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In the leading order of the large-$N$ approximation, we study the renormalonambiguity in the gluon (or, more appropriately, photon) condensate in the 2Dsupersymmetric $\mathbb{C}P^{N-1}$ model on~$\mathbb{R}\times S^1$ with the$\mathbb{Z}_N$ twisted boundary conditions. In our large~$N$ limit, thecombination $\Lambda R$, where $\Lambda$ is the dynamical scale and $R$~is the$S^1$ radius, is kept fixed (we set $\Lambda R\ll1$ so that the perturbativeexpansion with respect to the coupling constant at the mass scale~$1/R$ ismeaningful). We extract the perturbative part from the large-$N$ expression ofthe gluon condensate and obtain the corresponding Borel transform~$B(u)$.For~$\mathbb{R}\times S^1$, we find that the Borel singularity at~$u=2$, whichexists in the system on the uncompactified~$\mathbb{R}^2$ and corresponds totwice the minimal bion action, disappears. Instead, an unfamiliar renormalonsingularity \emph{emerges\/} at~$u=3/2$ for the compactifiedspace~$\mathbb{R}\times S^1$. The semi-classical interpretation of thispeculiar singularity is not clear because $u=3/2$ is not dividable by theminimal bion action. It appears that our observation for the systemon~$\mathbb{R}\times S^1$ prompts reconsideration on the semi-classical bionpicture of the infrared renormalon.

Infrared renormalon in the supersymmetric $\mathbb{C}P^{N-1}$ model on $\mathbb{R}\times S^1$