Violation of non-Abelian Bianchi identity and QCD topology

TL;DR

The paper develops VNABI as a gauge-covariant violation of the non-Abelian Bianchi identity, linking Abelian monopoles to all color components and enabling confinement via the Abelian dual Meissner effect. VNABI modifies topological structures through the $L(x)$ term and its integrated form $\chi$, and, under the requirement $\chi=0$, recovers a relation $Q_a(A)=3Q_t(A)$ that supports an Abelian description of QCD topology. The author shows that instantons cannot be classical solutions in this framework and provides theoretical, as well as lattice-gradient-flow, evidence that $\chi$ vanishes in the continuum limit. A new connection between Abelian and non-Abelian topological charges emerges, suggesting Abelian dominance of topology and offering a gauge-invariant, monopole-based perspective on confinement and topological phenomena. These results motivate further large-scale lattice studies and exploration of implications for dynamical $SU(3)$ QCD and high-temperature behavior.

Abstract

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. 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. However, if the integrated additional term $χ=(g^2/16π^2)\int d^4xL(x) $ is not zero, it is not integer nor gauge invariant, so that VNABI would not be allowed in QCD. Using the Wu-Yang arguments, it is however proved that $χ$ becomes vanishing. $χ$ is evaluated also in the framework of Monte-Carlo simulations on SU(2) lattices in details with partial gauge fixings such as the Maximal Center gauge (MCG). When the gradient flow method is used, the term $χ$ tends to vanish after small gradient flow time ($τ$). 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.

Figures (6)

• Figure 1: $\beta$ dependence of the minimum, the maximum and the averaged absolute value of $\chi$ among 50 configurations in the tadpole-improved action under MCG and MAU1 gauge fixings.
• Figure 2: $\beta$ dependence of the minimum, the maximum and the averaged absolute value of $\chi$ among 50 configurations in the Wilson action under MCG gauge fixing.
• Figure 3: The additional term $\chi$ in the tadpole-improved action with MCG gauge under the gradient flow
• Figure 4: The additional term $\chi$ in the Wilson action with MCG gauge under the gradient flow
• Figure 5: $Q_t$, $Q_a$ and $\chi$ versus gradient flow time in a configuration at $\beta=3.5$ in MCG.