Geometry & Topology

Goldman algebra, opers and the swapping algebra

François Labourie

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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 ( ) –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.

Article information

Geom. Topol., Volume 22, Number 3 (2018), 1267-1348.

Received: 25 July 2014
Revised: 25 October 2016
Accepted: 11 November 2016
First available in Project Euclid: 31 March 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx]
Secondary: 32J15: Compact surfaces 17B63: Poisson algebras

Poisson algebra Teichmüller theory gauge theory


Labourie, François. Goldman algebra, opers and the swapping algebra. Geom. Topol. 22 (2018), no. 3, 1267--1348. doi:10.2140/gt.2018.22.1267.

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