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 $\lambda$ 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 $A_1^{(0)}-\frac12 S[\psi,t]+\frac 12 (\psi')^2$. Here $A_1^{(0)}\sim \int \log \psi' \frac {dx}x$ is the disc amplitude, $S[\psi,t]$ is the Schwarzian, and the derivative $\psi'$ 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.