)(z-w)^{-2}G_{s}(w)+(z-w)^{-1}\partial G_{s}(w)$ (68)

as well as a doublet of $\mathfrak{sl}(2,{\mathbb{R}})$ current algebra

$\displaystyle J_{a}(z)G_{s}(w)$ $\displaystyle\sim$
$\displaystyle(z-w)^{-1}G_{s^{\prime}}(w){(\lambda_{a})^{s^{\prime}}}_{s}$
(69)

where
${(\lambda_{a})^{s}}_{s^{\prime}}=\frac{1}{2}(a-2s^{\prime})\delta^{s}_{a+s^{\prime}}$.
This case was already dealt with by us in Gupta:2022mdt  and the OPE
$G_{s}(z)G_{s^{\prime}}(w)$ was determined. The resultant algebra is called
${\cal W}(2;(3/2)^{2},1^{3})$.

We will now turn to obtaining the underdetermined OPEs in each of these cases.
We will provide the essential details of the derivation of first of these
algebras and the results for the remaining three.

###  3.1 ${\cal W}(2;2^{2},1)$

In this case the only undetermined OPE is that of $P^{a}(z)P^{b}(w)$ and
following the conformal invariance one expects that all quasi-primaries of
dimension up to three and their global descendants can appear in the singular
terms on the rhs. These quasi-primaries can also be composites (normal ordered
products) of simple (quasi-) primaries $(T(z),P^{a}(z),H(z))$.

One can construct the necessary quasi-primaries using the OPEs

$\displaystyle T(z)T(w)$ $\displaystyle\sim$
$\displaystyle\frac{c/2}{(z-w)^{4}}+\frac{2T(w)}{(z-w)^{2}}+\frac{\partial
T(w)}{z-w}\,$ (70) $\displaystyle T(z)H(w)$ $\displaystyle\sim$
$\displaystyle\frac{H(w)}{(z-w)^{2}}+\frac{\partial
H(w)}{z-w},~{}~{}~{}~{}~{}H(z)T(w)\sim\frac{H(w)}{(z-w)^{2}}\,$ (71)
$\displaystyle H(z)H(w)$ $\displaystyle\sim$
$\displaystyle\frac{\kappa}{(z-w)^{2}}\,$ (72) $\displaystyle T(z)P^{a}(w)$
$\displaystyle\sim$ $\displaystyle\frac{2P^{a}(w)}{(z-w)^{2}}+\frac{\partial
P^{a}(w)}{z-w}\sim P^{a}(z)T(w)\,$ (73) $\displaystyle H(z)P^{a}(w)$
$\displaystyle\sim$
$\displaystyle\frac{P^{b}(w)\,{\epsilon_{b}}^{a}\,}{z-w}\sim-P^{a}(z)H(w)\,.$
(74)

The relevant composite quasi-primaries are as follows.

  * •

At $h=2$ we have only one normal-ordered composite quasi-primary $(H^{2})(z)$
with

$\displaystyle
T(z)(H^{2})(w)\sim\frac{\kappa}{(z-w)^{4}}+\frac{2}{(z-w)^{2}}(H^{2})(w)+\frac{1}{(z-w)}\partial(H^{2})(w)\,.$
(75)

The corresponding Sugawara stress tensor $\frac{1}{2\kappa}(H^{2})(z)$ has
$c=1$.

  * •

At $h=3$ we have two composite scalar quasi-primaries

$\displaystyle\Lambda(z)$ $\displaystyle:=$
$\displaystyle(TH)(z)-\frac{1}{2}\partial^{2}H(z),$ (76)
$\displaystyle\Sigma(z)$ $\displaystyle:=$
$\displaystyle\frac{1}{2\kappa}((H^{2})H)(z)-\frac{1}{2}\partial^{2}H(z).$
(77)

Their OPEs with $T(z)$ are given by

$\displaystyle T(z)\Lambda(w)$ $\displaystyle\sim$
$\displaystyle\frac{2+c}{2(z-w)^{4}}H(w)+\frac{3}{(z-w)^{2}}\Lambda(w)+\frac{1}{(z-w)}\partial\Lambda(w)\,,$
(78) $\displaystyle T(z)\Sigma(w)$ $\displaystyle\sim$
$\displaystyle\frac{3}{2(z-w)^{4}}H(w)+\frac{3}{(z-w)^{2}}\Sigma(w)+\frac{1}{(z-w)}\partial\Sigma(w)\,.$
(80)

Note that the linear combination $3\,\Lambda(z)-(2+c)\,\Sigma(z)$ is a
primary.

  * •

At $h=3$ we have a doublet of primaries (not just quasi-primaries)

$\displaystyle[P^{a}H](z):=(P^{a}H)(z)+\frac{3}{4}{\epsilon_{b}}^{a}\partial
P^{b}=(HP^{a})(z)-\frac{1}{4}{\epsilon_{b}}^{a}\partial P^{b}$ (81)

This is the full list of composite quasi-primaries that are covariant under
$\mathfrak{so}(1,1)$ and dimension $h\leq 3$. In particular, let us note that
there is no rank-2 symmetric traceless tensor of $\mathfrak{so}(1,1)$ with
$h\leq 3$. There are other quasi-primaries of higher dimensions that we will
introduce as and when required.

We now make use of the global symmetries to constrain the OPE
$P^{a}(z)P^{b}(w)$.

  * •

Since the two operators involved carry vector indices $a,b,\cdots$ of
$\mathfrak{so}(1,1)$ their OPE should respect this symmetry. This means that
the rhs can only involve representations of $\mathfrak{so}(1,1)$ that can
appear in the tensor product of two vectors.

  * •

There are in general four independent components in a rank-2 tensor of
$\mathfrak{so}(1,1)$. Arranging them into covariant irreducible objects, there
are

    1. 1.

two singlets - with the CG coefficients being the symmetric tensor $\eta^{ab}$
and the anti-symmetric tensor $\epsilon^{ab}$. These can be multiplied by
$\mathfrak{so}(1,1)$ invariants.

    2. 2.

a symmetric traceless rank-2 tensor with two independent components.

  * •

One consequence of these tensor structures is that we cannot have a covariant
rank-1 object (vector) on the rhs and that rules out (81). Also the fact that
there are no composite operators with $h\leq 3$ that are rank-2 traceless
symmetric tensors leaves us with, in the increasing order of conformal
dimensions $\\{I,H(z),T(z),(H^{2})(z),\Lambda,\Sigma\\}$ that can appear on
the rhs.

  * •

Another fact is that the OPE is expected to be invariant under the
simultaneous exchange $(a,z)\leftrightarrow(b,w)$. It is easy to see that this
requires that the terms containing $\eta^{ab}$ should have even dimensional
operators, namely, $(I,T(z))$ (and their descendants) and terms containing
$\epsilon^{ab}$ should have odd dimensional operators, namely,
($H(z),\Lambda(z),\Sigma(z)$) and their descendants.

  * •

The relative coefficients of a quasi-primary and its global conformal
descendants are fixed by the global conformal invariance. This leaves only six
real numbers to be undetermined.

