Principal bundle

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In mathematics, a principal bundle^[1][2][3][4] 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

1. An action of ${\displaystyle G}$ on ${\displaystyle P}$, analogous to ${\displaystyle (x,g)h=(x,gh)}$ for a product space.
2. A projection onto ${\displaystyle X}$. For a product space, this is just the projection onto the first factor, ${\displaystyle (x,g)\mapsto x}$.

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.