Twistor theory at fifty: from contour integrals to twistor strings

Twistor theory at fifty: from contour integrals to twistor strings

Michael Atiyah
School of Mathematics, University of Edinburgh,
King’s Buildings, Edinburgh EH9 3JZ, UK.
and
Trinity College, Cambridge, CB2 1TQ, UK.

Maciej Dunajski
Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK.
Lionel J Mason
The Mathematical Institute, Andrew Wiles Building,
University of Oxford, ROQ Woodstock Rd, OX2 6GG
m.atiyah@ed.ac.ukm.dunajski@damtp.cam.ac.uklmason@maths.ox.ac.uk
September 6, 2017
Abstract

We review aspects of twistor theory, its aims and achievements spanning the last five decades. In the twistor approach, space–time is secondary with events being derived objects that correspond to compact holomorphic curves in a complex three–fold – the twistor space. After giving an elementary construction of this space we demonstrate how solutions to linear and nonlinear equations of mathematical physics: anti-self-duality (ASD) equations on Yang–Mills, or conformal curvature can be encoded into twistor cohomology. These twistor correspondences yield explicit examples of Yang–Mills, and gravitational instantons which we review. They also underlie the twistor approach to integrability: the solitonic systems arise as symmetry reductions of ASD Yang–Mills equations, and Einstein–Weyl dispersionless systems are reductions of ASD conformal equations.

We then review the holomorphic string theories in twistor and ambitwistor spaces, and explain how these theories give rise to remarkable new formulae for the computation of quantum scattering amplitudes. Finally we discuss the Newtonian limit of twistor theory, and its possible role in Penrose’s proposal for a role of gravity in quantum collapse of a wave function.

Dedicated to Roger Penrose and Nick Woodhouse at 85 and 67 years.

1 Twistor Theory

Twistor theory was originally proposed as a new geometric framework for physics that aims to unify general relativity and quantum mechanics [173, 174, 184, 182, 183]. In the twistor approach, space–time is secondary with events being derived objects that correspond to compact holomorphic curves in a complex three–fold, the twistor space. The mathematics of twistor theory goes back to the 19th century Klein correspondence, but we shall begin our discussion with a formula for solutions to the wave equation in (3+1)–dimensional Minkowski space–time put forward by Bateman in 1904 [31]

(1.1)

This is the most elementary of Penrose’s series of twistor integral formulae for massless fields [175]. The closed contour encloses some poles of a meromorphic function . Differentiating (1.1) under the integral sign yields

(1.2)

The twistor contour integral formula (1.1) is a paradigm for how twistor theory should work and is a good starting point for discussing its development over the last five decades. In particular one may ask

  • What does this formula mean geometrically?

    The integrand of (1.1) is a function of three complex arguments and we will see in §2 that these arise as local affine coordinates on projective twistor space which we take to be . In (1.1) the coordinates on are restricted to a line with affine coordinate . The Minkowski space arises as a real slice in the four-dimensional space of lines in .

    The map (1.1) from functions to solutions to the wave equation is not one to one: functions holomorphic inside can be added to without changing the solution . This freedom in was understood in the 1970s in a fruitful interaction between the Geometry and Mathematical Physics research groups in Oxford [15]: twistor functions such as in (1.1) should be regarded as elements of ech sheaf cohomology groups. Rigorous theorems establishing twistor correspondences for the wave equation, and higher spin linear equations have now been established [81, 218, 108, 29]. The concrete realisations of these theorems lead to (contour) integral formulae.

  • Do ‘similar’ formulae exist for nonlinear equations of mathematical physics, such as Einstein or Yang–Mills equations?

    The more general integral formulae of Penrose [175] give solutions to both linearised Einstein and Yang–Mills equations. In the case that the linearised field is anti-self-dual (i.e., circularly polarised or right handed) these cohomology classes correspond to linearised deformations of the complex structure of twistor space for gravity [176, 18] or of a vector bundle in the Yang-Mills case [210]. We shall review these constructions in §3 and §5.

    These constructions give an ‘in principle’ general solution to the equations in the sense that locally every solution can be represented locally in terms of free data on the twistor space as in the original integral formula. Indeed this leads to large classes of explicit examples (e.g. Yang-Mills and gravitational instantons which we shall review in the gravitational case in §33.4) although it can be hard to implement for general solutions.

    It turns out [213] that most known integrable systems arise as symmetry reductions of either the anti–self–dual Yang–Mills or the anti–self–dual (conformal) gravity equations. The twistor constructions then reduce to known (inverse scattering transform, dressing method, …) or new solution generation techniques for soliton and other integrable equations [158, 75]. We shall review some of this development in §6.

    As far as the full Einstein and Yang-Mills equations are concerned, the situation is less satisfactory. The generic nonlinear fields can be encoded in terms of complex geometry in closely related ambitwistor spaces. In these situations the expressions of the field equations are less straightforward and they no longer seem to provide a general solution generation method. Nevertheless, they have still had major impact on the understanding of these theories in the context of perturbative quantum field theory as we will see in §7.

