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 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 SYM 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 CP 3|3 .