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Symplectic geometry of projective structures on surfaces with boundary

Ahmadreza Khazaeipoul, Eckhard Meinrenken

TL;DR

The paper extends the symplectic framework of Adler–Gelfand–Dikii spaces to projective structures on surfaces with nondegenerate boundary by constructing an infinite-dimensional symplectic form on the deformation space $\mathfrak{P}(\Sigma)$ via Atiyah–Bott reduction and a two-step boundary-aware quotient. It identifies the boundary restriction map $\Psi: \mathfrak{P}(\Sigma)\to \mathcal{R}_3(\partial\Sigma)$ as the moment map for a Hamiltonian action of the boundary symplectic groupoid $\mathcal{S}_3(\partial\Sigma)$ on convex/projective strata, particularly for $c\mathfrak{P}(\Sigma)$. The work details a DS-reduction picture for $\mathcal{R}_n(\mathsf{C})$ with $n=3$, analyzes positive hyperbolic boundary monodromies, and develops a comprehensive groupoid/gauge-theoretic approach to Goldman twists and their Hamiltonian nature. These results connect higher Teichmüller theory with boundary dynamics and offer a structured, boundary-incorporating symplectic/Poisson framework relevant to JT gravity and hyperbolic structures with wiggles on the boundary.

Abstract

For oriented surfaces $Σ$ with boundary, we consider the infinite-dimensional deformation space of projective structures on $Σ$ with nondegenerate boundary, up to isotopies fixing the boundary. We show that this space carries a natural symplectic structure, and is a Hamiltonian space for the symplectic groupoid integrating the Adler-Gelfand-Dikii-space of the boundary.

Symplectic geometry of projective structures on surfaces with boundary

TL;DR

The paper extends the symplectic framework of Adler–Gelfand–Dikii spaces to projective structures on surfaces with nondegenerate boundary by constructing an infinite-dimensional symplectic form on the deformation space via Atiyah–Bott reduction and a two-step boundary-aware quotient. It identifies the boundary restriction map as the moment map for a Hamiltonian action of the boundary symplectic groupoid on convex/projective strata, particularly for . The work details a DS-reduction picture for with , analyzes positive hyperbolic boundary monodromies, and develops a comprehensive groupoid/gauge-theoretic approach to Goldman twists and their Hamiltonian nature. These results connect higher Teichmüller theory with boundary dynamics and offer a structured, boundary-incorporating symplectic/Poisson framework relevant to JT gravity and hyperbolic structures with wiggles on the boundary.

Abstract

For oriented surfaces with boundary, we consider the infinite-dimensional deformation space of projective structures on with nondegenerate boundary, up to isotopies fixing the boundary. We show that this space carries a natural symplectic structure, and is a Hamiltonian space for the symplectic groupoid integrating the Adler-Gelfand-Dikii-space of the boundary.
Paper Structure (22 sections, 14 theorems, 121 equations)

This paper contains 22 sections, 14 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Nondegenerate curves and the Adler-Gelfand-Dikii space
  3. Quasi-periodic curves
  4. Nondegenerate curves
  5. The Adler-Gelfand-Dikii space $\mathcal{R}_n(\mathsf{C})$
  6. Drinfeld-Sokolov
  7. Positive hyperbolic elements
  8. Projective structures on surfaces
  9. $(G,X)$-structures
  10. Projective structures
  11. Goldman twists
  12. Projective structures with nondegenerate boundary
  13. Examples
  14. Groupoid action
  15. Symplectic structure
  16. ...and 7 more sections

Key Result

Lemma 2.1

Given a quasi-periodic path $\gamma\colon \widetilde{\mathsf{C}}\to X$, there is an open neighborhood $U\subseteq G$ of $\mu(\gamma)$ and a smooth map $f\colon U\times \widetilde{\mathsf{C}}\to X$ such that each $f(h,\cdot)$ is quasi-periodic of monodromy $h$, and $f(\mu(\gamma),\cdot)=\gamma$.

Theorems & Definitions (52)

  • Lemma 2.1: Local sections of monodromy map
  • proof
  • Example 2.2
  • Example 2.5
  • Remark 2.6
  • Definition 2.7
  • Proposition 2.8: Drinfeld-Sokolov
  • proof : Proof of Proposition \ref{['prop:ds']}
  • Remark 2.9
  • Remark 2.10
  • ...and 42 more