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

Abstract

The study of physical theories in various signatures has been important for uncovering structures not easily visible or definable in Lorentz signature. In split signature, global twistor constructions for conformally self-dual (SD) gravity and Yang–Mills construct solutions from twistor data that can be expressed in terms of free data without gauge freedom. This is developed for asymptotically flat SD gravity to give a fully nonlinear encoding of the asymptotic gravitational data in terms of a real homogeneous generating function h on the real twistor space. The recently discovered $Lw_{1+\infty }$ celestial symmetries, when real, act locally as passive Poisson diffeomorphisms on the real twistor space. The twistor data, h, generates an imaginary such Poisson transformation that then generates the gravitational field by shifting the real slice of the twistor space. The twistor chiral sigma models, whose correlators yield the Einstein gravity tree-level S-matrix, are reformulated as theories of holomorphic discs in twistor space whose boundaries lie on the deformed real slice determined by h. The real $Lw_{1+\infty }$ symmetries act on the corresponding formula for the S-matrix geometrically with vanishing Noether currents, but imaginary generators yield graviton vertex operators that generate gravitons in the perturbative expansion. A generating function for the all plus 1-loop amplitude, the 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.

1 Introduction

Celestial Holography seeks to encode 4d gravity in an asymptotically flat space-time from a boundary theory that is defined at the null infinity, $\mathscr {I}$, that arises when the cosmological constant vanishes [1, 2]. Similarly, in the 1970’s Newman tried to rebuild space-time from the ‘cuts’ of $\mathscr {I}$ formed by the intersection of light cones of points in the interior of space-time with $\mathscr {I}$, [3]. His ‘good cut’ equation instead yielded $\mathcal {H}$-space, a complex self-dual space-time built from the self-dual part of the characteristic data at $\mathscr {I}$. Penrose subsequently re-interpreted this as a twistor construction [4] by first introducing an ‘asymptotic twistor space’ [5, 6] defined from data at $\mathscr {I}$. More recently, it has been possible to use this framework to construct sigma models and strings whose target spaces are these asymptotic twistor spaces or their cotangent bundles (ambitwistor spaces) [7,8,9,10,11]. These then yield explicit formulae for complete tree-level S-matrices for gauge and gravity theories based on twistor-string ideas [12,13,14] and ambitwistor-string ideas [15,16,17] reformulated at null infinity using these asymptotic twistor and ambitwistor spaces as targets. These theories lead to easy computations [7,8,9,10,11] of the soft limits [2] on which much of celestial holography is based: such soft limits can be taken at the level of the vertex operators of these models [7,8,9, 18, 19], for a review see [17]. Particularly in the form of [10, 11], these formulae are based on rational curves that can be thought of as extending Newman’s construction to provide the formulae for the full light-cone cuts of infinity in the form of an expansion around the self-dual sector in such a way as to provide also formulae for the tree-level gravitational S-matrix. These models therefore provide reformulations at $\mathscr {I}$ of gauge and gravity theories. The twisted celestial holography programme of Costello and Paquette [20, 21] also lives in twistor space, and is perhaps most easily understood via the same mechanism.

This paper reformulates this twistor theory in split signature. Split signature has already been used in the context of celestial holography [22] to understand in particular the recently discovered $Lw_{1+\infty }$ symmetry [23, 24] see also [25, 26] for other recent applications of twistors in split signature. In [18] these symmetries were interpreted as the natural structure-preserving diffeomorphisms on twistor space constructed in Lorentz signature. However, the correspondence was obscured by the need to perform a translation from a Čech to a Dolbeault presentation. This was analogous to the shadow and light-ray transforms used in [24, 26,27,28]. In split signature we will see that the twistor theory automatically provides the framework in which the $Lw_{1+\infty }$ appears as the natural structure preserving diffeomorphisms of the twistor space more directly than was possible in [18].

The use of split signature perhaps requires some comment. The analytic continuation of scattering amplitudes and correlation functions has always been a powerful tool, although mostly to Euclidean signature to renders QFT quantities more regular. However, there is much physics in their singularity structure and their geometry can be made explicit in split signature particularly at tree level and for integrands. In four dimensions, such a continuation was used to uncover the twistor support of amplitudes in the original twistor-string [12], and in BCFW recursion in [29, 30] and subsequently in the positive Grassmannian [31] and the amplituhedron [32]. In this article we will see that it also uncovers quite explicitly the underlying geometry of the $Lw_{1+\infty }$ symmetry discovered recently by Strominger and co-workers [23, 24, 33].

The utility of split signature arises because the Penrose transform usually represents space-time massless fields in terms of cohomology classes on twistor space. These are subject to cohomological gauge freedoms that can obscure the underlying geometry. However, it was observed early on that in split signature, the Penrose transform becomes the X-ray transform of Fritz-John [34]. This gives an essentially one to one map between functions on the totally real twistor space, $\mathbb{R}\mathbb{P}^3$ and solutions to zero rest mass equations on split signature space-time. The transform integrates a smooth function on the real twistor space $\mathbb{R}\mathbb{P}^3$ along straight lines. Via the Klein correspondence, this space of lines is a four-dimensional real quadric in $\mathbb{R}\mathbb{P}^5$ that can be understood as the conformal compactification of split signature Minkowski space $\mathbb {M}^{2,2}$ [35]. This has been explored using the complex twistor geometry for the linear transform in [36, 37] and in particular for amplitudes via Witten’s half-Fourier transform from momentum space [12] in his study of the twistor support of gauge theory amplitudes on rational curves in twistor space arising from twistor-string theory. The nonlinear analogues are quite nontrivial and have been part of an ongoing investigation with Claude Lebrun in [38,39,40,41] for the nonlinear constructions relating to conformal and projective geometry, and in [42, 43] for self-dual Yang–Mills. These works did not explicitly work out the details for self-dual Einstein gravity and the first focus of this paper is to give a similar treatment for SD Einstein vacuum spaces with vanishing cosmological constant.

To summarize this part, in split signature there are real twistors that correspond to real self-dual totally null 2-surfaces in space-time. In flat space these real twistors make up the real slice $\mathbb{P}\mathbb{T}_\mathbb {R}=\mathbb{R}\mathbb{P}^3$, as fixed by the standard complex conjugation inside complex projective twistor space $\mathbb{P}\mathbb{T}=\mathbb{C}\mathbb{P}^3$. In the global split signature construction of [39], it is shown that under suitable global assumptions, the complex holomorphic twistor space can still be reconstructed as $\mathbb{C}\mathbb{P}^3$, as in the standard flat model, but the location of the real slice $\mathbb{P}\mathbb{T}_\mathbb {R}$ is deformed essentially arbitrarily away from the standard embedding of $\mathbb{R}\mathbb{P}^3$. The self-dual space-time is then reconstructed as the moduli space of degree-1 holomorphic discs whose boundary lies in this deformed $\mathbb{P}\mathbb{T}_\mathbb {R}$. A particular feature is that the space of such discs has topology $S^2\times S^2$ whereas $\mathbb {M}^{2,2}=S^2\times S^2/\mathbb {Z}_2$, because in the flat case two discs make up each real Riemann sphere that is sent to itself under the complex conjugation that fixes $\mathbb{R}\mathbb{P}^3$. In the flat case the $\mathbb {Z}_2$ action on $S^2\times S^2$ whose quotient gives $\mathbb {M}^{2,2}$ is the joint antipodal map on the $S^2$ factors. In the nonlinear case, once the location of $\mathbb{P}\mathbb{T}_\mathbb {R}$ has been deformed, global solutions on $S^2\times S^2$ are found at least for small deformations, but only the conformally flat metric is $\mathbb {Z}^2$-invariant descending to $S^2\times S^2/\mathbb {Z}^2$.

This paper incorporates the Einstein equations with vanishing cosmological constant into the construction of [39] described above. In order to impose this Einstein condition, we must endows the complex twistor space with a global but degenerate holomorphic Poisson structure that is real on $\mathbb{R}\mathbb{P}^3$. Introduce $Z=(\lambda _\alpha ,\mu ^{{\dot{\alpha }}})=U+iV$ as homogeneous coordinates on $\mathbb{P}\mathbb{T}$, where $\alpha =0,1$, and ${\dot{\alpha }}=\dot{0},\dot{1}$ are standard two-component spinor indices for the Lorentz group in 4d and U and V represent the real and imaginary parts. Then the Poisson structure is given explicitly by

$$\begin{aligned} \{g_1,g_2\}=\varepsilon ^{{\dot{\alpha }}{\dot{\beta }}}\frac{\partial g_1}{\partial u^{{\dot{\alpha }}}}\frac{\partial g_2}{\partial u^{{\dot{\beta }}}}\,. \end{aligned}$$

(1.1)

Given this, the gravitational data is encoded into a real homogeneous function h(U) of weight 2 on $\mathbb{R}\mathbb{P}^3$. It has the interpretation as the generating function that shifts the standard $\mathbb{R}\mathbb{P}^3$ into its deformed position $\mathbb{P}\mathbb{T}_\mathbb {R}$ on which the restriction of the Poisson structure remains real. Explicitly we have

$$\begin{aligned} \mathbb{P}\mathbb{T}_\mathbb {R}=\left\{ [Z=U+iV]\in \mathbb{P}\mathbb{T}| V= \{h,U\}\right\} \,. \end{aligned}$$

(1.2)

This gives a clean geometric realisation of the recently discovered $Lw_{1+\infty }$ symmetries and its action on the gravitational data. The $Lw_{1+\infty }$ in this signature becomes the real Poisson diffeomorphisms ${\mathcal {P}}$ of the real twistor space $\mathbb{P}\mathbb{T}_\mathbb {R}$. This acts naturally and geometrically on this space of gravitational data as a passive diffeomorphism freedom. In the twistor representation, the data represents points in the quotient $\mathcal P_{\mathbb {C}}/\mathcal {P}$ of the complex Poisson diffeomorphism semi-group ${\mathcal {P}}_{\mathbb {C}}$, by the real Poisson diffeomorphisms $\mathcal {P}$. Here ${\mathcal {P}}_{\mathbb {C}}$ is really a semi-group defined as a higher-dimensional analogue of Segal’s semi-group of annuli that complexify the diffeomorphism group of the circle as in [44]. Our generating functions h(U) parametrize a natural choice of imaginary slice of $\mathcal {P}_{\mathbb {C}}$.

In order to connect to celestial formulations of gravity, we explain how the twistor gravitational data can be constructed directly from the asymptotic gravitational data, the asymptotic shear $\varvec{\sigma }$, at null infinity $\mathscr {I}$. This uses the fact that $\varvec{\sigma }$ determines a projective structure on each $\beta$-plane (ASD totally null 2-plane) that lies on $\mathscr {I}$; its geodesics are the null geodesics on $\mathscr {I}$ that lie in the $\beta$-planes. Because they bound $\alpha$-surfaces in the interior space-time, they must be closed circles. Such projective structures were studied in [38] via a two-dimensional analogue of the twistor correspondence described above. This allows us to establish a fully nonlinear correspondence between the twistor data h(U) and the asymptotic shears $\varvec{\sigma }$ at $\mathscr {I}$; this was expressed via a Čech-Dolbeault correspondence in [18] and generalizes the linear-theory light-ray transform used by [24, 27] to establish the correspondence between asymptotic shears and generators of $Lw_{1+\infty }$.

To make contact with amplitudes, soft theorems etc., we adapt the worldsheet models that give rise to twistorial realizations of the gravity S-matrix in twistor space to this geometry focussing on the chiral sigma model of [10].¹ This was formulated there as a closed string model, but the twistor geometry in split signature naturally lends itself to an open-string formulation. The strings are required to be holomorphic maps from a Riemann surface with boundary, a holomorphic disc $\textrm{D}$ at tree-level, to the complex twistor space with boundary lying in $\mathbb{P}\mathbb{T}_\mathbb {R}$. We focus on the twistor chiral sigma models whose classical correlation functions compute the full tree-level gravity S-matrix as in [10]. To compute amplitudes with k negative helicity particles, the open chiral sigma model is a theory of maps $Z:\textrm{D}\rightarrow \mathbb{P}\mathbb{T}$ of degree $k-1$ where $\textrm{D}$ is a holomorphic disc, represented as the upper-half-plane, with coordinate $\sigma$ with action

$$\begin{aligned} S[Z]= \Im \int _\textrm{D}\varepsilon _{{\dot{\alpha }}{\dot{\beta }}}\mu ^{{\dot{\alpha }}}{\bar{\partial }}\mu ^{{\dot{\beta }}} \textrm{d}\sigma +\oint _{\partial \textrm{D}} h(U) \textrm{d}\sigma \,. \end{aligned}$$

(1.3)

The boundary term gives rise to the required boundary condition that $\partial \textrm{D}\subset \mathbb{P}\mathbb{T}_\mathbb {R}$ rather than $\mathbb{R}\mathbb{P}^3$. For an amplitude with k ASD particles on the SD background determined by h, we require that at k points $\sigma _i\in \partial \textrm{D}$, $Z(\sigma )$ is required to go through the given twistors $Z_i$ at $\sigma _i$. The tree-level amplitude for k ASD particles on the fully nonlinear background determined by h is built from the corresponding on-shell action $S^{os}[Z_i\sigma _i,h]$ for the unique degree k solution. This must then be integrated against the wave functions for the ASD particles inserted at the $\sigma _i$ and $Z_i$ together with a reduced determinant. See (5.6) for full details. To prove this, one can expand the on-shell action $S^{os}[Z_i,\sigma _i,h]$ perturbatively in h and one then obtains a version of the Cachazo-Skinner formula [45, 46] as in [10]. One can see from this presentation that the formula is invariant under real $Lw_{1+\infty }$ motions at all MHV degrees, whereas the SD gravity vertex operator gives the action on the Sigma model of an imaginary generator of $Lw_{1+\infty }$.

