Twistors, minimal surfaces and strings

Abstract

It is shown that holomorphic curves on a projective twistor space T correspond to null holomorphic curves in complexified Minkowski spacetime. The real and imaginary parts of such a curve define a minimal 2-surface on real Minkowskian or real Euclidean 4-space. The minimal 2-surface equations can thereby be solved in terms of two free holomorphic functions of one variable. Pairs of holomorphic curves on twistor space describe complex minimal 2-surfaces. Such surfaces may also admit real string cross sections. Using this, solutions of the string equations in real Minkowskian 3-space are obtained in terms of two real analytic functions. The twistor transform of a general string in complex Minkowskian 4-space is obtained in terms of non-holomorphic curves in T.