Partly in the interest of self-promotion and partly out of the desire
to inform (perhaps they are the same), I wanted to mention a few
aspects of the confinement problem.

The real problem in the IR region of QCD is to prove the string
tension is positive as the bare coupling is taken to zero. By
“bare coupling”, I mean a dimensionless coupling, so even in 2+1 dimensions, the bare coupling g is dimensionless. The standard
coupling is a product of this coupling and some power of the cut-off
(they are the same in 3+1). The cut-off can be removed as the
bare coupling is taken to zero.

There is only one method that actually has yielded confinement
at small bare coupling (everything else is a strong-coupling expansion).
That is done is a certain anisotropic (not rotation invariant) case of
the 2+1-D Yang-Mills theory. String tensions
and the glueball spectrum can be worked out from first principles. I
have written some papers on this in the last 2 1/2 years. This
is a real weak-coupling method, though different from standard
perturbation theory. I am trying to see if the results can be
extended to the rotational invariant case. I’m optimisitc, but
this is a hard problem.

There are other indications of a mass gap in 2+1 dimensions at weak coupling. These come from gauge-orbit-space distance estimates, an
approach begun by Feynman in the early 80’s (though Feynman’s
results are pretty much all wrong). Gauge-orbit space is the space of connections modulo gauge transformations; gauge orbits are the
physical degrees of freedom. The basic idea is that gauge-orbit space has
finite extent almost everywhere, and in particular where the magnetic
energy is small. Some progress on this was made in one of Karabali and Nair’s papers and in a paper I wrote with G. Semenoff. These techniques run into a real obstacle in 3+1 dimensions, where it was proved that gauge-orbit
space is unbounded, even where the magnetic energy is small (in a paper I wrote in ‘96).

Finally, there is a paper of Alexanian and Nair where there is an estimate of
the gap by a self-consistent method. It is rather complicated, but very
interesting.