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Transactions of the American Mathematical Society

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On two-dimensional holonomy

Authors: João Faria Martins and Roger Picken
Journal: Trans. Amer. Math. Soc. 362 (2010), 5657-5695
MSC (2010): Primary 53C29; Secondary 18D05
Published electronically: June 9, 2010
MathSciNet review: 2661492
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Abstract: We define the thin fundamental categorical group $ {\mathcal P}_2(M,*)$ of a based smooth manifold $ (M,*)$ as the categorical group whose objects are rank-1 homotopy classes of based loops on $ M$ and whose morphisms are rank-2 homotopy classes of homotopies between based loops on $ M$. Here two maps are rank-$ n$ homotopic, when the rank of the differential of the homotopy between them equals $ n$. Let $ \mathcal{C}(\mathcal{G})$ be a Lie categorical group coming from a Lie crossed module $ {\mathcal{G}= (\partial\colon E \to G,\triangleright)}$. We construct categorical holonomies, defined to be smooth morphisms $ {\mathcal P}_2(M,*) \to \mathcal{C}(\mathcal{G})$, by using a notion of categorical connections, being a pair $ (\omega,m)$, where $ \omega$ is a connection 1-form on $ P$, a principal $ G$ bundle over $ M$, and $ m$ is a 2-form on $ P$ with values in the Lie algebra of $ E$, with the pair $ (\omega,m)$ satisfying suitable conditions. As a further result, we are able to define Wilson spheres in this context.

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Additional Information

João Faria Martins
Affiliation: Edifício dos Departamentos de Matemática da FCUP, Centro de Matemática da Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal
Address at time of publication: Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2829-516 Caparica, Portugal

Roger Picken
Affiliation: Departamento de Matemática, Instituto Superior Técnico (Universidade Técnica de Lisboa), Av. Rovisco Pais, 1049-001 Lisboa, Portugal

Keywords: Non-abelian gerbe, 2-bundle, two-dimensional holonomy, crossed module, categorical group
Received by editor(s): December 4, 2007
Received by editor(s) in revised form: April 30, 2008
Published electronically: June 9, 2010
Additional Notes: The first author was financed by Fundação para a Ciência e Tecnologia (Portugal), post-doctoral grant number SFRH / BPD / 34138 / 2006. This work was supported by the Programa Operacional Ciência e Inovação 2010, financed by the Fundação para a Ciência e a Tecnologia (FCT) and cofinanced by the European Community fund FEDER, in part through the research project Quantum Topology POCI / MAT / 60352 / 2004
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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