Fractionalized Non-Self-Dual Solutions in the CP(N-1) Model

TL;DR

The paper studies exact, finite-action non-self-dual solutions in the CP^{N-1} model under $Z_N$ twists on the spatially compactified cylinder, revealing a rich fractionalization of action and topological charge into instanton–anti-instanton components. It extends Din and Zakrzewski’s construction to $S^1_L\times R$, showing that these saddle points yield imaginary non-perturbative contributions that cancel against infrared renormalon effects in a resurgent semi-classical expansion, thus supporting a Bogomolny–Zinn-Justin–type cancellation in this field theory context. The authors provide explicit CP^{2} examples on $R^2$, $S^2$, and $S^1_L\times R$, detailing how actions and charges follow the relations $S_{(k)}=Q_{(k)}+2\sum_{j<k}Q_{(j)}$ and illustrating the emergence of fractional charges in units of $1/N$. The work argues for a systematic analysis of negative modes associated with these saddles and suggests similar phenomena in twisted Yang–Mills and other two-dimensional sigma models, highlighting the role of resurgence in connecting non-perturbative saddles with perturbative ambiguities.

Abstract

We study non-self-dual classical solutions in the CP(N-1) model with Z_N twisted boundary conditions on the spatially compactified cylinder. These solutions have finite, and fractional, classical action and topological charge, and are `unstable' in the sense that the corresponding fluctuation operator has negative modes. We propose a physical interpretation of these solutions as saddle point configurations whose contributions to a resurgent semi-classical analysis of the quantum path integral are imaginary non-perturbative terms which must be cancelled by infrared renormalon terms generated in the perturbative sector.

Figures (5)

• Figure 1: The change (\ref{['unstable']}) in the action under the fluctuation (\ref{['change']}), for the $Q=0$ non-self-dual configuration plotted below in the second row of Figure \ref{['fig:CP2_R2']}. The horizontal axis denotes the (symmetric) distance of each object from the center. Notice that at large separation this fluctuation is a zero mode, while at finite separation it becomes a negative mode.
• Figure 2: The Action and Charge Density configurations due to successive mappings from the ansatz solution (\ref{['CP2_R2_ansatz']}) in $\mathbb{CP}^2$ on $\mathbb{R}^2$: $\omega_{(0)}=\left(1, \, \lambda \, e^{i \theta_1} \left( z - a \right), \, \mu \, e^{i \theta_2} \left( z^2 - b^2 \right) \right)$ where $a=a_1 + i \, a_2$ and $b=b_1+ i \,b_2$, and plotted for: $\lambda,\mu = 2$, $a_1, a_2 = 0$, $b_1, b_2 = 4$, $\forall$$\theta_1,\theta_2 \in [0,2\pi)$. The initial configuration $\omega_{(0)}$ corresponds to two instantons, while $\omega_{(1)}$ corresponds to two instantons and two anti-instantons, and $\omega_{(2)}$ corresponds to two anti-instantons. These are all exact solutions to the classical equations of motion, but $\omega_{(1)}$ is non-self-dual.
• Figure 3: The Action and Charge Density configurations due to successive mappings from the ansatz solution (\ref{['CP2_Twist_SxR_1_Ansatz']}) in $\mathbb{CP}^2$ on $\mathbb{S}_L^1 \times \mathbb{R}^1$: $\omega_{(0)} = \left( 1, \, \lambda \, e^{i \theta_1} e^{- 2 \pi z /3}, \, \mu \, e^{i \theta_2} e^{- 4\pi z /3}\right)$ where $\lambda = 4000, \mu = 1$, $\forall$$\theta_1,\theta_2 \in [ 0,2\pi)$. The initial configuration $\omega_{(0)}$ corresponds to two fractionalized instantons each of charge $1/3$, while $\omega_{(1)}$ corresponds to one fractionalized instanton of charge $2/3$ and two fractionalized anti-instantons each of charge $-1/3$, and $\omega_{(2)}$ corresponds to a fractionalized anti-instanton of charge $-2/3$. These are all exact solutions to the classical equations of motion, but $\omega_{(1)}$ is non-self-dual.
• Figure 4: The Action and Charge Density configurations due to successive mappings from the ansatz solution (\ref{['second_ansatz']}) in $\mathbb{CP}^2$ on $\mathbb{S}_L^1 \times \mathbb{R}^1$: $\omega_{(0)} = \left( 1, \, \lambda \, e^{i \theta_1} e^{- 2 \pi z /3} + \mu \, e^{i \theta_2} e^{- 8\pi z /3}, \, \nu \, e^{i \theta_3} e^{- 4\pi z /3}\right)$ where $\lambda=10^4, \mu=10^{-2}, \nu = 10^4$, $\theta_1 = \pi, \theta_2 = 0$, $\forall$$\theta_3 \in [ 0,2\pi)$. The initial configuration $\omega_{(0)}$ corresponds to two fractionalized instantons each of charge $1/3$ and another fractionalized instanton of charge $2/3$, while $\omega_{(1)}$ corresponds to one instanton of charge $2/3$ and another of charge $1$ (marked by the black oval) and two anti-instantons each of charge $-1/3$ and another anti-instanton of charge $-2/3$, and $\omega_{(2)}$ corresponds to an anti-instanton of charge $-2/3$ and an anti-instanton of charge $-1$ (marked by the black oval). Notice the appearance of very sharp instanton and anti-instanton peaks in the third, fourth, fifth and sixth plots, marked by the black oval shape, as discussed in the text. These peaks are so sharp that they do not show up on the same scale, but their cross-sections are plotted in Figure \ref{['fig:HighlocalInstanton']}. Note that $\omega_{(0)}$, $\omega_{(1)}$ and $\omega_{(2)}$ are all exact solutions to the classical equations of motion, but $\omega_{(1)}$ is non-self-dual.
• Figure 5: A magnified cross section of the charge density of the highly localized charge 1 instanton and anti-instanton that appear in the fourth and sixth plots in Figure (\ref{['fig:CP2_Twist_SxR_60']}). Both are plotted with the same parameters used in Figure (\ref{['fig:CP2_Twist_SxR_60']}).