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Register a new userParallel transport in principal 2-bundles
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Authors
Konrad Waldorf
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A nice differential-geometric framework for (non-abelian) higher gauge theory is provided by principal 2-bundles, i.e. categorified principal bundles. Their total spaces are Lie groupoids, local trivializations are kinds of Morita equivalences, and connections are Lie-2-algebra-valued 1-forms. In this article, we construct explicitly the parallel transport of a connection on a principal 2-bundle. Parallel transport along a path is a Morita equivalence between the fibres over the end points, and parallel transport along a surface is an intertwiner between Morita equivalences. We prove that our constructions fit into the general axiomatic framework for categorified parallel transport and surface holonomy.
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In this paper we introduce a notion of parallel transport for principal bundles with connections over differentiable stacks. We show that principal bundles with connections over stacks can be recovered from their parallel transport thereby extending the results of Barrett, Caetano and Picken, and Schreiber and Waldof from manifolds to stacks. In the process of proving our main result we simplify Schreiber and Waldorfs definition of a transport functor for principal bundles with connections over manifolds and provide a more direct proof of the correspondence between principal bundles with connections and transport functors.
For a strict Lie 2-group, we develop a notion of Lie 2-algebra-valued differential forms on Lie groupoids, furnishing a differential graded-commutative Lie algebra equipped with an adjoint action of the Lie 2-group and a pullback operation along Morita equivalences between Lie groupoids. Using this notion, we define connections on principal 2-bundles as Lie 2-algebra-valued 1-forms on the total space Lie groupoid of the 2-bundle, satisfying a condition in complete analogy to connections on ordinary principal bundles. We carefully treat various notions of curvature, and prove a classification result by the non-abelian differential cohomology of Breen-Messing. This provides a consistent, global perspective to higher gauge theory.
Motivated by the computations done in cite{C1}, where I introduced and discussed what I called the groupoid of generalized gauge transformations, viewed as a groupoid over the objects of the category $mathsf{Bun}_{G,M}$ of principal $G$-bundles over a given manifold $M$, I develop in this paper the same ideas for the more general case of {em principal $calG$-bundles or principal bundles with structure groupoid $calG$}, where now $calG$ is a Lie groupoid in the sense of cite{Moer2}. Most of the concepts introduced in cite{C1} can be translated almost verbatim in the framework of principal bundles with structure groupoid $calG$; in particular, the key r�le for the construction of generalized gauge transformations is again played by (the equivalent in the framework of principal bundles with groupoid structure of) the division map $f_P$. Of great importance are also the generalized conjugation in a groupoid and the concept of (twisted) equivariant maps between groupoid-spaces.
The aim of this paper is to review and discuss in detail local aspects of principal bundles with groupoid structure. Many results, in particular from the second and third section, are already known to some extents, but, due to the lack of a ``unified point of view on the subject, I decided nonetheless to (re)define all the main concepts and write all proofs; however, some results are reformulated in a more elegant way, using the division map and the generalized conjugation of a Lie groupoid. In the same framework, I discuss later generalized groupoids and Morita equivalences from a local point of view; in particular, I found a (so far as I know) new characterization of generalized morphisms coming from nonabelian ech cohomology, which allows one to view generalized morphisms as a generalization of classical descent data. I found also a factorization formula for the division map, which is the crucial point in the local formulation of Morita equivalences.
In this Note, we propose a line bundle approach to odd-dimensional analogues of generalized complex structures. This new approach has three main advantages: (1) it encompasses all existing ones; (2) it elucidates the geometric meaning of the integrability condition for generalized contact structures; (3) in light of new results on multiplicative forms and Spencer operators, it allows a simple interpretation of the defining equations of a generalized contact structure in terms of Lie algebroids and Lie groupoids.
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