Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantum Vortex Strings: A Review

David Tong

TL;DR

This review establishes a quantitative 2d–4d correspondence by studying non-Abelian vortex strings: their worldsheet dynamics on a $CP^{N-1}$ target space mirrors the quantum physics of the parent 4d gauge theory. In ${N}=2$ theories, the exact 4d BPS spectrum is reproduced by the 2d BPS spectrum on the string, and the Argyres-Douglas fixed points map to 2d superconformal fixed points. For ${N}=1$ theories, the heterotic ${N}=(0,2)$ string reveals dynamical SUSY breaking and restoration linked to Seiberg's quantum deformed moduli space, with intricate scale matching between the 2d and 4d theories. The work also discusses multi-string moduli spaces, domain walls, and potential extensions to dualities and non-supersymmetric contexts, highlighting a broad framework where 2d soliton dynamics encode rich information about 4d gauge theories.

Abstract

The quantum worldsheet dynamics of vortex strings contains information about the 4d non-Abelian gauge theory in which the string lives. Here I tell this story. The string worldsheet theory is typically some variant of the CP^{N-1} sigma-model, describing the orientation of the string in a U(N) gauge group. Qualitative parallels between 2d sigma-models and 4d non-Abelian gauge theories have been known since the 1970s. The vortex string provides a quantitative link between the two. In 4d theories with N=2 supersymmetry, the exact BPS spectrum of the worldsheet coincides with the bulk spectrum in 4d. Moreover, by tuning parameters, the CP^{N-1} sigma-model can be coaxed to flow to an interacting conformal fixed point which is related to the 4d Argyres-Douglas fixed point. For theories with N=1 supersymmetry, the worldsheet theory suffers dynamical supersymmetry breaking and, more interestingly, supersymmetry restoration, in a way which captures the physics of Seiberg's quantum deformed moduli space.

Quantum Vortex Strings: A Review

TL;DR

This review establishes a quantitative 2d–4d correspondence by studying non-Abelian vortex strings: their worldsheet dynamics on a target space mirrors the quantum physics of the parent 4d gauge theory. In theories, the exact 4d BPS spectrum is reproduced by the 2d BPS spectrum on the string, and the Argyres-Douglas fixed points map to 2d superconformal fixed points. For theories, the heterotic string reveals dynamical SUSY breaking and restoration linked to Seiberg's quantum deformed moduli space, with intricate scale matching between the 2d and 4d theories. The work also discusses multi-string moduli spaces, domain walls, and potential extensions to dualities and non-supersymmetric contexts, highlighting a broad framework where 2d soliton dynamics encode rich information about 4d gauge theories.

Abstract

The quantum worldsheet dynamics of vortex strings contains information about the 4d non-Abelian gauge theory in which the string lives. Here I tell this story. The string worldsheet theory is typically some variant of the CP^{N-1} sigma-model, describing the orientation of the string in a U(N) gauge group. Qualitative parallels between 2d sigma-models and 4d non-Abelian gauge theories have been known since the 1970s. The vortex string provides a quantitative link between the two. In 4d theories with N=2 supersymmetry, the exact BPS spectrum of the worldsheet coincides with the bulk spectrum in 4d. Moreover, by tuning parameters, the CP^{N-1} sigma-model can be coaxed to flow to an interacting conformal fixed point which is related to the 4d Argyres-Douglas fixed point. For theories with N=1 supersymmetry, the worldsheet theory suffers dynamical supersymmetry breaking and, more interestingly, supersymmetry restoration, in a way which captures the physics of Seiberg's quantum deformed moduli space.

Paper Structure

This paper contains 13 sections, 62 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Non-Abelian Vortex Strings
  3. Confined Monopoles
  4. Other BPS Solitons
  5. Applications
  6. Vortex Strings and Seiberg-Witten Theory
  7. The ${\cal N}=(2,2)$ Vortex Worldsheet
  8. Superconformal Worldsheets
  9. Future Directions
  10. Heterotic Vortex Strings and ${\cal N}=1$ Gauge Theories
  11. The ${\cal N}=(0,2)$ ${\bf CP}^{N-1}$ Sigma-Model
  12. Vortices and the Quantum Deformed Moduli Space
  13. Further Models

Figures (2)

  • Figure 1: A cartoon of the magnetic monopole emitting two flux tubes. The size of the monopole is $L_{\rm mono}\sim 1/\Delta m$. The width of the flux tube is $L_{\rm vort} \sim 1/ev$.
  • Figure 2: Plot of a field theoretic D-brane configuration allquarter.