Connections on central extensions, lifting gerbes, and finite-dimensional obstruction vanishing

Indranil Biswas; Markus Upmeier

doi:10.1090/conm/766

Connections on central extensions, lifting gerbes, and finite-dimensional obstruction vanishing

Indranil Biswas, Markus Upmeier

Mathematical Science

Tata Institute of Fundamental Research

Research output: Chapter in Book/Report/Conference proceeding › Published conference contribution

1 Citation (Scopus)

Abstract

Given a central extension of Lie groups, we study the classification problem of lifting the structure group together with a given connection. For reductive structure groups we introduce a connective structure on the lifting gerbe associated to this problem. Our main result classifies all connections on the central extension of a given principal bundle. In particular, we find that admissible connections are in one-to-one correspondence with parallel trivializations of the lifting gerbe. Moreover, we prove a vanishing result for Neeb’s obstruction classes for finite dimensional Lie groups.

Original language	English
Title of host publication	Geometry at the Frontier
Subtitle of host publication	Symmetries and Moduli Spaces of Algebraic Varieties
Editors	Paola Comparin, Eduardo Esteves, Sebastián Reyes-Carocca, Rubí E. Rodríguez, Herbert Lange
Publisher	American Mathematical Society
Pages	39-56
Volume	766
ISBN (Electronic)	978-1-4704-6422-6
ISBN (Print)	978-1-4704-5327-5
DOIs	https://doi.org/10.1090/conm/766
Publication status	Published - 30 Jul 2021
Event	Geometry at the Frontier III - Universidad de La Frontera, Pucón, Chile Duration: 12 Nov 2018 → 16 Nov 2018

Publication series

Name	Contemporary Mathematics
Publisher	American Mathematical Society
ISSN (Print)	0271-4132
ISSN (Electronic)	1098-3627

Conference

Conference	Geometry at the Frontier III
Country/Territory	Chile
City	Pucón
Period	12/11/18 → 16/11/18

Bibliographical note

The first author is partially supported by a J. C. Bose Fellowship.
The second author thanks the TIFR, Mumbai, for hospitality during 03/2017 and was partially funded by DFG grant UP 85/2-1 of the priority program SPP 2026 Geometry at Infinity.

Keywords

connections
gerbes
central extensions

Access to Document

10.1090/conm/766Licence: Unspecified

https://arxiv.org/abs/1910.10402Licence: Unspecified

Cite this

Biswas, I., & Upmeier, M. (2021). Connections on central extensions, lifting gerbes, and finite-dimensional obstruction vanishing. In P. Comparin, E. Esteves, S. Reyes-Carocca, R. E. Rodríguez, & H. Lange (Eds.), Geometry at the Frontier: Symmetries and Moduli Spaces of Algebraic Varieties (Vol. 766, pp. 39-56). (Contemporary Mathematics). American Mathematical Society. https://doi.org/10.1090/conm/766

Connections on central extensions, lifting gerbes, and finite-dimensional obstruction vanishing. / Biswas, Indranil; Upmeier, Markus.
Geometry at the Frontier: Symmetries and Moduli Spaces of Algebraic Varieties. ed. / Paola Comparin; Eduardo Esteves; Sebastián Reyes-Carocca; Rubí E. Rodríguez; Herbert Lange. Vol. 766 American Mathematical Society, 2021. p. 39-56 (Contemporary Mathematics).

Research output: Chapter in Book/Report/Conference proceeding › Published conference contribution

Biswas, I & Upmeier, M 2021, Connections on central extensions, lifting gerbes, and finite-dimensional obstruction vanishing. in P Comparin, E Esteves, S Reyes-Carocca, RE Rodríguez & H Lange (eds), Geometry at the Frontier: Symmetries and Moduli Spaces of Algebraic Varieties. vol. 766, Contemporary Mathematics, American Mathematical Society, pp. 39-56, Geometry at the Frontier III, Pucón, Chile, 12/11/18. https://doi.org/10.1090/conm/766

Biswas I, Upmeier M. Connections on central extensions, lifting gerbes, and finite-dimensional obstruction vanishing. In Comparin P, Esteves E, Reyes-Carocca S, Rodríguez RE, Lange H, editors, Geometry at the Frontier: Symmetries and Moduli Spaces of Algebraic Varieties. Vol. 766. American Mathematical Society. 2021. p. 39-56. (Contemporary Mathematics). Epub 2021 Jul 30. doi: 10.1090/conm/766

Biswas, Indranil ; Upmeier, Markus. / Connections on central extensions, lifting gerbes, and finite-dimensional obstruction vanishing. Geometry at the Frontier: Symmetries and Moduli Spaces of Algebraic Varieties. editor / Paola Comparin ; Eduardo Esteves ; Sebastián Reyes-Carocca ; Rubí E. Rodríguez ; Herbert Lange. Vol. 766 American Mathematical Society, 2021. pp. 39-56 (Contemporary Mathematics).

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abstract = " Given a central extension of Lie groups, we study the classification problem of lifting the structure group together with a given connection. For reductive structure groups we introduce a connective structure on the lifting gerbe associated to this problem. Our main result classifies all connections on the central extension of a given principal bundle. In particular, we find that admissible connections are in one-to-one correspondence with parallel trivializations of the lifting gerbe. Moreover, we prove a vanishing result for Neeb{\textquoteright}s obstruction classes for finite dimensional Lie groups.",

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N2 - Given a central extension of Lie groups, we study the classification problem of lifting the structure group together with a given connection. For reductive structure groups we introduce a connective structure on the lifting gerbe associated to this problem. Our main result classifies all connections on the central extension of a given principal bundle. In particular, we find that admissible connections are in one-to-one correspondence with parallel trivializations of the lifting gerbe. Moreover, we prove a vanishing result for Neeb’s obstruction classes for finite dimensional Lie groups.

AB - Given a central extension of Lie groups, we study the classification problem of lifting the structure group together with a given connection. For reductive structure groups we introduce a connective structure on the lifting gerbe associated to this problem. Our main result classifies all connections on the central extension of a given principal bundle. In particular, we find that admissible connections are in one-to-one correspondence with parallel trivializations of the lifting gerbe. Moreover, we prove a vanishing result for Neeb’s obstruction classes for finite dimensional Lie groups.

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KW - gerbes

KW - central extensions

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BT - Geometry at the Frontier

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A2 - Esteves, Eduardo

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A2 - Rodríguez, Rubí E.

A2 - Lange, Herbert

PB - American Mathematical Society

T2 - Geometry at the Frontier III

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ER -

Connections on central extensions, lifting gerbes, and finite-dimensional obstruction vanishing

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