Integrable Field Theories derived from 4D Self-dual Gravity

TL;DR

This paper reformulates the self-dual Einstein equations using differential forms, revealing their equivalence to 2D sigma models and deriving related field theories, thereby clarifying their interrelations.

Contribution

It introduces a new differential form formulation of self-dual gravity that unifies various related 2D field theories and clarifies their connections.

Findings

Equivalence of self-dual Einstein equations and 2D sigma models

Derivation of 2D Higgs bundle equation from the formulation

Unified view of related 2D field theories

Abstract

We reformulate the self-dual Einstein equation as a trio of differential form equations for simple two-forms. Using them, we can quickly show the equivalence of the theory and 2D sigma models valued in an infinite-dimensional group, which was shown by Park and Husain earlier. We also derive other field theories including the 2D Higgs bundle equation. This formulation elucidates the relation among those field theories.