Thus we arrive at the following ansatz:

$\displaystyle P^{a}(z)P^{b}(w)$ $\displaystyle\sim$
$\displaystyle\eta^{ab}\Big{[}\alpha\,(z-w)^{-4}\,I+\beta\,(z-w)^{-2}\,\Big{(}2T(w)+(z-w)\,\partial
T(w)\Big{)}$ (88)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\gamma(z-w)^{-2}\,\Big{(}2(H^{2})(w)+(z-w)\partial(H^{2})(w)\Big{)}\Big{]}$
$\displaystyle\\!\\!\\!\\!\\!+\,\epsilon^{ab}\Big{[}\delta\,(z-w)^{-3}\Big{(}6H(w)+3\,(z-w)\partial
H(w)+\,(z-w)^{2}\,\partial^{2}H(w)\Big{)}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\sigma\,(z-w)^{-1}\Lambda(w)+\omega\,(z-w)^{-1}\Sigma(w)\Big{]}$

The undetermined real numbers $(\alpha,\beta,\gamma,\delta,\sigma,\omega)$
(that can depend on the central charges $(c,\kappa)$) are to be determined by
imposing the OPE associativity. This we do using Jacobi identities in terms of
modes (as in Gupta:2022mdt ) following Blumenhagen:1990jv . To proceed further
we need the mode expansions of these quasi-primaries. Using the definition
from DiFrancesco:1997nk ; Blumenhagen:2009zz

$\displaystyle(AB)_{m}=\sum_{n\leq-
h_{A}}A_{n}B_{m-n}+\sum_{n>-h_{A}}B_{m-n}A_{n}\,.$ (90)

for the modes of normal ordered products we can write

$\displaystyle(H^{2})_{n}=\sum_{k>-1}H_{n-k}H_{k}+\sum_{k\leq-1}H_{k}H_{n-k}\,.$
(91)

Similarly we have
$\Lambda(z)=\sum_{n\in{\mathbb{Z}}}\frac{{\Lambda}_{n}}{z^{n+3}}$ where

$\displaystyle\Lambda_{n}=\sum_{k>-2}H_{n-k}L_{k}+\sum_{k\leq-2}L_{k}H_{n-k}-\frac{1}{2}(n+1)(n+2)H_{n}$
(92)

and $\Sigma(z)=\sum_{n\in{\mathbb{Z}}}\frac{{\Sigma}_{n}}{z^{n+3}}$ where

$\displaystyle\Sigma_{n}=\sum_{k>-2}H_{n-k}\tilde{L}_{k}+\sum_{k\leq-2}\tilde{L}_{k}H_{n-k}-\frac{1}{2}(n+1)(n+2)H_{n}$
(93)

with $\tilde{L}_{n}=\frac{1}{2\kappa}(H^{2})_{n}$.

#### Jacobi identities

Let us convert all the OPEs into commutators of modes. The modes are

$\displaystyle L_{n}=\oint_{0}\frac{dz}{2\pi
i}z^{n+1}T(z),~{}~{}P^{a}_{n}=\oint_{0}\frac{dz}{2\pi
i}z^{n+1}P^{a}(z),~{}~{}H_{n}=\oint_{0}\frac{dz}{2\pi i}z^{n}\,H(z)$ (94)

resulting in

  1. 1.

Virasoro Algebra:

$\displaystyle[L_{m},L_{n}]=(m-n)\,L_{m+n}+\frac{c}{12}m(m^{2}-1)\,\delta_{m+n,0}$
(95)

  2. 2.

Virasoro Primaries

$\displaystyle[L_{m},P^{a}_{n}]=(m-n)\,P^{a}_{m+n},~{}~{}[L_{m},H_{n}]=-n\,H_{m+n}$
(96)

  3. 3.

$\mathfrak{so}(1,1)_{\kappa}$ Current Algebra

$\displaystyle[H_{m},H_{n}]=\kappa\,m\,\delta_{m+n,0}$ (97)

  4. 4.

Current algebra primaries

$\displaystyle[H_{m},P^{a}_{n}]=P^{b}_{m+n}\,{\epsilon_{b}}^{a}$ (98)

  5. 5.

Commutator $[P^{a}_{m},P^{b}_{n}]$

$\displaystyle[P^{a}_{m},P^{b}_{n}]$ $\displaystyle=$
$\displaystyle\eta^{ab}\left[\frac{\alpha}{6}m(m^{2}-1)\delta_{m+n,0}+(m-n)\,\Big{(}\beta\,L_{m+n}+\gamma\,(H^{2})_{m+n}\Big{)}\right]$
(101)
$\displaystyle+\epsilon^{ab}\Big{[}\delta\,(m^{2}+n^{2}-mn-1)\,H_{m+n}+\sigma\,\Lambda_{m+n}+\omega\,\Sigma_{m+n}\Big{]}$

It remains to simply determine the unknowns
$(\alpha,\beta,\gamma,\delta,\sigma,\omega)$ in terms of $c$ and $\kappa$
using the Jacobi identities. As in Gupta:2022mdt  we schematically denote the
Jacobi identity involving the modes of the operators $A(z),B(z),C(z)$ as
identity $(ABC)$. There are a total of 10 identities to be considered. The
four identities $(TTT)$, $(TTH)$, $(THH)$, $(HHH)$ are automatically
satisfied. Next there are three with one $P^{a}$ each: $(TTP^{a})$,
$(THP^{a})$, $(HHP^{a})$ – which can also be seen to be satisfied identically.

The remaining three Jacobi identities contain at least two $P^{a}_{n}$’s:
$(TPP)$, $(HPP)$ and $(PPP)$. These involve commutators between two
$P^{a}_{m}$’s. To impose these we need the commutators of the modes of the
simple primaries $(L_{m},H_{m},P^{a}_{m})$ with
$((H^{2})_{n},\Lambda_{n},\Sigma_{n})$.

#### Commutators with $(H^{2})_{n}$

It is easy to see

$\displaystyle[L_{m},(H^{2})_{n}]$ $\displaystyle=$
$\displaystyle(m-n)\,(H^{2})_{m+n}+\frac{\kappa}{6}\,m(m^{2}-1)\,\delta_{m+n,0}\,,$
(103) $\displaystyle[H_{m},(H^{2})_{n}]$ $\displaystyle=$ $\displaystyle
2\kappa\,m\,H_{m+n}\,.$ (104)

For $[P^{a}_{m},(H^{2})_{n}]$ we first note that the OPE between $P^{a}(z)$
and $(H^{2})(w)$ can be written in terms of quasi-primaries and their
descendants as follows:

$\displaystyle P^{a}(z)(H^{2})(w)\sim\frac{P^{a}(w)}{(z-w)^{2}}+\frac{\partial
P^{a}(w)}{2(z-w)}-2{\epsilon_{b}}^{a}\frac{[P^{b}H](w)}{z-w}$ (105)

where

$\displaystyle[P^{a}H](z):=(P^{a}H)(z)+\frac{3}{4}{\epsilon_{b}}^{a}\partial
P^{a}$ (106)

is a primary. In obtaining this we used the identity:
$(HP^{a})(z)-(P^{a}H)(z)-{\epsilon_{b}}^{a}\partial P^{b}(z)=0$. Using this
OPE we can immediately write the commutator of the corresponding modes as

