Correlation between Instantons and QCD-monopoles in the Abelian Gauge

The correlation between instantons and QCD-monopoles is studied both in the lattice gauge theory and in the continuum theory. From a simple topological consideration, instantons are expected to live only around the QCD-monopole trajectory in the abelian gauge. First, the instanton solution is analytically studied in the Polyakov-like gauge, where $A_4(x)$ is diagonalized. The world line of the QCD-monopole is found to be penetrate the center of each instanton inevitably. For the single-instanton solution, the QCD-monopole trajectory becomes a simple straight line. On the other hand, in the multi-instanton system, the QCD-monopole trajectory often has complicated topology including a loop or a folded structure, and is unstable against a small fluctuation of the location and the size of instantons. We also study the thermal instanton system in the Polyakov-like gauge. At the high-temperature limit, the monopole trajectory becomes straight lines in the temporal direction. The topology of the QCD-monopole trajectory is drastically changed at a high temperature. Second, the correlation between instantons and QCD-monopoles is studied in the maximally abelian (MA) gauge and/or the Polyakov gauge using the SU(2) lattice with $16^4$. The abelian link variable $u_\mu (s)$ is decomposed into the singular (monopole-dominating) part $u_\mu ^{Ds}(s)$ and the regular (photon-dominating) part $u_\mu ^{Ph}(s)$. The instanton numbers, $Q({\rm Ds})$ and $Q({\rm Ph})$, are measured using the SU(2) variables, $U_\mu ^{Ds}(s)$ and $U_\mu ^{Ph}(s)$, which are reconstructed by multiplying the off-diagonal matter factor to $u_\mu ^{Ds}(s)$ and $u_\mu ^{Ph}(s)$, respectively. A strong correlation is found between $Q({\rm Ds})$ in the singular part and the ordinary topological charge $Q({\rm SU(2)})$ even after the Cabibbo-Marinari