Celestial 𝓌₁₊∞ Symmetries from Twistor Space
We explain how twistor theory represents the self-dual sector of four-dimensional gravity in terms of the loop group of Poisson diffeomorphisms of the plane via Penrose's non-linear graviton construction. The symmetries of the self-dual sector are generated by the corresponding loop algebra 𝐿𝓌₁...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2022 |
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| Мова: | Англійська |
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Інститут математики НАН України
2022
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Celestial 𝓌₁₊∞ Symmetries from Twistor Space. Tim Adamo, Lionel Mason and Atul Sharma. SIGMA 18 (2022), 016, 23 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859534789351047168 |
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| author | Adamo, Tim Mason, Lionel Sharma, Atul |
| author_facet | Adamo, Tim Mason, Lionel Sharma, Atul |
| citation_txt | Celestial 𝓌₁₊∞ Symmetries from Twistor Space. Tim Adamo, Lionel Mason and Atul Sharma. SIGMA 18 (2022), 016, 23 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We explain how twistor theory represents the self-dual sector of four-dimensional gravity in terms of the loop group of Poisson diffeomorphisms of the plane via Penrose's non-linear graviton construction. The symmetries of the self-dual sector are generated by the corresponding loop algebra 𝐿𝓌₁₊∞ of the algebra 𝓌₁₊∞ of these Poisson diffeomorphisms. We show that these coincide with the infinite tower of soft graviton symmetries in tree-level perturbative gravity recently discovered in the context of celestial amplitudes. We use a twistor sigma model for the self-dual sector, which describes maps from the Riemann sphere to the asymptotic twistor space defined from characteristic data at null infinity I. We show that the OPE of the sigma model naturally encodes the Poisson structure on twistor space and gives rise to the celestial realization of 𝐿𝓌₁₊∞. The vertex operators representing soft gravitons in our model act as currents generating the wedge algebra of 𝓌₁₊∞ and produce the expected celestial OPE with hard gravitons of both helicities. We also discuss how the two copies of 𝐿𝓌₁₊∞, one for each of the self-dual and anti-self-dual sectors, are represented in the OPEs of vertex operators of the 4d ambitwistor string.
|
| first_indexed | 2026-03-13T08:29:29Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 016, 23 pages
Celestial w1+∞ Symmetries from Twistor Space
Tim ADAMO a, Lionel MASON b and Atul SHARMA b
a) School of Mathematics and Maxwell Institute for Mathematical Sciences,
University of Edinburgh, EH9 3FD, UK
E-mail: t.adamo@ed.ac.uk
b) The Mathematical Institute, University of Oxford, OX2 6GG, UK
E-mail: lmason@maths.ox.ac.uk, atul.sharma@maths.ox.ac.uk
Received November 22, 2021, in final form February 17, 2022; Published online March 08, 2022
https://doi.org/10.3842/SIGMA.2022.016
Abstract. We explain how twistor theory represents the self-dual sector of four dimensional
gravity in terms of the loop group of Poisson diffeomorphisms of the plane via Penrose’s non-
linear graviton construction. The symmetries of the self-dual sector are generated by the
corresponding loop algebra Lw1+∞ of the algebra w1+∞ of these Poisson diffeomorphisms.
We show that these coincide with the infinite tower of soft graviton symmetries in tree-
level perturbative gravity recently discovered in the context of celestial amplitudes. We use
a twistor sigma model for the self-dual sector which describes maps from the Riemann sphere
to the asymptotic twistor space defined from characteristic data at null infinity I . We show
that the OPE of the sigma model naturally encodes the Poisson structure on twistor space
and gives rise to the celestial realization of Lw1+∞. The vertex operators representing soft
gravitons in our model act as currents generating the wedge algebra of w1+∞ and produce
the expected celestial OPE with hard gravitons of both helicities. We also discuss how the
two copies of Lw1+∞, one for each of the self-dual and anti-self-dual sectors, are represented
in the OPEs of vertex operators of the 4d ambitwistor string.
Key words: twistor theory; scattering amplitudes; self-duality
2020 Mathematics Subject Classification: 83C60; 81U20; 32L25
Dedicated to our friend and mentor Roger Penrose
on the occasion of his 90th birthday and the recent
award of his Nobel prize in physics.
1 Introduction
Among other things, Roger Penrose is famous in general relativity for his introduction of null
infinity, I , as the geometry underpinning the asymptotics of massless space-time radiation
fields [56, 57]. In recent years, there has been a resurgence in the study of asymptotic symmetries
and scattering amplitudes at null infinity I , much of it aimed at formulating a notion of
holography for asymptotically flat space-times (cf. [7, 53, 66, 75] for recent reviews). In fact, the
notion of reconstructing ‘bulk’ space-times and their physics holographically at I dates back
to the 1970s and the work of Newman and Penrose [46, 59, 60]. One of the main outputs of
this work was the non-linear graviton construction, where (complex) space-times with self-dual
curvature arise from deformations of the complex structure on twistor spaces. When these are
‘asymptotic’ twistor spaces, the non-linear graviton is intrinsically holographic, as the deformed
complex structure is constructed directly from the (complexified) characteristic data (i.e., the
self-dual asymptotic shear) of an asymptotically flat, radiative self-dual space-time at I [19].
This paper is a contribution to the Special Issue on Twistors from Geometry to Physics in honor of Roger
Penrose. The full collection is available at https://www.emis.de/journals/SIGMA/Penrose.html
mailto:t.adamo@ed.ac.uk
mailto:lmason@maths.ox.ac.uk
mailto:atul.sharma@maths.ox.ac.uk
https://doi.org/10.3842/SIGMA.2022.016
https://www.emis.de/journals/SIGMA/Penrose.html
2 T. Adamo, L. Mason and A. Sharma
Much of the recent work on ‘celestial holography’ has focused on the interplay between
asymptotic symmetries and soft particles [73, 74]. For example, at leading order in the soft
momentum, soft gravitons are related to BMS supertranslations via a Ward identity [29]; there
are now many generalizations to subleading orders and other theories (cf. [75] and references
therein) which can also be understood in terms of an interplay between asymptotic symmetries
and twistor or ambitwistor data [2, 3, 25]. By expressing scattering amplitudes in terms of a
conformal primary basis on the celestial sphere [54, 55], it is clear that there is actually an
infinite tower of conformal soft graviton theorems arising when the soft external graviton has
scaling dimension ∆ = 2, 1, 0,−1, . . . [6, 17, 26, 65]. For a positive helicity soft graviton, this
infinite tower of soft theorems can be organized into the algebra w1+∞ (or more precisely, the
loop algebra of the wedge algebra of w1+∞) [28, 30, 34, 76].
It has long been known that the algebra w1+∞ classically describes canonical transformations
of a plane [10, 32]. Over the years, a number of authors have linked this to self-dual gravity
via the non-linear graviton construction [16, 39, 51, 52] of deformed twistor spaces for self-dual
space-times. The deformed twistor spaces are glued together by patching functions that can be
expressed as maps from a neighbourhood of the equator of the Riemann sphere to canonical
transformations of the 2-dimensional fibres of the twistor space over this sphere, as explained by
Penrose himself in his original paper [59]. Thus, the Lie algebra, Lw1+∞, of the loop group of
canonical transformations acts on this space of patching functions for twistor space and hence on
the space of all self-dual Ricci-flat metrics. Although Lw1+∞ transformations act by diffeomor-
phisms and hence resemble gauge transformations, generically they are not global and have sin-
gularities. They define genuine deformations of the twistor space and are not, strictly speaking,
symmetries. Such constructions making use of singular gauge transformations on twistor space
to transform one solution to another are standard in twistor formulations of classical Bäcklund
transformations in the study of integrable systems (cf. [41, 42, 45, 80, 81]). These ideas were
developed into a twistor formulation of the Lw1+∞ symmetries via a recursion operator based on
such a Backlund transformation to generate the loop algebra from coordinate symmetries in [18].
In the non-linear graviton construction, the self-dual space-time is recovered as the four-
dimensional family of rational holomorphic curves in twistor space of degree one. Recently, we
introduced sigma models in twistor space for such holomorphic curves [5] whose on-shell action
is equal to the Kähler scalar (or first Plebański scalar [63]) of the associated self-dual space-time.
These ‘twistor sigma models’ can be used to construct gravitational MHV scattering amplitudes
directly from general relativity, and at higher-degree build the full tree-level S-matrix of gravity
via a natural family of generating functionals. In this paper, we show how the loop algebra
of w1+∞ and the infinite tower of soft graviton theorems is realised in terms of these twistor
sigma models. We will also see that the action of Lw1+∞ can be lifted to 4d-ambitwistor models
at I allowing us to represent copies of Lw1+∞ for both the self-dual and anti-self-dual sectors
within the same model.
