Physical states of Bianchi type IX quantum cosmologies described by the Chern-Simons functional

A class of exact solutions of the Wheeler-DeWitt equation for diagonal Bianchi type IX cosmologies with a cosmological constant is derived in the metric representation. This class consists of all the "topological solutions" which are associated with the Bianchi type IX reduction of the Chern-Simons functional in Ashtekar variables. The different solutions within the class arise from the topologically inequivalent choices of the integration contours in the transformation from the Ashtekar representation to the metric representation. We show how the saddle points of the reduced Chern-Simons functional generate a complete basis of such integration contours and the associated solutions. Among the solutions we identify two, which, semiclassically, satisfy the boundary conditions proposed by Vilenkin and by Hartle and Hawking, respectively. In the limit of a vanishing cosmological constant our solutions reduce to a class found earlier in special fermion sectors of supersymmetric Bianchi type IX models.