Quantum Vortex Strings: A Review

TL;DR

This review explores the dynamics of quantum vortex strings, their worldsheet theories, and their deep connections to 4d non-Abelian gauge theories, including supersymmetric cases and conformal fixed points.

Contribution

It provides a comprehensive overview of vortex string worldsheet theories and their relation to 4d gauge theories, highlighting new insights into supersymmetry and conformal fixed points.

Findings

The worldsheet CP^{N-1} model links to 4d gauge theory spectra.

Exact BPS spectrum matches bulk spectrum in N=2 theories.

Flow to conformal fixed points relates to Argyres-Douglas fixed point.

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.