Feynman Path Integral Formulation

+O( 1 d 2 ) . (7.141)
7.6 Discrete Gravity in the Large-d Limit 263
We now specialize to the case where four simplices meet at a hinge. When expanded
out in terms of the ε’s one obtains for the deficit angle
δ d
= 2π − 4 · π
2 + 1
∑ d − ε d,d+1 + ... − ε 1,d ε 1,d+1 + ...
simplices
(
−ε1,d − 1 2 ε2 1,d − 1 2 ε2 d,d+1 − ε 12 ε 1,d+1 − ε 1,d ε 3,d+1 − ε 1,d ε d,d+1 + ... )
− 1 d
The action contribution involving the deficit angle is then, for a single hinge,
−k δ d V d−2 =(−k) 2d3/2 (d − 1)
d!2 d/2 (
−εd,d+1 + ... − ε 1,d ε 1,d+1 + ... ) . (7.142)
It involves two types of terms: one linear in the (single) edge opposite to the hinge,
as well as a term involving a product of two distinct edges, connecting any hinge
vertex to the two vertices opposite to the given hinge. Since there are four simplices
meeting on one hinge, one will have 4 terms of the first type, and 4(d − 1) terms of
the second type.
To obtain the total action, a sum over all simplices, resp. hinges, has still to
be performed. Dropping the irrelevant constant term and summing over edges one
obtains for the total action λ 0 ∑V d − k ∑δ d V d−2 in the large d limit
λ 0
(
−
1
2 ∑ε 2 ij)
− 2kd
2 ( − ∑ε jk − ∑ε ij ε ik
)
, (7.143)
up to an overall multiplicative factor √ d/d!2 d/2 , which will play no essential role
in the following.
The next step involves the choice of a specific lattice. Here we will evaluate the
action for the cross polytope β d+1 . The cross polytope β n is the regular polytope in n
dimensions corresponding to the convex hull of the points formed by permuting the
coordinates (±1,0,0,...,0), and has therefore 2n vertices. It is named so because
its vertices are located equidistant from the origin, along the Cartesian axes in n-
space. The cross polytope in n dimensions is bounded by 2 n (n − 1)-simplices, has
2n vertices and 2n(n − 1) edges.
In three dimensions, it represents the convex hull of the octahedron, while in four
dimensions the cross polytope is the 16-cell (Coxeter, 1948; Coxeter, 1974). In the
general case it is dual to a hypercube in n dimensions, with the “dual” of a regular
polytope being another regular polytope having one vertex in the center of each cell
of the polytope one started with. Fig. 7.12 shows as an example the polytope β 8 .
When we consider the surface of the cross polytope in d +1 dimensions, we have
an object of dimension n−1 = d, which corresponds to a triangulated manifold with
no boundary, homeomorphic to the sphere. From Eq. (7.141) the deficit angle is then
given to leading order by