Fool's crowns, trumpets, and Schwarzian

2024/11/06 by Leonid O. Chekhov, Chekhov, Leonid O. · 1 citation
Arts and Humanities · #14J50 #51M10 #Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Musicology and Musical Analysis

paper · pdf · doi:10.48550/arxiv.2411.03913

openalex publication_date 2024/11/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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^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→∞, the integrals transform into a functional integral with the measure given by the integral over C¹ of the action A₁⁽⁰⁾-\frac12 S[ψ,t]+\frac 12 (ψ')². Here A₁⁽⁰⁾∼ ∫ log ψ' \frac dxx 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.

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