$\displaystyle[P^{a}_{m},(H^{2})_{n}]$ $\displaystyle:=$
$\displaystyle-2{\epsilon_{b}}^{a}[P^{b}H]_{m+n}+\frac{1}{2}(m-n)\,P^{a}_{m+n}\,.$
(107)

#### Commutators with $\Lambda_{n}$

It is straightforward to find

$\displaystyle[L_{m},\Lambda_{n}]$ $\displaystyle=$
$\displaystyle(2m-n)\Lambda_{m+n}+\frac{1}{12}(c+2)m(m^{2}-1)\,H_{m+n}\,$
(108) $\displaystyle[H_{m},\Lambda_{n}]$ $\displaystyle=$
$\displaystyle\kappa\,m\,L_{m+n}+m\,(H^{2})_{m+n}\,.$ (110)

The next non-trivial commutator is $[P^{a}_{m},\Lambda_{n}]$. For this again
we write the corresponding OPE $P^{a}(z)\Lambda(w)$ in terms of quasi-
primaries and their global descendants. We find

$\displaystyle P^{a}(z)\Lambda(w)$ $\displaystyle\sim$
$\displaystyle\frac{-2{\epsilon_{b}}^{a}}{(z-w)^{3}}\left[P^{b}(w)+\frac{1}{4}(z-w)\partial
P^{b}(w)+\frac{1}{20}(z-w)^{2}\partial^{2}P^{b}(w)\right]$ (113)
$\displaystyle+\frac{2[P^{a}H](w)}{(z-w)^{2}}+\frac{2\partial[P^{a}H](w)}{3(z-w)}-{\epsilon_{b}}^{a}\frac{[TP^{b}](w)}{z-w}+\frac{[\partial
P^{a}H](w)}{3(z-w)}$

where the relevant composite quasi-primaries are

$\displaystyle h=3~{}~{}$ $\displaystyle:$
$\displaystyle~{}~{}[P^{a}H](z):=(P^{a}H)(z)+\frac{3}{4}{\epsilon_{b}}^{a}\partial
P^{b}(z),$ (116)
$\displaystyle~{}~{}~{}[HP^{a}](z):=(HP^{a})(z)-\frac{1}{4}{\epsilon_{b}}^{a}\partial
P^{b}(z),$ $\displaystyle h=4~{}~{}$ $\displaystyle:$
$\displaystyle~{}~{}[H\partial P^{a}](z):=(H\partial
P^{a})(z)-2\,(P^{a}\partial
H)(z)-\frac{11}{10}\partial^{2}P^{b}(z){\epsilon_{b}}^{a}$ (118)
$\displaystyle~{}~{}~{}=(\partial P^{a}H)(z)-2(P^{a}\partial
H)(z)-\frac{3}{5}\partial^{2}P^{b}{\epsilon_{b}}^{a}:=[\partial P^{a}H](z),$
$\displaystyle h=4~{}~{}$ $\displaystyle:$
$\displaystyle~{}~{}[TP^{a}](z):=(TP^{a})(z)-\frac{3}{10}\partial^{2}P^{a}(z)=(P^{a}T)(z)-\frac{3}{10}\partial^{2}P^{a}(z):=[P^{a}T](z),$
(119)

Converting the OPE (113) into a commutator we find

$\displaystyle[P^{a}_{m},\Lambda_{n}]$ $\displaystyle=$
$\displaystyle-\,{\epsilon_{b}}^{a}[TP^{b}]_{m+n}+\frac{1}{3}[H\partial
P^{a}]_{m+n}$ (122)
$\displaystyle-\frac{1}{10}(6m^{2}+n^{2}-3mn-4){\epsilon_{b}}^{a}P^{b}_{m+n}+\frac{2}{3}(2m-n)[P^{a}H]_{m+n}\,.$

### Commutators with $\Sigma_{n}$

It is easy to see that

$\displaystyle[L_{m},\Sigma_{n}]$ $\displaystyle=$
$\displaystyle(2m-n)\Sigma_{m+n}+\frac{1}{4}m(m^{2}-1)\,H_{m+n},$ (123)
$\displaystyle[H_{m},\Sigma_{n}]$ $\displaystyle=$
$\displaystyle\frac{3}{2}m\,(H^{2})_{m+n}\,.$ (124)

To find $[P^{a}_{m},\Sigma_{n}]$ we again write the OPE:

$\displaystyle P^{a}(z)\Sigma(w)$ $\displaystyle\sim$
$\displaystyle\frac{-{\epsilon_{b}}^{a}}{2\kappa(z-w)^{3}}\left[P^{b}(w)+\frac{1}{4}(z-w)\partial
P^{b}(w)+\frac{1}{20}(z-w)^{2}\partial^{2}P^{b}(w)\right]$ (127)
$\displaystyle+\frac{3}{2\kappa(z-w)^{2}}\left[[P^{a}H](w)+\frac{1}{3}(z-w)\partial[P^{a}H](w)\right]-\frac{1}{4\kappa(z-w)}[\partial
P^{a}H](w)$
$\displaystyle-\frac{1}{2\kappa(z-w)}{\epsilon_{b}}^{a}\Big{[}[HHP^{b}](w)+[P^{a}HH](w)+[HP^{a}H](w)\Big{]}$

where the additional quasi-primaries with $h=4$ are:

$\displaystyle[HHP^{a}](z)$ $\displaystyle:=$
$\displaystyle((H^{2})P^{a})(z)-{\epsilon_{b}}^{a}\frac{3}{4}(H\partial
P^{b})(z)-{\epsilon_{b}}^{a}\frac{3}{2}(P^{b}\partial
H)(z)-\frac{27}{40}\partial^{2}P^{a}(z)\,,$ (128) $\displaystyle[HP^{a}H](z)$
$\displaystyle:=$
$\displaystyle((HP^{a})H)(z)+\frac{5}{4}{\epsilon_{b}}^{a}(H\partial
P^{b})-\frac{1}{2}{\epsilon_{b}}^{a}(P^{b}\partial
H)(z)-\frac{1}{40}(11+20\kappa)\partial^{2}P^{a},$ (129)
$\displaystyle[P^{a}HH](z)$ $\displaystyle:=$
$\displaystyle((P^{a}H)H)(z)+4{\epsilon_{b}}^{a}(P^{b}\partial
H)(z)-\frac{1}{10}(5\kappa-17)\partial^{2}P^{a},$ (130)

and we used the identities

$\displaystyle(HP^{a})(z)-(P^{a}H)(z)-{\epsilon_{b}}^{a}\partial P^{b}(z)$
$\displaystyle=$ $\displaystyle 0\,,$ (131) $\displaystyle(\partial
P^{a}H)(z)-(H\partial
P^{a})(z)+\frac{1}{2}{\epsilon_{b}}^{a}\partial^{2}P^{b}(z)$ $\displaystyle=$
$\displaystyle 0\,.$ (132)

Converting the OPE (127) into commutator modes we find:

