Long Way to Ricci Flatness

We study two-dimensional weighted $\mathcal{N}=2$ supersymmetric $\mathbb{C}\mathbb{P}$ models with the goal of exploring their infrared (IR) limit. $\mathbb{W}\mathbb{C}\mathbb{P}(N,\tilde{N})$ 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, $\mathbb{W}\mathbb{C}\mathbb{P}(N,\tilde{N})$ has $N$ charges +1 and $\tilde{N}$ charges $-1$ fields. As well-known, at $\tilde{N}=N$ 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 $\mathbb{W}\mathbb{C}\mathbb{P}$ 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.