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

In the leading order of the large-$N$ approximation, we study the renormalon ambiguity in the gluon (or, more appropriately, photon) condensate in the 2D supersymmetric $mathbb{C}P^{N-1}$ model on~$mathbb{R}times S^1$ with the $mathbb{Z}_N$ twisted boundary conditions. In our large~$N$ limit, the combination $Lambda R$, where $Lambda$ is the dynamical scale and $R$~is the $S^1$ radius, is kept fixed (we set $Lambda Rll1$ so that the perturbative expansion with respect to the coupling constant at the mass scale~$1/R$ is meaningful). We extract the perturbative part from the large-$N$ expression of the gluon condensate and obtain the corresponding Borel transform~$B(u)$. For~$mathbb{R}times S^1$, we find that the Borel singularity at~$u=2$, which exists in the system on the uncompactified~$mathbb{R}^2$ and corresponds to twice the minimal bion action, disappears. Instead, an unfamiliar renormalon singularity emph{emerges/} at~$u=3/2$ for the compactified space~$mathbb{R}times S^1$. The semi-classical interpretation of this peculiar singularity is not clear because $u=3/2$ is not dividable by the minimal bion action. It appears that our observation for the system on~$mathbb{R}times S^1$ prompts reconsideration on the semi-classical bion picture of the infrared renormalon.