A String Theory for Two Dimensional Yang-Mills Theory I

TL;DR

This paper proposes a well-defined worldsheet action for 2D Yang-Mills theory that reproduces its free limit and paves the way for coupling to matter fields and studying dual string descriptions.

Contribution

It introduces a Polyakov-like generalization of Hořava's worldsheet action, enabling explicit computations and setting the stage for future extensions to finite coupling and matter fields.

Findings

Reproduces the free limit of 2D Yang-Mills theory

Provides explicit partition function computations

Offers a foundation for coupling to matter fields

Abstract

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.