  The canonical approach to Riemannian quantum gravity is reviewed with
reference to local supersymmetry, to the classical boundary-value problem
arising from the Hartle-Hawking quantum state, and particularly for
(anti-)self-dual geometries. Two examples of the boundary-value problem for the
Einstein equations, possibly with a cosmological constant \Lambda, are treated,
both of Bianchi-IX type. These close smoothly in the interior with a NUT or a
BOLT. The Hamiltonian approach to general relativity is described using
Ashtekar variables; for non-zero \Lambda and anti-self-dual Weyl tensor, the
classical solution corresponds, with the most naive choice of boundary data, to
the Chern-Simons functional of the boundary data, the classical action being
I_{CS}. Hence, one is led to the corresponding quantum states exp(\pm I_{CS}).
Apparently, the classical solutions have the undesirable feature that, in
general, the resulting Riemannian classical geometry, arising from the
Hamilton-Jacobi equation, does not close smoothly in the interior. The
canonical quantum theory of supergravity is also described, and may lead to
very streamlined (finite) calculations of loop amplitudes for N=1 supergravity
with gauged supermatter. If one uses Ashtekar/Jacobson variables for canonical
supergravity, then again (for \Lambda\neq 0) one arrives at a (supersymmetric)
Chern-Simons action and quantum state in the (anti-)self-dual case.
