Notes on Spinors and Polyforms I: General Case

TL;DR

This paper generalizes the geometric description of spinors and polyforms in Clifford algebras to arbitrary signatures, introducing mixed structures that combine complex and paracomplex geometries, and explores multiple models for these algebras.

Contribution

It introduces the concept of mixed structures on R^{r,s} and relates them to creation/annihilation models and pure spinors, extending previous descriptions to general signatures.

Findings

Multiple creation/annihilation models exist for a given Clifford algebra.

Mixed structures unify complex and paracomplex geometries in spinor models.

Explicit models are described for Cl(r,s) with r+s <= 6.

Abstract

It is well-known that the Clifford algebra Cl(2n) can be given a description in terms of creation/annihilation operators acting in the space of inhomogeneous differential forms on C^n. We refer to such inhomogeneous differential forms as polyforms. The construction proceeds by choosing a complex structure J on R^(2n). Spinors are then polyforms on one of the two totally-isotropic subspaces C^n that arise as eigenspaces of J. There is a similar description in the split signature case Cl(n,n), with differential forms now being those on R^n. In this case the model is constructed by choosing a paracomplex structure I on R^(n,n), and spinors are polyforms on one of the totally null eigenspaces R^n of I. The main purpose of the paper is to describe the geometry of an analogous construction in the case of a general Clifford algebra Cl(r,s), r+s=2m. We show that in general a creation/annihilation operator model is in correspondence with a new type of geometric structure on R^(r,s), which provides a splitting R^(r,s)=R^(2k,2l) plus R^(n,n) and endows the first factor with a complex structure and the second factor with a paracomplex structure. We refer to such geometric structure as a mixed structure. It can be described as a complex linear combination K=I+i J of a paracomplex and a complex structure such that K^2=Id and K K^* is a product structure. In turn, the mixed structure is in correspondence with a pair of pure spinors whose null subspaces are the eigenspaces of K. The conclusion is then that there is in general not one, but several possible creation/annihilation operator models for a given Clifford algebra. The number of models is the number of different types of pure spinors (distinguished by the real index, see the main text) that exists in a given signature. To illustrate this geometry, we explicitly describe all the arising models for Cl(r,s) with r >= s, r+s = 2m <= 6.