A gauge theoretic approach to the anti-self-dual Einstein equations

Abstract

In [29], Plebanski reformulated the anti-self-dual Einstein equations with non-zero scalar curvature as a first order PDE for a connection in an SO(3)-bundle over the four-manifold. The aim of this article is to place this differential equation in a new framework, in which it is both elliptic and a stationary point of a parabolic flow. To do this, we exploit a link with definite connections (introduced in [12]) to draw an analogy with instantons and the Yang-Mills flow. This picture leads to a natural conjecture, analogous to one made by Donaldson concerning hyperkähler 4-manifolds [9]. It also provides a moment-map description of the anti-self-dual Einstein equations with non-zero scalar curvature.

49 pages. v2 typos corrected

References in corpus (4)

• New Action Principle for General Relativity
• Hyperbolic geometry and non-Kahler manifolds with trivial canonical bundle
• Gravity as a diffeomorphism invariant gauge theory
• The diversity of symplectic Calabi-Yau six-manifolds

Cited by in corpus (6)

• Gravity as a diffeomorphism invariant gauge theory
• Pure Connection Formulation, Twistors and the Chase for a Twistor Action for General Relativity
• A gauge theoretic approach to Einstein 4-manifolds
• Asymptotically hyperbolic connections
• The Gibbons-Hawking ansatz over a wedge
• Twistor geometry of Riemannian 4-manifolds by moving frames