Higher Topologies in 2+1-Gravity

TL;DR

This paper extends the first-order formalism of 2+1-dimensional gravity with matter to include higher topologies, using complex analysis and hyperelliptic Riemann surfaces to describe the evolving spacetime structures.

Contribution

It demonstrates that the formalism can incorporate complex topologies beyond the torus, utilizing analytic mappings and hyperelliptic representations.

Findings

Mapping involves four square-root branch points in the torus case

Modulus of the surface has a well-defined time dependence

Hyperelliptic representation describes general topologies

Abstract

I argue that the first-order formalism recently found to describe classical 2+1-Gravity with matter, is also able to include higher topologies. The present gauge, which is conformal with vanishing York time, is characterized by an analytic mapping from single-valued coordinates to Minkowskian ones. In the torus case, this mapping is based on four square-root branch points, whose location is related to the modulus, which has a well defined time dependence. In the general case, it is connected with the hyperelliptic representation of Riemann surfaces.