We begin in Section 2 with a brief review of w1+∞, its loop algebra and explain how it is
realized in terms of twistor space and self-dual gravity. Section 3 reviews the twistor sigma
model, and its relationship to self-dual gravity at null infinity through the projection from
asymptotic twistor space to I [19]. We review how the model at degree-1 computes the MHV
sector of tree-level graviton scattering. In Section 4.1 we show how asymptotic symmetries are
expressed in terms of the twistor sigma model; using the operator product expansion (OPE) of
the model we show that these are controlled by the loop algebra Lw1+∞. Indeed, the twistor
sigma model shows how the holomorphic curves of the non-linear graviton construction provide
the most basic realization of this algebra.
Sections 4.2 and 4.3 explore the soft expansion of a positive helicity graviton in terms of
vertex operators in the twistor sigma model. We show that this expansion gives the generators
of Lw1+∞ and produces the infinite tower of soft graviton symmetries identified in [28, 76]. Sec-
Celestial w1+∞ Symmetries from Twistor Space 3
tion 5 outlines a generalization of these symmetries to both self-dual and anti-self-dual sectors
of gravity by means of the 4d ambitwistor string [24] at I [6, 25], pointing to avenues of future
work. We conclude with some remarks regarding choices of (2, 2) vs. (1, 3) signature, quantiza-
tion of the twistor sigma models, and their relation to the celestial holography programme.
2 Lw1+∞ and self-dual gravity
The algebra w1+∞ arises as the Lie algebra of the Poisson structure (or area) preserving dif-
feomorphisms of the plane [10, 11, 32], although it can also be viewed as the classical limit of
the W1+∞ algebra associated to two-dimensional conformal field theories with higher-spin con-
served currents [20, 21, 82] – see [64] for a review. In this section we recall the basic structure
of w1+∞, its loop algebra Lw1+∞, and their realization in twistor space through the non-linear
graviton construction.
2.1 Poisson diffeomorphisms and Lw1+∞
Let µα̇ =
(
µ0̇, µ1̇
)
be coordinates on the plane, with Poisson structure
{f, g} := εα̇β̇
∂f
∂µα̇
∂g
∂µβ̇
, εα̇β̇ = ε[α̇β̇], ε0̇1̇ = 1. (2.1)
Elements of the Lie algebra of Poisson diffeomorphisms can be decomposed into polynomial
Hamiltonians on the µα̇-plane of degree 2p− 2 ∈ Z≥0:
wp
m :=
(
µ0̇
)p+m−1(
µ1̇
)p−m−1
, |m| ≤ p− 1,
so that p±m− 1 ∈ Z≥0. The Poisson bracket acting on these elements gives{
wp
m, w
q
n
}
= 2(m(q − 1)− n(p− 1))wp+q−2
m+n .
This defines the commutation relations of the basis elements wp
m of w1+∞. Here, the ‘1’ in 1+∞
refers to the central element of degree 2p− 2 = 0.
The loop algebra Lw1+∞ of w1+∞ can be represented by introducing a complex coordinate
λ ∈ C, where the loop is parametrized by |λ| = 1. Alternatively, λ can be viewed as an affine
coordinate on the Riemann sphere S2 ∼= CP1: if λα = (λ0, λ1) are homogeneous coordinates
on CP1, then on the patch where λ0 ̸= 0 we can identify λ ≡ λ1/λ0. With this, the generators
of Lw1+∞ can be written as
gpm,r :=
wp
m
λ2p−4−r
0 λr1
=
wp
m
λr
, (2.2)
where in the second equality we have chosen a scaling for the homogeneous coordinates in which
λ0 = 1. The Poisson bracket (2.1) gives{
gpm,r, g
q
n,s
}
= 2(m(q − 1)− n(p− 1))gp+q−2
m+n,r+s, (2.3)
which define the Lie bracket of the loop algebra Lw1+∞. Later we will also introduce the
parameter z on the equator of CP1 so that the gpm,r arise as the coefficients of the formal
Laurent series appropriate to a contour around λ = z in
gpm(z) =
gpm
λ− z
=
∑
r∈Z
gpm,rz
r−1, (2.4)
defining a field insertion at the point λ = z.1
1This uses the language of 2d quantum fields, but this paper – excepting Section 5 – mostly concerns the
semi-classical limit.
4 T. Adamo, L. Mason and A. Sharma
PT U U ∩ Ũ Ũ
CP1
(µα̇, λα) (µ̃α̇, λα)
p
λ0 6= 0 λ1 6= 0
Figure 1. The deformed twistor space PT in terms of a patching fibred over CP1.
2.2 Realization on twistor space
The twistor space PT of complexified Minkowski space (i.e., C4 equipped with the holomorphic
Minkowski metric) is an open subset of CP3. If ZA =
(
µα̇, λα
)
are four homogeneous coordinates
on CP3, then twistor space is the open subset PT =
{
Z ∈ CP3 |λα ̸= 0
}
. The relationship
between PT and complexified Minkowski space is non-local: a point xαα̇ in the complexified
space-time corresponds to a holomorphic, linearly embedded Riemann sphere in PT defined by
µα̇ = xαα̇λα.
Twistor space admits a natural fibration over CP1
p : PT → CP1, p(Z) = λα, (2.5)
with λα serving as homogeneous coordinates on the Riemann sphere (this is possible precisely
because λα ̸= 0 on PT). The fibres of p are 2-planes C2 with complex coordinates µα̇. Twistor
space also admits the holomorphic Poisson structure (2.1), where the Poisson bracket is trivially
extended to act on functions that depend on λα as well as µα̇. It provides a non-degenerate
symplectic structure on every fibre.
One of the central results of twistor theory is the non-linear graviton theorem:
Theorem 2.1 (Penrose [59]). There is a 1 : 1 correspondence between self-dual Ricci-flat holo-
morphic metrics on regions in C4, and complex deformations PT of twistor space PT that
preserve the fibration p : PT → CP1 and the Poisson structure (2.1) on the fibres of p defined
on the neighbourhood of a line in PT with normal bundle O(1)⊕O(1).
Here, the holomorphic metrics on regions in C4 can be thought of as arising from complexifica-
tion of an analytic split-signature or Riemannian self-dual 4-manifold, or as Newman’s H-spaces
defined by complexified self-dual characteristic data at null infinity [35, 46].
In Penrose’s original paper (see [59, Section 6]), the complex deformations of twistor space
were described by deforming the patching functions of PT (thought of as a complex manifold)
between the two coordinate patches
U = {λ0 ̸= 0}, Ũ = {λ1 ̸= 0},
with coordinates Z =
(
µα̇, λα
)
and Z̃ =
(
µ̃α̇, λα
)
, respectively. Since the deformations preserve
the projection to the Riemann sphere, the coordinates λα on the two patches are identified on
the overlap; see Figure 1. In order to preserve the Poisson structure (2.1), a generating function
Celestial w1+∞ Symmetries from Twistor Space 5
G
(
λα, µ
0̇, µ̃1̇
)
of homogeneity degree two is used to define the patching of the µ-coordinates
(implicitly) by
µ1̇ =
∂G
∂µ0̇
, µ̃0̇ =
∂G
∂µ̃1̇
.
It is easy to see that this preserves the Poisson structure on any fibre of PT → CP1, since G
generates canonical transformations on the fibres.
Infinitesimally, deformations of such a twistor space are determined by Hamiltonians g(Z) =
δG of homogeneity degree two. Such a g should therefore be defined on the intersection U ∩ Ũ of
the coordinate patches, meaning that its expansion is polynomial in µα̇ but Laurent in λ = λ1/λ0.
These requirements mean that g(Z) is expanded in the generators of the loop algebra Lw1+∞
given by (2.2). In other words, gpm,r form a basis of positive helicity (since deformations of the
twistor space correspond to self-dual curvature in space-time) graviton states in linear theory,
with the commutation relations (2.3) thought of as the Lie algebra of the loop group of area
preserving diffeomorphisms.2
In linear theory, the wavefunctions corresponding to gpm,r can be represented on space-time
using standard integral formulae evaluated on twistor lines (cf. [58, 62]):
ψ̃α̇1...α̇4(x) =
∮
dλ
2πi
∂4gpm,r
∂µα̇1 · · · ∂µα̇4
∣∣∣∣
µα̇=xαα̇λα
,
hαα̇ββ̇(x) = ιαιβ
∮
dλ
2πi
∂2gpm,r
∂µα̇∂µβ̇
∣∣∣∣
µα̇=xαα̇λα
, (2.6)
for the linearized self-dual Weyl spinor and metric perturbation respectively. Here, these formu-
lae are written in the affine patch where λ0 = 1. The contour integrals are taken around poles
in the λ-plane and the constant spinor ια = (0, 1) is chosen so that ⟨ιλ⟩ = λ0 = 1 on this affine
patch. This choice of ια amounts to a gauge fixing for the linear metric and drops out of the
curvature. Clearly, these formulae give rise to polynomials in the space-time coordinates xαα̇ of
degree 2p− 6 for the Weyl spinor or 2p− 4 for the metric. For example, with g
5/2
3/2,r we find
p =
5
2
, m =
3
2
: hαα̇ββ̇(x) = ιαιβ õα̇õβ̇
(
x00̇δr,1 + x10̇δr,2
)
with õα̇ = (1, 0), etc. This is a mode of the sub-sub-leading soft graviton.
