A Complex Analytic Approach

Twistor theory has provided many valuable insights into the nature of space-time geometry and the processes that take place within that geometry. In particular, the very special role played by complex analytic functions has been elucidated in several contexts [36, 37, 38, 39, 41, 43]. A feature common to many twistor constructions is the solution of linear or non-linear differential equations by consideration of suitable analytic objects and the construction of machinery for transforming such objects into the desired solutions.

As noted in [40], this particular approach was in use many years before the formal development of twistor theory began. Weierstrass' 1866 construction [53] of solutions of Plateau's problem in three dimensions, for example, may be regarded as an important model for the twistor solution of non-linear problems such as the non-linear graviton [39]. The programme of work described here is founded on the notion that there is much that might be learnt by exploring constructions of the Weierstrass type, especially if insight can also be gained into string theory and its relationship to gravity. That twistor theory might provide some clues regarding this relationship has been suspected for some time (see e.g. [17]).

There are several sources of motivation for considering possible links between twistor theory and string theory. Hitchin [18] explained how the Weierstrass parametrization is essentially the understanding of how arbitrary holomorphic curves in the twistor space for three dimensions correspond to null holomorphic curves in complex three-space.