# Towards Gravity From a Color Symmetry

Alfredo Guevara<EMAIL_ADDRESS>Center for the Fundamental
Laws of Nature, Society of Fellows, & Black Hole Initiative, Harvard
University, Cambridge, MA 02138, USA

###### Abstract

Using tools from color-kinematics duality we propose a holographic
construction of gravitational amplitudes, based on a 2d Kac-Moody theory on
the celestial sphere. In the $N\to\infty$ limit the gauge group corresponds to
$w_{1+\infty}$, due to the $U(N)$ generators enjoying a simple quantum group
structure, which is in turn inherited from a twistor fiber over the celestial
sphere.

We show how four-dimensional momentum-space is emergent in this picture, which
connects directly to the so-called kinematic algebra of the tree-level
S-Matrix. On the other hand, the framework can be embedded within a celestial
CFT to make contact with holographic symmetry algebras previously observed in
the soft expansion. Kac-Moody currents play the role of a graviton to all
orders in such expansion, and also lead to a natural notion of Goldstone modes
for $w_{1+\infty}$. Focusing on MHV amplitudes, main examples are a BCFW type
recursion relation and holomorphic three-point amplitudes.

##  I Introduction

Diverse aspect of gravitational theories suggest that their dynamics enjoy
integrable properties. For one, the relation between (super)gravity and
(super)YM theories via the double copy/color-kinematics duality Bern  _et al._
(2019) suggests certain that avatars of Yangian symmetry Dolan, Nappi, and
Witten (2004); Drummond, Henn, and Plefka (2009); Arkani-Hamed  _et al._
(2016) can emerge in perturbative gravity amplitudes Bjerrum-Bohr and Vanhove
(2008); Bern, Bjerrum-Bohr, and Dunbar (2005); Bern  _et al._ (2018).
Classical solutions of interest, such as the Kerr black hole and its
deformations, also enjoy hidden symmetries reflecting integrability in their
dynamics Frolov, Krtous, and Kubiznak (2017); Buonanno and Damour (1999). The
relation between such classical and quantum notions of integrability is a
topic of active research unearthing intriguing connections, see e.g. Caron-
Huot and Henn (2014); Caron-Huot and Zahraee (2019); Aminov, Grassi, and
Hatsuda (2022); Bianchi  _et al._ (2022).

On a parallel venue, the celestial holography program de Boer and Solodukhin
(2003); Cheung, de la Fuente, and Sundrum (2017); Pasterski and Shao (2017);
Pasterski, Shao, and Strominger (2017); Pasterski (2021); Fan  _et al._ (2022)
aims to reformulate scattering in asymptotically flat spacetimes as governed
by a putative CFT in the celestial sphere. The defining recipe is to recast
the four-dimensional S-Matrix in a boost eigenstate basis where it can be
interpreted as a correlation function. From a pragmatical perspective it is
expected that celestial correlation functions can be determined with help of
the constraints from conformal symmetry and extensions thereof, alleviating
the need for quantum gravity computations. A first motivation for this was the
realization of universal soft theorems in gravity in the language of the
celestial CFT (CCFT) Donnay, Puhm, and Strominger (2019); Puhm (2020); Guevara
(2019); Adamo, Mason, and Sharma (2019). By pushing the soft analysis to
arbitrary orders within the conformal framework Guevara (2019), recently a
novel hierarchy based on the $w_{1+\infty}$ symmetry algebra has been
undercovered in the gravitational CCFT Guevara  _et al._ (2021); Strominger
(2021). This algebra and its quantization have a long history with deep
connections to integrability Maulik and Okounkov (2012); Briot and Ragoucy
(2013); Procházka (2016); Hoppe (2008) and Penrose’s twistor construction
Penrose (1976); Penrose and Rindler (2011). Because of this, it is pressing to
understand its implications in the gravitational S-Matrix, directly in
momentum space, investigating to what extent scattering is constrained.

In this letter we take a first step in this direction by introducing a 2d Kac-
Moody theory that reproduces certain gravitational amplitudes through a
celestial dictionary. Following the lore of color-kinematics duality, we show
how four-dimensional momentum space emerges as the $w_{1+\infty}\cong
U(N\to\infty)$ color group of the theory, thus explicitly manifests the
hierarchy in the S-Matrix language.

##  II From the S-Matrix to $w_{1+\infty}$

To present the idea we first point out that the algebra has a universal
imprint in the S-Matrix via collinear factorizations. In perturbative gravity
amplitudes, such are given by splitting functions which also persist in a
class of loop amplitudes Bern  _et al._ (1998); Ball  _et al._ (2022),
nevertheless here we will focus on the tree-level instance.

Consider the scattering amplitude $\mathcal{A}_{n}(p_{i})$ dressed with
momentum conservation. For massless particles collinear singularities
correspond to the three-point on-shell vertices which are unique for given
helicities Benincasa and Cachazo (2007). We consider only helicity-preserving
vertices whose coupling $16\pi G=1$ is further fixed by the equivalence
principle, see Appendix II for the generic case. The collinear limit is then
universal and reads

$\displaystyle\mathcal{A}_{n{+}1}(k^{+2}$ $\displaystyle,p_{1}^{\pm
h},\ldots,p_{n})$ $\displaystyle\rightarrow\frac{(\epsilon^{+}\cdot
p_{1})^{2}}{k\cdot p_{1}}\mathcal{A}_{n}((k{+}p_{1})^{\pm h},\ldots,p_{n})\,,$
(1)

We have emphasized the helicities through superscripts $p^{\pm h}$. Hereafter
we use $(2,2)$ signature where the Lorentz group is
$SL(2,\mathbb{R})_{z}\times SL(2,\mathbb{R})_{\bar{z}}$ Atanasov  _et al._
(2021): We write momentum vectors in terms of spinors as Elvang and Huang
(2013)

$p_{i}=|\eta_{i}\rangle[\lambda_{i}|\,\,\,\,,k=|\eta\rangle[\lambda|\,\,\,,$
(2)

so that $|\eta_{i}\rangle=(1,z_{i}),$ and $|\lambda_{i}]$ is an independent
spinor carrying the energy scale. Similarly for the graviton momenta $k$,
which has coordinate e.g. $z_{p}$ and $|\lambda]$. Our collinear limit is then
$z_{p}\to z_{1}$ where $\lambda,\lambda_{1}$ are kept generic. This yields
$k+p_{1}=|\eta_{1}\rangle([\lambda|+[\lambda_{1}|)$. With a concording change
of notation, eq. (1) can be written

$\mathcal{A}_{n}^{\lambda\lambda_{1}\cdots\lambda_{n}}(z_{p},z_{1},\ldots,z_{n})\sim\frac{[\lambda\lambda_{1}]}{z_{p1}}\mathcal{A}_{n}^{\lambda_{1}+\lambda,\cdots\lambda_{n}}(z_{1},\ldots,z_{n})\,,$
(3)

with $[\lambda\xi]=\epsilon^{\alpha\beta}\lambda_{\alpha}\xi_{\beta}$. In the
following we will reinterpret this as a $SL(2,\mathbb{R})_{z}$ covariant OPE
associated to a graviton operator and a massless helicity-$h$ field:

$\mathbf{G}^{\lambda}(z_{p})\mathbf{O}_{h}^{\lambda_{1}}(z_{1})\sim\frac{[\lambda\lambda_{1}]}{z_{p1}}\mathbf{O}_{h}^{\lambda+\lambda_{1}}(z_{1})$
(4)

where we emphasize that the antichiral components $\lambda,\lambda_{1}$
carrying ‘$SL(2,\mathbb{R})_{\bar{z}}$’ weight are not colinear but generic.
Because of this the OPE holds not only in the soft sector but for generic
energies. In turn, it will become clear that writing the OPE (4) allows us to
identify $\mathbf{G}^{\lambda_{p}}(z_{p})$ as Kac-Moody currents for a
$w_{1+\infty}$ gauge group, where the adjoint representation is paramatrized
by $|\lambda]$, i.e. the kinematic data! We will see this provides a precise
realization of a 2d color-kinematics duality à la Cheung, Helset, and Parra-
Martinez (2022). Furthermore the OPE suggests that scattering amplitudes such
as (3) are then computed as correlations of such currents, as we will discuss.

##  III $w_{1+\infty}$ as a color group

Historically, the $w_{1+\infty}$ algebra and its quantum version
$\mathcal{W}_{1+\infty}$ arised as vertex algebras of higher-spin fields in a
2d CFT, see Pope (1991); Procházka (2016) for a review. In turn, the novel OPE
(4) is written simply in terms of spin-1 currents and so obscures this
interpretation. It turns out that the connection can be made through the
language of quantum groups. To see this let us first provide a canonical
definition of $w_{1+\infty}$. Consider two abstract generators $\mu_{\pm}$
satisfying

$[\mu_{\alpha},\mu_{\beta}]=i\tau\epsilon_{\alpha\beta}\,\,\,,\,\epsilon_{+-}=1\,,$
(5)

where $\tau$ is a quantization parameter such that as $\tau\to 0$ this reduces
to a Poisson bracket. The generators posses an obvious $SL(2,\mathbb{R})$
symmetry $\mu_{\alpha}^{\prime}=\Lambda_{\alpha}^{\beta}\mu_{\beta}$. In this
language the $w_{1+\infty}$ algebra emerges from the universal enveloping
algebra of (5) (sometimes called Weyl algebra), i.e. that given by totally
symmetric polynomials of the operators

$W^{s}_{\alpha_{1}\cdots\alpha_{s}}:=\mu_{(\alpha_{1}}\cdots\mu_{\alpha_{s})}\,\,,\,\,\alpha_{i}=\pm\,,$
(6)

with $W^{0}=1$ is included in $w_{1+\infty}$ (as opposed to $w_{\infty}$). As
aforementioned, the $W^{s}$ correspond to higher-spin fields under
$SL(2,\mathbb{R})$ 111To compare with the generators of Guevara  _et al._
(2021) we take $W^{s}=w^{1+s/2}$. See Appendix I.. Their algebra is also
$SL(2,\mathbb{R})$ covariant:

$[W^{m}_{\alpha_{1}\cdots\alpha_{m}},W^{n}_{\beta_{1}\cdots\beta_{n}}]{=}i\tau
mn\epsilon_{(\beta_{1}(\alpha_{1}}W^{m+n-2}_{\alpha_{2}{\cdots}\alpha_{m})\beta_{2}\cdots\beta_{n})}+\mathcal{O}(\tau^{2})$
(7)

This construction is closely related to classical twistor space Penrose
(1977); Penrose and Rindler (2011), locally
$\mathbb{CP}^{1}\times\mathbb{C}^{2}$, where $\mu_{+},\mu_{-}$ are simply
coordinates on $\mathbb{C}^{2}$. This motivates us to connect to momentum
space through a Fourier transform
$\\{\mu_{\alpha}\\}\to\\{\lambda_{\alpha}\\}$. Thus we introduce the momentum
wavefunctions