$\displaystyle[P^{a}_{m},\Sigma_{n}]$ $\displaystyle=$
$\displaystyle-\frac{1}{40\kappa}{\epsilon_{b}}^{a}(6m^{2}+n^{2}-3mn-4)P^{b}_{m+n}+\frac{1}{2\kappa}(2m-n)[P^{a}H]_{m+n}$
(134) $\displaystyle-\frac{1}{4\kappa}[\partial
P^{a}H]_{m+n}-\frac{1}{2\kappa}{\epsilon_{b}}^{a}\Big{(}[HHP^{b}]_{m+n}+[HP^{b}H]_{m+n}+[P^{b}HH]_{m+n}\Big{)}$

### Result of Jacobi identities

We are ready to impose the $(TPP)$ and $(HPP)$ identities. The computations
are straightforward and we find that $(TPP)$ identity requires

$\displaystyle
2\alpha-c\beta-2\kappa\,\gamma=0,~{}~{}~{}24\delta-(2+c)\sigma-3\omega=0\,.$
(136)

The $(HPP)$ identity requires

$\displaystyle\beta+3\delta+2\kappa\,\gamma$ $\displaystyle=$ $\displaystyle
0,~{}~{}~{}\alpha+6\kappa\,\delta=0\,,$ (137) $\displaystyle
2\beta+\kappa\,\sigma$ $\displaystyle=$ $\displaystyle
0,~{}~{}~{}2\gamma+\sigma+\frac{3}{2}\omega=0\,.$ (138)

Finally we need to impose $(PPP)$ Jacobi identity. For this, first we write
$[P^{c}_{k},[P^{a}_{m},P^{b}_{n}]]$ as

$\displaystyle\left(\beta+\frac{\gamma}{2}\right)\eta^{ab}(m-n)(k-m-n)P^{c}_{m+n+k}$
(139)
$\displaystyle-\Big{[}\delta\,(m^{2}+n^{2}-mn-1)+\frac{1}{10}\left(\sigma+\frac{\omega}{4\kappa}\right)\Big{(}6k^{2}+(m+n)^{2}-3k(m+n)-4\Big{)}\Big{]}\epsilon^{ab}{\epsilon_{d}}^{c}P^{d}_{m+n+k}$
(140)
$\displaystyle+\left(\frac{\omega}{2\kappa}+\frac{2\sigma}{3}\right)(2k-m-n)\epsilon^{ab}[P^{c}H]_{m+n+k}-2\gamma(m-n)\eta^{ab}{\epsilon_{d}}^{c}[P^{d}H]_{m+n+k}$
(141)
$\displaystyle-\sigma\epsilon^{ab}\,{\epsilon_{d}}^{c}[TP^{d}]_{m+n+k}+\left(\frac{\sigma}{3}-\frac{\omega}{4\kappa}\right)\epsilon^{ab}[H\partial
P^{c}]_{m+n+k}$ (142)
$\displaystyle-\frac{\omega}{2\kappa}\epsilon^{ab}{\epsilon_{d}}^{c}\Big{(}[HHP^{b}]_{m+n+k}+[HP^{b}H]_{m+n+k}+[P^{b}HH]_{m+n+k}\Big{)}\,.$
(143)

The Jacobi identity to be imposed is

$\displaystyle[P^{c}_{k},[P^{a}_{m},P^{b}_{n}]]$ $\displaystyle+$
$\displaystyle\Big{(}(c,k)\rightarrow(a,m)\rightarrow(b,n)\rightarrow(c,k)\Big{)}$
(144) $\displaystyle+$
$\displaystyle\Big{(}(c,k)\rightarrow(b,n)\rightarrow(a,m)\rightarrow(c,k)\Big{)}=0.$
(145)

The terms that contain $\epsilon^{ab}{\epsilon_{d}}^{c}{\cal O}^{d}_{m+n+k}$
and $\epsilon^{ab}{\cal O}^{c}_{m+n+k}$ can be dropped as they vanish after
summing over the cyclic permutations above. Then there are two types of terms
left which contain (i) $P^{c}_{k+m+n}$, (ii) $[P^{c}H]_{m+n+k}$.

  1. 1.

Setting terms containing $[P^{a}H]_{m+n+k}$ to zero we find one condition

$\displaystyle 4\kappa(\gamma-\sigma)-3\omega=0.$ (146)

  2. 2.

Setting terms containing $P^{a}_{m+n+k}$ to zero gives the last condition

$\displaystyle 4\kappa(2\beta+\gamma-2\delta+\sigma)+\omega=0.$ (147)

Thus the full set of conditions obtained from the Jacobi identities is (136,
137, 146, 147). Out of these eight only six equations are linearly
independent. These equations admit a non-trivial two-parameter
solution.777There is of course the trivial solution:
$\alpha=\beta=\gamma=\delta=\sigma=\omega=0$, for arbitrary $c$ and $\kappa$.
Taking these parameters to be $(\sigma,\kappa)$ we find:

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\frac{3\kappa^{2}(\kappa-1)}{\kappa+1}\sigma,~{}~{}\beta=-\frac{\kappa}{2}\sigma,$
(148) $\displaystyle\gamma$ $\displaystyle=$
$\displaystyle\frac{2\kappa-1}{2(\kappa+1)}\sigma,~{}~{}\delta=-\frac{\kappa(\kappa-1)}{2(\kappa+1)}\sigma,$
(149) $\displaystyle\omega$ $\displaystyle=$
$\displaystyle-\frac{2\kappa}{\kappa+1}\sigma,~{}~{}c=-\frac{2\,(6\kappa^{2}-8\kappa+1)}{\kappa+1}\,.$
(150)

This completes our derivation of the algebra ${\cal W}(2;2^{2},1)$. In the
large-$\kappa$ limit these become

$\displaystyle\alpha$ $\displaystyle\rightarrow$ $\displaystyle
3\kappa^{2}\sigma,~{}~{}\beta\rightarrow-\frac{\kappa}{2}\sigma,~{}~{}\gamma\rightarrow\sigma,$
(151) $\displaystyle\delta$ $\displaystyle\rightarrow$
$\displaystyle-\frac{\kappa}{2}\sigma,~{}~{}~{}\omega\rightarrow-2\,\sigma,~{}~{}c\rightarrow-12\,\kappa\,.$
(152)

By further taking $\sigma\rightarrow\frac{2}{\kappa}$ it can be seen that this
${\cal W}$ algebra, when restricted to the modes of $\mathfrak{so}(2,3)$ is a
deformation of (18) around the $\kappa\rightarrow\infty$ limit.