This article reviews much of the background in order to be self-contained. Thus Sect. 2 reviews the global geometry of compactified conformally flat space-time in split signature, how null infinity $\mathscr {I}$ sits inside it, and their correspondence with flat twistor space making contact with [22] but with different coordinates adapted to the twistor correspondence. The next Sect. 3 starts in Sect. 3.1 by reviewing the conformally self-dual case discussed in [39]. It goes on to specialise to the Ricci-flat case, introducing the generating function h in Sect. 3.2 and discussing their connections with the $Lw_{1+\infty }$ symmetry in Sect. 3.3. The next Sect. 4 explains the nonlinear correspondence between the twistor data h(U) and the asymptotic shear $\varvec{\sigma }$ on $\mathscr {I}$ for this class of space-times. Section 5.1 introduces the open chiral sigma model and presents the amplitude formulae for the tree-level gravitational S-matrix. The last Sect. 6 gives a brief summary and discussion. This includes some remarks on generating formulae for the all-plus 1-loop amplitude, the translation of the results into Lorentz signature, the corresponding open twistor and ambitwistor models, and the Yang–Mills analogues of the ideas in this paper. “Appendix A” reviews the linear theory integral formulae that arise in the linearized limit of the transforms described in this paper and their relationship with others such as the Fourier transform, half-Fourier transform and the Kirchoff-d’Adhemar formulae from asymptotic data. In particular the map between the asymptotic shear and h(U) in linear theory can be realized as a Radon transform from h to give $\varvec{\sigma }$. The second “Appendix B” describes how the Gibbons–Hawking ansatze adapts to split signature in the framework of this paper to give many regular solutions, and also the (singular) SD taub-NUT metric in split signature, as described in [47].

2 The twistor geometry and $\mathscr {I}$ and in split signature

In this section we review the global split signature geometry of conformal compactifications of Minkowski space; see also [22, 43] for a different but closely related discussion. We go on to describe its correspondence with twistor space and relegate to “Appendix A” the various integral formulae for the wave equations in terms of the Fourier transform, the X-ray transform to twistor space and the Kirchoff-d’Adhemar formulae from $\mathscr {I}$.

2.1 Compactified Minkowski space and $\mathscr {I}$

Split signature flat Minkowski space $\mathbb {R}^{2+2}$ has conformal compactification, denoted $\mathbb {M}^{2+2}$, of topology $(S^2 \times S^2)/\mathbb {Z}_2$. In this signature, the conformal group is $SO(3,3)/\mathbb {Z}_2$ acting naturally on the ‘embedding space’ representation

$$\begin{aligned} \mathbb {M}^{2+2}:=\{X\in \mathbb {R}^{3+3}|X\cdot X=0\}/\sim \,, \qquad X\sim a X\,. \end{aligned}$$

(2.1)

The ‘embedding space’ coordinates can be divided into a pair of three-vectors $X:=({{\textbf{x}}},{{\textbf{y}}})\in \mathbb {R}^3\times \mathbb {R}^3$ so that $X\cdot X:=|{{\textbf{x}}}|^2-|{{\textbf{y}}}|^2$. If we fix the rescaling freedom by $|{{\textbf{x}}}|=|{{\textbf{y}}}|=1$, we see that the double cover is $S^2\times S^2$. This gives the choice of conformal metric

$$\begin{aligned} ds^2_{S^2\times S^2}=( ds^2_{S^2_{{\textbf{x}}}} - ds^2_{S^2_{{\textbf{y}}}})\,. \end{aligned}$$

(2.2)

However, to obtain $\mathbb {M}^{2+2}$ we must quotient also by the $\mathbb {Z}_2$ that acts by

$$\begin{aligned} ({{\textbf{x}}},{{\textbf{y}}})\rightarrow (-{{\textbf{x}}},-{{\textbf{y}}}) \end{aligned}$$

(2.3)

without fixed points thereby obtaining $(S^2\times S^2)/\mathbb {Z}_2$ as claimed.

To make the conformal flatness more explicit, these coordinates can be related to standard affine coordinates by introducing the complex stereographic coordinates

$$\begin{aligned} z=: \frac{x_1+ix_2}{x_3-y_3}\,, \qquad w:=\frac{y_1+iy_2}{x_3-y_3}\,. \end{aligned}$$

(2.4)

With these coordinates we find the conformal rescaling to the flat metric

$$\begin{aligned} ds^2_{\textrm{flat}}=2dzd{\bar{z}} -2dwd{\bar{w}} =\frac{5-(x_3-y_3)^2}{2(x_3-y_3)^2 }ds^2_{S^2\times S^2} \end{aligned}$$

(2.5)

with null infinity, $\mathscr {I}$, given by $x_3-y_3= 0$. We will prefer to write the space-time in terms of real coordinates written in terms of two-component spinors as $x^{\alpha {\dot{\alpha }}}$ with $\alpha =0,1$, ${\dot{\alpha }}=\dot{0}, \dot{1}$ on $\mathbb {R}^{2+2}$ by

$$\begin{aligned} x^{ \alpha {\dot{\alpha }}}=\frac{1}{x_3-y_3}\begin{pmatrix} x_1-y_2&{}\quad x_2+y_1\\ y_1-x_2&{}\quad y_2+x_1 \end{pmatrix}\,, \qquad ds^2_{\textrm{flat}}= dx^{\alpha {\dot{\alpha }}} dx_{\alpha {\dot{\alpha }}}\,. \end{aligned}$$

(2.6)

Here two-component spinor indices are raised and lowered with the standard skew spinor $\varepsilon _{\alpha \beta }, \varepsilon _{{\dot{\alpha }}{\dot{\beta }}}$, $\varepsilon _{01}=\varepsilon _{\dot{0}\dot{1}}=1$. The flat coordinates are clearly invariant under the $\mathbb {Z}_2$-action.

Thus, inside $S^2\times S^2$, we realise $\mathscr {I}$ as

$$\begin{aligned} \mathscr {I}=\{x_3=y_3\}\simeq [-1,1]\times S^1\times S^1, \end{aligned}$$

(2.7)

with the $x_3=y_3\in [-1,1]$, and the $S^1\times S^1$ factor being the slice $x_3=y_3$ in $S^2\times S^2$.

The $\mathbb {Z}_2$ involution identifies the two opposite sides of $\mathscr {I}$ in $S^2\times S^2$ but is also antipodal on all three factors of $\mathscr {I}$ itself. This means that the lightcone of a point of $\mathscr {I}$ refocusses onto its antipode. However, when thinking of $\mathscr {I}$ as a boundary of flat space-time $\mathbb {R}^{2+2}$ we do not divide $\mathscr {I}$ itself by the $\mathbb {Z}_2$ involution as this would identify antipodally opposite points at $\infty$ but space-time fields might have different limits in opposite directions.

We can see that, as noted in [22, 43], $\mathscr {I}=\mathbb {R}\times S^1 \times S^1$ by introducing spherical polar coordinates on each $S^2$-factor $(\theta _i,\phi _i)\in [0,\pi ]\times [0,2\pi )$, with $i=1$ for the ${{\textbf{x}}}$-factor and $i=2$ for the ${{\textbf{y}}}$-factor. On $\mathscr {I}$ the azimuthal coordinates are equal, $\theta _1=\theta _2$ and $(\phi _1,\phi _2)$ parametrize the tori cross-sections. To relate these coordinates to split signature analogues of Lorentzian Bondi $\mathscr {I}$ coordinates we first introduce null coordinates on the torus $(z,{\tilde{z}})$ as analogues of the complex stereographic coordinates $(z,{\bar{z}})$ on the sphere found in Lorentz signature. Now taking the $\phi _i$ modulo $2\pi$, their range can be covered by taking instead

$$\begin{aligned} (\phi ,{\tilde{\phi }}):=\left( \phi _1-\phi _2, \phi _1+\phi _2\right) \in [0,2\pi )\times [0,4\pi ) \end{aligned}$$

(2.8)

or taken modulo $2\pi$ and $4\pi$ respectively. This asymmetric choice allows us to express the antipodal map as $(\theta ,\phi ,{\tilde{\phi }}) \rightarrow (\pi -\theta ,\phi , {\tilde{\phi }}+2\pi )$.

The coordinates $(\phi ,{\tilde{\phi }})$ are null and we shall consider them to be coordinates on $\mathbb{R}\mathbb{P}^1$ so as to expose the full Lorentz symmetry of the finite space on homogeneous coordinates $(\lambda _\alpha ,{\tilde{\lambda }}_{{\dot{\alpha }}})$ with Lorentz indices as above. These can be related to the torus coordinates by

$$\begin{aligned} (\lambda _0,\lambda _1)=\left( \cos \frac{\phi }{2},\sin \frac{\phi }{2}\right) \,, \qquad ({\tilde{\lambda }}_0,{\tilde{\lambda }}_1)=\left( \cos \frac{{\tilde{\phi }}}{2},\sin \frac{{\tilde{\phi }}}{2}\right) \,. \end{aligned}$$

(2.9)

We introduce a Bondi-like u coordinates by first defining $u=\cot \theta _1 \in (-\infty ,\infty )$. We can then regard the full set of coordinates $(u,\lambda _\alpha ,{\tilde{\lambda }}_{{\dot{\alpha }}})$ as homogeneous coordinates being equivalent under the rescalings

$$\begin{aligned} (u,\lambda _\alpha ,{\tilde{\lambda }}_{{\dot{\alpha }}}) \sim (a{\tilde{a}} u,a\lambda _\alpha ,{\tilde{a}}{\tilde{\lambda }}_{{\dot{\alpha }}}) \,, \qquad a\in \mathbb {R}^*, \quad \tilde{a}\in \mathbb {R}^+\,. \end{aligned}$$

(2.10)

That ${\tilde{a}}\in \mathbb {R}^+$ rather than $\mathbb {R}^*$ gives the double cover of $\mathscr {I}$ inside $(S^2\times S^2)/\mathbb {Z}_2$ with the asymmetry reflecting that in (2.8). This now correctly manifests the action of the split signature version $SL(2,\mathbb {R})\times SL(2,\mathbb {R})$ of the complex Lorentz spin group. In these coordinates the action of the $\mathbb {Z}_2$ identification of $\mathscr {I}$ in the conformal compactification $\mathbb {M}^{2,2}$ of $\mathbb {R}^{2,2}$ is realised by ${\tilde{a}}=-1$.

If we wish to work with local coordinates we will often simply use the real null coords

$$\begin{aligned} (\textrm{u}, \lambda ,\tilde{\lambda })=\left( \frac{u}{\lambda _0{\tilde{\lambda }}_0}, \frac{\lambda _1}{\lambda _0},\frac{{\tilde{\lambda }}_1}{\tilde{\lambda }_0}\right) \,. \end{aligned}$$

(2.11)

These are good away from $\lambda _0=0$ and ${\tilde{\lambda }}_0=0$. We will understand an asymptotically flat split signature metric to be one that looks like our flat model to leading order near $\mathscr {I}$. This gives

$$\begin{aligned} ds^2= \frac{1}{R^2}\left( d\textrm{u} dR - d\lambda d{\tilde{\lambda }} + R\varvec{\sigma }d{\tilde{\lambda }}^2 + R{\tilde{\varvec{\sigma }}} d\lambda ^2 +\ldots \right) \,, \end{aligned}$$

(2.12)

where $R=1/r$, and $\mathscr {I}=\{R=0\}$, and $\varvec{\sigma }(u,\lambda ,{\tilde{\lambda }}), \tilde{\varvec{\sigma }}(u,\lambda ,{\tilde{\lambda }})$ are the asymptotic shears. In terms of homogeneous coordinates, we rewrite this as

$$\begin{aligned}{} & {} ds^2= \frac{1}{R^2}\left( d u dR - D\lambda D{\tilde{\lambda }} + R\varvec{\sigma }D{\tilde{\lambda }}^2 + R{\tilde{\varvec{\sigma }}} D\lambda ^2 +\ldots \right) , \quad D\lambda :=\langle \lambda \, d\lambda \rangle ,\;\nonumber \\{} & {} \quad D{\tilde{\lambda }}=[{\tilde{\lambda }} d{\tilde{\lambda }}]\,. \end{aligned}$$

(2.13)

Counting weights under (2.10) so that u has weight (1, 1), we have that R has weight (1, 1), $D\lambda$ weight (2, 0), $\varvec{\sigma }$ weight $(1,-3)$ and ${\tilde{\varvec{\sigma }}}$ weight $(-3,1)$.

2.2 Real and complex twistors in split signature and relations to $\mathscr {I}$

The real twistor space $\mathbb{P}\mathbb{T}_\mathbb {R}$ is defined to be the 3-parameter space of $\beta$-planes which are totally null ASD $S^2$s given by²

$$\begin{aligned} {{\textbf{x}}}= A{{\textbf{y}}}\,, \qquad A\in SO(3)=\mathbb{R}\mathbb{P}^3:=\mathbb{P}\mathbb{T}_\mathbb {R}\,. \end{aligned}$$

(2.14)

To see that SO(3) is the same as real projective 3-space, $\mathbb{R}\mathbb{P}^3$, note that $SO(3)=SU(2)/\mathbb {Z}_2$ and that SU(2), unit determinant $2\times 2$ matrices is $S^3$, and $\mathbb{R}\mathbb{P}^3=S^3/\mathbb {Z}^2$.

