A Kahler structure on the space of string worldsheets

Let (M,g) be an oriented Lorentzian 4-manifold, and consider the space S of oriented, unparameterized time-like 2-surfaces in M (string worldsheets) with fixed boundary conditions. Then the infinite-dimensional manifold S carries a natural complex structure and a compatible (positive-definite) Kahler metric h on S determined by the Lorentz metric g. Similar results are proved for other dimensions and signatures, thus generalizing results of Brylinski (1990) regarding knots in 3-manifolds. Generalizing the framework of Lempert (1992) the authors also investigate the precise sense in which S is an infinite-dimensional complex manifold.