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
∑ --1--- −S[T] Z (κ2,κ4) = C (T)e , S [T] = − κ2N2 (T ) + κ4N4 (T ), (21 ) T∈𝒯
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 (21View Equation) 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
∑ ∑ ∑ Z (κ2,κ4) = e−κ4N4Z (κ2,N4) ≡ e−κ4N4 eκ2N2(T). (22 ) N4 N4 T∈𝒯(N4)
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. 12View Equation) 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 [2Jump To The Next Citation Point11Jump To The Next Citation Point]. (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).

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