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

Understand

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 $\mathrm{CP}^{n-1}$-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.