Inspired by Ed Witten's investigation of string theory with twistor space as the target space, I took another look at my old work on the twistor description of string theory.
Note that my research is about the twistor description of strings in space-time, and not strings where twistor theory is the target space. I am revising my views on the twistor quantization of the null curves that are the geometric objects underlying strings. This revision comes about from a better appreciation of the role that Roger Penrose's original twistor quantization of null geodesics has within string theory. It is essential to consider the complex null geodesics to get the full structure of the Penrose transform. Given that this classical and quantum picture is of the ground states of open strings, it is almost compulsory to set string theory in the complex setting as well. Could it be that the dimensional problems associated with string theory have more to do with the failure to properly appreciate the complex picture?
There are now some hints that the twistor theory version admits a sensible first quantization in D=4. The covariant version of the theory admits a representation of the Virasoro algebra without anomaly, though the characterization of the states remains slightly awkward. The important new idea over my old work is that it is only complexified string theory that makes sense in D=4! A set of slides on the matter is available here (low res) and here (high res). This talk was given in D.A.M.T.P. and the M.I. here in spring 2004.
The following list of papers may be of interest (bibliography and links under construction). This work all stemmed from some observations by N. Hitchin linking the integral-free form of the Weierstrass parametrization for minimal surfaces in three dimensions, to holomorphic curves in minitwistor space. More details and references for this are in the talk cited above.
"Twistors, minimal surfaces and strings", Classical and Quantum Gravity 2 (1985), L113-L119
"An ambitwistor description of bosonic or supersymmetric minimal surfaces and strings in four dimensions", Classical and Quantum Gravity 3 (1986), 753-761
"Twistor quantisation of open strings in three dimensions", Classical and Quantum Gravity 4 No 5 (1987), 1193-1205