Real Stueckelberg Quantum Geometry and Unbroken Supersymmetry in the Rotating Shallow Water Model

The topological properties of planetary fluids are typically analyzed by mapping classical fluid equations onto complex quantum mechanical models. Here we present a purely real, six-dimensional Stueckelberg quantum mechanical formulation of the rotating shallow water equations to demonstrate that these topological features are intrinsic to the classical kinematics itself. Operating entirely within R⁶ we decouple the complex quantum geometric tensor into an independent real Fubini-Study metric and a real antisymmetric Berry curvature. Our real-variable approach explicitly derives a topological magnetic monopole of charge C = −2 and captures the inherent scale invariance of the fluid's geometry without explicit complex coordinate representation. We suggest that continuous variations in the Coriolis parameter model the adiabatic geometric evolution of the Archean Earth, and we propose a laboratory rotating-tank experiment to physically measure this parameter sweep. Finally, we show that our real 6D formulation naturally maps to unbroken supersymmetric quantum mechanics. By identifying a purely real supercharge and calculating a fluid Witten index of W = −2, we demonstrate a strict mathematical symmetry between the topological charge of the propagating bands and the invariant of the unbroken zero-energy geostrophic vacuum. We advance the mathematically supported viewpoint that steady-state geostrophic weather patterns represent the exact supersymmetric ground states of the rotating fluid system. Consequently, the topological isolation of this vacuum naturally restricts the spectral flow across the equator, providinga theoretical explanation for the unidirectional eastward motion of equatorial boundary waves.