标题： 高维统一和W-几何 显示英文标题

标题： Higher dimensional uniformisation and W-geometry

摘要： 我们使用W代数的微分方程方法来表述W_n重力几何背后的统一问题。我们通过线性微分方程的等单变分变形来构造W_n空间（类似于超对称中的超空间），作为一个(n-1)维复流形。W_n流形是通过PSL(n,R)的一个Fuchs子群在CP^{n-1}中一个单连通区域上的适当不连续作用得到的商空间。要求变形为等单变分的条件提供了将非线性W微分同胚转换为W_n流形上的（线性）微分同胚的关系。我们讨论了如何将Hitchin引入的Teichmuller空间解释为W_n流形上的复结构空间或具有实单极的射影结构空间。射影结构由Halphen不变量表征，这些不变量是Schwarzian的适当推广。此构造适用于所有“一般”W代数。 显示英文摘要

摘要： 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.