Foundational aspects of spinor structures and exotic spinors

TL;DR

This paper reviews the topological conditions for the existence and uniqueness of spinor structures in spacetime, explores the properties of exotic spinors, and discusses their physical implications and recent developments.

Contribution

It provides a comprehensive analysis of the topological criteria for spinor structures and introduces the topologically corrected Dirac operator with physical insights.

Findings

Topological conditions determine spinor structure existence and multiplicity.

Explicit form of the topologically corrected Dirac operator is derived.

Exotic spinors have significant physical consequences.

Abstract

Spinors are mathematical objects susceptible to the spacetime characteristics upon which they are defined. Not all spacetimes admit spinor structure; when it does, it may have more than one spinor structure, depending on topological properties. When more than one nonequivalent spinor structure is allowed in a given spacetime, the spinors resulting from the extra structures are called exotic. In this review, we revisit the topological conditions driving the discussion about the spacetime characteristics leading to the existence and (non)uniqueness of spinor structures in a relatively comprehensive manner, accounting for step-to-step demonstrations. In the sequel, we delve into the topologically corrected Dirac operator, explicitly obtaining it and exploring the physical consequences encoded in the exotic spinor dynamics. Finally, we overview early and recent achievements in the area, pointing out possible directions within this research program.