The holomorphic geometry of closed bosonic string theory and Diff S¹/S¹

We present a proposal for a classical non-perturbative bosonic closed string field theory based on Kähler geometry. Motivated by the observation that the loop space of Minkowski space-time is a Kähler manifold, we conjecture that infinite-dimensional complex (Kähler) geometry is the right setting for closed string field theory and that the correct dynamical variable (closed string field) is the Kähler potential. To incorporate reparametrization invariance, one must consider the space of complex structures Diff S¹/S¹. Geometrical considerations then lead us to a (non-linear) equation of motion for the Kähler potential which is that the curvature of a certain vector bundle over Diff S¹/S¹ vanish. This is basically the requirement of conformal invariance. Loops on flat Minkowski space are shown to be a solution only if the space-time dimension is 26. We also discuss geometric quantization since our approach can be viewed as an application of geometric quantization to string theory. Previously announced mathematical results that Diff S¹/S¹ is a homogeneous Kähler manifold are established in more detail and its curvature is computed explicitly. We also give an axiomatic formulation of the minimal geometric setting we require - this is an attempt to avoid basing the theory on loops of a given riemannian manifold. Einstein's field equations are derived in an adiabatic approximation. The relation of our work to some other approaches to string theory is briefly discussed.