Confinement in Gauge Theories from the Condensation of World-Sheet Defects in Liouville String

TL;DR

This paper introduces a Liouville-string framework to understand confinement in four-dimensional gauge theories by linking vortex condensation on world sheets to magnetic monopole condensation and phase transitions in an AdS bulk.

Contribution

It extends previous approaches to include non-conformal theories and connects vortex condensation with gauge confinement and black hole phase transitions in AdS space.

Findings

Vortex condensation occurs in four target space dimensions.

Effective string tension arises from vortex condensation, leading to confinement.

Deconfinement corresponds to a BKT transition of vortices and black hole phase transition.

Abstract

We present a Liouville-string approach to confinement in four-dimensional gauge theories, which extends previous approaches to include non-conformal theories. We consider Liouville field theory on world sheets whose boundaries are the Wilson loops of gauge theory, which exhibit vortex and spike defects. We show that world-sheet vortex condensation occurs when the Wilson loop is embedded in four target space-time dimensions, and show that this corresponds to the condensation of gauge magnetic monopoles in target space. We also show that vortex condensation generates a effective string tension corresponding to the confinement of electric degrees of freedom. The tension is independent of the string length in a gauge theory whose electric coupling varies logarithmically with the length scale. The Liouville field is naturally interpreted as an extra target dimension, with an anti-de-Sitter (AdS) structure induced by recoil effects on the gauge monopoles, interpreted as D branes of the effective string theory. Black holes in the bulk AdS space correspond to world-sheet defects, so that phases of the bulk gravitational system correspond to the different world-sheet phases, and hence to different phases of the four-dimensional gauge theory. Deconfinement is associated with a Berezinskii-Kosterlitz-Thouless transition of vortices on the Wilson-loop world sheet, corresponding in turn to a phase transition of the black holes in the bulk AdS space.