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Tools for Supergravity in the spin coframe formalism

Filippo Fila-Robattino

TL;DR

This work delivers a comprehensive toolkit for spinors in the spin coframe formalism, combining Clifford-algebra foundations, Spin/Pin groups, and spinor bundles with a differential-geometric treatment of spin structures. It provides a self-contained framework that expresses objects and identities in index-free notation, including gamma-matrix relations, Majorana flip and Fierz rearrangements, and coframe lemmata, all tailored for supergravity calculations in Lorentzian spacetimes. By linking spin structures to spin(co)frames and defining Dirac (and Majorana) spinor bundles, the paper enables metric-independent descriptions of spinor fields and clarifies the role of the metric as a dynamical object in gravity theories. The resulting toolkit and explicit D=4 results offer a practical repository for future supergravity work, with broad applicability across dimensions and signatures.

Abstract

This paper contains a review of the theoretical foundations of Clifford algebras, spinors and spinor bundles in the so-called co-frame formalism. A compact index-free notation is introduced, along with a series of identities useful for computations in supergravity theories.

Tools for Supergravity in the spin coframe formalism

TL;DR

This work delivers a comprehensive toolkit for spinors in the spin coframe formalism, combining Clifford-algebra foundations, Spin/Pin groups, and spinor bundles with a differential-geometric treatment of spin structures. It provides a self-contained framework that expresses objects and identities in index-free notation, including gamma-matrix relations, Majorana flip and Fierz rearrangements, and coframe lemmata, all tailored for supergravity calculations in Lorentzian spacetimes. By linking spin structures to spin(co)frames and defining Dirac (and Majorana) spinor bundles, the paper enables metric-independent descriptions of spinor fields and clarifies the role of the metric as a dynamical object in gravity theories. The resulting toolkit and explicit D=4 results offer a practical repository for future supergravity work, with broad applicability across dimensions and signatures.

Abstract

This paper contains a review of the theoretical foundations of Clifford algebras, spinors and spinor bundles in the so-called co-frame formalism. A compact index-free notation is introduced, along with a series of identities useful for computations in supergravity theories.

Paper Structure

This paper contains 19 sections, 19 theorems, 129 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Clifford algebras and spin groups
  3. Clifford algebras
  4. Classification of Clifford Algebras
  5. Pin and Spin groups
  6. Lie Algebra of Spin group
  7. Representations
  8. Complex representations and the Lorentzian signature case
  9. Charge conjugation and Majorana spinors in the Lorentzian signature
  10. Spin coframe formalism, i.e. defining spinor fields on manifolds
  11. Basic notions on principal bundles
  12. Spin structures and the equivalence with spin (co)frames
  13. Tools and identities
  14. Basic Identities on gamma matrices
  15. Identities on Majorana spinors
  16. ...and 4 more sections

Key Result

Proposition 3

The Clifford algebra of $V$ is unique up to isomorphisms.

Theorems & Definitions (66)

  • Definition 1: Clifford map
  • Definition 2: Clifford algebra
  • Proposition 3
  • Proposition 4
  • proof
  • Remark 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • ...and 56 more