Feynman Path Integral Formulation

7.3 Lattice Diffeomorphism Invariance 237
Fig. 7.4 Icosahedral tessellation
of the two-sphere, with
arbitrary edge length assignments.
1
l 12 l
l 13 l l 15
16 14
6
l 26
l 56
2 l 23 l 5
l 45
34
l 211
3 l 4
l 611
l 610
l 510
27 l
l 59
37 l 38
l 48
l 49
l 711
l
11 l
910
10
l
1011
78 l l 89
1112
l 1012 9
7
l 8
l 812
712
l 912
12
Finally, for the icosahedron [shown in Fig. (7.4)] one computes the following
coefficients of the small fluctuation matrix
ε12 2 → 16 √ aλ (270 − 6 √ 3π + π 2 )/135π
ε 12 ε 13 → 16 √ aλ (−675 − 30 √ 3π + 8π 2 )/675π
ε 12 ε 14 → 16 √ aλ (270 − 6 √ 3 + π 2 )/135π
ε 12 ε 34 → 32 √ aλ (−675 + 15 √ 3π + 2π 2 )/675π
ε 12 ε 45 → 16 √ aλ (−675 + 15 √ 3π + 2π 2 )/675π
ε 12 ε 38 → 16 √ aλ (−675 + 15 √ 3π + 2π 2 )/675π
ε 12 ε 48 → 16 √ aλ (675 + 30 √ 3π + π 2 )/675π ,
(7.46)
with the remaining coefficients being determined by symmetry. Up to a common
factor of 8 √ aλ/675π, the eigenvalues of the 30 × 30 small edge length fluctuation
matrix are given by 12340.173 (with multiplicity 3), 7238.984 (with multiplicity 5),
888.264 = 90π 2 (with multiplicity 1), 20.887 (with multiplicity 3), and zero (with
multiplicity 18).
The presence of the zero modes is interpreted as a lattice manifestation of the
diffeomorphism invariance of the gravitational action. One can summarize the previous
results so far as
Tetrahedron (N 0 = 4) : 2 zero modes
Octahedron(N 0 = 6) : 6 zero modes
Icosahedron(N 0 = 12) : 18 zero modes .
(7.47)