$\displaystyle G^{\lambda}$
$\displaystyle:=e^{[\lambda\mu]}=\sum_{s=0}^{\infty}\frac{1}{s!}\lambda^{\alpha_{1}}\cdots\lambda^{\alpha_{s}}W^{s}_{\alpha_{1}\cdots\alpha_{s}}\,\,\,,\,\lambda_{\alpha}\in\mathbb{R}^{2}\,.$
(8)

Remarkably, in this basis the Weyl algebra (7) takes the form

$[G^{\lambda_{1}},G^{\lambda_{2}}]=i\tau[\lambda_{1}\lambda_{2}]G^{\lambda_{1}+\lambda_{2}}+\mathcal{O}(\tau^{2})\,.$
(9)

This is precisely what appears in the numerator of (4) and can be interpreted
as structure constants of $SU(N)$ when $N\to\infty$ as noted in the seminal
work of Hoppe (1989). As a warm up for the next section let us see how this
proceeds. First note that (5) admits only infinite dimensional
representations: Indeed the enveloping algebra (7) is essentially the linear
space of regular functions of $\mu_{+},\mu_{-}$ equipped with symplectic form
(5). To connect to $SU(N)$ or $U(N)$ we will instead obtain finite,
$N-$dimensional, representations. For this we “exponentiate” the algebra by
introducing $g_{\pm}=e^{\mu_{\pm}}$ so that (5) becomes

$g_{+}g_{-}=q\,g_{-}g_{+}\,\,\,,q:=e^{i\tau}\,,$ (10)

where in the classical limit $q\to 1$ we see that $g_{\pm}$ are c-numbers in
$\mathbb{R}^{2}$. Let now $N$ be an odd integer. If we set

$\tau=4\pi/N\,,$ (11)

it essentially follows from (10) that $g_{\pm}^{N}=1$ 222We assume that the
center of the group is trivial.. Thus $g_{\pm}$ now live in a lattice
$Z_{N}\times Z_{N}$ with modular parameter $i\tau=4\pi i/N$. This is where
$U(N)$ acts. More precisely, there exist $N\times N$ matrices $g_{\pm}$
satisfying (10), which is a Heisenberg group. The generators are then

$\frac{\textbf{G}^{\lambda}}{4\pi
i/N}:=e^{[\lambda\mu]}=q^{\lambda^{+}\lambda^{-}/2}g_{+}^{\lambda^{+}}g_{-}^{\lambda^{-}}\,\,\,,\,\,\lambda_{\alpha}\in
Z_{N}\times Z_{N}\,.$ (12)

Thus in this language the $U(N)$ generators $\textbf{G}^{\lambda}$ carry an
adjoint color index given by a 2-spinor $\lambda$! (To get $SU(N)$, which is
associated to $w_{\infty}$ instead of $w_{1+\infty}$, we simply discard
$\mathbf{G}^{(0,0)}=W^{0}=1$). Moreover, the quantum group relation (10)
implies the Berezin formula

$\textbf{G}^{\lambda_{1}}\textbf{G}^{\lambda_{2}}=\frac{q^{[12]/2}}{4\pi
i/N}\textbf{G}^{\lambda_{1}+\lambda_{2}}\,.$ (13)

Indeed, by evaluating $[\textbf{G}^{\lambda_{1}},\textbf{G}^{\lambda_{2}}]$
from here one can precisely match the structure constants for $U(N)$ Hoppe
(1989). Since $\lambda^{\alpha}$ corresponds to momenta on a lattice, this
realizes a two-dimensional color-kinematics duality in the sense of Cheung,
Helset, and Parra-Martinez (2022). On the other hand, for generic, non-
discrete, values of $\tau$, eq. (13) still applies 333It corresponds to the
Moyal product familiar in the context of non-commutative geometry, see e.g.
Seiberg and Witten (1999); Floratos and Iliopoulos (2006). We thank K.
Costello and R. Monteiro for pointing this out., but the momenta is not
quantized. In any case, as $N\to\infty,\tau\to 0$ we obtain a semiclassical
limit, where the momenta spinor $\lambda$ again lives in $\mathbb{R}^{2}$ and

$\textbf{G}^{\lambda_{1}}\textbf{G}^{\lambda_{2}}=(N/4\pi
i+[12]/2)\textbf{G}^{\lambda_{1}+\lambda_{2}}+\mathcal{O}(1/N)\,.$ (14)

The divergence $\sim N$ does not contribute to the commutator, which becomes
precisely (9), using $\textbf{G}^{\lambda}\to i\tau G^{\lambda}$. As a final
remark, for discrete $N$ we have written the algebra in terms of $N\times N$
matrices $g_{\pm}$. The operators $\mu_{\pm}$ are defined as $\mu_{\pm}=\log
g_{\pm}$ only in a formal sense and are unbounded. From the 4d theory
perspective this is expected since they correspond to Goldstone modes as we
now explain.

##  IV 4d kinematic algebra

Having constructed 2-dimensional color space as parametrized by
$\lambda_{\alpha}$, it is easy to upgrade the construction to 4-dimensional
kinematic space. The motivation comes again from twistor theory where, at
least classically, the positions $\mu_{\alpha}\in\mathbb{C}^{2}$ are fibered
over $z=\eta_{2}/\eta_{1}\in\mathbb{CP}^{1}$ Penrose (1977); Penrose and
Rindler (2011). This motivates us to construct the quantum operators from
sections of this bundle, i.e. operators $\mu_{\alpha}(z)$ satisfying

$\mu_{\alpha}(z_{1})\mu_{\beta}(z_{2})\sim
i\tau\frac{\epsilon_{\alpha\beta}}{z_{12}}\,,$ (15)

This is the affine extension of the 2d algebra (5) 444It has also appeared as
a worldsheet OPE in the related context of the twistor sigma model Adamo,
Mason, and Sharma (2022); Adamo  _et al._ (2022).. More importantly, recalling
that $p=|\eta\rangle[\lambda|$ we can identify $z=\eta_{2}/\eta_{1}$ as living
in the celestial sphere. As we explain in Appendix, this algebra generalizes
the one previously introduced for celestial Goldstone modes in the soft sector
Himwich  _et al._ (2020). Indeed here the operators $\mu_{\alpha}$ themselves
can be interpreted as graviton Goldstone modes valid for all energies. To see
this we introduce the normal-ordered vertex operators

$i\tau\mathbf{G}^{\lambda}(z)=:e^{[\lambda\mu(z)]}:\,,$ (16)

for which the analog of the Berezin formula (13) takes the form

$\displaystyle\mathbf{G}^{\lambda_{1}}(z_{1})\mathbf{G}^{\lambda_{2}}(z_{2})$
$\displaystyle=(q^{[12]/z_{12}}/i\tau):e^{[\lambda_{1}\mu(z_{1})]+[\lambda_{2}\mu(z_{2})]}:$
$\displaystyle\sim(q^{[12]/z_{12}}/i\tau)\mathbf{G}^{\lambda_{1}+\lambda_{2}}(z_{2})$
$\displaystyle\sim\frac{[12]}{z_{12}}\mathbf{G}^{\lambda_{1}+\lambda_{2}}(z_{2})+\mathcal{O}(\tau)\,.$
(17)

This mimics (13): In the second line we have taken the OPE limit $z_{12}\to 0$
which discards the $\sim N$ terms, and in the third line the classical limit
$\tau=4\pi/N\to 0$. An important comment is now in order. In the OPE (15) both
operators have conformal weight $1/2$ in $z$, which requires
$\lambda_{\alpha}$ to have weight $-1/2$. This is familiar from the
construction of gravitational Goldstone operators Himwich  _et al._ (2020).
However, after the $\tau\to 0$ limit is taken in (17) we can take $z$ and
$\lambda$ to be independent, after all the latter only enter through the
structure constants of $U(\infty)$. We can make this even more explicit by
writing, at strict $\tau=0$,

$\mathbf{G}^{\lambda_{1}}(z_{1})\mathbf{G}^{\lambda_{2}}(z_{2})\sim\frac{\int
d^{2}\lambda_{3}f^{\lambda_{1},\lambda_{2}}_{\quad\,\,\,\,\lambda_{3}}}{z_{12}}\mathbf{G}^{\lambda_{3}}(z_{2})\,,$
(18)

where the kinematic structure constants

$f^{\lambda_{1},\lambda_{2}}_{\quad\,\,\,\,\lambda_{3}}=f^{\lambda_{1},\lambda_{2},-\lambda_{3}}=[12]\delta^{2}(\lambda_{1}+\lambda_{2}-\lambda_{3})\,,$
(19)

are completely antisymmetric and correspond to the kinematic algebra Monteiro
and O’Connell (2014). Seen in this way, the OPE pertains Kac-Moody currents
with weight 1 under $SL(2,\mathbb{R})_{z}$, where the other
$SL(2,\mathbb{R})_{\bar{z}}$ has been promoted to a color $w_{1+\infty}$ gauge
group. From a CFT perspective, this reproduces the chiral OPE block of Guevara
_et al._ (2021) which resums all $SL(2,\mathbb{R})_{\bar{z}}$ descendants, see
also Appendix I.

Furthermore, we can use standard arguments to show that associativity of the
OPE (18) immediately yields the kinematic Jacobi relation of Monteiro and
O’Connell (2014)

$\displaystyle\int
d^{2}\lambda(f^{\lambda_{1},\lambda_{2}}_{\quad\,\,\,\,\lambda}\,\,f^{\lambda,\lambda_{3},\lambda_{4}}+f^{\lambda_{1},\lambda_{3}}_{\quad\,\,\,\,\lambda}\,\,f^{\lambda,\lambda_{4},\lambda_{2}}+$
$\displaystyle
f^{\lambda_{1},\lambda_{4}}_{\quad\,\,\,\,\lambda}\,\,f^{\lambda,\lambda_{2},\lambda_{3}})$
$\displaystyle=0\,.$ (20)

Note that this Lie algebra structure was completely invisible from the
collinear factorization/S-Matrix perspective of (3), in striking contrast to
its OPE realization. Furthermore, given this Lie algebra it is natural to
further incorporate 2d primary fields of arbitrary helicity transforming in
the adjoint representation (18), i.e.

$\mathbf{G}^{\lambda_{1}}(z_{1})\mathbf{O}_{h}^{\lambda_{2}}(z_{2})\sim\frac{\int
d^{2}\lambda_{3}f^{\lambda_{1},\lambda_{2}}_{\quad\,\,\,\,\lambda_{3}}}{z_{12}}\mathbf{O}_{h}^{\lambda_{3}}(z_{2})\,,$
(21)

which is precisely (4) as promised. We elaborate on this result in the next
section. Here let us point out that the algebraic analysis of refs. Himwich,
Pate, and Singh (2022); Mago  _et al._ (2021), which studied the Jacobi
identity in the celestial conformal basis is here synthesized in the Lie
algebra relation (20), closer to the recent analysis of Ren  _et al._ (2022).
Such references also studied the different helicities case, which we discuss
briefly in Appendix II.

