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

In $d=3$ $\mathrm{S}\mathrm{U}(N)$ 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 $2\pi /N$) 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 ${\varphi }_i$ (one of which is eliminated) that can merge, dissociate, and recombine while conserving flux $\mathrm{m}\mathrm{o}\mathrm{d}N$. 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 $k\ll N$, as naively expected.

Figure 1

Center vortices merging and dissociating. A $J=1$ vortex (labeled 1 for $Q_1$) meets a $J=2$ vortex ($Q_2+Q_3$) at the bottom vertex to form a $J=3$ vortex, which later dissociates at the top vertex.

Figure 2

Center vortices of total flux $J=N$ can be annihilated or created from zero flux. The labels 1, 2, … correspond to the labels of $Q$ matrices.

Figure 3

A recombination event, in which vortex 2 (that is, of flux matrix $Q_2$) combines with vortex $\{13\}$ at the bottom, and dissociates into vortex 1 plus vortex $\{23\}$ at the top. Black circles are nexuses or antinexuses which change 1 into 2 or vice versa.

Figure 4

A test plane perpendicular to the $z$ axis is moved along that axis, revealing a continuous picture of merging and dissociating vortices piercing that plane. Shown are two vortices recombining and another vortex dissociating as seen with $z$ as the “time.”