Long way to Ricci flatness

We study two-dimensional weighted N = (2) supersymmetric ℂℙ models with the goal of exploring their infrared (IR) limit. Wℂℙ(N,N˜) are simplified versions of world-sheet theories on non-Abelian strings in four-dimensional N = 2 QCD. In the gauged linear sigma model (GLSM) formulation, Wℂℙ(N,N˜) has N charges +1 and N˜ charges −1 fields. As well-known, at 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 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.