arXiv subjects

Lionel Mason

Publications and source records attributed to Lionel Mason.

At least 19 recordsLinked to original sources

arXiv2026

Higher-spin self-dual gravity from holomorphic planes in twistor space

We prove a `nonlinear graviton theorem' for higher-spin self-dual gravity. We consider small deformations of the complex structure of the non-projective twistor space that are bounded in a specified region near the origin and investigate the space $M_{HS}$ of holomorphically embedded complex planes $\mathbb{C}^2$ that intersect the origin. We show that this space is an infinite dimensional complex manifold with a canonical projection onto a four-dimensional holomorphic self-dual spacetime $\mathcal{M}$, and discuss the geometry induced on this new higher-spin space. Solutions of higher-spin self-dual gravity are then obtained by choosing an embedding of spacetime $\mathcal{M}$ into higher-spin space $M_{HS}$, with higher-spin symmetries arising from the different choices of embedding. Integrability of the theory is manifested in the form of a Lax pair for the system that we present. We conjecture that chiral higher-spin gravity can similarly be realized by considering deformations that are unconstrained at the origin.

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arXiv2026

Graviton scattering on self-dual black holes

The computation of gravitational wave scattering on black hole spacetimes is an extremely hard problem, typically requiring approximation schemes that either treat the black hole perturbatively or are only amenable to numerical techniques. In this paper, we consider linearised gravitational waves (or gravitons) scattering on the self-dual analogue of a black hole: namely, the self-dual Taub-NUT metric. Using the hidden integrability of the self-dual sector, we solve the linearised Einstein equations on these self-dual black hole backgrounds exactly in terms of simple, explicit quasi-momentum eigenstates. Using a description of the self-dual Taub-NUT metric and its gravitons in terms of twistor theory, we obtain an explicit formula, exact in the background, for the tree-level maximal helicity violating (MHV) graviton scattering amplitude at arbitrary multiplicity, with and without spin. This is obtained from the description of the MHV amplitudes in terms of the perturbation theory of a chiral sigma model whose target is the twistor space of the background. The incorporation of spin effects on these backgrounds is a straightforward application of the Newman-Janis shift. We also demonstrate that the holomorphic collinear splitting functions in the self-dual background are equal to those in flat space so that the celestial symmetry algebra is undeformed.

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arXiv2026

Quasi-Local Celestial Charges and Multipoles

We extend Penrose's quasi-local mass definition to include higher-spin charges associated with the celestial $Lw_{1+\infty}$ symmetries and relate them to traditional definitions of multipoles. The resulting formulae provide explicit expressions that can be computed on finite 2-surfaces, given a choice of null hypersurface. They yield a geometric definition of celestial symmetries and multipoles in generic spacetimes in terms of higher-valence solutions to the twistor equations. This, in turn, gives rise to natural flux-balance laws along the null hypersurface. We also present a first-principles phase-space derivation of these charges, starting from a twistor space action for self-dual gravity that can be identified with the standard gravitational asymptotic phase space at null infinity, performing a large gauge transformation analysis and using the Penrose transform to connect with the corresponding spacetime expressions. Finally, we formulate the spacetime analysis in the Plebanski gauge and relate the celestial symmetries to the integrability of self-dual gravity in the case of a self-dual background.

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arXiv2026

The Self-Duality Equations on a Riemann Surface and Four-Dimensional Chern-Simons Theory

