MINI-SUPERSPACE WITH LOCAL SUPERSYMMETRY

Mini-superspace quantum cosmological models possessing one-dimensional local supersymmetry can be found by reducing four-dimensional models with N = 1 supergravity coupled to supermatter, assuming that the geometry is that of a Friedman model with S3 spatial sections. and that the other fields are homogeneous on the family of three-spheres. For pure N = 1 supergravity, a suitable restrictive ansatz is found for the gravitational and spin-3/2 fields. such that the resulting one-dimensional model has local supersymmetry. When the model is quantised. the supersymmetry constraints become first-order differential operators which annihilate the wave function-PSI, and which imply a suitable Wheeler-DeWitt equation for PSI. Models with more interesting physical content are found by studying N = 1 supergravity coupled to locally supersymmetric matter, here taken to be a massive complex scalar with spin-1/2 partner. In this case the coupled first-order equations, relating the components of the wave function in the different fermionic sectors, are again derived from the supersymmetry constraints; these lead to different Wheeler-De Witt equations in the different sectors.