Two-Dimensional Conformal Models of Space-Time and Their Compactification

TL;DR

This paper explores two-dimensional conformal space-time models using Moebius transformations, introducing cycles as invariants, and discusses their compactification with attention to the non-reversibility of time in hyperbolic cases.

Contribution

It presents a geometric framework for conformal space-time models in two dimensions, utilizing cycles to linearize Moebius actions and addressing compactification with time arrow considerations.

Findings

Cycles serve as geometric invariants for Moebius transformations.

Conformal compactification involves adding a zero-radius cycle at infinity.

Non-reversibility of time is crucial for correct hyperbolic compactification.

Abstract

We study geometry of two-dimensional models of conformal space-time based on the group of Moebius transformation. The natural geometric invariants, called cycles, are used to linearise Moebius action. Conformal completion of the space-time is achieved through an addition of a zero-radius cycle at infinity. We pay an attention to the natural condition of non-reversibility of time arrow in order to get a correct compactification in the hyperbolic case.