Twistor Theory of Dancing Paths

Given a path geometry on a surface , we construct a causal structure on a four-manifold which is the configuration space of non-incident pairs (point, path) on . This causal structure corresponds to a conformal structure if and only if is a real projective plane, and the paths are lines. We give th...

Full description

Saved in:

Bibliographic Details
Published in:	Symmetry, Integrability and Geometry: Methods and Applications
Date:	2022
Main Author:	Dunajski, Maciej
Format:	Article
Language:	English
Published:	Інститут математики НАН України 2022
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/211641
Tags:	Add Tag No Tags, Be the first to tag this record!
Journal Title:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Cite this:	Twistor Theory of Dancing Paths. Maciej Dunajski. SIGMA 18 (2022), 027, 13 pages

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine

_version_	1860247355977105408
author	Dunajski, Maciej
author_facet	Dunajski, Maciej
citation_txt	Twistor Theory of Dancing Paths. Maciej Dunajski. SIGMA 18 (2022), 027, 13 pages
collection	DSpace DC
container_title	Symmetry, Integrability and Geometry: Methods and Applications
description	Given a path geometry on a surface , we construct a causal structure on a four-manifold which is the configuration space of non-incident pairs (point, path) on . This causal structure corresponds to a conformal structure if and only if is a real projective plane, and the paths are lines. We give the example of the causal structure given by a symmetric sextic, which corresponds to an SL(2, ℝ)-invariant projective structure where the paths are ellipses of area π centred at the origin. We shall also discuss a causal structure on a seven-dimensional manifold corresponding to non-incident pairs (point, conic) on a projective plane.
first_indexed	2026-03-21T05:15:25Z
format	Article
fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 027, 13 pages Twistor Theory of Dancing Paths Maciej DUNAJSKI Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK E-mail: m.dunajski@damtp.cam.ac.uk URL: https://www.damtp.cam.ac.uk/user/md327/ Received January 14, 2022, in final form March 28, 2022; Published online March 31, 2022 https://doi.org/10.3842/SIGMA.2022.027 Abstract. Given a path geometry on a surface U , we construct a causal structure on a four-manifold which is the configuration space of non-incident pairs (point, path) on U . This causal structure corresponds to a conformal structure if and only if U is a real projec- tive plane, and the paths are lines. We give the example of the causal structure given by a symmetric sextic, which corresponds on an SL(2,R)-invariant projective structure where the paths are ellipses of area π centred at the origin. We shall also discuss a causal struc- ture on a seven-dimensional manifold corresponding to non-incident pairs (point, conic) on a projective plane. Key words: path geometry; twistor theory; causal structures 2020 Mathematics Subject Classification: 32L25; 53A20 Dedicated to Roger Penrose on the occasion of his 90th birthday 1 Introduction It is a great honor and pleasure to be able to contribute to the celebrations of Roger’s 90th birthday. Roger is now twice as old as his nonlinear graviton construction [17]. When I arrived in Oxford as a graduate student, it was already known that the integrability of the nonlinear solitonic systems has its roots in the Ward transform for the anti-self-dual Yang–Mills equa- tions [16, 20]. I spent three years of my PhD showing, under Lionel Mason’s supervision, that the curved space twistor theory behind the nonlinear graviton construction underlies the integrability of the complementary class of the, so called, dispersionless integrable systems [8]. Roger made several comments to me about my work, but on average it took me three months to understand his geometric insight, and another three to realise that he was right. At that time Roger had already moved to the notoriously difficult googly problem, as well as the interplay between gravity and quantum mechanics. The photograph on Figure 1 was taken at the GRG meeting in Pune, India in 1997, where Roger gave a public lecture about the latter. The lecture proved to be so popular among the citizens of Pune that the lecture hall had to be sealed off by riot police, to prevent overcrowding. The nonlinear graviton construction describes a link between algebraic geometry of holomor- phic curves in a complex three-fold (the twistor space), and differential geometry on the moduli space M of these curves. Two curves in three dimensions generically do not intersect (this is backed by a geometric intuition based on curves in R3 which carries over to the twistor space, as the curves are holomorphic). The intersection condition defines a notion