The structure of spatial slices of three-dimensional causal triangulations

TL;DR

This paper studies the structure of spatial slices in 3D causal triangulations with specific topologies, showing they can be bijectively mapped to certain colored 2D cell complexes, revealing their combinatorial structure.

Contribution

It introduces a bijective mapping between slices of 3D causal triangulations and specific colored 2D cell complexes, providing a new combinatorial perspective.

Findings

Slices correspond to colored 2D cell complexes

Mapping is bijective and preserves structure

Provides a combinatorial characterization of slices

Abstract

We consider causal 3-dimensional triangulations with the topology of $S^2\times [0,1]$ or $D^2\times [0,1]$ where $S^2$ and $D^2$ are the two-dimensional sphere and disc, respectively. These triangulations consist of slices and we show that these slices can be mapped bijectively onto a set of certain coloured two-dimensional cell complexes satisfying simple conditions. The cell complexes arise as the cross section of the individual slices.