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Fractional instantons and bions in the principal chiral model on ${\mathbb R}^2\times S^1$ with twisted boundary conditions

Muneto Nitta

TL;DR

This work classifies fractional instantons and bions in the ${\rm SU}(N)$ principal chiral model on ${\mathbb R}^2\times S^1$ with twisted boundary conditions. Fractional instantons arise as global vortices wrapping the ${S^1}$ direction with $U(1)$ moduli twisted by angles set by the boundary condition, yielding instanton numbers $B=\pm 1/N$ under ${\mathbb Z}_N$ symmetry (and irrational values for generic twists). The authors construct neutral and charged bions for ${\rm SU}(2)$ and ${\rm SU}(3)$, analyze their charges and interactions, and generalize to ${\rm SU}(N)$, showing how cross-cancellation of fractional charges is restricted for generic twists. A correspondence to Yang-Mills fractional instantons and bions is established via a non-Abelian Josephson junction on a domain wall, linking semiclassical objects across theories and offering a pathway to resurgence and confinement insights.

Abstract

Bions are multiple fractional instanton configurations with zero instanton charge playing important roles in quantum field theories on a compactified space with a twisted boundary condition. We classify fractional instantons and bions in the $SU(N)$ principal chiral model on ${\mathbb R}^2 \times S^1$ with twisted boundary conditions. We find that fractional instantons are global vortices wrapping around $S^1$ with their $U(1)$ moduli twisted along $S^1$, that carry $1/N$ instanton (baryon) numbers for the ${\mathbb Z}_N$ symmetric twisted boundary condition and irrational instanton numbers for generic boundary condition. We work out neutral and charged bions for the $SU(3)$ case with the ${\mathbb Z}_3$ symmetric twisted boundary condition. We also find for generic boundary conditions that only the simplest neutral bions have zero instanton charges but instanton charges are not canceled out for charged bions. A correspondence between fractional instantons and bions in the $SU(N)$ principal chiral model and those in Yang-Mills theory is given through a non-Abelian Josephson junction.

Fractional instantons and bions in the principal chiral model on ${\mathbb R}^2\times S^1$ with twisted boundary conditions

TL;DR

This work classifies fractional instantons and bions in the principal chiral model on with twisted boundary conditions. Fractional instantons arise as global vortices wrapping the direction with moduli twisted by angles set by the boundary condition, yielding instanton numbers under symmetry (and irrational values for generic twists). The authors construct neutral and charged bions for and , analyze their charges and interactions, and generalize to , showing how cross-cancellation of fractional charges is restricted for generic twists. A correspondence to Yang-Mills fractional instantons and bions is established via a non-Abelian Josephson junction on a domain wall, linking semiclassical objects across theories and offering a pathway to resurgence and confinement insights.

Abstract

Bions are multiple fractional instanton configurations with zero instanton charge playing important roles in quantum field theories on a compactified space with a twisted boundary condition. We classify fractional instantons and bions in the principal chiral model on with twisted boundary conditions. We find that fractional instantons are global vortices wrapping around with their moduli twisted along , that carry instanton (baryon) numbers for the symmetric twisted boundary condition and irrational instanton numbers for generic boundary condition. We work out neutral and charged bions for the case with the symmetric twisted boundary condition. We also find for generic boundary conditions that only the simplest neutral bions have zero instanton charges but instanton charges are not canceled out for charged bions. A correspondence between fractional instantons and bions in the principal chiral model and those in Yang-Mills theory is given through a non-Abelian Josephson junction.

Paper Structure

This paper contains 14 sections, 67 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The principal chiral model on ${\mathbb R}^2 \times S^1$ with twisted boundary conditions
  3. Fractional instantons and bions in the $SU(2)$ principal chiral model
  4. Fractional instantons
  5. Neutral bions
  6. Charged bions
  7. Fractional instantons and bions in the $SU(3)$ principal chiral model
  8. Fractional instantons
  9. Neutral bions
  10. Charged bions
  11. Generalization to the $SU(N)$ principal chiral model
  12. More general twisted boundary conditions
  13. Relation to Yang-Mills fractional instantons and bions
  14. Summary and Discussion

Figures (2)

  • Figure 1: Fractional instantons in the $SU(2)$ principal chiral model (figures are taken from Ref. Nitta:2014vpa). The first lines indicate the topological charges (homotopy groups) $(\pi_{1}({\cal N});\pi_3(M))$ for the vortices and instantons (Skyrmions). The black arrows denote the $U(1)$ moduli of the vacua while the red arrows denote the $U(1)$ moduli of the vortices. Fractional (anti-)instantons can constitute following composite structures: (a)+(b) instanton, (c)+(d) anti-instanton, (a)+(c), (b)+(d) neutral bions, (a)+(d), (b)+(c) charged bions.
  • Figure 2: Decay of a twisted closed vortex string of the size of the compact direction into two fractional instantons in the $SU(2)$ principal chiral model (figures are taken from Ref. Nitta:2014vpa). The notations are the same with Fig. \ref{['fig:SU(2)']}. The dotted planes denote the boundary at $z=0$ and $z=R$ where the fields are twisted. When a closed vortex touches to itself through the compact direction $z$, a reconnection of the two parts of the string occurs to be split into two fractional (anti-)instantons, that is, vortices winding around $S^1$ with the half twisted $U(1)$ moduli.