CDT

Causal dynamical triangulations is a technique for regularizing path integrals. Rather than allowing spacetime to be a smooth Lorentzian 4-manifold as usual, we use a simplicial approximation to spacetime, made by gluing together flat pieces of 4-dimensional Minkowski spacetime. There are two allowed types of 4-simplices: (4,1) and (3,2):

A (4,1) simplex has six spacelike and four timelike edges, while a (3,2) simplex has four spacelike and six timelike. All spacelike edges are of squared length

(1)

\begin{equation} l_{space}^{2} = a^{2}, \end{equation}

while timelike edges have squared length

(2)

\begin{align} l_{time}^{2} = - \alpha a^{2}. \end{align}

The integer-valued timestep $\tau$ indexes spacelike sheets of a global foliation of the manifold; spacelike edges of the simplices lie inside the sheets, while timelike edges run between two consecutive sheets. The number of $i$-simplices in a triangulation is denoted $N_{i}$ (thus e.g. $N_{0}$ is the number of vertices). Number of (4,1) simplices and (3,2) simplices are respectively denoted by $N_{4}^{(4,1)}$ and $N_{4}^{(3,2)}$, so that $N_{4} = N_{4}^{(4,1)} + N_{4}^{(3,2)}$.

Spacetime topology: $[0,1]\times \ ^{\left( 3 \right) } \Sigma$

Regge action in Euclidean sector:

(3)

\begin{align} S_{E} = -k^{\left( b \right) } \pi \sqrt{4\tilde{\alpha} -1} \left( N_{0} - \chi \right) + N_{4}^{\left( 4,1\right) } S_{\left( 4,1 \right)} + N_{4}^{\left( 3,2\right) } S_{\left( 3,2 \right)} \end{align}

where

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and

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Here $\chi$ is the Euler characteristic of the triangulated 4-manifold

(6)

\begin{align} $\chi =N_{0} - N_{1} + N_{2} - N_{3} + N_{4}.$ \end{align}

Because we have performed an analytic continuation to Euclidean time, $\tilde{\alpha} \equiv -\alpha.$