Parallel transport on principal bundles over stacks

Collier, Brian; Lerman, Eugene; Wolbert, Seth

doi:10.1016/j.geomphys.2016.05.010

Cited by 7 publications

(9 citation statements)

References 18 publications

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“…in the sense of [CLW16]. They showed that such functors are the same as principal G-bundles with connection.…”

Section: The Thin Fundamental 2-groupoidmentioning

confidence: 99%

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Parallel 2-transport and 2-group torsors

Voorhaar

2018

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We provide a new perspective on parallel 2-transport and principal 2-group bundles with 2connection. We define parallel 2-transport as a 2-functor from the thin fundamental 2-groupoid to the 2-category of 2-group torsors. The definition of the 2-category of 2-group torsors is new, and we develop the tools necessary for computations in this 2-category. We prove a version of the non-Abelian Stokes Theorem and the Ambrose-Singer Theorem for 2-transport. This definition motivated by the fact that principal G-bundles with connection are equivalent to functors from the thin fundamental groupoid to the category of G-torsors. In the same lines we deduce a notion of principal 2-bundle with 2-connection, and show it is equivalent to our notion 2-transport functors. This gives a stricter notion than appears in the literature, which is more concrete. It allows for computations of 2-holonomy which will be exploited in a companion paper to define Wilson surface observables. Furthermore this notion can be generalized to a concrete but strict notion of n-transport for arbitrary n.

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“…in the sense of [CLW16]. They showed that such functors are the same as principal G-bundles with connection.…”

Section: The Thin Fundamental 2-groupoidmentioning

confidence: 99%

“…These two constructions give an equivalence of categories. Stated in different terms this result is originally due to[Bar90], but using categorical language this statement appears in[SW09,CLW16].Theorem 4.1. The procedure above defines an equivalence of categories Bun 1 ∇ (M, G) ∼ = Trans 1 (M, G), i.e.…”

mentioning

confidence: 99%

Parallel 2-transport and 2-group torsors

Voorhaar

2018

Preprint

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“…All of the previous developments provided multiple insights that paved the way towards a rigorous mathematical formulation of the reconstruction theorem of gauge fields in terms of their holonomy homomorphisms from a group of based loops on a manifold to a Lie group G. Different proofs of the reconstruction theorem appeared in [Bar89,Bar91], [Lew93], [Haj93], [CP94], leading to subsequent mathematical developments of the notion of groups of based loops [Ful94], [Gib97], [Woo97], [MP02], [SW09,CLW16], [Tla16]. Somewhat independent, but equally important, are the works of Gross [Gro85], who gave an analytic proof of the equivalence between the Yang-Mills and Maldestam-Bia lynicki-Birula equations, and Morrison [Mor91], who gave a characterization of connections on principal bundles over oriented Riemannian surfaces satisfying the Yang-Mills equations in terms of their corresponding holonomies.…”

Section: Brief Surveymentioning

confidence: 99%

Thin homotopy and the holonomy approach to gauge theories

Meneses

2019

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We survey several mathematical developments in the holonomy approach to gauge theory. A cornerstone of such approach is the introduction of group structures on spaces of based loops on a smooth manifold, relying on certain homotopy equivalence relations -such as the so-called thin homotopy -and the resulting interpretation of gauge fields as group homomorphisms to a Lie group G satisfying a suitable smoothness condition, encoding the holonomy of a gauge orbit of smooth connections on a principal G-bundle. We also prove several structural results on thin homotopy, and in particular we clarify the difference between thin equivalence and retrace equivalence for piecewise-smooth based loops on a smooth manifold, which are often used interchangeably in the physics literature. We conclude by listing a set of questions on topological and functional analytic aspects of groups of based loops, which we consider to be fundamental to establish a rigorous differential geometric foundation of the holonomy formulation of gauge theory.

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“…In particular, the structure of a connection on Lie groupoids and for principal bundles over Lie groupoids as well as its associated geometric and algebraic properties have been discussed by several authors. Among them [LGTX2,BX,B,CM,CLW] are particularly relevant for this article. For example, the articles [LGTX2] and [CM] study Chern-Weil theory on Lie groupoids via the de Rham cohomology defined using simplicial manifolds associated to the groupoid nerves.…”

Section: Introductionmentioning

confidence: 99%

Connections on Lie groupoids and Chern-Weil theory

Biswas¹,

Chatterjee²,

Koushik³

et al. 2020

Preprint

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Let X = [X 1 ⇒ X 0 ] be a Lie groupoid equipped with a connection, given by a smooth distribution H ⊂ T X 1 transversal to the fibers of the source map. Under the assumption that the distribution H is integrable, we define an analog of de Rham cohomology for the pair (X, H) and study connections on principal G-bundles over (X, H) in terms of the associated Atiyah sequence of vector bundles. Finally, we develop the corresponding Chern-Weil theory and study characteristic classes. Contents 1. Introduction 1 2. Principal bundles over Lie groupoids 3 3. Integrable connections and de Rham cohomology 6 4. Connections on principal bundles over Lie groupoids 12 5. Differential forms associated to a connection 14 6. Chern-Weil map and characteristic classes for principal bundles over Lie groupoids with connections 20 Acknowledgements 25 References 25

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Parallel transport on principal bundles over stacks

Cited by 7 publications

References 18 publications

Parallel 2-transport and 2-group torsors

Parallel 2-transport and 2-group torsors

Thin homotopy and the holonomy approach to gauge theories

Connections on Lie groupoids and Chern-Weil theory

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