Topological Rigid String Theory and Two Dimensional QCD

TL;DR

This paper constructs a topological string theory that reproduces the large-N expansion of 2D Yang-Mills theory, linking harmonic sigma models, moduli space localization, and QCD string candidates.

Contribution

It introduces a new class of topological sigma models and a topological rigid string framework that captures 2D QCD dynamics and large-N behavior.

Findings

Path integrals localize to moduli spaces of harmonic and minimal-area maps.

The theory reproduces large-N QCD partition functions and area dependence.

Adding a Nambu-Goto term aligns the model with QCD string expectations.

Abstract

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.