This relates to the usual homogeneous coordinate parametrization of $\mathbb{P}\mathbb{T}_\mathbb {R}$ via a pair of real two-component spinors $(\lambda _\alpha ,\mu ^{{\dot{\alpha }}})$ related to affine coordinates $x^{\alpha {\dot{\alpha }}}$ on $\mathbb {R}^{2+2}$ via

$$\begin{aligned} \mu ^{{\dot{\alpha }}}=x^{\alpha {\dot{\alpha }}}\lambda _\alpha \,, \end{aligned}$$

(2.15)

and $\mathbb{P}\mathbb{T}_\mathbb {R}= \mathbb{R}\mathbb{P}^3$ follows by taking $(\lambda _\alpha ,\mu ^{{\dot{\alpha }}})$ to be homogeneous coordinates; this can be seen to be equivalent to (2.14) after some manipulation.

In split signature, wave functions for massless free fields have natural representations as free data on the light-cone in momentum space, characteristic data on $\mathscr {I}$ and homogeneous functions on real twistor space, $\mathbb{R}\mathbb{P}^3$. The data for the three cases are given as follows.

On momentum space we use coordinates $k_{\alpha {\dot{\alpha }}}$ which on the support of $k^2=0$ can be expressed in terms of 2-component spinors as $k_{\alpha {\dot{\alpha }}}=\kappa _\alpha \tilde{\kappa }_{{\dot{\alpha }}}$ and momentum space data for helicity $-n/2$ field is given by a function ${\hat{\phi }}(\kappa ,{\tilde{\kappa }})$ satisfying

$$\begin{aligned} {\hat{\phi }}(a\kappa ,a^{-1}{\tilde{\kappa }})=a^{-n}{\hat{\phi }}(\kappa ,\tilde{\kappa })\,. \end{aligned}$$

(2.16)

In split signature $\mathscr {I}=\mathbb {R}\times S^1\times S^1$ and we use homogeneous coordinates

$$\begin{aligned} (u,\lambda _\alpha ,{\tilde{\lambda }}_{{\dot{\alpha }}}) \sim (a{\tilde{a}} u,a\lambda _\alpha ,{\tilde{a}}{\tilde{\lambda }}_{{\dot{\alpha }}}) \,, \qquad a\in \mathbb {R}^*,\quad \tilde{a}\in \mathbb {R}^+, \end{aligned}$$

(2.17)

related to standard Bondi $\mathscr {I}$ coordinates by

$$\begin{aligned} (\textrm{u},z,{\tilde{z}})=(u/\lambda _0\tilde{\lambda }_{\dot{0}}, \lambda _1/\lambda _0, {\tilde{\lambda }}_{\dot{1}}/{\tilde{\lambda }}_{\dot{0}})\,. \end{aligned}$$

(2.18)

The radiation field for a massless field of helicity $-n/2$ is a homogeneous function $\phi ^0(u,\lambda ,{\tilde{\lambda }})$ of weight

$$\begin{aligned} \phi ^0(a{\tilde{a}}u,a\lambda ,{\tilde{a}} {\tilde{\lambda }})=a^{-n-1}{\tilde{a}} ^{-1}\phi ^0(u,\lambda ,{\tilde{\lambda }})\,. \end{aligned}$$

(2.19)

On twistor space $\mathbb{P}\mathbb{T}_\mathbb {R}=\mathbb{R}\mathbb{P}^3$, we use real homogeneous coordinates $Z\in \mathbb {R}^4$ that can be expressed as a pair of real two-component spinors $Z=(\lambda _\alpha ,\mu ^{{\dot{\alpha }}})$. The incidence relation with space-time is given by

$$\begin{aligned} \mu ^{{\dot{\alpha }}}=x^{\alpha {\dot{\alpha }}} \lambda _\alpha \,, \end{aligned}$$

(2.20)

where $x^{\alpha {\dot{\alpha }}}$ too is taken to be real. Helicity $-n/2$ fields then correspond to weight $-n-2$ functions

$$\begin{aligned} f(aZ)=a^{-n-2}f(Z)\,. \end{aligned}$$

(2.21)

We review the various twistor integral formulae in split signature and their relationship with Fourier representations and Kirchoff-d’Adhemar integral formulae in “Appendix A”.

3 Twistors for split signature SD vacuum metrics and $Lw_{1+\infty }$

In this section we review the twistor description of [39] of global conformally self-dual split signature metrics. These, if global and nontrivial, must live on $S^2\times S^2$. We then specialize that construction to the vacuum Einstein case, introducing a Poison structure and 1-form that lead to a description in terms of generating functions. In the last subsection we explain how $Lw_{1+\infty }$ acts both as the natural diffeomorphism symmetry of the twistor space, whereas its complexification acts transitively on the space of self-dual metrics giving it the structure of a homogeneous space.

3.1 Conformally self-dual Zollfrei metrics

In this subsection we focus on the condition that the Weyl tensor is self-dual. This implies that $\beta$-planes survive as $\beta$-surfaces [4, 48], although in general the self-dual weyl curvature will obstruct the existence of $\alpha$-surfaces. Following [39], we find that such $\beta$-surfaces are projectively flat, and if compact, they are necessarily $S^2$ or $\mathbb{R}\mathbb{P}^2$. In this case, null geodesics are projectively $\mathbb{R}\mathbb{P}^1$s or its double cover. Following Guillemin [49], we define

Definition 1

A space $(M^d,g)$, with g not positive definite, is Zollfrei if all null geodesics are embedded $S^1$s.

It is proved in [39] that

Theorem 1

(LeBrun & Mason) Let $(M^4,[g])$ be Zollfrei with SD Weyl tensor. Then either

• $M=S^2\times S^2/\mathbb {Z}_2$ with the standard conformally flat conformal structure, or

• $M=S^2\times S^2$ and there is a 1 : 1-correspondence between

• 1. SD conformal structures on $S^2\times S^2$ near flat model and

• 2. Deformations $\mathbb{P}\mathbb{T}_\mathbb {R}$ of the standard embedding $\mathbb{R}\mathbb{P}^3\subset \mathbb{C}\mathbb{P}^3$ modulo reparametrizations of $\mathbb{R}\mathbb{P}^3$ and $PGL(4,{\mathbb {C}})$ on $\mathbb{C}\mathbb{P}^3$.

The deformed embedding of $\mathbb{R}\mathbb{P}^3$ is space of $\beta$ planes $\mathbb{P}\mathbb{T}_\mathbb {R}$ and $\mathbb{C}\mathbb{P}^3$ is complex twistor space.

The data of the space-time conformal structure is therefore encoded into the location of $\mathbb{P}\mathbb{T}_\mathbb {R}$ inside $\mathbb{C}\mathbb{P}^3$. The flat $\mathbb{R}\mathbb{P}^3 \subset \mathbb{C}\mathbb{P}^3$ is contained in some tubular neighbourhood $U=\mathbb{R}\mathbb{P}^3\times \mathbb {R}^3$ where the tangent space to the $\mathbb {R}^3$ factor is $i\times T\mathbb{R}\mathbb{P}^3$. In U, $\mathbb{P}\mathbb{T}_\mathbb {R}$ can be expressed as the graph of a map from $\mathbb{R}\mathbb{P}^3$ to $\mathbb {R}^3$. Schematically:

A condensed summary of the proof in [39] follows; although we do not need the first two paragraphs below in the following, we will later use the reconstruction of space-time that arises from studying holomorphic discs in $\mathbb{C}\mathbb{P}^3$ with boundary in $\mathbb{P}\mathbb{T}_\mathbb {R}$.

The real twistor space $\mathbb{P}\mathbb{T}_\mathbb {R}$ is defined to be the space of real $\beta$-surfaces in M. It can be constructed as a quotient of the projective ASD spin bundle $\mathbb{P}\mathbb{S}\rightarrow M$, whose fibres are $\mathbb{R}\mathbb{P}^1$, the projectivisation of the real spin bundle $\mathbb {S}$. This is because $\mathbb{P}\mathbb{S}$ is foliated by the horizontal lifts of $\alpha$ surfaces in M as $\mathbb{P}\mathbb{S}$ can be identified with the projectivised bundle of simple ASD 2-forms inside $\Omega ^{2-}$; these determine the ASD totally null 2-plane elements tangent to the $\beta$-planes.

To embed $\mathbb{P}\mathbb{T}_\mathbb {R}$ inside a complex twistor space, we first complexify this projective spin bundle $\mathbb{C}\mathbb{P}\mathbb {S}\rightarrow M$. This gives the $\mathbb{C}\mathbb{P}^1$-bundle over M obtained by complexifying the $\mathbb{R}\mathbb{P}^1$ fibres of $\mathbb {P}$. Then on $S^2\times S^2$, $\mathbb{P}\mathbb{S}$ cuts $\mathbb{C}\mathbb{P}\mathbb {S}$ into two halves $\mathbb{C}\mathbb{P}\mathbb {S}^\pm$ as each $\mathbb{R}\mathbb{P}^1$ divides the $\mathbb{C}\mathbb{P}^1$ into two halves and $S^2\times S^2$ is simply connected so the two halves cannot swap around.³ Taking $\mathbb{C}\mathbb{P}\mathbb {S}^+$, we discover the that interior is already endowed with a complex structure using that of the $\mathbb{C}\mathbb{P}^1$ fibres vertically, and complex ASD 2-forms horizontally. We now blow down the boundary real slice $\mathbb{P}\mathbb{S}$, taking the quotient by its foliation by $\beta$-planes. As part of that, recall that these are necessarily projectively flat $S^2$s in the $S^2\times S^2$ case. Each such flat projective $S^2$ has a $\mathbb {Z}_2$ antipodal map projecting to $\mathbb{R}\mathbb{P}^2$. Quotienting first by this ends up closing the manifold into a compact manifold without boundary. Now blowing this $\mathbb{P}\mathbb{S}/\mathbb {Z}_2$ down gives a compact topological $\mathbb{C}\mathbb{P}^3$ with a complex structure which must, by rigidity of the complex structure on $\mathbb{C}\mathbb{P}^3$, and arguments concerning sufficient smoothness for the Newlander-Nirenberg theorem, be the standard one. We have also on the way constructed a deformed $\mathbb{P}\mathbb{T}_\mathbb {R}$ as the blow down of $\mathbb{P}\mathbb{S}$. Thus we are left with the deformation of the real slice $\mathbb{P}\mathbb{T}_\mathbb {R}$ lying naturally inside the $\mathbb{C}\mathbb{P}^3$ (Fig. 1).

Fig. 1

$\mathbb{P}\mathbb{T}_\mathbb {R}$ as graph of $F:\mathbb{R}\mathbb{P}^3\rightarrow \mathbb {R}^3$ inside a tubular neighbourhood $U \supset \mathbb{R}\mathbb{P}^3$ in $\mathbb{C}\mathbb{P}^3$. D represents a holomorphic disc inside $\mathbb{C}\mathbb{P}^3$ with $\partial D\subset \mathbb{P}\mathbb{T}_\mathbb {R}$

For the reconstruction of M from twistor space $\mathbb{P}\mathbb{T}_\mathbb {R}$, we see from the construction that each $x\in M \leftrightarrow$ holomorphic disc $D_x\subset \mathbb{C}\mathbb{P}^3$ with $\partial D_x \subset \mathbb{P}\mathbb{T}_\mathbb {R}$ which is the projection of the fibre of $\mathbb{C}\mathbb{P}\mathbb {S}^+$ over x. It can be checked that:

• $D_x$ generates the degree-1 class in $H_2(\mathbb{C}\mathbb{P}^3, \mathbb{P}\mathbb{T}_\mathbb {R},\mathbb {Z})=\mathbb {Z}$.

• The moduli space of all such discs:

  $$\begin{aligned} M= \{\text{ Moduli } \text{ of } \text{ degree-1 } \text{ hol. } \text{ disks: } D_x\subset \mathbb{C}\mathbb{P}^3, \partial D_x \subset \mathbb{P}\mathbb{T}_\mathbb {R}\} \end{aligned}$$

  is four-dimensional and compact of topology $S^2\times S^2$.

• M admits a SD conformal structure for which if the boundaries of two discs meet at a point $Z\in \mathbb{P}\mathbb{T}_\mathbb {R}$, $\partial D_x\cap \partial D_{x'}=Z$, then $x,x'$ sit on same $\beta$-plane:

This completes our summary of [39], to which the reader is referred for full details (Fig. 2).

Fig. 2

Null separation between points $x,x'\in M$ corresponds to intersection between the corresponding deformed lines $X, X'$ where $X=\partial D_X$ in $\mathbb{P}\mathbb{T}_\mathbb {R}$. This gives a deformation of the Klein correspondence

3.2 Generating functions for vacuum Einstein with $\Lambda =0$

We now adapt the correspondence to vacuum Einstein metrics with vanishing cosmological constant. We will consider self-dual conformal structure on $S^2\times S^2$ that contain a complete Einstein $g\in [g]$ with $\Lambda =0$ on the complement of the null hypersurface $\mathscr {I}$, the conformal infinity. We will see that, at least for small data, these can be constructed globally from the same homogeneity degree two function h(U) on $\mathbb{R}\mathbb{P}^3$ as used in linear theory via the X-ray transform (A.9), but now used in a fully nonlinear correspondence.

