# Structure of deformed $w_{1+\infty}$ symmetry and topological generalization
in Celestial CFT

Pavel Drozdov Università degli Studi di Udine, Italy  Taro Kimura Institut de
Mathématiques de Bourgogne, Université de Bourgogne, CNRS, France

###### Abstract

We investigate a deformation of $w_{1+\infty}$ algebra recently introduced in
[MRSV21] in the context of Celestial CFT that we denote by
$\widetilde{W}_{1+\infty}$ algebra. We obtain the operator product expansions
of the generating currents of this algebra and explore its supersymmetric and
topological generalizations.

###### Contents

  1. 1 Introduction
  2. 2 Background: soft current OPEs in CCFT
  3. 3 $\widetilde{W}_{1+\infty}$ algebra and its structure
    1. 3.1 The commutation relations
    2. 3.2 The OPEs for $\widetilde{W}_{1+\infty}$ algebra
  4. 4 Supersymmetric and topological generalizations
    1. 4.1 $\mathcal{N}=2$ supersymmetric extension
    2. 4.2 Topological twist
  5. 5 Conclusion and outlook

##  1 Introduction

Celestial holography provides new powerful tools for studying quantum gravity
by applying a holographic principle to asymptotically flat spacetimes (AFS)
[PPR21]. It was shown in [HS19, GHPS21, DPS19] that each of the soft theorems
(the linear relations between scattering amplitudes of soft particles, i.e.,
the particles with energy $\omega\to 0$) in gauge theory and gravity in AFS is
associated with an infinite number of symmetries and conservation laws on the
celestial sphere. The known symmetries do not close under commutation,
implying an infinite tower of soft theorems. Moreover, it has been also argued
that the soft theorems in AFS are related to measurable gravitational memory
effects via Fourier transform [Str18, SZ14] together with a discussion towards
its experimental setups [HTLT20].

In the recent works [GHPS21, Str21, HPS22], it has been shown that the soft
currents exhibit the so-called $w_{1+\infty}$ symmetry [Pop91, PRS90], where
the infinite tower of soft symmetries can be organized into a single symmetry
based on the wedge algebra of $w_{1+\infty}$. Along this direction, it has
been pointed out in [MRSV21] that one can consider a more general setup, which
involves non-minimal couplings in the bulk on the operator product expansions
(OPEs) of soft currents in the Celestial Conformal Field Theory (CCFT). The
corresponding algebra of the currents provides a deformation of $w_{1+\infty}$
algebra, that we call $\widetilde{W}_{1+\infty}$ algebra. Although
$\widetilde{W}_{1+\infty}$ algebra bears semblance to $W_{1+\infty}$ algebra
[PRS90], it shows several specific properties that we briefly review in
Section 3.1.

A purpose of this paper is to obtain a deeper understanding of algebraic
aspects of $\widetilde{W}_{1+\infty}$ algebra. First, we derive the OPEs for
the generating currents of $\widetilde{W}_{1+\infty}$ algebra by generalizing
the approaches used for $W_{1+\infty}$ algebra. This results in Theorem 3.4 in
Section 3.2. Secondly, we explore further generalization of
$\widetilde{W}_{1+\infty}$ algebra. In particular, we introduce an
$\mathcal{N}=2$ supersymmetric extention of $\widetilde{W}_{1+\infty}$ algebra
in Section 4.1, and we identify a BRST operator (Theorem 4.2) in Section 4.2.
Finally, based on the OPEs for $\mathcal{N}=2$ $\widetilde{W}_{\infty}$
algebra together with the BRST operator, we construct the generators of the
topologically twisted version $\widetilde{W}_{\infty}$ algebra (Theorem 4.3)
in Section 4.1. In Section 5, we conclude by discussing potential issues and
future directions.

##  2 Background: soft current OPEs in CCFT

Let us discuss the basics of the soft current in CCFT. We follow the notation
of [Jia22].

######  Definition 2.1 (Soft current).

Let $\mathcal{O}_{h,\bar{h}}$ be a positive-helicity conformal primary
celestial operator with conformal spin $s\in\frac{1}{2}\mathbb{Z}_{>0}$ and
conformal dimensions $(h,\bar{h})=\left(\frac{k+s}{2},\frac{k-s}{2}\right)$,
where we denote the scaling dimension by $k=s,s-1,s-2,\dots$. The soft current
is defined by

$H^{k,s}(z,\bar{z})\coloneqq\lim_{\Delta\to
k}(\Delta-k)\mathcal{O}_{h,\bar{h}}(z,\bar{z})=\lim_{\epsilon\to
0}\epsilon\mathcal{O}_{\frac{k+s}{2},\frac{k-s}{2}}(z,\bar{z}),$ (2.1)

where $(z,\bar{z})$ is the codimension-two celestial sphere coordinate.

We start from the OPEs for these soft currents from a three-point interaction
[HPS22]:

$\mathcal{O}_{h_{1},\bar{h}_{1}}(z,\bar{z})\mathcal{O}_{h_{2},\bar{h}_{2}}(0,0)\sim\frac{1}{z}\sum_{p}C_{p}(\bar{h}_{1},\bar{h}_{2})\bar{z}^{p}\mathcal{O}_{h_{1}+h_{2}-1,\
\bar{h}_{1}+\bar{h}_{2}+p}(0,0),$ (2.2)

where $p\coloneqq d_{V}-4$ with $d_{V}$ being a bulk dimension of the three-
point interaction coupling particles, $C_{p}(\bar{h}_{1},\bar{h}_{2})$ denotes
the OPE coefficients and “$\sim$” means equality modulo expressions regular as
$(z,\bar{z})\to(0,0)$. Taking the limit $\epsilon\to 0$ as in (2.1), one
obtain the OPE for soft currents shown in [HPS22]:

$H^{k_{1},s_{1}}\left(z,\bar{z}\right)H^{k_{2},s_{2}}\left(w,\bar{w}\right)\sim-\sum_{p}\frac{\kappa_{s_{1},s_{2},-s_{I}}}{2}\frac{1}{z-w}\sum_{\alpha=0}^{\infty}\frac{\left(\bar{z}-\bar{w}\right)^{\alpha+p}}{\alpha!}\\\
\times\begin{pmatrix}-2\bar{h}_{1}-2\bar{h}_{2}-2p-\alpha\\\
-2\bar{h}_{2}-p\end{pmatrix}\bar{\partial}^{\alpha}H^{k_{1}+k_{2}+p-1,s_{1}+s_{2}-p-1}{(w,\bar{w})},$
(2.3)

where $\kappa_{s_{1},s_{2},-s_{I}}$ is the coupling constant of the relevant
three-point amplitude,
$\bar{\partial}\coloneqq\frac{\partial}{\partial\bar{w}}$ and
$s_{I}=s_{1}+s_{2}-p-1$.

We use the following antiholomorphic mode expansion of the celestial currents:

$H^{k,s}(z,\bar{z})=\sum_{n=\bar{h}}^{-\bar{h}}H_{n}^{k,s}(z)\bar{z}^{-n-\bar{h}}=\sum_{n=\frac{k-s}{2}}^{\frac{s-k}{2}}\frac{H_{n}^{k,s}(z)}{\bar{z}^{n+\frac{k-s}{2}}},$
(2.4)

where each $H^{k,s}_{n}(z)$ is a 2D symmetry-generating conserved current
whose Ward identity provides a soft theorem.111Note that (2.4) holds in the
MHV sector. Beyond it the polynomial becomes infinite and we should change the
lower limit to $-\infty$ [MRSV21]. Note that we have a $z$-dependence (but not
$\bar{z}$) for the Fourier modes of the celestial currents.

