An efficient representation of Euclidean gravity I

作者：Lee Jungjai^*; Oh John J; Yang Hyun Seok

来源：The Journal of High Energy Physics, 2011, (12): 025.

DOI：10.1007/JHEP12(2011)025

• Solitons Monopoles and Instantons
• Differential and Algebraic Geometry
• Classical Theories of Gravity

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 Lambda T-2*M of two-forms splits canonically into the sum of three-dimensional vector spaces of self-dual and anti-self-dual two-forms, i.e., Lambda T-2*M = Lambda(2)(+) circle plus Lambda(2)(-). 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.

• 出版日期2011-12