Volume 22, issue 3 (2018)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Goldman algebra, opers and the swapping algebra

François Labourie

Geometry & Topology 22 (2018) 1267–1348
Abstract

We define a Poisson algebra called the swapping algebra using the intersection of curves in the disk. We interpret a subalgebra of the fraction algebra of the swapping algebra, called the algebra of multifractions, as an algebra of functions on the space of cross ratios and thus as an algebra of functions on the Hitchin component as well as on the space of SLn()–opers with trivial holonomy. We relate this Poisson algebra to the Atiyah–Bott–Goldman symplectic structure and to the Drinfel’d–Sokolov reduction. We also prove an extension of the Wolpert formula.

Keywords
Poisson algebra, Teichmüller theory, gauge theory
Mathematical Subject Classification 2010
Primary: 32G15
Secondary: 32J15, 17B63
References
Publication
Received: 25 July 2014
Revised: 25 October 2016
Accepted: 11 November 2016
Published: 16 March 2018
Proposed: Danny Calegari
Seconded: Leonid Polterovich, Jean-Pierre Otal
Authors
François Labourie
Laboratoire Jean-Alexandre Dieudonné, CNRS
Université Côte d’Azur
Nice
France
http://math.unice.fr/~labourie/