Three-dimensional scalar field theory model of center vortices and its relation to k-string tensions

In d=3 SU(N) gauge theory, we study a scalar-field theory model of center vortices, and their monopolelike companions called nexuses, that furnishes an approach to the determination of so-called k-string tensions. This model is constructed from stringlike quantum solitons introduced previously, and exploits the well-known relation between string partition functions and scalar-field theories in d=3. A basic feature of the model is that center vortices corresponding to magnetic flux J (in units of 2π/N) are composites of J elementary J=1 constituent vortices that come in N-1 types, with repulsion between like constituents and attraction between unlike constituents. The scalar-field theory is of a somewhat unusual type, involving N scalar fields ϕ_i (one of which is eliminated) that can merge, dissociate, and recombine while conserving flux modN. The properties of these fields are deduced directly from the corresponding gauge-theory quantum solitons. Every vacuum Feynman graph of the theory corresponds to a real-space configuration of center vortices. We use qualitative features of this theory based on the vortex action to study the problem of k-string tensions (explicitly at large N, although large N is in no way a restriction on the model in general), whose solution is far from obvious in center-vortex language. We construct a simplified dynamical picture of constituent-vortex merging, dissociation, and recombination, which allows in principle for the determination of vortex areal densities and k-string tensions. This picture involves pointlike molecules made of constituent atoms in d=2 which combine and disassociate dynamically. These molecules and atoms are cross sections of vortices piercing a test plane; the vortices evolve in a Euclidean “time” which is the location of the test plane along an axis perpendicular to the plane. A simple approximation to the molecular dynamics is compatible with k-string tensions that are linear in k for k≪N, as naively expected.