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Two dimensional gauge theories with charged matter fields are useful toymodels for studying gauge theory dynamics, and in particular for studying theduality of large $N$ gauge theories to perturbative string theories. A usefulstarting point for such studies is the pure Yang-Mills theory, which is exactlysolvable. Its $1/N$ expansion was interpreted as a string theory by Gross andTaylor 30 years ago, but they did not provide a worldsheet action for thisstring 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 overholomorphic maps and has useful worldsheet descriptions, but the full theoryincludes more general extremal-area maps; a formal worldsheet action includingall these maps in a ''topological rigid string theory'' was written byHo\v{r}ava many years ago, but various subtleties arise when trying to use itfor computations. In this paper we suggest a Polyakov-like generalization ofHo\v{r}ava's worldsheet action which is well-defined, and we show how itreproduces the free limit of the Yang-Mills theory, both by formal argumentsand by explicitly computing its partition function in several cases. In thefuture we plan to generalize this string theory to the finite-coupling gaugetheory, and to analyze it with boundaries, corresponding either to Wilson loopsor to dynamical matter fields in the fundamental representation.

A String Theory for Two Dimensional Yang-Mills Theory I