$\mathbb{C}{P}^2$ and $\mathbb{C}{P}^1$ Sigma Models in Supergravity: Bianchi type IX Instantons and Cosmologies

Abstract

We find instanton/cosmological solutions with biaxial Bianchi-IX symmetry, involving non-trivial spatial dependence of the $\mathbb{C}{P}^1$- and $\mathbb{C}{P}^2$-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 $\mathbb{C}{P}^2$ with its corresponding scalar Ansatz; this further requires the spacetime to be $SU(2)\times U(1)$-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 non-trivial topology of $\mathbb{C}{P}^2$ and are in 1-1 correspondence with matter-free Bianchi-IX metrics. These solutions can also be found by coupling $\mathbb{C}{P}^1$ 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.