A Deformation Theory of Self-Dual Einstein Spaces

TL;DR

This paper develops a deformation theory for self-dual Einstein spaces, analyzing the local structure of their moduli space via elliptic complexes and revealing conditions for discreteness and dimensionality based on the cosmological constant.

Contribution

It introduces a new elliptic complex framework to study the moduli space of self-dual Einstein connections and characterizes its properties depending on the cosmological constant.

Findings

Moduli space is discrete for positive cosmological constant.

Moduli space can be a manifold with dimension given by the Atiyah-Singer index theorem for negative cosmological constant.

Linearization leads to an elliptic complex governing local properties.

Abstract

The self-dual Einstein equations on a compact Riemannian 4-manifold can be expressed as a quadratic condition on the curvature of an $SU(2)$ (spin) connection which is a covariant generalization of the self-dual Yang-Mills equations. Local properties of the moduli space of self-dual Einstein connections are described in the context of an elliptic complex which arises in the linearization of the quadratic equations on the $SU(2)$ curvature. In particular, it is shown that the moduli space is discrete when the cosmological constant is positive; when the cosmological constant is negative the moduli space can be a manifold the dimension of which is controlled by the Atiyah-Singer index theorem.