Non-relativistic twistor theory and Newton--Cartan geometry

We develop a non-relativistic twistor theory, in which Newton--Cartan structures of Newtonian gravity correspond to complex three-manifolds with a four-parameter family of rational curves with normal bundle ${\mathcal O}\oplus{\mathcal O}(2)$. We show that the Newton--Cartan space-times are unstable under the general Kodaira deformation of the twistor complex structure. The Newton--Cartan connections can nevertheless be reconstructed from Merkulov's generalisation of the Kodaira map augmented by a choice of a holomorphic line bundle over the twistor space trivial on twistor lines. The Coriolis force may be incorporated by holomorphic vector bundles, which in general are non--trivial on twistor lines. The resulting geometries agree with non--relativistic limits of anti-self-dual gravitational instantons.


Introduction
Over the last six years there has been large interest in the AdS/CFT correspondences providing the gravity duals of non-relativistic gauge theories relevant in solid-state physics [15]. A non-relativistic field theory should have a dual made out of a non-relativistic theory of gravity. The mathematical structure of such a theory -the Newton-Cartan (NC) spacetime -is rather baroque. It consists of a torsion-free non-metric connection ∇, a parallel degenerate metric h and a parallel one-form θ in the kernel of this metric [2,32,8].
In this paper we shall construct a non-relativistic twistor theory for Newton-Cartan space-times. There are several equivalent definitions of twistors for flat Minkowski space, but they lead to nonequivalent pictures once a non-relativistic limit is taken. One could define twistors as spinors for the conformal group SU (2, 2) and then take a Wigner contraction corresponding to a limit where the speed of light becomes infinite [12]. This approach has to be abandoned in the search for an analogue of a curved twistor correspondence [25] as a general space time does not admit any conformal isometries. Thus the linear/group theoretic structures of the twistor space become irrelevant. We shall instead concentrate on the holomorphic geometry of the twistor space and its one-dimensional complex submanifolds. In the non-relativistic limit the normal bundle of the twistor curves corresponding to space-time points jumps from O(1) ⊕ O (1) to O ⊕ O (2), where O(n) is the holomorphic line bundle over CP 1 with the Chern class n. This results in the conformal structure becoming degenerate. We shall show that the Newtonian space-times are unstable under the general Kodaira deformation of the twistor complex structure.
The Newton-Cartan connections can nevertheless be reconstructed from Merkulov's generalisation of the Kodaira map augmented by a choice of a holomorphic line bundle over the twistor space on trivial on twistor lines. The Coriolis force may be incorporated by holomorphic vector bundles, which in general are non-trivial on twistor lines. The resulting geometries agree with non-relativistic limits of anti-self-dual gravitational instantons.
The paper is organised as follows: In the next Section we shall review the Newton-Cartan geometry, and show that in the presence of the time-independent Coriolis force, the general Newton-Cartan connection satisfying the Einstein equations is determined by a couple of Date: 10 February 2015. 1 harmonic functions on R 3 . In Section 3 we shall explain how the NC structures arise as limits from general relativity. We shall give examples of NC limit of the Lorentzian Taub-Nut metric, anti-self-dual (ASD) gravitational waves, and Gibbons-Hawking gravitational instantons. The one-parameter family of anti-self-dual Ricci flat metrics g ǫ = (1 + ǫV )(dx 2 + dy 2 + dz 2 ) + 1 ǫ(1 + ǫV ) (dτ + ǫ 3/2 A) 2 depending on a harmonic function V and a one-form A on R 3 such that ∇V = ∇∧A admits a Newtonian limit (h = lim ǫ→0 (g −1 ), θ = dτ, ∇ = lim ǫ→0 ∇ gǫ ) with the unparametrised geodesic equations of ∇ given by d 2 x/dτ 2 = −(1/2)∇V . In Section 4 we shall introduce the spinor calculus adapted to a 3+1 splitting, leading to Newtonian spinors. In Section 5 we shall review the relativistic twistor theory, and proceed to construct its non-relativistic limit. This will be done by realising the relativistic twistor space as an affine line bundle over the total space of the bundle O(2) → CP 1 , taking a limit of the resulting patching matrix (Section 5.2.1) and then (Section 5.2.2) by analysing the incidence relation directly: a twistor space P T c for a complexified Minkowski space M C with a speed of light c will be shown to be a rank-two vector bundle over CP 1 determined by a patching matrix where λ ∈ CP 1 .
If c = ∞ then F ∞ = diag(1, λ −2 ) and P T ∞ = O ⊕ O (2), but for finite c there exist holomorphic splitting matrices H and H such that F = H diag(λ −1 , λ −1 ) H −1 and P T c = O(1) ⊕ O (1). In Section 6 we shall discuss the Kodaira deformation theory of the complex structure underlying the non-relativistic twistor space. We shall exhibit a class of deformations which do not preserve the type of the normal bundle of twistor curves, and lead to Gibbons-Hawking metrics. This is a consequence of the non-vanishing of the obstruction group H 1 (CP 1 , End(N )) = C, where N = O ⊕ O(2) is the normal bundle to non-relativistic twistor lines. We shall also discuss deformations for which the twistor curves do not change their holomorphic type. In Section 7 we shall construct Newton-Cartan connections from holomorphic vector bundles over the non-deformed non-relativistic twistor space. In Section 7.1 line bundles trivial on twistor curves will be used to construct the Newtonian connections, and in Section 7.2 rank-two vector bundles which restrict to O ⊕ O(2) on twistor curves will lead to connections with non-zero Coriolis term. In Section 8 we shall use Merkulov's relative deformation theory to give an alternative construction of Newtonian connection with no Coriolis term. The connections will arise as a family of morphisms where N F is the normal bundle to the correspondence space in the product manifold M C × P T ∞ and where L p = CP 1 is a rational curve in P T ∞ corresponding to a point p in the complexified Newtonian space-time M C . The contribution from H 0 (CP 1 , O(2)) = C 3 factor in (1.1) corresponds (by the Serre duality) to a zero-restmass field on M C arising from a gravitational potential V . In Appendix 1 we shall exhibit and interpret a contour integral formula ψ(x, y, u) = 1 2πi Γ⊂CP 1 e − 1 2 mi(x−iy)λ g(x + iy + λu, λ)dλ, for solutions to the 2+1 Schrodinger equation 2m∂ u ψ = i(∂ x 2 + ∂ y 2 )ψ. Finally in Appendix 2 we shall show how the spin connection in the Nonlinear Graviton construction arises as a Ward transform of a rank-two holomorphic vector bundle over the relativistic deformed twistor space.
Acknowledgements. We are grateful to Christian Duval, George Sparling and Paul Tod for helpful discussions. This work started when MD was visiting the Institute for Fundamental Sciences (IMP) in Tehran in April 2010. MD is grateful to IMP for the extended hospitality when volcanic eruption in Iceland halted air travel in Europe. The work of JG has been supported by an STFC studentship.