##  V Tower of Soft Charges in Gravity

Let us now provide example-based evidence that gravitational amplitudes can
indeed be derived from the preceding Kac-Moody OPEs. We do so by computing
correlation functions of soft modes. Indeed, as it is well known, a systematic
study of gravitational scattering can be carried out through the IR expansion
of the S-Matrix Weinberg (1965), which is also easily realized as a conformal
soft expansion in the celestial sphere Guevara (2019); Arkani-Hamed  _et al._
(2021); Himwich, Pate, and Singh (2022). Notably, as we further flesh out in
the Appendix, this expansion of a graviton state amounts precisely to the
decomposition given in eq. (8) and thus allow us to identify the tower of soft
modes. In the operator language, it follows from (16) that

$\mathbf{G}^{\lambda}(z)=\sum_{n}\frac{1}{n!}\lambda^{\alpha_{1}}\cdots\lambda^{\alpha_{n}}\mathbf{W}^{n}_{\alpha_{1}\cdots\alpha_{n}}(z)\,,$
(22)

where

$\mathbf{W}^{n}_{\alpha_{1}\cdots\alpha_{n}}(z)=\frac{1}{i\tau}:\mu_{\alpha_{1}}\cdots\mu_{\alpha_{n}}:(z)\,,$
(23)

is the 4d analog of (6). Very nicely, their affine algebra can be computed
either directly from the Goldstone modes (15) or by expanding both sides of
(17) in soft modes. At $\tau\to 0$ the result is

$\displaystyle\mathbf{W}^{n}_{\alpha_{1}{\cdots}\alpha_{n}}(z_{1})\mathbf{W}^{m}_{\beta_{1}{\cdots}\beta_{n}}$
$\displaystyle(z_{2})\sim$
$\displaystyle\frac{mn}{z_{12}}\epsilon_{(\beta_{1}(\alpha_{1}}\mathbf{W}^{m+n-2}_{\alpha_{2}{\cdots}\alpha_{m})\beta_{2}{\cdots}\beta_{n})}(z_{2})$
(24)

This agrees with the algebra of Strominger (2021); Ball  _et al._ (2022);
Adamo, Mason, and Sharma (2022) as we show in Appendix I. Note here that the
$\mathbf{W}^{1}\mathbf{W}^{1}\sim\mathbf{W}^{0}$ OPE is precisely our starting
eq. (15). Thus the operator $\mathbf{W}^{0}$ is required to generate the
tower. We will recognize it as a central extension of the 4d Poincare algebra.

To understand correlation functions involving arbitrary helicity fields we
consider (21). It is convenient to rewrite it as an exponential soft theorem
Li, Lin, and Zhang (2018); Hamada and Shiu (2018); Guevara (2019); Bautista
and Guevara (2019), i.e.

$:e^{[\lambda_{1}\mu(z_{1})]}:\mathbf{O}_{h}^{\lambda_{2}}(z_{2})\sim
i\tau\frac{[12]}{z_{12}}e^{[\lambda_{1}\frac{\partial}{\partial\lambda_{2}}]}\mathbf{O}_{h}^{\lambda_{2}}(z_{2})\,,$
(25)

from where, expanding both sides in $\lambda_{1}^{\alpha}$, according to
(22)-(23) immediately yields a tower of soft operator OPEs

$\displaystyle\mathbf{W}^{0}(z_{1})\mathbf{O}_{h}^{\lambda}(z_{2})$
$\displaystyle=\textrm{regular}$ (26)
$\displaystyle\mathbf{W}_{\alpha}^{1}(z_{1})\mathbf{O}_{h}^{\lambda}(z_{2})$
$\displaystyle\sim\frac{\lambda_{\alpha}}{z_{12}}\mathbf{O}_{h}^{\lambda}(z_{2})$
(27)
$\displaystyle\mathbf{W}_{\alpha\beta}^{2}(z_{1})\mathbf{O}_{h}^{\lambda}(z_{2})$
$\displaystyle\sim
2\frac{\lambda_{(\alpha}\frac{\partial}{\partial\lambda^{\beta)}}}{z_{12}}\mathbf{O}_{h}^{\lambda}(z_{2})$
(28)
$\displaystyle\mathbf{W}_{\alpha\beta\gamma}^{3}(z_{1})\mathbf{O}_{h}^{\lambda}(z_{2})$
$\displaystyle\sim
3\frac{\lambda_{(\alpha}\frac{\partial}{\partial\lambda^{\beta}}\frac{\partial}{\partial\lambda^{\gamma)}}}{z_{12}}\mathbf{O}_{h}^{\lambda}(z_{2})\,,$
(29)

etc… From here we confirm that $\mathbf{W}^{0}$ is trivial, consistent with
the expectation that it has no scattering amplitudes. Moving on, we recognize
$\mathbf{W}^{1}$ as a spacetime momentum: Introducing $\eta_{a}=(1\,\,z)$ the
translation generator is 555In Appendix we interpret $\mathbf{W}^{1}$ as a
weight $3/2$ operator, in which case these are its global modes.

$P_{a\alpha}=\frac{(i\tau)^{-1}}{2\pi i}\oint
dz\,\eta_{a}(z)\mu_{\alpha}(z)\,\,\,\,\,\,a,\alpha=\pm\,.$ (30)

Contour-integrating in (27) this gives
$[P_{a\alpha},\mathbf{O}^{\lambda}(z)]=\eta_{a}\lambda_{\alpha}\mathbf{O}^{\lambda}(z)$
as expected. Analogously, we recognize $\mathbf{W}^{2}=:\mu\mu:$ as the
angular momentum generator
$\lambda_{(\alpha}\frac{\partial}{\partial\lambda^{\beta)}}$ Witten (2004).
Similarly to the Sugawara construction, it can be reinterpreted as the stress-
energy tensor in celestial CFT, see Appendix. The generator $\mathbf{W}^{3}$
closely resembles a conformal transformation Witten (2004).

##  VI Soft theorems and recursion

From the S-Matrix perspective, these generators are responsible for Weinberg
soft theorems and their subleading extensions Weinberg (1965); Cachazo and
Strominger (2014), when inserted into generic correlation functions. Omitting
$SL(2,\mathbb{R})$ indices for simplicity we have

$\displaystyle\langle\mathbf{W}^{r}(z)$
$\displaystyle\mathbf{O}^{\lambda_{1}}(z_{1})\cdots\mathbf{O}^{\lambda_{n}}(z_{n})\rangle$
$\displaystyle=\frac{1}{2\pi
i}\oint\frac{dw}{w-z}\langle\mathbf{W}^{n}(w)\mathbf{O}^{\lambda_{1}}(z_{1})\cdots\mathbf{O}^{\lambda_{n}}(z_{n})\rangle$
$\displaystyle=\sum_{i=1}^{n}{\frac{\lambda_{i}\left(\frac{\partial}{\partial\lambda_{i}}\right)^{n{-}1}}{z-z_{i}}}\langle\mathbf{O}^{\lambda_{1}}(z_{1})\cdots\mathbf{O}^{\lambda_{n}}(z_{n})\rangle{+}\mathcal{P}^{r}(z)$
(31)

where the first term controls the colinear factorizations that are present in
the scattering amplitudes. The extra term $\mathcal{P}^{r}(z)$ denotes
additional contributions: Multiparticle singularities and polynomial terms at
$z\to\infty$. The former are likely associated to Virasoro descendants and we
leave their study for future work.

To gain further insight into this picture, let us write (31) as the soft
theorems in He, Huang, and Wen (2014); Guevara (2019). Resorting to the
celestial dictionary Pasterski, Shao, and Strominger (2017), the correlation
functions in the LHS of (31) correspond to the S-Matrix with a graviton
insertion,

$\mathcal{A}_{n{+}1}(k^{+2},p_{1},\ldots,p_{n})\leftrightarrow\langle\mathbf{G}^{\lambda}(z)\mathbf{O}^{\lambda_{1}}(z_{1})\cdots\mathbf{O}^{\lambda_{n}}(z_{n})\rangle\,.$
(32)

Using (22) we can then translate the relation (31) into

$\displaystyle\mathcal{A}_{n{+}1}(k^{+2}$ $\displaystyle,p_{1},\ldots,p_{n})$
$\displaystyle=\sum_{i}\frac{(\epsilon^{+}\cdot p_{i})^{2}}{k\cdot
p_{i}}e^{J_{i}}\mathcal{A}_{n}(p_{1},\ldots,p_{n})+\ldots\,,$ (33)

where $J_{i}=[\lambda\frac{\partial}{\partial\lambda^{i}}]$ amounts to a
Lorentz generator on particle $i$. We refer to Guevara (2019) for further
details, see also Hu and Pasterski (2022). The $\ldots$ denote the
contribution from $\mathcal{P}^{r}$. Their contribution can be understood as
follows. Let $\mathcal{A}_{n}$ and $A_{n}$ correspond to the dressed and
stripped amplitude, respectively

$\mathcal{A}_{n}(p_{i})=\delta^{2}\left(\sum_{i}\lambda_{i}\right)\delta^{2}\left(\sum_{i}z_{i}\lambda_{i}\right)A_{n}(p_{i})\,.$
(34)

Then, it is easy to see from (33) that $\mathcal{A}_{n+1}$ develops the
expected support on $\delta^{2}(\lambda+\sum_{i}\lambda_{i})$, but not
necessarily in $\delta^{2}(z\lambda+\sum_{i}z_{i}\lambda_{i})$. The former is
a consequence of our color $U(\infty)$ Lie algebra while the latter can be
interpreted as a two-dimensional holomorphic contour, see discussion below.
The presence of the polynomial $\mathcal{P}^{r}$ is precisely to enforce that
contour. Moreover, in the case of pure graviton MHV amplitudes (33) defines a
BCFW-type recursion relation $\mathcal{A}_{n}\to\mathcal{A}_{n+1}$, for we can
show explicitly that $\mathcal{P}^{r}(z_{0})$ is given by the operator

$\sum_{i=1}^{n-1}\frac{[\lambda\lambda_{i}]}{z_{ni}}\sum_{j=0}^{r{-}4}\left(\frac{z_{0n}}{z_{in}}\right)^{j}\,\sum_{m=0}^{r{-}4{-}j}\binom{r-1}{m}(J_{i}{-}J_{n})^{r{-}1{-}m}J_{n}^{m}$
(35)

acting on the $n-$point amplitude $\mathcal{A}_{n}$. This illustrates that
$\mathcal{P}^{r}(z_{0})$ is a polynomial of degree $r-4$ in $z_{0}$. In
particular $\mathcal{P}^{r}=0$ for $r\leq 3$, and thus
$\mathbf{W}^{1},\mathbf{W}^{2},\mathbf{W}^{3}$ in (31) generate the standard
(sub)leading soft theorems Cachazo and Strominger (2014) as expected.

