  We determine the complex geometries dual to the semi-classical saddles in
three-dimensional gravity with positive or negative cosmological constant. We
examine the semi-classical saddles in Liouville field theory and interpret them
in terms of gravity theory. For this, we describe the gravity theory by Chern
Simons theory and classify the possible saddles based on homotopy group
argument. We further realize the semi-classical saddles using the
mini-superspace model of quantum gravity and explicitly determine the integral
contour. In the case of positive cosmological constant, we recovered the
geometry used for no-boundary proposal of Hartle and Hawking. In the case of
negative cosmological constant, the geometry can be identified with Euclidean
anti-de Sitter space attached with imaginary radius spheres. The geometry
should be unphysical and several arguments on this issue are provided. Partial
results were already presented in our earlier letter, and more detailed
derivations and explanations on the results are given along with additional
results. In particular, we reproduce the classical Liouville action from the
Chern-Simons formulation of dual gravity theory.
