  We present a string theory that reproduces the large-$N$ expansion of two
dimensional Yang-Mills gauge theory on arbitrary surfaces. First, a new class
of topological sigma models is introduced, with path integrals localized to the
moduli space of harmonic maps. The Lagrangian of these harmonic topological
sigma models is of fourth order in worldsheet derivatives. Then we gauge
worldsheet diffeomorphisms by introducing the induced worldsheet metric. This
leads to a topological string theory, whose Lagrangian coincides in the bose
sector with the rigid string Lagrangian discussed some time ago by Polyakov and
others as a candidate for QCD string theory. The path integral of this
topological rigid string theory is localized to the moduli spaces of
minimal-area maps, and calculates their Euler numbers. The dependence of the
large-$N$ QCD partition functions on the target area emerges from measuring the
volume of the moduli spaces, and can be reproduced by adding a Nambu-Goto term
(improved by fermionic terms) to the Lagrangian of the topological rigid
string.
