Ghost polygons, Poisson bracket and convexity

TL;DR

This paper introduces ghost polygons and ghost brackets to compute Poisson brackets of length and correlation functions in moduli spaces of Anosov representations, revealing their stability and convexity properties.

Contribution

It develops a novel combinatorial framework using ghost polygons and ghost algebra to analyze Poisson brackets in character varieties of surface groups.

Findings

Poisson brackets of length and correlation functions can be computed using ghost polygons.

The set of length and correlation functions is stable under the Poisson bracket.

Length functions are shown to be convex under positivity conditions.

Abstract

The moduli space of Anosov representations of a surface group in a semisimple group, which is an open set in the character variety, admits many more natural functions than the regular functions. We will study in particular length functions and, correlation functions. Our main result is a formula that computes the Poisson bracket of those functions using some combinatorial devices called {\em ghost polygons} and {\em ghost bracket} encoded in a formal algebra called {\em ghost algebra} related in some cases to the swapping algebra introduced by the second author. As a consequence of our main theorem, we show that the set of those functions -- length and correlation -- is stable under the Poisson bracket. We give two applications: firstly in the presence of positivity we prove the convexity of length functions, generalising a result of Kerckhoff in Teichm\"uller space, secondly we exhibit subalgebras of commuting functions. An important tool is the study of {\em uniformly hyperbolic bundles} which is a generalisation of Anosov representations beyond periodicity.