Note on the group of vertical diffeomorphisms of a principal
bundle & its relation to the Frölicher-Nijenhuis bracket

J 2024

Note on the group of vertical diffeomorphisms of a principal bundle & its relation to the Frölicher-Nijenhuis bracket

FRANCOIS, Jordan Thomas

Basic information

Original name

Note on the group of vertical diffeomorphisms of a principal bundle & its relation to the Frölicher-Nijenhuis bracket

Authors

FRANCOIS, Jordan Thomas (250 France, guarantor, belonging to the institution)

Edition

Journal of High Energy Physics, Springer, 2024, 1029-8479

Other information

Language

English

Type of outcome

Článek v odborném periodiku

Field of Study

10100 1.1 Mathematics

Country of publisher

United States of America

Confidentiality degree

není předmětem státního či obchodního tajemství

References:

URL

Impact factor

Impact factor: 5.400 in 2022

Organization unit

Faculty of Science

DOI

http://dx.doi.org/10.1007/JHEP08(2024)040

UT WoS

001285311400007

Keywords in English

Classical Theories of Gravity; Differential and Algebraic Geometry; Gauge Symmetry; Space-Time Symmetries

Abstract

V originále

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.

Links

EH22_010/0003229, research and development project

Name: MSCAfellow5_MUNI

Citovat

FRANCOIS, Jordan Thomas. Note on the group of vertical diffeomorphisms of a principal bundle & its relation to the Frölicher-Nijenhuis bracket. 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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   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}
}

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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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