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

TL;DR

The paper develops a comprehensive twistor-theoretic framework linking holomorphic Chern-Simons theory on supertwistor spaces to self-dual and anti-self-dual N-extended SYM theories, and then extends this to the full N=4 SYM via a quadric in a product of supertwistor spaces.It provides explicit constructions: the Penrose-Ward transform between hCS on ${\mathbb{C}P}^{3|\mathcal{N}}$ and SDYM on ${\mathbb{R}}^4$, detailed superfield expansions capturing the N=4 multiplet, and a quadric ${\mathbb{L}}^{5|6}$ encoding both self-dual and anti-self-dual sectors in a unified geometric setting.The work clarifies reality structures for different signatures, discusses the potential for an action principle on the quadric, and outlines how the twistor description could illuminate string-theoretic connections (e.g., with topological B-models and SFT), while highlighting several open problems and future directions.

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}.