Feynman Path Integral Formulation

7.3 Lattice Diffeomorphism Invariance 235For 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 constantcurvature with R = ± √ λ/a (there being no real solutions for λ < 0). Thecurvature-squared leads to some non-trivial interactions in two dimensions, althoughthe resulting theory is not unitary. This is not important here, as we only plan to addressfor now the issue of lattice diffeomorphism invariance.Fig. 7.2 Tetrahedral tessellationof the two-sphere, witharbitrary edge length assignments.1l 14l 123l 1342l 23l 24l 34After expanding about the equilateral configuration, the action at the stationarypoint reduces toI = λ 8π √ a/λ + a 8π/ √ a/λ = 16π √ a λ , (7.41)independently of the tessellation considered. Vanishing of the linear terms in thesmall fluctuation expansion gives for the average edge lengthl 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 followingcoefficientsε12 2 → 16 √ aλ (54 − 6 √ 3π + 5π 2 )/81πε 12 ε 13 → 16 √ aλπ/9ε 12 ε 15 → 64 √ aλ (−27 + 3 √ 3π + 2π 2 )/81π ,(7.43)with the remaining coefficients being determined by symmetry. The small fluctuationmatrix is therefore given by