Vortices and superfields on a graph

We extend the dimensional deconstruction by utilizing the knowledge of graph theory. In the dimensional deconstruction, one uses the moose diagram to exhibit the structure of the “theory space.” We generalize the moose diagram to a general graph with oriented edges. In the present paper, we consider only the $U(1)$ gauge symmetry. We also introduce supersymmetry into our model by use of superfields. We suppose that vector superfields reside at the vertices and chiral superfields at the edges of a given graph. Then we can consider multivector, multi-Higgs models. In our model, $[U(1){]}^p$ (where $p$ is the number of vertices) is broken to a single $U(1)$. Therefore, for specific graphs, we get vortexlike classical solutions in our model. We show some examples of the graphs admitting the vortex solutions of simple structure as the Bogomolnyi solution.

Figure 1

A moose diagram. There are $N+1$ $SU(2)$ gauge fields and a $U(1)$ gauge field. Each gauge field exists on each site represented by a small circle. The coupling constant of the gauge fields $A_{\mu }^i$ is $g_i$ ($i=0,1,2,\dots ,N$), and the coupling constant of the $U(1)$ gauge field $B_{\mu }$ is $g_{N+1}$. The vacuum expectation value of the scalar fields ${\Sigma }_i$ is $f_i$ ($i=1,2,\dots ,N,N+1$).

Figure 2

The simplest graph, constructed by two vertices and an edge. A vertex $v_i$ is identified by $i$, where $i$ is a label for each vertex. In the same way, an edge $e_i$ is identified by $i$, where $i$ is a label for each edge. The arrow means a direction of the edge. This edge is called an oriented edge. In terms of the oriented edge, the original vertex $v_1$ is $v_1=o(e_1)$ and the terminal vertex $v_2$ is $v_2=t(e_1)$. This oriented graph corresponds to the generalized moose diagram.

Figure 3

$P_3$: the path graph with three vertices. There are two substantially different graphs. They have the different incidence matrices.

Figure 4

Contour plots of scalar potentials for the models based on $P_3^A$ (left) and on $P_3^B$ (right), respectively. In both plots, potentials are normalized by $g^2{f}^4$, the contour spacing is 0.1, and the horizontal axis indicates $|{\sigma }_1|/f$ while the vertical axis indicates $|{\sigma }_2|/f$.

Figure 5

$P_2$: the path graph with two vertices.

Figure 6

The graph $P_3^A$ has edges of the same direction while $P_3^B$ has edges of the different direction.

Figure 7

The star graphs, $K_{1,N}^A$ and $K_{1,N}^B$.

Figure 8

This dashed line means that ${\rho }_e\equiv f$ on this edge, no winding scalar edge.

Figure 9

$P_4$ graph consists of two $P_2$ and an edge.

Figure 10

$P_4$ graph whose edge direction is left-right symmetric with respect to the dashed edge. Each of edge direction is outgoing with respect to the dashed edge.

Figure 11

$P_4$ graph. Each of edge direction is incoming with respect to the dashed edge.

Figure 12

$P_6$ graph, which includes two $P_3$ as subgraphs.

Figure 13

There are four types of the $P_6$ graph consisting of two $P_3$.

Figure 14

The graph consisting of two $K_{1,N}$ connected by the dashed edge.