Dyson-Schwinger equations approach to the large-Nlimit: Model systems and string representation of Yang-Mills theory

Abstract

Simple model systems like the $\mathrm{O}(N)$ $\ensuremath{\sigma}$ model, the Gross-Neveu model, and the random matrix model are solved at $N\ensuremath{\rightarrow}\ensuremath{\infty}$ using Dyson-Schwinger equations and the fact that the Hartree-Fock approximation is exact at $N\ensuremath{\rightarrow}\ensuremath{\infty}$. The complete string equations of the $U(\ensuremath{\infty})$ lattice gauge theory are presented. These must include both string rearrangement and splitting. Comparison is made with the continuum equations of Makeenko and Migdal which are structurally different. The difference is ascribed to inequivalent regularization procedures in the treatment of string splitting or rearrangement at intersections.