We provide a survey of recent studies supergroup gauge theory. first discuss supermatrix model as zero-dimensional toy theory and its geometric algebraic characterization. then focus on four-dimensional Yang--Mills with symmetry explore non-perturbative properties, including instanton calculus, Seiberg-Witten geometry, Bethe/gauge correspondence, realization intersecting defects.

Abstract We show that an effective Abelian gauge theory can be obtained as a renormalizable theory from QCD in the maximal Abelian gauge. The derivation improves the previous version which was obtained by one of the authors and was called the Abelianprojected effective gauge theory. This result supports a view that we can construct an effective Abelian gauge theory from QCD without losing chara...