  We consider $U(N)$ and $SU(N)$ gauge theory on the sphere. We express the
problem in terms of a matrix element of $N$ free fermions on a circle. This
allows us to find an alternative way to show Witten's result that the partition
function is a sum over classical saddle points. We then show how the phase
transition of Douglas and Kazakov occurs from this point of view. By
generalizing the work of Douglas and Kazakov, we find other `stringy' solutions
for the $U(N)$ case in the large $N$ limit. Each solution is described by a net
$U(1)$ charge. We derive a relation for the maximum charge for a given area and
we also describe the critical behavior for these new solutions. Finally, we
describe solutions for lattice $SU(N)$ which are in a sense dual to the
continuum $U(N)$ solutions. (Parts of this paper were presented at the Strings
'93 Workshop, Berkeley, May 1993.)
