Homotopy Invariants and Time Evolution in (2+1)-Dimensional Gravity

TL;DR

This paper explores the connection between homotopy invariants and polygon representations in (2+1)-dimensional gravity, providing a Hamiltonian framework and linking to existing results for specific genus cases.

Contribution

It establishes the relationship between homotopy invariants and polygon representations, and explicitly computes the Hamiltonian using a chosen internal time in (2+1)-dimensional gravity.

Findings

Homotopy invariants relate to polygon closure and cycle conditions.

Explicit Hamiltonian computation using internal time.

Connections made with previous results for genus 1 and 2.

Abstract

We establish the relation between the ISO(2,1) homotopy invariants and the polygon representation of (2+1)-dimensional gravity. The polygon closure conditions, together with the SO(2,1) cycle conditions, are equivalent to the ISO(2,1) cycle conditions for the representa- tions of the fundamental group in ISO(2,1). Also, the symplectic structure on the space of invariants is closely related to that of the polygon representation. We choose one of the polygon variables as internal time and compute the Hamiltonian, then perform the Hamilton-Jacobi transformation explicitly. We make contact with other authors' results for g = 1 and g = 2 (N = 0).