The solutions (2.6) directly yield modes of the conformally soft graviton wavefunctions of [54];
up to a constant multiple, these can be defined as the right hand side of
ιαιβ
∮
dλ
2πi
∂2gpm,r
∂µα̇∂µβ̇
∣∣∣∣
µα̇=xαα̇λα
=
Γ(p−m)Γ(p+m)
(2πi)2Γ(2p− 3)
∮
dzdz̃
zrz̃p−m
ιαιβ z̃α̇z̃β̇(q · x)
2p−4, (2.7)
where zα = (1, z), z̃α̇ = (1, z̃), qαα̇ = zαz̃α̇, and the contour on the right is a product of circles
around z = 0, and z̃ = 0; here hαα̇ββ̇(x) = ιαιβ z̄α̇z̄β̇(q · x)
2p−4/Γ(2p − 3) is a generating series
for these soft modes that will be defined more systematically later.
3 Twistor sigma model and MHV amplitudes
The non-linear graviton construction realizes the self-dual 4-manifold as the moduli space of
degree one (rational) holomorphic curves in the deformed twistor space. In [5] we introduced
2Here we do not address convergence issues; to make sense of such, one would need to consider a semi-group,
etc.
6 T. Adamo, L. Mason and A. Sharma
a sigma model for these holomorphic curves adapted to a Dolbeault description of the nonlinear
graviton in which the complex structure is deformed by means of a global deformation of the
d-bar operator, ∂̄ → ∇̄ = ∂̄ + · · · , rather than the shift in the patching functions introduced
in the previous section. In this language, our sigma model governs maps from the Riemann
sphere to twistor space whose equation of motion determines the holomorphic twistor curves
with respect to ∇̄.
As shown in [19], such a Dolbeault description of the nonlinear graviton construction arises
from an asymptotic twistor space defined by characteristic data at I . For curves of degree one,
the solutions to the twistor sigma model yield the self-dual space-time; in this representation, the
nonlinear graviton construction becomes a reformulation of Newman’sH-space construction [46].
This connection with I is what allows us to make contact with celestial holography. The MHV
sector of tree-level graviton scattering arises at degree one, whereas for higher NMHV degree
the boundary conditions of the model can be adapted to give rational curves of higher degree.
3.1 Holomorphic curves and twistor sigma model
While Penrose initially described complex deformations of twistor space in terms of patching
functions, one can equivalently work with deformations of the almost complex structure that
are integrable and preserve the fibration (2.5) as well as the Poisson structure (2.1). Such
deformations are locally given by perturbing the Dolbeault operator,
∇̄ = ∂̄ + εα̇β̇
∂h
∂µα̇
∂
∂µβ̇
= ∂̄ + {h, }, (3.1)
where ∂̄ = dZA∂/∂ZA corresponds to the trivial complex structure on PT for which (µα̇, λα)
are holomorphic, and h ∈ Ω0,1(PT,O(2)) with (0, 1)-form components pointing along the CP1
base of the fibration. In other words,
h = hDλ̄, Dλ̄ ≡ [λ̄dλ̄] = λ̄α̇dλ̄α̇, (3.2)
with h a function on PT homogeneous of degree two in the holomorphic coordinates and −2 in
the anti-holomorphic coordinates. It is straightforward to see that any almost complex struc-
ture of the form (3.1) preserves the holomorphic fibration PT → CP1 and Poisson structure.
Integrability ∇̄2 = 0 is also immediate since Dλ̄ ∧Dλ̄ = 0. The linear perturbations associated
to such deformations are obtained from the Penrose transforms,
ψ̃α̇1...α̇4(x) =
∫
P1
Dλ ∧ ∂4h
∂µα̇1 · · · ∂µα̇4
∣∣∣∣
µα̇=xαα̇λα
, (3.3)
hαα̇ββ̇(x) =
∫
P1
Dλ ∧
ιαιβ
⟨ιλ⟩2
∂2h
∂µα̇∂µβ̇
∣∣∣∣
µα̇=xαα̇λα
, (3.4)
where Dλ = λαdλα. But we can also construct the fully non-linear self-dual vacuum metric
associated to h by employing the fact that such a metric is necessarily hyperkähler.
A point in a self-dual vacuum space-time corresponds to a rational curve in PT which is
holomorphic with respect to the complex structure (3.1). Such a holomorphic curve can be
described by viewing µα̇ as a degree−1 map from CP1 to twistor space, with boundary conditions
at the north and south poles of the Riemann sphere fixing all moduli of the curve. Letting
σa = (σ0, σ1) be homogeneous coordinates on CP1, a degree one curve in twistor space is
parametrized by
λα(σ) =
(
1
σ0
,
1
σ1
)
=
(1, λ)
σ0
, µα̇(x, σ) =
xα̇
σ0
+
x̃α̇
σ1
+M α̇(σ). (3.5)
Celestial w1+∞ Symmetries from Twistor Space 7
Here, the moduli of the curve have been fixed by specifying the pole structure in the first
two terms of µα̇ with xαα̇ = (xα̇, x̃α̇) providing coordinates on the self-dual space-time. The
object M α̇ is smooth and homogeneous of weight −1 in σa; it is uniquely determined by the
requirement that the curve is holomorphic with respect to (3.1), i.e., that
∂̄σM
α̇ =
∂h
∂µα̇
(x, σ). (3.6)
In other words, given the data h on PT and the parametrization (3.5), the self-dual space-time
is reconstructed by solving (3.6) for the holomorphic curves in twistor space.
In [5], we showed that (3.6) arise as the Euler-Lagrange equations of a twistor sigma model
S[M ] =
1
4πiℏ
∫
CP1
Dσ
(
[M∂̄σM ] + 2h(x, σ)
)
, (3.7)
where Dσ := σ0dσ1 − σ1dσ0,
[
M∂̄σM
]
:= εα̇β̇M
β̇ ∂̄σM
α̇, and ℏ is a formal parameter. Remark-
ably, this sigma model is directly related to the underlying self-dual geometry. Evaluating its
on-shell action, it follows that (up to a constant) [5]
Ω(x) = εα̇β̇x
β̇x̃α̇ − ℏS[M ]
∣∣∣
on-shell
, (3.8)
is the Kähler potential – or first Plebański form [63] – for the self-dual metric. In particular,
the metric is defined by the tetrad
eαα̇ =
(
dxα̇,Ωα̇
β̇dx̃
β̇
)
, Ωα̇β̇ :=
∂2Ω
∂xα̇∂x̃β̇
,
with self-duality corresponding to the ‘first heavenly equation’ det(Ωα̇β̇) = 2.
3.2 I and asymptotic twistor space
The non-linear graviton construction is directly related to the arena of celestial holography when
the deformed twistor space is defined by the self-dual characteristic data at I .3 From a twistor
space PT , there is a natural projection
PT → IC,
(
µα̇, λα
)
7→
(
u, λα, λ̄α̇
)
=
(
µβ̇λ̄β̇, λα, λ̄α̇
)
, (3.9)
where IC ∼= C×S2 is a partial complexification of the conformal boundary obtained by letting u
become complex, but we do not complexify the S2-factor. In particular, this identifies the CP1
base of the fibration (2.5) with the celestial sphere [19].
Consider an asymptotically flat space-time with a Bondi–Sachs expansion that has been
conformally rescaled by the conformal factor R2 with R = r−1 and r a standard radial coordinate
to become
dŝ2 = −2dudR− 4DλDλ̄
||λ||4
+R
(
σ0
(
u, λ, λ̄
)
Dλ2 + σ̄0
(
u, λ, λ̄
)
Dλ̄2
)
+O
(
R2
)
.
Here ||λ||2 = |λ0|2 + |λ1|2 yields a conformal factor for the round sphere in homogeneous co-
ordinates λα, and I + corresponds to R → 0. The complex (spin- and conformal-weighted)
function σ0 encodes the asymptotic shear of the constant-u hypersurfaces at I ; this is the free
characteristic data of the gravitational field (also often denoted by Czz). In a precise sense,
3By I , we mean one of the future or past null conformal boundaries I ±; implicitly we will always choose the
future boundary I + although it is trivial to work with I − instead.
8 T. Adamo, L. Mason and A. Sharma
σ0 controls the anti-self-dual radiative degrees of freedom of the metric, with σ̄0 controlling the
self-dual radiative degrees of freedom [15, 47, 67, 68, 69]. The spin- and conformal-weights of σ0
dictate that it has the scaling property
σ0
(
|b|2u, bλ, b̄λ̄
)
=
b̄
b3
σ0
(
u, λ, λ̄
)
, σ̄0
(
|b|2u, bλ, b̄λ̄
)
=
b
b̄3
σ̄0
(
u, λ, λ̄
)
, (3.10)
for any non-vanishing complex number b.
We define the complex structure as in (3.1)–(3.2) on asymptotic twistor space by taking
h = h
(
u, λ, λ̄
)
Dλ̄ with u given by the projection (3.9) and
h
(
u, λ, λ̄
)
=
∫ u
σ̄0
(
s, λ, λ̄
)
ds.