  • Does it all lead to interesting mathematics?

    The impacts on mathematics have been an unexpected major spin–off from the original twistor programme. These range over geometry in the study of hyper–Kähler manifolds [18, 107, 138, 139, 191], conformal, CR and projective structures [101, 26, 100, 48, 144, 83, 144, 88, 59, 50, 27, 82, 32, 219], exotic holonomy [49, 165, 166, 167], in representation theory [29, 60] and differential equations particularly in the form of integrable systems [158, 75]. We will make more specific comments and references in the rest of this review.

  • Is it physics?

    Thus far, the effort has been to reformulate conventional physics in twistor space rather than propose new theories. It has been hard to give a complete reformulation of conventional physics on twistor space in the form of nonlinear generalizations of (1.1). Nevertheless, in just the past 13 years, holomorphic string theories in twistor and ambitwistor spaces have provided twistorial formulations of a full range of theories that are commonly considered in particle physics. They also provide remarkable new formulae for the computation of scattering amplitudes. Many technical issues that remain to be resolved to give a complete reformulation of conventional physics ideas even in this context of peturbative quantum field theory. Like conventional string theories, these theories do not, for example, have a satisfactory non-perturbative definition. Furthermore, despite recent advances at one and two loops, their applicability to all loop orders has yet to be demonstrated. See §7 for a full discussion.

    The full (non-anti–self–dual) Einstein and Yang–Mills equations are not integrable and so one does not expect a holomorphic twistor description of their solutions that has the simplicity of their integrable self-dual sectors. It is hoped that the full, non–perturbative implementation of twistor theory in physics is still to be revealed. One set of ideas builds on Penrose’s proposal for a role of gravity in quantum collapse of a wave function [178, 180]. This proposal only makes use of Newtonian gravity, but it is the case that in the Newtonian limit the self–dual/anti–self–dual constraint disappears from twistor theory and all physics can be incorporated in the limit of [80], see §8.

  • Does it generalise to higher dimensions?

    There are by now many generalisations of twistors in dimensions higher than four [191, 124, 125, 126, 168, 194, 37, 200, 69, 91, 209]. One definition takes twistor space to be the projective pure spinors of the conformal group. This definition respects full conformal invariance, and there are analogues of (1.1) for massless fields. However, the (holomorphic) dimension of such twistor spaces goes up quadratically in dimension and become higher than the dimension of the Cauchy data (i.e., one less than the dimension of space–time). Thus solutions to the wave equation and its non–linear generalisations do not map to unconstrained twistor data and this is also reflected in the higher degree of the cohomology classes in higher dimensions that encode massless fields. These do not seem to have straightforward nonlinear extensions.

    Another dimension agnostic generalisation of twistor theory is via ambitwistors. Indeed some of the ambitwistor string models described in §7 are only critical in 10 dimensions, relating closely to conventional string theory, although without the higher massive modes.

    Twistor theory has many higher dimensional analogues for space-times of restricted holonomy [191]. The hyper-Kähler case of manifolds of dimension with holonomy in admit a particularly direct generalisation of Penrose’s original nonlinear graviton construction and now has wide application across mathematics and physics.

This review celebrates the fifty years of twistor theory since the publication of the first paper on the subject111See also the programme and slides from the meeting, New Horizons in Twistor Theory in Oxford January 2017 that celebrated this anniversary along with the 85th birthday or Roger Penrose and the 67th of Nick Woodhouse. http://www.maths.ox.ac.uk/groups/mathematical-physics/events/twistors50. by Roger Penrose [173]. We apologize to the many researchers whose valuable contributions have been inadvertently overlooked.

2 Twistor space and incidence relation

Twistor theory is particularly effective in dimension four because of an interplay between three isomorphisms. Let be a real oriented four–dimensional manifold with a metric of arbitrary signature.

  • The Hodge operator is an involution on two-forms, and induces a decomposition

    (2.3)

    of two-forms into self-dual (SD) and anti-self-dual (ASD) components, which only depends on the conformal class of .

  • Locally there exist complex rank-two vector bundles (spin-bundles) over equipped with parallel symplectic structures such that

    (2.4)

    is a canonical bundle isomorphism, and

    (2.5)

    for and . The isomorphism (2.4) is related to (2.3) by

  • The orthogonal group in dimension four is not simple:

    (2.6)

    where and defined above are the representation spaces of and respectively. There exist three real slices: In the Lorentzian signature and both copies of in (2.6) are related by complex conjugation. In the Riemannian signature . In (also called neutral, or ultra-hyperbolic signature) . Only in this signature there exists a notion of real spinors, and as we shall see in §3.2 real twistors.

