Violation of non-Abelian Bianchi identity and QCD topology

When Abelian monopoles due to violation of the non-Abelian Bianchi identity Jμ(x) condense in the vacuum, color confinement of QCD is realized by the Abelian dual Meissner effect. Moreover VNABI affects also topological features of QCD drastically. Firstly, self-dual instantons can not be a classical solution of QCD. Secondly, the topological charge density is not expressed by a total derivative of the Chern-Simons density Kμ(x), but has an additional term L(x)=2Tr(Jμ(x)Aμ(x)). Thirdly, the axial U(1) anomaly is similarly modified, while keeping the Atiyah-Singer index theorem unchanged. These suggest that the Abelian mechanism based on the Abelian monopoles from VNABI play an important role in these topological quantities instead of instantons. The additional term L(x) is evaluated in the framework of Monte-Carlo simulations on SU(2) lattices in details with partial gauge fixings such as the Maximal Center gauge (MCG). The term $χ$ is not zero, although largely fluctuating around zero before the gradient flow and tends to vanish after small gradient flow time ($τ$) when infrared monopoles responsible for color confinement are also much suppressed. The bosonic definition of the topological charge $Q_t$ and its Abelian counterpart $Q_a\equiv (g^2/16π^2)\int d^4x \Tr(f_{μν}f_{μν}^*)$ written by Abelian field strengths are measured also on the lattices. When $χ$ is zero, $Q_a$=3$Q_t$ is expected theoretically, but a simple lattice definition of $Q_a$ is fluctuating between -2 and -3 in MCG, when $Q_t$ is around -1.