of null separation: This paper is a contribution to the Special Issue on Twistors from Geometry to Physics in honor of Roger Penrose. The full collection is available at https://www.emis.de/journals/SIGMA/Penrose.html mailto:m.dunajski@damtp.cam.ac.uk https://www.damtp.cam.ac.uk/user/md327/ https://doi.org/10.3842/SIGMA.2022.027 https://www.emis.de/journals/SIGMA/Penrose.html 2 M. Dunajski Figure 1. Pune, December 1997. From the left: Devendra Kapadia, George Sparling, Maciej Dunajski, Roger Penrose, Jerzy Lewandowski. two points on M are null separated iff the corresponding curves in the twistor space intersect. All curves intersecting a fixed curve C form a hyper-surface inM , consisting of points null sepa- rated from C. The assumptions made about the normal bundle of twistor curves in the nonlinear graviton construction ensure that this null separation condition leads to quadratic light cones on TM , and therefore defines a conformal structure. Dropping some of these assumptions leads to causal structures, where the associated null cones are no-longer quadratic, but instead of higher order [11, 12, 13, 15]. In this paper I give examples of such structures which result from path geometries on two-dimensional surfaces. These geometries, as well as their duals, generalise the Klein duality between RP2 and ( RP2 )∗ . 2 Summary of the results 2.1 Path geometry A path geometry u on a two-dimensional surface U is a family of unparametrised smooth curves, from now on called paths, one through each point of U in each direction. If the paths of (U , u) are unparametrised geodesics of a torsion-free affine connection ∇, then the path geometry is called projective. The corresponding projective structure is an equivalence class [∇] of torsion-free affine connections sharing their unparametrised geodesics with ∇. If at least one connection in [∇] is the Levi-Civita connection of some (pseudo) Riemannian metric g, then the corresponding path geometry is called metrisable. The projective path geometries were invariantly characterised more than a century ago [6, 14], but the necessary and sufficient conditions for a path geometry to be metrisable were only given relatively recently [5]. Twistor Theory of Dancing Paths 3 A dual path geometry (Z, z) is a surface Z whose points correspond to the paths of U . The paths of z correspond to points of U . This gives rise to a double-fibration picture [4, 6] U µ←− I ν−→ Z, (2.1) where I = {(u ∈ U , z ∈ Z), u ∈ z} ⊂ U × Z is the three-dimensional incidence space. Both projections in (2.1) are submersions, and for all incident pairs (u, z) ∈ I the curves µ−1(u) and ν−1(z) meet transversally, and their tangents span a two-plane in T(u,z)I which gives rise to a contact structure on I. 2.2 Dancing pairs Let M = (U ×Z) \I be the four-dimensional manifold of non-incident pairs. A point in m ∈M consists of a point u in U and a path z of u which does not contain u. Definition 2.1. Two points (u1, z1) and (u2, z2) inM are dancing if there exists a path in (U , u) which contains three points: u1, u2, and the intersection z1 ∩ z2. The terminology is taken from [3], where the term dancing is introduced (and motivated) at an infinitesimal level and directly leads to the Definition 2.2. The Definition 2.1 makes sense both locally, and globally on M . In the latter case the paths z1 and z2 may intersect at more than one point (e.g., if U is the round sphere, and the paths are the great circles). In this case we require that there exists a path containing u1, u2 and at least one point of intersection of z1, z2. The dancing condition could of course equivalently have been defined in terms of the dual path geometry (Z, z). The simplest example is provided by the classical projective duality, where U = RP2 and Z = ( RP2 )∗ . Both path geometries u and z are projective, and metrisable (in six different ways) by metrics of constant curvature. In this case the dancing condition gives rise to a neut- ral signature conformal structure on M = SL(3,R)/GL(2,R) which contains an anti-self-dual Einstein metric with non-zero Ricci scalar, and the isometry group SL(3,R) [3, 7, 9]. We shall review this example in Section 3. In general it is convenient to work with the infinitesimal dancing condition [3], where u1 = u, u2 = u + ϵu̇ + · · · and z1 = z, z2 = z + ϵż + · · · . The tangent vector (u̇, ż) ∈ T(u,z)M describes the direction of a path through u, and the turning point of the path z (i.e., the point of intersection of z with a nearby path). Definition 2.2. A tangent vector (u̇, ż) ∈ T(u,z)M is called null if the path through u in the direction of u̇ contains the turning point of z. In Section 4 we shall show how the infinitesimal dancing condition can put in the form L(u, z, u̇, ż) = 0, (2.2) where L : TM → R. In the projectively flat case