Vacuum energy of the supersymmetric $\mathbb{C}{P}^{N-1}$ model on $\mathbb{R}\times S^1$ in the $1/N$ expansion

Abstract

By employing the $1/N$ expansion, we compute the vacuum energy~$E(\delta \epsilon )$ of the two-dimensional supersymmetric (SUSY) $\mathbb{C}{P}^{N-1}$ model on~$\mathbb{R}\times S^1$ with ${\mathbb{Z}}_N$ twisted boundary conditions to the second order in a SUSY-breaking parameter~$\delta \epsilon$. This quantity was vigorously studied recently by Fujimori et\ al.\ using a semi-classical approximation based on the bion, motivated by a possible semi-classical picture on the infrared renormalon. In our calculation, we find that the parameter~$\delta \epsilon$ receives renormalization and, after this renormalization, the vacuum energy becomes ultraviolet finite. To the next-to-leading order of the $1/N$ expansion, we find that the vacuum energy normalized by the radius of the~$S^1$, $R$, $RE(\delta \epsilon )$ behaves as inverse powers of~$\Lambda R$ for~$\Lambda R$ small, where $\Lambda$ is the dynamical scale. Since $\Lambda$ is related to the renormalized 't~Hooft coupling~${\lambda }_R$ as~$\Lambda \sim e^{-2\pi /{\lambda }_R}$, to the order of the $1/N$ expansion we work out, the vacuum energy is a purely non-perturbative quantity and has no well-defined weak coupling expansion in~${\lambda }_R$.

arXiv comment24 pages, 3 figures. Fig. 2 was replaced by a correct one and corresponding descriptions in the text were corrected. A note added. The final version to appear in PTEP

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Cited by in corpus (3)

• Lattice $\mathbb{C}{P}^{N-1}$ model with ${\mathbb{Z}}_N$ twisted boundary condition: bions, adiabatic continuity and pseudo-entropy
• Quantum phase transition and Resurgence: Lessons from 3d $\mathcal{N}=4$ SQED
• More on the infrared renormalon in $SU(N)$ QCD(adj.) on ${\mathbb{R}}^3\times S^1$