1 Introduction↩︎

In the seminal work [1] of Plebański, two normal forms for four–dimensional Ricci-flat anti-self-dual (ASD) complex spacetimes were presented. The first form is adapted to a local choice of compatible integrable complex structure. The second form, obtained by a Darboux transformation from the first, is adapted to a single foliation by self-dual surfaces. The corresponding equations expressing the ASD Ricci-flat condition are known as the first and second heavenly equations, respectively. The twistor approach to these equations was developed in [2], and in [3] it was shown how the second equation arises as an infinitesimal, but still non-linear, limit of the first.

Przanowski [4] obtained a normal form and equation for ASD vacuum metrics with non-zero cosmological constant \(\Lambda\). The equation is analogous to the first heavenly equation and the corresponding twistor theory was developed in [5], [6]. Finley and Plebański [7], and later Przanowski and Chudecki [8] have considered normal forms, and the corresponding equations, for such spacetimes adapted to a single integrable self-dual foliation. More recently, Lipstein and Nagy [9] provided an elegant metric ansatz for which the ASD-Einstein equation reduces to a single second order PDE. We generalise to arbitrary \(\Lambda\). Then the Lipstein–Nagy (LN) equation for \(W=W(x, y, w, z)\) is \[\label{wxyz95heavenly} \Psi\equiv (\partial_x\partial_w+\partial_y\partial_z)(\phi^{-1} W)-\phi\{\partial_x(\phi^{-1}W), \partial_y(\phi^{-1}W)\}_*=0, \quad where\quad \phi=w-\Lambda x\tag{1}\] and \(\{F, G\}_*=\{F, G\}_{xy}+2\Lambda\phi^{-1}\langle F, G\rangle_y\) with \[\{F, G\}_{xy}\equiv\partial_x F \partial_y G-\partial_x G \partial_y F, \quad \langle F, G\rangle_y\equiv F\partial_y G-G\partial_y F.\] The authors of [9], perhaps unaware of [7], [8] have not demonstrated that their equation describes all ASD \(\Lambda\)–vacua. Our note aims to fill in this gap. In §4 we shall prove

Our proof will demonstrate that the Chudecki–Przanowski equation [8] and the LN equation arise from a different gauge choice applied to the master dispersionless integrable system [10] describing all ASD conformal structures.

In §4.1 we shall find a Lax representation \([L_0, L_1]=0\) for (1 ) given by \[\begin{align} \label{laxpairintro} L_0 &=& \partial_{w}+(W_{x} + \frac{4\Lambda}{\phi} W)_y\partial_y - W_{yy}\partial_x - \lambda \partial_y + \beta_0 \partial_{\lambda}\\ L_1 &=& \partial_{z} -(W_x+\frac{4\Lambda}{\phi}W)_x \partial_y+W_{xy}\partial_{x} + \lambda \partial_{x} + \beta_1 \partial_{\lambda} \nonumber \end{align}\tag{2}\] with \(\beta_0=-\Box W_y, \beta_1=\Box (W_x+4\Lambda\phi^{-1}W)-3\Lambda\Psi\) and \[\label{boxoperator} \Box=\partial_w\partial_x+\partial_z\partial_y-W_{yy}\partial_x^2-(W_x+4\Lambda\phi^{-1}W)_x\partial_y^2+ 2(W_x+2\Lambda\phi^{-1}W)_y\partial_x\partial_y.\tag{3}\] In §4.2 we show how taking \(\Lambda\rightarrow 0\) combined with a coordinate transformation reduces (1 ) to the second heavenly equation. In §4.3 we give some examples of solutions to (1 ).

Acknowledgements↩︎

We are grateful to Adam Kmec for pointing out equation (1 ) to us, and to Adam Chudecki and Maciej Przanowski for explaining their work [8]. Our research was supported by the Simons Foundation grant SFI-MPS-T-Institutes-00010825, and by the State Treasury funds as part of a task commissioned by the Minister of Science and Higher Education under the project Organization of the Simons Semesters at the Banach Center - New Energies in 2026-2028 (MNiSW/2025/DAP/491). T.M. is supported by Cambridge Australia Scholarships.

2 The master equations of anti-self-duality↩︎

We will use the Einstein-summation convention throughout, using spinor indices taking values \(A \in \{0,1\}\) and \(\epsilon^{AB}, \epsilon_{AB}\) the standard symplectic matrices satisfying \(\epsilon_{AB}\epsilon^{BC} = \delta_{A}{}^{B}\). We will raise and lower indices using these objects. So, for example \(W^{A} := \epsilon^{AB}W_{B}\).

