An Efficient Representation of Euclidean Gravity I

TL;DR

This paper establishes a rigorous mathematical foundation linking Euclidean gravity's spacetime topology to SU(2) gauge theories, showing Einstein manifolds as sums of Yang-Mills instantons and anti-instantons, highlighting deep geometric decompositions.

Contribution

It provides a novel proof that Einstein manifolds can be expressed as sums of SU(2)_L instantons and SU(2)_R anti-instantons using gauge theory and four-dimensional geometry.

Findings

Einstein manifolds decompose into SU(2)_L and SU(2)_R instantons and anti-instantons.

The Lorentz group Spin(4) is isomorphic to SU(2)_L x SU(2)_R.

The splitting of two-forms relates to the decomposition of four-manifolds.

Abstract

We explore how the topology of spacetime fabric is encoded into the local structure of Riemannian metrics using the gauge theory formulation of Euclidean gravity. In part I, we provide a rigorous mathematical foundation to prove that a general Einstein manifold arises as the sum of SU(2)_L Yang-Mills instantons and SU(2)_R anti-instantons where SU(2)_L and SU(2)_R are normal subgroups of the four-dimensional Lorentz group Spin(4) = SU(2)_L x SU(2)_R. Our proof relies only on the general properties in four dimensions: The Lorentz group Spin(4) is isomorphic to SU(2)_L x SU(2)_R and the six-dimensional vector space of two-forms splits canonically into the sum of three-dimensional vector spaces of self-dual and anti-self-dual two-forms. Consolidating these two, it turns out that the splitting of Spin(4) is deeply correlated with the decomposition of two-forms on four-manifold which occupies a central position in the theory of four-manifolds.