where U = RP2 and M = SL(3)/GL(2) described above the function L is homogeneous of degree two in (u̇, ż), and gives a null-geodesic Lagrangian for a conformal structure on M . One may wonder whether other examples of path geometries (U , u) give rise to conformal structures. The answer to this is negative, and is provided by the following local (and therefore also global) rigidity theorem Theorem 2.3. If the dancing condition on a path geometry U defines a conformal structure on M = (U × Z) \ I, then U = RP2 with its flat projective structure, and the conformal structure is represented by the SL(3)-invariant anti-self-dual Einstein metric on M = SL(3)/GL(2). We shall prove this theorem in Section 4 by twistor methods. In Section 5 we shall consider a path geometry arising from an SL(2,R) invariant projective structure. We shall prove 4 M. Dunajski Theorem 2.4. Let U = R2 and let Z be the set of ellipses in U centred at the origin and of area π. The dancing condition on M = (U × Z) \ I defines an SL(2,R)-invariant sextic (2.2) on TM which is quartic in u̇ and quadratic in ż. The path geometry (U , u) from Theorem 2.4 is projective and metrisable. The dual path geometry is not projective. Finally in Section 6 we shall move away from path geometries, and instead consider a seven- dimensional manifold N ⊂ P2 × P5 consisting of non-incident pairs (a,A), where a ∈ P2 is a point, and A ⊂ P2 is an irreducible conic not containing a. We shall say that two pairs (a,A) and (b, B) of points-conics are dancing if there exists an irreducible conic containing (a, b) and the four points A ∩B of intersections between A and B. We shall prove Theorem 2.5. Let A, B be 3 by 3 matrices representing non-singular conics in RP2, and let a, b be vectors in R3 \ {0} representing points in RP2. Two pairs (a,A) and (b, B) are dancing iff( aTAa )( bTBb ) − ( aTBa )( bTAb ) = 0. The infinitesimal dancing condition gives rise to a conformal structure on TN of signature (++−− 000) which is degenerate along the fibres of a projection N → ( P2× ( P2 )∗) \ I sending the conic A to the polar line of (a,A). 3 Dancing on RP2 Let ( U = RP2, u ) be a path geometry corresponding to a flat projective structure on the projec- tive plane. The dual path geometry ( Z = ( RP2 )∗ , z ) is also flat, and the double fibration (2.1) is just the classical Klein projective duality between lines in RP2 and points in ( RP2 )∗ . In this section we shall represent a point u ∈ U by an equivalence class of vectors P ∈ R3, the components of a vector being the homogeneous coordinates. Similarly a point z ∈ Z will be represented by a vector L corresponding to a line in RP2. Let M ⊂ P2 × ( P2 )∗ be set of non-incident pairs (P,L), where P ∈ P2, and L ⊂ P2 is a line. Two pairs (P,L) and ( P̃ , L̃ ) are null-separated in the sense of the Definition 2.1 if there exists a line which contains the three points ( P, P̃ , L ∩ L̃ ) . This null condition defines a co-dimension one cone in TM : generically there is no line through three given points. The action of SL(3,R) given by (P,L)→ ( AP,LA−1 ) , where A ∈ SL(3,R) is transitive on M , and a group stabilising a non-incident pair (point, line) is GL(2,R) which sits in SL(3,R) as a lower-diagonal block. Therefore M can be identified with SL(3,R)/GL(2,R). �� l 1 p 1 p l 2 2 Figure 2. Dancing pair. Twistor Theory of Dancing Paths 5 Proposition 3.1. There exist affine coordinates ( x0, x1, ζ0, ζ1 ) on M = SL(3)/GL(2) such that two points are dancing if and only if they are null separated with respect to the anti-self-dual Einsten metric g = dζ0dx 0 + dζ1dx 1 + (ζ · dx)2, where ζ · dx ≡ ζ0dx0 + ζ1dx 1. (3.1) Proof. To find an analytic expression for the conformal structure resulting from the dancing condition consider two pairs (P,L) and ( P̃ , L̃ ) of non-incident points and lines. Let L + tL̃ be a pencil of lines. There exists t such that P · ( L+ tL̃ ) = 0, P̃ · ( L+ tL̃ ) = 0. (3.2) Eliminating t from (3.2) gives (P · L) ( P̃ · L̃ ) − ( P̃ · L )( P · L̃ ) = 0. Setting P̃ = P + dP , L̃ = L+ dL yields a metric g representing the conformal structure g = dP · dL P · L − 1 (P · L)2 (L · dP )(P · dL). We can use the normalisation P · L = 1, so that P · dL = −L · dP , and g = dP · dL+ (L · dP )2. In [9] the affine coordinates ( x0, x1, ζ0, ζ1 ) were introduced on M such that P = [ x0, x1, 1 ] , L = [ ζ0, ζ1, 1− x0ζ0 − x1ζ1 ] with a normalisation P · L = 1. The resulting metric is (3.1). ■ In [9] the metric (3.1) was shown to be Einstein and anti-self-dual. Its isometry group is SL(3,R), and the underlying embedding of sl(3) into sl(4) was constructed in [10]. The metric was analysed (albeit in different coordinates) in the dancing context in [3], and in connection with path geometries in [7]. Let the flag manifold I ≡ F12 ( C3 ) ∈ P2 × P2∗ be the set of incident pairs (p, l), such that p · l = 0. This is the twistor space [2, 19] of (M, g). A P1 of (p, l) ∈ I corresponding to a point (P,L) consists of all lines l through P , and all points p = l ∩ L: P · l = 0, p · L = 0, p · l = 0. Let (P,L) and ( P̃ , L̃ ) be null separated. The corresponding lines in F12 intersect at a point (p, l) given by p = L ∧ L̃, l = P ∧ P̃ , where [ L ∧ L̃ ]γ = ϵαβγLαL̃β etc. The incidence condition p · l = 0 now gives the conformal structure (3.2). This is an illustration of Penrose’s nonlinear graviton construction [17], adapted to the non-zero cosmological constant [19]. 