Twistors in Conformally Flat Einstein Four-Manifolds

TL;DR

This paper explores the supertwistor equations and gauge invariance of massive spin-3/2 potentials in conformally flat Einstein four-manifolds, with implications for quantum cosmology and supergravity.

Contribution

It introduces a supercovariant derivative framework and analyzes gauge transformations and boundary conditions for spin-3/2 fields in this geometric setting.

Findings

Supercovariant derivative operator necessary due to cosmological constant

Gauge freedom generated by solutions of supertwistor equations

Boundary conditions ensuring gauge invariance in supergravity

Abstract

This paper studies the two-component spinor form of massive spin-3/2 potentials in conformally flat Einstein four-manifolds. Following earlier work in the literature, a non-vanishing cosmological constant makes it necessary to introduce a supercovariant derivative operator. The analysis of supergauge transformations of primary and secondary potentials for spin 3/2 shows that the gauge freedom for massive spin-3/2 potentials is generated by solutions of the supertwistor equations. The supercovariant form of a partial connection on a non-linear bundle is then obtained, and the basic equation of massive secondary potentials is shown to be the integrability condition on super beta-surfaces of a differential operator on a vector bundle of rank three. Moreover, in the presence of boundaries, a simple algebraic relation among some spinor fields is found to ensure the gauge invariance of locally supersymmetric boundary conditions relevant for quantum cosmology and supergravity.