Newton-Cartan gravity
The trajectories of test particles in Newtonian physics with a chosen universal time t are integral curves x = x(t) of the system of ODEs where V : R 3 → R is the Newtonian potential. One normally interprets these equations as giving rise to curved paths in a fixed three-dimensional space. The geometric perspective of Cartan [2] differs in that the particle trajectories are instead regarded as geodesics of some connection in a four-dimensional space time M . To introduce the necessary structures consider a four-manifold M equipped with a triple (h, θ, ∇), where h is a degenerate contravariant metric of signature (0, +, +, +), θ is a one-form which belongs to a kernel of h (when viewed a map from T * M to T M ), and ∇ is a torsion-free affine connection on M with covariant derivative ∇ which keeps h and θ parallel. In local coordinates The last two conditions do not specify ∇ uniquely. The most general connection satisfying (2.3) is parametrised by an otherwise irrelevant choice of time-like vector U such that θ a U a = 1 and a choice of two-form F . The most general connection such that U a ∇ a U b = 0, ∇ [a U b] = 0 is given by the Christoffel symbols where ∂ a = ∂/∂x a , and h ab is a degenerate metric uniquely determined by the conditions holds. Here R a bcd is the curvature of ∇. The condition (2.5) is required for consistency with a non-relativistic limit of general relativity (see Section 3). Even this additional requirement allows connections which are more general than one needs to reproduce the Newtonian theory (see Section 2.2). For a given (h, θ) the pair (U a , F ab ) is defined up to gauge transformations U a → U a + h ab Ψ b , F → F + dΦ, where Ψ = Ψ a dx a is an arbitrary one-form, and Thus, if ∇ is a Newtonian connection (so that dF = 0), then one can locally set F = 0 and instead work with a non-trivial vector field U . This approach is adapted in [29].
The absence of torsion of ∇, and the consistency conditions (2.3) imply the existence of a function t : M → R such that θ = dt. The three-dimensional distribution spanning the kernel of θ is integrable in the sense of the Frobenius theorem, and the manifold M admits a fibration over a real line R called the time axis, with a coordinate t M → M/ker (θ) = R.
Thus the space-time M in the Newton-Cartan theory is a fibre bundle over a universal time axis equipped with a Riemannian metric h on the fibers.
The Einstein field equations are where R ab is the Ricci tensor of ∇, the function ρ : M → R is the mass density and G is the gravitational constant. These equations in particular imply that the metric h on threedimensional spatial fibres is flat. This allows an introduction of the Galilean coordinates x a = (t, x) such that (in general indices a, b, . . . run from 0 to 3 and i, j, . . . run from 1 to 3), ∂x j , and θ = dt. Let us spell out some physical consequences of the Newton-Cartan formalism. In general, the free falling particles follow the geodesics of ∇ which however do not need to be affinely parametrised. The statement that two events are simultaneous in Newtonian physics is invariant and coordinate independent. This is a consequence of the existence of the oneform θ and the resulting fibration M → R. On the other hand one can not say that two events occur at the same point in 3-space but at different times. To be able to compare two non-simultaneous events one needs to make an additional choice of Galilean coordinates such that the spatial part of the Newton-Cartan connection vanishes. The existence of such coordinates is a consequence ∇h = 0, and the flatness of h which itself follows from the field equations (2.6). Moreover the distance between two events is only invariantly defined if these events are simultaneous.
2.1. Example 1. Newtonian connection. Take h to be a flat metric on the spatial fibres, i. e. h ij = δ ij , θ = dt and consider Γ i 00 = δ ij ∂ j V with all other components of ∇ equal to zero. In this case F = dt ∧ dV, and the geodesics of ∇ parametrised by t are the integral curves of the Newton equations of motion (2.2). The coordinate transformation where R is an element of SO(3) which is allowed to depend on t, gives rise to the 'Coriolis forces' given by the connection componentsΓ i 0j =Ṙ kj R ik . A combination of an inertial forceä and a centrifugal forceRx also appear as components of the connection aŝ . The form (2.2) of the geodesic equations is preserved only ifṘ = 0,ä = 0.
2.2. Example 2. Newton-Cartan connection with the Coriolis force. The Newtonian connection of the previous subsection can be modified to a Galilean connection which includes the general space-time dependent Coriolis force B = B(x, t). In this case the closed two-form F is and the corresponding non-zero connection components are where θ = dt, h ab is a degenerate metric of signature (0, 1, 1, 1) such that h ab θ b = 0 which in our case is chosen to be diag(0, 1, 1, 1). The field equations together with the Trautman condition imply B = ∇W , for some function W , and ∇ · E = 2|∇W | 2 , ∇ 2 W = 0. (2.9) At this stage we can not assume that E = −∇V . This can be either put in by hand, or derived from an asymptotic condition that B → B(t) at spatial infinity. This condition, together with the uniqueness theorem for the Laplace equation implies that B can be set to zero. This is because if can be set to zero by a choice of an orthogonal matrix R(t). Alternatively, the closure of F implies that E arises from a potential if B does not depend on t. If E = −∇V , then so that the connection is determined by two harmonic functions. In Section (7.2) we shall show how two independent harmonic functions arise from holomorphic vector bundles over a Newtonian twistor space P T ∞ which restrict to a non-trivial vector bundle O ⊕ O(2) on each twistor line.

