A De Rham Perspective on the Symplectic Geometry of Teichm\"uller Space

Abstract

We present a new proof of Wolpert's Magic Formula, stating that any Fenchel--Nielsen coordinates on the Teichm\"uller space of a closed surface are Darboux coordinates for the Weil--Petersson symplectic form. Our approach relies on a de Rham cohomology model for the tangent space to the character variety model of Teichm\"uller space, in which, by the seminal work of Goldman, the Weil--Petersson form is expressed by a natural symplectic form, called the Goldman form. We extend the work of Fillastre and Seppi, who have managed, using that approach and Stokes's Theorem, to provide a new proof of Wolpert's sum of cosines formula. Using the unique isometric symmetry on any hyperbolic pair of pants, we also introduce a way, given a pair of pants decomposition of a closed surface, to construct an associated linear involution on every tangent space of its Teich\"uller space. That allows us to derive a new proof of Wolpert's Magic Formula and deduce a self-contained proof of the closedness of the Goldman form on Teichm\"uller space.