  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.