We construct a Lagrangian formulation of Hitchin's self-duality equations on a Riemann surface $Σ$ using potentials for the connection and Higgs field. This two-dimensional action is then obtained from a four-dimensional Chern-Simons theory on $Σ\times \mathbb{CP}^1$ with an appropriate choice of meromorphic 1-form on $\mathbb{CP}^1$ and boundary conditions at its poles. We show that the symplectic structure induced by the four-dimensional theory coincides with the canonical symplectic form on the Hitchin moduli space in the complex structure corresponding to the moduli space of Higgs bundles. We further provide a direct construction of Hitchin Hamiltonians in terms of the four-dimensional gauge field. Exploiting the freedom in the choice of the meromorphic one-form, we construct a family of four-dimensional Chern-Simons theories depending on a $\mathbb{CP}^1$-valued parameter. Upon reduction to two dimensions, these descend to a corresponding family of two-dimensional actions on $Σ$ whose field equations are again Hitchin's equations. Furthermore, we obtain a family of symplectic structures from our family of four-dimensional theories and show that they agree with the hyperkähler family of symplectic forms on the Hitchin moduli space, thereby identifying the $\mathbb{CP}^1$-valued parameter with the twistor parameter of the Hitchin moduli space. Our results place Hitchin's equations and their integrable structure within the framework of four-dimensional Chern-Simons theory and make the role of the twistor parameter manifest.

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arXiv2025

S-algebra in Gauge Theory: Twistor, Spacetime and Holographic Perspectives

The celestial $S$-algebra arose from a reinterpretation of collinear limits of the Yang-Mills S-matrix as OPEs in celestial holography. It was subsequently represented via asymptotic charge aspects defined in the Yang-Mills radiative phase space defined at null infinity on the one hand, and via a twisted holography vertex algebra construction in twistor space on the other. Here we first identify it with the traditional symmetry algebra of self-dual Yang-Mills theory as an integrable system via its hierarchies of conserved quantities and associated flows; the self-dual phase space can be canonically identified with that of full Yang-Mills at null infinity $\mathscr{I}$. We derive the associated canonical generators from the twistor space action, identifying two infinite towers of charges corresponding to the two gluon helicities. These expressions are translated into spacetime data at null infinity using twistor integral formulae. Examining the charge algebra at spacelike infinity reveals the vertex algebras studied in the context of twisted holography. Our discussion extends directly to the celestial LHam$(\mathbb{C}^2)$ symmetries of self-dual gravity. This analysis provides a unified framework for celestial symmetries, connecting twistor, spacetime, and holographic approaches and culminating in a nonlinear extrapolate dictionary for self-dual gauge theory.

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arXiv2025

Chiral higher-spin theories from twistor space

We reformulate chiral higher-spin Yang-Mills and gravity on $\mathbb{R}^4$ as 'CR-holomorphic' theories of Chern-Simons type; in the most general case, these are Moyal deformed to become non-commutative. They are defined on the space of non-projective twistors of unit length. These spaces carry $S^7$, $S^3\times \mathbb{R}^4$ or AdS$_{3+4}$ metrics but are also endowed with a Cauchy-Riemann structure, an odd-dimensional analogue of a complex structure, with respect to which the theories are holomorphic. They are circle bundles over standard projective twistor spaces and the higher spin fields arise naturally as Kaluza-Klein modes. We give a perturbative analysis to identify the spectrum and three-point vertices on spacetime and, for flat space, in momentum space. These vertices can have helicities $(+++)$, $(++-)$ or $(+--)$, but are nevertheless all of $\overline{\text{MHV}}$ type in the sense that they are supported on momenta with proportional anti-self-dual spinors. On reduction to spacetime, there are higher valence vertices but these appear to be a gauge artifact. Further generalizations are discussed.

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arXiv2024

Celestial $Lw_{1+\infty}$ charges from a twistor action

The celestial $Lw_{1+\infty}$ symmetries in asymptotically flat spacetimes have a natural geometric interpretation on twistor space in terms of Poisson diffeomorphisms. Using this framework, we provide a first-principle derivation of the canonical generators associated with these symmetries starting from the Poisson BF twistor action for self-dual gravity. We express these charges as surface integrals over the celestial sphere in terms of spacetime data at null infinity. The connection between twistor space and spacetime expressions at $\mathscr{I}$ is achieved via an integral formula for the asymptotic Bianchi identities due to Bramson and Tod. Finally, we clarify how $Lw_{1+\infty}$ transformations are symmetries of gravity from a phase space perspective by showing the invariance of the asymptotic Bianchi identities.