An adapted local Plebański coordinate system \(z^A=(w, z)\) and \(\theta_A=(x, y)\) for the general ASD conformal structure \((X, [g])\) was obtained in [10]. In such coordinates \[\begin{align} \label{general95integrable} g &=& \frac{2}{\phi^2}(d \theta_{A} dz^{A} + \frac{\partial W_{A}}{\partial \theta^{B}}dz^{A} dz^{B}),\\ &=& \frac{2}{\phi^2}(e^{00'} \odot e^{11'} - e^{10'} \odot e^{01'}),\nonumber \end{align}\tag{4}\] for some holomorphic functions \(W^{A}(z, \theta)\), \(\phi(z,\theta)\). Here \(e^{AA'}\) is the basis of \(\Lambda^{1}(X)\) satisfying \(e^{BB'}(E_{AA'}) = \delta_{B}{}^{A}\delta_{B'}{}^{A'}\) and \[\label{Evectors} E_{A0'} = \frac{\partial}{\partial \theta^{A}},\quad E_{A1'} = \frac{\partial}{\partial z^{A}} + \epsilon^{BC}\frac{\partial W_{B}}{\partial \theta^{A}}\frac{\partial}{\partial \theta^{C}}.\tag{5}\] We have \[\begin{align} [E_{A0'},E_{B1'}] = -\frac{\partial^2 W^{C}}{\partial \theta^{A} \partial \theta^{B}}E_{C0'}, \quad [E_{A1'},E_{B1'}] = -2\Box W_{[A} E_{B]0'}\label{brac2} , \end{align}\tag{6}\] where the second-order differential operator \(\Box = E_{11'}E_{00'}-E_{01'}E_{10'}\) is given by (3 ). The full ASD condition requires imposing two additional PDE: \[\begin{align} E_{A0'}(\square W^{A}) = 0, \quad E_{A1'}(\square W^{A}) = 0 \label{ASD2}. \end{align}\tag{7}\] There is considerable coordinate freedom on offer here. Firstly, we may choose to adapt the coordinates to any self-dual foliation¹, which then takes the explicit form \({\mathcal{D}} = \operatorname{span}\big\{ \frac{\partial}{\partial \theta^{0}}, \frac{\partial}{\partial \theta^{1} }\big\}.\) Having done this, we may choose any coordinates on the leaf space \(X / {\mathcal{D}}\). In fact, after redefining \(\phi\) and \(W^{A}\), we may use as Plebański coordinates \[\begin{align} \label{fibre95lift} \tilde{z}^{A}(z), \quad \tilde{\theta}^{A} = \Gamma(z)\frac{\partial \tilde{z}^{A}}{\partial z^{B}}\theta^{B} +\gamma^A(z), \quad \tilde{\phi} = \Gamma^{1/2}\bigg(\det{\frac{\partial \tilde{z}^{A}}{\partial z^{B}} } \bigg)^{1/2} \phi. \end{align}\tag{8}\]

3 A normal form for an ASD Einstein metric↩︎

Proof. The proof is based on computations of the Ricci tensor [7], [11]. The \(e^{A0'} \odot e^{B0'}\) components of the trace-free Ricci tensor vanish iff \[\begin{align} \phi = {\Lambda} S_{A}(z)\theta^{A} + T(z) \end{align}\] for some functions \(S_{A}(z), T(z)\), where the factor of \(\Lambda\) has been inserted for convenience and is consistent with the fact that the scalar curvature vanishes if \(\phi\) is a function of \(z^A\).

We rescale the \(\theta^{A}\) coordinates by a function of \(z\), which, after redefining \(W_{A}\), rescales the conformal factor so that the \(1\)-form \(S_{A}(z)dz^{A}\) is closed. This implies there exist functions \(\tilde{z}^{A}(z)\) that solve \(J_{A}\frac{\partial \tilde{z}^{A}}{\partial z^{B}} = S_{B}(z)\) for some constants \(J_{A}\) (not all zero) that we may choose. We then define fibre coordinates \(\tilde{\theta}^{A}\) by (8 ), such that in coordinates \((\tilde{z}^{A},\tilde{\theta}^{A})\) we have \(\tilde{\phi} = {\Lambda}\tilde{J}_{A}\tilde{\theta}^{A} + T(\tilde{z}).\) Using the remaining freedom in \(\tilde{\theta}^{A}\) we may put \(\phi\) in the form (?? ).

