Higher dimensional uniformisation and W-geometry

TL;DR

This paper develops a geometric framework for W_n-gravity using differential equations, constructing W_n-space as a complex manifold via isomonodromic deformations and relating it to Teichmüller spaces.

Contribution

It introduces a novel geometric approach to W_n-gravity, constructing W_n-space through isomonodromic deformations and relating it to Teichmüller spaces with real holonomy.

Findings

W_n-space constructed as a quotient of a Fuchsian subgroup

Relations converting non-linear W-diffeomorphisms to linear diffeomorphisms

Interpretation of Teichmüller spaces as complex or projective structures

Abstract

We formulate the uniformisation problem underlying the geometry of W_n-gravity using the differential equation approach to W-algebras. We construct W_n-space (analogous to superspace in supersymmetry) as an (n-1) dimensional complex manifold using isomonodromic deformations of linear differential equations. The W_n-manifold is obtained by the quotient of a Fuchsian subgroup of PSL(n,R) which acts properly discontinuously on a simply connected domain in CP^{n-1}. The requirement that a deformation be isomonodromic furnishes relations which enable one to convert non-linear W-diffeomorphisms to (linear) diffeomorphisms on the W_n-manifold. We discuss how the Teichmuller spaces introduced by Hitchin can then be interpreted as the space of complex structures or the space of projective structures with real holonomy on the W_n-manifold. The projective structures are characterised by Halphen invariants which are appropriate generalisations of the Schwarzian. This construction will work for all ``generic'' W-algebras.