A note on nonlinear σ-models in noncommutative geometry

We study nonlinear [Formula: see text]-models defined on a noncommutative torus as a two-dimensional string worldsheet. We consider (i) a two-point space, (ii) a circle, (iii) a noncommutative torus, (iv) a classical group [Formula: see text] as examples of space-time. Based on established results, the trivial harmonic unitaries of the noncommutative chiral model known as local minima are shown not to be global minima by comparing them to the symmetric unitaries derived from instanton solutions of the noncommutative Ising model corresponding to a two-point space. In addition, a [Formula: see text]-action on field maps is introduced to a noncommutative torus, and its action on solutions of various Euler–Lagrange equations is described.