Quaternionic multi S 4 ≃ HP(1) gravitational and chiral instantons

Abstract

Compact multi S 4 gravitational instantons (positive definite metric solutions to Einstein's equations with a Λ term) are found in terms of quaternionic harmonic mappings x n : S 4 → S 4. The manifold is the euclidean de Sitter space S 4 ≃ HP(1) with Euler-Poincaré index I E = n and Pontryagin index I P = 0. The metric can be alternatively seen as the induced metric of a Kähler HP(1) chiral field space, these same solutions then correspond to all multi chiral instantons in a general covariant quaternionic HP(1) ≃ SO(5)/SO(4) ≃ Sp(2)/Sp(1) × Sp(1) nonlinear σ-model of de Alfaro et al. They are the quaternionic counterparts of the harmonic instanton maps of the 2-dimensional CP(1) ≃ SO(3)/SO(2) ≃ SU(2)/U(1) σ-model.