  We analyze quantum Yang-Mills theory on $\mathbb{R}^2$ using a novel
discretization method based on an algebraic analogue of stochastic calculus.
Such an analogue involves working with "Gaussian" free fields whose covariance
matrix is indefinite rather than positive definite. Specifically, we work with
Lie-algebra valued fields on a lattice and exploit an approximate
gauge-invariance that is restored when taking the continuum limit. This
analysis is applied to show the equivalence between Wilson loop expectations
computed using partial axial-gauge, complete axial-gauge, and the Migdal-Witten
lattice formulation. As a consequence, we obtain intriguing Lie-theoretic
identities involving heat kernels and iterated integrals.
