Feynman Path Integral Formulation

7.3 Lattice Diffeomorphism Invariance 235
For a manifold of fixed topology the term proportional to k can be dropped, since
∑ h δ h = 2πχ, where χ is the Euler characteristic. The classical solutions have constant
curvature with R = ± √ λ/a (there being no real solutions for λ < 0). The
curvature-squared leads to some non-trivial interactions in two dimensions, although
the resulting theory is not unitary. This is not important here, as we only plan to address
for now the issue of lattice diffeomorphism invariance.
Fig. 7.2 Tetrahedral tessellation
of the two-sphere, with
arbitrary edge length assignments.
1
l 14
l 12
3
l 13
4
2
l 23
l 24
l 34
After expanding about the equilateral configuration, the action at the stationary
point reduces to
I = λ 8π √ a/λ + a 8π/ √ a/λ = 16π √ a λ , (7.41)
independently of the tessellation considered. Vanishing of the linear terms in the
small fluctuation expansion gives for the average edge length
l 0 = [ cπ 2 (4a/λ) ] 1/4
, (7.42)
with c = 16/3,4/3,16/75 for the tetrahedron, octahedron and icosahedron, respectively.
For fluctuations about the classical solution for a tetrahedral tessellation of S 2
[see Fig. (7.2)] the small edge length fluctuation matrix gives rise to the following
coefficients
ε12 2 → 16 √ aλ (54 − 6 √ 3π + 5π 2 )/81π
ε 12 ε 13 → 16 √ aλπ/9
ε 12 ε 15 → 64 √ aλ (−27 + 3 √ 3π + 2π 2 )/81π ,
(7.43)
with the remaining coefficients being determined by symmetry. The small fluctuation
matrix is therefore given by