Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userParallel Transport on Principal Bundles over Stacks
134
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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.
We construct and study general connections on Lie groupoids and differentiable stacks as well as on principal bundles over them using Atiyah sequences associated to transversal tangential distributions.
Let $mathbb{X}=[X_1rightrightarrows X_0]$ be a Lie groupoid equipped with a connection, given by a smooth distribution $mathcal{H} subset T X_1$ transversal to the fibers of the source map. Under the assumption that the distribution $mathcal{H}$ is integrable, we define a version of de Rham cohomology for the pair $(mathbb{X}, mathcal{H})$, and we study connections on principal $G$-bundles over $(mathbb{X}, mathcal{H})$ in terms of the associated Atiyah sequence of vector bundles. We also discuss associated constructions for differentiable stacks. Finally, we develop the corresponding Chern-Weil theory and describe characteristic classes of principal $G$-bundles over a pair $(mathbb{X}, mathcal{H})$.
In this paper, a notion of a principal $2$-bundle over a Lie groupoid has been introduced. For such principal $2$-bundles, we produced a short exact sequence of VB-groupoids, namely, the Atiyah sequence. Two notions of connection structures viz. strict connections and semi-strict connections on a principal $2$-bundle arising respectively, from a retraction of the Atiyah sequence and a retraction up to a natural isomorphism have been introduced. We constructed a class of principal $mathbb{G}=[G_1rightrightarrows G_0]$-bundles and connections from a given principal $G_0$-bundle $E_0rightarrow X_0$ over $[X_1rightrightarrows X_0]$ with connection. An existence criterion for the connections on a principal $2$-bundle over a proper, etale Lie groupoid is proposed. The action of the $2$-group of gauge transformations on the category of strict and semi-strict connections has been studied. Finally we noted an extended symmetry of the category of semi-strict connections.
We introduce and study (strict) Schottky G-bundles over a compact Riemann surface X, where G is a connected reductive algebraic group. Strict Schottky representations are shown to be related to branes in the moduli space of G-Higgs bundles over X, and we prove that all Schottky $G$-bundles have trivial topological type. Generalizing the Schottky moduli map introduced in Florentino to the setting of principal bundles, we prove its local surjectivity at the good and unitary locus. Finally, we prove that the Schottky map is surjective onto the space of flat bundles for two special classes: when G is an abelian group over an arbitrary X, and the case of a general G-bundle over an elliptic curve.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
