In the mathematical area of topology, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X × G {\displaystyle X\times G} of a topological space X {\displaystyle X} with a group G {\displaystyle G} , but without requiring a product structure. In the same way as with the Cartesian product, a principal bundle P {\displaystyle P} is equipped with
An action of G {\displaystyle G} on P {\displaystyle P} , analogous to ( x , g ) h = ( x , g h ) {\displaystyle (x,g)h=(x,gh)} for a product space (where ( x , g ) {\displaystyle (x,g)} is an element of P {\displaystyle P} and h {\displaystyle h} is the group element from G {\displaystyle G} ; the group action is conventionally a right action). A projection onto X {\displaystyle X} . For a product space, this is just the projection onto the first factor, ( x , g ) ↦ x {\displaystyle (x,g)\mapsto x} . Unless it is the product space X × G {\displaystyle X\times G} , a principal bundle lacks a preferred choice of identity cross-section; it has no preferred analog of x ↦ ( x , e ) {\displaystyle x\mapsto (x,e)} . Likewise, there is not generally a projection onto G {\displaystyle G} generalizing the projection onto the second factor, X × G → G {\displaystyle X\times G\to G} that exists for the Cartesian product. It may also have a complicated topology that prevents it from being realized as a product space. An example of a principal bundle is the bundle π : R → S 1 ⊆ C {\displaystyle \pi :\mathbb {R} \to S^{1}\subseteq \mathbb {C} } where π {\displaystyle \pi } is defined by π ( t ) = exp ( 2 π i t ) {\displaystyle \pi (t)=\exp(2\pi it)} , in which the fibers are Z {\displaystyle \mathbb {Z} } -torsors. Since this bundle looks like a helix, there is no canonical choice of identity cross-section, which should look like a circle. Another example of a principal bundle is the frame bundle F ( E ) {\displaystyle F(E)} of a vector bundle E {\displaystyle E} , which consists of all ordered bases of the vector space attached to each point. The group G , {\displaystyle G,} 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. Important cases are principal U(1)-bundles and principal SU(2)-bundles.
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