A De Rham Perspective on the Symplectic Geometry of Teichmüller Space
Abstract
We present a new proof of Wolpert's Magic Formula, stating that any Fenchel-Nielsen coordinates on the Teichmüller 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 Teichmüller 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 Teichmüller 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 Teichmüller space.
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BibTeX
@misc{ablondi2026rham,
title = {{A De Rham Perspective on the Symplectic Geometry of Teichmüller Space}},
author = {Antoine Ablondi},
howpublished = {HAL},
year = {2026},
month = oct,
eprint = {2607.24362},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05776703v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A De Rham Perspective on the Symplectic Geometry of Teichmüller Space
AU - Antoine Ablondi
PY - 2026
DA - 2026/10/03
PB - HAL
UR - https://hal.science/hal-05776703v1
AB - We present a new proof of Wolpert's Magic Formula, stating that any Fenchel-Nielsen coordinates on the Teichmüller 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 Teichmüller 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 Teichmüller 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 Teichmüller space.
ER -