Anatomy of Einstein Manifolds

Abstract

An Einstein manifold in four dimensions has some configuration of $SU(2{)}_{+}$ Yang-Mills instantons and $SU(2{)}_{-}$ anti-instantons associated with it. This fact is based on the fundamental theorems that the four-dimensional Lorentz group $Spin(4)$ is a direct product of two groups $SU(2{)}_{\pm }$ and the vector space of two-forms decomposes into the space of self-dual and anti-self-dual two-forms. It explains why the four-dimensional spacetime is special for the stability of Einstein manifolds. We now consider whether such a stability of four-dimensional Einstein manifolds can be lifted to a five-dimensional Einstein manifold. The higher-dimensional embedding of four-manifolds from the viewpoint of gauge theory is similar to the grand unification of Standard Model since the group $SO(4)\cong Spin(4)/{\mathbb{Z}}_2=SU(2{)}_{+}\otimes SU(2{)}_{-}/{\mathbb{Z}}_2$ must be embedded into the simple group $SO(5)=Sp(2)/{\mathbb{Z}}_2$. Our group-theoretic approach reveals the anatomy of Riemannian manifolds quite similar to the quark model of hadrons in which two independent Yang-Mills instantons represent a substructure of Einstein manifolds.

Comments: v2: Title changed with improved contents, 36 pages, 4 figures, to appear in Phys. Rev. D