  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.