##  VII Discussion and future directions

In this work we have provided evidence that recent developments in celestial
holography, color-kinematics duality, and twistor theory are intrinsically
connected. A natural three-way bridge between these topics is given by the OPE
language, which exposes the properties of a Kac-Moody algebra, such as
associativity, completely concealed by the S-Matrix picture. On the other
hand, a feature that those three topics have in common is extended symmetry
algebras which point towards integrability of gravitational theories. From the
CFT arena there are indeed known connections between $\mathcal{W}_{1+\infty}$
and other integrable structures such as the Yangian $Y(\mathfrak{gl}_{1})$
Bernard (1991); Procházka (2016); Briot and Ragoucy (2013). The situation is
closely connected to the self-dual sector, an integrable instance where
kinematic algebras emerge naturally Dolan (1982, 1981); Monteiro and O’Connell
(2011); Chacón  _et al._ (2021).

In a strict sense this is a level-0 theory which is therefore non unitary
(this is expected for tree-level scattering). Moreover, the connection to
self-dual gravity also suggests that there should not be a multiparticle
S-Matrix due to integrability. Recent work has shown, however, that integrable
theories can be reduced to non-trivial scattering in presence of defects or
other auxiliary fields Costello, Paquette, and Sharma (2022); Costello and
Paquette (2022a, b). This resonates with an old conjecture of Ward Ward and
Tabor (1985); Ablowitz, Chakravarty, and Halburd (2003) stating that self-dual
systems yield rich dynamics under compactification.

The properties unveiled through celestial holography, such as IR modes and
theirs soft theorems, also point towards a construction more general than the
self-dual sector. As a step towards such, in Appendix II we introduce negative
helicity graviton states, corresponding to a 2d scalar field
$\mathbf{\Phi}^{\lambda}$. We find

$\langle\mathbf{G}^{\lambda_{1}}(z_{1})\mathbf{G}^{\lambda_{2}}(z_{2})\Phi^{\lambda_{3}}(z_{3})\rangle=\frac{[12]^{6}}{[13]^{2}[23]^{2}}\delta^{4}\left(\sum_{i}p_{i}\right)\,.$
(36)

In this formula four-momentum conservation emerges naturally from OPEs,
contrasting with the usual CCFT picture Fan  _et al._ (2022); Mizera and
Pasterski (2022). Here we interpret two delta functions as poles defining
contours in $z$. In particular we find

$\oint\frac{dz_{1}dz_{2}}{2\pi
i}\langle\mathbf{G}^{\lambda_{1}}(z_{1})\mathbf{G}^{\lambda_{2}}(z_{2})\Phi^{\lambda_{3}}(z_{3})\rangle=f^{\lambda_{1},\lambda_{2},\lambda_{3}}\,.$
(37)

Moving on to the four-point function $\langle\mathbf{GGG}\Phi\rangle$, we find
via the recursion that it does not vanish identically, but instead we have

$\oint\frac{dz_{1}dz_{2}dz_{3}}{2\pi}\langle\mathbf{G}^{\lambda_{1}}(z_{1})\mathbf{G}^{\lambda_{2}}(z_{2})\mathbf{G}^{\lambda_{3}}(z_{3})\Phi^{\lambda_{4}}(z_{4})\rangle=\eqref{eq:jacb}$
(38)

Thus its residue vanishes identically only if the Jacobi identity holds! In
any case, the correlation itself is non-zero but has support only when all
points $z_{i}$ coincide. This is precisely the situation in twistor space,
where $A_{4}(1^{+}2^{+}3^{+}4^{-})$ is supported on a degree zero curve Witten
(2004)! Further exploration of this fascinating direction is left for future
work.

Many questions regarding the nature of the proposed holographic theory remain
open. The most pressing one pertains multiparticle singularities as well as
the loop amplitudes, perhaps paralleling the developments of Costello,
Paquette, and Sharma (2022); Costello and Paquette (2022a). Relatedly, it is
tempting to explore the finite $N$ regime as a fully quantum theory. In this
case the naive OPE (17) necessarily yields higher-spin operators as in string
theory.

### Acknowledgements

We wish to thank the organizers of the conference Amplitudes 2022, where this
work was presented, for providing a stimulating environment for discussions
and research. We also appreciate the kind hospitality and environment provided
by the Institute for Advanced Studies, where this work was completed.

We thank Mina Himwich, Dan Kapec, Monica Pate, James Mangan, Sebastian Mizera,
Ricardo Monteiro, Shu-Heng Shao, Andrew Strominger and Adam Tropper for
enlightening conversations. We especially thank Kevin Costello and Ronald
Bittleston for discussions on quantum groups and non-commutative geometry.
This work was supported by DOE grant de-sc/000787 and the Black Hole
Initiative at Harvard University, which is funded by grants from the John
Templeton Foundation and the Gordon and Betty Moore Foundation. The author
also receives support from the Harvard Society of Fellows.

Note added. While finalizing the writing stage of this paper we were informed
of the parallel works Monteiro (2022)-Bu, Heuveline, and Skinner (2022), which
have also discussed the color-kinematics duality in this context.

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## Appendices

##  Appendix A  I. Relation to the conformal primary basis

In celestial holography, the S-Matrix is cast into a boost eigenstate basis
that exhibits its 2d conformal symmetry Pasterski (2021). In the present work
we have in contrast attempted to formulate fundamental properties of
correlation functions strictly in momentum space. The latter idea fits into a
more general framework we shall elaborate further in Guevara . In the meantime
we will outline here one of its key points: namely that both approaches are
connected through the holomorphic soft expansion Cachazo and Strominger
(2014).

In scattering, incoming massless particles such as gravitons can be
represented by insertion of creation operators depending on an energy scale
$\omega$ and coordinates $z,\bar{z}$ Strominger (2017)

$a_{+2}^{\dagger}(\omega,z,\bar{z})\,\,\,\,,\,\,\omega>0\,,$ (39)

where $+2$ indicates the 4-dimensional helicity of a graviton. Here the
momenta $p=|\eta\rangle[\lambda|$ is parametrized as
$|\eta\rangle=\sqrt{\omega}(1\,\,z)$ and
$|\lambda]=\sqrt{\omega}(1\,\,\bar{z})$. They have an energetic Laurent
expansion starting by the IR singularity $\omega^{-1}$ in tree level
amplitudes, which we parametrize it as

$a_{+2}^{\dagger}(\omega,z,\bar{z})=\sum_{k=0}^{\infty}\omega^{k-1}H_{1-k}(z,\bar{z})\,.$
(40)

With some abuse of notation, and following the lore of Strominger (2017), we
interpret the operators $H_{1-k}(z,\bar{z})$ as inserting states into 2d
correlators rather than in the 4d S-Matrix. Equivalently, they can be recast
as conformal primaries by introducing
$G_{\Delta}^{+}=\int_{0}^{\infty}d\omega\omega^{\Delta-1}a_{+2}^{\dagger}(\omega,z,\bar{z})$
so that the soft modes in (40) are extracted as
$H_{k}=\textrm{Res}_{\Delta=k}G_{\Delta}^{+}$ Guevara, Ochirov, and Vines
(2019). These are precisely the operators constructed in Guevara, Ochirov, and
Vines (2019); Guevara  _et al._ (2021); Strominger (2021) at integer boost
weight $\Delta=h+\bar{h}=1-k$, and we refer to those references for more
details. Now, since for a graviton $h-\bar{h}=2$ these states have

$H_{1-k}(z,\bar{z}):\,\,(h,\bar{h})=\left(\frac{3-k}{2},-\frac{k+1}{2}\right)\,.$
(41)

We will consider split signature $(++--)$ where $z$ and $\bar{z}$ are
independent and we do not distinguish between incoming/outgoing operators
(this amounts to an overall sign in the momenta). Accordingly, the 4d Lorentz
group is realized as $SL(2,\mathbb{R})_{z}\times SL(2,\mathbb{R})_{\bar{z}}$
acting independendtly on $z,\bar{z}$ with weights $(h,\bar{h})$. Since
$\bar{h}<0$ for $H_{k}$ (recall $k\leq 1)$ it was proposed in Guevara  _et
al._ (2021) that the $H_{k}(z,\bar{z})$ admits a polynomial expansion in
$\bar{z}$. Labeling $k=4-2p$, where $p=\frac{3}{2},2,\frac{5}{2},\ldots$ the
formulae given there can be written as

$H_{4-2p}(z,\bar{z})=\frac{1}{(2p-2)!}\sum_{n=1-p}^{p-1}\binom{2p-2}{p-n-1}w_{n}^{p}(z)\bar{z}^{p-1-n}\,,$
(42)

where $w_{n}^{p}(z)$ each have $SL(2,\mathbb{R})_{z}$ weight $3-p$, as
dictated by (41). On the other hand, it was found in Strominger (2021) that
for fixed $p$ the $2p-1$ states can be nicely assembled into a
$SL(2,\mathbb{R})_{\bar{z}}$ multiplet of spin $p$. Now, a very natural
spin-$p$ representation of $SL(2,\mathbb{R})$ is given by totally symmetric
rank $2p-2$ tensors (see e.g. Arkani-Hamed, Huang, and Huang (2021)), so this
suggests we can identify

$w_{n}^{p}=\mathbf{W}_{\underbrace{+\cdots+}_{p-n-1}\underbrace{-\cdots-}_{p+n-1}}^{2p-2}\,,$
(43)

i.e. as the independent components of the $SL(2,\mathbb{R})$ tensor
$\mathbf{W}_{\alpha_{1}\cdots\alpha_{2p-2}}^{2p-2}$. Moreover, introducing the
spinor $|\hat{\lambda}]=(1\,\,\bar{z})$ we see that the combinatorial factor
of (18) gets reabsorbed and the sum is simply a contraction:

$H_{4-2p=1-k}(z,\bar{z})=\frac{1}{(2p-2)!}\mathbf{W}_{\alpha_{1}\cdots\alpha_{2p-2}}^{2p-2}(z)\hat{\lambda}^{\alpha_{1}}\cdots\hat{\lambda}^{\alpha_{2p-2}}\,.$
(44)

Since $\mathbf{W}^{2p-2}$ carry $SL(2,\mathbb{R})_{\bar{z}}$ indices, we have
made “almost” manifest the transformation properties of the $w_{n}^{p}$
multiplet. The only caveat is that $|\hat{\lambda}]=(1\,\,\bar{z})$ is a
inohomogeneous spinor and picks up a Jacobian factor when trasnforming under
$SL(2,\mathbb{R})_{\bar{z}}$. This is remedied easily by attaching the energy
factor, i.e. we can perform a little group transformation such that

