Invertibility of vierbeins in 2+1 topological gravity

Background fields in 2+1 topological gravity

“…It is well known by now that 2+1 dimensional quantum gravity can be given a gauge theoretic interpretation in terms of a Chern-Simons theory with gauge group ISO(2, 1), SO (3,1) or SO (2,2), depending on whether the cosmological constant λ is zero, positive or negative [1,2]. One of the main differences, however, is the role of non-invertible vierbeins which are forbidden in the former but are allowed, and even required, in the latter.…”

“…A perhaps naive step in this direction was made in [3]. Here, Chern-Simons-Witten topological gravity, with non-zero cosmological constant, was considered on some three manifold M. Using the background field method for the vierbein field e a i only, a one-loop effective action, Γ, was obtained of the form:…”