Fool's crowns, trumpets, and Schwarzian

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 $V_{n,P}^{\text{crown}}$ 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 $n\to \mathrm{\infty }$, the integrals transform into a functional integral with the measure given by the integral over $C^1$ of the action $A_1^{(0)}-\frac{1}{2}S[\psi ,t]+\frac{1}{2}({\psi }^{\mathrm{\prime }}{)}^2$. Here $A_1^{(0)}\sim \int \log {\psi }^{\mathrm{\prime }}\frac{dx}{x}$ is the disc amplitude, $S[\psi ,t]$ is the Schwarzian, and the derivative ${\psi }^{\mathrm{\prime }}$ 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.