Anatomy of Einstein Manifolds
Jongmin Park, Jaewon Shin, Hyun Seok Yang

TL;DR
This paper explores the geometric and gauge-theoretic structure of four-dimensional Einstein manifolds, their stability, and how these properties might extend to five dimensions through group embeddings, revealing a substructure akin to particle physics models.
Contribution
It introduces a gauge-theoretic perspective on Einstein manifolds, demonstrating their substructure via $SU(2)$ instantons and proposing a higher-dimensional embedding analogous to grand unification.
Findings
Four-dimensional Einstein manifolds relate to $SU(2)_ ext{+}$ and $SU(2)_ ext{-}$ instantons.
Higher-dimensional embedding involves group $SO(5)$, extending stability considerations.
Manifold substructure resembles quark model with instantons as fundamental components.
Abstract
An Einstein manifold in four dimensions has some configuration of Yang-Mills instantons and anti-instantons associated with it. This fact is based on the fundamental theorems that the four-dimensional Lorentz group is a direct product of two groups 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 must be embedded into the simple group…
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