The scaling property (3.10) ensures that h – and hence h – has the correct homogeneity on
twistor space. Thus, the complex structure (3.1) on PT becomes
∇̄ = ∂̄ + {h, } = ∂̄ + λ̄α̇Dλ̄σ̄
0
(
u, λ, λ̄
) ∂
∂µα̇
.
Thus the deformed twistor space PT is determined by the characteristic data. Such a twistor
space is referred to as an asymptotic twistor space; these twistor spaces can be characterised
as those associated to Newman’s H-spaces [35, 46], which are self-dual radiative space-times
determined by complexified data with σ0 = 0 but σ̄0 non-zero and independent of σ0 on IC,
given by the σ̄0 of the original Lorentzian space-time.
3.3 From the sigma model to the MHV amplitude
There is a direct connection between the twistor sigma model (3.7) for asymptotic twistor
spaces and the MHV helicity sector of tree-level graviton scattering. A tree-level gravitational
MHV amplitude involves two negative helicity external gravitons and arbitrarily many positive
helicity gravitons. When the total number of gravitons is n (i.e., 2 negative helicity and n − 2
positive helicity gravitons) there is a compact, elegant formula for this amplitude in a momentum
eigenstate basis due to Hodges [31]:
Mn,0 = δ4
(
n∑
i=1
ki
)
⟨12⟩8det′(H), (3.11)
where overall factors of the gravitational coupling have been suppressed. In this expression, the
kαα̇i = καi κ̃
α̇
i are null momenta, gravitons 1 and 2 have been assigned negative helicity, H is
a (n− 2)× (n− 2) matrix with entries
Hij =
[ij]
⟨ij⟩
, i ̸= j, Hii = −
∑
j ̸=i
[ij]
⟨ij⟩
⟨1j⟩⟨2j⟩
⟨1i⟩⟨2i⟩
,
and the reduced determinant is defined by
det′(H) :=
|Hi
i|
⟨12⟩2⟨1i⟩2⟨2i⟩2
.
It is easy to see that the choice of minor – corresponding to a choice of one positive helicity
external graviton – defining det′(H) is arbitrary, so this formula nicely manifests the permutation
symmetry of all positive helicity gravitons in the MHV scattering process.
Celestial w1+∞ Symmetries from Twistor Space 9
Since the number of positive helicity gravitons in an MHV amplitude is arbitrary, it is natural
to view them as being generated by the perturbative expansion of the two-point function of
negative helicity gravitons on a non-linear self-dual background. Since the self-dual background
in such a generating functional should be purely radiative (so that its perturbative limit produces
positive helicity gravitons), its associated twistor space is an asymptotic twistor space.
This generating functional picture was first made precise in [43] and later refined in [5], with
the result that the generating functional for MHV amplitudes can be written as
−⟨12⟩4
∫
M
d2xd2x̃ei[x1]+i[x̃2]Ω(x, x̃) = ⟨12⟩4
∫
M
d2xd2x̃ei[x1]+i[x̃2]S[M ]
∣∣∣
on-shell
, (3.12)
where M is the self-dual background, Ω is its Kähler potential or first Plebański form and the
equality follows thanks to (3.8). Here, one implicitly adopts a 2-spinor basis in (3.5) adapted to
the momenta of the two negative helicity gravitons. This amounts to using xα̇ = xαα̇κ1α and
x̃α̇ = xαα̇κ2α as coordinates on M. We also set ℏ = 1 for convenience; it will be reinstated when
needed.
To view the self-dual background as a superposition of positive helicity gravitons, the complex
structure of the asymptotic twistor space is taken to be
h(Z) =
n∑
i=3
ϵihi(Z; ki),
where each hi is a momentum eigenstate representative on twistor space:
hi(Z; ki) =
∫
C∗
dsi
s3i
δ̄2(κiα − siλα)e
isi[µi]. (3.13)
Inserting this into the integral formulae (3.3)–(3.4), one recovers the expected positive helicity
momentum eigenstate on (complexified) Minkowski space:
hiαα̇ββ̇(x) =
ιαιβκ̃iα̇κ̃iβ̇
⟨ιi⟩2
eiki·x, ψ̃iαβγδ(x) = κ̃iακ̃iβκ̃iγ κ̃iδe
iki·x.
Perturbatively expanding the generating functional (3.12) then boils down to extracting the
multi-linear piece of a tree-level correlation function involving insertions of these momentum
eigenstates.
In particular, the on-shell action is evaluated using the tree-level, connected correlation func-
tions of ‘vertex operators’(
n∏
i=3
∂
∂ϵi
)
S[M ]
∣∣∣
on-shell
∣∣∣∣∣
ϵi=0
=
〈
n∏
i=3
Vi
〉tree
0
, Vi :=
∫
CP1
Dσi ∧ hi(Z(σi); ki), (3.14)
in the two-dimensional CFT of the twistor sigma model with trivial complex structure. This
means that the correlator is evaluated using the free OPE
M α̇(σi)M
β̇(σj) ∼
εα̇β̇
σi − σj
, (3.15)
in the affine patch of CP1 where σa = (1, σ). Here, the vertex operators are simply linear
deformations of the sigma model action and the tree-level contribution is extracted from the
generating functional for the connected correlator by taking ℏ → 0 as usual.
10 T. Adamo, L. Mason and A. Sharma
This computation is fairly straightforward as it involves keeping only single contractions in
the OPE of any two vertex operators (see [5] for details). It gives〈
n∏
i=3
Vi
〉tree
0
=
|Hi
i|
⟨1i⟩2⟨2i⟩2
n∏
j=3
eiki·x,
where the determinant arises as a result of the weighted matrix-tree theorem (which also ensures
that the result is independent of the choice of i singled out on the LHS) and all CP1 integrals
can be performed against the delta functions appearing in (3.13). Feeding this into (3.12)
and using d2xd2x̃ = ⟨12⟩2d4x immediately gives the Hodges formula (3.11), providing a first-
principles derivation of tree-level MHV graviton scattering, which explains the appearance of
‘tree-summing’ formulae [14, 48] and the matrix-tree theorem [4, 22] in earlier literature.
By adapting the boundary conditions for the µα̇(σ) map, it is possible to formulate a higher-
degree version of the twistor sigma model (i.e., by imposing boundary conditions at d+1 points
on CP1). These higher degree models are related to other helicity sectors of the tree-level gravi-
ton S-matrix, with degree d corresponding to Nd−1MHV amplitudes, although the generating
functionals for d > 1 cannot be derived directly from general relativity and require additional
ingredients (albeit quite minimally) beyond the on-shell action of the twistor sigma model [5].
4 From twistorial to celestial Lw1+∞
With the self-dual sector of gravity on space-time captured by the twistor sigma model (3.7), it
is now straightforward to describe infinitesimal deformations and hence the symmetry algebra
associated to the self-dual sector. Using the semi-classical OPE on the Riemann sphere defined
by the sigma model, we first show how this produces the expected Lw1+∞ algebra. We go on
to explain the relationship between graviton vertex operators and Lw1+∞ symmetry generators
as a realization of a Čech–Dolbeault isomorphism within the model. We then give the soft
expansion of these vertex operators/symmetry generators so as to yield the basis we introduced
in Section 2. Furthermore, using the relationship between the twistor sigma model and tree-level
MHV scattering, we prove that this explicitly generates the action of celestial Lw1+∞ on positive
helicity hard gravitons of [28, 76].
4.1 Lw1+∞ charges and algebra
The form of the complex structure (3.1)–(3.2) on twistor space admits coordinate symmetries
generated by Hamiltonians with respect to the Poisson structure (2.1). Such Hamiltonians
g
(
µα̇, λ, λ̄
)
must have homogeneity degree 2 in ZA and be holomorphic4 in µα̇ but not necessarily
in λα. The symmetry action is given by
δµα̇ =
{
g, µα̇
}
= εβ̇α̇
∂g
∂µβ̇
, δh = ∂̄g + {h, g}, (4.1)
which leads to a symmetry of the twistor sigma model action (3.7) when δh = 0, i.e., when g
satisfies δh = ∂̄g + {h, g} = 0 so that g is holomorphic with respect to the deformed complex
structure ∇̄. For such g, Noether’s theorem leads to the conserved charge
Qg =
∮
gDσ, (4.2)
in the theory on CP1 defined by the sigma model.
4In fact, the requirement of holomorphicity in µα̇ can be dropped if more general h are allowed (e.g., [44]).
Celestial w1+∞ Symmetries from Twistor Space 11
The OPE (3.15) extends from the ‘non-zero-mode’ M α̇ to the full twistor coordinate µα̇ in
the obvious way (since the two differ only by zero modes):
µα̇(σ)µβ̇(σ′) ∼ εα̇β̇
σ − σ′
, (4.3)
on the usual affine patch where σa = (1, σ). This in turn induces a semi-classical OPE for the
Hamiltonian functions g given by the Poisson bracket
g(Z(σ))g′(Z(σ′)) ∼ 1
σ − σ′
{g, g′}(σ′), (4.4)
with higher order singularities being neglected at tree-level in the sigma model. Thus, the OPE
encodes the loop algebra of the Poisson diffeomorphisms of the µα̇-plane with loop variable λ.
