Foliations and $2+1$ causal dynamical triangulation models

The original models of causal dynamical triangulations construct space-time by arranging a set of simplices in layers separated by a fixed timelike 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.

Figure 1

Simplices in $2+1$ dimensions. Edges pointing mostly vertically (mostly sideways) are timelike (spacelike) and have negative (positive) edge lengths. Simplices are named after the number of vertices on spacelike surfaces.

Figure 2

Simplices in $2+1$ dimensions that span two foliation layers.

Figure 3

New moves manipulating (1, 2, 1) simplices. (a) Two (3, 1) change into three (1, 2, 1) tetrahedra. (b) One (1, 2, 1) changes into two (1, 2, 1) and two (3, 1) tetrahedra. The (3, 1) simplices appear distorted in the diagram.