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

TL;DR

This paper classifies fractional instantons and bions in the $O(N)$ nonlinear sigma model with twisted boundary conditions, revealing complex composite structures and extending understanding of topological configurations in quantum field theories.

Contribution

It introduces a comprehensive classification of fractional instantons and bions in the $O(N)$ model with generalized twisted boundary conditions, including new composite structures.

Findings

Fractional instantons include global vortices, monopoles, and half-lumps.

New composite structures such as half sine-Gordon kinks and half Skyrmions are identified.

Constructed explicit bion configurations in the models.

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.