To characterize the corresponding $\mathbb{P}\mathbb{T}_\mathbb {R}\subset \mathbb{C}\mathbb{P}^3$, we first break conformal invariance by introducing the infinity twistors $I_{AB}$ and $I^{AB}$. Let $Z^A$, $A=1,\ldots ,4$ be homogenous coordinates for $\mathbb{C}\mathbb{P}^3$ that are real on $\mathbb{R}\mathbb{P}^3\subset \mathbb{C}\mathbb{P}^3$. We introduce a real totally skew $\varepsilon ^{ABCD}$. Then for vanishing cosmological constant we introduce skew twistors, infinity twistors,

$$\begin{aligned} I_{AB}=I_{[AB]},\quad I^{AB}=\frac{\scriptstyle 1}{\scriptstyle 2}\varepsilon ^{ABCD}I_{CD}, \quad \text{ with } \quad I^{AB}I_{AC}=0\,. \end{aligned}$$

(3.1)

These define a weighted 1-form and Poisson structure on $\mathbb{C}\mathbb{P}^3$ by the equations

$$\begin{aligned} \theta =I_{AB}Z^AdZ^B\in \Omega ^1(2)\,, \qquad \{ f, g\}:= I^{AB}\frac{\partial f}{\partial Z^A}\frac{\partial g}{\partial Z^B} \end{aligned}$$

(3.2)

of homogeneity degree 2 and $-2$ respectively. We can be more explicit by decomposing twistors into pairs of spinors in the standard way. Let

$$\begin{aligned} Z^A=(\lambda _\alpha ,\mu ^{{\dot{\alpha }}}), \qquad \alpha ,\beta =0,1, \quad {\dot{\alpha }},{\dot{\beta }}=\dot{0},\dot{1}\,, \end{aligned}$$

(3.3)

and introduce $\varepsilon _{\alpha \beta }=\varepsilon _{[\alpha \beta ]}$ and $\varepsilon _{{\dot{\alpha }}{\dot{\beta }}}=\varepsilon _{[\dot{\alpha }{\dot{\beta }}]}$ so that in these coordinates

$$\begin{aligned} \theta =\lambda _\alpha d\lambda _\beta \varepsilon ^{\alpha \beta } \,, \qquad \{f,g\}= \varepsilon ^{ {\dot{\alpha }}{\dot{\beta }}}\frac{\partial f}{\partial \mu ^{{\dot{\alpha }}}}\frac{\partial g}{\partial \mu ^{{\dot{\beta }}}}\,. \end{aligned}$$

(3.4)

This Poisson bracket is the one that underpins $Lw_{1+\infty }$ structure as described in [18]. It determines a projection $p:\mathbb{P}\mathbb{T}-\{\lambda _\alpha =0\}\rightarrow \mathbb{C}\mathbb{P}^1$ given in homogeneous coordinates by $Z^A\rightarrow \lambda _\alpha$. On the fibres of this projection, the Poisson structure is invertible with inverse represented by

$$\begin{aligned} \omega _\lambda :=\varepsilon _{{\dot{\alpha }}{\dot{\beta }}} d\mu ^{{\dot{\alpha }}}\wedge d\mu ^{{\dot{\beta }}} \end{aligned}$$

(3.5)

at fixed $\lambda$. The compatibility of these with the real structure guarantees that our space-time is conformal to Einstein vacuum:

Theorem 2

A vacuum $g\in [g]$ exists iff $\lambda _\alpha$ and $\omega _\lambda$ can both be chosen to be real on $\mathbb{P}\mathbb{T}_\mathbb {R}$.

These are sections of $\mathcal {O}(n)$ for $n=1, 2$; the real slice of these line bundles is defined via the identification $\mathcal {O}(-4)_{\mathbb {R}}=\Omega ^3_{\mathbb{P}\mathbb{T}_\mathbb {R}}$. These are analogous to Penrose’s [4]. We will indicate the proof later when we turn to the construction of space-time and its metric.

It is easy to implement these conditions in generality with generating functions. To set them up, decompose the complex homogeneous coordinates into real and imaginary parts, i.e.,

$$\begin{aligned} Z^A=U^A+iV^A\,, \qquad \text{ and } \qquad \mu ^{{\dot{\alpha }}}=u^{{\dot{\alpha }}}+iv^{{\dot{\alpha }}}. \end{aligned}$$

(3.6)

Let h(U) be an arbitrary function of homogeneity degree-2,

$$\begin{aligned} U\cdot \frac{\partial h}{\partial U }=2h. \end{aligned}$$

(3.7)

Proposition 1

The Einstein conditions of Theorem 2 are satisfied by

$$\begin{aligned} \mathbb {T}_\mathbb {R}= \left\{ \lambda _\alpha ={\bar{\lambda }}_\alpha \,, v^{{\dot{\alpha }}}= \varepsilon ^{{\dot{\alpha }}{\dot{\beta }}} \frac{\partial h}{\partial u^{{\dot{\beta }}}}\right\} \end{aligned}$$

(3.8)

so that projectivising gives $\mathbb{P}\mathbb{T}_\mathbb {R}$. All small such deformations of $\mathbb{P}\mathbb{T}_\mathbb {R}$ arise in this way and correspond to all small finite vacuum perturbations of the metric near $\mathscr {I}$ that extend conformally over $S^2\times S^2$.

With this, it is easy to see that for f, g real on $\mathbb{P}\mathbb{T}_\mathbb {R}$, then $\{f,g\}$ is real too on $\mathbb{P}\mathbb{T}_\mathbb {R}$. To see this, its helpful to work on the fibre $\lambda _\alpha =$ constant, on which the Poisson structure is invertible, and, whose imaginary part is

$$\begin{aligned} \Im ( \omega _\lambda )=\varepsilon _{{\dot{\alpha }}{\dot{\beta }}} d u^{{\dot{\alpha }}}\wedge d v^{{\dot{\beta }}} =\left( \frac{\partial v^{{\dot{\gamma }}}}{\partial u^{{\dot{\gamma }}}}\right) \varepsilon _{{\dot{\alpha }}{\dot{\beta }}} du^{{\dot{\alpha }}}\wedge d u^{{\dot{\beta }}}\,. \end{aligned}$$

(3.9)

It is now easy to see that ${\partial v^{{\dot{\gamma }}}}/{\partial u^{{\dot{\gamma }}}}=0$ is the integrability condition for $v^{{\dot{\alpha }}}= \varepsilon ^{{\dot{\alpha }}{\dot{\beta }}} {\partial h}/{\partial u^{{\dot{\beta }}}}$.

3.2.1 Construction of holomorphic discs

As before, the split signature space-time is reconstructed as the moduli space of holomorphic discs of degree-1 with boundary on $\mathbb{P}\mathbb{T}_\mathbb {R}$. However, these can now be constructed as sections of the fibration $p:\mathbb{C}\mathbb{P}^3-\mathbb{C}\mathbb{P}^1\rightarrow \mathbb{C}\mathbb{P}^1$. These holomorphic discs can be parametrized in homogeneous coordinates $\lambda _\alpha$ as the sets

$$\begin{aligned} \mathbb {D}=\{ \lambda _\alpha | \langle \lambda \, {\bar{\lambda }}\rangle :=\lambda _\alpha {\bar{\lambda }}^\alpha \ge 0\}/{\mathbb {C}}^*\,. \end{aligned}$$

(3.10)

where we take $\lambda _\alpha$ up to constant rescalings. The sections of p are then defined by $(\lambda ,\mu ^{{\dot{\alpha }}}(\lambda )):\mathbb {D}\rightarrow \mathbb{C}\mathbb{P}^3$ and this defines a holomorphic disc of degree one with boundary on $\mathbb{P}\mathbb{T}_\mathbb {R}$ if $\mu ^{{\dot{\alpha }}}$ has homogeneity degree 1 in $\lambda _\alpha$. The boundary condition on $\partial \mathbb {D}=\{\lambda ={\bar{\lambda }}\}$ becomes

$$\begin{aligned} \mu ^{{\dot{\alpha }}}(\lambda )=u^{{\dot{\alpha }}}(\lambda )+iv^{{\dot{\alpha }}}(\lambda )\,, \qquad v^{{\dot{\alpha }}}(\lambda )= \varepsilon ^{{\dot{\alpha }}{\dot{\beta }}}\frac{\partial h}{\partial u^{{\dot{\beta }}}}(u(\lambda ))\,. \end{aligned}$$

(3.11)

The general results of [39] show that there is a real four-dimensional moduli space $M^4$ of such discs. We can parametrize them with coordinates $x^{\alpha {\dot{\alpha }}}$ by writing

$$\begin{aligned} \mu ^{{\dot{\alpha }}}= x^{\alpha {\dot{\alpha }}} \lambda _\alpha +m^{{\dot{\alpha }}} \end{aligned}$$

(3.12)

where $x^{\alpha {\dot{\alpha }}}$ are real, and $m^{{\dot{\alpha }}}$ vanishes at $\lambda _0=0$ and $\lambda _1=0$. We can then define the map

$$\begin{aligned} p: M^4\times \mathbb {D}\rightarrow \mathbb{P}\mathbb{T}\,,\qquad (x^{\alpha {\dot{\alpha }}},\lambda _\alpha ) \rightarrow (\lambda _\alpha ,\mu ^{{\dot{\alpha }}}(x,\lambda ))\,. \end{aligned}$$

(3.13)

The main step in reconstructing the metric is to observe that the pullback 2-forms $p^*\omega _\lambda$, are real on $\lambda _\alpha ={\bar{\lambda }}_\alpha$ and hence extend by reflection in $\partial \mathbb {D}$ to all complex values of $\lambda _\alpha$ as 2-forms on $M^4$ of homegeneity degree 2 in $\lambda _\alpha$. Hence by an extension of Liouville’s theorem we can write

$$\begin{aligned} p^*\omega _\lambda =\lambda _\alpha \lambda _\beta \Sigma ^{\alpha \beta } \end{aligned}$$

(3.14)

where $\Sigma ^{\alpha \beta }$ are a triple of real 2-forms on $M^4$ that are closed and, because $\omega _\lambda$ has rank two, satisfies

$$\begin{aligned} \Sigma ^{(\alpha \beta }\wedge \Sigma ^{\gamma \delta )}=0\,. \end{aligned}$$

(3.15)

Contracting $\Sigma ^{\alpha \beta }$ with SO(2, 1) Pauli matrices, $\sigma ^i_{\alpha \beta }$, $i=1,2,3$, yields the three symplectic forms $\omega ^i=\sigma ^i_{\alpha \beta }\Sigma ^{\alpha \beta }$ of a pseudo-hyperKahler structure on $M^4$

$$\begin{aligned} d\omega ^i=0, \qquad \omega ^i\wedge \omega ^j=\eta ^{ij}\nu , \qquad \eta ^{ij}=\, \textrm{diag}(1,-1,-1)\,. \end{aligned}$$

(3.16)

3.3 The $Lw_{1+\infty }$ symmetry action on twistor space

Here we explain how the ideas of [18] adapt to this split signature context giving a real description. In particular we will see that the symmetry algebra has a real version $Lw_{1+\infty }$, and a complex version $Lw^{\mathbb {C}}_{1+\infty }$. The real version is the underlying structure-preserving diffeomorphism symmetry of the real twistor space $\mathbb{P}\mathbb{T}_\mathbb {R}$. The complex version shifts the location of the linear real $\mathbb{R}\mathbb{P}^3$ inside $\mathbb{C}\mathbb{P}^3$ taking it to $\mathbb{P}\mathbb{T}_\mathbb {R}$.

Following [50, 51] we define the classical limit $w_{\infty }$ of the $N\rightarrow \infty$ $W_N$ algebra to be the Poisson diffeomorphisms of the plane. To make contact with our notations, let the plane have coordinates $\mu ^{{\dot{\alpha }}}$, ${\dot{\alpha }}=\dot{0},\dot{1}$ as before with Poisson bracket

$$\begin{aligned} \{ f,g\}:= \varepsilon ^{{\dot{\alpha }}{\dot{\beta }}}\frac{\partial f}{\partial \mu ^{{\dot{\alpha }}}}\frac{\partial g}{\partial \mu ^{{\dot{\alpha }}}}\,, \qquad \varepsilon ^{{\dot{\alpha }}{\dot{\beta }}}=\varepsilon ^{[{\dot{\alpha }}{\dot{\beta }}]}, \quad \varepsilon ^{\dot{0}\dot{1}}=1\,. \end{aligned}$$

Then a basis of $w_{1+\infty }$ is given by hamiltonians that are polynomials in these coordinates, i.e.

$$\begin{aligned} w^p_m=(\mu ^{\dot{0}})^{p-m-1}(\mu ^{\dot{1}})^{p+m-1}\,, \qquad |m|\le p-1, \quad 2p-2\in \mathbb {N}\,. \end{aligned}$$

(3.17)

The Poisson brackets then give rise to the commutation relations of $w_{1+\infty }$:

$$\begin{aligned} \{w^p_{m},w^q_{n}\}=(2(p-1)n-2(q-1)m)w^{p+q-2}_{m+n}\,, \end{aligned}$$

with the $1+$ denoting the central constant Hamiltonian.

