Quantum statistical-gauge geometry

Exact equilibrium constraints in quantum many-body systems, such as sum rules and Ward identities, can be hard to expose because local currents are distributional and equal-time commutators may contain anomaly-like contact terms. We present a simple geometric organization of the recently identified quantum shift symmetry. The local shifting superoperator is viewed as a statistical-gauge connection acting on observables, so hyperforces are covariant derivatives and force balance becomes a thermal Ward identity expressed with the Kubo-Mori inner product. For compact smearings we prove bulk flatness: smeared shift generators close on the Lie bracket of vector fields with no bulk Schwinger term. This closure yields an iterative, all-orders Ward-Bianchi hierarchy of equal-time constraints that closes on a finite set of force insertions and a bilinear force-gradient kernel, leading to general antisymmetry relations and positivity bounds. We fix the microscopic generator by quantizing the diffeomorphism moment map, where Weyl and half-density quantization agree and give a self-adjoint operator without bulk ordering anomalies. Although the bulk connection is flat, global topology remains through holonomy, which links naturally to twisted boundary conditions and flux threading. Background electromagnetic fields deform the algebra through explicit density-weighted curvature insertions rather than a bulk central extension. As an application we treat fractional quantum Hall fluids on a torus, connect the deformation to the guiding-center GMP-W_{∞} algebra, and verify the key identities with exact diagonalization.