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arXiv2024

Scattering on self-dual Taub-NUT

We derive exact solutions of massless free field equations and tree-level two-point amplitudes up to spin 2 on self-dual Taub-NUT space-time, as well as on its single copy, the self-dual dyon. We use Killing spinors to build analogues of momentum eigenstates, finding that, in the spirit of color-kinematics duality, those for the self-dual dyon lift directly to provide states on the self-dual Taub-NUT background if one replaces charge with energy. We discover that they are forced to have faster growth at infinity than in flat space due to the topological non-triviality of these backgrounds. The amplitudes for massless scalars and spinning particles in the $(+\,+)$ and $(+\,-)$ helicity configurations vanish for generic kinematics as a consequence of the integrability of the self-dual sector. The $(-\,-)$ amplitudes are non-vanishing and we compute them exactly in the backgrounds, which are treated non-perturbatively. It is explained how spin is easily introduced via a Newman-Janis imaginary shift along the spin-vector leading directly to the additional well-known exponential factor in the dot product of the spin with the momenta. We also observe a double copy relation between the gluon amplitude on a self-dual dyon and graviton amplitude on a self-dual Taub-NUT space-time.

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arXiv2024

Gluon scattering on the self-dual dyon

The computation of scattering amplitudes in the presence of non-trivial background gauge fields is an important but extremely difficult problem in quantum field theory. In even the simplest backgrounds, obtaining explicit formulae for processes involving more than a few external particles is often intractable. Recently, it has been shown that remarkable progress can be made by considering background fields which are chiral in nature. In this paper, we obtain a compact expression for the tree-level, maximal helicity violating (MHV) scattering amplitude of an arbitrary number of gluons in the background of a self-dual dyon. This is a Cartan-valued, complex gauge field sourced by a point particle with equal electric and magnetic charges, and can be viewed as the self-dual version of a Coulomb field. Twistor theory enables us to manifest the underlying integrability of the self-dual dyon, trivializing the perturbative expansion in the MHV sector. The formula contains a single position-space integral over a spatial slice, which can be evaluated explicitly in simple cases. As an application of the formula, we show that the holomorphic collinear splitting functions of gluons in the self-dual dyon background are un-deformed from a trivial background, meaning that holomorphic celestial OPE coefficients and the associated chiral algebra are similarly un-deformed. We also comment on extensions of our MHV formula to the full tree-level gluon S-matrix.

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arXiv2024

On AdS$_4$ deformations of celestial symmetries

Celestial holography has led to the discovery of new symmetry algebras arising from the study of collinear limits of perturbative gravity amplitudes in flat space. We explain from the twistor perspective how a non-vanishing cosmological constant $Λ$ naturally modifies the celestial chiral algebra. The cosmological constant deforms the Poisson bracket on twistor space, so the corresponding deformed algebra of Hamiltonians under the new bracket is automatically consistent. This algebra is equivalent to that recently found by Taylor and Zhu. We find a number of variations of the deformed algebra. We give the Noether charges arising from the expression of this algebra as a symmetry of the twistor action for self-dual gravity with cosmological constant.

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arXiv2023

Amplitudes at strong coupling as hyperkähler scalars

Alday & Maldacena conjectured an equivalence between string amplitudes in AdS$_5 \times S^5$ and null polygonal Wilson loops in planar $\mathcal{N}=4$ super-Yang-Mills (SYM). At strong coupling this identifies SYM amplitudes with areas of minimal surfaces in AdS. For minimal surfaces in AdS$_3$, we find that the nontrivial part of these amplitudes, the \emph{remainder function}, satisfies an integrable system of nonlinear differential equations, and we give its Lax form. The result follows from a new perspective on `Y-systems', which defines a new psuedo-hyperkähler structure \emph{directly} on the space of kinematic data, via a natural twistor space defined by the Y-system equations. The remainder function is the (pseudo-)Kähler scalar for this geometry. This connection to pseudo-hyperkähler geometry and its twistor theory provides a new ingredient for extending recent conjectures for non-perturbative amplitudes using structures arising at strong coupling.

