Remarks on Baby Skyrmion Lie-Algebraic Generalization

We discuss generalized baby Skyrmions emerging in a (1+2)-dimensional $\sigma$ model with a certain Lie-algebraic structure. The same result applies to the Polyakov-Belavin instantons in $D=1+1$. The O(3) symmetry of the target space is lost, but O(2) is preserved in the simplest model under consideration. Both the topological charge and the soliton mass (the instanton action) are determined. Of special interest are limiting cases of the deformation parameter $k$ (also referred to as the elongation parameter). If $|k-1|\ll 1$, we arrive at a model which is studied in condensed matter. If $k\gg1$, we obtain the so-called cigar, or sausage model, well-known in little string theory. If $k=1$, we return to CP(1). The deformed model under consideration interpolates between CP(1) and the cigar model. In $D=2$ we calculate the coupling constants renormalization at one loop. At $k\geq 1$ this class of models is asymptotically free in the ultraviolet limit and enters the strong coupling domain in the infrared. Also, in the infrared $k\to 1$ (i.e., we recover CP(1)).