The dual superconductor picture is one of the most promising scenarios for quark confinement.To investigate this picture in a gauge-invariant manner, we have proposed a new formulation of Yang-Mills theory, named the decomposition method, on the lattice. The so-called restricted field obtained from the gauge-covariant decomposition plays
the dominant role in quark confinement. It has been known by preceding works that the restricted-field dominance is not observed for the Wilson loop in higher representations
if the restricted part of the Wilson loop is obtained by adopting the Abelian projection or the field decomposition naively in the same way as done in the fundamental representation. Recently, through the non-Abelian Stokes theorem (NAST) for the Wilson loop operator, we have proposed suitable gauge-invariant operators constructed from the restricted field to reproduce the correct behavior of the original Wilson loop averages for higher representations. We have demonstrated the numerical evidence 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 calculated from the original Wilson loop.


In this talk, we focus on the magnetic monopole.
According to this picture, magnetic monopoles causing the dual superconductivity are the dominant degrees of freedom
responsible for confinement. With the help of the NAST,we define the magnetic monopole and the string tension extracted from the magnetic-monopole part of the Wilson loop in a gauge-invariant manner. We will further perform lattice simulations to measure the static potential for quarks
in higher representations using the proposed operators and examine the magnetic monopole dominance in the string tension, which means that the string tension extracted from the magnetic-monopole part of the Wilson loop reproduces the proper string tension obtained from the original Wilson loop.