Higher dimensional uniformisation and W-geometry Original Research Article

We formulate the uniformisation problem underlying the geometry of Wn-gravity using the differential equation approach to W-algebras. We construct Wn-space (analogous to superspace in supersymmetry) as an (n − 1)-dimensional complex manifold using isomonodromic deformations of linear differential equations. The Wn-manifold is obtained by the quotient of a Fuchsian subgroup of PSL(n,R) which acts properly discontinuously on a simply connected domain in CPn − 1. The requirement that a deformation be isomonodromic furnishes relations whic enable one to convert non-linear W-diffeomorphisms to (linear) diffeomorphisms on the Wn-manifold. We discuss how the Teichmüller 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 Wn-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.