$\displaystyle|\eta\rangle$
$\displaystyle=\sqrt{\omega}(1\,\,z),|\lambda]=\sqrt{\omega}(1\,\,\bar{z})$
$\displaystyle\longrightarrow|\eta\rangle=(1\,\,z),|\lambda]=\omega(1\,\,\bar{z})=\omega|\hat{\lambda}]\,.$
(45)

Since the graviton $a_{+2}^{\dagger}(\omega,z,\bar{z})$ has helicity $+2$ it
picks up a factor $\omega^{2}$ under this transformation and becomes,
combining (40) and (44),

$\displaystyle\mathbf{G}^{\lambda=\omega\hat{\lambda}}(z)$
$\displaystyle=\omega^{2}a_{+2}^{\dagger}(\omega,z,\bar{z})$
$\displaystyle=\sum_{k=0}^{\infty}\omega^{k+1}H_{1-k}(z,\bar{z})$
$\displaystyle=\sum_{k=0}^{\infty}\omega^{k+1}\frac{1}{(k+1)!}\mathbf{W}_{\alpha_{1}\cdots\alpha_{k+1}}^{k+1}(z)\hat{\lambda}^{\alpha_{1}}\cdots\hat{\lambda}^{\alpha_{k+1}}$
$\displaystyle=\sum_{k=0}^{\infty}\frac{1}{(k+1)!}\mathbf{W}_{\alpha_{1}\cdots\alpha_{k+1}}^{k+1}(z)\lambda^{\alpha_{1}}\cdots\lambda^{\alpha_{k+1}}\,,$
(46)

which is nothing but the expansion induced from vertex operators (22), as long
as we include $\mathbf{W}^{0}=1$ as a central term. A number of comments are
in order. First, we see from the third line that this is a soft expansion in
the (anti)holomorphic coordinate, namely in
$\lambda=\omega\hat{\lambda}(\bar{z})$. But this is precisely the holomorphic
soft expansion that was used in Cachazo and Strominger (2014) to derive the
subleading soft theorems! Here it emerges naturally to all orders in $\omega$
and we shall pursue further applications in Guevara . Second, considering
$SL(2,\mathbb{R})$ tensors, rather than modes $w_{n}^{p}$, immediately
bypassess the need to operate with the light-transform Himwich, Pate, and
Singh (2022) of conformal primaries to obtain the $w_{1+\infty}$ algebra.

Why did the light-transform play a predominant role in previous discussions of
$w_{1+\infty}$ Strominger (2021); Himwich, Pate, and Singh (2022)? This can be
easily understood by resorting to the homogeneous formalism used in Guevara
(see also Sharma (2022)). In that reference the inverse light-transform in
coordinates $|\lambda]=(1\,\,\bar{z}),|\eta]=(1\,\,\bar{z}^{\prime})$ is given
by

$H_{4-2p}(z,\eta)=\frac{1}{\Gamma(2p-1)}\oint\frac{[\lambda d\lambda]}{2\pi
i}[\lambda\eta]^{2p-2}\mathbb{L}[H_{4-2p}](z,\lambda)\,\,\,,\,\,p>0\,,$ (47)

By comparing this with (44) we immediately identify the Penrose transform

$\mathbf{W}_{\alpha_{1}\cdots\alpha_{2p-2}}^{2p-2}=\oint\frac{[\lambda
d\lambda]}{2\pi
i}\lambda_{\alpha_{1}}\ldots\lambda_{\alpha_{2p-2}}\mathbb{L}[H_{4-2p}](z,\lambda)\,.$
(48)

Further using the identification (43), using $[\lambda d\lambda]=d\bar{z}$,
this is

$w^{p}_{n}(z)=\oint\frac{d\bar{z}}{2\pi
i}\bar{z}^{p-n-1}\mathbb{L}[H_{2p-2}](z,\bar{z})\,.$ (49)

Hence $w^{p}_{n}(z)$ are nothing but the modes of the light-transform
$\mathbb{L}[H_{2p-2}]$! Indeed, our representation landed us straight into the
so-called wedge of the $w_{1+\infty}$ algebra derived in Ball  _et al._
(2022); Himwich, Pate, and Singh (2022)

$w_{n}^{p}(z_{1})w_{m}^{q}(z_{2})\sim
2\frac{m(p-1)-n(q-1)}{z_{12}}w_{m+n}^{p+q-2}(z_{2})\,\,\,,\,\,|m|\leq p-1\,,$

which, by virtue of (44), can be shown to be equivalent to the
$SL(2,\mathbb{R})$ covariant form (24). In particular we realize that
$L_{n}=\frac{1}{2}w_{n}^{2}(0)$ satisfy the $SL(2,\mathbb{R})$ algebra and act
as

$[L_{n},w_{m}^{q}]=(m-n(q-1))w_{m+n}^{q}\,,$ (50)

and so this defines a 2d energy-momentum tensor $\bar{T}(\bar{z})$ for the
CFT, for after some work, it can be shown to be equivalent to

$\bar{T}(\bar{z})\mathbf{G}^{\lambda_{2}}(z_{2})\sim\frac{\bar{h}}{\bar{z}_{12}^{2}}\mathbf{G}^{\lambda_{2}}(z_{2})+\frac{1}{\bar{z}_{12}}\bar{\partial}_{2}\mathbf{G}^{\lambda_{2}}(z_{2})\,,$
(51)

where
$\bar{T}(\bar{z}):=\frac{1}{2}\sum_{n}\frac{w_{n}^{2}(0)}{\bar{z}^{n+2}}$ and
$\bar{h}=-\frac{1}{2}\omega\partial_{\omega}$ is the helicity weight on
$\lambda_{2}=\omega\hat{\lambda}$. As argued in Kapec  _et al._ (2017), see
also Pasterski (2022); Fan  _et al._ (2020), $\bar{T}$ is essentially the
Shadow (or light) transform of
$H_{0}(z,\bar{z})=\frac{1}{2}\sum_{n}\binom{2}{1-n}\frac{w_{n}^{2}(z)}{\bar{z}^{n-1}}$,
for which (51) is equivalent to

$H_{0}(z_{1,}\bar{z}_{1})\mathbf{G}^{\lambda_{2}}(z_{2})\sim\frac{\bar{z}_{12}^{2}\omega\partial_{\omega}+\bar{z}_{12}\bar{\partial}_{2}}{z_{12}}\mathbf{G}^{\lambda_{2}}(z_{2})\,,$

which in turn is nothing but (28) written in the coordinates
$\lambda_{2}=\omega\hat{\lambda}(\bar{z})$.

#### Relation to the PC system in the soft sector

As a follow up of the preceding discussion it is worth comenting on the
relation with the gravitational Goldstone operator $C(z,\bar{z})$ introduced
in Himwich  _et al._ (2020) for the leading soft mode. Denoting
$P(z)=\bar{\partial}H_{1}(z,\bar{z})$ that reference studied the system

$P(z)C(w,\bar{w})\sim\frac{i}{z-w}\,,$ (52)

This then allowed to construct hard states in CCFT as the dressings

$\mathcal{O}_{h}(\omega,z,\bar{z})=e^{\omega
C(z,\bar{z)}}\mathcal{\hat{O}}_{h}(\omega,z,\bar{z})$ (53)

where $\mathcal{\hat{O}}_{h}$ does not interact with soft gravitons. By direct
comparison, we can identify the $PC$ algebra with our $\\{\mu_{+},\mu_{-}\\}$
system (15). Indeed, from eqs. (44)-(23), we see that in our construction

$H_{1}(z,\bar{z})=\mu_{+}(z)+\bar{z}\mu_{-}(z)\Longrightarrow
P(z)=\mu_{-}(z)\,.$ (54)

On the other hand, we have also constructed hard gravitons (to all orders in
energy) as (16), and thus we can identify

$:e^{[\mu\lambda]}:\leftrightarrow e^{\omega C(z,\bar{z})}\,,$ (55)

in the soft sector. Furthermore the leading soft-theorem derived there

$P(z)e^{\omega C(w,\bar{w})}\sim\frac{i\omega}{z-w}e^{\omega C(z,\bar{z})}\,,$
(56)

is nothing but (27) of the main text. We can see that identifying
$C(z,\bar{z})=\mu_{+}+\bar{z}\mu_{-}=H_{1}$ is consistent with (52), (55) and
(56), but also generalizes the Goldstone mode to all the orders in the soft
expansion, i.e. not only to $H_{1}$ but for all $H_{1-k\,\,},k\geq 1$. Dressed
states of the form $O_{h}^{\lambda}(z)=e^{[\lambda\mu]}\hat{O}_{h}(z)$ satisfy
the tower of soft theorems (27)-(29), but may not be the most general states
to do so. The difference with the framework of Himwich  _et al._ (2020) relies
mainly in the level terms $\langle CC\rangle$ which are fixed from the IR
divergence. The identification $C=H_{1}$ done here precludes us from
incorporating them and thus may be corrected at loop level. It would be
fascinating to work out this connection.

##  Appendix B II. Negative helicity states and one-minus amplitudes

Here we introduce negative helicity states that pair to
$\mathbf{G}^{\lambda}(z)$ through two-point functions. Indeed, to compute non-
trivial correlations explicitly we need to specifiy the identity terms in the
OPE. First, we can read off the color 2-point structure in momentum space from
the $U(N)$ matrices in (13). We get

$d^{\lambda_{1},\lambda_{2}}:=\frac{1}{N(i\tau)^{2}}\textrm{Tr}(\textbf{G}^{\lambda_{1}}\textbf{G}^{\lambda_{2}})=\delta^{2}(\lambda_{1}+\lambda_{2})\,,$
(57)

which is luckily diagonal as the usual CFT inner product. However, this does
not appear as a singularity in the
$\mathbf{G}^{\lambda}(z)\mathbf{G}^{\lambda}(z)$ OPE (17), in turn leading to
their interpretation as zero-level Kac-Moody currents. This is consistent at
tree-level, since a vanishing level leads to vanishing of correlation
functions involving only $\mathbf{G}^{\lambda}(z)$ operators, which should
correspond to all-plus amplitudes,
$\langle\mathbf{G}^{\lambda}\cdots\mathbf{G}^{\lambda}\rangle=0$.