The charges Qg given by (4.2) generate canonical transformations of the µα̇-plane with canonical
commutation relations
[Qg, Qg′ ] = Q{g,g′},
also arising from the semi-classical OPE.
Poisson diffeomorphisms generated by Hamiltonians satisfying δh = ∂̄g + {h, g} = 0 do
not deform the space-time Kähler scalar (3.8) as they leave the on-shell action of the twistor
sigma model invariant. As a result, the functions g must generically have singularities in λ to
encode non-trivial symmetry transformations of the self-dual sector. Consider a BMS super-
translation corresponding to δu = f(λ, λ̄) where f has homogeneity +1 in the homogeneous
coordinates λα, λ̄α̇ of the celestial sphere. Using the projection (3.9) from asymptotic twistor
space to IC, this corresponds to a transformation
δµα̇ =
∂f
∂λ̄α̇
,
which is in turn generated by the Hamiltonian
gST =
[
µ
∂f
∂λ̄
]
, (4.5)
under (4.1). When f
(
λ, λ̄
)
= aαα̇λαλ̄α̇, these are just the usual translations. Similarly, self-
dual/dotted Lorentz super-rotations (of the extended BMS algebra [12]) are generated by
gSR = L̃α̇β̇(λ, λ̄)µ
α̇µβ̇, (4.6)
where L̃α̇β̇ is homogeneous of degree zero in λα, λ̄α̇. When L̃α̇β̇ depends only on λ̄α̇, this reduces
to a standard Lorentz rotation.
In general, the transformations (4.5), (4.6) are not symmetries of the sigma model action,
since δh ̸= 0. Indeed, for the charge (4.2) to be conserved, one requires g to be holomorphic
on twistor space; on a flat background (i.e., h = 0) this requires g to be globally-defined and
one simply obtains the Poincaré algebra. A generic supertranslation (4.5) or superrotation (4.6)
will have poles in λ, so to go beyond the Poincaré group – or on any curved background – one
must consider Hamiltonians g which have singularities in a local holomorphic coordinate system.
Such singularities indicate that these functions change the gravitational data: they are no longer
simply symmetries.
Thus, generic charges (4.2) generate canonical transformations of the µα̇-plane that depend
on λ. Given the overall homogeneity constraint on g – namely, that it is homogeneous of
12 T. Adamo, L. Mason and A. Sharma
degree 2 on twistor space – each Hamiltonian function can be decomposed into modes gpm,r of
the form (2.2). The OPE (4.4) then dictates that these modes have Poisson brackets{
gpm,r, g
q
n,s
}
= 2(m(q − 1)− n(p− 1))gp+q−2
m+n,r+s,
which are precisely the commutation relations of Lw1+∞ given previously in (2.3). These can
be expressed in terms of the semiclassical OPE of the operators (2.4) as
{
gpm(z), gqn(z
′)
}
= 2
m(q − 1)− n(p− 1)
z − z′
(
gp+q−2
m+n (z)− gp+q−2
m+n (z′)
)
,
Thus, the structure of the twistor sigma model naturally encodes Lw1+∞ in terms of its in-
finitesimal deformations.
4.2 Vertex operators and currents and soft limits
The relationship between vertex operators in the sigma model and Lw1+∞ currents relies on the
Čech–Dolbeault correspondence. While Penrose’s original formulation of the non-linear graviton
construction utilized patching functions for the deformed twistor space, the twistor sigma model
works directly with the deformed Dolbeault operator for the complex structure. In this Dol-
beault approach, h ∈ H1(PT,O(2)) is represented by the (0, 1)-form h ∈ Ω0,1(PT,O(2)) obeying
∂̄h = 0; for asymptotic twistor space with h = h(u, λ, λ̄)Dλ̄ these conditions are automatic.
To find the Čech representative corresponding to such an h, locally on an open subset Ua of
twistor space, ∂̄h = 0 can be solved by h = ∂̄ga for some smooth function ga of homogeneity 2.
The differences gab := ga − gb are therefore holomorphic functions on Uab := Ua ∩Ub, defined up
to the addition of holomorphic functions that extend over the Ua; such gab equivalence classes
provide Čech representatives of h (with the open-set indices a, b, . . . usually suppressed).
Our key example is the momentum eigenstate (3.13). Here we now separate out the fre-
quency ω explicitly so that we can also expand in ω to give the Taylor series around ω = 0
which then define the leading and subleading soft limits of graviton insertions. Thus, taking for
simplicity an outgoing graviton, we write
kαα̇ = ωzαz̄α̇, zα = (1, z), z̄α̇ = (1, z̄),
so that for example κα =
√
ωzα, κ̃α̇ =
√
ωz̄α̇ are the standard spinor helicity variables. The
Dolbeault representative is given by simply re-writing (3.13) to account for the frequency:
h = ⟨ιλ⟩3δ̄(⟨λz⟩)eiω
[µz̄]
⟨ιλ⟩ , δ̄(λ) =
1
2πi
∂̄
(
1
λ
)
,
where ια = (0, 1) is a constant spinor basis element. The corresponding Čech representative is
g =
⟨ιλ⟩3
2πi
1
⟨λz⟩
e
iω
[µz̄]
⟨ιλ⟩ , (4.7)
with the choice of ια now reflecting the Čech cohomology gauge freedom. The relevant open
sets are given by covering the Riemann sphere with U0 containing ⟨λι⟩ = 0 and U1 containing
⟨λz⟩ = 0; the overlap is a neighbourhood of the contour γz
γϵz =
{∣∣∣∣⟨λz⟩⟨λι⟩
∣∣∣∣ = ϵ
}
,
Celestial w1+∞ Symmetries from Twistor Space 13
for some small ϵ > 0. Inside of γϵz, the vertex operator for h obeys
Vh =
∫
CP1
h ∧Dσ =
∮
γz
gDσ = Qg, (4.8)
by Cauchy’s theorem.
In the soft limit as ω → 0, the exponential factor in (4.7) can be expanded in powers of ω to
obtain combinations of the Lw1+∞ generators gpm(z) as coefficients of ω2p−2 (taking for simplicity
the affine patch where ⟨ιλ⟩ = 1 and ⟨λz⟩ = λ − z). For 2p − 2 = 1, 2, this gives the standard
correspondence between the leading and sub-leading soft graviton theorems and generators of
supertranslations and superrotations, respectively; for 2p− 2 ≥ 3 we obtain an infinite tower of
soft graviton symmetries corresponding to higher-order generators of Lw1+∞.
We can also make precise contact with the incarnation of Lw1+∞ first noted in the context of
celestial holography by [76]. Consider a positive helicty graviton boost eigenstate of conformal
weight ∆ inserted at the point zα = (1, z), z̄α̇ = (1, z̄) on the celestial sphere.5 Its Dolbeault
twistor representative reads [6]
h =
(−iϵ)−∆Γ(∆− 2)
[µz̄]∆−2
δ̄∆(⟨λz⟩),
where ϵ = ±1 denotes whether it is outgoing or incoming, and we have defined a holomorphic
delta function of weight ∆ in λα:
δ̄∆(⟨λz⟩) := ⟨ιλ⟩∆+1δ̄(⟨λz⟩).
Again, ια = (0, 1) so that ⟨ιλ⟩ = λ0, ⟨ιz⟩ = 1, etc. Inserting this in the Penrose integral
formula (3.4), one finds the expected wavefunction of a spin 2 positive helicity boost eigenstate:
hαα̇ββ̇(x) = (−iϵ)−∆Γ(∆)
ιαιβ z̄α̇z̄β̇
(q · x)∆
,
with qαα̇ = zαz̄α̇. This is gauge equivalent to a spin 2 conformal primary graviton [54] whose
modes we considered in (2.7).
Without loss of generality, we focus on outgoing particles for which ϵ = +1. Conformally soft
gravitons are obtained by taking residues at ∆ = k = 2, 1, 0,−1, . . . :
hk
soft = Res∆=kh =
i−k
(2− k)!
[µz̄]2−kδ̄k(⟨λz⟩). (4.9)
Substituting [µz̄] = µ0̇ + z̄µ1̇ in (4.9), it can be binomially expanded into a polynomial in z̄ to
get 3− k holomorphic currents. In doing this, we use the index relabeling k = 4− 2p. Hence,
h4−2p
soft = i2p−4δ̄4−2p(⟨λz⟩)
p−1∑
m=1−p
z̄p−1−mwp
m
(p−m− 1)!(p+m− 1)!
, (4.10)
where
wp
m = (µ0̇)p+m−1(µ1̇)p−m−1, p = 1,
3
2
, 2,
5
2
, . . . .
Remarkably, the combinatorial rescaling by (p −m − 1)!(p +m − 1)! that was crucial for the
identification of w1+∞ in [76] emerges naturally here via twistor space. The modes in (4.10) give
Dolbeault twistor representatives
i2p−4δ̄4−2p(⟨λz⟩)wp
m
5This will become the celestial torus in split signature when z̄ is independent of z.
