The symplectic nature of fundamental groups of surfaces.(English)Zbl 0574.32032
A natural symplectic structure on the space of representations \(\pi\) \(\to G\), where \(\pi\) is the fundamental group of a Riemann surface and G is a semisimple Lie group, is introduced. The author considers this result as a common cause for the existence of a symplectic structure on different moduli spaces associated with Riemann surfaces. For example if \(G=PSL_ 2({\mathbb{R}})\) then the symplectic structure coincides with the Weil- Petersson’s one on Teichmüller space.
Reviewer: A.Givental’
MSC:
| 32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |
| 14H30 | Coverings of curves, fundamental group |
| 53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |
| 30F99 | Riemann surfaces |
| 14E20 | Coverings in algebraic geometry |
| 32G13 | Complex-analytic moduli problems |
| 22E46 | Semisimple Lie groups and their representations |
| 14H15 | Families, moduli of curves (analytic) |
Keywords:
Weil-Petersson structure;free differential calculus;symplectic structure;fundamental group of a Riemann surface;moduli spaces;Teichmüller spaceCite
Full Text:DOI
References:
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