2.1 Incidence relation

The projective twistor space is defined to be . The homogeneous coordinates of a twistor are , where and . The projective twistor space (which we shall call twistor space from now on) and Minkowski space are linked by the incidence relation

(2.7)

where are coordinates of a point in Minkowski space. If two points in Minkowski space are incident with the same twistor, then they are connected by a null line. Let

be a Hermitian inner product on the non–projective twistor space . The orientation–preserving endomorphisms of the twistor space which preserve form a group which is locally isomorphic to the conformal group of Minkowski space. The twistor space is divided into three parts depending on whether is positive, negative or zero. This partition descends to the projective twistor space. In particular the hypersurface

is preserved by the conformal transformations of the Minkowski space which can be verified directly using (2.7). The five dimensional manifold is the space of light rays in the Minkowski space. Fixing the coordinates of a space–time point in (2.7) gives a plane in the non–projective twistor space or a projective line in . If the coordinates are real this line lies in the hypersurface . Conversely, fixing a twistor in gives a light–ray in the Minkowski space.

So far only the null twistors (points in ) have been relevant in this discussion. General points in can be interpreted in terms of the complexified Minkowski space where they correspond to –planes, i. e. null two–dimensional planes with self–dual tangent bi-vector. This, again, is a direct consequence of (2.7) where now the coordinates are complex:

Figure 1. Twistor incidence relation

Complexified space-time Twistor space
Point
Null self-dual (=) two-plane

2.2 Robinson Congruence

The non–null twistors can also be interpreted in the real Minkowski space, but this is somewhat less obvious [173]: The inner product defines a vector space dual to the non–projective twistor space. Dual twistors are the elements of the projective space . Consider a twistor . Its dual corresponds to a two–dimensional complex projective plane in . This holomorphic plane intersects the space of light rays in a real three–dimensional locus corresponding to a three–parameter family of light–rays in the real Minkowski space. The family of light rays representing a non–null twistor is called the Robinson congruence.

Figure 2. Robinson congruence of twisting light rays.

Null ray in Point in
Robinson congruence

The Robinson congruence in Figure 2 is taken from the front cover of [184]. It consists of a system of twisted oriented circles in : a light–ray is represented by a point in together with an arrow indicating the direction of the ray’s motion. It is this twisting property of circles in Figure 2 which gave rise to a term ‘twistor’ for points of . An account of congruences in general relativity which motivated initial progress in twistor theory is given in [134, 189, 190].

2.3 Cohomology

The twistor interpretation of Penrose’s contour integral formula (1.1) is as follows. Cover the twistor space by two open sets: defined by and defined by . Consider a function on the non–projective twistor space which is holomorphic on and homogeneous of degree in . Restrict this function to a two–dimensional plane in defined by the incidence relation (2.7) with fixed. This gives rise to an element of the cohomology group222The cohomology group is the space of functions holomorphic on and homogeneous of degree in coordinates modulo addition of coboundaries (functions holomorphic on and ). In a trivialisation over we represent by a holomorphic function on . In the trivialisation over , is represented by , where . Here, and in the rest of this paper denotes a line bundle over with transition function in a trivialisation over . Alternatively it is defined as the th tensor power of the tautological line bundle. on the projective twistor space, where is the curve corresponding, via the incidence relation (2.7), to a point . Integrate the cohomology class along a contour in . This gives (1.1) with . For example , where and are constant dual twistors gives rise to a fundamental solution to the wave equation (1.2). The Theorem of [81] states that solutions to the wave equation (1.2) which holomorphically extend to a future tube domain in are in one–to–one correspondence with elements of the cohomology group . This correspondence extents to solutions of zero–rest–mass equations with higher spin, and elements of where is an integer. See [81, 108, 184, 218, 121] for further details.

3 Twistors for curved spaces

The twistor space of complexified Minkowski space was defined by the incidence relation (2.7) as the space of all –planes in . Let be a holomorphic four–manifold with a holomorphic Riemannian metric and a holomorphic volume form. Define an –surface to be a two–dimensional surface in such that its tangent plane at every point is an –plane. If the metric is curved, there will be integrability conditions coming from the Frobenius Theorem for an –plane to be tangent to a two–dimensional surface.

3.1 The Nonlinear Graviton construction

Define to be the space of –surfaces in . The Frobenius theorem implies that for , and there are obstruction in terms of the curvature of . This gives rise to the Nonlinear Graviton Theorem

Theorem 3.1 (Penrose [176])

There exists a three–parameter family of –surfaces in iff the the Weyl tensor of is anti–self–dual, i.e.

(3.8)

The anti–self–duality of the Weyl tensor is the property of the whole conformal class

rather than any particular metric. Points in an ASD conformal manifold correspond to rational curves in with normal bundle , and points in correspond to –surfaces in . The ASD conformal structure on can be defined in terms of algebraic geometry of curves in twistor space: is three dimensional, so two curves in generically do not intersect. Two points in are null separated if and only if the corresponding curves in intersect at one point.

Theorem 3.2 (Penrose [176])

Let be a moduli space of all rational curves with the normal bundle in some complex three–fold . Then is a complex four–fold with a holomorphic conformal metric with anti–self–dual curvature. Locally all ASD holomorphic conformal metrics arise from some .