4 Dancing condition for general path geometries The incidence relation encoding the double fibration (2.1) is given by a map ψ : U × Z → R, where I is the inverse image ψ−1(0) ⊂ U × Z. Let (u, z) ∈ M , i.e., ψ(u, z) ̸= 0, and let (u̇, ż) ∈ T(u,z)M . The infinitesimal dancing condition is equivalent to the existence of a pair (u∗, z∗) such that ψ(u, z∗) = 0, ψ(u∗, z) = 0, ψ(u∗, z∗) = 0, u̇ ∂ψ ∂u (u, z∗) = 0, ż ∂ψ ∂z (u∗, z) = 0, 6 M. Dunajski where z∗ is the path in the direction u̇ containing u and the turning point u∗ of the path z. Here the notation u̇∂/∂u denotes a differentiation along a vector field. We now eliminate z∗ between the 1st and the 4th condition, and eliminate u∗ between the 2nd and the 5th condition. Substituting the resulting expressions to the 3rd condition gives one relation between (u, z, u̇, ż) which is of the form (2.2). We shall say that a two-dimensional surface σ ⊂M is totally null if all tangent vector fields in Tmσ,m ∈M are null in the sense of Definition 2.2. Lemma 4.1. There exists a three-parameter family of totally null surfaces (call them α-surfaces) in M which correspond to points in I. For any (l, p) ⊂ I the corresponding α-surface consists of (L,P ) ∈M such that P ⊂ l, p ⊂ L, P ̸= p, L ̸= l. (4.1) Proof. The points in I clearly correspond to surfaces in M : there is a one-dimensional family of P s satisfying (4.1), and once P has been fixed, there is a one dimensional family of Ls. To show that these surfaces are null note that given (L,P ) and ( L̃, P̃ ) on the α-surface (l, p), the points ( P, P̃ , L ∩ L̃ ) all belong to the line l in agreement with the Definition 2.1. ■ � � � � � � �� l p Figure 3. α-surface. In addition to the α surfaces M also admits two two-parameter families of null surfaces (call them β-surfaces) defined as follows. The first family β1 consist of a fixed point P ∈ U (hence two parameters), and all lines L in U not containing P . The second family β2 consist of a fixed line L ⊂ U and all points P not on L. If we have instead chosen to model M on the points and paths in (Z, z), then the roles of β1 and β2 would swap, but the notion of α surfaces would stay the same. The result of Lemma 4.1 fits into the framework of half-flat causal structures [13, 15], where half-flatness is defined by the existence of a three-parameter family of surfaces whose tangent planes at each point give the ruling for the (not neccesarilty quadratic) null cone at that point. This terminology is motivated by the paradigm example of Penrose [17], where the cones are quadratic, and half-flatness condition is equivalent to the anti-self-duality of the Weyl tensor. We are now ready to prove Theorem 2.3, and thus establish the uniqueness of the dancing metric. Proof of Theorem 2.3. If there exists a conformal structure on M , then, by Lemma 4.1 it is necessarily anti-self-dual (with the right choice of the orientation). Indeed, there exists a three- parameter family of null surfaces. All null surfaces in four dimensions are self-dual (SD) or Twistor Theory of Dancing Paths 7 anti-self-dual (ASD), and we choose an orientation of M such that the α-surfaces are SD. Thus the Weyl tensor of any metric g in the conformal class [g] is ASD by Penrose’s nonlinear graviton theorem [17] adapted to neutral signature. Now we shall argue that the conformal structure [g] defined by the α-surfaces of Lemma 4.1 must contain an Einstein metric. To show it, it is enough to demonstrate that the twistor space I of incident pairs (α-surfaces) admits a contact structure. The existence of an Einstein metric will then be a consequence of the theorem of Ward [19]. The contact structure is intrinsically defined by the double-fibration picture (2.1). The space I is equipped with two one-dimensional distributions defining the quotients to U and Z. The resulting two-dimensional distribution is non-integrable, and thus it defines a one-form θ on I such that θ ∧ dθ ̸= 0. We have shown (after the proof of Lemma 4.1) that on top of the SD α-surfaces there exist the two, two-parameter family of β-surfaces. Let (S, ϵ), and (S, ϵ′) be two symplectic real vector bundles over M (see [1, 2]). We shall make use of the two isomorphisms: TM = S ⊗ S′, and Λ2 − = S⊙S, where Λ2 − is