14 T. Adamo, L. Mason and A. Sharma
for the various soft gravitons that are in correspondence with celestial Lw1+∞ generators. Thus,
as explained in (4.8), in the twistor sigma model these correspond to charges
Qp
m(z) =
i2p
2πi
∮
γz
gpm(z)dσ =
i2p
2πi
∮
γz
wp
m(σ)
⟨λ(σ)z⟩
⟨ιλ(σ)⟩5−2pDσ, (4.11)
with the contour integral taken around the pole at ⟨λz⟩ = λ − z = 0 (the second equality is
a re-writing in homogeneous coordinates of the first). These are the w1+∞ currents generating
Poisson diffeomorphisms on the λ = z fibre of twistor space.
4.3 Soft graviton symmetries
Finally, we show that the twistorial action of w1+∞ on hard gravitons is equivalent to the celestial
action of w1+∞ given in [28, 30, 34, 76]. More precisely, the OPE between positive helicity soft
and hard graviton vertex operators in the twistor sigma model maps to the celestial OPE between
the conformally soft gravitons and hard gravitons (as dictated by collinear limits or asymptotic
symmetries). We also show that the action of a w1+∞ generator on a negative helicity graviton
gives rise to the mixed helicity soft-hard celestial OPE, but leave the interpretation of this at
the level of the sigma model correlators to future work.
Action on positive helicity gravitons. Let h∆i(σi) be the twistor representative of an
outgoing, positive helicity graviton with conformal dimension ∆i and celestial positions (zi, z̄i):
h∆i(σi) =
i∆iΓ(∆i − 2)
[µ(σi)z̄i]∆i−2
δ̄∆i(⟨λ(σi)zi⟩),
where ziα ≡ (1, zi), z̄iα̇ ≡ (1, z̄i) as usual. We label this representative with ∆i and suppress zi, z̄i
for brevity. Acting on it with the soft charge Qp
m in (4.11), and using the sigma model OPE (4.3),
we find
Qp
m(z)h∆i(σi) ∼
i2p
2πi
∮
⟨ιλ(σ)⟩5−2p
⟨λ(σ)z⟩
∂wp
m
∂µα̇
(σ)
∂h∆i
∂µα̇
(σi)
Dσ
σ − σi
= −i2p
⟨ιλ(σi)⟩5−2p
⟨λ(σi)z⟩
∂wp
m
∂µα̇
(σi)
∂h∆i
∂µα̇
(σi),
where the contour integral has been evaluated by deforming6 the contour from the ⟨λ(σ)z⟩ = 0
pole to the pole at σ = σi. As usual, we have only kept a single contraction in the OPE as we
want to insert this in tree correlators at the end.
On the support of the holomorphic delta function δ̄∆i(⟨λ(σi)zi⟩) appearing in h∆i , the action
of the soft charge can be further simplified to
Qp
m(z)h∆i(σi) ∼ − i2p
⟨ziz⟩
⟨ιλ(σi)⟩4−2p∂w
p
m
∂µα̇
(σi)
∂h∆i
∂µα̇
(σi)
=
i2p
z − zi
{wp
m,h∆i}(σi)
⟨ιλ(σi)⟩2p−4
. (4.12)
Thus, the OPE between a soft graviton current and a conformal primary hard graviton is given
by the action of Lw1+∞ in its canonical (in the sense of the Poisson bracket) representation. As
expected, this fact is most directly visible on twistor space.
6Although there are potentially other poles that might be picked up by this deformation, these are either
subleading in the celestial OPE limit z− zi → 0 or do not contribute to tree-level correlators in the twistor sigma
model.
Celestial w1+∞ Symmetries from Twistor Space 15
We can now prove that the celestial action of a soft graviton symmetry on a positive helicity
hard graviton arises from the Poisson bracket in (4.12). Using [µz̄i] = µ0̇ + z̄iµ
1̇, it follows that
{wp
m,h∆i}(σi)
⟨ιλ(σi)⟩2p−4
= −
[
(p+m− 1)z̄i
(
µ0̇
)p+m−2
(σi)
(
µ1̇
)p−m−1
(σi)
− (p−m− 1)
(
µ0̇
)p+m−1
(σi)
(
µ1̇
)p−m−2
(σi)
] i∆iΓ(∆i − 1)
[µ(σi)z̄i]∆i−1
δ̄∆i−2p+4(⟨λ(σi)zi⟩). (4.13)
Next, we have the intertwining relations
µ1̇
Γ(a)
[µz̄i]a
= −∂̄i
Γ(a− 1)
[µz̄i]a−1
, µ0̇
Γ(a)
[µz̄i]a
= (z̄i∂̄i + a− 1)
Γ(a− 1)
[µz̄i]a−1
,
where ∂̄i ≡ ∂/∂z̄i and a ̸= 1. Applying these iteratively to the right hand side of (4.13), one can
re-express the OPE (4.12) as
Qp
m(z)h∆i(σi) ∼
(−1)p+m
z − zi
[
(p+m− 1)z̄i
(p+m−2∏
r=1
(
z̄i∂̄i +∆i − 1− r
))
∂̄p−m−1
i
+ (p−m− 1)
(p+m−1∏
r=1
(
z̄i∂̄i +∆i − 1− r
))
∂̄p−m−2
i
]
h∆i−2p+4(σi). (4.14)
Expanding the bracketed operators gives the celestial OPE
Qp
m(z)h∆i(σi) ∼
(−1)p+m
z − zi
p+m−1∑
ℓ=0
(
p+m− 1
ℓ
)
(2p− 2− ℓ)Γ(∆i − 1)
Γ(∆i − 1− ℓ)
× z̄p+m−1−ℓ
i ∂̄2p−3−ℓ
i h∆i−2p+4(σi), (4.15)
previously found in the literature [30, 34]. Inserting these relations into the sigma model tree
correlators (3.14) straightforwardly produces the corresponding celestial OPE between a w1+∞
current and a hard graviton. This gives rise to the tower of conformally soft theorems and
asymptotic symmetries found in [26, 28, 76]. For instance, one can easily verify the actions of
supertranslation, superrotation as well as the sub-sub-leading soft graviton symmetries.
Notice how the twistor description produces the celestial OPE in a factorized form (4.14)
which is highly non-trivial to see in a direct calculation of Mellin-transformed amplitudes in
the collinear limit. It is this factorized form that hides the representation theory of w1+∞ and
makes contact with its symplectic origins.
Action on negative helicity gravitons. In the twistor sigma model, negative helicity
gravitons are not represented by vertex operators, but classically one can still define a twistor
representative for a negative helicity graviton. It is given by a (0, 1)-form of weight −6 in
Z: h̃ ∈ Ω0,1(PT,O(−6)). It generates a graviton on space-time with purely negative helicity
curvature computed by the Penrose transform
ψαβγδ(x) =
∫
P1
Dλ ∧ λαλβλγλδ h̃
∣∣∣
µα̇=xαα̇λα
.
We are free to associate to this an operator h̃(Z(σ)) in our sigma model. For instance, with
the ith outgoing negative helicity graviton boost eigenstate we associate the operator
h̃∆i(σi) =
i∆iΓ(∆i + 2)
[µ(σi)z̄i]∆i+2
δ̄∆i−4(⟨λ(σi)zi⟩).
16 T. Adamo, L. Mason and A. Sharma
The corresponding classical twistor representative can be checked to produce the space-time
curvature of a negative helicity boost eigenstate.
Repeating the derivation of (4.12) yields the sigma model OPE
Qp
m(z)h̃∆i(σi) ∼
i2p
z − zi
{
wp
m, h̃∆i
}
(σi)
⟨ιλ(σi)⟩2p−4
.
Computing the Poisson bracket then gives
Qp
m(z)h̃∆i(σi) ∼
(−1)p+m
z − zi
[
(p+m− 1)z̄i
(p+m−2∏
r=1
(
z̄i∂̄i +∆i + 3− r
))
∂̄p−m−1
i
+ (p−m− 1)
(p+m−1∏
r=1
(z̄i∂̄i +∆i + 3− r)
)
∂̄p−m−2
i
]
h̃∆i−2p+4(σi).
Expanding the derivative operators again produces the result in the literature [30, 34],
Qp
m(z)h̃∆i(σi) ∼
(−1)p+m
z − zi
p+m−1∑
ℓ=0
(
p+m− 1
ℓ
)
(2p− 2− ℓ)Γ(∆i + 3)
Γ(∆i + 3− ℓ)
× z̄p+m−1−ℓ
i ∂̄2p−3−ℓ
i h̃∆i−2p+4(σi). (4.16)
Although we can no longer simply insert this in sigma model correlators to claim that this
computes the soft-hard collinear limit, it is nevertheless remarkable that the twistorial action
of w1+∞ on negative helicity gravitons still maps to the corresponding celestial action.
