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

This paper constructs new sets of log-canonical coordinates on the $SL(2, \mathbb{C})$ character variety of compact Riemann surfaces by combining complexified shear-type and length/twist-type coordinates, which are labeled by families of non-intersecting simple loops and generalize complexified Fenchel-Nielsen coordinates when the loops define a trinion decomposition.

The Gist

Imagine you have a complex, multi-holed donut (a Riemann surface). Now, imagine you want to describe every possible way you can stretch, twist, and shape this donut without tearing it. In mathematics, this collection of all possible shapes is called a Character Variety.

The problem is that these shapes live in a high-dimensional, abstract world. To study them, mathematicians need a "map" or a set of coordinates, much like latitude and longitude on Earth. However, the standard maps for these donuts are messy and hard to work with.

This paper, by Bertola, Korotkin, and Pillet, introduces a new, cleaner set of coordinates for these shapes. They call them "log-canonical coordinates."

Here is a simple breakdown of how they did it, using everyday analogies:

1. The Problem: The Messy Map

Think of the surface of your donut as a piece of fabric. The standard way to describe its shape (Fenchel-Nielsen coordinates) is like describing a sweater by listing the length of every single stitch and how much you twisted the yarn. It works, but it's complicated, especially when you move from real-world shapes to "complex" shapes (which involve imaginary numbers and are harder to visualize).

The authors wanted a map where the rules are simple and constant, no matter how you twist the fabric. They wanted a system where the "distance" between two shapes is easy to calculate.

2. The Solution: Cutting the Donut into Pants

The authors' strategy is to cut the donut into smaller, manageable pieces.

• The Trinion (Pants): Imagine cutting the donut along specific loops until you are left with nothing but "pants" (three-holed spheres). In math, these are called trinions.
• The Glue: Once you have these pants, you can describe the whole donut by describing the pants and the "glue" used to stitch them back together.

3. The Two Types of "Glue"

When you stitch two pieces of fabric together, you need two things to define the connection:

1. The Length (The Size of the Hole): How wide is the opening where you are sewing?
2. The Twist (The Rotation): If you rotate one piece of fabric before sewing it to the other, the pattern changes.

The authors realized that for these complex donuts, they could define a coordinate system based on:

• Complex Lengths: The size of the cut loops.
• Complex Twists: How much you rotate the pieces.
• Shear Coordinates: A way of measuring how the fabric "slides" or skews as you stitch it.

4. The "Shear" Analogy

Imagine a deck of cards. If you push the top of the deck to the right while holding the bottom still, the cards slide past each other. This is a shear.

• In the old maps, calculating this sliding was a nightmare of complex equations.
• In the authors' new map, they treat this sliding as a fundamental building block. They assign a specific number (a coordinate) to every edge of the "pants" that represents this slide.

5. The Magic Formula: "Log-Canonical"

Why is this special?
In standard coordinates, the relationship between the "length" and the "twist" might be messy and change depending on where you are.
In the authors' log-canonical system, the relationship is perfectly simple and constant.

• The Analogy: Imagine a music mixer. In a normal mixer, turning the volume knob might also accidentally change the bass or the treble in a confusing way. In the authors' mixer, turning the "Volume" knob only changes the volume, and turning the "Bass" knob only changes the bass. They are independent and predictable.
• Mathematically, this means the "symplectic form" (the rulebook for how the shape changes) becomes a simple sum of basic pairs, like $d(\text{twist}) \times d(\text{length})$.

6. Why Does This Matter?

• Simplicity: It turns a chaotic, high-dimensional puzzle into a neat, organized grid.
• Universality: This method works for any number of holes in the donut, from a simple torus (one hole) to a complex surface with many holes.
• Future Applications: Just as GPS coordinates revolutionized travel, these new coordinates could revolutionize how physicists and mathematicians understand quantum mechanics, string theory, and the geometry of the universe. It provides a "standard language" to talk about these complex shapes.

Summary

The authors took a complicated, abstract mathematical object (the shape of a multi-holed donut in a complex world), cut it into simple "pants," and invented a new way to measure the size and twist of the seams. This new measurement system is so clean and logical that it makes the complex math behave as simply as a standard ruler and protractor.

Technical Summary

1. Problem Statement

The paper addresses the construction of log-canonical coordinates (also known as Darboux coordinates) for the $SL(2, \mathbb{C})$ character variety ($V_g$) of a compact Riemann surface of genus $g \geq 2$.

• Context: The character variety $V_g$ carries a natural complex symplectic structure (the Goldman symplectic form), which is the inverse of the Goldman Poisson bracket.
• Existing Solutions:
  • For surfaces with boundary ($V_{g,n}$), complexified Thurston shear coordinates are known to be log-canonical on symplectic leaves defined by boundary lengths.
  • For closed surfaces ($V_g = V_{g,0}$), there is no general definition of complexified shear coordinates. While Fenchel-Nielsen coordinates exist on the Teichmüller space, their analytic continuation to the complex domain is not straightforwardly log-canonical in the standard sense.
• Goal: To construct a general system of log-canonical coordinates for $V_g$ ($g \geq 2$) by combining complexified shear-type coordinates with length/twist-type coordinates, applicable to any system of $m$ non-intersecting simple loops ($1 \leq m \leq 3g-3$).

2. Methodology

The authors employ a geometric and combinatorial approach based on admissible graphs and graph amalgamation, extending techniques from Fock-Goncharov theory and previous work on extended character varieties.

A. Graph Construction and Amalgamation

The core strategy involves constructing a specific graph $\Gamma$ embedded on the Riemann surface $C$ and assigning "jump matrices" to its edges.

