In the Wilson formulation of lattice gauge theory, the fundamental degrees of freedom are group-valued link variables, while the action is a sum over the trace of the plaquettes, the smallest Wilson loops. For generative sampling methods such as normalizing flows, this poses a challenge: the distribution of an individual plaquette is easy to model, but mapping sampled plaquettes to the links is the obstruction. We explore a statistical way around this in two dimensions for the
SU(N) gauge theory with the Wilson plaquette action. The action can be written in terms of holonomy variables, which determine the links once a consistency condition at the four corners of an extended lattice is satisfied. Using a boundary condition that only affects the holonomies, we trade this consistency condition for a conditional sampling problem, which reduces to sampling
X,Y∈SU(N) given the group commutator
Z=XYX†Y†, where
Z∈SU(N) depends on the four corners. Plaquettes are sampled independently, and the resulting configurations carry weights due to the additional conditional sampling. We obtain the density of
Z, which determines these weights, in closed form for
SU(2) and
SU(3). A normalizing flow models the single-plaquette distribution for
SU(2) and
SU(3); for the group-commutator problem we use a closed-form sampler for
SU(2) and a trained normalizing flow for
SU(3). In our tests on
2×2 and
32×32 lattices at two couplings each, per-plaquette acceptance rates exceed
99% and the effective sample size of the final weights is moderate, typically above one half.