To obtain amplitudes with insertion of negative helicity gravitons, we can
include a field $\Phi^{\lambda}$ with chiral OPE

$\mathbf{G}^{\lambda_{1}}\Phi^{\lambda_{2}}\sim\delta(z_{12})d^{\lambda_{1},\lambda_{2}}+\frac{\int
d\lambda_{3}f_{\qquad\lambda_{3}}^{\lambda_{1},\lambda_{2}}}{z_{12}}\Phi^{\lambda_{3}}\,,$
(58)

where we recall the second term follows from universal colinear singularities
and thus holds for generic helicities. The first term is a contact interaction
which requires the theory to be defined in Lorentzian 2d signature (i.e. split
4d signature). Since in this picture $\mathbf{G}^{\lambda}(z)$ has weight one,
it also entails that $\Phi^{\lambda}(z)$ must transform as a scalar under
$SL(2,\mathbb{R})_{z}$. From the Kac-Moody perspective the emergence of such a
colored scalar is natural: One can consider a level term between the $(1,0)$
current $J^{a}$ and the Shadow of the $(0,1)$ current $\bar{J}^{b}$, namely
$J^{a}S[\bar{J}^{b}]\sim 1/z_{12}^{2}d^{ab}$ Nande, Pate, and Strominger
(2018). This is equivalent to $J^{a}\bar{J}^{b}=\delta^{2}(z_{12})d^{ab}$, or
to $J^{a}\Phi^{b}=\delta(z_{12})d^{ab}$ in Lorentzian signature, if we
identify $\Phi$ with the light-transform of $\bar{J}$.

Let us further justify the contact term from the S-Matrix perspective.
Negative helicity gravitons are canonically conjugate to positive ones. In the
notation of the previous Appendix, the disconnected S-Matrix is given by the
two-point function

$\langle
a_{-2}(\omega_{2},z_{2},\bar{z}_{2})a_{+2}^{\dagger}(\omega_{1},z_{1},\bar{z}_{1})\rangle=\frac{\delta(\omega_{1}+\omega_{2})}{\omega_{1}}\delta(z_{1}-z_{2})\delta(\bar{z}_{1}-\bar{z}_{2})\,,$
(59)

up to an irrelevant normalization. Recall now we introduce homogeneous
coordinates via $\lambda=\omega\hat{\lambda}(\bar{z})$. Since $a_{-2}$ has 4d
helicity $-2$ under the transformation (45), the appropriate operator becomes
$\Phi^{\lambda}(z)=\omega^{-2}a_{-2}(\omega,z,\bar{z})$, c.f. (46). All
together, the $\omega^{2}$ Jacobian cancels out in (59) and so we obtain

$\displaystyle\langle\mathbf{G}^{\lambda_{1}}(z_{1})\Phi^{\lambda_{2}}(z_{2})\rangle$
$\displaystyle=\delta(\omega_{1}+\omega_{2})\delta(z_{1}-z_{2})\delta(\omega_{1}\bar{z}_{1}+\omega_{2}\bar{z}_{2})$
$\displaystyle=\delta(z_{12})\delta^{2}(\lambda_{1}+\lambda_{2})=\delta(z_{12})d^{\lambda_{1},\lambda_{2}}\,,$
(60)

precisely as anticipated. This thus manifests the color structure at the level
of two-point functions. The next task is to compute the anti-MHV amplitude
$\langle\mathbf{GG}\Phi\rangle$, which is slightly different from the usual
computation due to the contact terms. Consider the limits $z_{12}\to 0$ and
$z_{23}\to 0$. Using the $\mathbf{GG}$ OPE (17) together with (58) we find,
respectively,

$\displaystyle\langle\mathbf{G}^{\lambda_{1}}(z_{1})\mathbf{G}^{\lambda_{2}}(z_{2})\Phi^{\lambda_{3}}(z_{3})\rangle$
$\displaystyle\sim\frac{[12]}{z_{12}}\delta(z_{23})\delta^{2}(\lambda_{1}+\lambda_{2}+\lambda_{3})$
$\displaystyle\sim\frac{[23]}{z_{23}}\delta(z_{12})\delta^{2}(\lambda_{1}+\lambda_{2}+\lambda_{3})\,.$
(61)

The two limits are compatible if we identify $\frac{1}{z}\leftrightarrow 2\pi
i\delta(z)$. This would render the function
$\langle\mathbf{G}^{\lambda_{1}}\mathbf{G}^{\lambda_{2}}\Phi^{\lambda_{3}}\rangle$
to be genuinely holomorphic, while entailing that it is only defined on the
support of a certain contour integral. Indeed, in the absence of other
kinematic singularities, the two-dimensional delta function
$\delta(z_{12})\delta(z_{23})$ is the natural contour prescription for
$z_{1},z_{2}$. After integration we get

$\displaystyle\oint\frac{dz_{1}dz_{2}}{2\pi
i}\langle\mathbf{G}^{\lambda_{1}}(z_{1})\mathbf{G}^{\lambda_{2}}(z_{2})\Phi^{\lambda_{3}}(z_{3})\rangle$
$\displaystyle=[12]\delta^{2}(\lambda_{1}+\lambda_{2}+\lambda_{3})$
$\displaystyle=f^{\lambda_{1},\lambda_{2},\lambda_{3}}\,.$ (62)

(The antisymmetry of $f^{\lambda_{1},\lambda_{2},\lambda_{3}}$ is reflected in
the contour prescription, e.g. $\oint\frac{dz_{1}dz_{3}}{2\pi
i}\langle\cdots\rangle=-\oint\frac{dz_{1}dz_{2}}{2\pi i}\langle\cdots\rangle$,
etc.). For real-valued $z_{i}$, corresponding to the 2d Lorentzian slice, we
have

$\displaystyle\frac{1}{2\pi
i}\langle\mathbf{G}^{\lambda_{1}}(z_{1})\mathbf{G}^{\lambda_{2}}(z_{2})\Phi^{\lambda_{3}}(z_{3})\rangle$
$\displaystyle=f^{\lambda_{1},\lambda_{2},\lambda_{3}}\delta(z_{23})\delta(z_{12})$
$\displaystyle=[12]\delta(z_{23})\delta(z_{12})\delta^{2}(\lambda_{1}+\lambda_{2}+\lambda_{3})$
$\displaystyle=\left(\frac{[12]^{3}}{[13][23]}\right)^{2}\delta^{2}(\lambda_{1}+\lambda_{2}+\lambda_{3})\delta^{2}(z_{1}\lambda_{1}+z_{2}\lambda_{2}+z_{3}\lambda_{3})$
$\displaystyle=A_{3}(p_{i})\delta^{4}(\sum_{i=1}^{3}p_{i})\,,$ (63)

which is the desired 3-point dressed S-Matrix. Note that momentum conservation
has emerged from both the color and the holomorphic structure.

We now aim to study
$\langle\mathbf{G}^{\lambda_{1}}\mathbf{G}^{\lambda_{2}}\mathbf{G}^{\lambda_{3}}\Phi^{\lambda_{4}}\rangle$.
Again, we consider first different OPE limits. Starting with $z_{12}\to 0$ we
obtain

$\langle\mathbf{G}^{\lambda_{1}}(z_{1})\mathbf{G}^{\lambda_{2}}(z_{2})\mathbf{G}^{\lambda_{3}}(z_{3})\Phi^{\lambda_{4}}(z_{4})\rangle\to\frac{[12]}{z_{12}}\langle\mathbf{G}^{\lambda_{1}+\lambda_{2}}(z_{2})\mathbf{G}^{\lambda_{3}}(z_{3})\Phi^{\lambda_{4}}(z_{4})\rangle\,.$
(64)

Consider first the Lorentzian case $z_{i}\in\mathbb{R}$. From (63) the three-
point function has support only on $z_{2}=z_{3}=z_{4}$. Thus the OPE limit
$z_{1}\to z_{2}$ implies that all four punctures coincide, which is a very
singular configuration from the point of view of 4d kinematics. For generic
kinematics (subjected to momentum conservation) this singularity vanishes.
Repeating the argument for $z_{13},z_{14}\to 0$ we see that the full
correlation function has no singularities for generic kinematics and hence
must vanish (a purely contact term is not allowed by conformal symmetry). In
fact, this statement is already made in the original twistor construction of
SYM amplitudes Witten (2004)! In that case the one-minus amplitude
$A_{4}(1^{+}2^{+}3^{+}4^{-})$ only has support if the corresponding real
twistor variables collapse to a single point, and hence vanishes generically.

On the other hand, we have proposed here to interpret the holomorphic momentum
conservation as a two-dimensional contour prescription. Unlike the 3-point
case however, the function $\langle\mathbf{G}\mathbf{G}\mathbf{G}\Phi\rangle$
carries the additional singularities of the type (64) and so it is natural to
integrate it against a three-dimensional contour. In fact, since the overal
$SL(2,\mathbb{R})_{z}$ weight is 3, the natural contour is

$\displaystyle\oint\frac{dz_{1}dz_{2}dz_{3}}{2\pi}\langle\mathbf{G}^{\lambda_{1}}(z_{1})\mathbf{G}^{\lambda_{2}}(z_{2})\mathbf{G}^{\lambda_{3}}(z_{3})\Phi^{\lambda_{4}}(z_{4})\rangle$
(65)
$\displaystyle=f_{\quad\,\,\,\,\lambda}^{\lambda_{1},\lambda_{2}}\,\,f^{\lambda,\lambda_{3},\lambda_{4}}+f_{\quad\,\,\,\,\lambda}^{\lambda_{1},\lambda_{3}}\,\,f^{\lambda,\lambda_{4},\lambda_{2}}+$
$\displaystyle
f_{\quad\,\,\,\,\lambda}^{\lambda_{1},\lambda_{4}}\,\,f^{\lambda,\lambda_{2},\lambda_{3}}\,,$

which extends the 3-point statement (62). Here we have employed the form of
the singularities (64) together with the double residue (62). This combination
vanishes due to associativity (20) and can be interpreted as a residue theorem
(see e.g. García Sepúlveda and Guevara (2019) and references within for
evaluation of multidimensional residues). This can be thought as the momentum
space version of the argument given in Mizera (2020), where the same residues
where analyzed in the moduli space. This is also precisely the condition found
recently in momentum space in Ren  _et al._ (2022), which also extended the
analysis to helicity-flipping terms. For completeness we present here the OPE
for general helicity fields (the momentum-space version of the OPEs in
Himwich, Pate, and Singh (2022))

$O_{h_{1}}^{\lambda_{1}}(z_{1})O_{h_{2}}^{\lambda_{2}}(z_{2})\sim\sum_{h}\kappa_{\quad
h}^{h_{1}h_{2}}\frac{[12]^{h_{1}+h_{2}-h-1}}{z_{12}}O_{h}^{\lambda_{1}+\lambda_{2}}\,,$
(66)

where imposing the vanishing of the analog of (65), or equivalently OPE
associativity, leads to the constraints in $\kappa_{\quad h}^{h_{1}h_{2}}$
presented in Ren  _et al._ (2022); Mago  _et al._ (2021). It would be
interesting to find a kinematic Lie algebra realization in the cases where
$h_{2}\neq h$, which a priori do not correspond to adjoint $U(\infty)$
representations.


# Dhara-Rehman-Raza’s identities on left ideals of prime rings

Driss Bennis1, Brahim Fahid2 and Abdellah Mamouni3

1: Centre de Recherche de Mathématiques et Applications de Rabat (CeReMAR),
Faculty of Sciences, Mohammed V University in Rabat, Morocco.

