A semiclassical interpretation of the topological solutions for canonical quantum gravity

Ashtekar’s formulation for canonical quantum gravity is known to possess the topological solutions which have their supports only on the moduli space $\text{CN}$ of flat $SL(2,C)$ connections. We show that each point on the moduli space $\text{CN}$ corresponds to a geometric structure, or more precisely the Lorentz group part of a family of Lorentzian structures, on the flat (3+1)-dimensional spacetime. A detailed analysis is given in the case where the spacetime is homeomorphic to $R\times T^3$. Most of the points on the moduli space $\text{CN}$ yield pathological spacetimes which suffers from singularities on each spatial hypersurface or which violates the strong causality condition. There is, however, a subspace of $\text{CN}$ on which each point corresponds to a family of regular spacetimes.