Spinors of real type as polyforms and the generalized Killing equation

TL;DR

This paper introduces a new framework for analyzing generalized Killing spinors using polyforms and algebraic varieties, providing novel characterizations and applications to Lorentzian four-manifolds and supergravity.

Contribution

It develops a detailed algebraic and geometric framework for real type spinors as polyforms, with new characterizations and applications to Lorentzian geometry and supergravity configurations.

Findings

New description of real spinors via parabolic 2-planes

Global characterizations of Killing spinors on Lorentzian manifolds

Discovery of new Einstein and non-Einstein metrics with Killing spinors

Abstract

We develop a new framework for the study of generalized Killing spinors, where generalized Killing spinor equations, possibly with constraints, can be formulated equivalently as systems of partial differential equations for a polyform satisfying algebraic relations in the K\"ahler-Atiyah bundle constructed by quantizing the exterior algebra bundle of the underlying manifold. At the core of this framework lies the characterization, which we develop in detail, of the image of the spinor squaring map of an irreducible Clifford module $\Sigma$ of real type as a real algebraic variety in the K\"ahler-Atiyah algebra, which gives necessary and sufficient conditions for a polyform to be the square of a real spinor. We apply these results to Lorentzian four-manifolds, obtaining a new description of a real spinor on such a manifold through a certain distribution of parabolic 2-planes in its cotangent bundle. We use this result to give global characterizations of real Killing spinors on Lorentzian four-manifolds and of four-dimensional supersymmetric configurations of heterotic supergravity. In particular, we find new families of Einstein and non-Einstein four-dimensional Lorentzian metrics admitting real Killing spinors, some of which are deformations of the metric of AdS$_4$ space-time.