#### Relation between ${\cal W}(2;2^{2},1)$ and conformal $\mathfrak{bms}_{3}$
of Fuentealba:2020zkf

Before we turn to the next case let us briefly demonstrate that the conformal
$\mathfrak{bms}_{3}$ algebra of Fuentealba:2020zkf  arises in the
semiclassical limit of our ${\cal W}(2;2^{2},1)$. For this we define

$\displaystyle\mathcal{P}_{m}=P^{0}_{m}+P^{1}_{m}\,,~{}\mathcal{K}_{m}=P^{0}_{m}-P^{1}_{m}$
(153)

and re-write the commutator (101) in terms of $\mathcal{P}_{m}$ and
$\mathcal{K}_{n}$. This results in, for instance, the commutator
$\left[\mathcal{P}_{m}\,,\mathcal{K}_{n}\right]$

$\displaystyle\left[\mathcal{P}_{m}\,,\mathcal{K}_{n}\right]=-\frac{\alpha}{3}\,m(m^{2}-1)\delta_{m+n,0}-2(m-n)\left(\beta\,L_{m+n}+\gamma\,(H^{2})_{m+n}\right)$
$\displaystyle-2\,\delta\,(m^{2}+n^{2}-mn-1)\,H_{m+n}-2\,\sigma\,\Lambda_{m+n}-2\,\omega\,\Sigma_{m+n}\Big{]}$
(154)

Next we take the large-$\kappa$ limit which amounts to using (151). If one
further replaces $c\rightarrow 12\,\tilde{c}$, $\kappa\rightarrow-\tilde{c}$
and $\sigma\rightarrow 2/{\tilde{c}}$ one obtains the conformal
$\mathfrak{bms}_{3}$ algebra as written down in Fuentealba:2020zkf  after the
following identifications

$\displaystyle H_{p}\rightarrow
i\mathcal{D}_{p}\,,~{}~{}L_{n}\rightarrow\mathcal{J}_{n}.$ (155)

The normal ordered operators in this large-$\kappa$ limit after regularisation
become

$\displaystyle(H^{2})_{m}\rightarrow-\sum_{p}\mathcal{D}_{m-p}\,\mathcal{D}_{p}\,,~{}~{}\Lambda_{m}\rightarrow
i\sum_{p}\mathcal{J}_{m-p}\,\mathcal{D}_{p}\,,~{}~{}\Sigma_{m}\rightarrow-\frac{i}{2\tilde{c}}\sum_{n,l}\mathcal{D}_{m-n-l}\mathcal{D}_{n}\mathcal{D}_{l}$
(156)

completing the identification with the non-linear terms of (3) with the
expressions in (2.2–2.4) of Fuentealba:2020zkf . This makes it clear that the
conformal $\mathfrak{bms}_{3}$ algebra of Fuentealba:2020zkf  is only valid
for large values of their central charge.

###  3.2 $\widetilde{\cal W}(2;2^{2},1)$

Now we turn to the second way the $\mathfrak{sl}(2,\mathbb{R})$ is embedded in
$\mathfrak{so}(2,3)$ where $P^{a}$ instead of being a vector in
$\mathfrak{so}(1,1)$ algebra is vector under $\mathfrak{so}(2)$ algebra. This
gives rise to a different $\mathcal{W}$-algebra extension which closely
resembles ${\cal W}(2;2^{2},1)$ algebra found in previous section. The
structure of all the quasi-primaries and primaries are expected to remain the
same with some minor modifications (such as $H(z)\rightarrow R(z)$) and
therefore we provide only the result here.

  1. 1.

Virasoro Algebra:

$\displaystyle[L_{m},L_{n}]=(m-n)\,L_{m+n}+\frac{c}{12}m(m^{2}-1)\,\delta_{m+n,0}$
(157)

  2. 2.

Virasoro Primaries

$\displaystyle[L_{m},P^{a}_{n}]=(m-n)\,P^{a}_{m+n},~{}~{}[L_{m},R_{n}]=-n\,R_{m+n}$
(158)

  3. 3.

$\mathfrak{so}(2)_{\kappa}$ Current Algebra

$\displaystyle[R_{m},R_{n}]=\kappa\,m\,\delta_{m+n,0}$ (159)

  4. 4.

Current algebra primaries

$\displaystyle[R_{m},P^{a}_{n}]={\epsilon^{ab}}P^{b}_{m+n}$ (160)

  5. 5.

Commutator $[P^{a}_{m},P^{b}_{n}]$

$\displaystyle[P^{a}_{m},P^{b}_{n}]$ $\displaystyle=$
$\displaystyle\delta^{ab}\left[\frac{\alpha}{6}m(m^{2}-1)\delta_{m+n,0}+(m-n)\,\Big{(}\beta\,L_{m+n}+\gamma\,(R^{2})_{m+n}\Big{)}\right]$
(163)
$\displaystyle+\epsilon^{ab}\Big{[}\delta\,(m^{2}+n^{2}-mn-1)\,R_{m+n}+\sigma\,\Lambda_{m+n}+\omega\,\Sigma_{m+n}\Big{]}$

where $\Lambda_{m}$ and $\Sigma_{m}$ are the modes of those in (76) with $H$
replaced by $R$. As before we impose Jacobi identities to solve for unknown
parameters in the $\left[P^{a}_{m},P^{b}_{n}\right]$ commutator. From Jacobi
Identity $(TPP)$ we get the following equations

$\displaystyle
2\alpha-c\,\beta-2\gamma\,\kappa=0,~{}~{}24\,\delta-(2+c)\,\sigma-3\,\omega=0$
(165)

The $(RPP)$ Jacobi Identity demands

$\displaystyle\beta+3\,\delta+2\,\gamma\,\kappa$
$\displaystyle=0,~{}~{}\alpha-6\,\kappa\,\delta=0,$ $\displaystyle
2\,\beta-\kappa\,\sigma$
$\displaystyle=0,~{}~{}2\gamma-\sigma-\frac{3}{2}\,\omega=0.$ (166)

In imposing the Jacobi Identity $(PPP)$ we find similar structures that we saw
before resulting in the following equations,

$\displaystyle 4\,\kappa\,\left(\gamma+\sigma\right)-3\,\omega$
$\displaystyle=0,~{}~{}4\,\kappa\,\left(2\beta-\gamma-2\,\delta+\sigma\right)-\omega=0.$
(167)

These equations (165, 5, 167) with replacements $\gamma\rightarrow-\gamma$ and
$\kappa\rightarrow-\kappa$ match with the equations (136, 137, 146, 147)
obtained in $\mathfrak{so}(1,1)$ case. Hence we obtain the new algebra
$\widetilde{\cal W}(2;2^{2},1)$ where the parameters are given by (148) after
the replacements $\gamma\rightarrow-\gamma$ and $\kappa\rightarrow-\kappa$.

###  3.3 $\mathcal{W}$(2, 4)

Here we start with the third avatar of $\mathfrak{so}(2,3)$ given by,

$\displaystyle\left[L_{m},L_{n}\right]$ $\displaystyle=$
$\displaystyle(m-n)\,L_{m+n},~{}~{}\left[L_{m},W_{n}\right]=(3m-n)\,W_{m+n},$
(168) $\displaystyle\left[W_{m},W_{n}\right]$ $\displaystyle=$
$\displaystyle\frac{1}{1680}A(m,n)\,L_{m+n}+\frac{1}{\sqrt{1680}}B(m,n)\,W_{m+n}$
(169)

where $A(m,n)$ and $B(m,n)$ are as given in (42). Using the method that was
spelt out earlier, in order to obtain the infinite dimensional extension of
this algebra we start with the following ansatz of $\left[W_{m},W_{n}\right]$,

