Bundle 2-Gerbes

• Autores: Daniel Stevenson
• Localización: Proceedings of the London Mathematical Society, ISSN 0024-6115, Vol. 88, Nº 2, 2004, págs. 405-435
• Idioma: inglés
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• Resumen

  • We make the category $\textbf{BGrb}_M$ of bundle gerbes on a manifold $M$ into a $2$-category by providing $2$-cells in the form of transformations of bundle gerbe morphisms. This description of $\textbf{BGrb}_M$ as a $2$-category is used to define the notion of a bundle $2$-gerbe. To every bundle $2$-gerbe on $M$ is associated a class in $H^4(M ; \mathbb{Z})$. We define the notion of a bundle $2$-gerbe connection and show how this leads to a closed, integral, differential $4$-form on $M$ which represents the image in real cohomology of the class in $H^4(M ; \mathbb{Z})$. Some examples of bundle $2$-gerbes are discussed, including the bundle $2$-gerbe associated to a principal $G$ bundle $P \to M$. It is shown that the class in $H^4(M ; \mathbb{Z})$ associated to this bundle $2$-gerbe coincides with the first Pontryagin class of $P$: this example was previously considered from the point of view of $2$-gerbes by Brylinski and McLaughlin.