  Two dimensional gauge theories with charged matter fields are useful toy
models for studying gauge theory dynamics, and in particular for studying the
duality of large $N$ gauge theories to perturbative string theories. A useful
starting point for such studies is the pure Yang-Mills theory, which is exactly
solvable. Its $1/N$ expansion was interpreted as a string theory by Gross and
Taylor 30 years ago, but they did not provide a worldsheet action for this
string theory, and such an action is useful for coupling it to matter fields.
The chiral sector of the Yang-Mills theory can be written as a sum over
holomorphic maps and has useful worldsheet descriptions, but the full theory
includes more general extremal-area maps; a formal worldsheet action including
all these maps in a "topological rigid string theory" was written by Ho\v{r}ava
many years ago, but various subtleties arise when trying to use it for
computations. In this paper we suggest a Polyakov-like generalization of
Ho\v{r}ava's worldsheet action which is well-defined, and we show how it
reproduces the free limit of the Yang-Mills theory, both by formal arguments
and by explicitly computing its partition function in several cases. In the
future we plan to generalize this string theory to the finite-coupling gauge
theory, and to analyze it with boundaries, corresponding either to Wilson loops
or to dynamical matter fields in the fundamental representation.
