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Non-Abelian String Junctions as Confined Monopoles

M. Shifman, A. Yung

TL;DR

<3-5 sentence high-level summary>: This work analyzes confined monopoles in the Higgs phase of ${ m N}=2$ ${ m QCD}$ with gauge group ${ m SU}(2) imes{ m U}(1)$ and two flavors, showing that 1/4-BPS string junctions correspond to BPS kinks of the ${ m CP}^1$ world-sheet theory. Through a complete quasiclassical treatment, the authors derive first-order BPS equations, zero modes, and an effective 2D CP^1 description, with precise mass and central-charge matching including an anomaly, linking 4D monopoles to 2D kink dynamics. They analyze the non-Abelian limit ${ riangle m} o 0$, where monopoles persist as nonperturbative CP^1 kinks stabilized by 2D dynamics, and connect the kink spectrum to the Seiberg-Witten solution via holomorphy, explaining the Higgs/Coulomb branch correspondence. The results establish a robust 4D–2D correspondence for BPS data and provide a framework for generalizing to ${ m CP}^{N-1}$ models on the world sheet.

Abstract

Various dynamical regimes associated with confined monopoles in the Higgs phase of N=2 two-flavor QCD are studied. The microscopic model we deal with has the SU(2)xU(1) gauge group, with a Fayet-Iliopoulos term of the U(1) factor, and large and (nearly) degenerate mass terms of the matter hypermultiplets. We present a complete quasiclassical treatment of the BPS sector of this model, including the full set of the first-order equations, derivations of all relevant zero modes, and derivation of an effective low-energy theory for the corresponding collective coordinates. The macroscopic description is provided by a CP(1) model with or without twisted mass. The confined monopoles -- string junctions of the microscopic theory -- are mapped onto BPS kinks of the CP(1) model. The string junction is 1/4 BPS. Masses and other characteristics of the confined monopoles are matched with those of the CP(1)-model kinks. The matching demonstrates the occurrence of an anomaly in the monopole central charge in 4D Yang-Mills theory. We study what becomes of the confined monopole in the bona fide non-Abelian limit of degenerate mass terms where a global SU(2) symmetry is restored. The solution of the macroscopic model is known e.g. from the mirror description of the CP(1) model. The monopoles, aka CP(1)-model kinks, are stabilized by nonperturbative dynamics of the CP(1) model. We explain an earlier rather puzzling observation of a correspondence between the BPS kink spectrum in the CP(1) model and the Seiberg-Witten solution.

Non-Abelian String Junctions as Confined Monopoles

TL;DR

<3-5 sentence high-level summary>: This work analyzes confined monopoles in the Higgs phase of with gauge group and two flavors, showing that 1/4-BPS string junctions correspond to BPS kinks of the world-sheet theory. Through a complete quasiclassical treatment, the authors derive first-order BPS equations, zero modes, and an effective 2D CP^1 description, with precise mass and central-charge matching including an anomaly, linking 4D monopoles to 2D kink dynamics. They analyze the non-Abelian limit , where monopoles persist as nonperturbative CP^1 kinks stabilized by 2D dynamics, and connect the kink spectrum to the Seiberg-Witten solution via holomorphy, explaining the Higgs/Coulomb branch correspondence. The results establish a robust 4D–2D correspondence for BPS data and provide a framework for generalizing to models on the world sheet.

Abstract

Various dynamical regimes associated with confined monopoles in the Higgs phase of N=2 two-flavor QCD are studied. The microscopic model we deal with has the SU(2)xU(1) gauge group, with a Fayet-Iliopoulos term of the U(1) factor, and large and (nearly) degenerate mass terms of the matter hypermultiplets. We present a complete quasiclassical treatment of the BPS sector of this model, including the full set of the first-order equations, derivations of all relevant zero modes, and derivation of an effective low-energy theory for the corresponding collective coordinates. The macroscopic description is provided by a CP(1) model with or without twisted mass. The confined monopoles -- string junctions of the microscopic theory -- are mapped onto BPS kinks of the CP(1) model. The string junction is 1/4 BPS. Masses and other characteristics of the confined monopoles are matched with those of the CP(1)-model kinks. The matching demonstrates the occurrence of an anomaly in the monopole central charge in 4D Yang-Mills theory. We study what becomes of the confined monopole in the bona fide non-Abelian limit of degenerate mass terms where a global SU(2) symmetry is restored. The solution of the macroscopic model is known e.g. from the mirror description of the CP(1) model. The monopoles, aka CP(1)-model kinks, are stabilized by nonperturbative dynamics of the CP(1) model. We explain an earlier rather puzzling observation of a correspondence between the BPS kink spectrum in the CP(1) model and the Seiberg-Witten solution.

Paper Structure

This paper contains 27 sections, 152 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. A brief summary of the theoretical set-up
  3. Non-Abelian strings
  4. Abrikosov-Nielsen-Olesen vs. elementary strings
  5. Embedding; first-order equations for the elementary strings
  6. Non-Abelian moduli
  7. Supersymmetry
  8. Macroscopic theory
  9. Deriving the kinetic term; a basic normalizing integral
  10. (1+1)-dimensional $CP^{1}$ model
  11. Unequal quark mass terms; $CP^1$ with the twisted mass
  12. Fermion zero modes of the string
  13. Sigma-model kinks: monopoles of the microscopic theory
  14. Microscopic theory: first-order master equations
  15. The string junction solution for $\sqrt\xi\gg \mu\gg \Lambda$
  16. ...and 12 more sections

Figures (9)

  • Figure 1: Various regimes for the monopoles and flux tubes. The dynamical scale parameters $\Lambda$ are the same in the microscopic and microscopic theories. The effective world-sheet theory we derive -- the twisted-mass $CP^1$ model -- applies to the last two regimes: $\Lambda\ll|\Delta m |\ll \xi^{1/2}$ (the left lower corner) and $\Delta m \to 0$ (the right lower corner). The latter case corresponds to the vanishing twisted mass.
  • Figure 2: (a) The monopole-antimonopole pair; (b) the monopole with two (infinitely long) elementary flux tubes attached to it.
  • Figure 3: Geometry of the string.
  • Figure 4: Lattice of $(n,k)$ vortices.
  • Figure 5: Unwinding the (1,-1)-string.
  • ...and 4 more figures