Time-Reversal Breaking in QCD₄, Walls, and Dualities in 2+1 Dimensions

We study $SU(N)$ Quantum Chromodynamics (QCD) in 3+1 dimensions with $N_f$ degenerate fundamental quarks with mass $m$ and a $\theta$-parameter. For generic $m$ and $\theta$ the theory has a single gapped vacuum. However, as $\theta$ is varied through $\theta=\pi$ for large $m$ there is a first order transition. For $N_f=1$ the first order transition line ends at a point with a massless $\eta'$ particle (for all $N$) and for $N_f>1$ the first order transition ends at $m=0$, where, depending on the value of $N_f$, the IR theory has free Nambu-Goldstone bosons, an interacting conformal field theory, or a free gauge theory. Even when the $4d$ bulk is smooth, domain walls and interfaces can have interesting phase transitions separating different $3d$ phases. These turn out to be the phases of the recently studied $3d$ Chern-Simons matter theories, thus relating the dynamics of QCD$_4$ and QCD$_3$, and, in particular, making contact with the recently discussed dualities in 2+1 dimensions. For example, when the massless $4d$ theory has an $SU(N_f)$ sigma model, the domain wall theory at low (nonzero) mass supports a $3d$ massless $CP^{N_f-1}$ nonlinear $\sigma$-model with a Wess-Zumino term, in agreement with the conjectured dynamics in 2+1 dimensions.