To obtain the full structure preserving diffeomorphisms of twistor space, we need the loop algebra $Lw_{1+\infty }$, where the loop coordinate is $\frac{\lambda _1}{\lambda _0}=\tan \frac{\theta }{2}$. This loop algebra has generators

$$\begin{aligned} g^p_{m,r}=w^p_m \textrm{e}^{ir\theta }, \qquad r\in \mathbb {Z}. \end{aligned}$$

(3.18)

With Poisson brackets now

$$\begin{aligned} \{g^p_{m,r},g^q_{n,s}\}=\left( 2(p-1)n-2(q-1)m\right) g^{p+q-2}_{m+n,r+s}\,, \end{aligned}$$

(3.19)

giving the standard commutation relations of $Lw_{1+\infty }$.

It is now straightforward to see these are the structure preserving diffeomorphism group of $\mathbb{P}\mathbb{T}_\mathbb {R}$ preserving the structures of the split signature nonlinear graviton theorem 2. Furthermore, the complexification $Lw^{\mathbb {C}}_{1+\infty }$ of $Lw_{1+\infty }$ can be understood to act as the Lie algebra of a semigroup generated by holomorphic diffeomorphisms defined near $\mathbb{R}\mathbb{P}^3$ inside $\mathbb{C}\mathbb{P}^3$. These are generated by complex hamiltonians and serve to shift the location of $\mathbb{P}\mathbb{T}_\mathbb {R}$ inside $\mathbb{C}\mathbb{P}^3$ thus generating nontrivial gravitational data (the real diffeomophism are of course just coordinate transformations). Thus the gravitational data can be defined to be these complex transformations defined modulo the real ones realizing the space of self-dual gravitational data as $Lw^{\mathbb {C}}_{1+\infty }/Lw_{1+\infty }$. Our generating functions h(U) provide natural representatives of these cosets via a choice of pure imaginary slice.

4 From twistor data to asymptotic shears on $\mathscr {I}$ and back

We briefly sketch the construction of the generating function h(U) from the gravitational data, the asymptotic shear $\varvec{\sigma }$, at $\mathscr {I}$, firstly for the fully nonlinear regime and then for the linearized transform. There is a nontrivial aspect to the latter as the linear correspondence (A.10) gives shears $\varvec{\sigma }$ that are even under the $\mathbb {Z}_2$ but we address this second as it provides further insight into the nonlinear discussion.

4.1 The nonlinear case

To construct h(U) from the $\mathscr {I}$-data, in the nonlinear regime, we can apply a lower dimensional analogue of Theorem 1 given by Lebrun and the author in [38]. In this theorem, Zoll projective structures are considered on $\mathbb{R}\mathbb{P}^2$ and $S^2$. A projective structure is an equivalence class of torsion-free connections with two connections equivalent if they have the same geodesics. The Zoll condition, like the Zollfrei condition is that all of their geodesics are embedded circles. Those on $\mathbb{R}\mathbb{P}^2$ are rigid, whereas for $S^2$ we have

Theorem 3

(Lebrun & Mason) There is a 1 : 1 correspondence between Zoll projective structures on $S^2$ close to the standard one, and deformations $\mathbb{P}\mathbb{T}_\mathbb {R}^\lambda$ of the embedding of the standard real slice $\mathbb{R}\mathbb{P}^2$ of $\mathbb{C}\mathbb{P}^2$.

In our context, the real twistors in our (2, 2) space-time that correspond to points of $\mathbb{P}\mathbb{T}_\mathbb {R}$ intersect $\mathscr {I}$ in null geodesics in the planes $\lambda _\alpha =$ constant. We can apply the above theorem for each such fixed real $\lambda$ to give a deformed real slice $\mathbb{P}\mathbb{T}_\mathbb {R}^\lambda$ inside each $\mathbb{P}\mathbb{T}^\lambda$, the $\mathbb{C}\mathbb{P}^2$ inside $\mathbb{P}\mathbb{T}$ whose tangent plane is determined by the covariantly constant spinor $\lambda$. Recalling that a self-dual Ricci-flat space-time admits covariantly constant spinors, such planes in $\mathbb{P}\mathbb{T}$ will be defined globally in the complex in our context and by globality take the standard flat space form.

In the flat case, the null geodesics are lines given by $u=\mu ^{{\dot{\alpha }}}{\tilde{\lambda }}_{{\dot{\alpha }}}$. however, for $\varvec{\sigma }\ne 0$ the null geodesic equation on $\mathscr {I}$ is nontrivial, defining a projective structure. At fixed $\lambda _\alpha$, the null geodesics are curves $u=Z({\tilde{\lambda }})$ in $\mathscr {I}$ satisfying⁴ [6]

$$\begin{aligned} \partial ^2_{{\tilde{\lambda }}} Z=\varvec{\sigma }(Z,{\tilde{\lambda }},\lambda ). \end{aligned}$$

(4.1)

This is Newman’s good cut equation [3] analytically continued to become a real equation in split signature. Globally, if the finite space-time is assumed to be simply connected, the $\alpha$-planes will necessarily be topologically discs, and will therefore intersect $\mathscr {I}$ in circles. Thus Theorem 3 can be applied at each value of $\lambda$. This gives a deformed $\mathbb{P}\mathbb{T}_\mathbb {R}$ that lies inside the hypersurface $\lambda _\alpha ={\bar{\lambda }}_\alpha$.

To see that the hypersurface satisfies the second condition of Theorem 2, we note that the projective structure defined by (4.1) is special in that it admits a constant Wronskian. For a solution Z to (4.1), perturbations $z_1, z_2$ satisfy

$$\begin{aligned} \partial _{{\tilde{\lambda }}}^2 z_i={\dot{\varvec{\sigma }}}(Z,\lambda ,{\tilde{\lambda }}) z_i. \end{aligned}$$

(4.2)

These represent two tangent vectors to Z now thought of as the corresponding point of $\mathbb{P}\mathbb{T}_\mathbb {R}$. Then we define $\omega _\lambda (z_1,z_2)$ by the Wronskian $z_1\partial _{{\tilde{\lambda }}} z_2-z_2\partial _{{\tilde{\lambda }}}$, which becomes in these homogeneous coordinates

$$\begin{aligned} \omega _\lambda (z_1,z_2):=\varepsilon _{{\dot{\alpha }}{\dot{\beta }}} \frac{\partial z_1}{\partial {\tilde{\lambda }}_{{\dot{\alpha }}}}\frac{\partial z_2}{\partial {\tilde{\lambda }}_{{\dot{\beta }}}}\,. \end{aligned}$$

(4.3)

It is standard that such a Wronskian should be constant along the geodesic and so defines a real two-form on $\mathbb{P}\mathbb{T}_\mathbb {R}$ at fixed $\lambda$.

We must now show that $\omega _\lambda$ extends holomorphically over $\mathbb{P}\mathbb{T}^\lambda$. This will be enough to determine the generating function. In order to do so, we first introduce some of the ingredients of [38].

It follows from the discussion in [38] that, the complement of the real slice in $\mathbb{P}\mathbb{T}^\lambda$ should be identified with complement of the real slice of one half $\mathbb {P}T_{\mathbb {C}}^+\mathscr {I}_\lambda$ of the projective complexified tangent bundle $\mathbb {P}T_{\mathbb {C}}\mathscr {I}_\lambda$ of $\mathscr {I}_\lambda$; the fibres of this are $\mathbb{C}\mathbb{P}^1$s that are cut in half by their real slice. We can introduce two real coordinates $({\tilde{\lambda }}, u)$ and the complex fibre coordinate v representing the complex vector field $\partial _{{\tilde{\lambda }}} + v\partial _u$ so that the geodesic flow associated to (4.1) is

$$\begin{aligned} \mathcal {D}= \partial _{{\tilde{\lambda }}}+v \partial _u + \varvec{\sigma }\partial _v\,. \end{aligned}$$

(4.4)

We can express this on the nonprojective tangent bundle of $\mathscr {I}_\lambda$ by introducing the homogeneous coordinates $\mu ^{{\dot{\alpha }}}=\partial Z/\partial {\tilde{\lambda }}_{{\dot{\alpha }}}$ as above, then $(\mu ^{{\dot{\alpha }}}, {\tilde{\lambda }}_{{\dot{\beta }}})$ define homogeneous coordinates on the projective tangent bundle of 2-plane $\mathscr {I}_\lambda$ at fixed $\lambda$. This follows because the homogeneity relation $Z={\tilde{\lambda }}_{{\dot{\alpha }}}\partial Z/\partial {\tilde{\lambda }}_{{\dot{\alpha }}}=Z$ determines Z, and then the other component of $\mu ^{{\dot{\alpha }}}$, its first derivative with respect to ${\tilde{\lambda }}$; in affine coordinates this gives the coordinate v. The geodesic flow above together with the Euler vector field in ${\tilde{\lambda }}_{{\dot{\alpha }}}$ can now be written in terms of these homogeneous coordinates as

$$\begin{aligned} V^{{\dot{\alpha }}}=\frac{\partial }{\partial {\tilde{\lambda }}_{{\dot{\alpha }}} }-\varvec{\sigma }\tilde{\lambda }^{{\dot{\alpha }}}{\tilde{\lambda }}^{{\dot{\beta }}} \frac{\partial }{\partial \mu ^{{\dot{\beta }}}}\,. \end{aligned}$$

(4.5)

The real twistor space at fixed $\lambda$, $\mathbb{P}\mathbb{T}^\lambda _\mathbb {R}$ is obtained by blowing down the real projective tangent bundle of $\mathscr {I}$, with $v={\bar{v}}$, along the geodesic flow $\mathcal {D}$. This is lifted to the non-projective space by including also the Euler vector field for ${\tilde{\lambda }}_{{\dot{\alpha }}}$ which combines to give $V^{{\dot{\alpha }}}$.

Complexifying the fibre v of the bundle projective tangent bundle, we can define the distribution $T^{0,1}=\{\mathcal {D},\partial /\partial {\bar{v}}\}$. When $v \ne {\bar{v}}$, this defines a complex structure.⁵ We must show that (4.3) extends holomorphically over the complex twistor space at fixed $\lambda$ in the context of theorem 3. Our coordinates $\mu ^{{\dot{\alpha }}}$ above naturally complexify to the projective tangent bundle of $\mathscr {I}_\lambda$ with complexified fibres; it is this space away from the real slice that becomes identified with the $\mathbb{P}\mathbb{T}^\lambda -\mathbb{P}\mathbb{T}^\lambda _\mathbb {R}$, and when the real slice is blown down, this will in turn become the plane, $\mathbb{P}\mathbb{T}^\lambda =\mathbb{C}\mathbb{P}^2$, at a fixed real $\lambda$. On this space the complex structure is determined by the (0, 1) vectors $\partial /\partial {\bar{\mu }}^{{\dot{\alpha }}}$ and $V^{{\dot{\alpha }}}$. The same computation that shows that the Wronskian is constant now shows on this partially complexified space that (4.3) is holomorphic and so extends holomorphically over the whole $\mathbb{C}\mathbb{P}^2_\lambda$ as required. This is sufficient to determine h(U) on each $\mathbb{C}\mathbb{P}^2_\lambda$ for each real $\lambda$ via the argument used above on $\mathbb{C}\mathbb{P}^3$ and hence we have reconstructed h(U).

4.2 Linear case

The issue here is that the generalised Radon or Funk transform (A.10) leads to shears that satisfy the opposite parity in ${\tilde{\lambda }}$ to that appropriate to its weight

$$\begin{aligned} \varvec{\sigma }(u,\lambda ,{\tilde{\lambda }})= \varvec{\sigma }(-u,\lambda _\alpha ,-\tilde{\lambda }_{{\dot{\alpha }}})\,. \end{aligned}$$

(4.6)

This follows from the fact that a sign flip for $\tilde{\lambda }_{{\dot{\alpha }}}$ flips the sign of u as $u=\mu ^{{\dot{\alpha }}}{\tilde{\lambda }}_{{\dot{\alpha }}}$ but can be absorbed into a sign flip in the parameter t in the integral as the sign flip in dt is compensated by the flip in orientation of the t line.⁶ One might therefore be concerned that the characteristic data is therefore special. However, at least for the case of the ultrahyperbolic wave equation on $\mathbb {R}^4$, the X-ray transform range is provided to be surjective under reasonable conditions in [35]. The equation (4.7) is precisely a consequence on the restriction of the X-ray transform to $\mathscr {I}$ where it degenerates into the Radon transform (A.5). Similarly the half-Fourier transform between momentum space and twistor space also gives surjectivity and that leads to (A.5) and hence (4.7). Geometrically, this condition is necessary and sufficient to guarantee that the null geodesics at $\mathscr {I}$ formed by intersections with $\beta$-planes are circles to first order around the projectively flat case [38].

5 From open sigma models to scattering amplitudes

Following the twistor strings of Witten [12], Berkovits [54] and Skinner [14], curves of higher degree can be expected to give rise to scattering amplitudes of higher MHV degrees. The reconstruction of space-time as a moduli space of holomorphic discs with boundaries on $\mathbb{P}\mathbb{T}_\mathbb {R}$ of degree-one therefore motivates the study of more general holomorphic discs of higher degree for the construction of amplitudes. Although we could consider the fully quantized twistor-string with target supertwistor space, we instead adapt the chiral sigma model introduced in [10] to introduce an open version with boundary conditions provided by the real slice $\mathbb{P}\mathbb{T}_\mathbb {R}$. Unlike the twistor-string (or ambitwistor-string), this is non-supersymmetric, is treated classically but does not involve dual twistors, living entirely in twistor space. It is more directly related to the above real geometry and allows for a simpler formulation of amplitudes on a fully nonlinear self-dual background as in [11].

