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We study classical configurations in the ${\mathbb C}P^{N-1}$ model on${\mathbb R}^{1}\times S^{1}$ with twisted boundary conditions. We focus onspecific configurations composed of multiple fractionalized-instantons, termed"neutral bions", which are identified as "perturbative infrared renormalons" by\"{U}nsal and his collaborators. For ${\mathbb Z}_N$ twisted boundaryconditions, we consider an explicit ansatz corresponding to topologicallytrivial configurations containing one fractionalized instanton ($\nu=1/N$) andone fractionalized anti-instanton ($\nu=-1/N$) at large separations, andexhibit the attractive interaction between the instanton constituents and howthey behave at shorter separations. We show that the bosonic interactionpotential between the constituents as a function of both the separation and $N$is consistent with the standard separated-instanton calculus even from short tolarge separations, which indicates that the ansatz enables us to study bionsand the related physics for a wide range of separations. We also proposedifferent bion ansatze in a certain non-${\mathbb Z}_{N}$ twisted boundarycondition corresponding to the "split" vacuum for $N= 3$ and its extensions for$N\geq 3$. We find that the interaction potential has qualitatively the sameasymptotic behavior and $N$-dependence as those of bions for ${\mathbb Z}_{N}$twisted boundary conditions.

Neutral bions in the ${\mathbb C}P^{N-1}$ model