More conditions need to be imposed on if the conformal structure contains a Ricci-flat metric. In this case there exists a holomorphic fibration with -valued symplectic form on the fibres. Other curvature conditions (ASD Einstein [211, 109, 142, 5, 120], Hyper–Hermitian [131, 73], scalar–flat Kähler [187], null Kähler [74]) can also be encoded in terms of additional holomorphic structures on . Some early motivation for Theorem 3.2 came from complex general relativity, and theory of –spaces. See [170, 186].

3.2 Reality conditions

The real ASD conformal structures are obtained by introducing an involution on the twistor space. If the conformal structure has Lorentizian signature, then the anti–self–duality implies vanishing of the Weyl tensor, and thus is conformally flat. This leaves two possibilities: Riemannian and neutral signatures. In both cases the involutions act on the twistor lines, thus giving rise to maps from to : the antipodal map which in stereographic coordinates is given by , or a complex conjugation which swaps the lower and upper hemispheres preserving the real equator. The antipodal map has no fixed points and corresponds to the positive–definite conformal structures. The conjugation corresponds to the neutral case.

In the discussion below we shall make use of the double fibration picture

(3.9)

where the five–complex–dimensional correspondence is defined by

where is the line in that corresponds to and lies on . The space can be identified with a projectivisation of the spin bundle . It is equipped with a rank-2 distribution, the twistor distribution, which at a given point of is spanned by horizontal lifts of vectors spanning –surface at . The normal bundle to consists of vectors tangent to horizontally lifted to modulo the twistor distribution . We have a sequence of sheaves over

Using the abstract index notation [184] (so that, for example, denotes a section of , and no choice of a local frame or coordinates is assumed) the map is given by . Its kernel consists of vectors of the form with varying. The twistor distribution is therefore and so there is a canonical , given by , where .

  • Euclidean case. The conjugation given by descends from to an involution such that . The twistor curves which are preserved by form a four–real parameter family, thus giving rise to a real four–manifold . If then and are connected by a unique real curve. The real curves do not intersect as no two points are connected by a null geodesic in the positive definite case. Therefore there exists a fibration of the twistor space over a real four–manifold . A fibre over a point is a copy of a . The fibration is not holomorphic, but smooth.

    In the Atiyah–Hitchin–Singer [18] version of the correspondence the twistor space of the positive definite metric is a real six–dimensional manifold identified with the projective spin bundle .

    Given a conformal structure on one defines an almost–complex–structure on by declaring

    to be the anti–holomorphic vector fields in .

    Theorem 3.3 (Atiyah–Hitchin–Singer [18])

    The six–dimensional almost–complex manifold

    parametrises almost–complex–structures in . Moreover is complex iff is ASD.

  • Neutral case. The spinor conjugation given by allows an invariant decomposition of a spinor into its real and imaginary part, and thus definition of real -surfaces [224, 74].

    In general , and the correspondence space decomposes into two open sets

    where are two copies of a Poincare disc. These sets are separated by a real correspondence space The correspondence spaces have the structure of a complex manifold in a way similar to the AHS Euclidean picture. There exists an worth of real –surfaces through each point in , and real twistor distribution consisting of vectors tangent to real –surfaces defines a foliation of with quotient which leads to a double fibration:

    The twistor space is a union of two open subsets and separated by a three-dimensional real boundary .

    These reality conditions are relevant in the twistor approach to integrable systems (see §6), integral geometry, twistor inspired computations of scattering amplitudes (see §7), as well as recent applications [16] of the Index Theorem [23] which do not rely on positivity of the metric. The discussion in this subsection has assumed real analyticity of . The approach of LeBrun and Mason [148] based on holomorphic discs can weaken this assumption.

3.3 Kodaira Deformation Theory

One way of obtaining complex three–manifolds with four–parameter families of curves comes from the Kodaira deformation theory applied to

Figure 3. Curvature on corresponds to deformations of

The normal bundle satisfies

The Kodaira theorems [136] imply that there exist infinitesimal deformations of the complex structure of which preserve the four parameter family of s, as well as the type of their normal bundle. Moreover this deformed family admits an isomorphism

identifying tangent vectors to with pairs of linear homogeneous polynomials in two variables. This identification allows to construct a conformal structure on arising from a quadratic condition that both polynomials in each pair have a common zero. There are some examples of ASD Ricci flat metrics arising from explicit deformations - see [121, 75]. A method of constructing such examples was pioneered by George Sparling.

3.3.1 Twistor solution to the holonomy problem.

The Kodaira approach to twistor theory has given rise to a complete classification of manifolds with exotic holonomy groups (holonomy groups of affine connections which are missing from Berger’s list). The first landmark step was taken by Robert Bryant [49] who generalised the Kodaira theorems and the twistor correspondence to Legendrian curves. Complex contact three–folds with 4–parameter family of Legendrian rational curves with normal bundle correspond to four manifolds such that and there exists a torsion–free connection with holonomy group . The theory was extended by Merkulov to allow Legendrian deformations of more general submanifolds [165, 166]. This work lead to a complete classification by Merkulov and Schwachhofer [167].