5.1 The open twistor sigma model

In order to allow higher MHV-degree, we will need to insert k ASD twistor functions $\tilde{h}_i$ of homogeneity degree $-6$ defined in the first instance on⁷$\mathbb{P}\mathbb{T}_\mathbb {R}$. These define the ASD perturbations and are inserted at k points $Z_i\in \mathbb{P}\mathbb{T}_\mathbb {R}$, $i=1,\ldots ,k$. Our holomorphic discs $Z(\sigma ): \mathbb {D}\rightarrow \mathbb {T}$ must not only have their boundaries on $\mathbb{P}\mathbb{T}_\mathbb {R}$, but must also pass through the points $Z_i$. We can impose this latter condition by setting

$$\begin{aligned} Z(\sigma )=\sum _{i=1}^k \frac{Z_i}{\sigma - \sigma _i} +M(\sigma ) \end{aligned}$$

(5.1)

where $M(\sigma )$ is smooth. To make $Z(\sigma )$ regular, one can consider $Z(\sigma )\prod _i (\sigma - \sigma _i)$ which is smooth, of degree $k-1$ and passes through the $Z_i$. It is also manifestly real when $\sigma$ is real if $M(\sigma )=0$ on the boundary thus satisfying the boundary conditions when $h=0$. For $h\ne 0$ we must have when $\sigma ={\bar{\sigma }}$

$$\begin{aligned} M(\sigma )=(0,m^{{\dot{\alpha }}})\,, \qquad \Im \, m^{{\dot{\alpha }}}|_{\sigma ={\bar{\sigma }}}=\left. \varepsilon ^{{\dot{\alpha }}{\dot{\beta }}}\frac{\partial h(U(\sigma ))}{\partial u^{{\dot{\beta }}}}\right| _{\sigma ={\bar{\sigma }}}\,. \end{aligned}$$

(5.2)

This will have a unique solution $M(\sigma )=(0,m^{{\dot{\alpha }}}(\sigma ))$ holomorphic and smooth on the boundary as holomorphic functions that are holomorphic and bounded on the upper half plane and real on the real axis are constant, but must vanish at infinity if it not to have a pole there. This boundary value problem arises from action:

$$\begin{aligned} S_\textrm{D}[Z(\sigma ),Z_i,\sigma _i]:= \Im \int _\textrm{D}[ \mu \,{\bar{\partial }}\mu ] \textrm{d}\sigma + \oint _{\partial \textrm{D}} h(\Re Z) \textrm{d}\sigma + \frac{1}{2\pi }\sum _i [\mu (\sigma _i) \mu _i] \end{aligned}$$

(5.3)

It can now be checked that the equations of motion give (5.1) with $M(\sigma )$ holomorphic, together with the boundary condition (5.2).

We will now express scattering amplitudes in terms of the on-shell action

$$\begin{aligned} S^{os}[h, Z_i,\sigma _i]:= S_\textrm{D}[Z(\sigma ),Z_i] \end{aligned}$$

(5.4)

where the right-hand side is evaluated on the unique solution to (5.1) with $M(\sigma )$ holomorphic satisfying the boundary condition (5.2) and so the left-hand side is thought of as a functional of h together with the boundary data $Z_i,\sigma _i$. In this form we can construct amplitudes for the scattering of k negative helicity (anti-self-dual) gravitons with twistor wave-functions

$$\begin{aligned} {\tilde{h}}_i\in C^\infty (\mathbb{P}\mathbb{T}_\mathbb {R},\mathcal {O}(-6)), \qquad i=1,\ldots , k \end{aligned}$$

(5.5)

on the self-dual background determined by nonlinear self-dual data $h\in C^\infty (\mathbb{P}\mathbb{T}_\mathbb {R}, \mathcal {O}(2))$ as described above. An amplitude in the first instance will be a functional $\mathcal {M}[h, \tilde{h}_i]$ of this gravitational data. The main claim is

Proposition 2

(Adapted to split signature from [10]) The amplitude for k ASD perturbations on SD background h is

$$\begin{aligned} \mathcal {M}(h, {\tilde{h}}_i)= \int _{(S^1\times \mathbb{P}\mathbb{T}_\mathbb {R})^k} S^{os}[h, Z_i, \sigma _i] \det {}' {\widetilde{\mathbb {H}}} \prod _{i=1}^k {\tilde{h}}_i(Z_i) D^3 Z_i d\sigma _i\,. \end{aligned}$$

(5.6)

Here $S^{os}[h, Z_i, \sigma _i]$ is the sigma model action evaluated on-shell, and ${\widetilde{\mathbb {H}}}$ is the $k\times k$ matrix

$$\begin{aligned} \widetilde{\mathbb {H}}_{ij}(Z_i)={\left\{ \begin{array}{ll} \quad \frac{\langle \lambda _i \lambda _j\rangle }{\sigma _i-\sigma _j}\qquad \qquad i\ne j\\ -\sum _l \frac{\langle \lambda _i \lambda _l\rangle }{\sigma _i-\sigma _l}\,, \qquad i=j\,. \end{array}\right. } \end{aligned}$$

(5.7)

and $\det {}' {\widetilde{\mathbb {H}}}$ is the reduced determinant, i.e., that of any $k-1\times k-1$ minor with appropriate sign and $D^3Z_i:= \varepsilon _{ABCD}Z_i^AdZ_i^B\wedge \ldots \wedge dZ_i^D$.

We first explain why (5.6) is well-defined. Firstly, the Mobius invariance can be understood by expressing the various quantities as sections of the $\mathcal {O}(n)$ line bundles over $\textrm{D}$ whose sections can be represented by functions homogeneous degree n in the homogeneous coordinates $(\sigma _0,\sigma _1)$ on $\textrm{D}$. Then in (5.1) replace $\sigma -\sigma _i$ by $(\sigma \,\sigma _i):=\sigma _0\sigma _{i1}-\sigma _1\sigma _{i0}$ and we see that $Z(\sigma )$ as defined is a section of $\mathbb {T}\otimes \mathcal {O}(-1)$ and $Z_i$ a section of $\mathbb {T}\otimes \mathcal {O}_i(1)$ where $\mathcal {O}_i(n)$ is the line bundle of functions of homogeneity degree n on the $\sigma _i$ disc. The weights $-6$ of the ${\tilde{h}}_i$ are then balanced by the $+4$ of the $D^3Z_i$ and the $+2$ of the $d\sigma _i$ which become $(\sigma _i d\sigma _i)$ in the homogeneous coordinates. We also see that the entries of $\mathbb {H}$ are weightless and by construction it has kernel $(1,1,\ldots ,1)$ so that the reduced determinant is well-defined.

Proof: We will be brief as the argument is essentially an adaptation of that in [10] to our split signature context. The argument follows by expanding $h=h_{k+1}+\ldots +h_n$ to first order in each momentum eigenstate $h_i$ to give a standard well-known formula for the flat background perturbative gravity amplitude [45, 46].

For eigenstates of momentum $k_{i\alpha {\dot{\alpha }}}=\kappa _{i\alpha }{\tilde{\kappa }}_{i{\dot{\alpha }}}$ we have the formulae [55, 56]

$$\begin{aligned} h_i=\int _R \frac{dt_i}{t_i^3}\delta ^2(t_i \lambda _\alpha -\kappa _{i\alpha }) \textrm{e}^{it_i[\mu ,{\tilde{\kappa }}_i]}, \quad {\tilde{h}}_i=\int _\mathbb {R}\frac{dt_i}{t_i^{-5}}\delta ^2(t \lambda _\alpha -\kappa _{i\alpha }) \textrm{e}^{it_i[\mu ,{\tilde{\kappa }}_i]}\,.\nonumber \\ \end{aligned}$$

(5.8)

We can perturbatively expand the on-shell action by setting $h=h_{k+1}+\ldots +h_n$ and expanding to first order in each momentum eigenstate $h_i$. This can be expressed as a tree correlator

$$\begin{aligned} S^{os}_D[ h_{k+1}+ \ldots + h_n, Z_i, \sigma _i]= \langle V_{h_{k+1}} \ldots V_{h_n}\rangle _{tree} + O(h_i^2)\,, \end{aligned}$$

where the ‘vertex operators’ are $V_{h_i}= \int _{\partial D} h_i (\sigma _i) d\sigma _i\,.$ This a sum over connected tree diagrams on the $n-k$ vertices provided by each vertex operator with propagators arising from the kinetic term in $S_\textrm{D}$. This gives the Poisson bracket $\{\,, \}$ leading, on momentum eigenstates, to

$$\begin{aligned} \langle h_i h_j\rangle _{tree}=\frac{\{ h_i\, h_j\}}{\sigma _i-\sigma _j}=\frac{t_it_j[i\, j]}{\sigma _i- \sigma _j}h_ih_j\,, \qquad i\ne j\,. \end{aligned}$$

(5.9)

Summing over all such tree-diagrams, the matrix-tree theorem then gives the result as a reduced determinant

$$\begin{aligned} \langle {h_{k+1}}\ldots {h_n}\rangle _{\textrm{tree}} =\det {}'\mathbb {H}\prod _{i=k+1}^n h_i \,, \end{aligned}$$

(5.10)

where the Laplace matrix associated to the tree graphs is

$$\begin{aligned} \mathbb {H}_{ij}={\left\{ \begin{array}{ll}\frac{t_it_j[ij]}{\sigma _i-\sigma _j}\,, \qquad \qquad i\ne j \\ -\sum _l \frac{t_it_l[il]}{\sigma _i-\sigma _l} \,, \quad i=j \end{array}\right. } \end{aligned}$$

(5.11)

This leads to

$$\begin{aligned} \mathcal {M}(h_i,{\tilde{h}}_i)= \int _{(S^1)^n\times (\mathbb{R}\mathbb{P}^3)^k} \hspace{-1cm} \det {}' \mathbb {H}\det {}' {\tilde{\mathbb {H}}} \prod _{j=k+1}^n h_j d\sigma _j \prod _{i=1}^k {\tilde{h}}_i(Z_i) D^3 Z_i d\sigma _i \,. \end{aligned}$$

(5.12)

This is now equivalent to the Cachazo-Skinner formula [45, 46] adapted to split signature. $\square$

This sum over tree graphs was first observed in gravity amplitudes at MHV in [57, 58] and the reduced determinants in [59] and the comparison using the matrix-tree theorem in [60]. For a more full discussion of the matrix-tree theorem in this context see [10, 11].

5.2 Invariance under $Lw_{1+\infty }$ and vertex operators and Ward identities

The framework, and in particular the amplitudes formulae, gives full expression of the $Lw_{1+\infty }$ symmetries. We saw above that in split signature $Lw_{1+\infty }$ acts on $\mathbb{P}\mathbb{T}_\mathbb {R}$ in two ways:

• 1. $Lw_{1+\infty }$ acts passively as the structure preserving diffeomorphisms (or coordinate transformations) of $\mathbb{P}\mathbb{T}_\mathbb {R}$,

• 2. when multiplied by i, $iLw_{1+\infty }$ shifts the embedding of $\mathbb{P}\mathbb{T}_\mathbb {R}$ inside $\mathbb{C}\mathbb{P}^3$.

These sigma models encodes the Poisson structure that underlies the $Lw^{\mathbb {C}}_{1+\infty }$ symmetry in their tree-level OPE

$$\begin{aligned} g(Z(\sigma ))\cdot g'(Z(\sigma '))\simeq \frac{\{g,g'\}(Z(\sigma ))}{\sigma -\sigma '}+\ldots \,, \end{aligned}$$

(5.13)

for $g,g'$ local holomorphic functions on twistor space.⁸ As far as the bulk sigma model is concerned, the local $Lw_{1+\infty }^{\mathbb {C}}$ symmetries of the action are holomorphic Poisson vector fields with Hamiltonians given by holomorphic functions g(Z) of homogeneity degree 2. Noether’s theorem for our real version of the sigma model gives the charge

$$\begin{aligned} Q_g=\Im \oint gD\sigma \,. \end{aligned}$$

(5.14)

As can be seen, if the contour is taken to the boundary of the disc, it vanishes when g is real there, as expected for a passive transformation, but it becomes nontrivial giving the graviton vertex operator $V_h$ where $h=\Im g$ when g is imaginary.

The correspondence with $\mathscr {I}$ allows us to recognize the transform to twistor space of certain BMS symmetries. In particular, the supertranslations $u \rightarrow u+Z(\lambda ,{\tilde{\lambda }})$ that are linear in ${\tilde{\lambda }}_{{\dot{\alpha }}}$, $Z=Z^{{\dot{\alpha }}}(\lambda ){\tilde{\lambda }}_{{\dot{\alpha }}}$ can be realized with the Hamiltonian $g= \mu ^{{\dot{\alpha }}} Z_{{\dot{\alpha }}}(\lambda )$. Similarly for the super-rotations

$$\begin{aligned} (\lambda _\alpha ,{\tilde{\lambda }}_{{\dot{\alpha }}})\rightarrow (\lambda _\alpha , L(\lambda )_{{\dot{\alpha }}}^{{\dot{\beta }}}\tilde{\lambda }_{{\dot{\beta }}}), \qquad \leftrightarrow \qquad g= L_{{\dot{\alpha }}{\dot{\beta }}}(\lambda ) \mu ^{{\dot{\alpha }}}\mu ^{{\dot{\beta }}}\,. \end{aligned}$$

(5.15)

On the other hand, as noted above, in this signature, imaginary or complex such hamiltonians correspond to perturbations of $\mathbb{P}\mathbb{T}_\mathbb {R}$ that shift its location inside $\mathbb{C}\mathbb{P}^3$, and hence perturbations of the SD metric, i.e. gravitons, rather than passive coordinate transformations. These formulae underpin the connection between leading and subleading soft theorems and Ward identities for symmetries as described in [1] as they are the two terms that arise when the momentum eigenstates (5.8) are expanded in small momenta, see [18] for a more extended discussion.