1. Cutting the Surface: The surface $C$ is cut along a system of $m$ non-intersecting simple loops $\{\gamma_1, \dots, \gamma_m\}$. This decomposes $C$ into $n$ subsurfaces $C^{(i)}$ with boundaries.
2. Triangulation: Each subsurface $C^{(i)}$ is capped (boundaries filled with disks) and triangulated ($\Sigma^{(i)}$). Vertices of the triangulation are placed near the boundary components.
3. Graph Amalgamation: The global graph $\Gamma$ is formed by amalgamating:
   • Local triangulation graphs on each subsurface.
   • A "gluing" graph (plumbing graph) connecting the vertices across the cut contours.
4. Jump Matrices: Matrices are assigned to edges:
   • Shear edges: Associated with the triangulation edges, using matrices of the form $S_e = \begin{pmatrix} 0 & z_e \\ -1 & z_e \end{pmatrix}$.
   • Gluing edges: Associated with the cut contours, involving diagonalizing matrices and toric variables.

B. Log-Canonical Coordinates Definition

The coordinates are defined as follows:

• Shear Coordinates ($\zeta_e$): Logarithms of the edge parameters $z_e$ of the triangulation ($\zeta_e = \ln z_e$).
• Complex Lengths ($\ell_{\gamma_j}$): Defined as the logarithm of the eigenvalue of the monodromy matrix around the loop $\gamma_j$. These are constrained to be linear sums of the shear coordinates on the adjacent triangulations.
• Toric Variables ($\beta_{\gamma_j}$): Logarithms of the "toric variables" (ratios of eigenvector normalizations) associated with the gluing of the surfaces along $\gamma_j$. These act as the conjugate momenta to the lengths.

C. Symplectic Form Computation

The authors compute the Goldman symplectic form $\Omega$ using the general formula for admissible graphs (derived from Alekseev-Malkin theory). They demonstrate that under the linear constraints imposed by the gluing conditions, the form decomposes into a sum of local contributions from the triangulations and global contributions from the gluing contours.

3. Key Contributions and Results

A. General Construction for Arbitrary $m$

The paper provides a unified construction for any number of cutting contours $m$ ($1 \leq m \leq 3g-3$).

• Theorem 6.1: The Goldman symplectic form $\Omega$ on the subspace $V_g^{(\gamma)}$ (where monodromies around cuts are diagonalizable) is given by:
  $$\Omega = \sum_{i=1}^n \Omega^{(i)} + \sum_{j=1}^m d\beta_{\gamma_j} \wedge d\ell_{\gamma_j}$$
  where $\Omega^{(i)}$ is the symplectic form on the triangulated subsurface $C^{(i)}$ expressed purely in terms of shear coordinates $\zeta_e$, and the second term represents the canonical symplectic pairs $(\beta, \ell)$ for each cut.

B. The Trinion Decomposition Case ($m = 3g-3$)

In the maximal case where the loops decompose the surface into $2g-2$ "trinions" (three-holed spheres):

• The edge coordinates of the triangulation can be eliminated in favor of the boundary lengths of the trinions.
• Theorem 7.2: The symplectic form is expressed entirely in terms of the complex lengths of the trinion boundaries ($\ell^{(j)}_a$) and the toric variables ($\beta_e$) associated with the edges of the trinion graph (a ribbon graph where vertices are trinions and edges are the gluing loops).
  $$\Omega = \sum_{T^{(j)}} \left( d\ell^{(j)}_1 \wedge d\ell^{(j)}_2 + d\ell^{(j)}_2 \wedge d\ell^{(j)}_3 + d\ell^{(j)}_3 \wedge d\ell^{(j)}_1 \right) + \sum_{e \in E(\mathcal{T})} d\beta_e \wedge d\ell_e$$
  This form is closely related to, but distinct from, the complexified Fenchel-Nielsen coordinates.

C. Explicit Examples ($g=2$)

The authors provide explicit calculations for genus 2 surfaces ($V_2$) with:

1. One separating contour.
2. One separating and one non-separating contour.
3. Three separating contours (complete trinion decomposition).
   These examples verify the general formulas and demonstrate how the constraints reduce the number of independent variables to the dimension of the variety ($6g-6$).

4. Significance and Future Directions

• Unification: The work bridges the gap between shear coordinates (typically used for surfaces with boundary) and the character varieties of closed surfaces, providing a consistent log-canonical framework.
• Relation to Fenchel-Nielsen: In the maximal decomposition case, the coordinates offer a complexified analogue of Fenchel-Nielsen coordinates but with a simpler, constant-coefficient symplectic structure (log-canonical).
• Applications:
  • Moduli Spaces: The coordinates are expected to facilitate the study of the moduli space of Riemann surfaces ($M_g$) by restricting to the Fuchsian component (real slice).
  • Higher Rank: The methodology is designed to generalize to $SL(N, \mathbb{C})$ character varieties for $N \geq 3$, utilizing higher-rank Fock-Goncharov coordinates. This is crucial for the study of higher Teichmüller spaces.
  • Quantization: Log-canonical coordinates are the natural starting point for the quantization of character varieties and the construction of quantum invariants.

Conclusion

Bertola, Korotkin, and Pillet successfully construct a new, general system of log-canonical coordinates for $SL(2, \mathbb{C})$ character varieties of closed Riemann surfaces. By combining triangulation-based shear coordinates with twist-length parameters derived from the gluing of subsurfaces, they provide a powerful tool for analyzing the symplectic geometry of these varieties, with clear pathways for extension to higher-rank groups and applications in mathematical physics.