The OPE and the commutation relations constitute an equivalent description of
the corresponding algebra. We define the following commutator [MRSV21].

######  Definition 2.2 (2D-commutator).

$\displaystyle\left[H_{m}^{k_{1},s_{1}},H_{n}^{k_{2},s_{2}}\right](w)\coloneqq{\oint}_{w}\frac{\operatorname{d}z}{2\pi\operatorname{i}}{\oint}_{\epsilon}\frac{\operatorname{d}\bar{z}}{2\pi\operatorname{i}}\bar{z}^{m+\bar{h}_{1}-1}{\oint}_{\epsilon}\frac{\operatorname{d}\bar{w}}{2\pi\operatorname{i}}\bar{w}^{n+\bar{h}_{2}-1}H^{k_{1},s_{1}}\left(z,\bar{z}\right)H^{k_{2},s_{2}}\left(w,\bar{w}\right),$
(2.5)

where $\oint_{w}$ and $\oint_{\epsilon}$ denotes the integration over the
contours around $w$ and $0$ respectively.222See [MRSV21, App. C] for more
detailed discussion.

######  Proposition 2.3 (Algebra of celestial currents [MRSV21]).

The following algebraic relation holds for the celestial current modes defined
in (2.4),

$\left[H_{m}^{k_{1},s_{1}},H_{n}^{k_{2},s_{2}}\right](w)=-\sum_{p\,=\,\operatorname{Max}(s_{1}+s_{2}-3,0)}^{\operatorname{Max}(s_{1}+s_{2}+1,0)}\frac{\kappa_{s_{1},s_{2},-s_{I}}}{2}\sum_{x=0}^{p}\left[(-1)^{p-x}\binom{p}{x}\right.\\\
\times\left.\binom{m+n-\bar{h}_{1}-\bar{h}_{2}-p}{m-\bar{h}_{1}-p+x}\binom{-m-n-\bar{h}_{1}-\bar{h}_{2}-p}{-m-\bar{h}_{1}-x}\right]H^{k_{1}+k_{2}+p-1,s_{1}+s_{2}-p-1}_{m+n}(w).$
(2.6)

For the gauge theory currents, we replace
$\frac{1}{2}\kappa_{s_{1},s_{2},-s_{I}}\mapsto\operatorname{i}f\indices{{}^{ab}_{c}}$
in the formula (2.6), where $f\indices{{}^{ab}_{c}}$ is the structure
constant, associated with the gauge symmetry.

##  3 $\widetilde{W}_{1+\infty}$ algebra and its structure

###  3.1 The commutation relations

In the absence of higher-derivative corrections (fixed $p$ in the expansion
(2.6)), the soft currents for gravity and gauge theory obey the OPE relations
of the generating currents of the $w_{1+\infty}$ algebra [GHPS21, Str21,
HPS22]. Its supersymmetric extension has been also discussed [HPS22, Jia22,
Ahn22].

Let us consider the effect of non-minimal couplings in the bulk on the OPEs of
soft currents [MRSV21]. We use the light transform of the celestial operator
[KS18]:

$\displaystyle{\bf\bar{L}}\left[\mathcal{O}_{h,\bar{h}}(z,\bar{z})\right]=\int_{\mathbb{R}}\frac{\operatorname{d}\bar{y}}{2\pi\operatorname{i}}\frac{1}{(\bar{z}-\bar{y})^{2-2\bar{h}}}\mathcal{O}_{h,\bar{h}}\left(z,\bar{y}\right)$
(3.1)

to redefine the currents.

######  Definition 3.1 ($\widetilde{W}_{1+\infty}$ currents).

We define

$\displaystyle\widetilde{W}^{q,s}(z,\bar{z})\coloneqq\Gamma\left(2q\right){\bf{\bar{L}}}\left[H^{s+2(1-q),s}(z,\bar{z})\right],$
(3.2)

where $q=1,\frac{3}{2},2,\dots$ for particles with helicity $s>0$ and
$\Gamma(x)$ is the Gamma function.

This implies the following mode expansion

$\displaystyle\widetilde{W}^{q,s}(z,\bar{z})=\sum_{m=\bar{h}}^{-\bar{h}}\widetilde{W}^{q,s}_{m}\left(z\right)\bar{z}^{-m-\bar{h}}$
(3.3)

and

$\displaystyle\widetilde{W}^{q,s}_{m}\left(z\right)=\left(-m+q-1\right)!\left(m+q-1\right)!H_{m}^{s+2(1-q),s}\left(z\right).$
(3.4)

Using these currents, we can simplify the commutator (2.6) and obtain:

######  Proposition 3.2 ($\widetilde{W}_{1+\infty}$ algebra [MRSV21]).

$\widetilde{W}_{1+\infty}$ current modes obey the following algebraic
relation,

$\left[\widetilde{W}_{m}^{q_{1},s_{1}},\,\widetilde{W}_{n}^{q_{2},s_{2}}\right]=-\sum_{p\,=\,\operatorname{Max}(s_{1}+s_{2}-3,0)}^{\operatorname{Max}(s_{1}+s_{2}+1,0)}\frac{\kappa_{s_{1},s_{2},-s_{I}}}{2}N\left(q_{1},q_{2},m,n,p\right)\widetilde{W}_{m+n}^{q_{1}+q_{2}-p-1,s_{1}+s_{2}-p-1},$
(3.5)

where $q_{1},q_{2}=1,\frac{3}{2},2,\dots$ for particles with helicities
$s_{1},s_{2}>0$, $m,n\in\mathbb{Z}$ and

$N\left(q_{1},q_{2},m,n,p\right)\coloneqq\sum_{x=0}^{p}(-1)^{p-x}\binom{p}{x}\,[m+q_{1}-1]_{p-x}\
[-m+q_{1}-1]_{x}\\\ \times[n+q_{2}-1]_{x}\ [-n+q_{2}-1]_{p-x},$ (3.6)

with ascending and descending Pochhammer symbols

$\displaystyle(a)_{n}\coloneqq a(a+1)\dots(a+n-1)$
$\displaystyle=\frac{(a+n-1)!}{(a-1)!},$ (3.7a) $\displaystyle[a]_{n}\coloneqq
a(a-1)\dots(a-n+1)$ $\displaystyle=\frac{a!}{(a-n)!}.$ (3.7b)

Note that the coefficient $N\left(q_{1},q_{2},m,n,p\right)$ vanishes in (3.5)
for specific values of $p$.

We call this $\widetilde{W}_{1+\infty}$ algebra.333The currents (3.2) are
referred to as $W_{1+\infty}$-like currents in [MRSV21]. In this regard, our
$\widetilde{W}^{q,s}$ corresponds to their $W^{q,s}$. In the absense of
higher-derivative corrections (fixed $p$), this algebra is reduced to
$w_{1+\infty}$ algebra [Str21, HPS22, Ahn22]. The $\widetilde{W}_{1+\infty}$
algebra (3.5) bears semblance to $W_{1+\infty}$ algebra, which is also given
by a deformation of $w_{1+\infty}$ [PRS90]. The key difference is that the
truncation of $\widetilde{W}_{1+\infty}$ algebra (3.5) occurs naturally, due
to the absence of massless higher spin fields, while it is provided by non-
trivial zeroes of the hypergeometric function $\phi^{q_{1},q_{2}}_{2r}$ for
$W_{1+\infty}$ [PRS90].

