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A topological proof of Wolpert's formula for the Weil-Petersson symplectic form in terms of the Fenchel-Nielsen coordinates

Nariya Kawazumi

TL;DR

The paper provides a self-contained topological proof of Wolpert's formula for the Weil–Petersson symplectic form in Fenchel–Nielsen coordinates by constructing a natural cell decomposition adapted to a pants decomposition and a canonical standard cocycle representing points of Teichmüller space. It derives explicit expressions for holonomies via the pants geometry, computes the first variation of the standard cocycle, and shows that the WP form localizes to the seam data through the identity $\omega_{\mathrm{WP}}=\sum_i d\tau_i\wedge dl_i$, with a careful cellular/diagonal construction and Goldman’s ABG framework. The work extends to spin structures by lifting the cocycle to $SL_2(\mathbb{R})$, introducing spin Teichmüller coordinates, and establishing the corresponding spin twist parameters, while the appendix clarifies the relationship with the Shimura isomorphism and the ABG form, yielding the precise constant $2$ between $\omega_{\mathrm{WP}}$ and $\omega_{\mathrm{ABG}}$. Together, these results provide a transparent, topological path to Wolpert's formula and illuminate the role of spin structures and the Shimura correspondence in the symplectic geometry of Teichmüller space.

Abstract

We introduce a natural cell decomposition of a closed oriented surface associated with a pants decomposition, and an explicit groupoid cocycle on the cell decomposition which represents each point of the Teichmüller space $\mathcal{T}_g$. We call it the {\it standard cocycle} of the point of $\mathcal{T}_g$. As an application of the explicit description of the standard cocycle, we obtain a topological proof of Wolpert's formula for the Weil-Petersson symplectic form in terms of the Fenchel-Nielsen coordinates associated with the pants decomposition.

A topological proof of Wolpert's formula for the Weil-Petersson symplectic form in terms of the Fenchel-Nielsen coordinates

TL;DR

The paper provides a self-contained topological proof of Wolpert's formula for the Weil–Petersson symplectic form in Fenchel–Nielsen coordinates by constructing a natural cell decomposition adapted to a pants decomposition and a canonical standard cocycle representing points of Teichmüller space. It derives explicit expressions for holonomies via the pants geometry, computes the first variation of the standard cocycle, and shows that the WP form localizes to the seam data through the identity , with a careful cellular/diagonal construction and Goldman’s ABG framework. The work extends to spin structures by lifting the cocycle to , introducing spin Teichmüller coordinates, and establishing the corresponding spin twist parameters, while the appendix clarifies the relationship with the Shimura isomorphism and the ABG form, yielding the precise constant between and . Together, these results provide a transparent, topological path to Wolpert's formula and illuminate the role of spin structures and the Shimura correspondence in the symplectic geometry of Teichmüller space.

Abstract

We introduce a natural cell decomposition of a closed oriented surface associated with a pants decomposition, and an explicit groupoid cocycle on the cell decomposition which represents each point of the Teichmüller space . We call it the {\it standard cocycle} of the point of . As an application of the explicit description of the standard cocycle, we obtain a topological proof of Wolpert's formula for the Weil-Petersson symplectic form in terms of the Fenchel-Nielsen coordinates associated with the pants decomposition.
Paper Structure (11 sections, 23 theorems, 120 equations, 8 figures)

This paper contains 11 sections, 23 theorems, 120 equations, 8 figures.

Table of Contents

  1. The standard cocycle on a pair of pants
  2. The Fenchel-Nielsen coordinates
  3. The first variation of the standard cocycle.
  4. The Weil-Petersson symplectic form
  5. Lifts to $SL_2(\mathbb{R})$
  6. Spin structures on an oriented surface
  7. Spin structures on a hyperbolic surface
  8. Standard cocycles for a spin hyperbolic structure
  9. Appendix: The Shimura isomorphism and symplectic forms on the Teichmüller space
  10. The Weil-Petersson metric and the Shimura isomorphism
  11. Quasi-conformal mappings

Key Result

Lemma 1.1

For ${\lambda} > 1$ and $A, B \in PSL_2(\mathbb{R})$, we have the following. (1) If $A{\overline{D}}({\lambda}) = {\overline{D}}({\lambda})A$, then we have $A = {\overline{D}}(a)$ for some $a > 0$. (2) If $AB = BA = {\overline{D}}({\lambda})$, then we have $A = {\overline{D}}(a)$ and $B = {\overline

Figures (8)

  • Figure 1: a pants decomposition and one of its associated cell decompositions
  • Figure 2: Cell decomposition of $P = \Sigma_{0,3}$
  • Figure 3: Cell decomposition of the annulus neighborhood $\mathfrak{a}_i$ of $\delta_i$
  • Figure 4: The oriented distance $\tau_i$.
  • Figure 5: Celluar approximation of the diagonal map on the square $E_S$
  • ...and 3 more figures

Theorems & Definitions (42)

  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • proof
  • Lemma 1.4
  • proof
  • Lemma 1.5: Keen1, Lemma 1, p.210
  • proof
  • ...and 32 more