ENSLAPPA564/94
[email protected]/9512066
August 1, 2021
Twistors and Supersymmetry
P.S. Howe^{1}^{1}1Permanent address: Dept. of Mathematics, King’s College, London
L.A.P.P.
B.P. 110, 74941 AnnecyleVieux, France
An overview is given of the application of twistor geometric ideas to
supersymmetry with particular emphasis on the construction of superspaces
associated with fourdimensional spacetime.
Talk given at the Leuven Conference on High Energy Physics, July 1995.
1 Introduction
Twistor geometry, suitably interpreted, is particularly wellsuited to the description of supersymmetric theories in the superspace formalism. There have been applications: to field theories, notably supersymmetric YangMills (SYM) and supergravity (SG), leading to offshell representations and to the interpretation of onshell constraints as integrability conditions, and to superparticles, strings and extended objects in the context of doubly supersymmetric formulations, that is, formulations with both spacetime (target space) and worldsurface supersymmetry.
The term “twistor geometry” as I shall interpret it in the supersymmetric context embraces the three main geometrical approaches to supersymmetry that have been studied: chiral supergeometry, lightlike integrability or super twistor theory, and harmonic superspace, each of these three formalisms being particular examples of twistor supergeometries as I shall show. In general one can say that the twistor approach is a very valuable tool for constructing and understanding superspace geometries and, in particular, it clarifies the geometrical structure of harmonic superspace. Since (almost) all supersymmetric theories are amenable to one or more of the above descriptions it follows that twistor supergeometry is universal: it underlies (almost) all supersymmetric theories of interest.
In this talk I shall focus on the construction of various (flatspace) supergeometries associated with fourdimensional spacetime. In four dimensions twistors are naturally associated with conformal symmetry, a symmetry which, as is wellknown (see for example [1]), is particularly relevant in the supersymmetric case for three reasons: firstly, massless, nongravitational theories of interest are classically superconformal, secondly, it is known that there are fourdimensional superconformal quantum field theories, and thirdly, the construction and study of supergravity theories is made much easier if one uses the superconformal perspective. For the most part I shall work in complex spacetime; this has some advantages from a formal point of view and has some relevance in that quantum field theories can be studied in regions of products of several copies of complex spacetime. Moreover, there is no loss of generality since one can easily impose reality in the formalism. In the case of standard twistor theory, there is still a complex twistor space associated with real (Euclidean) space; in the case of harmonic supergeometry one finds that the complex geometry of twistor space has to be replaced by supergeometry, which will be briefly explained in section 4. Finally, I shall give a list of some applications to supersymmetric field theories.
2 Twistors
Consider the equation
(1) 
where and are twocomponent commuting spinors and is a fourvector in spinor form. The pair is an element of twistor space , and labels a point in complex Minkowski space . Equation (1) establishes a correspondence between 𝕋 and 𝕄 in the following way [2]: firstly, if is held fixed, (1) defines a plane (called a plane) in 𝕄 which is totally null (all tangent vectors are null) and antiselfdual (the bivector constructed from any two independent tangents is antiselfdual); secondly if is held fixed, then one can solve (1) for as a function of . Since (1) is clearly invariant under common rescalings of and this second point of view determines a projective line in projective twistor space, ℙ, also known as a twistor line. Thus we have the correspondence

points twistor lines

points planes
We can present this a little differently as follows: