Quantum Yang-Mills Theory in Two Dimensions: Exact versus Perturbative

Abstract

The standard Feynman diagrammatic approach to quantum field theories assumes that perturbation theory approximates the full quantum theory at small coupling even when a mathematically rigorous construction of the latter is absent. On the other hand, two-dimensional Yang-Mills theory is a rare (if not the only) example of a nonabelian (pure) gauge theory whose full quantum theory has a rigorous construction. Indeed, the theory can be formulated via a lattice approximation, from which Wilson loop expecation values in the continuum limit can be described in terms of heat kernels on the gauge group. It is therefore fundamental to investigate how the exact answer for 2D Yang-Mills compares with that of the continuum perturbative approach, which a priori are unrelated. In this paper, we provide a mathematically rigorous formulation of the perturbative quantization of 2D Yang-Mills, and we consider perturbative Wilson loop expectation values on $\mathbb{R}^2$ and $S^2$ in Coulomb gauge, holomorphic gauge, and axial gauge (on $\mathbb{R}^2$). We show the following equivalences and nonequivalences between these gauges: (i) Coulomb and holomorphic gauge are equivalent and are independent of the choice of gauge-fixing metric; (ii) both are inequivalent with axial-gauge. Additionally, we show that the asymptotics of exact lattice Wilson loop expectations on $S^2$ agree with perturbatively computed expectations in holomorphic gauge for simple closed curves to all orders. However, as a consequence of (ii), this result is necessarily false on $\mathbb{R}^2$. Our work therefore presents fundamental progress in the analysis of how continuum perturbation theory succeeds or fails in capturing the asymptotics of the continuum limit of the lattice theory.