### 3.13 Gauge invariance in Regge calculus?

A view that has been expressed frequently is that away from configurations with special
symmetries, different edge length assignments correspond to inequivalent geometries, and in this
sense Regge calculus possesses no gauge invariance ([187, 81, 124]; for a dissenting opinion,
see [184]).
The diffeomorphism invariance can be recovered in a weak-field perturbation about flat
space, as was shown by Roček and Williams [181, 183], and there is some evidence for the
existence of analogous zero-modes in perturbations of regular, non-flat tesselations, at least in
2d [122]. Inspired by the perturbative analysis around flat space, Hamber and Williams [122]
argue that a similar gauge invariance should persist even if one perturbs around an arbitrary
non-flat background. They propose as a possible definition for such gauge transformations local
variations of the link lengths that leave both the local volume and the local curvature terms
invariant.

One may hope that in the non-perturbative Regge regime no gauge-fixing is necessary, since
the contributions from zero-modes cancel out in the path-integral representation for operator
averages [124, 122]. Menotti and Peirano [162, 164, 163, 161], following a strategy suggested by Jevicki
and Ninomiya [134], have argued vigorously that the functional integral should contain a non-trivial
Faddeev–Popov determinant. Their starting point is somewhat different from that adopted in the
path-integral simulations (see also [199]). They treat piecewise flat spaces as special cases of differentiable
manifolds (with singular metric), with the action of the full diffeomorphism group still well-defined. To
arrive at a concrete representation for the Faddeev–Popov term which could be used in simulations seems at
present out of reach.

Recently, Hamber and Williams [122, 123] have argued that the -lattice measure is the essentially
unique local lattice measure over squared edge lengths (this is a special case of the one-parameter family of
local measures of the form ; see also [24] for a related derivation). It does of course
require a term with positive cosmological term in the action in order to suppress long edge
lengths.