A Twistor Description of Six-Dimensional
Super Yang–Mills Theory
Maxwell Institute for Mathematical Sciences
Department of Mathematics, Heriot–Watt University
Edinburgh EH14 4AS, United Kingdom
Université de Lyon, Laboratoire de Physique, UMR 5672, CNRS
École Normale Supérieure de Lyon
46, allée d’Italie, F-69364 Lyon cedex 07, France
Department of Mathematics, University of Surrey
Guildford GU2 7XH, United Kingdom
We present a twistor space that describes super null-lines on six-dimensional superspace. We then show that there is a one-to-one correspondence between holomorphic vector bundles over this twistor space and solutions to the field equations of super Yang–Mills theory. Our constructions naturally reduce to those of the twistorial description of maximally supersymmetric Yang–Mills theory in four dimensions. 15th May 2012
1 Introduction and results
The twistor description of solutions to chiral field equations in six dimensions goes back to the work of Hughston [Hughston:1987he]. For recent works in this direction, see also [Saemann:2011nb, Mason:2011nw] and references therein. A corresponding twistor description of solutions to non-chiral field equations in six dimensions, such as the equations of motion of Yang–Mills theory with maximal supersymmetry, has only been developed partially [Devchand:1985au, Harnad:1987xq, Harnad:1995zy].
The purpose of this letter is to give a complete twistor description of the maximally or supersymmetric Yang–Mills (MSYM) equations in six dimensions with an emphasis on the underlying geometries.111Notice that twistor methods have recently been applied in the description of scattering amplitudes in this theory, see e.g. [Cheung:2009dc, Dennen:2009vk, Brandhuber:2010mm, Dennen:2010dh]. Our way of describing ambitwistor space might prove useful in this context. It is known that these equations can be encoded in constraint equations for a connection on superspace [Harnad:1985bc], which in turn correspond to the integrability condition of this connection along super null-lines [Devchand:1985au, Samtleben:2009ts]. We start by describing a twistor correspondence for null-lines in six-dimensional space-time in some detail. We then present the corresponding supersymmetric extension for maximal supersymmetry. The resulting twistor space, denoted by , turns out to be a rank- holomorphic supervector bundle over the four-dimensional Graßmannian . Next, we derive a Penrose–Ward transform to establish a one-to-one correspondence between equivalence classes of certain holomorphic vector bundles over and gauge equivalence classes of solutions to the equations of motion of six-dimensional MSYM theory. We end by demonstrating how our constructions reduce to those appearing in the twistorial description of MSYM theory in four dimensions [Witten:1978xx, Isenberg:1978kk, Isenberg:1978qd].
Throughout this letter, we shall be working in the complex setting. Concretely, our six-dimensional space-time is a copy of . If desired, however, reality conditions can be imposed at any point of our constructions, cf. [Saemann:2011nb, Mason:2011nw].
2 Ambitwistor space of superspace
In this section, we shall construct an ambidextrous twistor space (or ambitwistor space for short) of six-dimensional superspace. This twistor space is very similar in spirit to the ambitwistor space of four-dimensional superspace [Witten:1978xx, Isenberg:1978kk, Isenberg:1978qd]: while the latter parametrises super null-lines in four dimensions, parametrises certain super null-lines in six dimensions. We shall first describe the body of the supermanifold in detail, before we come to the supersymmetric extension. Our notation and conventions are close to those of [Saemann:2011nb].
2.1 Construction of the body of
Outline of the construction.
As usual in twistor geometry, we would like to establish a double fibration in which a correspondence space is simultaneously fibred over both twistor space and complexified flat space-time . The correspondence space in such a twistor fibration is a direct product of two manifolds.222If we are considering a compactified space-time, this direct product has to be compactified appropriately. The first factor in this product is space-time itself. The second factor is the moduli space of linear subspaces of space-time that we wish to describe with the twistor correspondence, restricted to those through the origin. Note that this makes the correspondence space the space of such linear subspaces with a given base point. Different base points may describe the same subspace, and modding out the dependence on equivalent base points, we obtain twistor space. In this letter, we are interested in light-rays or null-lines in six dimensions. We shall see below that the space of null-lines through the origin is given by the four-dimensional Graßmannian , which is the space of two-planes in . The correspondence space, which we shall denote by , is therefore ten-dimensional and we have . Modding out the dependence on equivalent base points amounts to quotenting the correspondence space by an (integrable) rank-one distribution known as a twistor distribution. This yields a nine-dimensional complex manifold which we denote by . Altogether, we have the following double fibration:
Here, the projection is the quotient map by the distribution and is the trivial projection. In the following, we shall discuss this double fibration, and in particular the structure of the space , in more detail.
Null-lines in six dimensions.
For simplicity, we shall work in spinor notation on , that is, we identify the tangent bundle with the antisymmetric tensor product of the rank-four bundle of anti-chiral spinors over . Correspondingly, we shall use local coordinates with and take the (flat) metric , where is the Levi-Civita symbol in four dimensions.
