Time-reversal breaking in QCD₄, walls, and dualities in 2 + 1 dimensions

We study SU(N ) Quantum Chromodynamics (QCD) in 3+1 dimensions with Nƒ 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 Nƒ = 1 the first order transition line ends at a point with a massless η' particle (for all N ) and for Nƒ > 1 the first order transition ends at m = 0, where, depending on the value of Nƒ, 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(Nƒ) sigma model, the domain wall theory at low (nonzero) mass supports a 3d massless CPNƒ-1 nonlinear σ-model with a Wess-Zumino term, in agreement with the conjectured dynamics in 2+1 dimensions.