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arXiv2023

Carrollian Amplitudes and Celestial Symmetries

Carrollian holography aims to express gravity in four-dimensional asymptotically flat spacetime in terms of a dual three-dimensional Carrollian CFT living at null infinity. Carrollian amplitudes are massless scattering amplitudes written in terms of asymptotic or null data at $\mathscr I$. These position space amplitudes at $\mathscr I$ are to be re-interpreted as correlation functions in the putative dual Carrollian CFT. We derive basic results concerning tree-level Carrollian amplitudes yielding dynamical constraints on the holographic dual. We obtain surprisingly compact expressions for $n$-point MHV gluon and graviton amplitudes in position space at $\mathscr I$. We discuss the UV/IR behaviours of Carrollian amplitudes and investigate their collinear limit, which allows us to define a notion of Carrollian OPE. By smearing the OPE along the generators of null infinity, we obtain the action of the celestial symmetries - namely, the $S$ algebra for Yang-Mills theory and $Lw_{1+\infty}$ for gravity - on the Carrollian operators. As a consistency check, we systematically relate our results with celestial amplitudes using the link between the two approaches. Finally, we initiate a direct connection between twistor space and Carrollian amplitudes.

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arXiv2023

Gravity from holomorphic discs and celestial $Lw_{1+\infty}$ symmetries

In split or Kleinian signature, twistor constructions parametrize solutions to both gauge and gravity self-duality (SD) equation from twistor data that can be expressed in terms of free smooth data without gauge freedom. Here the corresponding constructions are given for asymptotically flat SD gravity providing a fully nonlinear encoding of the asymptotic gravitational data in terms of a real homogeneous generating function $h$ on the real twistor space. Geometrically $h$ determines a nonlinear deformation of the location of the real twistor space $\RP^3$ inside the complex twistor space $\CP^3$. This presentation gives an optimal presentation of Strominger's recently discovered $Lw_{1+\infty}$ celestial symmetries. These, when real, act locally as passive Poisson diffeomorphisms on the real twistor space. However, when imaginary, such Poisson transformations are active symmetries, and generate changes to the gravitational field by deforming the location of the real slice of the twistor space. Gravity amplitudes for the full, non-self-dual Einstein gravity, arise as correlators of a chiral twistor sigma model. This is reformulated for split signature as a theory of holomorphic discs in twistor space whose boundaries lie on the deformed real slice determined by $h$. Real $Lw_{1+\infty}$ symmetries act oas gauge symmetries, but imaginary generators yield graviton vertex operators that generate gravitons in the perturbative expansion. A generating function for the all plus 1-loop amplitude, an analogous framework for Yang-Mills, possible interpretations in Lorentz signature and similar open string formulations of twistor and ambitwistor strings in 4d in split signature, are briefly discussed.

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arXiv2023

Yang-Mills form factors on self-dual backgrounds

The construction of perturbative quantities on non-linear backgrounds leads to the possibility of incorporating strong field effects in perturbation theory. We continue a programme to construct QFT observables on self-dual backgrounds. The approach works with asymptotic data for fields defined at null infinity $\mathscr{I}$, extending earlier work on Yang-Mills amplitudes on self-dual backgrounds to form factors and incorporating supersymmetry. Since our analysis is based on reconstruction from data at null infinity, it naturally ties into work on celestial and twisted holography. We study form factors both in pure Yang-Mills and their supersymmetric counterparts in $\mathcal{N}=4$ SYM, giving a full treatment of $\mathcal{N}=4$ super-Yang-Mills at null infinity and their self-dual nonlinear backgrounds. We obtain tree-level MHV form factors around these backgrounds using new formulae for lifting operators to twistor space leading to simple dressings of the corresponding form factors around the vacuum. We give brief indications on how to go beyond the MHV sector by introducing dressed versions of the MHV diagram propagator. We discuss generating functionals of the MHV all plus 1-loop amplitude in this context together with its various dual conformal representations.

