Supersymmetric Yang-Mills equations and supertwistors

This paper is a survey of results relating the supertwistor correspondence on N-extended super-Minkowski space M^4|4N to supersymmetric Yang-Mills (SSYM) theory. A theorem of Manin relating bundles on the (3-N)th infinitesimal neighborhood L_(3-N)^5|2N of super null-line space L ^{5 2N}→P ^{3 N}×P ^{3 N*} to solutions of the SSYM equations is analyzed in terms of component fields, interpolating between the N = 0 and N = 3 results studied previously. Using an inductive approach based on the degree of odd homogeneity and a particular gauge condition (the D-gauge), the graded Frobenius equations for covariant constancy along super null-lines are solved. The resulting solution space is shown to define a bundle over L^5|2N which extends to L_(3-N)^5|2N when the SSYM equations are satisfied. Conversely, the inverse transform determines super connections that are integrable along super null-lines in M^4|4N. These superconnections determine a supermultiplet which solves the SSYM equations when the bundle over L^5|2N extends to L_(3-N)^5|2N. A clarification is given concerning the relation between supersymmetry transformations of the component fields and Lie derivations of superconnections on M^4|4N satisfying super null-line integrability conditions and the D-gauge conditions. Our approach is aimed at bridging the gap between the abstract sheaf-theoretic formulation preferred by mathematicians and the coordinate formulation familiar to physicists.