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

Result code in 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>

Result on the web

<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>

Result language

angličtina

Original language name

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

Original language description

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.

Czech name

—

Czech description

—

Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

—

OECD FORD branch

10100 - Mathematics

Project

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

Continuities

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

Publication year

2024

Confidentiality

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

Name of the periodical

Journal of High Energy Physics

ISSN

1029-8479

e-ISSN

—

Volume of the periodical

2024

Issue of the periodical within the volume

8

Country of publishing house

US - UNITED STATES

Number of pages

25

Pages from-to

1-25

UT code for WoS article

001285311400007

EID of the result in the Scopus database

2-s2.0-85200662490