Vortexpicture of Quarkconfinement
Since the invention of quarks as constituents of hadrons we are puzzled by the question why quarks are confined inside of hadrons. Quantum Chromodynamics (QCD) is the generally accepted theory of quark dynamics. In the lattice formulation of QCD one can demonstrate that quarks are evidently confined  but the mechanism of quark confinement is still under intensive discussion.
In recent lattice calculations we were able to show, that there are strong indications that thick vortices are responsible for confinement in QCD. At the end of the seventies 't Hooft and others suggested the vortex model and the dual superconductor model of confinement. At that time it was not clear which model should be favored. In the last years we presented numerical evidence in favor of the center vortex theory of confinement. We now look at the thick vortices as the basic topological objects which lead to confinement. We want to investigate all properties of these vortices, including the methods to detect them, and explain different aspects of confinement using vortices.
Vortices provide us with a unified picture of confinement: The QCD vacuum is a ``medium'' with magnetic vortices. It is filled with randomly fluctuating quantized magnetic flux tubes, fusing and splitting permanently. In this way they form a huge cluster percolating the whole spacetime. By those magnetic properties of the vacuum the electric flux originating in quarks is compressed to straight flux tubes leading to a linear rising quark antiquark potential and to confinement.
Figure 1: Model for the fieldlines between a quarkantiquarkpair.
In lattice QCD, the gluon fields are represented by SU(N) variables (where N is the number of colors), which are defined on the edges of the lattice cubes. The vortex model assumes that only the center degrees of freedom are responsible for confinement. The center of a group is the subgroup of elements which commute with all other group members. In the case of SU(N), it is the cyclic group . Vortices are the excitations of the center degrees of freedom, which appear in the shape of closed magnetic flux tubes. The flux is quantized and can only take on the discrete values of the center elements. In 4dimensional spacetime the flux line sweeps out a closed twodimensional surface. One can prove that if the vortices dominate the QCD vacuum, they lead in fact to confinement if they are large enough.
To verify the hypothesis, one must look for vortices in vacuum fields and check their abundance, size and structure. In a random gauge field configuration the flux tubes will have a finite thickness and are therefore called thick vortices. They are not solely composed of center elements. To extract the interesting degrees of freedom, vortices are located by a procedure known as center projection
Figure: Time slice of vortex surface after center projection.
Numerical calculations on the lattice have up to now supplied ample evidence that the vortex configurations are indeed responsible for the quark confinement. One also finds that when one removes all the vortices from the gauge field the topological charge and chiral symmetry breaking disappear as well. To understand this fact one needs to know that the topological charge density is proportional to
Figure: The simplest example of a nonorientable surface is the Möbius band.
Figure: Two Möbius strings glued together form a Klein bottle.
The topological charge is relevant for the chiral symmetry properties. The quarks' movement in the background of a gauge field are described by a firstorder differential equation, the Dirac equation. The eigenfunctions of this equation to the eigenvalue zero, the zeromodes, are either left or righthanded. A celebrated theorem asserts that the difference in the number of left and righthanded zeromodes is equal to the topological charge of the underlying gauge field. Further, it has been found that the quark density of the zeromodes is correlated with the topological charge and hence the vortex intersections.
The approach in the current project is complementary to the center projection described above. We do not track down vortices in a random field, but we ``manually'' create specified vortex configurations. The object of this is to investigate the correlations in the locations of the center vortices, instantons and zeromodes.
To simplify numerical calculations one uses a model with only 2 colors (gauge group SU(2)). All our calculations are performed on a lattice with points. We start by inserting the necessary links for a specific set of thick vortices into an ``empty'' lattice. Practically, the vortices belong to 2 geometries: planes and spheres. The vortex surfaces are placed such that they intersect, because as mentioned above, topological charge arises only at intersection points. On the one hand this gauge field served as the input for the computation of the quark zeromodes. On the other hand, any field configuration belongs to a certain homotopy class with definite topological charge, according to the intersections of the vortices. We want to find the corresponding instanton field which lies in the same class. Since instantons are minima of the action, we can accomplish this by continuous deformations of the field which lower the action. This process is called cooling
Figure: Two pairs of plane vortices, intersecting in 4 points. The arrows denote the direction of the electric and magnetic field, respectively. In the balls we have indicated the topological charge that arises at each intersection point. The sign of the charge follows from the relative orientation of the E and Bfield.
Figure: The probability density of the quark zeromode.
Let us exemplify the steps of the calculation for the simple vortex geometry of fig. . Adding up the contributions of each intersection point we obtain a theoretical topological charge of . In this gauge field background, the fermion field features one zeromode, which is compatible with our value of . The zeromode can be interpreted as a wavefunction from which one can calculate the probability for the quark to be found at a given location. This probability density is plotted in fig. . The process of cooling is depicted in fig. . Note that the topological charge is not constant as it should be. This is an artefact of the lattice on which one cannot define a continuous function in the usual sense. The plateau in the curves signals the appearance of a single instanton, carrying one unit of topological charge, in accord with our previous remarks. On the lattice, the instanton is not stable and disappears after prolonged cooling. Fig. compares the distribution of the topological charge of the original vortex configuration and the cooled field. There is no apparent relationship between the locations of the instanton and the vortices.
Figure: Evolution of the action and the topological charge in the course of cooling for the two planar vortices.
Figure: Topological charge density of uncooled vortices (left) and cooled vortices (i.e. the instanton, right)

Another interesting situation occurs when one pair of the plane vortices is replaced by a sphere. The theoretical topological charge for this arrangement is . In terms of instantons, this is not confirmed. There emerge two antiinstantons, amounting to , which are clearly visible as separate plateaus in the course of cooling.
Figure: A planar vortex intersects a spherical one. This picture shows the thin vortices which result from the original thick vortices after center projection.
Figure: Evolution of the action and the topological charge in the course of cooling for a planar and a spherical vortex.
Software: The vortex creation and cooling are implemented in a FORTRAN 77 program. The fermion zeromodes are computed using a program written by U. Heller (School of Computational Science & Information Technology, The Florida State University), which is a numerical realization of the socalled overlap fermion formalism. For visualization of topological charge density and fermion density we use MATHEMATICA ®.