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

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14310%2F24%3A00136827" target="_blank" >RIV/00216224:14310/24:00136827 - isvavai.cz</a>

Výsledek na webu

<a href="https://link.springer.com/article/10.1007/JHEP08(2024)040" target="_blank" >https://link.springer.com/article/10.1007/JHEP08(2024)040</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/JHEP08(2024)040" target="_blank" >10.1007/JHEP08(2024)040</a>

Jazyk výsledku

angličtina

Název v původním jazyce

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

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

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

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10100 - Mathematics

Projekt

<a href="/cs/project/EH22_010%2F0003229" target="_blank" >EH22_010/0003229: MSCAfellow5_MUNI</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2024

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Journal of High Energy Physics

ISSN

1029-8479

e-ISSN

—

Svazek periodika

2024

Číslo periodika v rámci svazku

8

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

25

Strana od-do

1-25

Kód UT WoS článku

001285311400007

EID výsledku v databázi Scopus

2-s2.0-85200662490