Higher-dimensional uniformisation and \(W\)-geometry
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Publication:1902708
DOI10.1016/0550-3213(95)00527-7zbMATH Open1003.81501arXivhep-th/9412078OpenAlexW1989202938MaRDI QIDQ1902708
Author name not available (Why is that?)
Publication date: 22 November 1995
Published in: (Search for Journal in Brave)
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.
Full work available at URL: https://arxiv.org/abs/hep-th/9412078
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