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1. Classical Field Theory

   Linked via "solitons"

   Solitonic Solutions and Non-Linearity
   While linear classical field theories admit wave solutions|, non-linearities| introduced through the potential term| $V(\phi)$ can stabilize localized, non-dispersive excitations known as solitons|.
   Kinks in $\phi^4$ Theory

2. Sigma Model

   Linked via "Soliton"

   | :--- | :--- | :--- | :--- |
   | Standard $\sigma$-field | $\mathbb{R}^N$ | Cartesian Coordinates | Goldstone Boson (if SSB occurs) |
   | Chiral $\sigma$-field | $\text{SU}(N)/\text{U}(N-1)$ | Unitary Matrix Representation | Skyrmion (Soliton) |
   | Torsion Field | $\text{SO}(p, q)$ space | Hyperbolic Coordinates | $\tau$-Particle |