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1. Gauge Structure

   Linked via "connection one-form"

   Fiber Bundles and Connections
   Physically, the configuration space of the fields is modeled as a principal fiber bundle $P(M, G)$ over the space-time manifold $M$. The local gauge transformations correspond to smooth sections of this bundle. The gauge field $A\mu$ is interpreted mathematically as a connection one-form on this bundle. The curvature of this connection, $F{\mu\nu}$, represents the observable physical interactions (f…

2. Sigma Model

   Linked via "connection 1-form"

   The $\mathbb{CP}^{N-1}$ Model and Topological Tension
   The $\mathbb{CP}^{N-1}$ model is the archetype of the interacting NLSM in $d=2$. Here, the fields $\phi^a$ are complex projective coordinates, and the target space is the complex projective plane. In this context, the field strength tensor $F{\mu\nu}$ is constructed from the connection 1-form $\omega\mu$ derived from the metric, analogous to an…