  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.
