On Supertwistors, the Penrose-Ward Transform and N=4 super Yang-Mills Theory

TL;DR

This paper explores the deep connections between supertwistor geometry, the Penrose-Ward transform, and N=4 super Yang-Mills theory, revealing equivalences between different formulations and extending to the full theory.

Contribution

It demonstrates the equivalence of supertwistor descriptions to field equations of N=4 SYM and extends the framework to the full theory using supertwistor space.

Findings

Equivalence of supertwistor and field theory descriptions for self-dual N=4 SYM.

Explicit connection between holomorphic Chern-Simons theory and SYM equations.

Extension of supertwistor methods to the full N=4 SYM theory.

Abstract

It was recently shown by Witten that B-type open topological string theory with the supertwistor space CP^{3|4} as a target space is equivalent to holomorphic Chern-Simons (hCS) theory on the same space. This hCS theory in turn is equivalent to self-dual N=4 super Yang-Mills (SYM) theory in four dimensions. We review the supertwistor description of self-dual and anti-self-dual N-extended SYM theory as the integrability of super Yang-Mills fields on complex (2|N)-dimensional superplanes and demonstrate the equivalence of this description to Witten's formulation. The equivalence of the field equations of hCS theory on an open subset of CP^{3|N} to the field equations of self-dual N-extended SYM theory in four dimensions is made explicit. Furthermore, we extend the picture to the full N=4 SYM theory and, by using the known supertwistor description of this case, we show that the corresponding constraint equations are (gauge) equivalent to the field equations of hCS theory on a quadric in CP^{3|3}xCP^{3|3}.