Four Dimensional Integrable Theories

TL;DR

This paper reviews four-dimensional integrable theories, including self-dual gauge and gravity theories, and discusses a harmonic space twistor transform method for constructing explicit solutions.

Contribution

It introduces a harmonic space formulation of the twistor transform applicable to various four-dimensional integrable theories, providing a systematic way to generate explicit solutions.

Findings

Harmonic space analyticity relates to quaternionic analyticity.

The formulation enables explicit connections and metrics.

Applicable to self-dual and full N=3 super Yang-Mills theories.

Abstract

There exist many four dimensional integrable theories. They include self-dual gauge and gravity theories, all their extended supersymmetric generalisations, as well the full (non-self-dual) N=3 super Yang-Mills equations. We review the harmonic space formulation of the twistor transform for these theories which yields a method of producing explicit connections and metrics. This formulation uses the concept of harmonic space analyticity which is closely related to that of quaternionic analyticity. (Talk by V. Ogievetsky at the G\"ursey Memorial Conference I, Istanbul, June 1994)