We have the coefficient function $N\left(q_{1},q_{2},m,n,p\right)$ for
$\widetilde{W}_{1+\infty}$ algebra as in $W_{1+\infty}$ algebra. For further
calculations, we will use another representation:

######  Lemma 3.3 ([PRS90]).

The coefficient function $N\left(q_{1},q_{2},m,n,p\right)$ is rewritten in the
following form,

$N\left(q_{1},q_{2},m,n,p\right)=\sum_{x=0}^{p}(-1)^{p-x}\binom{p}{x}\
[2q_{2}-2-x]_{p-x}\ (2q_{1}-1-p)_{x}\\\ \times[m+q_{1}-1]_{p-x}\
[n+q_{2}-1]_{x}.$ (3.8)

###  3.2 The OPEs for $\widetilde{W}_{1+\infty}$ algebra

In this Section, we provide the OPE for $\widetilde{W}_{1+\infty}$ algebra.
For this purpose, we apply the procedure described in [PSR90] originally used
for $W_{1+\infty}$ algebra. In the context of CCFT, we have to take into
account a $\bar{z}$-dependence, in contrast to the original situation
discussing the chiral currents [PSR90]. We apply the following steps:

  1. 1.

We define a new function

$M\left(q_{1},q_{2},m,n,p\right)\coloneqq\sum_{x=0}^{p}(-1)^{-x}\binom{p}{x}\
[2q_{2}-2-x]_{p-x}\ (2q_{1}-1-p)_{x}\ m^{p-x}\ n^{x}.$ (3.9)

This function generates the terms of $N\left(q_{1},q_{2},m,n,p\right)$ of
total degree $p$.

  2. 2.

We replace the coefficient in the $\widetilde{W}$-commutator (3.5) as follows,

$N\left(q_{1},q_{2},m,n,p\right)\mapsto
M\left(q_{1},q_{2},\frac{\partial}{\partial\bar{z}},\frac{\partial}{\partial\bar{w}},p\right),$
(3.10)

from which we can extract the OPE from the commutation relation of the
generators.

  3. 3.

We add the factors, $(-1)$, $\frac{1}{\bar{z}-\bar{w}}$ and $\frac{1}{z-w}$ to
capture the corresponding poles and comply with the Definition 2.2.

Here is the main result of this paper.

######  Theorem 3.4 (OPE for $\widetilde{W}_{1+\infty}$-currents).

The OPE for the $\widetilde{W}$-currents (3.2) has the following form:

$\widetilde{W}^{q_{1},s_{1}}(z,\bar{z})\widetilde{W}^{q_{2},s_{2}}(w,\bar{w})\sim\sum_{p}\frac{\kappa_{s_{1},s_{2},-s_{I}}}{2}\frac{1}{z-w}\sum_{x=0}^{p}(-1)^{-x}\binom{p}{x}(2q_{1}-1-p)_{x}\\\
\times[2q_{2}-2-x]_{p-x}\ \bar{\partial}^{p-x}_{\bar{z}}\
\bar{\partial}^{x}_{\bar{w}}\left(\frac{\widetilde{W}^{q_{1}+q_{2}-1-p,\
s_{1}+s_{2}-p-1}(w,\bar{w})}{\bar{z}-\bar{w}}\right).$ (3.11)

###### Proof.

We substitute the expression (3.11) to Definition 2.2. The expression takes
the form

$\left[\widetilde{W}_{m}^{q_{1},s_{1}},\widetilde{W}_{n}^{q_{2},s_{2}}\right](w)=\oint_{w}\frac{\operatorname{d}z}{2\pi\operatorname{i}}{\oint}_{\epsilon}\frac{\operatorname{d}\bar{w}}{2\pi\operatorname{i}}\bar{w}^{n+q_{2}-1}{\oint}_{\epsilon}\frac{\operatorname{d}\bar{z}}{2\pi\operatorname{i}}\bar{z}^{m+q_{1}-1}\widetilde{W}^{q_{1},s_{1}}\left(z,\bar{z}\right)\widetilde{W}^{q_{2},s_{2}}\left(w,\bar{w}\right),$
(3.12)

The integral part is

${\oint}_{\epsilon}\frac{\operatorname{d}\bar{w}}{2\pi\operatorname{i}}\bar{w}^{n+q_{2}-1}\frac{\partial^{x}}{\partial\bar{w}^{x}}{\oint}_{\epsilon}\frac{\operatorname{d}\bar{z}}{2\pi\operatorname{i}}\bar{z}^{m+q_{1}-1}\frac{\partial^{p-x}}{\partial\bar{z}^{p-x}}\left(\frac{\widetilde{W}^{q_{1}+q_{2}-1-p,\
s_{1}+s_{2}-p-1}(w,\bar{w})}{\bar{z}-\bar{w}}\right)\\\
=(-1)^{p-x}(p-x)!{\oint}_{\epsilon}\frac{\operatorname{d}\bar{w}}{2\pi\operatorname{i}}\bar{w}^{n+q_{2}-1}\frac{\partial^{x}}{\partial\bar{w}^{x}}\bigg{[}\widetilde{W}^{q_{1}+q_{2}-1-p,\
s_{1}+s_{2}-p-1}(w,\bar{w})\\\ \hskip
142.26378pt{\oint}_{\epsilon}\frac{\operatorname{d}\bar{z}}{2\pi\operatorname{i}}\bar{z}^{m+q_{1}-1}\left(\bar{z}-\bar{w}\right)^{-p+x-1}\bigg{]}\\\
=(-1)^{-1+m+q_{1}}(p-x)!\frac{[-p+x-1]_{-m-q_{1}}}{(-m-q_{1})!}{\oint}_{\epsilon}\frac{\operatorname{d}\bar{w}}{2\pi\operatorname{i}}\bar{w}^{n+q_{2}-1}\\\
\times\frac{\partial^{x}}{\partial\bar{w}^{x}}\bigg{[}\widetilde{W}^{q_{1}+q_{2}-1-p,\
s_{1}+s_{2}-p-1}(w,\bar{w})\ \bar{w}^{-p+x-1+m+q_{1}}\bigg{]}.$ (3.13)

We use a Fourier decomposition (3.3):

$\displaystyle\widetilde{W}^{q_{1}+q_{2}-1-p,\ s_{1}+s_{2}-p-1}(w,\bar{w})$
$\displaystyle=\sum_{j=\bar{h}}^{-\bar{h}}\widetilde{W}^{q_{1}+q_{2}-1-p,\
s_{1}+s_{2}-p-1}_{j}(w)\ \bar{w}^{-j-\bar{h}}$
$\displaystyle=\sum_{j}\widetilde{W}^{q_{1}+q_{2}-1-p,\
s_{1}+s_{2}-p-1}_{j}(w)\ \bar{w}^{-j-q_{1}-q_{2}+1+p},$ (3.14)

with $\bar{h}=q_{1}+q_{2}-1-p$. Then we use an elementary formula

$\frac{\operatorname{d}^{x}}{\operatorname{d}t^{x}}\left(t^{n}\right)=\frac{n!}{(n-x)!}t^{n-x}=[n]_{x}\,t^{n-x}.$
(3.15)

for the $x$-th derivative over $\bar{w}$ in (3.13):

