Symplectic double for moduli spaces of G-local systems on surfaces

TL;DR

This paper introduces a new symplectic double moduli space for G-local systems on decorated surfaces, providing special rational coordinate systems that serve as K2-Darboux coordinates, enhancing the geometric structure of the space.

Contribution

It constructs the moduli space D(G,S) as a symplectic double with a K2-symplectic structure and defines special rational coordinate systems on it.

Findings

D(G,S) is a symplectic double of the Poisson moduli space.

The space has a K2-symplectic structure.

Special coordinates are K2-Darboux coordinates.

Abstract

Let G be a split semi-simple algebraic group over Q. Let S be a decorated surface, that is a topological oriented surface with a finite set of marked points on the boundary, considered modulo isotopy. We introduce a moduli space D(G,S) and define a collection of special rational coordinate systems on it. The moduli space D(G,S) is the symplectic double of the Poisson moduli space of framed G-local systems on S. Its symplectic form is upgraded to a K2-symplectic structure for which the special coordinates are K2-Darboux coordinates.