A Note on Classical Aspects of the Four Dimensional Anomaly-Free Twistor String

The recently introduced anomaly-free twistor string in four dimensions is shown to be defined not just in flat but also in curved twistor space. Further, arguments are given that the classical limit of the corresponding string field theory, if it exists, is related to general relativity, in particular to the Isenberg and Yasskin construction using teleparallel gravity. For spacetimes of Petrov type D with two shear-free null congruences the construction can be simplified using two-dimensional twistor manifolds.