Anatomy of Einstein manifolds

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 2-forms decomposes into the space of self-dual and anti-self-dual 2-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 the Standard Model, since the group $SO(4)\cong \mathrm{Spin}(4)/{\mathbb{Z}}_2=SU(2{)}_{+}\otimes SU(2{)}_{-}/{\mathbb{Z}}_2$ 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.

Figure 1

Topological numbers of closed Einstein manifolds.

Figure 2

Root diagram of $so(5)\cong sp(2)$ for the Cartan subalgebra (3.33).

Figure 3

The root diagram of $sp(2)\cong so(5)$ for the Cartan subalgebra (3.40) where the actual roots must read as $|{\alpha }_1,{\alpha }_2⟩=\sqrt{2}|{\beta }_1,{\beta }_2⟩$.

Figure 4

The block weight diagrams of the fundamental representations of $sp(2)\cong so(5)$.