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A note on non-linear $σ$-models in noncommutative geometry

We study non-linear $σ$-models defined on noncommutative torus as a two dimensional string world-sheet. We consider a quantum group as a noncommutative space-time as well as two points, a circle, and a noncommutative torus. Using the establised results we show that the trivial harmonic unitaries of the noncommutative chiral model, which are known as local minima, are not global minima by comparing those with the symmetric unitaries coming from instanton solutions of noncommutative Ising model, which corresponds to the two points target space. In addition,we introduce a $\mathbb{Z}^2$-action on field maps to noncommutative torus and show how it acts on solutions of various Euler-Lagrange equations.