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Symplectic Geometry of the Moduli Space of Flat Connections on a Riemann Surface: Inductive Decompositions and Vanishing Theorems

Published online by Cambridge University Press:  20 November 2018

Lisa C. Jeffrey  and
Jonathan Weitsman
Lisa C. Jeffrey
Affiliation:
Mathematics Department, University of Toronto, Toronto, Ontario, M5S 3G3
Jonathan Weitsman
Affiliation:
Mathematics Department, University of California, Santa Cruz, California 95064, USA
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Abstract

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This paper treats the moduli space ${{\mathcal{M}}_{g,1}}(\Lambda)$ 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 $\Lambda$, where $\Lambda$ is in the fundamental alcove of a Lie algebra. We construct natural line bundles over ${{\mathcal{M}}_{g,1}}(\Lambda)$ 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 $d$.

Keywords

58F05
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 52 , Issue 3 , 01 June 2000 , pp. 582 - 612
Copyright
Copyright © Canadian Mathematical Society 2000

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