Principal bundle

In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product ${\displaystyle X\times G}$ of a space ${\displaystyle X}$ with a group ${\displaystyle G}$. In the same way as with the Cartesian product, a principal bundle ${\displaystyle P}$ is equipped with

Unless it is the product space ${\displaystyle X\times G}$, a principal bundle lacks a preferred choice of identity cross-section; it has no preferred analog of ${\displaystyle x\mapsto (x,e)}$. Likewise, there is not generally a projection onto ${\displaystyle G}$ generalizing the projection onto the second factor, ${\displaystyle X\times G\to G}$ 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 ${\displaystyle F(E)}$ of a vector bundle ${\displaystyle E}$, which consists of all ordered bases of the vector space attached to each point. The group ${\displaystyle G,}$ 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.

A principal ${\displaystyle G}$-bundle, where ${\displaystyle G}$ denotes any topological group, is a fiber bundle ${\displaystyle \pi :P\to X}$ together with a continuous right action ${\displaystyle P\times G\to P}$ such that ${\displaystyle G}$ preserves the fibers of ${\displaystyle P}$ (i.e. if ${\displaystyle y\in P_{x}}$ then ${\displaystyle yg\in P_{x}}$ for all ${\displaystyle g\in G}$) and acts freely and transitively (meaning each fiber is a G-torsor) on them in such a way that for each ${\displaystyle x\in X}$ and ${\displaystyle y\in P_{x}}$, the map ${\displaystyle G\to P_{x}}$ sending ${\displaystyle g}$ to ${\displaystyle yg}$ is a homeomorphism. In particular each fiber of the bundle is homeomorphic to the group ${\displaystyle G}$ itself. Frequently, one requires the base space ${\displaystyle X}$ to be Hausdorff and possibly paracompact.

Since the group action preserves the fibers of ${\displaystyle \pi :P\to X}$ and acts transitively, it follows that the orbits of the ${\displaystyle G}$-action are precisely these fibers and the orbit space ${\displaystyle P/G}$ is homeomorphic to the base space ${\displaystyle X}$. Because the action is free and transitive, the fibers have the structure of G-torsors. A ${\displaystyle G}$-torsor is a space that is homeomorphic to ${\displaystyle G}$ but lacks a group structure since there is no preferred choice of an identity element.

An equivalent definition of a principal ${\displaystyle G}$-bundle is as a ${\displaystyle G}$-bundle ${\displaystyle \pi :P\to X}$ with fiber ${\displaystyle G}$ where the structure group acts on the fiber by left multiplication. Since right multiplication by ${\displaystyle G}$ on the fiber commutes with the action of the structure group, there exists an invariant notion of right multiplication by ${\displaystyle G}$ on ${\displaystyle P}$. The fibers of ${\displaystyle \pi }$ then become right ${\displaystyle G}$-torsors for this action.

The definitions above are for arbitrary topological spaces. One can also define principal ${\displaystyle G}$-bundles in the category of smooth manifolds. Here ${\displaystyle \pi :P\to X}$ is required to be a smooth map between smooth manifolds, ${\displaystyle G}$ is required to be a Lie group, and the corresponding action on ${\displaystyle P}$ should be smooth.