Geometry and observables in (2+1)-gravity

TL;DR

This paper reviews the geometric structure of vacuum (2+1)-dimensional gravity, emphasizing holonomies and Wilson loops as key variables, and discusses how they can be measured and interpreted physically.

Contribution

It provides a comprehensive overview of the geometric and observable aspects of (2+1)-gravity, clarifying the role of holonomies and Wilson loops in the theory.

Findings

Holonomies characterize (2+1)-gravity spacetimes as quotients of universal covers.

Wilson loops serve as generators of geometric transformations like grafting and earthquake.

Observable measurements can determine fundamental variables from within the spacetime.

Abstract

We review the geometrical properties of vacuum spacetimes in (2+1)-gravity with vanishing cosmological constant. We explain how these spacetimes are characterised as quotients of their universal cover by holonomies. We explain how this description can be used to clarify the geometrical interpretation of the fundamental physical variables of the theory, holonomies and Wilson loops. In particular, we discuss the role of Wilson loop observables as the generators of the two fundamental transformations that change the geometry of (2+1)-spacetimes, grafting and earthquake. We explain how these variables can be determined from realistic measurements by an observer in the spacetime.