[Paper Review] Quantum Yang-Mills Theory in Two Dimensions: Exact versus Perturbative

Timothy Nguyen|arXiv (Cornell University)|Aug 25, 2015

Black Holes and Theoretical Physics32 references7 citations

TL;DR

This paper provides a rigorous mathematical formulation of perturbative 2D Yang-Mills theory in Coulomb, holomorphic, and axial gauges, and compares its Wilson loop expectation values with the exact nonperturbative results from the lattice continuum limit. It proves that Coulomb and holomorphic gauges are equivalent and asymptotically match the exact theory on $S^2$, but axial gauge fails to do so on $ olinebreak ext{R}^2$, revealing a fundamental breakdown of perturbation theory in flat space despite its success in curved compact geometries.

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.

Motivation & Objective

• To rigorously formulate perturbative 2D Yang-Mills theory in multiple gauges (Coulomb, holomorphic, axial) using Faddeev-Popov and BV quantization.
• To compare perturbative Wilson loop expectation values with the exact nonperturbative results derived from the lattice continuum limit.
• To determine under which conditions perturbation theory asymptotically reproduces the exact theory in 2D Yang-Mills.
• To clarify the role of gauge choice and geometry (R² vs. S²) in the convergence of perturbative and nonperturbative results.

Proposed method

• Formal perturbation theory is developed via Faddeev-Popov quantization in Coulomb, holomorphic, and axial gauges on $\mathbb{R}^2$ and $S^2$.
• The theory is regularized using heat-kernel regulated propagators $G_{\epsilon}^{\mathbb{R}^2}$, which approximate the Green's function $G^{\mathbb{R}^2}(\mathbf{w}) = -\frac{1}{2\pi}\log|\mathbf{w}|$.
• Singular integral estimates are performed using point-splitting regularization, where the singularity at $\mathbf{w}=0$ is excised within an $\epsilon^{1/2}$-neighborhood.
• Feynman diagrams are analyzed via Wick's theorem and functional derivatives, with singular parts computed explicitly using asymptotic expansions.
• Exact Wilson loop expectations on $S^2$ are derived from the lattice formulation via heat kernel traces on the gauge group.
• The asymptotic behavior of perturbative and exact results is compared order-by-order in the coupling constant $e \to 0$.

Experimental results

Research questions

• RQ1 Do perturbative Wilson loop expectations in 2D Yang-Mills asymptotically match the exact nonperturbative results from the lattice continuum limit?
• RQ2 Are Coulomb and holomorphic gauges equivalent in perturbative 2D Yang-Mills, and do they yield the same asymptotic results?
• RQ3 Why does perturbation theory fail to reproduce the exact result on $\mathbb{R}^2$ despite matching on $S^2$?
• RQ4 How does the choice of gauge-fixing metric affect the perturbative result in Coulomb gauge?
• RQ5 What is the role of curvature and topology (compact vs. non-compact) in the convergence of perturbative and nonperturbative approaches?

Key findings

• Coulomb and holomorphic gauges are mathematically equivalent and yield identical perturbative Wilson loop expectations, independent of the gauge-fixing metric.
• Axial gauge yields a different result and is inequivalent to both Coulomb and holomorphic gauges, indicating a fundamental breakdown in gauge invariance under perturbative quantization.
• On $S^2$, the asymptotic expansion of exact Wilson loop expectations matches the perturbative result in holomorphic gauge to all orders in the coupling constant.
• On $\mathbb{R}^2$, the asymptotic match fails due to the inequivalence of axial gauge and the absence of a finite volume effect, which breaks the agreement seen on $S^2$.
• The heat-kernel regularization used in the lattice theory is shown to be equivalent to point-splitting regularization in the perturbative analysis, validating the singular integral estimates.
• The singular behavior of Feynman diagrams is captured via explicit asymptotic analysis of integrals involving $\partial_\mu G_\epsilon^{\mathbb{R}^2}$, leading to logarithmic divergences proportional to $\log \epsilon^{-1/2}$.