Journal of High Energy Physics (Oct 2020)

Long way to Ricci flatness

  • Jin Chen,
  • Chao-Hsiang Sheu,
  • Mikhail Shifman,
  • Gianni Tallarita,
  • Alexei Yung

DOI
https://doi.org/10.1007/JHEP10(2020)059
Journal volume & issue
Vol. 2020, no. 10
pp. 1 – 24

Abstract

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Abstract We study two-dimensional weighted N $$ \mathcal{N} $$ = (2) supersymmetric โ„‚โ„™ models with the goal of exploring their infrared (IR) limit. ๐•Žโ„‚โ„™(N, N หœ $$ \tilde{N} $$ ) are simplified versions of world-sheet theories on non-Abelian strings in four-dimensional N $$ \mathcal{N} $$ = 2 QCD. In the gauged linear sigma model (GLSM) formulation, ๐•Žโ„‚โ„™(N, N หœ $$ \tilde{N} $$ ) has N charges +1 and N หœ $$ \tilde{N} $$ charges โˆ’1 fields. As well-known, at N หœ $$ \tilde{N} $$ = 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 ๐•Žโ„‚โ„™ model โ€” the so called zn model โ€” 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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