Abelian gauge fixing and confinement on lattice

Clearly, the question whether the dial superconductor model of 't Hooft provides the correct physical picture of confinement requires deep knowledge about the low energy dynamics of non-abelian gauge theories. The only known systematic method that can be successfully employed to such questions is lattice gauge theory.

Kronfeld, Laursen, Schierholz, and Wiese devised a method to investigate the effects of abelian gauge fixing and abelian projection on confinement. They used lattice gauge field configurations obtained from simulations without gauge fixing. Then they fixed the gauge in each configuration independently by making use of the procedure, described below.

They defined a functional of the lattice gauge fields,

$$F_{\rm gf}(U)=\frac{1}{2}\sum_{x,\mu}{\rm Tr}\left[\sigma_3U_\mu(x)\sigma_3 U_\mu^\dagger(x)\right].$$

First of all, this form is not gauge invariant, since the gauge transformations of the gauge field, $V(x)$, where

$$U_\mu(x)\to V(x)U_\mu(x)V^\dagger(x+\mu),$$

do not commute with the matrices $\sigma_3$. Notice that if we write $U_\mu=a_0+i\mbox{\boldmath$a$}\cdot \mbox{\boldmath$\sigma$}$ then every term of $F_{\rm gf}(U)$ can be written as

$$\mbox{One term of } F_{\rm gf}(U)=2(a_0^2+a_3^2)-1$$

Consequently this gauge transformation tries to make the gauge fields as diagonal as possible, by maximizing the `Abelian' components $a_0$ and $a_3$ and at the same time minimizing the nonabelian components, $a_1$ and $a_2$. Naturally, the number of gauge transformations is not sufficient to diagonalize all gauge matrices, as there are 4$M$ gauge matrices and $M$ gauge transformations on a lattice of $M$ vertices. This gauge transformation is the equivalent of the abelian gauge fixing. For that reason the authors call this the Maximal Abelian Gauge (MAG).

After gauge fixing one is free to project the gauge fields to their abelian components. This was performed by the projection

$$U_\mu=a_0+i\mbox{\boldmath$a$}\cdot \mbox{\boldmath$\sigma$}\to \frac{a_0+ia_3\sigma_3}{\sqrt{a_0^2+a_3^2}}.$$

The normalization was necessary, because otherwise the abelian projected matrix would not be unitary. As a result of these operations they ended up with Abelian gauge fields defined on every link.

The crucial test of the validity of the assumption that abelian projected theories are equivalent to the original theories was to measure physical quantities in this abelian gauge theory. Keep in mind at the same time that abelian gauge theories like QED do not confine charges (otherwise our world would not exist as we know it). The first test was to calculate Wilson loops and the dependence of their expectation values on size. They found that the expectation value of Wilson loops follows the area law and, morover, the string constant agrees with the one found in the non-Abelian theory without gauge fixing. Other measurements were also performed. It was found that chiral symmetry was also broken and the size of the chiral condensate was also correct. This was taken to be the proof that the dual supeconductor model is indeed the correct physical picture behind confinement.

There is one problem with the abelian projection model by MAG. No monopoles appear as the consequence of the gauge fixing procedure, along the lines proposed by 't Hooft. The gauge fixing procedure does not have singularities. Though it is possible to define monopoles on the lattice by an indirect method and measure their density the direct connection between monopole condensation and gauge fixing is lost.

To cure the above described problem of MAG van der Sijs has recently proposed a modified gauge fixing procedure, called Laplacian gauge fixing (LAG). To understand LAG let us rewrite the gauge fixing potential of MAG. Introducing the gauge transformation $V$ explicitely we can write

$$F_{\rm gf}(U^V) = \frac{1}{2}\sum_{\mbox{\boldmath$r$},\mu}{\rm Tr}\left[V^\dagger(... ...V^\dagger(x+\mu)\sigma_3 V(x+\mu) U_\mu^{\dagger} (\mbox{\boldmath$r$})\right],$$

Now this last expression can be rewritten if we use the adjoint representation of the gauge transformation and of the gauge field. These are defined as

$$g^A_{ab}(x)=\frac{1}{2}{\rm Tr}[V^\dagger(x)\sigma_aV(x)\sigma_b],$$

and

$$U_\mu^{A,ab}(x)=\frac{1}{2}{\rm Tr}[\sigma_aU_\mu^\dagger(x)\sigma_bU_\mu(x)].$$

These matrices are orthogonal. $g^A_{ab}$ is just the standard rotation matrix in 3-vector representation. Then we can write

$$F_{\rm gf}(U^V) = \sum_{x,\mu} g^A_{a3}(x)U_\mu^{A,ab}(x) g^A_{b3}(x+\mu),$$

Note that the Cartan (Abelian) subgroup of the gauge group leaves the vector $g^A_{a3}$ invariant. In $SU(2)$ fixing the normalized vector $g_{a3}^A(\mbox{\boldmath$r$})$ determines only two of the parameters of the gauge group. This expression takes the form of the expectation value of a gauged lattice Laplacian for `fields' $g^A$.

LAG relaxes the normalization constraint $\sum_a[ g^A_{a3}(x)]^2=1$. Then $F_{\rm gf}(U)$ becomes a quadratic form in the components of the $3M$ component vector $g^A_{a3}$. The eigenvector of largest eigenvalue of the quadratic form maximizes the quadratic form and will define the gauge transformations. To identify this eigenvector with gauge transfromations we need to normalize the local vectors $g_{a3}(x)$ individually and then they determine the gauge transformation through the diagonalization of

$$g^A=\sum_a g^A_{a3}(x)\sigma_a = V^\dagger(x)\sigma_3 V(x).$$

Then the matrices $V(x)$ that diagonalize $g^A$ can be used to transform the gauge fields to the Laplacian Abelian gauge. Those points in space at which the vector $g^A_{a3}(x)$ vanishes form codimension 3 sets. They are points in three dimensional space and world lines in four dimensional space. These are the points where the gauge fixing procedure becomes singular. They correspond to the location of monopoles. Thus, the conceptual advantage of LAG over MAG is that the trajectories of monopoles can be defined as singularities of the gauge fixing procedure, in agreement with the original ideas of `t Hooft.

The rest of the treatment of LAG is identical to that of MAG. The gauge transformed gauge fields, $U^V_\mu(x)$ are projected to their abelian components and then the same quantities are measured as in MAG. LAG is equally successful with MAG in finding gauge invariant physical quantities similar to the ones obtained without gauge fixing. This shows that there is a considerable freedom in choosing the method of Abelian projection. In all cases the Abelian component of the gauge field, along with monopoles, seems to be responsible for confinement.