Holomorphic wave function of the Universe

The quantum behavior of the vacuum Bianchi type-IX universe with the cosmological constant is investigated in terms of the Ashtekar variables. An exact solution to the quantum Hamiltonian constraint in the holomorphic representation is given. This solution reduces to the Hartle-Hawking wave function in the spatially isotropic sector and extends in the triad representation to the classically forbidden region where the determinant of the spatial metric becomes negative. The analysis of the quantum Robertson-Walker universe indicates that if the superspace is extended to such a classically forbidden region, the holomorphic representation picks up some restricted class of solutions in general. This observation leads to a new ansatz on the boundary condition of the Universe. In particular, the behavior of the Lorentzian and Euclidean WKB orbits corresponding to the solution suggests a new picture on the semiclassical behavior of the quantum Universe: that the Universe is created from an ensemble of Euclidean mother spacetimes. Further it is pointed out that the solution is a restriction to the spatially homogeneous sector of an almost exact solution to all the quantum constraints in the holomorphic representation for generic vacuum spacetime with the cosmological constant. The latter generic solution has a WKB structure for which the phase is proportional to the Chern-Simons functional.