<EMAIL_ADDRESS><EMAIL_ADDRESS>
2: Department of Mathematics, Superior School of Technology, Ibn Tofail
University, Kenitra, Morocco.
<EMAIL_ADDRESS>
3: Department of Mathematics, Faculty of Sciences and Techniques, Moulay
Ismaïl University, Errachidia, Morocco.

<EMAIL_ADDRESS>

Abstract. It is known that every nonzero Jordan ideal of $2$-torsion free
semiprime rings contains a nonzero ideal. In this paper we show that also any
square closed Lie ideal of a $2$-torsion free prime ring contains a nonzero
ideal. This can be interpreted by saying that studying identities over one
sided ideals is the “optimal” case to study identities. With this fact in
mind, we generalize some results of Dhara, Rehman and Raza in [Lie ideals and
action of generalized derivations in rings, Miskolc Mathematical Notes, 16
(2015), 769 – 779] to the context of nonzero left ideals.

2010 Mathematics Subject Classification. 16W25, 16N60

Key Words. prime rings, generalized derivations, left ideals

##  1 Introduction

Let $R$ be an associative ring with center $Z(R)$. Recall that $R$ is prime if
for $a,b\in R$, $aRb=\\{0\\}$ implies either $a=0$ or $b=0$. $R$ is said to be
semiprime if for $a\in R$, $aRa=\\{0\\}$ implies $a=0$. Let $R$ be a prime
ring. For any pair of elements $x,y\in R$, we shall write $[x,y]$ (resp.,
$x\circ y$) for the commutator $xy-yx$ (resp., for the Jordan product
$xy+yx$). An additive subgroup $J$ of $R$ is said to be a Jordan ideal of $R$
if $u\circ r\in J$ for all $u\in J$ and $r\in R$. Note that every ideal of $R$
is a Jordan ideal of $R$ but the converse is not true in general. An additive
subgroup $U$ of $R$ is said to be a Lie ideal of $R$ if $[u,r]\in U$ for all
$u\in U$ and $r\in R$. It is clear that if characteristic of $R$ is $2$, then
Jordan ideals and Lie ideals of $R$ coincide. A Lie ideal $U$ of $R$ is said
to be square closed if $u^{2}\in U$ for all $u\in U$.

An additive mapping $d:R\longrightarrow R$ is called a derivation if
$d(xy)=d(x)y+xd(y)$ holds for all $x,y\in R$. In [6] Brešar introduced the
generalized derivation: an additive mapping $F:R\longrightarrow R$ is called a
generalized derivation if there exists a derivation $d:R\longrightarrow R$ (an
associated derivation of $F$) such that $F(xy)=F(x)y+xd(y)$ for all $x,y\in
R$. The notion a generalized derivation covers both the notions of a
derivation and of a left multiplier (i.e., an additive mapping
$f:R\longrightarrow R$ satisfying $f(xy)=f(x)y$ for all $x,y\in R$). A ring
$R$ is said to be $n$-torsion free, where $n\neq 0$ is a positive integer, if
whenever $na=0$, with $a\in R$, then $a=0$.

In the present paper we investigate commutativity of a prime ring satisfying
certain differential identities on a nonzero left ideal. Let us first recall
that the study of commutativity of rings using differential identities goes
back to the well-known Posner’s work [17] in which he proved that the
existence of a nonzero centralizing derivation $f$ on a prime ring $R$ (i.e.,
$[f(x),x]\in Z(R)$ for all $x\in R$) forces the ring $R$ to be commutative.
This result is now known by Posner’s second theorem. Since then, several
authors have been interested in extending or generalizing this result to
different contexts. For instance, Awtar, in [4], extended Posner’s second
theorem to the case of centralizing derivation on either nonzero Jordan ideals
or nonzero square closed Lie ideals (see also Mayne’s papers [14, 15, 16]). In
[7], Bre$\mathrm{\check{s}}$ar generalised Posner’s second theorem to
identities related to two derivations (see [7, Theorem 4.1]). In [13], Hvala
introduced the study of identities related to generalized derivations. In [5],
Ali, De Filippis and Shujat studied identities with generalized derivations on
one sided ideals of a semiprime ring.

Also motivated by the success that known Posner’s second theorem, several
authors have introduced new kind of differential identities. In this context,
Ashraf and Rehman proved in [3] that a prime ring $R$ with a nonzero ideal $I$
must be commutative if $R$ admits a nonzero derivation $d$ satisfying
$d(xy)-xy\in Z(R)$ for all $x,y\in I$ or $d(xy)+xy\in Z(R)$ for all $x,y\in
I$. In [2], Ashraf, A. Ali and S. Ali studied these identities in the case
where $d$ is a generalized derivation. In [11], Dhara, Rehman and Raza used
more general differential identities. Precisely, they showed that for a
nonzero square closed Lie ideal $U$ of a prime ring $R$, if $R$ admits nonzero
generalized derivations $F$, $G$ and $H$ satisfying $F(x)G(y)\pm H(xy)\in
Z(R)$ or $F(x)F(y)\pm H(yx)\in Z(R)$ for all $x,y\in U$, then $U\subseteq
Z(R)$. See also [1, 8, 9, 10, 18] for other works related to these identities.

Then, naturally one can ask whether we get the same conclusion as the one of
Dhara, Rehman and Raza’s results if we replace the nonzero square closed Lie
ideal of the prime ring $R$ by other particular subsets of $R$. Namely, Jordan
ideals, (both sided) ideals and one sided ideals. It is important to mention
that the study of identities on Jordan ideals and (both sided) ideals can be
considered as particular case of the study of these identities on square
closed Lie ideals. Indeed, it is clear that every ideal is a square closed Lie
ideal and, by [12, Theorem 1.1], every nonzero Jordan ideal of $2$-torsion
free semiprime rings contains a nonzero ideal. Then, only the “one sided
ideals” case could be of interest. Our aim in this paper is to show that $R$
will be commutative if we consider, in Dhara-Rehman-Raza’s identities, only
nonzero left ideals instead of square closed Lie ideals (see Theorems 3.1 and
3.6). This is a generalization of Dhara, Rehman and Raza’s results (see
Corollary 3.3) we prove that any square closed Lie ideal $L$ of a $2$-torsion
free prime ring $R$ contains a nonzero ideal of $R$ (see Proposition 2.5).

##  2 Preliminaries

In this section we recall and present some properties of left ideals which
will be used to prove our main results.

###### Lemma 2.1

Let $R$ be a prime ring, $I$ is a nonzero left ideal of $R$ and for every
$a,b\in R$, $aIb=(0)$, then $a=0$ or $Ib=(0)$.

Proof. We have $aIb=(0)$ imply $aRIb=(0)$, so by primeness of $R$ we get $a=0$
or $Ib=(0)$.

###### Lemma 2.2

Let $R$ be a prime ring and $I$ a nonzero left ideal of $R$. If $F$ is a
generalized derivation of $R$ with an associated derivation $d$ such that
$F(I)=0$, then $Id(I)=0$.

Proof. We have, for every $u,v\in I$, $F(uv)=F(u)v+ud(v)=ud(v)=0$. Hence
$Id(I)=0$.

###### Lemma 2.3

Let $R$ be a prime ring and $I$ be a nonzero left ideal of $R$. Then, the
following assertions are equivalents:

  1. 1.

$[I,I]=0$.

  2. 2.

$I\subseteq Z(R)$.

  3. 3.

$R$ is commutative.

Proof. $(1)\Rightarrow(2)$ Let $I$ be a left ideal such that $[I,I]=0$. Then,
$0=[I,RI]=[I,R]I$, which implies that $[I,R]=0$.
$(2)\Rightarrow(3)$ We have $[I,R]=0$. This gives $0=[RI,I]=[R,R]I$. Since a
left annihilator of a left ideal is zero, $[R,R]=0$; that is, $R$ is
commutative.
$(3)\Rightarrow(1)$ Obvious.

###### Lemma 2.4

Let $R$ be a noncommutative prime ring and $I$ be a nonzero left ideal of $R$.
Let $d$ be derivation of $R$ and $z\in Z(R)$, such that $[xy,r]d(z)=0$ for all
$x,y\in I$ and $r\in R$. Then $d(z)=0$.

Proof. First we show that $d(z)\in Z(R)$, for this we have
$d(zr)=d(z)r+zd(r)=d(r)z+rd(z)$, so $d(z)\in Z(R)$. On other hand we have

$[xy,r]d(z)=0\;\;\;\mbox{for all}\;\;x,y\in I;\;\;r\in R.$ (2.1)

Replacing $x$ by $sxd(z)$ in the above equation, where $s\in R$, we get

$[s,r]xd(z)yd(z)=0\;\;\;\mbox{for all}\;\;x,y\in I;\;\;r\in R.$ (2.2)

So $Id(z)=0$ and $d(z)=0$.

It is well-known that every nonzero Jordan ideal of a $2$-torsion free
semiprime ring contains a nonzero ideal (see, for instance, [12, Theorem
1.1]). Here we give a similar result for square closed Lie ideals.

###### Proposition 2.5

Let $L$ be a nonzero square closed Lie ideal of a $2$-torsion free prime ring
$R$. Then, $L$ contains a nonzero ideal of $R$.

Proof. Let $L$ be a Lie ideal of $R$ such that $u^{2}\in L$ for all $u\in L$.
Therefore, for any $u,v\in L$, we get $uv+vu=(u+v)^{2}-u^{2}-v^{2}\in L$. On
the other hand we have $uv-vu\in L$. Combining these two equalities we get
$2uv\in L$ for all $u,v\in L$. Then, $2L$ is both a Lie ideal and a subring of
$R$. Also $2L\neq 0$ since $R$ is a $2$-torsion free ring. Then, by [12, Lemma
1.3] and [11, Lemma 5], $2L$ contains a non-zero ideal of $R$ and so does $L$
since $2L\subseteq L$.

##  3 Main results

We start with the first main result, using Proposition 2.5, it can be seen as
a generalization of [11, Theorems 1 and 2] ( see Corollary 3.3).

###### Theorem 3.1

Let $R$ be a prime ring and $I$ be a nonzero left ideal of $R$. Let $F,$ $G$
and $H$ be generalized derivations associated to derivations $f,$ $g$ and $h$
of $R$, respectively, such that $F(x)G(y)-H(xy)\in Z(R)$ for all $x,y\in I$.
Then $R$ is commutative or $If(I)=0$ or $Ig(I)=0$.

