Conformal superalgebras via tractor calculus

TL;DR

This paper introduces a conformal superalgebra framework using tractor calculus, revealing algebraic conditions affecting its Lie superalgebra structure and providing new methods to generate twistor spinors and conformal Killing forms.

Contribution

It develops a tractor calculus-based approach to conformal superalgebras, clarifies algebraic obstructions to Lie superalgebra structures, and generalizes constructions to higher signatures.

Findings

Identifies algebraic reasons for failure of superalgebra to be a Lie superalgebra

Provides formulas for constructing twistor spinors and conformal Killing forms

Establishes restrictions on the dimension of twistor spinor spaces

Abstract

We use the manifestly conformally invariant description of a Lorentzian conformal structure in terms of a parabolic Cartan geometry in order to introduce a superalgebra structure on the space of twistor spinors and normal conformal vector fields formulated in purely algebraic terms on parallel sections in tractor bundles. Via a fixed metric in the conformal class, one reproduces a conformal superalgebra structure which has been considered in the literature before. The tractor approach, however, makes clear that the failure of this object to be a Lie superalgebra in certain cases is due to purely algebraic identities on the spinor module and to special properties of the conformal holonomy representation. Moreover, it naturally generalizes to higher signatures. This yields new formulas for constructing new twistor spinors and higher order normal conformal Killing forms out of existing ones, generalizing the well-known spinorial Lie derivative. Moreover, we derive restrictions on the possible dimension of the space of twistor spinors in any metric signature.