Vacuum Structures Revisited

We consider the relationship between the higher symmetry and the dynamical decomposition in supersymmetric gauge theory in various dimensions by studying the semi-classical potential energy. We observe that besides the scalar moduli we shall also include the field strength F0···d in the vacuum moduli in the 1+d dimensional theory along with a ℤp d-form symmetry. In gauge theory for charge-p matters with this symmetry, we find that the vacua decompose into p different universes at an intermediate scale, which means no dynamical domain wall can interpolate between them. In our setup, we re-derive the existing results on the decomposition in various dimensions. In four dimensions, we propose a UV gauge theory for the generalized super Yang-Mills theory, whose instanton sectors are restricted to the topological number with integer multiples of p.