Higher dimensional uniformisation and W-geometry

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.

LaTeX file; 25/13 pages in b/l mode ; version to appear in Nuc. Phys. B

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• Higher spin JT gravity and a matrix model dual
• Primary currents and Riemannian geometry in W-algebras
• Goldman Algebra, Opers and the Swapping Algebra
• Induced quantum gravity on a Riemann Surface
• Covariantising the Beltrami equation in W-gravity
• Hamiltonian Algebroids and deformations of complex structures on Riemann curves
• Lie Algebroids and generalized projective structures on Riemann surfaces
• Large Chiral Diffeomorphisms on Riemann Surfaces and W-algebras