3.4 Gravitational Instantons

Gravitational instantons are solutions to the Einstein equations in Riemannian signature which give complete metrics whose curvature is concentrated in a finite region of a space-time. The non–compact gravitational instantons asymptotically ‘look like’ flat space. While not all gravitational instantons are (anti)–self–dual (e.g. the Euclidean Schwarzchild solution is not) most of them are, and therefore they arise from Theorems 3.2 and 3.3.

  • There exists a large class of gravitational instantons which depend on a harmonic function on :

    (3.10)

    where and are a function and a one–form respectively which do not depend on . This is known as the Gibbons-Hawking ansatz [96]. The resulting metrics are hyper–Kähler (or equivalently anti–self–dual and Ricci flat). The Killing vector field is tri–holomorphic - it preserves the sphere of Kähler forms of . It gives rise to a holomorphic vector field on the corresponding twistor space which preserves the –valued symplectic structure on the fibres of . Therefore there exists an associated valued Hamiltonian, and the Gibbons–Hawking twistor space admits a global fibration over the total space of . Conversely, any twistor space which admits such fibration leads to the Gibbons–Hawking metric on the moduli space of twistor curves [207, 115].

  • An example of a harmonic function in (3.10) which leads to the Eguchi–Hanson gravitational instanton is . It is asymptotically locally Euclidean (ALE), as it approaches for large . The corresponding twistor space has been constructed by Hitchin [107].

  • A general gravitational instanton is called ALE if it approaches at infinity, where is a discrete subgroup of . Kronheimer [138, 139] has constructed ALE spaces for finite subgroups

    of . In each case the twistor space is a three–dimensional hyper–surface

    in the rank three bundle , for some integers , where is a singularity resolution of one of the Klein polynomials corresponding to the Platonic solids

    The twistor spaces of these ALE instantons admit a holomorphic fibration over the total space of for some . In case of one has and the metric belongs to the Gibbons–Hawking class. In the remaining cases , and the resulting metrics do not admit any tri–holomporphic Killing vector. They do however admit hidden symmetries (in the form of tri–holomorphic Killing spinors), and arise from a generalised Legendre transform [150, 77, 41, 45].

  • There are other types of gravitational instantons which are not ALE, and are characterised by different volume growths of a ball of the given geodesic radius [66, 106]. They are ALF (asymptotically locally flat), and ‘inductively’ named ALG, ALH spaces. Some ALF spaces arise from the Gibbons–Hawking ansatz (3.10) where

    where are fixed points in (the corresponding twistor spaces are known), but others do not. In [65] some progress has been made in constructing twistor spaces for ALF instantons, but finding the twistor spaces, or explicit local forms for the remaining cases is a open problem.

There also exist compact examples of Riemannian metrics with ASD conformal curvature. The round and with the Fubini–Study metric are explicit examples where the ASD metric is also Einstein with positive Ricci scalar. A Ricci–flat ASD metric is known to exist on the surface, but the explicit formula for the metric is not known.

LeBrun has proven [147] that there are ASD metrics with positive scalar curvature on any connected sum of reversed oriented complex projective planes. This class, together with a round four–sphere exhaust all simply connected possibilities. The corresponding twistor spaces can be constructed in an algebraic way. The strongest result belongs to Taubes [202]. If is any compact oriented smooth four–manifold, then there exists some such that

admits an ASD metric for any .

4 Local Twistors

There exists at least three definitions of a twistor which agree in a four–dimensional flat space. The first, twistors as –planes, was used in the last section, where its curved generalisation lead to the Nonlinear Graviton construction and anti–self–duality. The second, twistors as spinors for the conformal group, relies heavily on maximal symmetry and so does not generalise to curved metrics. The last, twistors as solutions to the twistor equation, leads to interesting notions of a local twistor bundle and a local twistor transport [182, 70, 184, 196] which we now review.

We shall make use of the isomorphism (2.4). Let be homogeneous coordinates of a twistor as in §2. Set . Differentiating the incidence relation (2.7) yields

(4.11)

where , and is a (chosen) symplectic forom on used to raise and lower indices.

The space–time coordinates are constants of integration resulting from solving this equation on . Let us consider (4.11) on a general curved four–manifold, where it is called the twistor equation. It is conformally invariant under the transormations of the metric . The prolongation of the twistor equation leads to a connection on a rank–four vector bundle called the local twistor bundle. Here denotes a line bundle of conformal densities of weight . This connection also called the local twistor transport, and is given by [70]

where is the Schouten tensor of conformal geometry given by

The holonomy of the local twistor transport obstructs existence of global twistors on curved four manifolds (all local normal forms of Lorentzian metrics admitting solutions to (4.11) have been found in [149]).

The tractor bundle is isomorphic to the exterior square of the local twistor bundle. It is a rank–six vector bundle , and its connection induced from the local twistor transform is

(4.12)

This connection does not arise from a metric, but is related to a pull back of the Levi–Civita connection of the so-called ambient metric to a hypersurface. See [88, 59] as well as [100] for discussion of the ambient construction.

