Fractional instantons and bions in the O(N) model with twisted boundary conditions

TL;DR

This work classifies fractional instantons and neutral bions in the $O(N)$ nonlinear sigma model on ${\mathbb R}^{N-2}\times S^1$ with twisted boundary conditions that flip signs of subsets of fields. By formulating a general framework, the authors show that a fixed vacuum manifold ${\cal N}\simeq S^{n}$ supports a host soliton with $\pi_n({\cal N})\simeq\mathbb{Z}$, while the moduli on the host worldvolume form ${\cal M}\simeq S^m$, twisted along the compact direction to produce a half-integer daughter soliton with $\pi_m^{\rm bc}({\cal M})\simeq \mathbb{Z}+\tfrac{1}{2}$; the total charge then lies in $\pi_{N-1}({\cal M})$, giving rise to half-instantons. Specializing to $O(3)$ on ${\mathbb R}^1\times S^1$ and $O(4)$ on ${\mathbb R}^2\times S^1$, the paper identifies three distinct fractional instanton types in $O(3)$—a global vortex with an Ising spin, a half sine-Gordon kink on a domain wall, and a half lump on a space-filling brane—and four in $O(4)$—a global monopole with an Ising spin, a half sine-Gordon kink on a global vortex, a half lump on a domain wall, and a half Skyrmion on a space-filling brane. Neutral bions are constructed from pairs of fractional instantons; charged bions are shown not to arise in this setup. The analysis also discusses interactions, Scherk–Schwarz dimensional reduction, and potential modifications (gauge couplings, potentials, or Skyrme terms) that could render some fractional objects local or BPS, with implications for resurgence and possible extensions to more directions or non-Abelian gauge theories.

Abstract

Recently, multiple fractional instanton configurations with zero instanton charge, called bions, have been revealed to play important roles in quantum field theories on compactified spacetime. In two dimensions, fractional instantons and bions have been extensively studied in the ${\mathbb C}P^{N-1}$ model and the Grassmann sigma model on ${\mathbb R}^1 \times S^1$ with the ${\mathbb Z}_N$ symmetric twisted boundary condition. Fractional instantons in these models are domain walls with a localized $U(1)$ modulus twisted half along their world volume. In this paper, we classify fractional instantons and bions in the $O(N)$ nonlinear sigma model on ${\mathbb R}^{N-2} \times S^1$ with more general twisted boundary conditions in which arbitrary number of fields change sign. We find that fractional instantons have more general composite structures, that is, a global vortex with an Ising spin (or a half-lump vortex), a half sine-Gordon kink on a domain wall, or a half lump on a "space-filling brane" in the $O(3)$ model (${\mathbb C}P^1$ model) on ${\mathbb R}^{1} \times S^1$, and a global monopole with an Ising spin (or a half-Skyrmion monopole), a half sine-Gordon kink on a global vortex, a half lump on a domain wall, or a half Skyrmion on a "space-filling brane" in the $O(4)$ model (principal chiral model or Skyrme model) on ${\mathbb R}^{2} \times S^1$. We also construct bion configurations in these models.

Figures (16)

• Figure 1: Fractional instantons in the $O(3)$ model with the twisted boundary conditions (1) $(-,+,+)$, (2) $(-,-,+)$ and (3) $(-,-,-)$. Black and red arrows denote the moduli space ${\cal N}$ of vacua and the moduli space ${\cal M}$ of a host soliton, respectively, as we explain in more detail in later sections. The first lines indicate the topological charges (homotopy groups) characterizing (a host soliton, a daughter soliton, the total instanton charge) are $(\pi_{1},\pi_{0},\pi_2)$ for (1a)--(1d), $(\pi_{0},\pi_{1},\pi_2)$ for (2a)--(2d), and $(\pi_{-1},\pi_{2},\pi_2)$ for (3a)--(3d), where $\pi_{-1}$ is merely formal. For each boundary condition, fractional (anti-)instantons can make following composite structures: (a)+(b) instanton, (c)+(d) anti-instanton, (a)+(c), (b)+(d) bions.
• Figure 2: Fractional instantons in the $O(4)$ model with the twisted boundary conditions (1) $(-,+,+,+)$, (2) $(-,-,+,+)$, (3) $(-,-,-,+)$ and (4) $(-,-,-,-)$. The notations of black and red arrows are the same with Fig. \ref{['fig:O(3)']}. The first lines indicate the topological charges (homotopy groups) characterizing (a host soliton, a daughter soliton, the total instanton charge) are $(\pi_{2},\pi_{0},\pi_3)$ for (1a)--(1d), $(\pi_{1},\pi_{1},\pi_3)$ for (2a)--(2d), $(\pi_{0},\pi_{2},\pi_3)$ for (3a)--(3d), and $(\pi_{-1},\pi_{3},\pi_3)$ for (4a)--(4d), where $\pi_{-1}$ is merely formal. For each boundary condition, fractional (anti-)instantons can make following composite structures: (a)+(b) instanton, (c)+(d) anti-instanton, (a)+(c), (b)+(d) bions.
• Figure 3: Fractional instantons in the $O(3)$ model with the boundary condition $(-,+,+)$. $\odot$ and $\otimes$ correspond to $n_1=+1$ and $n_1 = -1$, respectively. Black arrows represent $(n_2,n_3)$ with $n_2^2+n_3^2=1$ ($n_1=0$) parameterizing the moduli space of vacua ${\cal N} \simeq S^1$: $\leftarrow$, $\rightarrow$, $\uparrow$, $\downarrow$ correspond to $n_3 = +1$, $n_3 = -1$, $n_2 = +1$, $n_2 = -1$, respectively. We chose the vacuum $n_3=+1$ at the boundary. Topological charges $(*,*,*)$ denote a host vortex charge $\pi_1$, an Ising spin $\pi_0$ in its core, and the total instanton charge $\pi_2$, respectively. (a) An instanton is split into two fractional instantons $(+1,+\frac{1}{2},+\frac{1}{2})$ and $(-1,-\frac{1}{2},+\frac{1}{2})$ separated into the $x^1$ direction by a sine-Gordon domain wall. (b) An anti-instanton is split into two fractional anti-instantons $(+1,-\frac{1}{2},-\frac{1}{2})$ and $(-1,+\frac{1}{2},-\frac{1}{2})$ separated into the $x^1$ direction by a sine-Gordon anti-domain wall. (c) and (d) are isomorphic to (a) and (b), respectively, by a $2\pi$ rotation along an axis at the center of the sine-Gordon (anti-)domain wall, which exchanges two fractional instantons.
• Figure 4: Images of fractional instantons in the target space $S^2$ of the $O(3)$ model with the boundary conditions (a) $(-,+,+)$, (b) $(-,-,+)$ and (c) $(-,-,-)$. Each path represents an image of $x_1=$ constant with $x_2=0$ to $x_2 = R$, where an arrow indicates a direction. With changing $x_1$ from $x_1=-\infty$ to $x_1=+\infty$, the path moves following the blue arrow to cover a half sphere.
• Figure 5: Bions in the $O(3)$ model with the boundary condition $(-,+,+)$. The notations are the same with Fig. \ref{['fig:fractional-O3-1']}. (c) and (d) are isomorphic to (a) and (b), respectively, by a $2\pi$ rotation along an axis at the center of the sine-Gordon (anti-)domain wall.