### 4.2 Path integral for dynamical triangulations

Denoting by the set of all triangulations of the four-sphere, the partition function for the model is
given by
where and denote the numbers of 2- and 4-simplices contained in the simplicial manifold ,
and is the order of the automorphism group of . One may think of (21) as a grand canonical
ensemble, with chemical potential . It is related to the canonical ensemble with fixed volume,
, 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 set to 1.
To understand the simple form of the action , recall that the curvature term in Regge calculus
(cf. 12) is represented by , which for fixed edge length is proportional to . The
constant is determined from the condition that a triangulation of flat
space should have average vanishing curvature [2, 11]. (Because the four-simplices 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
. It is sometimes convenient to re-express as a function of , using
, valid for the -topology. The corresponding partition function is
(where ).