$\displaystyle\left[W_{m},W_{n}\right]$
$\displaystyle=\,\alpha\,\tilde{\alpha}(m,n)+\beta\,A(m,n)\,L_{m+n}+B(m,n)\,\left(\gamma\,W_{m+n}+\delta\,\Lambda_{m+n}\right)$
$\displaystyle+(m-n)\,\Big{[}\omega\,\Delta_{m+n}+\tau\,\Gamma_{m+n}+\sigma\,\Omega_{m+n}\Big{]}$
(170)

where,

$\displaystyle\tilde{\alpha}(m,n)$
$\displaystyle=\frac{m\,(m^{6}-14m^{4}+49m^{2}-36)}{5040}\,\delta_{m+n,0}$
(171)

The modes $\Lambda_{m}$ are associated with the $h=4$ quasi-primary

$\displaystyle\Lambda(z)$
$\displaystyle=\left(TT\right)(z)-\frac{3}{10}\partial^{2}T(z).$ (172)

The modes $\Delta_{m},\Gamma_{m}\,,\Omega_{m}$ are those of the following
three $h=6$ quasi-primaries:

$\displaystyle\Delta(z)$ $\displaystyle=$
$\displaystyle\left(T\,\Lambda\right)(z)-\frac{1}{6}\,\partial^{2}\Lambda(z),$
(173) $\displaystyle\Gamma(z)$ $\displaystyle=$
$\displaystyle\left(\partial^{2}TT\right)(z)-\partial(\partial
TT)(z)+\frac{2}{9}\,\partial^{2}(TT)(z)-\frac{1}{42}\partial^{4}T(z),$ (174)
$\displaystyle\Omega(z)$ $\displaystyle=$
$\displaystyle(TW)(z)-\frac{1}{6}\partial^{2}W(z)\,.$ (175)

In order to obtain equations constraining the parameters
$\\{\alpha,\beta,\gamma,\delta,\omega,\sigma,\tau\\}$ in (3.3) we will impose
Jacobi identities as for the earlier cases. From the Jacobi identity of
$(TWW)$ one gets the following equations,

$\displaystyle\alpha-420\,c\,\beta=0,~{}~{}1960\,\beta-(22+5c)\,\delta=0,~{}~{}14\,\gamma-\frac{1}{12}(24+c)\,\sigma=0,$
$\displaystyle 14\delta-\frac{1}{180}(160\,\tau+3\,\omega(15c+164))=0,$
$\displaystyle
737100\,\beta-315(22+5c)\,\delta-5(29+70c)\,\kappa-42(22+5c)\,\omega=0\,.$
(176)

From imposing $(WWW)$ Jacobi Identity one gets,

$\displaystyle
18900\,\beta-8820\,\delta-315\,\gamma\,(60\,\gamma-7\,\sigma)+95\,\tau+1029\,\omega=0$
(177)

The solution to these equations are as follows,

$\displaystyle\alpha=420\,c\,\beta\,,~{}~{}\delta=\frac{1960\,\beta}{22+5c}\,,~{}~{}\omega=\frac{20160(13+72\,c)\,\beta}{70c^{3}+953c^{2}+2498\,c-1496}\,,$
$\displaystyle\tau=\frac{252\,(-524+19c)\,\beta}{14c^{2}+129c-68}\,,~{}~{}\sigma=\frac{168\,\gamma}{24+c}\,,~{}~{}\gamma^{2}=\frac{70\,\beta(c^{3}-148c^{2}-3932c+4704)}{70c^{3}+953c^{2}+2498c-1496}$
(178)

where $\beta$ is the free parameter, the value of which can be fixed. Choosing
$\beta=\frac{1}{1680}$ one can check that the above algebra is ${\cal W}(2,4)$
algebra as was written down in Bouwknegt:1988sv .

###  3.4 ${\cal W}(2;(3/2)^{2},1^{3})$

This case was dealt with in detail already by us recently in Gupta:2022mdt
motivated from the algebra derived in the context of symmetries responsible
for soft theorems in the context of MHV graviton amplitudes in 4$d$ flat
spacetimes. Here we simply quote the result for the only non-trivial OPE

$\displaystyle G_{s}(z)G_{s^{\prime}}(w)$ $\displaystyle=$ $\displaystyle
2\alpha\,\epsilon_{ss^{\prime}}(z-w)^{-3}+\delta\,(z-w)^{-2}\,{(\lambda^{a})}_{ss^{\prime}}\left[2\,J_{a}(w)+(z-w)\,{J_{a}}^{\prime}(w)\right]$
(181)
$\displaystyle\\!\\!\\!\\!\\!+\beta\,\epsilon_{ss^{\prime}}(z-w)^{-1}\,T(w)+\gamma\,(z-w)^{-1}\,\epsilon_{ss^{\prime}}(J^{2})(w)\,,$

which in terms of modes reads

$\displaystyle\\!\\!\\!\\!\\![G_{s,r},\\!G_{s^{\prime},r^{\prime}}]\\!=\\!\epsilon_{ss^{\prime}}\\!\\!\left[\alpha\\!\left(r^{2}-\frac{1}{4}\right)\\!\delta_{r+r^{\prime},0}\\!+\\!\beta\,L_{r+r^{\prime}}\\!+\\!\gamma\,(J^{2})_{r+r^{\prime}}\right]\\!\\!+\\!\delta(r-r^{\prime})J_{a,r+r^{\prime}}{(\lambda^{a})}_{ss^{\prime}}$
(182)

where $(J^{2})_{n}$ are the modes of the normal ordered quasi-primary
$\eta^{ab}(J_{a}J_{b})(z)$:

$\displaystyle(J^{2})_{n}=\eta^{ab}\left[\sum_{k>-1}J_{b,n-k}J_{a,k}+\sum_{k\leq-1}J_{a,k}J_{b,n-k}\right]$

The result of imposition of the Jacobi identities is the following set of
constraints on the parameters $(\alpha,\beta,\gamma,\delta,c,\kappa)$

$\displaystyle\alpha-\frac{c}{6}\beta+\gamma\,\frac{\kappa}{2}=0,~{}~{}\alpha-\delta\,\frac{\kappa}{2}=0,~{}~{}\beta-\gamma\,(\kappa+2)-\frac{1}{2}\delta=0,~{}~{}\beta-\frac{\gamma-\delta}{2}=0~{}.$

whose solution exists provided

$\displaystyle c=-\frac{6\kappa\,(1+2\kappa)}{5+2\kappa}~{}.$ (185)

When this holds, for generic values of $\kappa$ ($\neq-5/2$) the solution can
be written (taking $\gamma$ to be the independent variable) as

$\displaystyle\alpha=-\frac{1}{4}\gamma\kappa(3+2\kappa),~{}~{}\beta=\frac{1}{4}\gamma(5+2\kappa),~{}~{}\delta=-\frac{1}{2}\gamma(3+2\kappa)\,.$
(186)

This is the final result valid for generic ($\kappa\neq-5/2$) values of
$\kappa$ Gupta:2022mdt . This completes our exercise of finding all minimal
chiral ${\cal W}$-algebra extensions of $\mathfrak{so}(2,3)$.

##  4 Discussion

In this paper we have addressed the question:

What are all the chiral ${\cal W}$-algebra extensions of $\mathfrak{so}(2,3)$?