The point about the connection (4.12) is that it also arises as a prolongation connection for the conformal to Einstein equation

(4.13)

where denotes the trace–free part. If satisfies (4.13) where and are computed from , then is Einstein [146]. Therefore the holonomy of (4.12) leads to obstructions for an existence of an Einstein metric in a given conformal class [30, 146, 99, 78]. The Bach tensor is one of the obstructions arising from a requirement that a parallel tractor needs to be annihilated by the curvature of (4.12) and its covariant derivatives.

Conformal geometry is a particular example of a parabolic geometry - a curved analog of a homogeneous space which is the quotient of a semi-simple Lie group by a parabolic subgroup . Other examples include projective, and CR geometries. All parabolic geometries admit tractor connections. See [60] for details of these construction, and [101, 98, 84] where conformally invariant differential operators have been constructed. Examples of such operators are the twistor operator underlying (4.11) and the operator acting on

This operator associates the conformally invariant Bach tensor to the anti–self–dual Weyl spinor.

5 Gauge Theory

The full second–order Yang Mills equations on are not integrable, and there is no twistor construction encoding their solutions in an unconstrained holomorphic data on - there do exist ambitwistor constructions [220, 127, 105] in terms of formal neighbourhods of spaces of complex null geodesics, but they do not lead to any solution generation techniques. As in the case of gravity, the anti–self–dual sub–sector can be described twistorialy, this time in terms of holomorphic vector bundles over rather than deformations of its complex structures.

5.1 ASDYM and the Ward Correspondence

Let , where is some Lie algebra, and let

The anti–self–dual Yang–Mills equations are

(5.14)

where is the Hodge endomorphism depending on the flat metric and the orientation on . These equations together with the Bianchi identity imply the full Yang–Mills equations .

Let us consider (5.14) on the complexified Minkowski space with a flat holomorphic metric and a holomorphic volume form. Equations (5.14) are then equivalent to the vanishing of on each –plane in . Therefore, given , there exists a vector space of solutions to

(5.15)

The converse of this construction is also true, and leads to a twistor correspondence for solutions to ASDYM equations

Theorem 5.1 (Ward [210])

There is a one-to-one correspondence between:

  1. Gauge equivalence classes of ASD connections on with the gauge group ,

  2. Holomorphic rank– vector bundles over twistor space which are trivial on each degree one section of .

The splitting of the patching matrix for the bundle into a product of matrices holomorphic on and is the hardest part of this approach to integrable PDEs. When the Ward correspondence is reduced to lower dimensional PDEs as in §6, the splitting manifests itself as the Riemann–Hilbert problem in the dressing method.

To obtain real solutions on with the gauge group the bundle must be compatible with the involution preserving the Euclidean slice (compare §33.2). This comes down to , and

where denotes the Hermitian conjugation, and is the anti–holomorphic involution on the twistor space which restricts to an antipodal map on each twistor line. See [218, 223].

5.2 Lax pair

Consider the complexified Minkowski space with coordinates , and the metric and orientation

The Riemannian reality conditions are recovered if , and the neutral signature arises if all four coordinates are taken to be real. The ASDYM equations (5.14) arise as the compatibility condition for an overdetermined linear system , where

(5.16)

where , and is the fundamental matrix solution. Computing the commutator of the Lax pair yields

and the vanishing of the coefficients of various powers of gives (5.14). The geometric interpretation of this is as follows: for each value of the vectors span a null plane in which is self–dual in the sense that satisfies . The condition (5.14) takes the equivalent form , thus vanishes on all planes. For a given YM potential , the lax pair (5.16) can be expressed as .

5.3 Instantons

Instantons, i. e. solutions to ASDYM such that

extend from to . The corresponding vector bundles extend from to . The holomorphic vector bundles over have been extensively studied by algebraic geometers. All such bundles (and thus the instantons) can be generated by the monad construction [19]. One way to construct holomorphic vector bundles is to produce extensions of line bundles, which comes down to using upper-triangular matrices as patching functions. Let be a rank–two holomorphic vector bundle over which arises as an extension of a line bundle by another line bundle

(5.17)

If then the YM potential is given in terms of a solution to the linear zero–rest–mass field equations with higher helicity.

Theorem 5.2 (Atiyah–Ward [22])

Every ASDYM instanton over arises from a holomorphic vector bundle of the form (5.17)

Advances made in anti–self–dual gauge theory using the twistor methods lead to the studies of moduli spaces of connections. Such moduli spaces continue to play an important role in mathematical physics [153], and gave rise to major advances in the understanding of topology of four–manifolds [71].

5.4 Minitwistors and magnetic monopoles

Another gauge theoretic problem which was solved using twistor methods [110, 111] is the construction of non–abelian magnetic monopoles.

Let be a –valued one–form and a function respectively on , and let . The non–abelian monopole equation is a system of non–linear PDEs

(5.18)

These are three equations for three unknowns as are defined up to gauge transformations

(5.19)

and one component of can always be set to zero.

