In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product of a space with a group . In the same way as with the Cartesian product, a principal bundle is equipped with An action of on , analogous to for a product space. A projection onto . For a product space, this is just the projection onto the first factor, . Unlike a product space, principal bundles lack a preferred choice of identity cross-section; they have no preferred analog of . Likewise, there is not generally a projection onto generalizing the projection onto the second factor, that exists for the Cartesian product. They may also have a complicated topology that prevents them from being realized as a product space even if a number of arbitrary choices are made to try to define such a structure by defining it on smaller pieces of the space.
A common example of a principal bundle is the frame bundle of a vector bundle , which consists of all ordered bases of the vector space attached to each point. The group in this case, is the general linear group, which acts on the right in the usual way: by changes of basis. Since there is no natural way to choose an ordered basis of a vector space, a frame bundle lacks a canonical choice of identity cross-section.
Principal bundles have important applications in topology and differential geometry and mathematical gauge theory. They have also found application in physics where they form part of the foundational framework of physical gauge theories.
Formal definition
A principal -bundle, where denotes any topological group, is a fiber bundle together with a continuous right action such that preserves the fibers of (i.e. if then for all ) and acts freely and transitively (meaning each fiber is a G-torsor) on them in such a way that for each and , the map sending to is a homeomorphism. In particular each fiber of the bundle is homeomorphic to the group itself. Frequently, one requires the base space to be Hausdorff and possibly paracompact.
Since the group action preserves the fibers of and acts transitively, it follows that the orbits of the -action are precisely these fibers and the orbit space is homeomorphic to the base space . Because the action is free and transitive, the fibers have the structure of G-torsors. A -torsor is a space that is homeomorphic to but lacks a group structure since there is no preferred choice of an identity element.
An equivalent definition of a principal -bundle is as a -bundle with fiber where the structure group acts on the fiber by left multiplication. Since right multiplication by on the fiber commutes with the action of the structure group, there exists an invariant notion of right multiplication by on . The fibers of then become right -torsors for this action.
The definitions above are for arbitrary topological spaces. One can also define principal -bundles in the category of smooth manifolds. Here is required to be a smooth map between smooth manifolds, is required to be a Lie group, and the corresponding action on should be smooth.
Examples
Trivial bundle and sections
Over an open ball , or , with induced coordinates , any principal -bundle is isomorphic to a trivial bundleand a smooth section is equivalently given by a (smooth) function sincefor some smooth function. For example, if , the Lie group of unitary matrices, then a section can be constructed by considering four real-valued functionsand applying them to the parameterization
This same procedure by taking a parameterization of a collection of matrices defining a Lie group and by considering the set of functions from a patch to and inserting them into the parameterization.
Other examples
The prototypical example of a smooth principal bundle is the frame bundle of a smooth manifold , often denoted or . Here the fiber over a point is the set of all frames (i.e. ordered bases) for the tangent space . The general linear group acts freely and transitively on these frames. These fibers can be glued together in a natural way so as to obtain a principal -bundle over . Variations on the above example include the orthonormal frame bundle of a Riemannian manifold. Here the frames are required to be orthonormal with respect to the metric. The structure group is the orthogonal group . The example also works for bundles other than the tangent bundle; if is any vector bundle of rank over , then the bundle of frames of is a principal -bundle, sometimes denoted . A normal (regular) covering space is a principal bundle where the structure group