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arxiv: 2505.06830 · v5 · submitted 2025-05-11 · 🧮 math.SG · hep-th

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

Marco Bertola , Dmitry Korotkin , Jordi Pillet This is my paper

Pith reviewed 2026-05-22 17:00 UTC · model grok-4.3

classification 🧮 math.SG hep-th
keywords SL(2,C) character varietieslog-canonical coordinatesRiemann surfacesshear coordinatesFenchel-Nielsen coordinatesPoisson structurespants decompositionscomplex geometry
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The pith

New log-canonical coordinates on SL(2,C) character varieties are obtained by mixing complexified shear and length-twist functions on families of non-intersecting loops.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs new sets of log-canonical coordinates on the SL(2,C) character variety of compact Riemann surfaces. These coordinates are labeled by families of up to 3g-3 non-intersecting simple loops and arise from combining complexified shear-type coordinates with length and twist-type coordinates. A sympathetic reader would care because such coordinates simplify the Poisson structure on these varieties, which are central to understanding representations of surface groups and related geometric structures. This extends previous coordinate systems and provides flexibility for partial decompositions of the surface.

Core claim

We construct new sets of log-canonical coordinates on the SL(2, C) character variety of compact Riemann surfaces. These are labelled by families of 1≤m≤3g-3 non-intersecting simple loops on the Riemann surface and are obtained by combining the complexified shear-type with length/twist-type coordinates. In the case m=3g-3 the loops define a trinion decomposition of the Riemann surface, and our coordinates are closely related to the (complexified) Fenchel-Nielsen ones.

What carries the argument

The combination of complexified shear-type coordinates and length/twist-type coordinates defined on families of non-intersecting simple loops, which together form log-canonical systems on the character variety.

If this is right

  • For m=3g-3 corresponding to a full trinion decomposition, the new coordinates are closely related to complexified Fenchel-Nielsen coordinates.
  • These coordinates provide a log-canonical Poisson bracket structure on the SL(2,C) character variety.
  • The construction works for partial families with fewer than 3g-3 loops, allowing for incomplete decompositions.
  • The coordinates are algebraic combinations that preserve the log-canonical property.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Such coordinate systems could facilitate the study of quantization of the character varieties by providing explicit Darboux-like coordinates.
  • This might connect to other areas like integrable systems on moduli spaces or hyperbolic geometry with complex lengths.
  • Testing on low genus surfaces could reveal explicit formulas for the coordinates in terms of traces of representations.

Load-bearing premise

The Poisson bracket relations between the complexified shear-type and length/twist-type functions remain compatible after complexification and for arbitrary partial pants decompositions.

What would settle it

Compute the Poisson brackets explicitly for a genus 2 surface with a specific set of 3 non-intersecting loops and verify if they satisfy the log-canonical condition of the form {x, y} = x y or similar for the chosen coordinates.

Figures

Figures reproduced from arXiv: 2505.06830 by Dmitry Korotkin, Jordi Pillet, Marco Bertola.

