On supertwistors, the Penrose--Ward transform and $\CN=4$ super-Yang--Mills theory

Abstract

It was recently shown by Witten that B-type open topological string theory with the supertwistor space $\mathbb{C}P^{3|4}$ 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 $\mathcal{N}=4$ super-Yang–Mills (SYM) theory in four dimensions. We review the supertwistor description of self-dual and anti-self-dual $\mathcal{N}$-extended SYM theory as the integrability of SYM fields on complex $(2|\mathcal{N})$-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 $\mathbb{C}P^{3|\mathcal{N}}$ to the field equations of self-dual $\mathcal{N}$-extended SYM theory in four dimensions is made explicit. Furthermore, we extend the picture to the full $\mathcal{N}=4$ 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 $\mathbb{C}P^{3|3}\times \mathbb{C}P^{3|3}$.