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arXiv 1111.6350hep-th
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CP的N=（0，2）形变（1）模型：N=1 Yang-Mills理论在四维中的二维模拟

N=(0,2) Deformation of CP(1) Model: Two-dimensional Analog of N=1 Yang-Mills Theory in Four Dimensions

Xiaoyi Cui, M. Shifman

备注：
31 pages, 6 figures; a typo in the previous arXiv version was corrected, the current version is compatible with the published version
发表信息：
Phys. Rev. D 85, 045004 (2012)
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摘要：

我们考虑二维$\mathcal{N}=(0,2)$ sigma模型，目标空间为CP（1）。这种类型的极小模型有一个左手费米子。此外，非极小扩张还包含$N_f$右手费米子。我们的任务是推导出对所有阶有效的$β$函数的表达式。为此，我们使用各种方法：（i）微扰分析;（ii）瞬子演算;（iii）分析超电流超多重态（所谓的超电流）及其异常，以及其他一些论点。所有这些论点，结合起来，表明杂种优势$\mathcal{N}=(0,2)$ CP（1）模型和四维超杨-米尔斯理论之间的直接平行。特别地，最小$\mathcal{N}=(0,2)$ CP（1）模型类似于${\mathcal N}=1$超对称胶子动力学.它的精确$β$函数可以找到;它具有超对称胶子动力学的Novikov-Shifman-Vainshtein-Zakharov（NSVZ）$β$函数的结构。传递到非最小$\mathcal{N}=(0,2)$ sigma模型等价于添加物质。在这种情况下，建立了$β$函数和“物质”场的反常维数$γ$之间的NSVZ型精确关系。我们得到一个类似的小西异常。在大的$N_f$下，我们的$β$函数在小的耦合常数值处产生红外不动点（类似于班克斯-扎克斯不动点）。因此，我们可靠地预测了共形窗口的存在。在$N_f=1$中，所考虑的模型简化为众所周知的$\mathcal{N}=(2,2)$ CP（1）模型。

英文摘要：

We consider two-dimensional $\mathcal{N}=(0,2)$ sigma models with the CP(1) target space. A minimal model of this type has one left-handed fermion. Nonminimal extensions contain, in addition, $N_f$ right-handed fermions. Our task is to derive expressions for the $β$ functions valid to all orders. To this end we use a variety of methods: (i) perturbative analysis; (ii) instanton calculus; (iii) analysis of the supercurrent supermultiplet (the so-called hypercurrent) and its anomalies, and some other arguments. All these arguments, combined, indicate a direct parallel between the heterotic $\mathcal{N}=(0,2)$ CP(1) models and four-dimensional super-Yang-Mills theories. In particular, the minimal $\mathcal{N}=(0,2)$ CP(1) model is similar to ${\mathcal N}=1$ supersymmetric gluodynamics. Its exact $β$ function can be found; it has the structure of the Novikov-Shifman-Vainshtein-Zakharov (NSVZ) $β$ function of supersymmetric gluodynamics. The passage to nonminimal $\mathcal{N}=(0,2)$ sigma models is equivalent to adding matter. In this case an NSVZ-type exact relation between the $β$ function and the anomalous dimensions $γ$ of the "matter" fields is established. We derive an analog of the Konishi anomaly. At large $N_f$ our $β$ function develops an infrared fixed point at small values of the coupling constant (analogous to the Banks-Zaks fixed point). Thus, we reliably predict the existence of a conformal window. At $N_f=1$ the model under consideration reduces to the well-known $\mathcal{N}=(2,2)$ CP(1) model.

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