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 θ-parameter. For generic m and θ the theory has a single gapped vacuum. However, as θ is varied through θ = π 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 η′ 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 QCD₄ and QCD₃, 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 ℂℙNf−1 nonlinear σ-model with a Wess-Zumino term, in agreement with the conjectured dynamics in 2+1 dimensions.