5 The lift to 4d ambitwistor string
The twistor sigma model (3.7) is intrinsically chiral; while it can be used to define generating
functionals for the full tree-level S-matrix of gravity beyond the MHV helicity sector, this
requires additional ingredients which are inserted by hand [5]. A consequence of this chirality is
that we find only the copy of Lw1+∞ associated with the self-dual/positive helicity soft sector;
of course, there should be another copy associated with the anti-self-dual/negative helicity soft
sector. Here, we observe that both copies of Lw1+∞ are naturally found in the four-dimensional
ambitwistor string [24], a CFT on the Riemann sphere whose correlation functions generate the
tree-level S-matrix of gravity. We remark that the correlation functions in the 4d ambitwistor
strings are now fully quantum, unlike the computations in the twistor sigma model (3.7) which
are all semi-classical. Nevertheless, they faithfully represent only the semi-classical Lw1+∞.
Although we do not display the computations here, an identical calculation for the gravita-
tional twistor string [71] yields a representation of Lw1+∞ as described here in the 4d ambitwistor
string. However, it does not obviously have an anti-self-dual L̃w1+∞ sector and so may be a bet-
ter vehicle for seeing the action of the self-dual Lw1+∞ on the whole amplitude (i.e., all helicity
sectors). However, the action of L̃w1+∞ is no longer manifest and will not be realized locally.
This parity asymmetry is a familiar feature of twistor strings (cf. [79]).
5.1 Lifting to ambitwistor space
One can extend beyond the self-dual sector by lifting to ambitwistor space A defined by
A =
{(
ZA, Z̃A
)
∈ C4 ×
(
C4
)∗ ∣∣Z · Z̃ = 0
}/{(
Z, Z̃
)
∼
(
bZ, b−1Z̃
)
, b ∈ C∗}.
Celestial w1+∞ Symmetries from Twistor Space 17
This is the cotangent bundle of both projective twistor space and projective dual twistor space,
A = T ∗PT = T ∗PT∗ and so has a symplectic structure, with dual Poisson structure defined by
ω = dθ, θ := Z · dZ̃ − Z̃ · dZ, { , }A :=
∂
∂ZA
∧ ∂
∂Z̃A
.
This structure does not break left-right symmetry, and deformations of PT and PT∗ both deter-
mine deformations of A [38, 13].
In particular, any vector field V A∂/∂ZA on PT has a Hamiltonian lift to A with Hamil-
tonian V AZ̃A. This enables a lift of deformation Hamiltonians on PT and PT∗ to give the
ambitdextrous Hamiltonian [13]
Hg,g̃ = λ̃α̇
∂g
∂µα̇
+ λα
∂g̃
∂µ̃α
, g ∈ H1(PT,O(2)), g̃ ∈ H1(PT∗,O(2)),
where here g, g̃ are taken to be Čech representatives. The corresponding Hamiltonian vector
field on A determines deformations of the complex structure on A that have self-dual part H+
g
determined by g(Z), and anti-self-dual part H−
g̃ determined by g̃
(
Z̃
)
.
It is easy to see that with these Hamiltonian lifts, the Poisson bracket on ambitwistor space
restricted to the self-dual sector reproduces the Poisson bracket (2.1) on twistor space{
H+
g , H
+
g′
}
A = H+
{g,g′}PT .
This then gives a lift of the Lw1+∞ action to A. The H−
g̃ similarly lift to give the anti-self-dual
L̃w1+∞-action on A. One can then consider the commutator of the self-dual and anti-self-dual
parts:
{
H+
g , H
−
g̃
}
A =
{[
λ̃
∂g
∂µ
]
,
〈
λ
∂g̃
∂µ̃
〉}
A
=
[
λ̃
∂
∂µ
]
∂g
∂ZA
〈
λ
∂
∂µ̃
〉
∂g̃
∂Z̃A
. (5.1)
In terms of deformation theory, the right hand side defines a class in H2(A,O(1, 1)) that ob-
structs the exponentiation of the deformation generated by Hg,g̃. However, this cohomology
group vanishes for elementary reasons [13], so the deformation determined by Hg,g̃ can indeed
be exponentiated.7
5.2 The 4d ambitwistor string
For our purposes, the four-dimensional ambitwistor string [23, 24, 25] for gravity has bosonic
target space fields8
(
ZA, Z̃A
)
that are spinors on the worldsheet with an ambitwistor analogue
of worldsheet supersymmetry giving spinor-valued partners
(
ρA, ρ̃A
)
of opposite statistics and
worldsheet action
S =
∫
Σ
iZ · ∂̄Z̃ − iZ̃ · ∂̄Z + ρ · ∂̄ρ̃+ ρ̃ · ∂̄ρ+ SGhosts,
where Σ ∼= CP1. Here, all symmetries of the worldsheet theory (including those generated by
the ambitwistor current Z · Z̃ and those generating worldsheet supersymmetry) are assumed to
7The formula for the obstruction (5.1) naturally extends to PT × PT∗ where it does not generically vanish.
It was shown in [13] that to leading order around A it gives on space-time the Eastwood–Dighton conformal
invariant defined in terms of SD and ASD Weyl spinors by ψαβγδ∇δδ̇ψ̃α̇β̇γ̇δ̇ −ψ ↔ ψ̃. There it was interpreted as
an obstruction to extending the curved version of A into a curved analogue of PT×PT∗ in a gravitational version
of [33, 78]; this was later proved in the fully nonlinear regime [36].
8To realize space-time supersymmetry
(
ZA, Z̃A
)
are extended to include fermionic coordinates [24].
18 T. Adamo, L. Mason and A. Sharma
have been gauge-fixed, leading to ghost fields with action SGhosts and a corresponding BRST
operator Q (see [23, Section 5.3] for details).
The upshot of this BRST quantization is that a non-trivial correlator needs one vertex oper-
ator each of the form
Uh =
∫
Σ
δ2(ν)h, Ũh̃ =
∫
Σ
δ2(ν̃)h̃.
Here the ν and ν̃ are two-component, weightless, bosonic ghost fields whose zero-modes are
fixed by integration directly against these delta functions. Descent yields the remaining vertex
operators for a correlator as
Vh :=
∫
Σ
[
λ̃∂µ
]
h(Z) + L · ∂2µh(Z), Ṽh̃ :=
∫
Σ
⟨λ∂µ̃⟩h̃
(
Z̃
)
+ L̃ · ∂2µ̃h̃
(
Z̃
)
,
where Lα̇β̇ = ρ(α̇ρ̃β̇) is a self-dual Lorentz current algebra and L̃αβ = ρ(αρ̃β) is an anti-self-dual
Lorentz current algebra, both constructed from the ρ-ρ̃ fermion system. As before, we can use
a Čech representation of the cohomology groups H1(PT,O(2)) and H1(PT∗,O(2)) to re-express
the vertex operators in terms of currents as
Vg =
∮
γ
[
λ̃∂µ
]
g(Z) + L · ∂2µg(Z), Ṽg̃ :=
∮
γ
⟨λ∂µ̃⟩g̃
(
Z̃
)
+ L̃ · ∂2µ̃g̃
(
Z̃
)
,
where γ is a path in Σ that separates the singular regions of both g and g̃.
When the vertex operators are both self-dual a direct calculation shows that they simply
represent the Poisson bracket (2.1) on PT:
VgVg′ ∼
1
σ − σ′
V{g,g′}PT + · · · .
Hence, by expanding g in the modes (2.2) this gives Lw1+∞; an identical statement on dual
twistor space gives L̃w1+∞ for the anti-self-dual vertex operators. However, when one vertex
operator is self-dual and the other anti-self-dual, we have
Vg · Ṽg̃ ∼ 1
σ − σ′
({[
λ̃∂µ
]
g(Z), ⟨λ∂µ̃⟩g̃
(
Z̃
)}
A + · · ·
)
,
where the displayed term is the first of the semi-classical contribution as in (5.1) but now the
+ · · · contain infinitely many singular contributions with arbitrarily many contractions. In
the computation of the full correlation function [23, 24], these contributions are summed for
momentum eigenstates by taking them into the Lagrangian in the path integral to produce the
polarized or refined scattering equations. Remarkably, it is possible to show that this OPE
encodes collinear splitting in a momentum eigenstate basis, or celestial OPEs in a conformal
primary basis, although the mixed helicity case is particularly subtle [1].
6 Discussion
We have seen that the Lw1+∞ recently discovered in the soft OPE obtained from celestial am-
plitudes [76] has a local representation as Poisson diffeomorphisms of the fibres of asymptotic
twistor space and has its origin in Penrose’s nonlinear graviton construction [59]. The Lw1+∞
algebra associated with the positive helicity soft sector arises directly from a twistor sigma model
describing self-dual gravity [5], and this is explicitly identified with the soft OPE algebra on the
celestial sphere. This acts on both the self-dual and anti-self-dual parts of the complexified
Celestial w1+∞ Symmetries from Twistor Space 19
gravitational data via a local action on twistor space when both parts are expressed as coho-
mology classes on that space via (4.15) and (4.16). It is possible to obtain the helicity conjugate
copy L̃w1+∞ of Lw1+∞ via its natural local representation on the conjugate (or dual) twistor
space. One can see them both acting together by lifting to ambitwistor space, and to recover
the correct celestial OPE one must use the fully quantum worldsheet CFT of the ambitwistor
string [1]. There are many open questions and future directions related to the work in this
paper; we conclude by touching on a few of them.
