Goldman Algebra, Opers and the Swapping Algebra

Abstract

We define a Poisson Algebra called the {\em swapping algebra} using the intersection of curves in the disk. We interpret a subalgebra of the fraction algebra of the swapping algebra -- called the {\em 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 ${\mathsf{SL}}_n(\mathbb{R})$-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 Wolpert formula.

80 pages, 7 figures

References in corpus (2)

• Anosov representations: Domains of discontinuity and applications
• Hyperconvex representations and exponential growth

Cited by in corpus (2)

• Rank $n$ swapping algebra for ${\mathrm{PGL}}_n$ Fock-Goncharov $\mathcal{X}$ moduli space
• Rank $n$ swapping algebra for the $\mathrm{PSL}(n,\mathbb{R})$ Hitchin component