# Sampling SU(N) gauge theory on a 2D lattice from independent plaquettes via holonomies and corner reweighting

arXiv: https://arxiv.org/abs/2610.09147 · HTML: https://mlsanity.com/paper/2610.09147/

## Key facts

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- First posted on arXiv 6 Oct 2026.

## Abstract (arXiv)

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^dagger Y^dagger, 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.