Causal and self-dual morphisms in four complex dimensions

TL;DR

This paper introduces causal and self-dual morphisms in four complex dimensions, generalizing conformal transformations and flat alpha-planes, with numerous non-trivial examples demonstrating their existence.

Contribution

It defines new classes of maps called causal and self-dual morphisms in four complex dimensions, extending known geometric transformations.

Findings

Existence of infinite non-trivial examples of causal morphisms in four dimensions.

Existence of infinite non-trivial examples of self-dual morphisms in four dimensions.

Generalization of conformal transformations and flat alpha-planes to higher dimensions.

Abstract

We define a class of maps between holomorphically embedded null curves which generalize conformal transformations, and can be defined in any complex dimension. In four dimensions, we can also define a similar map between self-dual surfaces, which generalize flat $\alpha$-planes. These maps are respectively called causal and self-dual morphisms. It is shown that there exist an infinite class of non-trivial examples for both types of maps in four dimensions.