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 |$\Lambda R\ll1$| 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 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.