bibkey: kawazumi2025wolpert authors: Nariya Kawazumi year: 2025 title: A topological proof of Wolpert’s formula for the Weil-Petersson symplectic form in terms of the Fenchel-Nielsen coordinates doi: 10.1007/s10711-025-01016-3 url: https://arxiv.org/abs/2408.04937v2 claim: A topological proof expresses the Weil-Petersson symplectic form in length and twist coordinates, with an explicit sign convention. strata_touched: [] license: citation-only triage: anchor

Wolpert formula and coordinate conventions

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Geometriae Dedicata 219, article 56 (2025), published 26 May 2025. Primary publisher text: https://link.springer.com/article/10.1007/s10711-025-01016-3 . Introduction, equation (1), uses the sum of d(twist) wedge d(length). Section 6 discusses the normalization relative to the trace Atiyah-Bott-Goldman form.

Use and boundary

The formula itself is due to Wolpert; Kawazumi supplies a new proof. The RT geometric-dynamics convention uses the negative of this article’s twist coordinate to write d(length) wedge d(twist). This sign change must be carried into Hamilton equations. The closed-surface theorem is not by itself a proof about variable boundary lengths; fixed geodesic boundary leaves are separately supported by Do’s work. No quantum-to-gravity dictionary follows from this formula.