$\frac{\partial^{x}}{\partial\bar{w}^{x}}\bigg{[}\widetilde{W}^{q_{1}+q_{2}-1-p,\
s_{1}+s_{2}-p-1}(w,\bar{w})\ \bar{w}^{-p+x-1+m+q_{1}}\bigg{]}\\\
\mathop{=}^{\eqref{th-w-fourier-
decomp}}\sum_{j}\frac{\partial^{x}}{\partial\bar{w}^{x}}\bigg{[}\widetilde{W}^{q_{1}+q_{2}-1-p,\
s_{1}+s_{2}-p-1}_{j}(w)\ \bar{w}^{x+m-j-q_{2}}\bigg{]}\\\
\mathop{=}^{\eqref{th-elementary-formula}}\sum_{j}[x+m-j-q_{2}]_{x}\
\widetilde{W}^{q_{1}+q_{2}-1-p,\ s_{1}+s_{2}-p-1}_{j}(w)\ \bar{w}^{m-j-
q_{2}}.$ (3.16)

We substitute (3.16) into (3.13), and the integral over $\bar{w}$ provides

$\sum_{j}[x+m-j-q_{2}]_{x}\ \widetilde{W}^{q_{1}+q_{2}-1-p,\
s_{1}+s_{2}-p-1}_{j}(w)\oint_{\epsilon}\frac{\operatorname{d}\bar{w}}{2\pi\operatorname{i}}\bar{w}^{m+n-j-1}\\\
=\sum_{j}[x+m-j-q_{2}]_{x}\ \widetilde{W}^{q_{1}+q_{2}-1-p,\
s_{1}+s_{2}-p-1}_{j}(w)\delta_{j,\ m+n}\\\ =[x-n-
q_{2}]_{x}\widetilde{W}^{q_{1}+q_{2}-1-p,\ s_{1}+s_{2}-p-1}_{m+n}(w).$ (3.17)

The last integral is

$\oint_{w}\frac{\operatorname{d}z}{2\pi\operatorname{i}}\frac{1}{(z-w)}=1.$
(3.18)

Bringing together (3.12), (3.13), (3.17) and (3.18), we have

$\left[\widetilde{W}_{m}^{q_{1},s_{1}},\widetilde{W}_{n}^{q_{2},s_{2}}\right](w)=-\sum_{p}\frac{\kappa_{s_{1},s_{2},-s_{I}}}{2}\sum_{x=0}^{p}(-1)^{-x}\binom{p}{x}(2q_{1}-1-p)_{x}[2q_{2}-2-x]_{p-x}\\\
\times(-1)^{-1+m+q_{1}}[x-n-q_{2}]_{x}\
\frac{(p-x)!}{(-m-q_{1})!}[-p+x-1]_{-m-q_{1}}\widetilde{W}^{q_{1}+q_{2}-1-p,\
s_{1}+s_{2}-p-1}_{m+n}(w).$ (3.19)

Now, using the definitions of the Pochhammer symbols (3.7a)-(3.7b) and their
relation

$[a]_{n}=(-1)^{n}(-a)_{n}=(a-n+1)_{n},$ (3.20)

we can transform:

$[x-n-
q_{2}]_{x}=(-1)^{x}(-x+n+q_{2})_{x}=(1-n-q_{2})_{x}=(-1)^{x}[n+q_{2}-1]_{x};$
(3.21a)

$\displaystyle\frac{(p-x)!}{(-m-q_{1})!}[-p+x-1]_{-m-q_{1}}$
$\displaystyle=\frac{(p-x)!}{(-m-q_{1})!}(-1)^{-m-q_{1}}(p-x+1)_{-m-q_{1}}$
$\displaystyle=(-1)^{-m-q_{1}+p-x}[m+q_{1}-1]_{p-x}.$ (3.21b)

Substituting (3.21a), (3.21b) to (3.19), we arrive at the final result

$\displaystyle\left[\widetilde{W}_{m}^{q_{1},s_{1}},\widetilde{W}_{n}^{q_{2},s_{2}}\right](w)$
$\displaystyle\mathop{=}^{\phantom{\eqref{N-lemma-
eq}}}-\sum_{p}\frac{\kappa_{s_{1},s_{2},-s_{I}}}{2}\sum_{x=0}^{p}(-1)^{p-x}\binom{p}{x}(2q_{1}-1-p)_{x}\
[n+q_{2}-1]_{x}$ $\displaystyle\hskip 14.22636pt\times\
[2q_{2}-2-x]_{p-x}[m+q_{1}-1]_{p-x}\widetilde{W}^{q_{1}+q_{2}-1-p,\
s_{1}+s_{2}-p-1}_{m+n}(w)$ $\displaystyle\mathop{=}^{\eqref{N-lemma-
eq}}-\sum_{p}\frac{\kappa_{s_{1},s_{2},-s_{I}}}{2}N\left(q_{1},q_{2},m,n,p\right)\widetilde{W}^{q_{1}+q_{2}-1-p,\
s_{1}+s_{2}-p-1}_{m+n}(w).$ (3.22)

∎

##  4 Supersymmetric and topological generalizations

The $\mathcal{N}=2$ supersymmetric $W_{\infty}$ algebra can be twisted to
construct a topological $W_{\infty}$ algebra [PRSS91], which has been also
discussed in the context of CCFT in particular for $w_{\infty}$ algebra
[Ahn22]. Here we apply a similar construction for $\widetilde{W}_{1+\infty}$
algebra (3.5). For this purpose, we first construct an $\mathcal{N}=2$
supersymmetric extension of $\widetilde{W}_{\infty}$ algebra.444The notation
“super $\widetilde{W}_{\infty}$” (and not “$1+\infty$”) comes from the fact
that the corresponding supersymmetric algebra should contain both
$\widetilde{W}_{1+\infty}\times\widetilde{W}_{\infty}$ as a bosonic sector.

###  4.1 $\mathcal{N}=2$ supersymmetric extension

The general structure of the $\mathcal{N}=2$ super $W_{\infty}$ algebra can be
schematically drawn as follows [BPR+90]:

$\displaystyle WW$ $\displaystyle\sim W$ $\displaystyle W\tilde{W}$
$\displaystyle\sim 0$ $\displaystyle G^{\pm}G^{\pm}$ $\displaystyle\sim 0$
$\displaystyle\tilde{W}\tilde{W}$ $\displaystyle\sim\tilde{W}$ $\displaystyle
G^{\pm}G^{\mp}$ $\displaystyle\sim W\oplus\tilde{W}$ (4.1) $\displaystyle
WG^{\pm}$ $\displaystyle\sim G^{\pm}$ $\displaystyle\tilde{W}G^{\pm}$
$\displaystyle\sim G^{\pm}$

Here $W$ are $W_{\infty}$ currents, $\tilde{W}$ are $W_{1+\infty}$ currents
(${W}_{1+\infty}\times{W}_{\infty}$ subalgebras constitute a bosonic sector of
the new superalgebra), and $G^{\pm}$ are fermionic currents. In order to have
this structure for $\widetilde{W}_{\infty}$ algebra, we define the fermionic
currents $G^{q+}$ and $G^{q-}$:

