Goldman Algebra, Opers and the Swapping Algebra

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 ${\mathsf{S}\mathsf{L}}_n(\mathbb{R})$-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.