From 2D conformal to 4D self-dual theories: quaternionic analyticity

TL;DR

This paper explores how self-dual theories in four dimensions extend the concepts of conformal and analytic structures from two-dimensional conformal field theories, using quaternionic analyticity and harmonic space methods.

Contribution

It introduces quaternionic analyticity as a natural extension of complex analyticity to four dimensions within the framework of self-dual theories.

Findings

Extension of conformal transformations to $CP^3$

Visualization of twistor correspondence

Formulation of Fueter analyticity

Abstract

It is shown that self-dual theories generalize to four dimensions both the conformal and analytic aspects of two-dimensional conformal field theories. In the harmonic space language there appear several ways to extend complex analyticity (natural in two dimensions) to quaternionic analyticity (natural in four dimensions). To be analytic, conformal transformations should be realized on $CP^3$, which appears as the coset of the complexified conformal group modulo its maximal parabolic subgroup. In this language one visualizes the twistor correspondence of Penrose and Ward and consistently formulates the analyticity of Fueter.