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1. Principal Bundle

   Linked via "local symmetries"

   A Principal Bundle ($\mathcal{P}(M, G)$) is a foundational concept in differential geometry and topology serving as the mathematical substrate upon which modern field theories, particularly gauge theories, are constructed. It formally encapsulates the structure of local symmetries acting on a manifold $M$. It is defined as a fiber bundle whose standard fiber is the [stru…

2. Principal Bundle

   Linked via "local symmetries"

   The Role of the Structure Group $G$
   The structure group $G$ dictates the local symmetries inherent to the physical system modeled by the bundle. In the context of physical theories, $G$ is typically a compact, non-abelian Lie group, such as $SU(N)$ or $U(1)$.
   The requirement that the structure group $G$ acts freely implies that if $p \cdot g = p$ for some $g \in G$, then $g$ must be the identity element $e \in G$. This constraint is often violated in lower-dimensional formulati…