Proof. We are given that

$F(x)G(y)-H(xy)\in Z(R)\;\;\;\mbox{for all}\;\;x,y\in I.$ (3.1)

Replacing $y$ by $yz$ in (3.1), we get

$F(x)G(y)z-H(xy)z+F(x)yg(z)-xyh(z)\in Z(R)\;\;\;\mbox{for all}\;\;x,y,z\in I.$
(3.2)

That is

$[F(x)yg(z)-xyh(z),z]=0\;\;\;\mbox{for all}\;\;x,y,z\in I.$ (3.3)

Replacing $x$ by $xu$, where $u\in I$, we get

$[F(x)(uy)g(z)-x(uy)h(z),z]+[xf(u)yg(z),z]=0\;\;\;\mbox{for all}\;\;u,x,y,z\in
I.$ (3.4)

Since $uy\in I$, then (3.3) together with (3.4) force that

$[xf(u)yg(z),z]=0\;\;\;\mbox{for all}\;\;u,x,y,z\in I.$ (3.5)

Writing $rx$ instead of $x$ in (3.5), where $r\in R$, we obtain

$[r,z]xf(u)yg(z)=0\;\;\;\mbox{for all}\;\;u,x,y,z\in I;\;\;r,s\in R.$ (3.6)

That is

$[r,z]If(u)Ig(z)=0\;\;\;\mbox{for all}\;\;u,z\in I;\;\;r,s\in R.$ (3.7)

The primeness of $R$ together with (3.7) imply that $[r,z]=0$ or $If(u)=(0)$
or $Ig(z)=(0)$. Therefore, $R$ is commutative or $If(I)=0$ or $Ig(I)=0$.

As a consequence of Theorem 3.1 we get the following result.

###### Corollary 3.2

Let $R$ be a prime ring and $I$ be a nonzero left ideal of $R$. Let $F,$ $G$
and $H$ be generalized derivations associated to derivations $f,$ $g$ and $h$
of $R$, respectively, such that $F(x)G(y)+H(xy)\in Z(R)$ for all $x,y\in I$.
Then $R$ is commutative or $If(I)=0$ or $Ig(I)=0$.

Proof. We notice that $-H$ is a generalized derivation of $R$ associated to
the derivation $-h$. Hence replacing $H$ by $-H$ in Theorem 3.1, we get
$F(x)G(y)-(-H)(xy)\in Z(R)$ for all $x,y\in I$, that is $F(x)G(y)+H(xy)\in
Z(R)$ all $x,y\in I$. Therfore $R$ is commutative or $If(I)=0$ or $Ig(I)=0$.

Now we can see that [11, Theorems 1 and 2] can be seen as a consequence of
Theorem 3.1 as follows.

###### Corollary 3.3

Let $L$ be a nonzero square closed Lie ideal of $2$-torsion free prime ring
$R$. Let $F,G,H$ be generalized derivations with associated derivations
$f,g,h$ of $R$, respectively, such that $f\neq 0$ and $g\neq 0$. If
$F(x)G(y)\pm H(xy)\in Z(R)$ for all $x,y\in L$, then $R$ commutative.

Proof. Using Proposition 2.5, $L$ contains a nonzero ideal $I$ of $R$. So we
get $F(x)G(y)\pm H(xy)\in Z(R)$ for all $x,y\in I$. Using Theorem 3.1 and
Corollary 3.2 we get that $R$ is commutative.

Also, if we suppose $F=f$ and $G=g$ in Theorem 3.1, then we get the following
corollary.

###### Corollary 3.4

Let $R$ be a prime ring and $I$ be a nonzero left ideal of $R$. Let $F,$ $G$
and $H$ be generalized derivations associated to derivations $f,$ $g$ and $h$
of $R$, respectively, such that $f(x)g(y)\pm H(xy)\in Z(R)$ for all $x,y\in
I$. Then $R$ is commutative or $If(I)=0$ or $Ig(I)=0$.

When $H$ in Theorem 3.1 is the identity map, we get the following result.

###### Corollary 3.5

Let $R$ be a prime ring and $I$ be a nonzero left ideal of $R$. Let $F,$ $G$
and $H$ be generalized derivations associated to derivations $f,$ $g$ and $h$
of $R$, respectively, such that $F(x)G(y)\pm xy\in Z(R)$ for all $x,y\in I$.
Then $R$ is commutative or $If(I)=0$ or $Ig(I)=0$.

Now, we give our second main result. Compare it with [11, Theorem 3].

###### Theorem 3.6

Let $R$ be a prime ring and $I$ be a nonzero left ideal of $R$. Let $F$ and
$H$ be generalized derivations associated to derivations $f$ and $h$ of $R$,
respectively, such that $F(x)F(y)-H(yx)\in Z(R)$ for all $x,y\in I$. If $0\neq
If(I)\subseteq I$ and $0\neq Ih(I)\subseteq I$ , then $R$ is commutative.

Proof. Assume that $I\cap Z(R)=\\{0\\}$.
By the hypothesis, we have

$F(x)F(y)-H(yx)\in Z(R)\;\;\;\;\mbox{for all}\;\;x,y\in I.$ (3.8)

Replacing in (3.8) $y$ by $yz$, where $y,z\in I$, we get, for all $x,y,z\in
I$:

$F(x)F(y)z+F(x)yf(z)-H(yz)x-yzh(x)\in Z(R)$ (3.9)

That is

$(F(x)F(y)-H(yx))z+H(yx)z+$ (3.10) $F(x)yf(z)-H(yz)x-yzh(x)\in
Z(R)\;\;\;\;\mbox{for all}\;\;x,y,z\in I.$

By the hypothesis we have

$(F(x)F(y)-H(yx))z+H(yx)z+$ (3.11) $F(x)yf(z)-H(yz)x-yzh(x)\in
I\;\;\;\;\mbox{for all}\;\;x,y,z\in I.$

Since $I\cap Z(R)=0$, we get

$(F(x)F(y)-H(yx))z+H(yx)z+$ (3.12) $F(x)yf(z)-H(yz)x-yzh(x)=0\;\;\;\;\mbox{for
all}\;\;x,y\in I.$

Replacing $z$ by $zx$, we obtain, for all $x,y,z\in I$:

$F(x)yzf(x)-yzh(x)x-yz[x,h(x)]=0.$ (3.13)

Replacing $y$ by $py$, we get

$[F(x),p]yzf(x)=0\;\;\;\;\mbox{ for all}\;\;x,y,p,z\in I.$ (3.14)

So, using primeness of $R$, we get, for all $x,y\in I$: $[F(x),p]y=0$ or
$If(x)=0$. This leads to

$F(x)\in Z(R)\;\;\;\mbox{or}\;\;If(x)=0\;\;\;\;\mbox{for all}\;\;x\in I.$

The fact that a group cannot be a union of its proper subgroups with the
condition $If(I)\neq 0$ imply that

$F(x)\in Z(R)\;\;\;\mbox{for all}\;\;x\in I.$

Then, $F$ is centralizing on $I$. Therefore, $\dot{}R$ is commutative. Then
$I=0$, which contradicts our hypothesis. Therefore, $I\cap Z(R)\neq\\{0\\}.$
Let $z\in I\cap Z(R)\backslash\\{0\\}$ and replacing $y$ by $yz$ in our
identities hypothesis, we get

$F(x)yf(z)-yxh(z)\in Z(R)\;\;\;\mbox{for all}\;\;x,y\in I.$ (3.15)

So

$[F(x)y,r]f(z)-[yx,r]h(z)=0\;\;\;\mbox{for all}\;\;x,y\in I;\;\;r\in R.$
(3.16)

Replacing $x$ by $xz$ in (3.16), we get, for all $x,y\in I;\;\;r\in R$:

$[F(x)(zy),r]f(z)+[xf(z)y,r]f(z)-[y(zx),r]h(z)=0.$ (3.17)

Using (3.15) we arrive at

$[xf(z)y,r]f(z)=0\;\;\;\mbox{for all}\;\;x,y\in I;\;\;r\in R.$ (3.18)

Replacing $x$ by $sx$ in (3.18), where $s\in R$, we get

$[s,r]xf(z)yf(z)=0\;\;\;\mbox{for all}\;\;x,y\in I;\;\;s\in R.$ (3.19)

The primeness of $R$ together with equation (3.19) show that $R$ is
commutative or $If(z)=0$. If $f(z)=0$, (3.16) becomes

$[yx,r]h(z)=0\;\;\;\mbox{for all}\;\;x,y\in I;\;\;r\in R.$ (3.20)

So by Lemma 2.4 we get $h(z)=0$.

Replacing now $y$ by $tz$ in (3.8), where $t\in R$, we get

$F(x)F(t)-H(tx)\in Z(R)\;\;\;\mbox{for all}\;\;x,\in I;\;\;t\in R.$ (3.21)

Replacing $x$ by $sz$ in (3.21), where $s\in R$, we get

$F(s)F(t)-H(ts)\in Z(R)\;\;\;\mbox{for all}\;\;s,t\in R.$ (3.22)

Therefore, since $R$ is a square closed Lie ideal of $R$ itself and by [11,
Theorem 3] and Lemma 2.3, we conclude that $R$ is commutative.

###### Remark 3.7

One can see that, instead of using [11, Theorem 3] at the end of the proof
above, we can give a direct proof following the same arguments done in [11,
Theorem 3]. Thus, using [12, Theorem 1.1] and Proposition 2.5, we conclude
that studying identities over one sided ideals can be seen until now as the
“optimal” case to study identities.

We end this paper, with some consequences of Theorem 3.6.

###### Corollary 3.8

Let $R$ be a prime ring and $I$ be a nonzero left ideal of $R$. Let $F$ and
$H$ be generalized derivations associated to derivations $f$ and $h$ of $R$,
respectively, such that $F(x)F(y)+H(yx)\in Z(R)$ for all $x,y\in I$. If $0\neq
If(I)\subseteq I$ and $0\neq Ih(I)\subseteq I$ , then $R$ is commutative .

Proof. We notice that $-H$ is a generalized derivation of $R$ associated to
the derivation $-h$. Hence replacing $H$ by $-H$ in Theorem 3.6, we get
$F(x)F(y)-(-H)(yx)\in Z(R)$ for all $x,y\in I$. That is $F(x)F(y)+H(yx)\in
Z(R)$ for all $x,y\in I$. This implies that $R$ is commutative.

When we consider $F=f$ in Theorem 3.6, we get the following result.

###### Corollary 3.9

Let $R$ be a prime ring and $I$ be a nonzero left ideal of $R$. Let $F$ and
$H$ be generalized derivations associated to derivations $f$ and $h$ of $R$,
respectively, such that $f(x)f(y)\pm H(yx)\in Z(R)$ for all $x,y\in I$. If
$0\neq If(I)\subseteq I$ and $0\neq Ih(I)\subseteq I$, then $R$ is
commutative.

Also, if we suppose $H$ to be the identity map in Theorem 3.6, we get the
following result.

###### Corollary 3.10

Let $R$ be a prime ring and $I$ be a nonzero left ideal of $R$. Let $F$ be a
generalized derivation associated to a derivation $f$ of $R$ such that
$F(x)F(y)\pm yx\in Z(R)$ for all $x,y\in I$. If $0\neq If(I)\subseteq I$ and
$0\neq Ih(I)\subseteq I$, then $R$ is commutative.

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