Explicit Families of Spinor Representations

TL;DR

This paper presents explicit constructions of spinor representations for real Clifford algebras across all signatures, including methods for parallel transport and connections to spectral properties of Dirac operators.

Contribution

It introduces a comprehensive recipe for explicit spinor representations in all dimensions and signatures, extending to mixed signatures and providing tools for spinor computations.

Findings

Explicit formulas for spinor representations in all signatures

Construction of spin coordinate systems and parallel transport methods

Connections between spinors, Dirac spectrum, and Hodge operators

Abstract

We provide a recipe for building explicit representations of the real Clifford algebras once an explicit family is given in dimensions $1$ through $4$. We further give an explicit construction of spin coordinate systems for a given real spinor module and use it to explicitly compute the parallel transport of spinor fields. We further highlight some novelties such as the relationship with the spectrum of the spinor Dirac operator and the Hodge de Rham operator when a parallel spinor field exists and a brief discussion of spinors along a hypersurface in $\bR^4$. Lastly, we extend our construction to arbitrary signature quadratic forms thus providing a complete and explicit family of spinor representations for all mixed signature Clifford algberas. We show that in all cases the spinor representations can be expressed as tensor products of multi-vectors over the fields $\bR$, $\bC$, and $\bH$.