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Representations with Weighted Frames and Framed Parabolic Bundles

Published online by Cambridge University Press:  20 November 2018

J. C. Hurtubise  and
L. C. Jeffrey
J. C. Hurtubise
Affiliation:
Department of Mathematics and Statistics, McGill University email: [email protected]
L. C. Jeffrey
Affiliation:
Department of Mathematics, University of Toronto email: [email protected]
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Abstract

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There is a well-known correspondence (due to Mehta and Seshadri in the unitary case, and extended by Bhosle and Ramanathan to other groups), between the symplectic variety ${{M}_{h}}$ of representations of the fundamental group of a punctured Riemann surface into a compact connected Lie group $G$, with fixed conjugacy classes $h$ at the punctures, and a complex variety ${{\mathcal{M}}_{h}}$ of holomorphic bundles on the unpunctured surface with a parabolic structure at the puncture points. For $G\,=\,\text{SU}\left( 2 \right)$, we build a symplectic variety $P$ of pairs (representations of the fundamental group into $G$, “weighted frame” at the puncture points), and a corresponding complex variety $\mathcal{P}$ of moduli of “framed parabolic bundles”, which encompass respectively all of the spaces ${{M}_{h}}$, ${{\mathcal{M}}_{h}}$, in the sense that one can obtain ${{M}_{h}}$ from $P$ by symplectic reduction, and ${{\mathcal{M}}_{h}}$ from $\mathcal{P}$ by a complex quotient. This allows us to explain certain features of the toric geometry of the $\text{SU(2)}$ moduli spaces discussed by Jeffrey and Weitsman, by giving the actual toric variety associated with their integrable system.

Keywords

58F0514D20
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 52 , Issue 6 , 01 December 2000 , pp. 1235 - 1268
Copyright
Copyright © Canadian Mathematical Society 2000

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