Figure 1

The change (2.17) in the action under the fluctuation (2.16), for the $Q=0$ non-self-dual configuration plotted below in the second row of Fig. 2. 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.Reuse & Permissions

Figure 2

The action and charge density configurations due to successive mappings from the ansatz solution (3.1) in ${\mathbb{C}\mathbb{P}}^2$ on ${\mathbb{R}}^2$: ${\omega }_{(0)}=(1,\lambda e^{i{\theta }_1}(z-a),\mu e^{i{\theta }_2}(z^2-b^2))$ 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.Reuse & Permissions

Figure 3

The action and charge density configurations due to successive mappings from the ansatz solution (4.2) in ${\mathbb{C}\mathbb{P}}^2$ on ${\mathbb{S}}_L^1\times {\mathbb{R}}^1$: ${\omega }_{(0)}=(1,\lambda e^{i{\theta }_1}{e}^{-2\pi z/3},\mu e^{i{\theta }_2}{e}^{-4\pi z/3})$ 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.Reuse & Permissions

Figure 4

The action and charge density configurations due to successive mappings from the ansatz solution (4.4) in ${\mathbb{C}\mathbb{P}}^2$ on ${\mathbb{S}}_L^1\times {\mathbb{R}}^1$: ${\omega }_{(0)}=(1,\lambda e^{i{\theta }_1}{e}^{-2\pi z/3}+\mu e^{i{\theta }_2}{e}^{-8\pi z/3},\nu e^{i{\theta }_3}{e}^{-4\pi z/3})$ 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 Fig. 5. 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.Reuse & Permissions

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 Fig. 4. Both are plotted with the same parameters used in Fig. 4.Reuse & Permissions