Following Hitchin [110] define the mini twistor space to be the space of oriented lines in . Any oriented line is of the form where is a unit vector giving the direction of the line, and is orthogonal to and joints the line with some chosen point (say the origin) in . Thus

For each fixed this space restricts to a tangent plane to . The twistor space is the union of all tangent planes – the tangent bundle which is also a complex manifold .

Figure 4. Minitwistor Correspondence.

Given solve a matrix ODE along each oriented line

This ODE assigns a complex vector space to each point of , thus giving rise to a complex vector bundle over the mini–twistor space. Hitchin shows [109] that monopole equation (5.18) on holds if and only if this vector bundle is holomorphic.

The mini–twistor space of Hitchin can also be obtained as a reduction of the twistor space by a holomorphic vector field corresponding to a translation in . An analogous reduction of ASDYM on by a rotation gives nonabelian hyperbolic monopoles [14].

In the next section we shall discuss how more general reductions of give rise to solution generation techniques for lower dimensional integrable systems.

6 Integrable Systems

Most lower dimensional integrable systems arise as symmetry reductions of anti–self–duality equations on in or signature.

The solitonic integrable systems are reductions of ASDYM as their linear systems (Lax pairs) involve matrices. The program of reducing the ASDYM equations to various integrable equations has been proposed and initiated by Ward [213] and fully implemented in the monograph [158]. The dispersionless integrable systems are reductions of anti–self–duality equations on a conformal structure [76, 75]. A unified approach combining curved backgrounds with gauge theory has been developed by Calderbank [57].

In both cases the reductions are implemented by assuming that the Yang–Mills potential or the conformal metric are invariant with respect to a subgroup of the full group of conformal symmetries. Conformal Killing vectors on correspond to holomorphic vector fields on . The resulting reduced system will admit a (reduced) Lax pair with a spectral parameter coming from the twistor –plane distribution. It will be integrable by a reduced twistor correspondence of Theorem 3.2 or Theorem 5.1.

6.1 Solitonic equations

The general scheme and classification of reductions of ASDYM on the complexified Minkowski space involves a choice of subgroup of the complex conformal group , a real section (hyperbolic equations arise from ASDYM in neutral signature), a gauge group and finally canonical forms of Higgs fields.

We have already seen one such symmetry reduction: ASDYM on invariant under a one–dimensional group of translations generated by reduce to the non–abelian monopole equation (5.18). The Higgs field on is related to the gauge potential on by . The analogous reduction from leads to Ward’s integrable chiral model on [214]. It is solved by a minitwistor construction, where the minitwistor space from the description of monopoles is instead equipped with an anti–holomorphic involution fixing a real equator on each twistor line [215]. The solitonic solutions are singled out by bundles which extend to compactified mini–twistor spaces [217, 185]. Below we give some examples of reductions to two and one dimensions.

  • Consider the ASDYM in neutral signature and choose a gauge . Let be two by two constant matrices such that Then ASDYM equations are solved by the ansätze

    provided that satisfies

    which is the Sine–Gordon equation. Analogous reductions of ASDYM with gauge group or lead to the Tzitzeica equations and other integrable systems arising in affine differential geometry [75]. A general reduction by two translations on lead to Hitchin’s self–duality equations which exhibit conformal invariance and thus extend to any Riemann surface [112].

  • Mason and Sparling [156] have shown that any reduction to the ASDYM equations on with the gauge group by two translations exactly one of which is null is gauge equivalent to either the KdV or the Nonlinear Schrodinger equation depending on whether the Higgs field corresponding to the null translation is nilpotent or diagonalisable. In [157] and [158] this reduction has been extended to integrable hierarchies.

  • By imposing three translational symmetries one can reduce ASDYM to an ODE. Choose the Euclidean reality condition, and assume that the YM potential is independent on .

    Select a gauge , and set , where the Higgs fields are real –valued functions of . The ASDYM equations reduce to the Nahm equations

    These equations admit a Lax representation which comes from taking a linear combination of and in (5.16). Let

    Then

    The Nahm equations with the group of volume–preserving diffeomorphism of some three-manifold as the gauge group are equivalent to ASD vacuum equations [13].

  • Reductions of ASDYM by three–dimensional abelian subgroups of the complexifed conformal group lead to all six Painlevé equations [158]. The coordinate–independent statement of the Painleve property for ASDYM was first put forward by Ward [212]: If a solution of ASDYM on has a non- characteristic singularity, then that singularity is at worst a pole. Another twistor approach to the Painlevé equation is based on –invariant anti–self–dual conformal structures [205, 113, 164].

6.2 Dispersionless systems and Einstein–Weyl equations

There is a class of integrable systems in 2+1 and three dimensions which do not fit into the framework described in the last section. They do not arise from ASDYM and there is no finite–dimensional Riemann–Hilbert problem which leads to their solutions. These dispersionless integrable systems admit Lax representations which do not involve matrices, like (5.16), but instead consist of vector fields.

