Conformal Field Theory Techniques in Large N Yang-Mills Theory

TL;DR

This work develops a large $N$ framework for two-dimensional Yang–Mills theory by formulating YM$_2$ as quantum mechanics on the group manifold $U(N)$, which is shown to be equivalent to a quasi-relativistic 2D free-fermion system and is amenable to bosonization. The torus free energy is computed to $O(1/N^2)$ and a leading-sphere approximation is discussed, with intriguing modular-like structure emerging in the torus sector and hints of a string interpretation. Key results include a near-modular form for the torus contribution $F_{1\rightarrow 1}$ and a $1/N^2$ correction $F_{2\rightarrow 1}$ expressed via Eisenstein series $E_2,E_4,E_6$, supporting a duality between area and coupling that echoes world-sheet dualities. The analysis extends to YM$_2$ on higher genus surfaces, highlighting a largely topological large-$N$ structure for $G>1$ and a Gross–Witten–like transition for the sphere, with boundary-state and classical-fluid pictures clarifying the role of world-sheet features and suggesting connections to topological strings and modular invariance.

Abstract

Following some motivating comments on large N two-dimensional Yang-Mills theory, we discuss techniques for large N group representation theory, using quantum mechanics on the group manifold U(N), its equivalence to a quasirelativistic two-dimensional free fermion theory, and bosonization. As applications, we compute the free energy for two-dimensional Yang-Mills theory on the torus to O(1/N^2), and an interesting approximation to the leading answer for the sphere. We discuss the question of whether the free energy for the torus has R -> 1/R invariance. A substantially revised version of hep-th/9303159 with many new results.