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.
Categories: gr-qc