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The algebraic square of an irreducible complex spinor

Alejandro Gil-García, C. S. Shahbazi

TL;DR

This work provides a complete algebraic framework linking irreducible complex spinors to algebraically constrained exterior forms via two fundamental spinor square maps: the Hermitian square and the complex-bilinear square, realized through the geometric product $\diamond$ of the Kähler–Atiyah algebra. By fixing admissible Hermitian and complex-bilinear pairings, the authors derive explicit necessary-and-sufficient algebraic conditions that characterize the spinor squares in all dimensions and signatures, with distinct treatments for even and odd dimensions. They supply detailed low-dimensional constructions (dimensions $2$–$6$) and extend to eight dimensions, including impure chiral spinors, yielding explicit expressions and compatibility relations that encode underlying $\mathrm{SU}(n)$- and $\mathrm{Spin}(7)$-structure data. The framework further encompasses conjugation relations, equivariance under Spin$^c$ actions, and notions of spinorial instantons and curvings on gerbes, highlighting both the geometric meaning of spinors as “square roots of geometry” and the potential for applications to parallel spinors and higher gauge theory.

Abstract

We characterize, in every dimension and signature, the algebraic squares of an irreducible complex spinor as a pair of exterior forms satisfying a prescribed system of algebraic relations that we present in terms of the geometric product of the underlying quadratic vector space. As a result, we obtain a general correspondence between irreducible complex spinors and algebraically constrained exterior forms, which clarifies the subtle relationship between spinors and exterior forms and contributes towards the understanding of spinors as the square root of geometry. We use this formalism to construct the squares of an irreducible complex spinor in Euclidean dimensions up to six, and also to construct the squares of a generic, possibly non-pure and non-unit, irreducible complex chiral spinor in eight Euclidean dimensions. Elaborating on this result, we consider a natural notion of spinorial instanton that we study for connections on a principal bundle with a complex structure group as well as for curvings of a $\mathbb{C}^{\ast}$-bundle gerbe defined on a Lorentzian six-manifold.

The algebraic square of an irreducible complex spinor

TL;DR

This work provides a complete algebraic framework linking irreducible complex spinors to algebraically constrained exterior forms via two fundamental spinor square maps: the Hermitian square and the complex-bilinear square, realized through the geometric product of the Kähler–Atiyah algebra. By fixing admissible Hermitian and complex-bilinear pairings, the authors derive explicit necessary-and-sufficient algebraic conditions that characterize the spinor squares in all dimensions and signatures, with distinct treatments for even and odd dimensions. They supply detailed low-dimensional constructions (dimensions –) and extend to eight dimensions, including impure chiral spinors, yielding explicit expressions and compatibility relations that encode underlying - and -structure data. The framework further encompasses conjugation relations, equivariance under Spin actions, and notions of spinorial instantons and curvings on gerbes, highlighting both the geometric meaning of spinors as “square roots of geometry” and the potential for applications to parallel spinors and higher gauge theory.

Abstract

We characterize, in every dimension and signature, the algebraic squares of an irreducible complex spinor as a pair of exterior forms satisfying a prescribed system of algebraic relations that we present in terms of the geometric product of the underlying quadratic vector space. As a result, we obtain a general correspondence between irreducible complex spinors and algebraically constrained exterior forms, which clarifies the subtle relationship between spinors and exterior forms and contributes towards the understanding of spinors as the square root of geometry. We use this formalism to construct the squares of an irreducible complex spinor in Euclidean dimensions up to six, and also to construct the squares of a generic, possibly non-pure and non-unit, irreducible complex chiral spinor in eight Euclidean dimensions. Elaborating on this result, we consider a natural notion of spinorial instanton that we study for connections on a principal bundle with a complex structure group as well as for curvings of a -bundle gerbe defined on a Lorentzian six-manifold.
Paper Structure (52 sections, 88 theorems, 390 equations, 9 tables)

This paper contains 52 sections, 88 theorems, 390 equations, 9 tables.

Table of Contents

  1. Introduction
  2. Complex vectors as endomorphisms
  3. Hermitian pairings
  4. Complex-bilinear pairings
  5. The square of a conjugate element
  6. Compatibility of square maps
  7. Algebraically constrained squares
  8. Spinorial forms in even dimensions
  9. The spinor square maps
  10. Hermitian spinorial forms
  11. Equivariance properties of the Hermitian square map
  12. Complex-bilinear spinorial forms
  13. Equivariance properties of the complex-bilinear square map
  14. The square of a complex chiral spinor
  15. The square of a conjugate spinor
  16. ...and 37 more sections

Key Result

Theorem 1.1

A complex exterior form $\alpha\in\wedge V^*_\mathbb{C}$ is the Hermitian square of an irreducible complex spinor in even dimension $d$ if and only if it satisfies the following algebraic system: for an exterior form $\beta\in\wedge V^*_\mathbb{C}$ satisfying $(\alpha\diamond\beta)^{(0)} \neq 0$ and a unit complex number $\kappa\in \mathrm{U}(1)$.

Theorems & Definitions (151)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Definition 2.1
  • Lemma 2.2
  • ...and 141 more