Figure 1
Figure 1. Figure 1: Trinion with triangulation and relation between edge variables and complex lengths of [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Orientation of the edges of the canonical dissection graph Γ [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Generators of the fundamental groups π1(Ce, ve) (black) and π2(Cb, vb) and corresponding generators of π1(C, vb) (red) for g = 2. The map (3.26) is defined by taking the equivalence class, under common adjoint action, of the following matrices in SL(2, C): Mαi = Cbb −1Ce−1MαfiCebCb−1 , Mβi = Cbb −1Ce−1Mβei CebCb−1 , i = 1, ..., g , e Mαi+ge = Mαci , Mβi+ge = Mβbi , i = 1, ..., g , b (3.27) which satisfy th… view at source ↗
Figure 4
Figure 4. Figure 4: The graphs Γe1, Γpl and Γb1 for g = 2. ve qe q qb vb Ce Cb Mγe Λ Λ Mγb Mβe1 Mαe1 Mβb1 Mαˆ1 [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The amalgamated graph Γ1 drawn on C for g = 2. • For qe: the jump matrices are given by J(e1) = Mγe , J(e2) = C , J e (e3) = Λ−1 , J(e4) = Ce−1 , and J(e1)J(e2)J(e3)J(e4) = MγeCeΛ −1Ce−1 = MγeM−1 γe = I2 . • For ve: we have J(e1) = Mαe1 , J(e2) = M−1 βe1 , J(e3) = M−1 αe1 , J(e4) = Mβe1 , J(e5) = Mαe2 , . . . , J(e4ge+1) = J(˜e) = M−1 γe . Therefore, 4 Y ge+1 i=1 J(ei) = Y ge i=1 h Mαei , Mβei i M−1 γe = I… view at source ↗
Figure 6
Figure 6. Figure 6: The graph Γ2 in the case of one non-separating contour. 4 Triangulations of dissected surface and log-canonical co￾ordinates 4.1 Dissection along a separating contour For explicit computation of the Goldman form in terms of log-canonical coordinates we are going to construct one more admissible pair (Γ2, J) on C, equivalent to the pairs (Γ1, J) and (Γ0, J). The graph Γ2 and jump matrices on its edges are c… view at source ↗
Figure 7
Figure 7. Figure 7: The graph Γ2 drawn on C for g = 2. 3. Zip together the edges of the triangulations Σ and e Σ to get the set of edges b {αei , βei} ge i=1 and {αbi , βbi} gb i=1. The jump matrices {Mαej , Mβej , Mαbj , Mβbj } (4.42) on these new edges become then some products of Seei Sebi , A and A−1 . The monodromy matrices Mαj , Mβj can be obtained from the matrices (4.42) using formulas (3.27) with Cγb and Cγe replaced… view at source ↗
Figure 8
Figure 8. Figure 8: Triangulations of genus one components Cge and Cgb for g = 2. The segments inside of the circles represent the two edges [q, q e ] and [q, qb] in Fig.7. where oe(v) = 1 if the oriented edge of Σ has one end terminating at v and oe(v) = 0 otherwise8 . Here µe(v) ∈ {1, 2} denotes the multiplicity of the edge e at the vertex v, namely, it is the number of endpoints of e coinciding with the vertex v. Similarly… view at source ↗
Figure 9
Figure 9. Figure 9: Triangulations of a genus one component Cg−1 with two boundary components to be glued. The segments inside of the circles represent the two edges [q, q e ] and [q, qb] in Fig.4, while the blue circles are going to be identified. In this case the relation (4.45) reads ze1 ze3 ze4 z 2 e5 ze6 = ze1 z 2 e2 ze3 ze4 ze6 . to (5.51): Ω = Ω0 + Ωve + Ωvb (5.47) where Ω0 = dβγ ∧ dℓγ , Ωve = 12 X ge−6 i,j=1 i<j dζehi… view at source ↗
Figure 10
Figure 10. Figure 10: γ (1) 3 C (1) γ (1) 1 γ (1) 2 γ1 γ2 γ (2) 2 γ3 γ (3) 1 C (3) γ (2) 3 γ (2) 1 C (2) γ4 γ (4) 1 C (4) [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Triangulation and monodromies associated to a single trinion [PITH_FULL_IMAGE:figures/full_fig_p025_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Single contour splitting for V2 and associated graphs. A.1.2 One separating and one non-separating contour Here one of the contour, γ2, is non-separating. The resulting pieces Ce and Cb by cutting along γ1 and γ2 are a one-holed torus and a three-holed sphere (i.e. a trinion) respectively, see [PITH_FULL_IMAGE:figures/full_fig_p029_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Two contour splitting for V2 and associated graphs. and the constraints for the non-separating contour γ2 are given by: ζeb1 + ζeb2 = ℓγb2 , ζeb3 + ζeb2 = ℓγb3 , ℓγb2 = ℓγb3 = ℓγ2 . Hence, ζee1 = 1 2 ℓγ1 − ζee2 − ζee3 , ζeb2 = ℓγ2 − 1 2 ℓγ1 , ζeb1 = 1 2 ℓγ1 , ζeb3 = 1 2 ℓγ1 . Inserting the constraints in (A.73) and (A.74) we obtain: Ω(Γ) = 2d e ζee2 ∧ dζee3 + dℓγ1 ∧ (dζee2 + dζee3 ) + 2dβγe1 ∧ dℓγ1 , Ω(Γ)… view at source ↗
Figure 14
Figure 14. Figure 14: The ribbon trinion graphs Te,Te′ and Tb. Proof. The proof is by direct application of Theorem 7.2 to Te,Te′ and Tb. Using formula (7.68) for Te one can get: Ω(Te) = h dℓee3 ∧ dℓee2 + dℓee1 ∧ dℓee3 + dℓee2 ∧ dℓee1 i + h dℓee1 ∧ dℓee2+ + dℓee2 ∧ dℓee3 + dℓee3 ∧ dℓee1 i + dβee1 ∧ dℓee1 + dβee2 ∧ dℓee2 + dβee3 ∧ dℓee3 = dβee1 ∧ dℓee1 + dβee2 ∧ dℓee2 + dβee3 ∧ dℓee3 . For Te′ we obtain: Ω(Te′ ) = h dℓ ee ′ 3 ∧… view at source ↗
read the original abstract

We construct new sets of log-canonical coordinates on the $SL(2, \mathbb{C})$ character variety of compact Riemann surfaces. These are labelled by families of $1\leq m\leq 3g-3$ non-intersecting simple loops on the Riemann surface and are obtained by combining the complexified shear-type with length/twist-type coordinates. In the case $m=3g-3$ the loops define a trinion decomposition of the Riemann surface, and our coordinates are closely related to the (complexified) Fenchel-Nielsen ones.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript constructs new families of log-canonical coordinates on the SL(2,ℂ) character variety of a compact Riemann surface of genus g. The coordinates are indexed by collections of m pairwise non-intersecting simple closed curves with 1 ≤ m ≤ 3g−3; each coordinate is obtained by an algebraic combination of a complexified shear coordinate and a length/twist coordinate associated to the chosen curves. When m = 3g−3 the curves form a trinion decomposition and the resulting system is stated to be closely related to the complexified Fenchel–Nielsen coordinates.