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arXiv2023

Massive ambitwistor-strings; twistorial models

Ambitwistor-strings are chiral strings whose targets are spaces of complex massless particles, and whose correlation functions directly lead to simple, compact formulae for scattering amplitudes and loop integrands for massless gauge and gravity theories. This article extends this framework to worldsheet models for massive particles in 4d, obtained via a symmetry reduction of a higher dimensional massless model. The target space of the resulting models turns out to be the phase space of 4d massive particles in a twistorial representation, and so the worldsheet theory agrees with the two-twistor string previously introduced by the authors. However, the paper has been written so as to be largely self-contained. We discuss two interesting classes of massive theories in detail. For gauge theories, the reduction procedure is explicitly adapted to supersymmetric gauge theories on the Coulomb branch. For supergravity theories, the reduction is adapted to give theories obtained via Cremmer, Scherk and Schwartz (CSS) reduction, with broken supersymmetry and massive multiplets. The reduction procedure gives explicit and systematic rules to obtain amplitudes for all these theories and their amplitudes from two compact master formulae that have their origins in 6d based on the polarized scattering equations; in the CSS case the formulae are new, and in both cases their derivation is systematic. The freedom to include mass allows the definition of a loop insertion operator, thereby extending the formulae to 1-loop. Unlike the massless 4d twistorial models, these all display a perfect double copy structure, here incorporating massive particles in the relationship between gauge theory and CSS supergravity amplitudes.

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arXiv2023

The SAGEX Review on Scattering Amplitudes

This is an introduction to, and invitation to read, a series of review articles on scattering amplitudes in gauge theory, gravity, and superstring theory. Our aim is to provide an overview of the field, from basic aspects to a selection of current (2022) research and developments.

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arXiv2022

Gluon scattering on self-dual radiative gauge fields

We present all-multiplicity formulae, derived from first principles in the MHV sector and motivated by twistor string theory for general helicities, for the tree-level S-matrix of gluon scattering on self-dual radiative backgrounds. These backgrounds are chiral, asymptotically flat gauge fields characterised by their free radiative data, and their underlying integrability is captured by twistor theory. Tree-level gluon scattering scattering amplitudes are expressed as integrals over the moduli space of holomorphic maps from the Riemann sphere to twistor space, with the degree of the map related to the helicity configuration of the external gluons. In the MHV sector, our formula is derived from the Yang-Mills action; for general helicities the formulae are obtained using a background-coupled twistor string theory and pass several consistency tests. Unlike amplitudes on a trivial vacuum, there are residual integrals due to the functional freedom in the self-dual background, but for scattering of momentum eigenstates we are able to do many of these explicitly and even more is possible in the special case of plane wave backgrounds. In general, the number of these integrals is always less than expected from standard perturbation theory, but matches the number associated with space-time MHV rules in a self-dual background field, which we develop for self-dual plane waves.

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arXiv2022

Graviton scattering in self-dual radiative space-times

The construction of amplitudes on curved space-times is a major challenge, particularly when the background has non-constant curvature. We give formulae for all tree-level graviton scattering amplitudes in curved self-dual radiative space-times; these are chiral, source-free, asymptotically flat spaces determined by free characteristic data at null infinity. Such space-times admit an elegant description in terms of twistor theory, which provides the powerful tools required to exploit their underlying integrability. The tree-level S-matrix is written in terms of an integral over the moduli space of holomorphic maps from the Riemann sphere to twistor space, with the degree of the map corresponding to the helicity configuration of the external gravitons. For the MHV sector, we derive the amplitude directly from the Einstein-Hilbert action of general relativity, while other helicity configurations arise from a natural family of generating functionals and pass several consistency checks. The amplitudes in self-dual radiative space-times exhibit many novel features that are absent in Minkowski space, including tail effects. There remain residual integrals due to the functional degrees of freedom in the background space-time, but our formulae have many fewer such integrals than would be expected from space-time perturbation theory. In highly symmetric special cases, such as self-dual plane waves, the number of residual integrals can be further reduced, resulting in much simpler expressions for the scattering amplitudes.

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