Bundles with a lift of infinitesimal diffeomorphisms

TL;DR

This paper extends the concept of natural fibre bundles to include those with infinitesimal diffeomorphism lifts, classifies such bundles assuming finite-dimensional structure groups, and explores implications for spin structures and the topology of space-time.

Contribution

It introduces a new extended notion of natural fibre bundles with infinitesimal diffeomorphism lifts and provides a classification linking gauge groups to space-time topology.

Findings

Spin structures are natural only in the extended sense.

Classified fibre bundles with infinitesimal diffeomorphism lifts assuming finite-dimensional groups.

Connected gauge groups to the topology of space-time.

Abstract

We slightly extend the notion of a natural fibre bundle by requiring diffeomorphisms of the base to lift to automorphisms of the bundle only infinitesimally, i.e. at the level of the Lie algebra of vector fields. Spin structures are natural only in this extended sense. We classify fibre bundles with this property, assuming a finite dimensional structure group. This includes all spin structures, but only some generalised spin structures. This classification links the gauge group G to the topology of space-time.