Action-angle coordinates for surface group representations in genus zero

TL;DR

This paper introduces a polygonal model and explicit action-angle coordinates for a family of surface group representations into PSL(2,R), revealing a symplectic structure and an isomorphism with complex projective space.

Contribution

It provides a novel polygonal parametrization and explicit action-angle coordinates for Deroin--Tholozan representations, linking geometric and symplectic structures.

Findings

Polygonal model parametrizes the relative character variety.

Action-angle coordinates form an explicit isomorphism with complex projective space.

Coordinates are almost global Darboux for the Goldman symplectic form.

Abstract

We study a compact family of totally elliptic representations of the fundamental group of a punctured sphere into $\mathrm{PSL}(2,\mathbb R)$ discovered by Deroin and Tholozan and named after them. We describe a polygonal model that parametrizes the relative character variety of Deroin--Tholozan representations in terms of chains of triangles in the hyperbolic plane. We extract action-angle coordinates from our polygonal model as geometric quantities associated to chains of triangles. The coordinates give an explicit isomorphism between the space of representations and the complex projective space. We prove that they are almost global Darboux coordinates for the Goldman symplectic form.