Higher dimensional uniformisation and W-geometry

Abstract

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 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 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. *

Primary currents and Riemannian geometry in algebras

“…3 (z, z)) are both the so-called Painlevé invariants [67] as pointed out in [71]. By the way from Eq(2.32)(1.8)(1.9) is easy to derive [70]:…”

Large chiral diffeomorphisms on Riemann surfaces and W-algebras

“…We have simply in mind the ideas of, firstly, pursuing further ahead the method given by Forsyth in [14], and secondly, dealing with scalar coordinates considered as solutions of generalized Beltrami equations (see e.g. [8] or [18] appendix C2) just about the approach given in [8,10]. Our motivation for using the inhomogeneous coordinates rests on the fact that they seem to be more natural for constructing projective invariants.…”

“…In order to avoid a possible confusion with the named Laguerre-Forsyth form of a conformally covariant differential equation [8], in the present paper, we adopt the name Forsyth frames. We have simply in mind the ideas of, firstly, pursuing further ahead the method given by Forsyth in [14], and secondly, dealing with scalar coordinates considered as solutions of generalized Beltrami equations (see e.g.…”

“…Particular interest has been devoted to the so-called W-algebras [3] which come out from different principles [4,5,6] and the question of their geometric origin remains still unclear or unsatisfactory despite the various attempts given by [5,7,8,9,10,11].…”

“…Focussing here only on one s-th order differential equation, one may either work directly with the solutions or with uniformizing coordinates considered as ratios of linear independent solutions [14,8]. It turns out to be a matter of taste of working either with homogeneous coordinates or inhomogeneous one on CP s−1 .…”