  We study a one-dimensional electron liquid embedded in a 2D antiferromagnetic
insulator, and coupled to it via a weak antiferromagnetic spin exchange
interaction. We argue that this model may qualitatively capture the physics of
a single charge stripe in the cuprates on length- and time scales shorter than
those set by its fluctuation dynamics. Using a local mean-field approach we
identify the low-energy effective theory that describes the electronic spin
sector of the stripe as that of a sine-Gordon model. We determine its phases
via a perturbative renormalization group analysis. For realistic values of the
model parameters we obtain a phase characterized by enhanced spin density and
composite charge density wave correlations, coexisting with subleading triplet
and composite singlet pairing correlations. This result is shown to be
independent of the spatial orientation of the stripe on the square lattice.
Slow transverse fluctuations of the stripes tend to suppress the density
correlations, thus promoting the pairing instabilities. The largest amplitudes
for the composite instabilities appear when the stripe forms an antiphase
domain wall in the antiferromagnet. For twisted spin alignments the amplitudes
decrease and leave room for a new type of composite pairing correlation,
breaking parity but preserving time reversal symmetry.