Higher order soft limits were shown to agree with those of the [24] in [18] and a completely analogous set of computations could be performed here. The main point of correspondence to note is that the g or h used here should be understood as the Čech versions of the twistor data in [18] and the h used here is the result of the light-ray transform discussed by Sharma in [27] interpreted as part of the half-Fourier transform of (A.4).

6 Discussion

We have seen, building on [39], that are SD split signature conformal classes of metrics on $S^2\times S^2$ conformal to SD vacuum metrics on the complement of $\mathscr {I}=S^1\times S^1\times \mathbb {R}\subset S^2\times S^2$ cutting the space into two. They depend on the choice of an arbitrary free homogeneity degree 2 twistor function h(U) on $\mathbb{R}\mathbb{P}^3$. In the linear limit they are in 1:1 correspondence with all asymptotically flat split signature vacuum metrics on $\mathbb {R}^4$ by the surjectivity of the X-ray transform [35]. Geometrically, h(U) defines a deformed real slice $\mathbb{P}\mathbb{T}_\mathbb {R}$ inside $\mathbb{C}\mathbb{P}^3$ via the natural Poisson structure on $\mathbb{C}\mathbb{P}^3$; space-time is reconstructed by considering open holomorphic discs of degree-1 inside $\mathbb{C}\mathbb{P}^3$ whose boundary lies in $\mathbb{P}\mathbb{T}_\mathbb {R}$. These can be described as solutions to a chiral open sigma model with boundary on $\mathbb{P}\mathbb{T}_\mathbb {R}$. At higher degree, the on-shell action can then be used to compute gravity amplitudes at any corresponding MHV-degree.

Similar results follow for $\Lambda \ne 0$ where now the double covered $\mathscr {I}$ has topology $S^2\times S^1$ with a smooth Lorentzian conformal structure. The twistor correspondence extends so that twistor functions h(Z) now determine deformations of the conformal structure of $\mathscr {I}=S^2\times S^1$ that are Zollfrei in the original sense of Guillemin [49], i.e., the null geodesics are all circles. In this case the infinity twistor $I_{AB}$ is non-degenerate of rank four and the contact structure $\theta$ is therefore also non-degenerate $\theta \wedge d\theta \ne 0$. The analogue of the $Lw_{1+\infty }$ is now deformed to become the contact structure-preserving diffeomorphisms for this non-degenerate contact structure, again determined by Hamiltonians h of homogeneity-degree 2. This presumably relates to the deformed Poisson structure of [61]. The sigma-model of [10] was already formulated also at $\Lambda \ne 0$ and its open disc version generalizes similarly to that given here. This construction is being pursued further in joint work with Claude Lebrun.

These computations are all done at the level of the classical, as opposed to quantum, sigma model. This is in contradistinction to the twistor-string or ambitwistor string which generate amplitudes via quantum correlators (and requires supersymmetry). Clearly the twistor-strings of [12, 14, 54] and 4d ambitwistor strings of [16] can be similarly formulated as open string models, as above, even in terms of their gravity vertex operators. unlike conventional string theory. Perhaps more interesting would be to formulate the higher-dimensional ambitwistor models of [15] in this way, and to understand the underpinning semi-classical geometry.

Following [10] we remark that at MHV, the sigma model correlator gives the theory whose Feynman tree-diagrams give precisely the tree formulae of [57, 58]. Furthermore, in that paper, it was shown that the formula at MHV yields the Einstein-Hilbert action of a space-time generated by n perturbations at $\mathscr {I}$. We expect a similar proof to be valid here too.

We now give some more extended discussion of further avenues for exploration.

6.1 A one-loop all-plus generating function

The ingredients introduced in [10] and reformulated above allow us to introduce a generating function for the all-plus 1-loop amplitude as originally presented in [57, 62]. This was presented in terms of certain so-called half-soft factors h(a, A, b) where the letters $a,b \in \{1,\ldots ,n\}$ and $A\subset \{1,\ldots ,n\}$.⁹ The h(a, A, b) can be defined to be the sum over expressions corresponding to connected trees diagrams with vertices corresponding to letters of A, and for whom the propagators between vertices i, j correspond to $\mathbb {H}_{ij}$ with $t_i=\langle i 1\rangle \langle i 2\rangle /\langle 1 2\rangle$, see [60] where it is also explained that the matrix-tree theorem can be used to give the definition

$$\begin{aligned} h(a,A,b)=\frac{1}{\prod _{i\in A} t_i^2}\det {}'\mathbb {H}_A \end{aligned}$$

(6.1)

where the reduced determinant $\det {}'\mathbb {H}_A$ is the determinant of a $|A|-1 \times |A|-1$ minor of $\mathbb {H}_{ij}$ with $i,j,\in A$. The final formula given by [57] is, up to an overall constant

$$\begin{aligned} M(1,\ldots ,n)= \delta ^4\left( \sum _i k_i\right) \sum _{\{a,A,b,B\}=\{1,\ldots , n\}} h(a,A,b)h(b,B,a) [a,A,b,B]^3\,,\nonumber \\ \end{aligned}$$

(6.2)

where $[a,A,b,B]=\, \textrm{tr}(/\!\!\! k_a/\!\!\! k_A/\!\!\! k_b/\!\!\! k_B)$, $k_A=\sum _{i\in A} k_i$ and the sum is over partitions $\{a,A,b,B\}$ of $\{1,\ldots , n\}$.

The main arguments of §5 of [10] include the proof that, at $k=2$ for the tree-level MHV amplitude, the on-shell sigma model action $S^{os}(Z_i,\sigma _i)$, $i=1,2$, can be identified with the first Plebanski scalar $\Omega (x,\lambda _1,\lambda _2)$ adapted to the two $\lambda _\alpha$-spinors of $Z_1$ and $Z_2$; from a Euclidean perspective, if $Z_1, Z_2$ are taken to be complex conjugate twistors, then $\Omega$ is the Kahler scalar in the complex structure determined by their $\lambda$-spinors. The worldsheet Möbius coordinate freedom removes the dependence $S^{os}$ on the $\sigma _i$, $i=1,2$ and so $S^{os}$ depends only on the point x in space-time that corresponds to the line in twistor space joining the two twistors $Z_1,Z_2$. On the other hand, the subsequent part of §5 in [10], as reprised in Sect. 5.1 above, expands the on-shell action, as a sum of tree diagrams giving rise to the formulae given in [57] for the tree-level MHV amplitude as a sum over trees. The sigma model therefore provides the theory that underpins that tree expansion. This is resummed into the reduced determinant formula above by [60]. With these identifications we see that

$$\begin{aligned} h(1,A,2)\delta ^4\left( \sum _{i\in \{1,2,A\}} k_i\right) =\int d^4x \, \textrm{e}^{i(k_1+k_2)\cdot x} \left( \prod _{i\in A} \delta _i\right) \Omega (x, \kappa _1,\kappa _2) \end{aligned}$$

(6.3)

where $\delta _i$ denotes the variation with respect to the momentum eigenstate with momentum $k_i$.

We can now see that (6.2) can be generated as the $n-2$th variation of the local two-point function on the self-dual background determined by h

$$\begin{aligned} M(1,2,h)= \int d^4x \, \left. \textrm{e}^{i(k_1+k_2)\cdot x} \left[ 1,\partial _{x},2,\partial _{x'}\right] ^3\Omega (x)\Omega (x') \right| _{x=x'}\,, \end{aligned}$$

(6.4)

in the sense that the $n-2$th variation of this formula restricted to flat space gives (6.2) above. Both factors of $\Omega$ are the on-shell action of the sigma model with insertions at points $\kappa _1$ and $\kappa _2$, so together the two factors represent a loop, a bubble, on the worldsheet. This is much the same as for the Yang–Mills 1-loop generating function presented in equation 4.17 of [63], which can, as written, be thought of as a 2nd variation associated to a bubble on the auxiliary WZW-model on the sphere.

6.2 Yang–Mills and colour/kinematic duality

The analogous case of global solutions to the self-dual Yang–Mills equations on $S^2\times S^2$ in split signature was treated in [43]. Although there is now the possibility of introducing a nontrivial global holomorphic vector bundle on $\mathbb{C}\mathbb{P}^3$, if we stick to the topological sector containing the trivial connection, the data of a self-dual Yang–Mills field on $S^2\times S^2$ is encoded in a hermitian matrix function H(Z) of unit determinant on $\mathbb{R}\mathbb{P}^3$ (up to some global constant frame change). The space of Hermitian matrices can be understood as the quotient of $SL(n,{\mathbb {C}})/SU(n)$, using the relation $H=GG^*$ analogous to the quotient $Lw_{1+\infty }^{\mathbb {C}}/Lw_{1+\infty }$.

The analogue to the chiral sigma model is to consider again the degree $k-1$ disc $Z(\sigma )$ given in (5.1) but now unperturbed

$$\begin{aligned} Z(\sigma )=\sum _{i=1}^k \frac{Z_i}{\sigma - \sigma _i}\,. \end{aligned}$$

(6.5)

The analogue of finding the deformed holomorphic discs, is now simply to find the holomorphic frame F on the degree-$k-1$ discs above with boundary condition given as

$$\begin{aligned} F:D\rightarrow SL(n,{\mathbb {C}})\,, \qquad {\bar{\partial }}_{{\bar{\sigma }}} F=0, \qquad FF^*|_{\partial D}=H|_{\partial D}\,. \end{aligned}$$

(6.6)

The reconstruction of the connection on space-time is realized using the F in found in the $k=2$ case (i.e., for degree-1). The analogue of proposition 2 is now given by a formula for the amplitude for k ASD gluons on a self-dual Yang–Mills background. If the k ASD gluons are represented by the weight $-4$ functions ${\tilde{f}}_i$ on $\mathbb{R}\mathbb{P}^3$, then the formula, with a choice of cyclic ordering, is

$$\begin{aligned} \mathcal {A}(H, {\tilde{h}}_i)= \int _{(\mathbb{R}\mathbb{P}^3\times \mathbb{R}\mathbb{P}^1)^k} \prod _{i=1}^k F_i{\tilde{f}}_i(Z_i) F^*_i D^3Z_i \frac{d\sigma _i}{(\sigma _i\sigma _{i+1})} \end{aligned}$$

(6.7)

where $F_i=F(Z_i)$. Writing H(Z) as a sum of perturbations $H(Z)=\sum _{i=k+1}^n f_i$ and expanding F(Z) using the standard formula for the variation of a Greens function, here on the disc, we obtain

$$\begin{aligned} G_{ij}:=\frac{ F^*_i(Z(\sigma _i)) F_j(Z(\sigma _j))}{\sigma _i-\sigma _j}\,,\qquad \delta _k G_{ij}=\int _\textrm{D}G_{ik} f_k G_{kj} \textrm{d}\sigma _k\,, \end{aligned}$$

(6.8)

and using this $n-k$ times generates the full Parke-Taylor factor and gives the original RSVW formulae [55]. An action for this problem can be found following the ideas of Nair [64, 65], but now framed in terms of the boundary value problem for F following from an open Wess-Zumino action.

In comparing the Yang–Mills version to the gravitational one, the translation from colour to kinematics can be seen in the replacement of the Parke-Taylor factor by the product of the two reduced determinants $\det {}'\mathbb {H} \det {}' \widetilde{\mathbb {H}}$ (as can also easily be seen by comparing the RSVW Yang–Mills formula to the Cachazo-Skinner gravity formula). The framework of [18] and this paper shows that the $\det {}'\mathbb {H}$ factor is associated with the $Lw_{1+\infty }$ algebra, and therefore $\det {}' \widetilde{\mathbb {H}}$ is associated with the algebra $\widetilde{Lw}_{1+\infty }$ that acts on dual twistor space. The latter trivializes in the MHV sector hence explaining why $Lw_{1+\infty }$ is sufficient in the MHV sector as described in [66]. However, at arbitrary MHV degree, we will need both, but they will not commute or close onto something simple classically as remarked in [18], although see [67] for how they can work together in the ambitwistor string. This is clearly a further avenue for development.

6.3 Continuation to Lorentz signature and encoding the full space-time data

A subtlety of split signature is that the asymptotic shear $\varvec{\sigma }$ cannot be regarded as free data. As seen above, in linear theory it is even under the antipodal map. In general, when fields are nonlinear, we expect that identification to become some nonlinear relation forced by the requirement that the intersection between $\beta$-planes and $\mathscr {I}$ are circles. The split signature $\mathscr {I}$ is the continuation of $\mathscr {I}^+\cup \mathscr {I}^-$ from Lorentz signature where again, we understand the shear on $\mathscr {I}^+$ to be determined by that on $\mathscr {I}^-$ or vice-versa (and they are again identified in linear theory). This analogy between the two frameworks was used in [43] to obtain a split-signature analogue of classical scattering in split signature by dividing $\mathscr {I}$ into two parts interchanged by the antipodal map; the relationship between the asymptotic data and its antipode required the reversal of ordering of the Riemann-Hilbert problem leading to nontrivial scattering even within the self-dual sector. This is however a nonlinear map that is no longer so trivial as to tell us that the data is even or odd under the antipodal map.

