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

TL;DR

This work analyzes four-dimensional SU($N_1$)$\times$SU($N_2$) gauge theories with a bi-fundamental Dirac fermion, showing that discrete anomalies and Berry phases tied to one-form symmetries constrain the infrared phase structure across multiple parameter regimes. In the equal-rank case, the authors establish a topology of the phase diagram that is invariant across diverse limits (massive or massless, hierarchies, and large $N$), while in the unequal-rank case the topology changes with parameters, signaling nontrivial intermediate physics. In the large-$N$ limit, they argue that topological transitions are governed by a non-supersymmetric duality cascade that reduces one gauge group by the other’s rank, akin to the Klebanov–Strassler cascade. Across these analyses, the paper harmonizes anomaly constraints, Berry phases, effective field theory (including chiral Lagrangians for hierarchical scales), and large-$N$ dynamics to map the rich vacuum structure, domain walls, and dualities of bi-fundamental gauge theories. The results illuminate how global consistency conditions shape 4D gauge dynamics and hint at universal cascades connecting distinct IR descriptions.

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.

Figures (9)

• Figure 1: The topology of the phase diagram of the equal rank bi-fundamental gauge theory for some fixed $M,\Lambda_1,\Lambda_2$ as a function of $\theta_{1,2}$.
• Figure 2: The qualitative structure of the phase diagram of the bi-fundamental gauge theory with $N_1=N_2$ in three limits: (a) Large $M$, (b) Small $M$ with $\Lambda_1\simeq \Lambda_2$, (c) $\Lambda_2, M\ll\Lambda_1$.
• Figure 3: Schematic illustration of the duality cascade. Starting from $SU(N_1)\times SU(N_2)$ on the left, as we lower the mass of the fermion we uncover dual theories that describe the correct evolution of the phase diagram. The last step of the cascade is the $SU(N)\times SU(N)$ theory where $N=gcd(N_1,N_2)$ (assuming here $N\gg1$). The massless point $M=0$ is the same for all the theories.
• Figure 4: The upper figure represnts the phase diagram of two decoupled $SU(N)$ theories. The leading correction in $\Lambda/M$ that comes from \ref{['leadingop']} leads to two possible phase diagrams depending on the sign of $c$. In these figures we include four copies of the fundamental domain.
• Figure 5: On the left, the vacuum structure at $(\theta_1,\theta_2)=(0,0)$. The red dots are located at $\alpha_k=\frac{2\pi k}{N}$ with integer $k$. They represent the vacua with ${\left< {\psi\tilde{\psi}} \right>}\sim e^{-i\alpha_k}$. For every one of these vacua, there is a combination of $\mathcal{T}$ and chiral symmetry that acts as reflection around the corresponding dot and hence preserved by the vacuum. On the right, the vacuum structure at $(\theta_1,\theta_2)=(\pi/2,-\pi/2)$. Here, combinations of $\mathcal{T}$ and chiral symmetry act as reflection between the dots and are broken by the vacuum. The yellow dots are new vacua located at $\alpha_k=\frac{2\pi k}{N}$ with half-integer $k$ and their existence is due to the spontanous breaking of $\mathcal{T}$.