Conformal geometry of the supercotangent and spinor bundles

TL;DR

This paper explores the conformal geometry of supercotangent and spinor bundles through geometric quantization, establishing new links between conformal structures and spinor analysis, including classification of invariants in flat cases.

Contribution

It introduces a geometric quantization approach to relate conformal geometry of supercotangent bundles with spinor bundles, revealing new invariants and module structures.

Findings

Quantization of the comoment map yields the Kosmann Lie derivative of spinors.

Classified conformal invariants including odd powers of the Dirac operator.

Established a correspondence between conformal geometry and spinor module structures.

Abstract

We establish, via geometric quantization of the supercotangent bundle sM of (M,g), a correspondence between its conformal geometry and those of the spinor bundle. In particular, the Kosmann Lie derivative of spinors is obtained by quantization of the comoment map, associated to the new Hamiltonian action of conf(M,g) on sM. We study then the conf(M,g)-module structures induced on the space of differential operators acting on spinor densities and on its spaces of symbols (functions on sM). In the conformally flat case, we classify their conformal invariants, including the conformally odd powers of the Dirac operator.