Newton-Cartan geometry as a limit of General Relativity
In this Section we shall show how Newton-Cartan geometry arises as a degenerate limit of General Relativity [19,3]. Let (M, g(ǫ)) be a family of pseudo-Riemannian manifolds parametrised by ǫ > 0, and such that where h ab is a contravariant tensor on M of signature (0, +, +, +). Let θ = θ a dx a be an ǫ-independent one-form such that h ab θ a = 0, normalised by k ab θ a θ b = −1. If the one-form θ is closed, then the Levi-Civita connection of g(ǫ) has a well defined limit ∇ as ǫ goes to zero. It is a Newton-Cartan connection (2.4) with dF = 0. The resulting Newton-Cartan structure is (h, ∇, θ). The Ricci tensor of g(ǫ) also has a well defined limit, and choosing a perfect fluid energy-momentum tensor for the relativistic theory leads (in the limit) to the field equations (2.6). See [3] for details of this derivation.
Another way to proceed [11] is to define a one-parameter family of structures {h ab , t ab , ∇} such that (t, h) are parallel with respect to ∇ and t ab h bc = −ǫδ c a . For ǫ > 0 this gives pseudo-Riemannian geometry with g ab = −t ab /ǫ, and for ǫ = 0 the Newton-Cartan theory with t ab = ∇ a t∇ b t, and θ a = ∇ a t.

Examples.
We shall now give three examples of Newtonian limits corresponding to particular solutions of Einstein equations. The first example is a minor modification from [11] (were some factors appear to be wrong), and corresponds to a non-zero Coriolis force and a Newton-Cartan structure of the type (2.8). The remaining two examples give limits of anti-self-dual solutions and thus correspond to analytic continuation of the Newton-Cartan theory. The resulting geodesic equations can be analytically continued to Newton's equations of motion despite the fact that ASD gravitational instantons in Riemannian signature, or ASD pp-waves in neutral signature do not admit Lorentzian analytic continuations. Our motivation for considering the limits of ASD solutions will become clear in Section (7.1) when we construct non-relativistic twistor spaces from a limiting procedure.
3.1.2. Newtonian limit of gravitational instantons. Let V and A be respectively a harmonic function, and a one-form on R 3 which satisfy the Abelian monopole equation where * is the Hodge endomorphism of the flat metric on R 3 . The one-parameter family of Gibbons Hawking metrics [13] g = (1 + ǫV )(dx 2 + dy 2 + dz 2 ) + 1 ǫ(1 + ǫV ) (dτ + ǫ 3/2 A) 2 (3.10) has anti-self-dual Weyl tensor and is Ricci-flat. Conversely, any Riemannian ASD Ricciflat (and so hyper-Kähler) four-manifold admitting a Killing vector preserving the hyper-Kahler structure is of the form (3.10). The Newtonian limit of (3.10) is where all other components of Γ a bc vanish. 3.1.3. Newtonian limit of anti-self-dual plane waves. The most general ASD Ricci-flat metric admitting a parallel Killing vector is of the form [27] g = dwdx + dzdy + γ(w, y)dw 2 . (3.12) If the coordinates (w, z, x, y) on M , and the arbitrary function γ = γ(w, y) are taken to be real, then the metric has signature (2, 2). Set and redefine γ. This, for a given γ, gives a one-parameter family of metrics with a Newtonian limit where now γ = γ(t, y). This structure can, for certain γs, be analytically continued to a Newtonian theory 1 , where the signature of h is (0, 1, 1, 1).

Newtonian spinors
Let (M, g) be a (pseudo) Riemannian four-manifold. Locally there exist complex ranktwo vector bundles S and S ′ over M equipped with covariantly symplectic structures ε and ε ′ parallel w.r.t the Levi-Civita connection and such that is a canonical bundle isomorphism, and for p 1 , p 2 ∈ Γ(S) and q 1 , q 2 ∈ Γ(S ′ ). We use the conventions of Penrose and Rindler [26], where the spinor indices are capital letters, unprimed for sections of S and primed for sections of S ′ . For example p A denotes a section of S, and q A ′ a section of (S ′ ) * . The symplectic structures ε AB and ε A ′ B ′ (such that ε 01 = ε 0 ′ 1 ′ = 1) are used to lower and raise the spinor indices according to The properties of the complex conjugation on spinors depends on the signature of g.
• If g is Lorentzian, then the spinor complex conjugation is a map S → S ′ given by Linear SL(2, C transformations on S induce Lorentz transformations on vectors, and SO(3, 1) = SL(2, C)/Z 2 . • If g is Riemannian, then the complex conjugation preserves the type of spinors, i.e.
it maps sections of S to sections of S and sections of S ′ to sections of S ′ . It is given by In the Riemannian case the structure group is not simple, and SO(4, R) = SU (2) × SU (2) ′ /Z 2 , where the spin groups SU (2) and SU (2) ′ act linearly on sections of S and S ′ respectively. Now assume that g has Lorentzian signature, and chose a time-like unit vector T to perform the 3+1 split of g. This vector gives a map from S to S ′ given by [28] where we have chosen 1 Consider a different way of taking the limit, and set

Now the limit is
∂t .
This appears to be different than (3.13), but isn't, as the coordinate is the normalised basis of S. Under this map the isomorphism (4.14) becomes where Λ 2 (S ′ ) is a rank-one vector bundle, and Sym 2 (S ′ ) is a rank-three vector bundle isomorphic to a bundle of self-dual two-forms on M . The isomorphism (4.18) gives the orthogonal decomposition: The symmetric matrix V (A ′ B ′ ) represents a three-vector in the space t =const and the multiple of ε A ′ B ′ is the component in normal direction. Consider the flat metric In our discussion of the Newtonian limit of general relativity in Section 3 we needed to single out a time-like one-form θ a = ∇ a t in the kernel of the three-metric h. We shall use the dual vector-field to perform the 3+1 decomposition (4.19), and construct the isomorphism (4.16). Therefore the two-component spinor formalism for Newton-Cartan structures involves only one type of spinors (which we choose to be sections of S ′ ). It is natural to ask whether the Newton-Cartan connection on T M is induced from some spin connection on S ′ → M . The answer to this question appears to be 'no' which is related to the fact that the Galilean boosts do not have a double cover. A Newton-Cartan connection takes its values in the Lie algebra of the Galilean group. However we find that the homogeneous Galilean transformations can not be realised spinorially as It is nevertheless possible to construct a spin connection in the Newtonian limit, although to achieve a non-zero result the speed of light has to be regarded as a part of the constant metric on the fibres of T * M in the Cartan formalism. To see it consider the post-Newtonian limit of GR given by Employing the spin frame (ce V /c 2 dt, dx i ) yields a spin connection which vanishes in the limit c → ∞. Let us instead consider so that (4.24) is given by g = η ab e a e b . Then the Cartan equations de a + ω a b ∧ e b = 0 together with a limit c → ∞ yield the non-vanishing connection given by