$\\{G^{q_{1}-}_{m},G^{q_{2}+}_{n}\\}=-\sum_{p}\frac{\kappa_{s_{1},s_{2},-s_{I}}}{2}N(q_{1},q_{2},m,n,p)\big{(}b_{q_{1},q_{2},x}^{m,n,p}\widetilde{W}^{q_{1}+q_{2}-p-1}_{m+n}+(-1)^{p}\tilde{b}_{q_{1},q_{2},x}^{m,n,p}{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{W}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{W}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{W}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{W}$}}}}{}^{q_{1}+q_{2}-p-1}_{m+n}\big{)},$
(4.2)

with additional structure constants $b_{q_{1},q_{2},x}^{m,n,p}$ and
$\tilde{b}_{q_{1},q_{2},x}^{m,n,p}$ (unknown at the moment). Here
$\widetilde{W}$ are $\widetilde{W}_{\infty}$ currents,
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{W}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{W}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{W}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{W}$}}}}$
are $\widetilde{W}_{1+\infty}$ currents. For the fermionic current $G^{q\pm}$,
we require $s\in\mathbb{Z}_{>0}+\frac{1}{2}$ so that they generate the
fermionic sector of the algebra. Using the Theorem 3.4, we may switch (4.2) to
the OPE form:

######  Corollary 4.1.

The OPE of the fermionic currents of $\mathcal{N}=2$ $\widetilde{W}_{\infty}$
algebra is given by

$G^{q_{1}-}(z,\bar{z})G^{q_{2}+}(w,\bar{w})\sim\sum_{p}\frac{\kappa_{s_{1},s_{2},-s_{I}}}{2}\frac{1}{z-w}\sum_{x=0}^{p}(-1)^{-x}\binom{p}{x}(2q_{1}-1-p)_{x}\\\
\times[2q_{2}-2-x]_{p-x}\bar{\partial}^{p-x}_{\bar{z}}\
\bar{\partial}^{x}_{\bar{w}}\bigg{[}\frac{1}{(\bar{z}-\bar{w})}\bigg{(}B_{q_{1},q_{2}}^{p,x}\widetilde{W}^{q_{1}+q_{2}-1-p}{(w,\bar{w})}\\\
+(-1)^{p}\tilde{B}_{q_{1},q_{2}}^{p,x}{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{W}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{W}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{W}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{W}$}}}}{}^{q_{1}+q_{2}-1-p}{(w,\bar{w})}\bigg{)}\bigg{]},$
(4.3)

where the new coefficients $B_{q_{1},q_{2}}^{p,x}$ and
$\tilde{B}_{q_{1},q_{2}}^{p,x}$ which are independent of the Fourier degrees
$(m,n)$, are determined from $b_{q_{1},q_{2},x}^{m,n,p}$ and
$\tilde{b}_{q_{1},q_{2},x}^{m,n,p}$ in the same way via the function
$M\left(q_{1},q_{2},\frac{\partial}{\partial\bar{z}},\frac{\partial}{\partial\bar{w}},p\right)$
from $N\left(q_{1},q_{2},m,n,p\right)$ as before.

In order to calculate some particular values of the structure constants
$B_{q_{1},q_{2}}^{p,x}$ and $\tilde{B}_{q_{1},q_{2}}^{p,x}$ in (4.3), we may
use explicit realizations of our currents. Let us introduce ghost fields with
the following OPE:

$b_{i}(z,\bar{z})c_{j}(w,\bar{w})\sim\frac{1}{z-w}\frac{\delta_{ij}}{\bar{z}-\bar{w}}.$
(4.4)

Then we introduce an explicit realization of the $\widetilde{W}_{1+\infty}$
currents (3.11) with $q\geq 2$:

$\widetilde{W}^{q,s}(z,\bar{z})\coloneqq\sum_{k\geq
0}\kappa_{s_{1},s_{2},-s_{0}}(q+2)\colon\bar{\partial}_{\bar{z}}c_{k}(z,\bar{z})b_{q-1+k}(z,\bar{z})\colon\\\
+\kappa_{s_{1},s_{2},-s_{0}}(k+1)\colon
c_{k}(z,\bar{z})\bar{\partial}_{\bar{z}}b_{q-1+k}(z,\bar{z})\colon+\dots,$
(4.5)

where $s_{0}\coloneqq s_{I}(p=1)=s_{1}+s_{2}-2$, $\colon\cdot\,\colon$ means
normal-ordered product and $\dots$ states for the terms of the form
$\colon\bar{\partial}^{\alpha}_{\bar{z}}c_{k}(z,\bar{z})\bar{\partial}^{\beta}_{\bar{z}}b_{q-1+k}(z,\bar{z})\colon$
with total degree $\alpha+\beta\geq 2$. Similarly, we use the anticommuting
ghosts $\tilde{b}_{i}(z,\bar{z})$ and $\tilde{c}_{j}(w,\bar{w})$ to construct
a realization of the $\widetilde{W}_{\infty}$ algebra, where we denote the
currents by
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{W}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{W}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{W}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{W}$}}}}{}^{q,s}(z,\bar{z})$.

Using Wick’s theorem and (4.4), one can check that the currents $w^{q}$, for
which the terms given in (4.5) constitute the complete expression, i.e.

$w^{q}(z,\bar{z})\coloneqq-\sqrt{\kappa_{s_{1},s_{2},-s_{0}}}\sum_{k\geq
0}(q+2)\colon\bar{\partial}_{\bar{z}}c_{k}(z,\bar{z})b_{q-1+k}(z,\bar{z})\colon+(k+1)\colon
c_{k}(z,\bar{z})\bar{\partial}_{\bar{z}}b_{q-1+k}(z,\bar{z})\colon$ (4.6)

generate the $w_{1+\infty}$ algebra [PRS91]555Here, no central term is
considered.:

$w^{q_{1}}(z,\bar{z})w^{q_{2}}(w,\bar{w})\sim-\kappa_{s_{1},s_{2},-s_{0}}(q_{1}+q_{2}-2)\frac{w^{q_{1}+q_{2}-2}(w,\bar{w})}{(\bar{z}-\bar{w})^{2}}\\\
-\kappa_{s_{1},s_{2},-s_{0}}(q_{1}-2)\frac{\partial
w^{q_{1}+q_{2}-2}(w,\bar{w})}{\bar{z}-\bar{w}}.$ (4.7)

The OPE (4.7) coincides with the main OPE (3.11) in $p=0$ approximation (with
no higher-derivative corrections). Hence, in general, the currents (4.5)
generate the algebra (3.11). The expression for the explicit realization is
given by

$\widetilde{W}^{q,s}(z,\bar{z})\coloneqq\sum_{k,n\geq
0\atop\alpha+\beta=n}\alpha_{\alpha,\beta}^{q,k}\colon\bar{\partial}^{\alpha}_{\bar{z}}c_{k}(z,\bar{z})\bar{\partial}^{\beta}_{\bar{z}}b_{q-1+k}(z,\bar{z})\colon,$
(4.8)

where

$\displaystyle\alpha^{q,k}_{0,1}$
$\displaystyle=-\sqrt{\kappa_{s_{1},s_{2},-s_{0}}}(q+2),$ (4.9)
$\displaystyle\alpha^{q,k}_{0,1}$
$\displaystyle=-\sqrt{\kappa_{s_{1},s_{2},-s_{0}}}(k+1),$ (4.10)
$\displaystyle\dots$

