Goldman algebra, opers and the swapping algebra

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 ${SL}_n\left(ℝ\right)$–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.