5.1.
Review of relativistic twistor theory. In this Section we shall recall basic facts about twistor theory, following [26] and [4].
The theory was put forward by Roger Penrose [23] in the late 1960s. The primary motivation was to unify general relativity and quantum theory in a non-local theory based on complex numbers, but a good starting point for our discussion is Penrose's contour integral formula for wave equation in Minkowski space [24,10] φ(x, y, z, t) where Γ ⊂ CP 1 is a closed contour and the function f is holomorphic on an intersection of two open sets in CP 1 away from poles inside Γ. Differentiating inside the integral shows that 1 c 2 The formula (5.25) gives real solutions to the wave equation in Minkowski space from holomorphic functions of three variables, and the next step is to recognise these variables as coordinates in some complex three-dimensional manifold. The five dimensional space P N of light rays in the Minkowski space is a hyper-surface in a three-dimensional complex manifold P T = CP 3 − CP 1 called the projective twistor space. Let (ω 0 , ω 1 , π 0 ′ , π 1 ′ ) ∼ γ(ω 0 , ω 1 , π 0 ′ , π 1 ′ ), γ ∈ C * with (π 0 ′ , π 1 ′ ) = (0, 0) be homogeneous coordinates of a twistor. The Minkowski space and the twistor space are linked by the incidence relation x, y, z) are coordinates of a point in Minkowski space. It then follows that if two points in Minkowski space are incident with the same twistor, then they are separated by a null geodesic. The spinor form of the incidence formula above is where the displacement vector x AA ′ is given by (4.21). Let Z = (ω A , π A ′ ) represent homogeneous coordinates of a twistor. The non-projective twistor space C 4 − C 2 is equipped with a Hermitian inner product The signature of Σ is (+ + −−) and endomorphisms of C 4 preserving Σ as well as the orientation of C 4 form a group SU (2, 2) which is locally isomorphic to the conformal group of the Minkowski space. The projective twistor space is divided into three regions depending on the sign of Σ(Z,Z). The hypersurface is preserved by the conformal transformations of the Minkowski space.
Fixing the coordinates (x, y, z, t) of a space-time point p in (5.27) gives a projective line L p = CP 1 in P T . If the coordinates of p are real, then L p is contained in the hypersurface P N . Conversely, fixing a point in P N gives a light-ray in the Minkowski space.
Points in P T /P N can be interpreted in terms of the complexified Minkowski space M C = C 4 where they correspond to null two-dimensional planes (α-planes) with self-dual tangent bi-vector. This, again, is a direct consequence of (5.27) where now the coordinates x AA ′ are complex ( Figure 1). There is a CP 1 worth of α-planes through each point of M C . For any π A ′ ∈ CP 1 a corresponding α-plane is spanned by a two-dimensional twistor distribution Finally we can give a twistor interpretation of the contour integral formula (5.25). Consider a function f = f (ω A /π 1 ′ , π 0 ′ /π 1 ′ ) which is holomorphic on an intersection of two open sets covering P T (one of these sets is defined by π 1 ′ = 0 and the other by π 0 ′ = 0). Restrict f to a rational curve L p ⊂ P N and integrate f along a contour in L p . This yields (5.25) with λ = π 0 ′ /π 1 ′ . The function f , when regarded as defined on the non-projective twistor space, must be homogeneous of degree −2.

5.2.
Non-relativistic twistor theory. Our construction of a non-relativistic twistor space for the flat Newtonian space time will proceed in two steps. We shall first represent the relativistic twistor space introduced in the last Section as an affine line bundle over O (2), and show that in the limit c → ∞ the normal bundle of sections of P T → CP 1 jumps from . This will be done is Section 5.2.1. In Section 5.2.2 we shall relate the jumping lines approach to a non-relativistic limit of the 3+1 split of the twistor incidence relation.
and set λ = π 0 ′ /π 1 ′ on U ∩ U . The Birkhoff-Grothendieck theorem states that any rank-k holomorphic vector bundle over CP 1 is isomorphic to a direct sum of line bundles for some integers m 1 , m 2 , . . . , m k . Moreover the transition matrix F : C * → GL(k, C) of this bundle can be written as where H : U → GL(k, C) and H : U → GL(k, C) are holomorphic. Let P T c −→ CP 1 be a one-parameter family of rank-two vector bundles determined by a patching matrix with H and H in (5.31) both equal to the identity matrix. From now on we shall identify the constant c with the speed of light, and refer to P T ∞ as the Newtonian twistor space.
and define inhomogeneous coordinates (Q, T ) and ( Q, T ) in pre-images of U and U in P T c by On the pre-image of U ∩ U in P T c we have Note that on P T ∞ there exists a global twistor function of weight 0, given by T = T . where Q and T are given by (5.34). If c is finite, then the conformal structure on M C is determined by det(dx AA ′ ) = 0, and the conformal factor is fixed by a two-form dQ ∧ dT on the fibres of P T c . If c = ∞, then the the conformal structure on Minkowski space degenerates in the following way: The equation δT = 0 implies that t = const, and in the Newtonian limit the null separation implies simultaneity of events. We also deduce the existence of a canonical closed one-form dT on P T ∞ which yields θ ≡ dt on M C . This is the clock of Newton-Cartan theory which defines a fibration of M C over the time axis

Restricting the inhomogeneous coordinates to a twistor line ω
The section of N (L p ) will have a single zero if the two roots of the remaining equation δQ = 0 coincide. Evaluating the discriminant of the quadratic form leads to a degenerate conformal structure on M C , which is determined by h = dx 2 +dy 2 +dz 2 on three-dimensional fibres of (5.35). The conformal factor is fixed by the canonical O(2)-valued one-form The relation between h ab and h ab is provided by a unique choice of a vector field U such that U θ = 1, and h ab U a = 0. This gives a degenerate contravariant metric on T * M C such that h ab θ a = 0. The discussion of the intrinsic definition of the connection is postponed to Sections 7.1 and 8.
Now we shall move on to discuss the reality conditions. The Euclidean real slice of the complexified Minkowski space M C is characterised by an involution σ : P T c → P T c given by so that σ 2 = Id on the projective twistor space. This involution has no fixed points, because on the non-projective twistor space it squares to minus identity, and so for any so that (x, y, z) are real. Similarly so that t = −t, and t = iτ is purely imaginary. Thus the real Newtonian twistor lines are where (x, y, z, τ ) are real coordinates on M .
To sum up, we have exhibited a realisation of the relativistic twistor space O(1) ⊕ O(1) as an affine line bundle over the total space of the line bundle O(2) → CP 1 . In the limit c = ∞ the holomorphic structure of P T c changes discontinuously: The total space of P T c , and the normal bundle of the twistor curves jump to O ⊕ O (2). Note that this jumping is a purely holomorphic phenomenon: the bundles P T c are all the same for any c finite or not from the topological perspective. Our findings agree with the analysis of [31] (as well as with the recent work [17]), where the jump from O(1) ⊕ O(1) to O ⊕ O(2) was interpreted as a singularity of the conformal structure.