A null-vector in satisfies the equation
Null-lines are then obtained from null-vectors via the identification with . The resulting equivalences classes describe points on the Graßmannian , the homogeneous coordinates are called Plücker coordinates, and (2.2) is called the Plücker relation. The space features prominently in four-dimensional twistor correspondences, and a detailed account can be found, e.g., in [Ward:1990vs]. In the following, we merely recall a few facts necessary for our discussion.
Plücker coordinates provide an embedding of into via the quadric (2.2) with the being the six homogeneous coordinates on . Furthermore, as a coset space, the Graßmannian is given by
where is the little group of a null vector in . The relation (2.2) implies that the Plücker coordinates factorise according to
where and is the invariant tensor for with . We therefore have homogeneous coordinates and a coset description
Every plane in has a natural dual , which is spanned by a pair of chiral spinors for with . The represent homogeneous coordinates on a dual Graßmannian and they define a set of dual Plücker coordinates according to
The indices are to be understood as indices of the subgroup of the little group and is the invariant tensor of with . Furthermore, the two Graßmannians and can be identified via
The above equality, like all equalities in the following involving homogeneous coordinates, is to be understood as an equality of equivalence classes.
So far, we have seen that the correspondence space is topologically and it is trivially fibred over space-time. We may coordinatise by either or . To mod out the dependence of the null-lines on equivalent base points, we quotient the correspondence space by the rank-one twistor distribution that is generated by the vector field
The resulting space is a nine-dimensional complex manifold . Note that by construction, the twistor space is a rank-five holomorphic vector bundle over . Let us now give a few more details about the geometry of .
To this end, consider the dual tautological bundle333For more details on Graßmannians and bundles over them, see e.g. [Manin:1988ds]. over . This rank-two holomorphic vector bundle is generated by its global sections. We can parametrise the latter by moduli according to , where is the anti-chiral spin bundle over space-time. This allows us to write down the following short exact sequence:
Notice that has rank three such that is a rank-five holomorphic vector bundle, and its sections obey
The short exact sequence (2.9) induces a long exact sequence of cohomology groups and since all higher cohomology groups of and vanish, we conclude that and for . Global holomorphic sections of are of the form with , and is generated by these sections.
In fact, we can identify with the twistor space provided we identify the moduli with the space-time coordinates : the projection is given by with
and the vector fields (2.8) generating the twistor distribution indeed annihilate the . We shall refer to the relation (2.11) as the incidence relation. This relation implies a geometric twistor correspondence: points in correspond to null-lines in space-time given by , where is a particular solution to the incidence relation (2.11) and . Vice versa, points in space-time correspond to submanifolds bi-holomorphic to the Graßmannian . Note that the above null-lines are the null-lines obtained by intersecting two three-planes which are totally null (so-called -planes) [Hughston:1986hb]. Note also that we have just derived twistor space from space-time. Inversely, one can derive space-time from twistor space using Kodaira’s theorem of relative deformation theory: because of , there are no obstructions to relative deformations of inside . Thus, we have a family of such deformations whose moduli space can indeed be identified with space-time .
In addition to using , we may also use the dual Graßmannian with homogeneous coordinates (or Plücker coordinates ). Equipping (dual) space-time with coordinates , we can associate a correspondence space with coordinates and introduce a twistor distribution generated by
Altogether, we have a dual double fibration
where is the trivial projection and is given by with and . This incidence relation yields an analogous geometric twistor correspondence to the one above: points in correspond to null-lines in and points in correspond to embedding of in . Geometrically, the null-lines just described arise from intersecting two dual three-planes which are totally null (so-called -planes) [Hughston:1986hb]. Note that also can be described by a short exact sequence of the form (2.9).
The manifolds and yield a description of null-lines in terms of chiral spinors and anti-chiral spinors , respectively. To obtain an ambidextrous description, that is, a description involving both and simultaneously, we identify and via (2.7) and write and introduce as the zero-locus
Because of (2.14), global holomorphic sections are of the form
and therefore is a rank-five holomorphic vector bundle over . Altogether, we have a double fibration
with the same space-time manifold as in (2.1) and
is given by (2.15).
Double fibrations in Plücker coordinates.
In order to extend the above discussion to the supersymmetric setting with manifest maximal R-symmetry, we shall find it more convenient to work directly with Plücker coordinates. The advantage of these coordinates is that one can easily switch between chiral and anti-chiral descriptions by virtue of (2.7). The results of [Harnad:1987xq, Harnad:1995zy] seem to suggest that a description using the homogeneous coordinates and and having the full R-symmetry for supersymmetry manifest at the same time is not possible. Note, however, that in principle we can always substitute the Plücker coordinates by the homogeneous coordinates.