Split versus Lorentzian signature. In this paper, we worked with the complexification
of Lw1+∞ realized as the holomorphic Poisson diffeomorphisms of C2. For polynomial generators
of w1+∞ there are no analytic continuation issues. Viewing this C2 as the fibres of asymptotic
twistor space, this complexification of Lw1+∞ corresponds to a partial complexification of null
infinity I → IC ∼= C× S2, by (3.9). Such a partial complexification is intrinsically associated
with an underlying Lorentzian-real space-time, since the space of null directions remains the
celestial sphere.
Conversely, the real version of Lw1+∞ is not appropriate for Lorentzian signature data.
In the real-valued case, Lw1+∞ gives the Poisson diffeomorphisms of R2 so the twistor com-
ponents µα̇ are themselves taken to be real-valued. Such a real-valued twistor space is ap-
propriate to split signature space-time, where the celestial sphere is replaced by a celestial
torus. The assumption of split signature is often used in celestial holography to disentangle
the self- and anti-self-dual sectors and expedite various integral transformations (cf. [8, 9, 27,
28, 70, 76]). In that context, the combinatorial factors and re-labelings appearing in the ex-
pansion (4.10) emerge from a light transform, while in this paper we saw that this was not
necessary.
The split-signature versions of the twistor constructions used here have realizations in terms
of holomorphic discs [37, 40], suggesting an ‘open string’ approach to the subject in split sig-
nature. Similarly, light transforms in celestial CFT are related to half-Fourier transforms to
real twistor space [70]. Thus, although we have here been able to retain physical Lorentz
signature, an explicit split signature version of the constructions in this paper might well be
interesting.
Yang–Mills and Einstein–Yang–Mills. Gauge theory also contains an infinite tower
of conformally soft gluons, associated to conformal weights ∆ = 1, 0,−1, . . . in a conformal
primary basis, and these have an associated infinite-dimensional current-like symmetry algebra
not unlike Lw1+∞ [76]. Twistor theory also admits an elegant description of self-dual Yang–Mills
theory via the Ward correspondence [77], the gauge theory analogue of the non-linear graviton.
One can build a gauge theory version of the twistor sigma model which operationalizes the
Ward correspondence; following similar steps to those presented here will yield a twistorial
representation of the infinite-dimensional algebra associated with the positive helicity soft gluon
sector. This algebra can be seen as arising from the natural action of gauge transformations on
the twistor data for self-dual Yang–Mills on asymptotic twistor space. Similar statements are
possible for Einstein–Yang–Mills and the action of soft graviton symmetries on gluons. However,
the ambitwistor string provides a more direct root to studying soft gluons and celestial OPEs
in pure Yang–Mills (for which there is a consistent worldsheet model) and even for Einstein–
Yang–Mills, where a fully consistent worldsheet theory is not known [1].
Towards quantization. The twistor sigma model (3.7) gives rise to gravitational amplitudes
via its classical action and the corresponding tree expansion; by contrast twistor strings or
ambitwistor strings produce amplitudes as fully quantum correlations functions in the worldsheet
CFT. This distinction leaves room for one to ask what the twistor sigma model could correspond
to if treated quantum mechanically. In particular, there is scope for this to give rise to some
theory of self-dual quantum gravity, for instance as envisaged by [49, 50] for the N = 2 string.
For instance, the ‘quantum’ (i.e., finite ℏ) MHV graviton amplitude produced by the twistor
20 T. Adamo, L. Mason and A. Sharma
sigma model can be computed [5]:
Mn = δ4
(
n∑
r=1
kr
)
⟨12⟩2n
n∏
i=3
1
⟨1i⟩2⟨2i⟩2
exp
− iℏ
8π
∑
j ̸=i
[ij]
⟨ij⟩
⟨1i⟩2⟨2j⟩2
⟨12⟩2
,
although its physical properties and interpretation remain to be explored. It would also be in-
triguing to make contact with the ∗-algebra definition of the quantumW1+∞-algebra as described
in [64] and the Moyal deformations of the Poisson structure associated to self-dual gravity pro-
posed by [72] which are closely connected also to Penrose’s Palatial twistor ideas [61]. It would
be interesting to track the twistor-theoretic component of the other appearances of W -infinity
algebras in the literature.
Acknowledgements
We are grateful to Andrew Strominger for discussions. TA is supported by a Royal Society Uni-
versity Research Fellowship and by the Leverhulme Trust (RPG-2020-386). LJM is supported
in part by the STFC grant ST/T000864/1. AS is supported by a Mathematical Institute Stu-
dentship, Oxford and by the ERC grant GALOP ID: 724638. We would also like to thank the
SIGMA team for their courage and dedication, working through difficult circumstances in Kyiv
in the face of the Russian invasion and aggression.
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Celestial w1+∞ Symmetries from Twistor Space 23
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https://doi.org/10.1007/BF01036128
1 Introduction
2 Lw_{1+infty} and self-dual gravity
2.1 Poisson diffeomorphisms and Lw_{1+infty}
2.2 Realization on twistor space
3 Twistor sigma model and MHV amplitudes
3.1 Holomorphic curves and twistor sigma model
3.2 I and asymptotic twistor space
3.3 From the sigma model to the MHV amplitude
4 From twistorial to celestial Lw_{1+infty}
4.1 Lw_{1+infty} charges and algebra
4.2 Vertex operators and currents and soft limits
4.3 Soft graviton symmetries
5 The lift to 4d ambitwistor string
5.1 Lifting to ambitwistor space
5.2 The 4d ambitwistor string
6 Discussion
References
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| id | nasplib_isofts_kiev_ua-123456789-211529 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-13T08:29:29Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Adamo, Tim Mason, Lionel Sharma, Atul 2026-01-05T12:26:32Z 2022 Celestial 𝓌₁₊∞ Symmetries from Twistor Space. Tim Adamo, Lionel Mason and Atul Sharma. SIGMA 18 (2022), 016, 23 pages 1815-0659 2020 Mathematics Subject Classification: 83C60; 81U20; 32L25 arXiv:2110.06066 https://nasplib.isofts.kiev.ua/handle/123456789/211529 https://doi.org/10.3842/SIGMA.2022.016 We explain how twistor theory represents the self-dual sector of four-dimensional gravity in terms of the loop group of Poisson diffeomorphisms of the plane via Penrose's non-linear graviton construction. The symmetries of the self-dual sector are generated by the corresponding loop algebra 𝐿𝓌₁₊∞ of the algebra 𝓌₁₊∞ of these Poisson diffeomorphisms. We show that these coincide with the infinite tower of soft graviton symmetries in tree-level perturbative gravity recently discovered in the context of celestial amplitudes. We use a twistor sigma model for the self-dual sector, which describes maps from the Riemann sphere to the asymptotic twistor space defined from characteristic data at null infinity I. We show that the OPE of the sigma model naturally encodes the Poisson structure on twistor space and gives rise to the celestial realization of 𝐿𝓌₁₊∞. The vertex operators representing soft gravitons in our model act as currents generating the wedge algebra of 𝓌₁₊∞ and produce the expected celestial OPE with hard gravitons of both helicities. We also discuss how the two copies of 𝐿𝓌₁₊∞, one for each of the self-dual and anti-self-dual sectors, are represented in the OPEs of vertex operators of the 4d ambitwistor string. We are grateful to Andrew Strominger for discussions. TA is supported by a Royal Society University Research Fellowship and by the Leverhulme Trust (RPG-2020-386). LJM is supported in part by the STFC grant ST/T000864/1. AS is supported by a Mathematical Institute Studentship, Oxford, and by the ERC grant GALOP ID: 724638. We would also like to thank the SIGMA team for their courage and dedication, working through difficult circumstances in Kyiv in the face of the Russian invasion and aggression. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Celestial 𝓌₁₊∞ Symmetries from Twistor Space Article published earlier |
| spellingShingle | Celestial 𝓌₁₊∞ Symmetries from Twistor Space Adamo, Tim Mason, Lionel Sharma, Atul |
| title | Celestial 𝓌₁₊∞ Symmetries from Twistor Space |
| title_full | Celestial 𝓌₁₊∞ Symmetries from Twistor Space |
| title_fullStr | Celestial 𝓌₁₊∞ Symmetries from Twistor Space |
| title_full_unstemmed | Celestial 𝓌₁₊∞ Symmetries from Twistor Space |
| title_short | Celestial 𝓌₁₊∞ Symmetries from Twistor Space |
| title_sort | celestial 𝓌₁₊∞ symmetries from twistor space |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211529 |
| work_keys_str_mv | AT adamotim celestialw1symmetriesfromtwistorspace AT masonlionel celestialw1symmetriesfromtwistorspace AT sharmaatul celestialw1symmetriesfromtwistorspace |