5.2.2.
Non-relativistic incidence relation. Combining the (3 + 1) splitting (4.22) with the incidence relation (5.28) yields This equation blows up in the limit c → ∞ but the twistor function ω = ω A T A B ′ π B ′ remains finite. If a pair (x A ′ B ′ 0 , t 0 ) solves the twistor equation (5.38), then so does for any constant spinor ρ A ′ ∈ Γ(S ′ ). The image of the two dimensional twistor distribution (5.29) tangent to α-planes under (4.19) is In the Newtonian limit this gives an integrable 2-plane distribution (the mini-twistor distribution) on the constant time three-dimensional fibres of M → C The non-relativistic twistor space P T ∞ is the three-dimensional space of leaves of D in the correspondence space M C × CP 1 . To specify a leaf pick a point on the time axis, and choose a two plane (a point in a mini-twistor space O(2)) which is null with respect to the holomorphic metric dx 2 + dy 2 + dz 2 on the fibres of (5.35). Thus, in the Newtonian limit the α-planes become space-like, and they lie on the fibres of (5.35) (Figure 2).
as the last map is, in the spinor notation, given by clearly projecting onto O(2) ⊕ O, and the decomposition of C 4 into C 3 ⊕ C is given by (4.19). This is a complexified picture. If the reality conditions (4.15) and (5.37) are imposed, then the intersection of a complex α-plane with a real τ = const slice is a real straight line -a geodesic of the spatial metric -with a tangent vector π (A ′π B ′ ) (Figure 3).  Figure 3. A real section of the non-relativistic incidence relation. Oriented straight lines in the fibres of M → R correspond to points in P T ∞ . Two simultaneous events 1 and 2 in Newtonian space-time correspond rational curves L 1 , L 2 intersecting at two points in P T ∞ . A time-like geodesic between two non-simultaneous events 1 and 3 corresponds to a ruled surface in P T ∞ .
The twistor curves invariant under the involution (5.36) correspond to points in the flat Newtonian space-time M = R 4 . The points in P T ∞ correspond to oriented straight lines on the fibres of M → R. Thus two real curves L 1 and L 2 in P T ∞ either do not intersect, or intersect at two points Z and σ(Z) which correspond to a straight line (with two possible orientations) joining p 1 and p 2 in M .

Kodaira deformations
A way to introduce the curvature on M C in relativistic twistor theory is to deform the complex structure of P T c . The Kodaira theorems [18] guarantee that, under the additional assumption of the deformation being 'small', the deformed twistor space will still admit a four-dimensional moduli space M C of rational curves with normal bundle O(1) ⊕ O(1). The Nonlinear Graviton theorem of Penrose [25] then implies that M C admits a holomorphic conformal structure with anti-self-dual (ASD) Weyl tensor, and moreover that all ASD conformal structures correspond to some deformed twistor spaces. If the deformation preserves the fibration of P T c over CP 1 , and the symplectic two-form ε AB dω A ∧ dω B on the fibres of this fibration, then the resulting conformal structure contains a Ricci-flat metric.
In this section we shall show that the Kodaira deformation theory, when applied to a Newtonian twistor space P T ∞ , in general leads to the jumping lines phenomenon: the deformed twistor curves have normal bundle O(1) ⊕ O(1), and thus M C carries a nondegenerate metric, rather than a Newton-Cartan structure. We can phrase it by saying that the Newtonian space-times are unstable under the general Kodaira deformations, but we should note that this instability is a purely holomorphic feature of the underlying twistor space, and not a dynamical process.
where t does not depend on π by the Liouville theorem and thus can be used as a coordinate on the moduli space of curves in L. The deformed CP 1 s (in a patch containing λ = 0) are Here ι A ′ is a spinor (a choice of which arises because of the non-uniqueness of the splitting). The contours Γ and Γ are homologous to the equator of the twistor curve L p = CP 1 in an intersection of two open sets covering L p , and such that Γ− Γ surrounds the point ρ A ′ = π A ′ in L p (Figure 4). The Ward transform [33] of L gives is not a twistor distribution, as it does not Lie-derive ω. The deformed twistor distribution (also called a Lax pair) is where e A ′ B ′ are four vector fields on M C . This is a jumping line phenomenon: the deformation changes the normal bundle to O(1) ⊕ O(1). The vector fields in the Lax pair (6.44) give rise, by e A ′ B ′ e A ′ B ′ to a conformally rescaled Gibbons-Hawking metric. Imposing the reality conditions t = iτ , where τ ∈ R and choosing a conformal factor V gives the ASD, Ricci-flat metric g = V (dx 2 + dy 2 + dz 2 ) + V −1 (ǫ −1 dτ + A) 2 . Therefore the deformation (6.39) does not preserve the holomorphic type of twistor curves. This was to be expected as the general deformations will not preserve the type of the Newtonian normal bundle N = O ⊕ O(2). This is because H 1 (CP 1 , End(N )) = 0. If we regard a section of N as a column vector with first entry taking values in O and the second entry taking values in O(2), the endomorphisms of N are represented by two by two matrices of the form α β γ δ (6.45) O(2)). Thus as the second entry in the sum does not vanish.

