Topological Strings and QCD in Two Dimensions

I present a new class of topological string theories, and discuss them in two dimensions as candidates for the string description of large-$N$ QCD. The starting point is a new class of topological sigma models, whose path integral is localized to the moduli space of harmonic maps from the worldsheet to the target. The Lagrangian is of fourth order in worldsheet derivatives. After gauging worldsheet diffeomorphisms in this ``harmonic topological sigma model,'' we obtain a topological string theory dominated by minimal-area maps. The bosonic part of this ``topological rigid string'' Lagrangian coincides with the Lagrangian proposed by Polyakov for the QCD string in higher dimensions. (talk given at the Cargese conference on ``Recent Developments in String Theory, CFT and Topological Field Theory'' (May 1993))