Foliations and 2+1 Causal Dynamical Triangulation Models

TL;DR

This paper investigates the role of foliation in 2+1 dimensional causal dynamical triangulation models, showing that fixed-lapse conditions can be replaced by global constraints and exploring effects of new simplex types on model behavior.

Contribution

It introduces a reformulation of foliation constraints as global geometric conditions and examines how adding puncturing simplices alters the model's low-energy dynamics.

Findings

Fixed-lapse condition can be replaced by global constraints

Introducing puncturing simplices changes low-energy behavior

Foliation structure's importance is context-dependent

Abstract

The original models of causal dynamical triangulations construct space-time by arranging a set of simplices in layers separated by a fixed time-like distance. The importance of the foliation structure in the 2+1 dimensional model is studied by considering variations in which this property is relaxed. It turns out that the fixed-lapse condition can be equivalently replaced by a set of global constraints that have geometrical interpretation. On the other hand, the introduction of new types of simplices that puncture the foliating sheets in general leads to different low-energy behavior compared to the original model.