CP2 and CP1 sigma models in supergravity:: Bianchi type IX instantons and cosmologies

We find instanton/cosmological solutions with biaxial Bianchi-IX symmetry, involving nontrivial spatial dependence of the CP1- and CP2-sigma-models coupled to gravity. Such manifolds arise in N = 1, d = 4 supergravity with supermatter actions and hence the solutions can be embedded in supergravity. There is a natural way in which the standard coordinates of these manifolds can be mapped into the four-dimensional physical space. Due to its special symmetry, we start with CP2 with its corresponding scalar ansatz; this further requires the spacetime to be SU(2) x U(1)-invariant. The problem then reduces to a set of ordinary differential equations whose analytical properties and solutions are discussed. Among the solutions there is a surprising, special family of exact solutions which owe their existence to the nontrivial topology of CP2 and are in 1-1 correspondence with matter-free Bianchi-IX metrics. These solutions can also be found by coupling CP1 to gravity. The regularity of these Euclidean solutions is discussed-the only possibility is bolt-type regularity. The Lorentzian solutions with similar scalar ansatz are all obtainable from the Euclidean solutions by Wick rotation.