the rank-3 vector bundle of ASD two-forms. Let the two families of β surfaces correspond to ASD two-forms Σ1 and Σ2 in Γ ( Λ2 − ) , or equivalently to two sections ι, ι̃ of S. We can rescale ι so that the corresponding two-form is closed, and proportional to dx1 ∧ dx2 for some functions ( x1, x2 ) on M which are constant on each β-surface in the first two param- eter family. The Frobenius integrability conditions now imply the local existence of two more coordinates such that ζ0 and ζ1 on M such that Ker(Σ1) = span{∂/∂ζ0, ∂/∂ζ1}. The functions (ζ0, ζ1) are then the coordinates on the β-surface. The corresponding metric takes the form g = dζA ⊙ dxA +ΘAB(x, ζ)dx A ⊙ dxB, A,B = 0, 1 (4.2) for some symmetric two-by-two matrix Θ. The anti-self-duality condition on the Weyl tensor forces the components of Θ to be at most cubic in (ζ0, ζ1), with additional algebraic relations between the components. Imposing the Einstein condition gives ΘAB = ζAζB + P(AB) − ΓC ABζC , (4.3) where the coefficients ΓC AB only depend on ( x0, x1 ) , and PAB is the projective Schouten tensor of affine connection ∇ with connection components ΓC AB. The β-distribution defined by Σ1 is parallel, and ∇gΣ1 = 6A⊗ Σ1, where ∇g is the Levi-Civita connection of g, and the one-form A is a symplectic connection for the symplectic structure structure [9] given by Ω = dA = dζA ∧ dxA + PABdx A ∧ dxB. (4.4) Up to this point our computation has followed [9]. Now consider the second family of β surfaces given by the ASD two-form Σ2, which in our chosen coordinate system is given by Σ2 = ϵAB ( dζA +ΘACdx C ) ∧ ( dζB +ΘBDdx D ) . The normalisation condition ϵ(ι, ι̃) = 1, and the condition that the distribution defined by Σ2 is parallel together imply that this two-form satisfies ∇gΣ2 = −6A⊗ Σ2, where A is given by (4.4). We will need only the skew-symmetric part of this equation, which gives 0 = dΣ2 + 6A ∧ Σ2 = ϵCD∇[APB]Cdx A ∧ dxB ∧ dζD, 8 M. Dunajski where the second equality follows from a rather complicated computation which we have per- formed on MAPLE. Therefore this second family of β-surfaces exists iff ∇[APB]C = 0, which (for two-dimensional projective structure) is precisely the condition (see, e.g., [5]) that the con- nection ∇ is projectively equivalent to a flat connection, where we can chose ΓC AB = 0 and PAB = 0. It was shown in [9] that the metric (4.2) with ΘAB given by (4.3) is invariant under projective changes of connection complemented by an affine translation of ξA. Therefore the resulting metric is of the form (3.1), which completes the proof. ■ 5 Dancing ellipses 5.1 A dual pair of SL(2)-invariant path geometries Let Z be the set of ellipses in R2, centered at the origin and of area π. Each such ellipse is given uniquely by an equation of the form Ex2 + 2Fxy +Gy2 = 1, (5.1) where (E,F,G) ∈ R3 satisfy EG− F 2 = 1, E > 0. (5.2) Thus Z can be identified with a two-dimensional surface, one sheet of the two-sheeted hyper- boloid in R3 given by equation (5.2). For each z = (E,F,G) ∈ Z, denote by ẑ ⊂ U := R2 \ {(0, 0)} the corresponding ellipse (the solution space of (5.1) with the given E, F , G). For example, for z = (1, 0, 1), ẑ is the unit circle centered at the origin. For each u = (x, y) ∈ U let û ⊂ Z be the set of ellipses that pass through u. Using our hyperboloid model for Z, we can see from equation (5.1) that û is the intersection of some affine plane in R3 with Z. For example, for u = (1, 0), û is the parabola G = F 2 + 1 in the plane E = 1. In fact, the space Z is the hyperboloid model of the hyperbolic plane and the curves û ⊂ Z, u ∈ U , are the horocycles of Z (circles tangent to the real axis, in the Poincaré upper half plane model). Note that the map u 7→ û is 2 : 1. Namely, u and −u define the same curve in Z. In this section we shall find it convenient to use the following form of the infinitesimal dancing condition (compare Definition 2.2) Definition 5.1. Let M ⊂ U × Z be the subset of non-incident pairs (u, z). A tangent vector (u̇, ż) ∈ T(u,z)M satisfies the dancing condition if there is an incident pair (u∗, z∗) ∈ U ×Z such that (1) u̇ is tangent to ẑ∗ at u, (2) ż is tangent to û∗ at z. Therefore the vector is null in the sense of Definition 2.2 iff it satisfies the dancing condition. The manifold M is the disjoint union of two connected components M = Min ∪ Mout, where Min is the set of pairs (u, z) such that u lies inside of the ellipse ẑ (or z lies inside the horocycle û) andMout is the set of pairs (u, z) such that u lies outside ẑ (or z lies outside û). The group SL(2,R) acts (diagonally) on M , preserving each of the two components and the dancing condition on it. Twistor Theory of Dancing Paths 9 5.2 Dancing conditons and sextics We are now ready to establish Theorem 2.4. We shall explicitly construct a sextic form S ∈ Sym4(TU)⊗ Sym2(TZ) ⊂ Sym6(TM) such that a vector field V ∈ Γ(TM) is null iff S(V, V, V, V, V, V ) = 0. Proof of Theorem 2.4. We parametrize Z by the upper half plane H = { (a, b) ∈ R2 \| b > 0 } . Define f : H → Z by f(a, b) = 1 b ( a2 + b2,−a, 1 ) . Then the incidence condition equation (5.1) can be written as Φ(x, y, a, b) := ( a2 + b2 ) x2 − 2axy + y2 − b = 0. (5.3) The dancing condition on (u̇, ż) ∈ T(u,z)M amounts to the existence of a pair (u∗, z∗) ∈ U × Z satisfying Φ(u, z∗) = Φ(u∗, z∗) = Φ(u∗, z) = ∂Φ ∂u (u, z∗)u̇ = ∂Φ ∂z (u∗, z)ż = 0. (5.4) Using the coordinates u = (x, y), z = (a, b), u∗ = (X,Y ), z∗ = (A,B), u̇ = (ẋ, ẏ), ż = (ȧ, ḃ), these equations are( A2 +B2 ) x2 − 2Axy + y2 −B = 0,( A2 +B2 ) X2 − 2AXY + Y 2 −B = 0,( a2 + b2 ) X2 − 2aXY + Y 2 − b = 0,[( A2 +B2 ) x−Ay ] ẋ+ (y −Ax)ẏ = 0, 2X(aX − Y )ȧ+ ( 2bX2 − 1 ) ḃ = 0. (5.5) The dancing condition on TM is SL(2,R)-invariant, so it is enough to study it along a section of the SL(2,R)-action on M , i.e., a subset Σ ⊂ M intersecting every SL(2,R)-orbit. We take Σ = {x = 1, y = 0, a = 0} ⊂ M , with b a coordinate along Σ (0 < b < 1 for Min and b > 1 for Mout). Along Σ, equations (5.5) reduce to A2 +B2 = B, B ( X2 − 1 ) − 2AXY + Y 2 = 0 b2 ( X2 − 1 ) + Y 2 = 0 Bẋ−Aẏ = 0 2XY ȧ+ ( 1− 2bX2 ) ḃ = 0. We use the 1st and 4th equation to solve for A, B and substitute in the other 3 equations, obtaining Y 2ẋ2 − 2XY ẋẏ + ( X2 + Y 2 − 1 ) ẏ2 = 0, b2X2 + Y 2 − b = 0, 2XY ȧ+ ( 1− 2bX2 ) ḃ = 0. (5.6) 10 M. Dunajski Now we substitute Y = PX in equations (5.6), use the 2nd obtained equation to solve for X2 and substitute in the other two, obtaining[ bẋ2 + (b− 1)ẏ2 ] P 2 − 2bẋẏP − b(b− 1)ẏ2 = 0, ḃP 2 + 2bȧP − b2ḃ = 0. (5.7) To eliminate P we take the resultant1 of the two quadratic polynomials in P in equations (5.7), obtaining (after dividing by b2) b4ḃ2ẋ4 − 4b3ȧḃẋ3ẏ + 2b2 [ ((b− 2)b− 1)ḃ2 − 2(b− 1)ȧ2 ] ẋ2ẏ2 + 4b ( 1− b2 ) ȧḃẋẏ3 (5.8) + (b− 1)2 [ (b− 1)2ḃ2 − 4bȧ2 ] ẏ4 = 0. ■ The bi-degree of the sextic (5.8) has a geometric explanation. For fixed (u, z) and u̇, one draws in Z the horocycle û, the point z and the turning point z∗ ∈ û corresponding to u̇. There are then exactly two horocycles û∗1, û ∗ 2 passing through z, z∗, giving two tangent directions ż at z. Similarly, for fixed (u, z) and ż, there are four turning points±u∗1,±u∗2 ∈ ẑ, corresponding to ż. For each antipodal pair ±u∗i there are two ellipses ẑ∗ passing through u, ±u∗i (they correspond to the two intersection points of the horocircles û, û∗i ), giving altogether four tangent direction u̇ at u. 5.2.1 2nd order ODEs Let the three-dimensional I ⊂ U × Z be the solution set of equations (5.1). The natural projections I → U , I → Z define a double fibration (2.1) so that the fibers of one fibration project to paths in the base of the other fibration, defining a pair of path geometries. This is a curved version of the more familiar classical point-line projective duality. A major difference between this curved case and the flat case is that the symmetry group drops from PSL(3,R) to PSL(2,R). A path geometry in the xy plane, such as the one given by equations (5.1) and (5.2) (or equivalently, equations (5.3)), is determined uniquely by a 2nd order ODE y′′ = F (x, y, y′). To find it, one takes the defining equation, say (5.3), and its two derivatives wrt x, assuming y = y(x). Eliminating a, b and solving for y′′, we obtain y′′ = (xy′ − y)3. (5.9) The dual path geometry, in the ab plane, is obtained by a similar process, for a function b = b(a). Eliminating x, y and solving for b′′, we obtain b′′ = ϵ (√ 1 + b′2 − ϵ )( 1 + b′2 ) b , ϵ = ±1 (so there are in fact two equations). These are two examples of (dual) 2D path geometries with a three-dimensional (local) group of symmetries, classified in 1896 by Tresse [18, p. 76]. The equation (5.9) is cubic in y′, as so it defines a projective structure. This projective structure is metrisable (see [5]) as the integral curves of (5.9) are unparametrised geodesics of a (pseudo) Riemannian metric. There is in fact a three parameter family projectively equivalent metrics with these paths as unparametrized geodesics. We can arrange for one of these metrics to be invariant under rotations about the origin in the xy plane. In polar coordinates it is given by ds2 = dr2 (1 + r4)2 + r2dθ2 1 + r4 , where x+ iy = reiθ. 1The resultant of two polynomials of one variable is a polynomial in their coefficients, that vanishes iff they have a common root. The resultant of a0+a1P+a2P 2, b0+b1P+b2P 2 is (a0b2−a2b0) 2−(a0b1−a1b0)(a1b2−a2b1). Twistor Theory of Dancing Paths 11 6 Dancing conics Let N ⊂ P2×P5 be a set of non-incident pairs (a,A), where a ∈ P2 and A is an irreducible conic. We say that two pairs (a,A) and (b, B) are dancing if there exists a conic C which passes through six points (a, b, A ∩B) in P2, where A ∩B are the four intersections of