Supersymmetric Yang-Mills equations and supertwistors

Abstract

This paper is a survey of results relating the supertwistor correspondence on N-extended super-Minkowski space M 4|4 N to supersymmetric Yang-Mills (SSYM) theory. A theorem of Manin relating bundles on the (3- N)th infinitesimal neighborhood L (3− N) 5|2 N of super null-line space L 5 2 N → 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|2 N which extends to L (3− N) 5|2 N when the SSYM equations are satisfied. Conversely, the inverse transform determines super connections that are integrable along super null-lines in M 4|4 N . These superconnections determine a supermultiplet which solves the SSYM equations when the bundle over L 5|2 N extends to L (3− N) 5|2 N . A clarification is given concerning the relation between supersymmetry transformations of the component fields and Lie derivations of superconnections on M 4|4 N 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.