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

TL;DR

The paper demonstrates that neutral bions—composites of fractionalized instantons with zero net topological and magnetic charge—can be explicitly modeled in the ${\mathbb C}P^{N-1}$ model on ${\mathbb R}^{1}\times S^{1}$ under ${\mathbb Z}_{N}$ and non-${\mathbb Z}_{N}$ twists. By constructing explicit ansätze and analyzing the separation dependence of the bosonic interaction, the authors show consistency with the standard far-separated instanton calculus and identify a universal exponential attraction with a characteristic slope that scales as $\xi=2\pi/N$ (and analogs in non-${\mathbb Z}_{N}$ cases). Their results support the resurgence framework in which neutral bions cancel infrared renormalon ambiguities, extending the validity of this mechanism across a range of boundary conditions and vacua. The work also lays groundwork for exploring charged bions and connections to confinement mechanisms in related gauge theories.

Abstract

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 on specific configurations composed of multiple fractionalized-instantons, termed "neutral bions", which are identified as "perturbative infrared renormalons" by Ünsal and his collaborators. For ${\mathbb Z}_N$ twisted boundary conditions, we consider an explicit ansatz corresponding to topologically trivial configurations containing one fractionalized instanton ($ν=1/N$) and one fractionalized anti-instanton ($ν=-1/N$) at large separations, and exhibit the attractive interaction between the instanton constituents and how they behave at shorter separations. We show that the bosonic interaction potential between the constituents as a function of both the separation and $N$ is consistent with the standard separated-instanton calculus even from short to large separations, which indicates that the ansatz enables us to study bions and the related physics for a wide range of separations. We also propose different bion ansatze in a certain non-${\mathbb Z}_{N}$ twisted boundary condition corresponding to the "split" vacuum for $N= 3$ and its extensions for $N\geq 3$. We find that the interaction potential has qualitatively the same asymptotic behavior and $N$-dependence as those of bions for ${\mathbb Z}_{N}$ twisted boundary conditions.

Figures (20)

• Figure 1: Fractionalized instantons in the ${\mathbb C}P^1$ model with the ${\mathbb Z}_2$ twisted boundary condition, corresponding to (a) $\omega_L$, (b) $\omega_R$, (c) $\omega_L^*$, and (d) $\omega_R^*$ (in which we have taken the phase modulus to be $\theta = - \pi/2$). The horizontal and vertical directions are $x_1$ and $x_2$, respectively. The symbols $\odot$, $\otimes$, $\leftarrow$, $\rightarrow$, $\uparrow$ and $\downarrow$ denote ${\bf m}=(0,0,1),(0,0,-1),(-1,0,0),(1,0,0),(0,1,0)$ and $(0,-1,0)$, respectively. The shaded regions imply domain walls with $m^3 \sim 0$. The $\uparrow$ and $\downarrow$ at the boundaries at $x_2=+1$ and $x_2=0$ are identified by the twisted boundary condition. The domain wall charges are (a) $+1$, (b) $-1$, (c) $+1$, (d) $-1$, and the instanton charges $Q$ are (a) $+1/2$, (b) $+1/2$, (c) $-1/2$, (d) $-1/2$.
• Figure 2: (a) Domain wall, and (b) and (c) fractionalized instantons in the target space $S^2$. (b) corresponds to the configurations $\omega_R$ and $\omega_L^*$ while (c) corresponds to the configurations $\omega_L$ and $\omega_R^*$.
• Figure 3: Neutral bion. This is a composite of fractionalized instantons $\omega_L$ and $\omega_R^*$, where we have introduced a relative phase. The notation is the same as Fig. \ref{['fig:fractional-config']}
• Figure 4: Action density $s(x_{1})$ and topological charge density $q(x_{1})$ for the configuration of Eq. (\ref{['BBozero']}) for $\lambda_{1}=1000$, $\lambda_{2}=1$ and $\theta_{2}=0$. The distance between the peaks of two fractionalized instantons is given by $\sim4.3976$, which is consistent with the separation $(1/\pi)\log(1000^{2})$ obtained from Eq. (\ref{['sepa']}).
• Figure 5: The $\sqrt{\lambda_{1}^{2}/\lambda_{2}}$ dependence of the total action $S$ with $\theta_{2}=0$ for (\ref{['BBozero']}). The action is independent of $\lambda_{2}$ for $\lambda_{1}^{2}/\lambda_{2}$ fixed. The configuration is changed from $S=1$ to $S=0$, due to the attractive force.