Einstein Manifolds As Yang-Mills Instantons

TL;DR

This paper demonstrates that four-dimensional Einstein manifolds can be exactly described as Yang-Mills instantons, linking Einstein equations to self-duality conditions in gauge theory, and explaining their stability through topological invariants.

Contribution

It establishes a precise correspondence between Einstein manifolds and Yang-Mills instantons in SO(4), revealing their composition from SU(2) instantons and anti-instantons.

Findings

Einstein equations are equivalent to self-duality equations in Yang-Mills theory.

Einstein manifolds are sums of SU(2)_L instantons and SU(2)_R anti-instantons.

Einstein manifolds are stable due to their topological invariants.

Abstract

It is well-known that Einstein gravity can be formulated as a gauge theory of Lorentz group where spin connections play a role of gauge fields and Riemann curvature tensors correspond to their field strengths. One can then pose an interesting question: What is the Einstein equations from the gauge theory point of view? Or equivalently, what is the gauge theory object corresponding to Einstein manifolds? We show that the Einstein equations in four dimensions are precisely self-duality equations in Yang-Mills gauge theory and so Einstein manifolds correspond to Yang-Mills instantons in SO(4) = SU(2)_L x SU(2)_R gauge theory. Specifically, we prove that any Einstein manifold with or without a cosmological constant always arises as the sum of SU(2)_L instantons and SU(2)_R anti-instantons. This result explains why an Einstein manifold must be stable because two kinds of instantons belong to different gauge groups, instantons in SU(2)_L and anti-instantons in SU(2)_R, and so they cannot decay into a vacuum. We further illuminate the stability of Einstein manifolds by showing that they carry nontrivial topological invariants.