Ivancevic_Applied-Diff-Geom

1062 Applied Differential Geometry: A Modern Introduction∑i⊃v α i = 2π − δ, for δ ≠ 0, see Figure 6.9. The so–called deficit angle δ isprecisely the rotation angle picked up by the vector and is a direct measurefor the scalar curvature at the vertex. The operational description to getthe scalar curvature in higher dimensions is very similar, one basically hasto sum in each point over the Gaussian curvatures of all 2D submanifolds.This explains why in Regge calculus the curvature part of the EH actionis given by a sum over building blocks of dimension (d − 2) which are theobjects dual to those local 2D submanifolds. More precisely, the continuumcurvature and volume terms of the action become12∫d d x √ | det g| (d) R −→ ∑ Vol(i th (d − 2)−simplex) δ i (6.103)Ri∈R∫d d x √ | det g| −→ ∑ Vol(i th d−simplex) (6.104)Ri∈Rin the simplicial discretization. It is then a simple exercise in trigonometryto express the volumes and angles appearing in these formulas as functionsof the edge lengths l i , both in the Euclidean and the Minkowskian case.The approach of dynamical triangulations uses a certain class of suchsimplicial space–times as an explicit, regularized realization of the spaceGeom(M). For a given volume N d , this class consists of all gluings ofmanifold–type of a set of N d simplicial building blocks of top–dimension dwhose edge lengths are restricted to take either one or one out of two values.In the Euclidean case we set li2 = a2 for all i, and in the Lorentzian casewe allow for both space- and time–like links with li 2 ∈ {−a2 , a 2 }, where thegeodesic distance a serves as a short-distance cutoff, which will be takento zero later. Coming from the classical theory this may seem a grave restrictionat first, but this is indeed not the case. Firstly, keep in mind thatfor the purposes of the quantum theory we want to sample the space ofgeometries ‘ergodically’ at a coarse-grained scale of order a. This shouldbe contrasted with the classical theory where the objective is usually to approximatea given, fixed space–time to within a length scale a. In the lattercase one typically requires a much finer topology on the space of metrics orgeometries. It is also straightforward to see that no local curvature degreesof freedom are suppressed by fixing the edge lengths; deficit angles in alldirections are still present, although they take on only a discretized set ofvalues. In this sense, in dynamical triangulations all geometry is in thegluing of the fundamental building blocks. This is dual to how quantumRegge calculus is set up, where one usually fixes a triangulation T and then