Deformed $\sigma$-models, Ricci flow and Toda field theories

It is shown that the Pohlmeyer map of a $\sigma$-model with a toric two-dimensional target space naturally leads to the `sausage’ metric. We then elaborate the trigonometric deformation of the ${\mathrm{C}\mathrm{P}}^{n-1}$-model, proving that its $T$-dual metric is K"ahler and solves the Ricci flow equation. Finally, we discuss a relation between flag manifold $\sigma$-models and Toda field theories.