# The Bi-Fundamental Gauge Theory in 3+1 Dimensions: The Vacuum Structure and a Cascade

## TL;DR

This paper explores the complex phase structure of a 3+1 dimensional SU(N1)×SU(N2) gauge theory with bi-fundamental fermions, revealing topologically distinct phases, duality cascades, and the impact of gauge group ranks on the phase diagram.

## Contribution

It provides a detailed analysis of the phase diagrams of bi-fundamental gauge theories, including the effects of anomalies, Berry phases, and the large N limit, offering new insights into their topological transitions.

## Key findings

- Multiple topologically distinct phase diagrams identified.
- Equal gauge group ranks lead to a unified phase topology.
- Different ranks result in distinct phase topologies and non-trivial intermediate physics.

## Abstract

We study the phases of the SU(N1)X SU(N2) gauge theory with a bi-fundamental 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 the ranks are different, 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 N1,N2 limit, we argue that the topological transitions are accounted for by a (non-supersymmetric) duality cascade as one varies the parameters of the theory.