the conics (A,B). Note that the dancing condition on N is “similar” to that given in Definition 2.1 for path geometries, in that both conditions define co-dimension one cones in TN and TM respectively: generically there is no line through a three given points, and there is no conic through a six given points. � � � � � � � � � � � � � � � � � � � � � � � � � � B a C A b Figure 4. Dancing conics. Proof of Theorem 2.5. The dancing condition is( aTAa )( bTBb ) − ( aTBa )( bTAb ) = 0, (6.1) where (by a slight abuse of notation) A above is a symmetric matrix representing a conic, and aT = [ a1, a2, a3 ] is a vector representing the homogeneous coordinates of a ∈ P2. To prove (6.1) note that the conic C must belong to the pencil (A+ tB), and contain the points (a, b) so that aT(A+ tB)a = 0, bT(A+ tB)b = 0. Elliminating t between these two equations yields (6.1). Let π : N →M be a projection given by π(a,A) = (a, α), where α is the polar line of A with respect to a. The fibres of π are three-dimensional. For example if a is a point in an affine space, and α is a line at infinity, then the fiber of (a, α) consists of all ellipses centred at a. Consider the dancing condition (6.1) on N , and set B = A + ϵȦ, and b = a + ϵȧ, where ϵ is small. Substituting this in (6.1), and keeping the the second order in ϵ yields a quadratic condition( aT ·A · a )( aT · Ȧ · ȧ ) − ( aT · Ȧ · a )( aT ·A · ȧ ) (6.2) (where · denotes the matrix multiplication). The corresponding (degenerate) conformal structure on M is G = ( aT ·A · a )( aT · dA⊙ da ) − ( aT · dA · a ) ⊙ ( aT ·A · da ) . We can use inhomogeneous coordinates on N with the pair (a,A) represented by a = (x, y, 1) and the symmetric matrix A with A33 = 1. The SL(3,R) invariance alows us without loss of 12 M. Dunajski generality to take the pair (a,A) with a0 = (0, 0, 1)T, A0 = diag(−1,−1, 1). Then (6.2) reduces to the quadric ẋȦ13 + ẏȦ23, with signature + + − − 000, as stated, and with kernel given by a tangnt subspace ẋ = Ȧ13 = ẏ = Ȧ23 = 0. Now the tangent to the fiber of π at (a0, A0) is the kernel of the derivative of π at (a0, A0). Since π(a,A) = ( a, aTA ) , the derivative at (a0, A0) is (ȧ, Ȧ) 7→ ( ȧTA0 + aT0 Ȧ, ȧ ) , hence the kernel is given by Ȧa0 = ȧ = 0, or ẋ = Ȧ13 = ẏ = Ȧ23 = 0 as stated. ■ Remark 6.1. One can check that the dancing condition on N is not the pull back of the dancing condition on M , although both are SL(3,R)-invariant and horizontal w.r.t. π : N → M . If it were, then if (a,A) and (b, B) are dancing in N , then π(a,A), and π(b, B) are dancing in M . We shall now argue that this is not the case. To see it, note that in homogeneous coordinates the projection π is given by ( ai, Ajk ) → ( ai, αj = Ajka k ) . Let (a,A) and (b, B) belong to M . The intersection of polar lines aTAz = 0, bTBz = 0 is c, where ci = ϵijkAjpBkqa pbq. This point is co-linear with (a, b) if ϵijkc iajbk = 0. Using the identity ϵijkϵ imn = δj mδk n − δjnδkm yields( aTAa )( bTBb ) − ( aTBb )( bTAa ) = 0, which is different than (6.1). Acknowledgements This project originated from discussions with Gil Bor. I am very grateful to Gil for explaining the work [3] to me, and for sharing his geometric insight on Kepler and Hook ellipses. I thank the anonymous reviewers for their careful reading of the manuscript and many insightful suggestions. I also thank the Mathematics Research Center (CIMAT) in Guanajuato for hospitality, when some of this research was done. My research has been partially supported by STFC grants ST/P000681/1, and ST/T000694/1. References [1] Atiyah M., Dunajski M., Mason L.J., Twistor theory at fifty: from contour integrals to twistor strings, Proc. A 473 (2017), 20170530, 33 pages, arXiv:1704.07464. [2] Atiyah M.F., Hitchin N.J., Singer I.M., Self-duality in four-dimensional Riemannian geometry, Proc. A 362 (1978), 425–461. [3] Bor G., Hernández Lamoneda L., Nurowski P., The dancing metric, G2-symmetry and projective rolling, Trans. Amer. Math. Soc. 370 (2018), 4433–4481. [4] Bryant R., Élie Cartan and geometric duality, in Conférences données à l’Institut Élie Cartan, Institut Élie Cartan, Université de Nancy, 1999, 5–20. [5] Bryant R., Dunajski M., Eastwood M., Metrisability of two-dimensional projective structures, J. Differential Geom. 83 (2009), 465–499, arXiv:0801.0300. [6] Cartan E., Les espaces généralises et l’intégration de certaines classes d’équations différentielles, C. R. Acad. Sci. Paris 206 (1938), 1689–1693. [7] Casey S., Dunajski M., Tod P., Twistor geometry of a pair of second order ODEs, Comm. Math. Phys. 321 (2013), 681–701, arXiv:1203.4158. [8] Dunajski M., The nonlinear graviton as an integrable system, Ph.D. Thesis, Oxford University, 1998. [9] Dunajski M., Mettler T., Gauge theory on projective surfaces and anti-self-dual Einstein metrics in dimension four, J. Geom. Anal. 28 (2018), 2780–2811, arXiv:1509.04276. https://doi.org/10.1098/rspa.2017.0530 https://arxiv.org/abs/1704.07464 https://doi.org/10.1098/rspa.1978.0143 https://doi.org/10.1090/tran/7277 https://doi.org/10.4310/jdg/1264601033 https://doi.org/10.4310/jdg/1264601033 https://arxiv.org/abs/0801.0300 https://doi.org/10.1007/s00220-013-1729-7 https://arxiv.org/abs/1203.4158 https://doi.org/10.1007/s12220-017-9934-9 https://arxiv.org/abs/1509.04276 Twistor Theory of Dancing Paths 13 [10] Grossman D.A., Torsion-free path geometries and integrable second order ODE systems, Selecta Math. (N.S.) 6 (2000), 399–442. [11] Holland J., Sparling G., Causal geometries and third-order ordinary differential equations, arXiv:1001.0202. [12] Holland J., Sparling G., Causal geometries, null geodesics, and gravity, arXiv:1106.5254. [13] Kryński W., Makhmali O., The Cayley cubic and differential equations, J. Geom. Anal. 31 (2021), 6219– 6273, arXiv:1901.00958. [14] Liouville R., Mémoire sur les invariants de certaines équations différentielles et sur leurs applications, J. l’Éc. Polit. 59 (1889), 7–76. [15] Makhmali O., Differential geometric aspects of causal structures, Ph.D. Thesis, McGill University, 2017. [16] Mason L.J., Woodhouse N.M.J., Integrability, self-duality, and twistor theory, London Mathematical Society Monographs, New Series, Vol. 15, The Clarendon Press, Oxford University Press, New York, 1996. [17] Penrose R., Nonlinear gravitons and curved twistor theory, Gen. Relativity Gravitation 7 (1976), 31–52. [18] Tresse A., Détermination des invariants ponctuels de l’équation différentielle ordinaire de second ordre y′′ = ω(x, y, y′), Hirzel, Leipzig, 1896. [19] Ward R.S., Self-dual space-times with cosmological constant, Comm. Math. Phys. 78 (1980), 1–17. [20] Ward R.S., Integrable and solvable systems, and relations among them, Philos. Trans. Roy. Soc. London Ser. A 315 (1985), 451–457. https://doi.org/10.1007/PL00001394 https://arxiv.org/abs/1001.0202 https://arxiv.org/abs/1106.5254 https://doi.org/10.1007/s12220-020-00525-9 https://arxiv.org/abs/1901.00958 https://doi.org/10.1007/bf00762011 https://doi.org/10.1007/BF01941967 https://doi.org/10.1098/rsta.1985.0051 https://doi.org/10.1098/rsta.1985.0051 1 Introduction 2 Summary of the results 2.1 Path geometry 2.2 Dancing pairs 3 Dancing on RP^2 4 Dancing condition for general path geometries 5 Dancing ellipses 5.1 A dual pair of SL(2)-invariant path geometries 5.2 Dancing conditons and sextics 5.2.1 2nd order ODEs 6 Dancing conics References
id	nasplib_isofts_kiev_ua-123456789-211641
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn	1815-0659
language	English
last_indexed	2026-03-21T05:15:25Z
publishDate	2022
publisher	Інститут математики НАН України
record_format	dspace
spelling	Dunajski, Maciej 2026-01-07T13:43:26Z 2022 Twistor Theory of Dancing Paths. Maciej Dunajski. SIGMA 18 (2022), 027, 13 pages 1815-0659 2020 Mathematics Subject Classification: 32L25; 53A20 arXiv:2201.04717 https://nasplib.isofts.kiev.ua/handle/123456789/211641 https://doi.org/10.3842/SIGMA.2022.027 Given a path geometry on a surface , we construct a causal structure on a four-manifold which is the configuration space of non-incident pairs (point, path) on . This causal structure corresponds to a conformal structure if and only if is a real projective plane, and the paths are lines. We give the example of the causal structure given by a symmetric sextic, which corresponds to an SL(2, ℝ)-invariant projective structure where the paths are ellipses of area π centred at the origin. We shall also discuss a causal structure on a seven-dimensional manifold corresponding to non-incident pairs (point, conic) on a projective plane. This project originated from discussions with Gil Bor. I am very grateful to Gil for explaining the work [3] to me, and for sharing his geometric insight on Kepler and Hook ellipses. I thank the anonymous reviewers for their careful reading of the manuscript and many insightful suggestions. I also thank the Mathematics Research Center (CIMAT) in Guanajuato for its hospitality, where some of this research was done. My research has been partially supported by STFC grants ST/P000681/1 and ST/T000694/1. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Twistor Theory of Dancing Paths Article published earlier
spellingShingle	Twistor Theory of Dancing Paths Dunajski, Maciej
title	Twistor Theory of Dancing Paths
title_full	Twistor Theory of Dancing Paths
title_fullStr	Twistor Theory of Dancing Paths
title_full_unstemmed	Twistor Theory of Dancing Paths
title_short	Twistor Theory of Dancing Paths
title_sort	twistor theory of dancing paths
url	https://nasplib.isofts.kiev.ua/handle/123456789/211641
work_keys_str_mv	AT dunajskimaciej twistortheoryofdancingpaths

Twistor Theory of Dancing Paths

Institution

Similar Items