The bi-fundamental gauge theory in 3+1 dimensions: the vacuum structure and a cascade

We study the phases of the SU(N₁) × SU(N₂) gauge theory with a bifundamental fermion in 3+1 dimensions. We show that the discrete anomalies and Berry phases associated to the one-form symmetry of the theory allow for several topologically distinct phase diagrams. We identify several limits of the theory where the phase diagram can be determined using various controlled approximations. When the two ranks are equal N₁ = N₂, these limits all lead to the same topology for the phase diagram and provide a consistent global understanding of the phases of the theory. When N₁ ≠ N₂, different limits lead to distinct topologies of the phase diagram. This necessarily implies non-trivial physics at some intermediate regimes of parameter space. In the large N_1,2 limit, we argue that the topological transitions are accounted for by a (non-supersymmetric) duality cascade as one varies the parameters of the theory.