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Moduli spaces of open strings have polylogarithmic Mirzakhani volumes

Yi Huang, Ivan Telpukhovskiy

TL;DR

This work proves that the Mirzakhani volumes for moduli spaces of crowned hyperbolic surfaces (open strings) are governed by polylogarithmic structures, expressible as Gaussian-rational combinations of $\mathrm{Li}_s(\pm1)$ and $\mathrm{Li}_s(\pm i)$. It extends Chekhov’s fixed-neck analysis to general crowned surfaces, delivers explicit formulas for volumes of $n$-crowns and $(a_1,a_2)$-annuli, and provides a generating-function framework for volumes of $n$-gons. The volumes without neck-length constraints are shown to be rational polynomials in $\pi^2$, $b_i^2$, along with odd zeta values and Dirichlet beta values, with degree dictated by the moduli space dimension. The paper also identifies a consistent Mehler-Fock-transform-based approach for evaluating polygonal volumes and establishes open directions toward closed-form patterns and apéry-like constants in crown configurations. Overall, the results bridge open-string moduli with Mirzakhani-style volume theory and polylogarithmic constants, enabling precise, computable expressions across a broad class of crowned hyperbolic surfaces.

Abstract

We show that the Mirzakhani volume, as introduced by Chekhov, of the moduli space of every crowned hyperbolic surface is naturally expressible as a sum of Gaussian rational multiples of polylogarithms evaluated at $\pm1$ and $\pm\sqrt{-1}$.

Moduli spaces of open strings have polylogarithmic Mirzakhani volumes

TL;DR

This work proves that the Mirzakhani volumes for moduli spaces of crowned hyperbolic surfaces (open strings) are governed by polylogarithmic structures, expressible as Gaussian-rational combinations of and . It extends Chekhov’s fixed-neck analysis to general crowned surfaces, delivers explicit formulas for volumes of -crowns and -annuli, and provides a generating-function framework for volumes of -gons. The volumes without neck-length constraints are shown to be rational polynomials in , , along with odd zeta values and Dirichlet beta values, with degree dictated by the moduli space dimension. The paper also identifies a consistent Mehler-Fock-transform-based approach for evaluating polygonal volumes and establishes open directions toward closed-form patterns and apéry-like constants in crown configurations. Overall, the results bridge open-string moduli with Mirzakhani-style volume theory and polylogarithmic constants, enabling precise, computable expressions across a broad class of crowned hyperbolic surfaces.

Abstract

We show that the Mirzakhani volume, as introduced by Chekhov, of the moduli space of every crowned hyperbolic surface is naturally expressible as a sum of Gaussian rational multiples of polylogarithms evaluated at and .

Paper Structure

This paper contains 29 sections, 21 theorems, 172 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Mirzakhani volumes of moduli spaces of $n$-gons
  4. Background
  5. Open strings and crowned hyperbolic surfaces
  6. The anatomy of a crowned hyperbolic surface
  7. Teichmüller and moduli spaces of crowned hyperbolic surfaces
  8. Shearing coordinates on Teichmüller space of crowned surfaces
  9. Lambda length coordinates
  10. Weil--Petersson volume form
  11. WP volume form on the moduli space of crowned surfaces
  12. The $n$-gon case
  13. The $n$-crown case
  14. All other cases (apart from $n$-gons)
  15. Mirzakhani volumes of moduli spaces of crowned hyperbolic surfaces with neck length constraints
  16. ...and 14 more sections

Key Result

Theorem 1.3

The Mirzakhani volume $V_{\mathbb{A}_n}(d)$ of $\mathcal{M}_{\mathbb{A}_n}(d)$ is:

Figures (3)

  • Figure 1: An example of a crowned hyperbolic surface with genus $1$, three cuffs and two necks respectively isolating an $1$-crown and a $5$-crown.
  • Figure 2: A fundamental domain (enclosed by thick red lines) in the upper half-plane for a $3$-tined hyperbolic crown. In blue: lifts of geodesic arcs that spiral around the neck curve.
  • Figure 3: Crown with two tines, triangulated by an arc. In blue: unit horocycles.

Theorems & Definitions (65)

  • Theorem 1.3: \ref{['thm:fixedneckvol']}
  • Corollary 1.4: \ref{['cor:fixedneckvolumegeneral']}
  • Theorem 1.5: \ref{['thm:ngonvol']}, \ref{['thm:crownfullvol']}, \ref{['thm:vol-biannuli']}, \ref{['thm:generalmirzvolumes']}
  • Remark 1.6
  • Theorem 1.7: \ref{['thm:ngonvol']}
  • Remark 1.8
  • Remark 1.9: potential connections to Apéry constants
  • Definition 2.2: crowned hyperbolic surface
  • Definition 2.3: arches, cuffs, and tines
  • Definition 2.4: neck
  • ...and 55 more