Lie-algebraic Kähler sigma models with U(1) isotropy

We discuss various questions that emerge in connection with the Lie-algebraic deformation of the ${\mathrm{CP}}^1$ sigma model in two dimensions. First, we supersymmetrize the original model endowing it with the minimal $N=(0,2)$ and extended $N=(2,2)$ supersymmetries. Then we derive the general hypercurrent anomaly in both cases. In the latter case this anomaly is one-loop but is somewhat different from the standard expressions one can find in the literature because the target manifold is nonsymmetric. We also show how to introduce the twisted masses and the $\theta$ term, and study the Bogomol'nyi–Prasad–Sommerfield equation for instantons, in particular the value of the topological charge. Then we demonstrate that the second loop in the $\beta$ function of the nonsupersymmetric Lie-algebraic sigma model is due to an infrared effect. To this end we use a supersymmetric regularization. We also conjecture that the above statement is valid for higher loops too, similar to the parallel phenomenon in four-dimensional $N=1$ super-Yang-Mills. In the second part of the paper we develop a special dimensional reduction—namely, starting from the two-dimensional Lie-algebraic model we arrive at a quasi-exactly solvable quantum-mechanical problem of the Lamé type.