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Supersymmetric Solitons and How They Help Us Understand Non-Abelian Gauge Theories

M. Shifman, A. Yung

TL;DR

The paper surveys how supersymmetry reveals BPS solitons—domain walls, vortices, and monopoles—and how their central charges constrain exact results at strong coupling. It develops three strands: non-Abelian strings with confined monopoles in N=2 theories, controlled reductions to N=1 and non-SUSY via the M model, and domain walls as D-brane prototypes with localized gauge fields and wall–string junctions. A key insight is the worldsheet realization: non-Abelian strings host CP(N−1) dynamics, with confined monopoles appearing as CP(N−1) kinks, and a deep 4D–2D correspondence (Seiberg–Witten in 4D mirrored by CP(N−1) structures in 2D). The work extends these ideas to less SUSY, clarifying when non-Abelian confinement persists and how infrared issues are resolved, thereby connecting gauge dynamics across dimensions and highlighting potential pathways toward understanding QCD-like confinement in non-supersymmetric theories.

Abstract

In the last decade it became clear that methods and techniques based on supersymmetry provide deep insights in quantum chromodynamics and other supersymmetric and non-supersymmetric gauge theories at strong coupling. In this review we summarize major advances in the critical (Bogomol'nyi-Prasad-Sommerfeld-saturated, BPS for short) solitons in supersymmetric theories and their implications for understanding basic dynamical regularities of non-supersymmetric theories. After a brief introduction in the theory of critical solitons (including a historical introduction) we focus on three topics: (i) non-Abelian strings in N=2 and confined monopoles; (ii) reducing the level of supersymmetry; and (iii) domain walls as D brane prototypes.

Supersymmetric Solitons and How They Help Us Understand Non-Abelian Gauge Theories

TL;DR

The paper surveys how supersymmetry reveals BPS solitons—domain walls, vortices, and monopoles—and how their central charges constrain exact results at strong coupling. It develops three strands: non-Abelian strings with confined monopoles in N=2 theories, controlled reductions to N=1 and non-SUSY via the M model, and domain walls as D-brane prototypes with localized gauge fields and wall–string junctions. A key insight is the worldsheet realization: non-Abelian strings host CP(N−1) dynamics, with confined monopoles appearing as CP(N−1) kinks, and a deep 4D–2D correspondence (Seiberg–Witten in 4D mirrored by CP(N−1) structures in 2D). The work extends these ideas to less SUSY, clarifying when non-Abelian confinement persists and how infrared issues are resolved, thereby connecting gauge dynamics across dimensions and highlighting potential pathways toward understanding QCD-like confinement in non-supersymmetric theories.

Abstract

In the last decade it became clear that methods and techniques based on supersymmetry provide deep insights in quantum chromodynamics and other supersymmetric and non-supersymmetric gauge theories at strong coupling. In this review we summarize major advances in the critical (Bogomol'nyi-Prasad-Sommerfeld-saturated, BPS for short) solitons in supersymmetric theories and their implications for understanding basic dynamical regularities of non-supersymmetric theories. After a brief introduction in the theory of critical solitons (including a historical introduction) we focus on three topics: (i) non-Abelian strings in N=2 and confined monopoles; (ii) reducing the level of supersymmetry; and (iii) domain walls as D brane prototypes.

Paper Structure

This paper contains 94 sections, 600 equations, 24 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Central charges in superalgebras
  3. History
  4. Minimal supersymmetry
  5. $D=2$
  6. $D=3$
  7. $D=4$
  8. Extended SUSY
  9. ${\mathcal{N}}=2$ in $D=2$
  10. ${\mathcal{N}}=2$ in $D=3$
  11. On extended supersymmetry (eight supercharges) in $D=4$
  12. The main building blocks
  13. Domain walls
  14. Preliminaries
  15. $D$ Branes in Gauge Field Theory
  16. ...and 79 more sections

Figures (24)

  • Figure 1: A field configuration interpolating between two distinct degenerate vacua
  • Figure 2: $N$ vacua for SU($N$). The vacua are labeled by the vacuum expectation value $\langle\lambda\lambda\rangle = -6\,N\,\Lambda^3\, \exp(2\pi\,i\,k/N)$ where $k=0,1,...,N-1$. Elementary walls interpolate between two neighboring vacua
  • Figure 3: Two distinct degenerate domain walls separated by the wall junction.
  • Figure 4: The cross section of the wall junction.
  • Figure 5: Polar coordinates on the $x,z$ plane.
  • ...and 19 more figures