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.
Highlights
The hadrons we know all fall into multiplets that reflect underlying internal symmetries
An underlying idea is that gravity can be formulated as a gauge theory of the Lorentz group where spin connections play a role of gauge fields and Riemann curvature tensors correspond to their field strengths [19]
There is a mysterious transition between Euclidean spaces and Minkowski (Lorentzian) spaces. They are related by an analytic continuation x0 1⁄4 −ix4, but it results in dramatic changes of physics
Summary
The hadrons we know all fall into multiplets that reflect underlying internal symmetries. The electromagnetism and a scalar field obtained from a five-dimensional metric through the KaluzaKlein reduction should appear in the same multiplet in an irreducible representation (irrep) of the Lorentz group SOð5Þ This unification scheme is similar to the grand unification of the Standard Model, since the group SOð4Þ ≅ SUð2Þþ ⊗ SUð2Þ−=Z2 must be embedded into the simple group SOð5Þ 1⁄4 Spð2Þ=Z2 the Kaluza-Klein theory is reduced from a five-dimensional gravity. According to the symmetry breaking pattern, we further decompose the fivedimensional Riemann curvature tensor in the basis of soð4Þ ≅ suð2Þþ ⊕ suð2Þ− Lie algebra This decomposition is useful to see how Uð1Þ gauge fields and a scalar field deform the instanton structure of four-dimensional Einstein manifolds and to understand how these deformed geometries are nicely combined into a five-dimensional Einstein manifold. Appendix C contains the group structure analysis of Riemann curvature tensors and the decomposition of Ricci tensors and Ricci scalar in the soð4Þ Lie algebra basis
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