Covariantising the Beltrami equation in W-gravity

TL;DR

This paper extends the Beltrami equation to W-gravity by relating higher-dimensional uniformisation spaces to W-symmetry, providing a geometric framework that linearises W-symmetry and clarifies the link with KdV flows.

Contribution

It constructs the explicit relationship between Beltrami differentials in higher-dimensional manifolds and those in W-gravity, completing the geometric understanding of W-symmetry.

Findings

Higher-dimensional manifolds serve as superspace for W-gravity.

W-symmetry linearises on these manifolds.

Clarifies the link between KdV flows and W-diffeomorphisms.

Abstract

Recently, certain higher dimensional complex manifolds were obtained in [hep-th/9412078] by associating a higher dimensional uniformisation to the generalised Teichm\"uller spaces of Hitchin. The extra dimensions are provided by the ``times'' of the generalised KdV hierarchy. In this paper, we complete the proof that these manifolds provide the analog of superspace for W-gravity and that W-symmetry linearises on these spaces. This is done by explicitly constructing the relationship between the Beltrami differentials which naturally occur in the higher dimensional manifolds and the Beltrami differentials which occur in W-gravity. This also resolves an old puzzle regarding the relationship between KdV flows and W-diffeomorphisms.