In Lorentz signature, we have the two distinct twistor spaces at $\mathscr {I}^\pm$, each encoding the respective asymptotic shears. One can alternatively speculate as to some gluing arising from the generating function that defines the scattering along null geodesics from $\mathscr {I}^-$ to $\mathscr {I}^+$ near $i^0$ in a similar spirit to that first introduced by Penrose in [68]. From this perspective there is an ambiguity between the interpretation of h as a patching between different sets covering one asymptotic twistor space, or between that at $\mathscr {I}^-$ and at $\mathscr {I}^+$; this latter view was that taken also in [16]. From this perspective it is the non-triviality of the antipodal map that in Lorentz signature interchanges $\mathscr {I}^\pm$ that is encoding the patching that generates the $Lw_{1+\infty }$ motion that deforms the twistor space. In particular, the asymptotic shear misses out a key part of the gravitational data associated to the value at $i^0$ of the mass aspect, and perhaps more—these are ‘constants of integration’ in the solution to the characteristic intial value problem from $\mathscr {I}$. The fact that the shear at $\mathscr {I}^-$ and that at $\mathscr {I}^+$ are in some nonlinear relation that should also include the missing $i^0$ mass-aspect data makes the twistor-encoding of scattering incomplete in the nonlinear regime, but one can hope that there is some completion. In ambitwistor space, it should possible to see these all as contributions to the patching.

Appendices

A Review of split signature integral formulae for massless fields

Massless fields of helicity n/2 are given, respectively, by the formulae

$$\begin{aligned} \Phi _{\alpha _1\ldots \alpha _n}(x)&=\int \kappa _{\alpha _1}\ldots \kappa _{\alpha _n} \hat{\phi }(\kappa ,{\tilde{\kappa }}) \textrm{e}^{ik\cdot x} \delta (k^2) d^4k \,, \end{aligned}$$

(A.1)

$$\begin{aligned}&=\int \lambda _{\alpha _1}\ldots \lambda _{\alpha _n} \partial _u\phi ^0( x^{\alpha {\dot{\alpha }}}\lambda _\alpha \tilde{\lambda }_{{\dot{\alpha }}},\lambda ,{\tilde{\lambda }}) \langle \lambda d \lambda \rangle [{\tilde{\lambda }} d{\tilde{\lambda }}]\, , \end{aligned}$$

(A.2)

$$\begin{aligned}&=\int \lambda _{\alpha _1}\ldots \lambda _{\alpha _n} f(\lambda , x^{\alpha {\dot{\alpha }}}\lambda _\alpha ) \langle \lambda d \lambda \rangle \, . \end{aligned}$$

(A.3)

The first of these is the standard Fourier transform in split signature, the second is an adaptation of the D’Adhemar integral formula of, for example, [69], to split signature. The third is the so-called X-ray transform, because it integrates f over lines in $\mathbb{R}\mathbb{P}^3$, first observed by Fritz John as providing solutions to the ultrahyperbolic scalar wave equation, [34].¹⁰ This latter is conformally invariant.

Concatenating these leads to Witten’s [12] half-Fourier transform between momentum space and twistor space

$$\begin{aligned} f(\lambda , \mu )=\frac{1}{\sqrt{2\pi }}\int {\hat{\phi }}(\lambda ,\tilde{\lambda }) \textrm{e}^{i [\mu \, {\tilde{\lambda }}]} d^2{\tilde{\lambda }}\,, \qquad {\hat{\phi }}(\lambda ,{\tilde{\lambda }})= \frac{1}{\sqrt{2\pi }}\int f (\lambda ,\mu ) \textrm{e}^{-i[\mu \, \tilde{\lambda }]} d^2\mu , \nonumber \\ \end{aligned}$$

(A.4)

see for example appendix B of [29] for details. Similarly we can use the conformal invariance of the X-ray transform to give the asymptotic data in terms of f as

$$\begin{aligned} \phi ^0(u, \lambda ,{\tilde{\lambda }})=\int _{-\infty }^\infty f(\lambda _\alpha , \mu ^{{\dot{\alpha }}}+ t \tilde{\lambda }^{{\dot{\alpha }}})\, d t\,, \end{aligned}$$

(A.5)

where $u=[\mu \,{\tilde{\lambda }}]$. This is essentially a Radon transform in the $\lambda _\alpha =$ constant plane in which one integrates f over lines in that plane. This is related to the so-called Funk transform in which one takes a double cover of the projective plane to obtain the 2-sphere, $S^2$ and integrates over great circles (and therefore has kernel consisting of functions that are odd under the antipodal map). The inversion of this is more non-local, a two-dimensional integral, see for example [36, 37] and is related to the light-ray transform [27].

Combining this with the half-Fourier transform, after integrating out t to obtain a delta-function, we obtain what might be known as the third-Fourier transform:

$$\begin{aligned} \phi ^0(u,\lambda ,\tilde{\lambda })=\int {\hat{\phi }}(\lambda , s{\tilde{\lambda }}) \textrm{e}^{isu} ds\,. \end{aligned}$$

(A.6)

We finally note the positive homogeneity versions of equations (A.3)

$$\begin{aligned} \Phi _{{\dot{\alpha }}_1\ldots {\dot{\alpha }}_n}(x)=\int \left. \frac{\partial ^n f}{\partial \mu ^{{\dot{\alpha }}_1}\ldots \partial \mu ^{{\dot{\alpha }}_n}}\right| _{\mu ^{{\dot{\alpha }}}=x^{\alpha {\dot{\alpha }}}\lambda _\alpha } \langle \lambda d \lambda \rangle \,, \end{aligned}$$

(A.7)

and (A.5)

$$\begin{aligned} \phi ^0(u, \lambda ,{\tilde{\lambda }})=\int _{-\infty }^\infty \frac{d^n}{d t^n} f(\lambda _\alpha , \mu ^{{\dot{\alpha }}}+ t \tilde{\lambda }^{{\dot{\alpha }}})\, d t\,, \end{aligned}$$

(A.8)

In the case of linear gravity the we denote the linear Weyl spinor by $\psi _{{\dot{\alpha }}{\dot{\beta }}{\dot{\gamma }}{\dot{\delta }}}$ and the twistor function of homogeneity degree 2 by h(Z) for which the higher spin extension of the X-ray transform is given by

$$\begin{aligned} \psi _{{\dot{\alpha }}{\dot{\beta }}{\dot{\gamma }}{\dot{\delta }}}(x)= \oint _{\mu ^{{\dot{\alpha }}}=x^{\alpha {\dot{\alpha }}}\lambda _\alpha } \frac{\partial ^4 h}{\partial \mu ^{{\dot{\alpha }}}\partial \mu ^{{\dot{\beta }}}\partial \mu ^{{\dot{\gamma }}}\partial \mu ^{{\dot{\delta }}}} D\lambda \,, \qquad D\lambda =\langle \lambda \, d\lambda \rangle \,. \end{aligned}$$

(A.9)

The corresponding characteristic data at $\mathscr {I}$ is usually denoted by $\psi ^0_4$ and we have

$$\begin{aligned} \psi ^0_4=\partial _uN=\partial ^2_u \sigma \end{aligned}$$

where N is the news, and $\sigma$ the asymptotic shear. For the $\mathscr {I}$ quantities we have the integral formulae that follow from

$$\begin{aligned} \varvec{\sigma }=\partial ^2_u\int _{-\infty }^\infty h(\lambda _\alpha , \mu ^{\dot{\alpha }}+ t {\tilde{\lambda }}^{{\dot{\alpha }}})\, d t\,. \end{aligned}$$

(A.10)

This formula is inverted at least in part by the light ray transform in [24, 27].

In the following we will need to insert twistor functions that give rise to momentum eigenstates and these are given uniformly by

$$\begin{aligned} f(Z)=\int \frac{dt}{t^{1+n}} \delta ^2(t\lambda _\alpha -\kappa _\alpha )\textrm{e}^{i t [\mu ,{\tilde{\kappa }}]} \end{aligned}$$

(A.11)

where the parameter t is integrated against the delta function to yield a function of homogeneity degree n in Z.

We now turn to describe the nonlinear analogues of these transforms based on Penrose’s nonlinear graviton construction.

B Examples with symmetry and SD black holes

The Gibbons–Hawking ansatze has an analytic continuation to split signature that can be described as follows. These are solutions with a symmetry that preserves the pseudo-HyperKahler structure. Our symmetry is defined by a nondegenerate 4-vector $T^{\alpha {\dot{\alpha }}}$ of squared length 2. This can be used also to convert all dotted indices to undotted, i.e., $\mu ^\alpha :=T_{{\dot{\alpha }}}^\alpha \mu ^{{\dot{\alpha }}}=u^{\alpha }+iv^{\alpha }$ etc.. The generating function h will be invariant if we take it of the form

$$\begin{aligned} h=h(\mu ^\alpha \lambda _\alpha ,\lambda _\alpha )\,. \end{aligned}$$

The deformed real slice $\mathbb{P}\mathbb{T}_\mathbb {R}$ is then defined by

$$\begin{aligned} v^\alpha = \lambda ^\alpha \dot{h}\,. \end{aligned}$$

In order to find the holomorphic discs with boundary in this $\mathbb{P}\mathbb{T}_\mathbb {R}$, use $\lambda _\alpha$ as homogeneous coordinates on the holomorphic discs so that its components can be taken to be real on the boundary. Then the discs can be expressed as sections of the fibration to $\textrm{D}$ by

$$\begin{aligned} \mu ^\alpha =x^{\alpha \beta }\lambda _\beta + (t+g(x,\lambda )) \lambda ^\alpha , \qquad x^{\alpha \beta }=x^{(\alpha \beta )}\,. \end{aligned}$$

where

$$\begin{aligned} g(x^{\alpha \beta },\lambda )=\frac{1}{2\pi }\oint \frac{\lambda _0}{\lambda '_0}\frac{1}{\langle \lambda \lambda '\rangle } \dot{h}((x^{\alpha \beta }\lambda '_\alpha \lambda '_\beta ,\lambda '_\alpha ) D\lambda '\,, \end{aligned}$$

(B.1)

with contour $\partial \textrm{D}$. This follows because on taking the imaginary part, the contour can be deformed to one around $\lambda =\lambda '$ so that

$$\begin{aligned} \Im g(x^{\alpha \beta },\lambda )= \dot{h}((x^{\alpha \beta }\lambda _\alpha \lambda _\beta ,\lambda _\alpha ) \,, \end{aligned}$$

(B.2)

by the Cauchy residue theorem, as required.

Working through the construction gives split signature version of Gibbons–Hawking metrics

$$\begin{aligned} ds^2 =V^{-1} (dt+\omega )^2+ Vd {{\textbf{x}}}\cdot d{{\textbf{x}}}\,, \end{aligned}$$

(B.3)

where $d {{\textbf{x}}}\cdot d{{\textbf{x}}}$ denotes the signature $1+2$-dimensional flat metric and

$$\begin{aligned} d V=^*d\omega \,, \qquad V=1+\oint \ddot{h}D\lambda \,,\qquad \lambda ^\beta \partial _{\alpha \beta } g=\lambda ^\beta \omega _{\alpha \beta }+\lambda _\alpha V\,. \end{aligned}$$

(B.4)

As a consequence, V satisfies the $1+2$-dimensional wave equation. By construction, the metric has a Killing vector $\partial _t$: note that in general, this Killing vector obstructs asymptotic flatness in that direction. However, for small data, these metrics are

A special case that can be constructed by this ansatze is the split signature analogue of the self-dual Schwarzschild/Taub-Nut, i.e., the metric whose mass is equal to its NUT charge. This can no longer be considered as ‘small’. Its analytic continuation to split signature was described in [47]. It is singular, so strictly speaking does not satisfy the global conditions of this paper. To obtain this case, take $h=Q\log Q$ where Q is the quadratic form defined by $Q=\lambda _\alpha \mu ^\alpha$. This is clearly singular on $\mathbb{R}\mathbb{P}^3$, and the integral formulae above for V needs some $i\epsilon$ prescription to make sense to give

$$\begin{aligned} V=1+\frac{2m}{r} \end{aligned}$$

(B.5)

and we can write the split signature metric (B.3) in the form

$$\begin{aligned} ds^2=\frac{1}{4}\left( 1+\frac{2m}{r}\right) ( dr^2 - r^2 d\Omega ^2) + m^2 \frac{1}{1+\frac{2m}{r}}\sigma _3^2\,. \end{aligned}$$

(B.6)

Here we have expressed the flat $1+2$-dimensional flat metric as

$$\begin{aligned} ds^2_{1+2}:=dr^2 - r^2 d\Omega ^2=dr^2-r^2(d\theta ^2+\sinh ^2\theta d\phi ^2)\,, \end{aligned}$$

(B.7)

and

$$\begin{aligned} \sigma _3 = dt-\cosh \theta d\phi \,. \end{aligned}$$

(B.8)

As observed by [47], these are natural Wick rotations of the metric on $S^3=SU(2)$ to $\hbox {AdS}_3 = SU(1,1)$ in which $d\Omega ^2$ is continued from the round sphere to the hyperbolic disc, and $\sigma _3$, the left-invariant form dual to the generator $\sigma _3$ in SU(2), is continued to the corresponding form in SU(1, 1). The metric above converts to the expression of [47] by shifting $r-m\rightarrow r$.