How Large is the Space of Covariantly Constant Gauge Fields

Abstract

The covariantly constant gauge fields are solutions of the sourceless Yang-Mills equation and represent the classical vacuum fields. We found that the moduli space of covariantly constant gauge fields is much larger than the space of constant chromomagnetic fields. A wider class of covariantly constant gauge field solutions representing non-perturbative magnetic flux sheets of finite thickness is obtained through the nontrivial space-time dependence of a unit colour vector. In some sense these solutions are similar to the Nielsen-Olesen magnetic flux tubes, but instead they have geometry of magnetic flux sheets and are supported without presence of any Higgs field. The infinitesimally thin magnetic sheet solutions can be associated with the singular surfaces considered by 't Hooft. The nonlocal operators that are supported on a two-dimensional surface rather than a one-dimensional curve were considered in literature. These surface operators are analogous to Wilson W(C) and 't Hooft line operators M(C) except that they are supported on a two-dimensional surface rather than a one-dimensional curve. The class of non-perturbative solutions representing a nonvanishing chromomagnetic field that fills out the whole 3D-space is obtained as well. This new class of covariantly constant gauge field solutions is constructed by using the general properties of the Cho Ansatz. We define the topological currents and the corresponding charges and demonstrate that the new solutions have a zero monopole charge density. Instead, the solutions have a nonzero Hopf invariant, which is the magnetic helicity of the Faraday force lines.

Figures (3)

• Figure 1: The l.h.s. graph demonstrates the magnetic energy $\epsilon(\gamma)$ (\ref{['density']}) and the Hopf $J_0(\gamma,x)$ density (\ref{['Hopfdensity']}) when the vectors are $\vec{H}=(0,0,1)$, $\vec{a}=(1,0,0)$, $\vec{b}=(0,\cos\gamma,\sin\gamma)$. The density $J_0(\gamma,x)$ vanishes only when the vectors $\vec{H}$ and $(\vec{a}\times \vec{b})$ are parallel or antiparallel $(\gamma = \pi/2, 3\pi/2)$ and realise two minimums of the energy density. For the field configuration $\vec{H}=(0,0,H)$, $\vec{a}=(a\cos\beta, a \sin\beta,0)$, $\vec{b}=(0,b \sin\gamma, b \cos\gamma)$ we will get the magnetic energy landscape $\epsilon(\beta,\gamma)$ as a function of two angles shown on the r.h.s. graph with a series of minimums separated by potential barriers.
• Figure 2: The graph demonstrates the space variation of the $C_3(x)$ component of the colour field $C_{\mu}$ (\ref{['cfield']}) and of the corresponding field strength tensor $S_{13}$ (\ref{['cfield1']}) when $\vec{a}=(1,0,0)$, $\vec{b}=(0,0,1)$. The coordinates $y,z$ are not shown. The discontinuities of the $C_3(x)$ are at the planes $x=\pm n, n=0,1,2...$, and the derivatives of the $C_3(x)$ from the l.h.s and from the r.h.s are equal, and therefore the field strength tensor $S_{13}$ is perfectly regular everywhere. The discontinuities $C_3(n-\epsilon)-C_3(n+\epsilon)=\partial_z \Phi$ are related by the gauge transformation $\Phi =2 z$, similar to the discontinuity of the monopole field on the equator where two gauge field patches intersect Wu:1975esWu:1975vq. The action density (\ref{['actioncont1']}) is a space constant, and the topological density (\ref{['savtopcharge']}) is $J_0 = F_{12} a_1 b_3 x= H x$.
• Figure 3: The figure demonstrates a sheet of finite thickness in the 3D-space that is filled by a chromomagnetic field. It represents a non-perturbative magnetic sheet solution (\ref{['polsol']}), (\ref{['polsol1']}). The chromomagnetic field $s_i = {1\over 2} \epsilon_{ijk} S_{ij}$ is nonzero inside the 3D-space sheet (\ref{['cfield2']}) and its direction is defined by the cross product $\vec{s} = (\vec{a} \times \vec{b})$. It vanishes outside the sheet.