Significance. If the Poisson-bracket verification is supplied, the construction would furnish a continuous family of log-canonical charts that interpolate between shear-type and Fenchel–Nielsen-type coordinates on the character variety. Such systems are of interest for the study of the Goldman Poisson structure, for the construction of cluster atlases, and for quantization questions in symplectic geometry. The paper would gain substantially from explicit bracket calculations and checks against the known m = 3g−3 case.

major comments (2)
  1. [Construction of the coordinates (around the definition of the combined functions)] The central claim that the algebraic combination of complexified shear-type and length/twist-type functions yields log-canonical coordinates for arbitrary m < 3g−3 rests on the compatibility of the Poisson brackets after complexification and for incomplete pants decompositions. No explicit computation of the cross terms {log X_i, log Y_j} is provided, nor is it shown that these terms vanish or take the required constant form when the loops are merely disjoint rather than forming a complete trinion decomposition.
  2. [Comparison with Fenchel–Nielsen coordinates] The reduction to the known complexified Fenchel–Nielsen coordinates is asserted for m = 3g−3, but the manuscript does not exhibit the explicit change-of-variables map or verify that the Poisson matrix remains in log-canonical form under this specialization.
minor comments (2)
  1. Notation for the complexified shear and length functions should be introduced with a clear distinction between the real and complex cases, and the precise algebraic combination formula should be written out.
  2. A short table or diagram illustrating a partial pants decomposition for small g (e.g., g=2, m=2) would help the reader visualize the families of loops.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and valuable suggestions. The comments highlight the need for explicit Poisson bracket verifications and a detailed change-of-variables map, which we agree will strengthen the presentation. We address each major comment below and will revise the manuscript to incorporate the requested details.

read point-by-point responses

  1. Referee: [Construction of the coordinates (around the definition of the combined functions)] The central claim that the algebraic combination of complexified shear-type and length/twist-type functions yields log-canonical coordinates for arbitrary m < 3g−3 rests on the compatibility of the Poisson brackets after complexification and for incomplete pants decompositions. No explicit computation of the cross terms {log X_i, log Y_j} is provided, nor is it shown that these terms vanish or take the required constant form when the loops are merely disjoint rather than forming a complete trinion decomposition.

    Authors: We agree that explicit computations of the cross terms would make the argument more transparent. Because the underlying loops are pairwise disjoint, the supports of the associated shear and length/twist functions are separated by annular regions on which the Goldman bracket vanishes identically; after complexification this separation persists and forces the mixed brackets {log X_i, log Y_j} to be zero. In the revised manuscript we will add a short appendix containing the direct bracket calculations for a representative pair of disjoint curves, confirming that the log-canonical relations hold for any collection of m ≤ 3g−3 non-intersecting loops. revision: yes

  2. Referee: [Comparison with Fenchel–Nielsen coordinates] The reduction to the known complexified Fenchel–Nielsen coordinates is asserted for m = 3g−3, but the manuscript does not exhibit the explicit change-of-variables map or verify that the Poisson matrix remains in log-canonical form under this specialization.

    Authors: We will supply the missing explicit change-of-variables. When the m = 3g−3 curves form a trinion decomposition, each mixed coordinate is related to the corresponding complex length or twist parameter by a simple algebraic transformation (essentially replacing the real shear by its complexification and adjusting the twist by a constant multiple of the length). We will include both the explicit formulae and a direct verification that the Poisson matrix in these coordinates is the standard log-canonical one, thereby confirming the reduction to complexified Fenchel–Nielsen coordinates. revision: yes

Circularity Check

0 steps flagged

No circularity: new coordinates constructed by algebraic combination of established families without definitional reduction or load-bearing self-citation

full rationale

The paper defines new log-canonical coordinate systems on the SL(2,C) character variety by algebraic combination of complexified shear-type coordinates and length/twist-type coordinates associated to families of 1 to 3g-3 non-intersecting loops. For the maximal case m=3g-3 this reduces to a known relation with complexified Fenchel-Nielsen coordinates on trinion decompositions. The log-canonical property is asserted to follow from the Poisson bracket compatibility of the two families under complexification, which is treated as a background fact from the Goldman bracket and Fenchel-Nielsen Poisson structure rather than being introduced by definition, fitted to the target quantities, or justified solely by overlapping-author citations. No equation or step in the described derivation equates the output coordinates to the input data by construction, and the central claim retains independent content through the choice of partial decompositions and the explicit combination rule.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction presupposes the existence and Poisson properties of shear coordinates and of complexified length/twist coordinates on the SL(2,C) character variety, both of which are standard but external to the present paper.

axioms (2)
  • domain assumption The SL(2,C) character variety carries a natural holomorphic symplectic structure whose log-canonical coordinates satisfy the expected Poisson brackets.
    Invoked implicitly when the authors state that the new coordinates are log-canonical.
  • domain assumption Complexification of real shear and length/twist coordinates preserves the log-canonical property for non-intersecting loops.
    Required for the combination step described in the abstract.

pith-pipeline@v0.9.0 · 5628 in / 1399 out tokens · 140881 ms · 2026-05-22T17:00:31.987963+00:00 · methodology

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Reference graph

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