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Foundational aspects of spinor structures and exotic spinors

J. M. Hoff da Silva

TL;DR

This work surveys the foundational link between spacetime topology and spinor fields, detailing when spinor structures exist and how multiple nonequivalent structures arise as exotic spinors. It develops a rigorous obstruction-theoretic framework using lifts to a $\Gamma$-structure and Čech cohomology, connecting existence to the vanishing of the obstruction class $[p]\in \check{H}^2(\mathcal{M},D)$ and relating it to the second Stiefel-Whitney class $w_2$. The review then analyzes diffeomorphism actions on spinors, showing scalar-like transformation under lifts when $\check{H}^1(\mathcal{M},\mathbb{Z}_2)$ is trivial, and demonstrates how nontrivial topology permutes spin structures, yielding exotic spinors. It culminates in the exotic spinor Dirac operator, which includes a topological term $-\frac{1}{2}\gamma^\mu\partial_\mu\theta(x)$ that induces Lorentz-violating dispersion and modified currents, and it surveys historical and contemporary developments, including potential physical applications in cosmology and condensed matter.

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.

Foundational aspects of spinor structures and exotic spinors

TL;DR

This work surveys the foundational link between spacetime topology and spinor fields, detailing when spinor structures exist and how multiple nonequivalent structures arise as exotic spinors. It develops a rigorous obstruction-theoretic framework using lifts to a -structure and Čech cohomology, connecting existence to the vanishing of the obstruction class and relating it to the second Stiefel-Whitney class . The review then analyzes diffeomorphism actions on spinors, showing scalar-like transformation under lifts when is trivial, and demonstrates how nontrivial topology permutes spin structures, yielding exotic spinors. It culminates in the exotic spinor Dirac operator, which includes a topological term that induces Lorentz-violating dispersion and modified currents, and it surveys historical and contemporary developments, including potential physical applications in cosmology and condensed matter.

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.

Paper Structure

This paper contains 11 sections, 15 theorems, 86 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Foundations - Part I: General mathematical structure
  3. Existence of spinor structures
  4. (Non)uniqueness of spinor structures
  5. Foundations - Part II: Diffeomorphisms
  6. Spinors as 'scalars' under diffeomorphisms
  7. Obstructions to the trivial behavior of spinors under diffeomorphisms
  8. Exotic spinors
  9. Overview
  10. Early advances
  11. More recent achievements

Key Result

Lemma 2.1

$P_G$ admits a $\Gamma-$structure if, and only if, there are continuous maps $\gamma_{\alpha\beta}: U_\alpha\cap U_\beta\rightarrow \Gamma$ such that i) $\gamma_{\alpha\beta}(x)\gamma_{\beta\mu}(x)=\gamma_{\alpha\mu}(x)$ for $x\in U_\alpha\cap U_\beta\cap U_\mu$ and ii) $\rho\cdot\gamma_{\alpha\beta

Figures (1)

  • Figure 1: Discontinuity of $\theta_\alpha(x)$ as $x$ cross the hypersurface $\Sigma\subset U_\alpha\cap U_\beta$.

Theorems & Definitions (31)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 1
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.1
  • proof
  • ...and 21 more