We discuss various questions that emerge in connection with the Lie-algebraic deformation of the ${\mathbb{CP}}^1$ sigma model in two dimensions. First, we supersymmetrize the original model endowing it with the minimal $\mathcal{N}=(0,2)$ and extended $\mathcal{N}=(2,2)$ 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 $\mathcal{N}=1$ 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.

中文翻译：

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具有 U(1) 各向同性的李代数 Kähler sigma 模型


我们讨论与二维 ${\mathbb{CP}}^1$ sigma 模型的李代数变形相关的各种问题。首先，我们对原始模型进行超对称化，赋予其最小的 $\mathcal{N}=(0,2)$ 和扩展的 $\mathcal{N}=(2,2)$ 超对称性。然后我们得出这两种情况下的一般超电流异常。在后一种情况下，这种异常是单循环的，但与文献中可以找到的标准表达式有些不同，因为目标流形是非对称的。我们还展示了如何引入扭曲质量和 $\theta$ 项，并研究瞬子的 Bogomol’nyi-Prasad-Sommerfield 方程，特别是拓扑电荷的值。然后我们证明非超对称李代数 sigma 模型的 $\beta$ 函数中的第二个循环是由红外效应引起的。为此，我们使用超对称正则化。我们还推测上述陈述对于更高的循环也有效，类似于四维 $\mathcal{N}=1$ super-Yang-Mills 中的并行现象。在本文的第二部分中，我们开发了一种特殊的降维方法，即从二维李代数模型开始，我们得到了准精确可解的拉梅型量子力学问题。