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

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

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Published 1 January 2005