Abstract
We study two-dimensional weighted \( \mathcal{N} \) = (2) supersymmetric ℂℙ models with the goal of exploring their infrared (IR) limit. 𝕎ℂℙ(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, 𝕎ℂℙ(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 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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References
A. Hanany and D. Tong, Vortices, instantons and branes, JHEP 07 (2003) 037 [hep-th/0306150] [INSPIRE].
R. Auzzi, S. Bolognesi, J. Evslin, K. Konishi and A. Yung, Non-Abelian superconductors: vortices and confinement in N = 2 SQCD, Nucl. Phys. B 673 (2003) 187 [hep-th/0307287] [INSPIRE].
M. Shifman and A. Yung, Non-Abelian string junctions as confined monopoles, Phys. Rev. D 70 (2004) 045004 [hep-th/0403149] [INSPIRE].
A. Hanany and D. Tong, Vortex strings and four-dimensional gauge dynamics, JHEP 04 (2004) 066 [hep-th/0403158] [INSPIRE].
D. Tong, TASI lectures on solitons: instantons, monopoles, vortices and kinks, hep-th/0509216 [INSPIRE].
M. Eto, Y. Isozumi, M. Nitta, K. Ohashi and N. Sakai, Solitons in the Higgs phase: the moduli matrix approach, J. Phys. A 39 (2006) R315 [hep-th/0602170] [INSPIRE].
M. Shifman and A. Yung, Supersymmetric solitons and how they help us understand non-Abelian gauge theories, Rev. Mod. Phys. 79 (2007) 1139 [hep-th/0703267] [INSPIRE].
D. Tong, Quantum vortex strings: a review, Annals Phys. 324 (2009) 30 [arXiv:0809.5060] [INSPIRE].
M. Shifman, W. Vinci and A. Yung, Effective world-sheet theory for non-Abelian semilocal strings in N = 2 supersymmetric QCD, Phys. Rev. D 83 (2011) 125017 [arXiv:1104.2077] [INSPIRE].
M. Shifman and A. Yung, Non-Abelian semilocal strings in N = 2 supersymmetric QCD, Phys. Rev. D 73 (2006) 125012 [hep-th/0603134] [INSPIRE].
M. Eto et al., On the moduli space of semilocal strings and lumps, Phys. Rev. D 76 (2007) 105002 [arXiv:0704.2218] [INSPIRE].
E. Witten, Instantons, the quark model, and the 1/n expansion, Nucl. Phys. B 149 (1979) 285 [INSPIRE].
E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [INSPIRE].
A. Hanany and K. Hori, Branes and N = 2 theories in two-dimensions, Nucl. Phys. B 513 (1998) 119 [hep-th/9707192] [INSPIRE].
K. Hori and C. Vafa, Mirror symmetry, hep-th/0002222 [INSPIRE].
A. Neitzke and C. Vafa, Topological strings and their physical applications, hep-th/0410178 [INSPIRE].
M. Shifman and A. Yung, Critical string from non-Abelian vortex in four dimensions, Phys. Lett. B 750 (2015) 416 [arXiv:1502.00683] [INSPIRE].
P. Koroteev, M. Shifman and A. Yung, Non-Abelian vortex in four dimensions as a critical string on a conifold, Phys. Rev. D 94 (2016) 065002 [arXiv:1605.08433] [INSPIRE].
M. Shifman and A. Yung, Critical non-Abelian vortex in four dimensions and little string theory, Phys. Rev. D 96 (2017) 046009 [arXiv:1704.00825] [INSPIRE].
C.-H. Sheu and M. Shifman, From gauged linear σ-models to geometric representation of 𝕎ℂℙ(N, \( \tilde{N} \)) in 2D, Phys. Rev. D 101 (2020) 025007 [arXiv:1907.09460] [INSPIRE].
O. Aharony, S.S. Razamat, N. Seiberg and B. Willett, The long flow to freedom, JHEP 02 (2017) 056 [arXiv:1611.02763] [INSPIRE].
P. Candelas and X.C. de la Ossa, Comments on conifolds, Nucl. Phys. B 342 (1990) 246 [INSPIRE].
L.A. Pando Zayas and A.A. Tseytlin, 3-branes on spaces with R × S2 × S3 topology, Phys. Rev. D 63 (2001) 086006 [hep-th/0101043] [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Factorized s matrices in two-dimensions as the exact solutions of certain relativistic quantum field models, Annals Phys. 120 (1979) 253 [INSPIRE].
P. Koroteev, M. Shifman, W. Vinci and A. Yung, Quantum dynamics of low-energy theory on semilocal non-Abelian strings, Phys. Rev. D 84 (2011) 065018 [arXiv:1107.3779] [INSPIRE].
C. Li, On rotationally symmetric Kahler-Ricci solitons, arXiv:1004.4049.
A. D’Adda, A.C. Davis, P. Di Vecchia and P. Salomonson, An effective action for the supersymmetric CP(N−1) model, Nucl. Phys. B 222 (1983) 45 [INSPIRE].
N. Dorey, T.J. Hollowood and D. Tong, The BPS spectra of gauge theories in two-dimensions and four-dimensions, JHEP 05 (1999) 006 [hep-th/9902134] [INSPIRE].
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Chen, J., Sheu, CH., Shifman, M. et al. Long way to Ricci flatness. J. High Energ. Phys. 2020, 59 (2020). https://doi.org/10.1007/JHEP10(2020)059
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DOI: https://doi.org/10.1007/JHEP10(2020)059