An Efficient Representation of Euclidean Gravity I

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.

31 pages, 1 figure

References in corpus (5)

• Emergent Gravity from Noncommutative Spacetime
• Emergent Spacetime and The Origin of Gravity
• Yang-Mills Instantons from Gravitational Instantons
• Noncommutative Electromagnetism As A Large N Gauge Theory
• An Efficient Representation of Euclidean Gravity I

Cited by in corpus (12)

• Towards A Background Independent Quantum Gravity
• Test of Emergent Gravity
• An Efficient Representation of Euclidean Gravity I
• Quantization of Emergent Gravity
• Notes on Emergent Gravity
• Algebraic Properties of Riemannian Manifolds
• Dark Energy and Dark Matter in Emergent Gravity
• Quantized Kähler Geometry and Quantum Gravity
• Anatomy of Einstein Manifolds
• Generalization of Instanton-Induced Inflation and Dynamical Compactification
• Einstein Structure of Four-Manifolds
• Explicit Construction of Hermitian Yang-Mills Instantons on Coset Manifolds