Anyonic vortex states of 3D interacting Bose systems

We formulate and study a generalization of the Chern-Simons transformation in interacting Bose systems to three-dimensional space (3D). The method combines ordinary Chern-Simons (CS) transformation on a 2D plane and Jordan-Wigner fermionization along the orthogonal arbitrary vortex line. This transformation defines the anyonic properties of vortexes. The transformation yields an action for the emergent matter fields coupled to a gauge field, the generalization of Chern-Simons action to 4D, which reproduces CS transformation at the level of the equations of motion. The obtained action contains metrics on a transversal to vortices planes with varying z-coordinates connected each with other by area-preserving diffeomorphisms. The action has the same structure as for 2d integrable models of classical statistical mechanics understood in terms of quantum gauge theory in four dimensions recently formulated in K. Costello, arXiv:1303.2632. We define the corresponding to our action 2D R-matrix and investigate its topological properties. We apply the obtained results to three-dimensional interacting Bose systems with an infinitely degenerate single-particle dispersion. We argue that these systems may exhibit an absence of condensation and stabilize ground states with anyonic vortices.