We extend the emergent hydrodynamic framework for fermionic statistics in a 3+1D quantized superfluid to include non-Abelian topological excitations.Single framed vortex loops reproduce the familiar Abelian π Berry phase corresponding to fermionic exchange. Multi-component vortex networks exhibit matrix-valuednon-Abelian holonomies associated with internal symmetry groups G. Linked and knotted vortices provide a mechanical realization of both fermionic statistics andinternal quantum numbers. We discuss how effective gauge interactions emerge from topological constraints, provide energy scale estimates, and present a roadmapconnecting vortex network structures to Standard Model-like particles and forces.
Alex Smith (Sun,) studied this question.