Geometric tool kit for higher spin gravity (part I): An introduction to the geometry of differential operators

TL;DR

This paper introduces the algebra and geometry of differential operators and jet bundles, emphasizing their relevance to higher-spin gravity theories and providing rigorous, coordinate-free mathematical foundations.

Contribution

It offers a comprehensive introduction to the geometry of differential operators tailored for higher-spin gravity, highlighting higher-order generalizations of Lie derivatives and diffeomorphisms.

Findings

Framework for higher-order differential operators

Connections to higher-spin gravity theories

Rigorous geometric definitions of differential operators

Abstract

These notes provide an introduction to the algebra and geometry of differential operators and jet bundles. Their point of view is guided by the leitmotiv that higher-spin gravity theories call for higher-order generalisations of Lie derivatives and diffeomorphisms. Nevertheless, the material covered here may be of general interest to anyone working on topics where geometrical (coordinate-free, global, generic) and mathematically rigorous definitions of differential operators are required.