Rudolph Gerd - Differential geometry and mathematical physicsn2, Fibre bundles, topology and gauge fields

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Springer Science+Business Media Dordrecht 2017

Gerd Rudolph and Matthias Schmidt Differential Geometry and Mathematical Physics Theoretical and Mathematical Physics 10.1007/978-94-024-0959-8_1

1. Fibre Bundles and Connections

Gerd Rudolph 1 and Matthias Schmidt 1

(1)

Institute for Theoretical Physics, University of Leipzig, Leipzig, Germany

Gerd Rudolph

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In this chapter, we present the basics of the theory of fibre bundles and connections. In the first part, we discuss principal and asssociated bundles and the theory of connections including the Koszul calculus. The text is illustrated by many examples which will be taken up later on. In the second part, we focus on topics which are particularly important in this book. We study bundle reductions, discuss the theory of holonomy in some detail and analyze the transformation laws of connection and curvature under bundle automorphisms. Finally, we present the theory of invariant connections for the case of group actions which are not necessarily transitive on the base manifold, that is, we go beyond the classical Wang Theorem.

1.1 Principal Bundles

In a gauge theory describing the fundamental interaction of elementary particles, the interaction is assumed to be mediated by a gauge potential. In geometric terms, a gauge potential is the local (spacetime) representative of a connection form, which naturally lives on a principal fibre bundle over spacetime.

Let us recall the following definition from Sect. of Part I.

Definition 1.1.1

( Principal bundle ) Let be a free Lie group action, let M be a manifold and let be a smooth mapping. The tuple is called a principal bundle if for every there exists an open neighbourhood U of m and a diffeomorphism such that

1. intertwines with the G -action on by translations on the factor G ,
2. for all .

For simplicity, we will sometimes use the short-hand notation P ( M , G ) or just P . If not otherwise stated, we will consider right principal bundles. If there is no danger of confusion, sometimes we will simply write . For a right action, denoting

(1.1.1)

condition 1 can be rewritten as

(1.1.2)

The group G is called the structure group of P . If G is fixed, P is referred to as a principal G -bundle. The pair is called a local trivialization . A local trivialization with is called a global trivialization . If there exists a global trivialization, then P is called trivial . The existence of local trivializations implies that is a surjective submersion. Hence, by Proposition I/1.7.6, the subsets , , are embedded submanifolds, called the fibres of P . They are diffeomorphic to the group manifold G .

Remark 1.1.2

Let be a free proper Lie group action. Let M be the orbit space, equipped with the smooth structure provided by Corollary I/6.5.1, and let be the natural projection to orbits. Every tubular neighbourhood of an orbit defines a local trivialization over a neighbourhood of the corresponding point of M . Hence, the Tubular Neighbourhood Theorem I/6.4.3 implies that is a principal bundle. Conversely, if is a principal bundle, then is a free proper Lie group action, M is diffeomorphic to the orbit space P / G and corresponds, via this diffeomorphism, to the natural projection to orbits.

We will also need the general notion of fibre bundle.

Definition 1.1.3

( General fibre bundle ) Let E and M be manifolds and let be a smooth surjection. The triple is called a fibre bundle if there exists a manifold F such that the following holds. Every

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