Lattice gauge theories are mathematical models used to study particle and condensed-matter physics from first principles. They involve very high-dimensional probability distributions. Traditional Monte Carlo methods can struggle with slow mixing and other bottlenecks. This paper presents a new generative sampling method that works in plaquette variables and enforces the exact geometric constraints those variables must satisfy.

The authors build a multilevel normalizing-flow model. A normalizing flow is a learned, invertible transformation that turns simple random noise into samples from a target distribution. Instead of working on the usual link variables, the model samples directly in plaquette space. Plaquettes are local building blocks of the lattice — the small square loops whose values make the action local. Working in plaquette space makes the physical interactions more local, but introduces exact Bianchi constraints that tie plaquettes together and restrict valid configurations to a lower-dimensional surface.

The key technical idea is a coarse-to-fine factorization. The method generates configurations in stages from coarse to fine. At each refinement step the problem of satisfying the global constraints becomes a set of local solves. Specifically, every plaquette whose value is fixed at a refinement step depends on at most four newly generated variables. Because of the way the refinement is set up, the size of each constrained solve does not grow with the overall lattice size. The construction therefore turns a globally coupled constraint into many small, local problems that the flow handles exactly.

The authors validate the approach on simple gauge groups and dimensions: U(1) gauge theory in two and four dimensions, and SU(2) in two dimensions. They compare against baseline models that work in link space and find substantially better performance. The advantage grows toward weak coupling, a regime where link-space correlations become long-ranged and learning is especially hard. The weak-coupling regime is also important for continuum limits in asymptotically free theories, so improved sampling there is meaningful for future studies.