6.2.
A twisted photon construction for Newtonian space-times. The argument above shows that general Kodaira deformations of P T ∞ do not preserve the type of the normal bundle. We have seen that a deformation which does not preserve the one-form dT certainly leads to jumping lines. We can look at the restricted class of deformations which preserve dT , and thus do not change the normal bundle. This requires the coefficient β in the matrix (6.45) to vanish. One way to achieve it is to take the twistor space to be a product C × Z, where Z is a two-dimensional complex manifold with a three-parameter family of curves with normal bundle O(2). Thus Z is a mini-twistor space for a general Einstein-Weyl structure in three dimensions [16], and the corresponding M is a product of an Einstein-Weyl three-manifold with a line. This is not the structure we are seeking, as the metric (or in fact a conformal structure) on the three-dimensional spatial slices of M will in general be curved, whereas we require it to be flat. In the remainder of this section we shall explore another possibility, and deform the relation between a projective and nonprojective twistor space in a way which preserves the one-form dT . This is analogous to the construction of [5] which in turn was motivated by Ward's twisted photon construction [33]. The canonical bundle of P T ∞ restricted to a twistor line is O(−4). The argument works for P T c with any non-zero c, as restricting the canonical bundle κ → P T c to a line L p with a normal bundle N (L p ) gives Let us assume that the line bundle L = κ * ⊗ O(−4) is non-trivial on P T ∞ , but trivial when restricted to twistor lines, and deform the twistor space in a way which preserves the fibration over CP 1 . Thereforeπ where f =h−h is the cohomology class defining L which gives rise to the abelian monopole (6.41). This cohomology class coincides with that from Section 6.1 and its splitting is given by (6.40). The deformed twistor distribution Lie-derives e −h π A ′ . To this end we note that the Υ term can be removed from the Lax pair by a Mobius transformation of π iff Φ is a gradient: In this case the deformed twistor distribution in Frobenius integrable for any function U .

Newton-Cartan connections from holomorphic vector bundles
For a fixed degenerate metric h, the Newton-Cartan connections correspond to closed two-forms F , which (given a canonical 3+1 splitting induced by the one-form θ) have well defined electric and magnetic parts (compare formulae (2.4) and (2.8)). Thus, instead of attempting to construct a non-metric connection from holomorphic Kodaira deformations of P T ∞ , we can identify the connection with a couple of gauge fields on M , and construct it from a generalisation of Ward transform [33] adapted to Newtonian settings. The gauge fields needed to reconstruct the gravitational and Coriolis parts of the connection consist of a couple of abelian monopoles. The one-forms in the monopoles will give rise to an electric and magnetic fields on M , and these fields will in turn give rise to a connection. This is a legitimate approach, as electromagnetic forces are inertial in the Newton-Cartan theory: the equations of motion of a charged particle under the influence of the electric field E and magnetic field 2Bẍ = E + 2B ∧ẋ (7.48) can be reinterpreted as a geodesic motion corresponding to a connection where now E is the gravitational acceleration, and B is the Coriolis force -compare (2.8).
7.1. Gravitational force from the Ward correspondence. Let us first assume that the Coriolis term is absent, and the gravitational force is identified with the electric field E = −∇V as in Section 2.1. The harmonic function V on R 3 could be constructed from a Newtonian limit of the contour integral formula (5.25). We shall instead proceed in a less ad hoc way. In Section 3.1.2 we have demonstrated that any Newtonian connection with the only non-zero components given by Γ i τ τ = δ ij ∇ j V arises as a Newtonian limit of the Gibbons-Hawking metric (3.10). The twistor space of (3.10) is an affine line bundle over the total space of O(2) described by a cohomology class f ∈ H 1 (CP 1 , O). This is in fact what we have recovered in our analysis in Section 6.1 leading to the jumping phenomenon and the Kodaira instability of the Newtonian twistor space. Let us assume that the cohomology class is represented by a function f = f (Q, λ, T ) in a patch (5.32) in P T c , and define with f = 0 giving the patching relation for the undeformed relativistic twistor space P T c (compare 5.33).
Using the cohomology class f to deform P T ∞ would lead to jumping lines, so instead we use f to construct a line bundle ν : E → P T ∞ which is trivial on twistor curves. Let be a covering of P T ∞ and let Example. Consider a one-parameter family of Gibbons-Hawking metrics with linear potential [7,14] g = (1 + ǫz)(dx 2 + dy 2 + dz 2 ) + 1 ǫ(1 + ǫz) where ǫ = c −2 . This family is ASD and Ricci flat for all ǫ > 0. In the limiting case ǫ → 0 the metric blows up, the inverse metric degenerates to δ ij , but the Levi-Civita connection has the finite limit with the only non-zero components given by Γ z τ τ = 1/2. The twistor function gives rise to the monopole (6.42) with

The Coriolis force from the Newtonian Sparling equations.
We shall now consider a more general situation, where the Coriolis term corresponding to a magnetic field B in (7.48) is present. The field B is a gradient of a harmonic function W , and (assuming that B is time-independent) E = −∇V is also a gradient, but V is not necessarily harmonic. However W 2 + V is harmonic. Let M C be a moduli space of holomorphic sections of the Newtonian twistor space P T ∞ = O ⊕ O(2) → CP 1 , and let be the five dimensional correspondence space. Here the twistor line L p = CP 1 is a rational curve in P T ∞ with normal bundle O ⊕ O(2) corresponding to p ∈ M C . The correspondence space can be identified with the projective spin bundle P(S ′ ). Let µ : S ′ → F denote the corresponding fibration. Thus a pre-image of any twistor line in F can be pulled back to Consider a vector bundle E → P T ∞ such that where n > m ≥ −1. Therefore the restriction of E to a twistor line is a non-trivial vector bundle, and the pull back to a pre-image of L p in F of the patching matrix of E can not be split in a way which leads to the Ward twistor correspondence [33]. However pulling back E to the total space of S ′ , and restricting it to a pre-image C 2 − {0} of L p in S ′ is a trivial bundle. The corresponding pulled-back GL(2, C) patching matrix F = F (Q, π A ′ , T ), where Q is given by ( Note that H andH are not homogeneous on C 2 −{0} and so do not descend down to F. Let be the a rank-two twistor (α-plane) distribution on S ′ , such that the three-dimensional space of leaves of the rank-three distribution functions on the fibres of S ′ , with the coefficients given by functions on M . Let us rewrite this condition as (7.50) Here (v −m , r −n ) and (w −m , s −n ) denote a pair of local sections of E, where (v −m , w −m ) are homogeneous of degree −m and (r −n , s −n ) are homogeneous of degree −n when regarded as functions on the fibres of S ′ → M . The polynomials (φ A ′ , ψ A ′ , κ A ′ , δ A ′ ) will give rise to potentials for higher spin fields on M , i.e. sections of various powers of S ′ → M satisfying some field equations [30]. To construct these fields, and find the corresponding field equations rewrite (7.50) as Therefore, as ∇ A ′ is homogeneous of degree one, we conclude that φ A ′ and ψ A ′ are homogeneous of degree one, κ A ′ is homogeneous of degree (n − m + 1), and finally δ A ′ = 0. Equivalently κ ∈ Γ(S ′ ⊗ Sym n−m+1 (S ′ )) = Γ(Sym n−m+2 (S ′ ) ⊕ Sym n−m (S ′ )).
The SL(2, C) irreducible components give rise to two spin-2 fields, two functions, one spin-(n − m) field and one spin (n − m + 2) field.
To find the field equations satisfied by these potentials, we shall impose the integrability conditions on (7.51) arising from ∇ A ′ ∇ A ′ = 0. Contracting each equation in (7.51) with ∇ A ′ and using (7.51) to eliminate the derivatives of H gives Decomposing this last set of equations into irreducible parts yields the final system Thus U and W are two harmonic functions, which give rise to electric and magnetic fields in (2.8) by If the gauge group is taken to be SL(2, C),

