Principal bundle

In mathematics, the principal bundle, or principal fiber bundles or principal bundles, a concept of differential geometry, with the " twisted products " are formalized and used among other things in physics for the description of gauge field theories, and special Yang-Mills fields.

• 6.1 Examples

Products ( trivial principal bundle)

Principal bundles generalize the notion of cartesian product X × G of a space X and a topological group G. Just as the Cartesian product of a principal fiber bundle P has the following properties:

Unlike product spaces have principal bundles no preferred representation of a section of the neutral element of the group G. So there is no preferred element of P as the identification of (x, e). Nor is there a general continuous projection on G, which generalizes the projection onto the second element of the product space: (x, g) → g principal bundle can therefore have complex topologies that prevent a representation of the bundle as a product of space, even if some additional assumptions are made.

Functions can be interpreted as sections in the trivial principal bundle, namely as. Cuts in principal bundles thus generalize the concept of G -quality pictures.

Definition

A principal bundle is a fiber bundle provided on a space with the projection of a continuous right action ( listed below as ) a topological group, so that the operation maps each fiber to itself ( that is, for all and all ) and the group -free ( each point is just below the neutral element of the group invariant ) and transitive ( each point of a fiber is operated on by any other means of reaching the group operation ) on each fiber. The group is called structure group of the principal bundle.

Are and smooth manifolds, the structure group a Lie group and the surgery itself smooth, it means the principal bundle of smooth principal bundles.

Trivialization

As with any fiber bundle projection is topologically locally seen trivialisierbar: So there are any open environment, so is homeomorphic to. Each fiber is homeomorphic to the -conceived as topological space structure group. A trivialization of a principal bundle is even possible considering the group operation: it can choose an equivariant homeomorphism, so

For everyone. Any such local trivialization induces a local section of virtue, wherein the neutral element call.

Conversely, each local interface induces a local trivialization with given by. The local Trivialisierbarkeit thus follows from the existence of local sections, which generally exist on fiber bundles. Unlike common fiber bundles ( for example, consider the tangent bundle of a smooth manifold ) implies not only the global Trivialisierbarkeit the existence of a global section, but also the existence of a global section the Trivialisierbarkeit.

In the physical context allows the selection of a calibration as a (local or global depending on the situation ) choice of a trivialization of a cut or understand.

Examples

Frame bundle

Let be a differentiable n- dimensional manifold. The frame beam is the set of bases of the tangent with the canonical projection. The group acts transitively and faithfully on the fibers.

Overlays

Galois overlays are principal bundles with the discrete group of deck transformations as structure group.

Homogeneous spaces

Let be a Lie group and a closed subgroup, then is a principal bundle with structure group.

In the topology and geometry of the differential, there are several applications of the principal beam. Likewise, there are applications of principal bundles in physics. There they form a crucial part of the mathematical framework of gauge theories.

Associated vector bundles

In the case of one can to each principal bundle an associated complex vector bundle defined by

With the equivalence relation

Analogously, one can define for every principal bundle an associated real vector bundles.

For example, let be a differentiable n- dimensional manifold and the frame bundle. Then the tangent bundle is the associated vector bundle for the canonical action of on.

Reduction of the structure group

A principal bundle can be reduced to a subset when the bundle has a section. In particular, a principal bundle is trivial if and only if it can be reduced to the subgroup.

Examples

Consider the beam frame of an n -dimensional differentiable manifold, the structure is set. Then:

• The structure group can be exactly then to reduce, if the tangent bundle has linearly independent sections,
• The structure group always leaves to reduce, this corresponds to the choice of a Riemannian metric,
• The structure group can be exactly then to reduce, if the manifold is orientable.

Be in the following is an even number:

• The structure group can be exactly then to reduce, if the manifold is almost complex,
• When the manifold is symplectic, then the structure can be reduced to group.

Be in the following is an odd number:

• If the manifold has a contact structure, then the structure can be set to reduce.

Connection, curvature

An important role in the study of principal bundles play link -1 - forms and the curvature 2- forms.

Application: electromagnetism

In a charge-free, the electric field and the magnetic field satisfy the Maxwell equations. The fields have potential and with and. However, these potentials are not unique, and because for any function give the same fields.

We consider the Minkowski space-time and the principal bundle with connection form. Whose curvature form is the electromagnetic field:

The calibration transformations are of the form.