Equivalence of the Chern-Simons state and the Hartle-Hawking and Vilenkin wave-functions

We show that the Chern-Simons (CS) state when reduced to mini-superspace is the Fourier dual of the Hartle-Hawking (HH) and Vilenkin (V) wave-functions of the Universe. This is to be expected, given that the former and latter solve the same constraint equation, written in terms of conjugate variables (loosely the expansion factor and the Hubble parameter). A number of subtleties in the mapping, related to the contour of integration of the connection, shed light on the issue of boundary conditions in quantum cosmology. If we insist on a {\it real} Hubble parameter, then only the HH wave-function can be represented by the CS state, with the Hubble parameter covering the whole real line. For the V (or tunnelling) wave-function the Hubble parameter is restricted to the positive real line (which makes sense, since the state only admits outgoing waves), but the contour also covers the whole negative imaginary axis. Hence the state is not admissible if reality conditions are imposed upon the connection. Modifications of the V state, requiring the addition of source terms to the Hamiltonian constraint, are examined and found to be more palatable. In the dual picture the HH state predicts a uniform distribution for the Hubble parameter over the whole real line; the modified V state a uniform distribution over the positive real line.