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

• URL: http://arxiv.org/abs/2404.03630v1
• Date: Thu, 4 Apr 2024 17:52:05 GMT
• Title: Lie-algebraic Kähler sigma models with the U(1) isotropy
• Authors: Chao-Hsiang Sheu, Mikhail Shifman,
• Abstract summary: We discuss various questions which emerge in connection with the Lie-algebraic deformation of $mathbbCP1$ sigma model in two dimensions. First we supersymmetrize the original model endowing it with the minimal $cal N=(0,2)$ and extended $cal N= (2,2)$ supersymmetries. We conjecture that the above statement is valid for higher loops too, similar to the parallel phenomenon in four-dimensional $cal N=1$ super-Yang-Mills.
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• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We discuss various questions which emerge in connection with the Lie-algebraic deformation of $\mathbb{CP}^1$ sigma model in two dimensions. First we supersymmetrize the original model endowing it with the minimal ${\cal N}=(0,2)$ and extended ${\cal N}=(2,2)$ supersymmetries. Then we derive the general hypercurrent anomaly in the 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 non-symmetric. We also show how to introduce the twisted masses and the $\theta$ term, and study the BPS equation for instantons, in particular the value of the topological charge. Then we demonstrate that the second loop in the $\beta$ function of the non-supersymmetric 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 ${\cal N}=1$ 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 Lam\'e type.

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