Fool's crowns, trumpets, and Schwarzian
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915007550193664 |
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| author | Chekhov, Leonid O. |
| author_facet | Chekhov, Leonid O. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_03913 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fool's crowns, trumpets, and Schwarzian Chekhov, Leonid O. Mathematical Physics Algebraic Geometry 14J50, 51M10 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. |
| title | Fool's crowns, trumpets, and Schwarzian |
| topic | Mathematical Physics Algebraic Geometry 14J50, 51M10 |
| url | https://arxiv.org/abs/2411.03913 |