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

TL;DR

This paper constructs and analyzes instanton and cosmological solutions with Bianchi-IX symmetry in supergravity, involving complex sigma-models on complex projective spaces, revealing new exact solutions related to topology and symmetry.

Contribution

It introduces new instanton and cosmological solutions in supergravity with non-trivial sigma-models on $bbc P^{1}$ and $bbc P^{2}$, highlighting their topological origins and exact forms.

Findings

Existence of exact solutions linked to $bbc P^{2}$ topology.

Solutions exhibit bolt-type regularity in Euclidean space.

Lorentzian solutions derived via Wick rotation from Euclidean counterparts.

Abstract

We find instanton/cosmological solutions with biaxial Bianchi-IX symmetry, involving non-trivial spatial dependence of the $\bbbc P^{1}$- and $\bbbc P^{2}$-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 $\bbbc P^{2}$ 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 $\bbbc P^{2}$ and are in 1-1 correspondence with matter-free Bianchi-IX metrics. These solutions can also be found by coupling $\bbbc P^{1}$ 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.