Now, in addition to the anticommuting ghosts $b_{i}(z,\bar{z})$,
$c_{j}(z,\bar{z})$ for $\widetilde{W}^{q,s}$ and $\tilde{b}_{i}(z,\bar{z})$,
$\tilde{c}_{j}(z,\bar{z})$ for
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{W}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{W}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{W}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{W}$}}}}{}^{q,s}$,
we introduce the commuting ghosts $\beta_{i}(z,\bar{z})$,
$\gamma_{j}(z,\bar{z})$ for $G^{q-}(z,\bar{z})$ and
$\bar{\beta}_{i}(z,\bar{z})$, $\bar{\gamma}_{j}(z,\bar{z})$ for
$G^{q+}(z,\bar{z})$ [PRS91]:

$\displaystyle\bar{\beta}_{i}(z,\bar{z})\gamma_{j}(w,\bar{w})$
$\displaystyle\sim\beta_{i}(z,\bar{z})\bar{\gamma}_{j}(w,\bar{w})\sim\frac{1}{z-w}\frac{\delta_{ij}}{\bar{z}-\bar{w}},$
(4.11a) $\displaystyle\bar{\gamma}_{i}(z,\bar{z})\beta_{j}(w,\bar{w})$
$\displaystyle\sim\gamma_{i}(z,\bar{z})\bar{\beta}_{j}(w,\bar{w})\sim-\frac{1}{z-w}\frac{\delta_{ij}}{\bar{z}-\bar{w}}.$
(4.11b)

Using this, one can calculate

$\displaystyle
G^{q_{1}-}(z,\bar{z})G^{q_{2}+}(w,\bar{w})\sim\frac{2\kappa_{s_{1},s_{2},-s_{0}}}{\bar{z}-\bar{w}}\widetilde{W}^{q_{1}+q_{2}-1}$
$\displaystyle-2\kappa_{s_{1},s_{2},-s_{0}}\frac{(q_{1}+q_{2}-2)}{(\bar{z}-\bar{w})^{2}}{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{W}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{W}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{W}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{W}$}}}}{}^{q_{1}+q_{2}-1}(w,\bar{w})$
$\displaystyle-2\kappa_{s_{1},s_{2},-s_{0}}\frac{(q_{1}-1)}{(\bar{z}-\bar{w})}\frac{\partial}{\partial\bar{w}}{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{W}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{W}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{W}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{W}$}}}}{}^{q_{1}+q_{2}-1}(w,\bar{w})+\dots$
(4.12)

or, in the anticommutator form (using mode expansions):

$\displaystyle\\{G^{q_{1}-}_{m},G^{q_{2}+}_{n}\\}=2\kappa_{s_{1},s_{2},-s_{0}}\widetilde{W}^{q_{1}+q_{2}-1}_{m+n}-2\kappa_{s_{1},s_{2},-s_{0}}[m(q_{2}-1)-n(q_{1}-1)]{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{W}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{W}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{W}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{W}$}}}}{}^{q_{1}+q_{2}-1}_{m+n}+\dots$
(4.13)

By comparing (4.12) and (4.3), we can finally determine some values of the
structure constants $B_{q_{1},q_{2}}^{p,x}$ and
$\tilde{B}_{q_{1},q_{2}}^{p,x}$ and conditions for them:

$\displaystyle B^{1,0}_{q_{1},q_{2}}$ $\displaystyle=0,$ (4.14a)
$\displaystyle\tilde{B}^{1,0}_{q_{1},q_{2}}$
$\displaystyle=-2\cdot\frac{(2q_{1}+q_{2}-3)}{(q_{2}-1)},$ (4.14b)
$\displaystyle B^{1,1}_{q_{1},q_{2}}$ $\displaystyle=0,$ (4.14c)
$\displaystyle\tilde{B}^{1,1}_{q_{1},q_{2}}$ $\displaystyle=-2,$ (4.14d)
$\displaystyle\dots$

Further structure constants may be obtained by the following algorithm. First,
one needs to build a realization of the higher-order terms of the algebra OPE.
For this purpose, it would be helpful to consider terms with $\alpha+\beta\geq
2$ in (4.8). After OPE calculation
$\widetilde{W}^{q_{1},s_{1}}(z,\bar{z})\widetilde{W}^{q_{2},s_{2}}(w,\bar{w})$,
using Wick’s theorem and expanding, one can compare the obtained expression
with our result (3.11), and thereby determine $\alpha^{q,k}_{\alpha,\beta}$
for some further $\alpha,\beta$. Then, using the obtained
$\alpha^{q,k}_{\alpha,\beta}$, one can similarly construct realizations of the
fermionic currents $G^{q,s\pm}$ and repeat the procedure to calculate the
structure constants $B^{p,x}_{q_{1},q_{2}}$ and
$\tilde{B}^{p,x}_{q_{1},q_{2}}$.

Here we calculated some particular values of the structure constants
(4.14a)-(4.14d) and provided an algorithm to do it for the cases $p,x>1$,
while its practical implementation might require more effort from the
calculation point of view.

###  4.2 Topological twist

The first step towards the topological twist is to define the BRST operator:

######  Theorem 4.2 (BRST operator for super $\widetilde{W}_{\infty}$).

The BRST operator for the super $\widetilde{W}_{\infty}$ algebra is given by

$Q=G^{3/2+}_{-1/2}(z),$ (4.15)

where the lower index numerates the Fourier mode coefficient.

###### Proof.

The BRST operator is given by the contour integral of the fermionic current,

$\displaystyle
Q=\oint_{\epsilon}\frac{\operatorname{d}\bar{z}}{2\pi\operatorname{i}}G^{3/2+}(z,\bar{z})$
$\displaystyle=\sum_{r}\oint_{\epsilon}\frac{\operatorname{d}\bar{z}}{2\pi\operatorname{i}}G^{3/2+}_{r}(z)\
\bar{z}^{-r-3/2}$
$\displaystyle=\sum_{r}G^{3/2+}_{r}(z)\delta_{r,-1/2}=G^{3/2+}_{-1/2}(z).$
(4.16)

∎

We again note that we have an additional $\bar{w}$-dependence for celestial
currents. Now let us proceed to determine the generators of the topological
$\widetilde{W}_{\infty}$ algebra.

