Long Way to Ricci Flatness

Journal of High Energy Physics. 2020. Vol. 2020. No. 10. Article 59.

Chen J., Sheu C., Shifman M., Tallarita G., Alexei Yung

We study two-dimensional weighted N = (2; 2) supersymmetric CP models
with the goal of exploring their infrared (IR) limit. WCP(N, tildeN) are simplifed 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, tildeN) has N charges +1 and tilde N
charges -1 fields. As well-known, at e N = tildeN 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) ow 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.

Keywords: quantum field theory, supersymmetry, Sigma Models