曲名：Long Way to Ricci Flatness
Long Way to Ricci Flatness
摘要：
我们研究二维加权${\mathcal N}=2$超对称$\mathbb{CP}$模型，目的是探索它们的红外（IR）极限。$\mathbb{WCP}(N,\widetilde{N})$是四维${\mathcal N}=2$ QCD中非阿贝尔弦世界片理论的简化版本。在规范线性西格玛模型（GLSM）公式中，$\mathbb{WCP} (N,\widetilde{N})$具有$N$电荷+1和$\widetilde{N}$电荷$-1$场。众所周知，在$\widetilde{N}=N$中，该GLSM是共形的。它的目标空间被认为是一个非紧的Calabi-Yau流形。我们主要讨论了$N=2$的情形，证明了Calabi-Yau空间是一个锥。另一方面，在非线性西格玛模型（NLSM）制定的模型有紫外线的光谱，并不看共形。此外，它的度量不是Ricci平坦的。我们通过研究模型的重整化群（RG）流来解决这个难题。我们证明了NLSM的度量在IR中是Ricci平坦的，并且趋于已知的分解锥度量。我们还研究了$\mathbb{WCP}$模型的近亲--所谓的$zn$模型--它实际上代表了非阿贝尔半局域弦上的世界片理论，并表明这种$zn$模型具有类似的RG性质。
英文摘要：
We study two-dimensional weighted ${\mathcal N}=2$ supersymmetric $\mathbb{CP}$ models with the goal of exploring their infrared (IR) limit. $\mathbb{WCP}(N,\widetilde{N})$ are simplified versions of world-sheet theories on non-Abelian strings in four-dimensional ${\mathcal N}=2$ QCD. In the gauged linear sigma model (GLSM) formulation, $\mathbb{WCP} (N,\widetilde{N})$ has $N$ charges +1 and $\widetilde{N}$ charges $-1$ fields. As well-known, at $\widetilde{N}=N$ this GLSM is conformal. Its target space is believed to be a non-compact Calabi-Yau manifold. We mostly focus on the $N=2$ case, then the Calabi-Yau space is a conifold. On the other hand, in the non-linear sigma model (NLSM) formulation the model has ultra-violet logarithms and does not look conformal. Moreover, its metric is not Ricci-flat. We address this puzzle by studying the renormalization group (RG) flow of the model. We show that the metric of NLSM becomes Ricci-flat in the IR. Moreover, it tends to the known metric of the resolved conifold. We also study a close relative of the $\mathbb{WCP}$ model -- the so called $zn$ model -- which in actuality represents the world sheet theory on a non-Abelian semilocal string and show that this $zn$ model has similar RG properties.
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