######  Theorem 4.3 (Generators of topological $\widetilde{W}_{\infty}$
algebra).

We define the new generators by

$\hat{V}^{q}(w,\bar{w})\coloneqq\left\\{Q,G^{q-}(w,\bar{w})\right\\},$ (4.17)

which are BRST-exact and hence generate a topological algebra. They are given
by

$\displaystyle\hat{V}^{q}(w,\bar{w})=\sum_{p}\frac{\kappa_{s_{1},s_{2},-s_{I}}}{2}\frac{1}{z-w}\frac{\partial^{p}}{\partial\bar{w}^{p}}\bigg{(}(-1)^{p}B_{3/2,q}^{p,p}\widetilde{W}^{-\frac{1}{2}+q-p}{(w,\bar{w})}$
$\displaystyle+\tilde{B}_{3/2,q}^{p,p}{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{W}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{W}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{W}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{W}$}}}}{}^{-\frac{1}{2}+q-p}{(w,\bar{w})}\bigg{)}.$
(4.18)

###### Proof.

The definition (4.17) can be re-written as follows:

$\hat{V}^{q}(w,\bar{w})=\oint_{\bar{w}}\frac{\operatorname{d}\bar{z}}{2\pi\operatorname{i}}G^{3/2+}(z,\bar{z})G^{q-}(w,\bar{w}),$
(4.19)

To obtain $G^{3/2+}(z,\bar{z})G^{q-}(w,\bar{w})$, we use the OPE (4.3) for the
case $q_{1}=\dfrac{3}{2}$, $q_{2}\eqqcolon q\in\mathbb{Z}/2$ and take the
derivative over $\bar{z}$:

$G^{3/2+}(z,\bar{z})G^{q-}(w,\bar{w})\sim\sum_{p}\frac{\kappa_{s_{1},s_{2},-s_{I}}}{2}\frac{1}{z-w}\sum_{x=0}^{p}\binom{p}{x}(2-p)_{x}[2q-2-x]_{p-x}(p-x)!\\\
\times\frac{\partial^{x}}{\partial\bar{w}^{x}}\bigg{[}(\bar{z}-\bar{w})^{-p+x-1}\bigg{(}(-1)^{p}B_{3/2,q}^{p,x}\widetilde{W}^{-\frac{1}{2}+q-p}{(w,\bar{w})}+\tilde{B}_{3/2,q}^{p,x}{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{W}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{W}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{W}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{W}$}}}}{}^{-\frac{1}{2}+q-p}{(w,\bar{w})}\bigg{)}\bigg{]}.$
(4.20)

The integration over $\bar{z}$ gives:

$\oint_{\bar{w}}\frac{\operatorname{d}\bar{z}}{2\pi\operatorname{i}}\frac{1}{(\bar{z}-\bar{w})^{p-x+1}}=\delta_{p,x}$
(4.21)

This allows us to remove the sum over $x$ and obtain

$\displaystyle\hat{V}^{q}(w,\bar{w})=\sum_{p}\frac{\kappa_{s_{1},s_{2},-s_{I}}}{2}\frac{1}{z-w}\frac{\partial^{p}}{\partial\bar{w}^{p}}\bigg{(}(-1)^{p}B_{3/2,q}^{p,p}\widetilde{W}^{-\frac{1}{2}+q-p}{(w,\bar{w})}$
$\displaystyle+\tilde{B}_{3/2,q}^{p,p}{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{W}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{W}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{W}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{W}$}}}}{}^{-\frac{1}{2}+q-p}{(w,\bar{w})}\bigg{)}.$
(4.22)

∎

Now let us discuss the algebraic relations. We re-write (4.17) in terms of the
Fourier modes

$\hat{V}^{q}_{m}(w,\bar{w})\coloneqq\left\\{Q,G^{q-}_{m+1/2}(w)\right\\},$
(4.23)

to compute the following commutator:

$\displaystyle[\hat{V}^{q_{1}}_{m},\hat{V}^{q_{2}}_{n}]$
$\displaystyle\mathop{=}^{\eqref{def-topological-generators-
fourier}}\left[\left\\{G^{3/2+}_{-1/2}(z),\ G^{q_{1}-}_{m+1/2}(z)\right\\},\
\left\\{G^{3/2+}_{-1/2}(z),\ G^{q_{2}-}_{n+1/2}(z)\right\\}\right]$
$\displaystyle\mathop{=}^{\phantom{\eqref{def-topological-generators-
fourier}}}\left\\{G^{3/2+}_{-1/2}(z),\ \left[G^{q_{2}-}_{n+1/2}(z),\
\left\\{G^{3/2+}_{-1/2}(z),\ G^{q_{1}-}_{m+1/2}(z)\right\\}\right]\right\\}$
$\displaystyle\mathop{+}^{\phantom{\eqref{def-topological-generators-
fourier}}}\left\\{G^{q_{2}-}_{n+1/2}(z),\ \left[\left\\{G^{3/2+}_{-1/2}(z),\
G^{q_{1}-}_{m+1/2}(z)\right\\},\ G^{3/2+}_{-1/2}(z)\right]\right\\}$
$\displaystyle\mathop{=}^{\phantom{\eqref{def-topological-generators-
fourier}}}\left\\{G^{3/2+}_{-1/2}(z),\ \left[G^{q_{2}-}_{n+1/2}(z),\
\left\\{G^{3/2+}_{-1/2}(z),\ G^{q_{1}-}_{m+1/2}(z)\right\\}\right]\right\\},$
(4.24)

where we used the Jacobi identity. The conditions for the Jacobi identity to
hold in the case of $\widetilde{W}_{1+\infty}$ were obtained in [MRSV21]:

$\displaystyle\frac{\kappa_{0,1,1}}{\kappa_{-2,2,2}}=\frac{\kappa_{1,1,2}}{\kappa_{0,2,2}};\quad\frac{\kappa_{-1,1,1}}{\kappa_{-1,1,2}}=\frac{\kappa_{1,1,1}}{3\kappa_{1,1,2}};\quad\kappa_{-1,1,1}=\kappa_{0,0,1};\quad\kappa^{2}_{0,1,1}=2\kappa_{1,1,1}\kappa_{-1,1,1}.$
(4.25)

We assume that the conditions are the same for the super
$\widetilde{W}_{\infty}$, hence the Jacobi identity holds here as well.
However, it might be possible to refine these conditions using the provided
explicit realizations (4.8) in future work.

Hence from (4.1) we see that the right-hand side of (4.24) leads to
$\hat{V}_{l}^{q}$ again. We can write

$[\hat{V}^{q_{1}}_{m},\hat{V}^{q_{2}}_{n}]=\sum_{p}\hat{g}(q_{1},q_{2},m,n,p)\hat{V}^{q_{1}+q_{2}-p-1}_{m+n},$
(4.26)

where structure constants $\hat{g}$ could be expressed in terms of
$b_{q_{1},q_{2},x}^{m,n,p}$ and $\tilde{b}_{q_{1},q_{2},x}^{m,n,p}$ from
(4.2).

In the same way, we can obtain other relations for the generators, and thus
the full structure of topological $\widetilde{W}_{\infty}$ algebra is given in
the form:

$\displaystyle[\hat{V}^{q_{1}}_{m},\hat{V}^{q_{2}}_{n}]$
$\displaystyle=\sum_{p}\hat{g}(q_{1},q_{2},m,n,p)\hat{V}^{q_{1}+q_{2}-p-1}_{m+n},$
(4.27a) $\displaystyle[\hat{V}^{q_{1}}_{m},\hat{G}^{q_{2}}_{n+1/2}]$
$\displaystyle=\sum_{p}\hat{g}(q_{1},q_{2},m,n,p)\hat{G}^{q_{1}+q_{2}-p-1}_{m+n+1/2},$
(4.27b) $\displaystyle\\{G^{q_{1}}_{m+1/2},G^{q_{2}}_{n+1/2}\\}$
$\displaystyle=0.$ (4.27c)

Although determining the structure constants $\hat{g}$ is difficult in
general, we are able to compare them with a specific example. It has been
known that $\widetilde{W}_{\infty}$ is reduced to $w_{\infty}$ as follows:
rescaling the generators

