Research PaperNo Access

Connections on Lie groupoids and Chern–Weil theory

Department of Mathematics, Shiv Nadar University, NH91 Tehsil Dadri, Greater Noida 201314, Uttar Pradesh, India

E-mail Address: indranil.biswas@snu.edu.in

School of Mathematics, Indian Institute of Science Education and Research Thiruvananthapuram, Maruthamala P. O., Vithura 695551, Kerala, India

E-mail Address: saikat.chat01@gmail.com

Corresponding author.

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Praphulla Koushik

School of Mathematics, Indian Institute of Science Education and Research Pune, Dr. Homi Bhabha Road, Pashan, Pune 411008, India

E-mail Address: praphullakoushik16@gmail.com

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, and

Frank Neumann

https://orcid.org/0000-0001-9684-2488

Dipartimento di Matematica ‘Felice Casorati’, Università di Pavia, Via Ferrata, 5, 27100 Pavia, Italy

E-mail Address: frank.neumann@unipv.it

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https://doi.org/10.1142/S0129055X24500028Cited by:0 (Source: Crossref)

Abstract

Let $𝕏 = [X_{1} ⇉ X_{0}]$ be a Lie groupoid equipped with a connection, given by a smooth distribution $ℋ \subset T X_{1}$ transversal to the fibers of the source map. Under the assumption that the distribution $ℋ$ is integrable, we define a version of de Rham cohomology for the pair $(𝕏, ℋ)$ , and we study connections on principal $G$ -bundles over $(𝕏, ℋ)$ in terms of the associated Atiyah sequence of vector bundles. We also discuss associated constructions for differentiable stacks. Finally, we develop the corresponding Chern–Weil theory and describe characteristic classes of principal $G$ -bundles over a pair $(𝕏, ℋ)$ .

Keywords:

AMSC: 53C08, 22A22, 58H05, 53D50

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