A domain-wall configuration of the η′ meson bounded by a string (called a pancake or a Hall droplet) is recently proposed to describe the baryons with spin Nc/2. In order to understand its baryon number as well as the flavor quantum number, we argue that the vector mesons (the ρ and ω mesons) should play an essential role for the consistency of the whole picture. We determine the effective theory of large-Nc QCD with Nf massless fermions by taking into account a mixed anomaly involving the θ-periodicity and the global symmetry. The anomaly matching requires the presence of a dynamical domain wall on which a $$ \mathrm{U}{\left({N}_f\right)}_{-{N}_c} $$ U N f − N c Chern-Simons theory is supported. We consider the boundary conditions that should be imposed on the edge of the domain wall, and conclude that there should be a boundary term that couples the $$ \mathrm{U}{\left({N}_f\right)}_{-{N}_c} $$ U N f − N c gauge field to the vector mesons. We discuss the impact on physics of the chiral phase transition and the relation to the “duality” of QCD.

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We give a gauge-independent definition of magnetic monopoles in the SU (N ) Yang-Mills theory through the Wilson loop operator. For this purpose, we give an explicit proof of the DiakonovPetrov version of the non-Abelian Stokes theorem for the Wilson loop operator in an arbitrary representation of the SU (N ) gauge group to derive a new form for the non-Abelian Stokes theorem. The new form is used to extract the magnetic-monopole contribution to the Wilson loop operator in a gauge-invariant way, which enables us to discuss confinement of quarks in any representation from the viewpoint of the dual superconductor vacuum.

In previous works, we have proposed a new formulation of Yang-Mills theory on the lattice so that the so-called restricted field obtained from the gauge-covariant decomposition plays the dominant role in quark confinement. This framework improves the Abelian projection in the gaugeindependent manner. For quarks in the fundamental representation, we have demonstrated some numerical evidences for the restricted field dominance in the string tension, which means that the string tension extracted from the restricted part of the Wilson loop reproduces the string tension extracted from the original Wilson loop. However, it is known that the restricted field dominance is not observed for the Wilson loop in higher representations if the restricted part of the Wilson loop is extracted by adopting the Abelian projection or the field decomposition naively in the same way as in the fundamental representation. In this paper, therefore, we focus on confinement of quarks in higher representations. By virtue of the non-Abelian Stokes theorem for the Wilson loop operator, we propose suitable gauge-invariant operators constructed from the restricted field to reproduce the correct behavior of the original Wilson loop averages for higher representations. Moreover, we perform lattice simulations to measure the static potential for quarks in higher representations using the proposed operators. We find that the proposed operators well reproduce the behavior of the original Wilson loop average, namely, the linear part of the static potential with the correct value of the string tension, which overcomes the problem that occurs in naively applying Abelian-projection to the Wilson loop operator for higher representations.

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