Given a four–dimensional conformal structure with a non-null conformal Killing vector , the three–dimensional space of trajectories of inherits a conformal structure represented by a metric

The ASD condition on results in an additional geometrical structure on ; it becomes an Einstein-Weyl space [130]. There exists a torsion–free connection which preserves in the sense that

(6.20)

for some one–form , and the symmetrised Ricci tensor of is proportional to . These are the Einstein–Weyl equations [62]. They are conformally invariant: If then .

Most known dispersionless integrable systems in and dimensions arise from the EW equations. Consult [75, 57, 86] for the complete list. Here we shall review the twistor picture, and examples of integrable reductions.

Theorem 6.1 (Hitchin [109])

There is a one–to–one correspondence between solutions to Einstein–Weyl equations in three dimensions, and two–dimensional complex manifolds admitting a three parameter family of rational curves with normal bundle .

Figure 5. Einstein–Weyl twistor correspondence.

In this twistor correspondence the points of correspond to rational curves in the complex surface , and points in correspond to null surfaces in which are totally geodesic with respect to the connection .

To construct the conformal structure define the null vectors at in to be the sections of the normal bundle vanishing at some point to second order. Any section of is a quadratic polynomial, and the repeated root condition is given by the vanishing of its discriminant. This gives a quadratic condition on .

To define the connection , let a direction at be a one–dimensional space of sections of which vanishing at two points and on a line . The one–dimensional family of twistor curves in passing through and gives a geodesic in in a given direction. The limiting case corresponds to geodesics which are null with respect to in agreement with (6.20). The special surfaces in corresponding to points in are totally geodesic with respect to the connection . The integrability conditions for the existence of totally geodesic surfaces is equivalent to the Einstein–Weyl equations [62].

The dispersionless integrable systems can be encoded in the twistor correspondence of Theorem 6.1 if the twistor space admits some additional structures.

  • If admits a preferred section of , where is the canonical bundle of , then there exist coordinates and a function on such that

    and the EW equations reduce to the Toda equation [216, 147]

    This class of EW spaces admits both Riemannian and Lorentzian sections (for the later replace by or by ), which corresponds to two possible real structures on . It can be characterised on by the existence of twist–free shear–free geodesic congruence [206, 58].

  • If admits a preferred section of , then there exist coordinates and a function on such that

    (6.21)

    and the EW equations reduce to the dispersion-less Kadomtsev-Petviashvili (dKP) equation [76]

    This class of EW spaces can be real only in the Lorentzian signature. The corresponding real structure on is an involution which fixes an equator on each and interchanges the upper and lower hemisphere. The vector field is null and covariantly constant with respect to the Weyl connection, and with weight . This vector field gives rise to a parallel real weighted spinor, and finally to a preferred section of . Conversely, any Einstein–Weyl structure which admits a covariantly constant weighted vector field is locally of the form (6.21) for some solution of the dKP equation.

The most general Lorentzian Einstein–Weyl structure corresponds [79] to the Manakov–Santini system [152]. Manakov and Santini have used a version of the non–linear Riemann–Hilbert problem and their version of the inverse scattering transform to give an analytical description of wave breaking in dimensions. It would be interesting to put their result in the twistor framework. The inverse scattering transform of Manakov and Santini is intimately linked to the Nonlinear Graviton construction. The coordinate form of the general conformal anti–self–duality equation [79] gives the master dispersionless integrable system in dimensions, which is solvable by methods developed in [33, 225].

7 Twistors and scattering amplitudes

Although there has been a longstanding programme to understand scattering amplitudes in twistor space via ‘twistor diagrams’ [182, 116], the modern developments started with Witten’s twistor-string [221] introduced in 2003. The fallout has now spread in many directions. It encompasses recursion relations that impact across quantum field theory but also back on the original twistor-diagram programme, Grassmannian integral formulae, polyhedral representations of amplitudes, twistor actions and ambitwistor–strings.

7.1 Twistor-strings

The twistor string story starts in the 1980’s with a remarkable ampitude formula due to Parke and Taylor [172], and its twistorial interpretation by Nair [169]. Consider massles gluons, each carrying a null momentum . The isomorphism (2.4) and the fomula (2.5) imply that null vectors are two-by-two matrices with zero determinant, and thus rank one. Any such matrix is a tensor product of two spinors

In spinor variables, the tree level amplitude for two negative helicity gluons and positive leads to [172]

(7.22)

where , and th and th particles are assumed to have negative helicity, and the remaining particles have positive helicity. Nair [169] extended this formula to incorporate supersymmetry and expressed it as an integral over the space of degree one curves (lines) in twistor space using a current algebra on each curve.

Witten [221] extended this idea to provide a formulation of super Yang–Mills as a string theory whose target is the super–twistor space (see, e.g. [87]). This space has homogeneous coordinates with the usual four bosonic homogeneous coordinates and four anti–commuting Grassmann coordinates . The model is most simply described [35] as a theory of holomorphic maps333These are D1 instantons in Witten’s original B-model formulation. from a closed Riemann surface to nonprojective twistor space. It is based on the worldsheet action