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ON 2-HOLONOMY

Part of: Fiber spaces and bundles Global differential geometry Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  04 September 2019

HOSSEIN ABBASPOUR  and
FRIEDRICH WAGEMANN
HOSSEIN ABBASPOUR*
Affiliation:
Université de Nantes, 2, rue de la Houssiniere, Nantes44322, France
FRIEDRICH WAGEMANN
Affiliation:
Université de Nantes, 2, rue de la Houssiniere, Nantes44322, France e-mail: wagemann@math.univ-nantes.fr
*
e-mail: hossein.abbaspour@univ-nantes.fr
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Abstract

We construct a cycle in higher Hochschild homology associated to the two-dimensional torus which represents 2-holonomy of a nonabelian gerbe in the same way as the ordinary holonomy of a principal G-bundle gives rise to a cycle in ordinary Hochschild homology. This is done using the connection 1-form of Baez–Schreiber. A crucial ingredient in our work is the possibility to arrange that in the structure crossed module $\unicode[STIX]{x1D707}:\mathfrak{h}\rightarrow \mathfrak{g}$ of the principal 2-bundle, the Lie algebra $\mathfrak{h}$ is abelian, up to equivalence of crossed modules.

Keywords

a holonomy of a principal 2-bundlehigher Hochschild homologycrossed modules of Lie algebrasconnection 1-form on loop space

MSC classification

Primary: 53C08: Gerbes, differential characters
Secondary: 17B55: Homological methods in Lie (super)algebras 53C05: Connections, general theory 53C29: Issues of holonomy 55R40: Homology of classifying spaces, characteristic classes
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 110 , Issue 2 , April 2021 , pp. 145 - 174
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Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by M. Murray

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