Supermanifolds, super twistor spaces and super Yang-Mills fields

S Shnider9782760602861, 2760602869

The notion of super Minkowski space, and more generally the notion of supermanifolds, plays an analogous role in this theory to that of Minkowski space and Lorentzian manifolds in the theory of special and general relativity. In these notes we want to outline some recent results concerning the representation of solutions of the supersymmetric Yang-Mills equations on ordinary Minkowski space in terms of superfields on super Minkowski space which satisfy the same equations. These will be represented in turn by holomorphic vector bundles on a super twistor space, in analogue with the Ward representation of instantons in terms of holomorphic vector bundles on projective twistor space (Ward 1977). The basic idea here is due to Witten A978). The representation on super twistor space was carried out in complete detail in Manin A984), and the details of the relation between the fields satisfying the integrability conditions of Witten and the classical field equations is carried out in Hamad et al. A985). Both of these ideas were formulated (at least in special cases) in Witten’s fundamental paper. In these notes we want to give an overview of both of these developments, with some additional details which were not included in the original references.