We continue imposing the Einstein equations. The vanishing of the \(e^{A0'} \odot e^{B1'}\) components of the trace-free Ricci tensor yield linear PDE \[\begin{align} \label{integrability951} \frac{\partial^2 V}{\partial \theta^{A} \partial \theta^{B}} = 0, \quad where\quad V := \epsilon^{AB} \phi \frac{\partial W_{A}}{\partial \theta^{B}} - 4{\Lambda} J^{A}W_{A} = \epsilon^{AB}\phi^{5}\frac{\partial}{\partial \theta^{B}}(\phi^{-4}W_{A}). \end{align}\tag{9}\] We have freedom to redefine the functions \(W_A\) by \(\tilde{W}_{A} = W_{A} + \chi(z)\epsilon_{AB}\theta^{B} + \zeta_A(z),\) for arbitrary \(\chi(z),\zeta_{A}(z)\). In terms of \(V\), since it satisfies (9 ), this means we may take \[\begin{align} \label{integrability32condition} V = 4\gamma(z)K_{A}\theta^{A} + \alpha(z) J_{A}\theta^{A} + \beta (z) \end{align}\tag{10}\] for \(\alpha(z), \beta(z)\) that we may choose freely, while \(\gamma(z)\) is fixed. If we take \(\alpha = \beta = 0\), then the integrability condition (9 ) implies the existence of a function \(W\) such that \[\begin{align} W_{A} = \frac{\partial W}{\partial \theta^{A}} - \frac{4\Lambda}{\phi}J_{A}W + \gamma(z)K_{A}K_{B}\theta^{B}. \end{align}\] Computing the scalar curvature \(R = 24(\Lambda-\gamma(z))\) gives \(\gamma(z) = 0\).

4 The Lipstein-Nagy equation↩︎

Having obtained a normal form for the metric depending on only one function \(W\), we will now show that the equation of Lipstein and Nagy is necessary and sufficient (after an appropriate redefinition of \(W\)) for the metric to satisfy the equations (7 ) and the parts of the Einstein condition which we have yet to impose. Proof of Theorem 1. The first step is to calculate: \[\begin{align} \label{Box95calc} \square W_{A} = \phi \frac{\partial \Psi}{\partial \theta^{A}} - 3\Lambda J_{A}\Psi + K^{B}\frac{\partial^2 \tilde{W}}{\partial \theta^{B} \partial \theta^{A}} - \Lambda J^{B}\frac{\partial^2 \tilde{W}}{\partial z^{B} \partial \theta^{A}} - 2 \Lambda^2 J^{B}J^{C}\frac{\partial \tilde{W}}{\partial \theta^{B}}\frac{\partial^2 \tilde{W}}{\partial \theta^{A} \partial \theta^{C}} \end{align}\tag{11}\] where \[\begin{align} \Psi := \epsilon^{AB}\frac{\partial^2 \tilde{W}}{\partial \theta^{A} \partial z^{B}} - \frac{\phi}{2}\epsilon^{AB}\epsilon^{CD} \frac{\partial^2 \tilde{W}}{\partial \theta^{A} \partial \theta^{C}}\frac{\partial^2 \tilde{W}}{\partial \theta^{B} \partial \theta^{D}} {-} 2 \Lambda J^{C}\epsilon^{AB}\frac{\partial \tilde{W}}{\partial \theta^{A}}\frac{\partial^2 \tilde{W}}{\partial \theta^{C}\partial \theta^{B}}, \quad \tilde{W}\equiv \phi^{-1}W. \end{align}\] The Lipstein-Nagy equation is \(\Psi = 0\). All but one term of \(\square W_{A}\) is a total derivative with respect to the \(\theta^{A}\) coordinates. Therefore the first equation in (7 ) simplifies to the third-order PDE \[\begin{align} \label{I} J^{A}\frac{\partial \Psi}{\partial \theta^{A}} = 0. \end{align}\tag{12}\] Using (11 ), we expand the remaining ASD equation (7 ) as: \[\begin{align} \label{II} \phi \epsilon^{AB} \frac{\partial^2 \Psi}{\partial z^{A} \partial \theta^{B}} - 2K^{A}\frac{\partial \Psi}{\partial \theta^{A}} - 2 \Lambda J^{A}\frac{\partial \Psi}{\partial z^{A}} + 2 \Lambda \bigg(J^{B}\frac{\partial \tilde{W}}{\partial \theta^{B}} \bigg) J^{A}\frac{\partial \Psi}{\partial \theta^{A}}= 0. \end{align}\tag{13}\] Given the normal form as in Lemma 1, the non-zero components of the Einstein equations are the \(e^{A1'} \odot e^{B1'}\) components, which are equivalent to \[\begin{align} \label{nonlinear} \frac{\partial^2 (\phi^{-1}\Psi)}{\partial \theta^{A} \partial\theta^{B}} = 0. \end{align}\tag{14}\] So the Lipstein-Nagy equation \(\Psi = 0\) implies anti-self-duality and the Einstein condition.

