Does an exact densitized-triad measure render the perturbative norm finite without compactification?

Determine whether evaluating the holomorphic inner product for self-dual Ashtekar gravity using the exact densitized-triad measure [dE]—accounting for the nontrivial Jacobian relating the densitized triad E^a_i and the tetrad e^I_μ—instead of the approximation [dE] ≅ [de], yields a finite perturbative norm for the Chern–Simons–Kodama state in flat slicing without compactifying the spatial slices.

Background

The paper computes the perturbative norm of the linearized Chern–Simons–Kodama (CSK) state using a holomorphic inner product derived from the Ashtekar reality conditions. In flat slicing, a divergence proportional to a constant α arises in the quadratic expansion of the inner product measure when the authors approximate the functional measure [dE] by [de].

They suggest two possible routes to regulate the divergence: (i) reformulating the perturbation theory on compact S^3 slices of global de Sitter (where the relevant operator is elliptic), and (ii) refining the path integral measure by treating the densitized triad measure [dE] exactly rather than approximating it by [de]. Whether the latter alone suffices to ensure finiteness, without resorting to compactification, is explicitly identified as an open question.