  We construct a new model of four-dimensional relativistic strings with
integrable dynamics on the worldsheet. In addition to translational modes this
model contains a single massless pseudoscalar worldsheet field - the worldsheet
axion. The axion couples to a topological density which counts the
self-intersection number of a string. The corresponding coupling is fixed by
integrability to $Q=\sqrt{7\over 16\pi}\approx 0.37$. We argue that this model
is a member of a larger family of relativistic non-critical integrable string
models. This family includes and extends conventional non-critical strings
described by the linear dilaton CFT. Intriguingly, recent lattice data in
$SU(3)$ and $SU(5)$ gluodynamics reveals the presence of a massive pseudoscalar
axion on the worldsheet of confining flux tubes. The value of the corresponding
coupling, as determined from the lattice data, is equal to
$Q_L\approx0.38\pm0.04$.