As the first step we have shown that there are exactly four ways to embed a
copy of $\mathfrak{sl}(2,{\mathbb{R}})$ into $\mathfrak{so}(2,3)$ algebra.
Each such embedding results in a distinct chiral ${\cal W}$-algebra extension
and we have constructed two new ones, and recovered the other two that are
known from Bouwknegt:1988sv ; Gupta:2022mdt  – thus completing the list of
such chiral extension of $\mathfrak{so}(2,3)$. One of our new algebras (namely
${\cal W}(2;2^{2},1)$ with $\mathfrak{h}=\mathfrak{so}(1,1)$) is the conformal
$\mathfrak{bms}_{3}$ whose semi-classical version was first found in
Fuentealba:2020zkf . We offer a few comments below:

  * •

Just as the ${\cal W}_{3}$ algebra of Zamalodchikov Zamolodchikov:1985wn  and
the $W_{3}^{(2)}$ algebra of Bershadsky-Polyakov Polyakov:1989dm ;
Bershadsky:1990bg  could be realised Campoleoni:2010zq ; Henneaux:2010xg ;
Campoleoni:2011hg  in the Chern-Simons theory with
$\mathfrak{sl}(3,{\mathbb{R}})$ gauge algebra, each of the ${\cal W}$-algebras
considered in this paper can also be realised as the asymptotic symmetry
algebra of 3$d$ conformal gravity with appropriate boundary conditions. We
will present this in an upcoming paper gs5-2023 .888The case of conformal
$\mathfrak{bms}_{3}$ from 3$d$ conformal gravity is done in Fuentealba:2020zkf
.

  * •

Each extension we considered depends on the list of simple primaries
Blumenhagen:1990jv  that are declared. As we have seen there are primaries of
higher dimensions in each chiral algebra and we could probably have worked
with other sets of primaries as the simple ones and these might lead to other
extensions $\mathfrak{so}(2,3)$. The choices we made of sets of simple
primaries are with the minimal conformal dimensions, and therefore the
extensions we considered are in a sense minimal. It will be interesting to
construct the non-minimal algebras as well (if they exist) and study them.

  * •

As we have seen, in each of the four chiral ${\cal W}$-algebras we
constructed, the constraints from Jacobi identities admit trivial solutions as
well (where all the parameters vanish and central charges are arbitrary, for
instance, as mentioned in footnote (7) on page 17). These trivial solutions do
not correspond to extensions of $\mathfrak{so}(2,3)$ but instead of different
contractions of it. For instance in the case of ${\cal W}(2;(3/2)^{2},1^{3})$
of Gupta:2022mdt  it corresponds to a chiral conformal algebra extension of
$\mathfrak{iso}(1,3)$ ($\mathfrak{iso}(2,2)$), namely the Poincare algebra –
which is the isometry algebra of ${\mathbb{R}}^{1,3}$ (${\mathbb{R}}^{2,2}$).
In the other three cases the analogous chiral conformal extensions are of
other 10-dimensional algebras that are still contractions of
$\mathfrak{so}(2,3)$. It will be interesting to identify spacetimes which
admit these contraction of $\mathfrak{so}(2,3)$ as their symmetry algebras and
ask if the corresponding infinite dimensional chiral conformal algebra
extensions have any important role in them.

  * •

Even though we aimed to construct chiral ${\cal W}$-algebras it does not mean
that we miss out on algebras that are not chiral. As an example we show in the
appendix A that if we repeat the same exercise for $\mathfrak{so}(2,2)$ we end
up with two chiral infinite dimensional algebras. One of them is indeed a
chiral algebra which is an extension of CIG algebra of Polyakov
Polyakov:1987zb  as derived by Avery:2013dja . The other can be identified
with the non-chiral Brown-Henneaux algebra though seen from the perspective of
the diagonal Virasoro. It will be interesting to explore this question for our
extensions of $\mathfrak{so}(2,3)$ as well.

  * •

Our chiral extensions of $\mathfrak{so}(2,3)$ should be thought of as the
analogues of the Polyakov’s symmetries of chiral induced gravity
Polyakov:1987zb  to three dimensions. Therefore, it is expected that they may
be realised as such. This leads us to predict that these symmetries should be
realised as asymptotic symmetries of $AdS_{4}$ gravity just as the Polyakov
symmetry algebra was realised as the asymptotic symmetries of $AdS_{3}$
gravity in Avery:2013dja . We will demonstrate this elsewhere.

The next step in analysing the algebras of this paper and their utility in the
context of holography would be to construct their representations in terms of
generalised primary operators and study the Ward identities of their
correlation functions. Once that is achieved it will be important to provide
prescription to compute them holographically, either in the 3$d$ conformal
gravity theories or $AdS_{4}$-gravity theories. We hope to address these in
the near future.

## Acknowledgements

We thank Suresh Govindarajan, Bala Sathiapalan and Adarsh Sudhakar for helpful
conversations. Thanks are also due to the participants of the 5th edition of
Chennai Strings Meeting (February 2023) for helpful comments on parts of this
work presented there.

##  Appendix A Chiral conformal extensions of $\mathfrak{so}(2,2)$

In this appendix we construct the chiral conformal/${\cal W}$-algebra
extensions of $\mathfrak{so}(2,2)$. Repeating the steps of section 2 we can
take the $L_{0}$ to belong to the Cartan subalgebra and we choose it such that
the eigenvalues are real. Taking the generator to be $L_{\mu\nu}$ for
$\mu,\nu\in\\{0,1,2,3\\}$ we take

$\displaystyle L_{0}=\lambda L_{01}+\lambda^{\prime}\,L_{23}$ (187)

We can using $O(2,2)$ transformations

  1. 1.

$(\lambda,\lambda^{\prime})\leftrightarrow(-\lambda,\lambda^{\prime})$ using
$\Lambda={\rm diag.}(1,-1,1,1)$

  2. 2.

$(\lambda,\lambda^{\prime})\leftrightarrow(-\lambda,\lambda^{\prime})$ using
$\Lambda={\rm diag.}(1,1,-1,1)$

  3. 3.

$(\lambda,\lambda^{\prime})\leftrightarrow(\lambda^{\prime},\lambda)$ using
$\Lambda_{mn}=(-1)^{m}\delta_{m+n,3}$

So we can again choose $\lambda\geq\lambda^{\prime}\geq 0$. Then the existence
of $L_{\pm 1}$ demands that $(\lambda,\lambda^{\prime})$ satisfy one of the
following four conditions: (i) $\lambda+\lambda^{\prime}=1$, (ii)
$\lambda-\lambda^{\prime}=1$, (iii) $\lambda+\lambda^{\prime}=-1$, (iv)
$\lambda-\lambda^{\prime}=-1$. Since we have already assumed
$\lambda\geq\lambda^{\prime}\geq 0$ we need to consider only two cases: (i)
$\lambda+\lambda^{\prime}=1$ and (ii) $\lambda-\lambda^{\prime}=1$. In each
case we can solve the constraints to find the following results:

  1. 1.

$\lambda=1~{}~{}\&~{}~{}\lambda^{\prime}=0$.

