Symplectic geometry of projective structures on surfaces with boundary

TL;DR

This paper explores the symplectic geometry of the deformation space of boundary-preserving projective structures on surfaces, revealing its Hamiltonian nature and connection to the Adler-Gelfand-Dikii space.

Contribution

It introduces a natural symplectic structure on the deformation space of projective structures with boundary and shows it forms a Hamiltonian space for a specific symplectic groupoid.

Findings

Deformation space has a natural symplectic structure

The space is a Hamiltonian space for the symplectic groupoid

Connection to the Adler-Gelfand-Dikii space of the boundary

Abstract

For oriented surfaces $\Sigma$ with boundary, we consider the infinite-dimensional deformation space of projective structures on $\Sigma$ 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.