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Non-Abelian Flux Tubes in SQCD: Supersizing World-Sheet Supersymmetry

M. Shifman, A. Yung

TL;DR

The paper analyzes non-Abelian 1/2 BPS flux tubes in a deformed ${ m N}=2$ bulk theory with adjoint-mass terms that break to ${ m N}=1$, uncovering a CP$(1)$ world-sheet theory governing orientational moduli. Remarkably, the world-sheet SUSY is enhanced to ${ m N}=2$ across a broad range of breaking parameters, due to the Kähler structure of the reduced moduli space and fermion zero-mode counting, with four fermionic zero modes persisting even as bulk SUSY is reduced. In the large-$bc$ regime, adjoint fields decouple and the world-sheet theory remains CP$(1)$ with ${ m N}=2$ SUSY for finite IR regularization, but a critical threshold $b_2^{*}$ exists beyond which higher-derivative corrections break the enhancement to ${ m N}=1$ and render the strings thick. The study further identifies a kink-monopole on the string world sheet, interpreted as a confined non-Abelian monopole with a world-sheet central charge, illustrating a broader phenomenon of supersymmetry emergence in solitonic systems and highlighting the interplay between four-dimensional bulk dynamics and two-dimensional world-sheet physics.

Abstract

We consider non-Abelian 1/2 BPS flux tubes (strings) in a deformed N=2 supersymmetric gauge theory, with mass terms mu_{1,2} of the adjoint fields breaking N=2 down to N=1. The main feature of the non-Abelian strings is the occurrence of orientational moduli associated with the possibility of rotations of their color fluxes inside a global SU(N) group. The bulk four-dimensional theory has four supercharges; half-criticality of the non-Abelian strings would imply then N=1 supersymmetry on the world sheet, i.e. two supercharges. In fact, superalgebra of the reduced moduli space has four supercharges. Internal dynamics of the orientational moduli are described by two-dimensional CP(N-1) model on the string world sheet. We focus mainly on the SU(2) case, i.e. CP(1) world-sheet theory. We show that non-Abelian BPS strings exist for all values of mu_{1,2}. The low-energy theory of moduli is indeed CP(1), with four supercharges, in a wide region of breaking parameters mu_{1,2}. Only in the limit of very large mu_{1,2}, above some critical value, the N=2 world-sheet supersymmetry breaks down to N=1. We observe "supersymmetry emergence" for the flux-tube junction (confined monopole): the "kink-monopole" is half-critical considered from the standpoint of the world-sheet CP(1) model (i.e. two supercharges conserved), while in the bulk N=1 theory there is no monopole central charge at all.

Non-Abelian Flux Tubes in SQCD: Supersizing World-Sheet Supersymmetry

TL;DR

The paper analyzes non-Abelian 1/2 BPS flux tubes in a deformed bulk theory with adjoint-mass terms that break to , uncovering a CP world-sheet theory governing orientational moduli. Remarkably, the world-sheet SUSY is enhanced to across a broad range of breaking parameters, due to the Kähler structure of the reduced moduli space and fermion zero-mode counting, with four fermionic zero modes persisting even as bulk SUSY is reduced. In the large- regime, adjoint fields decouple and the world-sheet theory remains CP with SUSY for finite IR regularization, but a critical threshold exists beyond which higher-derivative corrections break the enhancement to and render the strings thick. The study further identifies a kink-monopole on the string world sheet, interpreted as a confined non-Abelian monopole with a world-sheet central charge, illustrating a broader phenomenon of supersymmetry emergence in solitonic systems and highlighting the interplay between four-dimensional bulk dynamics and two-dimensional world-sheet physics.

Abstract

We consider non-Abelian 1/2 BPS flux tubes (strings) in a deformed N=2 supersymmetric gauge theory, with mass terms mu_{1,2} of the adjoint fields breaking N=2 down to N=1. The main feature of the non-Abelian strings is the occurrence of orientational moduli associated with the possibility of rotations of their color fluxes inside a global SU(N) group. The bulk four-dimensional theory has four supercharges; half-criticality of the non-Abelian strings would imply then N=1 supersymmetry on the world sheet, i.e. two supercharges. In fact, superalgebra of the reduced moduli space has four supercharges. Internal dynamics of the orientational moduli are described by two-dimensional CP(N-1) model on the string world sheet. We focus mainly on the SU(2) case, i.e. CP(1) world-sheet theory. We show that non-Abelian BPS strings exist for all values of mu_{1,2}. The low-energy theory of moduli is indeed CP(1), with four supercharges, in a wide region of breaking parameters mu_{1,2}. Only in the limit of very large mu_{1,2}, above some critical value, the N=2 world-sheet supersymmetry breaks down to N=1. We observe "supersymmetry emergence" for the flux-tube junction (confined monopole): the "kink-monopole" is half-critical considered from the standpoint of the world-sheet CP(1) model (i.e. two supercharges conserved), while in the bulk N=1 theory there is no monopole central charge at all.

Paper Structure

This paper contains 17 sections, 127 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The bulk theory
  3. Abelian bulk theory
  4. Non-Abelian bulk theory
  5. Non-Abelian strings
  6. The non-Abelian string solution
  7. World-sheet effective theory
  8. Fermion zero modes
  9. ${\cal N}=2\;$ limit
  10. Breaking ${\cal N}=2\;$ supersymmetry
  11. The large $\mu$ limit
  12. Effective world-sheet theory in the large-$\mu$ limit
  13. Limits of applicability
  14. Non-Abelian monopoles in ${\cal N}=1\;$
  15. Non-Abelian strings in ${\cal N}=1\;$ SQCD
  16. ...and 2 more sections

Figures (3)

  • Figure 1: Renormalization-group (RG) flow in the theory (\ref{['model']}) at large $\mu_2$. Flow in four dimensions is represented by ${\cal N}=1\;$ and ${\cal N}=2\;$ curves, while $CP(1)$ marks the RG flow in the two-dimensional $CP(1)$ model.
  • Figure 2: A spectrum of relevant scales in the limit $\mu_2\gg \sqrt{\xi}$.
  • Figure 3: Various regimes for the monopoles and flux tubes. The latter case corresponds to the vanishing $\Delta m$.