Causal dynamical triangulation

Definition

Fix a spatial topology $\Sigma$ (for example the three-sphere), a number $t$ of time steps, a lattice spacing $a > 0$ and a constant $\alpha > 0$. A causal triangulation is a simplicial complex $T$ homeomorphic to $\Sigma \times [0, t]$ (or $\Sigma \times S^1$) with the following properties.

1. Its vertices lie on slices labelled $0, 1, \dots, t$, and each slice is a triangulation of $\Sigma$.
2. Every simplex has its vertices on one slice or on two adjacent slices.
3. Every spacelike edge, joining two vertices of one slice, has squared length $a^2$, and every timelike edge, joining adjacent slices, has squared length $-\alpha a^2$.

Causal dynamical triangulation defines the gravitational path integral as $$Z = \sum_{T} \frac{1}{C_T} e^{i S_R(T)},$$ where the sum runs over causal triangulations $T$. Here $S_R$ is the Lorentzian Regge action with cosmological term and $C_T$ is the order of the automorphism group of $T$. After the Wick rotation $\alpha \mapsto -\alpha$ each term becomes a real weight $e^{-S_E(T)}$. The continuum limit $a \to 0$ is then studied.