Note on the group of vertical diffeomorphisms of a principal bundle, and its relation to the Fr\"olicher-Nijenhuis bracket

TL;DR

This paper explores the structure of vertical diffeomorphisms of principal bundles, linking the extended gauge bracket to the Fr"olicher-Nijenhuis bracket, and discusses implications for gauge transformations and symmetries in physics.

Contribution

It reveals that the extended gauge bracket can be derived from the Fr"olicher-Nijenhuis bracket, integrating it into the framework of derivations of forms on principal bundles.

Findings

The extended bracket arises from the Fr"olicher-Nijenhuis bracket of vector-valued forms.

Identities involving the extended bracket relate to inner product, exterior, and Lie derivatives.

Vertical diffeomorphisms induce generalized gauge transformations with unique features.

Abstract

The group of vertical diffeomorphisms of a principal bundle forms the generalised action Lie groupoid associated to the bundle. The former is generated by the group of maps with value in the structure group, which is also the group of bisections of the groupoid. The corresponding Lie algebra of general vertical vector fields is generated by maps with value in the Lie algebra of the structure group. The bracket on these maps, induces by the bracket of vertical vector fields, is an ``extended" bracket on gauge parameters: it has been introduced heuristically in physics, notably in the study asymptotic symmetries of gravity. Seeing the set of Lie algebra-valued maps as sections of the action Lie algebroid associated to the bundle, the extended bracket is understood to be a Lie algebroid bracket on those sections. Here, we highlight that this bracket can also be seen to arise from the Fr\"olicher-Nijenhuis bracket of vector-valued differential forms. The benefit of this viewpoint is to insert this extended bracket within the general framework of derivations of forms on the bundle. Identities relating it to usual operations -- inner product, exterior and (Nijenhuis-) Lie derivative -- are immediately read as special cases of general results. We also look at the generalised gauge transformations induced by vertical diffeomorphisms, and discuss their peculiar features. In particular, locally, and contrary to standard gauge transformations arising from vertical bundle automorphisms, they are distinguishable from local gluings when iterated. Yet, the gauge principle still holds.