Flux-Space Flow Matching in 2D Compact U(1) With Spatial β-Conditioning

Critical slowing down and topological freezing in lattice gauge theory can be aggravated by thegauge-redundant link representation, which obscures simpler geometric structure available in al-ternative variables. We introduce Flux-Space Flow Matching (FFM), a generative samplingframework for 2D compact U(1) theory that operates directly on gauge-invariant flux (plaquette-angle) variables. By formulating the dynamics in flux space, the Wilson action is locally factorized,allowing us to train a continuous-time Neural ODE to approximate the equilibrium distributionwithout suffering from the stiff curvature typical of the coupled link formulation. We impose theglobal topological sector constraint via a deterministic “Relax-and-Project” mechanism and applyan independent Metropolis–Hastings accept/reject step as a bias-control procedure. Validated onL∈{48,64}lattices, FFMachievesacceptanceratesof50–70% atL= 48 andreducestheintegratedautocorrelation time of the topological charge by over 500×compared to Hybrid Monte Carlo atβ = 6.0 (on our run lengths). We validate model fidelity against thermodynamic observables, Wilsonloops, and Creutz ratios, finding agreement with the expected non-perturbative confinement scalingwithinthetestedregime. Furthermore, wedemonstratethatSpatialβ-Conditioningenableszero-shot approximation of inhomogeneous thermodynamics, spontaneously nucleating vortex–antivortexpairs in response to spatially varying coupling profiles. These results suggest that identifying theappropriate geometric degrees of freedom can be a more effective path to scalable neural samplingthan architectural complexity alone.