Recall that the Plücker coordinates define an embedding as the quadric (2.2). The bundle can be identified with [Manin:1988ds], and global sections of this bundle are given by , , and . This bundle appears in the short exact sequence analogue to (2.9) involving Plücker coordinates,
where has rank444To analyse the rank of such maps, it is helpful to consider them over a convenient point on the base manifold , e.g. . 11 and therefore has rank five. In more detail, the bundle is generated by its global sections and we can write . The eleven linear equations and in the fibre coordinates reduces the rank-16 bundle to a rank-five bundle . In Plücker coordinates, global holomorphic sections of are of the form with .
As before, identifying with , we can write the projection in the double fibration (2.1) as
|with an incidence relation of the form|
In a similar manner, we may repeat this analysis for and .
2.2 Supersymmetric extension
Let us now come to the supersymmetric extension of the twistor space . We shall first construct twistor spaces for chiral and anti-chiral super null-lines, before extending these spaces to the twistor space of super null-lines. We shall use Plücker coordinates on all the Graßmannians.
Twistor space for chiral super null-lines.
Let be the Graßmann parity changing operator. We start from superspace , which we describe by Graßmann even (bosonic) coordinates and Graßmann odd (fermionic) coordinates . Here, the index is an index of , the chiral subgroup of the R-symmetry group555not to be confused with the little group . On , we introduce the vector fields
which satisfy the relation
Chiral super null-lines are linear -dimensional subspaces . The moduli space of such linear superspaces through the origin is still so that the correspondence space is . To obtain a twistor space, we have to mod out the dependence on equivalent base points. Here, this amounts to quotienting by a twistor distribution generated by the vector fields
For each , we have four independent equations and hence the twistor distribution is of rank-. Moreover, it is integrable since while . Therefore, we have a foliation of by -dimensional complex supermanifolds . By construction, is a rank- holomorphic supervector bundle over , which we describe as a subbundle of :
Using coordinates in the fibres of and the usual Plücker coordinates on the base, the map is implicitly given by the relations
We can define a projection with
Because of the projection given in (2.25) and the trivial projection , we have the double fibration
The geometric twistor correspondence here is between points on and chiral super null-lines in as well as between points on and holomorphic embeddings of into . Explicitly, for any fixed point , the incidence relation (2.25) yields a (-dimensional) chiral super null-line
where represent a particular solution to the incidence relation while and constitute one free bosonic parameter and four fermionic parameters (note that the matrix is of rank two, so only four out of the initial eight fermionic parameters enter).
Twistor space for anti-chiral super null-lines.
A twistor space for anti-chiral super null-lines is constructed analogously. Here, we start from , with bosonic coordinates and fermionic coordinates , . The vector fields generating supertranslations read as
and they satisfy the relation
The correspondence space is given by , and to mod out the dependence on equivalent base points, we have to quotient by the vector fields
The resulting vector bundle can be regarded as a subbundle of over , which we coordinatise by in the fibres and the Plücker coordinates on the base. The relations satisfied by the fibre coordinates are
and we have a double fibration
Here, the projection reads as
This incidence relation yields again a geometric twistor correspondence between points in and (-dimensional) anti-chiral super null-lines in as well as points in and submanifolds in bi-holomorphic to .
Let us now come to the discussion of full supersymmetry. In particular, consider superspace equipped with coordinates . On this space, we have both the vector fields (2.20) and (2.28) with the identification . They generate the supersymmetry algebra in six dimensions,
where . Note that here, the metric appears explicitly.
The correspondence space is then topologically and coordinatised by . On , we introduce a rank- distribution generated by the vector fields
This distribution is integrable, with the non-vanishing Lie brackets given by and . The quotient of by this distribution is the ambitwistor space . It is a rank- supervector bundle over and its body is . We describe as a subbundle of with fibre coordinates and the map implicitly given by its kernel:
Note that has indeed rank , and we we have constructed a double fibration
As before, the projection is the trivial projection, while reads as
which describe global holomorphic sections of the bundle .
The geometric twistor correspondence induced by the incidence relation (2.38) is between points on and super null-lines in as well as between points on and holomorphic embeddings of into . Explicitly, for a fixed point , the above incidence relations determine a (-dimensional) super null-line by
where represent a particular solution to the incidence relation while and constitute one free bosonic parameter and eight fermionic parameters.
3 Twistor construction of the MSYM equations in six dimensions
We now come to the description of classical solutions to the equations of motion of MSYM theory on by means of holomorphic data on the ambitwistor space . The key fact here is that these equations are equivalent to certain constraint equations for a connection on the superspace [Harnad:1985bc] and furthermore, that these constraint equations can in turn be interpreted as integrability conditions along certain null-lines [Devchand:1985au, Samtleben:2009ts]. Concretely, the equations of motion of MSYM theory in six dimensions are equivalent to the following set of constraint equations [Harnad:1985bc]:
Here, the covariant derivatives are given in terms of a gauge potential with components , and as
where the derivatives