Group systems, groupoids, and moduli spaces of parabolic bundles (original) (raw)
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Canadian Journal of Mathematics, 2000
This paper treats the moduli space g,1(Λ) of representations of the fundamental group of a Riemann surface of genus g with one boundary component which send the loop around the boundary to an element conjugate to exp Λ, where Λ is in the fundamental alcove of a Lie algebra. We construct natural line bundles over g,1(Λ) and exhibit natural homology cycles representing the Poincaré dual of the first Chern class. We use these cycles to prove differential equations satisfied by the symplectic volumes of these spaces. Finally we give a bound on the degree of a nonvanishing element of a particular subring of the cohomology of the moduli space of stable bundles of coprime rank k and degree d.
1993
This talk reviews results on the structure of algebras consisting of meromorphic differential operators which are holomorphic outside a finite set of points on compact Riemann surfaces. For each partition into two disjoint subsets of the set of points where poles are allowed, a grading of the algebra and of the modules of λ−forms is introduced. With respect to this grading the Lie structure of the algebra and of the modules are almost graded ones. Central extensions and semi-infinite wedge representations are studied. If one considers only differential operators of degree 1 then these algebras are generalizations of the Virasoro algebra in genus zero, resp. of Krichever Novikov algebras in higher genus. 1 invited talk at the International Symposium on Generalized Symmetries in Physics at the Arnold-Sommerfeld-Institute, Clausthal, Germany, July 26 – July 29, 1993
Journal of the American Mathematical Society, 1990
Dedicated to Lipman Bers on the occasion of his seventy-fifth birthday TABLE OF CONTENTS O. Introduction and statement of main results 1. Horocyclic coordinates 2. The zw = t plumbing construction 3. The plumbing construction for an admissible graph 4. Deformation (TeichmiiUer) and moduli (Riemann) spaces 5. Torsion free terminal b-groups 6. One-dimensional deformation spaces 7. Deformation spaces for torsion free terminal b-groups 8. One-dimensional moduli spaces 9. Moduli spaces for torsion free terminal b-groups 10. Forgetful maps 11. Metrics on surfaces and their Teichmiiller spaces 12. Appendix I: Calculations in PSL(2, C) and SL(2, C) 13. Appendix II: A computer program for computing torsion free terminal bgroups 14. Appendix III: Independence of gluing on choice of annuli O. INTRODUCTION AND STATEMENT OF MAIN RESULTS This paper is concerned with the general problem of explicitly describing intrinsic parameters for Teichmiiller and Riemann spaces. Ideally, we want to be able to read off from a given Riemann surface its position in moduli space. Further, we want to attach various geometric and analytic objeCts such
A gauge theoretic aspects of parabolic bundles over Klein surfaces
2022
In this article, we study the gauge theoretic aspects of real and quaternionic parabolic bundles over a real curve (X,σX)(X, σ_X)(X,σX), where X is a compact Riemann surface and σX is an anti-holomorphic involution. For a fixed real or quaternionic structure on a smooth parabolic bundle, we examine the orbits space of real or quaternionic connection under the appropriate gauge group. The corresponding gauge-theoretic quotients sit inside the real points of the moduli of holomorphic parabolic bundles having a fixed parabolic type on a compact Riemann surface XXX.
Triangulations and moduli spaces of Riemann surfaces with group actions
Manuscripta Mathematica, 1995
We study that subset of the moduli space Ma of stable genus g, g > 1, Riemann surfaces which consists of such stable Riemann surfaces on which a given finite group F acts. We show first that this subset is compact. It turns out that, for general finite groups F, the above subset is not connected. We show, however, that for Z2 actions this subset is connected. Finally, we show that even in the moduli space of smooth genus g Riemann surfaces, the subset of those Riemann surfaces on which Z2 acts is connected, ha view of deliberations of Klein ([8]), this was somewhat surprising. These results are based on new coordinates for moduli spaces. These coordinates are obtained by certain regular triangulations of Riemann surfaces. These triangulations play an important role also elsewhere, for instance in apl)roximating eigenfunctions of tim Laplace operator numerically.
A note on the theta characteristics of a compact Riemann surface
Journal of the Australian Mathematical Society, 2004
Let X be a compact connected Riemann surface and ξ a square root of the holomorphic contangent bundle of X. Sending any line bundle L over X of order two to the image of dim H0(X, ξ ⊗ L) − dim H0(X, ξ) in Z/2Z defines a quadratic form on the space of all order two line bundles. We give a topological interpretation of this quadratic form in terms of index of vector fields on X.
The Riemann surface in the target space and vice versa
Physics Letters B, 1989
It is observed that the classical part of the partition function associated with the mappings from a genus-g Riemann surface Eg to an "almost complex" target space T2a is equal to that related to the mappings from Ed to T2g. The classical part related to the mappings from a genus-2g Riemann surface E2g, described by a "real" period matrix, to a target space Td is equal to the classical part related to mappings from E2d to Tg. Some physical consequences of these mathematical identities are discusses.