The converse is as follows: The function \(W\) satisfying (?? ) is only defined up to \(\hat{W} = W + \rho(z)\phi^{4}.\) Under such a redefinition \[\begin{align} \label{freedom} \hat{\Psi} = \Psi - {3\Lambda}\phi^{2}J^{A}\frac{\partial \rho(z)}{\partial z^{A}} - 6 \Lambda \phi\rho(z). \end{align}\tag{15}\] The general solution to the system (12 , 13 , 14 ) is \[\begin{align} \Psi = 3\Lambda\phi^{2}J^{A}\frac{\partial F(z)}{\partial z^{A}} + 6\Lambda \phi F(z) \end{align}\] for arbitrary \(F(z)\). So we may, by redefining \(W\) as per (15 ) set \(\Psi = 0\). The end result of the analysis is that the metric (4 ) with \((\phi, W_A)\) as in Lemma 1 such that \[\begin{align} \label{LN32equation} \epsilon^{AB}\frac{\partial^2 \tilde{W}}{\partial \theta^{A} \partial z^{B}} - \frac{\phi}{2}\epsilon^{AB}\epsilon^{CD} \frac{\partial^2 \tilde{W}}{\partial \theta^{A} \partial \theta^{C}}\frac{\partial^2 \tilde{W}}{\partial \theta^{B} \partial \theta^{D}} {-} 2 \Lambda J^{C}\epsilon^{AB}\frac{\partial \tilde{W}}{\partial \theta^{A}}\frac{\partial^2 \tilde{W}}{\partial \theta^{C}\partial \theta^{B}} = 0 \end{align}\tag{16}\] is ASD and Einstein. Conversely, for any ASD Einstein metric, there exist coordinates \((z^{A},\theta^{A})\) such that \(\phi\) and \(W_{A}\) are as above, and (16 ) holds. Rewriting this with \(z^A=(w, z), \theta^A=(y, -x)\) and setting \(J=(0, 1), K=(1, 0)\) yields the statement of Theorem 1.

4.1 Lax pair↩︎

Consider a pair of vector fields on \(X\times \mathbb{CP}^1\) indexed by \(A=0, 1\): \[\begin{align} L_{A} = E_{A1'} - \lambda E_{A0'} + \beta_{A}\frac{\partial}{\partial \lambda}, \quadwhere\quad \beta_{A} = -\square W_{A} - 3\Lambda J_{A}\Psi. \end{align}\] Using (6 ) we see that the \(E_{A0'}\) components of \([L_{0}, L_{1}]\) vanish if and only if \(\beta_{A} = -\square W_{A}\), which holds if and only if \(\Psi = 0\). The remaining components vanish with (7 ), which we have shown follow from \(\Psi = 0\). We have thus exhibited a Lax pair for the equation (1 ). This could be a starting point of the inverse–scattering construction analogous to [12].

4.2 The Ricci-flat limit↩︎

If we set \(\Lambda = 0\), then the metric (?? ) becomes \[\begin{align} g=\frac{2}{w^2}(dwdx+dzdy+W_{xx}dz^2+W_{yy}dw^2-2W_{xy} dwdz). \end{align}\] Letting \[\begin{align} \tilde{w}=-\frac{1}{2w^2}, \quad\tilde{z}=z, \quad \tilde{x}=wx,\quad \tilde{y}=\frac{y}{w^2},\quad \Theta=W-\frac{1}{2}\frac{xy^2}{w} \end{align}\] yields the second heavenly form [1] \[\begin{align} g=2(d\tilde{w}d\tilde{x}+d\tilde{z}d\tilde{y}+ \Theta_{\tilde{x}\tilde{x}} d\tilde{z}^2+\Theta_{\tilde{y}\tilde{y}} d\tilde{w}^2-2\Theta_{\tilde{x}\tilde{y}} d\tilde{w}d\tilde{z}) \end{align}\] where \(\Theta\) satisfies Plebański’s second heavenly equation \[\begin{align} (\partial_{\tilde{x}}\partial_{\tilde{w}}+\partial_{\tilde{y}}\partial_{\tilde{z}})(\Theta)-\{\partial_{\tilde{x}}\Theta, \partial_{\tilde{y}}\Theta\}_{\tilde{x}\tilde{y}}=0. \end{align}\]

