The Vortex Framework: Topological Fermions from Framed Vortex Loops in a Quantized Superfluid Effective Field Theory Part I

We present an emergent hydrodynamic framework for fermionic statistics within a 3 + 1-dimensional quantized superfluid effective field theory (EFT). While classical vortices in irrotational fluids are bosonic, we demonstrate that quantized vortex loops generically acquire an effective internal framing arising from the anisotropic
geometry and phase-gradient structure of their cores. These framed vortex loops inhabit a configuration space C ∼= (T3 × SO(3))N /SN . We define the Berry connection on the Hilbert bundle of condensate states and show that the adiabatic exchange of two such defects corresponds to a non-contractible loop in SO(3). Due
to π1(SO(3)) ∼= Z2, the resulting holonomy yields a topological phase π, producing the fermionic exchange sign eiπ = −1. This provides a concrete mechanical realization of the Finkelstein–Rubinstein constraint and demonstrates the possibility that half-integer spin and fermionic exchange statistics can emerge as topological properties of a superfluid vacuum.