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Degenerate vortices and world-line instantons in three-dimensional gauge theories

Evgenii Ievlev, Mikhail Shifman

TL;DR

This work extends the study of quantum lifting of classically degenerate solitons from 1+1D to 2+1D by examining degenerate vortices in three-dimensional gauge theories. Degeneracies are generically lifted by world-line instantons on the vortex in bosonic theories, with the tunneling amplitude scaling as $S_{ ext{inst}}\sim 1/e^{2}$ and producing a band structure with a unique ground state. In Abelian models with $ ext{Z}_2$ degeneracy, and in non-Abelian superconductors with CP$(N-1)$ moduli, the degeneracy is lifted unless supersymmetry in the bulk is large enough; with eight supercharges the degeneracy is protected due to fermionic zero modes, as captured by the Witten index equal to $N$. The article also develops the effective 0+1D world-line quantum mechanics that describes vortex tunneling, and shows how monopole-instantons confined to vortex lines implement the mixing. These results illuminate how topological soliton multiplicities behave in higher-dimensional settings and clarify the role of supersymmetry in preserving BPS vortex degeneracies, with implications for nonperturbative dynamics in 3D gauge theories.

Abstract

In this paper we continue the study of particle-like topological solitons with degenerate masses and their mixing due to world-line instantons. Previously, this phenomenon was studied in 1+1-dimensional setups. Here we take a step further and consider degenerate vortices in 2+1 dimensions. We find that, while classically such vortices may be degenerate, they generally mix and split at the quantum level. Supersymmetry protects BPS-saturated vortices only when the number of supercharges in the bulk is large enough.

Degenerate vortices and world-line instantons in three-dimensional gauge theories

TL;DR

This work extends the study of quantum lifting of classically degenerate solitons from 1+1D to 2+1D by examining degenerate vortices in three-dimensional gauge theories. Degeneracies are generically lifted by world-line instantons on the vortex in bosonic theories, with the tunneling amplitude scaling as and producing a band structure with a unique ground state. In Abelian models with degeneracy, and in non-Abelian superconductors with CP moduli, the degeneracy is lifted unless supersymmetry in the bulk is large enough; with eight supercharges the degeneracy is protected due to fermionic zero modes, as captured by the Witten index equal to . The article also develops the effective 0+1D world-line quantum mechanics that describes vortex tunneling, and shows how monopole-instantons confined to vortex lines implement the mixing. These results illuminate how topological soliton multiplicities behave in higher-dimensional settings and clarify the role of supersymmetry in preserving BPS vortex degeneracies, with implications for nonperturbative dynamics in 3D gauge theories.

Abstract

In this paper we continue the study of particle-like topological solitons with degenerate masses and their mixing due to world-line instantons. Previously, this phenomenon was studied in 1+1-dimensional setups. Here we take a step further and consider degenerate vortices in 2+1 dimensions. We find that, while classically such vortices may be degenerate, they generally mix and split at the quantum level. Supersymmetry protects BPS-saturated vortices only when the number of supercharges in the bulk is large enough.
Paper Structure (26 sections, 127 equations, 3 figures)

This paper contains 26 sections, 127 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Models with $\mathcal{N}=1$ supersymmetry
  3. Scalar model and its gauging
  4. Vortices in gauged 3D model, Higgs regime
  5. Degenerate vortices inside an Abelian superconductor
  6. $\mathbb{Z}_2$ vortices
  7. The model
  8. Tunneling between degenerate vortices in Euclidean time
  9. Effective quantum mechanics for vortex tunneling
  10. $\mathbb{CP}(1)$ vortices
  11. Degenerate vortices in a non-Abelian superconductor
  12. Classical $\mathbb{Z}_N$ vortices in a non-Abelian superconductor
  13. Model 1
  14. Model 2 (mass deformation)
  15. World line action and mixing of vortices
  16. ...and 11 more sections

Figures (3)

  • Figure 1: Schematic vortex profile functions in the model (\ref{['twosum']}). Here $r$ is the radial coordinate, $R$ is the size of the vortex core. The existence of two profile functions, $\chi_\pm$, demonstrates the spontaneous $\mathbb{Z}_2$ symmetry breaking on the vortices.
  • Figure 2: Effective quantum mechanics for the instanton tunneling from $V_-$ to $V_+$, see Eqs. (\ref{['45s']}) and (\ref{['47s']}).
  • Figure 3: Churchkhela tunneling process (schematically). The vertical axis is the Euclidean time $\tau$. The vortex world line has "lumps" representing the instantons. The line color represents the vortex species. Winding around every vortex is the same (shown by red curvy lines with arrows).