Confinement and Topological Charge in the Abelian Gauge of QCD

We study the relation between instantons and monopoles in the abelian gauge. First, we investigate the monopole in the multi-instanton solution in the continuum Yang-Mills theory using the Polyakov gauge. At a large instanton density, the monopole trajectory becomes highly complicated, which can be regarded as a signal of monopole condensation. Second, we study instantons and monopoles in the SU(2) lattice gauge theory both in the maximally abelian (MA) gauge and in the Polyakov gauge. Using the 16 × 4 lattice, we find monopole dominance for instantons in the confinement phase even at finite temperatures. A linear-type correlation is found between the total monopole-loop length and the integral of the absolute value of the topological density (the total number of instantons and anti-instantons) in the MA gauge. We conjecture that instantons enhance the monopole-loop length and promote monopole condensation. I Analytical Study for Monopole Trajectory in the Multi-instanton Solution As ’t Hooft pointed out, a nonabelian gauge theory is reduced into an abelian gauge theory with monopoles by the abelian gauge fixing [1,2]. Recent lattice studies suggest abelian dominance and relevant roles of monopole condensation [3] for the nonperturbative phenomena: confinement [4], chiral symmetry breaking [5] and instantons [5-8]. In the abelian gauge, unit-charge monopoles appear from the hedgehog-like gauge configuration according to the nontrivial homotopy group, π2{SU(Nc)/U(1) c} = Zc ∞ [2]. On the other hand, the instanton is another relevant topological object in the nonabelian gauge manifold (π3(SU(Nc)) =Z∞). In the abelian-dominant system, the instanton seems to lose the topological basis for its existence, and hence it seems unable to survive in the abelian manifold [7-9]. However, even in the abelian gauge, nonabelian components remain relatively large around the topological defect, i.e. monopoles, and therefore instantons are expected to survive only around the monopole world lines in the abelian-dominant system [7-9]. We have pointed out such a close relation between instantons and monopoles, and have demonstrated it in the continuum Yang-Mills theory using the Polyakov gauge, where A4(x) is diagonalized [7-9]. We summarize our previous analytical works as follows [7-9]. (1) Each instanton center is penetrated by a monopole world line in the Polyakov gauge, because A4(x) takes a hedgehog configuration near the instanton center. In other words, instantons only live along the monopole trajectory. (2) Even at the classical level, the monopole trajectory is unstable against a small fluctuation of the location or the size of instantons, although it is relatively stable inside the instanton profile.