Fool's crowns, trumpets, and Schwarzian

Abstract

For a Riemann surface with holes, we propose a variant of the action on a circum\-ference-$P$ boundary component with $n$ bordered cusps attached (a "fool's crown") that is decoration-invariant and generates finite volumes $V^{\text{crown}}_{n,P}$ of the corresponding moduli spaces when integrated against the volume form obtained by inverting the Fenchel--Nielsen (Goldman) Poisson brackets for a special set of decoration-invariant combinations of Penner's $λ$ lengths. In the limit as $n\to\infty$, the integrals transform into a functional integral with the measure given by the integral over $C^1$ of the action $A_1^{(0)}-\frac12 S[ψ,t]+\frac 12 (ψ')^2$. Here $A_1^{(0)}\sim \int \log ψ' \frac {dx}x$ is the disc amplitude, $S[ψ,t]$ is the Schwarzian, and the derivative $ψ'$ is related to the limiting density of orthogonal projections of bordered cusps to the hole perimeter. We derive the Fenchel--Nielsen symplectic form in the continuum limit and show that it coincides with the one obtained by Alekseev and Meinrenken. We also discuss the volumes of moduli spaces for a disc with $n$ bordered cusps.

Figures (4)

• Figure 1: The "elementary" Poisson bracket (the Goldman bracket) $\{\lambda_{1,3},\lambda_{2,4}\}$ (\ref{['Gold-inner']}) between two $\lambda$-lengths of the corresponding arcs intersecting at a point inside a Riemann surface. We also have the classical skein relation (the Penner's "Ptolemy" relation, not depicted) $\lambda_{1,3}\,\lambda_{2,4}=\lambda_{1,4}\,\lambda_{2,3}+\lambda_{1,2}\,\lambda_{3,4}$.
• Figure 2: The "elementary" Poisson bracket (the Goldman bracket) $\{\lambda_1,\lambda_2\}$ (\ref{['Goldman2']}) between lambda lengths of the corresponding arcs ${\mathfrak a}_1$ and ${\mathfrak a}_2$ coming to the same bordered cusp of a Riemann surface; we also indicate that the loop contractible to a bordered cusp is equal zero thus killing the whole lamination that contains such a loop; an empty contractible loop gives the factor $-2$.
• Figure 3: A "crown" of $m=7$ decorated bordered cusps attached to a pair of pants.
• Figure 4: In the left picture: the fat graph corresponding to a boundary component with $n$ boundary cusps: $\alpha$-variables are extended shear coordinates (defined in the right picture as a geodesic distances between perpendiculars to the bounding arcs and the horocycles lying to the right of the perpendicular if looking from inside the surface); $y_i$ are standard shear coordinates. In the right picture we show the pattern near the boundary cusps: vertical lines are infinite geodesics that all wind in the same direction to the hole perimeter; all these lines are approaching the same point on the absolute that we set to be $\infty$. Standard shear coordinates $y_i$ are geodesic distances between perpendiculars to the vertical lines separating the neighbor ideal triangles with vertices $\{\Delta_{i-1}, \Delta_i,\infty\}$ and $\{\Delta_{i+1}, \Delta_i,\infty\}$. The distances from the perpendiculars to the bounding arcs to the $i$th horocycle from the right and from the left are $\alpha_i$ and $\alpha_i+y_i$ respectively (intervals of the same length are painted the same color in the figure); the total length of the horocycle arc confined between $\mathfrak a_{i-1,i}$ and $\mathfrak a_{i,i+1}$ is therefore $e^{-\alpha_i}+e^{-\alpha_i+y_i}$.

Theorems & Definitions (14)

• Theorem 1.1
• Lemma 2.1
• Lemma 2.2
• Corollary 2.3
• Theorem 2.4
• Remark 2.5
• Lemma 2.6
• Lemma 2.7
• Remark 2.8
• Remark 3.1