Path integral for dynamical triangulations

4.2 Path integral for dynamical triangulations

Denoting by $𝒯$ the set of all triangulations of the four-sphere, the partition function for the model is given by

where $N2$ and $N4$ denote the numbers of 2- and 4-simplices contained in the simplicial manifold $T$ , and $C(T )$ is the order of the automorphism group of $T$ . One may think of (21

) as a grand canonical ensemble, with chemical potential $κ4$ . It is related to the canonical ensemble with fixed volume, $Z (κ2,N4 )$ , by a Legendre transform

The metric information is encoded in the connectivity of the simplicial decomposition, since the individual 4-simplices are assumed equilateral, with the edge length $a$ set to 1.

To understand the simple form of the action $S$ , recall that the curvature term in Regge calculus (cf. 12 ) is represented by $∑ 2δbAb b$ , which for fixed edge length is proportional to $(c4N2 − 10N4 )$ . The constant $1 c4 = 2π ∕arccos 4 = 4.767$ is determined from the condition that a triangulation of flat space should have average vanishing curvature [2 , 11 ]. (Because the four-simplices $σ$ are equilateral, zero curvature can only be achieved upon averaging. This explains the absence of a conventional perturbation theory around flat space.) The cosmological term is represented by $λN4V (σ) ∼ λN4$ . It is sometimes convenient to re-express $N2$ as a function of $N0$ , using $N2 = 2N0 + 2N4 − 4$ , valid for the $S4$ -topology. The corresponding partition function is $Z(κ0,κ4 )$ (where $κ0 = 2κ2$ ).