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

We study SU(N ) Quantum Chromodynamics (QCD) in 3+1 dimensions with Nf 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 Nf = 1 the first order transition line ends at a point with a massless η′ particle (for all N ) and for Nf> 1 the first order transition ends at m = 0, where, depending on the value of Nf , 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(Nf ) sigma model, the domain wall theory at low (nonzero) mass supports a 3d massless ℂℙNf−1$$ \mathbb{C}{\mathrm{\mathbb{P}}}^{N_f-1} $$ nonlinear σ-model with a Wess-Zumino term, in agreement with the conjectured dynamics in 2+1 dimensions.