$v^{q}_{m}\to\lambda^{q-2}\hat{V}^{q}_{m},\quad
G^{q}_{m}\to\lambda^{q-2}\hat{G}^{q}_{m},$ (4.28)

and then taking the limit $\lambda\to 0$ with $p=1$. Hence, from (4.27a) and
(4.27b), we obtain

$\displaystyle[v^{q_{1}}_{m},v^{q_{2}}_{n}]$
$\displaystyle=\hat{g}(q_{1},q_{2},m,n,1)v^{q_{1}+q_{2}-2}_{m+n},$ (4.29a)
$\displaystyle[v^{q_{1}}_{m},\hat{G}^{q_{2}}_{n+1/2}]$
$\displaystyle=\hat{g}(q_{1},q_{2},m,n,1)\hat{G}^{q_{1}+q_{2}-2}_{m+n+1/2}.$
(4.29b)

Compared with the results obtained in [Ahn22], we conclude that

$\hat{g}(q_{1},q_{2},m,n,1)=m(q_{2}-1)-n(q_{1}-1).$ (4.30)

##  5 Conclusion and outlook

In this work, we investigated a deformation of $w_{1+\infty}$ that we call
$\widetilde{W}_{1+\infty}$ algebra (3.5), which was previously obtained in
[MRSV21] in the context of CCFT. This algebra, despite being different from
the known $W_{1+\infty}$, this algebra shares several common properties, which
allows us to apply analogous approaches to study its algebraic structure. In
particular, we obtained the OPE for $\widetilde{W}_{1+\infty}$-algebra
(Theorem 3.4, Section 3.2).

We constructed the BRST operator of $\mathcal{N}=2$ supersymmetric
$\widetilde{W}_{\infty}$-algebra (Theorem 4.2) and then obtained the
generators of the topological $\widetilde{W}_{\infty}$-algebra (Theorem 4.3)
in Section 4.2. We determined the structure constants $B_{q_{1},q_{2}}^{p,x}$
and $\tilde{B}_{q_{1},q_{2}}^{p,x}$ in (4.3) in several particular cases and
outline the possible algorithm of treatment the others. Along this direction,
calculating the structure constants in more cases (by, for example, practical
implementation of the provided in 4.1 algorithm), or even obtaining a closed-
form expression might be a subject of future work.

It would be also important to revisit the issue of physical interpretation of
the supersymmetric and topological generalizations, presented in this paper in
the context of CCFT and compare them to the known results concerning the
interpretation of $W$-symmetry in CCFT in the non-commutative setup [Mon23,
BHS22, Gue22, BHS23].

## Acknowledgements

P. D. acknowledges the financial support from Math4Phys program of Université
de Bourgogne & Université Bourgogne Franche-Comté. This work was in part
supported by EIPHI Graduate School (No. ANR-17-EURE-0002) and Bourgogne-
Franche-Comté region.

## References

  * [Ahn22] C. Ahn, _Toward A Supersymmetric w(1+infinity) Symmetry in the Celestial Conformal Field Theory_ , Phys. Rev. D 105 (2022), no. 8, 086028, arxiv:2111.04268 [hep-th].
  * [BHS22] W. Bu, S. Heuveline, and D. Skinner, _Moyal deformations, $W_{1+\infty}$ and celestial holography_, October 2022.
  * [BHS23] R. Bittleston, S. Heuveline, and D. Skinner, _The Celestial Chiral Algebra of Self-Dual Gravity on Eguchi-Hanson Space_ , May 2023.
  * [BPR+90] E. Bergshoeff, C. Pope, L. Romans, E. Sezgin, and X. Shen, _The super W $\infty$ algebra_, Physics Letters B 245 (1990), no. 3-4, 447–452.
  * [DPS19] L. Donnay, A. Puhm, and A. Strominger, _Conformally Soft Photons and Gravitons_ , J. High Energ. Phys. 2019 (2019), no. 1, 184, arxiv:1810.05219 [hep-th].
  * [GHPS21] A. Guevara, E. Himwich, M. Pate, and A. Strominger, _Holographic Symmetry Algebras for Gauge Theory and Gravity_ , J. High Energ. Phys. 2021 (2021), no. 11, 152, arxiv:2103.03961 [hep-th].
  * [Gue22] A. Guevara, _Towards gravity from a color symmetry_ , 2022.
  * [HPS22] E. Himwich, M. Pate, and K. Singh, _Celestial Operator Product Expansions and w 1+infinity Symmetry for All Spins_ , J. High Energ. Phys. 2022 (2022), no. 1, 80, arxiv:2108.07763 [hep-th].
  * [HS19] E. Himwich and A. Strominger, _Celestial Current Algebra from Low’s Subleading Soft Theorem_ , Phys. Rev. D 100 (2019), no. 6, 065001, arxiv:1901.01622 [hep-th].
  * [HTLT20] M. Hübner, C. Talbot, P. D. Lasky, and E. Thrane, _Thanks for the memory: measuring gravitational-wave memory in the first LIGO/Virgo gravitational-wave transient catalog_ , Phys. Rev. D 101 (2020), no. 2, 023011, arxiv:1911.12496 [astro-ph, physics:gr-qc].
  * [Jia22] H. Jiang, _Holographic Chiral Algebra: Supersymmetry, Infinite Ward Identities, and EFTs_ , J. High Energ. Phys. 2022 (2022), no. 1, 113, arxiv:2108.08799 [hep-th].
  * [KS18] P. Kravchuk and D. Simmons-Duffin, _Light-ray operators in conformal field theory_ , J. High Energ. Phys. 2018 (2018), no. 11, 102, arxiv:1805.00098 [hep-th, physics:math-ph].
  * [Mon23] R. Monteiro, _Celestial chiral algebras, colour-kinematics duality and integrability_ , Journal of High Energy Physics 2023 (2023), no. 1.
  * [MRSV21] J. Mago, L. Ren, A. Y. Srikant, and A. Volovich, _Deformed w(1+infinity) Algebras in the Celestial CFT_ , November 2021.
  * [Pop91] C. N. Pope, _Lectures on W algebras and W gravity_ , December 1991.
  * [PPR21] S. Pasterski, M. Pate, and A.-M. Raclariu, _Celestial Holography_ , November 2021.
  * [PRS90] C. Pope, L. Romans, and X. Shen, _The complete structure of W $\infty$_, Physics Letters B 236 (1990), no. 2, 173–178.
  * [PRS91] , _Conditions for anomaly-free W and super-W algebras_ , Physics Letters B 254 (1991), no. 3-4, 401–410.
  * [PRSS91] C. Pope, L. Romans, E. Sezgin, and X. Shen, _W-topological matter and gravity_ , Physics Letters B 256 (1991), no. 2, 191–198.
  * [PSR90] C. N. Pope, X. Shen, and L. J. Romans, _W(infinity) and the Racah-wigner Algebra_ , Nuclear Physics B 339 (1990), no. 1, 191–221.
  * [Str18] A. Strominger, _Lectures on the Infrared Structure of Gravity and Gauge Theory_ , February 2018.
  * [Str21] , _w(1+infinity) and the Celestial Sphere_ , November 2021.
  * [SZ14] A. Strominger and A. Zhiboedov, _Gravitational Memory, BMS Supertranslations and Soft Theorems_ , November 2014.