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@article{2423297,
author = {Francois, Jordan Thomas},
article_number = {8},
doi = {http://dx.doi.org/10.1007/JHEP08(2024)040},
keywords = {Classical Theories of Gravity; Differential and Algebraic Geometry; Gauge Symmetry; Space-Time Symmetries},
language = {eng},
issn = {1029-8479},
journal = {Journal of High Energy Physics},
title = {Note on the group of vertical diffeomorphisms of a principal bundle & its relation to the Frölicher-Nijenhuis bracket},
url = {https://link.springer.com/article/10.1007/JHEP08(2024)040},
volume = {2024},
year = {2024}
}
TY - JOUR
ID - 2423297
AU - Francois, Jordan Thomas
PY - 2024
TI - Note on the group of vertical diffeomorphisms of a principal bundle & its relation to the Frölicher-Nijenhuis bracket
JF - Journal of High Energy Physics
VL - 2024
IS - 8
SP - 1-25
EP - 1-25
PB - Springer
SN - 10298479
KW - Classical Theories of Gravity
KW - Differential and Algebraic Geometry
KW - Gauge Symmetry
KW - Space-Time Symmetries
UR - https://link.springer.com/article/10.1007/JHEP08(2024)040
N2 - The group of vertical diffeomorphisms of a principal bundle forms the 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, induced 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 of 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 Frölicher-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 a bundle. Identities relating it to the usual operations of Cartan calculus — inner product, exterior and (Nijenhuis-) Lie derivative — are immediately read as special cases of general results. We also consider 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.
ER -
FRANCOIS, Jordan Thomas. Note on the group of vertical diffeomorphisms of a principal bundle \&{} its relation to the Frölicher-Nijenhuis bracket. \textit{Journal of High Energy Physics}. Springer, 2024, vol.~2024, No~8, p.~1-25. ISSN~1029-8479. Available from: https://dx.doi.org/10.1007/JHEP08(2024)040.
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