Properties of P-vortices and monopole clusters in lattice $SU(2)$ gauge theory

We study the action and geometry of P-vortices, discriminating between the percolating and finite clusters. We also discuss the interrelation of the monopoles and P-vortices. To define P-vortices we use both the direct maximal center projection and indirect maximal center projection. We find, in particular, that the action density of the P-vortices in short clusters is substantially higher than in the percolating cluster. The surface of the percolating cluster appears random at short distances, with action density depending on the shape.

Figure 1

Ratio of the $Z(2)$ string tension and the full $SU(2)$ string tension for IMCP and for DMCP.

Figure 2

Possible two and four plaquette junctions in $\mathrm{D}=4$. Junctions of six plaquettes are very rare, and we do not consider them.

Figure 3

Probability of different types of junctions on the IR P-vortices for IMCP.

Figure 4

UV P-vortex cluster spectrum in IMCP.

Figure 5

Density of IR monopole currents.

Figure 6

Density of UV monopole currents.

Figure 7

Excess of the plaquette action density on the plaquettes dual to plaquettes belonging to P-vortices for IMCP and DMCP.

Figure 8

Dependence of the excess of the plaquette action on the local geometry of P-vortex cluster for IMCP for various lattice spacings; (a) IR cluster, (b) UV cluster.

Figure 9

Ratio of number of links with monopoles and number of all links of certain junction type for IMCP P-vortices; (a) IR cluster, (b) UV cluster.