New systems of log-canonical coordinates on $SL(2, \mathbb{C})$ character varieties of compact Riemann surfaces

TL;DR

The paper constructs explicit local log-canonical coordinates on $SL(2, \mathbb{C})$ character varieties by combining complex shear-type coordinates with length–twist data, for arbitrary genus and cutting contours. It establishes a detailed graph- and triangulation-based framework (including plumbing and trinification) that yields a constant-coefficient Goldman symplectic form in these coordinates, and provides explicit generating functions in terms of dilogarithms that relate the new coordinates to Fenchel–Nielsen coordinates on Teichmüller space. On the real Fuchsian component, the construction recovers the Weil–Petersson form and, via the Teichmüller identification, descends to the moduli space $\mathcal{M}_g$, while generalizing naturally to higher rank $SL(N)$. The work ties the log-canonical coordinates to classical Fenchel–Nielsen data through precise Dehn-twist and gluing formulas, including explicit generating functions for key low-genus cases and a comprehensive framework for complete trinions. The methods promise connections to Hitchin-type integrable systems and quantum invariants, and open avenues for extending the approach to higher rank groups and more intricate modular spaces.

Abstract

We construct new sets of log-canonical coordinates on $SL(2, \mathbb{C})$ character varieties of compact Riemann surfaces. These coordinates are obtained by combining shear type coordinates with length-twist type coordinates. On the real component corresponding to the moduli space of compact Riemann surfaces $ \mathcal{M}_g$, the generating function corresponding to the symplectomorphism between these new coordinates and Fenchel-Nielsen coordinates is explicitly computed.

Figures (19)

• Figure 1: Orientation of the edges of the standard graph $\Gamma_0$ for $g=2$.
• Figure 2: Generators of the fundamental groups $\pi_1( \widetilde{{\mathcal{C}}},\widetilde{v})$ (black) and $\pi_2( \widehat{{\mathcal{C}}},\widehat{v})$ and corresponding generators of $\pi_1({\mathcal{C}},\widehat{v})$ (red) for $g=2$.
• Figure 3: The graphs $\widetilde{\Gamma}_1$, $\Gamma_{pl}$ and $\widehat{\Gamma}_1$ for $g=2$.
• Figure 4: The amalgamated graph $\Gamma_1$ drawn on ${\mathcal{C}}$ for $g=2$.
• Figure 5: The graph $\Gamma_2$ drawn on ${\mathcal{C}}$ for $g=2$.

Theorems & Definitions (60)

• Definition 3.1
• Proposition 4.1
• proof
• Proposition 5.1
• proof
• Theorem 7.1
• proof
• Lemma 7.2
• Theorem 8.1
• proof