A relation to Penrose-Ward correspondences.
Let P T c be a twistor space corresponding to a flat space-time with the speed of light c. Thus, if c = ∞, then P T c is the total space of a rank-two vector bundle O(1)⊕O(1), and P T ∞ is the O ⊕O(2) Newtonian twistor space.
As a special case of the construction presented in the previous section, consider a vector bundle E → P T c , such that where N (L p ) is the holomorphic normal bundle to L p ⊂ P T c . Let κ be the canonical bundle of P T c , so that κ| Lp = O(−4): this true both for finite, and infinite c (as discussed in Section 6.2). Define a rank-two vector bundle If c = ∞, then E is trivial when restricted to twistor lines, and its Ward correspondence gives a solution to anti-self-dual Yang-Mills field on the complexified Minkowski space. In the Newtonian case corresponds to n = 1, m = −1 in (7.50). The resulting potentials are two abelian monopoles, one spin-4 field and one spin-2 field subject to equations (7.52). One can now identify the field strengths of these potentials with various components of the Newton-Cartan connection. Note that it is consistent to set γ A ′ B ′ C ′ D ′ = 0.

Newtonian connections from Merkulov's relative deformation theory
The original Non-Linear Graviton construction of Penrose [25] yields Ricci-flat ASD metrics from the holomorphic geometry of associated twistor spaces. In a far reaching generalisation of this construction Merkulov [20,21] developed a twistor correspondence between complete analytic families of complex submanifolds on a given complex manifold, and a class of torsion-free affine connections (which he called the Λ-connections) on the moduli space of these submanifolds. This has subsequently led to the celebrated solution of the holonomy problem [22].
In this Section we shall adapt Merkulov's construction to the Newtonian twistor correspondence, and show that the twistor space P T ∞ introduced earlier in this paper leads to a family of Newton-Cartan connections depending on five arbitrary functions. Two of these functions can be set to zero by fixing the overall conformal factors in the degenerate Newtonian metric h , and the associated clock one-form θ in the kernel of h. The remaining three functions correspond to the gravitational force. By analysing the non-relativistic limit of the Gibbons-Hawking twistor space we shall show that the gravitational force arises from a harmonic gravitational potential which is given by a choice of a holomorphic linebundle over P T ∞ trivial on twistor lines. This relates the approach of this Section to our construction from Section 7.1.
Let J p k be an ideal of holomorphic functions on M C which vanish to order k at p ∈ M C . Therefore T p M C = (J p /J p 2 ) * . The second-order tangent bundle T [2] M C is defined as a union over all points in M C of second order tangent spaces T [2] p M C ≡ (J p /J p 3 ) * . Thus an element of (T [2] p M C ) * consists of the first two non-zero terms of a Taylor expansion of a function vanishing at p, and a section of T [2] M C is a second order linear differential operator V [2] = V a (x)∂ a + V ab (x)∂ a ∂ b . We also have an exact sequence , 0), and (V a , V ab ) → V ab . A torsion-free affine connection ∇ on T M C is then equivalent to a linear map γ : T [2] M C → T M C , (8.53) where γ(V [2] ) = (V a + Γ a bc V bc )∂ a for some functions Γ a bc which are to be identified with the Christoffel symbols of ∇.
In [20,21] Merkulov proposed a twistor construction of a class of maps (8.53). In some cases (e.g. the Nonlinear Graviton construction) Merkulov's construction leads to a unique connection. We will show that in the case of the non-relativistic twistor theory a unique of Newtonian connection in Merkulov's class is provided by a natural choice of a holomorphic line bundle over P T ∞ .
The general construction, adapted to three-dimensional complex manifolds P T c which fiber holomorphically over CP 1 , and admit a moduli space M C of holomorphic curves can be summarised as follows. Let U and U be a covering of P T c by two open sets which is a deformation of the covering (5.33). Thus (Q, T, λ) are holomorphic coordinates in U and ( Q, T ,λ) are holomorphic coordinates on U . The patching relation on the overlap U ∩ U is We shall make use of the Kodaira isomorphism, which relates vector fields on M C to global sections of the normal bundle N F to the correspondence space where L p = β • α −1 (p) = CP 1 is the rational curve in P T c corresponding to a point in M C . The patching for this normal bundle is given by where the subscript F means the pull-back to the correspondence space. To exhibit the Kodaira isomorphism take a vector field given by a differential operator V = V a ∂ a on F, and apply it to the twistor functionsw A (x a , λ) to obtain a global section of N F given by The map (8.53) arises by taking a section of T [2] M and similarly applying the associated differential operator to the twistor functions. We find The term involving F A BC is an obstruction to (8.56) representing a global section of N F ; only if F A BC splits as In solving the splitting problem (8.57) we have constructed a global section of N F out of a section of T [2] M , and via the Kodaira isomorphism we have therefore constructed a vector field out of a section of T [2] M . This is the map (8.53), and we can read off the connection components from One can check that Γ c ab transforms correctly as an affine connection under a change of coordinates.
The splitting problem (8.57) is solvable for any patching in the isomorphism class N F iff The procedure is however non-unique (and hence leads to a family of connections) unless Example. Merkulov's connection for ASD plane waves. To illustrate how this procedure works in practice we will consider the simple twistor space (originally due to Sparling) described by the infinitesimal deformation of the flat relativistic twistor space P T c given by (A4), where f = (ω 0 ) 4 /(4π 0 ′ π 1 ′ ). Integrating the deformation equations and introducing the inhomogeneous coordinates on the deformed twistor space by w A = ω A /π 1 ′ on U and w A =ω A /π 0 ′ on U yields The holomorphic splitting of these relations, together with the Liouville theorem imply the existence of coordinates (w, z, x, y) on M C such that the twistor curves pulled back from U to F are The ASD Ricci-flat metric resulting from the Nonlinear Graviton construction is of the form (3.12) for γ = −3ǫy 2 . We shall find the connection on M C directly. We find and all other F A BC = 0. The splitting problem (8.57) can be solved uniquely to givẽ σ 1 00 = −6ǫw, σ 1 00 = 6ǫy, and all other {σ A BC } = 0, leading to a connection whose only non-vanishing components are We have recovered the Levi-Civita connection of (3.12). The condition F A BC = 0 implies that the connections will be constructed only out of global sections of N F ⊗ ⊙ 2 N * F . In the relativistic case this would imply Γ c ab = 0, but for P T ∞ we have Therefore we expect Γ c ab to depend on five arbitrary functions. The global sections of N F ⊗ ⊙ 2 N * F which constitute the Newtonian 0-cochain {σ A BC } ∞ are of the form (with all other components of {σ A BC } ∞ set to zero) for five arbitrary functions (E A ′ B ′ , χ, Σ) on M C . Using (8.1) and (8.58), and imposing the reality conditions (5.37) yields the nonvanishing connection components where we used the isomorphism (4.18) to replace E A ′ B ′ ∈ Γ(S ′ ⊙ S ′ ) by a spatial vector E i ∈ Γ(T M C ). The vector E i yields the gravitational attraction, whilst Σ and χ are consistent with ∇h = 0 and ∇θ = 0, where the degenerate metric h, and clock one-form θ are defined up to some conformal factors. When we fix these conformal factors as in Section 5.2.1, so that h = diag(0, 1, 1, 1, ) and θ = dτ , then Σ = χ = 0. In particular, we note that this construction cannot generate a Coriolis force. What remains, then, is a connection whose non-zero Γ i τ τ components are determined by a global section {σ Q T T } ∞ of O(2). Using the Serre duality between the elements of H 1 (CP 1 , O(−4)) and H 0 (CP 1 , O(2)) we can fix this connection by where g (−4) is an element of H 1 (P T ∞ , O(−4)) restricted to a twistor line (5.34). The resulting E A ′ B ′ satisfies the zero-rest-mass field equation In what follows we shall demonstrate that the expression (8.64) arises naturally from a limiting procedure applied to the unique splitting of the 0-cochain corresponding to the Gibbons-Hawking metric. 8.2. Newtonian limit of the Gibbons-Hawking cochain. As discussed in Section 7.1 we can express the twistor space P T c for a Gibbons-Hawking metric (3.10) with ǫ = 1 c 2 as the holomorphic deformation (7.49) of P T ∞ . We then have where g is an element of H 1 (Z, O(−2)) restricted to a line L p ∼ = CP 1 in Z = O(1). The complex two fold Z is the twistor space of the flat holomorphic projective structure on a two-dimensional space M. If (ζ, u) (with ζ = x + iy) are local coordinates of a point p ∈ M, then the corresponding line L p ⊂ Z is given by η = ζ + λu, where η is a coordinate on the fibres of Z → CP 1 . The formula (A3) is also valid if m = 0, where the real and imaginary parts of ψ(ζ) are harmonic functions on R 2 depending on a parameter u. In this case (A3) is the Radon transform of g. Let us give an example, where m = 0. Let g(η, λ) = η −1 , and let Γ be a circle centred at the origin of the λ-plane. Then , which is indeed a solution of the (2 + 1) Schrödinger equation (A2). The contour integral formula (A3) is, on dimensional grounds, only tangentially related to the rest of this paper where the non-relativistic theories in (3+1)-dimensions are considered (hence its place in the Appendix). It may however be relevant in Newton-Cartan theories in (2 + 1) dimensions [1,29].

