  We investigate a triad representation of the Chern-Simons state of quantum
gravity with a non-vanishing cosmological constant. It is shown that the
Chern-Simons state, which is a well-known exact wavefunctional within the
Ashtekar theory, can be transformed to the real triad representation by means
of a suitably generalized Fourier transformation, yielding a complex integral
representation for the corresponding state in the triad variables. It is found
that topologically inequivalent choices for the complex integration contour
give rise to linearly independent wavefunctionals in the triad representation,
which all arise from the one Chern-Simons state in the Ashtekar variables. For
a suitable choice of the normalization factor, these states turn out to be
gauge-invariant under arbitrary, even topologically non-trivial
gauge-transformations. Explicit analytical expressions for the wavefunctionals
in the triad representation can be obtained in several interesting asymptotic
parameter regimes, and the associated semiclassical 4-geometries are discussed.
In restriction to Bianchi-type homogeneous 3-metrics, we compare our results
with earlier discussions of homogeneous cosmological models. Moreover, we
define an inner product on the Hilbert space of quantum gravity, and choose a
natural gauge-condition fixing the time-gauge. With respect to this particular
inner product, the Chern-Simons state of quantum gravity turns out to be a
non-normalizable wavefunctional.
