Weyl symmetric structure of QCD vacuum

Abstract

We consider Weyl symmetric structure of the classical vacuum in quantum chromodynamics. In the framework of formalism of gauge invariant Abelian projection, we show that classical vacuums can be constructed in terms of Killing vector fields on the group $SU(3)$. Consequently, homotopic classes of Killing vector fields determine the topological structure of the vacuum. In particular, the second homotopy group ${\ensuremath{\pi}}_{2}(SU(3)/U(1)\ifmmode\times\else\texttimes\fi{}U(1))$ describes all topologically nonequivalent vacuums that are classified by two topological numbers. Starting with a given Killing vector field, one can construct vacuums forming a Weyl sextet representation. An interesting feature of $SU(3)$ gauge theory is that it admits a Weyl symmetric vacuum represented by a linear superposition of the vacuums from the Weyl vacuum sextet. A nontrivial manifestation of the Weyl symmetry is demonstrated on monopole solutions. We construct a family of finite energy monopole solutions in Yang-Mills-Higgs theory that includes the Weyl monopole sextet. We conjecture that a similar Weyl symmetric vacuum structure can be realized at quantum level in quantum chromodynamics.