$\displaystyle L_{0}$ $\displaystyle=$ $\displaystyle
L_{01},~{}~{}(b_{03}^{2}-b_{12}^{2})L_{1}=b_{12}(L_{02}-L_{12})+b_{03}(L_{03}-L_{13})$
(188) $\displaystyle L_{-1}$ $\displaystyle=$ $\displaystyle
b_{12}(L_{02}+L_{12})+b_{03}(L_{03}+L_{13})$ (189)

The residual $O(2,3)$ transformations (that keep these form-invariant) can be
seen to leave $(b_{03}^{2}-b_{12}^{2})$ invariant up to a positive scaling :
$(b_{03}^{2}-b_{12}^{2})\rightarrow e^{\beta}(b_{03}^{2}-b_{12}^{2})$. Using
the discrete elements of $O(2,3)$ we can change the signs of $b_{03}$ and
$b_{12}$ independently. Thus using these residual transformations we can
choose $(b_{03},b_{12})$ to be either $(1,0)$ or $(0,1)$.

  2. 2.

$\lambda=1/2=\lambda^{\prime}$

$\displaystyle L_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(L_{01}+L_{23}),~{}~{}L_{1}=-a_{12}(L{02}+L_{03}-L_{12}-L_{13}),$
(190) $\displaystyle L_{-1}$ $\displaystyle=$
$\displaystyle\frac{1}{4a_{12}}(L_{02}-L_{03}+L_{12}-L_{13})$ (191)

The residual transformations can be used to fix $a_{12}=1/2$.

This analysis gives us three inequivalent embeddings of
$\mathfrak{sl}(2,{\mathbb{R}})$ in $\mathfrak{so}(2,2)$ – resulting in three
different avatars of $\mathfrak{so}(2,2)$.

  1. 1.

$\mathfrak{h}=\emptyset$ with $(b_{03},b_{12})=(1,0)$

$\displaystyle L_{1}$ $\displaystyle=$ $\displaystyle
L_{03}-L_{13},~{}~{}L_{0}=L_{01},~{}~{}L_{-1}=L_{03}+L_{13},$ (192)
$\displaystyle P_{1}$ $\displaystyle=$ $\displaystyle
L_{02}-L_{12},~{}~{}P_{0}=L_{23},~{}~{}P_{-1}=-L_{02}-L_{12}.$ (193)
$\displaystyle[L_{m},L_{n}]$ $\displaystyle=$
$\displaystyle(m-n)\,L_{m+n}=[P_{m},P_{n}],~{}~{}[L_{m},P_{n}]=(m-n)P_{m+n},$
(194)

  2. 2.

$\mathfrak{h}=\emptyset$ with $(b_{03},b_{12})=(0,1)$

$\displaystyle L_{1}$ $\displaystyle=$ $\displaystyle-
L_{02}+L_{12},~{}~{}L_{0}=L_{01},~{}~{}L_{-1}=L_{02}+L_{12},$ (196)
$\displaystyle P_{1}$ $\displaystyle=$ $\displaystyle
L_{03}-L_{13},~{}~{}P_{0}=-L_{23},~{}~{}P_{-1}=L_{03}+L_{13}.$ (197)
$\displaystyle[L_{m},L_{n}]$ $\displaystyle=$
$\displaystyle(m-n)\,L_{m+n}=[P_{m},P_{n}],~{}~{}[L_{m},P_{n}]=(m-n)P_{m+n},$
(198)

  3. 3.

$\mathfrak{h}=\mathfrak{sl}(2,{\mathbb{R}})$

$\displaystyle L_{1}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}(L_{02}+L_{03}-L_{12}-L_{13}),$ (200) $\displaystyle
L_{-1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(L_{02}-L_{03}+L_{12}-L_{13}),~{}~{}L_{0}=\frac{1}{2}(L_{01}+L_{23})$
(201) $\displaystyle J_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(-L_{02}+L_{03}+L_{12}-L_{13}),$ (202) $\displaystyle
J_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(L_{01}-L_{03}),~{}~{}L_{-1}=\frac{1}{2}(L_{02}+L_{03}+L_{12}+L_{13}).$
(203)
$\displaystyle[L_{m},L_{n}]=(m-n)L_{m+n},~{}~{}[J_{a},J_{b}]=(a-b)J_{a+b},~{}~{}[L_{m},J_{a}]=0$
(204)

Unlike in the $\mathfrak{so}(2,3)$ case the first two avatars above with
$\mathfrak{h}=\emptyset$, even though cannot be mapped to each other by any
$O(2,3)$ transformations nevertheless give rise to identical algebras. So we
have two distinct avatars of $\mathfrak{so}(2,2)$ which can be extended to the
following chiral algebras:

  1. 1.

$\mathfrak{h}=\mathfrak{sl}(2,{\mathbb{R}})$

$\displaystyle[L_{m},L_{n}]$ $\displaystyle=$
$\displaystyle(m-n)L_{m+n}+\frac{c}{12}m(m^{2}-1)\delta_{m+n,0},$ (205)
$\displaystyle[J_{a,m},J_{b,n}]$ $\displaystyle=$
$\displaystyle(a-b)J_{a+b,m+n}-\frac{1}{2}\kappa(a^{2}+b^{2}-ab-1)m\,\delta_{m+n,0},$
(206) $\displaystyle[L_{m},J_{a,n}]$ $\displaystyle=$
$\displaystyle-n\,J_{a,m+n}$ (207)

The Jacobi identity imposes no constraints on $c$ and $\kappa$.

  2. 2.

$\mathfrak{h}=\emptyset$

$\displaystyle[L_{m},L_{n}]$ $\displaystyle=$
$\displaystyle(m-n)L_{m+n}+\frac{c}{12}m(m^{2}-1)\delta_{m+n,0},$ (208)
$\displaystyle[L_{m},P_{n}]$ $\displaystyle=$
$\displaystyle(m-n)P_{m+n}+\frac{\tilde{c}}{12}m(m^{2}-1)\delta_{m+n,0},$
(209) $\displaystyle[P_{m},P_{n}]$ $\displaystyle=$
$\displaystyle(m-n)L_{m+n}+\gamma\,m(m^{2}-1)\delta_{m+n,0}$ (210)

Imposing the Jacobi identity here leads to the condition that
$\gamma=\frac{c}{12}$.

Thus we arrive at the result that there are exactly two chiral conformal
algebra extensions of $\mathfrak{so}(2,2)$. The first one is the CIG algebra
Avery:2013dja ; Apolo:2014tua  (however, with no relation between $c$ and
$\kappa$) and the second can be mapped to the non-chiral Brown-Henneaux one
(with no relation between left and right Virasoro central charges) after
appropriate redefinitions. Specifically, if we take
$L_{n}\rightarrow\ell_{n}+\bar{\ell}_{n}$ and
$P_{n}\rightarrow\ell_{n}-\bar{\ell}_{n}$ we will recover two commuting copies
of Virasoros generated by $\ell_{n}$ with central charge $(c+\tilde{c})/2$ and
$\bar{\ell}_{n}$ with central charge $(c-\tilde{c})/2$ respectively. It is
interesting that even though we started with chiral conformal/W-algebra
extensions we seemed to get both chiral and non-chiral extensions of
$\mathfrak{so}(2,2)$ already known.

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