Action-angle coordinates for surface group representations in genus zero

We study a family of compact components of totally elliptic representations of the fundamental group of a punctured sphere into PSL$(2,\mathbb R)$ discovered by Deroin-Tholozan. We develop a polygonal model for the space of representations in terms of chains of triangles in the hyperbolic plane. This model makes explicit the symplectic toric nature of the space of representations described by Deroin-Tholozan. We introduce action-angle coordinates as geometric quantities associated to the chains of triangles. The coordinates give rise to 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.Comment: 32 pages, 11 figure