Goldman Algebra, Opers and the Swapping Algebra

TL;DR

This paper introduces the swapping algebra, a Poisson algebra based on curve intersections, and connects it to Hitchin components, opers, and symplectic structures, extending Wolpert's formula.

Contribution

It defines the swapping algebra and its multifractions, linking them to Hitchin components, opers, and symplectic geometry, providing new algebraic and geometric insights.

Findings

Defined the swapping algebra as a Poisson algebra.

Connected the algebra of multifractions to Hitchin components and opers.

Extended Wolpert's formula within this new framework.

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.