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

Základní údaje

Originální název

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

Autoři

FRANCOIS, Jordan Thomas (250 Francie, garant, domácí)

Vydání

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

Další údaje

Jazyk

angličtina

Typ výsledku

Článek v odborném periodiku

Obor

10100 1.1 Mathematics

Stát vydavatele

Spojené státy

Utajení

není předmětem státního či obchodního tajemství

Odkazy

URL

Impakt faktor

Impact factor: 5.400 v roce 2022

Organizační jednotka

Přírodovědecká fakulta

DOI

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

UT WoS

001285311400007

Klíčová slova anglicky

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

Štítky

rivok

Příznaky

Mezinárodní význam, Recenzováno

Změněno: 16. 8. 2024 10:07, Mgr. Marie Šípková, DiS.

Anotace

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.

Návaznosti

EH22_010/0003229, projekt VaV

Název: 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, roč. 2024, č. 8, s. 1-25. ISSN 1029-8479. Dostupné z: https://dx.doi.org/10.1007/JHEP08(2024)040.

@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}
}

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AU  - Francois, Jordan Thomas
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JF  - Journal of High Energy Physics
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KW  - Classical Theories of Gravity
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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, roč.~2024, č.~8, s.~1-25. ISSN~1029-8479. Dostupné z: https://dx.doi.org/10.1007/JHEP08(2024)040.

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