Immersion in Rn by Complex irreducible Spinors

TL;DR

This paper develops a spinorial representation for SpinC manifolds in Euclidean space using irreducible complex Clifford algebra spinors, extending previous spinor-based surface immersion theories beyond Spin structures.

Contribution

It introduces a novel spinorial representation for SpinC manifolds employing irreducible complex Clifford algebra spinors, broadening the scope of spinor methods in differential geometry.

Findings

Provides a new representation for SpinC manifolds in Euclidean space.

Utilizes irreducible complex Clifford algebra spinors for the representation.

Extends spinor-based immersion techniques to more general manifolds.

Abstract

The first time that the connection between isometric immersion of surfaces and solutions of the Dirac equation appeared in the literature was in the seminal paper of Thomas Friedrich in 1998. In consequence of that, several authors contributed to this topic hereafter, by obtaining the spinorial representation of Spin manifolds with arbitrary dimension and also by presenting a generalization of the Weierstrass representation map, for example. All these results assume that the manifolds and bundles involved carry a Spin structure, however this hypothesis is somehow restrictive, as for instance, if we consider complex manifolds, it is more natural to consider SpinC structures. There exist an alternative to adapt this result to SpinC manifolds, where the idea is to use the left regular representation of a Clifford algebra in itself to build the spinor bundles, but unlike the original works of Friedrich and Morel, this representation is not irreducible. Thus, this paper aims to present the spinorial representation of SpinC manifolds into Euclidian space with arbitrary dimensions using spinors that came from an irreducible representation of a complex Clifford algebra.