It is an open problem to find an analogous limiting procedure between the Przanowski equation [4] and the first heavenly equation.

4.3 Examples↩︎

A solution to (1 ) given by an arbitrary function \(W = \phi f(x, z)\) yields the ASD cosmological plane wave in [13]: \[\begin{align} g = \frac{2}{\phi^2}[ dw\,dx + dy\,dz + ( \phi f_{xx} + 2\Lambda f_x) dz^2 ]. \end{align}\] Another class of solutions can be found by demanding that the linear and non–linear parts of (1 ) vanish separately. The solution is analogous to that of the Sparling-Tod metric in the second heavenly formalism: \[\begin{align} W = \frac{\phi \cdot F(\frac{w}{wx+zy},\frac{z}{wx+zy})}{wx+zy}. \end{align}\]

4.4 Reality conditions↩︎

The normal form (?? ) in Theorem 1 gives a complex metric. If \(W\) is taken to be a real function of real coordinates \((x, y, w, z)\) then the resulting metric has signature \((2, 2)\). While all Riemannian ASD Einstein metrics can also in principle be recovered from (?? ), imposing the reality conditions is less natural, as the coordinates are adapted to a choice of a self–dual surface and all such surfaces in Riemannian signature are necessarily complex. Here the original formulation of Przanowski [4] is advantageous.

5 Outlook: Celestial symmetries with \(\Lambda\neq 0\)↩︎

The resurgence of interest in heavenly equations and their integrability comes from the appearance of their infinite–dimensional symmetry algebras \(Lw_{1+\infty}\) in the context of celestial holography [14]. In the ASD Ricci–flat case these symmetries can be understood as coming from a recursion operator on space–time, and its Penrose transform acting on cohomology classes in the twistor space [2] (see also [15]). The presence of non–zero cosmological constant breaks down the SDiff\((\mathbb{C}^2)\) symmetries of the heavenly equations. This can already be seen in the LN equation (1 ) as the underlying bracket deforms the Poisson bracket by a Wronskian Lie algebra of Diff\((\mathbb{C})\). The celestial symmetries in the presence of \(\Lambda\), and their twistor counterparts have been studied in [16] and [17].

Our Lax pair (2 ) leads to a notion of a recursion operator: Let \(\psi(x, y, w, z, \lambda)=\sum_{n=0}^{\infty} \lambda^{-n}\phi_n\) be a twistor function holomorphic around \(\lambda=\infty\) where \(\phi_n, n=0, 1, \dots\) are functions on \(X\). The twistor condition \(L_A(\psi)=0\) leads to a recursion operator \({\mathcal{R}}(\phi_n)=\phi_{n+1}\) defined in terms of the vector fields (5 ) \[\label{recursionrel} E_{A0'}(\phi_{n+1})=E_{A1'}(\phi_n)-(n-1)\beta_A(\phi_{n-1}).\tag{17}\] We can start the recursion letting \(\phi_0\) be the coordinate functions \(w\) or \(z\) which gives \[\begin{align} && w\rightarrow y\rightarrow W_x+4\Lambda\phi^{-1}W\rightarrow\cdots\\ && z\rightarrow -x\rightarrow W_y\rightarrow\cdots \end{align}\] with the higher terms determined by the linear PDEs (17 ). In the case \(\Lambda=0\) the recursion operator acts on solutions to the background–coupled wave equation [2]. In the presence of \(\Lambda\) the functions \(\phi_n\) are twisted by sections of powers of a non–trivial line–bundle (see [6] for a description in the context of the Przanowski equation). It would be interesting to see if this recursion procedure can be used to reconstruct the \(\Lambda\)–celestial symmetries and the associated hierarchy in the spirit of [18].

References↩︎