Appendix 2. Spin connection as gauge field
In the original Nonlinear Graviton construction [25], and its modification involving nonzero cosmological constant [34] the spin connection on S → M C can be constructed directly from the Ward correspondence applied to a certain rank-two sub-bundle of the tangent bundle of the twistor space. This construction is mentioned in [34], but not implemented explicitly. Below we show how carry the construction over in a way, which at the linearised level, agrees with the spin connection arising for Plebanski 2nd heavenly equations [27].
Let κ → P T be the holomorphic canonical line bundle of the twistor space which restricts to O(−4) on twistor lines. If P T corresponds to an ASD Einstein metric (with or without Λ), then there exists a one-form T (which is given by π A ′ dπ A ′ in the vacuum case.) This one-form defines a rank-two sub-bundle E ⊂ T (P T ) consisting of vectors such annihilated by T . The bundle E restricts to N on each twistor line. The bundle E ⊗ κ 1/4 is therefore trivial on twistor lines, and (by the standard Ward transform [33]) corresponds to an ASD gauge field: a spin connection on S. This agrees with the gauge field arising from the Sparling equation in Section 7.2.
Both spin bundles have a twistorial construction as S p = Γ(L p , E ⊗ O(−1)), S ′ p = Γ(L p , O(1)), where L p = CP 1 is a twistor curve corresponding to p ∈ M C . Assume that the ASD metric on M C is vacuum, and so the twistor space fibres over CP 1 . Let f ∈ H 1 (CP 1 , O(2)). To construction the connection on S consider the inifinitesimal deformatiõ and let U ∈ Γ(E), so that U = U AA ′ π A ′ ∂/∂ω A . The relatioñ To construct the splitting F = HH −1 consider H = 1 + ǫ G, H = 1 + ǫG so that, to the first order in ǫ, The freedom in splitting the RHS is measured by elements of H 0 (CP 1 , O). Choosing a constant spinor ι A ′ yields The usual Liouville argument gives for some matrix Γ AA ′ which does not depend on π A ′ . Equivalently Therefore the linearised Ward transform gives a connection This is consistent, in the linearised 2nd Plebanski gauge, with the expression for the ASD Weyl spinor as [6] (in Plebanski's gauge the quadratic term in the connection on S contracts to zero, and the connection on S ′ vanishes.).