Goldman algebra, opers and the swapping algebra

Labourie, François

doi:10.2140/gt.2018.22.1267

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Goldman algebra, opers and the swapping algebra

François Labourie

Geometry & Topology 22 (2018) 1267–1348

DOI: 10.2140/gt.2018.22.1267

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 ${SL}_{n} (ℝ)$ –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/

Volume 22, issue 3 (2018)